**The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes**

Alois Hoeld

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53372

## **1. Introduction**

The development of LWR Nuclear Power Plants (NPP) and the question after their safety behaviour have enhanced the need for adequate efficient theoretical descriptions of these plants. Thus thermal-hydraulic models and, based on them, effective computer codes played already very early an important role within the field of NPP safety research. Their objective is to describe both the steady state and transient behaviour of characteristic key parameters of a single- or two-phase fluid flowing along corresponding loops of such a plant and thus also along any type of heated or non-heated coolant channels being a part of these loops in an adequate way.

Due to the presence of discontinuities in the first principle of mass conservation of a two-phase flow model, caused at the transition from single- to two-phase flow and vice versa, it turned out that the direct solution of the basic conservation equations for mixture fluid along such a coolant channel gets very complicated. Obviously many discussions have and will continue to take place among experts as to which type of theoretical approach should be chosen for the correct description of thermal-hydraulic two-phase problems when looking at the wide range of applications. What is thus the most appropriate way to deal with such a special thermalhydraulic problem?

With the introduction of a 'Separate-Phase Model Concept' already very early an efficient way has been found how to circumvent these upcoming difficulties. Thereby a solution method has been proposed with the intention to separate the two-phases of such a mixture-flow in parts of the basic equations or even completely from each other. This yields a system of 4-, 5 or sometimes even 6-equations by splitting each of the conservation equations into two so-

© 2013 Hoeld; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Hoeld; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

called 'field equations'. Hence, compared to the four independent parameters characterising the mixture fluid, the separate-phase systems demand a much higher number of additional variables and special assumptions. This has the additional consequence that a number of speculative relations had to be incorporated into the theoretical description of such a module and an enormous amount of CPU-time has to be expended for the solution of the resulting sets of differential and analytical equations in a computer code. It is also clear that, based on such assumptions, the interfacial relations both between the (heated or cooled) wall but also between each of the two phases are completely rearranged. This raises the difficult question of how to describe in a realistic way the direct heat input into and between the phases and the movement resp. the friction of the phases between them. In such an approach this problem is solved by introducing corresponding exchange (=closure) terms between the equations based on special transfer (= closure) laws. Since they can, however, not be based on fundamental laws or at least on experimental measurements this approach requires a significant effort to find a correct formulation of the exchange terms between the phases. It must therefore be recognised that the quality of these basic equations (and especially their boundary conditions) will be intimately related to the (rather artificial and possibly speculative) assumptions adopted if comparing them with the original conservation laws of the 3-equation system and their constitutive equations as well. The problem of a correct description of the interfacial reaction between the phases and the wall remains. Hence, very often no consistency between different separate-phase models due to their underlying assumptions can be stated. Another problem arises from the fact that special methods have to be foreseen to describe the moving boiling boundary or mixture level (or at least to estimate their 'condensed' levels) in such a mixture fluid (see, for example, the 'Level Tracking' method in TRAC). Additionally, these methods show often deficiencies in describing extreme situations such as the treatment of single- and two-phase flow at the ceasing of natural circulation, the power situations if decreasing to zero etc. The codes are sometimes very inflexible, especially if they have to provide to a very complex physical system also elements which belong not to the usual class of 'thermalhydraulic coolant channels'. These can, for example, be nuclear kinetic considerations, heat transfer out of a fuel rod or through a tube wall, pressure build-up within a compartment, time delay during the movement of an enthalpy front along a downcomer, natural circulation along a closed loop, parallel channels, inner loops etc.

ter and main steam system, an own theoretical model has been derived. The resulting digital code UTSG could be used both in a stand-alone way but also as part of more comprehensive transient codes, such as the thermal-hydraulic GRS system code ATHLET. Together with a high level simulation language GCSM (General Control Simulation Module) it could be taken care of a manifold of balance-of-plant (BOP) actions too. Based on the experience of many years of application both at the GRS and a number of other institutes in different countries but also due to the rising demands coming from the safety-related research studies this UTSG theory and code has been continuously extended, yielding finally a very satisfactory and mature code

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

5

During the research work for the development of an enhanced version of the code UTSG-2 it arose finally the idea to establish an own basic element which is able to simulate the thermalhydraulic mixture-fluid situation within any type of cooled or heated channel in an as general as possible way. It should have the aim to be applicable for any modular construction of complex thermal-hydraulic assemblies of pipes and junctions. Thereby, in contrast to the above mentioned class of 'separate-phase' modular codes, instead of separating the phases of a mixture fluid within the entire coolant channel an alternative theoretical approach has been proposed, differing both in its form of application but also in its theoretical background. To circumvent the above mentioned difficulties due to discontinuities resulting from the spatial discretization of a coolant channel, resulting eventually in nodes where a transition from single- to two-phase flow and vice versa can take place, a special and unique concept has been proposed. Thereby it has been assumed that each coolant channel can be seen as a (basic) channel (BC) which can, according to their different flow regimes, be subdivided into a number of sub-channels (SC-s). It is clear that each of these SC-s can consist of only two types of flow regimes. A SC with just a single-phase fluid, containing exclusively either sub-cooled water, superheated steam or supercritical fluid, or a SC with a two-phase mixture. The theoretical considerations of this 'Separate-Region Approach' can then (within the class of mixture-fluid models) be restricted to only these two regimes. Hence, for each SC type, the 'classical' 3 conservation equations for mass, energy and momentum can be treated in a direct way. In case of a sub-channel with mixture flow these basic equations had to be supported by a drift flux correlation (which can take care also of stagnant or counter-current flow situations), yielding

an additional relation for the appearing fourth variable, namely the steam mass flow.

The main problem of the application of such an approach lies in the fact that now also varying SC entrance and outlet boundaries (marking the time-varying phase boundary positions) have to be considered with the additional difficulty that along a channel such a SC can even disappear or be created anew. This means that after an appropriate nodaliza‐ tion of such a BC (and thus also it's SC-s) a 'modified finite volume method' (among others based on the Leibniz Integration Rule) had to be derived for the spatial discretization of the fundamental partial differential equations (PDE-s) which represent the basic conservation equations of thermal-hydraulics for each SC. Furthermore, to link within this procedure the resulting mean nodal with their nodal boundary function values an adequate quadratic polygon approximation method (PAX) had to be established. The procedure should yield

version UTSG-2.

However, despite of these difficulties the 'Separate-Phase Models' have become increasingly fashionable and dominant in the last decades of thermal-hydraulics as demonstrated by the widely-used codes TRAC (Lilles et al.,1988, US-NRC, 2001a), CATHENA (Hanna, 1998), RELAP (US-NRC,2001b, Shultz,2003), CATHARE (Bestion,1990), ATHLET (Austregesilo et al., 2003, Lerchl et al., 2009).

Within the scope of reactor safety research very early activities at the Gesellschaft für Anlagenund Reaktorsicherheit (GRS) at Garching/Munich have been started too, developing thermalhydraulic models and digital codes which could have the potential to describe in a detailed way the overall transient and accidental behaviour of fluids flowing along a reactor core but also the main components of different Nuclear Power Plant (NPP) types. For one of these components, namely the natural circulation U-tube steam generator together with its feedwa‐ ter and main steam system, an own theoretical model has been derived. The resulting digital code UTSG could be used both in a stand-alone way but also as part of more comprehensive transient codes, such as the thermal-hydraulic GRS system code ATHLET. Together with a high level simulation language GCSM (General Control Simulation Module) it could be taken care of a manifold of balance-of-plant (BOP) actions too. Based on the experience of many years of application both at the GRS and a number of other institutes in different countries but also due to the rising demands coming from the safety-related research studies this UTSG theory and code has been continuously extended, yielding finally a very satisfactory and mature code version UTSG-2.

called 'field equations'. Hence, compared to the four independent parameters characterising the mixture fluid, the separate-phase systems demand a much higher number of additional variables and special assumptions. This has the additional consequence that a number of speculative relations had to be incorporated into the theoretical description of such a module and an enormous amount of CPU-time has to be expended for the solution of the resulting sets of differential and analytical equations in a computer code. It is also clear that, based on such assumptions, the interfacial relations both between the (heated or cooled) wall but also between each of the two phases are completely rearranged. This raises the difficult question of how to describe in a realistic way the direct heat input into and between the phases and the movement resp. the friction of the phases between them. In such an approach this problem is solved by introducing corresponding exchange (=closure) terms between the equations based on special transfer (= closure) laws. Since they can, however, not be based on fundamental laws or at least on experimental measurements this approach requires a significant effort to find a correct formulation of the exchange terms between the phases. It must therefore be recognised that the quality of these basic equations (and especially their boundary conditions) will be intimately related to the (rather artificial and possibly speculative) assumptions adopted if comparing them with the original conservation laws of the 3-equation system and their constitutive equations as well. The problem of a correct description of the interfacial reaction between the phases and the wall remains. Hence, very often no consistency between different separate-phase models due to their underlying assumptions can be stated. Another problem arises from the fact that special methods have to be foreseen to describe the moving boiling boundary or mixture level (or at least to estimate their 'condensed' levels) in such a mixture fluid (see, for example, the 'Level Tracking' method in TRAC). Additionally, these methods show often deficiencies in describing extreme situations such as the treatment of single- and two-phase flow at the ceasing of natural circulation, the power situations if decreasing to zero etc. The codes are sometimes very inflexible, especially if they have to provide to a very complex physical system also elements which belong not to the usual class of 'thermalhydraulic coolant channels'. These can, for example, be nuclear kinetic considerations, heat transfer out of a fuel rod or through a tube wall, pressure build-up within a compartment, time delay during the movement of an enthalpy front along a downcomer, natural circulation along

However, despite of these difficulties the 'Separate-Phase Models' have become increasingly fashionable and dominant in the last decades of thermal-hydraulics as demonstrated by the widely-used codes TRAC (Lilles et al.,1988, US-NRC, 2001a), CATHENA (Hanna, 1998), RELAP (US-NRC,2001b, Shultz,2003), CATHARE (Bestion,1990), ATHLET (Austregesilo et al.,

Within the scope of reactor safety research very early activities at the Gesellschaft für Anlagenund Reaktorsicherheit (GRS) at Garching/Munich have been started too, developing thermalhydraulic models and digital codes which could have the potential to describe in a detailed way the overall transient and accidental behaviour of fluids flowing along a reactor core but also the main components of different Nuclear Power Plant (NPP) types. For one of these components, namely the natural circulation U-tube steam generator together with its feedwa‐

a closed loop, parallel channels, inner loops etc.

4 Nuclear Reactor Thermal Hydraulics and Other Applications

2003, Lerchl et al., 2009).

During the research work for the development of an enhanced version of the code UTSG-2 it arose finally the idea to establish an own basic element which is able to simulate the thermalhydraulic mixture-fluid situation within any type of cooled or heated channel in an as general as possible way. It should have the aim to be applicable for any modular construction of complex thermal-hydraulic assemblies of pipes and junctions. Thereby, in contrast to the above mentioned class of 'separate-phase' modular codes, instead of separating the phases of a mixture fluid within the entire coolant channel an alternative theoretical approach has been proposed, differing both in its form of application but also in its theoretical background. To circumvent the above mentioned difficulties due to discontinuities resulting from the spatial discretization of a coolant channel, resulting eventually in nodes where a transition from single- to two-phase flow and vice versa can take place, a special and unique concept has been proposed. Thereby it has been assumed that each coolant channel can be seen as a (basic) channel (BC) which can, according to their different flow regimes, be subdivided into a number of sub-channels (SC-s). It is clear that each of these SC-s can consist of only two types of flow regimes. A SC with just a single-phase fluid, containing exclusively either sub-cooled water, superheated steam or supercritical fluid, or a SC with a two-phase mixture. The theoretical considerations of this 'Separate-Region Approach' can then (within the class of mixture-fluid models) be restricted to only these two regimes. Hence, for each SC type, the 'classical' 3 conservation equations for mass, energy and momentum can be treated in a direct way. In case of a sub-channel with mixture flow these basic equations had to be supported by a drift flux correlation (which can take care also of stagnant or counter-current flow situations), yielding an additional relation for the appearing fourth variable, namely the steam mass flow.

The main problem of the application of such an approach lies in the fact that now also varying SC entrance and outlet boundaries (marking the time-varying phase boundary positions) have to be considered with the additional difficulty that along a channel such a SC can even disappear or be created anew. This means that after an appropriate nodaliza‐ tion of such a BC (and thus also it's SC-s) a 'modified finite volume method' (among others based on the Leibniz Integration Rule) had to be derived for the spatial discretization of the fundamental partial differential equations (PDE-s) which represent the basic conservation equations of thermal-hydraulics for each SC. Furthermore, to link within this procedure the resulting mean nodal with their nodal boundary function values an adequate quadratic polygon approximation method (PAX) had to be established. The procedure should yield finally for each SC type (and thus also the complete BC) a set of non-linear ordinary differential equations of 1st order (ODE-s).

For the establishment of the corresponding (digital) module CCM, based on this theoretical model very specific methods had to be achieved. Thereby the following points had to be taken

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

7

**•** The code has to be easily applicable, demanding only a limited amount of directly available input data. It should make it possible to simulate the thermal-hydraulic mixture-fluid situation along any cooled or heated channel in an as general as possible way and thus describe any modular construction of complex thermal-hydraulic assemblies of pipes and junctions. Such an universally applicable tool can then be taken for calculating the steady state and transient behaviour of all the characteristic parameters of each of the appearing coolant channels and thus be a valuable element for the construction of complex computer codes. It should yield as output all the necessary time-derivatives and constitutive param‐ eters of the coolant channels required for the establishment of an overall thermal-hydraulic

**•** It was the intention of CCM that it should act as a complete system in its own right, requiring only BC (and not SC) related, and thus easily available input parameters (geometry data, initial and boundary conditions, parameters resulting from the integration etc.). The partitioning of BC-s into SC-s is done at the beginning of each recursion or time-step

**•** The quality of such a model is very much dependent on the method by which the problem of the varying SC entrance and outlet boundaries can be solved. Especially if they cross BC node boundaries during their movement along a channel. For this purpose a special 'modified finite element-method' has been developed which takes advantage of the

**•** For the support of the nodalized differential equations along different SC-s a 'quadratic polygon approximation' procedure (PAX) was constructed in order to interrelate the mean nodal with the nodal boundary functions. Additionally, due to the possibility of varying SC entrance and outlet boundaries, nodal entrance gradients are required too (See section 3.3).

**•** Several correlation packages such as, for example, packages for the thermodynamic properties of water and steam, heat transfer coefficients, drift flux correlations and singleand two-phase friction coefficients had to be developed and implemented (See sections 2.2.1

**•** Knowing the characteristic parameters at all SC nodes (within a BC) then the single- and two-phase parameters at all node boundaries of the entire BC can be determined. And also the corresponding time-derivatives of the characteristic averaged parameters of coolant temperatures resp. void fraction over these nodes. This yields a final set of ODE-s and

**•** In order to be able to describe also thermodynamic non-equilibrium situations it can be assumed that each phase is represented by an own with each other interacting BC. For these purpose in the model the possibility of a variable cross flow area along the entire channel

automatically within CCM, so no special actions are required of the user.

'Leibniz' rule for integration (see eq.(15)).

into account:

code.

to 2.2.4).

constitutive equations.

had to be considered as well.

It has to be noted that besides the suggestion to separate a (basic) channel into regions of different flow types this special PAX method represents, together with the very thoroughly tested packages for drift flux and single- and two-phase friction factors, the central part of the here presented 'Separate - Region Approach'. An adequate way to solve this essential problem could be found and a corresponding procedure established. As a result of these theoretical considerations an universally applicable 1D thermal-hydraulic drift-flux based separateregion coolant channel model and module CCM could be established. This module allows to calculate automatically the steady state and transient behaviour of the main characteristic parameters of a single- and two-phase fluid flowing within the entire coolant channel. It represents thus a valuable tool for the establishment of complex thermal-hydraulic computer codes. Even in the case of complicated single- and mixture fluid systems consisting of a number of different types of (basic) coolant channels an overall set of equations by determining automatically the nodal non-linear differential and corresponding constitutive equations needed for each of these sub- and thus basic channels can be presented. This direct method can thus be seen as a real counterpart to the currently preferred and dominant 'separate-phase models'.

To check the performance and validity of the code package CCM and to verify it the digital code UTSG-2 has been extended to a new and advanced version, called UTSG-3. It has been based, similarly as in the previous code UTSG-2, on the same U-tube, main steam and downcomer (with feedwater injection) system layout, but now, among other essential im‐ provements, the three characteristic channel elements of the code UTSG-2 (i.e. the primary and secondary side of the heat exchange region and the riser region) have been replaced by adequate CCM modules.

It is obvious that such a theoretical 'separate-region' approach can disclose a new way in describing thermal-hydraulic problems. The resulting 'mixture-fluid' technique can be regarded as a very appropriate way to circumvent the uncertainties apparent from the separation of the phases in a mixture flow. The starting equations are the direct consequence of the original fundamental physical laws for the conservation of mass, energy and momen‐ tum, supported by well-tested heat transfer and single- and two-phase friction correlation packages (and thus avoiding also the sometimes very speculative derivation of the 'closure' terms). In a very comprehensive study by (Hoeld, 2004b) a variety of arguments for the here presented type of approach is given, some of which will be discussed in the conclusions of chapter 6.

The very successful application of the code combination UTSG-3/CCM demonstrates the ability to find an exact and direct solution for the basic equations of a 'non-homogeneous driftflux based thermal-hydraulic mixture-fluid coolant channel model'. The theoretical back‐ ground of CCM will be described in very detail in the following chapters.

For the establishment of the corresponding (digital) module CCM, based on this theoretical model very specific methods had to be achieved. Thereby the following points had to be taken into account:

finally for each SC type (and thus also the complete BC) a set of non-linear ordinary

It has to be noted that besides the suggestion to separate a (basic) channel into regions of different flow types this special PAX method represents, together with the very thoroughly tested packages for drift flux and single- and two-phase friction factors, the central part of the here presented 'Separate - Region Approach'. An adequate way to solve this essential problem could be found and a corresponding procedure established. As a result of these theoretical considerations an universally applicable 1D thermal-hydraulic drift-flux based separateregion coolant channel model and module CCM could be established. This module allows to calculate automatically the steady state and transient behaviour of the main characteristic parameters of a single- and two-phase fluid flowing within the entire coolant channel. It represents thus a valuable tool for the establishment of complex thermal-hydraulic computer codes. Even in the case of complicated single- and mixture fluid systems consisting of a number of different types of (basic) coolant channels an overall set of equations by determining automatically the nodal non-linear differential and corresponding constitutive equations needed for each of these sub- and thus basic channels can be presented. This direct method can thus be seen as a real counterpart to the currently preferred and dominant 'separate-phase

To check the performance and validity of the code package CCM and to verify it the digital code UTSG-2 has been extended to a new and advanced version, called UTSG-3. It has been based, similarly as in the previous code UTSG-2, on the same U-tube, main steam and downcomer (with feedwater injection) system layout, but now, among other essential im‐ provements, the three characteristic channel elements of the code UTSG-2 (i.e. the primary and secondary side of the heat exchange region and the riser region) have been replaced by

It is obvious that such a theoretical 'separate-region' approach can disclose a new way in describing thermal-hydraulic problems. The resulting 'mixture-fluid' technique can be regarded as a very appropriate way to circumvent the uncertainties apparent from the separation of the phases in a mixture flow. The starting equations are the direct consequence of the original fundamental physical laws for the conservation of mass, energy and momen‐ tum, supported by well-tested heat transfer and single- and two-phase friction correlation packages (and thus avoiding also the sometimes very speculative derivation of the 'closure' terms). In a very comprehensive study by (Hoeld, 2004b) a variety of arguments for the here presented type of approach is given, some of which will be discussed in the conclusions of

The very successful application of the code combination UTSG-3/CCM demonstrates the ability to find an exact and direct solution for the basic equations of a 'non-homogeneous driftflux based thermal-hydraulic mixture-fluid coolant channel model'. The theoretical back‐

ground of CCM will be described in very detail in the following chapters.

differential equations of 1st order (ODE-s).

6 Nuclear Reactor Thermal Hydraulics and Other Applications

models'.

chapter 6.

adequate CCM modules.


**•** Within the CCM procedure two further aspects play an important role. These are, however, not essential for the development of mixture-fluid models but can help enormously to enhance the computational speed and applicability of the resulting code when simulating a complex net of coolant pipes:

**2. Thermal-hydraulic drift-flux based mixture fluid approach**

Thermal-hydraulic single-phase or mixture-fluid models for coolant channels or, as presented here, for each of the sub-channels are generally based on a number of fundamental physical laws, i.e., they obey genuine conservation equations for mass, energy and momentum. And they are supported by adequate constitutive equations (packages for thermo-dynamic and transport properties of water and steam, for heat transfer coefficients, for drift flux, for single-

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

9

In view of possible applications as an element in complex thermal-hydraulic ensembles outside of CCM eventually a fourth and fifth conservation law has to be considered too. The fourth law, namely the volume balance, allows then to calculate the transient behaviour of the overall absolute system pressure. Together with the local pressure differences then the absolute pressure profile along the BC can be determined. The fifth physical law is based on the (trivial) fact that the sum of all pressure decrease terms along a closed loop must be zero. This is the basis for the treatment of the thermal-hydraulics of a channel according to a 'closed channel concept' (See section 3.11). It refers to one of the channels within the closed loop where the BC entrance and outlet pressure terms have to be assumed to be fixed. Due to this concept then the necessary entrance mass flow term has be determined in order to fulfil the demand from

> A 1- r + W S { [( ) ]r G=0 } *t z* a

Containing the density terms ρW and ρS for sub-cooled or saturated water and saturated or superheated steam, the void fraction α and the cross flow area A which can eventually be changing along the coolant channel. It determines, after a nodalization, the total mass flow

¶ ¶ é ù ë û ¶ ¶ (2)

Containing the enthalpy terms hW and hS for sub-cooled or saturated water and saturated or superheated steam. As boundary values either the 'linear power qTWL', the 'heat flux qTWF' along the heated (or cooled) tube wall (with its heated perimeter UTW) or the local 'power density term qD' (being transferred into the coolant channel with its cross section A) are demanded to

¶ ¶ (1)

 a+ ¶ ¶

G=GW+GS at each node outlet in dependence of its node entrance value.

A 1- r h + r h -P + G h +G h = q = U q = A q WW SS W W S S TWL TW TWF D { ([ ) ]} *t z*

**2.1. Thermal-hydraulic conservation equations**

and two-phase friction coefficients etc.).

*2.1.1. Mass balance (Single- and two-phase flow)*

*2.1.2. Energy balance (Single- and two-phase flow)*

 a

momentum balance.

a


The application of a direct mixture-fluid technique follows a long tradition of research efforts. Ishii (1990), a pioneer of two-fluid modelling, states with respect to the application of effective drift-flux correlation packages in thermal-hydraulic models: 'In view of the limited data base presently available and difficulties associated with detailed measurements in two-phase flow, an advanced mixture-fluid model is probably the most reliable and accurate tool for standard two-phase flow problems'. There is no new knowledge available to indicate that this view is invalid.

Generally, the mixture-fluid approach is in line with (Fabic, 1996) who names three strong points arguing in favour of this type of drift-flux based mixture-fluid models:


and, it can be added,


A documentation of the theoretical background of CCM will be given in very condensed form in the different chapters of this article. For the establishment of the corresponding (digital) module CCM, based on this theoretical model, very specific methods had to be achieved.

The here presented article is an advanced and very condensed version of a paper being already published in a first Open Access Book of this INTECH series (Hoeld, 2011a). It is updated to the newest status in this field of research. An example for an application of this module within the UTSG-3 steam generator code is given in (Hoeld 2011b).

## **2. Thermal-hydraulic drift-flux based mixture fluid approach**

#### **2.1. Thermal-hydraulic conservation equations**

**•** Within the CCM procedure two further aspects play an important role. These are, however, not essential for the development of mixture-fluid models but can help enormously to enhance the computational speed and applicability of the resulting code when simulating

**•** The solution of the energy and mass balance equations at each intermediate time step will be performed independently from momentum balance considerations. Hence the heavy

**•** This decoupling allows then also the introduction of an 'open' and 'closed channel' concept (see section 3.11). Such a special method can be very helpful in describing complex physical systems with eventually inner loops. As an example the simulation of a 3D compartment

The application of a direct mixture-fluid technique follows a long tradition of research efforts. Ishii (1990), a pioneer of two-fluid modelling, states with respect to the application of effective drift-flux correlation packages in thermal-hydraulic models: 'In view of the limited data base presently available and difficulties associated with detailed measurements in two-phase flow, an advanced mixture-fluid model is probably the most reliable and accurate tool for standard two-phase flow problems'. There is no new knowledge available to indicate that this view is

Generally, the mixture-fluid approach is in line with (Fabic, 1996) who names three strong

**•** they do not require unknown or untested closure relations concerning mass, energy and momentum exchange between phases (thus influencing the reliability of the codes),

**•** discontinuities during phase changes can be avoided by deriving special solution proce‐

**•** the possibility to circumvent a set of 'stiff' ODE-s saves an enormous amount of CPU time

A documentation of the theoretical background of CCM will be given in very condensed form in the different chapters of this article. For the establishment of the corresponding (digital) module CCM, based on this theoretical model, very specific methods had to be achieved.

The here presented article is an advanced and very condensed version of a paper being already published in a first Open Access Book of this INTECH series (Hoeld, 2011a). It is updated to the newest status in this field of research. An example for an application of this module within

which means that the other parts of the code can be treated in much more detail.

points arguing in favour of this type of drift-flux based mixture-fluid models:

dures for the simulation of the movement of these phase boundaries,

the UTSG-3 steam generator code is given in (Hoeld 2011b).

CPU-time consuming solution of stiff equations can be avoided (Section 3.6).

by parallel channels can be named (Jewer et al., 2005).

**•** They are supported by a wealth of test data,

**•** they are much simpler to apply,

and, it can be added,

a complex net of coolant pipes:

8 Nuclear Reactor Thermal Hydraulics and Other Applications

invalid.

Thermal-hydraulic single-phase or mixture-fluid models for coolant channels or, as presented here, for each of the sub-channels are generally based on a number of fundamental physical laws, i.e., they obey genuine conservation equations for mass, energy and momentum. And they are supported by adequate constitutive equations (packages for thermo-dynamic and transport properties of water and steam, for heat transfer coefficients, for drift flux, for singleand two-phase friction coefficients etc.).

In view of possible applications as an element in complex thermal-hydraulic ensembles outside of CCM eventually a fourth and fifth conservation law has to be considered too. The fourth law, namely the volume balance, allows then to calculate the transient behaviour of the overall absolute system pressure. Together with the local pressure differences then the absolute pressure profile along the BC can be determined. The fifth physical law is based on the (trivial) fact that the sum of all pressure decrease terms along a closed loop must be zero. This is the basis for the treatment of the thermal-hydraulics of a channel according to a 'closed channel concept' (See section 3.11). It refers to one of the channels within the closed loop where the BC entrance and outlet pressure terms have to be assumed to be fixed. Due to this concept then the necessary entrance mass flow term has be determined in order to fulfil the demand from momentum balance.

#### *2.1.1. Mass balance (Single- and two-phase flow)*

$$\frac{\partial}{\partial t} \{ \mathbf{A} [(1 \text{-} a) \mathbf{r}\_W + a \mathbf{r}\_S] \} + \frac{\partial}{\partial z} \mathbf{G} = \mathbf{0} \tag{1}$$

Containing the density terms ρW and ρS for sub-cooled or saturated water and saturated or superheated steam, the void fraction α and the cross flow area A which can eventually be changing along the coolant channel. It determines, after a nodalization, the total mass flow G=GW+GS at each node outlet in dependence of its node entrance value.

#### *2.1.2. Energy balance (Single- and two-phase flow)*

$$\frac{\partial}{\partial t} \| \mathbf{A} [(1 \text{-} a) \mathbf{r}\_W \mathbf{h}\_W + a \mathbf{r}\_S \mathbf{h}\_S \cdot \mathbf{P}] \| + \frac{\partial}{\partial z} \Big[ \mathbf{G}\_W \mathbf{h}\_W + \mathbf{G}\_S \mathbf{h}\_S \Big] = \mathbf{q}\_{\text{TWL}} = \mathbf{U}\_{\text{TWL}} \mathbf{q}\_{\text{TWF}} = \mathbf{A} \cdot \mathbf{q}\_D \tag{2}$$

Containing the enthalpy terms hW and hS for sub-cooled or saturated water and saturated or superheated steam. As boundary values either the 'linear power qTWL', the 'heat flux qTWF' along the heated (or cooled) tube wall (with its heated perimeter UTW) or the local 'power density term qD' (being transferred into the coolant channel with its cross section A) are demanded to be known (See also sections 2.2.4 and 3.5). They are assumed to be directed into the coolant (then having a positive sign).

#### *2.1.3. Momentum balance (Single- and two-phase flow)*

$$\frac{\partial}{\partial t}(G\_F) + (\frac{\partial P}{\partial z}) = (\frac{\partial P}{\partial z})\_A + (\frac{\partial P}{\partial z})\_S + (\frac{\partial P}{\partial z})\_F + (\frac{\partial P}{\partial z}) \times \tag{3}$$

tions established by various authors, to find the best fitting correlations for the different fields of application and to get a smooth transfer from one to another of them special and effective correlation packages had to be developed. Their validities can be and has been tested out-ofpile by means of adequate driver codes. Obviously, my means of this method improved

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

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11

A short characterization of the main packages being applied within CCM is given below. For

The different thermodynamic properties for water and steam (together with their derivatives with respect to P and T, but also P and h) demanded by the conservation and constitutive equations have to be determined by applying adequate water/steam tables. This is, for light-

Then the time-derivatives of these thermodynamic properties which respect to their inde‐

( ) ( ) ( ) ( ) ( ) T P Mn Mn h z,t = h T z,t ,P z,t = h T z,t + h d d d d

Additionally, corresponding thermodynamic transport properties such as 'dynamic viscosity' and 'thermal heat conductivity' (and thus the 'Prantl number') are asked from some constit‐ utive equations too as this can be stated, for example, for the code packages MPPWS and MPPETA (Hoeld, 1996). All of them have been derived on the basis of tables given by (Schmidt

Obviously, the CCM method is also applicable for other coolant systems (heavy water, gas) if

The friction factor fR needed in eq.(6) can in case of single-phase flow be set, as proposed by

The corresponding coefficient for two-phase flow has to be extended by means of a two-phase

Usually, the three conservation equations (1), (2) and (3) demand for single-phase flow the three parameters G, P and T as independent variables. In case of two-phase flow, they are, however, dependent on four of them, namely G, P, α and GS. This means, the set has to be completed by an additional relation. This can be achieved by any two-phase correlation, acting

dt dt dt dt é ù P z,t ë û (7)

pendent local parameters (for example of an enthalpy term h) can be represented as

correlations can easily be incorporated into the existing theory.

*2.2.1. Thermodynamic and transport properties of water and steam*

water systems, realized in the code package MPP (Hoeld, 1996 and 2011a).

adequate thermodynamic tables for this type of fluids are available.

(Moody, 1994), equal to the Darcy-Weisbach single-phase friction factor.

2 as recommended by (Martinelli-Nelson, 1948).

more details see (Hoeld, 2011a).

and Grigull,1982) and (Haar et al., 1988).

*2.2.2. Single and two-phase friction factors*

For more details see again (Hoeld, 2011a).

multiplier Φ 2*PF*

*2.2.3. Drift flux correlation*

describing either the pressure differences (at steady state) or (in the transient case) the change in the total mass flux (GF =G/A) along a channel (See chapter 3.10).

The general pressure gradient ( <sup>∂</sup> *<sup>P</sup>* <sup>∂</sup> *<sup>z</sup>* ) can be determined in dependence of

**•** the mass acceleration

$$\left(\frac{\partial \mathcal{P}}{\partial \mathbf{z}}\right)\_{\mathbf{A}} = -\frac{\partial}{\partial \mathbf{z}} \Big[ \left( \mathbf{G}\_{\text{FW}} \mathbf{v}\_{\text{W}} \star \mathbf{G}\_{\text{FS}} \mathbf{v}\_{\text{S}} \right) \Big] \tag{4}$$

with vS and vW denoting steam and water velocities given by the eqs.(19) and (20),

**•** the static head

$$\mathbf{g}(\frac{\delta P}{\delta z})\mathbf{s} = \cos(\Phi\_{\mathbf{Z}\mathbf{C}}) \, \mathbf{g}\_{\mathbf{C}}[\alpha \rho\_{\mathbf{S}} + (1 \cdot \alpha)\rho \, \mathbf{w}] \tag{5}$$

with ΦZG representing the angle between z-axis and flow direction. Hence

cos(ΦZG)= ± ΔzEL/ΔzL, with Δz<sup>L</sup> denoting the nodal length and ΔzEL the nodal elevation height (having a positive sign at upwards flow).

**•** the single- and/or two-phase friction term

$$(\frac{\partial P}{\partial z})\_{\rm F} = -\mathbf{f}\_{\rm R} \frac{\mathbf{G}\_{\rm F} | \mathbf{G}\_{\rm F}|}{2 \ \mathbf{d}\_{\rm HW} \,\rho} \tag{6}$$

with a friction factor derived from corresponding constitutive equations (section 2.2.2) and finally

**•** the direct perturbations (∂*P* / ∂ *z*)X from outside, arising either by starting an external pump or considering a pressure adjustment due to mass exchange between parallel channels.

#### **2.2. Constitutive equations**

For the exact description of the steady state and the transient behaviour of single- or two-phase fluids a number of mostly empirical constitutive correlations are, besides the above mentioned conservation equations, demanded. To bring a structure into the manifold of existing correla‐ tions established by various authors, to find the best fitting correlations for the different fields of application and to get a smooth transfer from one to another of them special and effective correlation packages had to be developed. Their validities can be and has been tested out-ofpile by means of adequate driver codes. Obviously, my means of this method improved correlations can easily be incorporated into the existing theory.

A short characterization of the main packages being applied within CCM is given below. For more details see (Hoeld, 2011a).

## *2.2.1. Thermodynamic and transport properties of water and steam*

be known (See also sections 2.2.4 and 3.5). They are assumed to be directed into the coolant

( )( )( ) ( ) ( ) ( ) *<sup>A</sup> <sup>S</sup> FF PP P P P*

describing either the pressure differences (at steady state) or (in the transient case) the change

¶ ¶¶ ¶ ¶ ¶ = + + +´ ¶ ¶¶ ¶ ¶ ¶

*z z*

<sup>∂</sup> *<sup>z</sup>* ) can be determined in dependence of

+ (3)

ë û ¶ ¶ (4)

<sup>∂</sup> *<sup>z</sup>* )s=- cos(ΦZG) gC[α*ρ*S+(1-α)*ρ*w] (5)

¶ (6)

*t zz z*

( ) <sup>A</sup> GFW W FS S <sup>v</sup> +G) v( *<sup>P</sup>*

with vS and vW denoting steam and water velocities given by the eqs.(19) and (20),

¶ ¶ = - é ù

cos(ΦZG)= ± ΔzEL/ΔzL, with Δz<sup>L</sup> denoting the nodal length and ΔzEL the nodal elevation height

F F HW

r

in the total mass flux (GF =G/A) along a channel (See chapter 3.10).

*z z*

with ΦZG representing the angle between z-axis and flow direction. Hence

¶

F R

G |G <sup>=</sup><sup>|</sup> ( ) 2 d - f *P z*

with a friction factor derived from corresponding constitutive equations (section 2.2.2)

**•** the direct perturbations (∂*P* / ∂ *z*)X from outside, arising either by starting an external pump or considering a pressure adjustment due to mass exchange between parallel channels.

For the exact description of the steady state and the transient behaviour of single- or two-phase fluids a number of mostly empirical constitutive correlations are, besides the above mentioned conservation equations, demanded. To bring a structure into the manifold of existing correla‐

(then having a positive sign).

The general pressure gradient ( <sup>∂</sup> *<sup>P</sup>*

**•** the mass acceleration

**•** the static head

and finally

**2.2. Constitutive equations**

*2.1.3. Momentum balance (Single- and two-phase flow)*

10 Nuclear Reactor Thermal Hydraulics and Other Applications

*G*

( ∂ *P*

(having a positive sign at upwards flow).

**•** the single- and/or two-phase friction term

The different thermodynamic properties for water and steam (together with their derivatives with respect to P and T, but also P and h) demanded by the conservation and constitutive equations have to be determined by applying adequate water/steam tables. This is, for lightwater systems, realized in the code package MPP (Hoeld, 1996 and 2011a).

Then the time-derivatives of these thermodynamic properties which respect to their inde‐ pendent local parameters (for example of an enthalpy term h) can be represented as

$$\frac{d\mathbf{d}}{dt}\mathbf{h}(\mathbf{z},\mathbf{t}) = \frac{d\mathbf{d}}{dt}\mathbf{h}\Big[\mathbf{T}(\mathbf{z},\mathbf{t}),\mathbf{P}(\mathbf{z},\mathbf{t})\Big] = \mathbf{h}^{\mathrm{T}}\frac{d\mathbf{d}}{dt}\mathbf{T}\_{\mathrm{Mn}}(\mathbf{z},\mathbf{t}) + \mathbf{h}^{\mathrm{P}}\frac{d\mathbf{d}}{dt}\mathbf{P}\_{\mathrm{Mn}}(\mathbf{z},\mathbf{t})\tag{7}$$

Additionally, corresponding thermodynamic transport properties such as 'dynamic viscosity' and 'thermal heat conductivity' (and thus the 'Prantl number') are asked from some constit‐ utive equations too as this can be stated, for example, for the code packages MPPWS and MPPETA (Hoeld, 1996). All of them have been derived on the basis of tables given by (Schmidt and Grigull,1982) and (Haar et al., 1988).

Obviously, the CCM method is also applicable for other coolant systems (heavy water, gas) if adequate thermodynamic tables for this type of fluids are available.

#### *2.2.2. Single and two-phase friction factors*

The friction factor fR needed in eq.(6) can in case of single-phase flow be set, as proposed by (Moody, 1994), equal to the Darcy-Weisbach single-phase friction factor.

The corresponding coefficient for two-phase flow has to be extended by means of a two-phase multiplier Φ 2*PF* 2 as recommended by (Martinelli-Nelson, 1948).

For more details see again (Hoeld, 2011a).

#### *2.2.3. Drift flux correlation*

Usually, the three conservation equations (1), (2) and (3) demand for single-phase flow the three parameters G, P and T as independent variables. In case of two-phase flow, they are, however, dependent on four of them, namely G, P, α and GS. This means, the set has to be completed by an additional relation. This can be achieved by any two-phase correlation, acting thereby as a 'bridge' between GS and α. For example, by a slip correlation. However, to take care of stagnant or counter-current flow situations too an effective drift-flux correlation seemed here to be more appropriate. For this purpose an own package has been established, named MDS (Hoeld, 2001 and 2002a). Due to the different requirements in the application of CCM it turned out that it has a number of advantages if choosing the 'flooding-based full-range' Sonnenburg correlation (Sonnenburg, 1989) as basis for MDS. This correlation combines the common drift-flux procedure being formulated by (Zuber-Findlay, 1965) and expanded by (Ishii-Mishima, 1980) and (Ishii, 1990) etc. with the modern envelope theory. The correlation in the final package MDS had, however, to be rearranged in such a way that also the special cases of α → 0 or α → 1 are included and that, besides their absolute values and corresponding slopes, also the gradients of the approximation function can be made available for CCM. Additionally, an inverse form had to be installed (needed, for example, for the steady state conditions) and, eventually, also considerations with respect to possible entrainment effects be taken care.

For the case of a vertical channel this correlation can be represented as

$$\mathbf{v\_D = 1.5 v\_{WLM} C\_0 C\_{VD} [(1 + C\_{VD}^2)^3]^2 \ - (1.5 + C\_{VD}^2) C\_{VD}]} \\ \text{with } \mathbf{v\_D \to} \\ \mathbf{v\_D = \frac{9}{16} C\_0 v\_{WLM}} \quad \text{if } \mathbf{a} \to \mathbf{0} \tag{8}$$

where the coefficient CVD is given by

$$\mathbf{C\_{VD}} = \frac{2}{3} \frac{\mathbf{v\_{SLM}}}{\mathbf{v\_{WLM}}} \frac{\mathbf{1} \cdot \mathbf{C\_0} \alpha}{\mathbf{C\_0} \alpha} \tag{9}$$

CGC= 1-(1- *<sup>ρ</sup>* //

the steam mass flow gradient

→G*<sup>S</sup>* <sup>1</sup>

*2.2.4. Heat transfer coefficients*

*<sup>α</sup>* <sup>=</sup> *<sup>ρ</sup>* //

*<sup>α</sup>* =A *<sup>ρ</sup>* //

*<sup>ρ</sup>* / (C00G+Ar/

*<sup>ρ</sup>* / (1+C01(

procedure of the mixture-fluid mass and energy balance.

inverse (INV) form of this drift-flux correlation (with GS now as input).

Fourier heat conduction equation, demanding as boundary condition

qF=αTW(TTW-T) =

conduction through a U-tube wall (See also section 3.5).

G*S <sup>α</sup>*)→G*<sup>S</sup>* <sup>0</sup> *<sup>ρ</sup>* / )αC0→<sup>1</sup> if α→ *<sup>ρ</sup>* //

a)) (G -r/ /vSLIM) = Ar/

will play (as shown, for example, in eq.(52)) an important part, if looking to the special situation that the entrance or outlet position of a SC is crossing a BC node boundary (α → 0 or → 1). This possibility makes the drift-flux package MDS to an indispensable part in the nodalization

At a steady state situation as a result of the solution of the basic (algebraic) set of equations the steam mass flow term GS acts as an independent variable (and not the void fraction α). The same is the case after an injection of a two-phase mixture coming from a 'porous' channel or an abrupt change in steam mass flux GFS, as this can take place after a change in total mass flow or in the cross flow area at the entrance of a following BC. Then the total and the steam mass flow terms G and GS have to be taken as the basis for further two-phase considerations. The void fraction α and other two-phase parameters (vD, C0) can now be determined from an

The nodal BC heat power terms QBMk into the coolant are needed (as explained in section 3.5) as boundary condition for the energy balance equation (2). If they are not directly available (as this is the case for electrically heated loops) they have to be determined by solving an adequate

> qLTW UTW

Such a procedure is, for example, presented in (Hoeld, 2002b, 2011) for the case of heat

<sup>=</sup> <sup>A</sup> UTW

qD (13)

Knowing now the fourth variable then, by starting from their definition equations, relations for all the other characteristic two-phase parameters can be established. Such two-phase parameters could be the 'phase distribution parameter C0', the 'water and steam mass flows GW and GS', 'drift, water, steam and relative velocities vD, vW,vS and vR' (with special values for vS if α -> 0 and vW if α -> 1) and eventually the 'steam quality X'. Their interrelations are shown, for example, in the tables of (Hoeld, 2001 and 2002a). Especially the determination of

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

vD0) =Ar//vS0 or =0 if *<sup>α</sup>* <sup>→</sup>0 and LHEATD= 0 or 1

vW1 if α→1

*<sup>ρ</sup>* / if α→<sup>1</sup> (11)

http://dx.doi.org/10.5772/53372

(12)

13

with (in case of a heated or non-heated channel) C0 → 0 or 1 if α → 0 resp. C0 → 1 if α → 1 and corresponding drift velocity terms vD according to eq.(8.)

The resulting package MDS yields in combination with an adequate correlation for the phase distribution parameter C0 relations for the limit velocities vSLIM and vWLIM and thus (independ‐ ently of the total mass flow G which is important for the theory below) relations for the drift velocity vD with respect to the void fraction α. All of them are dependent on the given 'system pressure P', the 'hydraulic diameter dHY' (with respect to the wetted surface ATW and its inclination angle ΦZG), on specifications about the geometry type (LGTYPE) and, for low void fractions, the information whether the channel is heated or not.

The drift flux theory can thus be expressed in dependence of a (now already on G dependent) steam mass flow (or flux) term

$$\mathbf{G}\_{\rm S} \frac{\rho^{\prime \prime}}{\rho^{\prime}} \frac{\alpha}{\mathbf{C}\_{\rm GC}} (\mathbf{C}\_{0} \mathbf{G} + \mathbf{A} \rho^{\prime} \mathbf{v}\_{\rm D}) = \mathbf{A} \mathbf{G}\_{\rm FS} \tag{10}$$

with the coefficient

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 13

$$\mathbf{C\_{CC}} = \mathbf{1} \cdot (\mathbf{1} \cdot \frac{\boldsymbol{\rho}^{\prime\prime}}{\boldsymbol{\rho}^{\prime}}) \boldsymbol{\alpha} \mathbf{C\_0} \rightharpoonup \mathbf{1} \,\text{if } \boldsymbol{\alpha} \to \frac{\boldsymbol{\rho}^{\prime\prime}}{\boldsymbol{\rho}^{\prime}} \text{ if } \boldsymbol{\alpha} \to \mathbf{1} \tag{11}$$

Knowing now the fourth variable then, by starting from their definition equations, relations for all the other characteristic two-phase parameters can be established. Such two-phase parameters could be the 'phase distribution parameter C0', the 'water and steam mass flows GW and GS', 'drift, water, steam and relative velocities vD, vW,vS and vR' (with special values for vS if α -> 0 and vW if α -> 1) and eventually the 'steam quality X'. Their interrelations are shown, for example, in the tables of (Hoeld, 2001 and 2002a). Especially the determination of the steam mass flow gradient

$$\begin{aligned} \mathbf{G}\_{S}^{\alpha} &\rightarrow \mathbf{G}\_{S0}^{\alpha} = \frac{\rho^{\prime\prime}}{\rho^{\prime}} (\mathbf{C}\_{00} \mathbf{G} + \mathbf{A} \mathbf{r}^{\prime} \mathbf{v}\_{\rm D0}) = \mathbf{A} \mathbf{r}^{\prime\prime} \mathbf{v}\_{S0} \quad \text{or} \ \mathbf{0} &\quad \text{if } \alpha \rightharpoonup 0 \text{ and } \mathbf{L}\_{\text{HEATD}} \mathbf{n} = 0 \text{ or } 1 \\\ \mathbf{-G}\_{S1}^{\alpha} &\equiv \mathbf{A} \frac{\rho^{\prime\prime}}{\rho^{\prime}} (\mathbf{1} + \mathbf{C}\_{01} \mathbf{r}^{\prime}) \text{ (G - r}^{\prime} \mathbf{v}\_{\rm SLM} \mathbf{)} = \mathbf{A} \mathbf{r}^{\prime} \mathbf{v}\_{\rm W1} \text{ if } \alpha \to 1 \end{aligned} \tag{12}$$

will play (as shown, for example, in eq.(52)) an important part, if looking to the special situation that the entrance or outlet position of a SC is crossing a BC node boundary (α → 0 or → 1). This possibility makes the drift-flux package MDS to an indispensable part in the nodalization procedure of the mixture-fluid mass and energy balance.

At a steady state situation as a result of the solution of the basic (algebraic) set of equations the steam mass flow term GS acts as an independent variable (and not the void fraction α). The same is the case after an injection of a two-phase mixture coming from a 'porous' channel or an abrupt change in steam mass flux GFS, as this can take place after a change in total mass flow or in the cross flow area at the entrance of a following BC. Then the total and the steam mass flow terms G and GS have to be taken as the basis for further two-phase considerations. The void fraction α and other two-phase parameters (vD, C0) can now be determined from an inverse (INV) form of this drift-flux correlation (with GS now as input).

#### *2.2.4. Heat transfer coefficients*

thereby as a 'bridge' between GS and α. For example, by a slip correlation. However, to take care of stagnant or counter-current flow situations too an effective drift-flux correlation seemed here to be more appropriate. For this purpose an own package has been established, named MDS (Hoeld, 2001 and 2002a). Due to the different requirements in the application of CCM it turned out that it has a number of advantages if choosing the 'flooding-based full-range' Sonnenburg correlation (Sonnenburg, 1989) as basis for MDS. This correlation combines the common drift-flux procedure being formulated by (Zuber-Findlay, 1965) and expanded by (Ishii-Mishima, 1980) and (Ishii, 1990) etc. with the modern envelope theory. The correlation in the final package MDS had, however, to be rearranged in such a way that also the special cases of α → 0 or α → 1 are included and that, besides their absolute values and corresponding slopes, also the gradients of the approximation function can be made available for CCM. Additionally, an inverse form had to be installed (needed, for example, for the steady state conditions) and, eventually, also considerations with respect to possible entrainment effects

® ® α 0 (8)

(10)

C0<sup>α</sup> (9)

For the case of a vertical channel this correlation can be represented as

2 3/2 2 D WLIM 0 VD VD VD VD D D0 0 WLIM v = 1.5 v C C 1+C - 1.5+C C wit <sup>9</sup> [( ) ( ) ] h v v = C v if <sup>16</sup>

> CVD <sup>=</sup> <sup>2</sup> 3 vSLIM vWLIM

and corresponding drift velocity terms vD according to eq.(8.)

fractions, the information whether the channel is heated or not.

α CGC

(C0G+A*<sup>ρ</sup>* /

GS= *ρ* // *ρ* /

1-C0α

with (in case of a heated or non-heated channel) C0 → 0 or 1 if α → 0 resp. C0 → 1 if α → 1

The resulting package MDS yields in combination with an adequate correlation for the phase distribution parameter C0 relations for the limit velocities vSLIM and vWLIM and thus (independ‐ ently of the total mass flow G which is important for the theory below) relations for the drift velocity vD with respect to the void fraction α. All of them are dependent on the given 'system pressure P', the 'hydraulic diameter dHY' (with respect to the wetted surface ATW and its inclination angle ΦZG), on specifications about the geometry type (LGTYPE) and, for low void

The drift flux theory can thus be expressed in dependence of a (now already on G dependent)

vD) = AGFS

*α CGC*

be taken care.

where the coefficient CVD is given by

12 Nuclear Reactor Thermal Hydraulics and Other Applications

steam mass flow (or flux) term

with the coefficient

The nodal BC heat power terms QBMk into the coolant are needed (as explained in section 3.5) as boundary condition for the energy balance equation (2). If they are not directly available (as this is the case for electrically heated loops) they have to be determined by solving an adequate Fourier heat conduction equation, demanding as boundary condition

$$\mathbf{q}\_{\rm F} = \alpha\_{\rm TVW} \left( \mathbf{T}\_{\rm TV} \cdot \mathbf{T} \right) = \frac{\mathbf{q}\_{\rm L\_{\rm TVW}}}{\mathbf{U}\_{\rm TVW}} = \frac{\mathbf{A}}{\mathbf{U}\_{\rm TVW}} \mathbf{q}\_{\rm D} \tag{13}$$

Such a procedure is, for example, presented in (Hoeld, 2002b, 2011) for the case of heat conduction through a U-tube wall (See also section 3.5).

For this purpose adequate heat transfer coefficients are demanded. This means a method had to be found for getting these coefficients αTW along a coolant channel at different flow regimes. In connection with the development of the UTSG code (and thus also of CCM) an own very comprehensive heat transfer coefficient package, called HETRAC (Hoeld 1988a), has been established.

and k (n=k-NBCE with n=1, NCT), the corresponding positions (zNn, zELCE, zELNn), their lengths (ΔzNn=zNn-zNn-1), elevations (ΔzELNn=zELNn-zELNn-1) and volumes (VMn=zNnAMn) and nodal boun‐

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

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15

**Figure 1.** Subdivision of a 'basic channel (BC)' into 'sub-channels (SC-s)' according to their flow regimes and their dis‐

Based on this nodalization the spatial discretization of the fundamental eqs.(1) to (3) can be performed by means of a 'modified finite element method'. This means that if a partial differential equation (PDE) of 1-st order having the general form with respect to a general

**3.2. Spatial discretization of PDE-s of 1-st order (Modified finite element method)**

f z,t + H f z,t = R f z,t ( ) ( ) ( ) *t z*

¶ ¶ é ùé ù ë ûë û ¶ ¶ (14)

dary and mean nodal flow areas (ANn, AMn).

cretization

solution function f(z,t)

This classic method is different to the 'separate-phase' models where it has to be taken into account that the heat is transferred both directly from the wall to each of the two possible phases but also exchanged between them. There arises then the question how the correspond‐ ing heat transfer coefficients for each phase should look like.

## **3. Coolant channel module CCM**

#### **3.1. Channel geometry and finite-difference nodalization**

The theoretical considerations take advantage of the fact that, as sketched in fig.1, a 'basic' coolant channel (BC) can, as already pointed-out, according to their flow regimes (character‐ ized by the logical LFTYPE = 0, 1, 2 or 3) be subdivided into a number (NSCT) of sub-channels (SCs), each distinguished by their characteristic key numbers (NSC). Obviously, it has to be taken into account that their entrance and outlet SC-s can now have variable entrance and/or outlet positions.

The entire BC, with its total length zBT = zBA-zBE, can then, for discretization purposes, be also subdivided into a number of (not necessarily equidistant) NBT nodes. Their nodal positions are zBE, zBk (with k=1,NBT), the elevation heights zELBE, zELk, the nodal length ΔzBk=zBk-zBk-1, the nodal elevations ΔzELBk=zELBk-zELBk-1, with eventually also locally varying cross flow and average areas ABk and ABMk=0.5(ABk+ABk-1) and their slopes A *Bk <sup>z</sup>* = (ABk-ABk-1)/ΔzBk, a hydraulic diameter dHYBk and corresponding nodal volumes VBMk = ΔzBkABMk. All of them can be assumed to be known from input.

As a consequence, each of the sub-channels (SC-s) is then subdivided too, now into a number of NCT SC nodes with geometry data being identical to the corresponding BC values, except, of course, at their entrance and outlet positions. The SC entrance position zCE and their function fCE are either identical with the BC entrance values zBE and fBE or equal to the outlet values of the SC before. The SC outlet position (zCA) is either limited by the BC outlet (zBA) or character‐ ized by the fact that the corresponding outlet function has reached an upper or lower limit (fLIMCA). This the term represents either a function at the boiling boundary, a mixture level or the start position of a supercritical flow. Such a function follows from the given BC limit values and will, in the case of single-phase flow, be equal to the saturation temperature TSATCA or saturation enthalpies (h/ or h// if LFTYPE=1 or 2). In the case of two-phase flow (LFTYPE=0) it has to be equal to a void fraction of α = 1 or = 0. The moving SC inlet and outlet positions zCE and zCA can (together with their corresponding BC nodes NBCE and NBCA = NBCE+NCT) be determined according to the conditions (zBNk-1 ≤ zCE < zBNk at k = NBCE) and (zBNk-1 ≤ zCA < zBNk at k = NBCA). Then also the total number of SC nodes (NCT=NBCA-NBCE) is given, the connection between n and k (n=k-NBCE with n=1, NCT), the corresponding positions (zNn, zELCE, zELNn), their lengths (ΔzNn=zNn-zNn-1), elevations (ΔzELNn=zELNn-zELNn-1) and volumes (VMn=zNnAMn) and nodal boun‐ dary and mean nodal flow areas (ANn, AMn).

For this purpose adequate heat transfer coefficients are demanded. This means a method had to be found for getting these coefficients αTW along a coolant channel at different flow regimes. In connection with the development of the UTSG code (and thus also of CCM) an own very comprehensive heat transfer coefficient package, called HETRAC (Hoeld 1988a), has been

This classic method is different to the 'separate-phase' models where it has to be taken into account that the heat is transferred both directly from the wall to each of the two possible phases but also exchanged between them. There arises then the question how the correspond‐

The theoretical considerations take advantage of the fact that, as sketched in fig.1, a 'basic' coolant channel (BC) can, as already pointed-out, according to their flow regimes (character‐ ized by the logical LFTYPE = 0, 1, 2 or 3) be subdivided into a number (NSCT) of sub-channels (SCs), each distinguished by their characteristic key numbers (NSC). Obviously, it has to be taken into account that their entrance and outlet SC-s can now have variable entrance and/or outlet

The entire BC, with its total length zBT = zBA-zBE, can then, for discretization purposes, be also subdivided into a number of (not necessarily equidistant) NBT nodes. Their nodal positions are zBE, zBk (with k=1,NBT), the elevation heights zELBE, zELk, the nodal length ΔzBk=zBk-zBk-1, the nodal elevations ΔzELBk=zELBk-zELBk-1, with eventually also locally varying cross flow and average areas

dHYBk and corresponding nodal volumes VBMk = ΔzBkABMk. All of them can be assumed to be

As a consequence, each of the sub-channels (SC-s) is then subdivided too, now into a number of NCT SC nodes with geometry data being identical to the corresponding BC values, except, of course, at their entrance and outlet positions. The SC entrance position zCE and their function fCE are either identical with the BC entrance values zBE and fBE or equal to the outlet values of the SC before. The SC outlet position (zCA) is either limited by the BC outlet (zBA) or character‐ ized by the fact that the corresponding outlet function has reached an upper or lower limit (fLIMCA). This the term represents either a function at the boiling boundary, a mixture level or the start position of a supercritical flow. Such a function follows from the given BC limit values and will, in the case of single-phase flow, be equal to the saturation temperature TSATCA or

be equal to a void fraction of α = 1 or = 0. The moving SC inlet and outlet positions zCE and zCA can (together with their corresponding BC nodes NBCE and NBCA = NBCE+NCT) be determined according to the conditions (zBNk-1 ≤ zCE < zBNk at k = NBCE) and (zBNk-1 ≤ zCA < zBNk at k = NBCA). Then also the total number of SC nodes (NCT=NBCA-NBCE) is given, the connection between n

or h// if LFTYPE=1 or 2). In the case of two-phase flow (LFTYPE=0) it has to

*<sup>z</sup>* = (ABk-ABk-1)/ΔzBk, a hydraulic diameter

ing heat transfer coefficients for each phase should look like.

**3.1. Channel geometry and finite-difference nodalization**

ABk and ABMk=0.5(ABk+ABk-1) and their slopes A *Bk*

**3. Coolant channel module CCM**

14 Nuclear Reactor Thermal Hydraulics and Other Applications

established.

positions.

known from input.

saturation enthalpies (h/

**Figure 1.** Subdivision of a 'basic channel (BC)' into 'sub-channels (SC-s)' according to their flow regimes and their dis‐ cretization

#### **3.2. Spatial discretization of PDE-s of 1-st order (Modified finite element method)**

Based on this nodalization the spatial discretization of the fundamental eqs.(1) to (3) can be performed by means of a 'modified finite element method'. This means that if a partial differential equation (PDE) of 1-st order having the general form with respect to a general solution function f(z,t)

$$\frac{\partial}{\partial t}\mathbf{f}(\mathbf{z},\mathbf{t}) + \frac{\partial}{\partial \mathbf{z}}\mathbf{H}\Big[\mathbf{f}(\mathbf{z},\mathbf{t})\Big] = \mathbf{R}\Big[\mathbf{f}(\mathbf{z},\mathbf{t})\Big] \tag{14}$$

is integrated over the length of a SC node three types of discretization elements can be expected:

fMn <sup>s</sup> =2 (fMn-fNn-1) ΔzNn

have to fulfil the following requirements:

equations could be hurt).

→fCEI

cross BC node boundaries) in an appropriate and exact way.

SC input gradient (for the special case n = NCT =1). Thus

( )

= 2f - 3f + f if

Nn+1 Nn

Nn

<sup>2</sup> f =f =f = ( ) Δz

f -f f =( ) ( Δz +Δz

Nn

z zz Nn-1 CE CEI z Nn

=

*f z* ¶ ¶

Nn

z Mn+1 Mn

*3.3.1. Establishment of an effective and adequate approximation function*

<sup>z</sup> (at n=1) =input or→fNn-1

Hence, for this purpose a special 'quadratic polygon approximation' procedure, named 'PAX', had to be developed. It plays (together with the Leibniz rule presented above) an outstanding part in the development of the here presented 'mixture-fluid model' and helps, in particular, to solve the difficult task of how to take care of varying SC boundaries (which can eventually

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

The PAX procedure is based on the assumption that the solution function f(z) of a PDE (for example temperature or void fraction) is split into a number of NCT nodal SC functions fn(z,t). Each of them has then to be approximated by a specially constructed quadratic polygon which

**•** The node entrance functions (fNn-1) must be either equal to the SC entrance function (fNn-1 = fCE) (if n = 1) or to the outlet function of the node before (if n > 1). This is obviously not demanded for gradients of the nodal entrance functions (except for the last node at n = NCT). **•** The mean function values fMn over all SC nodes have to be preserved (otherwise the balance

**•** With the objective to guarantee stable behaviour of the approximated functions (for example by excluding 'saw tooth-like behaviour') it will, in an additional assumption, be demanded that the outlet gradients of the first NCT -1 nodes should be set equal to the slopes between their neighbour mean function values. The entrance gradient of the last node (n= NCT) should be either equal to the outlet gradient of the node before (if n = NCT > 1) or equal to a given

> (z) N

® D

Nn CT CT

<sup>2</sup> f 0z n = N , ( Δz if N > 1)

Nn Mn Nn n-1 -1 CA CT CT

( ) ( )

This means, the corresponding approximation function reaches not only over the node n. Its next higher one (n+1) has to be included into the considerations too (except, of course, for the last node). This assumption makes the PAX procedure very effective (and stable). It is a conclusive onset in this method since it helps to smooth the curve, guarantees that the gradients at the upper or lower SC boundary do not show abrupt changes if these boundaries cross a BC

of the node before n = N > 1

Mn CA CE CT

3f –f - 2f n = N =1

( )

CT

= 2 n=1, N -1, if N >1)

<sup>z</sup> (at n=NCT>1) if ΔzNn→<sup>0</sup> (17)

http://dx.doi.org/10.5772/53372

17

(18)

(19)


and, finally,

**•** integrating over a time-derivative of a function (by applying the 'Leibniz' rule) yields

$$\begin{aligned} \int\_{\gamma\_{\rm Mn}(t)}^{\gamma\_{\rm Mn}(t)} \mathbf{f}(\mathbf{z}, \mathbf{t}) d\mathbf{z} &= \Delta \mathbf{z}\_{\rm Mn}(\mathbf{t}) \frac{\mathbf{d}}{\mathrm{d}t} \mathbf{f}\_{\mathrm{Mn}}(\mathbf{t}) \cdot \mathbf{f}\_{\mathrm{Mn}}(\mathbf{t}) \cdot \mathbf{f}\_{\mathrm{Mn}}(\mathbf{t}) \frac{\mathbf{d}}{\mathrm{d}t} \mathbf{z}\_{\rm Mn}(\mathbf{t}) \\ &\quad \cdot \left[ \mathbf{f}\_{\mathrm{Mn}}(\mathbf{t}) \cdot \mathbf{f}\_{\mathrm{N}\mathrm{n}-1}(\mathbf{t}) \frac{\mathbf{d}}{\mathrm{d}t} \mathbf{z}\_{\mathrm{Mn}-1}(\mathbf{t}) \right. \end{aligned} \tag{15}$$

This last rule plays for the here presented 'separate-region mixture-fluid approach' an outstanding part. It allows (together with PAX) to determine in a direct way the timederivatives of parameters which represent either a boiling boundary, mixture or a supercritical level. This procedure differs considerably from some of the 'separate-phase methods' where, as already pointed out, very often only the collapsed levels of a mixture fluid can be calculated.

#### **3.3. Quadratic polygon approximation procedure PAX**

According to the above described three different types of possible discretization elements the solution of the set of algebraic equations will in the steady state case (as shown later-on) yield function values (fNn) at the node boundaries (zNn), the also needed mean nodal functions (fMn) will then have to be determined on the basis of fNn. On the other hand, the solution of the set of ordinary differential equations will in the transient case now yield the mean nodal functions fMn as a result, the also needed nodal boundary values fNn will have to be estimated on the basis of fMn.

It is thus obvious that appropriate methods had to be developed which can help to establish relations between such mean nodal (fMn) and node boundary (fNn) function values. Different to the 'separate-phase' models where mostly a method is applied (called 'upwind or donor cell differencing scheme') with the mean parameter values to be shifted (in flow direction) to the node boundaries in CCM a more detailed mixture-fluid approach is asked. This is also demanded because, as to be seen later-on from the relations of the sections 3.7 to 3.9, not only absolute nodal SC boundary or mean nodal function values are required but as well also their nodal slopes f *Nn <sup>s</sup>* and f *Mn <sup>s</sup>* together with their gradients f *Nn z* since according to this approach the length of SC nodes can tend also to zero.

$$\mathbf{f}\_{\text{Nn}}^{s} = \frac{(\mathbf{f}\_{\text{Nn}} \cdot \mathbf{f}\_{\text{Nn}-1})}{\Delta \mathbf{z}\_{\text{Nn}}} \to \mathbf{f}\_{\text{Cell}}^{x} \text{(at } \mathbf{n} = 1\text{)}\text{ =input}\text{ or } \to \mathbf{f}\_{\text{Nn-1}}^{s} \text{(at } \mathbf{n} = \mathbf{N}\_{\text{CT}} > 1\text{)}\text{ if } \mathbf{D}\mathbf{z}\_{\text{Nn}} \to 0\tag{16}$$

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 17

$$\mathbf{f}\_{\text{Mn}}^{\text{s}} = 2 \frac{(\mathbf{f}\_{\text{Mn}} - \mathbf{f}\_{\text{Mn}-1})}{\Delta \mathbf{z}\_{\text{Mn}}} \rightarrow \mathbf{f}\_{\text{CI}}^{\text{x}} \text{(at n=1)} \text{ =input or } \rightarrow \mathbf{f}\_{\text{Mn}-1}^{\text{x}} \text{ (at n=N}\_{\text{CI}} \text{>1)} \text{ if } \Delta \mathbf{z}\_{\text{Mn}} \rightarrow 0 \tag{17}$$

Hence, for this purpose a special 'quadratic polygon approximation' procedure, named 'PAX', had to be developed. It plays (together with the Leibniz rule presented above) an outstanding part in the development of the here presented 'mixture-fluid model' and helps, in particular, to solve the difficult task of how to take care of varying SC boundaries (which can eventually cross BC node boundaries) in an appropriate and exact way.

#### *3.3.1. Establishment of an effective and adequate approximation function*

is integrated over the length of a SC node three types of discretization elements can be expected:

**•** integrating over the gradient of a function f(z,t) yields a difference of functions values (fNn

dt fMn(t) - fNn(t) -fMn(t) <sup>d</sup>

This last rule plays for the here presented 'separate-region mixture-fluid approach' an outstanding part. It allows (together with PAX) to determine in a direct way the timederivatives of parameters which represent either a boiling boundary, mixture or a supercritical level. This procedure differs considerably from some of the 'separate-phase methods' where, as already pointed out, very often only the collapsed levels of a mixture fluid can be calculated.

According to the above described three different types of possible discretization elements the solution of the set of algebraic equations will in the steady state case (as shown later-on) yield function values (fNn) at the node boundaries (zNn), the also needed mean nodal functions (fMn) will then have to be determined on the basis of fNn. On the other hand, the solution of the set of ordinary differential equations will in the transient case now yield the mean nodal functions fMn as a result, the also needed nodal boundary values fNn will have to be estimated on the basis

It is thus obvious that appropriate methods had to be developed which can help to establish relations between such mean nodal (fMn) and node boundary (fNn) function values. Different to the 'separate-phase' models where mostly a method is applied (called 'upwind or donor cell differencing scheme') with the mean parameter values to be shifted (in flow direction) to the node boundaries in CCM a more detailed mixture-fluid approach is asked. This is also demanded because, as to be seen later-on from the relations of the sections 3.7 to 3.9, not only absolute nodal SC boundary or mean nodal function values are required but as well also their

*<sup>s</sup>* together with their gradients f *Nn*

<sup>f</sup> at n=1 =input or f at n=N >1 i (f -f ) <sup>f</sup> f D 0

® ®= z ® (16)

( ) ( ) sz s Nn Nn-1 Nn CEI Nn-1 CT Nn

dt zNn(t)

(15)

dt zNn-1(t) (n=1, NCT)

*z* since according to this approach

**•** Integrating a function f(z,t) over a SC node n yields the nodal mean function values fMn,

**•** integrating over a time-derivative of a function (by applying the 'Leibniz' rule) yields


– fNn-1) at their node boundaries

16 Nuclear Reactor Thermal Hydraulics and Other Applications

<sup>∂</sup> *<sup>t</sup>* f(z,t)dz =ΔzNn(t) <sup>d</sup>

**3.3. Quadratic polygon approximation procedure PAX**

*∫ zNn*−1(*t*)

*zNn*(*t*) ∂

and, finally,

of fMn.

nodal slopes f *Nn*

*<sup>s</sup>* and f *Mn*

Nn

Δz

the length of SC nodes can tend also to zero.

The PAX procedure is based on the assumption that the solution function f(z) of a PDE (for example temperature or void fraction) is split into a number of NCT nodal SC functions fn(z,t). Each of them has then to be approximated by a specially constructed quadratic polygon which have to fulfil the following requirements:


$$\begin{aligned} \mathbf{f}\_{\text{Nn}}^{x} &= (\frac{\varepsilon \ell}{\varepsilon z})\_{\text{Nn}} = 2 \frac{\mathbf{f}\_{\text{Mn}+1} \cdot \mathbf{f}\_{\text{Mn}}}{\Delta z\_{\text{Nn}+1} + \Delta z\_{\text{Nn}}} & \text{(n=1, N\_{\text{CT}}-1, if N\_{\text{CT}}>1)}\\ &= \frac{2}{\Delta z\_{\text{Nn}}} (\mathbf{2f}\_{\text{Nn}} \cdot \mathbf{3f}\_{\text{Mn}} + \mathbf{f}\_{\text{Nn}-1}) \to \mathbf{f}\_{\text{Nn-1}}^{(x)} \text{ if} \Delta z\_{\text{CA}} \, 0 \, (\text{n=N}\_{\text{CT}}, \text{if N}\_{\text{CT}} > 1) \end{aligned} \tag{18}$$

$$\begin{aligned} \mathbf{f}\_{\text{Nn-1}}^x &= \mathbf{f}\_{\text{CE}}^x = \mathbf{f}\_{\text{CEI}}^x = \frac{2}{\Delta \mathbf{z}\_{\text{Nn}}} (\mathbf{\hat{z}} \mathbf{f}\_{\text{Mn}} - \mathbf{f}\_{\text{CA}} \cdot \mathbf{2} \mathbf{f}\_{\text{CE}}) & \quad \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} = 1\right) \\ &= \mathbf{f}\_{\text{Nn}}^x \left(\text{of the node before}\right) \qquad \qquad \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} > 1\right) \end{aligned} \tag{19}$$

This means, the corresponding approximation function reaches not only over the node n. Its next higher one (n+1) has to be included into the considerations too (except, of course, for the last node). This assumption makes the PAX procedure very effective (and stable). It is a conclusive onset in this method since it helps to smooth the curve, guarantees that the gradients at the upper or lower SC boundary do not show abrupt changes if these boundaries cross a BC node boundary and has the effect that perturbations at channel entrance do not directly affect corresponding parameters of the upper BC nodes.

available from the integration. Thereby, as input data needed for the use in the PAX procedure

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

**•** if the now known SC outlet position is identical with the BC outlet (zCA=zBA) the mean nodal

**•** if the SC outlet position moves still within the BC (zCA < zBA) the mean nodal function values fMn of only NCT-1 nodes (n=1, NCT-1) are demanded. Additionally, the nodal function limit values fLIMNn (usually saturation temperature values at single-phase flow resp. void fraction values limited by 1 or 0 at mixture flow conditions) are asked. Then the missing mean nodal function fMn of the last SC is not any longer needed, since this function can be determined from eq.(20) in dependence of the known outlet function fNn = fCA = fLIMCA at zNn = zCA.

These nodal input function values can then be taken (together with its input parameter fCE and the nodal positions zBE and zBn at n=1, NCT) as basic points for the PAX procedure, yielding, after rearranging the definition equations of the approximation function in an adequate way,

Hence, it follows for the special situation of a SC being the last one within the BC (zCA = zBA)

=fCA = 2(fMn– fMn-1) + fNn-2 (n=NCT > 1 if zCA= zBA)

Mn Nn-1 LIMCA Nn-1 CT CT CA BA

= n = N if z < z ( CT CA BA )

The last mean nodal function fMn (at n=NCT) which is, for this case (zCA < zBA), not available from

The corresponding time-derivative of the last mean node function which is needed for the determination of the SC boundary time-derivative (see section 3.9) follows (as long as zCA < zBA)

<sup>1</sup> 3f - f + f - f n =N -1 with N > 1 if z < z

= ()

1 1 f = 3f - f + f - f n =1, N –2 with N > 2 if z < z 2 2

the integration, follows now directly from eq.(21) if replacing there fCA by fLIMCA.

Finally, from the eqs.(16), (17) and (19) the slopes and gradients can be determined.

ΔzNn+1+ΔzNn (fMn+1- fMn) (n=1, NCT-1 with NCT>1 if zCA= zBA)

Δz ( ) Δz Δz

<sup>z</sup> (n=NCT = 1 if zCA= zBA)

n CT CT CA BA

(21)

http://dx.doi.org/10.5772/53372

19

(22)

function values fMn for all NCT nodes (n=1,NCT) should be provided as input

all the other not directly known nodal function parameters of the SC.

it has now to be distinguished between two cases:

or

fNn= 1

<sup>2</sup> (3fMn- fNn-1) <sup>+</sup>

resp. for the case zCA < zBA

1

=3fMn- 2fCE-

CA LIMC

f = f

1

2

A

by differentiating the relation above

1 2

<sup>2</sup> ΔzCAfCEI

( ) ( )

Nn Mn Nn-1 + Mn+1 M

4

Nn Nn+1 Nn Nn Nn+1 Nn

z

( ) ( )

Δz Δz +Δ

ΔzNn

For the special case of a SC having shrunk to a single node (n=NCT=1) the quadratic approxi‐ mation demands as an additional input to PAX (instead of the now not available term fMn) the gradient f *CEI <sup>z</sup>* at SC entrance. It represents thereby the gradient of either the coolant tempera‐ ture T *CEI <sup>z</sup>* or void fraction <sup>α</sup> *CEI <sup>z</sup>* (in case of single- or two-phase flow entrance conditions). If this parameter is not directly available it can, for example, be estimated by combining the mass and energy balance equations at SC entrance in an adequate way (See Hoeld, 2005). This procedure allows to take care not only of SC-s consisting of only one single node but also of situations where during a transient either the first or last SC of a BC starts to disappear or to be created anew (i.e. zCA → zBE or zCE → zBA), since now the nodal mean value fMn at n = NCT (for both NCT = 1 or > 1) is no longer or not yet known.

#### *3.3.2. Resulting nodal parameters due to PAX*

It can be expected that in the steady state case after having solved the basic set of non-linear algebraic equations (as presented later-on in the sections 3.7, 3.8 and 3.9) as input to PAX the following parameters will be available:


and finally

**•** the nodal boundary functions fNn (n=1,NCT) with fCA = fNn at n = NCT and fCA= fLIMCA if zCA < zBA.

Based on these inputs PAX yields then the nodal mean function values fMn (at n=1,NCT)

$$\begin{aligned} \mathbf{f}\_{\text{Mn}} &= \frac{(\Delta \mathbf{z}\_{\text{Mn}+1} + \Delta \mathbf{z}\_{\text{Mn}})(2\mathbf{f}\_{\text{Mn}} + \mathbf{f}\_{\text{Mn}-1}) \cdot \Delta \mathbf{z}\_{\text{Mn}} \mathbf{f}\_{\text{Mn}+1}}{3\Delta \mathbf{z}\_{\text{Mn}+2} \Delta \mathbf{z}\_{\text{Mn}}}\\ &\quad \text{ (n=1, N\_{\text{CT}}-1, N\_{\text{CT}}>1 \text{ if } \mathbf{z}\_{\text{CA}} = \mathbf{z}\_{\text{BA}} \text{ or } \mathbf{n} = \mathbf{1}, N\_{\text{CT}}-2, N\_{\text{CT}}>2 \text{ if } \mathbf{z}\_{\text{CA}} \text{ or } \mathbf{z}\_{\text{RA}})\\ &\quad \mathbf{-\frac{1}{3}} (\mathbf{f}\_{\text{CA}} + 2 \mathbf{f}\_{\text{CE}}) + \frac{1}{6} \mathbf{D} \mathbf{z}\_{\text{CA}} \mathbf{f}\_{\text{CH}}^{\text{2}} &\quad \text{ (n=N\_{\text{CT}}=1)}\\ &\quad \mathbf{-\frac{1}{3}} (\mathbf{f}\_{\text{CA}} + 2 \mathbf{f}\_{\text{CE}}) + \frac{1}{6} \Delta \mathbf{z}\_{\text{CA}} \mathbf{f} (\mathbf{f}\_{\text{CA}} \cdot \mathbf{f}\_{\text{N} \text{m}}) \end{aligned} \tag{20}$$

functions which are needed as initial values for the transient case.

In the transient case it can be expected that after having integrated the set of non-linear ordinary differential equations (ODE-s) (as to be shown again in the sections 3.7, 3.8 and 3.9) the SC outlet position zCA (=zNn) and thus also the total number NCT of SC nodes will now be directly available from the integration. Thereby, as input data needed for the use in the PAX procedure it has now to be distinguished between two cases:

**•** if the now known SC outlet position is identical with the BC outlet (zCA=zBA) the mean nodal function values fMn for all NCT nodes (n=1,NCT) should be provided as input

or

node boundary and has the effect that perturbations at channel entrance do not directly affect

For the special case of a SC having shrunk to a single node (n=NCT=1) the quadratic approxi‐ mation demands as an additional input to PAX (instead of the now not available term fMn) the gradient f *CEI <sup>z</sup>* at SC entrance. It represents thereby the gradient of either the coolant tempera‐

this parameter is not directly available it can, for example, be estimated by combining the mass and energy balance equations at SC entrance in an adequate way (See Hoeld, 2005). This procedure allows to take care not only of SC-s consisting of only one single node but also of situations where during a transient either the first or last SC of a BC starts to disappear or to be created anew (i.e. zCA → zBE or zCE → zBA), since now the nodal mean value fMn at n = NCT

It can be expected that in the steady state case after having solved the basic set of non-linear algebraic equations (as presented later-on in the sections 3.7, 3.8 and 3.9) as input to PAX the

**•** Geometry data such as the SC entrance (zCE) and node positions (zNn) (and thus also the SC outlet boundary position zCA as explained in section 3.9) determining then in PAX the

**•** the SC entrance function fN0 = fCE and (at least for the special case n=NCT=1) its gradient f

**•** the nodal boundary functions fNn (n=1,NCT) with fCA = fNn at n = NCT and fCA= fLIMCA if zCA < zBA.

(z) (n=NCT= 1)

(20)

<sup>6</sup> ΔzCAf(fCA- fNn-2) (fCA- fNn-2)

In the transient case it can be expected that after having integrated the set of non-linear ordinary differential equations (ODE-s) (as to be shown again in the sections 3.7, 3.8 and 3.9) the SC outlet position zCA (=zNn) and thus also the total number NCT of SC nodes will now be directly

Based on these inputs PAX yields then the nodal mean function values fMn (at n=1,NCT)

(n =1, NCT-1, NCT> 1 ifzCA=zBAor n =1, NCT-2, NCT> 2 ifzCA<zBA)

functions which are needed as initial values for the transient case.

*CEI <sup>z</sup>* (in case of single- or two-phase flow entrance conditions). If

corresponding parameters of the upper BC nodes.

18 Nuclear Reactor Thermal Hydraulics and Other Applications

(for both NCT = 1 or > 1) is no longer or not yet known.

*3.3.2. Resulting nodal parameters due to PAX*

following parameters will be available:

fMn <sup>=</sup> (ΔzNn+1+ΔzNn)(2fNn+fNn-1)-ΔzNnfMn+1 3ΔzNn+1+2ΔzNn

<sup>6</sup> DzCAfCEI

number of SC nodes (NCT),

*CEI* (*z*)

and finally

= 1

= 1

<sup>3</sup> (fCA+2 fCE)+ <sup>1</sup>

<sup>3</sup> (fCA+2 fCE)+ <sup>1</sup>

ture T *CEI <sup>z</sup>* or void fraction <sup>α</sup>

**•** if the SC outlet position moves still within the BC (zCA < zBA) the mean nodal function values fMn of only NCT-1 nodes (n=1, NCT-1) are demanded. Additionally, the nodal function limit values fLIMNn (usually saturation temperature values at single-phase flow resp. void fraction values limited by 1 or 0 at mixture flow conditions) are asked. Then the missing mean nodal function fMn of the last SC is not any longer needed, since this function can be determined from eq.(20) in dependence of the known outlet function fNn = fCA = fLIMCA at zNn = zCA.

These nodal input function values can then be taken (together with its input parameter fCE and the nodal positions zBE and zBn at n=1, NCT) as basic points for the PAX procedure, yielding, after rearranging the definition equations of the approximation function in an adequate way, all the other not directly known nodal function parameters of the SC.

Hence, it follows for the special situation of a SC being the last one within the BC (zCA = zBA)

fNn= 1 <sup>2</sup> (3fMn- fNn-1) <sup>+</sup> 1 2 ΔzNn ΔzNn+1+ΔzNn (fMn+1- fMn) (n=1, NCT-1 with NCT>1 if zCA= zBA) =3fMn- 2fCE-1 <sup>2</sup> ΔzCAfCEI <sup>z</sup> (n=NCT = 1 if zCA= zBA) =fCA = 2(fMn– fMn-1) + fNn-2 (n=NCT > 1 if zCA= zBA) (21)

resp. for the case zCA < zBA

$$\begin{aligned} \text{f.\_{Nin}} &= \frac{1}{2} \Big( 3 \mathbf{f}\_{\text{Mn}} \cdot \mathbf{f}\_{\text{Nin}-1} \Big) + \frac{1}{2} \frac{\Delta \mathbf{z}\_{\text{Na}}}{\Delta \mathbf{z}\_{\text{Na}+1} \Delta \mathbf{z}\_{\text{Na}}} (\mathbf{f}\_{\text{Mn}+1} \cdot \mathbf{f}\_{\text{Mn}}) \text{ (n=1, N}\_{\text{CT}} - 2 \text{ with N}\_{\text{CT}} > 2 \text{ if } \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}}) \\ &= \frac{1}{2} \Big( 3 \mathbf{f}\_{\text{Mn}} \cdot \mathbf{f}\_{\text{Nin}-1} \Big) + \frac{1}{4} \frac{\Delta \mathbf{z}\_{\text{Na}}}{\Delta \mathbf{z}\_{\text{Na}+1} + \Delta \mathbf{z}\_{\text{Na}}} (\mathbf{f}\_{\text{LIMCA}} \cdot \mathbf{f}\_{\text{Nin}-1}) \quad \text{(n=N}\_{\text{CT}} - 1 \quad \text{with } \mathbf{N}\_{\text{CT}} > 1 \text{ if } \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}}) \\ &= \mathbf{f}\_{\text{CA}} = \mathbf{f}\_{\text{LIMCA}} \quad \text{(n=N}\_{\text{CT}} \qquad \text{if } \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}}) \end{aligned} \tag{22}$$

The last mean nodal function fMn (at n=NCT) which is, for this case (zCA < zBA), not available from the integration, follows now directly from eq.(21) if replacing there fCA by fLIMCA.

Finally, from the eqs.(16), (17) and (19) the slopes and gradients can be determined.

The corresponding time-derivative of the last mean node function which is needed for the determination of the SC boundary time-derivative (see section 3.9) follows (as long as zCA < zBA) by differentiating the relation above

$$\frac{\text{d}}{\text{d}t}\mathbf{f}\_{\text{Mn}} = \mathbf{f}\_{\text{PAXCA}}^{\text{t}} + \mathbf{f}\_{\text{PAXCA}}^{\text{x}} \frac{\text{d}}{\text{d}t}\mathbf{z}\_{\text{CA}} \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} \text{if } \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}}\right) \tag{23}$$

Before incorporating the subroutine into the overall coolant channel module the validity of the presented PAX procedure has been thoroughly tested. By means of a special driver code (PAXDRI) different characteristic and extreme cases have been calculated. The resulting curves of such a characteristic example are plotted in fig.2. It represents the two approximation curves of an artificially constructed void fraction distribution f(z) = α(z) along a SC with two-phase flow both for the steady state but also transient situation. Both curves (on the basis of fMn and

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

For the start of the transient calculations adequate steady state parameters have to be available

As boundary conditions for both the steady state and especially for transient calculations the following input parameters are expected to be known. Thereby demanding only easily available BC values (They will, within CCM, then be automatically translated into the

**•** Power profile along the entire BC. This means that either the nodal heat flux terms qFBE and qFBk (at BC entrance and each node k=1,NBT) or qFBE and the nodal power terms QBMk are expected to be known, either directly from input or (as explained in section 2.2.4) by solving

UTWBE

then the other BC nodal terms (qFBk or QBMk, qLBk and qDBk) can be determined too. (Hoeld,

**•** For normalization purposes at the starting calculation (i.e., at the steady state situation) as an additional parameter the total nominal (steady state) heat power QNOM,0 is asked.

**•** System pressure PSYS and its time-derivative (dPSYS/dt), situated at a fixed position either along the BC (entrance, outlet) or even outside of the ensemble. Due to the fast pressure wave propagation each local pressure time-derivative can then be set equal to the change

**•** Total mass flow GBEIN at BC entrance together with pressure terms at BC entrance PBEIN and outlet PBAIN. These three parameters are needed for steady state considerations (and partially

<sup>2</sup> ΔzBk(UTWBkqFBk+UTWBk-1qFBk-1) = VBMkqBMk

ABE qFBE (k=1,NBT)

(25)

http://dx.doi.org/10.5772/53372

21

the appropriate 'Fourier heat conduction equation'. From the relation

1

fNn) should be (and are) identical.

**3.4. Needed input parameters**

*3.4.1. Initial conditions*

as initial conditions.

*3.4.2. Boundary conditions*

corresponding SC values):

QBMk= <sup>1</sup>

2002b, 2004a and 2011).

<sup>2</sup> ΔzBk(qLBk+qLBk-1) =

with qLBE= UTWBEqFBEand qDBE=

**•** Channel entrance temperature TBEIN (or enthalpy hBEIN)

in system pressure (as described in section 3.6).

with the coefficients

$$\mathbf{f}\_{\rm PMAX}^{\rm t} = \frac{1}{3} (\frac{\rm d}{\rm dt} \mathbf{f}\_{\rm IIMCA} + 2 \frac{\rm d}{\rm dt} \mathbf{f}\_{\rm CE}) - \frac{1}{6} (\mathbf{f}\_{\rm CE}^{x} \frac{\rm d}{\rm dt} \mathbf{z}\_{\rm CE} \cdot \rm D \mathbf{z}\_{\rm CA} \frac{\rm d}{\rm dt} \mathbf{f}\_{\rm CE}^{x}) (\mathbf{n} \cdot \mathbf{N}\_{\rm CT} = 1 \text{ if } \mathbf{z}\_{\rm CA} < \mathbf{z}\_{\rm BA})$$

$$= \frac{\rm d}{\rm dt} \mathbf{f}\_{\rm Me-1} + \frac{1}{2} \frac{\rm d}{\rm dt} \mathbf{f}\_{\rm IIMCA} \cdot \frac{1}{2} \frac{\rm d}{\rm dt} \mathbf{f}\_{\rm Ne-2} \qquad\qquad\qquad \text{(n = N}\_{\rm CT} > 1 \text{ if } \mathbf{z}\_{\rm CA} < \mathbf{z}\_{\rm BA})\tag{24}$$

$$\mathbf{f}\_{\rm PACA}^{x} = \frac{1}{6} \mathbf{f}\_{\rm CE}^{(x)} \text{or} = 0 \qquad\qquad\qquad\qquad \left(\mathbf{n} = \mathbf{N}\_{\rm CT} = 1 \text{ or } > 1 \text{ if } \mathbf{z}\_{\rm CA} < \mathbf{z}\_{\rm BA}\right)$$

The differentials *<sup>d</sup> dt* fMn-1, *<sup>d</sup> dt* zCA, *<sup>d</sup> dt* fLIMCA are directly available from CCM and, if NCT=2, the term *d dt* fNn-2 = *<sup>d</sup> dt* fCE from input too. For the case that a SC contains more than two nodes only their corresponding mean values are known, the needed term *<sup>d</sup> dt* fNn-2 has thus to be estimated by establishing the time-derivatives of all the boundary functions at the nodes below NCT < 2.

#### *3.3.3. Code package PAX*

Based on the above established set of equations a routine PAX had to be developed. Its objective was to calculate automatically either the nodal mean or nodal boundary values (in case of an either steady state or transient situation). The procedure should allow also determining the gradients and slopes at SC entrance and outlet (and thus also outlet values characterizing the entrance parameters of an eventually subsequent SC). Additionally, contributions needed for the calculation of the time-derivatives of the boiling boundary or mixture level can be gained (See later-on the eqs.(66) and (67)).

**Figure 2.** Approximation function f(z) along a SC for both steady state and transient conditions after applying PAX (Example)

Before incorporating the subroutine into the overall coolant channel module the validity of the presented PAX procedure has been thoroughly tested. By means of a special driver code (PAXDRI) different characteristic and extreme cases have been calculated. The resulting curves of such a characteristic example are plotted in fig.2. It represents the two approximation curves of an artificially constructed void fraction distribution f(z) = α(z) along a SC with two-phase flow both for the steady state but also transient situation. Both curves (on the basis of fMn and fNn) should be (and are) identical.

#### **3.4. Needed input parameters**

## *3.4.1. Initial conditions*

( ) t z Mn PAXCA PAXCA CA CT CA BA d d +f dt <sup>d</sup> f = f z n =N if z < z

LIMCA CE CE CA CT CA BA

<sup>1</sup> f +2 f - (f z -Dz n =N =1 if z < z <sup>6</sup>

<sup>1</sup> dd d d f =( ) f )( ) <sup>3</sup> dt dt dt dt

dd d <sup>1</sup> <sup>=</sup> ( ) dt 2 dt dt

<sup>6</sup> n = N = 1 or > 1 if z < z ( CT CA BA )

by establishing the time-derivatives of all the boundary functions at the nodes below NCT < 2.

Based on the above established set of equations a routine PAX had to be developed. Its objective was to calculate automatically either the nodal mean or nodal boundary values (in case of an either steady state or transient situation). The procedure should allow also determining the gradients and slopes at SC entrance and outlet (and thus also outlet values characterizing the entrance parameters of an eventually subsequent SC). Additionally, contributions needed for the calculation of the time-derivatives of the boiling boundary or mixture level can be gained

**Figure 2.** Approximation function f(z) along a SC for both steady state and transient conditions after applying PAX

Nn-2 CT CA BA

*dt* fCE from input too. For the case that a SC contains more than two nodes only

*dt* fLIMCA are directly available from CCM and, if NCT=2, the

with the coefficients

z (z) PAXC

The differentials *<sup>d</sup>*

*3.3.3. Code package PAX*

(See later-on the eqs.(66) and (67)).

term *d dt* fNn-2 = *<sup>d</sup>*

(Example)

A CEI

f =f or = 0

1

Mn-1 LIMCA

20 Nuclear Reactor Thermal Hydraulics and Other Applications

*dt* fMn-1, *<sup>d</sup>*

t z z PAXCA CEI CEI

2

*dt* zCA, *<sup>d</sup>*

their corresponding mean values are known, the needed term *<sup>d</sup>*

<sup>1</sup> f + f - f n =N

<sup>t</sup> (23)

> 1 if z < z

*dt* fNn-2 has thus to be estimated

(24)

For the start of the transient calculations adequate steady state parameters have to be available as initial conditions.

## *3.4.2. Boundary conditions*

As boundary conditions for both the steady state and especially for transient calculations the following input parameters are expected to be known. Thereby demanding only easily available BC values (They will, within CCM, then be automatically translated into the corresponding SC values):

**•** Power profile along the entire BC. This means that either the nodal heat flux terms qFBE and qFBk (at BC entrance and each node k=1,NBT) or qFBE and the nodal power terms QBMk are expected to be known, either directly from input or (as explained in section 2.2.4) by solving the appropriate 'Fourier heat conduction equation'. From the relation

$$\mathbf{Q\_{BMk}} = \frac{1}{2} \Delta \mathbf{z\_{Bk}} (\mathbf{q\_{LBk}} + \mathbf{q\_{LBk-1}}) = \frac{1}{2} \Delta \mathbf{z\_{Bk}} (\mathbf{U\_{TWBk}} \mathbf{q\_{FBk}} + \mathbf{U\_{TWBk-1}} \mathbf{q\_{FBk-1}}) = \mathbf{V\_{BMk}} \mathbf{q\_{BMk}} \tag{25}$$
 
$$\text{with } \mathbf{q\_{LBE}} = \mathbf{U\_{TWBE}} \mathbf{q\_{FBE}} \text{ and } \mathbf{q\_{DBE}} = \frac{\mathbf{U\_{TWB}}}{A\_{\text{BE}}} \mathbf{q\_{FBE}} \tag{25}$$

then the other BC nodal terms (qFBk or QBMk, qLBk and qDBk) can be determined too. (Hoeld, 2002b, 2004a and 2011).


used for normalization purposes). In the transient case only two of them are demanded as input. The third one will be determined automatically by the model. These allows then to distinguish between the situation of an 'open' or 'closed channel' concept as this will be explained in more detail in section 3.11.

( ) ( ) ( )

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

( )

( ) ( )

k-1 CT BCA CA BA

LNn LCE LBE LCA of the last node of the SC before CE BE LBk CT BCE FTYPE

= q n=1,N -1and, if z = z , n= N with k=n+N

Δz

=q = q + q – q n=N and k=N if z Δz

= q n=1,N and k=n+N if L =2

LBk CT CA BA CT BCE

CA Bk

<sup>2</sup> ΔzNn(qLNn+qLNn-1) (n=1,NCT) = QBMk– (QMCA)of the last node of the SC before (n=1 with k=1+NBCE)

and the 'mean nodal' and 'nodal boundary SC power density' terms (qMn and qNn)

Mn Mn Mn LNn-1 LNn CT Nn CE BE CA of the last node of the SC befo CE BE

to be expected that ΔzNn -> 0 (as this can be seen later-on by the eqs.(32) and (49)).

**3.6. Decoupling of mass and energy balance from momentum balance equations**

q = and independently of the node length = q +q n =1,N q = q = q or = q n=0 if z = or > z

<sup>2</sup> (qLBk– qLBk-1) (qLBk– qLBk-1)

Q 1

manner, ΔzCA,,0 can be calculated (See eq.(65) in section 3.9).

= QBMk (n=2, NCT-1and k=n+NBCE)

ΔzCA

( ) ( )

2 A

re

n =1,N ( CT )

These last parameters are, since independent of ΔzNn, very useful for equations where it has

The SC length zCA (and thus that of its last node ΔzCA) are (for the case zCA,0 < zBA) only known in the transient case (as a result of the integration procedure). For steady state conditions the term QMCA,0 follows from energy balance considerations (eqs.(40) and (59)). Then, in a reverse

Treating the conservation equations in a direct way produces due to elements with fast pressure wave propagation (which are responsible for very small time constants) a set of 'stiff' ODE-s. This has the consequence that their solution turns out to be enormously CPU-time consuming. Hence, to avoid this costly procedure CCM has been developed with the aim to decouple the mass and energy from their momentum balance equations. This can be achieved by determining the thermodynamic properties of water and steam in the energy and mass balance equations on the basis of an estimated pressure profile P(z,t). Thereby the pressure difference terms from a recursive (or a prior computational time step) will be added to an eventually time-varying system pressure PSYS(t), known from boundary conditions. After having solved the two conservation equations for mass and energy (now separately from and not simultaneously with the momentum balance) the different nodal pressure gradient terms

( ) ( )

q = q = q =BC entr. or = q n=0 if z = or > z

LCA LBk-1 LBk LB

for the 'nodal SC power term'

= QMCA =DzCA[qLBk-1+ <sup>1</sup>

Mn Nn-

1

Mn

= 2q - q

V

QMn= <sup>1</sup>

( )

ΔzBk (n=NCTand k=NBCAif zCA< zBA)

( )

http://dx.doi.org/10.5772/53372

(26)

23

(27)

(28)

< z

**•** Steam mass flow GSBEIN at BC entrance (=0 or = GBEIN at single- or 0 < GSBEIN < GBEIN at twophase flow conditions). The corresponding entrance void fraction αBE will then be deter‐ mined automatically within the code by applying the inverse drift-flux correlation.

Eventually needed time-derivatives of such entrance functions can either be expected to be known directly from input or be estimated from their absolute values.

By choosing adequate boundary conditions then also thermal-hydraulic conditions of other situations can be simulated. For example that of several channel assembles (nuclear power plants, test loops etc.) which can consist of a complex web of pipes and branches (represented by different BC-s, all of them distinguished by their key numbers KEYBC). Even the case of an ensemble consisting of inner loops (for example describing parallel channels) can be treated in an adequate way according to the concept of a 'closed' channel (see section 3.11).

## *3.4.3. Solution vector*

The characteristic steady state parameters are determined in a direct way, i.e. calculated by solving the non-linear set of algebraic equations for SC-s (as being presented in the chapters 3.7, 3.8 and 3.9). Thereby, due to the nonlinearities in the set of the (steady state) constitutive equations a recursive procedure in combination with and controlled by the main program has to be applied until a certain convergence in the solution vector can be stated. The results are then combined to BC parameters and transferred again back to the main (= calling) program.

For the transient case, as a result of the integration (performed within the calling program and thus outside of CCM) the solution parameters of the set of ODE-s are transferred after each intermediate time step to CCM. These are (as described in detail also in chapter 4) mainly the mean nodal SC and thus BC coolant temperatures, mean nodal void fractions and the resulting boiling or superheating boundaries. These last two parameters allow then to subdivide the BC into SC-s yielding the corresponding constitutive parameters and the total and nodal length (zNn and ΔzNn) of these SC-s and thus also their total number (NCT) of SC nodes. Finally, the needed SC (and thus BC) time-derivatives can be determined within CCM (as described in the sections 3.7, 3.8 and 3.9) and then transmitted again to the calling program where the integra‐ tion for the next time step can take place.

#### **3.5. SC power profile**

Knowing (as explained in section 3.4.2) the nodal BC power QBMk together with the linear power qLBE at BC entrance the corresponding SC power profile can now be determined too.

Hence, since linear behaviour of the linear nodal power terms within the corresponding BC nodes can be assumed it follows for the 'linear SC power' term

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 23

$$\begin{aligned} \mathbf{q}\_{\text{LKB}} &= \mathbf{q}\_{\text{LCE}} = \mathbf{q}\_{\text{LBE}} \left( \mathbf{=BC} \text{ ent.r.} \right) \text{ or } = \left( \mathbf{q}\_{\text{LCA}} \right)\_{\text{of the last node of the SC before}} \left( \mathbf{n} = 0 \text{ if } \mathbf{z}\_{\text{CE}} = \text{or } \mathbf{z}\_{\text{BE}} \right) \\ &= \mathbf{q}\_{\text{LBK}} \qquad \left( \mathbf{n} = \mathbf{1}, \mathbf{N}\_{\text{C}} \text{and } \mathbf{k} = \mathbf{n} + \mathbf{N}\_{\text{BCE}} \text{if } \mathbf{I}\_{\text{FTPE}} = \mathbf{2} \right) \\ &= \mathbf{q}\_{\text{LBK}} \qquad \left( \mathbf{n} = \mathbf{1}, \mathbf{N}\_{\text{C}} - \mathbf{1} \text{and}, \text{ if } \mathbf{z}\_{\text{CA}} = \mathbf{z}\_{\text{BA}}, \mathbf{n} = \mathbf{N}\_{\text{C}T} \text{with } \mathbf{k} = \mathbf{n} + \mathbf{N}\_{\text{BCE}} \right) \\ &= \mathbf{q}\_{\text{LCA}} = \mathbf{q}\_{\text{LBK}-1} + \left( \mathbf{q}\_{\text{LBK}-1} \mathbf{q}\_{\text{LBK}-1} \right) \frac{\Delta \mathbf{z}\_{\text{CA}}}{\Delta \mathbf{z}\_{\text{BK}}} \left( \mathbf{n} = \mathbf{N}\_{\text{C}} \text{and } \mathbf{k} = \mathbf{N}\_{\text{BCA}} \text{if } \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}} \right) \end{aligned} \tag{26}$$

#### for the 'nodal SC power term'

used for normalization purposes). In the transient case only two of them are demanded as input. The third one will be determined automatically by the model. These allows then to distinguish between the situation of an 'open' or 'closed channel' concept as this will be

**•** Steam mass flow GSBEIN at BC entrance (=0 or = GBEIN at single- or 0 < GSBEIN < GBEIN at twophase flow conditions). The corresponding entrance void fraction αBE will then be deter‐

Eventually needed time-derivatives of such entrance functions can either be expected to be

By choosing adequate boundary conditions then also thermal-hydraulic conditions of other situations can be simulated. For example that of several channel assembles (nuclear power plants, test loops etc.) which can consist of a complex web of pipes and branches (represented by different BC-s, all of them distinguished by their key numbers KEYBC). Even the case of an ensemble consisting of inner loops (for example describing parallel channels) can be treated

The characteristic steady state parameters are determined in a direct way, i.e. calculated by solving the non-linear set of algebraic equations for SC-s (as being presented in the chapters 3.7, 3.8 and 3.9). Thereby, due to the nonlinearities in the set of the (steady state) constitutive equations a recursive procedure in combination with and controlled by the main program has to be applied until a certain convergence in the solution vector can be stated. The results are then combined to BC parameters and transferred again back to the main (= calling) program.

For the transient case, as a result of the integration (performed within the calling program and thus outside of CCM) the solution parameters of the set of ODE-s are transferred after each intermediate time step to CCM. These are (as described in detail also in chapter 4) mainly the mean nodal SC and thus BC coolant temperatures, mean nodal void fractions and the resulting boiling or superheating boundaries. These last two parameters allow then to subdivide the BC into SC-s yielding the corresponding constitutive parameters and the total and nodal length (zNn and ΔzNn) of these SC-s and thus also their total number (NCT) of SC nodes. Finally, the needed SC (and thus BC) time-derivatives can be determined within CCM (as described in the sections 3.7, 3.8 and 3.9) and then transmitted again to the calling program where the integra‐

Knowing (as explained in section 3.4.2) the nodal BC power QBMk together with the linear power

Hence, since linear behaviour of the linear nodal power terms within the corresponding BC

qLBE at BC entrance the corresponding SC power profile can now be determined too.

nodes can be assumed it follows for the 'linear SC power' term

in an adequate way according to the concept of a 'closed' channel (see section 3.11).

mined automatically within the code by applying the inverse drift-flux correlation.

known directly from input or be estimated from their absolute values.

explained in more detail in section 3.11.

22 Nuclear Reactor Thermal Hydraulics and Other Applications

*3.4.3. Solution vector*

tion for the next time step can take place.

**3.5. SC power profile**

$$\begin{aligned} \text{Q}\_{\text{Mn}} &= \frac{1}{2} \text{L}\_{\text{Na}} \text{(}\text{l}\_{\text{LiNa}} \star \text{l}\_{\text{LiNa}}\text{)} & \text{(}\text{m}\text{-l}\_{\text{Cl}}\text{)} \\ &\quad \text{-Q}\_{\text{BRA}} - \text{(}\text{Q}\_{\text{kCA}}\text{)}\_{\text{of the last node of the SC before}} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \\ &\quad \text{-Q}\_{\text{BRA}} \text{} & \text{(}\text{m}\text{-l, N}\_{\text{Cl}}\text{-}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{(}\text{l}\text{-}\text{R}^{\text{-}}\text{)} \text{$$

and the 'mean nodal' and 'nodal boundary SC power density' terms (qMn and qNn)

$$\begin{aligned} \mathbf{q}\_{\text{Mn}} &= \frac{\mathbf{q}\_{\text{Mn}}}{\mathbf{V}\_{\text{Mn}}} \text{and (independent) of the node length)} = \frac{1}{2\,\text{A}\_{\text{Mn}}} \left( \mathbf{q}\_{\text{LNa}-1} + \mathbf{q}\_{\text{LNa}} \right) \text{ (n = 1, \text{N}\_{\text{CT}})} \\ \mathbf{q}\_{\text{Mn}} &= \mathbf{q}\_{\text{CE}} = \mathbf{q}\_{\text{BE}} \text{ or } = \left( \mathbf{q}\_{\text{CA}} \right)\_{\text{of the last node of the SC before}} \qquad \left( \text{n=0 if } \mathbf{z}\_{\text{CE}} = \text{or } > \mathbf{z}\_{\text{BE}} \right) \\ &= 2\mathbf{q}\_{\text{Mn}} \cdot \mathbf{q}\_{\text{Nn-1}} \qquad \qquad \left( \text{n=1}, \text{N}\_{\text{CT}} \right) \end{aligned} \tag{28}$$

These last parameters are, since independent of ΔzNn, very useful for equations where it has to be expected that ΔzNn -> 0 (as this can be seen later-on by the eqs.(32) and (49)).

The SC length zCA (and thus that of its last node ΔzCA) are (for the case zCA,0 < zBA) only known in the transient case (as a result of the integration procedure). For steady state conditions the term QMCA,0 follows from energy balance considerations (eqs.(40) and (59)). Then, in a reverse manner, ΔzCA,,0 can be calculated (See eq.(65) in section 3.9).

#### **3.6. Decoupling of mass and energy balance from momentum balance equations**

Treating the conservation equations in a direct way produces due to elements with fast pressure wave propagation (which are responsible for very small time constants) a set of 'stiff' ODE-s. This has the consequence that their solution turns out to be enormously CPU-time consuming. Hence, to avoid this costly procedure CCM has been developed with the aim to decouple the mass and energy from their momentum balance equations. This can be achieved by determining the thermodynamic properties of water and steam in the energy and mass balance equations on the basis of an estimated pressure profile P(z,t). Thereby the pressure difference terms from a recursive (or a prior computational time step) will be added to an eventually time-varying system pressure PSYS(t), known from boundary conditions. After having solved the two conservation equations for mass and energy (now separately from and not simultaneously with the momentum balance) the different nodal pressure gradient terms can (by the then following momentum balance considerations) be determined according to the eqs.(4), (5) and (6).

It can additionally be assumed that according to the very fast (acoustical) pressure wave propagation along a coolant channel all the local pressure time-derivatives can be replaced by a given external system pressure time-derivative, i.e.,

$$\frac{\text{d}}{\text{dt}}\text{P}(\text{z},\text{t}) \equiv \frac{\text{d}}{\text{dt}}\text{P}\_{\text{SYS}}\tag{29}$$

TTn

<sup>t</sup> <sup>=</sup> qMn - qGn <sup>+</sup> qPn ρMn hMn <sup>T</sup> CTMn

> <sup>=</sup> <sup>1</sup> VMn ρMn hMn

+ T <sup>d</sup>

<sup>T</sup> CTMn

dt zNn-1 with <sup>d</sup>

r

r

t

Zn Nn Mn Nn Nn-1 M

s s TCE n Nn

2(1- )T ] <sup>2</sup> A C

Mn TM

n

r

r

<sup>1</sup> <sup>A</sup> T = [T -

z s Nn TCA Nn

<sup>1</sup> <sup>A</sup> T T 2 A C

follow also the SC nodal terms TNn, T *Nn*

are fixed.

CE CE

n

M

= 0 or =

z

dt zNn-1= <sup>d</sup>

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

T T Mn Nn Mn Mn T Mn Mn Mn TMn Nn Mn h -h C = 1 - =1- Th

> Nn-1 Mn G s

Pn RHPn SYS RH ( ) P P Pn Mn <sup>M</sup> <sup>M</sup> Nn Mn nn q = C P and C = 1- <sup>d</sup> rh + h- <sup>d</sup><sup>h</sup>

<sup>1</sup> d d <sup>=</sup> {A <sup>r</sup> <sup>h</sup> <sup>z</sup> +A [ ( q ) )] } 2A dt dt r

(s) (s) (s) Nn n Mn Mn Nn-1 Nn

Mn Mn TMn FTYPE

if either z >

CT CA BA CT CA BA E

<sup>M</sup> or = 0 at n = 1 and L > 0

( )

In the transient case the mean nodal coolant temperature value TMn is at the begin of each (intermediate) time step known. This either, at the first time step, from steady state consider‐ ations (in combination with PAX) or as a result of the integration procedure. Hence, additional parameters needed in the relations above can be determined too. From the PAX procedure it

(*s*) and the slopes T *Mn*

their gradients. Finally, using the water/steam tables (Hoeld, 1996), also their nodal enthalpies

From the integration procedure then also the SC outlet position zCA is provided, allowing to determine the total number of SC nodes (NCT) too. The situation that zCA = zBT means the SC nodal boundary temperature values have, within the entire BC, not yet reached their limit values (TLIMNn=TSATNn). Thus NCT= NBCA with NBCA=NBT–NBCE. Otherwise, if zCA < zBT this limit is

n < N and z < z or n = N and z =

[QMn - GNn-1 (hNn - hNn-1) + VMn qPn]+ T+ TTCE

 r

 r

dt zCE or = 0 if n = 1 or > 1

( )

r

 rr

T

Gn <sup>A</sup> Nn q h= (34)

CE BE CE

= z( )

(37)

z or z

z d dt zNn-1

http://dx.doi.org/10.5772/53372

(33)

(35)

Nn-1

B

FTYP

(*s*) resp., for the case that ΔzNn → 0,

z , L > 0

E

h -2 - h z (36)

(32)

25

(38)

By applying the above explained 'intelligent' (since physically justified) simplification in CCM the small, practically negligible, error in establishing the thermodynamic properties on the basis of such an estimated pressure profile can be outweighed by the enormous benefit substantiated by two facts:


#### **3.7. Thermal-hydraulics of a SC with single-phase flow (LFTYPE > 0)**

The spatial integration of the two PDE-s of the conservation eqs.(1) and (2) over a (single-phase) SC node n yields (by taking into account the rules from section 3.2, the relations from the eqs. (7) and (29), the possibility of a locally changing nodal cross flow area along the BC and the fact that eventually VMn -> 0 ) for the transient case the relation

**•** a relation for the total nodal mass flow

$$\mathbf{C\_{Nn}} = \mathbf{C\_{Nn-1}} \cdot \mathbf{V\_{Mn}}(\mathbf{q\_{Mn}^T} \frac{\mathbf{d}}{\mathbf{dt}} \mathbf{T\_{Mn}} + \mathbf{q\_{Mn}^P} \frac{\mathbf{d}}{\mathbf{dt}} \mathbf{P\_{SYS}}) + (\mathbf{r\_{Nn}} \cdot \mathbf{r\_{Mn}}) \mathbf{A\_{Nn}} \mathbf{P\_{SYS}} + (\mathbf{r\_{Nn}} \cdot \mathbf{r\_{Mn}}) \mathbf{A\_{Nn}} \frac{\mathbf{d}}{\mathbf{dt}} \mathbf{z\_{Nn}} \tag{30}$$
 
$$+ (\mathbf{r\_{Mn}} \cdot \mathbf{r\_{Nn-1}}) \mathbf{A\_{Nn-1}} \frac{\mathbf{d}}{\mathbf{dt}} \mathbf{z\_{Nn-1}} \tag{31}$$

and

**•** the time-derivative for the mean nodal coolant temperature (if eliminating the term GNn in the resulting equation by inserting from the equation above):

$$\begin{aligned} \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{T}\_{\mathrm{Mn}} &= \mathbf{T}\_{\mathrm{Tn}}^{\mathrm{t}} + \mathbf{T}\_{\mathrm{CCA}}^{\mathrm{x}} \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\mathrm{Mn}} \text{ with } \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\mathrm{Mn}} = \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\mathrm{CA}} \text{ or } = 0 \text{ if } \mathbf{n} = \mathbf{N}\_{\mathrm{CT}} \text{ or } \mathbf{Q} \mathbf{N}\_{\mathrm{CT}}\\ \left(\mathbf{n} = \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} \text{ and } \mathbf{z}\_{\mathrm{CA}} = \mathbf{z}\_{\mathrm{BA}}\right) \text{ or } \left(\mathbf{n} = \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} \text{ -1 and } \mathbf{z}\_{\mathrm{CA}} < \mathbf{z}\_{\mathrm{BA}}\right), \mathbf{L}\_{\mathrm{FT} \text{TE}} > 0 \end{aligned} \tag{31}$$

with the coefficients

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 25

$$\begin{aligned} \mathbf{T}\_{\mathrm{Tn}}^{\mathrm{t}} &= \frac{\mathbf{q\_{\mathrm{Mn}}} \cdot \mathbf{q\_{\mathrm{Zn}}} + \mathbf{q\_{\mathrm{Pn}}}}{\mathbf{q\_{\mathrm{Mn}}} \mathbf{h\_{\mathrm{Mn}}^{\mathrm{T}}} \mathbf{C\_{\mathrm{TMn}}}} + \mathbf{T}\_{\mathrm{dT}}^{\mathrm{d}} \mathbf{z}\_{\mathrm{Mn-1}} \text{ with} \frac{\mathrm{d}}{\mathrm{d}\mathrm{T}} \mathbf{z}\_{\mathrm{Mn-1}} \mathbf{r} \frac{\mathrm{d}}{\mathrm{d}\mathrm{T}} \mathbf{z}\_{\mathrm{CE}} \text{ or } = 0 \text{ if } \mathrm{n} = 1 \text{ or } > 1 \end{aligned} \tag{32}$$
 
$$= \frac{1}{\mathbf{V\_{Mn}} \mathbf{q\_{Mn}} \mathbf{h\_{Mn}^{\mathrm{T}}} \mathbf{C\_{\mathrm{Tm}}}} \left[ \mathbf{Q\_{Mn}} \cdot \mathbf{G\_{\mathrm{Mn-1}}} (\mathbf{h\_{\mathrm{Mn}}} \cdot \mathbf{h\_{\mathrm{Mn-1}}}) + \mathbf{V\_{Mn}} \mathbf{q\_{\mathrm{Pn}}} \right] + \mathbf{T} + \mathbf{T}\_{\mathrm{TCE}}^{\mathrm{Z}} \frac{\mathrm{d}}{\mathrm{d}\mathrm{T}} \mathbf{z}\_{\mathrm{Mn-1}}$$

can (by the then following momentum balance considerations) be determined according to the

It can additionally be assumed that according to the very fast (acoustical) pressure wave propagation along a coolant channel all the local pressure time-derivatives can be replaced by

> ( ) SYS d d dt d P z

By applying the above explained 'intelligent' (since physically justified) simplification in CCM the small, practically negligible, error in establishing the thermodynamic properties on the basis of such an estimated pressure profile can be outweighed by the enormous benefit

**•** the calculation of the mass flow distribution into different channels resulting from pressure balance considerations can, in a recursive way, be adapted already within each integration time step, i.e. there is no need to solve the entire set of differential equations for this purpose

The spatial integration of the two PDE-s of the conservation eqs.(1) and (2) over a (single-phase) SC node n yields (by taking into account the rules from section 3.2, the relations from the eqs. (7) and (29), the possibility of a locally changing nodal cross flow area along the BC and the

**•** the time-derivative for the mean nodal coolant temperature (if eliminating the term GNn in

( ) ( )

T T

dt

Mn Nn Nn CA CT C

T = T +T z with z = z or = 0 if n =N or <N d d dd <sup>d</sup>

n=1, N and z =z or n=1, N -

dt PSYS) + (rNn-rMn)ANnPSYS) + (rNn-rMn)ANn

CA BA CT CA BA FTYPE

1 and z < z , L

dt zNn-1 (n=1,NCT), LFTYPE> 0

d dt zNn

> 0

(30)

(31)

<sup>t</sup> ,t P @ (29)

eqs.(4), (5) and (6).

substantiated by two facts:

a given external system pressure time-derivative, i.e.,

24 Nuclear Reactor Thermal Hydraulics and Other Applications

(See 'closed channel' concept in section 3.11).

**•** a relation for the total nodal mass flow

T d

+ (rMn-rNn-1)ANn-1

CT

t z

n TCA

t dt dt

GNn= GNn-1- VMn(ρMn

with the coefficients

and

**•** Avoidance of the very time-consuming solution of stiff equations,

**3.7. Thermal-hydraulics of a SC with single-phase flow (LFTYPE > 0)**

fact that eventually VMn -> 0 ) for the transient case the relation

P d

d

the resulting equation by inserting from the equation above):

dt TMn+ρMn

$$\mathbf{C}\_{\text{TMn}} = \mathbf{1} \cdot \frac{\rho\_{\text{Mn}}^{\text{T}}}{\rho\_{\text{Mn}}} \frac{\mathbf{h}\_{\text{Nn}} \cdot \mathbf{h}\_{\text{Mn}}}{\mathbf{h}\_{\text{Mn}}^{\text{T}}} = \mathbf{1} \cdot \frac{\rho\_{\text{Mn}}^{\text{T}}}{\rho\_{\text{Mn}}} \left(\mathbf{T}\_{\text{Nn}} \cdot \mathbf{T}\_{\text{Mn}}\right) \tag{33}$$

$$\mathbf{q}\_{\rm Con} = \frac{\mathbf{G}\_{\rm Nm-1}}{\mathbf{A}\_{\rm Mn}} \mathbf{h}\_{\rm Nm}^{\rm s} \tag{34}$$

$$\mathbf{q}\_{\rm Pn} = \mathbf{C}\_{\rm RH\rm Pn} \frac{\mathbf{d}}{\mathbf{d}t} \mathbf{P}\_{\rm SY} \quad \text{and} \; \mathbf{C}\_{\rm RH\rm Pn} = \mathbf{1} \cdot \mathbf{r}\_{\rm Mn} \mathbf{h}\_{\rm Mn}^{\rm P} + \rho\_{\rm Mn}^{\rm P} \left(\mathbf{h}\_{\rm Mn} \cdot \mathbf{h}\_{\rm Mn}\right) \tag{35}$$

$$\mathbf{q\_{Zn}} = \frac{1}{2\mathbf{A\_{Mn}}} \{ \mathbf{A\_{Mn}} \mathbf{r\_{Mn}} \mathbf{h\_{Mn}^{(s)}} \frac{\mathbf{d}}{\mathbf{d}t} \mathbf{z\_{Mn}} + \mathbf{A\_{Mn-1}} [\boldsymbol{\rho}\_{\mathbf{Mn}} \mathbf{h\_{Mn}^{(s)}} \mathbf{2} (\boldsymbol{\rho}\_{\mathbf{Mn}} \cdot \boldsymbol{\rho}\_{\mathbf{Mn-1}}) \mathbf{h\_{Mn}^{(s)}}] \frac{\mathbf{d}}{\mathbf{d}t} \mathbf{z\_{Mn-1}} \} \tag{36}$$

$$\mathbf{T}\_{\rm CE}^{x} = \frac{1}{2} [\mathbf{T}\_{\rm Mn}^{s} \cdot \mathbf{2} (1 - \frac{\rho\_{\rm CE}}{\rho\_{\rm Mn}}) \mathbf{T}\_{\rm Mn}^{s}] \frac{\mathbf{A}\_{\rm CE}}{\mathbf{A}\_{\rm Mn} \mathbf{C}\_{\rm Mn}} \text{or } = 0 \text{ at } \mathbf{n} = 1 \text{ and } \mathbf{L}\_{\rm FTYPE} > 0 \tag{37}$$
 
$$\text{ (if either } \mathbf{z}\_{\rm CE} > \mathbf{z}\_{\rm BE} \text{ or } \mathbf{z}\_{\rm CE} = \mathbf{z}\_{\rm BE})$$

$$\begin{aligned} \mathbf{T\_{TA}^z} = 0 \text{ or } &= \frac{1}{2} \frac{\mathbf{A\_{Nn}}}{\mathbf{A\_{Mn}} \mathbf{C\_{TMn}}} \quad \mathbf{T\_{Nn}^s} \\ & \quad \left(\mathbf{n < N\_{CT}} \text{ and } \mathbf{z\_{CA}} < \mathbf{z\_{BA}} \text{ or } \mathbf{n = N\_{CT}} \text{and } \mathbf{z\_{CA}} = \mathbf{z\_{BA}}\right) / \text{ L}\_{\text{FTTPE}} > 0 \end{aligned} \tag{38}$$

In the transient case the mean nodal coolant temperature value TMn is at the begin of each (intermediate) time step known. This either, at the first time step, from steady state consider‐ ations (in combination with PAX) or as a result of the integration procedure. Hence, additional parameters needed in the relations above can be determined too. From the PAX procedure it follow also the SC nodal terms TNn, T *Nn* (*s*) and the slopes T *Mn* (*s*) resp., for the case that ΔzNn → 0, their gradients. Finally, using the water/steam tables (Hoeld, 1996), also their nodal enthalpies are fixed.

From the integration procedure then also the SC outlet position zCA is provided, allowing to determine the total number of SC nodes (NCT) too. The situation that zCA = zBT means the SC nodal boundary temperature values have, within the entire BC, not yet reached their limit values (TLIMNn=TSATNn). Thus NCT= NBCA with NBCA=NBT–NBCE. Otherwise, if zCA < zBT this limit is reached (at node n), then NCT = n. Obviously, the procedure above yields also the timederivative of the SC outlet position moving within this channel (As described in section 3.9).

The steady state part of the total nodal mass flow (charaterized by the index 0) follows from the basic non-linear algebraic equation (30) if setting there the time-derivative equal to 0:

$$\mathbf{G}\_{\text{Nin},0} = \mathbf{G}\_{\text{CA},0} = \mathbf{G}\_{\text{CE},0} = \mathbf{G}\_{\text{BA},0} = \mathbf{G}\_{\text{BE},0} \qquad \left(\text{n=1}, \text{N}\_{\text{CT}}\right) \text{ } \mathbf{L}\_{\text{FTYP}} \gg 0 \tag{39}$$

And

d dt

then relations for

αGZn <sup>t</sup> =αCE z d dt zCE<sup>=</sup> <sup>1</sup> 2 ACE AMn αMn s d

and

= αCA z d

n=1,N a

with the coefficients

**•** the total mass flow term

saturation values, the coefficients

GNn = GNn-1+VMn(*ρ*'-*<sup>ρ</sup>* '')Mn( <sup>d</sup>

αGPn <sup>t</sup> <sup>=</sup> <sup>1</sup>

dt zCA<sup>=</sup> ACA AMn

**•** the mean nodal void fraction time-derivative

(αCA <sup>s</sup> - 1 <sup>2</sup> αMn <sup>s</sup> ) <sup>d</sup>

Mn Nn Nn-1

α = α z +α z

tt z z ASn APn CA CE

d d d


( ) ( )

nd z =z or n=1, N -1, N >1

CT CA BA CT CT CA B

( )

1, N , if L =1 or = 2

CT FTY

n=

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

The spatial integration of the two PDE-s of the conservation) eqs.(1) and (2) (now over the mixture-phase SC nodes n) can be performed by again taking into account the rules from section 3.2, the relations from the eqs.(7) and (29), by considering the possibility of locally changing nodal cross flow areas along the BC) and the fact that eventually VMn -> 0. This yields

dt αMn-αGPn

(ρ'-ρ'')Mn [(1-α)*<sup>ρ</sup>* /P+α*<sup>ρ</sup>* //P

<sup>t</sup> -αGZn <sup>t</sup> )

> Mn d

dt zCE (n = 1 and zCE> zBE)

dt zCA (n=NCT, NCT>1 and zCA< zBA)

with, if neglecting thereby small differences between mean and nodal thermodynamic

=0 (1<n<NCT)

h

SWNn,0 Mn Nn Nn,0 Mn,0 Nn,0

<sup>h</sup> α = 0 , α = α = α = 0 or = 1 and X =

**3.8. Thermal-hydraulics of a SC with two-phase flow (LFTYPE = 0)**

/ Nn,0 Nn,0

PE

(n=1,NCT, LFTYPE= 0) (45)

dt PSYS (46)

A FTYPE

and z <z , L =0

(44)

27

http://dx.doi.org/10.5772/53372

(47)

(48)


Treating eq.(31) in a similar way and multiplying the resulting relation by VMn,0 yields the steady state nodal temperature resp. enthalpy terms

$$\begin{aligned} \mathbf{h}\_{\text{Nn},0} = \mathbf{h}\_{\text{Nn}-1,0} + \frac{\mathbf{q}\_{\text{Nn},0}}{\mathbf{G}\_{\text{B},0}} \le \mathbf{h}\_{\text{Nn},0}^{\prime} \text{ or } \mathbf{z} \ge \mathbf{h}\_{\text{Nn},0}^{\prime\prime} \qquad \left(\text{with } \mathbf{h}\_{\text{Nn}-1,0} = \mathbf{h}\_{\text{CE},0} \text{at } \mathbf{n} = 1\right) \\ \quad \left(\text{if } \mathbf{L}\_{\text{FYPE}} = 1 \text{ or } = 2 \text{ at } \mathbf{n} = 1, \mathbf{N}\_{\text{BT}} \cdot \mathbf{N}\_{\text{BCE}}\right) \end{aligned} \tag{40}$$

restricted by their saturation values. Then, with regard to eq.(27) which results from energy balance considerations, the nodal power term for the last SC node has to obey the relation

$$\begin{aligned} \mathbf{Q\_{MCA,0}} = \mathbf{Q\_{Mn,0}} &= \begin{pmatrix} \mathbf{h\_{CA,0}^{\prime}} \cdot \mathbf{h\_{Nn-1,0}} \end{pmatrix} \mathbf{G\_{BE,0}} & \quad \begin{pmatrix} \text{if } \mathbf{n=N\_{CT}} \le \mathbf{N\_{BCA}} \text{at } \mathbf{L\_{FTYE}} = 1 \end{pmatrix} \\ &= \begin{pmatrix} \mathbf{h\_{Nn-1,0}} - \mathbf{h\_{CA,0}^{\prime\prime}} \end{pmatrix} \mathbf{G\_{BE,0}} & \quad \begin{pmatrix} \text{if } \mathbf{n=N\_{CT}} < \mathbf{N\_{BCA}} \text{at } \mathbf{L\_{FTYE}} = 2 \end{pmatrix} \end{aligned} \tag{41}$$

Thus, for the steady state case, NCT is fixed too:

$$\begin{aligned} \mathbf{N\_{CT} = n \times N\_{BCA} = N\_{BT} - N\_{BCE} + 1 \text{ and } \mathbf{z\_{CA,0}} \left(\mathbf{w\_{Nn,0}}\right) < \mathbf{z\_{BA}} \text{ (if } \mathbf{h\_{Nn,0} = h\_{Nn,0}^{\prime} \text{ at } \mathbf{L\_{FTPE} = 1}\text{)} \\ \mathbf{N\_{CT} = n = N\_{BCA} \text{ and } \mathbf{z\_{CA,0}} = \mathbf{z\_{BA}} \text{ (if } \mathbf{h\_{Nn,0} < h\_{Nn,0}^{\prime} \text{ at } \mathbf{L\_{FTPE} = 1}\text{)} \end{aligned} \tag{42}$$

with similar relations for the case LFTYPE = 2.

From the resulting steady state enthalpy value hNn,0 follows then (by using the thermodynamic water/steam tables) the corresponding coolant temperature value TNn,0 (with TNn,0 = TSATNn,0 if n = NCT and zCA < zBA) and, by applying the PAX procedure (see section 3.3), their mean nodal temperature and enthalpy values TMn,0 and hMn,0, parameters which are needed as start values for the transient calculations. Obviously, due to the non-linearity of the basic steady state equations, this procedure has to be done in a recursive way.

It can additionally be stated that both the steady state and transient two-phase mass flow parameters get the trivial form

$$\begin{aligned} \mathbf{G\_{SNn}} &= \mathbf{G\_{SNn,0}} = \mathbf{0} \text{ resp. } \mathbf{G\_{WNn}} = \mathbf{G\_{CE}} \text{ and } \mathbf{G\_{WNn,0}} = \mathbf{G\_{BE,0}} \quad \text{(n=1, N\_{CT}, if L\_{FTPE} = 1)}\\ \mathbf{G\_{SNn}} &= \mathbf{G\_{CE}} \text{ and } \mathbf{G\_{SNn,0}} = \mathbf{G\_{BE,0}} \quad \text{resp. } \mathbf{G\_{WNn}} = \mathbf{G\_{WNn,0}} = \mathbf{0} \text{ (n=1, N\_{CT}, if L\_{FTPE} = 2)} \end{aligned} \tag{43}$$

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 27

And

reached (at node n), then NCT = n. Obviously, the procedure above yields also the timederivative of the SC outlet position moving within this channel (As described in section 3.9).

The steady state part of the total nodal mass flow (charaterized by the index 0) follows from the basic non-linear algebraic equation (30) if setting there the time-derivative equal to 0:

Treating eq.(31) in a similar way and multiplying the resulting relation by VMn,0 yields the

F

T

restricted by their saturation values. Then, with regard to eq.(27) which results from energy balance considerations, the nodal power term for the last SC node has to obey the relation

N

MCA,,0 Mn,,0 Nn-1,0 BE,0 CT BCA FTYPE

CT BCA BT BCE CA,0 Nn,0 BA Nn,0 ( ) FTYPE

N = n < N = N - N +1 and z =z < z if h = h at L = 1)

From the resulting steady state enthalpy value hNn,0 follows then (by using the thermodynamic water/steam tables) the corresponding coolant temperature value TNn,0 (with TNn,0 = TSATNn,0 if n = NCT and zCA < zBA) and, by applying the PAX procedure (see section 3.3), their mean nodal temperature and enthalpy values TMn,0 and hMn,0, parameters which are needed as start values for the transient calculations. Obviously, due to the non-linearity of the basic steady state

It can additionally be stated that both the steady state and transient two-phase mass flow

G = G and G = G resp. G = G = 0 n=1, N , if L = 2 (43)

SNn SNn,0 WNn CE WNn,0 BE,0 CT FTYPE SNn CE SNn,0 BE,0 WNn WNn,0 CT FTYPE

G = G = 0 resp. G = G and G = G n=1, N , if L = 1

Q = Q = h - h G if n =N < N at L = 1 = h –h G if n =N < N

<sup>G</sup> <sup>0</sup> h = h + with h = h at n=1

n,

steady state nodal temperature resp. enthalpy terms

26 Nuclear Reactor Thermal Hydraulics and Other Applications

Mn,0

Nn-1,0

( )

/ CA,0

Thus, for the steady state case, NCT is fixed too:

with similar relations for the case LFTYPE = 2.

parameters get the trivial form

BE,0 Nn,0 Nn-1,0 Nn-1,0 Q / //

h rh

// CA,0

CT BCA CA,0 BA Nn,0

equations, this procedure has to be done in a recursive way.

N = n = N and z = z if h <

£ ³ o

G = G = G = G = G n=1,N , L >0 Nn,0 CA,0 CE,0 BA,0 BE,0 ( CT FTYPE ) (39)

Nn,0 CE,0

( )

( ) ( )

> / N

n,0 /

( ) ( )

FTY

Nn,0 PE

( h at L = 1) (42)

N

(40)

( )

BE,0 CT BCA FTYPE

(

( ) at L = 2 (41)

if L = 1 or = 2 at n = 1, N -

YPE BT BCE

$$\begin{aligned} \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \boldsymbol{\alpha}\_{\text{Mn}} = 0 \; \boldsymbol{\alpha}\_{\text{Nn}} = \boldsymbol{\alpha}\_{\text{Nn},0} = \boldsymbol{\alpha}\_{\text{Mn},0} = 0 \quad \text{or} = 1 \; \text{and} \; \mathbf{X}\_{\text{Nn},0} = \frac{\mathbf{h}\_{\text{Nn},0} \cdot \mathbf{h}'\_{\text{Nn},0}}{\mathbf{h}\_{\text{SNn},0}} \\ & \qquad \qquad \qquad \qquad \left(\mathbf{n} = \mathbf{1}, \mathbf{N}\_{\text{CI}}, \text{if } \mathbf{L}\_{\text{FTYP}} = \mathbf{1} \; \text{or} = 2\right) \end{aligned} \tag{44}$$

#### **3.8. Thermal-hydraulics of a SC with two-phase flow (LFTYPE = 0)**

The spatial integration of the two PDE-s of the conservation) eqs.(1) and (2) (now over the mixture-phase SC nodes n) can be performed by again taking into account the rules from section 3.2, the relations from the eqs.(7) and (29), by considering the possibility of locally changing nodal cross flow areas along the BC) and the fact that eventually VMn -> 0. This yields then relations for

**•** the total mass flow term

$$\begin{aligned} \mathbf{C\_{Nn}} = \mathbf{G\_{Nn-1}} + \mathbf{V\_{Mn}}(\boldsymbol{\rho}' \cdot \boldsymbol{\rho}'')\_{\mathrm{Mn}} (\frac{\mathrm{d}}{\mathrm{d}t} \mathbf{a\_{Mn}} \mathbf{-} \mathbf{a\_{CPn}^{\mathrm{t}}} \mathbf{-} \mathbf{a\_{GZn}^{\mathrm{t}}}) \\ & \quad \{\mathbf{n=1}, \mathbf{N\_{C\mathcal{T}}}, \mathbf{L\_{FT\mathcal{P}\mathcal{E}}} = 0\} \end{aligned} \tag{45}$$

with, if neglecting thereby small differences between mean and nodal thermodynamic saturation values, the coefficients

$$
\alpha\_{\rm GPn}^{\rm t} = \frac{1}{(\rm 0^{\circ} \, \rm \overline{\cdot})\_{\rm Mn}} \mathbf{I} (1 \text{-} \alpha) \rho^{\prime \overline{\cdot} \mathbf{P}} + \alpha \rho^{\prime \prime \mathbf{P}} \mathbf{J}\_{\rm Mn} \stackrel{\rm d}{\rm dt} \mathbf{P}\_{\rm SYS} \tag{46}
$$

$$\begin{aligned} \alpha\_{\text{CZn}}^{\text{t}} &= \alpha\_{\text{CE}}^{\text{2}} \frac{\text{d}}{\text{dt}} \text{z}\_{\text{CE}} = \frac{1}{2} \frac{\text{A}\_{\text{CE}}}{\text{A}\_{\text{Mn}}} \alpha\_{\text{Mn}}^{\text{s}} \frac{\text{d}}{\text{dt}} \text{z}\_{\text{CE}} & \text{ (n = 1 and } \text{z}\_{\text{CE}} > \text{z}\_{\text{BE}})\\ &= 0 & \text{ (1 < n < N\_{\text{CT}})} \end{aligned} \tag{47}$$
 
$$\mathbf{=} \alpha\_{\text{CA}}^{\text{2}} \frac{\text{d}}{\text{dt}} \mathbf{z}\_{\text{CA}} = \frac{\mathbf{A}\_{\text{CA}}}{\mathbf{A}\_{\text{Mn}}} (\alpha\_{\text{CA}}^{\text{s}} \frac{1}{2} \alpha\_{\text{Mn}}^{\text{s}}) \frac{\text{d}}{\text{dt}} \text{z}\_{\text{CA}} \left\{ \mathbf{n} = \mathbf{N}\_{\text{CT}}, \mathbf{N}\_{\text{CT}} > \text{1 and } \text{z}\_{\text{CA}} < \text{z}\_{\text{BA}} \right\}$$

and

**•** the mean nodal void fraction time-derivative

$$\begin{array}{l} \frac{\mathbf{d}}{\mathbf{d}} \mathbf{a}\_{\mathrm{Mn}} = \boldsymbol{\alpha}\_{\mathrm{ASn}}^{\mathrm{t}} \boldsymbol{\alpha}\_{\mathrm{APh}}^{\mathrm{t}} + \boldsymbol{\alpha}\_{\mathrm{CA}}^{\mathrm{x}} \frac{\mathbf{d}}{\mathbf{d}} \mathbf{z}\_{\mathrm{Mn}} + \boldsymbol{\alpha}\_{\mathrm{CE}}^{\mathrm{x}} \frac{\mathbf{d}}{\mathbf{d}} \mathbf{z}\_{\mathrm{Mn}-1} \\\ \mathbf{n} = \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} \text{and } \mathbf{z}\_{\mathrm{CA}} = \mathbf{z}\_{\mathrm{BA}} \end{array} \tag{48}$$
  $\begin{array}{l} \mathbf{n} = \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} \text{and } \mathbf{z}\_{\mathrm{CA}} = \mathbf{1} \\\end{array} \text{ or } \begin{array}{l} \mathbf{n} = \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} - \mathbf{1}, \mathbf{N}\_{\mathrm{CT}} > \mathbf{1} \text{ and } \mathbf{z}\_{\mathrm{CA}} < \mathbf{z}\_{\mathrm{BA}} \end{array} \}, \begin{array}{l} \mathbf{L}\_{\mathrm{FT} \text{PE}} = \mathbf{0} \end{array} \tag{49}$ 

with the coefficients

$$\begin{aligned} \boldsymbol{\alpha}\_{\text{ASn}}^{\text{t}} &= \boldsymbol{\alpha}\_{\text{AMn}}^{\text{t}} \cdots \boldsymbol{\alpha}\_{\text{ASn}}^{\text{t}} = \frac{1}{\boldsymbol{\alpha}\_{\text{Mn}}^{\text{U}} \cdot \mathbf{h}\_{\text{SWMn}}} \left[ \mathbf{q}\_{\text{Mn}} \text{ - } \mathbf{h}\_{\text{SWMn}} \frac{\mathbf{G}\_{\text{SNn}}^{\text{s}}}{\mathbf{A}\_{\text{Mn}}} \right] \\ &= \frac{1}{\boldsymbol{\chi}\_{\text{Mn}} \boldsymbol{\alpha}\_{\text{Mn}}^{\text{U}} \cdot \mathbf{h}\_{\text{SWMn}}} \left[ \mathbf{Q}\_{\text{Mn}} \text{ - } \left( \mathbf{G}\_{\text{SNn}} \text{ - } \mathbf{G}\_{\text{SNn}-1} \right) \left( \mathbf{h}\_{\text{SWMn}} \right) \right] \end{aligned} \tag{49}$$

$$\boldsymbol{\alpha}\_{\text{APh}}^{\text{t}} = \mathbf{C}\_{\text{RPh}n} \frac{\mathbf{d}\_{\text{P}}}{\mathbf{d}\mathbf{f}} \mathbf{e}\_{\text{SYS}} \\ \text{with } \mathbf{C}\_{\text{RHPh}} = \frac{1}{(\boldsymbol{\rho}^{\text{t}} \mathbf{h}\_{\text{SW}})\_{\text{Mn}}} [\boldsymbol{\mathbb{I}} \text{-a}] \boldsymbol{\rho} \mathbf{\dot{h}}^{\text{T}} + \boldsymbol{a} (\boldsymbol{\rho} \mathbf{\dot{h}}^{\text{T}} + \boldsymbol{\rho}^{\text{T}} \mathbf{h}\_{\text{SW}}) \cdot \mathbf{1} \mathbf{J}\_{\text{Mn}} \tag{50}$$

$$\mathfrak{a}\_{\text{AZn}}^{\text{t}} = \mathfrak{a}\_{\text{GZn}}^{\text{t}} \quad \left(\text{see eq.(47)}\right) \tag{51}$$

Nn Mn SNn Xn ( CT FTYP )

QMn

before knowing GNn). This term (GNn) results then by combining the eqs.(54) and (10)

1+(α C0 CDC)Nn

/ DCNn Mn

r

C = -1 r

*Nn*

and are thus only dependent on the local resp. system pressure value.

(48) (or eq.(45)) the mean nodal void fraction time-derivative *<sup>d</sup>*

// /

Nn ( )( ) <sup>C</sup>

From the drift flux correlation package (Hoeld et al., 1992) and (Hoeld, 1994) follow then all the other characteristic two-phase parameters. These are especially the nodal steam mass flow

Obviously, at a mixture flow situation the mean nodal temperature and enthalpy terms are

( ) ( ) ( ) ( ) / // Mn SAT Mn Mn Mn Mn CT FTYPE T = T P resp. h = h P or = h P n= 1, N and L = 0 (58)

Relations for the steady state case can be derived if setting in the eqs.(45) and (48) (resp. the eq.(49)) the time-derivatives equal to 0. For the total mass flow parameters a similar relation

as already given for the single-phase flow (see eq.(48)) is valid, yielding GNn,0 = GBE,0.

 r

GXn-(A <sup>α</sup> vD *<sup>ρ</sup>* /

The 'auxiliary' mass flow term GXn is directly available since it refers only to already known

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

hSWMn )) - VMn(

A similar relation can be established if starting from the drift flux correlation (eq.(10)) and taking advantage of the fact that the needed drift velocity vDNn and the phase distribution parameter C0Nn are independent from the total mass flow GNn (and can thus be determined

CDC)Nn

//

 r

GC

(*z*) resp., according to eq.(53), G *SNn* (*s*) . Then, finally, from eq.

*ρ* / *ρ* // ) Mn(αAPn

<sup>t</sup> +αGPn <sup>t</sup> )

(n=1,NCT) (56)

*dt* αMn will result, needed for the

(n=1,NCT, LFTYPE=0)

(54)

29

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(55)

(57)

// E G + -1 G = G n=1,N , L ( ) =0

parameters (for example taken from the node before or the power profile)

*<sup>ρ</sup>* // -1)Mn(GSNn-1+

/

r

r

GNn =

GXn= GNn-1+ ( *<sup>ρ</sup>* /

with the coefficient

GSNn and, eventually, the slope α

equal to their saturation values

next integration step.

$$\mathbf{G}\_{\rm SNn}^{s} = \frac{\Delta \mathbf{G}\_{\rm SNn}}{\Delta \mathbf{z}\_{\rm Nn}} \rightharpoonup \mathbf{G}\_{\rm SNn}^{z} = \mathbf{G}\_{\rm SNn}^{\alpha} \alpha\_{\rm Nn}^{s} \text{ if } \alpha\_{\rm Nn} \rightharpoonup \alpha\_{\rm CE} = 0 \text{ (at } \mathbf{n} = 1\text{)}\tag{52}$$
 
$$\text{or } \alpha\_{\rm Nn} \rightharpoonup \alpha\_{\rm CA} = 1 \text{ (at } \mathbf{n} = \mathbf{N}\_{\rm CT}\text{)}$$

It can again be expected that at the begin of each (intermediate) time step the mean nodal void fraction values αMn are known. This either from steady state considerations (at the start of the transient calculations) or as a result of the integration procedure. Hence, the additional parameters needed in the relations above can be determined too. From the PAX procedure it follow their nodal boundary void fraction terms αNn together with their slopes α *Nn* (*s*) and α *Mn* (*s*) resp. gradient α *Nn* (*z*) and thus, as shown both in section 2.2.3 but also in the tables given by Hoeld (2001 and 2002a), all the other characteristic two-phase parameters (steam, water or relative velocities, steam qualities etc). Obviously, due to the non-linearity of the basic equations the steady state solution procedure has to be performed in a recursive way.

If, in the transient case, the SC nodal boundary void fraction αNn does (within the entire BC) not reach its limit value (αLIMNn=1 or 0) it follows as total number of SC nodes NCT=NBT–NBCE and zNn (at n=NCT) =zCA=zBT. Otherwise, if this limit is reached (at node n), NCT = n, αNn=1 (or =0) with zCA (< zBT) resulting from the integration. Then, from the procedure above also the time-derivative of the boiling boundary, moving within BC, can be established (as this will be discussed in section 3.9).

Hence, it follows a relation for the steam mass flow gradients

$$\begin{aligned} \mathbf{G\_{SNn}^{(\mathbf{a})}} &= \mathbf{A\_{CE}} \mathbf{v\_{S0}} \rho\_{\mathrm{Nn}}^{(\parallel)} \text{with } \mathbf{v\_{S0}} = \mathbf{v}\_{\mathrm{S}} \text{ (at } a\_{\mathrm{Nn}} = 0) \quad \text{(n=1 } \quad \text{and } a\_{\mathrm{Nn}} \rightharpoonup a\_{\mathrm{CE}} = 0) \\ &= \mathbf{A\_{CA}} \mathbf{v\_{W1}} \rho\_{\mathrm{Nn}}^{(\parallel)} \text{with } \mathbf{v\_{W1}} = \mathbf{v}\_{\mathrm{W}} \text{ (at } a\_{\mathrm{Nn}} = 1) \quad \text{(n=1}\_{\mathrm{CT}} \text{ and } a\_{\mathrm{Nn}} \rightharpoonup a\_{\mathrm{CA}} = 1) \end{aligned} \tag{53}$$

If eliminating the term *<sup>d</sup> dt* αMn in eq.(45) by inserting from eq.(48) yields a relation between GSNn and GNn

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 29

$$\mathbf{G\_{Nn}} + (\frac{\rho^{\prime}}{\rho^{\prime\prime}} \mathbf{-1})\_{\text{Mn}} \mathbf{G\_{SNn}} = \mathbf{G\_{Nn}} \quad \text{( $n=1,N$ }\_{\text{CT}^{\prime}} \text{ L}\_{\text{FT}\text{YP}} = \mathbf{0} \text{)}\tag{54}$$

The 'auxiliary' mass flow term GXn is directly available since it refers only to already known parameters (for example taken from the node before or the power profile)

$$\mathbf{G\_{Nn}=G\_{Nn-1}+(\frac{\rho^{'}}{\rho^{'/\prime}}-1)\_{\mathrm{Mn}}(\mathbf{G\_{SNn-1}}+\frac{\mathbf{Q\_{Mn}}}{\mathbf{h\_{SWMn}}}))-\mathbf{V\_{Mn}}{}\_{\rho^{'}}{}^{\rho^{'/\mathrm{Mn}}}(\mathbf{a\_{APn}^{\dagger}+\mathbf{a\_{GPn}^{\dagger}}}) \tag{55}$$

$$\{\mathfrak{n}=1,\mathrm{N}\_{\mathrm{CP}},\mathrm{L}\_{\mathrm{FTPPE}}=\mathbf{0}\}$$

A similar relation can be established if starting from the drift flux correlation (eq.(10)) and taking advantage of the fact that the needed drift velocity vDNn and the phase distribution parameter C0Nn are independent from the total mass flow GNn (and can thus be determined before knowing GNn). This term (GNn) results then by combining the eqs.(54) and (10)

$$\mathbf{G}\_{\rm Nn} = \frac{\mathbf{G}\_{\rm Nn} \text{--} (\mathbf{A} \,\alpha \,\mathbf{v}\_{\rm D} \,\rho \,' \,\mathbf{C}\_{\rm DC})\_{\rm Nn}}{1 + (\alpha \,\mathbf{C}\_{\rm O} \,\mathbf{C}\_{\rm DC})\_{\rm Nn}} \tag{10.1} \text{--} \,\text{N}\_{\rm CT} \text{} \tag{56}$$

with the coefficient

( )

SNn SNn-1 SW n ρ h M

Q - G -

= [ G h ]

APn SW Mn <sup>d</sup> <sup>1</sup> [( ) ( ) ] ( h) α = C P with C = 1-a h + h + h - 1 dt

( ( )) t t

It can again be expected that at the begin of each (intermediate) time step the mean nodal void fraction values αMn are known. This either from steady state considerations (at the start of the transient calculations) or as a result of the integration procedure. Hence, the additional parameters needed in the relations above can be determined too. From the PAX procedure it

Hoeld (2001 and 2002a), all the other characteristic two-phase parameters (steam, water or relative velocities, steam qualities etc). Obviously, due to the non-linearity of the basic

If, in the transient case, the SC nodal boundary void fraction αNn does (within the entire BC) not reach its limit value (αLIMNn=1 or 0) it follows as total number of SC nodes NCT=NBT–NBCE and zNn (at n=NCT) =zCA=zBT. Otherwise, if this limit is reached (at node n), NCT = n, αNn=1 (or =0) with zCA (< zBT) resulting from the integration. Then, from the procedure above also the time-derivative of the boiling boundary, moving within BC, can be established (as this will be

// with vS0= v<sup>S</sup> (at <sup>α</sup>Nn= 0) (n=1 and αNn→αCE= 0)

/ with vW1= vW(at <sup>α</sup>Nn=1) (n=NCT and <sup>α</sup>Nn→αCA= 1)

*dt* αMn in eq.(45) by inserting from eq.(48) yields a relation between

Mn SWMn Mn

r

<sup>s</sup> if αNn→αCE= 0 (at n=1)

(*z*) and thus, as shown both in section 2.2.3 but also in the tables given by

or αNn→αCA= 1 (at n= NCT)

q - h

//

ttt 1 G ASn AQn AGn ρ h Mn A

> // SW Mn

r

α =α -α = [ ]

// Mn Mn SWMn

RHPn SYS RHPn

28 Nuclear Reactor Thermal Hydraulics and Other Applications

ΔGSNn ΔzNn

→GSNn

Hence, it follows a relation for the steam mass flow gradients

<sup>z</sup> =GSNn

<sup>α</sup> <sup>α</sup>Nn

follow their nodal boundary void fraction terms αNn together with their slopes α *Nn*

equations the steady state solution procedure has to be performed in a recursive way.

GSNn <sup>s</sup> =

resp. gradient α *Nn*

discussed in section 3.9).

GSNn

GSNn and GNn

If eliminating the term *<sup>d</sup>*

(α) = ACEvS0*ρ*Nn

==ACAvW1*ρ*Nn

t

1 Mn V s SNn

' 'P ' ''P ''P

AZn GZn α =α eq. 47 see (51)

 r

(50)

ar

(49)

(52)

(*s*) and α *Mn* (*s*)

(53)

SWMn

$$\mathbf{C}\_{\rm DCNn} = (\frac{\rho^{\prime}}{\rho^{\prime \prime}} \text{-1})\_{\rm Mn} (\frac{\rho^{\prime \prime}}{\rho^{\prime} \mathbf{C}\_{\rm GC}})\_{\rm Nm} \tag{57}$$

From the drift flux correlation package (Hoeld et al., 1992) and (Hoeld, 1994) follow then all the other characteristic two-phase parameters. These are especially the nodal steam mass flow GSNn and, eventually, the slope α *Nn* (*z*) resp., according to eq.(53), G *SNn* (*s*) . Then, finally, from eq. (48) (or eq.(45)) the mean nodal void fraction time-derivative *<sup>d</sup> dt* αMn will result, needed for the next integration step.

Obviously, at a mixture flow situation the mean nodal temperature and enthalpy terms are equal to their saturation values

$$\mathbf{T\_{Mn}} = \mathbf{T\_{SAT}}\left(\mathbf{P\_{Mn}}\right)\text{ resp. } \mathbf{h\_{Mn}} = \mathbf{h}^{\prime}\left(\mathbf{P\_{Mn}}\right)\text{ or } = \mathbf{h}^{\prime\prime}\left(\mathbf{P\_{Mn}}\right) \quad \left(\mathbf{n=1}, \mathbf{N\_{CT}}\text{and } \mathbf{L\_{FTPE}}=\mathbf{0}\right) \tag{58}$$

and are thus only dependent on the local resp. system pressure value.

Relations for the steady state case can be derived if setting in the eqs.(45) and (48) (resp. the eq.(49)) the time-derivatives equal to 0. For the total mass flow parameters a similar relation as already given for the single-phase flow (see eq.(48)) is valid, yielding GNn,0 = GBE,0.

Hence, one obtains for the (steady state) nodal steam mass flow

$$\mathbf{G}\_{\text{SNn},0} = \mathbf{G}\_{\text{SNn-1,0}} + \frac{\mathbf{Q}\_{\text{Mn,0}}}{\mathbf{h}\_{\text{SNn,0}}} \le \mathbf{G}\_{\text{Ni,0}} = \mathbf{G}\_{\text{BE,0}} \quad \left(\text{n=1, N}\_{\text{BT}} - \text{N}\_{\text{BCE}} \text{and } \text{L}\_{\text{FTYPE}} = 0\right) \tag{59}$$

if knowing the term QMCA,,0 = QMn,,0 from eq.(27).

Then the total number (NCT) of SC nodes is given too

$$\begin{aligned} \mathbf{N\_{CT} = n < N\_{\rm BT} - N\_{\rm BCE}} \text{ and } \mathbf{z\_{CA,0}} \begin{pmatrix} \mathbf{w\_{Nn,0}} \\ \end{pmatrix} < \mathbf{z\_{BA}} \quad & \quad \begin{pmatrix} \text{if } \mathbf{G\_{SNn,0}} = \mathbf{G\_{BE,0}} \text{and } \mathbf{L\_{FTSPE} = 0} \end{pmatrix} \\ \mathbf{N\_{CT} = N\_{\rm BT} - N\_{\rm BCE}} \quad & \quad \text{and } \mathbf{z\_{CA,0}} = \mathbf{z\_{BA}} \quad \quad \quad \text{(if } \mathbf{G\_{SNn,0}} < \mathbf{G\_{BE,0}} \text{and } \mathbf{L\_{FTSPE} = 0)} \end{aligned} \tag{60}$$

and also the corresponding steam quality parameter

$$\mathbf{X}\_{\text{Nn},0} = \frac{\mathbf{G}\_{\text{Slu},0}}{\mathbf{G}\_{\text{Nn},0}} \Big(\mathbf{n} = \mathbf{1} \text{ } \mathbf{N}\_{\text{CT}} \text{ if } \mathbf{L}\_{\text{FTYPE}} = \mathbf{0}\Big) \tag{61}$$

1 2

yielding finally as solution

can be calculated.

qLBk,0 - qLBk-1,0 qLBk-1,0 ( ΔzCA,0 ΔzBk )2+

= ΔzBk

<sup>→</sup> QMCA,0

boundary has to be replaced by a gradient (determined in PAX).

boundary time derivative can be expressed by

PAXCA TCA z z TCA PAXCA T -T

PAXCA ACA z z CA PAXCA α -α

If zCA < zBA, the corresponding time-derivatives *<sup>d</sup>*

total number NCT of SC nodes are given.

second one follows then from the PAX procedure after the integration.

ΔzCA,0 ΔzBk - QMCA,0

qLBk-1,0

qLBk-1,0 [1 + (1 - qLBk,0

<sup>z</sup>Bk qLBk-1,0 = 0

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

ΔzCA,0=ΔzBk (n=NCTand k= NBCA if NCT= NBT)

qLBk-1,0 ) QMCA,0

Then, from the relations in section 3.5, also the other characteristic steady state power terms

In the transient case the SC outlet boundary zCA (=boiling boundary or mixture level) follows (and thus also ΔzCA and NCT ), as already pointed-out, directly from the integration procedure. During the transient this boundary can move along the entire BC (and thereby also cross BC node boundaries). A SC can even shrink to a single node (NCT =1), start to disappear or to be created anew. Then, if ΔzCA -> 0, in the relations above the slope in the vicinity of such a

The mean nodal coolant temperature or, if LFTYPE=0, void fraction of the last SC node is interrelated by the PAX procedure with the locally varying SC outlet boundary zCA. Hence, in a transient situation the time-derivative of only one of these parameters is demanded. The

If combining (in case of single-phase flow) the eqs.(23) and (31), the wanted relation for the SC

<sup>T</sup> CA BB CT CA BA CA BA FTYPE -T z = z = or = 0 n = N , z < z or z = d d

<sup>α</sup> CA ML CT CA BA CA BA FTYPE -α z = z = or = 0 n = N , z < z or z = d d

and if taking for the case of a mixture flow the eqs. (23) and (48) into account

( ) t t

( ) t t

*dt* TMn or

*d*

*dt* αMn of the last SC node (at n=NCT)

dt dt z if L = 0 (67)

follow by inserting the terms above into the eqs.(31) or (48). After the integration procedure then the SC outlet boundary zCA (= boiling boundary zBB or mixture level zML) and thus also the

dt dt z if L > 0 (66)

qLBk-1,0-qLBk,0 [1 - 1 - 2(1 - qLBK,0

(n=NCTandk= NBCAif NCT< NBCA)

qLBk-1,0 ) QCMA,0

<sup>z</sup>Bk qLBk-1,0 if qLBk,0 →qLBk-1,0

ΔzBk qLBMk-1,0 (k= NBCA if NCT< NBT) (64)

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(65)

The nodal boundary void fraction values αNn,0 can now be determined by applying the inverse drift-flux correlation, the mean nodal void fraction value αMn,0 from the PAX procedure. All of them are needed as starting values for the transient calculation.

#### **3.9. SC boundaries**

The SC entrance position zCE (= zNn at n=0) is either (for the first SC within the BC) equal to BC entrance zBE or equal to the SC outlet boundary of the SC before.

In the steady state case the SC outlet boundary (= boiling boundary zBB,0 or mixture level zML, 0) can be represented as

$$\mathbf{z}\_{\text{CA},0} = \mathbf{z}\_{\text{BA}} \cdot \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} \text{and } \mathbf{z}\_{\text{CA},0} = \mathbf{z}\_{\text{BA}}\right) \tag{62}$$

Or

$$\mathbf{z}\_{\text{CA},0} = \mathbf{z}\_{\text{Nn-1}} + \mathbf{D}\mathbf{z}\_{\text{CA},0} \qquad \quad \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} \text{and } \mathbf{z}\_{\text{CA},0} < \mathbf{z}\_{\text{BA}}\right) \tag{63}$$

The (steady state) numbers (NCT) of (single- or two-phase) SC-s are already determined by the eqs.(42) or (60), the nodal power terms QMCA,0 by the eqs.(41) or (59). Hence, after dividing eq. (27) by ΔzBkqLBk-1,0 one gets an algebraic quadratic equation of the form

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 31

$$\begin{aligned} \frac{1}{2} \frac{\mathbf{q}\_{\text{LIR},0} \cdot \mathbf{q}\_{\text{LIR-1,0}}}{\mathbf{q}\_{\text{LIR-1,0}}} (\frac{\Delta \mathbf{x}\_{\text{CA},0}}{\Delta \mathbf{z}\_{\text{R}}}) \mathbf{2} + \frac{\Delta \mathbf{z}\_{\text{CA},0}}{\Delta \mathbf{z}\_{\text{R}}} \cdot \frac{\mathbf{Q}\_{\text{bCA},0}}{\mathbf{z}\_{\text{R}} \cdot \mathbf{q}\_{\text{LIR-1,0}}} &= \mathbf{0} \\ \{\mathbf{n} = \mathbf{N}\_{\text{CT}} \text{and} \mathbf{k} = \mathbf{N}\_{\text{BCA}} \text{if } \mathbf{N}\_{\text{CT}} \le \mathbf{N}\_{\text{BCA}}\} \end{aligned} \tag{64}$$

yielding finally as solution

Hence, one obtains for the (steady state) nodal steam mass flow

Mn,

if knowing the term QMCA,,0 = QMn,,0 from eq.(27).

30 Nuclear Reactor Thermal Hydraulics and Other Applications

Then the total number (NCT) of SC nodes is given too

and also the corresponding steam quality parameter

**3.9. SC boundaries**

0) can be represented as

Or

SWMn,0 SNn,0 SNn-1,0 Nn,0 BE,0 BT BCE FTYP

CT BT BCE CA,0 Nn,0 BA ( ) ( SNn,0 BE,0 FTYPE ) CT BT BCE CA,0 BA SNn,0 BE,0 FTYPE

N = N - N and z = z if G < G a ( nd L = 0) (60)

( ) SMn,0

The nodal boundary void fraction values αNn,0 can now be determined by applying the inverse drift-flux correlation, the mean nodal void fraction value αMn,0 from the PAX procedure. All of

The SC entrance position zCE (= zNn at n=0) is either (for the first SC within the BC) equal to BC

In the steady state case the SC outlet boundary (= boiling boundary zBB,0 or mixture level zML,

The (steady state) numbers (NCT) of (single- or two-phase) SC-s are already determined by the eqs.(42) or (60), the nodal power terms QMCA,0 by the eqs.(41) or (59). Hence, after dividing eq.

(27) by ΔzBkqLBk-1,0 one gets an algebraic quadratic equation of the form

N = n < N - N and z =z < z if G = G and L = 0

X = n = 1, Nn,0 CT FTYPE

<sup>0</sup> G = G + G = G n= 1, N –N a <sup>Q</sup>

Nn,0

G

them are needed as starting values for the transient calculation.

entrance zBE or equal to the SC outlet boundary of the SC before.

( )

<sup>G</sup> N if L = 0 (61)

z = z n = N and z = z CA,0 BA CT CA,0 BA ( ) (62)

z = z +Dz n = N and z < z CA,0 Nn-1 CA,0 ( CT CA,0 BA ) (63)

<sup>h</sup> £ nd L = 0 (59)

E

ΔzCA,0=ΔzBk (n=NCTand k= NBCA if NCT= NBT) = ΔzBk qLBk-1,0 qLBk-1,0-qLBk,0 [1 - 1 - 2(1 - qLBK,0 qLBk-1,0 ) QCMA,0 ΔzBk qLBMk-1,0 (k= NBCA if NCT< NBT) <sup>→</sup> QMCA,0 qLBk-1,0 [1 + (1 - qLBk,0 qLBk-1,0 ) QMCA,0 <sup>z</sup>Bk qLBk-1,0 if qLBk,0 →qLBk-1,0 (65)

Then, from the relations in section 3.5, also the other characteristic steady state power terms can be calculated.

In the transient case the SC outlet boundary zCA (=boiling boundary or mixture level) follows (and thus also ΔzCA and NCT ), as already pointed-out, directly from the integration procedure. During the transient this boundary can move along the entire BC (and thereby also cross BC node boundaries). A SC can even shrink to a single node (NCT =1), start to disappear or to be created anew. Then, if ΔzCA -> 0, in the relations above the slope in the vicinity of such a boundary has to be replaced by a gradient (determined in PAX).

The mean nodal coolant temperature or, if LFTYPE=0, void fraction of the last SC node is interrelated by the PAX procedure with the locally varying SC outlet boundary zCA. Hence, in a transient situation the time-derivative of only one of these parameters is demanded. The second one follows then from the PAX procedure after the integration.

If combining (in case of single-phase flow) the eqs.(23) and (31), the wanted relation for the SC boundary time derivative can be expressed by

$$\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\text{CA}} = \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\text{BB}} = \frac{\mathbf{T}\_{\text{PACA}}^{\mathbf{t}} \cdot \mathbf{T}\_{\text{CA}}^{\mathbf{t}}}{\mathbf{T}\_{\text{CA}}^{\mathbf{x}} \cdot \mathbf{T}\_{\text{PACA}}^{\mathbf{x}}} \text{or} = 0 \quad \left(\mathbf{n} = \mathbf{N}\_{\text{CT}} \cdot \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}} \text{or} \; \mathbf{z}\_{\text{CA}} = \mathbf{z}\_{\text{BA}} \text{if } \mathbf{L}\_{\text{FTTPE}} > 0\right) \tag{66}$$

and if taking for the case of a mixture flow the eqs. (23) and (48) into account

$$\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\text{CA}} = \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}} \mathbf{z}\_{\text{ML}} = \frac{a\_{\text{PACA}}^{\dagger} \cdot a\_{\text{ACA}}^{\dagger}}{a\_{\text{CA}}^{\dagger} \cdot a\_{\text{PACA}}^{\dagger}} \text{or} = 0 \quad \left(\mathbf{n} = \mathbf{N}\_{\text{CT}'} \, \mathbf{z}\_{\text{CA}} < \mathbf{z}\_{\text{BA}} \, \text{or} \, \mathbf{z}\_{\text{CA}} = \mathbf{z}\_{\text{BA}} \, \text{if } \, \mathbf{L}\_{\text{FTPPE}} = 0\right) \tag{67}$$

If zCA < zBA, the corresponding time-derivatives *<sup>d</sup> dt* TMn or *d dt* αMn of the last SC node (at n=NCT) follow by inserting the terms above into the eqs.(31) or (48). After the integration procedure then the SC outlet boundary zCA (= boiling boundary zBB or mixture level zML) and thus also the total number NCT of SC nodes are given.

#### **3.10. Pressure profile along a SC (and thus also BC)**

After having solved the mass and energy balance equations, separately and not simultaneously with the momentum balance, the now exact nodal SC and BC pressure difference terms (ΔPNn = PNn - PNn-1 and ΔPBNn) can be determined for both single- or two-phase flow situations by discretizing the momentum balance eq.(3) and, if applying a modified 'finite element method', integrating the eqs.(4) to (6) over the corresponding SC nodes. The total BC pressure difference ΔPBT = PBA - PBE between BC outlet and entrance follows then from the relation

$$
\Delta \mathbf{P\_{BT}} = \Delta \mathbf{P\_{BBT}} - \Delta \mathbf{P\_{GBT}} \qquad \text{(with} \\
\Delta \mathbf{P\_{GBT,0}} = 0 \text{at steady state conditions)} \tag{68}
$$

Looking at the available friction correlations, there arises the problem how to consider correctly contributions from spacers, tube bends, abrupt changes in cross sections etc. as well. The entire friction pressure decrease (ΔPFBT) along a BC can thus never be described in a satisfactory manner solely by analytical expressions. To minimize these uncertainties a further friction

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

The corresponding steady state additive pressure difference term ΔPDBT,0 is (according to the eqs.(69) and (70)) fixed since it can be assumed that the total steady state BC pressure difference ΔPBT,0 = ΔPPBT,0 is known from input. It seems, however, to be reasonable to treat this term as a 'friction' (or at least the sum with ΔPFBT) and not as a 'driving' force. Thus it must be demanded that these terms should remain negative. Otherwise, input terms such as the entire pressure difference along the BC or corresponding friction factors have to be adjusted in an

means that ΔPDBT is either supplemented with a direct additive term (index FADD) or the friction part is provided with a multiplicative factor (fFMP,0-1). For the additive part it will be assumed to be the (1-εDPZ)-th part of the total additional pressure difference term and to be

FADD ADD BT BE DPZ DBT <sup>2</sup>ρ<sup>d</sup> DPZ DBT 0, ΔP = - f z ( )( ) = 1 -ε P and = 0 i.e., ε =1 if P 0 D D ( ) > (74)

with the input coefficient εDPZ = εDPZI governing from outside which of them should prevail.

From the now known steady state total 'additional' term ΔPDBT,0 the corresponding additive friction factor fADD,0 follows then directly from the equation above, from eq.(73) then the

For the steady state description of a 3D compartment with (identical) parallel channels in a first step a factor should be derived for a 'mean' channel. Then this factor can be assumed to

For the transient case there arises the question how the validity of both friction factors could be expanded to this situation too. In the here presented approach this is done by assuming that these factors should remain time-independent, i.e., that fADD = fADD,0 and fFMP = fFMP,0. This allows finally also to determine the wanted nodal pressure decrease term ΔPDBT of eq.(69) for

D DD P = f - 1 P + P DBT FMP,0 FBT FADD ( ) (73)

http://dx.doi.org/10.5772/53372

33

ΔPFBT,0 (setting εDPZ=1 if *<sup>Δ</sup>*PDBT,0> 0) (75)

Assuming the general additive pressure difference term ΔPDBT may have the form

proportional to the square of the total coolant mass flow, e.g., at BC entrance

ΔPDBT,0

be valid for all the other (identical) parallel channels too (See Jewer et al., 2005).

F F H G G

fFMP,0 =1+ εDPZ

term, ΔPDBT, had to be included into these considerations.

adequate way.

multiplicative one

the transient case.

with the part

$$\begin{aligned} \mathsf{\sf AP}\_{\mathsf{PBT}} & \mathsf{\sf AP}\_{\mathsf{SBT}} + \mathsf{\sf AP}\_{\mathsf{ABT}} + \mathsf{\sf AP}\_{\mathsf{XBT}} + \mathsf{\sf AP}\_{\mathsf{FBT}} + \mathsf{\sf AP}\_{\mathsf{DBT}}\\ & \text{ (with} \,\mathsf{AP}\_{\mathsf{PBT},0} = \mathsf{\sf AP}\_{\mathsf{BTN},0} \text{ at steady state conditions} \end{aligned} \tag{69}$$

comprising, as described in section 2.1.3, terms from static head (ΔPSBT), mass acceleration (ΔPABT), wall friction (ΔPFBT) and external pressure accelerations (ΔPXBT due to pumps or other perturbations from outside) and an (only in the transient situation needed) term ΔPGBT, being represented as

$$\begin{aligned} \, ^\text{AD}\mathbf{C}\_\text{GBT} &= \int\_0^\text{z\_{\text{BT}}} \frac{\text{d}}{\text{dt}} \mathbf{G}\_\text{FB} \left( \mathbf{z}, \mathbf{t} \right) \, \text{dz} = \mathbf{z}\_\text{BT} \frac{\text{d}}{\text{dt}} \mathbf{G}\_\text{FBMT} & \quad \text{at transient conditions} \\ &= \mathbf{0} & \quad \text{at steady state} \end{aligned} \tag{70}$$

This last term describes the influence of time-dependent changes in total mass flux along a BC (caused by the direct influence of changing nodal mass fluxes) and can be estimated by introducing a 'fictive' mean mass flux term GFBMT (averaged over the entire BC)

$$\begin{split} \mathbf{G\_{FBMT}} &= \frac{1}{\mathbf{z\_{BT}}} \int\_{0}^{\mathbf{z\_{BT}}} \frac{\mathbf{d}}{\text{d}t} \mathbf{G\_{FB}} \Big( \mathbf{z}, \mathbf{t} \Big) \, d\mathbf{z} \\ &\equiv \frac{1}{\mathbf{z\_{BT}}} \sum\_{\mathbf{z\_{BT}}}^{\mathbf{N\_{ST}}} \sum\_{\mathbf{n}=1}^{\mathbf{N\_{ST}}} \Delta \mathbf{z\_{Nn}} \mathbf{G\_{FBn}} = \frac{1}{2} \frac{1}{\mathbf{z\_{BT}}} \sum\_{\mathbf{l}}^{\mathbf{N\_{ST}}} \frac{\Delta \mathbf{z\_{Bk}}}{\mathbf{A\_{Bk}}} (\mathbf{G\_{Bk}} + \mathbf{G\_{BK-1}}) \end{split} \tag{71}$$

Its time derivative can then be represented by

$$\begin{aligned} \stackrel{\text{d}}{\text{d}} \stackrel{\text{d}}{\text{G}} \text{G}\_{\text{FBMT}} & \equiv \frac{\text{G}\_{\text{FBMT}} \cdot \text{G}\_{\text{FBMT}}}{\text{At}} \\ & \quad \stackrel{\text{d}}{\text{g}} \stackrel{\text{d}}{\text{G}} \text{G}\_{\text{FBMT}} \quad \text{if } \, \Delta \text{t-t} \text{-} \text{t}\_{\text{B}} \text{ 0 (Index B = begin of time-step)} \end{aligned} \tag{72}$$

Looking at the available friction correlations, there arises the problem how to consider correctly contributions from spacers, tube bends, abrupt changes in cross sections etc. as well. The entire friction pressure decrease (ΔPFBT) along a BC can thus never be described in a satisfactory manner solely by analytical expressions. To minimize these uncertainties a further friction term, ΔPDBT, had to be included into these considerations.

**3.10. Pressure profile along a SC (and thus also BC)**

32 Nuclear Reactor Thermal Hydraulics and Other Applications

with the part

represented as

After having solved the mass and energy balance equations, separately and not simultaneously with the momentum balance, the now exact nodal SC and BC pressure difference terms (ΔPNn = PNn - PNn-1 and ΔPBNn) can be determined for both single- or two-phase flow situations by discretizing the momentum balance eq.(3) and, if applying a modified 'finite element method', integrating the eqs.(4) to (6) over the corresponding SC nodes. The total BC pressure difference

BT PBT GBT GBT,0 P = P - P with P = 0at steady state condition )s DD D D ( (68)

ò (70)

Bk

) if Δt=t-t<sup>B</sup> <sup>0</sup> (Index B = begin of time-step) (72)

BK-1

(71)

D D (69)

(with P = P at steady state conditions)

comprising, as described in section 2.1.3, terms from static head (ΔPSBT), mass acceleration (ΔPABT), wall friction (ΔPFBT) and external pressure accelerations (ΔPXBT due to pumps or other perturbations from outside) and an (only in the transient situation needed) term ΔPGBT, being

> d d ΔP = G z,t dz = z G at transient conditions <sup>0</sup> dt dt FBMT =0 at steady state

introducing a 'fictive' mean mass flux term GFBMT (averaged over the entire BC)

SCT CT BT

åå å

dz

N N N Δz 1 1 1

BT FBMn BT BMk

z 2z A k

@ G

( ) BT

FB

n=1

GFBMT - GFBMTB Δt

This last term describes the influence of time-dependent changes in total mass flux along a BC (caused by the direct influence of changing nodal mass fluxes) and can be estimated by

Nn B

1

Δz G = (G + )

ΔPBT = PBA - PBE between BC outlet and entrance follows then from the relation

PBT,0 BTIN,0

PBT SBT ABT XBT FBT DBT

P =P +P +P +P +P

DDD DDD

( ) BT <sup>z</sup> GBT FB BT

BT

Its time derivative can then be represented by

( d dt GFBMT

FBMT

d dt GFBMT≅

<sup>z</sup> <sup>1</sup> <sup>d</sup> z dt 0

G = G z,t

ò

The corresponding steady state additive pressure difference term ΔPDBT,0 is (according to the eqs.(69) and (70)) fixed since it can be assumed that the total steady state BC pressure difference ΔPBT,0 = ΔPPBT,0 is known from input. It seems, however, to be reasonable to treat this term as a 'friction' (or at least the sum with ΔPFBT) and not as a 'driving' force. Thus it must be demanded that these terms should remain negative. Otherwise, input terms such as the entire pressure difference along the BC or corresponding friction factors have to be adjusted in an adequate way.

Assuming the general additive pressure difference term ΔPDBT may have the form

$$
\Delta \mathbf{P}\_{\rm DBT} = \left( \mathbf{f}\_{\rm FMP,0} \mathbf{-1} \right) \Delta \mathbf{P}\_{\rm FBT} + \Delta \mathbf{P}\_{\rm FADD} \tag{73}
$$

means that ΔPDBT is either supplemented with a direct additive term (index FADD) or the friction part is provided with a multiplicative factor (fFMP,0-1). For the additive part it will be assumed to be the (1-εDPZ)-th part of the total additional pressure difference term and to be proportional to the square of the total coolant mass flow, e.g., at BC entrance

$$
\Delta \mathbf{P}\_{\text{FADD}} = -\mathbf{f}\_{\text{ADD}} \mathbf{z}\_{\text{BT}} (\frac{\mathbf{c}\_{\text{F}} | \mathbf{c}\_{\text{T}}}{2 \text{qd}\_{\text{H}}})\_{\text{BE}} = (1 \cdot \mathbf{c}\_{\text{DPZ}}) \Delta \mathbf{P}\_{\text{DBT}} \text{ and } \mathbf{=0} \text{ (i.e., } \mathbf{c}\_{\text{DPZ}} = \mathbf{1}) \text{ if } \Delta \mathbf{P}\_{\text{DBT},0} > 0 \tag{74}
$$

with the input coefficient εDPZ = εDPZI governing from outside which of them should prevail.

From the now known steady state total 'additional' term ΔPDBT,0 the corresponding additive friction factor fADD,0 follows then directly from the equation above, from eq.(73) then the multiplicative one

$$\mathbf{f}\_{\text{FMP},0} = \mathbf{1} + \varepsilon\_{\text{DPZ}} \frac{\Lambda \mathbf{P}\_{\text{DRT},0}}{\Lambda \mathbf{P}\_{\text{FIT},0}} \text{(setting } \varepsilon\_{\text{DPZ}} = \mathbf{1} \text{ if } \Lambda \mathbf{P}\_{\text{DRT},0} > 0\text{)}\tag{75}$$

For the steady state description of a 3D compartment with (identical) parallel channels in a first step a factor should be derived for a 'mean' channel. Then this factor can be assumed to be valid for all the other (identical) parallel channels too (See Jewer et al., 2005).

For the transient case there arises the question how the validity of both friction factors could be expanded to this situation too. In the here presented approach this is done by assuming that these factors should remain time-independent, i.e., that fADD = fADD,0 and fFMP = fFMP,0. This allows finally also to determine the wanted nodal pressure decrease term ΔPDBT of eq.(69) for the transient case.

By adding the resulting nodal BC pressure difference terms to the (time-varying) system pressure PSYS(t), given from outside as boundary condition with respect to a certain position (in- or outside of the BC), then finally also the absolute nodal pressure profile PBk along the BC can be established (This term is needed at the begin of the next time step within the constitutive equations).

There exist, obviously, different methods how to deal with this complicated problem. One of them could be to determine the entrance mass flow GBE by changing this value in a recursive

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

Another possibility is to find a relation between the time-derivatives of the mean and of special local mass flux values (e.g., at BC entrance), i.e., to establish for example a relation between

be determined directly from eq.(76). One practicable method to establish such a relation could follow if considering that a change of the mass flux is propagating along the channel so fast that the time derivative of its mean values could be set (in a first step) almost equal to the timederivative at its entrance value. This term can, eventually, be provided with a form factor which can be adapted by an adequate recursion procedure until the condition of eq.(75) is fulfilled. The so won entrance mass flow is then governing the mass flow behaviour of the entire loop. This last and very provisional method has been tested by applying the UTSG-3 code (Hoeld, 2011b) for the simulation of the natural-circulation behaviour of the secondary steam generator loop. Similar considerations have been undertaken for a 3D case where the automatic mass flow distribution into different entrances of a set of parallel channels had to be found (See e.g. Hoeld, 2004a and Jewer et al., 2005). The experience of such calculations should help to decide

The 'open/closed channel concept' makes sure that measures with regard to the entire closed loop do not need to be taken into account simultaneously but (for each channel) separately. Its application can be restricted to only one 'characteristic' channel of a sequence of channels of a complex loop. This additional tool of CCM can in such cases help to handle the variety of

Starting from the above presented 'drift-flux based mixture-fluid theory' a (1D) thermalhydraulic coolant channel module, named CCM, could be established. A module which was derived with the intention to provide the authors of different and sometimes very complex thermal-hydraulic codes with a general and easily applicable tool needed for the simulation of the steady state and transient behaviours of the most important single- and two-phase

For more details about the construction and application of the module CCM see (Hoeld,

During the course of development of the different versions of the code combination UTSG-3/ CCM the module has gone through an appropriate verification and validation (V&V) proce‐

which of the different possible procedures should finally be given preference.

closed loops within a complex physical system in a very comfortable way.

parameters along any type of heated or cooled coolant channel.

**5. Verification and Validation (V & V) procedures**

*dt* GFBMT agrees with the criterion above.

*dt* GFBE. Then the wanted mass flow time-derivative at BC entrance can

http://dx.doi.org/10.5772/53372

35

way until the resulting term *<sup>d</sup>*

*dt* GFBMT and *<sup>d</sup>*

**4. Code package CCM**

2005, 2007a and b and 2011b).

the terms *<sup>d</sup>*

#### **3.11. BC entrance mass flow ('Open and closed channel concept')**

As to be seen from the sections above in order to be able to calculate the characteristic nodal and total single- and two-phase parameters along a BC the BC entrance mass flow must be known. This term is for the steady state case given by input (GBE = GBEIN), together with the two pressure entrance and outlet values (PBE = PBEIN, PBA = PBAIN). In the transient situation it can, however, be expected that for the normal case of an 'open' channel besides the entrance mass flow only one of these two pressure terms is known by input, either at BC entrance or outlet. The missing one follows then from the calculation of the pressure decrease part.

Such a procedure can not be applied without problems if the channels are part of a complex set of closed loops. Loops consisting of more than one coolant channel (and not driven by an outside source, such as a pump). Then the mass flow values (and especially the entrance term of at least one of the channels) has to be adjusted with respect to the fact that the sum of the entire pressure decrease terms along such a closed circuit must be zero. Usually, in the common thermal-hydraulic codes (see for example the 'separate-phase' approaches) this problem is handled by solving the three (or more) fundamental equations for the entire complex system simultaneously, a procedure which affords very often immense computational times and costs. In the here applied module (based on a separate treatment of momentum from mass and energy balance) a more elegant method could be found by introducing an additional aspect into the theory of CCM. It allows, different to other approaches, to take care of this situation by solving this problem by means of a 'closed channel concept' (in contrast to the usual 'open channel' method).

Choosing for this purpose a characteristic 'closed' channel within such a complex loop it can be expected that its pressure difference term ΔPBT = ΔPBA - ΔPBE over this channel is fix (= negative sum of all the other decrease terms of the remaining channels which can be calculated by the usual methods). Thus also the outlet and entrance BC pressure values (PBAIN, ΔPBEIN) are now available as inputs to CCM. Since, according to eq.(69), the term ∆PPBT is known too, then from eq.(68) a 'closed channel concept criterion' can be formulated

$$\mathbf{z\_{BT}} \frac{d}{dt} \mathbf{G\_{FBMT}} = \boldsymbol{\Delta P\_{GBT}} = \boldsymbol{\Delta P\_{BBT}} \cdot \boldsymbol{\Delta P\_{BTN}} \quad \text{(at \text{\textquotedbl{}closed channel\textquotedbl{}} conditions\textquotedbl{})} \tag{76}$$

demanding that in order to fulfil this criterion the total mass flow along a BC (and thus also at its entrance) must be adapted in such a way that the above criterion which is essential for the 'closed channel' concept remains valid at each time step, i.e., that the actual time derivative of the mean mass flux GFBMT averaged over the channel must, as recommended by eq.(71), agree in a satisfactory manner with the required one from the equation above.

There exist, obviously, different methods how to deal with this complicated problem. One of them could be to determine the entrance mass flow GBE by changing this value in a recursive way until the resulting term *<sup>d</sup> dt* GFBMT agrees with the criterion above.

Another possibility is to find a relation between the time-derivatives of the mean and of special local mass flux values (e.g., at BC entrance), i.e., to establish for example a relation between the terms *<sup>d</sup> dt* GFBMT and *<sup>d</sup> dt* GFBE. Then the wanted mass flow time-derivative at BC entrance can be determined directly from eq.(76). One practicable method to establish such a relation could follow if considering that a change of the mass flux is propagating along the channel so fast that the time derivative of its mean values could be set (in a first step) almost equal to the timederivative at its entrance value. This term can, eventually, be provided with a form factor which can be adapted by an adequate recursion procedure until the condition of eq.(75) is fulfilled. The so won entrance mass flow is then governing the mass flow behaviour of the entire loop.

This last and very provisional method has been tested by applying the UTSG-3 code (Hoeld, 2011b) for the simulation of the natural-circulation behaviour of the secondary steam generator loop. Similar considerations have been undertaken for a 3D case where the automatic mass flow distribution into different entrances of a set of parallel channels had to be found (See e.g. Hoeld, 2004a and Jewer et al., 2005). The experience of such calculations should help to decide which of the different possible procedures should finally be given preference.

The 'open/closed channel concept' makes sure that measures with regard to the entire closed loop do not need to be taken into account simultaneously but (for each channel) separately. Its application can be restricted to only one 'characteristic' channel of a sequence of channels of a complex loop. This additional tool of CCM can in such cases help to handle the variety of closed loops within a complex physical system in a very comfortable way.

## **4. Code package CCM**

By adding the resulting nodal BC pressure difference terms to the (time-varying) system pressure PSYS(t), given from outside as boundary condition with respect to a certain position (in- or outside of the BC), then finally also the absolute nodal pressure profile PBk along the BC can be established (This term is needed at the begin of the next time step within the constitutive

As to be seen from the sections above in order to be able to calculate the characteristic nodal and total single- and two-phase parameters along a BC the BC entrance mass flow must be known. This term is for the steady state case given by input (GBE = GBEIN), together with the two pressure entrance and outlet values (PBE = PBEIN, PBA = PBAIN). In the transient situation it can, however, be expected that for the normal case of an 'open' channel besides the entrance mass flow only one of these two pressure terms is known by input, either at BC entrance or outlet. The missing one follows then from the calculation of the pressure decrease part.

Such a procedure can not be applied without problems if the channels are part of a complex set of closed loops. Loops consisting of more than one coolant channel (and not driven by an outside source, such as a pump). Then the mass flow values (and especially the entrance term of at least one of the channels) has to be adjusted with respect to the fact that the sum of the entire pressure decrease terms along such a closed circuit must be zero. Usually, in the common thermal-hydraulic codes (see for example the 'separate-phase' approaches) this problem is handled by solving the three (or more) fundamental equations for the entire complex system simultaneously, a procedure which affords very often immense computational times and costs. In the here applied module (based on a separate treatment of momentum from mass and energy balance) a more elegant method could be found by introducing an additional aspect into the theory of CCM. It allows, different to other approaches, to take care of this situation by solving this problem by means of a 'closed channel concept' (in contrast to the usual 'open

Choosing for this purpose a characteristic 'closed' channel within such a complex loop it can be expected that its pressure difference term ΔPBT = ΔPBA - ΔPBE over this channel is fix (= negative sum of all the other decrease terms of the remaining channels which can be calculated by the usual methods). Thus also the outlet and entrance BC pressure values (PBAIN, ΔPBEIN) are now available as inputs to CCM. Since, according to eq.(69), the term ∆PPBT is known too, then

demanding that in order to fulfil this criterion the total mass flow along a BC (and thus also at its entrance) must be adapted in such a way that the above criterion which is essential for the 'closed channel' concept remains valid at each time step, i.e., that the actual time derivative of the mean mass flux GFBMT averaged over the channel must, as recommended by eq.(71), agree

dt GFBMT = ΔPGBT= ΔPPBT- ΔPBTIN (at 'closed channel' conditions) (76)

from eq.(68) a 'closed channel concept criterion' can be formulated

in a satisfactory manner with the required one from the equation above.

**3.11. BC entrance mass flow ('Open and closed channel concept')**

34 Nuclear Reactor Thermal Hydraulics and Other Applications

equations).

channel' method).

zBT d Starting from the above presented 'drift-flux based mixture-fluid theory' a (1D) thermalhydraulic coolant channel module, named CCM, could be established. A module which was derived with the intention to provide the authors of different and sometimes very complex thermal-hydraulic codes with a general and easily applicable tool needed for the simulation of the steady state and transient behaviours of the most important single- and two-phase parameters along any type of heated or cooled coolant channel.

For more details about the construction and application of the module CCM see (Hoeld, 2005, 2007a and b and 2011b).

## **5. Verification and Validation (V & V) procedures**

During the course of development of the different versions of the code combination UTSG-3/ CCM the module has gone through an appropriate verification and validation (V&V) proce‐ dure (with continuous feedbacks being considered in the continual formulation of the theoretical model).

**•** the quality of the fundamental equations (basic conservation equations following directly from physical laws supported by experimentally based constitutive equations vs. split 'field'

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

37

**•** the special solution methods due to the detailed interpolation procedure from PAX allowing to calculate the exact movement of boiling boundaries and mixture (or dry-out) levels (different to the 'donor-cell averaging' methods yielding mostly only 'condensed' levels), **•** the easy replacement of new and improved correlations within the different packages without having to change the basic equations of the theory (for example the complicated

**•** the possibility to take advantage of the 'closed-channel concept' (needed for example for thermal- hydraulic 3D considerations) allowing thus to decouple a characteristic ('closed')

**•** the derivation of the theory in close connection with the establishment of the code by taking

**•** the considerable effort that has been made in verifying and checking CCM (besides an extensive V & V procedure), with respect to the applicability and adjustment and also for

**•** the maturity of the module, which is continuously enhanced by new application cases, **•** taking advantage of the fact that most of the development work for the coolant channel thermal-hydraulics has already been shifted to the establishment of the here presented module (including the special provisions for extreme situations such as stagnant flow, zero power or zero sub-cooling, test calculations for the verification and validation of the code

The existence of the resulting widely verified and validated module CCM represents an important basic element for the construction of a variety of other comprehensive thermalhydraulic models and codes as well. Such models and modules can be needed for the simu‐ lation of the steady state and transient behaviour of different types of steam generators, of 3D thermal-hydraulic compartments consisting of a number of parallel channels (reactor cores, VVER steam generators etc.). It shows special advantages in view of the determination of the mass flow distribution into different coolant channels after non-symmetric perturbations see (Hoeld, 2004a) or (Jewer et al., 2005), a problem which is far from being solved in many of the

The introduction of varying cross sections along the z axis allows to take care also of thermal non-equilibrium situations by simulating the two separate phases by two with each other interacting basic channels (for example if injecting sub-cooled water rays into a steam dome). The resulting equations for different channels appearing in a complex physical system can be combined with other sets of algebraic equations and ODE-s coming from additional parts of

equations with artificial closure terms),

**•** the speed of the computation,

very extreme situations,

**•** its easy applicability,

etc.).

newest 3D studies.

exchange terms of a 'separate-phase' approach),

advantage of feedbacks coming from both sides,

channel from other parts of a complex system of loops,

CCM is (similar as done in the separate-phase models) constructed with the objective to be used only as an element within an overall code. Hence, further V&V steps could be performed only in an indirect way, i.e. in combination with such overall codes. This has been done in a very successful way by means of the U-tube steam generator code UTSG-3. Thereby the module CCM could profit from the experiences been gained in decades of years work with the construction of an effective non-linear one-dimensional theoretical model and, based on it, corresponding digital code UTSG-2 for vertical, natural-circulation U-tube steam generators (Hoeld, 1978, 1988b, 1990a and 1990b), (Bencik et al., 1991) and now also the new advanced code version UTSG-3 (Hoeld 2002b, 2004a).

The good agreement of the test calculations with similar calculations of earlier versions applied to the same transient cases demonstrates that despite of the continuous improvements of the code UTSG and the incorporation of CCM into UTSG-3 the newest and advanced version has still preserved its validity.

A more detailed description over these general V&V measures demonstrated on one charac‐ teristic test case can be found in (Hoeld, 2011b).

## **6. Conclusions**

The universally applicable coolant channel module CCM allows describing the thermalhydraulic situation of fluids flowing along up-, horizontal or downwards channels with fluids changing between sub-cooled, saturated and superheated conditions. It must be recognized that CCM represents a complete system in its own right, which requires only BC-related, and thus easily available input values (geometry data, initial and boundary conditions, resulting parameters from integration). The partitioning of a basic channel into SC-s is done automati‐ cally within the module, requiring no special actions on the part of the user. At the end of a time-step the characteristic parameters of all SC-s are transferred to the corresponding BC positions, thus yielding the final set of ODE-s together with the parameters following from the constitutive equations of CCM.

In contrast to the currently very dominant separate-phase models, the existing theoretical inconsistencies in describing a two-phase fluid flowing along a coolant channel if changing between single-phase and two-phase conditions and vice versa can be circumvented in a very elegant way in the 'separate-region' mixture-fluid approach presented here. A very unique technique has been established built on the concept of subdividing a basic channel (BC) into different sub-channels (SC-s), thus yielding exact solutions of the basic drift-flux supported conservation equations. This type of approach shows, as discussed in (Hoeld, 2004b), distinct advantages vs. 'separate phase' codes, especially if taking into account


dure (with continuous feedbacks being considered in the continual formulation of the

CCM is (similar as done in the separate-phase models) constructed with the objective to be used only as an element within an overall code. Hence, further V&V steps could be performed only in an indirect way, i.e. in combination with such overall codes. This has been done in a very successful way by means of the U-tube steam generator code UTSG-3. Thereby the module CCM could profit from the experiences been gained in decades of years work with the construction of an effective non-linear one-dimensional theoretical model and, based on it, corresponding digital code UTSG-2 for vertical, natural-circulation U-tube steam generators (Hoeld, 1978, 1988b, 1990a and 1990b), (Bencik et al., 1991) and now also the new advanced

The good agreement of the test calculations with similar calculations of earlier versions applied to the same transient cases demonstrates that despite of the continuous improvements of the code UTSG and the incorporation of CCM into UTSG-3 the newest and advanced version has

A more detailed description over these general V&V measures demonstrated on one charac‐

The universally applicable coolant channel module CCM allows describing the thermalhydraulic situation of fluids flowing along up-, horizontal or downwards channels with fluids changing between sub-cooled, saturated and superheated conditions. It must be recognized that CCM represents a complete system in its own right, which requires only BC-related, and thus easily available input values (geometry data, initial and boundary conditions, resulting parameters from integration). The partitioning of a basic channel into SC-s is done automati‐ cally within the module, requiring no special actions on the part of the user. At the end of a time-step the characteristic parameters of all SC-s are transferred to the corresponding BC positions, thus yielding the final set of ODE-s together with the parameters following from the

In contrast to the currently very dominant separate-phase models, the existing theoretical inconsistencies in describing a two-phase fluid flowing along a coolant channel if changing between single-phase and two-phase conditions and vice versa can be circumvented in a very elegant way in the 'separate-region' mixture-fluid approach presented here. A very unique technique has been established built on the concept of subdividing a basic channel (BC) into different sub-channels (SC-s), thus yielding exact solutions of the basic drift-flux supported conservation equations. This type of approach shows, as discussed in (Hoeld, 2004b), distinct

advantages vs. 'separate phase' codes, especially if taking into account

theoretical model).

code version UTSG-3 (Hoeld 2002b, 2004a).

36 Nuclear Reactor Thermal Hydraulics and Other Applications

teristic test case can be found in (Hoeld, 2011b).

still preserved its validity.

**6. Conclusions**

constitutive equations of CCM.


The existence of the resulting widely verified and validated module CCM represents an important basic element for the construction of a variety of other comprehensive thermalhydraulic models and codes as well. Such models and modules can be needed for the simu‐ lation of the steady state and transient behaviour of different types of steam generators, of 3D thermal-hydraulic compartments consisting of a number of parallel channels (reactor cores, VVER steam generators etc.). It shows special advantages in view of the determination of the mass flow distribution into different coolant channels after non-symmetric perturbations see (Hoeld, 2004a) or (Jewer et al., 2005), a problem which is far from being solved in many of the newest 3D studies.

The introduction of varying cross sections along the z axis allows to take care also of thermal non-equilibrium situations by simulating the two separate phases by two with each other interacting basic channels (for example if injecting sub-cooled water rays into a steam dome).

The resulting equations for different channels appearing in a complex physical system can be combined with other sets of algebraic equations and ODE-s coming from additional parts of such a complex model (heat transfer or nuclear kinetics considerations, top plenum, main steam system and downcomer of a steam generator etc.). The final overall set of ODE-s can then be solved by applying an appropriate time-integration routine (Hoeld, 2004a, 2005, 2007a and b, 2011b).

NBT - Total number of BC nodes

NCT=NBCA-NBCE+1 - Total number of SC nodes

*W m* <sup>3</sup>

*W m* <sup>2</sup>

*W*

*W m* <sup>3</sup>

T, t C, s Temperature, time

*m*

α - Void fraction

*W*

Δ - Nodal differences

2 - Two-phase multiplier

*<sup>m</sup>* <sup>3</sup> , *kg <sup>J</sup>* , *kg m* <sup>3</sup>*C*

*<sup>m</sup>* <sup>3</sup> <sup>=</sup>*kg ms* <sup>2</sup>

QBT, QBMk W Total and nodal power into BC node k

UTW m Perimeter of a heated (single) tube wall

<sup>2</sup> (ANn+ANn-1)ΔzNn m³ Mean nodal SC volume

P, ΔPT = PA- PE Pa= *<sup>J</sup>*

qBMk = *<sup>U</sup> Bk*

qFBk=ABk *qBk UBk* = *qLBk UTWBk*

qLBk= *QBk*

qMn = *QMn V Mn*

= <sup>1</sup>

VMn= <sup>1</sup>

vS = *GFS*

X= *GS*

αTWk

<sup>Φ</sup>2*PF*

ρ, ρ<sup>P</sup>, ρ<sup>T</sup> *kg*

*ABMk* qLBMk

<sup>Δ</sup>*zBk* =ABkqBk

<sup>2</sup> *AMn* (qLNn+qLNn-1)

αρ*<sup>S</sup>* , vW = *GFW* (1 − α)ρ*<sup>W</sup>*

*G* or = *<sup>h</sup>* <sup>−</sup> *<sup>h</sup>* / *h SW*

NBCA = NCT+NBCE-1, NBCE - BC node numbers containing SC outlet or entrance

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

NSC = NSCE, NSCA - Characteristic number of SC, setting NSCE= 1, 2 or 3

if LFTYPE= 0, 1 or 2 and NSCA=NSCE+NSCT-1

Pressure and pressure difference (in flow direction)

http://dx.doi.org/10.5772/53372

39

Mean nodal BC power density into the fluid

Mean nodal SC power density into fluid (= volumetric0 heat transfer rate)


*<sup>m</sup>* <sup>2</sup>*<sup>C</sup>* Heat transfer coefficient along a BC wall surface

Density and their partial derivatives with respect to

Heat flux from (heated) wall to fluid at BC node k

(= volumetric heat transfer rate)

*<sup>m</sup>* Linear power along BC node k

*<sup>s</sup>* Steam and water velocity

zBB, zML, zSPC m Boiling boundary, mixture level resp. supercritical boundary within

a BC

εDPZ - Coefficient controlling the additional friction part

εTW m Abs. roughness of tube wall (εTW/dHW = relative value)

εQTW - Correction factor with respect to QNOM,0

ΦZG - Angle between upwards and flow direction

z, ΔzNn=zNn-zNn-1 m Local position, SC node length (zNn-1=zCE at n=0) zBA-zBE=zBT,zCA-zCE=zCT m BC and SC outlet and entrance positions, total length

The enormous efforts already made in the verification and validation of the codes UTSG-3, its application in a number of transient calculations at very extreme situations (fast opening of safety valves, dry out of the total channel with SC-s disappearing or created anew) brings the code and thus also CCM to a very mature and (what is important) easily applicable state. However, there is not yet enough experience to judge how the potential of the mixture-fluid models and especially of CCM can be expanded to other extreme cases (e.g., water and steam hammer). Is it justified to prefer separate-phase models versus the drift-flux based (and thus non-homogeneous) mixture fluid models? This depends, among other criteria, also on the quality of the special models and their exact derivation. Considering the arguments presented above it can, however, be stated that in general the here presented module can be judged as a very satisfactory approach.

## **7. Nomenclature**


#### The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 39

such a complex model (heat transfer or nuclear kinetics considerations, top plenum, main steam system and downcomer of a steam generator etc.). The final overall set of ODE-s can then be solved by applying an appropriate time-integration routine (Hoeld, 2004a, 2005,

The enormous efforts already made in the verification and validation of the codes UTSG-3, its application in a number of transient calculations at very extreme situations (fast opening of safety valves, dry out of the total channel with SC-s disappearing or created anew) brings the code and thus also CCM to a very mature and (what is important) easily applicable state. However, there is not yet enough experience to judge how the potential of the mixture-fluid models and especially of CCM can be expanded to other extreme cases (e.g., water and steam hammer). Is it justified to prefer separate-phase models versus the drift-flux based (and thus non-homogeneous) mixture fluid models? This depends, among other criteria, also on the quality of the special models and their exact derivation. Considering the arguments presented above it can, however, be stated that in general the here presented module can be judged as a

ABk m² BC cross sectional area (at BC node boundary k)

ATWMn m² Surface area of a (single) tube wall along a node n

fLIMCA - Upper or lower limit of the approx. function f(z,t) fADD0, fFMP0 - Additive and multiplicative friction coefficients

KEYBC - Characteristic key number of channel BC


dHY m Hydraulic diameter

*kg s kg sm* <sup>2</sup>

*kg* , *m* <sup>3</sup> *kg* , *<sup>J</sup> m* <sup>3</sup>*kg*

LFTBE (=LFTYPE of 1-st NSC) - LFTYPE of 1-st SC within BC

ANn, AMn m² SC cross sectional area (at SC node boundary n, mean value)

f(z,t), fNn, fMn - General and nodal (boundary and mean) solution functions

Mass flow Mass flux

LFTYPE= 0, 1 or 2 - SC with saturated water/steam mixture, sub-cooled water or superheated steam

Dimensionless constant Phase distribution parameter

Specific enthalpy and its partial derivatives with respect to pressure and temperature (= specific heat)

*kg* Latent heat, saturation steam and water enthalpy

2007a and b, 2011b).

38 Nuclear Reactor Thermal Hydraulics and Other Applications

very satisfactory approach.

**7. Nomenclature**

C C0

G = GW + GS

hSW = h// – h/

GF = *<sup>G</sup> <sup>A</sup>* = v ρ

h, hP

=Α[(1−α)ρWvW+αvSρS]

, cP = hT *<sup>J</sup>*

, h// ,h/ *<sup>J</sup>*



GCSM General Control Simulation Module (Part of ATHLET)

HETRAC Heat transfer coefficients package

HTC Heat transfer coefficients MDS Drift flux code package

NPP Nuclear power plant

**Author details**

Alois Hoeld\*

**References**

Vol.4.

GRS Gesellschaft für Anlagen- und Reaktorsicherheit

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

41

MPPWS, MPPETA Thermodynamic and transport property package

PAX (PAXDRI) Quadratic polygon approximation procedure (with driver code)

PDE, ODE Partial and ordinary differential equation

V & V Verification and validation procedure

Retired from GRS, Garching/Munich, Munich, Germany

[1] Austregesilo, H, et al. (2003). *ATHLET Mod 2.0, Cycle A, Models and methods*. GRS-P-1/

[2] Bencik, V, & Hoeld, A. (1991). *Experiences in the validation of large advanced modular system*

[3] Bencik, V, & Hoeld, A. (1993). *Steam collector and main steam system in a multi-loop PWR NPP representation.* Simulation MultiConf., March 29 April 1, Arlington/Washington

[4] Bestion, D. (1990). *The physical closure laws in CATHARE code*. Nucl.Eng.& Design Burwell, J. et al. 1989. *The thermohydraulic code ATHLET for analysis of PWR and SWR system*, Proc.of 4-th Int. Topical Meeting on Nuclear Reactor Thermal-Hydraulics,

[5] Fabic, S. (1996). *How good are thermal-hydraulics codes for analyses of plant transients*. International ENS/HND Conference, Oct. 1996, Proc.193-201, Opatija/Croatia, 7-9.

*codes.* 1991 SCS Simulation MultiConf., April 1-5New Orleans

NURETH-4, Oct.10-13, Karlsruhe, 124, 229-245.

UTSG U-tube steam generator code

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes http://dx.doi.org/10.5772/53372 41


## **Author details**

Alois Hoeld\*

(system) pressure and temperature

Θ s Time constant ∂ - Partial derivative

40 Nuclear Reactor Thermal Hydraulics and Other Applications

Mn, BMk Mean values over SC or BC nodes

0, 0 (=E or BE) Steady state or entrance to SC or BC (n or k =0) A, E, T (=AE) Outlet, entrance, total (i.e. from outlet to entrance) B, S Basic channel or sub-channel (=channel region)

> Acceleration, static head, direct and additional Friction, external pressure differences

(in connection with ΔP) and pressure differences

due to changes in mass flux

Nn, Bk SC or BC node boundaries (n = 0 or k = 0: Entrance)

P, T Derivative at constant pressure or temperature

Subscripts

A, S, F, D, X

Superscripts

Acronyms

TRAC

ATHLET, CATHARE; CATHENA, RELAP,

G

(P=A+S+F+D+X) and

D Drift

S, W Steam, water

TW Tube wall surface

/,// Saturated water or steam

P, T Partial derivatives with respect to P or T

GS, .α. or z (=gradient), slope

a separate-phase approach)

Well-known thermal-hydraulic codes (on the basis

into subchannels (=regions of different flow types)

(GS), (α), z, s Partial derivatives with respect to

BC, SC Basic(=coolant) channel subdivided

BWR, PWR Boiling and pressurized reactors

CCM Coolant channel model and module (on the basis

a separate-region approach)

CCF Counter-current flow

Retired from GRS, Garching/Munich, Munich, Germany

## **References**


[6] Haar, L, Gallagher, J. S, & Kell, G. S. (1988). *NBS/NRC Wasserdampftafeln*. London/Paris/ New York 1988

[22] Hoeld, A. (2007a). C*oolant channel module CCM. An universally applicable thermalhydraulic drift-flux based mixture-fluid 1D model and code.* Nucl. Eng. and Desg. 237(2007),

The Coolant Channel Module CCM — A Basic Element for the Construction of Thermal-Hydraulic Models and Codes

http://dx.doi.org/10.5772/53372

43

[23] Hoeld, A. (2007b). *Application of the mixture-fluid channel element CCM within the U-tube steam generator code UTSG-3 for transient thermal-hydraulic calculations.*th Int. Conf. on Nuclear Engineering (ICONE-15). Paper 10206. April 22-26, 2007, Nagoya, Japan, 15.

[24] Hoeld, A. (2011a). *Coolant channel module CCM. An universally applicable thermalhydraulic drift-flux based separate-region mixture-fluid model.* Intech Open Access Book 'Steam Generator Systems: Operational Reliability and Efficiency'', V. Uchanin (Ed.), March 2011. 978-9-53307-303-3InTech, Austria, Available as: http://www.intechop‐ en.com/articles/show/title/*the-thermal-hydraulic- u-tube-steam-generator- model-and-codeutsg-3-based-on-the-universally- applicab* or http://www.intechopen.com/search?q=hoeld

[25] Hoeld, A. (2011b). *The thermal-hydraulic U-tube steam generator model and code UTSG3 (Based on the universally applicable coolant channel module CCM).* See Intech Open Access

[26] Hoeld, A, & Jakubowska, E. Miro, J.E & Sonnenburg,H.G. (1992). *A comparative study*

[27] Ishii, M. (1990). *Two-Fluid Model for Two-Phase Flow*, Multiphase Sci. and Techn. 5.1,

[28] Ishii, M, & Mishima, K. (1980). *Study of two-fluid model and interfacial area.* NUREG/

[29] Jewer, S, Beeley, P. A, Thompson, A, & Hoeld, A. (2005). *Initial version of an integrated thermal-hydraulic and neutron kinetics 3D code X3D.* ICONE13, May Beijing. See also NED

[30] Lerchl, G, et al. (2009). *ATHLET Mod 2.2, Cycle B, User's Manual'*. GRS-Rev. 5, July, 1 [31] Liles, D. R, et al. (1988). *TRAC-PF1/MOD1 Correlations and Models*. NUREG/CR-5069,

[32] Martinelli, R. C, & Nelson, D. B. (1948). *Prediction of pressure drop during forced-circulation*

[33] Moody, N. L. F. (1994). *Friction factors for pipe flow*. Trans. ASME, 66, 671 (See also VDI-

[34] Schmidt, E, & Grigull, U. ed.) (1982). *Properties of water and steam in SI-Units*. Springer-

[35] Sonnenburg, H. G. (1989). *Full-range drift-flux model based on the combination of drift-flux*

[36] Shultz, R. R. (2003). *RELAP5/3-D code manual. Volume V: User's guidelines*. INEEL-

*theory with envelope theory*, NURETH-4, Oct.10-13, Karlsruhe., 1003-1009.

*on slip and drift- flux correlations*. GRS-A-1879, Jan. 1992

1952-1967

Book referred above.

Hemisphere Publ. Corp.

CR-1873 (ANL-80-111), Dec.

236(2006), 1533-1546., 16-20.

*of boiling water*. Trans. ASME 70, 695.

EXT-98-00084. Rev.2.2, October.

Wärmeatlas, 7.Auflage, 1994, VDI- Verlag).

LA-11208-MS, Dec.

Verlag.


[22] Hoeld, A. (2007a). C*oolant channel module CCM. An universally applicable thermalhydraulic drift-flux based mixture-fluid 1D model and code.* Nucl. Eng. and Desg. 237(2007), 1952-1967

[6] Haar, L, Gallagher, J. S, & Kell, G. S. (1988). *NBS/NRC Wasserdampftafeln*. London/Paris/

[7] Hanna, B. N. (1998). *CATENA. A thermalhydraulic code for CANDU analysis.* Nucl. Eng.

[8] Hoeld, A. (1978). *A theoretical model for the calculation of large transients in nuclear natural circulation U-tube steam generators (Digital code UTSG)*. Nucl. Eng. Des. (1978), 47, 1.

[10] Hoeld, A. (1988b). *Calculation of the time behavior of PWR NPP during a loss of a feedwater ATWS case*. NUCSAFE October, Proc. 1037-1047, Avignon, France., 88, 2-7.

[11] Hoeld, A. *UTSG-2. A theoretical model describing the transient behaviour of a PWR natural-*

[12] Hoeld, A. (1990b). *The code ATHLET as a valuable tool for the analysis of transients with and without SCRAM*, SCS Eastern MultiConf,April, Nashville, Tennessee., 23-26.

[13] Hoeld, A. (1996). *Thermodynamisches Stoffwertpaket MPP für Wasser und Dampf*. GRS,

[14] Hoeld, A. (1999). *An Advanced Natural-Circulation U-tube Steam Generator Model and Code*

[15] Hoeld, A. (2000). *The module CCM for the simulation of the thermal-hydraulic situation within a coolant channel*. Intern. NSS/ENS Conference, September Bled, Slovenia., 11-14.

[16] Hoeld, A. (2001). *The drift-flux correlation package MDS*. th International Conference on

[17] Hoeld, A. (2002a). *The consideration of entrainment in the drift-flux correlation package MDS*. th Int. Conference on Nuclear Engineering (ICONE-10), April 14-18, Arlington,

[18] Hoeld, A. (2002b). *A steam generator code on the basis of the general coolant channel module*

[19] Hoeld, A. (2004a). *A theoretical concept for a thermal-hydraulic 3D parallel channel core*

[20] Hoeld, A. (2004b). *Are separate-phase thermal-hydraulic models better than mixture-fluid approaches. It depends. Rather not*. Int. Conference on Nuclear Engineering for New

[21] Hoeld, A. *A thermal-hydraulic drift-flux based mixture-fluid model for the description of single- and two-phase flow along a general coolant channel.* The th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11). Paper 144. Oct. 2-6,

[9] Hoeld, A. *HETRACA heat transfer coefficients package.* GRS-A-1446, May.

*circulation U-tube steam generator*. Nuclear Techn. April., 90, 98-118.

*UTSG-3*, EUROTHERM-SEMINAR September Genoa, Italy.(63), 6-8.

Nuclear Engineering (ICONE-9), 8-12 April, Nice, France., 9.

*CCM*. PHYSOR 2002, October Seoul, Korea., 7-10.

*model.* PHYSOR 2004, April Chicago, USA., 25-29.

Europe 2004'. Sept. Portoroz, Slovenia., 6-9.

2005, Avignon, France., 11.

New York 1988

USA., 10.

Des. , 180(1998), 113-131.

42 Nuclear Reactor Thermal Hydraulics and Other Applications

Technische Notiz, TN-HOD-1/96, Mai.


[37] , U. S-N. R. C. 2001a. *TRAC-M FORTRAN90 (Version 3.0). Theory Manual,* NUREG/ CR-6724, April

**Chapter 2**

**Provisional chapter**

**Large Eddy Simulation of Thermo-Hydraulic Mixing in**

Unsteady heat transfer problems that are associated with non-isothermal flow mixing in pipe flows have long been the topic of concern in the nuclear engineering community because of the relation to thermal fatigue of nuclear power plant pipe systems. When turbulent flow streams of different velocity and density rapidly mix at the right angle, a contact interface between the two mixing streams oscillates and breaks down because of hydrodynamic instabilities, and large-scale unsteady flow structures emerge (Figure 1). These structures lead to low-frequency oscillations at the scale of the pipe diameter, *D*, with a period scaling as *O*(*D*/*U*), where *U* is the characteristic flow velocity. If the mixing flow streams are of different temperatures, the hydrodynamic oscillations are accompanied by thermal fluctuations (thermal striping) on the pipe wall. The latter accelerate thermal-mechanical

The importance of this phenomenon prompted the nuclear energy modeling and simulation community to establish a common benchmark to test the ability of computational fluid dynamics (CFD) tools to predict the effect of thermal striping. The benchmark is based on the thermal and velocity data measurements obtained in a recent Vattenfall experiment designed specifically for this purpose [1, 2]. Because thermal striping is associated with large-scale anisotropic mixing, standard engineering modeling tools based on time-averaging approaches, such as Reynolds-averaged Navier-Stokes (RANS) models, are badly suited. Consequently, one must consider unsteady modeling methods such as unsteady RANS (URANS), parabolised stability equations (PSE), or large eddy simulation (LES). Among these choices, the LES approach is most generic and modeling-assumption free, unlike the PSE and URANS methods, which, for instance, assume the existence of a spectral gap between the large and the small scales. On the other hand, owing to recent advances in computing, LES

> ©2012 Karabasov et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Obabko et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Obabko et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Large Eddy Simulation of Thermo-Hydraulic Mixing**

**a T-Junction**

**in a T-Junction**

Sergey A. Karabasov

**1. Introduction**

Anna E. Aksenova and Sergey A. Karabasov

http://dx.doi.org/10.5772/53143

Aleksandr V. Obabko, Paul F. Fischer, Timothy J. Tautges, Vasily M. Goloviznin, Mikhail A. Zaytsev, Vladimir V. Chudanov, Valeriy A. Pervichko, Anna E. Aksenova and

Aleksandr V. Obabko, Paul F. Fischer, Timothy J. Tautges, Vasily M. Goloviznin,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Mikhail A. Zaytsev, Vladimir V. Chudanov, Valeriy A. Pervichko,

fatigue, damage the pipe structure and, ultimately, cause its failure.

methods are becoming an increasingly affordable tool.


## **Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction**

Aleksandr V. Obabko, Paul F. Fischer, Timothy J. Tautges, Vasily M. Goloviznin, Mikhail A. Zaytsev, Vladimir V. Chudanov, Valeriy A. Pervichko, Anna E. Aksenova and Sergey A. Karabasov Aleksandr V. Obabko, Paul F. Fischer, Timothy J. Tautges, Vasily M. Goloviznin, Mikhail A. Zaytsev, Vladimir V. Chudanov, Valeriy A. Pervichko, Anna E. Aksenova and Sergey A. Karabasov

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53143

## **1. Introduction**

[37] , U. S-N. R. C. 2001a. *TRAC-M FORTRAN90 (Version 3.0). Theory Manual,* NUREG/

[38] , U. S-N. R. C. 2001b. *RELAP5/MOD3.3. Code Manual.* Volume I, NUREG/CR-5535,

[39] Zuber, N, & Findlay, J. A. (1965). Average volume concentration in two-phase flow

CR-6724, April

systems, J. Heat Transfer 87, 453

44 Nuclear Reactor Thermal Hydraulics and Other Applications

December

Unsteady heat transfer problems that are associated with non-isothermal flow mixing in pipe flows have long been the topic of concern in the nuclear engineering community because of the relation to thermal fatigue of nuclear power plant pipe systems. When turbulent flow streams of different velocity and density rapidly mix at the right angle, a contact interface between the two mixing streams oscillates and breaks down because of hydrodynamic instabilities, and large-scale unsteady flow structures emerge (Figure 1). These structures lead to low-frequency oscillations at the scale of the pipe diameter, *D*, with a period scaling as *O*(*D*/*U*), where *U* is the characteristic flow velocity. If the mixing flow streams are of different temperatures, the hydrodynamic oscillations are accompanied by thermal fluctuations (thermal striping) on the pipe wall. The latter accelerate thermal-mechanical fatigue, damage the pipe structure and, ultimately, cause its failure.

The importance of this phenomenon prompted the nuclear energy modeling and simulation community to establish a common benchmark to test the ability of computational fluid dynamics (CFD) tools to predict the effect of thermal striping. The benchmark is based on the thermal and velocity data measurements obtained in a recent Vattenfall experiment designed specifically for this purpose [1, 2]. Because thermal striping is associated with large-scale anisotropic mixing, standard engineering modeling tools based on time-averaging approaches, such as Reynolds-averaged Navier-Stokes (RANS) models, are badly suited. Consequently, one must consider unsteady modeling methods such as unsteady RANS (URANS), parabolised stability equations (PSE), or large eddy simulation (LES). Among these choices, the LES approach is most generic and modeling-assumption free, unlike the PSE and URANS methods, which, for instance, assume the existence of a spectral gap between the large and the small scales. On the other hand, owing to recent advances in computing, LES methods are becoming an increasingly affordable tool.

©2012 Karabasov et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Obabko et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Obabko et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Parameter Main Branch Hot Branch**

Inlet temperature (◦C) 19.0 36.0 Density *ρ* (kg/m3) 998.5 993.7 Dynamic viscosity (*N* × *s*/*m*3) 1.029e-3 7.063e-4 Kinematic viscosity *ν* (*m*2/*s*) 1.031e-6 7.108e-7

**Parameter Main Branch Hot Branch** Diameter *D*, *DH* 1.000 0.714 Average velocity *U*, *UH* 1.000 1.307 Inlet temperature 0.000 1.000 Density 1.000 0.9952 Reynolds number *Re*, *ReH* 79400 107000

The flow enters the cold (main) branch from a stagnation chamber located 80 *D* upstream of the T-junction and is assumed to be a fully developed turbulent flow by the time it reaches the T-junction. The hot branch flow enters at 20*DH* upstream and is not quite fully developed as it enters the T-junction. The inlet flow rates are 9 and 6 liters per second (l/s), respectively in the cold and hot branches, which corresponds to a Reynolds number of *Re* ≈ 80, 000 and 100, 000, respectively. The key dimensional parameters are summarized in Table 1 and their nondimensional counterparts in Table 2. More detailed description of the T-junction

The Nek5000 simulations are based on the spectral element method developed by Patera [3]. Nek5000 supports two formulations for spatial and temporal discretization of the Navier-Stokes equations. The first is the lP*<sup>N</sup>* − lP*N*−<sup>2</sup> method [17–19], in which the velocity/pressure spaces are based on tensor-product polynomials of degree *N* and *N* − 2,

domain consists of the union of *E* elements Ω*e*, each of which is parametrically mapped from Ωˆ to yield a body-fitted mesh. The second is the low-Mach number formulation due to Tomboulides and Orszag [20, 21], which uses consistent order-*N* approximation spaces for both the velocity and pressure. The low-Mach number formulation is also valid in the zero-Mach (incompressible) limit [22]. Both formulations yield a decoupled set of elliptic problems to be solved at each timestep. In *d*=3 space dimensions, one has

symmetric positive-definite Laplace operator, and *β* is an order-unity coefficient coming from a third-order backward-difference approximation to the temporal derivative. (For the lP*<sup>N</sup>* − lP*N*−<sup>2</sup> method, the Laplace operator *<sup>A</sup>* is replaced by a spectrally equivalent matrix arising from the unsteady Stokes equations [4, 19].) For marginally resolved LES cases, we find that the higher-order pressure approximation of the lP*<sup>N</sup>* − lP*<sup>N</sup>* formulation tends to yield improved skin-friction estimates, and this is consequently the formulation considered here.

*<sup>i</sup>* <sup>=</sup> *<sup>f</sup> <sup>n</sup> i*

*<sup>d</sup>*, for *d* = 2 or 3. The computational

, *i* = 1, . . . , *d*, and a pressure

*<sup>n</sup>*, *n* = 1, . . . . Here, *A* is the

*<sup>H</sup>* (m) 0.1400 0.1000

*<sup>H</sup>* (m/s) 0.585 0.764

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

http://dx.doi.org/10.5772/53143

47

*<sup>H</sup>* (l/s) 9.00 6.00

Diameter *<sup>D</sup>*∗, *<sup>D</sup>*<sup>∗</sup>

Average velocity *<sup>U</sup>*∗, *<sup>U</sup>*<sup>∗</sup>

Flow rate *<sup>q</sup>*∗, *<sup>q</sup>*<sup>∗</sup>

**Table 1.** Dimensional Parameters for Vatenfall T-junction Experiment

**Table 2.** Nondimensional Parameters for Vatenfall T-junction Experiment

benchmark experiment can be found in [1, 2].

respectively, in the reference element Ωˆ := [−1, 1]

three Helmholtz solves of the form (*βI* + *ν*∆*tA*)*u<sup>n</sup>*

Poisson solve of the form *Ap<sup>n</sup>* = *g<sup>n</sup>* at each timestep, *t*

**3. Nek5000 simulations**

**Figure 1.** Instantaneous wall temperature distribution from Nek5000 T-junction simulation.

In this chapter, we compare the results of three LES codes produced for the Vattenfall benchmark problem: Nek5000, developed at Argonne National Laboratory in the United States, and CABARET and Conv3D, developed at the Moscow Institute for Nuclear Energy Safety (IBRAE) in Russia. Nek5000 is an open source code based on the spectral-element method (SEM), a high-order weighted residual technique that combines the geometric flexibility of the finite-element method with the tensor-product efficiencies of spectral methods [3, 4]. CABARET, which stands for Compact Accurately Boundary Adjusting high REsolution Technique, is based on the scheme of [5]. CABARET was developed as a significant upgrade of the second-order upwind leapfrog scheme for linear advection equation [6, 7] to nonlinear conservation laws in one and two dimensions [8–10]. The CABARET scheme is second-order accurate on nonuniform grids in space and time, is nondissipative, and is low-dispersive. For nonlinear flow calculations, it is equipped with a conservative nonlinear flux correction that is directly based on the maximum principle. For 3D calculations, a new unstructured CABARET solver was developed (e.g. [11, 12]) that operates mainly with hexagonal meshes. Conv3D is a well-established solver of IBRAE that has been validated on various experimental and benchmark data [13, 14]. For ease of the grid generation in the case of complex geometries, Conv3D utilizes the immersed boundary method (IBM) on Cartesian meshes with a cut-cell approach. The underlying scheme of Conv3D [15] is a low-dissipative, second-order approximation in space and a first-order approximation in time.

This chapter is organized as follows. In Section 2 we briefly describe the experimental setup and data collection procedure. In Sections 3–5 we outline the computational models. The results are discussed in Section 6, including a comparison of data from the simulations and experiment. Concluding remarks and a brief discussion of future work are provided in Section 7.

## **2. Experimental configuration**

The Vatenfall experiment [16] is based on water flow in a main pipe of diameter *D*=140 mm with a side branch of diameter *DH*=100 mm adjoining the main at a 90 degree angle. The pipes and the T-junction, which is made from a plexiglass block, are transparent so that the velocity can be measured with laser Doppler anemometry (LDA). Velocity data was taken under isothermal conditions with both flows entering at 19 ◦C. In order to measure thermal striping, time-dependent temperature data was collected from thermocouples downstream of the T-junction with flow at 19 ◦C entering the main branch and flow at 36 ◦C entering the side branch.


**Table 1.** Dimensional Parameters for Vatenfall T-junction Experiment

2 Nuclear Reactor Thermal Hydraulics

approximation in time.

**2. Experimental configuration**

Section 7.

side branch.

**Figure 1.** Instantaneous wall temperature distribution from Nek5000 T-junction simulation.

In this chapter, we compare the results of three LES codes produced for the Vattenfall benchmark problem: Nek5000, developed at Argonne National Laboratory in the United States, and CABARET and Conv3D, developed at the Moscow Institute for Nuclear Energy Safety (IBRAE) in Russia. Nek5000 is an open source code based on the spectral-element method (SEM), a high-order weighted residual technique that combines the geometric flexibility of the finite-element method with the tensor-product efficiencies of spectral methods [3, 4]. CABARET, which stands for Compact Accurately Boundary Adjusting high REsolution Technique, is based on the scheme of [5]. CABARET was developed as a significant upgrade of the second-order upwind leapfrog scheme for linear advection equation [6, 7] to nonlinear conservation laws in one and two dimensions [8–10]. The CABARET scheme is second-order accurate on nonuniform grids in space and time, is nondissipative, and is low-dispersive. For nonlinear flow calculations, it is equipped with a conservative nonlinear flux correction that is directly based on the maximum principle. For 3D calculations, a new unstructured CABARET solver was developed (e.g. [11, 12]) that operates mainly with hexagonal meshes. Conv3D is a well-established solver of IBRAE that has been validated on various experimental and benchmark data [13, 14]. For ease of the grid generation in the case of complex geometries, Conv3D utilizes the immersed boundary method (IBM) on Cartesian meshes with a cut-cell approach. The underlying scheme of Conv3D [15] is a low-dissipative, second-order approximation in space and a first-order

This chapter is organized as follows. In Section 2 we briefly describe the experimental setup and data collection procedure. In Sections 3–5 we outline the computational models. The results are discussed in Section 6, including a comparison of data from the simulations and experiment. Concluding remarks and a brief discussion of future work are provided in

The Vatenfall experiment [16] is based on water flow in a main pipe of diameter *D*=140 mm with a side branch of diameter *DH*=100 mm adjoining the main at a 90 degree angle. The pipes and the T-junction, which is made from a plexiglass block, are transparent so that the velocity can be measured with laser Doppler anemometry (LDA). Velocity data was taken under isothermal conditions with both flows entering at 19 ◦C. In order to measure thermal striping, time-dependent temperature data was collected from thermocouples downstream of the T-junction with flow at 19 ◦C entering the main branch and flow at 36 ◦C entering the


**Table 2.** Nondimensional Parameters for Vatenfall T-junction Experiment

The flow enters the cold (main) branch from a stagnation chamber located 80 *D* upstream of the T-junction and is assumed to be a fully developed turbulent flow by the time it reaches the T-junction. The hot branch flow enters at 20*DH* upstream and is not quite fully developed as it enters the T-junction. The inlet flow rates are 9 and 6 liters per second (l/s), respectively in the cold and hot branches, which corresponds to a Reynolds number of *Re* ≈ 80, 000 and 100, 000, respectively. The key dimensional parameters are summarized in Table 1 and their nondimensional counterparts in Table 2. More detailed description of the T-junction benchmark experiment can be found in [1, 2].

## **3. Nek5000 simulations**

The Nek5000 simulations are based on the spectral element method developed by Patera [3]. Nek5000 supports two formulations for spatial and temporal discretization of the Navier-Stokes equations. The first is the lP*<sup>N</sup>* − lP*N*−<sup>2</sup> method [17–19], in which the velocity/pressure spaces are based on tensor-product polynomials of degree *N* and *N* − 2, respectively, in the reference element Ωˆ := [−1, 1] *<sup>d</sup>*, for *d* = 2 or 3. The computational domain consists of the union of *E* elements Ω*e*, each of which is parametrically mapped from Ωˆ to yield a body-fitted mesh. The second is the low-Mach number formulation due to Tomboulides and Orszag [20, 21], which uses consistent order-*N* approximation spaces for both the velocity and pressure. The low-Mach number formulation is also valid in the zero-Mach (incompressible) limit [22]. Both formulations yield a decoupled set of elliptic problems to be solved at each timestep. In *d*=3 space dimensions, one has three Helmholtz solves of the form (*βI* + *ν*∆*tA*)*u<sup>n</sup> <sup>i</sup>* <sup>=</sup> *<sup>f</sup> <sup>n</sup> i* , *i* = 1, . . . , *d*, and a pressure Poisson solve of the form *Ap<sup>n</sup>* = *g<sup>n</sup>* at each timestep, *t <sup>n</sup>*, *n* = 1, . . . . Here, *A* is the symmetric positive-definite Laplace operator, and *β* is an order-unity coefficient coming from a third-order backward-difference approximation to the temporal derivative. (For the lP*<sup>N</sup>* − lP*N*−<sup>2</sup> method, the Laplace operator *<sup>A</sup>* is replaced by a spectrally equivalent matrix arising from the unsteady Stokes equations [4, 19].) For marginally resolved LES cases, we find that the higher-order pressure approximation of the lP*<sup>N</sup>* − lP*<sup>N</sup>* formulation tends to yield improved skin-friction estimates, and this is consequently the formulation considered here.

yielding *Re* = 60, 000 in the outlet. Downstream of the T-junction, the first grid point away

Inlet flow conditions in the main branch are based on a recycling technique in which the inlet velocity at time *<sup>t</sup><sup>n</sup>* is given by *<sup>α</sup>***u**(*x*˜, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>n*−1), where *<sup>x</sup>*˜ = −3 and *<sup>α</sup>* is chosen to ensure that the mass flow rate at inflow is constant ( *un*(−9, *y*, *z*)*dy dz* ≡ *π*/4). Recycling is also

chosen so that the average inflow velocity is -*UH* and *z*˜ = 2.1. The 0.8 multiplier was added in order to give a flatter profile characteristic of the non-fully-developed flow realized in the experiment, but a systematic study of this parameter choice was not performed (see also

Initial conditions for all branches were taken from fully developed turbulent pipe flow simulations at *Re* = 40, 000. The timestep size was <sup>∆</sup>*<sup>t</sup>* = 2.5 × <sup>10</sup>−<sup>4</sup> in convective time units, or about 6 × <sup>10</sup>−<sup>5</sup> seconds in physical units corresponding to the experiment. The simulation was run to a time of *t*=28 convective time units prior to acquiring data. Data was then collected over the interval *t* ∈ [28, 58] (in convective time units) for the benchmark submission1 and over *t* ∈ [28, 104] subsequently (longer average) both with *N* = 7. In addition to *N* = 7 results (i.e., with *n* ≈ *EN*<sup>3</sup> ≈ 2.1 × 107 points), Nek5000 runs were conducted with *N* = 5 (*n* ≈ 7.7 × 106 points) and with *N* = 9 (*n* ≈ 4.5 × 107 points). The case with *N* = 5 started with the *N* = 7 results and was run and averaged over about 110 convective time units (i.e., about 26 seconds). The timestep size for *N* = 5 runs was twice as big (i.e, <sup>∆</sup>*<sup>t</sup>* = <sup>5</sup> × <sup>10</sup><sup>−</sup>4). The benchmark submission results required approximately 3.4 × 105 CPU hours on Intrepid (IBM BlueGene/P) while the follow-up study for a longer time average results took additional 5.9 × 105 CPU hours. Note that all Nek5000 results reported here were obtained with constant density equal to the nondimensional value of

The system of Navier-Stokes equations in a slightly compressible form (i.e., for a constant sound speed) is solved with a new code based on CABARET. In order to lessen the computational grid requirements, the code uses hybrid unstructured hexahedral/tetrahedral grids. CABARET is an extension of the original second-order, upwind leapfrog scheme [6] to nonlinear conservation laws [5, 24] and to multiple dimensions [9, 25]. In summary, CABARET is an explicit, nondissipative, conservative finite-difference scheme of second-order approximation in space and time. In addition to having low numerical dispersion, CABARET has a very compact computational stencil because of the use of separate variables for fluxes and conservation. The stencil is staggered in space and time and for advection includes only one computational cell. For nonlinear flows, CABARET uses a low-dissipative, conservative flux correction method directly based on the maximum principle [8]. In the LES framework, the nonlinear flux correction plays the role of implicit turbulence closure following the MILES approach of Grinstein and Fureby [26, ch.5], which

A detailed description of the CABARET code on a mixed unstructured grid will be the subject of a future publication; an outline of the method on a structured Cartesian grid is

*<sup>n</sup>*−1) <sup>−</sup> 0.2*UH*, with *<sup>β</sup>*

http://dx.doi.org/10.5772/53143

49

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

used for the hot branch, save that the inflow condition is 0.8*β***u**(*x*, *y*, *z*˜, *t*

from the wall is thus at *y*<sup>+</sup> ≈ 2.5 in wall units.

Section 6.1.1).

1.000 (see Table 2).

**4. CABARET simulations**

was discussed in the ocean modeling context in [10].

<sup>1</sup> Ranked #1 in temperature and #6 in velocity prediction out of 29 submissions [1]

**Figure 2.** Computational mesh for Nek5000 T-junction simulation comprising *E*=62176 elements.

Consisting of *E* = 62, 176 elements, the computational mesh for the Nek5000 simulations was generated by using CUBIT and read through the Nek-MOAB coupling interface [23]. Within each element, velocity and pressure are represented as Lagrange interpolating polynomials on tensor products of *N*th-order Gauss-Lobatto-Legendre points. Unless otherwise noted, all the simulations were run with polynomial order *N*=7, corresponding to a total number of mesh points *n* ≈ *EN*<sup>3</sup> ≈ 21 million. Figure 2 shows a closeup of the mesh in the vicinity of the origin, which is located at the intersection of the branch centerlines. The inlet for the main branch is at *x*=-9.2143 and for the hot branch at *z*=6.4286. These lengths permitted generation of fully developed turbulence upstream of the T-junction, as described below. The outlet at *x*=22 was chosen to allow downstream tracking of temperature data at locations provided in the experiment. Away from the origin, the axial extent of the spectral elements in the main branch is 0.18 *D*, corresponding to a maximum axial mesh spacing of *δx*max = 0.0377. At the wall, the wall-normal element size is 0.01222, corresponding to a minimum spacing of *δyn* ≈ 0.0008. The submitted simulations were run with *Re* = 40, 000 in the inlet branches,

yielding *Re* = 60, 000 in the outlet. Downstream of the T-junction, the first grid point away from the wall is thus at *y*<sup>+</sup> ≈ 2.5 in wall units.

Inlet flow conditions in the main branch are based on a recycling technique in which the inlet velocity at time *<sup>t</sup><sup>n</sup>* is given by *<sup>α</sup>***u**(*x*˜, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>n*−1), where *<sup>x</sup>*˜ = −3 and *<sup>α</sup>* is chosen to ensure that the mass flow rate at inflow is constant ( *un*(−9, *y*, *z*)*dy dz* ≡ *π*/4). Recycling is also used for the hot branch, save that the inflow condition is 0.8*β***u**(*x*, *y*, *z*˜, *t <sup>n</sup>*−1) <sup>−</sup> 0.2*UH*, with *<sup>β</sup>* chosen so that the average inflow velocity is -*UH* and *z*˜ = 2.1. The 0.8 multiplier was added in order to give a flatter profile characteristic of the non-fully-developed flow realized in the experiment, but a systematic study of this parameter choice was not performed (see also Section 6.1.1).

Initial conditions for all branches were taken from fully developed turbulent pipe flow simulations at *Re* = 40, 000. The timestep size was <sup>∆</sup>*<sup>t</sup>* = 2.5 × <sup>10</sup>−<sup>4</sup> in convective time units, or about 6 × <sup>10</sup>−<sup>5</sup> seconds in physical units corresponding to the experiment. The simulation was run to a time of *t*=28 convective time units prior to acquiring data. Data was then collected over the interval *t* ∈ [28, 58] (in convective time units) for the benchmark submission1 and over *t* ∈ [28, 104] subsequently (longer average) both with *N* = 7. In addition to *N* = 7 results (i.e., with *n* ≈ *EN*<sup>3</sup> ≈ 2.1 × 107 points), Nek5000 runs were conducted with *N* = 5 (*n* ≈ 7.7 × 106 points) and with *N* = 9 (*n* ≈ 4.5 × 107 points). The case with *N* = 5 started with the *N* = 7 results and was run and averaged over about 110 convective time units (i.e., about 26 seconds). The timestep size for *N* = 5 runs was twice as big (i.e, <sup>∆</sup>*<sup>t</sup>* = <sup>5</sup> × <sup>10</sup><sup>−</sup>4). The benchmark submission results required approximately 3.4 × 105 CPU hours on Intrepid (IBM BlueGene/P) while the follow-up study for a longer time average results took additional 5.9 × 105 CPU hours. Note that all Nek5000 results reported here were obtained with constant density equal to the nondimensional value of 1.000 (see Table 2).

## **4. CABARET simulations**

4 Nuclear Reactor Thermal Hydraulics

Y X

X

**Figure 2.** Computational mesh for Nek5000 T-junction simulation comprising *E*=62176 elements.

Consisting of *E* = 62, 176 elements, the computational mesh for the Nek5000 simulations was generated by using CUBIT and read through the Nek-MOAB coupling interface [23]. Within each element, velocity and pressure are represented as Lagrange interpolating polynomials on tensor products of *N*th-order Gauss-Lobatto-Legendre points. Unless otherwise noted, all the simulations were run with polynomial order *N*=7, corresponding to a total number of mesh points *n* ≈ *EN*<sup>3</sup> ≈ 21 million. Figure 2 shows a closeup of the mesh in the vicinity of the origin, which is located at the intersection of the branch centerlines. The inlet for the main branch is at *x*=-9.2143 and for the hot branch at *z*=6.4286. These lengths permitted generation of fully developed turbulence upstream of the T-junction, as described below. The outlet at *x*=22 was chosen to allow downstream tracking of temperature data at locations provided in the experiment. Away from the origin, the axial extent of the spectral elements in the main branch is 0.18 *D*, corresponding to a maximum axial mesh spacing of *δx*max = 0.0377. At the wall, the wall-normal element size is 0.01222, corresponding to a minimum spacing of *δyn* ≈ 0.0008. The submitted simulations were run with *Re* = 40, 000 in the inlet branches,

Z X Y

Y

(b) Top view

(a) Side view of surface mesh

X Y

Z

Z

(d) Slice at *x*=4.0

(c) Slice at *x*=0.0

Z

The system of Navier-Stokes equations in a slightly compressible form (i.e., for a constant sound speed) is solved with a new code based on CABARET. In order to lessen the computational grid requirements, the code uses hybrid unstructured hexahedral/tetrahedral grids. CABARET is an extension of the original second-order, upwind leapfrog scheme [6] to nonlinear conservation laws [5, 24] and to multiple dimensions [9, 25]. In summary, CABARET is an explicit, nondissipative, conservative finite-difference scheme of second-order approximation in space and time. In addition to having low numerical dispersion, CABARET has a very compact computational stencil because of the use of separate variables for fluxes and conservation. The stencil is staggered in space and time and for advection includes only one computational cell. For nonlinear flows, CABARET uses a low-dissipative, conservative flux correction method directly based on the maximum principle [8]. In the LES framework, the nonlinear flux correction plays the role of implicit turbulence closure following the MILES approach of Grinstein and Fureby [26, ch.5], which was discussed in the ocean modeling context in [10].

A detailed description of the CABARET code on a mixed unstructured grid will be the subject of a future publication; an outline of the method on a structured Cartesian grid is

<sup>1</sup> Ranked #1 in temperature and #6 in velocity prediction out of 29 submissions [1]

given below. Let us consider the governing equations written in the standard conservation form:

$$\frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{G}}{\partial y} + \frac{\partial \mathbf{W}}{\partial z} = \mathbf{Q}\_{\nu} \tag{1}$$

(a) (b) (c)

**Figure 3.** Computational grid used with the CABARET method: full domaim (a), pipe inlet (b,d), and mixed-grid elements in

For specifying inlet boundary conditions, a recycling technique is used similar to that for Nek5000. The outflow boundary is prescribed by using characteristic boundary conditions. For the inlet boundaries at the main and the hot branch, laminar inflow conditions are specified based on prescribing the mean flow velocity profiles. The length of the pipe upstream of the junction was sufficiently far from the junction to permit an adequate turbulent flow upstream of the junction. The outflow boundary is imposed at 20 jet diameters

In order to speed statistical convergence, the LES CABARET solution was started from a precursor RANS k-epsilon calculation. The CABARET simulation was then run for 10 seconds to allow the statistics to settle down. This was followed by the production run during which the required solution fields were stored for a duration of 5 and 10 seconds. The computations on grids with 0.53 and 4 million cells required approximately 1.8 × 104

Researchers at IBRAE have been developing a 3D, unified, numerical thermal-hydraulic technique for safety analysis of the nuclear power plants, which includes (1) methods,

downstream of the junction, where characteristic boundary conditions are set.

and 2.9 × 105 CPU hours on Lomonosov (Intel Xeon X5570 / X5670 processors).

the vicinity of the junction (c,e) for the 0.5 and 4 million cell grid, respectively.

**5. C**onv3**D simulations**

(d) (e)

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

http://dx.doi.org/10.5772/53143

51

where the sources in the right-hand side include viscous terms. By mapping the physical domain to the grid space-time coordinates (*x*, *y*, *z*, *t*) → (*i*, *j*, *k*, *n*) and referring control volumes to the cell centers (fractional indices) and fluxes to the cell faces (integer indices), the algorithm proceeds from the known solution at time level (n) to the next timestep (n+1) as follows:

• Conservation predictor step:

$$\frac{\mathbf{U}\_{i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}}^{n+\frac{1}{2}} - \mathbf{U}\_{i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}}^{n}}{0.5\,t} + \frac{\mathbf{F}\_{i+1}^{n} - \mathbf{F}\_{i}^{n}}{\Delta x} + \frac{\mathbf{G}\_{j+1}^{n} - \mathbf{G}\_{j}^{n}}{\Delta y} + \frac{\mathbf{W}\_{k+1}^{n} - \mathbf{W}\_{k}^{n}}{\Delta z} = \mathbf{Q}\_{\nu}^{n} \tag{2}$$

	- For each cell face, decompose the conservation and flux variables into local Riemann fields, **U** → *Rq*, *q* = 1, . . . , *N*, that correspond to the local cell-face-normal coordinate basis, where *N* is the dimension of the system.
	- For each cell face at the new timestep, compute a dual set of preliminary local Riemann variables that correspond to the upwind and downwind extrapolation of the characteristic fields, for example, *R*˜ *<sup>n</sup>*+<sup>1</sup> *<sup>q</sup>* = <sup>2</sup> *<sup>R</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> *<sup>q</sup>* upwind/downwind cell − *Rn q* local face

for the upwind/downwind conservation volumes.

◦ Correct the characteristic flux fields if they lie outside the monotonicity bounds

$$\begin{aligned} \left(\mathbb{R}\_q^{n+1}\right) &= \max(\mathbb{R}\_q)^n, \mathbb{R}\_q^{n+1} > \max(\mathbb{R}\_q)^n; \left(\mathbb{R}\_q^{n+1}\right) = \min(\mathbb{R}\_q)^n, \mathbb{R}\_q^{n+1} < \min(\mathbb{R}\_q)^n; \text{else} \\\left(\mathbb{R}\_q^{n+1}\right) &= \mathbb{R}\_q^{n+1}. \end{aligned}$$

For reconstructing a single set of flux variables at the cell face, use an approximate Riemann solver.

• Conservation corrector step:

$$\frac{\mathbf{U}\_{l+\frac{1}{2},l+\frac{1}{2},k+\frac{1}{2}}^{n+1} - \mathbf{U}\_{l+\frac{1}{2},l+\frac{1}{2},k+\frac{1}{2}}^{n+\frac{1}{2}}}{0.51} + \frac{\mathbf{F}\_{l+1}^{n+1} - \mathbf{F}\_{l}^{n+1}}{\Delta x} + \frac{\mathbf{G}\_{l+1}^{n+1} - \mathbf{G}\_{l}^{n+1}}{\Delta y} + \frac{\mathbf{W}\_{k+1}^{n+1} - \mathbf{W}\_{k}^{n+1}}{\Delta z} = 2\mathbf{Q}\_{\nu}^{n+\frac{1}{2}} - \mathbf{Q}\_{\nu\nu}^{n} \tag{3}$$

where a second-order central approximation is used for the viscous term.

For the T-junction problem with the CABARET method, two hybrid computational grids are used with 0.53 and 4 million cells. Figure 3 shows the layout of (a) the computational domain, (b,d) the hexahedral grid with a uniform Cartesian block in the pipe center, and (c,e) a small collar area of the pipe junction covered by the mixed hexahedral/tetrahedral elements for the smaller and larger grid, respectively.

**Figure 3.** Computational grid used with the CABARET method: full domaim (a), pipe inlet (b,d), and mixed-grid elements in the vicinity of the junction (c,e) for the 0.5 and 4 million cell grid, respectively.

For specifying inlet boundary conditions, a recycling technique is used similar to that for Nek5000. The outflow boundary is prescribed by using characteristic boundary conditions. For the inlet boundaries at the main and the hot branch, laminar inflow conditions are specified based on prescribing the mean flow velocity profiles. The length of the pipe upstream of the junction was sufficiently far from the junction to permit an adequate turbulent flow upstream of the junction. The outflow boundary is imposed at 20 jet diameters downstream of the junction, where characteristic boundary conditions are set.

In order to speed statistical convergence, the LES CABARET solution was started from a precursor RANS k-epsilon calculation. The CABARET simulation was then run for 10 seconds to allow the statistics to settle down. This was followed by the production run during which the required solution fields were stored for a duration of 5 and 10 seconds. The computations on grids with 0.53 and 4 million cells required approximately 1.8 × 104 and 2.9 × 105 CPU hours on Lomonosov (Intel Xeon X5570 / X5670 processors).

## **5. C**onv3**D simulations**

6 Nuclear Reactor Thermal Hydraulics

• Conservation predictor step:

**<sup>U</sup>***n*<sup>+</sup> <sup>1</sup> 2 *i*+ <sup>1</sup> <sup>2</sup> ,*j*+ <sup>1</sup> <sup>2</sup> ,*k*+<sup>1</sup> 2 − **U***<sup>n</sup> i*+ <sup>1</sup> <sup>2</sup> ,*j*+ <sup>1</sup> <sup>2</sup> ,*k*+<sup>1</sup> 2

form:

as follows:

 *Rn*+<sup>1</sup> *q* 

 *Rn*+<sup>1</sup> *q* 

given below. Let us consider the governing equations written in the standard conservation

where the sources in the right-hand side include viscous terms. By mapping the physical domain to the grid space-time coordinates (*x*, *y*, *z*, *t*) → (*i*, *j*, *k*, *n*) and referring control volumes to the cell centers (fractional indices) and fluxes to the cell faces (integer indices), the algorithm proceeds from the known solution at time level (n) to the next timestep (n+1)

> **F***n <sup>i</sup>*+<sup>1</sup> <sup>−</sup> **<sup>F</sup>***<sup>n</sup> i* ∆*x*

+ **G***<sup>n</sup>*

◦ For each cell face, decompose the conservation and flux variables into local Riemann fields, **U** → *Rq*, *q* = 1, . . . , *N*, that correspond to the local cell-face-normal coordinate

◦ For each cell face at the new timestep, compute a dual set of preliminary local Riemann variables that correspond to the upwind and downwind extrapolation of

*<sup>q</sup>* =

 *Rn*+<sup>1</sup> *q* 

For reconstructing a single set of flux variables at the cell face, use an approximate

**G***n*+<sup>1</sup> *j*+1 −**G***n*+<sup>1</sup> *j* <sup>∆</sup>*<sup>y</sup>* +

For the T-junction problem with the CABARET method, two hybrid computational grids are used with 0.53 and 4 million cells. Figure 3 shows the layout of (a) the computational domain, (b,d) the hexahedral grid with a uniform Cartesian block in the pipe center, and (c,e) a small collar area of the pipe junction covered by the mixed hexahedral/tetrahedral

◦ Correct the characteristic flux fields if they lie outside the monotonicity bounds

*<sup>q</sup>* > max(*Rq*)*n*;

*i*+1 −**F***n*+<sup>1</sup> *i* <sup>∆</sup>*<sup>x</sup>* +

where a second-order central approximation is used for the viscous term.

 <sup>2</sup> *<sup>R</sup>n*<sup>+</sup> <sup>1</sup> <sup>2</sup> *<sup>q</sup>* 

*<sup>j</sup>*+<sup>1</sup> <sup>−</sup> **<sup>G</sup>***<sup>n</sup> j*

+ **W***<sup>n</sup>*

upwind/downwind cell

= min(*Rq*)*n*, *R*˜ *<sup>n</sup>*+<sup>1</sup>

**W***n*+<sup>1</sup> *k*+1 −**W***n*+<sup>1</sup> *k* <sup>∆</sup>*<sup>z</sup>* <sup>=</sup> <sup>2</sup>**Q***n*<sup>+</sup> <sup>1</sup>

*<sup>k</sup>*+<sup>1</sup> <sup>−</sup> **<sup>W</sup>***<sup>n</sup> k* <sup>∆</sup>*<sup>z</sup>* <sup>=</sup> **<sup>Q</sup>***<sup>n</sup>*

> − *Rn q* local face

*<sup>q</sup>* < min(*Rq*)*n*; else

<sup>2</sup> *<sup>ν</sup>* − **<sup>Q</sup>***<sup>n</sup>*

*<sup>ν</sup>* , (3)

*<sup>ν</sup>* (2)

∆*y*

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> **<sup>Q</sup>***ν*, (1)

*∂***U** *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

0.5 *<sup>t</sup>* <sup>+</sup>

basis, where *N* is the dimension of the system.

for the upwind/downwind conservation volumes.

the characteristic fields, for example, *R*˜ *<sup>n</sup>*+<sup>1</sup>

= max(*Rq*)*n*, *R*˜ *<sup>n</sup>*+<sup>1</sup>

elements for the smaller and larger grid, respectively.

= *R*˜ *<sup>n</sup>*+<sup>1</sup> *<sup>q</sup>* .

Riemann solver. • Conservation corrector step:

> **U***n*+<sup>1</sup> *i*+ 1 <sup>2</sup> ,*j*<sup>+</sup> <sup>1</sup> <sup>2</sup> ,*k*<sup>+</sup> <sup>1</sup> 2 <sup>−</sup>**U***n*<sup>+</sup> <sup>1</sup> 2 *i*+ 1 <sup>2</sup> ,*j*<sup>+</sup> <sup>1</sup> <sup>2</sup> ,*k*<sup>+</sup> <sup>1</sup> 2 0.5 *<sup>t</sup>* <sup>+</sup> **<sup>F</sup>***n*+<sup>1</sup>

• Upwind extrapolation based on the characteristic decomposition:

*∂***F** *∂x* + *∂***G** *∂y* + *∂***W**

> Researchers at IBRAE have been developing a 3D, unified, numerical thermal-hydraulic technique for safety analysis of the nuclear power plants, which includes (1) methods,

algorithms, and software for automatically generating computing grids with local refinement near body borders; (2) methods and algorithms for solving heat and mass transfer compressible/incompressible problems for research of 3D thermal-hydraulic phenomena; and (3) approaches for modeling turbulence.

For grid construction at IBRAE an automatic technology using CAD systems for designing the computational domain has been developed. A generation of structured orthogonal/Cartesian grids with a local refinement near boundaries is incorporated into a specially developed program that has a user-friendly interface and can be utilized on parallel computers [27]. The computational technique is based on the developed algorithms with small-scheme diffusion, for which discrete approximations are constructed with the use of finite-volume methods and fully staggered grids. For modeling 3D turbulent single-phase flows, the LES approach (commutative filters) and a quasi-direct numerical simulation (QDNS) approach are used. For simulation of 3D turbulent two-phase flows by means of DNS, detailed grids and effective numerical methods developed at IBRAE for solving CFD problems are applied. For observing the interface of two-phase flow, a modified level set (LS) method and multidimensional advection/convection schemas of total variation diminishing (TVD) type with small scheme diffusion involving subgrid simulation (with local resolution) are used. The Conv3D code is fully parallelized and highly effective on high-performance computers. The developed modules were validated on a series of well-known tests with Rayleigh numbers ranging from 106 to 1016 and Reynolds numbers ranging from 10<sup>3</sup> to 105.

In order to simulate thermal-hydraulic phenomena in incompressible media, the time-dependent incompressible Navier-Stokes equations in the primitive variables [13] coupled with the energy equation are used:

$$\frac{d\vec{v}}{dt} = -\text{grad}\,p + \text{div}\,\frac{\mu}{\rho}\,\text{grad}\,\vec{v} + \text{g}\_{\prime} \tag{4}$$

$$\text{div}\,\vec{v} = 0\tag{5}$$

div*<sup>h</sup>* 1

following two-step procedure results:

as *<sup>A</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*(*un*) + *<sup>N</sup>*, where *<sup>N</sup>* <sup>=</sup> div *<sup>µ</sup>*

other (i.e., *<sup>A</sup>*<sup>∗</sup>

Equations 9–10.

a porous medium:

*∂vǫ*

*<sup>ρ</sup>* grad*<sup>h</sup> <sup>δ</sup>pn*+<sup>1</sup>

*<sup>v</sup>n*+<sup>1</sup> <sup>=</sup> *<sup>v</sup>n*+1/2 <sup>−</sup> *<sup>τ</sup>*

*hn*+1/2 − *h<sup>n</sup> τ*

*vn*+1/2 − *v<sup>n</sup> τ*

*vn*+<sup>1</sup> − *v<sup>n</sup> τ*

*∂τ* <sup>+</sup> *<sup>N</sup>*˜ (*vǫ*) *<sup>v</sup><sup>ǫ</sup>* <sup>−</sup> div *<sup>µ</sup>*

*hn*+<sup>1</sup> − *hn*+1/2

 <sup>=</sup> <sup>1</sup> *τ*

In order to construct the time-integration scheme for the energy equation, its operators are decomposed into two parts associated with the enthalpy and temperature, respectively; the

In the momentum equation, the operators are also split into two parts. The first part is associated with the velocity transport by convection/diffusion written in linearized form

gradient *A*<sup>2</sup> = grad. We note that the gradient and divergence operators are adjoints of each

<sup>2</sup> <sup>=</sup> <sup>−</sup>div). The additive scheme of splitting looks like the following:

*<sup>ρ</sup>* grad *v*

*A*∗

where *f <sup>n</sup>* is the right-hand side. This numerical scheme is used as the predictor-corrector procedure. That is, introducing the pressure correction in Equations 14–15 leads to the well-known Poisson equation and the equation for velocity correction in the form of

In computational mathematics two variants of fictitious domain methods are recognized: continuation of coefficients at lower-order derivatives and continuation of coefficients at the highest-order derivatives. Both approaches are commonly used in computational fluid dynamics involving phase change processes. Here the first variant is employed, which in a physical sense can be considered as inclusion into the momentum equations of the model of

*<sup>ρ</sup>* grad *<sup>v</sup><sup>ǫ</sup>*

div*<sup>h</sup> <sup>v</sup>n*<sup>+</sup>1/2, (9)

http://dx.doi.org/10.5772/53143

53

*<sup>ρ</sup>* grad*<sup>h</sup> <sup>δ</sup>pn*<sup>+</sup>1. (10)

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

+ *C*˜(*un*) *hn*+1/2 = 0, (11)

*<sup>τ</sup>* <sup>−</sup> *N T* ˜ *<sup>n</sup>*+<sup>1</sup> <sup>=</sup> 0. (12)

+ *<sup>A</sup>*1*vn*+1/2 + *<sup>A</sup>*<sup>2</sup> *<sup>p</sup><sup>n</sup>* = *<sup>f</sup> <sup>n</sup>*, (13)

+ *<sup>A</sup>*1*vn*+1/2 + *<sup>A</sup>*<sup>2</sup> *<sup>p</sup>n*+<sup>1</sup> = *<sup>f</sup> <sup>n</sup>*, (14)

<sup>2</sup> *<sup>v</sup>n*+<sup>1</sup> <sup>=</sup> 0, (15)

+ grad *p* + *cǫv<sup>ǫ</sup>* = *fǫ*, (16)

div *v<sup>ǫ</sup>* = 0. (17)

. The second part deals with the pressure

$$\frac{\partial h}{\partial t} + \operatorname{div}(\vec{v}\,h) = \operatorname{div}\left(\frac{k}{\rho}\operatorname{grad}\,T\right),\tag{6}$$

$$h = \int\_0^T \mathfrak{c}(\xi) d\xi \,\tag{7}$$

where *p* is pressure, normalized by the density. The basic features of the numerical algorithm [14, 28] are the following. An operator-splitting scheme for the Navier-Stokes equations is used as the predictor-corrector procedure with correction for the pressure *δp*:

$$\frac{v^{n+1/2} - v^n}{\tau} + \left(\mathbf{C}(v) - \operatorname{div} \frac{\mu}{\rho} \operatorname{grad} \right) v^{n+1/2} + \operatorname{grad} p^n - \mathbf{g} \tag{8}$$

$$\mathrm{div}\_{\hbar} \left( \frac{1}{\rho} \,\mathrm{grad}\_{\hbar} \,\delta p^{n+1} \right) = \frac{1}{\pi} \mathrm{div}\_{\hbar} \,\boldsymbol{v}^{n+1/2} \,, \tag{9}$$

$$v^{n+1} = v^{n+1/2} - \frac{\tau}{\rho} \operatorname{grad}\_h \delta p^{n+1}.\tag{10}$$

In order to construct the time-integration scheme for the energy equation, its operators are decomposed into two parts associated with the enthalpy and temperature, respectively; the following two-step procedure results:

8 Nuclear Reactor Thermal Hydraulics

and (3) approaches for modeling turbulence.

coupled with the energy equation are used:

*vn*+1/2 − *v<sup>n</sup> τ*

*dv*

*∂h*

+ 

*dt* <sup>=</sup> <sup>−</sup>grad *<sup>p</sup>* <sup>+</sup> div *<sup>µ</sup>*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> div(*v h*) = div

*h* = *<sup>T</sup>* 0

used as the predictor-corrector procedure with correction for the pressure *δp*:

*<sup>C</sup>*(*v*) <sup>−</sup> div *<sup>µ</sup>*

where *p* is pressure, normalized by the density. The basic features of the numerical algorithm [14, 28] are the following. An operator-splitting scheme for the Navier-Stokes equations is

*<sup>ρ</sup>* grad

*k*

*<sup>ρ</sup>* grad *<sup>T</sup>*

*<sup>ρ</sup>* grad*<sup>v</sup>* <sup>+</sup> *<sup>g</sup>*, (4)

, (6)

div*v* = 0 (5)

*c*(*ξ*)*dξ*, (7)

*vn*+1/2 + grad *p<sup>n</sup>* − *g*, (8)

algorithms, and software for automatically generating computing grids with local refinement near body borders; (2) methods and algorithms for solving heat and mass transfer compressible/incompressible problems for research of 3D thermal-hydraulic phenomena;

For grid construction at IBRAE an automatic technology using CAD systems for designing the computational domain has been developed. A generation of structured orthogonal/Cartesian grids with a local refinement near boundaries is incorporated into a specially developed program that has a user-friendly interface and can be utilized on parallel computers [27]. The computational technique is based on the developed algorithms with small-scheme diffusion, for which discrete approximations are constructed with the use of finite-volume methods and fully staggered grids. For modeling 3D turbulent single-phase flows, the LES approach (commutative filters) and a quasi-direct numerical simulation (QDNS) approach are used. For simulation of 3D turbulent two-phase flows by means of DNS, detailed grids and effective numerical methods developed at IBRAE for solving CFD problems are applied. For observing the interface of two-phase flow, a modified level set (LS) method and multidimensional advection/convection schemas of total variation diminishing (TVD) type with small scheme diffusion involving subgrid simulation (with local resolution) are used. The Conv3D code is fully parallelized and highly effective on high-performance computers. The developed modules were validated on a series of well-known tests with Rayleigh numbers ranging from 106 to 1016 and Reynolds numbers ranging from 10<sup>3</sup> to 105. In order to simulate thermal-hydraulic phenomena in incompressible media, the time-dependent incompressible Navier-Stokes equations in the primitive variables [13]

$$\frac{h^{n+1/2} - h^n}{\tau} + \tilde{\mathcal{C}}(u^n) \, h^{n+1/2} = 0,\tag{11}$$

$$\frac{h^{n+1} - h^{n+1/2}}{\tau} - \tilde{N} \, T^{n+1} = 0. \tag{12}$$

In the momentum equation, the operators are also split into two parts. The first part is associated with the velocity transport by convection/diffusion written in linearized form as *<sup>A</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup>*(*un*) + *<sup>N</sup>*, where *<sup>N</sup>* <sup>=</sup> div *<sup>µ</sup> <sup>ρ</sup>* grad *v* . The second part deals with the pressure gradient *A*<sup>2</sup> = grad. We note that the gradient and divergence operators are adjoints of each other (i.e., *<sup>A</sup>*<sup>∗</sup> <sup>2</sup> <sup>=</sup> <sup>−</sup>div). The additive scheme of splitting looks like the following:

$$\frac{v^{n+1/2} - v^n}{\tau} + A\_1 v^{n+1/2} + A\_2 p^n = f^n,\tag{13}$$

$$\frac{v^{n+1} - v^n}{\tau} + A\_1 v^{n+1/2} + A\_2 p^{n+1} = f^n,\tag{14}$$

$$A\_2^\* v^{n+1} = 0,\tag{15}$$

where *f <sup>n</sup>* is the right-hand side. This numerical scheme is used as the predictor-corrector procedure. That is, introducing the pressure correction in Equations 14–15 leads to the well-known Poisson equation and the equation for velocity correction in the form of Equations 9–10.

In computational mathematics two variants of fictitious domain methods are recognized: continuation of coefficients at lower-order derivatives and continuation of coefficients at the highest-order derivatives. Both approaches are commonly used in computational fluid dynamics involving phase change processes. Here the first variant is employed, which in a physical sense can be considered as inclusion into the momentum equations of the model of a porous medium:

$$\frac{\partial v\_{\varepsilon}}{\partial \tau} + \tilde{N}(v\_{\varepsilon}) \, v\_{\varepsilon} - \operatorname{div} \left( \frac{\mu}{\rho} \operatorname{grad} v\_{\varepsilon} \right) + \operatorname{grad} \, p + c\_{\varepsilon} v\_{\varepsilon} = f\_{\varepsilon \prime} \tag{16}$$

$$\operatorname{div} v\_{\mathfrak{E}} = 0.\tag{17}$$

Various formulae of *cǫ* can be employed for the flow resistance term in these equations. For Equations 16–17, the modified predictor-corrector procedure taking into account the fictitious domain method looks like the following:

$$\frac{v\_{\varepsilon}^{n+1/2} - v\_{\varepsilon}^{n}}{\tau} + A\_1 v\_{\varepsilon}^{n+1/2} + A\_2 p\_{\varepsilon}^{n} + c\_{\varepsilon} v\_{\varepsilon}^{n+1/2} = f^n,\tag{18}$$

*u*(*t* = *tn*), *n* = 1, . . . , *M*, are defined as usual:

**6.1. Comparison with experiment**

*6.1.1. Cold and Hot Inlets*

Nek5000 simulation.

*<sup>u</sup>* <sup>=</sup> <sup>1</sup> *M*

style as in [2, Figures 5 and 6 or Figures A5 and A8].

(*y* = 0) and *ξ* = *y* line (*x* = 0) for the hot inlet (Figure 4).

*M* ∑ *n*=1

*un*, *<sup>u</sup>*′ <sup>=</sup>

In this section, we compare the results of the three codes with experimental data. First we look at the inlet profiles in the cold and hot branches of the T-junction and then at the vertical and horizontal profiles of the mean and rms axial velocity downstream of the junction.

Figure 4 shows the profiles of the mean (top) and rms (bottom) streamwise velocity in the cold branch at *<sup>x</sup>* <sup>=</sup> <sup>−</sup>3.1 (left) and in the hot inlet branch at *<sup>z</sup>* <sup>=</sup> 2.14286 (i.e., *<sup>z</sup>*∗/*DH* <sup>=</sup> 3) (right). The experimental data are plotted with symbols, the blue solid line represents the Nek5000 simulation, and red dashed line are the inlet profiles from the CABARET simulation. Note that the streamwise component of velocity for cold inlet coincides with the *x*-direction, while the hot flow in the hot inlet is in the direction opposite the *z* coordinate; hence, −*w* is plotted for the hot inlet. Also note that experimental data is plotted in the same

We have observed that the normalized experimental data for the cold inlet does not integrate to unity, indicating a discrepancy between the reported flow rate and the LDA measurements of approximately 6%. On the contrary, using the trapezoidal rule, the integrals of Nek5000 profiles for the cold and hot inlets are equal to 1.0027 and 1.3080, respectively, once averaged over and normalized by the corresponding inlet cross-sections. The same integration procedure for CABARET profiles in Figure 4 gives the normalized inlet flow rates equal to 1.014 and 1.330 for the cold and hot inlets, respectively. These values are in a good agreement with the nondimensional values for inlet velocities *U* = 1.000 and *UH* = 1.307 given in Table 2. Note that the normalized inlet mass flow rates for Nek5000 were averaged over the *ξ* = *z* line (*y* = 0) and *ξ* = *y* line (*z* = 0) for the cold inlet and over the *ξ* = *x* line

Moreover, a comparison of the shape of the inlet simulation profiles with the experimental data reveals a few differences. The shape of the hot inlet profile for the mean and rms velocity for Nek5000 simulation is not as flat as the experimental or CABARET profiles. This difference can be attributed to a particular modeling of the non-fully-developed flow with the recycling technique described in Section 3. Note that the same technique works nearly perfectly for the cold inlet, where the profile of the mean velocity for Nek5000 is in excellent agreement with the experimental data points after multiplication by 0.94 factor to account for the mass conservation uncertainty of 6%. Similarly, the cold inlet rms velocity for Nek5000 data shows good agreement, diverging from the experimental data points only in the near-wall region, which can be explained by the lower Reynolds number of the simulation, *Re* = 4 × 10<sup>5</sup> (cf. Table 2). These results indicate that further investigation is needed into the effects of the non-fully-developed flow in the hot inlet for

 1 *M*

*M* ∑ *n*=1

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

(*un* − *u*)<sup>2</sup> . (21)

http://dx.doi.org/10.5772/53143

55

$$\mathrm{div}\_{\hbar}\left(\frac{1}{\rho}\,\mathrm{grad}\_{\hbar}\,\delta p^{s+1}\right) = \mathrm{div}\_{\hbar}\left(\frac{1}{\rho}\,\frac{\tau c\_{\varepsilon}}{1+\tau c\_{\varepsilon}}\,\mathrm{grad}\_{\hbar}\,\delta p^{s}\right) + \frac{1}{\tau}\mathrm{div}\_{\hbar}v^{n+1/2}\_{\varepsilon},\tag{19}$$

$$
v\_{\varepsilon}^{n+1} = v\_{\varepsilon}^{n+1/2} - \frac{1}{\rho} \frac{1}{1 + \tau c\_{\varepsilon}} \operatorname{grad}\_{\hbar} \delta p\_{\varepsilon}.\tag{20}$$

An iterative method with a Chebyshev set of parameters using a fast Fourier transform (FFT) solver for the Laplace operator as a preconditioner can serve as an alternative to the conjugate gradient method. The application of this approach for solving the elliptical equations with variable coefficients allows one to reach 50 times the acceleration of the commonly used method of conjugate gradients. For solving the convection problem, a regularized nonlinear monotonic operator-splitting scheme was developed [15]. A special treatment of approximation of convection terms *C*(*v*) results in the discrete convective operator, which is skew-symmetric and does not contribute to the kinetic energy (i.e., is energetically neutral [15]). The numerical scheme is second, and first-order accurate in space and time, correspondingly. The algorithm is stable at a large-enough integration timestep. Details of the validation for the approach on a wide set of both 2D and 3D tests are reported in [29].

In the next section, we present the numerical results computed with Conv3D on a uniform mesh with 40 million nodes. For the sensitivity study (Section 6.2), we provide results from computation on a uniform mesh with 12 million nodes and on a nonuniform mesh with 3 and 40 million nodes with near-wall refinement. The computations on a uniform mesh with 40 million nodes required approximately 4.3 × 104 CPU hours on Lomonosov (Intel Xeon X5570 / X5670 processors).

## **6. Results**

We focus here on a comparison of the experimental data with the numerical results from three simulation codes: Nek5000, CABARET, and Conv3D. In Section 6.2 we study the velocity field sensitivity to the computational mesh and integration time interval. A similar sensitivity study is undertaken in Section 6.3, where we investigate the effects of grid resolution and time integration interval on the reattachment region immediately downstream of the T-junction. A comparison of velocity and temperature spectra for these codes will be reported elsewhere [30].

All results presented here are nondimensionalized with the cold inlet parameters according to Table 1. For reference, the mean and rms quantities for a set of temporal values *un* =

*u*(*t* = *tn*), *n* = 1, . . . , *M*, are defined as usual:

$$u = \frac{1}{M} \sum\_{n=1}^{M} u\_{n\nu} \qquad \qquad u' = \sqrt{\frac{1}{M} \sum\_{n=1}^{M} (u\_n - u)^2}. \tag{21}$$

#### **6.1. Comparison with experiment**

In this section, we compare the results of the three codes with experimental data. First we look at the inlet profiles in the cold and hot branches of the T-junction and then at the vertical and horizontal profiles of the mean and rms axial velocity downstream of the junction.

#### *6.1.1. Cold and Hot Inlets*

10 Nuclear Reactor Thermal Hydraulics

div*<sup>h</sup>* 1

X5570 / X5670 processors).

reported elsewhere [30].

in [29].

**6. Results**

domain method looks like the following:

*<sup>v</sup>n*+1/2 *<sup>ǫ</sup>* − *<sup>v</sup><sup>n</sup>*

*τ*

*<sup>ρ</sup>* grad*<sup>h</sup> <sup>δ</sup>ps*+<sup>1</sup>

*ǫ*

= div*<sup>h</sup>*

*<sup>v</sup>n*+<sup>1</sup> *<sup>ǫ</sup>* <sup>=</sup> *<sup>v</sup>n*+1/2 *<sup>ǫ</sup>* <sup>−</sup> <sup>1</sup>

Various formulae of *cǫ* can be employed for the flow resistance term in these equations. For Equations 16–17, the modified predictor-corrector procedure taking into account the fictitious

<sup>+</sup> *<sup>A</sup>*1*vn*+1/2 *<sup>ǫ</sup>* <sup>+</sup> *<sup>A</sup>*<sup>2</sup> *<sup>p</sup><sup>n</sup>*

 1 *ρ*

*ρ*

An iterative method with a Chebyshev set of parameters using a fast Fourier transform (FFT) solver for the Laplace operator as a preconditioner can serve as an alternative to the conjugate gradient method. The application of this approach for solving the elliptical equations with variable coefficients allows one to reach 50 times the acceleration of the commonly used method of conjugate gradients. For solving the convection problem, a regularized nonlinear monotonic operator-splitting scheme was developed [15]. A special treatment of approximation of convection terms *C*(*v*) results in the discrete convective operator, which is skew-symmetric and does not contribute to the kinetic energy (i.e., is energetically neutral [15]). The numerical scheme is second, and first-order accurate in space and time, correspondingly. The algorithm is stable at a large-enough integration timestep. Details of the validation for the approach on a wide set of both 2D and 3D tests are reported

In the next section, we present the numerical results computed with Conv3D on a uniform mesh with 40 million nodes. For the sensitivity study (Section 6.2), we provide results from computation on a uniform mesh with 12 million nodes and on a nonuniform mesh with 3 and 40 million nodes with near-wall refinement. The computations on a uniform mesh with 40 million nodes required approximately 4.3 × 104 CPU hours on Lomonosov (Intel Xeon

We focus here on a comparison of the experimental data with the numerical results from three simulation codes: Nek5000, CABARET, and Conv3D. In Section 6.2 we study the velocity field sensitivity to the computational mesh and integration time interval. A similar sensitivity study is undertaken in Section 6.3, where we investigate the effects of grid resolution and time integration interval on the reattachment region immediately downstream of the T-junction. A comparison of velocity and temperature spectra for these codes will be

All results presented here are nondimensionalized with the cold inlet parameters according to Table 1. For reference, the mean and rms quantities for a set of temporal values *un* =

*τcǫ* 1 + *τc<sup>ǫ</sup>*

> 1 1 + *τc<sup>ǫ</sup>*

grad*<sup>h</sup> <sup>δ</sup>p<sup>s</sup>*

 + 1 *τ*

*<sup>ǫ</sup>* <sup>+</sup> *<sup>c</sup>ǫvn*+1/2 *<sup>ǫ</sup>* <sup>=</sup> *<sup>f</sup> <sup>n</sup>*, (18)

grad*<sup>h</sup> δpǫ*. (20)

div*<sup>h</sup> <sup>v</sup>n*+1/2 *<sup>ǫ</sup>* , (19)

Figure 4 shows the profiles of the mean (top) and rms (bottom) streamwise velocity in the cold branch at *<sup>x</sup>* <sup>=</sup> <sup>−</sup>3.1 (left) and in the hot inlet branch at *<sup>z</sup>* <sup>=</sup> 2.14286 (i.e., *<sup>z</sup>*∗/*DH* <sup>=</sup> 3) (right). The experimental data are plotted with symbols, the blue solid line represents the Nek5000 simulation, and red dashed line are the inlet profiles from the CABARET simulation. Note that the streamwise component of velocity for cold inlet coincides with the *x*-direction, while the hot flow in the hot inlet is in the direction opposite the *z* coordinate; hence, −*w* is plotted for the hot inlet. Also note that experimental data is plotted in the same style as in [2, Figures 5 and 6 or Figures A5 and A8].

We have observed that the normalized experimental data for the cold inlet does not integrate to unity, indicating a discrepancy between the reported flow rate and the LDA measurements of approximately 6%. On the contrary, using the trapezoidal rule, the integrals of Nek5000 profiles for the cold and hot inlets are equal to 1.0027 and 1.3080, respectively, once averaged over and normalized by the corresponding inlet cross-sections. The same integration procedure for CABARET profiles in Figure 4 gives the normalized inlet flow rates equal to 1.014 and 1.330 for the cold and hot inlets, respectively. These values are in a good agreement with the nondimensional values for inlet velocities *U* = 1.000 and *UH* = 1.307 given in Table 2. Note that the normalized inlet mass flow rates for Nek5000 were averaged over the *ξ* = *z* line (*y* = 0) and *ξ* = *y* line (*z* = 0) for the cold inlet and over the *ξ* = *x* line (*y* = 0) and *ξ* = *y* line (*x* = 0) for the hot inlet (Figure 4).

Moreover, a comparison of the shape of the inlet simulation profiles with the experimental data reveals a few differences. The shape of the hot inlet profile for the mean and rms velocity for Nek5000 simulation is not as flat as the experimental or CABARET profiles. This difference can be attributed to a particular modeling of the non-fully-developed flow with the recycling technique described in Section 3. Note that the same technique works nearly perfectly for the cold inlet, where the profile of the mean velocity for Nek5000 is in excellent agreement with the experimental data points after multiplication by 0.94 factor to account for the mass conservation uncertainty of 6%. Similarly, the cold inlet rms velocity for Nek5000 data shows good agreement, diverging from the experimental data points only in the near-wall region, which can be explained by the lower Reynolds number of the simulation, *Re* = 4 × 10<sup>5</sup> (cf. Table 2). These results indicate that further investigation is needed into the effects of the non-fully-developed flow in the hot inlet for Nek5000 simulation.

0 1 2 3 4 5 6

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

http://dx.doi.org/10.5772/53143

57

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6

*x*

**Figure 5.** T-junction experimental data (·), CABARET (red), and Conv3D (magenta) results, and Nek5000 simulation for the benchmark submission results (green) and for the calculation with a longer time integration (blue line). From top to bottom:

) velocity and their horizontal profiles. (black symbols) are contrasted with numerical simulations with CABARET (red), Conv3D (magenta), and Nek5000 for the benchmark submission results (green) and for longer time integration/averaging (blue). Note the T-junction geometry outline and the equal unit scale for the mean and rms axial velocity equal to the velocity scale, namely, the axial mean velocity in the cold inlet branch *U* = 1.000 (Table 2). For reference, we provide a detailed

−0.5

−0.5

−0.5

−0.5

*y*

0

0.5

*y*

0

0.5

*z*

0

0.5

1

*z*

0

0.5

1

u/U = 0 1 2

u'/U = 0 1 2

u/U = 0 1 2

u'/U = 0 1 2

vertical profiles of axial mean (*u*) and rms (*u*′

**Figure 4.** Mean and rms velocity profiles in the cold inlet branch and in the hot inlet branch vs a centerline coordinate *ξ* for the experimental data (symbols) and simulation results with Nek5000 (blue solid line) and with CABARET (red dashed line).

On the contrary, the CABARET simulation models the flat profile at the hot inlet well (Figure 4, right). The cold inlet profiles, however, deviate substantially from the experimental data, especially in the case of rms profiles (Figure 4, left).

#### *6.1.2. Downstream of the T-Junction*

We next look at the axial mean *<sup>u</sup>* and rms *<sup>u</sup>*′ velocity downstream of the T-junction. Figure <sup>5</sup> shows a "bird's-eye" view of the mean (*u*) and rms (*u*′ ) velocity profiles downstream of the T-junction at *x*=0.6, 1.6, 2.6, 3.6, and 4.6. Here the experimental data

12 Nuclear Reactor Thermal Hydraulics

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

*u*′

0.2

0.4

0.6

0.8

*u*

1

1.2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

*6.1.2. Downstream of the T-Junction*

*z*

data, especially in the case of rms profiles (Figure 4, left).

*z*

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

*y*

) velocity profiles

*y*

0

(a) Mean profile in the cold inlet (b) Mean profile in the hot inlet

0

(c) Rms profile in the cold inlet (d) Rms profile in the hot inlet

**Figure 4.** Mean and rms velocity profiles in the cold inlet branch and in the hot inlet branch vs a centerline coordinate *ξ* for the experimental data (symbols) and simulation results with Nek5000 (blue solid line) and with CABARET (red dashed line).

On the contrary, the CABARET simulation models the flat profile at the hot inlet well (Figure 4, right). The cold inlet profiles, however, deviate substantially from the experimental

We next look at the axial mean *<sup>u</sup>* and rms *<sup>u</sup>*′ velocity downstream of the T-junction.

downstream of the T-junction at *x*=0.6, 1.6, 2.6, 3.6, and 4.6. Here the experimental data

Figure <sup>5</sup> shows a "bird's-eye" view of the mean (*u*) and rms (*u*′

0.05

0.1

0.15

*u*′

0.2

0.2

0.4

0.6

0.8

*u*

1

1.2

1.4

1.6

**Figure 5.** T-junction experimental data (·), CABARET (red), and Conv3D (magenta) results, and Nek5000 simulation for the benchmark submission results (green) and for the calculation with a longer time integration (blue line). From top to bottom: vertical profiles of axial mean (*u*) and rms (*u*′ ) velocity and their horizontal profiles.

(black symbols) are contrasted with numerical simulations with CABARET (red), Conv3D (magenta), and Nek5000 for the benchmark submission results (green) and for longer time integration/averaging (blue). Note the T-junction geometry outline and the equal unit scale for the mean and rms axial velocity equal to the velocity scale, namely, the axial mean velocity in the cold inlet branch *U* = 1.000 (Table 2). For reference, we provide a detailed

**Figure 6.** Axial mean velocity and rms vertical profiles at *x*=1.6 for experimental data (triangles) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

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−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*z*

*z*

0.1

(a) Mean vertical profile at *x* = 3.6 (b) Rms vertical profile at *x* = 3.6

**Figure 8.** Axial mean velocity and rms vertical profiles at *x*=3.6 for experimental data (diamonds) and simulation with Nek5000

0.1

(a) Mean vertical profile at *x* = 4.6 (b) Rms vertical profile at *x* = 4.6

**Figure 9.** Axial mean velocity and rms vertical profiles at *x*=4.6 for experimental data (squares) and simulation with Nek5000

horizontal profiles are plotted similarly in Figures 10–13. Detailed comparison with Nek5000

It is encouraging that overall agreement of simulation results for the three codes with the experimential data is good for the both mean and rms axial velocity. The Nek5000 profiles (blue lines) of the rms velocity match experimental data points well (Figures 5 and 6–13,

benchmark submission results is reported in Section 6.2 (see Figure 14).

0.2

0.3

0.4

*u*′

0.5

0.6

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

(blue), CABARET (red), and Conv3D (magenta line).

*z*

(blue), CABARET (red), and Conv3D (magenta line).

*z*

−0.5

−0.5

0

0.5

1

*u*

1.5

2

2.5

0

0.5

1

*u*

1.5

2

2.5

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 7.** Axial mean velocity and rms vertical profiles at *x*=2.6 for experimental data (circles) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

comparison of numerical simulations with experimental data points of the mean and rms axial velocity for each cross-section separately (Figures 6–13). Each figure shows the results of numerical simulations with Nek5000 (blue), CABARET (red), and Conv3D (magenta) against the experimental data points (symbols). The vertical profiles of the mean *<sup>u</sup>* (left) and rms *<sup>u</sup>*′ (right) axial velocity are shown in Figures 6–9 at *x* = 1.6 . . . 4.6, correspondingly, while the

14 Nuclear Reactor Thermal Hydraulics

−0.5

−0.5

0

0.5

1

*u*

1.5

2

2.5

0

0.5

1

*u*

1.5

2

2.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

(blue), CABARET (red), and Conv3D (magenta line).

*z*

(blue), CABARET (red), and Conv3D (magenta line).

*z*

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*z*

*z*

0.1

(a) Mean vertical profile at *x* = 1.6 (b) Rms vertical profile at *x* = 1.6

**Figure 6.** Axial mean velocity and rms vertical profiles at *x*=1.6 for experimental data (triangles) and simulation with Nek5000

0.1

(a) Mean vertical profile at *x* = 2.6 (b) Rms vertical profile at *x* = 2.6

**Figure 7.** Axial mean velocity and rms vertical profiles at *x*=2.6 for experimental data (circles) and simulation with Nek5000

comparison of numerical simulations with experimental data points of the mean and rms axial velocity for each cross-section separately (Figures 6–13). Each figure shows the results of numerical simulations with Nek5000 (blue), CABARET (red), and Conv3D (magenta) against the experimental data points (symbols). The vertical profiles of the mean *<sup>u</sup>* (left) and rms *<sup>u</sup>*′ (right) axial velocity are shown in Figures 6–9 at *x* = 1.6 . . . 4.6, correspondingly, while the

0.2

0.3

0.4

*u*′

0.5

0.6

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 8.** Axial mean velocity and rms vertical profiles at *x*=3.6 for experimental data (diamonds) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

**Figure 9.** Axial mean velocity and rms vertical profiles at *x*=4.6 for experimental data (squares) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

horizontal profiles are plotted similarly in Figures 10–13. Detailed comparison with Nek5000 benchmark submission results is reported in Section 6.2 (see Figure 14).

It is encouraging that overall agreement of simulation results for the three codes with the experimential data is good for the both mean and rms axial velocity. The Nek5000 profiles (blue lines) of the rms velocity match experimental data points well (Figures 5 and 6–13,

**Figure 10.** Axial mean velocity and rms horizontal profiles at *x*=1.6 for experimental data (triangles) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

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−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*y*

*y*

0.1

(a) Mean horizontal profile at *x* = 3.6 (b) Rms horizontal profile at *x* = 3.6

0.1

(a) Mean horizontal profile at *x* = 3.6 (b) Rms horizontal profile at *x* = 3.6

**Figure 13.** Axial mean velocity and rms horizontal profiles at *x*=4.6 for experimental data (squares) and simulation with

12–13 can be attributed to the lower Reynolds number used in the simulation because of the

On the contrary, CABARET simulation results (red lines) agree with experiment remarkably well further downstream, at *x* = 3.6 (Figures 8 and 12) and *x* = 4.6 (Figures 9 and 13), with the notable exception of the best match of rms data with experimental points in the

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 12.** Axial mean velocity and rms horizontal profiles at *x*=3.6 for experimental data (diamonds) and simulation with

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

*y*

time constraints and will be the subject of a follow-up study.

*y*

0

0

0.5

1

1.5

*u*

2

2.5

0.5

1

1.5

*u*

2

2.5

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 11.** Axial mean velocity and rms horizontal profiles at *x*=2.6 for experimental data (circles) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

right). Moreover, the agreement between the simulation and experiment for the mean velocity (left figures) is best at *x* = 2.6 (Figures 7 and 11) and at *x* = 1.6 (Figures 6 and 10) apart of two near-wall data points close to at *z* = 0.5 (Figure 6). This discrepancy is the focus of an investigation described in Section 6.3. The deviation of the mean velocity for Nek5000 results from experimental data further downstream at *x* = 3.6 and 4.6 in Figures 5, 8–9, and

16 Nuclear Reactor Thermal Hydraulics

0

0

0.5

1

1.5

*u*

2

2.5

0.5

1

1.5

*u*

2

2.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

(blue), CABARET (red), and Conv3D (magenta line).

*y*

*y*

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*y*

*y*

0.1

(a) Mean horizontal profile at *x* = 1.6 (b) Rms horizontal profile at *x* = 1.6

0.1

(a) Mean horizontal profile at *x* = 2.6 (b) Rms horizontal profile at *x* = 2.6

**Figure 11.** Axial mean velocity and rms horizontal profiles at *x*=2.6 for experimental data (circles) and simulation with Nek5000

right). Moreover, the agreement between the simulation and experiment for the mean velocity (left figures) is best at *x* = 2.6 (Figures 7 and 11) and at *x* = 1.6 (Figures 6 and 10) apart of two near-wall data points close to at *z* = 0.5 (Figure 6). This discrepancy is the focus of an investigation described in Section 6.3. The deviation of the mean velocity for Nek5000 results from experimental data further downstream at *x* = 3.6 and 4.6 in Figures 5, 8–9, and

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 10.** Axial mean velocity and rms horizontal profiles at *x*=1.6 for experimental data (triangles) and simulation with

0.2

0.3

0.4

*u*′

0.5

0.6

**Figure 12.** Axial mean velocity and rms horizontal profiles at *x*=3.6 for experimental data (diamonds) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

**Figure 13.** Axial mean velocity and rms horizontal profiles at *x*=4.6 for experimental data (squares) and simulation with Nek5000 (blue), CABARET (red), and Conv3D (magenta line).

12–13 can be attributed to the lower Reynolds number used in the simulation because of the time constraints and will be the subject of a follow-up study.

On the contrary, CABARET simulation results (red lines) agree with experiment remarkably well further downstream, at *x* = 3.6 (Figures 8 and 12) and *x* = 4.6 (Figures 9 and 13), with the notable exception of the best match of rms data with experimental points in the

recirculation region at *x* = 1.6 and *z* > 0 (Figure 6). However, close to the T-junction, at x=1.6 and 2.6, the CABARET profiles of the mean axial velocity deviate from experimental points at *z* > 0 (Figures 6–7) and near the centerline *y* = 0 (Figures 10–11). Similar to CABARET profiles, the Conv3D results (magenta lines) show the most deviation from experimental data for the mean axial velocity at *z* > 0 (Figures 6–9) and near the centerline *y* = 0 (Figures 10–13). However, the agreement of Conv3D simulation with experiment is best at *x* = 4.6 and 0.4 < |*y*| < 0.5 (Figure 13).

## **6.2. Sensitivity study**

To study the effects of increasing resolution and time averaging interval, we performed an additional set of simulations with Nek5000, CABARET, and Conv3D.

The Figures 14–16 highlight the mesh sensitivity of velocity profiles extracted from results of numerical simulations with Nek5000, CABARET and Conv3D, correspondingly. Each figure compares the vertical and horizontal profiles of the mean axial velocity *<sup>u</sup>* and its rms *<sup>u</sup>*′ for different meshes with the experimental data. More detailed comparisons can be found in [31].

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−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*y*

*y*

0

(a) Nek5000 mean vertical profiles (b) Nek5000 mean horizontal profiles

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−0.5 −0.4 −0.3 −0.2 −0.1 <sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

**6.3. Study of reversed flow region**

*z*

*z*

−0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

submission results (dash-dotted).

*u*′

0

0.5

1

*u*

1.5

2

2.5

0.1

(c) Nek5000 rms vertical profiles (d) Nek5000 rms horizontal profiles **Figure 14.** Mean and rms profiles of axial velocity at *x*=0.6 (magenta), 1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and Nek5000 simulation with *N* = 5 (dashed) and *N* = 7 (solid) and for Nek5000 benchmark

To address the discrepancy between experimental data and simulations near the upper wall at *x* = 1.6, we have also undertaken a series of Nek5000 simulations to investigate the sensitivity of the reattachment of the recirculation region downstream of the T-junction.

Summarizing the effects of Reynolds number, grid resolution, and time averaging on the reversed flow region, Figure 17 shows time-averaged velocity profiles near the upper wall at *z*=0.005 and *y*=0 for Nek5000 simulations at *Re* = 9 × 104 with *N* = 11 (red), at *Re* = 6 × 104

0.2

0.3

0.4

*u*′

0.5

0.6

0.5

1

1.5

*u*

2

2.5

In addition to the benchmark submission results with *N* = 7 (i.e., *n* ≈ *EN*<sup>3</sup> ≈ 2.1 × 107 points), Nek5000 runs were conducted with *N* = 5 (*n* ≈ 7.7 × 106 points). This case started with the *<sup>N</sup>* = 7 results and was run with <sup>∆</sup>*<sup>t</sup>* = <sup>5</sup> × <sup>10</sup>−<sup>4</sup> and averaged over about 110 convective time units (i.e., about 26 seconds). Figure 14 shows the vertical (left) and horizontal (right) profiles of the mean (top) and rms (bottom) axial velocity profiles at *x*=0.6 (magenta), 1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) with *N* = 5 (dashed) and *N* = 7 (solid). The benchmark submission results (i.e., with *N*=7 and shorter time average) plotted with dash-dotted line at *x* = 1.6 . . . 4.6 are in the excellent agreement with the longer time average run (solid). All profiles agree well with the experimental data points (symbols), especially considering the fact that the Reynolds number of these simulations, *Re* = 4 × 105, is two times less than that of the experiment (see Table 2). In general, the profiles from the coarse mesh simulation (i.e., *N* = 5, dashed) are close to the solution for the finer mesh (solid and dash-dotted lines). Note the excellent agreement between the profiles for *N* = 5 and *N* = 7 at *x* = 0.6, where the strength of the reversed flow is close to its peak (Figure 17). The largest deviation in the profiles (up to about 0.2) is observed at *x* = 1.6 and 0.1 < *z* < 0.25; thus, a further study is warranted with even finer mesh, say, *N* = 8 or *N* = 9.

Similarly, for the CABARET simulations, the vertical and horizontal profiles of *<sup>u</sup>* and *<sup>u</sup>*′ are plotted in Figure 15 at *x* = 1.6 . . . 4.6. These figures show results from CABARET calculations on a coarser mesh with 0.5 million points averaged over a half time interval (red dotted) and the full time interval (blue dash-dotted) and on a finer mesh with 4 million points averaged over a half time interval (green dashed) and the full time interval (solid black line). Note the excellent convergence for the mean axial velocity profiles at *x* = 3.6 and *x* = 4.6.

For the Conv3D simulations, the vertical and horizontal profiles of *<sup>u</sup>* and *<sup>u</sup>*′ are plotted in Figure 16 at *x* = 1.6 . . . 4.6. This figure shows results from Conv3D simulations on a 40 million node uniform (black solid) and nonuniform (blue dash-dotted) mesh, on a 12 million node uniform mesh (green dashed), and on a 3 million node nonuniform mesh (red dotted line). Note the good convergence for the mean axial velocity profiles at *x* = 1.6 . . . 3.6 and *z* < 0.

<sup>62</sup> Nuclear Reactor Thermal Hydraulics and Other Applications Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction 19 Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction http://dx.doi.org/10.5772/53143 63

**Figure 14.** Mean and rms profiles of axial velocity at *x*=0.6 (magenta), 1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and Nek5000 simulation with *N* = 5 (dashed) and *N* = 7 (solid) and for Nek5000 benchmark submission results (dash-dotted).

#### **6.3. Study of reversed flow region**

18 Nuclear Reactor Thermal Hydraulics

**6.2. Sensitivity study**

[31].

*z* < 0.

at *x* = 4.6 and 0.4 < |*y*| < 0.5 (Figure 13).

recirculation region at *x* = 1.6 and *z* > 0 (Figure 6). However, close to the T-junction, at x=1.6 and 2.6, the CABARET profiles of the mean axial velocity deviate from experimental points at *z* > 0 (Figures 6–7) and near the centerline *y* = 0 (Figures 10–11). Similar to CABARET profiles, the Conv3D results (magenta lines) show the most deviation from experimental data for the mean axial velocity at *z* > 0 (Figures 6–9) and near the centerline *y* = 0 (Figures 10–13). However, the agreement of Conv3D simulation with experiment is best

To study the effects of increasing resolution and time averaging interval, we performed an

The Figures 14–16 highlight the mesh sensitivity of velocity profiles extracted from results of numerical simulations with Nek5000, CABARET and Conv3D, correspondingly. Each figure compares the vertical and horizontal profiles of the mean axial velocity *<sup>u</sup>* and its rms *<sup>u</sup>*′ for different meshes with the experimental data. More detailed comparisons can be found in

In addition to the benchmark submission results with *N* = 7 (i.e., *n* ≈ *EN*<sup>3</sup> ≈ 2.1 × 107 points), Nek5000 runs were conducted with *N* = 5 (*n* ≈ 7.7 × 106 points). This case started with the *<sup>N</sup>* = 7 results and was run with <sup>∆</sup>*<sup>t</sup>* = <sup>5</sup> × <sup>10</sup>−<sup>4</sup> and averaged over about 110 convective time units (i.e., about 26 seconds). Figure 14 shows the vertical (left) and horizontal (right) profiles of the mean (top) and rms (bottom) axial velocity profiles at *x*=0.6 (magenta), 1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) with *N* = 5 (dashed) and *N* = 7 (solid). The benchmark submission results (i.e., with *N*=7 and shorter time average) plotted with dash-dotted line at *x* = 1.6 . . . 4.6 are in the excellent agreement with the longer time average run (solid). All profiles agree well with the experimental data points (symbols), especially considering the fact that the Reynolds number of these simulations, *Re* = 4 × 105, is two times less than that of the experiment (see Table 2). In general, the profiles from the coarse mesh simulation (i.e., *N* = 5, dashed) are close to the solution for the finer mesh (solid and dash-dotted lines). Note the excellent agreement between the profiles for *N* = 5 and *N* = 7 at *x* = 0.6, where the strength of the reversed flow is close to its peak (Figure 17). The largest deviation in the profiles (up to about 0.2) is observed at *x* = 1.6 and 0.1 < *z* < 0.25;

thus, a further study is warranted with even finer mesh, say, *N* = 8 or *N* = 9.

excellent convergence for the mean axial velocity profiles at *x* = 3.6 and *x* = 4.6.

Similarly, for the CABARET simulations, the vertical and horizontal profiles of *<sup>u</sup>* and *<sup>u</sup>*′ are plotted in Figure 15 at *x* = 1.6 . . . 4.6. These figures show results from CABARET calculations on a coarser mesh with 0.5 million points averaged over a half time interval (red dotted) and the full time interval (blue dash-dotted) and on a finer mesh with 4 million points averaged over a half time interval (green dashed) and the full time interval (solid black line). Note the

For the Conv3D simulations, the vertical and horizontal profiles of *<sup>u</sup>* and *<sup>u</sup>*′ are plotted in Figure 16 at *x* = 1.6 . . . 4.6. This figure shows results from Conv3D simulations on a 40 million node uniform (black solid) and nonuniform (blue dash-dotted) mesh, on a 12 million node uniform mesh (green dashed), and on a 3 million node nonuniform mesh (red dotted line). Note the good convergence for the mean axial velocity profiles at *x* = 1.6 . . . 3.6 and

additional set of simulations with Nek5000, CABARET, and Conv3D.

To address the discrepancy between experimental data and simulations near the upper wall at *x* = 1.6, we have also undertaken a series of Nek5000 simulations to investigate the sensitivity of the reattachment of the recirculation region downstream of the T-junction.

Summarizing the effects of Reynolds number, grid resolution, and time averaging on the reversed flow region, Figure 17 shows time-averaged velocity profiles near the upper wall at *z*=0.005 and *y*=0 for Nek5000 simulations at *Re* = 9 × 104 with *N* = 11 (red), at *Re* = 6 × 104

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−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

*y*

*y*

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

(a) Conv3D mean vertical profiles (b) Conv3D mean horizontal profiles

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

*z*

node uniform mesh (dashed) and on 3 million node nonuniform mesh (dotted line).

simulations could be performed in order to resolve them.

*z*

−0.5

0.1

0.2

0.3

0.4

*u*′

0.5

0.6

0

0.5

1

*u*

1.5

2

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

(c) Conv3D rms vertical profiles (d) Conv3D rms horizontal profiles **Figure 16.** Mean and rms profiles of axial velocity at *x*=1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and Conv3D simulation on 40 milliom node uniform (solid) and nonuniform (dash-dotted) mesh, on 12 million

with longer integration/averaging time to resolve the discrepancy between the simulations and experimental data points near the upper wall at *x* = 1.6. For now, we quote that in the experimental measurements of velocity, ". . . the focus . . . is not the near-wall region" [2, p. 15]. This underscores a necessity of conducting concurrent experiments and validation simulations in which discrepancies like the extent of reciculation region or flow rate uncertainty arising here could be easier to detect and additional data acquisitions and

*u*′

*u*

**Figure 15.** Mean and rms profiles of axial velocity at *x*=1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and CABARET simulation on a coarser mesh with 0.5 million ponts averaged over a half (dotted) and full (dash-dotted) interval, and on a finer mesh with 4 million points averaged over a half (dashed) and full (solid line) time interval.

with *N* = 9 (cyan) and at *Re* = 4 × 104 with *N* = 9 (magenta), *N* = 5 (black) and *N* = 7 for benchmark submission results (green) and longer time averaging (blue). Note the dotted lines that correspond to experimental profile measurements and *u* = 0 value.

Based on this study, we conclude that the reattachment region is insensitive mainly to an increase in Reynolds number and grid resolution. Moreover, results at *Re* = 4 × 104 indicate that the recirculation region is not sensitive to the averaging/integration time interval. Ideally, one should conduct a further investigation at higher Reynolds number <sup>64</sup> Nuclear Reactor Thermal Hydraulics and Other Applications Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction 21 Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction http://dx.doi.org/10.5772/53143 65

20 Nuclear Reactor Thermal Hydraulics

−0.5

0.1

0.2

0.3

0.4

*u*′

0.5

0.6

0

0.5

1

*u*

1.5

2

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

*z*

*z*

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

*y*

*y*

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

(a) CABARET mean vertical profiles (b) CABARET mean horizontal profiles

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

(c) CABARET rms vertical profiles (d) CABARET rms horizontal profiles **Figure 15.** Mean and rms profiles of axial velocity at *x*=1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and CABARET simulation on a coarser mesh with 0.5 million ponts averaged over a half (dotted) and full (dash-dotted) interval, and on a finer mesh with 4 million points averaged over a half (dashed) and full (solid line) time interval. with *N* = 9 (cyan) and at *Re* = 4 × 104 with *N* = 9 (magenta), *N* = 5 (black) and *N* = 7 for benchmark submission results (green) and longer time averaging (blue). Note the dotted

Based on this study, we conclude that the reattachment region is insensitive mainly to an increase in Reynolds number and grid resolution. Moreover, results at *Re* = 4 × 104 indicate that the recirculation region is not sensitive to the averaging/integration time interval. Ideally, one should conduct a further investigation at higher Reynolds number

lines that correspond to experimental profile measurements and *u* = 0 value.

*u*′

*u*

**Figure 16.** Mean and rms profiles of axial velocity at *x*=1.6 (black), 2.6 (blue), 3.6 (green), and 4.6 (red) for the experimental data (symbols) and Conv3D simulation on 40 milliom node uniform (solid) and nonuniform (dash-dotted) mesh, on 12 million node uniform mesh (dashed) and on 3 million node nonuniform mesh (dotted line).

with longer integration/averaging time to resolve the discrepancy between the simulations and experimental data points near the upper wall at *x* = 1.6. For now, we quote that in the experimental measurements of velocity, ". . . the focus . . . is not the near-wall region" [2, p. 15]. This underscores a necessity of conducting concurrent experiments and validation simulations in which discrepancies like the extent of reciculation region or flow rate uncertainty arising here could be easier to detect and additional data acquisitions and simulations could be performed in order to resolve them.

**Acknowledgments**

during his work on this project.

Aleksandr V. Obabko1, Paul F. Fischer1, Timothy J. Tautges1, Vasily M. Goloviznin2,

4 Cambridge University, Cambridge, UK

expansion. *J. Comput. Phys.*, 54:468–488, 1984.

Cambridge University Press, Cambridge, 2002.

Bathesda, MD, 2010.

2012. To appear.

6(3):381–392, 1986.

Mikhail A. Zaytsev2, Vladimir V. Chudanov2, Valeriy A. Pervichko2,

2 Moscow Institute of Nuclear Energy Safety (IBRAE), Moscow, Russia

**Author details**

USA

UK

**References**

Anna E. Aksenova2 and Sergey A. Karabasov3,4

This work was partially supported by the National Institutes of Health, RO1 Research Project Grant (2RO1HL55296-04A2), by Whitaker Foundation Grant (RG-01-0198), and by the Office of Science, U.S. Department of Energy, under Contract DE-AC02-06CH11357. The work was completed as part of the SHARP reactor performance and safety code suite development project, which is funded under the auspices of the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program of the U.S. Department of Energy Office of Nuclear Energy. The authors are grateful to Gail Pieper for editing the manuscript. One of the authors also thanks Gary Leaf, Elia Merzari, and Dave Pointer for helpful discussions, Scott Parker and other staff at Argonne Leadership Computing Facility, and his wife Alla for her patience

Large Eddy Simulation of Thermo-Hydraulic Mixing in a T-Junction

http://dx.doi.org/10.5772/53143

67

1 Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL,

3 Queen Mary University of London, School of Engineering and Materials Science, London,

[1] J. H. Mahaffy and B. L. Smith. Synthesis of benchmark results. In *Experimental Validation and Application of CFD and CMFD Codes to Nuclear Reactor Safety Issues (CFD4NRS-3)*,

[2] B. L. Smith, J. H. Mahaffy, K. Angele, and J. Westin. Experiments and unsteady CFD-calculations of thermal mixing in a T-junction. *Nuclear Engineering and Design*,

[3] A.T. Patera. A spectral element method for fluid dynamics: laminar flow in a channel

[4] M.O. Deville, P.F. Fischer, and E.H. Mund. *High-order methods for incompressible fluid flow*.

[5] A.A. Samarskii and V.M. Goloviznin. Difference approximation of convective transport with spatial splitting of time derivative. *Mathematical Modelling*, 10(1):86–100, 1998.

[6] A. Iserles. Generalized leapfrog methods. *IMA Journal of Numerical Analysis*,

**Figure 17.** Axial mean velocity profile near the upper wall at *z*=0.0050 and *y*=0.0000 for Nek5000 simulation at *Re* = 9 × 104 with *N* = 11 (red), at *Re* = 6 × 104 with *N* = 9 (cyan) and at *Re* = 4 × 104 with *N* = 9 (magenta), *N* = 5 (black) and *N* = 7 for the benchmark submission results (green) and longer time averaging (blue).

## **7. Conclusions and future work**

Several fully unsteady computational models in the framework of large eddy simulations (LES) are implemented for a thermohydraulic transport problem relevant to the design of nuclear power plant pipe systems. Specifically, numerical simulations for the recent 2010 OECD/NEA Vattenfall T-junction benchmark problem concerned with thermal stripping in a T-junction have been conducted with three numerical techniques based on finite-difference implicit LES (CABARET code), finite-volume LES (CONV3D code) and spectral-element (Nek5000 code) approaches. The simulation results of all three methods, including the blind test submission results of Nek5000, show encouraging agreement with the experiment despite some differences in the operating conditions between the simulations and the experiment. In particular, the Nek5000 results tend to closely match the experiment data close to the T-junction, and the CABARET solution well captures the experimental profiles farther downstream of the junction. The differences in the operating conditions between the simulation and the experiment include the uncertainty of the mass flow rate reported in the experiment (about 6%), a reduction in the effective Reynolds number used in some of the simulations that was needed to speedup the computations (Nek5000), and some discrepancies in the inflow boundary conditions. Nevertheless, the good level of agreement between the current simulations and the experiment despite these discrepancies indicates that the flow dynamics in the T-junction experiment is driven mainly by large-scale mixing effects that were not very sensitive to the differences in the operating conditions.

For further investigation, the improvement of the quality of turbulent inflow conditions (most notably, at the hot inlet boundary for Nek5000 and at the cold inlet for CABARET) is planned as well as running additional high-resolution simulations to capture the high Reynolds number regimes typical of the experiment. A further study also will be devoted to a detailed analysis of the temporal spectra of velocity and temperature fluctuations obtained from all three LES solutions.

## **Acknowledgments**

22 Nuclear Reactor Thermal Hydraulics

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

−4 −3 −2 −1 0 1 2 3 4 5 6

**Figure 17.** Axial mean velocity profile near the upper wall at *z*=0.0050 and *y*=0.0000 for Nek5000 simulation at *Re* = 9 × 104 with *N* = 11 (red), at *Re* = 6 × 104 with *N* = 9 (cyan) and at *Re* = 4 × 104 with *N* = 9 (magenta), *N* = 5 (black) and *N* = 7

Several fully unsteady computational models in the framework of large eddy simulations (LES) are implemented for a thermohydraulic transport problem relevant to the design of nuclear power plant pipe systems. Specifically, numerical simulations for the recent 2010 OECD/NEA Vattenfall T-junction benchmark problem concerned with thermal stripping in a T-junction have been conducted with three numerical techniques based on finite-difference implicit LES (CABARET code), finite-volume LES (CONV3D code) and spectral-element (Nek5000 code) approaches. The simulation results of all three methods, including the blind test submission results of Nek5000, show encouraging agreement with the experiment despite some differences in the operating conditions between the simulations and the experiment. In particular, the Nek5000 results tend to closely match the experiment data close to the T-junction, and the CABARET solution well captures the experimental profiles farther downstream of the junction. The differences in the operating conditions between the simulation and the experiment include the uncertainty of the mass flow rate reported in the experiment (about 6%), a reduction in the effective Reynolds number used in some of the simulations that was needed to speedup the computations (Nek5000), and some discrepancies in the inflow boundary conditions. Nevertheless, the good level of agreement between the current simulations and the experiment despite these discrepancies indicates that the flow dynamics in the T-junction experiment is driven mainly by large-scale mixing

effects that were not very sensitive to the differences in the operating conditions.

For further investigation, the improvement of the quality of turbulent inflow conditions (most notably, at the hot inlet boundary for Nek5000 and at the cold inlet for CABARET) is planned as well as running additional high-resolution simulations to capture the high Reynolds number regimes typical of the experiment. A further study also will be devoted to a detailed analysis of the temporal spectra of velocity and temperature fluctuations obtained

for the benchmark submission results (green) and longer time averaging (blue).

**7. Conclusions and future work**

from all three LES solutions.

This work was partially supported by the National Institutes of Health, RO1 Research Project Grant (2RO1HL55296-04A2), by Whitaker Foundation Grant (RG-01-0198), and by the Office of Science, U.S. Department of Energy, under Contract DE-AC02-06CH11357. The work was completed as part of the SHARP reactor performance and safety code suite development project, which is funded under the auspices of the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program of the U.S. Department of Energy Office of Nuclear Energy.

The authors are grateful to Gail Pieper for editing the manuscript. One of the authors also thanks Gary Leaf, Elia Merzari, and Dave Pointer for helpful discussions, Scott Parker and other staff at Argonne Leadership Computing Facility, and his wife Alla for her patience during his work on this project.

## **Author details**

Aleksandr V. Obabko1, Paul F. Fischer1, Timothy J. Tautges1, Vasily M. Goloviznin2, Mikhail A. Zaytsev2, Vladimir V. Chudanov2, Valeriy A. Pervichko2, Anna E. Aksenova2 and Sergey A. Karabasov3,4

1 Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USA

2 Moscow Institute of Nuclear Energy Safety (IBRAE), Moscow, Russia

3 Queen Mary University of London, School of Engineering and Materials Science, London, UK

4 Cambridge University, Cambridge, UK

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**Chapter 3**

**CFD as a Tool for the Analysis of the Mechanical**

The analysis of the mechanical integrity of Light Water Nuclear Reactors has experienced a strong evolution in the two latest decades. Until the nineteen eighties, the structural design was practically based on static loads with amplifying factors to take dynamic effects into ac‐ count, (see e. g. Laheyand Moody, 1993), and on wide safety margins. Even if an incipient methodology for studying earthquake effects already existed, it was only at the end of the nineties when a complete analysis about dynamic loads, caused by a steam line guillotinebreak of a BWR, was carried out for the first time (Hermansson and Thorsson, 1997). This analysis, based on a three-dimensional Computational Fluid Dynamic (CFD) simulation of the break, revealed a discrepancy with former applied loads that had never been exposed in the past due to inappropriate experimental setup geometry (Tinoco, 2001). In addition, the analysis pointed out the importance of also studying the load frequencies. The complete is‐

Nonetheless, CFD has shown to be a much more versatile tool. Issues like checking the max‐ imum temperature of the core shroud of the reactor, when the power of the reactor is uprat‐ ed (Tinoco et al., 2008), or estimating the thermal loadings during different transient conditions when the Emergency Core Cooling System (ECCS) has been modified, (Tinoco et al., 2005), or studying the automatic boron injection during an Anticipated Transient With‐ out Scram (ATWS),(Tinoco et al., 2010a), have all been settled by means of CFD simulations.

Even more significantly, a rather complete CFD model of the Downcomer, internal Main Recir‐ culation Pumps (MRP) and Lower Plenum up to the Core Inlet, (Tinoco and Ahlinder, 2009), has shed some light concerning the thermal mixing in the different regions of the reactor and al‐ so about the appropriateness of using connected sub-models as constituting parts of a larger

> © 2013 Tinoco; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Tinoco; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

model. The complete matter of thermal mixing is discussed in detail in Section 4.

**Integrity of Light Water Nuclear Reactors**

Additional information is available at the end of the chapter

Hernan Tinoco

**1. Introduction**

http://dx.doi.org/10.5772/52691

sue is further discussed in Section 3.
