**Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber**

Ten-See Wang, Francisco Canabal, Yen-Sen Chen, Gary Cheng and Yasushi Ito

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52569

## **1. Introduction**

Nuclear thermal propulsion [1] can carry far larger payloads and reduce travel time for astro‐ nauts going to Mars than is now possible with chemical propulsion. The most feasible con‐ cept, extensively tested during the Rover/NERVA (Nuclear Engine for Rocket Vehicle Application) era, is the solid-core concept [2]. It features a solid-core nuclear reactor consist‐ ing of hundreds of heat generating prismatic flow elements. Each flow element contains tens of tubular flow channels through which the working fluid, hydrogen, acquires energy and ex‐ pands in a high expansion nozzle to generate thrust. Approximately twenty nuclear thermal engines with different sizes and designs were tested during that era. Using hydrogen as pro‐ pellant, a solid-core design typically delivers specific impulses (ISP) on the order of 850 to 1000 s, about twice that of a chemical rocket such as the Space Shuttle Main Engine (SSME).

With the announcement of the Vision for Space Exploration on January 14, 2004, NASA Marshall Space Flight Center started a two-year solid-core nuclear rocket development ef‐ fort in 2006. The tasks included, but not limited to, nuclear systems development, design methodology development, and materials development. In 2011, with the retirement of Space Shuttle fleets, and NASA's shifting focus to further out places such as Mars and aste‐ roids, nuclear thermal propulsion is likely to garner substantial interest again. It is therefore timely to discuss the design methodology development effort from 2006 to 2007, entitled "Multiphysics Thrust Chamber Modeling", which developed an advanced thermal hydraul‐ ics computational methodology and studied a solid-core, nuclear thermal engine designed in the Rover/NREVA era.

© 2013 Wang 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 Wang 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.

One of the impacts made by this thermal hydraulics design methodology is the considera‐ tion of chemical reacting flow that addresses the effect of hydrogen decomposition and re‐ combination. The advantage of using hydrogen as a propellant is well known in the chemical rocket due to its low molecular weight. However, molecular hydrogen decompos‐ es to atomic hydrogen during high-temperature heating in the thermal nuclear reactor. Since atomic hydrogen has half the weight of that of molecular hydrogen, it was speculated by some that the total thrust could be doubled if all of the hydrogen is dissociated at very high temperatures, therefore leaning to the high power density reactor design. In actuality, the hydrogen conversion is often not uniform across the solid-core since the reactor temperature depends on the hydrogen flow distribution and the actual power profile generated by the nuclear fuel. In addition, the hydrogen atoms recombine in the nozzle since temperature de‐ creases rapidly during the gas expansion, thereby negating the thrust gain. To the best of our knowledge, however, the effect of hydrogen decomposition in a thermal nuclear thrust chamber on the thrust performance has never been studied.

tational model [7] of the Small Engine that shows the surfaces of the hydrogen inlet duct, a pressure vessel that houses the solid-core reactor, and an exhaust nozzle that provides the

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber

http://dx.doi.org/10.5772/52569

109

thrust.

**Figure 1.** Computational Model of the Small Engine.

On the other hand, it is always desirable to decrease the reactor weight while one of the ideas is to reduce the reactor size, which increases the power density. One of the impacts of operating at the combination of high temperature and high power density is a phenomenon known as the mid-section corrosion [3], as reported during the legacy engine tests. It is the cracking of the coating layer deposited on the inner wall of the flow channel, coupled with an excessive mass loss of the material near the mid-section of a flow element. The purpose of the coating layer was to isolate the carbonaceous compound in the flow element matrix from the attack by hydrogen. The causes of mid-section corrosion were speculated as a mismatch in the thermal expansion of flow element and its coating material, high flow element web internal temperature gradients, and change of solid thermal property due to irradiation [3, 4]. Those are all possibilities related to the materials. We, however, wanted to trace the cause from a thermal-hydraulics design view point. That is, with the long, narrow flow channel design that is used to heat the hydrogen, the possibility of flow choking in the channel was never studied. One of the efforts in this task was therefore to investigate the possibility of chocked flow occurring in the long, narrow flow channel, and its implication on heat accu‐ mulation in the flow element and mid-section corrosion.

The objective of this study was to bridge the development of current design methods with those developed in the Rover/NERVA era, thereby providing a solid base for future devel‐ opment. This is because during the Rover/NERVA era, there was a wealth of lessons learned from those legacy engine tests. All those lessons learned culminated in the final design of the Small Engine [5], but was never built nor tested since it was designed near the end of that era. The Small Engine was therefore a 'paper engine' that bears the best features of les‐ sons learned. By simulating and comparing the computed environments with those of the Small Engine design analysis and available test information from the legacy tests, the les‐ sons learned during Rover/NERVA era may be revitalized and the effect of some important design features may be validated. This Chapter therefore reviews the thermal hydraulics computational methodology developed to help the design of a materials tester [4, 6], and the results reported in the design analyses of the Small Engine [7, 8]. Figure 1 shows the compu‐

tational model [7] of the Small Engine that shows the surfaces of the hydrogen inlet duct, a pressure vessel that houses the solid-core reactor, and an exhaust nozzle that provides the thrust.


**Figure 1.** Computational Model of the Small Engine.

One of the impacts made by this thermal hydraulics design methodology is the considera‐ tion of chemical reacting flow that addresses the effect of hydrogen decomposition and re‐ combination. The advantage of using hydrogen as a propellant is well known in the chemical rocket due to its low molecular weight. However, molecular hydrogen decompos‐ es to atomic hydrogen during high-temperature heating in the thermal nuclear reactor. Since atomic hydrogen has half the weight of that of molecular hydrogen, it was speculated by some that the total thrust could be doubled if all of the hydrogen is dissociated at very high temperatures, therefore leaning to the high power density reactor design. In actuality, the hydrogen conversion is often not uniform across the solid-core since the reactor temperature depends on the hydrogen flow distribution and the actual power profile generated by the nuclear fuel. In addition, the hydrogen atoms recombine in the nozzle since temperature de‐ creases rapidly during the gas expansion, thereby negating the thrust gain. To the best of our knowledge, however, the effect of hydrogen decomposition in a thermal nuclear thrust

On the other hand, it is always desirable to decrease the reactor weight while one of the ideas is to reduce the reactor size, which increases the power density. One of the impacts of operating at the combination of high temperature and high power density is a phenomenon known as the mid-section corrosion [3], as reported during the legacy engine tests. It is the cracking of the coating layer deposited on the inner wall of the flow channel, coupled with an excessive mass loss of the material near the mid-section of a flow element. The purpose of the coating layer was to isolate the carbonaceous compound in the flow element matrix from the attack by hydrogen. The causes of mid-section corrosion were speculated as a mismatch in the thermal expansion of flow element and its coating material, high flow element web internal temperature gradients, and change of solid thermal property due to irradiation [3, 4]. Those are all possibilities related to the materials. We, however, wanted to trace the cause from a thermal-hydraulics design view point. That is, with the long, narrow flow channel design that is used to heat the hydrogen, the possibility of flow choking in the channel was never studied. One of the efforts in this task was therefore to investigate the possibility of chocked flow occurring in the long, narrow flow channel, and its implication on heat accu‐

The objective of this study was to bridge the development of current design methods with those developed in the Rover/NERVA era, thereby providing a solid base for future devel‐ opment. This is because during the Rover/NERVA era, there was a wealth of lessons learned from those legacy engine tests. All those lessons learned culminated in the final design of the Small Engine [5], but was never built nor tested since it was designed near the end of that era. The Small Engine was therefore a 'paper engine' that bears the best features of les‐ sons learned. By simulating and comparing the computed environments with those of the Small Engine design analysis and available test information from the legacy tests, the les‐ sons learned during Rover/NERVA era may be revitalized and the effect of some important design features may be validated. This Chapter therefore reviews the thermal hydraulics computational methodology developed to help the design of a materials tester [4, 6], and the results reported in the design analyses of the Small Engine [7, 8]. Figure 1 shows the compu‐

chamber on the thrust performance has never been studied.

108 Nuclear Reactor Thermal Hydraulics and Other Applications

mulation in the flow element and mid-section corrosion.

A two-pronged approach was conducted to address the aforementioned efforts: a detailed modeling of a single, powered flow element that addresses possible, additional cause of the mid-section corrosion [8], and a global modeling approach that computes the entire thermal flowfield of the Small Engine thrust chamber [7]. The latter links the thrust performance with the effects of power profiles, chemical reactions, and overall heat transfer efficiency. The global approach solves the entire thrust chamber with a detailed modeling, except the thousands of flow channels in the hundreds of flow elements were lumped together as a po‐ rous media, for computational efficiency. The heat transfer between other supporting solid components and the working fluid is solved with the conjugate transfer methodology, which was developed in [4, 6]. Theoretical and neutronics provided power profiles were used in lieu of the coupled neutronics modeling. The computational methodology, the re‐ sults of the simulations of a single flow element and that of the entire thrust chamber, are presented and discussed herein.

## **2. Computational heat transfer methodology**

#### **2.1. Fluid dynamics**

The computational methodology was based on a multi-dimensional, finite-volume, viscous, chemically reacting, unstructured grid, and pressure-based, computational fluid dynamics formulation [9]. Time-varying transport equations of continuity, species continuity, momen‐ tum, total enthalpy, turbulent kinetic energy, and turbulent kinetic energy dissipation rate were solved using a time-marching sub-iteration scheme and are written as:

$$\frac{\partial \mathcal{D}}{\partial \mathbf{t}} + \frac{\partial}{\partial \mathbf{x}\_f} \left( \rho u\_f \right) = 0 \tag{1}$$

( ( ) *<sup>t</sup>*

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber

( ) ( ) <sup>2</sup>

A predictor and corrector solution algorithm was employed to provide coupling of the gov‐ erning equations. A second-order central-difference scheme was employed to discretize the diffusion fluxes and source terms. For the convective terms, a second-order upwind total variation diminishing difference scheme was used. To enhance the temporal accuracy, a sec‐ ond-order backward difference scheme was employed to discretize the temporal terms. De‐

An extended *k*-*ε* turbulence model [14] was used to describe the turbulent flow and turbu‐ lent heat transfer. A modified wall function approach was employed to provide wall boun‐ dary layer solutions that are less sensitive to the near-wall grid spacing. Consequently, the model has combined the advantages of both the integrated-to-the-wall approach and the conventional law-of-the-wall approach by incorporating a complete velocity profile and a

A solid heat conduction equation was solved with the gas-side heat flux distributions as its

The present conjugate heat transfer model [4] solves the heat conduction equation for the solid blocks separately from the fluid equations. The interface temperature between gas and solid, which is stored at interior boundary points, is calculated using the heat flux continuity condition. For solution stability and consistency, the gas and solid interface boundary tem‐

A two-temperature porosity model was formulated with separate thermal conductivities for the flow and the solid parts. The heat transfer between the flow and solid was modeled by using the empirical correlation of the heat transfer coefficient for circular pipes as a function of flow Reynolds numbers. Empirical multipliers for both the heat transfer and drag loss were determined by comparing solutions of flow passing through a porous flow element with those of a Small Engine 19-channel flow element using detailed conjugate heat transfer

*p s*

¶ ¶ æ ö ¶ = ++ ç ÷ ¶¶¶ è ø (7)

boundary conditions. The solid heat conduction equation can be written as:

perature is updated using the transient heat conduction equation (7).

**2.3. Flow and heat transfer in porous media**

*p s s*

*j i pC T <sup>T</sup> K QQ t xx*

 e  e

+ = + + P- + P ê ú ç ÷ ¶¶ ¶ ¶ ê ú ë û è ø (6)

2 3 / *<sup>t</sup>*

¶ ¶ ¶ é ù æ ö <sup>+</sup> ê ú ç ÷ + P- ç ÷ ¶¶ ¶ ê ú ë û è ø (5)

e

 e

http://dx.doi.org/10.5772/52569

111

*j j kj pk k pu <sup>p</sup> tx x* m

m

*j j*

e

*<sup>p</sup> pu pC C C tx x x k*

d

m e

*jj j*

¶ ¶¶ ¶ é ù æ ö

m

e

tails of the numerical algorithm can be found in [8-13].

e

universal temperature profile [10].

**2.2. Heat transfer in solids**

d

$$\frac{\partial \,\rho \,\alpha\_{i}}{\partial t} + \frac{\partial}{\partial \,\mathbf{x}\_{j}} \left(\rho u\_{j} \alpha\_{i}\right) = \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \left(\rho \,\mathbf{D} + \frac{\mu\_{i}}{\sigma\_{\alpha}} \right) \frac{\partial \alpha\_{i}}{\partial \mathbf{x}\_{j}} \right] + o\_{i} \tag{2}$$

$$\frac{\partial \mathcal{D} \boldsymbol{u}\_{i}}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \rho \boldsymbol{u}\_{j} \boldsymbol{u}\_{i} \right) = -\frac{\partial \boldsymbol{p}}{\partial \mathbf{x}\_{i}} + \frac{\partial \boldsymbol{\tau}\_{ij}}{\partial \mathbf{x}\_{j}} \tag{3}$$

$$\begin{aligned} \frac{\partial \rho H}{\partial t} + \frac{\partial}{\partial x\_j} \left( \rho u\_j H \right) &= \frac{\partial p}{\partial t} + \frac{\partial}{\partial x\_j} \left[ \left( \frac{\mathcal{K}}{\mathbb{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \frac{\partial H}{\partial x\_j} \right] + \frac{\partial}{\partial x\_j} \left[ \left( (\mu + \mu\_l) - \left( \frac{\mathcal{K}}{\mathbb{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \right) \frac{\partial \left( V^2 / 2 \right)}{\partial x\_j} \right] \\ &+ \frac{\partial}{\partial x\_j} \left[ \left( \frac{\mathcal{K}}{\mathbb{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \left( u\_k \frac{\partial u\_j}{\partial x\_k} - \frac{2}{3} u\_j \frac{\partial u\_k}{\partial x\_k} \right) \right] + \mathcal{Q}\_r \end{aligned} \tag{4}$$

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber http://dx.doi.org/10.5772/52569 111

$$\frac{\partial pk}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (p\mu\_j) \left[ \left( \mu \frac{\mu\_l}{\delta\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + p \left( \Pi - \kappa \right) \tag{5}$$

$$\frac{\partial p\boldsymbol{\varepsilon}}{\partial t} + \frac{\partial}{\partial \boldsymbol{\chi}\_{j}} \left( p\boldsymbol{\mu}\_{j\boldsymbol{\varepsilon}} \right) = \frac{\partial}{\partial \boldsymbol{\varepsilon}\_{j}} \left[ \left( \boldsymbol{\mu} + \frac{\boldsymbol{\mu}\_{j}}{\boldsymbol{\delta}\_{\boldsymbol{\varepsilon}}} \right) \frac{\partial \boldsymbol{\varepsilon}}{\partial \boldsymbol{\varepsilon}\_{j}} \right] + p \frac{\mathcal{E}}{k} \left( \mathbb{C}\_{\boldsymbol{\beta}} \boldsymbol{\Pi} - \mathbb{C}\_{2} \boldsymbol{\varepsilon} + \mathbb{C}\_{3} \boldsymbol{\Pi}^{2} / \boldsymbol{\varepsilon} \right) \tag{6}$$

A predictor and corrector solution algorithm was employed to provide coupling of the gov‐ erning equations. A second-order central-difference scheme was employed to discretize the diffusion fluxes and source terms. For the convective terms, a second-order upwind total variation diminishing difference scheme was used. To enhance the temporal accuracy, a sec‐ ond-order backward difference scheme was employed to discretize the temporal terms. De‐ tails of the numerical algorithm can be found in [8-13].

An extended *k*-*ε* turbulence model [14] was used to describe the turbulent flow and turbu‐ lent heat transfer. A modified wall function approach was employed to provide wall boun‐ dary layer solutions that are less sensitive to the near-wall grid spacing. Consequently, the model has combined the advantages of both the integrated-to-the-wall approach and the conventional law-of-the-wall approach by incorporating a complete velocity profile and a universal temperature profile [10].

#### **2.2. Heat transfer in solids**

A two-pronged approach was conducted to address the aforementioned efforts: a detailed modeling of a single, powered flow element that addresses possible, additional cause of the mid-section corrosion [8], and a global modeling approach that computes the entire thermal flowfield of the Small Engine thrust chamber [7]. The latter links the thrust performance with the effects of power profiles, chemical reactions, and overall heat transfer efficiency. The global approach solves the entire thrust chamber with a detailed modeling, except the thousands of flow channels in the hundreds of flow elements were lumped together as a po‐ rous media, for computational efficiency. The heat transfer between other supporting solid components and the working fluid is solved with the conjugate transfer methodology, which was developed in [4, 6]. Theoretical and neutronics provided power profiles were used in lieu of the coupled neutronics modeling. The computational methodology, the re‐ sults of the simulations of a single flow element and that of the entire thrust chamber, are

The computational methodology was based on a multi-dimensional, finite-volume, viscous, chemically reacting, unstructured grid, and pressure-based, computational fluid dynamics formulation [9]. Time-varying transport equations of continuity, species continuity, momen‐ tum, total enthalpy, turbulent kinetic energy, and turbulent kinetic energy dissipation rate

> ( ) 0 *<sup>j</sup> j <sup>u</sup> t x* r

*j i i jj j u D tx x x*

 r

( ) *ij <sup>i</sup> j i j i j <sup>u</sup> <sup>p</sup> u u t x xx*

> 2 3

*j t k k j*

*<sup>V</sup> Hp H K K u Hj t x txC x x C x <sup>j</sup> j jj <sup>j</sup>*

é ù æ ö é ù æ ö æ ö æ ö ¶ç ÷ ¶ ¶¶ ¶¶ ¶ ê ú æ ö ê ú è ø + =+ + + +- + ç ÷ ê ú ç ÷ ç ÷ ç ÷ ¶ ¶ ¶¶ ¶ ¶ è ø ê ú ç ÷ ç ÷ ç ÷ ¶ ê ú ê ú è ø è ø è ø ë û ê ú ë û

*<sup>K</sup> <sup>u</sup> <sup>u</sup> uuQ xC x x <sup>r</sup>*

*j pH k k*

¶ é ù æ öæ ö ¶ ¶ ++ - + ê ú ç ÷ç ÷ ¶ ¶¶ ê ú ç ÷ ë û è øè ø

m

s

m

s

a

t

( )

*t t t*

m m

*p H p H*

m a

s

¶ ¶ + = ¶ ¶ (1)

ë û è ø (2)

<sup>2</sup> / 2

(4)

w

¶ ¶ ¶ ¶ + =- + ¶¶ ¶¶ (3)

 m

 s

r

( ) *<sup>i</sup> t i*

¶ ¶ ¶ ¶ é ù æ ö + =++ ê ú ç ÷ ¶¶ ¶ ¶ ê ú

r

were solved using a time-marching sub-iteration scheme and are written as:

ra

presented and discussed herein.

110 Nuclear Reactor Thermal Hydraulics and Other Applications

**2.1. Fluid dynamics**

r

r

**2. Computational heat transfer methodology**

ra

> r

A solid heat conduction equation was solved with the gas-side heat flux distributions as its boundary conditions. The solid heat conduction equation can be written as:

$$\frac{\partial p\mathbf{C}\_pT\_s}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_f} \left( K \frac{\partial T\_s}{\partial \mathbf{x}\_i} \right) + \mathbf{Q}\_p + \mathbf{Q}\_s \tag{7}$$

The present conjugate heat transfer model [4] solves the heat conduction equation for the solid blocks separately from the fluid equations. The interface temperature between gas and solid, which is stored at interior boundary points, is calculated using the heat flux continuity condition. For solution stability and consistency, the gas and solid interface boundary tem‐ perature is updated using the transient heat conduction equation (7).

#### **2.3. Flow and heat transfer in porous media**

A two-temperature porosity model was formulated with separate thermal conductivities for the flow and the solid parts. The heat transfer between the flow and solid was modeled by using the empirical correlation of the heat transfer coefficient for circular pipes as a function of flow Reynolds numbers. Empirical multipliers for both the heat transfer and drag loss were determined by comparing solutions of flow passing through a porous flow element with those of a Small Engine 19-channel flow element using detailed conjugate heat transfer modeling [8]. The only affected fluid governing equations are Navier-Stokes and energy equations and can be rewritten as:

$$\frac{\partial \mathcal{D} \rho \mathbf{u}\_i}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \rho \mathbf{u}\_j \mathbf{u}\_i \right) = -\frac{\partial \mathcal{D}}{\partial \mathbf{x}\_i} + \frac{\partial \mathbf{r}\_{ij}}{\partial \mathbf{x}\_j} - \frac{\mathbf{L}}{\mathcal{J}} \tag{8}$$

**3. Small engine**

neering group.

**3.1. Power profiles**

The goals of this study were achieved by computing the thermal hydraulics of a flow el‐ ement and the entire thrust chamber of an engine designed near the end of the Rover/ NERVA era – the Small Engine. The thrust chamber of Small Engine composes of an in‐ let plenum, the solid-core nuclear reactor or heat exchanger, and an exhaust nozzle, as shown in Fig. 1. There are 564 flow elements and 241 support elements or tie-tubes de‐ signed for the thermal nuclear reactor. The flow element is shaped like a hexagonal prism, with a length of about 890 mm and a width of about 19 mm from flat to flat [5, 8]. The prismatic flow element contains 19 tubular coolant channels. Three coolant chan‐ nel diameters were designed for the Small Engine. Each flow element is held in position by three tie-tubes and the corresponding hot-end support system (not modeled). General geometry and operating conditions were obtained from [5], while certain specific operat‐ ing conditions and nozzle geometry were calculated and provided by the Systems Engi‐

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber

http://dx.doi.org/10.5772/52569

113

For the purpose of this study, theoretical and neutronics calculation provided power pro‐ files were imposed onto the solid-core domain in lieu of the coupled neutronics calcula‐ tions, for computational efficiency. Combinations of two axial and three radial power profiles, as shown in Figs. 2 and 3, were used to show the effect of which on the heat transfer and thrust performance. Figure 2 shows a Cosine profile and a clipped Cosine profile for the power distribution in the axial direction of the solid-core reactor, while Fig. 3 shows a Cosine profile, a flattened (Cosine) profile, and a flat profile for the pow‐ er distributions in the radial direction of the nuclear heat exchanger. Given an example for the thrust chamber computations, three combinations of these power profiles were as‐ sumed. The first combination uses the shape of the Cosine curve (shown in Figs. 2 and 3) in both the axial and radial directions. That combined power distribution resembles the thermal flux distribution in bare reactors [19], and is simply named as the Cosine-Cosine power profile. By definition, the combined Cosine-Cosine power profile peaks at the middle of the core and drops to zero at the core boundary due to escaping neutrons. The second combination was prescribed by a neutronics calculation with varied Uranium loading. It features the clipped Cosine profile (shown in Fig. 2) in the axial direction and the flattened (Cosine) profile (shown in Fig. 3) in the radial direction, and is dubbed as the clipped Cosine-flattened power profile. The varied fuel loading flattens the (Cosine) power profile in the radial direction, but the power rises near the boundary to show the effect of the reflector, as shown in Fig. 3. The idea of flattening the radial profile is such that the flow in the channels is heated more uniformly, thereby improving the heat transfer efficiency. It is envisioned that the clipped Cosine-flattened power profile is

probably the closest power profile to that intended for the Small Engine.

$$\begin{aligned} \frac{\partial \rho H}{\partial t} + \frac{\partial}{\partial x\_j} \left( \rho u\_j H \right) &= \frac{\partial p}{\partial t} + \frac{\partial}{\partial x\_j} \left[ \left( \frac{\mathcal{K}}{\mathcal{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \frac{\partial H}{\partial x\_j} \right] + \frac{\partial}{\partial x\_j} \left[ \left( (\mu + \mu\_l) - \left( \frac{\mathcal{K}}{\mathcal{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \right) \frac{\partial \left( V^2 / 2 \right)}{\partial x\_j} \right] \\ &+ \frac{\partial}{\partial x\_j} \left[ \left( \frac{\mathcal{K}}{\mathcal{C}\_p} + \frac{\mu\_l}{\sigma\_H} \right) \left( u\_k \frac{\partial u\_j}{\partial x\_k} - \frac{2}{3} u\_j \frac{\partial u\_k}{\partial x\_k} \right) \right] + Q\_r + \frac{1}{\rho} Q\_s \end{aligned} \tag{9}$$

For the solid heat conduction in porous media,

$$\frac{\partial \mathcal{D} \rho\_s \mathbb{C}\_{ps} T\_s}{\partial t} = \frac{\partial}{\partial \mathbf{x}\_f} \left( K\_s \frac{\partial T\_s}{\partial \mathbf{x}\_i} \right) + \frac{1}{1 - \beta} \left( Q\_p - Q\_s \right) \tag{10}$$

For the Small Engine 19-channel flow element heat-exchanger configuration, drag loss for flow in circular pipes can be used as a point of departure. That is,

$$\mathcal{L} = \frac{1}{2} \rho \left. f\_{\mathcal{L}} \right| \mathcal{U} \left| u\_i \right| d\tag{11}$$

where *cf* =0.0791*Re* <sup>−</sup>0.25 is the Blasius formula for turbulent pipe flow [15]. Typical Reynolds numbers in a flow channel range from 10,000 to 40,000.

For the heat exchange source term,

$$\mathcal{Q}\_s = \frac{1}{2} \rho \, f\_q \frac{c\_f}{p\_r^{2/3}} \left\| L \right\| \mathbb{C}\_p \left( T\_s - T \right) / d \tag{12}$$

For the purpose of this study, the conjugate heat transfer module for solids was bench‐ marked with the analysis of a cylindrical specimen heated by an impinging hot hydrogen jet [4]. The computed solid temperature profiles agreed well with those of a standard solid heat transfer code SINDA [16]. The methodology for flow through porous media was verified through a particle-bed nuclear flow element [17] and the Space Shuttle Main Engine (SSME) main injector assembly [18]. The numerical and physical models for predicting the thrust and wall heat transfer were benchmarked with an analysis of the SSME thrust chamber flowfield, in which the computed axial-thrust performance, flow features, and wall heat fluxes compared well with those of the test data [12].

## **3. Small engine**

modeling [8]. The only affected fluid governing equations are Navier-Stokes and energy

t

b

( )

 b

¶ ¶ æ ö ¶ = +- ç ÷ ¶ ¶¶- è ø (10)

(11)


2 1

( ) <sup>1</sup>

*t t t*

m m

*p H p H*

*kj s*

3

1

For the Small Engine 19-channel flow element heat-exchanger configuration, drag loss for

where *cf* =0.0791*Re* <sup>−</sup>0.25 is the Blasius formula for turbulent pipe flow [15]. Typical Reynolds

( ) 2/3 <sup>1</sup> / <sup>2</sup> *f s q ps r <sup>c</sup> Q f UC T T d*

For the purpose of this study, the conjugate heat transfer module for solids was bench‐ marked with the analysis of a cylindrical specimen heated by an impinging hot hydrogen jet [4]. The computed solid temperature profiles agreed well with those of a standard solid heat transfer code SINDA [16]. The methodology for flow through porous media was verified through a particle-bed nuclear flow element [17] and the Space Shuttle Main Engine (SSME) main injector assembly [18]. The numerical and physical models for predicting the thrust and wall heat transfer were benchmarked with an analysis of the SSME thrust chamber flowfield, in which the computed axial-thrust performance, flow features, and wall heat

*<sup>p</sup>* <sup>=</sup> r

*s p s*

b

*j t k*

*<sup>K</sup> <sup>u</sup> <sup>u</sup> u u QQ xC x x <sup>r</sup>*

*<sup>V</sup> Hp H K K u Hj t x t xC x x C x <sup>j</sup> j jj <sup>j</sup>*

¶ é ù æ öæ ö ¶ ¶ + + - ++ ê ú ç ÷ç ÷ ¶ ¶¶ ê ú ç ÷ ë û è øè ø

> *j i C T <sup>T</sup> K QQ t xx*

> > <sup>1</sup> / <sup>2</sup> *Lf i L f c Uu d* <sup>=</sup> r

é ù æ ö é ù æ ö æ ö æ ö ¶ç ÷ ¶ ¶¶ ¶¶ ¶ ê ú æ ö ê ú è ø + =+ + + +- + ç ÷ ê ú ç ÷ ç ÷ ç ÷ ¶ ¶ ¶¶ ¶ ¶ è ø ê ú ç ÷ ç ÷ ç ÷ ¶ ê ú ê ú è ø è ø è ø ë û ê ú ë û

*j pH k k*

*s ps s s*

¶ ¶ ¶ ¶ + =- + - ¶¶ ¶¶ (8)

 m

 s <sup>2</sup> / 2

(9)

( ) *ij <sup>i</sup> j i j i j <sup>u</sup> <sup>p</sup> <sup>L</sup> u u t x xx*

r

m

s

m

s

r

equations and can be rewritten as:

112 Nuclear Reactor Thermal Hydraulics and Other Applications

r

For the solid heat conduction in porous media,

r

numbers in a flow channel range from 10,000 to 40,000.

fluxes compared well with those of the test data [12].

For the heat exchange source term,

flow in circular pipes can be used as a point of departure. That is,

r The goals of this study were achieved by computing the thermal hydraulics of a flow el‐ ement and the entire thrust chamber of an engine designed near the end of the Rover/ NERVA era – the Small Engine. The thrust chamber of Small Engine composes of an in‐ let plenum, the solid-core nuclear reactor or heat exchanger, and an exhaust nozzle, as shown in Fig. 1. There are 564 flow elements and 241 support elements or tie-tubes de‐ signed for the thermal nuclear reactor. The flow element is shaped like a hexagonal prism, with a length of about 890 mm and a width of about 19 mm from flat to flat [5, 8]. The prismatic flow element contains 19 tubular coolant channels. Three coolant chan‐ nel diameters were designed for the Small Engine. Each flow element is held in position by three tie-tubes and the corresponding hot-end support system (not modeled). General geometry and operating conditions were obtained from [5], while certain specific operat‐ ing conditions and nozzle geometry were calculated and provided by the Systems Engi‐ neering group.

#### **3.1. Power profiles**

For the purpose of this study, theoretical and neutronics calculation provided power pro‐ files were imposed onto the solid-core domain in lieu of the coupled neutronics calcula‐ tions, for computational efficiency. Combinations of two axial and three radial power profiles, as shown in Figs. 2 and 3, were used to show the effect of which on the heat transfer and thrust performance. Figure 2 shows a Cosine profile and a clipped Cosine profile for the power distribution in the axial direction of the solid-core reactor, while Fig. 3 shows a Cosine profile, a flattened (Cosine) profile, and a flat profile for the pow‐ er distributions in the radial direction of the nuclear heat exchanger. Given an example for the thrust chamber computations, three combinations of these power profiles were as‐ sumed. The first combination uses the shape of the Cosine curve (shown in Figs. 2 and 3) in both the axial and radial directions. That combined power distribution resembles the thermal flux distribution in bare reactors [19], and is simply named as the Cosine-Cosine power profile. By definition, the combined Cosine-Cosine power profile peaks at the middle of the core and drops to zero at the core boundary due to escaping neutrons. The second combination was prescribed by a neutronics calculation with varied Uranium loading. It features the clipped Cosine profile (shown in Fig. 2) in the axial direction and the flattened (Cosine) profile (shown in Fig. 3) in the radial direction, and is dubbed as the clipped Cosine-flattened power profile. The varied fuel loading flattens the (Cosine) power profile in the radial direction, but the power rises near the boundary to show the effect of the reflector, as shown in Fig. 3. The idea of flattening the radial profile is such that the flow in the channels is heated more uniformly, thereby improving the heat transfer efficiency. It is envisioned that the clipped Cosine-flattened power profile is probably the closest power profile to that intended for the Small Engine.

It can be seen that if a radially flattened power profile improves the heat transfer efficiency, then a theoretically flat, radial power profile should reach even higher heat transfer efficien‐ cy. We therefore proposed a flat power profile design for the radial direction. A theoretically flat radial power distribution may be achieved with a combination of varied fuel loading and working fluid flow distribution. The third combination therefore employs the clipped Cosine curve (shown in Fig. 2) in the axial direction and a flat curve (shown in Fig. 3) in the radial direction, and is called the clipped Cosine-flat power profile.

Radial location

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>0</sup>

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High-temperature real-gas thermodynamic properties were obtained from [20]. These proper‐ ties were generated for temperatures up to 20,000 K. The peak gas temperature computed in this study did not exceed 10,000 K, hence is well within the applicable range. A 2-species, 2-re‐ action chemical kinetics mechanism was used to describe the finite-rate hydrogen dissociation and recombination reactions, as shown in Table 1. The first hydrogen recombination reaction is abridged from a large set of kinetics mechanism developed for kerosene combustion [21], while an irreversible, second reaction [22] is added to describe the hydrogen decomposition. The kinetics of the first hydrogen recombination step have been benchmarked through many kinetic mechanism studies, as described in Ref. [21], while the kinetic rates of the second hydro‐ gen recombination reaction were measured [22]. Note the first reaction is a reversible reaction.

**Reactiona** *A B E/R* **M Ref.** M + H + H = H2 + M 5.0E15 0 0 H2, H 21 M + H2 → H + H + M 8.8E14 0 48300 H2 22

= *ATB*exp(-*E*/*RT*).

Cosine Flattened Flat

Average radial power profile

0.2

**Figure 3.** Power profiles employed in the radial direction.

M is third-body collision partner and forward rate constant *Kf*

**Table 1.** Hydrogen reaction kinetics mechanism

**3.2. Thermal properties and kinetics**

a

0.4

0.6

0.8

1

1.2

1.4

1.6

**Figure 2.** Power profiles used in the axial direction

**Figure 3.** Power profiles employed in the radial direction.

## **3.2. Thermal properties and kinetics**

It can be seen that if a radially flattened power profile improves the heat transfer efficiency, then a theoretically flat, radial power profile should reach even higher heat transfer efficien‐ cy. We therefore proposed a flat power profile design for the radial direction. A theoretically flat radial power distribution may be achieved with a combination of varied fuel loading and working fluid flow distribution. The third combination therefore employs the clipped Cosine curve (shown in Fig. 2) in the axial direction and a flat curve (shown in Fig. 3) in the

Axial location

Cosine

Clipped Cosine

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>0</sup>

radial direction, and is called the clipped Cosine-flat power profile.

A

0.2

**Figure 2.** Power profiles used in the axial direction

0.4

0.6

0.8

1

1.2

1.4

1.6

114 Nuclear Reactor Thermal Hydraulics and Other Applications

v

era

g

e

a

xial p

o

w

er

profile

High-temperature real-gas thermodynamic properties were obtained from [20]. These proper‐ ties were generated for temperatures up to 20,000 K. The peak gas temperature computed in this study did not exceed 10,000 K, hence is well within the applicable range. A 2-species, 2-re‐ action chemical kinetics mechanism was used to describe the finite-rate hydrogen dissociation and recombination reactions, as shown in Table 1. The first hydrogen recombination reaction is abridged from a large set of kinetics mechanism developed for kerosene combustion [21], while an irreversible, second reaction [22] is added to describe the hydrogen decomposition. The kinetics of the first hydrogen recombination step have been benchmarked through many kinetic mechanism studies, as described in Ref. [21], while the kinetic rates of the second hydro‐ gen recombination reaction were measured [22]. Note the first reaction is a reversible reaction.


a M is third-body collision partner and forward rate constant *Kf* = *ATB*exp(-*E*/*RT*).

**Table 1.** Hydrogen reaction kinetics mechanism

The importance of finite-rate chemistry was demonstrated in [7], where thermal hydraulics analyses were conducted for the Small Engine with and without finite-rate chemistry. One of the results show that, when the Cosine-Cosine power profile was imposed on the solidcore, the maximum solid temperature calculated with the finite-rate chemistry case was 5369 K, while that for the frozen chemistry case was much higher at 9366 K. This is because the hydrogen decomposition is endothermic. The frozen chemistry freezes molecular hydrogen throughout the thrust chamber and does not allow the hydrogen decomposition to occur, hence an artificially high maximum solid temperature was calculated. It can be seen that without finite-rate chemistry involved, the computed thermal environment and thrust per‐ formance are not physical [7].

**4.1. Computational grid generation**

**Figure 4.** Sketch of the flow element geometry.

**Figure 5.** Cross-sectional computational grid layout for the flow channels and the solid web.

As mentioned above, the flow element is shaped like a hexagonal prism, containing 19 tubu‐ lar coolant channels for the propellant hydrogen to acquire heat from the neutronic reac‐ tions, as sketched in Fig. 4. For computational efficiency, a 60º pie-section of a single flow element was computed by taking advantage of the symmetrical nature of the prism geome‐ try. A hybrid volume grid was generated by extrusion [25, 26] from a 2-D cross-sectional mesh, as shown in Fig. 5. Structured grid cells were used for the inner and outer layers of the flow channel, and for the outer edges of the prismatic flow element, while unstructured grid cells were used for the rest of the internal web. The structured grid cells at the inner layer of the flow channel was to resolve the turbulent boundary layer, while the general strategy of the hybrid mesh was to minimize numerical diffusion and the number of cells.

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The solid-core flow element material is assumed to be the (U, Zr)C-graphite composite A, which was tested as flow element material in a legacy reactor [23]. Properties of thermal conductivity, density, and heat capacity as a function of temperature were obtained for (U, Zr)C-graphite composite A from Ref. [23]. Those properties of Beryllium [19] were used for the reflector and slat in the thrust chamber computation. Slat acts as a buffer between the solid-core and the reflector. The thermal properties for the coating layer deposited on the in‐ ner wall of flow channels were also obtained from [23].

## **4. Small engine single flow element modeling**

From the material properties point of view, the main cause of mid-section corrosion was speculated as a mismatch in the thermal expansion of flow element and its coating material [3, 4]. The solution is therefore to improve the material properties through materials devel‐ opment [4, 6]. In addition to the materials development, however, we feel further study through the thermal hydraulics analysis of the fundamental reactor design is also important. That is, reduction of the reactor size often started with reducing the diameter of the flow channels, which results in higher aspect ratio, or longer flow channels. According to the Rayleigh line theory, flow with continuous heat addition in a long tube could choke. When that happens, any further heat addition can only serve to reduce the mass flow rate in the tube or, in other words, to jump to another Rayleigh line of lower mass flow [24]. This phe‐ nomenon could cause unintended mass flow mal-distribution in the solid-core reactor, re‐ sulting in uneven and high local thermal load in the flow element matrix, thereby cracking the coating material. The goal is therefore to compute if choking could occur in one of the flow channels in the Small Engine. To achieve this goal, the worst case was pursued. That is, the Cosine-Cosine power profile which generates peak thermal load in the center of the reac‐ tor was assumed. As a result, the flow element located at the center of the solid-core reactor was selected for the computation. In addition, among the three diameters considered for the flow channel of the Small Engine [5], the smallest flow channel diameter was selected, for its highest aspect ratio. These choices were made such that the computed hot hydrogen flow inside one of the flow channels had the best chance to choke.

#### **4.1. Computational grid generation**

The importance of finite-rate chemistry was demonstrated in [7], where thermal hydraulics analyses were conducted for the Small Engine with and without finite-rate chemistry. One of the results show that, when the Cosine-Cosine power profile was imposed on the solidcore, the maximum solid temperature calculated with the finite-rate chemistry case was 5369 K, while that for the frozen chemistry case was much higher at 9366 K. This is because the hydrogen decomposition is endothermic. The frozen chemistry freezes molecular hydrogen throughout the thrust chamber and does not allow the hydrogen decomposition to occur, hence an artificially high maximum solid temperature was calculated. It can be seen that without finite-rate chemistry involved, the computed thermal environment and thrust per‐

The solid-core flow element material is assumed to be the (U, Zr)C-graphite composite A, which was tested as flow element material in a legacy reactor [23]. Properties of thermal conductivity, density, and heat capacity as a function of temperature were obtained for (U, Zr)C-graphite composite A from Ref. [23]. Those properties of Beryllium [19] were used for the reflector and slat in the thrust chamber computation. Slat acts as a buffer between the solid-core and the reflector. The thermal properties for the coating layer deposited on the in‐

From the material properties point of view, the main cause of mid-section corrosion was speculated as a mismatch in the thermal expansion of flow element and its coating material [3, 4]. The solution is therefore to improve the material properties through materials devel‐ opment [4, 6]. In addition to the materials development, however, we feel further study through the thermal hydraulics analysis of the fundamental reactor design is also important. That is, reduction of the reactor size often started with reducing the diameter of the flow channels, which results in higher aspect ratio, or longer flow channels. According to the Rayleigh line theory, flow with continuous heat addition in a long tube could choke. When that happens, any further heat addition can only serve to reduce the mass flow rate in the tube or, in other words, to jump to another Rayleigh line of lower mass flow [24]. This phe‐ nomenon could cause unintended mass flow mal-distribution in the solid-core reactor, re‐ sulting in uneven and high local thermal load in the flow element matrix, thereby cracking the coating material. The goal is therefore to compute if choking could occur in one of the flow channels in the Small Engine. To achieve this goal, the worst case was pursued. That is, the Cosine-Cosine power profile which generates peak thermal load in the center of the reac‐ tor was assumed. As a result, the flow element located at the center of the solid-core reactor was selected for the computation. In addition, among the three diameters considered for the flow channel of the Small Engine [5], the smallest flow channel diameter was selected, for its highest aspect ratio. These choices were made such that the computed hot hydrogen flow

formance are not physical [7].

116 Nuclear Reactor Thermal Hydraulics and Other Applications

ner wall of flow channels were also obtained from [23].

**4. Small engine single flow element modeling**

inside one of the flow channels had the best chance to choke.

As mentioned above, the flow element is shaped like a hexagonal prism, containing 19 tubu‐ lar coolant channels for the propellant hydrogen to acquire heat from the neutronic reac‐ tions, as sketched in Fig. 4. For computational efficiency, a 60º pie-section of a single flow element was computed by taking advantage of the symmetrical nature of the prism geome‐ try. A hybrid volume grid was generated by extrusion [25, 26] from a 2-D cross-sectional mesh, as shown in Fig. 5. Structured grid cells were used for the inner and outer layers of the flow channel, and for the outer edges of the prismatic flow element, while unstructured grid cells were used for the rest of the internal web. The structured grid cells at the inner layer of the flow channel was to resolve the turbulent boundary layer, while the general strategy of the hybrid mesh was to minimize numerical diffusion and the number of cells.

**Figure 4.** Sketch of the flow element geometry.

**Figure 5.** Cross-sectional computational grid layout for the flow channels and the solid web.

In this study, a coating layer, which has different thermal properties from those of the flow element, was also modeled for the flow channel. This was done to imitate the design of those flow elements described in the legacy engine test [23], in which the coating layer was added to protect the carbonaceous compound in the flow element from the alleged chemical attack by the heated, high temperature hydrogen. It is noted that, in the present simulation, the thermal properties of the flow element and the coating layer vary with temperatures. Lastly, in the computational model, an entrance region and an exit region were added to the upstream and downstream of the flow element, respectively, to simulate the environments above and below the solid-core. Total number of cells used was 4.5 million. The heat transfer between the fluid flow, coating layer and the solid fuel was simulated with the conjugate heat transfer model.

of propellant hydrogen, rise steadily but drop about 200 K to 2300 K near the end of the fuel port. That is because near the end of fuel port, the flows become choked in both channels. The predicted maximum temperature of both flow channels is about 2700 K. The most sig‐ nificant result from Fig. 7 is that the flow choked near the end of the fuel port, indicating the mass flow rates in these two channels could be reduced, resulting in higher flow element web internal temperature and higher thermal gradient at the coating-solid fuel interfaces and eventually, the possible cracking of the coating layer. Under the operating conditions and assumptions made in this study, the possibility of chocked flow occurring in the flow channels is therefore demonstrated. For future study, three-dimensional numerical simula‐ tions of multiple flow elements with a fixed upstream total pressure boundary condition, and a downstream nozzle to remove the fixed exit mass flow boundary condition are neces‐ sary to include the inter-element effect. Although the chocked flow occurring in the flow channel is a potential design flaw, the impact of which can readily be avoided by shortening the length of the flow channel to, say, 0.85 m (Fig. 7), without lowering the maximum gas

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber

T, K

1300 1400 1500 1600 1700 1800 <sup>0</sup>

Inner fuel port

Inner flow channel

Outer flow channel

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temperature.

R, m

2

4

6

8

Outer fuel port

**Figure 6.** Radial temperature profiles at the mid-section of the flow element.

Coating layer

m

### **4.2. Results and discussion**

In this study, various power generation levels were simulated, and it was found that flow choking occurred in some coolant channels as the power level reached about 80% of the maximum power level. Theoretically, once the flow choking occurs, further increase of heat addition would lead to the reduction of mass flow rate in the flow channel, amount to shift‐ ing from one Rayleigh line to another as described in Ref. [24]. This could cause mass flow maldistribution in the flow-element matrix, resulting in uneven thermal load in the solidcore. That is, the temperature of the internal web houses the flow channels starved with coolant hydrogen may rise unexpectedly, potentially leading to the cracking of the coating material. Unfortunately, further increase of the power led to unrealistic numerical solutions because of the boundary conditions employed (fixed mass flow rate at the inlet and mass conservation at the exit). This set of boundary conditions was used because only one flow element was simulated, and thus, mass flow reduction at higher power level was precluded. To allow for local mass flow reductions, at least 1/12 of a pie shaped cross-section of the sol‐ id-core has to be computed such that fixed mass flow boundary condition need not be used. However, that option was out of the scope of this study. Nevertheless, the current setup is still acceptable since our goal is to determine whether flow choking could occur in the flow channel.

Figure 6 shows the computed temperature distribution in the radial direction (across the sol‐ id-fluid interface) at the mid-section of the flow channel. The radial temperature profile re‐ veals that substantial thermal gradients occur at the fluid-coating interface and the coatingsolid fuel interface. This kind of high thermal gradient occurring at the coating-solid interfaces could be an issue already without flow choking in the flow channel. With flow choking and the coolant flow rate reduced, the thermal gradient at the coating-solid inter‐ face could be even higher since heat is not carried away at the design point, and eventually mid-section corrosion could develop.

Figure 7 shows the computed axial temperature and Mach number profiles along centerlines 1 and 2 (see locations of centerlines 1 and 2 in Fig. 5). The axial temperature profiles almost overlap for centerlines 1 and 2, so are the axial Mach number profiles. The peak Mach num‐ ber exceeds unity near the end of the flow channel. It can also be seen that the temperatures In this study, a coating layer, which has different thermal properties from those of the flow element, was also modeled for the flow channel. This was done to imitate the design of those flow elements described in the legacy engine test [23], in which the coating layer was added to protect the carbonaceous compound in the flow element from the alleged chemical attack by the heated, high temperature hydrogen. It is noted that, in the present simulation, the thermal properties of the flow element and the coating layer vary with temperatures. Lastly, in the computational model, an entrance region and an exit region were added to the upstream and downstream of the flow element, respectively, to simulate the environments above and below the solid-core. Total number of cells used was 4.5 million. The heat transfer between the fluid flow, coating layer and the solid fuel was simulated with the conjugate

In this study, various power generation levels were simulated, and it was found that flow choking occurred in some coolant channels as the power level reached about 80% of the maximum power level. Theoretically, once the flow choking occurs, further increase of heat addition would lead to the reduction of mass flow rate in the flow channel, amount to shift‐ ing from one Rayleigh line to another as described in Ref. [24]. This could cause mass flow maldistribution in the flow-element matrix, resulting in uneven thermal load in the solidcore. That is, the temperature of the internal web houses the flow channels starved with coolant hydrogen may rise unexpectedly, potentially leading to the cracking of the coating material. Unfortunately, further increase of the power led to unrealistic numerical solutions because of the boundary conditions employed (fixed mass flow rate at the inlet and mass conservation at the exit). This set of boundary conditions was used because only one flow element was simulated, and thus, mass flow reduction at higher power level was precluded. To allow for local mass flow reductions, at least 1/12 of a pie shaped cross-section of the sol‐ id-core has to be computed such that fixed mass flow boundary condition need not be used. However, that option was out of the scope of this study. Nevertheless, the current setup is still acceptable since our goal is to determine whether flow choking could occur in the flow

Figure 6 shows the computed temperature distribution in the radial direction (across the sol‐ id-fluid interface) at the mid-section of the flow channel. The radial temperature profile re‐ veals that substantial thermal gradients occur at the fluid-coating interface and the coatingsolid fuel interface. This kind of high thermal gradient occurring at the coating-solid interfaces could be an issue already without flow choking in the flow channel. With flow choking and the coolant flow rate reduced, the thermal gradient at the coating-solid inter‐ face could be even higher since heat is not carried away at the design point, and eventually

Figure 7 shows the computed axial temperature and Mach number profiles along centerlines 1 and 2 (see locations of centerlines 1 and 2 in Fig. 5). The axial temperature profiles almost overlap for centerlines 1 and 2, so are the axial Mach number profiles. The peak Mach num‐ ber exceeds unity near the end of the flow channel. It can also be seen that the temperatures

heat transfer model.

channel.

mid-section corrosion could develop.

**4.2. Results and discussion**

118 Nuclear Reactor Thermal Hydraulics and Other Applications

of propellant hydrogen, rise steadily but drop about 200 K to 2300 K near the end of the fuel port. That is because near the end of fuel port, the flows become choked in both channels. The predicted maximum temperature of both flow channels is about 2700 K. The most sig‐ nificant result from Fig. 7 is that the flow choked near the end of the fuel port, indicating the mass flow rates in these two channels could be reduced, resulting in higher flow element web internal temperature and higher thermal gradient at the coating-solid fuel interfaces and eventually, the possible cracking of the coating layer. Under the operating conditions and assumptions made in this study, the possibility of chocked flow occurring in the flow channels is therefore demonstrated. For future study, three-dimensional numerical simula‐ tions of multiple flow elements with a fixed upstream total pressure boundary condition, and a downstream nozzle to remove the fixed exit mass flow boundary condition are neces‐ sary to include the inter-element effect. Although the chocked flow occurring in the flow channel is a potential design flaw, the impact of which can readily be avoided by shortening the length of the flow channel to, say, 0.85 m (Fig. 7), without lowering the maximum gas temperature.

**Figure 6.** Radial temperature profiles at the mid-section of the flow element.

deg. cross-section was computed assuming symmetry. The strategy of using hybrid mesh was again for computational efficiency. Figure 8 shows a 240 deg. view of the computational grid and geometry layout of the Small Engine thrust chamber. The thrust chamber includes the hydrogen inlet duct, pressure vessel, and a nozzle with an expansion ratio of 100. Only tie-tubes in the pressure vessel are shown for clarity. Figure 9 shows a cross-sectional view of the solid-core computational grid, depicting flow elements, tie-tubes, slat, and reflector.

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**Figure 8.** Computational grid and geometry layout of the solid-core Small Engine thrust chamber.

**Figure 7.** Axial temperature and Mach number profiles along centerlines 1 and 2.

## **5. Small engine thrust chamber modeling**

The rationale of analyzing the entire thrust chamber was to bridge the Small Engine design in the Rover/NERVA era to the current modern thermal hydraulic computational methodol‐ ogy. Specifically, to relate the assigned power profiles to that of the Small Engine by com‐ paring the computed thrust performance with that of the design. Since there are 564 flow elements and 241 support elements or tie-tubes, and each flow element contains 19 tubular flow channels, it is computationally cost-prohibitive to solve detailed thermal hydraulics for each and every flow channel. The 19 flow channels in each flow element were therefore lumped as one porous flow channel for computational efficiency, and the effect of drag and heat transfer were modeled with flow and conjugate heat transfer through porous media, or Eq. (8) through Eq. (12). The medium coolant channel diameter, out of the three coolant channel diameters designed for the Small Engine [5], was selected to derive the porosity of the flow elements.

#### **5.1. Computational grid generation**

Hybrid computational grids were generated using a software package GRIDGEN V15.07 [27]. The layout of the flow elements and tie-tubes of the solid-core is such that the whole thrust chamber is symmetrical about a 30 deg. pie-shaped cross-section, hence only the 30 deg. cross-section was computed assuming symmetry. The strategy of using hybrid mesh was again for computational efficiency. Figure 8 shows a 240 deg. view of the computational grid and geometry layout of the Small Engine thrust chamber. The thrust chamber includes the hydrogen inlet duct, pressure vessel, and a nozzle with an expansion ratio of 100. Only tie-tubes in the pressure vessel are shown for clarity. Figure 9 shows a cross-sectional view of the solid-core computational grid, depicting flow elements, tie-tubes, slat, and reflector.

Z, m

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>0</sup>

The rationale of analyzing the entire thrust chamber was to bridge the Small Engine design in the Rover/NERVA era to the current modern thermal hydraulic computational methodol‐ ogy. Specifically, to relate the assigned power profiles to that of the Small Engine by com‐ paring the computed thrust performance with that of the design. Since there are 564 flow elements and 241 support elements or tie-tubes, and each flow element contains 19 tubular flow channels, it is computationally cost-prohibitive to solve detailed thermal hydraulics for each and every flow channel. The 19 flow channels in each flow element were therefore lumped as one porous flow channel for computational efficiency, and the effect of drag and heat transfer were modeled with flow and conjugate heat transfer through porous media, or Eq. (8) through Eq. (12). The medium coolant channel diameter, out of the three coolant channel diameters designed for the Small Engine [5], was selected to derive the porosity of

Hybrid computational grids were generated using a software package GRIDGEN V15.07 [27]. The layout of the flow elements and tie-tubes of the solid-core is such that the whole thrust chamber is symmetrical about a 30 deg. pie-shaped cross-section, hence only the 30

**Figure 7.** Axial temperature and Mach number profiles along centerlines 1 and 2.

**5. Small engine thrust chamber modeling**

Centerline 1 Centerline 2 Centerline 1 Centerline 2

Fuel Port

M

0

0.2

0.4

0.6

0.8

1

1.2

ach

N

o.

T, K

the flow elements.

**5.1. Computational grid generation**

500

1000

1500

2000

2500

3000

120 Nuclear Reactor Thermal Hydraulics and Other Applications

**Case Axial power shape Radial power shape** Cosine Cosine Clipped Cosine Flattened Clipped Cosine Flat

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Figure 10 shows the computed thrust chamber temperature contours with three different power profiles: the Cosine-Cosine, clipped Cosine-flattened, and clipped Cosine-flat pro‐ files. In the plotted temperature contours, those in the solid-core represent the solid temper‐ atures in the flow elements; the portions surrounding the solid-core depict the temperatures in the slat and reflector, while the rest are the gas temperature contours. It can be seen that the solid-core temperature contours on the left (case 1) reflect the effect of Cosine-Cosine power distribution, since the peak temperature is located near the middle of the reactor, ex‐ cept it is shifted downstream of the geometrical center in the axial direction. The hydrogen gas temperature contours in the solid core take the same shape as those of the solid tempera‐ ture, except lower. The heat transfer delay in the axial direction is a result of the energy bal‐ ance between the cooling from the incoming cold hydrogen in the flow channels and the heating from the nuclear material in the web of the flow elements. It is apparent that the ef‐ fect of cold hydrogen won the fight between the two counter-acting phenomena, and pushes the peak flow element temperature downstream that shows up as a delay in heat transfer.

**Figure 10.** Solid and gas temperature contours on the symmetry plane. From left to right: case 1, case 2, and case 3.

**Table 2.** Run matrix.

**5.2. Results and discussion**

*5.2.1. Temperature contours and profiles in the thrust chamber*

**Figure 9.** A cross-sectional view of the solid-core. For illustration, circles are imposed onto the flow elements to differ‐ entiate them from the tie-tubes.

No-slip boundary condition was applied to all solid walls, while supersonic outflow boun‐ dary condition was employed at the nozzle exit. A fixed total pressure and temperature con‐ dition was used at the inlet. Inlet hydrogen flow properties were obtained from a system model simulation. Since the minor thermal effects of tie-tubes were included in the system model simulation, an adiabatic wall boundary condition was used for tie-tube walls. The core surrounding components such as the slat and reflector were treated as heat conducting solids to provide accurate boundary conditions for the solid-core boundary. The heat con‐ ducted from the core through slat and reflector to the outer thrust chamber walls was dissi‐ pated to a far-field temperature assumed to be 310 K. Table 2 shows the run matrix with three combinations of the axial and radial power profiles: the Cosine-Cosine, clipped Co‐ sine-flattened (Cosine), and clipped Cosine-flat profiles. A series of mesh studies using grid sizes of 1,903,278, 1,925,742, 4,681,751, and 7,460,255 points were performed on case 1. It was found that the computed average pressure drops across the solid-core were very similar and the differences among the computed thrust values were less than 2.5%. Based on those find‐ ings and the consideration for computational efficiency, the grid size of 1,925,742 points (or 2,408,198 computational cells) was selected for the rest of the computations.

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber http://dx.doi.org/10.5772/52569 123


**Table 2.** Run matrix.

Tie-tube Slat

122 Nuclear Reactor Thermal Hydraulics and Other Applications

Flow element Reflector

2,408,198 computational cells) was selected for the rest of the computations.

entiate them from the tie-tubes.

**Figure 9.** A cross-sectional view of the solid-core. For illustration, circles are imposed onto the flow elements to differ‐

No-slip boundary condition was applied to all solid walls, while supersonic outflow boun‐ dary condition was employed at the nozzle exit. A fixed total pressure and temperature con‐ dition was used at the inlet. Inlet hydrogen flow properties were obtained from a system model simulation. Since the minor thermal effects of tie-tubes were included in the system model simulation, an adiabatic wall boundary condition was used for tie-tube walls. The core surrounding components such as the slat and reflector were treated as heat conducting solids to provide accurate boundary conditions for the solid-core boundary. The heat con‐ ducted from the core through slat and reflector to the outer thrust chamber walls was dissi‐ pated to a far-field temperature assumed to be 310 K. Table 2 shows the run matrix with three combinations of the axial and radial power profiles: the Cosine-Cosine, clipped Co‐ sine-flattened (Cosine), and clipped Cosine-flat profiles. A series of mesh studies using grid sizes of 1,903,278, 1,925,742, 4,681,751, and 7,460,255 points were performed on case 1. It was found that the computed average pressure drops across the solid-core were very similar and the differences among the computed thrust values were less than 2.5%. Based on those find‐ ings and the consideration for computational efficiency, the grid size of 1,925,742 points (or

#### **5.2. Results and discussion**

#### *5.2.1. Temperature contours and profiles in the thrust chamber*

Figure 10 shows the computed thrust chamber temperature contours with three different power profiles: the Cosine-Cosine, clipped Cosine-flattened, and clipped Cosine-flat pro‐ files. In the plotted temperature contours, those in the solid-core represent the solid temper‐ atures in the flow elements; the portions surrounding the solid-core depict the temperatures in the slat and reflector, while the rest are the gas temperature contours. It can be seen that the solid-core temperature contours on the left (case 1) reflect the effect of Cosine-Cosine power distribution, since the peak temperature is located near the middle of the reactor, ex‐ cept it is shifted downstream of the geometrical center in the axial direction. The hydrogen gas temperature contours in the solid core take the same shape as those of the solid tempera‐ ture, except lower. The heat transfer delay in the axial direction is a result of the energy bal‐ ance between the cooling from the incoming cold hydrogen in the flow channels and the heating from the nuclear material in the web of the flow elements. It is apparent that the ef‐ fect of cold hydrogen won the fight between the two counter-acting phenomena, and pushes the peak flow element temperature downstream that shows up as a delay in heat transfer.

**Figure 10.** Solid and gas temperature contours on the symmetry plane. From left to right: case 1, case 2, and case 3.

Figure 11 shows the computed solid and gas axial temperature profiles on the symmetry plane, along the centerline of each flow element. Only the temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elementfrom the center tie-tube are plotted for clarity. The solid tem‐ peratures (Ts shown as dashed lines) lead those of the gas temperatures (Tg, solid lines), showing the effect of the heat transfer lag between the flow element matrix and hydrogen gas. The gas temperatures appear to peak near the core exit, while the solid temperatures appear to peak more upstream for the 1st, 3rd, and 5th flow element, comparing to the peak temperature locations of the hydrogen. The lead decreases at the 7th flow element, and dis‐ appears completely at the 9th flow element. These axial temperature profiles reflect the effect of the Cosine-Cosine power profile that concentrates the power in the middle of the core, and drops to zero power at the core boundary. The peak solid temperature for the 9th flow element is the lowest at around 700 K, as expected. The gas temperature at the core-exit is about 4600 K for the 1st flow element, and again about 700 K for the 9th flow element. The 3900 K core-exit gas temperature spread between the 1st and 9th flow elements indicates a very non-uniform temperature distribution at the core-exit, or poor heat transfer efficiency, apparently caused by the power-centric Cosine-Cosine power distribution. Note the temper‐ ature gradient of the solid temperature is very steep for the 1st flow element occurring at a location between the core entrance and the peak temperature.

As discussed above, it is apparent that the Cosine-Cosine power distribution causes not only peak solid temperatures that are too high, but also very non-uniform core-exit gas tempera‐ tures, indicating inefficient heat transfer. Fortunately, by measures such as varying the fuel loading, a flattened (Cosine) distribution in the radial direction could be achieved, as indi‐ cated in Fig. 3. Note that this profile is not completely flat, since the normalized power is about 1.1 at the center, drops to a minimum of about 0.9 at two-thirds of the radius outward, then rise to a maximum power of 1.6 at the boundary. But comparing it to the Cosine-Cosine power profile, it is relatively flat. In fact, by our estimation, this clipped Cosine–flattened power profile is the most representative of that intended for the Small Engine. The middle picture of Fig. 10 shows the computed solid and gas temperatures, for this clipped Cosineflattened power profile. It can be seen that the solid and gas temperature contours, reflect the effect of the clipped Cosine-flattened power distribution. Comparing to the temperature contours in the left of Fig. 10, those in the middle are more uniform. The peak solid and gas temperatures in the middle of Fig. 10 at about 3149 and 3081 K., respectively, are much low‐

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Figure 12 shows the computed solid and gas axial temperature profiles on the symmetry plane. Comparing to those of Fig. 11, it can be seen that other than the 9th flow element, the solid temperature generally leads the gas temperature only slightly, since the clipped Co‐ sine-flattened power profile is more uniform than the Cosine-Cosine profile. The peak coreexit gas temperature of the 1st fuel element is about 3037 K., while that of the 9th fuel element is about 2514 K. The range of the gas temperatures at the core-exit is hence only about 523 K, much less than that of 3900 K in case 1, indicating the core-exit gas temperature of case 2 is much more uniform than that of case 1. The flattened radial power profile is therefore a vast

As mentioned before, it is possible to make the flattened radial power profile in Fig. 3 com‐ pletely flat, by further fine tuning the fuel loading and by shaping the radial hydrogen mass flow rate profile to match that of the shape of the radial power profile; hence, the proposed theoretical flat radial power profile. The hydrogen mass flow rate profile may be shaped with the installation of various sizes of orifices at the entrance of each flow element. The re‐ sult of the computed solid and gas temperature contours of such a case, or case 3, using the clipped Cosine-flat power profile, is shown in the right of Fig. 10. It can be seen in terms of the solid-core radial temperature distribution and the uniformity of the gas temperature be‐ neath the solid core, this result is the most uniform. Figure 13 shows the solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements in the solid-core for case 3. It can be seen that the temperature spread among the 1st, 3rd, 5th and 7th flow elements are much smaller than those of earlier cases, except those of the 9th flow element due to the heat loss to the slat and reflector. From the right picture of Fig. 10 and from Fig. 13, it can be seen this flat radial power profile is a very good power profile in terms of general uniformity of the temperatures for both solid and gas temperatures inside the core and for gas exit tem‐

er than those in the left of Fig. 10, or 5369 and 4596 K., respectively.

improvement over the Cosine radial power distribution.

peratures.

**Figure 11.** Solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetry plane for case 1.

Figure 11 shows the computed solid and gas axial temperature profiles on the symmetry plane, along the centerline of each flow element. Only the temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elementfrom the center tie-tube are plotted for clarity. The solid tem‐ peratures (Ts shown as dashed lines) lead those of the gas temperatures (Tg, solid lines), showing the effect of the heat transfer lag between the flow element matrix and hydrogen gas. The gas temperatures appear to peak near the core exit, while the solid temperatures appear to peak more upstream for the 1st, 3rd, and 5th flow element, comparing to the peak temperature locations of the hydrogen. The lead decreases at the 7th flow element, and dis‐ appears completely at the 9th flow element. These axial temperature profiles reflect the effect of the Cosine-Cosine power profile that concentrates the power in the middle of the core, and drops to zero power at the core boundary. The peak solid temperature for the 9th flow element is the lowest at around 700 K, as expected. The gas temperature at the core-exit is about 4600 K for the 1st flow element, and again about 700 K for the 9th flow element. The 3900 K core-exit gas temperature spread between the 1st and 9th flow elements indicates a very non-uniform temperature distribution at the core-exit, or poor heat transfer efficiency, apparently caused by the power-centric Cosine-Cosine power distribution. Note the temper‐ ature gradient of the solid temperature is very steep for the 1st flow element occurring at a

Z, cm

Axial Core Length


**Figure 11.** Solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetry plane

location between the core entrance and the peak temperature.

<sup>5000</sup> Tg, 1st FE

Ts, 1st FE Tg, 3rd FE Ts, 3rd FE Tg, 5th FE

, 5th FE Tg, 7th FE

, 7th FE Tg, 9th FE

, 9th FE

Ts

Ts

Ts

T, K

500

for case 1.

1000

1500

2000

2500

3000

3500

4000

4500

124 Nuclear Reactor Thermal Hydraulics and Other Applications

As discussed above, it is apparent that the Cosine-Cosine power distribution causes not only peak solid temperatures that are too high, but also very non-uniform core-exit gas tempera‐ tures, indicating inefficient heat transfer. Fortunately, by measures such as varying the fuel loading, a flattened (Cosine) distribution in the radial direction could be achieved, as indi‐ cated in Fig. 3. Note that this profile is not completely flat, since the normalized power is about 1.1 at the center, drops to a minimum of about 0.9 at two-thirds of the radius outward, then rise to a maximum power of 1.6 at the boundary. But comparing it to the Cosine-Cosine power profile, it is relatively flat. In fact, by our estimation, this clipped Cosine–flattened power profile is the most representative of that intended for the Small Engine. The middle picture of Fig. 10 shows the computed solid and gas temperatures, for this clipped Cosineflattened power profile. It can be seen that the solid and gas temperature contours, reflect the effect of the clipped Cosine-flattened power distribution. Comparing to the temperature contours in the left of Fig. 10, those in the middle are more uniform. The peak solid and gas temperatures in the middle of Fig. 10 at about 3149 and 3081 K., respectively, are much low‐ er than those in the left of Fig. 10, or 5369 and 4596 K., respectively.

Figure 12 shows the computed solid and gas axial temperature profiles on the symmetry plane. Comparing to those of Fig. 11, it can be seen that other than the 9th flow element, the solid temperature generally leads the gas temperature only slightly, since the clipped Co‐ sine-flattened power profile is more uniform than the Cosine-Cosine profile. The peak coreexit gas temperature of the 1st fuel element is about 3037 K., while that of the 9th fuel element is about 2514 K. The range of the gas temperatures at the core-exit is hence only about 523 K, much less than that of 3900 K in case 1, indicating the core-exit gas temperature of case 2 is much more uniform than that of case 1. The flattened radial power profile is therefore a vast improvement over the Cosine radial power distribution.

As mentioned before, it is possible to make the flattened radial power profile in Fig. 3 com‐ pletely flat, by further fine tuning the fuel loading and by shaping the radial hydrogen mass flow rate profile to match that of the shape of the radial power profile; hence, the proposed theoretical flat radial power profile. The hydrogen mass flow rate profile may be shaped with the installation of various sizes of orifices at the entrance of each flow element. The re‐ sult of the computed solid and gas temperature contours of such a case, or case 3, using the clipped Cosine-flat power profile, is shown in the right of Fig. 10. It can be seen in terms of the solid-core radial temperature distribution and the uniformity of the gas temperature be‐ neath the solid core, this result is the most uniform. Figure 13 shows the solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements in the solid-core for case 3. It can be seen that the temperature spread among the 1st, 3rd, 5th and 7th flow elements are much smaller than those of earlier cases, except those of the 9th flow element due to the heat loss to the slat and reflector. From the right picture of Fig. 10 and from Fig. 13, it can be seen this flat radial power profile is a very good power profile in terms of general uniformity of the temperatures for both solid and gas temperatures inside the core and for gas exit tem‐ peratures.

*5.2.2. Hydrogen atom mass fraction contours in the thrust chamber*

case 1 is 0.40, or 40% conversion from hydrogen molecule decomposition.

Figure 14 shows the computed thrust chamber hydrogen mass fraction contours with three aforementioned power profiles: the Cosine-Cosine, clipped Cosine-flattened, and clipped Co‐ sine-flat profiles. The hydrogen atom mass fractions are important because the amount of which is directly related to the eventual temperature contours. As a result, the hydrogen atom mass fraction contours look qualitatively similar to those of the solid-core temperature con‐ tours, with the peak conversion pushed to near the end of the core due to the same reason that pushed the peak solid temperature away from the center of the core. For the Cosine-Cosine power profile case, as shown in the left picture of Fig. 14, coolant hydrogen enters the thrust chamber at about 370 K, heats up to about 4800 K in the solid-core, then cools and expands into the diverging nozzle to generate the thrust. Molecular hydrogen decomposes to atomic hydro‐ gen at about 2400 K; hence, most of the hydrogen atoms are formed while in the flow elements. Once the local temperature starts to cool, i.e. during the expansion in the nozzle, hydrogen atoms recombine to become molecular hydrogen. The peak hydrogen atom mass fraction in

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The middle picture in Fig. 14 shows the computed hydrogen atom mass fraction contours for case 2. It surely reflects the effect of the clipped Cosine-flattened power distribution shown in Figs. 2 and 3. Since the peak solid and gas temperatures of the clipped Cosine-flattened power profile are lower than those of the Cosine-Cosine power profile, as displayed in the previous section, the peak hydrogen atom mass fraction also drops from a high of 0.40 in the Cosine-Co‐ sine power profile case to a lowly 0.02, or a 2% conversion in the clipped Cosine-flattened pow‐ er profile case. Note that the atomic hydrogen contours in case 2 exhibit a more pronounced striation in the pressure vessel beneath the core than those in case 1. This is caused by the high‐ er power profile near the core boundary, resulting in higher local temperatures, thereby higher

hydrogen molecule conversion near the core boundary for case 2 than that for case 1.

**Figure 14.** Hydrogen atom mass fraction contours on the symmetry plane. From left to right: case 1, case 2, and case 3.

**Figure 12.** Solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetry plane for case 2.

**Figure 13.** Solid and gas axial temperatures for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetric plane for case 3.

## *5.2.2. Hydrogen atom mass fraction contours in the thrust chamber*

Z, cm

Z, cm

Axial Core Length


**Figure 13.** Solid and gas axial temperatures for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetric plane for

Axial Core Length


**Figure 12.** Solid and gas axial temperature profiles for the 1st, 3rd, 5th, 7th, and 9th flow elements on the symmetry plane

Tg, 1st FE Ts, 1st FE Tg, 3rd FE Ts, 3rd FE Tg, 5th FE Ts , 5th FE Tg, 7th FE Ts , 7th FE Tg, 9th FE Ts , 9th FE

Tg, 1st FE Ts, 1st FE Tg, 3rd FE Ts, 3rd FE Tg, 5th FE Ts , 5th FE Tg, 7th FE Ts , 7th FE Tg, 9th FE Ts , 9th FE

T, K

T, K

500

1000

1500

2000

2500

3000

for case 2.

case 3.

500

1000

1500

2000

2500

3000

126 Nuclear Reactor Thermal Hydraulics and Other Applications

Figure 14 shows the computed thrust chamber hydrogen mass fraction contours with three aforementioned power profiles: the Cosine-Cosine, clipped Cosine-flattened, and clipped Co‐ sine-flat profiles. The hydrogen atom mass fractions are important because the amount of which is directly related to the eventual temperature contours. As a result, the hydrogen atom mass fraction contours look qualitatively similar to those of the solid-core temperature con‐ tours, with the peak conversion pushed to near the end of the core due to the same reason that pushed the peak solid temperature away from the center of the core. For the Cosine-Cosine power profile case, as shown in the left picture of Fig. 14, coolant hydrogen enters the thrust chamber at about 370 K, heats up to about 4800 K in the solid-core, then cools and expands into the diverging nozzle to generate the thrust. Molecular hydrogen decomposes to atomic hydro‐ gen at about 2400 K; hence, most of the hydrogen atoms are formed while in the flow elements. Once the local temperature starts to cool, i.e. during the expansion in the nozzle, hydrogen atoms recombine to become molecular hydrogen. The peak hydrogen atom mass fraction in case 1 is 0.40, or 40% conversion from hydrogen molecule decomposition.

The middle picture in Fig. 14 shows the computed hydrogen atom mass fraction contours for case 2. It surely reflects the effect of the clipped Cosine-flattened power distribution shown in Figs. 2 and 3. Since the peak solid and gas temperatures of the clipped Cosine-flattened power profile are lower than those of the Cosine-Cosine power profile, as displayed in the previous section, the peak hydrogen atom mass fraction also drops from a high of 0.40 in the Cosine-Co‐ sine power profile case to a lowly 0.02, or a 2% conversion in the clipped Cosine-flattened pow‐ er profile case. Note that the atomic hydrogen contours in case 2 exhibit a more pronounced striation in the pressure vessel beneath the core than those in case 1. This is caused by the high‐ er power profile near the core boundary, resulting in higher local temperatures, thereby higher hydrogen molecule conversion near the core boundary for case 2 than that for case 1.

**Figure 14.** Hydrogen atom mass fraction contours on the symmetry plane. From left to right: case 1, case 2, and case 3.

Shown in the right figure of Fig. 14 is the computed hydrogen atom mass fraction contours for the clipped Cosine-flat power profile case, or case 3. It can be seen that the hydrogen atom mass fraction contours are the most uniform throughout the thrust chamber, among the three cases. The effect of the flat radial power profile is apparently the driver. As a re‐ sult, the hydrogen atom conversion drops from 2% in case 2 to 1.5% in case 3.

From the last column comes with the comparison of the specific impulses. The best thrust performance belongs to case 3 that is much higher than the design value, followed by case2 which is within the design range, while case 1 has the lowest ISP. Yet the maximum atomic hydrogen conversions are 40%, 2% and 1.5%, for cases 1, 2, and 3, respectively. These results demonstrate that the notion of higher temperature, more atomic hydrogen, therefore higher thrust is not valid. Rather, it is the most uniform radial power profile that produces the

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In summary, the computed core pressure drop, core-exit gas temperature, and specific im‐ pulse, from the clipped Cosine-flattened power distribution agree well with those of the Small Engine. The result indicates the fluid, thermal, and hydrogen environments computed with the present computational thermal hydraulics methodology closely emulate what was intended for the original Small Engine design. In addition, examining the results of our pro‐ posed clipped Cosine-flat power distribution, not only the computed core-exit gas tempera‐ ture and core pressure drop agree equally well with those of the Small Engine, the computed specific impulse is 25 to 40 seconds higher than those of the Small Engine, dem‐ onstrating the present computational methodology can be used to guide the future design.

Thermal hydraulics computational design analyses were performed to investigate the midsection corrosion issue occurred during the legacy engine tests, and to predict the heat trans‐ fer efficiency and thrust performance for a virtual solid-core, nuclear thermal engine thrust chamber – the Small Engine. Multiphysics invoked include the turbulent flow and heat transfer, finite-rate chemistry, power generation, and conjugate heat transfer for solids and porous media. The results of a detailed single flow element modeling show that under the assumption of the worst design condition, the hot hydrogen flow could choke near the end of the narrow, long flow channels, which could lead to hydrogen flow reduction and local hot spots in the solid-core; hence, originating the mid-section corrosion. In addition, a uni‐ fied Small Engine thrust chamber thermal flow field constituting those of the inlet plenum, the pressure vessel, and the nozzle, was analyzed with three power profiles. It was found that the computed core pressure drop, core-exit gas temperature, and specific impulse for the clipped Cosine-flattened power profile closely agreed with those of the Small Engine de‐ sign values. It was also found that with our proposed clipped Cosine-flat power profile, not only the computed core-ext gas temperature and core pressure drop closely agreed with those of the Small Engine, the corresponding specific impulse was much higher than those of the original Small Engine numbers. Design lesson learned from this effort is that high hy‐ drogen molecule conversion to hydrogen atoms neither improves the heat transfer efficien‐ cy, nor increases the thrust performance. Rather, it is the more uniform radial power profile that produces lower peak solid temperature, higher heat transfer efficiency, and higher

highest thrust.

**6. Conclusion**

thrust performance.

## *5.2.3. Summary*

The computed heat transfer and thrust performance design parameters for all three cases are compared with those available of the Small Engine and summarized in Table 3. The comput‐ ed core pressure drops are shown in the second column. The values for all three cases are close to the design number, although that of the Cosine-Cosine power distribution case is slightly lower. The computed peak solid temperatures are shown in the third column, in which the highest peak solid temperature belongs to case 1, followed by case 2, with the lowest peak solid temperature comes from case 3, as expected.

The fourth column shows the average core-exit gas temperatures. These were obtained by averaging the temperatures at the core-exit for all nine elements on the symmetry plane. It can be seen that the average core-exit gas temperature for the Cosine-Cosine power distribu‐ tion at 3334 K, is much higher than that of the Small Engine design temperature at 2750 K. On the other hand, the average core-exit gas temperatures for the clipped Cosine-flattened and clipped Cosine-flat power distribution cases are both quite close to that of the Small En‐ gine at 2785 and 2782 K, respectively.


**Table 3.** A summary of the heat transfer and performance parameters.

The computed temperature spread, or the difference of the core-exit gas temperature be‐ tween the first and the 9th flow elements, is shown in the fifth column. Lower temperature spread represents more uniform temperature distribution amongst the flow elements, or better heat transfer efficiency. It can be seen that the clipped Cosine-flattened power profile produces best heat transfer efficiency, the clipped Cosine-flat power profile ranks the sec‐ ond, while the Cosine-Cosine power profile has the worst heat transfer efficiency. Note that if without the outlier of the temperature profile at the 9th fuel element for case 3, its tempera‐ ture spread from the rest is more uniform than that of case 2, as indicated in Figs. 12 and 13.

From the last column comes with the comparison of the specific impulses. The best thrust performance belongs to case 3 that is much higher than the design value, followed by case2 which is within the design range, while case 1 has the lowest ISP. Yet the maximum atomic hydrogen conversions are 40%, 2% and 1.5%, for cases 1, 2, and 3, respectively. These results demonstrate that the notion of higher temperature, more atomic hydrogen, therefore higher thrust is not valid. Rather, it is the most uniform radial power profile that produces the highest thrust.

In summary, the computed core pressure drop, core-exit gas temperature, and specific im‐ pulse, from the clipped Cosine-flattened power distribution agree well with those of the Small Engine. The result indicates the fluid, thermal, and hydrogen environments computed with the present computational thermal hydraulics methodology closely emulate what was intended for the original Small Engine design. In addition, examining the results of our pro‐ posed clipped Cosine-flat power distribution, not only the computed core-exit gas tempera‐ ture and core pressure drop agree equally well with those of the Small Engine, the computed specific impulse is 25 to 40 seconds higher than those of the Small Engine, dem‐ onstrating the present computational methodology can be used to guide the future design.

## **6. Conclusion**

Shown in the right figure of Fig. 14 is the computed hydrogen atom mass fraction contours for the clipped Cosine-flat power profile case, or case 3. It can be seen that the hydrogen atom mass fraction contours are the most uniform throughout the thrust chamber, among the three cases. The effect of the flat radial power profile is apparently the driver. As a re‐

The computed heat transfer and thrust performance design parameters for all three cases are compared with those available of the Small Engine and summarized in Table 3. The comput‐ ed core pressure drops are shown in the second column. The values for all three cases are close to the design number, although that of the Cosine-Cosine power distribution case is slightly lower. The computed peak solid temperatures are shown in the third column, in which the highest peak solid temperature belongs to case 1, followed by case 2, with the

The fourth column shows the average core-exit gas temperatures. These were obtained by averaging the temperatures at the core-exit for all nine elements on the symmetry plane. It can be seen that the average core-exit gas temperature for the Cosine-Cosine power distribu‐ tion at 3334 K, is much higher than that of the Small Engine design temperature at 2750 K. On the other hand, the average core-exit gas temperatures for the clipped Cosine-flattened and clipped Cosine-flat power distribution cases are both quite close to that of the Small En‐

**Case ΔPcore,atm** *Ts,max, K Tg,core exit, K* **ΔTg,core exit, K ISP**

1 8.9 5369 3334 3900 811

2 9.1 3149 2785 523 868

3 9.1 3066 2782 842 900

Small Engine 9.0 - 2750 - 860~875

The computed temperature spread, or the difference of the core-exit gas temperature be‐ tween the first and the 9th flow elements, is shown in the fifth column. Lower temperature spread represents more uniform temperature distribution amongst the flow elements, or better heat transfer efficiency. It can be seen that the clipped Cosine-flattened power profile produces best heat transfer efficiency, the clipped Cosine-flat power profile ranks the sec‐ ond, while the Cosine-Cosine power profile has the worst heat transfer efficiency. Note that if without the outlier of the temperature profile at the 9th fuel element for case 3, its tempera‐ ture spread from the rest is more uniform than that of case 2, as indicated in Figs. 12 and 13.

sult, the hydrogen atom conversion drops from 2% in case 2 to 1.5% in case 3.

lowest peak solid temperature comes from case 3, as expected.

**Table 3.** A summary of the heat transfer and performance parameters.

gine at 2785 and 2782 K, respectively.

128 Nuclear Reactor Thermal Hydraulics and Other Applications

*5.2.3. Summary*

Thermal hydraulics computational design analyses were performed to investigate the midsection corrosion issue occurred during the legacy engine tests, and to predict the heat trans‐ fer efficiency and thrust performance for a virtual solid-core, nuclear thermal engine thrust chamber – the Small Engine. Multiphysics invoked include the turbulent flow and heat transfer, finite-rate chemistry, power generation, and conjugate heat transfer for solids and porous media. The results of a detailed single flow element modeling show that under the assumption of the worst design condition, the hot hydrogen flow could choke near the end of the narrow, long flow channels, which could lead to hydrogen flow reduction and local hot spots in the solid-core; hence, originating the mid-section corrosion. In addition, a uni‐ fied Small Engine thrust chamber thermal flow field constituting those of the inlet plenum, the pressure vessel, and the nozzle, was analyzed with three power profiles. It was found that the computed core pressure drop, core-exit gas temperature, and specific impulse for the clipped Cosine-flattened power profile closely agreed with those of the Small Engine de‐ sign values. It was also found that with our proposed clipped Cosine-flat power profile, not only the computed core-ext gas temperature and core pressure drop closely agreed with those of the Small Engine, the corresponding specific impulse was much higher than those of the original Small Engine numbers. Design lesson learned from this effort is that high hy‐ drogen molecule conversion to hydrogen atoms neither improves the heat transfer efficien‐ cy, nor increases the thrust performance. Rather, it is the more uniform radial power profile that produces lower peak solid temperature, higher heat transfer efficiency, and higher thrust performance.

## **Nomenclature**

*A=*pre-exponential rate constant


*μt*

*ρ*=density

τ=shear stress

cl=centerline

r=radiation

t=turbulent flow

p=nuclear power source

**Acknowledgments**

**Author details**

, Francisco Canabal<sup>1</sup>

1 NASA Marshall Space Flight Center, Huntsville, Alabama, USA

2 Engineering Sciences Incorporated, Huntsville, Alabama, USA

3 University of Alabama at Birmingham, Birmingham, Alabama, USA

Ten-See Wang<sup>1</sup>

=turbulent eddy viscosity (=*ρCμk*<sup>2</sup>

*σ*=turbulence modeling constants

ω=chemical species production rate

**Subscripts and superscripts**

s=surface, solid, or porous media

Π=turbulent kinetic energy production

/*ε*)

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber

This study was partially supported by a Nuclear Systems Office task entitled "Multiphysics Thrust Chamber Modeling" with funding from the Prometheus Power and Propulsion Of‐ fice, NASA Headquarter. Ron Porter was the Marshall Space Flight Center Nuclear Systems Office Manager. Wayne Bordelon was the Nuclear Thermal Propulsion manager. Michael Houts was the Nuclear Research Manager. Steve Simpson and Karl Nelson provided the clipped Cosine-flattened power profile, nozzle geometry and system modeling results. Bill Emrich suggested the Cosine-Cosine power profile. Solid material thermal properties pro‐

vided by Panda Binayak, Robert Hickman, and Bill Emrich are also acknowledged.

, Yen-Sen Chen<sup>2</sup>

, Gary Cheng<sup>3</sup>

and Yasushi Ito<sup>3</sup>

http://dx.doi.org/10.5772/52569

131

*Cp*=heat capacity

*D*=diffusivity

*d*=flow channel diameter

E=activation energy

*fL, fq*=empirical multipliers

*H*=total enthalpy

*K*=thermal conductivity

*k*=turbulent kinetic energy

*L*=drag loss due to porous media

*P*=pressure

*pr*=Prandtl number

*Q*=volumetric heat source

R=radial distance

*R*=gas constant

Re=Reynolds number

*T*=temperature

*t*=time, s

*U*=flow speed

*ui* =mean velocities in three directions

*x*=Cartesian coordinates

Z=axial distance

*α*=species mass fraction

*β*=porosity or void of fraction

*ε*=turbulent kinetic energy dissipation rate

*μ*=viscosity

Thermal Hydraulics Design and Analysis Methodology for a Solid-Core Nuclear Thermal Rocket Engine Thrust Chamber http://dx.doi.org/10.5772/52569 131

*μt* =turbulent eddy viscosity (=*ρCμk*<sup>2</sup> /*ε*) Π=turbulent kinetic energy production *ρ*=density *σ*=turbulence modeling constants τ=shear stress ω=chemical species production rate

**Nomenclature**

*Cp*=heat capacity

*d*=flow channel diameter

*fL, fq*=empirical multipliers

*K*=thermal conductivity

*k*=turbulent kinetic energy

*Q*=volumetric heat source

*L*=drag loss due to porous media

=mean velocities in three directions

*ε*=turbulent kinetic energy dissipation rate

E=activation energy

*H*=total enthalpy

*P*=pressure

*pr*=Prandtl number

R=radial distance

Re=Reynolds number

*x*=Cartesian coordinates

*α*=species mass fraction

*β*=porosity or void of fraction

*R*=gas constant

*T*=temperature

*U*=flow speed

Z=axial distance

*μ*=viscosity

*t*=time, s

*ui*

*D*=diffusivity

*A=*pre-exponential rate constant

*B=*temperature power dependence

130 Nuclear Reactor Thermal Hydraulics and Other Applications

*C*1,*C*2,*C*3,*Cμ*=turbulence modeling constants, 1.15, 1.9, 0.25, and 0.09

## **Subscripts and superscripts**

cl=centerline p=nuclear power source r=radiation s=surface, solid, or porous media t=turbulent flow

## **Acknowledgments**

This study was partially supported by a Nuclear Systems Office task entitled "Multiphysics Thrust Chamber Modeling" with funding from the Prometheus Power and Propulsion Of‐ fice, NASA Headquarter. Ron Porter was the Marshall Space Flight Center Nuclear Systems Office Manager. Wayne Bordelon was the Nuclear Thermal Propulsion manager. Michael Houts was the Nuclear Research Manager. Steve Simpson and Karl Nelson provided the clipped Cosine-flattened power profile, nozzle geometry and system modeling results. Bill Emrich suggested the Cosine-Cosine power profile. Solid material thermal properties pro‐ vided by Panda Binayak, Robert Hickman, and Bill Emrich are also acknowledged.

## **Author details**

Ten-See Wang<sup>1</sup> , Francisco Canabal<sup>1</sup> , Yen-Sen Chen<sup>2</sup> , Gary Cheng<sup>3</sup> and Yasushi Ito<sup>3</sup>

1 NASA Marshall Space Flight Center, Huntsville, Alabama, USA

2 Engineering Sciences Incorporated, Huntsville, Alabama, USA

3 University of Alabama at Birmingham, Birmingham, Alabama, USA

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**Chapter 5**

**CFD Simulation of Flows in Stirred Tank Reactors**

The mixing and agitation of fluid in a stirred tank have raised continuous attention. Starting with Harvey and Greaves, Computational Fluid Dynamics (CFD) has been applied as a power‐ ful tool for investigating the detailed information on the flow in the tank [1-2]. In their work, the impeller boundary condition (IBC) approach has been proposed for the impeller modeling, in which the flow characteristics near the impeller are experimentally measured, and are speci‐ fied as the boundary conditions for the whole flow field computation [1-4]. Because it depends on the experimental data, IBC can hardly predict the flow in the stirred tank and its applicabili‐ ty is inherently limited. To overcome this drawback, the multiple rotating reference frames ap‐ proach (MRF) has been developed, in which the vessel is divided into two parts: the inner zone using a rotating frame and the outer zone associated with a stationary frame, for a steady state simulation. Although it can predict the flow field, the computation result is slightly lack of ac‐ curacy and needs a longer time for convergence [5]. Sliding mesh approach (SM) is another available approach, in which the inner grid is assumed to rotate with the impeller speed, and the full transient simulations are carried out [6-7]. SM approach gives an improved result, but it suffers from the large computational expense [8]. Moving-deforming grid technique was proposed by Perng and Murthy [9], in which the grid throughout the vessel moves with the im‐ peller and deforms. This approach requires a rigorous grid quality and the computational ex‐

Momentum source term approach adds momentum source in the computational cells to rep‐ resent the impeller propelling and the real blades are ignored. In the approach, the genera‐ tions of the vessel configuration and grids are simpler, and the computational time is shorter and the computational accuracy is higher. However, the determination of the momentum source depends on experimental data or empirical coefficients presently [12-13]. The ap‐

> © 2013 Huang and Li; 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 Huang and Li; 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.

**Through Prediction of Momentum Source**

Weidong Huang and Kun Li

http://dx.doi.org/10.5772/51754

penses are even higher than SM [10-11].

**1. Introduction**

Additional information is available at the end of the chapter
