**3.1.3 Influence of pipe diameter (D)**

In this analysis, the effect of pipe diameter on heat transfer in a helical coil is considered. The pipe diameters considered for analyses were, 10 mm, 20 mm, 30 mm and 40 mm. For all these cases, coil has a pitch of 45 mm and PCD of 300 mm and the coil consists of two turns.

For the coil with 10 mm diameter, Nusselt number in the top and bottom regions of the pipe are approximately equal. In the region of fully developed heat transfer, there is even uniform Nusselt number along the periphery of many planes. When the pipe diameter is low, the secondary flows are weaker and hence mixing is lesser. This produces nearly the same heat transfer in the upper half cross-section in a given plane.

When the diameter of the coil is changed to 20 mm, in contrast to the case where *d*=10 mm, heat transfer at the outer side of the coil remain the highest for all of the sections. As expected, the length of pipe needed for the heat transfer to attain a fully developed state has increased as the pipe diameter is increased. A straight line relationship is observed between Nusselt and pipe diameter. Regression analysis was carried out and the result verifies a linear relationship between *Nuav* and pipe diameter.

#### **3.1.4 Correlation for estimation of Nusselt number**

The correlation for Nusselt number already consists of pipe diameter in terms of Reynolds number and curvature ratio. Hence the correlation can be of the form,

$$N\mu = \mathbb{C}Re^n Pr^{0.4} \delta^m \,, \tag{7}$$

where *C, n* and *m* are to be evaluated. If we use Dean number in the formulation, the curvature ratio term needs to be included twice. Hence Reynolds number is chosen in the general form of the equation for estimation of Nusselt number. In order to cover a wide range of Reynolds number, Dean number and curvature ratio, eight more cases, apart from those given above have been analysed.

Multiple-regression analysis based on the data generated from above case studies has been done using MATLAB® . The correlation so developed for estimation of Nusselt number is given by:

Hence the effect of coil pitch on overall heat transfer for design purposes need not be considered for most of the practical applications with helical coils. However, it has implications in heat transfer in the developing region (ref. Fig. 7). The maximum difference in Nusselt number between the top and bottom locations is given in table 2. This clearly shows the extent of oscillatory behaviour. Another observation is the shift of the symmetry

between top and bottom locations 0 7 12 18 26

In this analysis, the effect of pipe diameter on heat transfer in a helical coil is considered. The pipe diameters considered for analyses were, 10 mm, 20 mm, 30 mm and 40 mm. For all these cases, coil has a pitch of 45 mm and PCD of 300 mm and the coil consists of two

For the coil with 10 mm diameter, Nusselt number in the top and bottom regions of the pipe are approximately equal. In the region of fully developed heat transfer, there is even uniform Nusselt number along the periphery of many planes. When the pipe diameter is low, the secondary flows are weaker and hence mixing is lesser. This produces nearly the

When the diameter of the coil is changed to 20 mm, in contrast to the case where *d*=10 mm, heat transfer at the outer side of the coil remain the highest for all of the sections. As expected, the length of pipe needed for the heat transfer to attain a fully developed state has increased as the pipe diameter is increased. A straight line relationship is observed between Nusselt and pipe diameter. Regression analysis was carried out and the result verifies a

The correlation for Nusselt number already consists of pipe diameter in terms of Reynolds

*n m* 0.4 *Nu CRe Pr*

where *C, n* and *m* are to be evaluated. If we use Dean number in the formulation, the curvature ratio term needs to be included twice. Hence Reynolds number is chosen in the general form of the equation for estimation of Nusselt number. In order to cover a wide range of Reynolds number, Dean number and curvature ratio, eight more cases, apart from

Multiple-regression analysis based on the data generated from above case studies has been done using MATLAB® . The correlation so developed for estimation of Nusselt number is

, (7)

Pitch, mm 0 15 30 45 60

plane of temperature and velocity profiles with the change in coil pitch.

same heat transfer in the upper half cross-section in a given plane.

number and curvature ratio. Hence the correlation can be of the form,

linear relationship between *Nuav* and pipe diameter.

**3.1.4 Correlation for estimation of Nusselt number** 

those given above have been analysed.

given by:

Max difference in values of Nuloc

**3.1.3 Influence of pipe diameter (D)** 

turns.

Table 2. Difference in values of Nusselt number.

$$Nu = 0.116 Re^{0.71} Pr^{0.4} \delta^{0.11} \,\text{.}\tag{8}$$

The applicable ranges of parameters for the equation 8 are: (i) 14000 < Re < 70000; (ii) 3000 < De < 22000; (iii) 3.0 < Pr < 5.0; and (iv) 0.05 < δ < 0.2.

Fig. 8 gives a comparison of the Nusselt numbers predicted by eqn. (8) with Roger & Mayhew (1964) and Mori&Nakayama (1967b). It is found that present correlation is fairly in agreement with Nusselt number predicted by the experimental correlations. The earlier correlations are found to be under predicting the Nusselt number. This is attributable to the approximations used by the authors in data reduction and conservative nature of their approach.

Fig. 8. Comparison of Nusselt number for the constant Tw B.C.

#### **3.2 Constant wall heat flux boundary condition**

This boundary condition is applicable to heat flux controlled surfaces such as electrically heated pipes, nuclear fuel elements etc. In these analyses, hot water at 330 K at a specified velocity of 0.8 ms-1 is entering the helical pipe at the top, where an inlet velocity boundary condition is specified. The fluid is made to cool down as it flows along the tube by specifying a wall heat flux of -150 kW m-2.

Influence of parameters such as *PCD*, coil pitch and pipe diameter has been studied in this case also. They are found to be behaving in a manner similar to those described in section 3.1 and are not repeated here. Hence in this case also a correlation of the form given by eqn. (7) will be applicable. In order to cover a wider range of parameters, analysis of eight additional cases were also done.

Multiple regression analysis of the data obtained from the above 20 runs was performed to get a best fit of eqn. (7). The correlation resulted is,

Helically Coiled Heat Exchangers 327

It has been shown that the heat transfer and hence the Nusselt number is not uniform along the periphery at any given cross-section of the helical pipe (Jayakumar, 2009). In section 3, the development of Nusselt number along the periphery was discussed. It will be useful to find out a relationship to predict the local values of Nusselt number *Nuloc*.as a function of the

Values of local Nusselt numbers in the fully developed heat transfer regime for the 20 simulations carried out in the previous section is used for this analysis. For each of the cases, a cross-sectional cut-plane in the fully developed heat transfer regime is created. Value of local Nusselt number has been calculated (at intervals of 3o around the periphery) for each of these cut-planes. The local Nusselt numbers are then normalised with the average Nusselt number *Nuavc* calculated using the equation (8) for that coil configuration and flow

Fig. 10. Variation of local Nusselt number as function of angular position and average

independent of the coil geometry and Dean number. Utilising these *loc*

Fig. 10 shows plot of ratio of local Nusselt number to the average Nusselt number

measured anti-clockwise, starting from the inner side of the coil. It is observed that except in the regions close to the inner side of the pipe, the distribution of *Nu Nu loc av* is almost

for the 20 simulations, the following correlation is developed for the prediction of local

 <sup>2</sup> 2 411 05 8 692 03 0.4215 *Nu Nu - . e - . e - loc av* 

) for the different cases. The angle

  *av*

*Nu Nu* -

(in degrees) is

pairs of data

(10)

**4. Estimation of local Nusselt number** 

parameters.

angular location and this is presented in this section.

**4.1 Constant wall temperature boundary condition** 

Nusselt number for constant *Tw* boundary condition.

( *Nu Nu loc av* ) as a function of angle (

Nusselt number.

$$Nu = 0.085 Re^{0.74} Pr^{0.4} \delta^{0.1} \,\text{.}\tag{9}$$

Nusselt number predicted by the correlation developed has been compared with the earlier works (Seban & McLaughlin, 1963 and Mori & Nakiyama, 1967a) and the results are presented in fig. 9. Seban and McLaughlin, (1963) have used constant values for transport and thermal properties of the working fluid. Also for data reduction, they considered the pipes to be straight. The authors themselves had stated that these approximations can lead to an error of 10% in the values of Nusselt number predicted. It has been shown that usage of constant properties for estimation of Nusselt number can lead to an error more than 20% (Jayakumar et. al 2008a). Thus the earlier correlations are found to be under-predicting the Nusselt number. A good match with the experimental results also verifies the simulation methodology, including the turbulence modelling.

Fig. 9. Comparison of Nusselt number for the constant wall flux B.C.

A comparison of the Nusselt numbers generated from the correlations given by equations (8) and (9) is done. It is found that both the correlations give almost the same value of Nusselt number at lower values of Reynolds number. However, they show marginal difference when Re > 50000.

#### **3.3 Conjugate heat transfer boundary condition**

A correlation was developed for estimation of Nusselt number considering conjugate heat transfer (Jayakumar et. al 2008a). It was found that the percentage difference between conjugate heat transfer and constant wall flux boundary conditions is about 8%. Thus use of heat flux boundary condition is a good engineering approximation for estimation of heat transfer for the case of conjugate heat transfer. Since the effort required for analysing heat transfer with conjugate heat transfer may not be worth from design point of view, results of constant wall heat flux boundary condition can be used for the conjugate case as well.

0.74 0.4 0.1 *Nu Re Pr* 0.085

Nusselt number predicted by the correlation developed has been compared with the earlier works (Seban & McLaughlin, 1963 and Mori & Nakiyama, 1967a) and the results are presented in fig. 9. Seban and McLaughlin, (1963) have used constant values for transport and thermal properties of the working fluid. Also for data reduction, they considered the pipes to be straight. The authors themselves had stated that these approximations can lead to an error of 10% in the values of Nusselt number predicted. It has been shown that usage of constant properties for estimation of Nusselt number can lead to an error more than 20% (Jayakumar et. al 2008a). Thus the earlier correlations are found to be under-predicting the Nusselt number. A good match with the experimental results also verifies the simulation

methodology, including the turbulence modelling.

Fig. 9. Comparison of Nusselt number for the constant wall flux B.C.

**3.3 Conjugate heat transfer boundary condition** 

difference when Re > 50000.

A comparison of the Nusselt numbers generated from the correlations given by equations (8) and (9) is done. It is found that both the correlations give almost the same value of Nusselt number at lower values of Reynolds number. However, they show marginal

A correlation was developed for estimation of Nusselt number considering conjugate heat transfer (Jayakumar et. al 2008a). It was found that the percentage difference between conjugate heat transfer and constant wall flux boundary conditions is about 8%. Thus use of heat flux boundary condition is a good engineering approximation for estimation of heat transfer for the case of conjugate heat transfer. Since the effort required for analysing heat transfer with conjugate heat transfer may not be worth from design point of view, results of constant wall heat flux boundary condition can be used for the conjugate case as well.

. (9)

#### **4. Estimation of local Nusselt number**

It has been shown that the heat transfer and hence the Nusselt number is not uniform along the periphery at any given cross-section of the helical pipe (Jayakumar, 2009). In section 3, the development of Nusselt number along the periphery was discussed. It will be useful to find out a relationship to predict the local values of Nusselt number *Nuloc*.as a function of the angular location and this is presented in this section.

#### **4.1 Constant wall temperature boundary condition**

Values of local Nusselt numbers in the fully developed heat transfer regime for the 20 simulations carried out in the previous section is used for this analysis. For each of the cases, a cross-sectional cut-plane in the fully developed heat transfer regime is created. Value of local Nusselt number has been calculated (at intervals of 3o around the periphery) for each of these cut-planes. The local Nusselt numbers are then normalised with the average Nusselt number *Nuavc* calculated using the equation (8) for that coil configuration and flow parameters.

Fig. 10. Variation of local Nusselt number as function of angular position and average Nusselt number for constant *Tw* boundary condition.

Fig. 10 shows plot of ratio of local Nusselt number to the average Nusselt number ( *Nu Nu loc av* ) as a function of angle () for the different cases. The angle (in degrees) is measured anti-clockwise, starting from the inner side of the coil. It is observed that except in the regions close to the inner side of the pipe, the distribution of *Nu Nu loc av* is almost independent of the coil geometry and Dean number. Utilising these *loc av Nu Nu* - pairs of data for the 20 simulations, the following correlation is developed for the prediction of local Nusselt number.

$$Nu\_{loc} = Nu\_{av} \left( -2.411e \cdot 05\phi^2 + 8.692e \cdot 03\phi + 0.4215 \right) \tag{10}$$

Helically Coiled Heat Exchangers 329

flow of air-water through helical pipes. However numerical investigation, which can give much insight into the physics of the problem, is lacking. No work is reported on detailed numerical study of hydrodynamics and heat transfer characteristics for air-water two-phase flow through such systems. A numerical study can give much more insight into the phenomena and further, the influence of various parameters can also be studied. Jayakumar et al. (2008b, 2010b) have carried out heat transfer studies of flow of two-phase air-water mixture through helical coils. This paper gives a clear picture on the influence of velocity, temperature and void fraction on the coil parameters. The work also gives details of benchmarking for hydrodynamic and heat transfer analysis of air-water mixture through

> *Hl dz Hg dP*

 

*Xf dz*

*Xf dz*

(12)

(13)

(14)

, ,

*dz dP*

<sup>2</sup>

The Lockhart-Martinelli parameter (*χ*) was estimated and two-phase pressure drop is calculated using the functional relationships of two-phase friction multipliers, *фl* and *фg*.

*dP*

 

*dP*

 

*HTP Hl l* 1 , , <sup>2</sup>

*HTP Hg g* 2 , , <sup>2</sup>

For modelling two-phase flows, one can use either Eulerian-Lagrangian model or Eulerian-Eulerian approach. The first one is generally used to trace the particles and hence is not appropriate to deal with gas-liquid flows in pipes. In the Eulerian-Eulerian approach, the phases are treated as interpenetrating and void fraction is used to distinguish the phases. There are 3 different schemes possible in this method, viz., Volume Of Fluid (VOF) model, the mixture model and the Eulerian model. The VOF is applicable when surface tracking is of importance and in the mixture model, pseudo properties of the mixture are used to solve a single set of conservation equations. However, in the Eulerian model, complete set of conservation equations are solved for each of the phases. Thus, either the mixture model or the Eulerian model may be applied for solution of two-phase flow through the helical coil system. Due to the centrifugal and corioli's forces generated during the flow in a helical pipe, a well-mixed gas-liquid region will not be probable and use of mixture properties for the flow will not yield a correct solution. Hence, it is decided to use the Euler scheme to be

Heat transfer to flow of air-water mixture through a helical pipe, where the pipe wall has been kept at a constant temperature, is analysed with Eulerian model of two-phase flows

The modified Lockhart-Martinelli parameter (*χ*) is defined as,

where, the subscript *H* refers to pressure drop in helical coils.

**5.1 Schemes for two-phase flow modelling** 

used for the modelling.

*dz dP*

*dz dP*

**5.2 Governing equations for modelling Eulerian two-phase scheme** 

helical coils.

In this relation the angular location is to be expressed in degrees and the average Nusselt number *Nuavc* is calculated using eqn. (8). Hence the applicability of equation 10 is same as that of eqn. (8).

#### **4.2 Constant wall heat flux boundary condition**

A similar exercise has been carried out to correlate the variation of local Nusselt number for the constant wall heat flux boundary condition. The plot of *Nu Nu loc av* as a function of for the 20 different cases is shown in Fig. 11. The following correlation can be used for the prediction of local Nusselt number as a function of average Nusselt number and angular location.

 <sup>2</sup> 2 331 05 8 424 03 0.4576 *Nu Nu - . e - . e - loc av* (11)

Fig. 11. Variation of local Nusselt number as function of angular position and average Nusselt number for constant *q"w* boundary condition.

In this relation the angular location is to be expressed in degrees and the average Nusselt number *Nuavc* is calculated using eqn. (9). Hence the applicability range of eqn. (11) is same as that of eqn. (9).

#### **5. Air-water two-phase flow and heat transfer**

Process requirements make some of the helically coiled heat exchangers to operate with airwater two-phase mixture as working fluid. As an example, there are situations of singlephase water and two-phase air-water mixture flowing through the helically coiled heat exchanger (Jayakumar and Grover, 1997). The characteristics of operation of such heat exchangers with two-phase working fluids are not well documented. There do exist a few experimental results on hydrodynamics of air-water flow through helical pipes. Experiments have been carried out to generate the pressure drop correlations for two-phase

number *Nuavc* is calculated using eqn. (8). Hence the applicability of equation 10 is same as

A similar exercise has been carried out to correlate the variation of local Nusselt number for the constant wall heat flux boundary condition. The plot of *Nu Nu loc av* as a function of

the 20 different cases is shown in Fig. 11. The following correlation can be used for the prediction of local Nusselt number as a function of average Nusselt number and angular

> <sup>2</sup> 2 331 05 8 424 03 0.4576 *Nu Nu - . e - . e - loc av*

Fig. 11. Variation of local Nusselt number as function of angular position and average

number *Nuavc* is calculated using eqn. (9). Hence the applicability range of eqn. (11) is same

Process requirements make some of the helically coiled heat exchangers to operate with airwater two-phase mixture as working fluid. As an example, there are situations of singlephase water and two-phase air-water mixture flowing through the helically coiled heat exchanger (Jayakumar and Grover, 1997). The characteristics of operation of such heat exchangers with two-phase working fluids are not well documented. There do exist a few experimental results on hydrodynamics of air-water flow through helical pipes. Experiments have been carried out to generate the pressure drop correlations for two-phase

Nusselt number for constant *q"w* boundary condition.

**5. Air-water two-phase flow and heat transfer** 

In this relation the angular location

as that of eqn. (9).

is to be expressed in degrees and the average Nusselt

 

is to be expressed in degrees and the average Nusselt

for

(11)

In this relation the angular location

**4.2 Constant wall heat flux boundary condition** 

that of eqn. (8).

location.

flow of air-water through helical pipes. However numerical investigation, which can give much insight into the physics of the problem, is lacking. No work is reported on detailed numerical study of hydrodynamics and heat transfer characteristics for air-water two-phase flow through such systems. A numerical study can give much more insight into the phenomena and further, the influence of various parameters can also be studied. Jayakumar et al. (2008b, 2010b) have carried out heat transfer studies of flow of two-phase air-water mixture through helical coils. This paper gives a clear picture on the influence of velocity, temperature and void fraction on the coil parameters. The work also gives details of benchmarking for hydrodynamic and heat transfer analysis of air-water mixture through helical coils.

The modified Lockhart-Martinelli parameter (*χ*) is defined as,

$$\mathcal{X}^2 = \left(\frac{dP}{dz}\right)\_{l,H} \bigg/ \left(\frac{dP}{dz}\right)\_{\mathcal{S},H} \tag{12}$$

where, the subscript *H* refers to pressure drop in helical coils.

The Lockhart-Martinelli parameter (*χ*) was estimated and two-phase pressure drop is calculated using the functional relationships of two-phase friction multipliers, *фl* and *фg*.

$$
\phi\_l^2 = \left(\frac{dP}{dz}\right)\_{IP,H} \left/ \left(\frac{dP}{dz}\right)\_{l,H} = f\_1(X) \tag{13}
$$

$$
\phi\_{\mathcal{g}}^2 = \left(\frac{dP}{dz}\right)\_{TP,H} \bigg/ \left(\frac{dP}{dz}\right)\_{\mathcal{g},H} = f\_2\left(X\right) \tag{14}
$$

#### **5.1 Schemes for two-phase flow modelling**

For modelling two-phase flows, one can use either Eulerian-Lagrangian model or Eulerian-Eulerian approach. The first one is generally used to trace the particles and hence is not appropriate to deal with gas-liquid flows in pipes. In the Eulerian-Eulerian approach, the phases are treated as interpenetrating and void fraction is used to distinguish the phases. There are 3 different schemes possible in this method, viz., Volume Of Fluid (VOF) model, the mixture model and the Eulerian model. The VOF is applicable when surface tracking is of importance and in the mixture model, pseudo properties of the mixture are used to solve a single set of conservation equations. However, in the Eulerian model, complete set of conservation equations are solved for each of the phases. Thus, either the mixture model or the Eulerian model may be applied for solution of two-phase flow through the helical coil system. Due to the centrifugal and corioli's forces generated during the flow in a helical pipe, a well-mixed gas-liquid region will not be probable and use of mixture properties for the flow will not yield a correct solution. Hence, it is decided to use the Euler scheme to be used for the modelling.

#### **5.2 Governing equations for modelling Eulerian two-phase scheme**

Heat transfer to flow of air-water mixture through a helical pipe, where the pipe wall has been kept at a constant temperature, is analysed with Eulerian model of two-phase flows

Helically Coiled Heat Exchangers 331

*<sup>f</sup> <sup>K</sup>* 

 

*pq*

and Re is the relative Reynolds number, which is evaluated using the expression,

**uu qp**

*p*

 

*vm pq*

of the secondary phase is much smaller than that of primary phase.

The energy balance equation for the phase *k* is expressed as,

*kkk kkk k*

**u**

*<sup>p</sup> <sup>h</sup> <sup>h</sup>*

where *f* is the drag function and p is the particulate relaxation time.

for the primary phase. The particulate relaxation time is defined as

In the present analysis,

computed using the relation,

primary phase (*q*), is estimated using,

**5.2.3 Conservation of energy** 

*t*

where *Dt D*

where *Cd* is evaluated from the expression,

 

*p ppq*

24

 44.0 1000 for Re 1000 for Re Re/Re15.0124 687.0

*q*

*q pp*

2

*d* 

18

Lift force on the secondary phase (droplets or bubbles) is due to the velocity gradient in the primary-phase flow field. The lift force acting on a secondary phase *p* in a primary phase *q* is

**F***lift*

Virtual mass force, which occurs when the secondary phase (*p*) accelerates relative to the

 *Dt*

**<sup>q</sup> uu <sup>p</sup> <sup>F</sup>** 5.0 

*Dt D*

is the material derivative of . Virtual mass effect is significant when the density

*glp*

,

**τ**

*t*

*k*

 

: **<sup>k</sup> <sup>k</sup> qu <sup>k</sup>**

*<sup>q</sup> d <sup>p</sup>* 

, (20)

Re *Cd <sup>f</sup>* , (21)

Re , (23)

*pq* **pq uuu <sup>q</sup>** 5.0 (25)

, (26)

 

(27)

*S*

.

 

*qp qp pq pq pq*

*hmhmQ*

*k k*

*D*

, (24)

*Cd* (22)

using the CFD package FLUENT. Conservation equations are solved for each of the phase *k*, viz., gas (*g*) and liquid (*l*). In describing the two-phase flow, which has been treated as interpenetrating continua, the concept of void fraction is used. The void fraction of any phase represents the fraction of volume (of the total volume) occupied by that phase. The volume of a phase *k* is defined as,

$$W\_k = \int\_V \alpha\_k dV\tag{15}$$

where, the summation of the void fractions is unity.

$$
\alpha\_g + \alpha\_l = 1 \tag{16}
$$

#### **5.2.1 Conservation of mass**

The continuity equation for the phase *k* is expressed as,

$$\frac{\partial}{\partial t}(\boldsymbol{\alpha}\_{k}\boldsymbol{\rho}\_{k}) + \nabla \cdot \left(\boldsymbol{\alpha}\_{k}\boldsymbol{\rho}\_{k}\mathbf{u}\_{k}\right) = \sum\_{p=l,g} \left(\dot{m}\_{pq} - \dot{m}\_{qp}\right) + S\_{k} \tag{17}$$

Here . . *m m pq qp* is the mass exchange between the liquid and gaseous phases. In the present case, since no mass exchange takes place between the phases, the source *Sk* is zero.

#### **5.2.2 Conservation of momentum**

The set of equations for conservation of momentum for the phase *k* is written as,

$$\begin{aligned} \frac{\partial}{\partial t} \left( \alpha\_k \rho\_k \mathbf{u}\_k \right) + \nabla \cdot \left( \alpha\_k \rho\_k \mathbf{u}\_k \mathbf{u}\_k \right) &= -\alpha\_k \nabla p + \nabla \cdot \mathbf{r}\_k + \alpha\_k \rho\_k \mathbf{g} \\ &+ \sum\_{p=l,g} \left( \mathbf{R}\_{pq} + m\_{pq} \mathbf{u}\_{pq} - m\_{qp} \mathbf{u}\_{qp} \right) \\ &+ \mathbf{F}\_\mathbf{k} + \mathbf{F}\_{lpt,\mathbf{k}} + \mathbf{F}\_{vu,\mathbf{k}} \end{aligned} \tag{18}$$

Here **upq** is the interphase velocity and **Rpq** is the interaction between the phases. **Fk** is external body force, **F***vm***<sup>k</sup>** virtual mass force and **F***lift,* **k** is the lift force. The stress – strain tensor is defined as:

$$\mathbf{T} = \alpha\_k \mu\_k \left(\nabla \mathbf{u}\_k + \nabla \mathbf{u}\_k^T\right) + \alpha\_k \left(\mathcal{A}\_k - \frac{2}{3}\mu\_k\right) \nabla \cdot \mathbf{u}\_k \mathbf{I} \tag{19}$$

In this relation, *k* and *<sup>k</sup>* are the shear and bulk viscosity of the phase *k*. The inter-phase force, **Rpq** is evaluated from , , *q p lg p lg* **R Kuu pq pq p** , where **Kpq** is the inter-phase momentum exchange coefficient. The coefficient **Kpq** can be estimated using the relation,

330 Heat Exchangers – Basics Design Applications

using the CFD package FLUENT. Conservation equations are solved for each of the phase *k*, viz., gas (*g*) and liquid (*l*). In describing the two-phase flow, which has been treated as interpenetrating continua, the concept of void fraction is used. The void fraction of any phase represents the fraction of volume (of the total volume) occupied by that phase. The

> *V k k*

. .

case, since no mass exchange takes place between the phases, the source *Sk* is zero.

The set of equations for conservation of momentum for the phase *k* is written as,

**u uu τ g**

Here **upq** is the interphase velocity and **Rpq** is the interaction between the phases. **Fk** is external body force, **F***vm***<sup>k</sup>** virtual mass force and **F***lift,* **k** is the lift force. The stress – strain

**uu <sup>k</sup> Iu <sup>T</sup> <sup>τ</sup> <sup>k</sup> <sup>k</sup>**

momentum exchange coefficient. The coefficient **Kpq** can be estimated using the relation,

*kk kk*

, ,

*p lg p lg*

 

*kk kk k kkk*

 

  1

, *pq qp kk kk k p lg*

*m m pq qp* is the mass exchange between the liquid and gaseous phases. In the present

*mm S*

**k k k**

, ,

*lift vm*

**FFF**

*<sup>k</sup>* are the shear and bulk viscosity of the phase *k*. The inter-phase

**R Kuu pq pq p** , where **Kpq** is the inter-phase

*q*

 *<sup>k</sup>* 3 2

 

,

*glp*

**pq**

**uuR**

*mm*

.

 

(18)

(19)

*qp qp pq pq*

*dVV* (15)

*lg* (16)

**uk** (17)

volume of a phase *k* is defined as,

**5.2.1 Conservation of mass** 

**5.2.2 Conservation of momentum** 

 

tensor is defined as:

*k* and 

In this relation,

Here . .

where, the summation of the void fractions is unity.

The continuity equation for the phase *k* is expressed as,

*t* 

force, **Rpq** is evaluated from

**k kk**

*<sup>p</sup> <sup>t</sup>*

$$K\_{\rho q} = \frac{\alpha\_q \alpha\_p \rho\_p f}{\pi\_p},\tag{20}$$

where *f* is the drag function and p is the particulate relaxation time. In the present analysis,

$$f = \frac{C\_d \text{ Re}}{24},$$

where *Cd* is evaluated from the expression,

$$C\_d = \begin{cases} 24 \left( 1 + 0.15 \,\text{Re}^{0.687} \right) / \,\text{Re} & \text{for } \text{Re} \le 1000\\ 0.44 & \text{for } \text{Re} > 1000 \end{cases} \tag{22}$$

and Re is the relative Reynolds number, which is evaluated using the expression,

$$\text{Re} = \frac{\rho\_q |\mathbf{u\_p} - \mathbf{u\_q}| d\_p}{\mu\_q},\tag{23}$$

for the primary phase. The particulate relaxation time is defined as

$$
\pi\_{\rho} = \frac{\rho\_{\rho} d\_{\rho}^{2}}{18\mu\_{q}}\tag{24}
$$

Lift force on the secondary phase (droplets or bubbles) is due to the velocity gradient in the primary-phase flow field. The lift force acting on a secondary phase *p* in a primary phase *q* is computed using the relation,

$$\mathbf{F}\_{\text{lift}} = -0.5 \rho\_q \alpha\_p \left(\mathbf{u\_q} - \mathbf{u\_p}\right) \times \left(\nabla \times \mathbf{u\_q}\right) \tag{25}$$

Virtual mass force, which occurs when the secondary phase (*p*) accelerates relative to the primary phase (*q*), is estimated using,

$$\mathbf{F}\_{\rm un} = 0.5 \rho\_q \alpha\_\rho \left( \frac{D\mathbf{u}\_q}{Dt} - \frac{D\mathbf{u}\_p}{Dt} \right) \tag{26}$$

where *Dt D* is the material derivative of . Virtual mass effect is significant when the density of the secondary phase is much smaller than that of primary phase.

#### **5.2.3 Conservation of energy**

The energy balance equation for the phase *k* is expressed as,

$$\begin{aligned} \frac{\hat{\mathcal{C}}}{\partial t} \left( a\_k \rho\_k h\_k \right) + \nabla \cdot \left( a\_k \rho\_k \mathbf{u\_k} h\_k \right) &= -\alpha\_k \frac{\partial p\_k}{\partial t} + \mathbf{r}\_k : \nabla \mathbf{u\_k} - \nabla \cdot \mathbf{q\_k} + S\_k \\ &+ \sum\_{p \sim l, g} \left( Q\_{pq} + m\_{pq} h\_{pq} - m\_{qp} h\_{qp} \right) \end{aligned} \tag{27}$$

Helically Coiled Heat Exchangers 333

In order to study the influence of PCD on heat transfer and pressure drop in two-phase flows, 4 cases were analysed. The results are presented in fig. 12(a). Values of two-phase heat transfer coefficient are estimated using the data extraction methods described in section 3. Mixture temperature and thermal conductivity are used in these computations. The figure also gives the values of single-phase heat transfer coefficient calculated as per the eq. 9. The single phase heat transfer coefficient is calculated assuming the entire flow (both liquid and

The ratio of two-phase heat transfer coefficient to single-phase heat transfer coefficient is presented in fig. 12(b). It is an important observation that the ratio is almost independent of the curvature ratio of the coil. Thus effect of curvature on two-phase heat transfer almost same as that for single-phase flow and it is well predicted by the single-phase heat transfer

The influence of coil pitch on heat transfer rates of two-phase flows is studied by analysing five values of pitch, viz., 0, 15, 30, 45 and 60 mm. The values of heat transfer coefficient obtained from the analysis are shown in fig. 13(a). The ratio htp/hs for different values of pitch are presented in fig 13(b). An almost constant value of the ratio indicates that the

influence of pitch is similar to both single-phase and two-phase flows.

(a) (b)

Fig. 12. (a)Variation of heat transfer coefficient and (b) Ratio htp/hs as function of pitch circle

In this study different coils having pipe diameters of 10, 20, 30 and 40 mm are analysed. In all of these analyses, the PCD was 300 mm and the coil pitch was 45 mm. The results of heat transfer coefficients are presented fig. 14(a). The ratio of two-phase to single-phase heat transfer coefficient for different values of pipe diameters is shown in fig. 14(b). It is clear from the figure that effects of pipe diameter for heat transfer in single-phase and two-phase flows are similar and the single-phase correlation is capable of predicting the heat transfer

**5.3.1 Influence of Pitch Circle Diameter (PCD)** 

gas) as water.

correlation.

diameter (*PCD*).

coefficient adequately.

**5.3.3 Influence of pipe diameter (2r)** 

**5.3.2 Influence of coil pitch (H)** 

In this equation, *hk* is the specific enthalpy, **qk** is heat flux and *Sk* is the heat generation for the phase. Qpq is the heat exchange between the phases and *hpq* is interphase enthalpy.

#### **5.2.4 Turbulence modelling**

Turbulence is modelled using multiphase "mixture *k-ε* model" based on the realizable *k-ε* model. It has already stated that realizable model is the most appropriate one for flows with rotation, adverse pressure gradient etc. Usage of other models, viz., dispersed model and per phase model are computationally very expensive. The transport equations for the mixture *k* and *ε* are as follows.

$$\frac{\partial}{\partial t}(\rho\_m k) + \nabla \cdot (\rho\_m \mathbf{u}\_m k) = \nabla \cdot \left(\frac{\mu\_{t,m}}{\sigma\_k} \nabla k\right) + G\_{k,m} - \rho\_m \varepsilon \tag{28}$$

and

$$\frac{\partial}{\partial t}(\rho\_m \boldsymbol{\varepsilon}) + \nabla \cdot (\rho\_m \mathbf{u}\_m \boldsymbol{\varepsilon}) = \nabla \cdot \left(\frac{\mu\_{\varepsilon, m}}{\sigma\_{\varepsilon}} \nabla \cdot \boldsymbol{\varepsilon}\right) + \frac{\boldsymbol{\varepsilon}}{k} (C\_1 G\_{k, m} - C\_2 \rho\_m \boldsymbol{\varepsilon}) \tag{29}$$

where, the subscript *m* stands for the mixture. Production of turbulent energy is calculated from,

$$G\_{k,m} = \mu\_{t,m} \left(\nabla \mathbf{u\_m} + \left(\nabla \mathbf{u\_m}\right)^T\right) \colon \nabla \mathbf{u\_m} \tag{30}$$

The mixture density and velocity are evaluated using , *m ii k lg* and , , *i i k lg i i k lg* **i m u u**

respectively. The turbulent viscosity is estimated using 2 *tm m* , *k C* . The model constants are same as those used for the single-phase realizable *k-ε* model.

#### **5.3 Estimation of two-phase heat transfer coefficient**

Hydrodynamics of air-water two-phase flow through helical pipes are validated against the experimental results generated by previous researchers. Heat transfer calculations for the two-phase flow are validated against experimental results of flow through an annular pipe (Jayakumar et al., 2010b). In the section 3 details of heat transfer characteristics along the length of the pipe for single-phase fluid have been presented. These give us qualitative picture of various phenomena at various flow sections of the pipe. Quantitative studies of heat transfer with an objective to derive a heat transfer correlation are taken-up in the present chapter. For this 11 coil configurations were analysed.

The analyses have been carried out with a constant wall heat flux boundary condition. The wall heat flux imposed was -150 kW m-2. In all cases, uniform inlet velocity of 0.8 m s-1 was specified for the phases. The air void fraction at the inlet was taken to be 0.2. The numerical schemes used in these analyses are same as those described in the previous chapter.

In this equation, *hk* is the specific enthalpy, **qk** is heat flux and *Sk* is the heat generation for the phase. Qpq is the heat exchange between the phases and *hpq* is interphase enthalpy.

Turbulence is modelled using multiphase "mixture *k-ε* model" based on the realizable *k-ε* model. It has already stated that realizable model is the most appropriate one for flows with rotation, adverse pressure gradient etc. Usage of other models, viz., dispersed model and per phase model are computationally very expensive. The transport equations for the

*k mt*

 

, 

**um** (29)

*mmk*

*mt <sup>m</sup> <sup>m</sup> CGC t k* 2,1

where, the subscript *m* stands for the mixture. Production of turbulent energy is calculated

 **<sup>m</sup> <sup>m</sup> uuu <sup>m</sup>** : ,, *<sup>T</sup> G*

Hydrodynamics of air-water two-phase flow through helical pipes are validated against the experimental results generated by previous researchers. Heat transfer calculations for the two-phase flow are validated against experimental results of flow through an annular pipe (Jayakumar et al., 2010b). In the section 3 details of heat transfer characteristics along the length of the pipe for single-phase fluid have been presented. These give us qualitative picture of various phenomena at various flow sections of the pipe. Quantitative studies of heat transfer with an objective to derive a heat transfer correlation are taken-up in the

The analyses have been carried out with a constant wall heat flux boundary condition. The wall heat flux imposed was -150 kW m-2. In all cases, uniform inlet velocity of 0.8 m s-1 was specified for the phases. The air void fraction at the inlet was taken to be 0.2. The numerical

schemes used in these analyses are same as those described in the previous chapter.

 

constants are same as those used for the single-phase realizable *k-ε* model.

 

*mk <sup>m</sup> k Gk*

*mk m*

*mtmk* (30)

, *m ii k lg*

 

*tm m* ,

 *C* . The model

and ,

,

*k lg*

*k lg*

**m**

**u**

2

*k*

*i i*

**i**

**u**

 

 

*i i*

 

, , **um** (28)

*t*

The mixture density and velocity are evaluated using

respectively. The turbulent viscosity is estimated using

present chapter. For this 11 coil configurations were analysed.

**5.3 Estimation of two-phase heat transfer coefficient** 

 

**5.2.4 Turbulence modelling** 

mixture *k* and *ε* are as follows.

and

from,

#### **5.3.1 Influence of Pitch Circle Diameter (PCD)**

In order to study the influence of PCD on heat transfer and pressure drop in two-phase flows, 4 cases were analysed. The results are presented in fig. 12(a). Values of two-phase heat transfer coefficient are estimated using the data extraction methods described in section 3. Mixture temperature and thermal conductivity are used in these computations. The figure also gives the values of single-phase heat transfer coefficient calculated as per the eq. 9. The single phase heat transfer coefficient is calculated assuming the entire flow (both liquid and gas) as water.

The ratio of two-phase heat transfer coefficient to single-phase heat transfer coefficient is presented in fig. 12(b). It is an important observation that the ratio is almost independent of the curvature ratio of the coil. Thus effect of curvature on two-phase heat transfer almost same as that for single-phase flow and it is well predicted by the single-phase heat transfer correlation.

#### **5.3.2 Influence of coil pitch (H)**

The influence of coil pitch on heat transfer rates of two-phase flows is studied by analysing five values of pitch, viz., 0, 15, 30, 45 and 60 mm. The values of heat transfer coefficient obtained from the analysis are shown in fig. 13(a). The ratio htp/hs for different values of pitch are presented in fig 13(b). An almost constant value of the ratio indicates that the influence of pitch is similar to both single-phase and two-phase flows.

Fig. 12. (a)Variation of heat transfer coefficient and (b) Ratio htp/hs as function of pitch circle diameter (*PCD*).

#### **5.3.3 Influence of pipe diameter (2r)**

In this study different coils having pipe diameters of 10, 20, 30 and 40 mm are analysed. In all of these analyses, the PCD was 300 mm and the coil pitch was 45 mm. The results of heat transfer coefficients are presented fig. 14(a). The ratio of two-phase to single-phase heat transfer coefficient for different values of pipe diameters is shown in fig. 14(b). It is clear from the figure that effects of pipe diameter for heat transfer in single-phase and two-phase flows are similar and the single-phase correlation is capable of predicting the heat transfer coefficient adequately.

Helically Coiled Heat Exchangers 335

Fig. 14. (a)Variation of heat transfer coefficient and (b) Ratio htp/hs as a function of pipe

The data generated from the analysis reported in sections 5.3.1 to 5.3.4 is used to develop a correlation for prediction of two-phase heat transfer coefficient. Kim et al. (1999) has prepared a comparison of 40 two-phase heat transfer correlations and recommended the ones matching with the experimental results. A generalised heat transfer correlation for nonboiling gas-liquid flow in horizontal pipes has been proposed by Kim et al. (2006). Based on

> *s tp C h <sup>h</sup>*

is proposed. The values of Martinelli parameter, , is calculated using the pressure drop

*n*

(31)

**5.3.5 Correlation for estimation of two-phase heat transfer coefficient** 

these results and discussion by Collier (2004), a correlation of the type,

relations provided by Czop et al. (1994).

Fig. 15. Inside heat transfer coefficients for air-water flow.

diameter (*D*).

Fig. 13. (a)Variation of heat transfer coefficient and (b)Ratio htp/hs as a function of coil pitch (*H*)

#### **5.3.4 Influence of inlet void fraction ()**

After establishing the influence of coil parameters on two-phase flow and heat transfer, it is necessary to understand the influence of inlet void fraction on the heat transfer. The details of the helical coil chosen for these analyses are: diameter=12.8mm, PCD=450mm, and pitch=24mm. In this analysis, the inlet velocities considered were 0.8, 1.0, 1.5, 2.0 and 3.0 m s-1. For each of the inlet velocities, air void fractions chosen were 0.01, 0.02, 0.04, 0.07, 0.1, 0.15 and 0.2. This leads to a total of 35 runs. Hot fluid (air-water mixture) flows into the coil at the upper face, where an inlet velocity boundary condition was specified. Inlet temperature of the fluid was taken as 360 K. A constant wall heat flux of -150 kW m-2 was used for inlet velocities 0.8 and 1.0 m s-1. For the other three inlet velocities, a constant wall temperature boundary condition, Tw = 300 K, was specified. The working fluid, after getting cooled, flows out through bottom face. An outlet pressure boundary condition is specified for this face. Temperature and pressure dependent properties of viscosity, density, thermal conductivity and specific heat were used for both air and water. In this analysis gravity effect was also taken into account. Each of these runs takes about 23 hrs of computer time on AMD Athlon X2 64 3.0 GHz computer and requires 4GB RAM and 1GB hard disk space. The system has an installed memory of 8 GB and runs on 64 bit Scientific Linux Operating System. There was difficulty in getting converged results for some of the runs. It had been found that the temperature equation is diverging after a few iterations. In order to overcome this problem, the analysis was started with an appropriate Diritchlet boundary condition, so that an approximate temperature field will be established. Then the analysis was restarted with the desired Neumann boundary condition (Jayakumar, 2009).

Fig. 15 shows the values of heat transfer coefficients. The two-phase (TP) heat transfer coefficient values are estimated by post-processing of the CGNS data file. The single phase (SP) heat transfer coefficient is calculated using the correlation developed in section 3. For estimation of single phase heat transfer coefficient, the entire flow is assumed to be liquid.

It is found that with an increase in void fraction, the two-phase heat transfer coefficient continuously decreases. A plot of the ratio of heat transfer coefficients as a function of inverse of Martinelli parameter is given in Fig. 16.

(a) (b)

with the desired Neumann boundary condition (Jayakumar, 2009).

inverse of Martinelli parameter is given in Fig. 16.

**5.3.4 Influence of inlet void fraction ()** 

pitch (*H*)

Fig. 13. (a)Variation of heat transfer coefficient and (b)Ratio htp/hs as a function of coil

After establishing the influence of coil parameters on two-phase flow and heat transfer, it is necessary to understand the influence of inlet void fraction on the heat transfer. The details of the helical coil chosen for these analyses are: diameter=12.8mm, PCD=450mm, and pitch=24mm. In this analysis, the inlet velocities considered were 0.8, 1.0, 1.5, 2.0 and 3.0 m s-1. For each of the inlet velocities, air void fractions chosen were 0.01, 0.02, 0.04, 0.07, 0.1, 0.15 and 0.2. This leads to a total of 35 runs. Hot fluid (air-water mixture) flows into the coil at the upper face, where an inlet velocity boundary condition was specified. Inlet temperature of the fluid was taken as 360 K. A constant wall heat flux of -150 kW m-2 was used for inlet velocities 0.8 and 1.0 m s-1. For the other three inlet velocities, a constant wall temperature boundary condition, Tw = 300 K, was specified. The working fluid, after getting cooled, flows out through bottom face. An outlet pressure boundary condition is specified for this face. Temperature and pressure dependent properties of viscosity, density, thermal conductivity and specific heat were used for both air and water. In this analysis gravity effect was also taken into account. Each of these runs takes about 23 hrs of computer time on AMD Athlon X2 64 3.0 GHz computer and requires 4GB RAM and 1GB hard disk space. The system has an installed memory of 8 GB and runs on 64 bit Scientific Linux Operating System. There was difficulty in getting converged results for some of the runs. It had been found that the temperature equation is diverging after a few iterations. In order to overcome this problem, the analysis was started with an appropriate Diritchlet boundary condition, so that an approximate temperature field will be established. Then the analysis was restarted

Fig. 15 shows the values of heat transfer coefficients. The two-phase (TP) heat transfer coefficient values are estimated by post-processing of the CGNS data file. The single phase (SP) heat transfer coefficient is calculated using the correlation developed in section 3. For estimation of single phase heat transfer coefficient, the entire flow is assumed to be liquid. It is found that with an increase in void fraction, the two-phase heat transfer coefficient continuously decreases. A plot of the ratio of heat transfer coefficients as a function of

Fig. 14. (a)Variation of heat transfer coefficient and (b) Ratio htp/hs as a function of pipe diameter (*D*).

#### **5.3.5 Correlation for estimation of two-phase heat transfer coefficient**

The data generated from the analysis reported in sections 5.3.1 to 5.3.4 is used to develop a correlation for prediction of two-phase heat transfer coefficient. Kim et al. (1999) has prepared a comparison of 40 two-phase heat transfer correlations and recommended the ones matching with the experimental results. A generalised heat transfer correlation for nonboiling gas-liquid flow in horizontal pipes has been proposed by Kim et al. (2006). Based on these results and discussion by Collier (2004), a correlation of the type,

$$\frac{h\_{\rm up}}{h\_s} = C\mathcal{X}^{\rm u} \tag{31}$$

is proposed. The values of Martinelli parameter, , is calculated using the pressure drop relations provided by Czop et al. (1994).

Fig. 15. Inside heat transfer coefficients for air-water flow.

Helically Coiled Heat Exchangers 337

It is observed that the use of constant values for the thermal and transport properties of the heat transport medium results in prediction of inaccurate heat transfer coefficients. Heat transfer characteristics of the heat exchanger with helical coil are also studied using the CFD code. The CFD predictions match reasonably well with the experimental results within experimental error limits. Based on the results a correlation was developed to calculate the

Necessary Python codes, which run in the framework of AnuVi visualisation package, have been developed for accurate estimation of Nusselt number at any point on the heat transfer surface. The research work also includes development of various C++ and MATLAB® codes. Characteristics of non-isothermal fluid flow and heat transfer under turbulent flow of single phase water through helical coils have been presented in detail. Analysis has been carried out both for the constant wall temperature and constant wall heat flux boundary conditions. Fluid particles are found to undergo oscillatory motion inside the pipe and this causes

Nusselt numbers at various points along the length of the pipe was estimated. Nusselt number on the outer side of the coil is found to be the highest among all other points at a specified cross-section, while that at the inner side of the coil is the lowest. Velocity profiles for the two boundary conditions were found to be matching, while the temperature profiles

A number of numerical experiments have been carried out to study influence of coil parameters, viz., pitch circle diameter, coil pitch and pipe diameter on heat transfer. The coil pitch is found to have significance only in the developing section of heat transfer. The torsional forces induced by the pitch causes oscillations in the Nusselt number. However, the average Nusselt number is not affected by the coil pitch. After establishing the parametric influence, a correlation has been developed for estimation of average Nusselt number. This correlation is compared with those available in the literature and the deviations are within reasonable limits. It is also observed that these correlations are applicable for either of the boundary conditions. For most of the engineering applications,

In the fully developed section, ratio Nuloc/Nuav is almost independent of coil parameters and Dean number. Correlations have been developed for prediction of local values of Nusselt number as a function of the average Nusselt number and the angular position of the

CFD simulations of heat transfer to air-water two-phase mixture flowing through a helically coiled heat exchanger has been carried out. Studies have been carried out by varying (i) coil pitch, (ii) pipe diameter (iii) pitch circle diameter. Their influence on heat transfer and

Unlike the flow through a straight pipe, the centrifugal force caused due to the curvature of the pipe causes heavier fluid (water-phase) to flow along the outer side of the pipe. High velocity and high temperature are also observed along the outer side. The torsion caused by pitch of the coil makes the flow unsymmetrical about the horizontal plane of coil. As the

the correlations are applicable for conjugate heat transfer as well.

**6. Conclusion** 

inside heat transfer coefficient of the helical coil.

fluctuations in heat transfer rates.

point along the circumference.

pressure drop has been brought out.

are different.

Regression analysis was carried out using the entire set of two-phase heat transfer data. This leads to a correlation,

$$\frac{h\_{tp}}{h\_s} = 0.7 \,\text{\AA}^{0.0424} \tag{32}$$

Fig. 17 shows the correlation along with the data points. The correlation is able to predict the data points within an error of 10%.

Fig. 16. Ratio htp/hs as a function of

Fig. 17. Correlation for estimation of htp/hs as a function of
