Seok-Ki Choi1 e-mail:
[email protected]
Seong-O Kim Fast Reactor Development Division, Korea Atomic Energy Research Institute, 150-1 Deokjin-dong, Yuseong-gu, Daejeon 305-353, Korea
Hoon-Ki Choi Department of Mechanical Engineering, Changwon National University, 7 Sarim-dong, Changwon, Gyeongnam 641-773, Korea
1
Computational Fluid Dynamics Evaluation of Heat Transfer Correlations for Sodium Flows in a Heat Exchanger A numerical study for the evaluation of heat transfer correlations for sodium flows in a heat exchanger of a fast breeder nuclear reactor is performed. Three different types of flows such as parallel flow, cross flow, and two inclined flows are considered. Calculations are performed for these three typical flows in a heat exchanger changing turbulence models. The tested turbulence models are the shear stress transport (SST) model and the SSG-Reynolds stress turbulence model by Speziale, Sarkar, and Gaski (1991, “Modelling the Pressure-Strain Correlation of Turbulence: An Invariant Dynamical System Approach,” J. Fluid Mech., 227, pp. 245–272). The computational model for parallel flow is a flow past tubes inside a circular cylinder and those for the cross flow and inclined flows are flows past the perpendicular and inclined tube banks enclosed by a rectangular duct. The computational results show that the SST model produces the most reliable results that can distinguish the best heat transfer correlation from other correlations for the three different flows. It was also shown that the SSG-RSTM high-Reynolds number turbulence model does not deal with the low-Prandtl number effect properly when the Peclet number is small. According to the present calculations for a parallel flow, all the old correlations do not match with the present numerical solutions and a new correlation is proposed. The correlations by Dwyer (1966, “Recent Developments in Liquid-Metal Heat Transfer,” At. Energy Rev., 4, pp. 3–92) for a cross flow and its modified correlation that takes into account of flow inclination for inclined flows work best and are accurate enough to be used for the design of the heat exchanger. 关DOI: 10.1115/1.4000707兴
Introduction
The heat exchanger of a liquid metal nuclear power reactor is a shell-and-tube heat exchanger and on the shell side of the heat exchanger a liquid metal such as sodium flows across the tube bundles in all directions from parallel to perpendicular. Thus, a proper heat transfer coefficient for these flow conditions should be provided for a better heat exchanger design. However, the experimental correlations for a heat transfer of a liquid metal are very rare in literature since the sodium experiment is very expensive and difficult. The most difficult thing is that the differences among the correlations are so grave that it is difficult to decide which correlation should be used for a particular flow situation. In the present study, we propose a computational fluid dynamics 共CFD兲 evaluation of the existing heat transfer correlations for sodium flows in a heat exchanger so that an appropriate heat transfer correlation is used for a particular flow situation. It is useful to outline the previous heat transfer correlations for parallel flow, cross flow, and inclined flows. For a parallel flow without a baffle, many correlations have been proposed in the past mainly for the design of a fuel assembly. Those are the Westinghouse and modified-Schad correlations by Tang et al. 关1兴, Graber and Rieger 关2兴, Borishansky et al. 关3兴, and Kisohara et al. 关4兴. The comparisons of these correlations with experimental data are given by Tang et al. 关1兴. It is shown that for P / D values of 1.1 and 1.2, the modified-Schad 共by Tang et al. 关1兴兲 and Borishansky et al. 关3兴 correlations agree best with the experimental data while the 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 3, 2008; final manuscript received November 3, 2009; published online March 4, 2010. Assoc. Editor: S. A. Sherif.
Journal of Heat Transfer
correlation by Graber and Rieger 关2兴 overpredicts the heat transfer coefficient when the P / D value is 1.1. They also reported that the Westinghouse correlation 共by Tang et al. 关1兴兲 underestimates the heat transfer coefficient when the P / D value is 1.2. These observations indicate that the performance of each correlation is strongly dependent on the value of P / D and care should be paid in choosing a proper correlation for a particular problem. For example, the P / D range of the Westinghouse correlation is 1.15 ⱕ P / D ⱕ 1.3 and that of the modified-Schad correlation is 1.05 ⱕ P / D ⱕ 1.15. Thus, the above conclusions cannot be applied to a parallel flow in an intermediate heat exchanger of a liquid metal reactor where the P / D values are relatively high; for example, the P / D value is 1.6 for the Korea Advanced Liquid Metal Reactor design. For a cross flow, Hsu 关5兴 published analytic solutions for a heat transfer coefficient based on the following assumptions: 共a兲 the physical properties are constant, 共b兲 the flow is two-dimensional and steady, 共c兲 the flow is incompressible, inviscid, and irrotational, 共d兲 there is no contact resistance at a solid-liquid interface, 共e兲 a negligible eddy conductivity compared with a molecular conduction, 共f兲 the hydrodynamic potential distribution on the surface of a tube is linear, and 共f兲 there is no interaction between the boundary layers of adjacent tubes. The assumption of a negligible eddy conductivity compared with a molecular conduction is a good assumption when the Peclet number is small. Using these assumptions, Hsu 关5兴 developed two correlations for the condition of a simple cosine surface temperature distribution around a tube and for the condition of a uniform heat flux from a tube. These correlations show linear profiles on the log-log plot of Nusselt number 共Nu兲 versus Peclet number 共Pe兲. 共Nu= a共Pe兲bv,max兲. Later, Dwyer 关6兴 presented the experimental results by Hoe et al. 关7兴, Rickard et al. 关8兴, and Subbotin et al. 关9兴 and claimed that the
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profiles should be concave upward and explained that the linear profile on a log-log plot of Nu versus Pe in the theoretical study by Hsu 关5兴 was due to the assumption of inviscid and irrotational flow in its derivation. Dwyer and co-worker 关6,7兴 modified Hsu’s correlation to remedy this problem. They derived another correlation for a concave upward type of correlation. 共Nu= a + b共Pe兲cv,max兲. In estimating the heat transfer coefficients for inclined flows, it is a good approximation to assume that they are the same as the cross flow through a bundle of elliptically shaped rods. Hsu 关11兴 developed such a correlation using the same assumptions as those for a cross flow. Thus, this correlation gives a straight line on a log-log plot of Nu versus Pe, whereas the true curve is presumably one of increasing slope as the Peclet number is increased, which gives a concave upward shape on a log-log plot of Nu versus Pe. Dwyer 关6兴 modified Hsu’s correlation to account for the flow inclination effect by a function of the inclined angle. Kalish and Dwyer 关10兴 later devised a different correlation from their NaK experimental data. It is evident that the heat transfer correlations for a liquid metal flow are not abundant compared with those of water or air. In the present study a CFD evaluation of a sodium heat transfer coefficient is proposed. The CFD at present is mature enough to calculate complex flows and the existing sodium heat transfer correlations for the design of a heat exchanger are evaluated by the CFD results and the compared results are presented. The CFD code used in the present study is CFX-11 code and four different turbulence models such as the shear stress transport model 共SST兲, the renormalization group 共RNG兲 k − model, the SSG-Reynolds stress turbulence model 共SSF-RSTM兲, and the omega-Reynolds stress turbulence model 共omega-RSTM兲, which are available in CFX-11 code are tested to see the impact of a turbulence model on the solutions. In what follows, we present the existing heat transfer correlations for parallel flow, cross flow, and inclined flows. Then, the numerical method is briefly explained and the numerical evaluation of the heat transfer coefficients is presented, followed by a conclusion.
冉冊冉
冊
P P + − 0.005 + 0.021 共Pe兲共0.8−0.024共 P/D兲兲 D D
Nu = 0.16 + 4.03
共6兲 2.2 Cross Flow. Three liquid metal heat transfer correlations for a cross flow are considered. 共1兲 Hsu 关5兴 correlation
冉 冊冉 冊
Nu = 0.958
1 D
0.5
P−D P
Heat Transfer Correlations for Liquid Metal Flows
2.1 Parallel Flow. The liquid metal heat transfer correlations for a parallel flow are relatively abundant, mainly due to the design of a fuel assembly, and in this case the range of P / D value is very small 共P / D = 1.15− 1.35兲 compared with that of a parallel flow in a heat exchanger. Among them, the following six correlations are considered in the present study: 共1兲 Graber–Rieger 关2兴 correlation
冉冊冉
Nu = 0.25 + 6.20
冊
P P P + − 0.007 + 0.032 共Pe兲共0.8−0.024 D 兲 D D
共1兲 共2兲 Lubarsky–Kaufman 关12兴 correlation 共2兲
Nu = 0.625共Pe兲0.4 共3兲 Seban–Shimazaki 关13兴 correlation
共3兲
Nu = 5.0 + 0.025共Pe兲0.8 共4兲 Kisohara et al. 关4兴 correlation
共4兲
Nu = 4.77 + 0.728共Pe兲0.454 共5兲 Westinghouse correlation by Tang et al. 关1兴:
冉冊冉 冊
Nu = 4.0 + 0.33
P D
3.8
Pe 100
共6兲 Present correlation 051801-2 / Vol. 132, MAY 2010
0.86
冉冊
+ 0.16
P D
5.0
共5兲
共Pev,max兲0.5
共7兲
共2兲 Kalish and Dwyer 关10兴 correlation Nu =
冉 冊冉 冊 1 D
0.5
P−D P
0.5
共6.19 + 0.2665关Pev,max兴0.635兲
共8兲
共5.36 + 0.1974关Pev,max兴0.682兲
共9兲
共3兲 Dwyer 关6兴 correlation Nu =
冉 冊冉 冊 1 D
0.5
P−D P
0.5
2.3 Inclined Flows. Only three liquid metal heat transfer correlations for an inclined flow exist within the present author’s knowledge and they are as follows: 共1兲 Kalish and Dwyer 关10兴 correlation Nu =
冉 冊冉 冊冋 1 D
0.5
P−D P
0.5
sin  + sin2  1 + sin2 
册
0.5
共5.44
+ 0.228关Pev,max兴0.614兲
共10兲
共2兲 Dwyer 关6兴兲 correlation
冉 冊冉 冊冋
Nu = 0.958
1 D
0.5
P−D P
0.5
sin  + sin2  1 + sin2 
册
0.5
关Pev,max兴0.5 共11兲
共3兲 Modified Dwyer 关6兴 correlation Nu =
冉 冊冉 冊冋 1 D
0.5
P−D P
0.5
sin  + sin2  1 + sin2 
+ 0.1974关Pev,max兴0.682兲
2
0.5
册
0.5
共5.36 共12兲
In the above equations,  is the inclined angle and 1 is a hydrodynamic potential function and its value for many different cases of the P / D are tabulated by Hsu 关5兴.
3
Turbulence Models
The selection of turbulence model is very important for accurate prediction of thermal-hydraulic problems. Four turbulence models, which are available in CFX-11 code 关14兴, are considered and they are the SST, the RNG k − model, the SSG-RSTM, and the omega-RSTM. The SST model and RNG k − model are commonly used two-equation turbulence models, which have been validated for various problems in literature and these two models are known to perform better than the conventional k − model, especially for the problems involving separation and curvature effect such as the present problem. The SSG-RSTM turbulence model is chosen here due to its theoretically sound derivation of the pressure-strain term 关15兴 and is also due to the best performance among the high-Reynolds number Reynolds stress models available in CFX-11 code. The omega-RSTM model is tested here to investigate the performance of the low-Reynolds number second-moment closure for the present problem. The SST and omega-RSTM models are the low-Reynolds number models in which calculations are carried out all the way to the wall, while the RNG k − and SSG-RSTM models are the high-Reynolds number models and use the wall functions near the wall. A preliminary test of the above four turbulence models was performed and it was shown that for a low-Reynolds number model the SST model gives nearly same solutions as the omega-RSTM. It was Transactions of the ASME
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also shown that the SSG-RSTM model outperforms the RNG k − model for the present problem. The results of these tests lead us to use the SST model and the SSG-RSTM for the present purpose. As details of these two models are given in CFX-11 code 关14兴 only a brief description of them is given here. The transport equations for the turbulence kinetic energy and its specific dissipation rate in the SST model can be written as follows:
共U jk兲 = 共k兲 + t xj xj 共U j兲 = 共兲 + t xj xj
冋冉 冊 册
T k + Pk − ⬘k 共13兲 k* x j
+
冋冉 冊 册 +
+ 共1 − F1兲
T + ␣ Pk − 2 k x j
2 k 2 x j x j
共14兲
In this model, a blending function 共F1兲 is used to activate the Wilcox k − model close to the wall and the k − model in the outer region. By this approach, the SST model overcomes the deficiencies of both k − and k − models. The advantage of the -equation is that it allows for a more accurate near wall treatment with an automatic switch from a wall function to a lowReynolds number formulation based on the grid spacing. The governing equations for the Reynolds stresses and their dissipation rate in the SSG-RSTM can be written as follows:
共 U ku iu j 兲 = 共 u iu j 兲 + t xk xk
冋冉
冊 册
2 k 2 u iu j + Cs + Pij + ij 3 xk
2 − ␦ij 3
共Uk兲 = 共兲 + t xk xk
共15兲
冋冉 冊 册 +
T + 共C1 Pk − C2兲 xk k 共16兲
In the SSG-RSTM, the pressure-strain term 共ij兲 is treated by that given by Speziale et al. 关15兴.
4
Results and Discussion
Let us consider a flow situation where the sodium 共liquid Na兲 flows inside a circular cylinder or rectangular duct in which several tubes are placed as shown in Fig. 1. The energy balance equation can be written as follows: ˙ C p共Tin − Tout兲 hAw共Tb − Tw兲 = m
共17兲
The overall Nusselt number can be obtained as follows if the outlet temperature 共Tout兲 and the bulk temperature 共Tb兲 are known from the numerical results: ˙ C pD共Tin − Tout兲 m h Nu = D = k kAw共Tb − Tw兲
共18兲
The inlet temperature of sodium 共Tin兲 is 818 K and the wall temperature of the tube 共Tw兲 is 598 K for all cases considered in the present study. The working fluid in the present study is a liquid Na and the physical properties of the liquid Na are based on the average temperature of inlet and tube wall 共=708 K兲. The outer diameter of the tube is 12.7 mm. The bulk temperature 共Tb兲 is obtained from the volume average of the temperature in the tube region. For boundary conditions, the temperature and mass flow rate are specified at the inlet and at the outlet the pressure boundary condition and the zero gradient temperature condition are imposed. The adiabatic wall condition was imposed for the outer cylinder in the parallel flow computation and the same condition was imposed at the top and bottom boundaries for the cross and Journal of Heat Transfer
Fig. 1 Numerical grids: „a… parallel flow and „b… inclined flow
inclined flows. The symmetry condition can be imposed at the top and bottom walls for the cross and inclined flows and it was shown that the calculated heat transfer coefficients were nearly same between two different impositions of boundary condition. The symmetry condition was imposed at the lateral 共left and right兲 boundaries Typically 350,000–560,000 numerical grids are generated using the WORKBENCH program in CFX-11 code. The shapes of typical numerical grids are shown in Fig. 1 for a parallel flow and an inclined flow where the inclined angle is 60 deg. In this figure, the pitch to tube diameter ratio 共P / D兲 is 1.85 for the parallel flow and is 1.6 for the cross and inclined flows. Calculations are performed changing the inlet mass flow rate. The inlet mass flow rate is divided by eight intervals in a range between 0.54 kg/s and 5.4 kg/s, and this corresponds to the Peclet number in a range of 40 ⱕ Peⱕ 400. In the initial phase of the present study, the grid sensitivity on the numerical solutions are performed. Figure 2 shows the predicted Nusselt number using two different numerical grids for the 60 deg inclined flow. This figure shows that the predicted heat transfer coefficient is not so sensitive to the numerical grid when the numerical grids are properly refined near the wall. All the numerical solutions presented in this paper are those using the fine grids. In order to check whether the first grid point from the tube wall is placed within the logarithmic layer when the wall function method is used in the calculations by the SSG-RSTM high-Reynolds number model, the value of y+ was checked for the whole tube walls. The maximum value of y+ for a cross flow when the Peclet number is maximum 共Pe= 400兲 was 62.25, indicating that the grid refinement near the wall is properly done in the calculations by the SSG-RSTM. In CFX-11 code, the typical convergence is declared when all the root-mean-square of the residuals of momentum equations and energy equation is less than 10−4 and we adopt this criterion for all calculations. Due to using MAY 2010, Vol. 132 / 051801-3
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Fig. 2 Grid dependency test „SST model, Pev,max = 400, and  = 60 deg…
the coupled method in CFX-11 code, the convergence is reached within 50 iterations for all cases. 4.1.1 Parallel Flow. Figure 3 shows the predicted Nusselt number profiles by the two turbulence models together with the previous correlations for a parallel flow. The Pe number in this figure is that based on the average inlet velocity. We can observe that the Graber and Rieger 关2兴 correlation predicts the Nusselt number much larger than the present solutions using the two different turbulence models. The differences in the magnitude of the Nusselt number between the Graber and Rieger 关2兴 correlation and the present numerical results are very large. It may be partly due to the different imposition of temperature boundary condition at the tube wall 共qw=constant by Graber and Rieger 关2兴 and Tw=constant in the present study兲. However, the trend of increase in the Nusselt number versus the Peclet number is nearly the same as the present predictions. The correlation by Kisohara et al. 关4兴 and the Westinghouse correlation by Tang et al. 关1兴 predicted the Nusselt number nearly the same and the predicted Nusselt numbers increases rapidly as the Peclet number increase when com-
Fig. 3 Comparison of correlations and CFD results for a parallel flow
051801-4 / Vol. 132, MAY 2010
Fig. 4 Comparison of correlations and CFD results for a cross flow „ = 90 deg…
pared with other correlations. Their predictions are nearly the same as the present numerical results only when the Peclet number is very small 共Pe⬇ 40兲 and the differences become more grave when the Peclet number becomes larger. It is observed that the correlations by Lubarsky–Kaufman 关12兴 and Seban–Shimazaki 关13兴 predicted the Nusselt number much smaller than the present numerical results in the whole range of the Peclet number considered 共40ⱕ Peⱕ 400兲. If the Peclet number is relatively small, the heat transfer is dictated by the molecular conduction due to the high conductivity of the sodium and the heat transfer curve should be flat or concave 关6兴 as is observed in the results by the SST model. It is noted that the SSG-RSTM is a high-Reynolds number model and this model use the wall functions near the wall, thus the near wall turbulence is not modeled properly especially when the Peclet number is small and that is observed in the Fig. 3. When the Peclet number is rather large 共Peⱖ 140兲 and the flow become fully turbulent, the two models result in nearly the same magnitude for the heat transfer coefficient. We can observe that the SSG-RSTM produce a little convex curve when the Peclet number is very small 共Peⱕ 60兲, which is a contradiction to the physical phenomenon explained before. The SST model predicts the proper behaviors and a new correlation based on the predictions by the SST model is proposed 共Eq. 共6兲兲 since no previous correlations match with our predictions. It is noted that the correlations by Graber and Rieger 关2兴, Seban–Shimazaki 关13兴, and the proposed correlation showed the same trends of increase in Nusselt number versus the Peclet number, as shown in Fig. 3, although the magnitudes of Nusselt number are different. It is noted that when one considers a conservative design based on the our proposed correlation, one may choose the correlation by Seban– Shimazaki 关13兴 and it was done recently by Mochizuki and Takano 关16兴 in their heat transfer analysis of heat exchanger in a sodium cooled fast reactor. 4.1.2 Cross Flow. Fig. 4 shows the log-log plot of the Nusselt number versus the Peclet number for a cross flow 共 = 90 deg兲. The Peclet number in this figure 共共Pe兲v,max兲 is the Peclet number based on the maximum inlet velocity. The prediction by the SSGRSTM follows the theoretical correlation by Hsu 关5兴 while that of the SST model follows the Dwyer’s correlation. The Hsu’s correlation is based on the assumptions such that the flow is twodimensional, incompressible, inviscid, and irrotational as mentioned earlier. The fact that the prediction by the SSG-RSTM matches with Hsu’s correlation means that the high Reynolds Transactions of the ASME
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Fig. 5 Comparison of correlations and CFD results for an inclined flow „ = 60 deg…
Fig. 6 Comparison of correlations and CFD results for an inclined flow „ = 30 deg…
number model SSG-RSTM with the wall function method behaves like an inviscid and irrotational flow when the Peclet number is small. The inaccurate behavior of the high Reynolds number turbulence model is due to the use of the wall function method near the wall and is also due to the fact that the high Reynolds number model does not properly deal with the low-Prandtl number effect when the Peclet number is small. The SST model predicts a little larger magnitude of the heat transfer coefficient than that by the SSG-RSTM when the Peclet number is small 共共Pe兲v,max ⱕ 100兲. The predictions by the SST model agrees well with the correlation by Dwyer 关6兴 in Eq. 共9兲, when the Peclet number is small and they also agree well with the correlation by Kalish and Dwyer 关10兴 in Eq. 共8兲, when the Peclet number is large. Note that the SST model calculates the momentum and energy equations all the way to the wall and properly model the near wall behavior. The difference between the present predictions by the SST model and the correlation by the Dwyer 关6兴 is small and the Dwyer’s correlation 共Eq. 共9兲兲 can be used for the design of a heat exchanger in a liquid metal nuclear reactor. Note that the correlation by Kalish and Dwyer 关10兴 overpredicts the heat transfer coefficient when the Peclet number is small.
haviors have been observed for cross flow. It should be noted that the correlation by Kalish and Dwyer 关6兴 severely underpredicted the heat transfer coefficient when the Peclet number is larger than 150. When the inclined angle is small 共 = 30 deg兲, the computational results for the heat transfer coefficient follow a very similar trend although some differences in the magnitude of Nusselt number exist, which can be seen in Fig. 6. The heat transfer coefficient by the SSG-RSTM follows the correlation by Dwyer 关6兴 in Eq. 共11兲 as expected. The predicted Nusselt number by the SST model generally follows that of the modified correlation by Dwyer 关6兴 in Eq. 共12兲 and a small difference in magnitude of heat transfer coefficient exists. This may be due to the numerical error originated from using strongly nonorthogonal numerical grids. It is observed that the correlation by Kalish and Dwyer 关10兴 in Eq. 共10兲 underpredicted the heat transfer coefficient when the Peclet number is large. All of these results recommend us to use the modified correlation by Dwyer 关6兴 in Eq. 共12兲 for inclined flows.
4.1.3 Inclined Flows. Figures 5 and 6 show the predicted Nusselt number versus the Peclet number on the log-log plot for the inclined flows where the inclined angles are 60 deg and 30 deg, respectively. The heat transfer correlations for the inclined flows of a liquid metal are very rare in literature. We could only find three correlations such as those by Kalish and Dwyer 关10兴 in Eq. 共10兲, and Dwyer 关6兴 in Eq. 共11兲, and the modified Dwyer correlation in Eq. 共12兲. The Dwyer’s correlation is based the theoretical study by Hsu 关11兴. Dwyer 关6兴 modified Hsu’s correlation to account for the flow inclination effect by multiplying a function based on the inclined angle. Thus, this correlation gives a straight line on the log-log plot of the Nusselt number versus the Peclet number whereas the true curve from the experimental data is a concave upward shape 关6兴. The correlation by Kalish and Dwyer 关10兴 is based on their NaK experimental data. Figure 5 shows the predictions by the two turbulence models and the experimental correlations for an inclined flow where the inclined angle is 60 deg. It is observed that the numerical result by the SSG-RSTM follows the correlation by Dwyer 关6兴 in Eq. 共11兲. The prediction by the SST model agrees well with the modified correlation by Dwyer 关6兴 in Eq. 共12兲 and they also agree well with the correlation by Dwyer 关6兴 when the Peclet number is large. Similar beJournal of Heat Transfer
5
Conclusions
The results of the present numerical study for the evaluation of heat transfer correlations for liquid metal fluid flows in a heat exchanger are presented. The computations were performed for parallel flow, cross flow, and inclined flows changing turbulence models. The turbulence models employed in the present study are the SST model and the SSG-Reynolds stress model. The following conclusions are drawn from the present study. 共1兲 The high-Reynolds model SSG-RSTM does not behave properly in a region where the Peclet number is small while the SST model results in reliable solutions for a whole region 共40ⱕ Peⱕ 400兲. 共2兲 For parallel flow, no previous correlations match well with the present computational results and a new correlation is proposed based on the present results by the SST model in Eq. 共6兲. 共3兲 For cross flow, the correlation by Dwyer 关6兴 in Eq. 共9兲 can be used for the design of the heat exchanger without a significant error. 共4兲 For inclined flows, the modified Dwyer model 关6兴 in Eq. 共12兲 was reliable and accurate enough to be used in a real design. The Kalish and Dywer correlation in Eq. 共10兲 underpredicts the heat transfer coefficient when the Peclet number is large. MAY 2010, Vol. 132 / 051801-5
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⫽ dissipation rate of turbulent kinetic energy ⫽ dynamic viscosity T ⫽ dynamic turbulent viscosity ⫽ density 1 ⫽ hydrodynamic potential function, which is a function of P / D ij ⫽ pressure-strain term ⫽ specific dissipation rate of turbulent kinetic energy
Acknowledgment This study has been supported by the Nuclear Research and Development Program of the Ministry of Education, Science, and Technology of Korea.
Nomenclature a,b,c Aw CP Cs C1 , C2
⫽ ⫽ ⫽ ⫽ ⫽
D F1 h k ˙ m Nu P Pe
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
Pev,max
Pij ⫽ Pk ⫽ Pr Re Tb Tin Tout Tw Ui u iu j
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
constants total rod heat transfer area specific heat constant in the Reynolds stress model constants in the -equation of the Reynolds stress model outer diameter of tube blending function in the SST model heat transfer coefficient turbulent kinetic energy or conductivity mass flow rate Nusselt number pitch, distance between tube centers Peclet number Peclet number based on maximum velocity production rate of Reynolds stress production rate of turbulent kinetic energy Prandtl number Reynolds number bulk temperature inlet temperature outlet temperature tube wall temperature time mean velocity components Reynolds stresses
Greek ␣ ,  ⬘ , , k*, , 2 ⫽ constants in the model transport equations  ⫽ inclined angle or turbulent model constant ␦ij ⫽ Kronecker delta
051801-6 / Vol. 132, MAY 2010
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