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Numerically simulating the thermal–hydraulic characteristics within the fuel rod bundle using CFD methodology
Nuclear Engineering and Design 240 (2010) 3078–3086
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Numerically simulating the thermal–hydraulic characteristics within the fuel rod bundle using CFD methodology C.C. Liu ∗ , Y.M. Ferng Institute of Nuclear Engineering and Science, National Tsing Hua University, 101, Section 2, Kuang-Fu Road, Hsingchu 30013, 325 Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 30 December 2009 Received in revised form 25 April 2010 Accepted 7 May 2010
a b s t r a c t With the dramatic progress in the computer processing power, computational fluid dynamics (CFD) methodology can be applied in investigating the detailed knowledge of thermal–hydraulic characteristics in the rod bundle, especially with the spacer grid. These localized information, including flow, turbulence, and heat transfer characteristics, etc., can assist in the design and the improvement of rod bundles for nuclear power plants. In this paper, a three-dimensional (3D) CFD model with the Reynolds stresses turbulence model is proposed to simulate these characteristics within the rod bundle and subsequently to investigate the effects of different types of grid on the turbulent mixing and heat transfer enhancement. Two types of grid designs are used herein, including the standard grid and split-vane pair one, respectively. Based on the CFD simulations, the secondary flow can be reasonably captured in the rod bundle with the grid. The split-vane pair grid would enhance both the flow mixing and the heat transfer capability more than the standard grid does, as clearly shown in the simulation results. In addition, compared with the results of experiment and correlation, the present predicted result for the Nusselt (Nu) number distribution downstream the grid shows reasonable agreement for the standard grid design. However, there is discrepancy in the decay trend of Nu number between the prediction and measurement for the split-vane pair gird. This would be improved by adopting the finer mesh (y+ < 1) simulation and Low-Reynolds form turbulence model, which is our future research work. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In most of nuclear reactors, rod bundles are the essential elements that consist of tightly packed rod arrays. These fuel rods are surrounded by the flowing coolant which plays a significant role in both heat removal and radiation shielding. Therefore, the investigation of thermal–hydraulic characteristics inside the rod bundles is important for optima design and safety operation of a nuclear reactor power plant. Spacer grids are one of the main components used on the rod bundles in order to maintain appropriate rod-to-rod clearance, secure flow passage, and prevent the bundle damage from flow-induced vibration (FIV), etc. Existence of the grids would significantly influence the flow and temperature distributions within the rod bundles. In order to promote the coolant mixing effect, vane structures are generally attached on the grids. These vanes would generate the directive cross-flows, churn the thermal energy, and subsequently enhance the turbulent mixing and the heat transfer for the rod bundles. However, it is difficult to carry out the related experiments to investigate these thermal–hydraulic
∗ Corresponding author. Tel.: +886988158698. E-mail address: [email protected] (C.C. Liu). 0029-5493/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2010.05.021
characteristics within the mixing-vane grids. Computational fluid dynamics (CFD) methodology then attracts more attention from the researcher to model these complicated phenomena, including the swirling secondary flow, turbulent mixing and heat transfer enhancement downstream a spacer with appropriated vanes. Yao et al. (1982) had analyzed the effect of mixing vanes and spacer grids on the heat transfer coefficient of the sub-channels. They developed two correlations for the local heat transfer coefficient in these geometries. Wu and Trupp (1993) clearly revealed that flow conditions inside the fuel bundles are very different from those in the typical pipes. The near-wall turbulence anisotropy results in the formation of secondary vortices inside the channel, causing the coolant to spiral through the bundle. Karoutas et al. (1995) investigated axial and lateral velocities downstream a split-vane pair grid by comparing LDV measurements to CFD simulations. Lee and Jang (1997) performed numerical simulations for a rod bundle using a nonlinear k–ε model without any adjustment and concluded that this approach strongly underestimated the strong azimuthally turbulence intensity. However, by adjusting model coefficients adopted in a quadratic k–ε model (Shih et al., 1993), Baglietto and Ninokata (2005) had shown its promising capability in adequate anisotropy modeling of the wall-shear stress distribution and the velocity field in tight lattice fuel bundles.
C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
Nomenclature C Dh d E g k kp P Pr Prt Tp Tw u ui up u (u ) xi y yp y+ Z
modeling coefficient equilibrium diameter, m normal distance to the wall, m empirical constant in turbulence model (9.793) gravity acceleration, m/s2 turbulence kinetic energy, m2 /s2 turbulence kinetic energy at point p, m2 /s2 mean pressure, N/m2 Prandtl number turbulence Prandtl number temperature at point p, K wall temperature, K x-direction velocity component, m/s velocity vector, m/s velocity at point p, m/s ensemble averaged quantity turbulent fluctuating quantity coordinate vector, m wall distance, m wall distance at point p, m dimensionless wall distance y+ = y/ distance form the grid, m
Greek symbols ıij Kronecker delta tensor ε turbulence dissipation rate, m2 /s3 εij turbulence dissipation rate tensor, m2 /s3 DT,ij diffusion term production term Pij ˚ij pressure–strain term von Karman constant dynamic viscosity, kg/m s t dynamic turbulent viscosity friction velocity = ω / density (kg/m3 ) modeling coefficient wall-shear stress ω ij Reynolds stress tensor
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rod, etc. Based on the reasonable simulation (Hàzi, 2005; Lee and Choi, 2007) in the bundle geometry, the RSM turbulence model is adopted in the present work. Two types of grid designs on the bundle are investigated in this paper: one is the standard grid; another is the split-vane pair one. In addition, the experimental data (Holloway et al., 2004) and the Yao’s correlation (Yao et al., 1982) for the Nusselt (Nu) number distribution are also used to assess the present CFD model. 2. Mathematical model A 3D CFD model is developed in this paper to investigate the thermal–hydraulic characteristics in the rod bundle with the different types of grid designs. The simulation mathematical model includes the continuity equation, momentum equation, energy equation, and turbulence model. Continuity equation:
∂ ui
=0
∂xi
(1)
Momentum equation:
∂ ui
uj
=−
∂xj
∂ P ∂xi
∂ + ∂xj
∂ ui ∂xj
+
∂ uj ∂xi
−
ui uj
+ gi
(2)
Energy equation: ∂ ∂ ( ui (E + P)) = ∂xi ∂xi
keff
∂T + uj (ij )eff ∂xi
(3)
Turbulence model: The RSM solves differential transport equations for each Reynolds stress component. Such transport equations are essentially derived from the Navier–Stokes equation and described as follows. ∂ ∂ ui uj + Cij = DT,ij + ∂t ∂xk
∂ ui uj ∂xk
+ Pij + ˚ij − εij (4)
where Cij = Yadigaroglu et al. (2003) conducted an in-depth review of rod bundle numerical simulations and concluded that the gradienttransport models, like the standard k–ε model, are not adequate to predict turbulent flow in the narrow gap regions. Hàzi (2005) had demonstrated the Reynolds stress model (RSM) could be accurately applied in simulating the rod bundle geometry. Lee and Choi (2007) also used the RSM turbulence model to compare the performance of grid designs between the small scale vortex flow (SSVF) mixing vane and the large scale vortex flow (LSVF) mixing vane. Recently, Conner et al. (2009) applied the CFD methodology in a 5 × 5 rod bundle with the mixing-vane grid using the renormalization group (RNG) k–ε model (Yakhot et al., 1992). This CFD model is shown to reasonably agree with the test data (Smith et al., 2002). Most of these CFD works were concentrated on the hydraulic characteristics in the bundle with the mixing-vane grids and fewer works were conducted to investigate the thermal behavior within this geometry. The majority of this paper is to simulate the thermal–hydraulic characteristics in the rod bundle with the different grid designs by way of the CFD methodology. These characteristics include the flow distribution, the secondary flow structure, wall temperature distribution, shear stress variation, and heat transfer behavior along the
∂ uk ui uj ∂xk
Pij = −
ui uk
∂ uj ∂xk
DT,ij = −
∂ ∂xk
+
⎛ ⎝ t k
∂ ui uj ∂xk
˚ij = ˚ij,1 + ˚ij,2 + ˚ij,W ˚ij,1 = −C1 ˚ij,2 = −C2 ˚ij,W = C1
ε
k
ε k
ui uj −
Pij − Cij −
(5)
uj uk
(6)
∂xk
⎞ ⎠
(7)
(8)
2 ı k , 3 ij
(9)
2 ı (P − C˚ ) 3 ij ˚
uk um nk nm ıij −
∂ ui
(10)
3 3 ui uk nj nk − uj uk ni nk 2 2
×
k3/2 3 3 + C2 ˚km,2nk nm ıij − ˚ik,2nj nk − ˚jk,2ni nk 2 2 C εd
×
k3/2 C εd
(11)
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C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
Table 1 Constants in the RSM turbulence model. C1
C1
Cε1
Cε2
C1
C2
C
k
ε
1.8
0.6
1.44
1.92
0.5
0.3
0.09
0.4187
0.82
1.0
3/4
C =
where
C
(12)
P˚ =
1 2Pkk
(13)
C˚ =
1 2Ckk
(14)
1/4 1/2
1/4 1/2
u+ ≡
uP C kP ω /
, y+ ≡
C kP yP
(19)
2.1. Energy
1/4 1/2
C k2 t = ε
(15)
ui ui
∂(εui ) ∂(ε) + = ∂t ∂xi ∂xj
+
t ∂ε ε
∂xj
1 ε ε2 + Cε1 Pii − Cε2 2 k k (16)
(Tω − TP )CpC kP qw
⎧ ⎪ ⎨
and the The turbulent kinetic energy (k) is given as (1/2) dissipation rate tensor can be expressed as εij = (2/3)(ıij ε) for incompressible flow. The scalar dissipation rate i can be found by solving its transport equation:
∂
T+ =
=
1/4 1/2
⎪ ⎩ Pr t
Pr y+ + 1 ln(Ey+ ) + Pt
C kP 1 Pr qw 2
u2P
(y+ < yT+ )
1/4 1/2
+
1 C kP Pr qw 2
{Pr t u2P + (Pr − Pr t )u2c }
(y+ > yT+ )
(20) where yT+ is computed from the intersection of the linear and logarithmic profiles.
3/4 Pr
− 1 [1 + 0.28e−0.007Pr/Prt ]
where all the empirical constants in above equations are illustrated in Table 1. In addition, a standard wall-function (Launder and Spalding, 1973) is adopted in the near-wall treatment for the momentum and energy equations. Momentum:
Pt = 9.24
u+ = y+
Fig. 1 shows the simulation geometry for (a) the end view of the test bundle with vane (b) the standard grid and (c) the splitvane pair one, respectively, from the top view. The corresponding boundary conditions, including the wall and the periodic ones,
u+ =
for y+ < 11.225
1 ln(Ey+ ) V
for y+ ≥ 11.225
(17) (18)
Prt
(21)
us = mean velocity magnitude at y+ = yT+ . 3. Mesh model and numerical treatment
Fig. 1. Rod bundle with a spacer grid model. (a) End view of the test bundle with vane, (b) standard type grid and (c) split-vane pair grid with vane.
C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
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Fig. 2. Computational geometry and mesh.
are also presented on plot (b) and plot (c). In this figure, green, grey, red, and blue colors represent fuel rods, grids, vanes, and weld nugget, respectively. As mentioned above, the spacer grid maintains a constant distance along the fuel rod to secure flow passage and prevent rod damage. The split-vane pair arranged on the support grid alternates between the up/down and left/right vane pattern, with an angle of approximately 30◦ along the axial flow direction. This mixing vane would guide the flow, which generates more turbulence and promotes the thermal mixing between the sub-channels. The present simulation domain is schematically shown on the right side of Fig. 2, which is referred to the experimental work of Holloway et al. (2004). Five support grids are used to position a rod in the bundle. The first support grid is a standard one. The second, third, and fourth grids in the rod bundle are those with the test vanes. The span between these grids is 50.8 cm. The rod diameter is 9.5 mm and its heated length is 104.1 cm. The test rod bundle is positioned on a square array with a pitch of 1.3 mm. The heated portion of rod bundle is located between the second and the fourth support grids. The fifth support grid, a standard one, is placed near the outlet of the test section. 3.1. Mesh model A typical mesh model for the present simulation of the rod bundle with split-vane pair grids is illustrated on the right portion of Fig. 2. The upper plot shows the top view of mesh distribution and the lower one is its side view. As shown in both plots, unstructured meshes are adopted for simulating this complicated geometry. There are 1,420,199 meshes for this typical simulation case and the values of near-wall y+ are taken between 31 and 48. Calculations with different mesh models are also performed to conform that the simulation results are mesh independent.
3.2. Numerical treatment The differential equations governing the thermal–hydraulic phenomena within the bundle geometry are discretized into the algebraic equations for numerical calculations based on the control-volume-based technique. The coupled equations for both the velocity and pressure are solved by the so-called SIMPLE scheme (Patankar, 1981). The numerical procedures to solve this 3D CFD model can be described as follows: 1. Set the boundary conditions, including the inlet distributions of flow velocity, turbulent property and temperature, outlet pressure, wall heat flux, etc. 2. Solve the momentum Eq. (2) to determine the flow field within the rod bundle. 3. Solve the pressure correction equation (Patankar, 1981) derived from the continuity Eq. (1) to eliminate the mass conservation error and then apply appropriate corrections to update the velocity and pressure fields. 4. Solve the energy Eq. (4) to obtain the temperature distribution. 5. Solve RSM turbulence model Eqs. (4)–(21) to obtain the distributions of Reynolds stress components. 6. Repeat step 2 through 5 until the convergent criteria are satisfied. The convergent criteria adopted in the numerical calculation are set as that summation of the relative residual in every control volume for each governing equation is smaller than 10−5 . In addition, the decay trend in the residual plot for each equation is also considered as an alternative criterion that the relative residual must be decreased by two orders at least. Both criteria should be met for the convergence of numerical simulations. The FLUENT code (FLUENT, 2005) is selected in the present simulation works that are performed on the PC with Intel Core I7 2.66 GHz. In the
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present simulation work, the typical computer time for modeling the thermal–hydraulic characteristics in the rod bundle with the split-vane pair grid is about 90,063 s. 4. Results and discussion The simulation conditions are summarized as that the inlet temperature is set to be 300 K, the inlet velocity is set to be 2.5 m/s (the
corresponding Reynolds number = 28,000), and the wall heat flux is set to be 1.1 MW/m2 . Two types of spacer grids are simulated in this study. Figs. 3 and 4 show the temperature contours on the horizontal cross-section at different axial locations (Z/Dh ) of the rod with the standard and split-vane pair grids, respectively. The vector distributions of secondary flow for both grid designs are also shown one the temperature contours. It can be clearly shown in Fig. 3(a) that the vortex pair occurs in the gap. These vortexes of secondary
Fig. 3. Standard support grid on local temperature and secondary flow vectors.
C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
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Fig. 4. Split-vane pairs support grid on local temperature and secondary flow vectors.
flow would gradually decay as the coolant flows downstream the grid. Above phenomenon related to the strong secondary flow would not appear in the rod bundle with the split-vane pair grid since this grid design would guide the flow in the axial direction, as clearly shown in Fig. 4. This guide function would result in the axial flow mixing, which is also confirmed in Fig. 5. This figure shows the comparison of stream line characteristics along the rod with (a) the standard grid and (b) the split-vane pair one, respectively. It is clearly seen in Fig. 5 that the stronger stream swirling phenomenon is revealed in the simulation results for the split-vane pair grid. The
comparison also indicates that the more anisotropic characteristic of thermal–hydraulic parameters would be induced for this type of grid design. The higher flow mixing would cause the higher heat removal and then lower the temperature distribution on the wall, which can be demonstrated in Fig. 6. This figure compares the calculated temperature distributions on the rod wall for (a) the standard grid design and (b) the split-vane pair one, respectively. The surface wall temperature for the split-vane pair grid is predicted to be lower than that for the standard one, resulting from the higher axial swirling mixing by the vane.
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Fig. 5. Characteristics of stream lines for (a) standard support grid and (b) split-vane pairs support grid.
At the different axial locations, Fig. 7 displays the calculated azimuthal variations of wall-shear stress along the rod bundle for both grid designs. These distributions are presented in the form of normalized shear stress that is the local shear stress divided by the averaged one. It can be clearly revealed in these four plots that the azimuthal dependence of shear stress shows more symmetry characteristics for the standard grid than for the split-vane pair one since the latter would guide the flow direction. In addition, the patterns of azimuthal dependence for the shear stress approach to be similar as the flow passes downstream the standard grid. However, for the split-vane pair grid, the anisotropic characteristic of shear stress distribution around the azimuthal angles would decay along the downstream side. According to the comparison of flow characteristics shown above, the grid with the split-vane would cause the more turbulent mixing, resulting in the enhancement the fluid heat transfer. This phenomenon can be demonstrated in the Nu number distribution downstream the grid. Fig. 8 shows the axial distributions of normalized Nu at the different azimuthal angles for (a) the standard grid and (b) the split-vane pair one, respectively. Similar to the definition of previous study (Yao et al., 1982), the normalized Nu number along the rod is the local Nu number divided by the fully developed (FD) Nu number. This FD Nu number is obtained by the addition calculation with the same conditions and rod arrangement, except for no grids attached on the fuel rod. Detailed observation of this
Fig. 6. 2D local temperature profiles for the rod bundle with the split-vane pairs grid.
Fig. 7. Azimuthal variations in normalized wall-shear stress for different grid designs. (a) Z/Dh = 0.677, (b) Z/Dh = 2.370, (c) Z/Dh = 4.064 and (d) Z/Dh = 5.757.
C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
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Fig. 8. Nusselt numbers downstream support grid. (a) Standard support grid and (b) split-vane pairs grid.
Fig. 9. Normalized Nusselt numbers downstream for Standard support grid.
figure reveals that the Nu number for the split-vane pair grid is calculated to be higher than that for the standard grid. In addition, the stronger axial flow swirling would cause more anisotropic characteristics of heat transfer distribution for the rod with the split-vane pair grid. With the experimental data, Figs. 9 and 10 compare the predicted axial distributions of averaged Nu number for the standard and split-vane pair girds, respectively. The dots in the figures are the experimentally measured data (Holloway et al., 2004), the solid lines are the present predicted results, and the dash ones are the results calculated using the correlation of Yao et al. (1982). The normalized Nu number, larger than 1.0 just near the outlet of grid, indicates the enhancement of heat transfer capability by the grid, which is reasonably revealed in both the measurements and the predictions.
Fig. 10. Normalized Nusselt numbers downstream for split-vane pairs grid.
It can be clearly shown in Fig. 9 that the distribution of normalized Nu number predicted by the present CFD model agrees reasonably well with the experimental data and Yao’s correlation prediction for the standard grid. For the grid with the split vane, the present CFD model under-predicts the decay trend of experiments, especially downstream the grid, as revealed in Fig. 10. The Yao’s correlation cannot also simulate this over-decay characteristic in the decline trend of Nu number. However, the present predictions agree with the experimental data more than the correlation. The discrepancy between the measurements and CFD predictions may be resulted from that the RSM turbulence model with the standard wall-function is incapable of completely capturing the sophisticated thermal–hydraulic characteristics within the rod bundle with the split-wane grid. In addition, the near-wall mesh (y+ = 31–48) is not fine enough to resolve the thermal wall behaviors in this type of geometry. 5. Conclusions A 3D CFD model is developed to investigate the thermal–hydraulic characteristics in the rod bundle with the different types of grid designs, including the standard grid and the split-vane pair one, respectively. Several important conclusions can be drawn from the simulation results and are presented as follows. 1. The secondary flow characteristics within the rod bundle can be reasonably captured by the present CFD model with the RSM turbulence model. This flow characteristic is predicted to gradually decay as the flow passes downstream the grid. With the addition of split-vane, this grid design would guide the flow and subsequently cause the axial swirling pattern. In addition, based on the comparison of simulation results, the thermal–hydraulic characteristics for the split-vane pair grid are calculated to be more anisotropic than those for the standard grid. 2. The enhancement of turbulent mixing and heat transfer capability by adding the split-vane on the standard grid can be reasonably simulated and shown in the comparison of the shear stress and the Nu number for both types of grid designs. 3. The Nu number gradually decreases downstream the grid and approaches to a fully developed value, which is confirmed in the experimental measurement and the predicted results by the Yao’s correlation and present CFD model, respectively. For the standard grid design, the present CFD model can reasonably reproduce the experimental data in the Nu number distribution downstream the grid. However, there is discrepancy in the decay trend of Nu number between the experiment and the CFD prediction for the split-vane pair grid. This may be improved by adopting the more sophisticated turbulence model and the finer meshes (y+ < 1.0), etc, which is our future investigation work.
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References Baglietto, E., Ninokata, H., 2005. A turbulence model study for simulating flow inside tight lattice rod bundles. Nuclear Engineering and Design 235, 773–784. Conner, M.E., Baglietto, E., Elmahdi, A.M., 2009. CFD methodology and validation for single-phase flow in PWR fuel assemblies. Nuclear Engineering and Design, published online 2009. FLUENT Inc., 2005. Fluent 6.2.16 User’s Guide. Hàzi, Gàbor, 2005. On turbulence models for rod bundle flow computations. Annals of Nuclear Energy 32, 755–761. Holloway, M.V., McClusky, H.L., Beasley, D.E., Conner, M.E., 2004. The effect of support grid features on local, single-phase heat transfer measurements in rod bundles. Journal of Heat Transfer 126, 43–53. Karoutas, Z., Gu, C.Y., Scholin, B., 1995. 3-D flow analyses for design of nuclear fuel spacer. In: Proceedings of the Seventh International Meeting on Nuclear Reactor Thermal–Hydraulics, New York, United States, September. Lee, C.M., Choi, Y.D., 2007. Comparison of thermo-hydraulic performances of large scale vortex flow (LSVF) and small scale vortex flow (SSVF) mixing vanes in 17 × 17 nuclear rod bundle. Nuclear Engineering and Design 237, 2322–2331. Lee, K.B., Jang, H.G., 1997. A numerical prediction on the turbulent flow in closely spaced bare rod arrays by a nonlinear k–i model. Nuclear Engineering and Design 172, 351–357.
Launder, B.E., Spalding, D.B., 1973. The numerical computational of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3, 269. Patankar, S.V., 1981. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corp., New York. Shih, T.-H., Zhu, J., Lumley, J.L., 1993. Arealizable Reynolds stress algebraic equation model, NASA TM 105993. Smith III, L.D., Conner, M.E., Liu, B., Dzodzo, M.B., Paramonov, D.V., Beasley, D.E., Langford, H.M., Holloway, M.V., 2002. Benchmarking computational fluid dynamics for application to PWR fuel. In: Proceedings of the 10th International Conference on Nuclear Engineering, Arlington, VI, USA, April 14–18. Wu, X., Trupp, A.C., 1993. Experimental study on the unusual turbulence intensity distribution in rod-to-wall gap regions. Experimental Thermal and Fluid Science 6 (4), 360–370. Yadigaroglu, G., Anderani, M., Dreier, J., Coddington, P., 2003. Trends and needs in experimentation and numerical simulation for LWR safety. Nuclear Engineering and Design 221, 205–223. Yakhot, V., Orszag, S.A., Thangam, S., Gatski, T.B., Speziale, C.G., 1992. Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids A4 (7), 1510–1520. Yao, S.C., Hochreiter, L.E., Leech, W.J., 1982. Heat-transfer augmentation in rod bundles near grid spacers. Journal of Heat Transfer 104, 76–81.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Numerically simulating the thermal–hydraulic characteristics within the fuel rod bundle using CFD methodology C.C. Liu ∗ , Y.M. Ferng Institute of Nuclear Engineering and Science, National Tsing Hua University, 101, Section 2, Kuang-Fu Road, Hsingchu 30013, 325 Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 30 December 2009 Received in revised form 25 April 2010 Accepted 7 May 2010
a b s t r a c t With the dramatic progress in the computer processing power, computational fluid dynamics (CFD) methodology can be applied in investigating the detailed knowledge of thermal–hydraulic characteristics in the rod bundle, especially with the spacer grid. These localized information, including flow, turbulence, and heat transfer characteristics, etc., can assist in the design and the improvement of rod bundles for nuclear power plants. In this paper, a three-dimensional (3D) CFD model with the Reynolds stresses turbulence model is proposed to simulate these characteristics within the rod bundle and subsequently to investigate the effects of different types of grid on the turbulent mixing and heat transfer enhancement. Two types of grid designs are used herein, including the standard grid and split-vane pair one, respectively. Based on the CFD simulations, the secondary flow can be reasonably captured in the rod bundle with the grid. The split-vane pair grid would enhance both the flow mixing and the heat transfer capability more than the standard grid does, as clearly shown in the simulation results. In addition, compared with the results of experiment and correlation, the present predicted result for the Nusselt (Nu) number distribution downstream the grid shows reasonable agreement for the standard grid design. However, there is discrepancy in the decay trend of Nu number between the prediction and measurement for the split-vane pair gird. This would be improved by adopting the finer mesh (y+ < 1) simulation and Low-Reynolds form turbulence model, which is our future research work. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In most of nuclear reactors, rod bundles are the essential elements that consist of tightly packed rod arrays. These fuel rods are surrounded by the flowing coolant which plays a significant role in both heat removal and radiation shielding. Therefore, the investigation of thermal–hydraulic characteristics inside the rod bundles is important for optima design and safety operation of a nuclear reactor power plant. Spacer grids are one of the main components used on the rod bundles in order to maintain appropriate rod-to-rod clearance, secure flow passage, and prevent the bundle damage from flow-induced vibration (FIV), etc. Existence of the grids would significantly influence the flow and temperature distributions within the rod bundles. In order to promote the coolant mixing effect, vane structures are generally attached on the grids. These vanes would generate the directive cross-flows, churn the thermal energy, and subsequently enhance the turbulent mixing and the heat transfer for the rod bundles. However, it is difficult to carry out the related experiments to investigate these thermal–hydraulic
∗ Corresponding author. Tel.: +886988158698. E-mail address: [email protected] (C.C. Liu). 0029-5493/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2010.05.021
characteristics within the mixing-vane grids. Computational fluid dynamics (CFD) methodology then attracts more attention from the researcher to model these complicated phenomena, including the swirling secondary flow, turbulent mixing and heat transfer enhancement downstream a spacer with appropriated vanes. Yao et al. (1982) had analyzed the effect of mixing vanes and spacer grids on the heat transfer coefficient of the sub-channels. They developed two correlations for the local heat transfer coefficient in these geometries. Wu and Trupp (1993) clearly revealed that flow conditions inside the fuel bundles are very different from those in the typical pipes. The near-wall turbulence anisotropy results in the formation of secondary vortices inside the channel, causing the coolant to spiral through the bundle. Karoutas et al. (1995) investigated axial and lateral velocities downstream a split-vane pair grid by comparing LDV measurements to CFD simulations. Lee and Jang (1997) performed numerical simulations for a rod bundle using a nonlinear k–ε model without any adjustment and concluded that this approach strongly underestimated the strong azimuthally turbulence intensity. However, by adjusting model coefficients adopted in a quadratic k–ε model (Shih et al., 1993), Baglietto and Ninokata (2005) had shown its promising capability in adequate anisotropy modeling of the wall-shear stress distribution and the velocity field in tight lattice fuel bundles.
C.C. Liu, Y.M. Ferng / Nuclear Engineering and Design 240 (2010) 3078–3086
Nomenclature C Dh d E g k kp P Pr Prt Tp Tw u ui up u (u ) xi y yp y+ Z
modeling coefficient equilibrium diameter, m normal distance to the wall, m empirical constant in turbulence model (9.793) gravity acceleration, m/s2 turbulence kinetic energy, m2 /s2 turbulence kinetic energy at point p, m2 /s2 mean pressure, N/m2 Prandtl number turbulence Prandtl number temperature at point p, K wall temperature, K x-direction velocity component, m/s velocity vector, m/s velocity at point p, m/s ensemble averaged quantity turbulent fluctuating quantity coordinate vector, m wall distance, m wall distance at point p, m dimensionless wall distance y+ = y/ distance form the grid, m
Greek symbols ıij Kronecker delta tensor ε turbulence dissipation rate, m2 /s3 εij turbulence dissipation rate tensor, m2 /s3 DT,ij diffusion term production term Pij ˚ij pressure–strain term von Karman constant dynamic viscosity, kg/m s t dynamic turbulent viscosity friction velocity = ω / density (kg/m3 ) modeling coefficient wall-shear stress ω ij Reynolds stress tensor
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rod, etc. Based on the reasonable simulation (Hàzi, 2005; Lee and Choi, 2007) in the bundle geometry, the RSM turbulence model is adopted in the present work. Two types of grid designs on the bundle are investigated in this paper: one is the standard grid; another is the split-vane pair one. In addition, the experimental data (Holloway et al., 2004) and the Yao’s correlation (Yao et al., 1982) for the Nusselt (Nu) number distribution are also used to assess the present CFD model. 2. Mathematical model A 3D CFD model is developed in this paper to investigate the thermal–hydraulic characteristics in the rod bundle with the different types of grid designs. The simulation mathematical model includes the continuity equation, momentum equation, energy equation, and turbulence model. Continuity equation:
∂ ui
=0
∂xi
(1)
Momentum equation:
∂ ui
uj
=−
∂xj
∂ P ∂xi
∂ + ∂xj
∂ ui ∂xj
+
∂ uj ∂xi
−
ui uj
+ gi
(2)
Energy equation: ∂ ∂ ( ui (E + P)) = ∂xi ∂xi
keff
∂T + uj (ij )eff ∂xi
(3)
Turbulence model: The RSM solves differential transport equations for each Reynolds stress component. Such transport equations are essentially derived from the Navier–Stokes equation and described as follows. ∂ ∂ ui uj + Cij = DT,ij + ∂t ∂xk
∂ ui uj ∂xk
+ Pij + ˚ij − εij (4)
where Cij = Yadigaroglu et al. (2003) conducted an in-depth review of rod bundle numerical simulations and concluded that the gradienttransport models, like the standard k–ε model, are not adequate to predict turbulent flow in the narrow gap regions. Hàzi (2005) had demonstrated the Reynolds stress model (RSM) could be accurately applied in simulating the rod bundle geometry. Lee and Choi (2007) also used the RSM turbulence model to compare the performance of grid designs between the small scale vortex flow (SSVF) mixing vane and the large scale vortex flow (LSVF) mixing vane. Recently, Conner et al. (2009) applied the CFD methodology in a 5 × 5 rod bundle with the mixing-vane grid using the renormalization group (RNG) k–ε model (Yakhot et al., 1992). This CFD model is shown to reasonably agree with the test data (Smith et al., 2002). Most of these CFD works were concentrated on the hydraulic characteristics in the bundle with the mixing-vane grids and fewer works were conducted to investigate the thermal behavior within this geometry. The majority of this paper is to simulate the thermal–hydraulic characteristics in the rod bundle with the different grid designs by way of the CFD methodology. These characteristics include the flow distribution, the secondary flow structure, wall temperature distribution, shear stress variation, and heat transfer behavior along the
∂ uk ui uj ∂xk
Pij = −
ui uk
∂ uj ∂xk
DT,ij = −
∂ ∂xk
+
⎛ ⎝ t k
∂ ui uj ∂xk
˚ij = ˚ij,1 + ˚ij,2 + ˚ij,W ˚ij,1 = −C1 ˚ij,2 = −C2 ˚ij,W = C1
ε
k
ε k
ui uj −
Pij − Cij −
(5)
uj uk
(6)
∂xk
⎞ ⎠
(7)
(8)
2 ı k , 3 ij
(9)
2 ı (P − C˚ ) 3 ij ˚
uk um nk nm ıij −
∂ ui
(10)
3 3 ui uk nj nk − uj uk ni nk 2 2
×
k3/2 3 3 + C2 ˚km,2nk nm ıij − ˚ik,2nj nk − ˚jk,2ni nk 2 2 C εd
×
k3/2 C εd
(11)
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Table 1 Constants in the RSM turbulence model. C1
C1
Cε1
Cε2
C1
C2
C
k
ε
1.8
0.6
1.44
1.92
0.5
0.3
0.09
0.4187
0.82
1.0
3/4
C =
where
C
(12)
P˚ =
1 2Pkk
(13)
C˚ =
1 2Ckk
(14)
1/4 1/2
1/4 1/2
u+ ≡
uP C kP ω /
, y+ ≡
C kP yP
(19)
2.1. Energy
1/4 1/2
C k2 t = ε
(15)
ui ui
∂(εui ) ∂(ε) + = ∂t ∂xi ∂xj
+
t ∂ε ε
∂xj
1 ε ε2 + Cε1 Pii − Cε2 2 k k (16)
(Tω − TP )CpC kP qw
⎧ ⎪ ⎨
and the The turbulent kinetic energy (k) is given as (1/2) dissipation rate tensor can be expressed as εij = (2/3)(ıij ε) for incompressible flow. The scalar dissipation rate i can be found by solving its transport equation:
∂
T+ =
=
1/4 1/2
⎪ ⎩ Pr t
Pr y+ + 1 ln(Ey+ ) + Pt
C kP 1 Pr qw 2
u2P
(y+ < yT+ )
1/4 1/2
+
1 C kP Pr qw 2
{Pr t u2P + (Pr − Pr t )u2c }
(y+ > yT+ )
(20) where yT+ is computed from the intersection of the linear and logarithmic profiles.
3/4 Pr
− 1 [1 + 0.28e−0.007Pr/Prt ]
where all the empirical constants in above equations are illustrated in Table 1. In addition, a standard wall-function (Launder and Spalding, 1973) is adopted in the near-wall treatment for the momentum and energy equations. Momentum:
Pt = 9.24
u+ = y+
Fig. 1 shows the simulation geometry for (a) the end view of the test bundle with vane (b) the standard grid and (c) the splitvane pair one, respectively, from the top view. The corresponding boundary conditions, including the wall and the periodic ones,
u+ =
for y+ < 11.225
1 ln(Ey+ ) V
for y+ ≥ 11.225
(17) (18)
Prt
(21)
us = mean velocity magnitude at y+ = yT+ . 3. Mesh model and numerical treatment
Fig. 1. Rod bundle with a spacer grid model. (a) End view of the test bundle with vane, (b) standard type grid and (c) split-vane pair grid with vane.
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Fig. 2. Computational geometry and mesh.
are also presented on plot (b) and plot (c). In this figure, green, grey, red, and blue colors represent fuel rods, grids, vanes, and weld nugget, respectively. As mentioned above, the spacer grid maintains a constant distance along the fuel rod to secure flow passage and prevent rod damage. The split-vane pair arranged on the support grid alternates between the up/down and left/right vane pattern, with an angle of approximately 30◦ along the axial flow direction. This mixing vane would guide the flow, which generates more turbulence and promotes the thermal mixing between the sub-channels. The present simulation domain is schematically shown on the right side of Fig. 2, which is referred to the experimental work of Holloway et al. (2004). Five support grids are used to position a rod in the bundle. The first support grid is a standard one. The second, third, and fourth grids in the rod bundle are those with the test vanes. The span between these grids is 50.8 cm. The rod diameter is 9.5 mm and its heated length is 104.1 cm. The test rod bundle is positioned on a square array with a pitch of 1.3 mm. The heated portion of rod bundle is located between the second and the fourth support grids. The fifth support grid, a standard one, is placed near the outlet of the test section. 3.1. Mesh model A typical mesh model for the present simulation of the rod bundle with split-vane pair grids is illustrated on the right portion of Fig. 2. The upper plot shows the top view of mesh distribution and the lower one is its side view. As shown in both plots, unstructured meshes are adopted for simulating this complicated geometry. There are 1,420,199 meshes for this typical simulation case and the values of near-wall y+ are taken between 31 and 48. Calculations with different mesh models are also performed to conform that the simulation results are mesh independent.
3.2. Numerical treatment The differential equations governing the thermal–hydraulic phenomena within the bundle geometry are discretized into the algebraic equations for numerical calculations based on the control-volume-based technique. The coupled equations for both the velocity and pressure are solved by the so-called SIMPLE scheme (Patankar, 1981). The numerical procedures to solve this 3D CFD model can be described as follows: 1. Set the boundary conditions, including the inlet distributions of flow velocity, turbulent property and temperature, outlet pressure, wall heat flux, etc. 2. Solve the momentum Eq. (2) to determine the flow field within the rod bundle. 3. Solve the pressure correction equation (Patankar, 1981) derived from the continuity Eq. (1) to eliminate the mass conservation error and then apply appropriate corrections to update the velocity and pressure fields. 4. Solve the energy Eq. (4) to obtain the temperature distribution. 5. Solve RSM turbulence model Eqs. (4)–(21) to obtain the distributions of Reynolds stress components. 6. Repeat step 2 through 5 until the convergent criteria are satisfied. The convergent criteria adopted in the numerical calculation are set as that summation of the relative residual in every control volume for each governing equation is smaller than 10−5 . In addition, the decay trend in the residual plot for each equation is also considered as an alternative criterion that the relative residual must be decreased by two orders at least. Both criteria should be met for the convergence of numerical simulations. The FLUENT code (FLUENT, 2005) is selected in the present simulation works that are performed on the PC with Intel Core I7 2.66 GHz. In the
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present simulation work, the typical computer time for modeling the thermal–hydraulic characteristics in the rod bundle with the split-vane pair grid is about 90,063 s. 4. Results and discussion The simulation conditions are summarized as that the inlet temperature is set to be 300 K, the inlet velocity is set to be 2.5 m/s (the
corresponding Reynolds number = 28,000), and the wall heat flux is set to be 1.1 MW/m2 . Two types of spacer grids are simulated in this study. Figs. 3 and 4 show the temperature contours on the horizontal cross-section at different axial locations (Z/Dh ) of the rod with the standard and split-vane pair grids, respectively. The vector distributions of secondary flow for both grid designs are also shown one the temperature contours. It can be clearly shown in Fig. 3(a) that the vortex pair occurs in the gap. These vortexes of secondary
Fig. 3. Standard support grid on local temperature and secondary flow vectors.
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Fig. 4. Split-vane pairs support grid on local temperature and secondary flow vectors.
flow would gradually decay as the coolant flows downstream the grid. Above phenomenon related to the strong secondary flow would not appear in the rod bundle with the split-vane pair grid since this grid design would guide the flow in the axial direction, as clearly shown in Fig. 4. This guide function would result in the axial flow mixing, which is also confirmed in Fig. 5. This figure shows the comparison of stream line characteristics along the rod with (a) the standard grid and (b) the split-vane pair one, respectively. It is clearly seen in Fig. 5 that the stronger stream swirling phenomenon is revealed in the simulation results for the split-vane pair grid. The
comparison also indicates that the more anisotropic characteristic of thermal–hydraulic parameters would be induced for this type of grid design. The higher flow mixing would cause the higher heat removal and then lower the temperature distribution on the wall, which can be demonstrated in Fig. 6. This figure compares the calculated temperature distributions on the rod wall for (a) the standard grid design and (b) the split-vane pair one, respectively. The surface wall temperature for the split-vane pair grid is predicted to be lower than that for the standard one, resulting from the higher axial swirling mixing by the vane.
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Fig. 5. Characteristics of stream lines for (a) standard support grid and (b) split-vane pairs support grid.
At the different axial locations, Fig. 7 displays the calculated azimuthal variations of wall-shear stress along the rod bundle for both grid designs. These distributions are presented in the form of normalized shear stress that is the local shear stress divided by the averaged one. It can be clearly revealed in these four plots that the azimuthal dependence of shear stress shows more symmetry characteristics for the standard grid than for the split-vane pair one since the latter would guide the flow direction. In addition, the patterns of azimuthal dependence for the shear stress approach to be similar as the flow passes downstream the standard grid. However, for the split-vane pair grid, the anisotropic characteristic of shear stress distribution around the azimuthal angles would decay along the downstream side. According to the comparison of flow characteristics shown above, the grid with the split-vane would cause the more turbulent mixing, resulting in the enhancement the fluid heat transfer. This phenomenon can be demonstrated in the Nu number distribution downstream the grid. Fig. 8 shows the axial distributions of normalized Nu at the different azimuthal angles for (a) the standard grid and (b) the split-vane pair one, respectively. Similar to the definition of previous study (Yao et al., 1982), the normalized Nu number along the rod is the local Nu number divided by the fully developed (FD) Nu number. This FD Nu number is obtained by the addition calculation with the same conditions and rod arrangement, except for no grids attached on the fuel rod. Detailed observation of this
Fig. 6. 2D local temperature profiles for the rod bundle with the split-vane pairs grid.
Fig. 7. Azimuthal variations in normalized wall-shear stress for different grid designs. (a) Z/Dh = 0.677, (b) Z/Dh = 2.370, (c) Z/Dh = 4.064 and (d) Z/Dh = 5.757.
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Fig. 8. Nusselt numbers downstream support grid. (a) Standard support grid and (b) split-vane pairs grid.
Fig. 9. Normalized Nusselt numbers downstream for Standard support grid.
figure reveals that the Nu number for the split-vane pair grid is calculated to be higher than that for the standard grid. In addition, the stronger axial flow swirling would cause more anisotropic characteristics of heat transfer distribution for the rod with the split-vane pair grid. With the experimental data, Figs. 9 and 10 compare the predicted axial distributions of averaged Nu number for the standard and split-vane pair girds, respectively. The dots in the figures are the experimentally measured data (Holloway et al., 2004), the solid lines are the present predicted results, and the dash ones are the results calculated using the correlation of Yao et al. (1982). The normalized Nu number, larger than 1.0 just near the outlet of grid, indicates the enhancement of heat transfer capability by the grid, which is reasonably revealed in both the measurements and the predictions.
Fig. 10. Normalized Nusselt numbers downstream for split-vane pairs grid.
It can be clearly shown in Fig. 9 that the distribution of normalized Nu number predicted by the present CFD model agrees reasonably well with the experimental data and Yao’s correlation prediction for the standard grid. For the grid with the split vane, the present CFD model under-predicts the decay trend of experiments, especially downstream the grid, as revealed in Fig. 10. The Yao’s correlation cannot also simulate this over-decay characteristic in the decline trend of Nu number. However, the present predictions agree with the experimental data more than the correlation. The discrepancy between the measurements and CFD predictions may be resulted from that the RSM turbulence model with the standard wall-function is incapable of completely capturing the sophisticated thermal–hydraulic characteristics within the rod bundle with the split-wane grid. In addition, the near-wall mesh (y+ = 31–48) is not fine enough to resolve the thermal wall behaviors in this type of geometry. 5. Conclusions A 3D CFD model is developed to investigate the thermal–hydraulic characteristics in the rod bundle with the different types of grid designs, including the standard grid and the split-vane pair one, respectively. Several important conclusions can be drawn from the simulation results and are presented as follows. 1. The secondary flow characteristics within the rod bundle can be reasonably captured by the present CFD model with the RSM turbulence model. This flow characteristic is predicted to gradually decay as the flow passes downstream the grid. With the addition of split-vane, this grid design would guide the flow and subsequently cause the axial swirling pattern. In addition, based on the comparison of simulation results, the thermal–hydraulic characteristics for the split-vane pair grid are calculated to be more anisotropic than those for the standard grid. 2. The enhancement of turbulent mixing and heat transfer capability by adding the split-vane on the standard grid can be reasonably simulated and shown in the comparison of the shear stress and the Nu number for both types of grid designs. 3. The Nu number gradually decreases downstream the grid and approaches to a fully developed value, which is confirmed in the experimental measurement and the predicted results by the Yao’s correlation and present CFD model, respectively. For the standard grid design, the present CFD model can reasonably reproduce the experimental data in the Nu number distribution downstream the grid. However, there is discrepancy in the decay trend of Nu number between the experiment and the CFD prediction for the split-vane pair grid. This may be improved by adopting the more sophisticated turbulence model and the finer meshes (y+ < 1.0), etc, which is our future investigation work.
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References Baglietto, E., Ninokata, H., 2005. A turbulence model study for simulating flow inside tight lattice rod bundles. Nuclear Engineering and Design 235, 773–784. Conner, M.E., Baglietto, E., Elmahdi, A.M., 2009. CFD methodology and validation for single-phase flow in PWR fuel assemblies. Nuclear Engineering and Design, published online 2009. FLUENT Inc., 2005. Fluent 6.2.16 User’s Guide. Hàzi, Gàbor, 2005. On turbulence models for rod bundle flow computations. Annals of Nuclear Energy 32, 755–761. Holloway, M.V., McClusky, H.L., Beasley, D.E., Conner, M.E., 2004. The effect of support grid features on local, single-phase heat transfer measurements in rod bundles. Journal of Heat Transfer 126, 43–53. Karoutas, Z., Gu, C.Y., Scholin, B., 1995. 3-D flow analyses for design of nuclear fuel spacer. In: Proceedings of the Seventh International Meeting on Nuclear Reactor Thermal–Hydraulics, New York, United States, September. Lee, C.M., Choi, Y.D., 2007. Comparison of thermo-hydraulic performances of large scale vortex flow (LSVF) and small scale vortex flow (SSVF) mixing vanes in 17 × 17 nuclear rod bundle. Nuclear Engineering and Design 237, 2322–2331. Lee, K.B., Jang, H.G., 1997. A numerical prediction on the turbulent flow in closely spaced bare rod arrays by a nonlinear k–i model. Nuclear Engineering and Design 172, 351–357.
Launder, B.E., Spalding, D.B., 1973. The numerical computational of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3, 269. Patankar, S.V., 1981. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corp., New York. Shih, T.-H., Zhu, J., Lumley, J.L., 1993. Arealizable Reynolds stress algebraic equation model, NASA TM 105993. Smith III, L.D., Conner, M.E., Liu, B., Dzodzo, M.B., Paramonov, D.V., Beasley, D.E., Langford, H.M., Holloway, M.V., 2002. Benchmarking computational fluid dynamics for application to PWR fuel. In: Proceedings of the 10th International Conference on Nuclear Engineering, Arlington, VI, USA, April 14–18. Wu, X., Trupp, A.C., 1993. Experimental study on the unusual turbulence intensity distribution in rod-to-wall gap regions. Experimental Thermal and Fluid Science 6 (4), 360–370. Yadigaroglu, G., Anderani, M., Dreier, J., Coddington, P., 2003. Trends and needs in experimentation and numerical simulation for LWR safety. Nuclear Engineering and Design 221, 205–223. Yakhot, V., Orszag, S.A., Thangam, S., Gatski, T.B., Speziale, C.G., 1992. Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids A4 (7), 1510–1520. Yao, S.C., Hochreiter, L.E., Leech, W.J., 1982. Heat-transfer augmentation in rod bundles near grid spacers. Journal of Heat Transfer 104, 76–81.