Sensitivity study on the variation of a shell side heat transfer coefficient with longitudinal pitch variation in a staggered tube bank ASHRAF ALFANDI University of Science and Technology, Advanced Nuclear System Engineering Department, 217 Gajeong-Ro Yuseong-Gu, Daejeon, 305-350, Republic of Korea
[email protected] Young In Kim, Hyungi Yoon, Namgyun Jeong and Juhyeon Yoon Korea Atomic Energy Research Institute, 989-111 Daedeok-Daero, Yuseong-Gu, Daejeon, 305-353, Republic of Korea
[email protected],
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[email protected] Abstract In designing compact heat exchangers, the tube bank arrangement is of high importance since the variation of the longitudinal and transverse pitches affects the heat transfer and pressure drop in a heat exchanger. Smaller pitches allow a high performance compact heat exchanger at the expense of a high pressure drop. Normally, the transverse tube pitch is determined by the given requirement on the pressure drop limit through the heat exchanger. The longitudinal pitch has a quite different effect on the heat transfer and pressure drop depending on the in-line and staggered tube banks, respectively. In this study, the effect on a shell-side heat transfer coefficient is investigated using the CFD code FLUENT with a variation in longitudinal pitch to diameter ratio, SL, in the range of 1.15 to 2.6 with a fixed transverse pitch to diameter ratio. For the benchmark purposes with the available empirical correlation, typical thermal-hydraulic conditions for the Zukauskas correlation are assumed. Many sensitivity calculations for different mesh sizes and turbulent models are performed to check the accuracy of the numerical solution. A realizable κ-ε turbulence model was found to be in good agreement with results of the Zukauskas correlation among the other turbulence models, at least for the staggered tube bank. It was found that the average heat transfer coefficient of a crossflow over a staggered tube bank calculated using FLUENT is in good agreement with the Zukauskas correlation-calculated heat transfer coefficient in the range of 1.15 – 2.6. For a staggered tube bank, using the Zukauskas correlation seems to be valid down to SL = 1.15. Keyword: Heat transfer coefficient, staggered tube bank, longitudinal pitch, crossflow, turbulence model.
1. Introduction The tube banks within the heat exchanger can be arranged in either staggered or in-line configurations according to the heat transfer and pressure drop design optimization analysis. For a compact design of a shell and tube heat exchanger, longitudinal and transvers pitches are the most important parameters from a thermal performance optimization point of view. Normally, using a smaller longitudinal pitch enables more heat transfer area density to be utilized and to design a more compact heat exchanger, whereas the transverse pitch is determined mainly for meeting a specified pressure drop requirement of the heat exchanger. In this study, the effect on a variation of a shell side heat transfer coefficient with the longitudinal pitch variation in a staggered tube bank is investigated. 1.1 Literature Review Many researchers have investigated the heat transfer characteristics in tube banks. Pierson [1] and Huge [2] have carried out many experiments on the heat transfer in in-line and staggered tube arrangements. Colburn [3] proposed an empirical correlation for the calculation of the heat transfer in a staggered tube bank with more than ten rows. Grimison [4] has correlated the experimental data done by Pierson [1] and Huge [2]. Zukauskas [5] suggested empirical correlations to estimate the average Nusselt number for a tube bank, as a function of Reynolds number and Prandtl number. Khan [6] developed an analytical model to investigate the heat transfer from tube banks in a crossflow for both in-line and staggered arrangements. Bassiouny and Wilson [7] developed a mathematical model to simulate the laminar and turbulent flow fields in in-line and staggered tube banks. Kim [8] investigated numerically the effect of the longitudinal pitch on the heat transfer characteristics of the crossflow over in-line tube banks. Lee [9] identified the effect of an uneven horizontal pitch in a tube bank heat exchanger and derived a general correlation that can predict the individual heat transfer coefficient of each row for an arbitrary longitudinal pitch distribution. Few researchers have conducted a numerical investigation to study the effect of longitudinal pitch variation on the crossflow heat transfer of over-staggered tube banks. In the present study, the effect of the longitudinal pitch variation on the heat transfer coefficient of a cross flow over staggered tube banks while fixing the transverse pitch is investigated numerically using the CFD code FLUENT [10]. The calculation is modeled as a conjugate heat transfer problem to impose a non-constant wall temperature boundary condition on the tube surface. 2. Numerical Modeling
2.1 Geometry and boundary conditions Figure 1 shows a staggered tube bank that has transverse and longitudinal pitches. The transverse and longitudinal pitch- to- tube diameter ratios, ST and SL, respectively, are defined as 𝑆𝑇 =
𝑃𝑇
𝑆𝐿 =
𝑃𝐿
(1)
𝐷
(2)
𝐷
To study the effect of the longitudinal pitch variation, the longitudinal pitch- todiameter ratio SL is changed in the range of 1.15 – 2.6, but the transverse pitch- todiameter ratio ST is kept constant at 1.4. The tube diameter, D, is set to 1.0 cm (see figure 1 below.) D = 10-2 m
PT = 1.4
PL Figure 1 Cross-sectional view of the tube bundle with the definition of the geometrical parameters
The red-dotted box shown in figure 1 represents the two-dimensional computational domain used in this study. To solve as a conjugate heat transfer problem, the three solid regions shown in figure 2 are modelled to describe a tube metal thickness of 1.5×10-5 m.
Figure 2 Computational domain and boundary conditions
In this study, typical thermal hydraulic parameters are taken from typical once-through steam generator design data [11]. These numerical values just represent a physically meaningful set of data. All numerical calculations are performed at a Reynolds number of 8.9×104. Having known ReD, the maximum velocity can be calculated by 𝐷
=
𝐷
(3)
In the range of 1.15 – 2.6 of SL, the flow will have a maximum velocity of 1.09 m/s at the transverse cross section [12] because √(𝑃𝑇 )
( 𝐿)
(𝑃𝑇 𝐷)
(4)
At the inlet boundary, the hot water flow rate is set to 1.59 kg/s and the upstream bulk temperature is assumed to be constant at 297.4ᵒC. Considering the repeated pattern of the flow at the inlet and outlet boundaries, a periodic boundary condition is prescribed. Because of the symmetry in the upper and lower parts of the computational domain, symmetric boundary conditions are applied, as shown in figure 2. The working fluid in the tube side is assumed to have a constant saturation temperature of 255.27ᵒC. 2.2 Mesh generation
Figure 3 Computational grid
An unstructured, the quadrilateral dominant method is used to generate a grid for the entire computational domain. Two examples of the meshes are shown in figures 3 (a) and 3 (b) for the two extreme cases at SL = 1.15 and 2.6, having a total number of elements of 58,040 and 109,615, respectively. A two-layer model is adapted to treat the wall boundary layer near the wall. Along the fluid – solid interface boundary, a maximum of 25 inflation layers, are used to have a maximum y = 2.5×10-6 m at the first grid so that y+ ~ 0.5. A mesh sensitivity study was conducted by investigating several cases of different grid numbers, as shown in figure 4. The mesh was continually refined until a variation in the heat transfer coefficient is small enough to be 0.15 %.
2.05E+04
2.00E+04
Heat transfer coefficient (W/m²-K)
1.95E+04
1.90E+04
SL = 1.15
SL = 2.6
1.85E+04
1.80E+04
1.75E+04
1.70E+04
1.65E+04
1.60E+04 0
20000
40000
60000
80000
100000
120000
Grid number
Figure 4 Mesh sensitivity study
For pressure-velocity coupling, the SIMPLE algorithm was utilized. A second-order upwind scheme has been applied for the convection terms of mass, momentum, and energy conservation equations. All calculations are considered to be converged when the heat transfer coefficient and mass flow rate reach a steady state value. 3. Results and discussion 3.1 Average heat transfer coefficient and turbulence model effect For the purpose of benchmarking, the Zukauskas correlation [5] has been used to estimate the heat transfer coefficient at different longitudinal pitch-to-diameter ratios in the range of 1.15 – 2.6. The Zukauskas correlation is as follows for a staggered tube bank: 𝑃
𝑁𝑢 = 0.35 × (𝑃𝑇 ) 𝐿
0.
0.6
𝑃𝑟
0. 5
𝑟 0.36 (𝑃𝑟 ) w
(5)
where the subscript w means that the fluid property is to be evaluated at the tube wall temperature. Other fluid properties are to be evaluated at the bulk fluid temperature. Figure 5 shows the results of the sensitivity study on the variation of the average heat transfer coefficient for different turbulence models including the standard κ-ε [13], Realizable κ-ε [14], Re-Normalization Group (RNG) κ-ε [15], and SST κ-ω. The results of the sensitivity study demonstrate that the Realizable κ-ε turbulence model gives the same variation trend of the average heat transfer coefficient with that of the Zukauskas correlation. In the staggered tube banks, the adverse pressure gradient field is not dominant, and the SST κ-ω turbulence model over-predicts the heat transfer coefficient values compared with the values of the Zukauskas correlation [12]. The
results also demonstrate that, as the longitudinal pitch-to-diameter ratio, SL, decreases, the flow speed becomes larger, and thus the heat transfer coefficient increases, as shown in figure 5. Zukausckas Realizable K-epsilon
22000
SST k-omega
Heat transfer coefficient Wm²-K
21000
RNG k-epsilon Standard k-epsilon
20000 19000 18000 17000 16000 15000 14000 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Longitudinal pitch to diameter ratio variation
Figure 5 FLUENT – calculated heat transfer coefficient at different turbulence models
3.2 local heat transfer coefficient The velocity contour in figure 6 shows that, for tube banks with a smaller longitudinal pitch, the fluid velocity impinging on the tube surface is higher compared to that of a widely spaced tube bank case. This high-speed impinging fluid velocity makes the boundary layer thickness on the head-on spot thinner in the smaller longitudinal pitch case. The thinner laminar boundary layer manifests the higher local heat transfer coefficient at the head-on spot, as shown in figure 7.
Figure 6 Flow velocity contour for different pitches
Figure 7 shows that the calculated local heat transfer coefficient around the tube surface is largest at the stagnation point located at the upstream region, and decreases with distance along the tube surface as the boundary layer thickness increases. The heat transfer coefficient reaches its minimum value after the separation point. Beyond the separation point, the local heat transfer coefficient decreases. However, at the rear portion of the tube, the heat transfer coefficient increases again because of the considerable fluid sweeping phenomena by alternative periodic vortex shedding eddies over the rear. [16] 3.5E+04
SL = 1.15
Heat transfer coefficient (W/m2-K)
3.0E+04
SL = 2.6
2.5E+04
2.0E+04
1.5E+04
1.0E+04
5.0E+03
0.0E+00 0
30
60
90
120
150
180
Angle (ϕ)
Figure 7 Local heat transfer coefficient
As shown in figure 7 above, the heat transfer to the tube is not uniform around the tube surface, which means that it is physically not correct to assume a constant wall temperature boundary condition for the shell-side surface. To take this issue into account, all calculations are performed as a conjugate heat transfer problem. Figure 8 shows that the shell-side surface temperature is changing around the tube surface rather than being constant. It is noticeable that the local temperature profile around the tube has the same trend as the heat transfer coefficient shown in figure 8.
290 SL = 1.15 SL = 2.6
288
Temperature (ᵒC)
286 284 282 280 278 276 0
30
60
90 Angle (ϕ)
120
150
180
Figure 8 Wall temperature profile for different pitches
4. Conclusion In the present study, a numerical model was developed to study the effect of longitudinal pitch variation on the shell-side heat transfer coefficient of a crossflow over a staggered tube bank. Many sensitivity studies were performed including different numbers of meshes and turbulence models to minimize the numerical simulation uncertainties. The Realizable κ-ε turbulence model was found to be in good agreement with the results of the Zukauskas correlation among the other turbulence models for a staggered tube bank case. The conjugate heat transfer principle is applied where the wall thickness is modelled as a separate tube metal zone. The heat transfer coefficient increases as the longitudinal pitch decreases owing to the increased fluid velocity and turbulence. The profile of the calculated heat transfer coefficient was found to be in a good agreement with the Zukauskas correlation heat transfer coefficient in the longitudinal pitch-to-diameter ratio range of 1.15 – 2.6. For a staggered tube bank, using the Zukauskas correlation seems to be valid down to SL = 1.15. 5. Acknowledgement This work has been carried out under the auspices of the Jordan Research and Training Reactor Project being operated by Korea Atomic Energy Research Institute. 6. References 1. O. L. Pierson, Experimental investigation of the influence of tube arrangement on convective heat transfer and flow resistance in cross flow of gases over tube banks, ASME 59, 563-572 (1937). 2. E.C. Huge, Experimental investigation of effects of equipment size on convection heat transfer and flow resistance in cross flow of gases over tube banks, ASME, 59, 573-581 (1937).
3. A.P. Colburn, A method of correlating forced convection heat transfer data and a comparison with fluid friction, Trans. Am. Inst. Chem. Eng. 29, 174-210 (1933). 4. E.D. Grimison, Correlation and utilization of new data on flow resistance and heat transferfor cross flow of gases over tube banks, ASME 59, 583-594 (1933). 5. A.A. Zukauskas, Heat Transfer from Tubes in Crossflow, Adv. Heat Transfer 8, 93-160 (1972). 6. W.A. Khan, J.R. Culham, M.M. Yovanovich, Convection heat transfer from tube banks in cross flow: Analytical approach, Int. J. Heat Mass Transfer 49, 4831-4838 (2006). 7. M. Khalil Bassiouny, A. Safwat Wilson, Modeling of heat transfer for flow across tube banks, Chem. Eng. Process 39, 1-14 (2000). 8. T. Kim, Effect of longitudinal pitch on convective heat transfer in crossflow over in-line tube banks, Ann. Nucl. Energy 57, 209-215 (2013). 9. D. Lee, A. Joon, S. Shin, Uneven longitudinal pitch effect on tube bank heat transfer in cross flow, Appl. Them. Eng 51, 937-947 (2013). 10. Inc. Fluent, Fluent User’s Guide (2006). 11. J. Yoon, J.-P. Kim, H.-Y. Kim, D. J. Lee, M. H. Chang, Development of a computer code, ONCESG, for thermal-hydraulic design of a once-through steam generator, J. Nucl. Sci. Technol 37, 445-454 (2000) 12. F. Kreth, M.S. Bohn, Principles of heat transfer (Books/Cole, Thomas Learning, 2001). 13. B.E. Launder, D.B. Spalding, Lectures in mathematical models of turbulence. (Academic Press, 1982). 14. T.-H. Shih, W.W. liou, A. Shibber, Z. Yang, J. Zhu, A new κ-ε eddy-viscosity model for high reynolds number turbulent flows-Model development and validation, Computer Fluids 24, 227-238 (1995). 15. V. Yakhot, S.A. Orszag, S. Thangma, T.B. Gatski, S.G. Speziale, Development of turbulent models for shear flows by a double expansion technique, phys. Fluids A 4, 1510-1520 (1992). 16. J.E. Bardina, P.G. Huang, T.J. Coakley, Turbulence modelling validation, testing, and development. NASA TM 110446, (1997).
