Published on: **Mar 4, 2016**

Published in:
Engineering

- 1. International Journal of Applied Engineering Research ISSN 0973-4562 Volume 9, Number 18 (2014) pp. 5125-5139 © Research India Publications http://www.ripublication.com Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts in a Parallel Plate Channel S. D. Mhaske #1 , S. P. Sunny #2 , S. L. Borse #3 , Y. B. Parikh#4 #1 Department of Mechanical Engineering, Symbiosis Institute of Technology, Pune, India. #2 Department of Mechanical Engineering, Symbiosis Institute of Technology, Pune, India. #3 Department of Mechanical Engineering, Rajarshi Shahu College of Engineering, Pune, India. #4 Department of Mechanical Engineering, Symbiosis Institute of Technology, Pune, India. 1 mhaskesiddharth@gmail.com, 2 sobzsunny@gmail.com, 3 sachinlb@yahoo.co.uk, 4 yash.parikh@sitpune.edu.in Abstract Flow and heat transfer for two-dimensional laminar flow at low Reynolds number for five in-line ducts of various cross-sections in a parallel plate channel are studied in this paper. The governing equations were solved using finite-volume method. A commercial CFD software, ANSYS Fluent 14.5 was used to solve this problem. A total of three different non-conventional cross-sections and their characteristics are compared with that of circular cross-section. Shape-1, Shape-2 and Shape-3 performed better for heat transfer rate than the circular cross-section, but also offered higher resistance to the flow. Shape-1 offered less resistance to flow at Re < 200 but post Re = 200 the resistance equalled to that of the Shape-3. In overall, Shape-2 performed better when the heat transfer and resistance to flow were considered. Keyword- CFD, Flow over non-circular ducts, Heat transfer, Pressure drop, Parallel Plates I. INTRODUCTION The quest for heat transfer enhancement in most of the engineering applications is a never ending process. The need for better heat transfer and low flow resistance has led to extensive research in the field of heat exchangers. Higher heat transfer and low pumping
- 2. 5126 S. D. Mhaske et al power are desirable properties of a heat exchanger. Tube shape and its arrangement highly influences flow characteristics in a heat exchanger. Flow past cylinders, esp. circular, flat, oval and diamond arranged in a parallel plate channel were extensively studied by Bahaidarah et al. [1-3]. They carried domain discretization in body-fitted co-ordinate system while the governing equations were solved using a finite-volume technique. Chhabra [4] studied bluff bodies of different shapes like circle, ellipse, square, semi-circle, equilateral triangle and square submerged in non-Newtonian fluids. Kundu [5] [6] investigated fluid flow and heat transfer coefficient experimentally over a series of in-line circular cylinders in parallel plates using two different aspect ratios for intermediate range of Re 220 to 2800. Grannis and Sparrow [7] obtained numerical solutions for the fluid flow in a heat exchanger consisting of an array of diamond-shaped pin fins. Implementation of the model was accomplished using the finite element method. Tanda [8] performed experiments on fluid flow and heat transfer for a rectangular channel with arrays of diamond shaped elements. Both in-line and staggered fin arrays were considered in his study. Jeng [9] experimentally investigated pressure drop and heat transfer of an in-line diamond shaped pin-fin array in a rectangular duct. Ota [10] studied the Heat transfer characteristics and flow behaviours around an elliptic cylinder at high Reynolds number. Chen et al. [11] [12] calculated flow and conjugate heat transfer in a high- performance finned oval tube heat exchanger element have been calculated for a thermally and hydrodynamically developing three-dimensional laminar flow. Computations have been performed with a finite volume method based on the SIMPLEC algorithm for pressure correction. Zdravistch [13] numerically predicted fluid flow and heat transfer around staggered and in-line tube banks and found out close agreement experimental test cases. Tahseen et al. [14] conducted numerical study of the two-dimensional forced convection heat transfer across three in-line flat tubes confined in a parallel plate channel, the flow under incompressible, steady-state conditions. They solved the system in the body fitted coordinates (BFC) using the finite volume method (FVM). Gautier and Lamballais [15] proposed a new set of boundary conditions to improve the representation of the infinite flow domain. Conventional tube/duct shapes have been investigated for fluid flow and heat transfer characteristics. In this paper two-dimensional, steady state, laminar flow over non- conventional non-circular (Shape-1, Shape-2 and Shape-3) in-line ducts confined in a parallel plate channel is considered. The different geometries investigated are Circular, Shape-1, Shape-2 and Shape-3. Only representative cases are discussed in this paper. NOMENCLATURE A area of heat transfer surfaces in a module (m2 ) Cp specific heat (J/kg.K) D blockage height of a tube inside the channel (m) f module friction factor f0 module friction factor for circular tubes H height of the channel (m) L module length (m) k thermal conductivity (W/m.K) l length of the tube cross-section along the flow direction (m) Nu module average Nusselt number (= 𝑚̇ 𝐶 𝑝 𝛥𝑇 𝑏 𝐻 𝑘𝐴(𝑇 𝑤−𝑇 𝑚) ) Nu0 module average Nusselt number for the reference case of circular tubes Nu+ heat transfer enhancement ratio (= Nu/Nu0)
- 3. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5127 Nu* heat transfer performance ratio ṁ mass flow rate (kg/s) p pressure (Pa) Re Reynolds number (= ρVH/µ) T temperature (K) Tm average of module inlet bulk temperature and module outlet bulk temperature (K) Tw temperature of heat transfer surfaces (K) Uin velocity at the channel inlet (m/s) 𝑉⃗ velocity field y vertical distance from the channel bottom (m) Δp pressure drop across a module (Pa) Δp* normalized pressure drop (=Δp /ρ 𝑈ⅈ𝑛 2 ) ΔTb change in bulk temperature across a module (K) µ dynamic viscosity (N.s/m2 ) ρ density 𝛻 del operator II. MATHEMATICAL FORMULATION The governing mass, momentum, and energy conservation equations [1] for steady incompressible flow of a Newtonian fluid are expressed as: Mass conservation: 𝛻 ⋅ 𝑉⃗ = 0 (1) Momentum conservation: 𝑉⃗ ⋅ (𝛻𝑉⃗ ) = −𝛻𝑝 + 𝑣𝛻2 𝑉⃗ (2) Energy Conservation: 𝜌𝐶 𝑝 𝑉⃗ ⋅ (𝛻𝑇) = 𝑘𝛻2 𝑇 (3) A commercial CFD software, ANSYS Fluent 14.5 was used to solve the governing equations. The pressure-velocity coupling is done by Semi-Implicit Method for Pressure Linked Equations (SIMPLE) scheme. III. GEOMETRIC CONFIGURATION A total of four different duct cross-sections namely Circular, Shape-1, Shape-2andShape-3 were analysed. The five in-line ducts were confined in a parallel-plate channel. The distance between two consecutive ducts was kept constant. H/D, L/D and l /D ratio (shown in Fig. 1) kept constant for all cross-sections of the ducts. The unobstructed length of the pipe before the first and the last module kept as one module length and three module lengths respectively so that the flow is fully developed. It is also ensured that the flow is fully developed when it enters the channel. The ducts and the parallel plate channel walls were assumed to be of infinite extent in the z-direction i. e. perpendicular to the paper. Hence, the flow could be considered as two- dimensional. Since, the geometry is symmetric along the x-axis, only the lower half portion is considered for the computation purpose.
- 4. 5128 S. D. Mhaske et al (a) (b) (c) (d) Fig. 1. Various duct cross-sections studied in this work: (a) Circular cross-section ducts, (b) Shape-1 cross-section ducts, (c) Shape-2 cross-section ducts, (d) Shape-3 cross-section ducts IV. GRID GENERATION The grid for all the cases is generated using the Automatic Method with all Quadrilateral elements in the Mesh Component System of the ANSYS Workbench 14.5 software. In the numerical simulation, the grid of 11,000 nodes is found to be fair enough for grid independent solution. Refining the grid proved to be unnecessary and resulting in utilizing more computer resources. Fig. 2 shows the grid generated for all the configurations. (a) (b)
- 5. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5129 (c) (d) Fig. 2. Grid for geometries considered: (a) Circular cross-section ducts, (b) Shape-1 cross-section ducts, (c) Shape-2 cross-section ducts, (d) Shape-3 cross-section ducts V. BOUNDARY CONDITIONS Fig. 3 represents the schematic of the boundary conditions for all the cross-sections used. A fully developed flow with velocity profile (u = Uin) and with temperature (T = Tin) was assigned for the flow at inlet. The temperature was kept different than that of the plate and duct walls. A no slip (u = v = 0) boundary condition was given to the plate and duct surface. The surface temperature of the plates and the ducts were taken to be constant (T = Tw). At the symmetry normal component of velocity and normal gradient of other velocity components were taken as zero. At the outflow boundary there is no change in velocity across boundary. Fig. 3. Flow domains studied in this study with the boundary condition VI. VALIDATION The parallel-plate channel problem corresponds to the rectangular duct with infinite aspect ratio. The calculated value of Nusselt number is 7.541. Comparing the calculated Nusselt number with that of the theoretical Nusselt number value of 7.54 [16], shows that our computed values are in good agreement with that of the theoretical values. Using the geometric parameters L/D = 3 and H/D = 2 for circular cross-section, the normalized pressure drop (Δp* ) and Nusselt number for third module of the parallel plate channel, when compared with the values of Kundu et al. [5] shows good agreement between both the values.
- 6. 5130 S. D. Mhaske et al Fig. 4. Validation with previous work Kundu et al. [5]: normalized pressure drop across third module TABLE I: Validation with previous work of Kundu et al. [6]: module average Nusselt number for L/D= 3, H/D= 2 Re Second Module Third Module Fourth Module Kundu et al. [6] 50 9.4 9.4 9.8 Present Work 9.51 9.48 9.48 Kundu et al. [6] 200 12.5 12.6 12.8 Present Work 12.68 12.63 12.64 VII. RESULTS AND DISCUSSIONS The numerical simulation was carried out for Reynolds number ranging from 25-350 and Prandtl number considered as 0.74. A. Onset of Recirculation: For any duct shape no separation of flow was observed below Re = 30. Circular duct displayed the signs of recirculation at lowest Re value (Re = 32) [1]. The Shape-1 and Shape-2 ducts displayed the onset of recirculation at Re value 45 while the Shape-3 showed the signs of recirculation just under Re = 35, almost similar to the value of Circular duct. As the values of Reynolds number is increased the recirculation region starts increasing. At Re = 150 the length of the recirculation region is slightly smaller than the distance between the two consecutive ducts in the flow direction while at Re = 350 the length of the recirculation region is greater than the distance between the two consecutive ducts. Since there is no obstruction to the flow downstream of the last duct the elliptic behaviour of the flow is observed as shown in Fig. 5-8.
- 7. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5131 (a) (b) (c) Fig. 5: Effect of Reynolds number on the streamline over Circular cross-section ducts: (a) Re = 25, (b) Re = 150 and (c) Re = 350 (a) (b) (c) Fig. 6: Effect of Reynolds number on the streamline over Shape-1 cross-section ducts: (a) Re = 25, (b) Re = 150 and (c) Re = 350 (a)
- 8. 5132 S. D. Mhaske et al (b) (c) Fig. 7: Effect of Reynolds number on the streamline over Shape-2 cross-section ducts: (a) Re = 25, (b) Re = 150 and (c) Re = 350 (a) (b) (c) Fig. 8: Effect of Reynolds number on the streamline over Shape-3 cross-section ducts: (a) Re = 25, (b) Re = 150 and (c) Re = 350 B. Fluid Flow Characteristics: From the Fig. 5-8 and the data from Table II we can observe that the streamlines past the ducts 1-4 are similar in nature. Also the Fig. 9 shows the normalized u-velocity profiles at module inlet for modules 2,3,4 and 5 for the lower portion of the channel for various cases at Re = 150. We can observe that the normalized u-velocity profiles at module inlet for modules 2, 3, 4 and 5 matches well with each other at all the Reynolds number. Hence, we can say that the flow is periodically developed for inner modules. As the flow is periodically developed, the pressure drop and Nusselt number values for inner modules are same we can say that the study of any module from 2-5 is sufficient. In present study we have selected module 3. Fig. 10 shows the normalized pressure drop (Δp* ) as a function of Reynolds number. We can observe that, as the value of Re increases there is a steep descend in normalized pressure drop for all the ducts presented in this study. All the ducts show higher normalized pressure drop than the Circular ducts. The Shape-3 duct shows highest normalized pressure drop. There is geometric similarity between Shape-2 and Shape-3. Hence there is no significant normalized pressure drop difference between Shape-2 and Shape-3 ducts until Re = 50, but as the Re value increases normalized pressure drop difference between Shape-2 and Shape-3 ducts starts increasing. Recirculation of flow plays a vital role in this behaviour of normalized pressure drop.
- 9. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5133 The ratio of normalized pressure drop for noncircular duct module (Δp* ) to that for the circular duct module (Δp0 * ) is shown in Fig. 11. The Shape-1 ducts presents lower pressure drop ratio than its counterparts for Re < 200. On the other hand the pressure drop ratio for Shape-2 ducts dips considerably. The pressure drop ratio shown in Fig. 11 of Shape-1, Shape- 2 and Shape-3 ducts is higher than 1 for all the values of Reynolds number which means that they offer higher flow resistance than Circular ducts. This will lead to higher pumping power for heat exchanger with any of the above duct cross-section. (a) (b) (c) (d) Fig. 9. Normalized u-velocity profiles at inlet of modules at Re=150 Fig. 10. Variation of normalized pressure drop (Δp*) with Re for third module
- 10. 5134 S. D. Mhaske et al Fig. 11.Variation of Δp*/Δp0 * with Re for third module C. Heat Transfer Characteristics: Fig. 12 presents the module average Nusselt number as a function of Re for module 3 for Shape- 1, Shape-2, Shape-3 and Circular ducts. The Nusselt number for Re < 50 increases rapidly as the Re increases. The duct shape has a very little effect on heat transfer rate at Re < 50. Post Re = 50 the Nusselt number varies exponentially with the cross-section of the duct. Shape-1, Shape-2 and Shape-3 ducts showed higher heat transfer rate than the circular ducts. At higher values of Re the recirculation of the flow takes place. This recirculation vortices do not transfer heat energy to the main stream of flow. Hence, their contribution to heat transfer is very less. Fig. 13 presents the heat transfer enhancement ratio (Nu+ ) as a function of Re for module 3 for Shape-1, Shape-2 and Shape-3 ducts. Shape-1, Shape-2 and Shape-3 ducts performed better than the Circular ducts. Shape-2 ducts showed highest heat transfer enhancement ratio (Nu+ ) post Re = 50. Fig. 14 presents the heat transfer performance ratio (Nu* ) as a function of Re for module 3 for Shape-1, Shape-2 and Shape-3 ducts. Nu* represents heat transfer enhancement per unit increase in pumping power. At Re < 40 Shape-1 ducts performed better than its counterparts, but post Re = 40 Shape-2 performs better. Fig. 12. Variation of Nu with Re for third module
- 11. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5135 Fig. 13. Variation of Nu+ with Re for third module Fig. 14. Variation of Nu* with Re for third module
- 12. 5136 S. D. Mhaske et alTABLEII:ComparisonofnormalizedpressuredropandmoduleaverageNusseltnumberforallthegeometrictubeshapes Shape-3 Nu* 0.976 0.977 0.977 0.976 0.980 1.019 1.020 1.019 1.020 1.021 1.043 1.031 1.033 1.032 1.039 1.062 1.029 1.032 1.031 1.034 Nu+ 1.047 1.047 1.047 1.047 1.050 1.087 1.085 1.084 1.085 1.084 1.100 1.092 1.089 1.090 1.089 1.111 1.090 1.084 1.089 1.076 Nu 7.207 7.216 7.205 7.207 7.252 10.313 10.330 10.281 10.283 10.311 12.473 12.254 12.128 12.128 12.166 13.979 13.150 12.984 13.000 12.900 Δp*/Δp0* 1.234 1.232 1.232 1.233 1.229 1.216 1.205 1.203 1.204 1.196 1.173 1.186 1.173 1.179 1.151 1.143 1.188 1.162 1.179 1.128 Δp* 9.896 9.997 9.992 9.989 9.992 5.519 5.432 5.428 5.419 5.378 3.761 3.191 3.200 3.180 3.064 3.282 2.400 2.406 2.391 2.264 Shape-2 Nu* 0.976 0.976 0.977 0.974 0.983 1.020 1.017 1.019 1.017 1.018 1.047 1.029 1.037 1.031 1.038 1.064 1.030 1.041 1.036 1.037 Nu+ 1.048 1.046 1.046 1.044 1.053 1.089 1.082 1.083 1.082 1.082 1.100 1.090 1.090 1.091 1.087 1.106 1.095 1.087 1.095 1.078 Nu 7.211 7.210 7.202 7.188 7.276 10.325 10.298 10.267 10.259 10.285 12.474 12.232 12.133 12.131 12.140 13.916 13.215 13.017 13.072 12.918 Δp*/Δp0* 1.238 1.231 1.230 1.232 1.231 1.215 1.205 1.201 1.206 1.199 1.158 1.189 1.160 1.182 1.147 1.122 1.200 1.138 1.181 1.121 Δp* 9.927 9.992 9.978 9.984 10.006 5.513 5.430 5.419 5.426 5.389 3.715 3.198 3.165 3.188 3.053 3.222 2.423 2.358 2.395 2.251 Shape-1 Nu* 0.996 0.996 0.996 0.995 0.977 1.016 1.013 1.014 1.012 1.015 1.026 1.010 1.012 1.004 1.024 1.034 1.007 1.007 1.001 1.023 Nu+ 1.026 1.025 1.025 1.025 1.005 1.045 1.042 1.042 1.043 1.042 1.052 1.044 1.044 1.047 1.042 1.061 1.048 1.045 1.056 1.031 Nu 7.059 7.063 7.057 7.059 6.945 9.915 9.912 9.882 9.883 9.908 11.934 11.716 11.622 11.648 11.638 13.360 12.647 12.506 12.609 12.361 Δp*/Δp0* 1.093 1.091 1.090 1.093 1.090 1.088 1.088 1.086 1.094 1.083 1.080 1.104 1.097 1.135 1.054 1.082 1.128 1.114 1.177 1.024 Δp* 8.766 8.854 8.846 8.858 8.858 4.939 4.902 4.901 4.924 4.870 3.463 2.970 2.994 3.061 2.806 3.106 2.279 2.309 2.387 2.057 Circular Nuo 6.882 6.890 6.883 6.885 6.908 9.486 9.517 9.482 9.479 9.509 11.340 11.224 11.134 11.123 11.172 12.588 12.070 11.973 11.937 11.988 Δp0* 8.020 8.118 8.113 8.104 8.128 4.538 4.507 4.510 4.500 4.496 3.208 2.690 2.729 2.697 2.662 2.871 2.020 2.071 2.028 2.008 Module Re=25 1 2 3 4 5 Re=50 1 2 3 4 5 Re=100 1 2 3 4 5 Re=150 1 2 3 4 5
- 13. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5137TABLEII:Continued Shape-3 Nu* 1.076 1.026 1.025 1.033 1.015 1.079 1.020 1.016 1.031 1.007 1.081 1.010 1.007 1.025 1.004 1.082 1.000 0.998 1.020 1.001 Nu+ 1.122 1.086 1.077 1.092 1.052 1.124 1.078 1.067 1.091 1.043 1.127 1.065 1.059 1.087 1.041 1.131 1.052 1.051 1.082 1.040 Nu 15.769 13.776 13.604 13.706 13.087 17.657 14.254 14.063 14.323 13.062 19.558 14.606 14.432 14.868 12.943 21.418 14.883 14.722 15.352 12.790 Δp*/Δp0* 1.133 1.187 1.159 1.183 1.115 1.131 1.180 1.160 1.188 1.111 1.134 1.172 1.163 1.192 1.113 1.139 1.163 1.168 1.196 1.121 Δp* 3.039 1.978 1.979 1.969 1.860 2.886 1.712 1.710 1.704 1.615 2.779 1.526 1.525 1.521 1.450 2.700 1.388 1.391 1.387 1.329 Shape-2 Nu* 1.074 1.035 1.036 1.043 1.024 1.081 1.032 1.029 1.046 1.016 1.088 1.026 1.023 1.043 1.019 1.095 1.025 1.015 1.040 1.015 Nu+ 1.112 1.105 1.079 1.104 1.061 1.118 1.106 1.071 1.109 1.056 1.126 1.103 1.063 1.107 1.065 1.136 1.106 1.055 1.107 1.067 Nu 15.622 14.011 13.625 13.858 13.199 17.560 14.629 14.108 14.550 13.230 19.539 15.136 14.490 15.147 13.243 21.515 15.648 14.774 15.699 13.120 Δp*/Δp0* 1.108 1.215 1.129 1.185 1.115 1.106 1.230 1.125 1.191 1.124 1.109 1.244 1.123 1.198 1.142 1.114 1.256 1.121 1.205 1.162 Δp* 2.975 2.026 1.928 1.973 1.860 2.821 1.784 1.659 1.710 1.634 2.716 1.620 1.472 1.529 1.486 2.642 1.499 1.334 1.398 1.378 Shape-1 Nu* 1.046 1.007 1.000 1.003 1.007 1.057 1.007 0.990 0.992 1.003 1.068 1.006 0.982 0.976 1.008 1.079 1.004 0.974 0.960 1.012 Nu+ 1.078 1.054 1.042 1.065 1.011 1.097 1.057 1.036 1.057 1.009 1.115 1.057 1.029 1.042 1.019 1.135 1.056 1.024 1.025 1.028 Nu 15.154 13.361 13.162 13.369 12.575 17.222 13.982 13.643 13.873 12.643 19.350 14.499 14.027 14.257 12.676 21.500 14.939 14.339 14.543 12.648 Δp*/Δp0* 1.096 1.146 1.130 1.200 1.014 1.116 1.156 1.144 1.211 1.019 1.139 1.161 1.154 1.216 1.033 1.163 1.162 1.161 1.219 1.050 Δp* 2.942 1.910 1.930 1.997 1.691 2.848 1.677 1.685 1.738 1.482 2.791 1.512 1.512 1.553 1.345 2.756 1.387 1.382 1.414 1.245 Circular Nu0 14.053 12.681 12.630 12.548 12.435 15.706 13.225 13.175 13.125 12.525 17.352 13.717 13.625 13.681 12.435 18.945 14.149 14.009 14.185 12.299 Δp0 * 2.684 1.667 1.707 1.664 1.668 2.551 1.450 1.474 1.435 1.454 2.450 1.302 1.311 1.277 1.302 2.370 1.194 1.190 1.160 1.186 Module Re=200 1 2 3 4 5 Re=250 1 2 3 4 5 Re=300 1 2 3 4 5 Re=350 1 2 3 4 5
- 14. 5138 S. D. Mhaske et al VIII. CONCLUSION In an environment wherein the heat transfer rate is critical, any of the above mentioned non- conventional cross-section of the ducts (Shape-1, Shape-2 and Shape-3) will perform better than that of the circular cross-section ducts. The duct shape has a very little influence on the heat transfer rate at low Reynolds number. For optimal design of a heat exchanger for heat transfer enhancement per unit increase in pumping power, Reynolds number and cross-section of the duct must be carefully chosen. In overall, Shape-2 performed better when both, the heat transfer rate and resistance to flow were considered. ACKNOWLEDGEMENT The authors would like to present their sincere gratitude towards the Faculty of Mechanical Engineering in Symbiosis Institute of Technology, Pune and Rajarshi Shahu College of Engineering, Pune. REFERENCES [1] Haitham M. S. Bahaidarah, M. Ijaz and N. K. Anand, Numerical Study of Fluid Flow and Heat Transfer over A Series of In-line Noncircular Tubes Confined In A Parallel-Plate Channel, Numerical Heat Transfer, Part B, vol. 50, pp. 97–119, 2006. [2] Haitham M. S. Bahaidarah, N. K. Anand and H. C. Chen, A Numerical Study of Fluid Flow and Heat Transfer over a Bank of Flat Tubes, Numerical Heat Transfer, Part A, vol. 48, pp. 359–385, 2005. [3] Haitham M. S. Bahaidarah, A Numerical Study of Heat and Momentum Transfer over a Bank of Flat Tubes, Ph. D. Thesis, Texas A&M University, College Station, Texas, USA, Aug. 2004. [4] R.P. Chhabra, Fluid Flow and Heat Transfer from Circular and Noncircular Cylinders Submerged in Non-Newtonian Liquids, Advances in Heat Transfer, vol. 43, pp. 289–417, 2011. [5] D. Kundu, A. Haji-Sheikh and D. Y. S. Lou, Heat Transfer Predictions in Cross Flow over Cylinders between Two Parallel Plates, Numerical Heat Transfer, Part A, vol. 19, pp. 361- 377, 1991. [6] D. Kundu, A. Haji-Sheikh and D. Y. S. Lou, Heat Transfer in Crossflow over Cylinders between Two Parallel Plates, J. Heat Transfer, vol. 114(3), pp. 558-564, 1992. [7] K B. Grannis and E. M. Sparrow, Numerical Simulation of Fluid Flow Through an Array of Diamond-Shaped Pin Fins, Numerical Heat Transfer, Part A, vol. 19, pp. 381-403, 1991. [8] Giovanni Tanda, Heat Transfer and Pressure Drop in a Rectangular Channel with Diamond Shaped Elements, Inter. J. Heat Mass Transfer, vol. 44, pp. 3529–3541, 2001. [9] Tzer-Ming Jeng, Thermal Performance of In-line Diamond-Shaped Pin Fins in a Rectangular Duct, International Communications in Heat and Mass Transfer, vol. 33, pp. 1139–1146, 2006. [10] Terukazu Ota, Hideya Nishiyama and Yukiyasu Taoka, Heat Transfer and Flow around an Elliptic Cylinder, Int. J. Heat Mass Transfer, vol. 27, no. 10, pp. 1771–1779, 1984. [11] Y. Chen, M. Fiebig, and N. K. Mitra, Conjugate Heat Transfer of A Finned Oval Tube Part A: Flow Patterns, Numerical Heat Transfer, Part A, vol. 33, pp.371-385, 1998. [12] Y. Chen, M. Fiebig, and N. K. Mitra, Conjugate Heat Transfer of a Finned Oval Tube Part B: Heat Transfer Behaviors, Numerical Heat Transfer, Part A, vol. 33, pp.387-401, 1998.
- 15. Pressure and Heat Transfer over a Series of In-line Non-Circular Ducts 5139 [13] Franz Zdravistch, Clive A. Fletcher and Masud Behnia, Numerical Laminar and Turbulent Fluid Flow and Heat Transfer Predictions in Tube Banks, Int. J. Num. Meth. Heat Fluid Flow, Vol. 5, pp. 717-733, 1995. [14] Tahseen A. Tahseen, M. Ishak and M. M. Rahman, A Numerical Study of Forced Convection Heat Transfer over a Series of Flat Tubes between Parallel Plates, Journal of Mechanical Engineering and Sciences (JMES), vol. 3, pp. 271-280, 2012. [15] R. Gautier, D. Biau and E. Lamballais, A Reference Solution of the Flow over a Circular Cylinder at Re = 40, Computers & Fluids, vol.75, pp. 103–111, 2013. [16] Yunus A. Cengel and Afshin J. Ghajar, Heat and Mass Transfer: Fundamentals and Applications, 4th ed., Tata McGraw Hill, 2011.