Study of flow and heat transfer in a 3D pin-fin channel

Transcription

1 Study of flow and heat transfer in a 3D pin-fin channel Jean-Eloi Lombard Swiss Institute of Technology Lausanne Supervisor : Dr. Mark Sawley Assistants : Laura Augello, Thierry Cornu December 27, 2011

2

3 Contents 1 Introduction 1 2 Problem definition 1 3 Techniques employed Mesh Turbulence model Numerical methods Results and discussion Convergence Conservation of heat Qualitative post-processing Contours Vector field Pressure and Temperature contours Velocity and Temperature profile Streamlines Velocity and Temperature plane-cut Quantitative post-processing Comparison with experimental extrapolation Total pressure drop Total heat flux Pin heat-flux and position Cooling Efficiency Proposed Improvements Numerical solution improvements Design improvement of the fin-pin channel geometry Conclusion 15

4

5 List of Figures 1 Rectangular pin-fin channel. The cross-flow enters through the front left side of the channel. The upper solid wall has been removed in the figure for visibility 1 2 Geometrical parameters. The array is centered on the origin, D = 12.5[mm], L x,y = 31.25[mm], X i = 125[mm], X c = 312.5[mm], Y c = 31.25, Z c = 12.5[mm] 1 3 Overview of the fine 59,964 hexahedral mesh from +Z View of the boundary layers around each pin in the fine 59,964 hexahedral mesh from +Z From left to right y+ values for each of the four full pins (Pin1, Pin2, Pin3 and Pin4) Convergence of the residuals of continuity, energy, k and ɛ as a function of iteration number for the three available wall functions : standard, unsteady and enhanced Convergence of the heat flux for all 10 pins as a function of iteration number for the three available wall functions : standard, unsteady and enhanced Relative pressure field distribution on the symmetry plane (z=0) Absolute temperature distribution on the symmetry plane (z=0) Velocity magnitude distribution on the symmetry plane (z=0) Velocity vector field in flow in near pin Enlarged view of the velocity vector field at the detachment zone on pin Relative pressure distribution on the pins and absolute temperature distribution on the symmetry plane Temperature and x component of velocity profile on line x = 0 z = 6.25[mm] Temperature and x component of velocity profile on line x = 0 y = A rake of 20 points (in z = 0[mm], x = 70[mm]) on the symmetry plane is defined from which the streamlines originate. They are computed both in the forward and backward directions x component of velocity in the plane (x = -20[mm]) The white interstice is Pin Absolute temperature in the plane (x = -20[mm]) The white interstice is Pin Aboslute pressure distribution on the symmetry plane. Each peak correspond to the point d arret of the pins Iso-surface of turbulence intensity. The down-wind pin keeps the high-turbulence flow near the upward pin. Note the difference in distance of the turbulence iso-surface for pins 3 and

6

7 1 Introduction The purpose of this study is to qualify and quantify the steady state flow and heat transfer capability of a 3D fin-pin array comprising of 32 pins arranged in a staggered arrangement located in a rectangular channel with solid walls. Figure 1 Rectangular pin-fin channel. The cross-flow enters through the front left side of the channel. The upper solid wall has been removed in the figure for visibility 2 Problem definition For the computations only a representative periodic section of the channel is considered the scheme of which is presented in (Fig. 2). The 3D model is obtained by extruding this 2D geometry along the z-axis. The cooling cross-flow is in the positive x direction. The pin-fin array consists of 7 stream-wise and 2 crosswise rows of equal-sized circular pins or radius (4 full pins and 6 half pins) in a uniformly spaced staggered arrangement. In this model, the two side surfaces (y = ±Y c /2) are considered to be periodic. The 3D geometry is obtained by extruding this 2D cross-section in the z direction. Furthermore since the flow is assumed symmetric in the z direction the flow is only computed within 0 z Z c /2. The cross-flow is air with the following properties : ρ [kg m 3 ] µ [P a s] C p [J kg 1 K 1 ] k [W m 1 K 1 ] Table 1 Properties of air Figure 2 Geometrical parameters. The array is centered on the origin, D = 12.5[mm], L x,y = 31.25[mm], X i = 125[mm], X c = 312.5[mm], Y c = 31.25, Z c = 12.5[mm] 1

8 Boundary conditions. At the inlet of the channel the flow velocity field U i = 5[m.s 1 ] is uniform and temperature is constant T i = 288[K]. The wall surface of each pin is at a fixed constant temperature T w = 398[K]. A uniform pressure is assumed at the outlet of the channel. The boundary at the side walls is defined as symmetric, the lower solid wall wall of the pin-fin array is adiabatic and has a no-slip boundary condition. Only half of the height in the z direction is computed so in z = 0[mm] correspond to a symmetric boundary condition. The effect of gravity is neglected. 3 Techniques employed 3.1 Mesh An automated mesh generator is used because it offers quick and descent meshes for computation use in simple flow cases. Both a coarse 33,006 and fine 59,964 hexahedral unstructured mesh are generated in two steps using : 1. first a 2D mesh is generated with a boundary layer around the pins (the radius of each pin is segmented into 51 uniform parts and the cells are formed with a uniform growth factor of 1.15 for 20 cells). For minimum computation error the regions before and after the array have been meshed by a cartesian grid. Hexahedral mapping is used for the region with the pins. Each pin has a boundary layer mesh. 2. the 2D mesh is extruded with the geometry along the z-direction. Figure 3 Overview of the fine 59,964 hexahedral mesh from +Z. Figure 4 View of the boundary layers around each pin in the fine 59,964 hexahedral mesh from +Z. 2

9 Mesh comparison. The coarse mesh is not able to correctly resolve the flow near the viscous sublayer even with a wall function (y+ max = 189 >> 30). For this reason the fine mesh is used to obtain all the significant results on flow and heat transfer in the remainder of this paper. Coarse Mesh Refined Mesh Cell # y+ max Cell quality % 80% Table 2 Mesh quality comparison Verification and validation For the non-equilibrium wall function to yield good results for the modeling of turbulence near the pin walls y+ should be smaller than 30 which is achieved with the fine mesh y+ [ ] x [m] Figure 5 From left to right y+ values for each of the four full pins (Pin1, Pin2, Pin3 and Pin4). 3.2 Turbulence model Although SST-k ω model has been shown to provide one of the most accurate results for heat and flow modeling amongst RANS methods in pin-fin configurations [6] the coarse cell limit imposed by the guidelines does not allow the construction of a mesh that guaranties y+ 1 at the walls. Due to this limitation it would not be feasible to use any 3 or more equation models or even the SST k ω which has been shown to provide most precise flow and heat transfer predictions. Hence the choice must be made between the k ɛ and 3

10 Spallart-Allmaras models. k ɛ being a two equation model is bound to better describe the flow. Even-though it requires solving an extra solution at every iteration the small grid used relative to computational power available renders this model the optimal choice. The choice of the k ɛ turbulence model is dictated by the size of the mesh and will not provide the most accurate prediction of flow since it doesn t predict flow detachment correctly [5] and since it is an important feature of cooling (an important fraction of cooling is done within the re-circulation zone) this model is not optimal. Cmu 0.09 C1-epsilon 1.44 C2-epsilon 1.92 TKE Prandtl Number 1 TDR Prandtl Number 1.3 Energy Prandtl Number 0.85 Wall Prandtl Number 0.85 Table 3 Parameters of k ɛ model Regarding wall treatment the non-equilibrium wall treatment is better suited for complex flows involving separation and strong pressure and temperature gradients (Fig. 6) for a mesh that does not guarantee a y+ 1 but still has y+ < Numerical methods A Pressure-Implicit with Splitting of Operator scheme 1 is used for the Pressure-Velocity coupling because it can take into account skewed cells and second-order spatial discretization is used for gradient, pressure, momentum, turbulent kinetic energy, turbulent dissipation rate and energy to minimize error. 1 the meshes are generated with ANSYS Gambit, the solutions computated with ANSYS Fluent and visualization with Paraview 4

11 4 Results and discussion 4.1 Convergence During the computations, monitor the residuals and the average heat flux at the surface of the pins. Provide detailed comments on the observed convergence history and whether it appears adequate to provide an accurate flow solution. A convergence of energy, continuity, k, and ɛ better than 5 orders of magnitude is obtained. Continuity Residual Std. k ε std. wall fct. Std. k ε enh. wall fct. Std. k ε unst. wall fct. Energy Residual Std. k ε std. Wall fct. Std. k ε Enhanced Wall fct. Std. k ε unst. wall fct Iterations Iterations Std. k ε std. Wall fct. Std. k ε Enhanced Wall fct. Std. k ε unst. wall fct Std. k ε std. Wall fct. Std. k ε Enhanced Wall fct. Std. k ε unst. wall fct. k Residual ε Residual Iterations Iterations Figure 6 Convergence of the residuals of continuity, energy, k and ɛ as a function of iteration number for the three available wall functions : standard, unsteady and enhanced. 5

12 Eventhough the k ɛ turbulence model with the unsteady wall function offers the best convergence properties in terms of residuals, there is no guarantee a priori it is converging to the correct physical solution. As can be seen in (Fig. 7) the three different wall functions result in completely different heat fluxes at convergence. There difference is of a factor 2 between unsteady wall function and the enhanced wall function while there is a 50% difference in predicted heat flux between the unsteady wall function and the standard one. The only to validate the correct wall function is by comparing the computed Nusselt number to the theoretical one using the Grimison correlation which give a 4% error for the unsteady wall function and more than 50% errors for both standard and enhanced wall functions Std. k ε std. Wall fct. Std. k ε Enhanced Wall fct. Std. k ε unst. wall fct. Total Pin Hit flux [W.m 2 ] Iterations Figure 7 Convergence of the heat flux for all 10 pins as a function of iteration number for the three available wall functions : standard, unsteady and enhanced. 4.2 Conservation of heat Thermal energy is conserved within in better than 5 orders of magnitude : Surface Heat transfer rate [W] Inlet Outlet Pin Pin Pin Pin Pin Pin Pin Pin Pin Pin Net Table 4 Energy conservation 6

13 5 Qualitative post-processing As expected the flow around a cylinder at Re 7000 exhibits a turbulent recirculation zone but the use of RANS does not allow for an appreciation of the vortex shedding taking place behind each pin. 5.1 Contours The relative pressure distribution betrays the recirculation zones behind the pins. Figure 8 Relative pressure field distribution on the symmetry plane (z=0). The absolute temperature distribution shows how the recirculation zones behind the pins works to advect the heat from the pins. Figure 9 Absolute temperature distribution on the symmetry plane (z=0). The velocity magnitude distribution shows a higher velocity of the flow between the rows of pins, with vanishing velocity at the boundary of the pins. Also the recirculation zones behind the pins have low velocity magnitude relative to the main flow. Figure 10 Velocity magnitude distribution on the symmetry plane (z=0). 7

14 5.2 Vector field Four main features of the flow can be seen in (Fig. 11), a broad view of the velocity vector field around pin 1. Firstly the impinging point in the bottom left-corner, secondly, the Figure 11 Velocity vector field in flow in near pin 1. recirculation zone in the down-flow of pin 1, thirdly the high-velocity main flow with a higher density of longer arrows and finally the detachment zone an enlarged view of which is provided in (Fig. 12). Figure 12 Enlarged view of the velocity vector field at the detachment zone on pin 1. 8

15 5.3 Pressure and Temperature contours As can be seen in (Fig. 13) the two components of heat dissipation in the flow are the horse shoe vortex on the leading edge of the pins with high temperature gradient and high pressure and the turbulent mixing in the recirculation behind the trailing-edge. Note the ripples in Figure 13 Relative pressure distribution on the pins and absolute temperature distribution on the symmetry plane. the back flow of Pin1 in the pressure distribution on the symmetry plane 5.4 Velocity and Temperature profile In (Fig. 14) it is interesting to note that temperature reaches the boundary condition temperature of the pins of 398[K] at the boundaries of pins 6 and 9 and also the recirculation zone downwind of pin 2 that betrays the a higher temperature relative to the high velocity region where it is minimal. The correlation is evident when looking at the x component of velocity on the same line. The velocity vanishes at the wall of pins 6 and 9 and even though the x component of velocity is barely negative one can still see the recirculation zone. Temperature [K] v x [K] y [mm] y [mm] Figure 14 Temperature and x component of velocity profile on line x = 0 z = 6.25[mm] In (Fig. 15) it is interesting to note that the temperature of the flow increases from the bottom to the symmetry plane but that the x component of the velocity profile is more 9

16 complex. The no-slip boundary condition is verified on the bottom plane the negative x component of the velocity betrays the recirculation zone down-wind of pin 2. Temperature [K] v x [K] z [mm] x z [mm] x 10 3 Figure 15 Temperature and x component of velocity profile on line x = 0 y = Streamlines (Fig. 16) presents the streamlines on the symmetry plane and allows for an appreciation of the size of the recirculation zones behind the full pins (1,2,3, and 4) and the relative proximity of adjacent streamlines. The biggest recirculation zone is the last pin (pin 4) and pins 1 through 3 have roughly the same recirculation geometry. The streamlines closest one to another near pins 2 and 3 yielding high velocity flow near the walls of pins 2 and 3 which allows for a delayed separation and better heat advection. Figure 16 A rake of 20 points (in z = 0[mm], x = 70[mm]) on the symmetry plane is defined from which the streamlines originate. They are computed both in the forward and backward directions. It is also interesting to note the recirculation that is clearly put in evidence behind pins 2 and 3 and that because of the turbulent flow it is not perfectly symmetric to the X Z plane. 10

17 5.6 Velocity and Temperature plane-cut The distribution of the x component of velocity in the plane (x = -20[mm] i.e. intersecting Pin2 ) in (Fig. 17) shows the vanishing velocity (in white) near the wall of the pin and the recirculation zone behind pins 5 and 8 with negative and zero x component of velocity (in black) and the high velocity flow between these regions (in light-grey). Figure 17 x component of velocity in the plane (x = -20[mm]) The white interstice is Pin 2. The temperature distribution somewhat follows the velocity distribution in the plane x = 20[mm] with the high velocity regions remaining at a temperature near the inlet conditions 288[K] because heat is advected by the thin region in red (temperature of 398[K] at the pin wall) near the pin wall were the velocity is much lower. Note in light blue the temperature of the recirculation zones behind pins 5 and 8. Figure 18 Absolute temperature in the plane (x = -20[mm]) The white interstice is Pin 2. 11

18 6 Quantitative post-processing 6.1 Comparison with experimental extrapolation For the fin-pin channel flow the Reynolds number is defined as : Re D,max = ρu maxd µ with U max = and the average Nusselt number 2 Nu D = qd AkT in U il y L y D = where q = 34.33[W ] it the total heat flow normal to the pin walls, k = [W m 1 K 1 ] si the thermal conductivity, A = π D/2 10 Z c /2 = π / [m 2 ] is the total surface area of the pins and T in = T o T ( i ) log Tw Ti T w T o with T i,t o and T w the inlet, outlet and pin-wall temperature respectively. The average Nusselt number for the fin-pin array can be approximated using the Grimison correlation : Nu D = C 1 Re m D,max with c 1 = 0.521, m = Hence the error in computed Nusselt number relative to the Grimison correlation is 4% : Nu err D = 3.94% One possible source of error is the k ɛ model that cannot (because of the isotropic assumption) accurately predict separation and suffers from over prediction of turbulent energy at the impinging point. 2 is the ratio of convective to conductive heat transfer (i.e. a small Nusselt number indicate heat is mainly transferred through conduction whereas a large Nusselt number indicates heat is mostly dissipated by convection). 12

19 6.2 Total pressure drop Pressure [Pa] x [m] Figure 19 Aboslute pressure distribution on the symmetry plane. Each peak correspond to the point d arret of the pins. The absolute pressure difference between inlet and outlet is computed to be : p = = 140 ± 1[P a] hence a relative low pressure drop of 0.14% ± Hence the energy required to push the air through the pin-fin array is small. 6.3 Total heat flux Pin Heat transfer rate [W] Pin Pin Pin Pin Pin Pin Pin Pin Pin Pin Total Pin Heat transfer rate per unit area [mw m 2 ] Pin Pin Pin Pin Pin Pin Pin Pin Pin Pin Table 5 Total heat flux computed from the heat transfer rate of each pin and heat transfer rate per unit area. The pin-fin array section is computed to have a heat transfer rate of 48.80±0.01[W ] with ± 0.01 [W] from the row of 4 pins and ± 0.01[W ] for the row with 3 pins. Hence the global design with 5 rows of 4 pins and 3 rows of 3 pins has total power dissipation of ± 0.01[W ]. 13

20 6.4 Pin heat-flux and position Pin1 is exposed to natural flow (i.e. identical to cylinder in free flow), whereas second, third, fourth pins are exposed to highly turbulent flow similarly to pins in the rows of 3. The extend and efficiency of the turbulence is due to two factors : upwind flow they are in the recirculation zones, and secondly the downwind pin limits the size of the recirculation zone. Hence the most efficient pins are Pins 2 and 3. Full Pin Heat transfer rate [W] Pin Pin Pin Pin Table 6 Heat transfer rate for the 4 pins in the row of 4. Figure 20 Iso-surface of turbulence intensity. The down-wind pin keeps the highturbulence flow near the upward pin. Note the difference in distance of the turbulence iso-surface for pins 3 and 4. 14

21 6.5 Cooling Efficiency Provide insights into the heat transfer between the cross-flow fluid and the pins. Comment on the cooling efficiency of the pin-fin channel. For efficient cooling a pin-fin array must maximize heat transfer augmentation while minimizing pressure drop (i.e. drop in velocity magnitude). Cooling efficiency can be computed as the ratio between heat transferred from the pins and the kinetic energy of the inlet air. v in = 5[m s 1 ] and v out = 4.87[m s 1 ] and T in = 288[K] whereas T out = [K]. The coefficient of performance (COP) [10] of the fin-pin array is : COP = Q flow W where Q flow is the difference between heat of flow and outlet and inlet and W the amount of work required for insuring the inlet conditions, hence : COP = cṁr T 2M ṁ in v 2 in 2 = 5R T 2Mv 2 in 1.17 where R = 8.314[J K 1 mol 1 ] is the ideal gas constant, M = 28.97[g mol 1 ] is molecular mass of air, v in = 5[m s 1 ] the inflow velocity. Hence the pin-fin array with the given boundary conditions evacuates 17% more power than is required cost to produce the flow. 7 Proposed Improvements 7.1 Numerical solution improvements Even-though a high order LES with a much finer grid would yield more accurate results, if the purpose of this work is to determine the efficiency of the proposed pin-fin array for engineering purposes, then limit in size of the grid is not a limiting factor for the quality of the flow and heat transfer properties required. 7.2 Design improvement of the fin-pin channel geometry. Since we have shown that the staggered array is efficient insofar that produces turbulent flow throughout the array it might be interesting to add the number of pins in each row. Also the velocity of the inlet flow might influence the overall efficiency by developing more turbulent flow in the array. 8 Conclusion The 32 pin-fin array is computed to dissipate ± 0.01[W ] with a COP of The high average Nusselt number indicates a high ratio of convective to conductive heat transfer. Correlation to experimental data by means of the Grimison correlation shows good agreement with 4% error. A parametric study with different inlet velocities and pin temperatures is required to optimize the operating efficiency. 15

22 References [1] W. L. Oberkampf and T. G. Trucano. Verification and validation in computational fluid dynamics. Progress in Aerospace Sciences, 38(3): , [2] S. Krajnović. Large eddy simulation of the flow over a three-dimensional hill. Flow, Turbulence and Combustion, [3] R. Webb and E. Eckert. Application of rough surfaces to heat exchanger design. International Journal of Heat and Mass Transfer, 15(9): , [4] M. E. Lyall, A. A. Thrift, K. A. Thole, and A. Kohli. Heat transfer from low aspect ratio pin fins. Journal of Turbomachinery, 133(1):011001, [5] R. Bahadur and A. Bar-Cohen. Orthotropic thermal conductivity effect on cylindrical pin fin heat transfer. International Journal of Heat and Mass Transfer, 50(5-6): , [6] W. Khan, J. Culham, and M. Yovanovich. Convection heat transfer from tube banks in crossflow: Analytical approach. International Journal of Heat and Mass Transfer, 49(25-26): , [7] Stephen B. Pope Turbulent Flows Cambridge University Press, 2000 [8] Incropera F.P., DeWitt D.P., Bergman T.L. & Lavine A.S. Fundamentals of Heat and Mass Transfer Wiley, 6th edition, [9] Lienhard J.H. & Lienhard J.H. A Heat Transfer Handbook 4th edition, 2011 [10] Hans U. Fuchs The Dynamics of Heat Springer-Verlag, 1996, p