s. Balachandar and A. M. Jacobi, Principal Investigators

Transcription

1 A Numerical Study of Heat Transfer Enhancement Mechanisms in Parallel-Plate Fin Heat Exchangers L. Zhang, S. Balachandar, F. M. Najjar, and D. K. Tafti ACRCTR-89 October 1995 For additional information: Air Conditioning and Refrigeration Center University of Illinois Mechanical & Industrial Engineering Dept West Green Street Urbana,IL (217) Prepared as part of ACRC Project 38 An Experimental and Numerical Study of Flow and Heat Transfer in Louvered-Fin Heat Exchangers s. Balachandar and A. M. Jacobi, Principal Investigators

2 The Air Conditioning and Refrigeration Center was founded in 1988 with a grant from the estate of Richard W. Kritzer, the founder of Peerless of America Inc. A State of Illinois Technology Challenge Grant helped build the laboratory facilities. The ACRC receives continuing supportfrom the Richard W. Kritzer Endowment and the National Science Foundation. The following organizations have also become sponsors of the Center. Acustar Division of Chrysler Amana Refrigeration, Inc. Brazeway, Inc. Carrier Corporation Caterpillar, Inc. Delphi Harrison Thermal Systems Eaton Corporation Electric Power Research Institute Ford Motor Company Frigidaire Company General Electric Company Lennox International, Inc. Modine Manufacturing Co. Peerless of America, Inc. U. S. Army CERL U. S. Environmental Protection Agency Whirlpool Corporation For additional information: Air Conditioning & Refrigeration Center Mechanical & Industrial Engineering Dept. University of Illinois 1206 West Green Street Urbana IL

3 A NUMERICAL STUDY OF HEAT TRANSFER ENHANCEMENT MECHANISMS IN PARALLEL-PLATE FIN HEAT EXCHANGERS L. Zhang, Department of Mechanical & Industrial Engineering S. Balachandar, Department of Theoretical and Applied Mechanics EM. Najjar, D.K. Tafti, National Center for Supercomputing Applications University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A. ABSTRACf The heat transfer enhancement mechanisms and the performance of parallel-plate-fin heat exchangers are studied numerically by solving the unsteady two-dimensional Navier Stokes and energy equations. Different fm arrangements are considered and the effect of boundary layer restart and self-sustained oscillatory mechanisms on heat transfer enhancement and overall performance have been compared. Results of grid dependence study showed satisfactory convergence of the solution. These computations were performed efficiently on the massively parallel connection machine (CM5). 1. INTRODUCTION It has been known from simple theory and from empirical experimental results [1-9]. that surface interruption can be used for enhancing heat transfer. Some examples which exploit surface interruption are the offset strip-fins and perforated-plate surfaces. The surface interruption prevents the continuous growth of the thermal boundary layer by periodically interrupting it Thus the thicker thermal boundary layer in continuous plate-fins, which offer higher thermal resistance to heat transfer, are maintained thin and their resistance to heat transfer is reduced. Previous experimental and numerical studies have shown that this heat transfer enhancement mechanism is operational even at low Reynolds numbers when the flow is steady and laminar [1,4,6]. At higher Reynolds numbers, above criticality, the interrupted surface offers additional mechanisms of heat transfer enhancement by inducing self-sustained oscillations in the flow in the form of shed vortices. In addition to heat transfer enhancement, the surface interruption also increases the pressure drop and thus requires higher pumping power. This is partly due to the higher skin friction associated with

4 the hydrodynamic boundary layer restarting and also due to the Stokes layer dissipation [9] and higher Reynolds stresses [10] in the unsteady regime. Thus the boundary layer restart and the self-sustained oscillatory mechanisms simultaneously influence both the overall heat transfer and the pumping power requirement. Therefore design optimization must take into account the impact of design parameters on the relative importance of the different heat transfer enhancement mechanisms and their attendant effect on pumping cost. Numerical investigation of the geometry effect on heat transfer enhancement mechanisms in compact heat exchangers is very limited and most past studies are limited to the steady laminar flows [4,6]. At higher Reynolds numbers, when the flow becomes unsteady, it is important to compute the flow in a time accurate manner and furthermore the additional length scales that appear at higher Reynolds numbers in the form of small eddies need to be accurately resolved as well. This increases the dynamic range of length and time scales to be computed and makes the problem computationally demanding. 2. MATHEMATICAL FORMULATION The three kinds of fin arrangements considered in this study are shown in Figure 1. The first case is the inline arrangement where flat-fins of thickness, t, and length, /, form a periodic pattern with a pitch, Lx, along the flow direction, x, and a fm separation of 2H between adjacent rows along the transverse, y, direction. Thus the basic elemental unit, indicated by the dashed line, contains a single fm. Here we consider a large periodic array of this basic unit and Figure l(a) shows only six basic elements of this large array. The next is the staggered arrangement, which is obtained from the inline arrangement by shifting alternate rows of fin elements by half a wavelength along the flow direction. The basic elemental unit, again marked by the dashed line in Figure 1 (b), is periodically repeated along the streamwise and transverse directions. The most common arrangement investigated in the past is the staggered arrangement due to its relevance to offset fins and it differs from the inline arrangement in that the transverse distance between the adjacent fm elements is doubled. Figure l(c) shows the staggered-it arrangement, which is obtained from the inline arrangement by shifting alternate colums of fms in the transverse direction by half a wavelenth and the basic unit is marked by the dashed line. In all these three cases the heat transfer surface area per unit volume is maintained the same and the actual sizes and lengths employed in the simulations are given in terms of H. The numerical simulations will assume periodicity of the velocity and temperature fields along both the streamwise and transverse directions, over one basic elemental unit and therefore the actual computation geometry will be limited to this elemental periodic unit Thus, in an attempt to model the flow and heat transfer in a large periodic array of fm elements, the present computation ignores the entrance and exit effects. Furthermore, the possibility of subharmonic effects along both the streamwise and transverse directions is ignored. These effects can be taken into account by employ-

5 ing a larger computational domain, which includes multiple elemental units, Mx and My respectively, along the x and y directions and assuming periodicity of the flow and temperature fields over this extended domain. At low Reynolds numbers to be considered in this study these subharmonic modes are not energetic and therefore for the sake of computational efficiency here we choose Mx=My = 1. The governing equations solved in two-dimensions for the non-dimensional velocity, u, and temperature, T, fields are the Navier-Stokes equations along with the incompressibility condition and the energy equation, as shown below: au + u.vu = e - Vp + _1_V2u at x Re'l' ind (1) at v 1 v2 at + U T = Re'l' Pr T ind (2) V u = 0 ind (3) Where D denotes the computational domain, indicated by the dashed line in Figure 1. for each of the three cases. In the above equations, the length and pressure scales are given by the half distance between adjacent fin rows along the transverse direction, H, and the applied pressure difference over a unit non-dimensional length along the streamwise direction, M. The corresponding velocity and time scales are then given by, u*=(m/e)l/2 and t=(fi2e/ap)l!2, where e is the density of the fluid. The temperature has been nondimensionalized by q"h/k, where q" is the specified constant heat flux on fm surfaces and k is the thermal conductivity of the fluid. The friction Reynolds number, Rec, is given by Hu* Iv and Pr is the Prandtl number. Furthermore, to enable periodicity of the flow field along the streamwise direction, the non-dimensional pressure gradient has been split into an imposed constant mean pressure gradient given by the unit vector, ex, and a fluctuating part, p, which can be considered periodic along x and y. Thus, in the present computations the streamwise pressure gradient is maintained a constant and therefore the flow rate, Q, fluctuates over time, but for all the cases considered the flow rate fluctuation is less than 1 % of its mean value. Under constant heat flux boundary condition, a modified temperature field, e, can be defined as: e(x,y, t) = T(x,y, t) - rx, where r is the mean temperature gradient along the flow direction. From a balance of the total rate of heat input along the fin surface, r can be computed from the following expression: r = ~ /(QRecPr), where ~ is the perimeter of the fin surface in thex-y plane. The modified temperature, e, can then be considered as the perturbation away from the linear temperature variation and can be considered to be periodic along both x and y directions. Since the temporal fluctuations in the flow rate are small, the corresponding fluctuations in r are also negligible in magnitude. On the surface of the fm, no-slip and no-penetration conditions are imposed on the velocity field. The corresponding boundary condition for the modified temperature is given by (ve) ;' = 1 - re~ ;' on adfin (4)

6 where n is the outward normal to the fm swiace denoted by ad fin. The numerical approach followed here is the direct simulation where the governing equations are solved faithfully with all the relevant length and time scales adequately resolved and no models are employed. A second-order accurate Harlow-Welch scheme is employed with a control-volume formulation on a staggered grid with central difference approximations for the convection terms. The equations are integrated explicitly in time until a steady or periodic state is reached. For the inline fin arrangement, the periodic domain with one fm element is resolved with a grid of 128 X 32 points, while in the staggered arrangements, the periodic domain with two fin elements is discretized with 256 X 64 grid points. A detailed description of the numerical methodology can be found in reference [11]. 3. RESULTS AND DISCUSSION Before the presentation of the results the following quantities will be defmed first. Although the computations were performed with Hand (API e) 1/2 as the length and velocity scales, in the results to be presented the Reynolds number, Re, is defined based on the hydraulic diameter, Dh, as: VD h Re=- v and D - 4Am h - A/Lx (5) where Am is the minimum flow cross-section area, V is the average velocity at this section and A is the heat transfer swiace area. Local heat transfer effectiveness will be expressed in terms of the instantaneous local Nusselt number based on hydraulic diameter, defined as: () D.IH Nu s,t = (}IJ ) _ () f ) J's, t ref's, t (6) where s measures the length along the periphery of the fm and the local reference temperature, () ref' is defined by taking into consideration the recirculating zones as [6]: f 8Iuldy () re/s,t) - f luldy (7) Following the above definition, the instantaneous global Nusselt number, <Nu>, can be expressed through an integration of heat flux and temperature difference around the fm swiace, as: < Nu > = (Dh/H) <5' (8) f ad, [(}/s,t) - (}re/s,t)]dl The overall Nusselt number, <Nu>, is then defmed as the average of the above over time. Furthermore we define the modified Colbumj-factor, and friction factor,/, as:

7 and J = LJP (Dh) 1QV24L (9) In Figure 2 the performance of the three fin arrangements is plotted in terms of the j and J factors against the Reynolds number based on the hydraulic diameter. Also plotted are the j and J factors resulting from a fully developed channel flow between continuous uninterrupted flat plates with identical transverse fin spacing as that of the inline arrangement. A comparison of theses various performance quantities will provide deeper understanding into the net effect of the various heat transfer enhancement mechanisms. Based on Figure 2 it is clear that the inline and staggered-ii arrangements have about the same performance suggesting that each column of fin has little impact on the flow and heat transfer of the following column of fin, provided that they are well separated by one fin length or more. This result seems to hold over the entire Reynolds number range investigated. The inline and staggered-ii geometries result in better heat transfer than the staggered arrangement, but this increase is accompanied by a corresponding increase in the friction factor as well. All three fin arrangements are seen to result in higher heat transfer and higher friction factor than the corresponding continuous parallel plate case. The difference is due to the combined effect of boundary layer restart mechanism and the self-sustained flow oscillations (vortex shedding). In the results presented above it is difficult to separate the importance of boundary layer restart mechanism and the impact of vortex shedding. Although heat transfer enhancement associated with the onset of flow unsteadiness has been observed qualitatively in many experimental studies [7], the separation and quantification of the individual effects has not been explored. Here we utilize the flexibility of the present numerical approach to separate these individual effects. For all the three geometries over the range of Reynolds numbers the flow and heat transfer were computed with appropriate symmetries imposed on the velocity and temperature fields about the wake centerline. This symmetrization of the flow removes all asymmetry in the wake associated with the shedding process and thus the flow is made steady. We shall refer to these simulations as symmetrized or steady solutions. Differences in the performance of the symmetrized and the non-symmetrized cases is solely due to the shedding process. On the other hand, the difference between the symmetrized case and the continuous flat plate arises mainly from the periodic restart of the hydrodynamic and thermal boundary layers. Geometry effects such as those arising from the finite thickness of the fin also contribute to this difference. For example, in the symmetrized case, in addition to the restart of the boundary layers, standing recirculating zones can also be seen in the wake, thus affecting the flow field and the overall heat transfer. The j and Jfactors and their ratio for the symmetrized steady and the non-symmetrized cases are presented in Figure 3 for both the inline and the staggered geometries. In the symmetrized cases, the j and J factors are seen to follow a power law of the form: j = 1O.5*Re-O 879 and J = 22.5*Re-O 824 for Inline Geometry

8 j = 6.0*Re-O 843 and f = 6.2*Re-O 743 for Staggered Geometry as shown by the linear behaviour in the log-log plot. This when compared with the theoretical result for the continuous flat plate, given by: j = 9.5*Re-1 and f= 24.0*Re-1 for Continuous Flat Plate clearly shows the expected result that the effects of boundary layer restart and the finite fm thickness are to increase both heat transfer and friction factor within the range of Reynolds numbers simulated. The effect of self-sustained oscillation is to further increase the heat transfer and friction factor above the power law behavior. This deviation from the power law is seen to occur above a Reynolds number of around 350 in the inline arrangement, and at around 650 in the staggered arrangement. Above the critical Reynolds number for the onset of self-sustained oscillations, the flow and temperature fields are observed to oscillate at a single frequency corresponding to the main shedding frequency. Figure 4 shows the instantaneous velocity vector and its corresponding local Nusselt number along the fm surface for the staggered arrangement at Reynolds number of The impact of the flow oscillation in terms of vortices travelling on the fm surface can be seen in the local Nusselt number as a sharp increase in its local value. The detailed effect of varying the Reynolds number on the temperature and flow fields will be considered here. Figure 5 shows the velocity variations of the velocity signal u at.x=o, y=o as a function of time, t* (non-dimensionalized by mean flow bulk velocity and fm thickness), for the inline geometry at three different Reynolds numbers: Re=546.3, and At Re=546.3 and , flow exhibits a single dominant frequency. The corresponding frequency spectra shows a single dominant frequency with its higher harmonics. At the highest Reynolds number, the frequency spectrum shows a lot more activity signifying departure from a smooth laminar flow. The inline arrangement has higher higher heat transfer than staggered arrangement for the same Reynolds number as shown in Figure 2. However the inline arrangement also has a higher friction factor. In order to gain a better understanding of the significant differences in friction factor of the two arrangements, we pursued to investigate the corresponding mean flows. We have considered mean flow for the inline and staggered arrangements at approximately the same Reynolds number (Re= for inline and Re= for staggered). Shown in Figure 6 are the mean flow velocity vectors and their corresponding time averaged skin friction distributions around the top and bottom surfaces of the fin element. From the local reversal of skin friction it can be seen that in the staggered arrangement, a recirculating zone is present in the mean flow at the leading edges, one on top and bottom of the fm surfaces. The recirculation bubble at Re=1465.3, extends over one third of the fm surface, resulting in the significant decrease in the skin friction contribution to the f factor over the corresponding inline arrangement. An added explanation for the higher heat transfer and friction fac-

9 tor in the inline arrangement is that, since the distance between adjacent fms along the y-direction is half of that in the staggered geometry, the thermal and hydrodynamic boundary layer are maintained thin. Finally we report results from a grid independence study conducted with three different resolutions of size 128x32, 256x64 and 512x128 for the inline geometry at a Re of about Figure 7 shows j and / factors for the three different grid resolutions. We observe that by doubling the grid in each direction to 256x64, the friction factor reduces by about 9% while the j factor reduces by 6%. Further doubling of the grid to 512x128, results in a nominal reduction of 1 % and 2% for the j and /factors, respectively. The present results are in very good agreement with experimental results [7] up to the Reynolds number where secondary frequencies begin to appear, after which the present calculations overpredict thej and/factors slightly. We suspect that this is possibly due to the onset of strong three-dimensional effects which are neglected in the present simulations. Overprediction of the mean and rms drag and lift coefficients in two-dimensional models has also been observed in a recent study by Mittal and Balachandar[lO]. Further details are are provided in reference [13]. 4. CONCLUSION Direct numerical simulation is a powerful tool which can be used to explore in detail the flow and heat transfer phenomena in heat exchangers. It provides detailed information about the flow pattern and temperature field over the entire computational domain. Here unsteady Navier-Stokes and energy equations are solved in two-dimensions to simulate flow over a large array of parallel-plate fin elements. 'Three different fin arrangements have been studied and their performance have been compared with theoretical results where the boundary layer restart and self-sustained oscillatory mechanisms are absent In all the arrangements, the heat transfer surface area per unit volume is maintained the same to facilitate comparison. It is shown that both the boundary layer restart and self-sustained oscillatory mechanisms are beneficial for increasing the heat transfer. Furthermore, steady flows in these geometries were computed with imposed symmetry boundary conditions and their performance was compared with that of unsteady simulation to quantify the effect of unsteadiness in heat. transfer enhancement. A complete grid dependence study showed satisfactory convergence of our solution. The limitations of 2-D simulations in predicting flow and heat transfer quantities at high Reynolds numbers has also been illustrated. 5. ACKNOWLEDGEMENTS AND REFERENCES The authors would like to thank the Air Conditioning and Refrigeration Center (ACRC) at the University of lllinois at Urbana-Champaign for their support. Dr. F. N. Najjar was partially supported

10 by a Division of Advanced Scientific Computing, National Science Foundation Post-Doctoral Fellowship. These computations were performed on the CM5 at National Center for Supercomputing Applications. 1. S.V. Patankar, C.H. Liu and E.M. Sparrow, Journal of Heat Transfer, Transaction of ASME., Vol. 99 (1977) pp E.M. Sparrow, B.R Baliga and S.Y. Patankar, Journal of Heat Transfer, Transaction of ASME, Vol. 99 (1977) pp N. Cur, and E.M. Sparrow, International Journal of Heat and Mass Transfer, Vol. 21 (1978) pp E.M. Sparrow, and C.H. Liu, International Journal of Heat and Mass Transfer, Vol. 22 (1979) pp E.M. Sparrow, and A. Hajiloo, Journal of Heat Transfer, Vol. 102 (1980) pp S.Y. Patankar and C. Prakash, International Journal of Heat and Mass Transfer, Vol., 24 (1981) pp R.S. Mullisen, and RI. Loehrke, Journal of Heat Transfer, Transaction of ASME, Vol. 108 (1986) pp N.K. Ghaddar, G.E. Karniadakis, and A.T. Patera, Numerical Heat Transfer, Vol. 9 (1986) pp C.H. Amon, and B.B. Mikic, Numerical Heat Transfer, Part A, Vol. 19 (1991) pp R Mittal, and S. Balachandar, Physics of Fluids, 7 (1995) pp D.K. Tafti, "A Study of High-Order Spatial Finite Difference Formulation for the Incompressible Navier-Stokes Equations, NCSA Pre print, 031, C.H.K. Williamson and A. Roshko, Z. Flugwiss. Weltrawnforch, 14, (1990) pp D.N. Tafti, L. Zhang, EM. Najjar and S. Balachandar, S., "A Tune-Dependent Calculation Procedure for Studying Heat Transfer in Parallel-Plate Fin Heat Exchangers on the Connection Machine-5", to be submitted to Numerical Heat Transfer, 1995.

11 1 =14.4H = ~ tzh!!i (al...!!!i!i!i!i!~. 1=6.4H,=11.7S,"._ 1=6.4H : _-_._ _._-----: (b) ~ ~/~.,=~/4~.4~H_~~... lzh!!..,,=0.75 ~... ~.. ~... ~.. ~... ~... ~~~~~~.... (el... ~!!!!! Figure 1. Three Fin Arrangements: (a) Inline (b) Staggered (c) Staggered-II...,... CurTent Simuiation-Inline _._ A Current Simulation-Staggered IC Current Simuiation.S Gftgered.n 10.3!:-r -'--_-'---'--'-... ~!'"'5r Re (a) A IC Continuous Par.dlel Plates CUrTent Simuiation-lnUne CUrTent Simulation-Staggered CUrTent Simuiation.Staggered.n 10.3 &,.,-----~~ ,..I,-,.---~~ Re (b) Figure 2. Overall Performance: (a) Colburn j Factor vs Reynolds Number (b) Friction Factor f vs Reynolds Number

12 10"! 'if til... ~jif 'S. "- 10" Inline Unsymmetrized --l(-j( - Inline Symmetrized IO~ Staggered Unsymmetrized --li-x - Staggered Symmetrized 10"1...,...---~-~-~~~~...L,:----~~ 10 2 Re 10' 10~OL' ~-~~-~~~I-'-:O':----~~ Re (a) (b) Figure 3. Impact of Vortex Shedding on Overall Perfonnance: (a) Inline (b) Staggered 15.~ IIIiliiiili :;:~ 05 la' (a) 10' o Fin Surface Location Figure 4. (a) Instantaneous Flow Field for Staggered Geometry at Re= (b) Corresponding Instantaneous Local Nusselt Number vs Fin Surface Location

13 4 3 2 ::--1 ci ~O o II x -1 > ' 10' 10 ' (a) 10" 1oS'------~--~~~10~------~~~ f* 6 10' 4 C"2 ci II :>:'-0 o II x > t* (b) 10 Hi 5!... ci II :;0 II x > -5 10' E 210' ~ 10 ' a. ~10 2 g 10'" III 5-10'" ~ IL 10" 10" 10 ' t* ' 10 f* (c) Figure 5. Velocity v Probes for Inline at (x=o,y=o) and Corresponding Frequency Spectrum at: (a) Re=546.3 (b) Re= (c) Re=2191.2

14 00.5 Skin Friction(Top Fin) Skin Friction(Bottom Fin) (a) Skin Friction(Top Fin) Skin Friction(BoUom Fin) 2 I.. (b) Figure 6. TIme Averaged Mean Flow Field and Corresponding Skin Friction Distribution on Top and Bottom Fin Surfaces: (a) Inline at Re=I407.2 (b) Staggered at Re=1465.3

15 ~ ~ ~ x x f ~ ~ A... j Figure 7. Grid Dependence Study: j and f at Different Grid Sizes