Large eddy simulation of turbulent flow and heat transfer in compact heat exchangers

Transcription

1 Large eddy simulation of turbulent flow and heat transfer in compact heat exchangers P. Tochon% J.F. Fourmigue*, P. Mercier* & B. Thonon* 'CEA-Grenoble, DTP/GRETh, Grenoble, France. Abstract Compact Heat Exchangers are often used for cryogenic, air cooling and automotive systems. Indeed, they allow to transfer high amount of heat while keeping low volume and weight. The design of new geometries involves precise knowledge of local phenomena such as momentum and energy transfer through turbulence. Since few predictive models issued from experiences are available and their accuracy limited, numerical simulation is more and more used. However, because the flow is dominated by swirl, separation, recirculation and reattachment, and unsteady laminar or low Reynolds turbulent flows are encountered, the precise simulation of the non-stationary phenomena is at the moment quite complex. The objective of this paper is to contribute to the determination of turbulent flow inside an Offset Strip Fin geometry used in compact heat exchanger under the aspect of Computational Fluid Dynamics (CFD) method. The approach is based on the Large Eddy Simulation (LES) of the turbulent flow using various meshes (coarse and refined, 2-D and 3-D), numerical schemes and wall functions. Results are qualitatively and quantitatively compared with results from (k,z) model and experimental data obtained from open literature. With the most accurate mesh and scheme, the Large Eddy Simulation gives satisfactory results compared with experimental ones. Recommendations for use for numerical simulations of Compact Heat Exchangers are proposed. 1 Introduction Compact plate-fin heat exchangers were initially developed in the 40's to provide

2 478 Advanced Computational Methods in Heat Transfer VI plain fins offset strip fins lowered fins Figure 1: Different type of fins used in compact heat exchangers. for the aerospace industry compact, light and high efficient heat exchangers for gas/gas applications. Plate fin heat exchangers offer high process integration possibilities (12 simultaneous different streams and more in one single heat exchanger) and high efficiency under close temperature approaches (1 to 2 C) in large variety of geometric configurations or applications (cryogenic, air cooling and automotive systems). The fluid flows in the passages created by two plain sheets between which the fins are inserted. These heat exchangers can provide secondary surface up to 90% of the overall heat transfer surface. Several types of fins are available, and the selection depends essentially on the application (figure 1). For air flows, louvered fins are extensively used, while for process applications (single and two-phase) plain or offset strip fins are used. A considerable amount of experimental results and databases are available in the literature for flow and heat transfer phenomena in compact heat exchanger surfaces (Shah [1], Kays [2], Bergles [3]). Starting with the description of some of the complex flows in compact heat exchanger surfaces, it is explained that flows in these surfaces are dominated by swirl and vortices in uninterrupted flow passages, and by boundary layer flows and wake regions (separation, recirculation and reattachment) for interrupted flow passages. While unsteady laminar flows are relatively easy to analyse, swirl and low Reynolds number turbulent flows are difficult to solve numerically because of the lack of appropriate turbulence models. Since a comprehensive experimental study of the performance of CHE surfaces is very expensive because of the high cost of the tools needed to produce a wide range of geometric variations, numerical modelling has the potential of offering aflexibleand cost-effective means for such parametric investigation. Thus the objective of this work is to provide specific comparisons to evaluate the accuracy of numerical turbulent models for describing flow inside Offset Strip Fins Heat Exchangers where experimental data are available. First, a description of some of the complex flows in a simple fin geometry is presented. Next, the numerical simulation of the flow field inside a compact plate heat exchanger is discussed. Finally, some highlights are presented for the numerical analysis of compact heat exchanger surfaces. 2. Geometries Two successive calculations are presented.

3 Advanced Computational Methods in Heat Transfer VI 479 "bulk Hf B Figure 2: Domain of calculation for a uniform flow around a single fin. First, a single fin of length B and width H is located in a uniform flow (figure 2) ; the upstream boundary condition of the computational domain is located at a distance B12 from the plate and the downstream end at B12. The aspect ratio AIL of each channel (where A is the height of the channel) is assumed to be very large so that the flow characteristics are mainly two dimensional (Kelkar [4], Xi [5]). So, a two dimensional grid, uniformly located in the computational domain, is used. Two cases were studied, with the wall grid points inside the viscous sub layer : a mesh (192 x 108) where the non dimensional wall distance of the computational points is about y* = 5 ; and a refined mesh (384 x 216) where / =2.5. Then, a real three-dimensional geometry of an offset strip fin heat exchanger is investigated. As shown in figure 3, three rows of fins (of length B ) and one-half row of fins of the same thickness H are placed in the flow direction. The geometry is described by a transverse spacing L and a fin offset equal to the half fin spacing. The mesh is uniform in all directions. Two cases were studied : a coarse mesh (32 x 48 x 35) where the non dimensional wall distance of the computational points is y* = 9.5 ; and a refined mesh (32 x 96 x 70) where /=19. For both studies, flat velocity u^ and temperature 7^ profiles are set at the inlet. At the outlet, the pressure distribution is imposed to be uniform and equal to zero. At the solid boundary, no-slip condition is used for the velocity components. The fin walls are assumed to be heated at constant temperature TW Figure 3: Domain of calculation for an array of fins.

4 480 Advanced Computational Methods in Heat Transfer VI 3. Numerical method and near wall treatment 3.1 Numerical method In the present study, a numerical computation is carried out. The unsteady flow and thermal fields are governed by energy, momentum and continuity balance equations, with the assumption of constant fluid properties. These equations are solved along the time axis by TRIO software (Mercier [6]). The finite volume method is used : partial differential equations are integrated, using the Gauss theorem, on control volumes to obtain macroscopic balance equations. A staggered grid technique is used, the main variables are not located at the same point. A semi-implicit time discretisation is chosen : explicit velocity and temperature variables are algebraically eliminated from momentum and continuity balances to produce a linear pressure system which is solved directly or iteratively. A Fourier analysis provides optimum stability time step for explicit convective and diffusive terms. Convective terms are computed by two different ways : a standard 1* order Upwind scheme and a third order QUICK scheme (Leonard [7]). 3.2 Turbulence models For the turbulence modelling, the Large Eddy Simulation model has been chosen. The main idea of this model is to compute the large scale turbulence, and to model the smaller scales. All large scale structures are determined by solving the governing equations. Numerical models are applied to subscale structures. As a result, a set of filtered equations with subscale correlations is obtained. The subscale structures have relatively low energy and their structure is expected to be rather universal. TRIO software uses the Structure Function Model (Metais [8]) to calculate the eddy viscosity. As an example, Ciofalo [9] used the LES method for a chevron trough plate geometry and obtained the best prediction of friction and heat transfer coefficient compared with other classical models. However, this kind of approach is still not widespread. LES methods have also been used with success for modelling turbulent flows in complex geometries (Rodi [10], Moin [11]). As a comparison, a classical (&,f) model (Launder [12]) has also been used to give averaged results. 4. Results and discussion In this section we consider local and global approaches of the heat transfer mechanisms and the flow behaviour inside a compact heat exchanger. Indeed, turbulent flows around a fin with rectangular cross section and inside a two dimensional idealisation of an offset strip fin heat exchanger have been

5 Advanced Computational Methods in Heat Transfer VI 481 Upwind scheme Quick scheme Figure 4: Temperature field around a single fin. computed. Hydraulic and thermal results are presented in terms of instantaneous and also time averaged values. 4.1 Local Approach The objective of this section is the investigation of the turbulent separated flow around a rectangular obstacle (figure 2) placed in a uniform flow of velocity Ufafc. This type of flow has been fully investigated in the past and can be characterised by two parameters : the aspect ratio (BIH) and the Reynolds number (Re#/%) based on the half width of the fin and the frontal velocity. The former is usually used to characterise the reattachment (Okajima [13]) while the latter is used to qualify the separation and reattachment regimes of the flow (Lane [14], Ota [15]). In the present study, Re^/% is equal to 350 and the aspect ratio is nearly equal to 16. According to the authors, for BIH greater than 6, the shear layer should reattach on the faces of the profile, and a vortex shedding should occur. Furthermore, for the present value of Reynolds number, the flow field should separate in a laminar way, and become turbulent before attachment on the fin surface. Shown on figure 4 is the temperature field around the fin for the Upwind and Quick scheme obtained with a L.E.S. method. Depending on the convection scheme, the numerical simulation yields to a steady or an unsteady flow. The higher order scheme, the higher unsteadiness. Indeed, the Upwind scheme is highly diffusive and removes the turbulent structures, like the (&, ) model does. Using the Quick scheme, all the described phenomena are in good accordance with the regime predicted by the two non-dimensional numbers (Re^/% and BlH). Indeed, the flow hits the front edge of the rectangular obstacle and separates immediately. The shear layer re attaches to the wall and splits in two parts : one part flows upstream, creating a recirculating and high shear area ; the other is convected downwards by the mean flow. The shear layer becomes unstable with time near the reattachment point, and oscillates ; it generates a

6 482 Advanced Computational Methods in Heat Transfer VI vortex production inside the bubble and a growth of the recirculating area. Due to impinging shear layer instabilities, the long bubble is broken, and an eddy is liberated and convected towards the trailing edge. At the same time, the bubble length decreases. This vortex shedding occurs on both faces of the obstacle, and a Von Karman street is formed in the wake of the fin. Moreover, the general behaviour of the flow shows that the heat transfer between the wall and the fluid is confined and convected near the wall, inside the thermal boundary layer. The vortices do not succeed in sucking the heat within the central part of them. This heat is mainly transferred to the bulk flow in the wake of the fin, through its trailing edge. The recirculating bubble is usually characterised by its non dimensional length x, IB where x, is the distance between the leading edge and the reattachment point. This length can be displayed by the maximum pressure, the zero shear stress point at wall or the maximum heat transfer position. Using these criteria, the results of the present simulations are summarised in table 1. Table 1. Comparison between experimental and computed reattachment length. Investigator(s) Ota [16] Sorensen [17] Present study (LES Quick) Present study (LES Upwind) Present study (k, ) (Quick and Upwind) *r IB Using L.E.S. with a Quick scheme yields to a reasonable prediction of the reattachment length (for both the two meshes). In the range of small y* (inside the viscous sublayer), the L.E.S. method seems to be less sensitive to the mesh size than to the convective scheme. Indeed, the same wall treatment is used for both mesh and is directly based on the computation of the friction velocity. Using Upwind scheme always yields to a high underestimation of the recirculating area, as the (k,s) model does. However, the mesh used for this study is not well adapted for the (k,s) model which recommends that the first computational point must be inside the logarithmic region. The Strouhal number, based on the chord plate B, is often defined for separated flow analysis. It also coincides with the number of vortices formed on one face of the plate (Ozono [18]). As shown in figure 4, the computed Strouhal number can be estimated between 2 and 3 which is in accordance with authors who obtained a value of 3 for the present aspect ratio. The blunt flat plate placed parallel to an uniform flow generates a high turbulence and heat transfer level. As described on figure 4, one part of the shear layer carries cold fluid inside the bubble at the reattachment point; at this point,

7 Advanced Computational Methods in Heat Transfer VI 483 Heatflux (MW/m=) 10. _*_LES 384x216 Quick -»_LES 384x216 Upwind -x-les 192 x 108 Upwind -*-(k,e) 192 x 108 Upwind Figure 5: Evolution of the local heat flux along the fin. the heat transfer increases. Integrating the mean heat flux at the wall along the fin (figure 5), the results show than 50% of the total heat flux is liberated by 30% of the plate surface. Due to the reattachment point oscillation a high heat flux area can be displayed on the faces. However, this point depends on the convective scheme. Indeed, using both the (k,s) or LES model with an Upwind scheme gives a constant heat flux along the fin due to the high numerical diffusion, even for very fine mesh. So, the L.E.S. Quick model is the only one able to describe qualitatively the non-steady phenomena. In conclusion, the Large Eddy Simulation, combined with a high order convective scheme, displays the main turbulence mechanisms for that geometry : fluid separation at the leading edge of the obstacle, reattachment on the faces, recirculating bubble oscillation, coherent structures shedding and Von-Karman streets. In a range of low y*, the simulation is not very sensitive to the mesh size because the same wall function is used. With a low order convective scheme, the L.E.S. method does not describe the non-steady behaviour of the flow and underestimates the heat transfer and the recirculating area because of the numerical diffusion combined with a fine mesh. The classical (k,s) model does not need a high order scheme but also underestimates the heat transfer due to an unsuitable mesh. 4.2 Global Approach Compact heat exchangers are generally characterised by complex geometries ; it is the reason why reliable predictions of heat transfer and pressure drop are difficult, restrictive and uncertain. Many different correlations in offset strip fin heat exchangers have been reported in the literature ; they concern average friction factor and Colburn factor (Wieting [19], Manglik [20]).

8 484 Advanced Computational Methods in Heat Transfer VI For an offset strip fin configuration in turbulent regime, available correlations have been used, as a function of geometric parameters (A, /?, L and//) and specific Reynolds numbers. The previous section has shown how a blunt flat plate acts as a turbulence generator and enhances heat transfer. The purpose of this section is to investigate the thermo-hydraulic behaviour of a turbulent flow inside a real 3-D configuration of an offset strip fin heat exchanger. For that, pressure drop and heat transfer coefficients are provided. The instantaneous velocity field is presented on figure 6 for L.E.S. model. This time, even for the Quick scheme, no instabilities are described. However, the recirculating area is well described (x, IB = 0.38 instead of 0.1 for the Upwind scheme). This fact is due to the coarse mesh used, which enhanced the numerical diffusion and increases the modelled part of the turbulent spectrum. The 3-D part of the flow is not really marked in this configuration due to the high aspect ratio. In order to qualify that kind of complex flow, the value of pressure drop and temperature difference between plates of successive ranks is computed and the fanning f^ and Colburn j factors have been estimated and compared to experimental correlations (table 2). Table 2: Comparison of calculated fanning and Colburn numbers with experimental correlations. Investigator(s) Manglik [20] Present study (LES 32 x 48 x 35) Present study (LES 32 x 96 x 70) Present study (&, s) (32 x48 X 35) fa j Despite the fact that the unsteadiness of the flow is not well described, the overall results for the friction factor and the heat exchange are in good accordance with literature data. However, for this 3-D computation, the mesh size has an influence on the results because a wall function is used. So the finer the mesh, the better the prediction. The coarse mesh is well adapted to the Figure 6: Velocity field in the middle height of the CHE.

9 Advanced Computational Methods in Heat Transfer VI 485 recommendation for use of the standard (k,s). So this model gives relatively good results. In conclusion, the Large Eddy Simulation with a coarse mesh does not display the main turbulence mechanisms. However, the overall predictions are in godd accordance with experimental correlations. The classical (k,s) model gives also rather good results. The 3-D modelling in that geometry is not really necessary. 5. Concluding remarks The present study has allowed us to improve our knowledge about turbulent flow and heat transfer inside an offset strip fin heat exchanger. By means of Large Eddy Simulation model, non-stationary results have been numerically produced by TRIO code : separation and re attachment of the flow, recirculating bubble oscillation, vortices creation and shedding. Time averaged friction factor and heat transfer coefficient have been also provided for a 3-D configuration and compared with experimental databases. Local as well as global results are in good accordance with the literature. This study has proven the good performance of L.E.S. simulation by a CFD code based on the finite volume method. The main physical phenomena are correctly described, the main geometrical parameters are accurately identified by their effect on turbulence and finally the transient behaviour of the flow is shown to have a great importance on thermal phenomena. 6. References [1] Shah, R.K., & Webb, R.L. Compact and enhanced heat exchangers. Heat exchangers: theory and practice, ed. Taborek, Hemisphere: Washington DC., pp , 1983 [2] Kays, W.M., & London, A.L. Compact Heat Exchangers. McGraw-Hill: New York, [3] Bergles, A.E. Techniques to augment heat transfer (Chapter 3). Handbook of Heat Transfer Appl., ed. Rohsenow, McGraw-Hill: New York, [4] Kelkar, S.M., & Patankar, S.V. Numerical prediction of heat transfer and fluid flow in rectangular offset-fin arrays. Numerical Heat Transfer - Part A, 15, pp , [5] Xi, G.N., Hagiwara, Y., & Suzuki, K. Flow instability and augmented heat transfer of fin arrays. J. ofenhan. Heat Transfer, 2 (1-2), pp , [6] Mercier, P., & Villand, M. The multi-dimensional thermohydraulics code TRIO - applications to heat exchangers. Eurotherm Seminar No. 18, Hamburg Feb March L, [7] Leonard, B.P. A stable and accurate convective modelling procedure base and quadratic upstream interpolation. Comp. Meth. Appl. Mech. Engng, 107, pp , 1979.

10 486 Advanced Computational Methods in Heal Transfer VI [8] Metais, O. & Lesieur, M. Spectral large eddy simulation of isotropic and stable stratified turbulence. J. Fluid Mech., 239, pp , [9] Ciofalo, M., Stasiek, J. & Collins, M.W. Investigation of flow and heat transfer in corrugated passages -II. Numerical results. Int. J. Heat Mass Transfer, 39, pp , [10] Rodi, W., Ferziger, J.H., Breuer, M. & Pourquie, M. Status of large-eddy simulation: Results of a workshop. J. Fluids Eng., 119, pp , [HJMoin, P. Numerical and physical issues in large-eddy simulation of turbulent flows. JSME Int. J., Series B, 41 (2), pp , [12] Launder, B.E. & Spalding, D.B. The numerical computation of turbulent flows. Computer Methods in Appl. Mech. and Eng., 3, pp , [13] Okajima, A., Ueno, H., & Sakai, H. Numerical simulation of laminar and turbulent flows around rectangular cylinders. Int. J. for Num. Methods in Fluids, 15, pp , [14] Lane, J.C., & Loehrke, R.J. Leading edge separation from a blunt flat plate at low Reynolds number. ASME J. of Fluids Eng., 102, pp , [15] Ota, T., Asano, Y., & Okawara, J.I. Reattachment length and transition of separated flow over blunt flat plates. Bulletin of JSME, 24, pp , [16] Ota, T., & Narita, M. A separated and reattached flow on a blunt flat plate. ASMEJ. of Fluids Eng., 98, pp , [17] Sorensen, A. Mass transfer coefficients on truncated slabs. Chem. Eng. Sci., 24, pp , [18] Ozono, S., Ohya, Y., & Nakamura, Y. Stepwise increase in the Strouhal number for flows around flat plates. Int. J. for Num. Methods in Fluids, 15, pp , [19] Wieting, A.R. Empirical correlations for heat transfer and flow friction characteristics of rectangular offset-fin plate-fin heat exchangers. Trans. ASME, J. Heat Transfer, 97, pp , [20] Manglik, R.M., & Bergles, A.E. Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger. Exp. Thermal and Fluid Science, 10, pp , 1995.