CFD calculation of convective eat transfer coefficients and validation Part I: Laminar flow Annex 41 Kyoto, April 3 rd to 5 t, 2006 Adam Neale 1, Dominique Derome 1, Bert Blocken 2 and Jan Carmeliet 2,3 1) Dep. of Building, Civil and Environmental Engineering, Concordia University, 1455 de Maisonneuve lvd West, Montreal, Qc, H3G 1M8, corresponding autor e-mail: aneale@sympatico.ca 2) Laoratory of Building Pysics, Department of Civil Engineering, Katolieke Universiteit Leuven, Kasteelpark Arenerg 40, 3001 Heverlee Belgium 3) Building Pysics and Systems, Faculty of Building and Arcitecture, Tecnical University Eindoven, P.O. ox 513, 5600 MB Eindoven, Te Neterlands Astract Computational Fluid Dynamics (CFD) simulations of convective eat transfer are considered to e particularly callenging to perform y te CFD community. In tis paper, te calculation of convective eat transfer coefficients ( c ) y te commercial CFD code Fluent is verified y studying two cases of laminar flow etween parallel infinite flat plates under different termal conditions. In te first case, te plates produce a constant eat flux, q w, wit a constant free stream temperature. In te second case, te walls are at a constant temperature, T w, wit a constant free stream temperature. A grid sensitivity analysis wit Ricardson extrapolation was performed for ot cases to determine te grid independent solutions for c. Te values for c reported y Fluent were ten compared wit analytical values from literature. Te percentage error etween te analytical and grid independent solutions for c is on te order of 10-2 %. 1. Introduction Te surface coefficients for eat and mass transfer ( c and m, respectively) are parameters tat are generally not easily calculated analytically and difficult to derive from experimental measurements. Te values of surface coefficients depend on many variales flow field, oundary conditions, material properties, etc. In addition, despite te fact tat te two transfer processes are mutually dependent, tey are often solved as uncoupled penomena. Finally, altoug existing 1
correlations relating c and m are valid for specific cases, suc correlations are applied widely trougout literature. Tis paper is te first of a two part study of te option to solve for c using Computational Fluid Dynamics (CFD). Part I validates te CFD code Fluent for eat transfer in te laminar regime using two cases: 1) parallel flat plates wit constant wall temperature and 2) parallel flat plates wit constant eat flux. In addition, te eat transfer coefficients calculated wit several grid refinements will e compared in a grid sensitivity analysis wit Ricardson extrapolation. Te grid independent solution is compared to analytical values. Part II is a comparative study of eat transfer coefficients calculated using te different turulence models implemented in Fluent. Te validity of using wall functions for natural convection cases is also examined. 2. Description of case studies and analytical solutions 2.1. Geometry and oundary conditions Te cases studied in tis paper assume tat te flow field as ecome fully developed efore te eated region. Tis assumption is valid wen u x = 0 (1) were u is te orizontal component of te velocity at any given eigt in te flow field (for orizontal plates). Aerodynamically developed flow is a requirement for analytical solution of te termal oundary layer (Lienard and Lienard 2006). Te material properties used in te simulations and te analytical solutions are sown in Tale 1. Te two cases studied are illustrated in Figure 1. Tale 1. Material properties for air Density ρ 1.225 kg/m 3 Dynamic Viscosity µ 1.7894 x 10-5 kg/m s Termal Conductivity k 0.0242 W/m K Heat Capacity c p 1006.43 J/kg K 2
Domain : Y X (Note : Not to scale) L = 3.0 m = 0.05 m Inlet Conditions U av = 0.1 m/s U(y) = 3/2*U av*[1 4*(y/) 2 ] m/s T = 283 K Wall oundaries q w = 10 W/m 2 T w = 293K (a) Constant Heat Flux (CHF) () Constant Wall Temperature (CWT) T w = 293K q w = 10 W/m 2 Figure 1. Scematic representation of te two case studies wit (a) constant eat flux or () constant wall temperature 2.2. Reference temperatures Te goal of te eat transfer simulations is to find te convective eat transfer coefficient cx at a particular location x. Tis relationsip is defined as: q wx cx ( T T ) = (2) wx f were q wx is te eat flux at te wall at x, T wx is te temperature of te wall at x, and T f is a reference temperature witin te fluid. Te actual value used for T f depends largely upon te geometry used in te prolem. An improperly assigned reference temperature can yield a significant error, as will e sown in te case studies presented. Tree reference temperatures are used in a comparison exercise to sow te effects on te calculation of c : a constant reference temperature T ref (as used in Fluent to report c values), te centerline temperature T c (taken at y=0 on Figure 1), and a ulk temperature T wic is defined as (Lienard and Lienard 2006): T = y ρc p mc & utdy p (3) 3
were ρ is te fluid density, c p is te specific eat, u is te orizontal velocity component, T is te temperature and m& is te mass flow rate. Equation (3) is derived from te rate of flow of entalpy troug a given cross section divided y te rate of eat flow troug te same cross section. For te cases sown in tis paper, te material properties may e considered constant, and Equation (3) can e simplified to te following form: T n ( u T ) i i i i= = 1 (4) U av were u i is te velocity of in te centre of a control volume (CV), i is te eigt of te CV, T i te temperature in te CV, U av is te velocity averaged over te eigt and is te eigt of te domain. It can e sown tat te energy alance troug any given cross-section wit a tickness dx can e derived to e, (Lienard and Lienard 2006): q w Pdx = mc & p dt wic can e rearranged as: dt dx q P mc & p w = (5) were q w is te eat flux at te wall, P is te eated perimeter, m& is te mass flow rate, and c p is te specific eat. Using te conditions specified in Figure 1 along wit te constant wall eat flux oundary condition, te rigt and side of Equation (5) ecomes a constant value. dt dx q = wp q = mc & ρ p w ( d ) 2d U c av p = ( 10)( 2) ( 1.225)( 0.05)( 0.1)( 1006.43) = 3.2444 K/m were d is te dept of te plates and U av is te average velocity (equal to 0.1 m/s for te cases studied). Integrating ot sides of Equation (6) wit respect to x results in T =. 2444x + C 3 (7) By imposing te oundary condition tat at x=0m te ulk temperature is equal to te inlet temperature (T = T = 283K), Equation (7) ecomes T = 3.2444x + 283 (8) (6) 4
Equation (8) will e used to verify tat ulk temperatures calculated from Fluent data are consistent wit te analytical equations. 2.3. Heat transfer coefficient Te eat transfer coefficient may e otained from analytically derived values of te Nusselt numer, wic sould e constant for termally developed flow etween parallel plates. Te values will differ sligtly ased upon te eating conditions as follows (Lienard and Lienard 2006): Nu D = c D k 7.541 = 8.235 for fixed platetemperatures for fixed wall eat fluxes were D is te ydraulic diameter (typically twice te distance etween parallel plates) and k is te termal conductivity of air. Te appropriate parameters may ten e input to yield te following analytical values for c : c = Nu D D k 1.825 = 1.993 for fixed platetemperatures for fixed wall eat fluxes (9) W/m 2 K (10) 3. CFD simulations 3.1. Geometry and oundary conditions Te geometry sown in Figure 1 was reproduced wit a mes tat was generated from a preliminary mes sensitivity analysis. Mes refinement was applied exponentially towards te wall surfaces. A uniformly spaced mes was used in te streamwise direction. Te initial mes used for te Constant Heat Flux (CHF) and Constant Wall Temperature (CWT) cases ad a total of 19,800 cells (33 in te vertical direction, 600 in te orizontal). A portion of te initial mes is sown in Figure 2 elow. = 0.05m L = 3m Figure 2. Initial mes used for te CFD simulations. 5
Te oundary conditions for te simulations were input as sown in Figures 3 and 4. Te Fluent solution parameters and model information are provided in Appendix A. Velocity Inlet B.C. U av = 0.1 m/s U(y) = 3/2*U av *[1 4*(y/) 2 ] m/s T = 283 K Wall B.C.: q w = 10 W/m 2 Wall B.C.: q w = 10 W/m 2 Pressure Outlet B.C. Note :Nottoscale. Figure 3. Boundary Conditions (B.C.) CHF Case Velocity Inlet B.C. U av = 0.1 m/s U(y) = 3/2*U av *[1 4*(y/) 2 ] m/s T = 283 K Wall B.C.: T w = 293K Wall B.C.: T w = 293K Pressure Outlet B.C. Note : Not to scale. Figure 4. Boundary Conditions (B.C.) CWT Case Te velocity profile used as te inlet condition is a paraolic profile commonly used to descrie te flow etween parallel plates [1]. Wen comparing te inlet and outlet profiles from te simulation results, te difference in velocity at a given eigt is on te order of 10-4 m/s. Terefore it can e assumed tat te flow is indeed already fully developed at te inlet. Te flow field was initialized to te inlet conditions (descried in Figure 4). Te simulations were iterated until a scaled residual of 10-7 (Fluent Inc. 2003) was acieved for all te solution parameters involved. 3.2. Simulation results Once te simulations were completed, te ulk temperatures were calculated wit Equation (4) using te cell temperature and velocity data. For te case of constant eat flux, te simulation data can e compared to te analytical equation derived in Equation (8). Te results from Fluent are plotted in Figure 5, and te resulting trendline equation is very close to te expected equation. Analytical Bulk Temperatures: = 3.2444x + 283 T 6
Fluent Bulk Temperatures: = 3.2423x + 283 Te error increases sligtly along te lengt of te plate. After 3m te difference etween analytical and Fluent ulk temperatures is on te order of 10-3 %. T 292 Bulk Temperature (K) 290 288 286 284 T = 3.2423x + 283 T Linear (T) 282 0 0.5 1 1.5 2 2.5 3 X Position (m) Figure 5. Bulk temperature calculated from Fluent output data Te convective eat transfer coefficients were calculated wit Equation (2), using te tree different fluid reference temperatures previously mentioned. Te parameters used to solve Equation (2) are outlined in Tale 2 elow. Te results for te convective eat transfer coefficient indicate tat te temperature value used to descrie te fluid (T f from Equation 2) can ave a significant effect on te result. Te cosen reference temperature must matc te one used in te derivation of te equation or correlation used for comparison. Te reported values in Fluent are calculated ased on a user specified constant reference value, wic results in non-constant convective coefficients after termally developed flow (Fluent Inc. 2003). Correlations tat were developed using any oter fluid temperature as a reference will not matc te results from Fluent. Terefore, care must e taken on wic values are used wen reporting information from Fluent. Te convective coefficients calculated from te centerline temperatures are more realistic and follow te expected trend, ut tey under-predict te c values y aout 20% for te CHF solution and y aout 24% for te CWT solution. Te ulk temperature yielded te est solution for te convective eat transfer coefficient, resulting in an error margin of less tan 0.5% for ot cases (after termal development). Since te ulk temperature calculation is dependent on te 7
grid used, a grid sensitivity and discretization error analysis was performed to determine wat te grid independent solution would e. Tale 2. Convective eat transfer coefficient solution parameters CHF Case (a) CWT Case () q w (x) q w = 10 W/m 2 q w (x) From Fluent T w (x) T W (x) From Fluent T w = 293 K T f (x) cx qw( x) = T ( x) T ( x) w f T f (x) = T ref = 283 K (Constant value specified in Fluent (Fluent Inc. 2003)) T f (x) = T c (x) (Horizontal temperature profile at te center of te flow (y = 0)) T f (x) = T (x) (Bulk Temperature calculated at different x positions from te Fluent Data) 10 q cref (x) = T w ( x) 283 cref (x) = w ( x) 293 283 cc (x) = T c (x) = T w w 10 ( x) T ( x) c 10 ( x) T ( x) qw ( x) cc (x) = 293 T ( x) qw ( x) c (x) = 293 T ( x) c 8
c (W/m 2 K) 4 3.5 3 2.5 2 1.5 cref c-ref cc c Analytical 1 0.5 0 0 0.5 1 1.5 2 2.5 3 X Position (m) Figure 6. Convective eat transfer coefficients for constant wall eat flux 4 c (W/m 2 K) 3.5 3 2.5 2 1.5 cref c-ref cc c Analytical 1 0.5 0 0 0.5 1 1.5 2 2.5 3 X Position (m) Figure 7. Convective eat transfer coefficients for constant wall temperature 9
3.3. Grid Sensitivity Analysis For te purposes of te grid sensitivity analysis, te convective eat transfer coefficients calculated are compared for different grid densities at x = 2.5 m. Te process was repeated for ot te CHF and CWT cases to compare te grid dependency for te two different oundary conditions. Only te coefficients calculated from te ulk temperature are part of tis comparison. Te initial grid used for te simulations ad a total of 19,800 cells. It was decided to proceed wit several coarser grids and one finer mes. Te details of te different meses are presented in Tale 3. Te notation φ is adopted to descrie te solution for te finest mes. Te susequent meses are all notated wit respect to te finest mes. Te next grid size as cell dimensions douled in ot directions, ence te notation φ 2. Tale 3. Mes dimensions Numer of cells in te Y Direction Numer of cells in te X Direction Smallest cell eigt (m) Smallest cell widt (m) Total numer of cells * Original mes φ (80400) φ 2 (19800)* φ 4 (5100) φ 8 (1200) φ 16 (300) 67 33 17 8 4 1200 600 300 150 75 4.202E-04 8.749E-04 1.775E-03 3.948E-03 9.147E-03 0.0025 0.005 0.01 0.02 0.04 80400 19800 5100 1200 300 It can e sown (Ferziger and Peric 1997) tat te discretization error of a grid is approximately d φ φ2 ε (11) a 2 1 were a is te order of te sceme and is given y φ 2 φ4 log φ φ2 a = log ( 2) (12) 10
In ot equations te 2 refers to te increase in dimensions of te mes. From Equation (12), it follows tat a minimum of tree meses are required to determine te discretization error. In order to prevent a calculation error from te logaritm of a negative numer, te tree solutions must e monotonically converging [2]. Te teory of Ricardson Extrapolation states tat te solution from te finest mes can e added to te discretization error found in Equation (11) to attain an approximate grid independent solution. In equation form tis can e stated as: Φ = φ + (13) d ε Te results from te grid sensitivity analysis are sown in Tale 4 and plotted elow in Figures 8 and 9. Tale 4. Discretization error and Ricardson Extrapolation Results Order of te sceme a Discretization Error d ε (W/m 2 K) Finest mes solution φ (W/m 2 K) Ricardson Solution Φ (W/m 2 K) Analytical solution c (W/m 2 K) CHF 1.460 2.297x10-3 1.990578 1.992875 1.992875 CWT 1.858 1.001x10-3 1.824089 1.825090 1.824922 2.02 9.0 Φ = 1.992875 1.97 1.975 1.987 1.991 8.0 7.0 c (W/m 2 K) 1.92 1.931 x=2.5m Ricardson Relative error 6.0 5.0 4.0 3.0 Relative Error (%) 1.87 1.848 2.0 1.0 1.82 φ16 (300) φ8 (1200) φ4 (5100) φ2 (19800) φ (80400) 0.0 Grid (#cells) Figure 8. Grid convergence of te eat transfer coefficient for constant eat flux and relative error compared wit Ricardson solution 11
1.85 9.0 Φ = 1.82509 1.80 1.812 1.821 1.824 8.0 7.0 c (W/m 2 K) 1.75 1.70 1.690 1.769 x=2.5m Ricardson Relative error 6.0 5.0 4.0 3.0 2.0 Relative Error (%) 1.0 1.65 φ16 (300) φ8 (1200) φ4 (5100) φ2 (19800) φ (80400) 0.0 Grid (#cells) Figure 9. Grid convergence of te eat transfer coefficient for constant wall temperature and relative error compared wit Ricardson solution Note tat te actual order of te sceme (a) is iger tan te discretization sceme used in te Fluent solver (sown in Appendix A). 4. Conclusions A validation exercise was performed y comparing te computed convective eat transfer coefficients ( c ) for laminar air flow etween parallel plates y Computational Fluid Dynamics to analytical solutions. Te CFD simulations were performed for constant wall temperature and constant eat flux conditions. Te importance of a correct reference temperature was confirmed. Te CFD results sowed a good agreement wit te analytical solutions, indicating a proper performance of te CFD code, at least for te cases studied. Finally, a grid sensitivity analysis was performed on te mes for ot wall oundary conditions. Te discretization error for c was calculated at a given location on te plate and Ricardson extrapolation was used to compute te grid independent solution. Te resulting c values ad good agreement wit analytical values from literature. Te percentage error etween te analytical and te grid independent solutions for c is on te order of 10-2 %. 12
Appendix A: Fluent solution parameters Model Space Time Viscous Heat Transfer Solidification and Melting Radiation Species Transport Coupled Dispersed Pase Pollutants Soot Settings 2D Steady Laminar Enaled Disaled None Disaled Disaled Disaled Disaled Equation Flow Energy Solved yes yes Numerics Asolute Velocity Formulation Enaled yes Relaxation: Variale Relaxation Factor Pressure 0.3 Density 1 Body Forces 1 Momentum 0.7 Energy 1 Solver Termination Residual Reduction Variale Type Criterion Tolerance Pressure V-Cycle 0.1 X-Momentum Flexile 0.1 0.7 Y-Momentum Flexile 0.1 0.7 Energy Flexile 0.1 0.7 Discretization Sceme Variale Pressure Pressure-Velocity Coupling Momentum Energy Sceme Standard SIMPLE First Order Upwind First Order Upwind Solution Limits Quantity Limit Minimum Asolute Pressure 1 Maximum Asolute Pressure 5000000 Minimum Temperature 1 Maximum Temperature 5000 13
Appendix B: Nomenclature a Order of te discretization error sceme (-) c p d D Distance etween parallel plates (m) Specific eat (J/kgK) Dept of te parallel plates (m) Hydraulic diameter (m) c Convective eat transfer coefficient (W/m 2 K) k L m& Termal conductivity (W/m-K) Lengt of domain (m) Mass flow rate (kg/s) Nu D Nusselt numer calculated wit te ydraulic diameter (-) P Heated perimeter of te domain (m) q Heat flux (W/m 2 ) u v T U Velocity component in te x-direction (m/s) Velocity component in te y-direction (m/s) Temperature (K) Velocity magnitude (m/s) Greek symols d ε φ Discretization error (units ased on parameter analyzed) Solution for finest mes (units ased on parameter analyzed) φ n Solution for a mes wit cell dimensions n times te finest mes Φ Ricardson Extrapolation solution (units ased on parameter analyzed) µ Dynamic viscosity (kg/m-s) ρ Density (kg/m 3 ) Suscripts AV c f i ref x Average property Bulk property Property taken at te centerline of te domain Fluid property Property of an element i Reference property Property taken at a location x Free stream property 14
References 1. Ferziger, J.H., Perić, M. Computational Metods for Fluid Dynamics. Springer, 3 rd Edition, 58-60, 2002. 2. Fluent 6.1 User s Guide, 2003. 3. Lienard IV, J.H., Lienard V, J.H. A Heat Transfer Textook, Plogiston Press, 2006. 15