CFD calculation of convective eat transfer coefficients and validation Part I: Laminar flow Neale, A.; Derome, D.; Blocken, B.; Carmeliet, J.E. Publised in: IEA Annex 41 working meeting, Kyoto, Japan Publised: 01/01/2006 Document Version Publiser s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please ceck te document version of tis publication: A submitted manuscript is te autor's version of te article upon submission and before peer-review. Tere can be important differences between te submitted version and te official publised version of record. People interested in te researc are advised to contact te autor for te final version of te publication, or visit te DOI to te publiser's website. Te final autor version and te galley proof are versions of te publication after peer review. Te final publised version features te final layout of te paper including te volume, issue and page numbers. Link to publication Citation for publised version (APA): Neale, A., Derome, D., Blocken, B. J. E., & Carmeliet, J. (2006). CFD calculation of convective eat transfer coefficients and validation Part I: Laminar flow. In IEA Annex 41 working meeting, Kyoto, Japan General rigts Copyrigt and moral rigts for te publications made accessible in te public portal are retained by te autors and/or oter copyrigt owners and it is a condition of accessing publications tat users recognise and abide by te legal requirements associated wit tese rigts. Users may download and print one copy of any publication from te public portal for te purpose of private study or researc. You may not furter distribute te material or use it for any profit-making activity or commercial gain You may freely distribute te URL identifying te publication in te public portal? Take down policy If you believe tat tis document breaces copyrigt please contact us providing details, and we will remove access to te work immediately and investigate your claim. Download date: 13. Sep. 2018
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 blvd West, Montreal, Qc, H3G 1M8, corresponding autor e-mail: aneale@sympatico.ca 2) Laboratory of Building Pysics, Department of Civil Engineering, Katolieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001 Heverlee Belgium 3) Building Pysics and Systems, Faculty of Building and Arcitecture, Tecnical University Eindoven, P.O. box 513, 5600 MB Eindoven, Te Neterlands Abstract Computational Fluid Dynamics (CFD) simulations of convective eat transfer are considered to be particularly callenging to perform by te CFD community. In tis paper, te calculation of convective eat transfer coefficients ( c ) by te commercial CFD code Fluent is verified by studying two cases of laminar flow between 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 bot cases to determine te grid independent solutions for c. Te values for c reported by Fluent were ten compared wit analytical values from literature. Te percentage error between 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 variables flow field, boundary 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 be 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 turbulence 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 boundary conditions Te cases studied in tis paper assume tat te flow field as become fully developed before 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 boundary layer (Lienard and Lienard 2006). Te material properties used in te simulations and te analytical solutions are sown in Table 1. Te two cases studied are illustrated in Figure 1. Table 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 b = 0.05 m Inlet Conditions U av = 0.1 m/s U(y) = 3/2*U av*[1 4*(y/b) 2 ] m/s T = 283 K Wall boundaries q w = 10 W/m 2 T w = 293K (a) Constant Heat Flux (CHF) (b) 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 (b) 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 problem. An improperly assigned reference temperature can yield a significant error, as will be 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 bulk temperature T b wic is defined as (Lienard and Lienard 2006): T b = 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 by te rate of eat flow troug te same cross section. For te cases sown in tis paper, te material properties may be considered constant, and Equation (3) can be simplified to te following form: T b n ( u b T ) i i i i= = 1 (4) U av b were u i is te velocity of in te centre of a control volume (CV), b i is te eigt of te CV, T i te temperature in te CV, U av is te velocity averaged over te eigt and b is te eigt of te domain. It can be sown tat te energy balance troug any given cross-section wit a tickness dx can be derived to be, (Lienard and Lienard 2006): q w Pdx = mc & p dt wic can be rearranged as: dt dx q P mc & p b b 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 boundary condition, te rigt and side of Equation (5) becomes a constant value. dtb dx q = wp q = mc & ρ p w ( bd ) 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 bot sides of Equation (6) wit respect to x results in T b =. 2444x + C 3 (7) By imposing te boundary condition tat at x=0m te bulk temperature is equal to te inlet temperature (T b = T = 283K), Equation (7) becomes T b = 3.2444x + 283 (8) (6) 4
Equation (8) will be used to verify tat bulk temperatures calculated from Fluent data are consistent wit te analytical equations. 2.3. Heat transfer coefficient Te eat transfer coefficient may be obtained from analytically derived values of te Nusselt number, wic sould be constant for termally developed flow between parallel plates. Te values will differ sligtly based 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 between parallel plates) and k is te termal conductivity of air. Te appropriate parameters may ten be 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 boundary 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 below. b = 0.05m L = 3m Figure 2. Initial mes used for te CFD simulations. 5
Te boundary 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/b) 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/b) 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 parabolic profile commonly used to describe te flow between 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 be assumed tat te flow is indeed already fully developed at te inlet. Te flow field was initialized to te inlet conditions (described 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 bulk temperatures were calculated wit Equation (4) using te cell temperature and velocity data. For te case of constant eat flux, te simulation data can be 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 b 6
Fluent Bulk Temperatures: = 3.2423x + 283 Te error increases sligtly along te lengt of te plate. After 3m te difference between analytical and Fluent bulk temperatures is on te order of 10-3 %. T b 292 Bulk Temperature (K) 290 288 286 284 T b = 3.2423x + 283 Tb Linear (Tb) 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 Table 2 below. Te results for te convective eat transfer coefficient indicate tat te temperature value used to describe 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 based 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 be 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, but tey under-predict te c values by about 20% for te CHF solution and by about 24% for te CWT solution. Te bulk temperature yielded te best solution for te convective eat transfer coefficient, resulting in an error margin of less tan 0.5% for bot cases (after termal development). Since te bulk 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 be. Table 2. Convective eat transfer coefficient solution parameters CHF Case (a) CWT Case (b) 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 b (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 cb (x) = T w w 10 ( x) T ( x) c 10 ( x) T b ( x) qw ( x) cc (x) = 293 T ( x) qw ( x) cb (x) = 293 T ( x) c b 8
c (W/m 2 K) 4 3.5 3 2.5 2 1.5 cref c-ref cc cb 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 cb 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 bot te CHF and CWT cases to compare te grid dependency for te two different boundary conditions. Only te coefficients calculated from te bulk 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 Table 3. Te notation φ is adopted to describe te solution for te finest mes. Te subsequent meses are all notated wit respect to te finest mes. Te next grid size as cell dimensions doubled in bot directions, ence te notation φ 2. Table 3. Mes dimensions Number of cells in te Y Direction Number of cells in te X Direction Smallest cell eigt (m) Smallest cell widt (m) Total number 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 be 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 by φ 2 φ4 log φ φ2 a = log ( 2) (12) 10
In bot 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 number, te tree solutions must be monotonically converging [2]. Te teory of Ricardson Extrapolation states tat te solution from te finest mes can be added to te discretization error found in Equation (11) to attain an approximate grid independent solution. In equation form tis can be stated as: Φ = φ + (13) d ε Te results from te grid sensitivity analysis are sown in Table 4 and plotted below in Figures 8 and 9. Table 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 by comparing te computed convective eat transfer coefficients ( c ) for laminar air flow between parallel plates by 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 bot wall boundary 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 between 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 Enabled Disabled None Disabled Disabled Disabled Disabled Equation Flow Energy Solved yes yes Numerics Absolute Velocity Formulation Enabled yes Relaxation: Variable Relaxation Factor Pressure 0.3 Density 1 Body Forces 1 Momentum 0.7 Energy 1 Solver Termination Residual Reduction Variable Type Criterion Tolerance Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Energy Flexible 0.1 0.7 Discretization Sceme Variable Pressure Pressure-Velocity Coupling Momentum Energy Sceme Standard SIMPLE First Order Upwind First Order Upwind Solution Limits Quantity Limit Minimum Absolute Pressure 1 Maximum Absolute Pressure 5000000 Minimum Temperature 1 Maximum Temperature 5000 13
Appendix B: Nomenclature a Order of te discretization error sceme (-) b c p d D Distance between 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 number 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 symbols d ε φ Discretization error (units based on parameter analyzed) Solution for finest mes (units based on parameter analyzed) φ n Solution for a mes wit cell dimensions n times te finest mes Φ Ricardson Extrapolation solution (units based on parameter analyzed) µ Dynamic viscosity (kg/m-s) ρ Density (kg/m 3 ) Subscripts AV b 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 Textbook, Plogiston Press, 2006. 15