Online proceedings - EDA Publishing Association

24-26 September 2008, Rome, Italy21 d pf Swu ∂ ⎛∂u ⎞ (1)0 =− + ff+ ⎜ m( z) ( ν + νT) ⎟ρfdx m( z)2 ∂z ⎝∂z⎠The terms on the right hand side represents pressure drop,internal drag (2 terms), viscous dissipation and Reynoldsstresses. Internal drag is a sum of friction and form drag, dueto the solid fluid interfaces inside of the channel. The turbulentkinetic energy of a porous media flow from Gratton et.al.(1996),2 3 22⎛∂u⎞ ∂ ⎡⎛ νT1 ⎞∂b⎤ffu2 ⎛∂b ⎞ b νT+ ⎢+ ⎥+ + = C (2)⎜ ⎟ μ∂x ∂x ⎢⎜PrT Re ⎟por∂x⎥m( z) Re ⎜por∂x⎟⎝ ⎠ ⎣⎝⎠ ⎦⎝ ⎠ νTis used to close the momentum equation. The relationshipbetween the eddy viscosity and the turbulent kinetic energy is νT= Clμ ( z)b(3)where l(z) is a turbulent scale function, defined by an assignedporous medium structure from Travkin and Catton, 1992 (a,b), Cµ is the turbulent exchange coefficient, and b(z) is themean turbulent fluctuation energy function.The energy equation for the fluid phase is given by∂T ⎛f∂T ⎞fcpf ρfm( z) u∂= ⎜ m( z)( kT+ kf ) ⎟ + αTSw ( Ts −Tf )(4)∂x ∂z⎜∂z⎟⎝⎠and for the solid phase by∂ ⎛ ∂Ts⎞ ∂ ⎛⎛ ∂Ts⎞⎞ (5)( mz ) kST ⎜ ( mz ) kST ⎟ αT Sw ( Ts Tf)0= 1 − ( ) + 1 − ( ) − −∂x ⎜∂x ⎟∂z ⎜∂z⎟⎝ ⎠ ⎝⎝ ⎠⎠Both the momentum and the energy equations needclosure; the friction, f for the momentum equation, and thefinternal heat transfer coefficient, αT, for the energy equations.The needed closure is obtained experimentally from availabledata for a specific morphology. The remaining parametersappearing in the VAT conservation equations, and S w ,are the porosity and the specific surface area. For engineereddevices, these are easily calculated.METHOD OF SOLUTIONThe energy and momentum equations are rewritten in finitedifference form using numerical schemes found in Samarskii[3] and rearranged into a form suitable for solving with anADI scheme. Fig. (1) shows the numerical simulation flowdiagram for flow and heat transfer in porous media using VATmodel.For ADI schemes, an implicit equation in each direction issolved at each psuedo-time step (or iteration). Hence, for thepresent two-dimensional case, each time pseudo-step isdivided into two steps. For example, the solid phase energyequation, the finite difference equation for the sweep in thevertical direction iss s+ 1/2 s s+ 1/2 s s+ 1/2 s s+1/2 sAjTSTS + Bi, j 1 jTSTS + Ci, j jTSTS + D T= F− i, j+1 i,jiTS(1)and for the horizontal direction,s s+ 1 s s+ 1 s s+ 1 s s+1 siTS S+i 1, j iTS S+i, j iTS S+ + 1/2=− i+1, j i,j jTSA T B T C T D T F(2)Similarly, for the fluid phase, the finite differenceequation for the sweep in the vertical direction isAssign morphology functions m, S wInitialize all the variables, b, u, TSolve the turbulent kinetic energy equation for b s+1T 1C lbSolve the solid temp. Eq. for T s+1s Solve the momentum equation for u s+1Using closure correlation, solve for c d andSolve the fluid temp. equation for T s+1 Solve the solid temp. Eq. for Ts+1s~~s1 sTTs1sTsTsε~s1 ~suuYSolve the fluid temp. equation for T s+1 s 1 sbsbsεFigure 1. VAT flow diagram.1/ 2 1/2 1/ 2 1/ 2A T + + + ++ B T+ C T+ D T = Fs s s s s s s s sjTF i, j− 1 jTF i, j jTF i, j+1 Si,j iTFand for the horizontal direction,s s 1 s s 1 s s 1 s s 1 sAiTFT + + + ++ Bi 1, j iTFT + Ci, j iTFT + 1/2+ D Ti 1, j S= F− +i,j jTF(4)In these four equations, the effects of the non-uniformgrid, relaxation, morphology, and thermo-physical propertiesof the working solid and liquid materials are grouped togetherand accounted for in the coefficients A, B, C, D, and F.CODE VALIDATIONCode validation is accomplished by comparing predictionswith measurements for two geometries; pin fin and plane fingeometries. Optimization of these two geometries will bedemonstrated in a later section.To illustrate the validity of the present mathematicalmodel and numerical scheme for pin fins, calculated localNusselt number is compared with the Nusselt number reportedby Zukauskas (1987) as it is the local Nusselt numberaveraged around a pin fin. To compare with Zukauskas, theNusselt number is defined as~α dpordpordT Nu = =1~ kfdsk k ΔΩST −T∫dx (1)ffw( )s ∂SwEquation (11) is simplified with the help of Eq. (4)~ αdporNupor= =kf~~T( )( x z)T x zc m u~∂ , ∂ ⎡ ∂ ,pfρfz − ⎢ m( z)kfdpor∂x∂z⎣ ∂z~kS z T x,z − T x,zfw[ ]( ) ( ) ( )si( )Using calculated temperature distributions, the aboveNu por is found by substituting temperature values into theabove equation. The pin fin surface heat transfer results werecompared with those obtained from Zukauskas (1987) and theend wall values with Rizzi and Catton (2002). Comparisonwith the results of Rizzi and Catton are shown in Fig. 2.⎤⎥⎦(3)(2)©EDA Publishing/THERMINIC 2008 179ISBN: 978-2-35500-008-9