强制对流过程中含局部BrinkmanCForchheimer多孔介质的复合通道中的传热及流动分析(英文)

1、 174A.V . KUZNETSOV Nu =Nusselt number, 2H q /k f (T W T m p =intrinsic average pressure, PaP =parameter dened by Equation (42q =wall heat ux, W/m2Q =parameter dened by Equation (40R =thermal conductivity ratio, k eff /kfs =half thickness of the clear uid region, mS =dimensionless half thickness of

2、the clear uid region, s/HT =dimensionless temperature, T T W /T m T W T i =dimensionless temperature at the clear uid/porousmedium interfaceT =intrinsic average temperature, KT m =mean temperature, (1/HU H 0 u f T d y , KT W =temperature at the wall (y =H , Ku =dimensionless velocity, u f µf /G

3、H2u 1=parameter dened by Equation (29u i =dimensionless velocity at the clear uid/porousmedium interfaceu f =ltration (seepagevelocity, m s 1u =ltration velocity in the portion of the porous layer which is outside the momentum boundary layers, m s 1u =dimensionless ltration velocity in the portion o

4、f the porous layer which isoutside the momentum boundary layers, u µf /GH2U =mean ow velocity, (1/H H 0 u f d y , m s 1U =dimensionless mean ow velocity, Uµf /GH2x =streamwise coordinate, my =transverse coordinate, my =dimensionless transverse coordinate, y/Hz 1=function of the coordinate

5、y dened by Equation (33z 2=function of the coordinate y dened by Equation (28Greek letters=adjustable coefcient in the stress jump boundary condition=constant, (µeff /µf 1/2µf =uid viscosity, kg m 1s 1µeff =effective viscosity in the Brinkman term for the porous region, kg m 1s 1

6、f =density of the uid, kg m 3=parameter dened by Equation (440=parameter dened by Equation (461=parameter dened by Equation (482=parameter dened by Equation (471. IntroductionForced convection in a composite region part of which is occupied by a clear uid and part by a uid-saturated porous medium ha

7、s recently attracted considerable attention and become the subject of numerous investigations. This interest is due to many important thermal engineering applications related to this problem. Solid matrix heat exchangers, the use of porous materials for heat transfer enhancement,ANALYTICAL STUDY OF

8、FLUID FLOW AND HEAT TRANSFER 175 fault zones in geothermal systems, and solidication of binary alloys are a few to mention in this respect. Fluid mechanics in the interface region between a clear uid and a porous medium has been recently investigated by Vafai and Kim 1. In this investigation the ow

9、in the porous region is modeled utilizing the so-called BrinkmanForchheimer-extended Darcy equation. The Brinkman term in this equation represents viscous effects and makes it possible to impose a no-slip boundary condition at the impermeable wall and also to match momentum equations at the porous m

10、edium/clearuid interface without having a jump in velocity. Many modern applications of porous media are characterized by high ow velocities. In such cases, it is necessary to account for deviation from linearity in the momentum equation for porous media. This deviation is accounted for by the Forch

11、heimer term representing the quadratic drag which is essential for large particle Reynolds numbers. From the physical point of view, quadratic drag appears in the momentum equation for porous media because for large ltration velocities, the form drag due to the solid obstacles becomes comparable wit

12、h the surface drag due to friction 2.In solving the problem, Vafai and Kim 1utilized the continuity of both the seepage velocity and the shear stress at the porous medium/clearuid interface. At the time when Vafai and Kim obtained their solution, no volume averaging analysis was available for the in

13、terface region. Such an analysis has been very recently carried out by Ochoa-Tapia and Whitaker 3,4,who have shown that matching the BrinkmanDarcy and Stokes equation retains continuity of velocity, but produces a jump in the shear stress. The solution obtained by Vafai and Kim was modied in 5to acc

14、ount for the jump in the stress at the interface. Fluid ow in ducts partly lled with a porous material was investigated in 6,where the jump in the stress at the interface was also accounted for.Heat transfer in the interface region between a clear uid and a Brinkman Forchheimer porous medium was inv

15、estigated by Vafai and Thiyagaraja 7uti-lizing the perturbation technique. Numerical analysis of forced convection in a channel with multiple emplaced porous blocks was presented by Huang and Vafai in 8.A thorough theoretical study of fully developed forced convection in a channel partly lled with a

16、 porous matrix utilizing the Brinkman ow model for the porous region was carried out by Poulikakos and Kazmierczak 9.In this paper, we concentrate on fully developed forced convection in a parallel-plate channel partly lled with a uid-saturated homogeneous porous material. The ow in the porous mater

17、ial is described by the BrinkmanForchheimer-extended Darcy equation. The case of the uniform heat ux at the walls of the channel is considered. Utilizing the boundary layer approximation, analytical solutions for the ow velocity, the temperature distribution, as well as for the Nusselt number are ob

18、tained.176A.V .KUZNETSOV Figure 1. Schematic diagram of the problem.2. Mathematical FormulationFigure 1depicts a schematic diagram of the problem. Fully developed forced con-vection in a composite channel bounded by two innite xed plates is considered. Since the problem is symmetrical, only half of

19、the channel is shown in Figure 1. A porous material is attached to the walls of the channel, while the center of the channel is occupied by a clear uid. To describe the ow in the porous region, a mo-mentum equation which accounts for both Brinkman and Forchheimer extensions of the Darcy law is utili

20、zed.Figure 1shows two momentum boundary layers in the porous region. The rst momentum boundary layer is near the wall of the channel, while the second boundary layer is near the clear uid/porousmedium interface. The thickness of the boundary layers depends on the value of the Darcy number. Since for

21、 most practical applications of porous media, the value of the Darcy number is small, in this research we proceed from the assumption that these boundary layers do not meet in the center of the porous region. It means that a part of the porous region is outside the momentum boundary layers. A simila

22、r assumption is utilized by Vafai and Kim 10,11in obtaining their analytical solution for the forced convection in a parallel-plate channel completely lled with a uid-saturated porous medium. The limits of validity of this assumption for our problem are discussed later on. A y -coordinate of a point

23、 which belongs to the porous region and is outside the momentum boundary layers is denotes as l . For practical computations l is taken to be the coordinate of the center of the porous layer.ANALYTICAL STUDY OF FLUID FLOW AND HEAT TRANSFER 177 It is assumed that there is a uniform heat ux at the wal

24、ls of the channel. The governing equations for this problem can be presented as: d p d x +µf d2u fd y 2=0, 0 y s, (1 d p d x +µeff d2u fd y 2 µfKu f f c FK 1/2u 2f =0, s y H, (2f c f u f T x=k f 2T y 2, 0 y s, (3f c f u f T x=k eff 2T y 2, s y H, (4where c f is the specic heat of the

25、uid, c F is the Forchheimer coefcient, k f is the thermal conductivity of the uid, k eff is the effective thermal conductiv-ity of the porous medium, K is the permeability of the porous medium, p is the pressure, T is the temperature, u f is the ltration (seepagevelocity, x is the streamwise coordin

26、ate, y is the transverse coordinate, µf is the uid viscosity,µeff is the effective viscosity in the Brinkman term for the porous region, and f is the density of the uid. Equation (1is a momentum equation for the clear uid region,while Equation (2is a momentum equation for the porous region

27、, the BrinkmanForchheimer-extended Darcy equation. Equations (3and (4are energy equations for the clear uid and porous regions, respectively. Following 1215, the longitudinal heat conduction term is neglected in Equations (3and (4.Also, in Equation (4the local thermal equilibrium assumption between

28、the uid and solid phases is utilized.Equations (14 are subject to the following boundary conditions: u f y =0, T y=0, at y =0, (5u f |y =s 0=u f |y =s +0, µeff u f yy =s +0 µf u f yy =s 0=µf K 1/2u f |y =s ,k eff T yy =s +0=k f T yy =s 0, at y =s, (6u f =0, k eff T y=q , at y =H, (7 w

29、here q is the wall heat ux.178A.V . KUZNETSOV The rst two equations in (6present the continuity of the ltration veloc-ity and a jump in the shear stress at the interface, respectively. These conditions are obtained in 3,4without accounting for the Forchheimer term represent-ing quadratic drag in the

30、 momentum equation for the porous region. However, these boundary conditions include an adjustable coefcient, , which permits the necessary exibility in the adjustment of these conditions to experimental data. Therefore, following 5,we utilize here these conditions to match the Stokes and BrinkmanFo

31、rchheimer-extended Darcy equations at the interface.Outside the momentum boundary layers 2u f / y 2 0and u f u and the momentum equation, Equation (2,reduces to d p d x µfKu f c FK 1/2u 2 =0, (8where u is the ltration velocity outside the momentum boundary layers. In the fully developed region

32、of a channel with uniform wall heat ux, T / x on the left-hand side of Equations (3and (4is constant 16.The value of T / x can then be found from the following energy balancef c f H U T x=q , (9where the mean ow velocity, U , is dened byU =1 HHu f d y. (10The Nusselt number is then dened asNu =2H qk

33、 f (T W T m , (11where the mean temperature, T m , is dened byT m =1H UHu f T d y, (12and T W is the wall temperature.Introducing dimensionless variables, the momentum and energy equations, Equations (14, can be recast into the following dimensionless form:1+ d 2ud y 2=0, 0 y S, (131+2 d 2ud y 2 1Da

34、u F u 2=0, S y 1, (14ANALYTICAL STUDY OF FLUID FLOW AND HEAT TRANSFER 179d 2T d y 2 = 12NuuU, 0 y S, (15R d 2Td y 2= 12NuuU, S y 1. (16and the equation for the velocity outside the momentum boundary layer, Equa-tion (8,can be recast into the dimensionless form11Dau F u 2 =0. (17The positive root of

35、Equation (17isu = 1+1+4Da 2F 1/22Da F. (18In Equations (1318 the following dimensionless variables are utilized:Da = KH 2, F =f c FK 1/2H 4µ2fG, R =k effk f, T =T TWTm TW, (19u = u f µfGH 2, u =u µfGH 2, y =yH, =µeffµf1/2, (20where G = d p/d x is the applied pressure gradien

36、t.Equations (1316 must be solved subject to the following dimensionless boundary conditions:u y =0, Ty=0, at y =0, (21u |y =S +0=u |y =S 0, 2 d ud yy =S +0 d ud yy =S 0=Da 1/2u |y =S ,R d Td yy =S +0=d Td yy =S 0, at y =S, (22u =0, T =0, at y =1. (23 Equations (15and (16determine the temperature dis

37、tribution in the channel asa function of the Nusselt number. After Equations (1316 are solved subject toboundary conditions given by Equations (2123, the value of the Nusselt number can be found from the condition that the temperature distribution, T (y, must obey the denition for the mean temperatu

38、re given by Equation (12.This results in the180A.V . KUZNETSOVfollowing compatibility condition which can be used for calculating the Nusselt number 16:1T u d y =U,(24where U =Uµf /GH2. 3. Boundary Layer SolutionVelocity distribution in the channel can be found by integrating the dimensionlessm

39、omentum equations, Equations (13and (14.3.1. V ELOCITYIN THE CLEAR FLUID REGION(0 y S u =u i +S 2 y 22, (25where u i is the dimensionless velocity at the clear uid/porousmedium interface. Exact solution for the velocity distribution in the porous region is not avail-able. However, it is possible to

40、obtain a boundary-layer solution. There are two momentum boundary layers in the porous region, one is near the channel wall and the other is near the clear uid/porousmedium interface. To obtain the boundary layer solution it is assumed that these boundary layers do not overlap in the center of the p

41、orous region. The following conditions are utilized outside the momentum boundary layers:u u ,d ud y 0, at y =L.(26As L the y -coordinate of any point which belongs to the porous region and which is outside the momentum boundary layers can be taken. For practical computations, we assume that L is th

42、e coordinate of the center of the porous layer L =S +(1 S/2. Utilizing this approach, the following velocity distribution in the channel is obtained. 3.2. V ELOCITYIN THE BOUNDARY LAYER2REGION (S y L u =(u +u 11 z 21+z 22 u 1,(27wherez 2=B exp D(y S , (28ANALYTICAL STUDY OF FLUID FLOW AND HEAT TRANS

43、FER 181u 1=2u +32Da F, (29B = 1u i +u 1u +u 11/21+u i +u 1u +u 11/2, (30andD = 12F (u +u 131/2(313.3. V ELOCITY IN THE BOUNDARY LAYER 1REGION (L y 1u =(u +u 1z 1 1z 1+12 u 1, (32wherez 1=A exp D(1 y (33 andA = 1+u 1u +u 11/21u 1u +u 11/2. (34The dimensionless velocity at the clear uid/porousmedium i

44、nterface, u i , can be found from the jump in the shear stress condition given by the second of Equations (22.This results in the following transcendental equation for u i : (ui u 23F (ui +u 11/2+S =Da 1/2u i . (353.4. M EAN VELOCITYThe dimensionless mean velocity, U , can then be found from the fol

45、lowing equation:U =Su d y +LSu d y +1Lu d y. (36182A.V . KUZNETSOVThis results in the following equation for U :U =u i S +S 33+u (L S +2 6(u +u 1 1/2F 1/2(1+B26(u +u 1F (u +u 1 1/2(1+B exp D(L S +u (1 L 26(u +u 1 1/2F 1/2(1+A +26(u +u 1F (u +u 1 1/2(1+A exp D(1 L . (37With the velocity distribution

46、found, the temperature distribution in the channel can be obtained by integrating the energy equations, Equations (15and (16.3.5. D IMENSIONLESS TEMPERATURE IN THE CLEAR FLUIDREGION (0 y S T =T i +12Nu U u i 2+S 24(S2 y 2 124(S4 y 4 .(383.6. D IMENSIONLESS TEMPERATUREIN THE BOUNDARY LAYER2REGION (S

47、y L T =T i 12R Nu U Q(y S +u 2(y S 2 62F ln 1+B exp D(y S 1+B, (39whereQ =u i S +S 33 2 6(u +u 1 1/2BF 1/2(1+B.(403.7. D IMENSIONLESS TEMPERATUREIN THE BOUNDARY LAYER1REGION (L y 1T =Nu 12RU (1 yP +u 2(y+1 2L(1 y+62F ln 1+A exp D(1 y 1+A, (41ANALYTICAL STUDY OF FLUID FLOW AND HEAT TRANSFER 183where

48、P =Q +u (L S 2 6(u +u 1 1/2F 1/2(1+B exp D(L S +2 6(u +u 1 1/2F 1/2(1+A exp D(1 L . (42The dimensionless temperature at the clear uid/porousmedium interface can be found by matching the temperature distributions given by Equations (39and (41at y =L :T i =Nu ,(43where =12RU u i S +S 33 (1 S +u 2(1 S

49、2 2 6(u +u 1 1/2B F 1/2(1+B (1 S 2 6(u +u 1 1/2F 1/2(1+B exp D(L S (1 L +2 6(u +u 1 1/2F 1/2(1+A exp D(1 L (1 L 62F ln 1+B exp D(L S 1+B +62F ln 1+A exp D(1 L 1+A . (443.8. N USSELT NUMBERThe Nusselt number can now be found from the compatibility condition, Equa-tion (24:Nu =U 0+ 2+ 1, (45where the

50、parameters 0, 2and 1are dened as the following integrals: 0=S0(T/Nu u d y, (46184A.V . KUZNETSOV 2=LS (T/Nu u d y, (47and1=1L (T/Nu u d y. (48With the velocity and temperature distributions found, these integrals can also be calculated analytically. The value of 0can be found:0=S 3 3+17S 7630U +S ui

51、 +2S 5u i 15U +S 3u 2i 6U. (49The value of 2can be found as2= Lu Su +3D2L 2u 2F RU L 2Qu 4RU +S 3u 2 12RU 32( 8u +D 2S 2u +BD 2S 2u 8u 1 2D F RU(1+B+LQ( 4u +DSu 4u 1 2DRU +4 (u +u 1 D(1+B 4exp DL (u +u 1 D(exp DL +B exp DS +2exp DL Q(L S(u +u 1 DRU(exp DL +B exp DS 122(u +u 1 D F RU(1+B exp D(L S +1

52、2DRU(1+B exp D(L S 1×u 12L 2u DL 3u BD exp D(L S L 3u 24LSu +3DL 2Su (1+B exp D(L S +12S 2u 3DLS 2u (1+B exp D(L S +12L 2u 1 24LSu 1+12S 2u 1 Q 4D 2RUD 2S 2u 8DS(u +u 1 2Q(u +u 1 D 2RU ln 1+B exp D(L S 1+B+D F RU(1+B exp D(L S 1ANALYTICAL STUDY OF FLUID FLOW AND HEAT TRANSFER 185×32ln 1+B

53、exp D(L S 1+B×DLu (1+B exp D(L S 4(u +u 1 +2Su (u +u 1 DRU L S +1D ln 1+B exp D(L S 1+B 4u (u +u 1 DRU L 22 S 22 S ln 1+B D+L ln 1+B exp D(L S D +Li 2 B D 2 Li 2 B exp D(L S D 2, (50where Li 2=0ln (1 d (51is the dilogarithm function. To compute values of Li 2for large negative values of the arg

54、ument the following correlation is utilized:Li 2 =Li 2 11+ +12ln 2 1+1 12ln 2 26, >0. (52Finally, 1can be found as 1=12RU 3D2u F (1 L 2 P u 2(1 L 2 +(2 3Lu 2 6+L 3u 2 6 P (4u Du +4u 1 D+L 2u (4u DLu +4u 1 2D 242(u +u 1 D F (1+A +242(u +u 1 D F (1+A exp D(1 L +4exp DL (1 LP (u +u 1D(Aexp D +exp DL +2(1 L 2u (u +u 1 D(1+A exp D(1 L +2L 2u (u +u 1 D +LP (4u Du +4u 1 D186A.V . KUZNETSOV Lu (Du +8Lu 2DLu +8Lu 1 2D+4P (u +u 1D 2 ln1+A exp D(1 L 1+A 6 2( 4u +DLu +ADLu exp D(1 L 4u 1 D F (1+A exp D(1