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C HAPTER 1 0 Convective Heat Transfer Through Porous Media 10.1 INTRODUCTION A porous medium basically consists of a bed of many relatively closely packed particles or some other fonn of solid matrix which remains at .rest and through which a fluid flows. I f the fluid .fills all the gaps between the particles, the porous medium is said to be saturated·with the fluid, i.e., with a saturated porous medium it is not possible to add more fluid to the porous medium without changing the conditions at which the fluid exists, e.g., its density i f the fluid is a gas. A porous medium is shown schematicaltyin Fig. 10.1. An example of heat transfer through a porous medium is heat transfer through a layer of granular insulating material. This material will be saturated with air, i.e.~ the space between the granules of insulating material is entirely filled with air, and . this air will flow through the insulation material as a result of the ~mperaturediffer­ ence imposed on tbe material, i.e., there will be a free convective flow in the porous mat~rial. Even when a fibrous insulation is used, the flow in ~e insulation can be Fluid Flow '- Between Particles . -.. "Particles" _ /-AtRest - .. @ )'® ' -" ~ .- .. (((( I.:.:Y'-" r;;;:-.. @®'-'"' \ ««) ~® - .. ~-~--~ -..-@~®@ . ( ((( ~- --- --®~® ~ rrrrr '-"~ ----.. ~~~. ~ ~ t{(((l -- '-" , FIGURE 10.1 A porous medium. 487 488 Introduction to Convective Heat Transfer Analysis Resistance t o Flow i n x-direction > Resistance t o Flow i n z-direction. F IGURE 10.2 Possible directional dependency of properties of a fibrous insulation material. FIGURE 10.3 Heat transfer to a pipe buried in water saturated soil. treated as flow i n a porous medium although, in this case, the fibers are often aligned with each other and as a result the properties o f the porous medium will vary with the direction considered. This is illustrated in Fig. 10.2. Another example :of heat transfer involving a porous medium is 'heat transfer from a pipe. o r cable bUrled i n soil or in a bed o f crushed stones which is saturated with ground water which is flowing through the soil or stones. This is illustrated in Fig. 10.3. Many geological flows such as flow in an oil reservoir or i n a geothermal power system involve convective heat transfer in a porous medium. The present chapter gives no more than a brief introduction to convective heat transfer i n a porous medium. I t is an area o f considerable practical importance and there is a large body o f literature on the topic to which the reader is referred for more detail, for example see [1] to [12]. Attention will b e restricted to steady flows in this chapter and multiphase flows will not b e considered. 10.2 AREA AVERAGED VELOCITY I f a plane drawn in a porous media flow is considered, the velocity will not be uniform over the plane. T here will be no flow where this plane intersects the solid particles, i.e., the velocity is zero over these areas, and even where the plane passes between ,/ CHAPIER 10: Convective Heat Transfer Through Porous Media 489 particles, the velocity will not, in general, be uniform. This is illustrated schematically in Fig. 10.4 in which the velocity variation along a line drawn in a porous media flow is shown. Because the size of the particles considered is small compared to the overall size of the system, the concern is usually not with the details of the local velocity variation but only with the mean velocity over an area of the plane that is small compared with the overall size of the system but large compared to the size of the particles [12]. The mean velocity over a rectangular ,area AyAz iIi size on the plane is then given by: U = AyAz Jo 1 r~z r~Y Jo u pdydz (10.1) where up is the local velocity in the porous material. Because Ay and Az are small compared to, the size of the system, u may be regarded as the area-averaged velocity component at a point in the flow. The areaaveraged velocity components in the other coordinate directions at a point in the flow are defined iIi the same way, i.e., by: v= ~d: W= AxAz Jo 1 1 r~x r~z Jo v pdzdx AxAy Jo r~Y r~x Jo w pdxdy . The analyses of convection in porous media presented in this chapter will all be based on .the use of these area-averaged velocity components. Their use is, of course, similar t athe use of time-averaged velocity components in turbulent flows. By definition, with the area-averaged velocity defined as above, the continuity equation for :flow in a porous medium will have the same form as that for the flow o f a pure fluid, i.e.; the continuity equation for flow through a porous medium is, i f density variations are negligible: au + av + aw a x ay az Distance A A longAA =0 (10.2) Mean Velocity o nAA / A Velocity o nAA F IGURE 10.4 Velocity distribution in a porous medium flow. 490 Introduction to Convective Heat Transfer Analysis 10.3 DARCY FLOW MODEL When the fluid flows through a porous medium, the solid particles exert a force on the fluid equal and opposite to the drag force on the solid particles. Thi,s force must be balanced by the pressure gradient in the flow, i.e., for flow through a control volume for any chosen direction: Difference between rate fluid 'd momentum Ieaves and the rate fl W momentum enters control volume . Net VISCOUS force on surface = f tr I I 0 con 0 vo ume + Net pressure force on control volume - Drag force on particles i n control volume + Net buoyancy force (10.3) In the Darcy model of ffow through a porous medium,. it is assumed that the flow velocities are low and that momentum changes and viscous forces in the fluid are consequently negligible compared to the drag force on the particles, i.e., i f flow through a control volume of the type shown in Fig. 10.5 is considered, then: . Drag force on Net pressure force arti I ' tr I on control volume = P Ic e~ l fl con 0 + Buoyancy force . directi vo ume i ll i ll any on direction considered (10.4) Thus, in the Darcy model, the rate of change of momentum through the control volume and the viscous forces acting on the surfaces of the control volume are assumed to be negligible compared to the drag force and the buoyancy force [13],[14]. Buoyancy forces will, for the present, be neglected. In this case, i f the x-direction is considered, Eq. (10.4) gives: p dydz - (p + ~~ d X) d ydz = D px where D px is the/net drag force. Hence: _u p = _ D px _D ux d xdydz - . x Dx . (10.5) being the drag force per unit volume in the x-direction. Control Volume / FIGURE 10.5 Control volume considered.. CHAPTER 10: Convective Heat Transfer Through Porous Media 491 Now because, by the basic assumptions being made, the fluid velocity over the particles is small, the Reynolds number based on particle size will be very· small . and the flow over the particles will be a "creeping" or "Stokes" type flow. I n such flows, the drag force on a body is proportional to the velocity over the body and to the viscosity o f the fluid, I Ll' Hence, Dx in Eq. ( lOA) will be proportional to u lLl' This means that Eq. (10.5) can be written as: . u lLl = - K ap ax (10~6) where K i s the constant o f proportionality. This is often referred to as D arcy's l aw or the Darcy· model. K is called the "permeability" o f the porous medium. I t will depend on the number o f particles p er unit volume o f the medium and on the shape and size o f the particles. K has the dimensions o f ( Lengthi, e.g., the units o f m2 • Eq. (10.6) can be written as u=--ILl ax K ap (10.7) Similarly, in the other flow directions, the Darcy model gives: V= - - - K ap ILl a y (10.8) and w=--ILl K ap az (10.9) In writing Eqs. (10.7) to (10.9), i thas been assumed that K has the same value i n e ach direction, i e., that the porous medium is isotropic. In general, as discussed earlier, because the particles are not spherical and because they can all b e aligned in some way as shown in Fig. 10.6 (see also Fig. 10.2), this will not be the case. In such a case, i.e., i n the case o f anisotropic porous materials, Eqs. (10.7) to (10.9) must be written: K z ap K yap Kx u p V=--W=--(10.10) u=--ILl ax' ILl a y' ILl UZ where K x , K y, a nd Kz are the permeabilities in the x, y, and z directions, respectively. "Particles" ~~ ~~~ ~~~ y ~ K in x-Direction ~ Kin y-Direction ~~ F IGURE 1 0.6 An anisotropic porous material. 492 Introduction to Convective Heat Transfer Analysis I I t will b~ a ssumed in the present chapter that the material being considered isotropic, i.e., that Kx = Ky = Kz = K. Hence, for a flow i n which the Darcy assumptions c an b e used, the governing equations for the velocity components are, i f buoyancy forces are neglected: au + av + aw ax ay az =0 (10.11) (10.12) ( 0.13) u = -- IL I ax a v = - .K - p - K ap 1 ILl ay w= - - - . IL I az K ap (10.14) This represents a set o f four equations in the four variables u, v, w, and p. I n solving these equations, i t m ust be noted that because the effects o f viscosity are being ignored, t he no-slip boundary condition cannot b e applied, i.e., the only boundary condition on velocity at a solid surface is that the normal component o f the velocity at the surface is O. T his is illustrated in Fig. 10.7. W hen t he Darcy assumptions are adopted and the effect o f buoyancy forces are neglected, no vorticity can be generated i n t he flow because the only forces on the surface o f a control volume being considered are pressure forces which act normal to the surface o f t he control volume. Forces with a component that is tangential to the surface are required in order to generate vorticity. Therefore, the velocity distribution i n D arcy flow will b e that for irrotational flow, i.e., exactly the same as that that would exist i n inviscid or potential flow in the situation being considered. To illustratethat this is the case, consider two-dimensional flow. T he governing equations for the velocity components in this case are: au + av ax ay =0 (10.15) (10.16) K ap u = -- IL I ax Component =0 at Surface Non-Zero Velocity Component at Surface / FIGURE 10.7 Boundary ,(ondition on velocity at surface. CHAPI'ER 10: Convective Heat Transfer Through Porous Media 493 v=--f.L f ay K ap (10.17) .The pressure can b e e liminated b y differentiatitig Eq. ( 10.16) with respect to y a nd Eq. (10.17) with respect to x a nd subtracting the first result from the second. This process gives: av _ au ax ay =0 (10.18) Therefore, the equations governing the velocity components i n D arcy flow are: . au + av = 0 ax ay ,~ (10.15) and: av _ au = 0 ax ay I f t he s tream function, (10.18) r/J, is introduced, it being defined as before by: ar/J ar/J u=-v=--·- ay' ax t hen i t w ill b e s een that Eq. (10.15) i s always satisfied and Eq. ( 10.18) becomes i n terms o f r/J: -+-=0 2 a2 r/J ax a2 r/J ay2 T his IS t he equation that has traditionally been used to obtain solutions for twodimensiorralpotential flows. T he methods used to obtain s uch solutions can therefore be used to find the velocity distributions i n Darcy flows. T he a bove equations for the pressure variation i n D arcy flow involve the permeability, K. T his quantity i s m ainly dependent on the porosity o f t he material, cp, Le., the ratio o f t he pore volume, Vp , t o the total volume, V, o f t he p orous material, and on the m ean diru;neter o f t he particles. The relation between these quantities depends on the shape o f t he particles and h ow they are arranged. I f, for example, the particles T ABLE 1 0.1 .Typical Values of Permeability and Porosity Material Brick Permeability. m2 4.8 X 2.4 X 2.0 x 4.8 X 5 .0 X L3 X 2.9 X 1 0- 14 to 2.2 X 1 0- 11 1 0- 11 to 5 .1.x 1 0- 11 1 0- 15 to 4.5 X 1 0- 14 1 0- 11 to 1.8 X 1 0- 10 1 0- 16 to 3 .0x 1 0- 12 1 0- 14 to 5.1 X 1 0- 14 1 0- 13 to 1.4 X 1 0- 11 P orosity· percent 12 to 35 88 to 93 4 to 1 0 35 t o 5 0 8 to 38 37 to 4 9 43 to 55 Fiberglass Limestone Sand (Loose Bed) Sandstone Silica Powder Soil , , 494 Introduction to Convective Heat Transfer Analysis are spherical, it.has been proposed that: K = cf>3d 2 150(1 - cf»2 (10.19) Some typical values. of permeability and porosity for a few materials are shown i n Table 10.1. Deviations from the Darcy model will b e considered i n a later section. E XAMPLE 1 0.1. I n some cases, unidirectional flow through a porous medium c an b e approximately modeled as flow through a bundle o f small diameter parallel tubes o f diameter, D, as shown in Fig. BIO.I. Assuming that the pressure gradient i n t he tubes is given b y the Hagen-Poiseuille equation, i.e., by: .w here U i s the mean velocity i n the tubes, derive a n expression for the permeability, K. Solution. I f U is the mean velocity through the porous medium, then the m ean velocity i n t he tubes is given by continuity considerations as: U=~ </> H ence, using the Hagen-Poiseui~le equation i t follows that: dp 64J.LfU d x = - </>D2 i.e.: J.LfU - (dpldx) = J.Lfu </>D2 64 B ut t he definition o f the permeability is such that: K = - (dpldx) so the model here being considered gives: K = </>VZ 64 Flow T hrough Series of Parallel Tubes / FIGURE EIO.I CHAPTER 10: Convective Heat Transfer Through Porous Media 495 10.4 E NERGY E QUATION The energy equation is derived, as before, by applying conservation of energy considerations'to flow through a control volume'of the type shown in Fig. 10:8. Viscous dissipation effects are neglected, in line with the neglect of visCous stresses in the consideration of momentum conservation that was given in the previous section. Hence, since steady flow is being assumed, conservation of energy requires that: Rate enthalpy leaves _ Rate enthalpy enters _ control volume control volume Rate of heat transfer Rate of heat transfer , (10.20) intO control volume' out of control volume I t w ill be assumed that the fluid and the particulate material are in thermal equi- librium, i.e., that locally the fl~d and the particulate matter in contact with it are at the same temperature. ' ,, . The enthalpy changes will first be considered. Consider the x-direction. Because the fluid properties are being assumed to be constant, the rate enthaipy enters the left-hand face of the control volume is given by: Hx = m xcpfT = P fudydz c pfT (10.21) Here~ P f and c pf are the density arid specific heat of the fluid. The fluid values are used because only the fluid is in motion. ' The rate that enthalpy leaves the right-hand face of the control volume is given by: Hence, the'difference between the rate that en:thalpy leaves and the rate i tenters the control volume in the x-direction is given 'by: ' . Hx+dx - Hx = : x(PiCPfUT)dxdydZ (10.22) F IGURE 10.8 Control volume used in deri-ying ene~gy equations. / , , I I i i , I", I i , ';' 496 Introduction to Convective Heat Transfer Analysis Since the properties of the fluid are being assumed constant, this gives: Hx+dx - Hx = P fCpfax(uT)dxdydZ a (10.23) Similarly, in the y - and z-directions, the differences between the rates that enthalpy leaves and enters the control volume are: Hy+dy - Hy = P fcpf a y ( vT)dxdydz a and Hence, the net difference between the rate that enthalpy leaves and the rate at which it enters the control volume is given by: ' 'a [a P fCpfax(uT) + a y(vT) a] + a /wT ) d xdydz (10.24) This can be written as: P fcf { T [~: + :~ + ~:] + [u~~ + v~~ + w ~:]} dxdydz aT P fcf [ u a x aT a T] (10.25) The first term in this equation is 0 by virtue of the continuity equation so Eq. (10.24) can be written as: . + v a y + w az d xdydz (10.26) Next, consider the heat transfer rate into the control volume. The coordinate directions will again be separately considered. In the x-direction, the rate of heat transfer into the control volume through the left-hand face is given by: aT Qx = - ka a xdydz (10.27) The thermal conductivity has to be carefully defined. The fluid and the particulate matter will, in general, have different thermal conductivities. The conductivity, ka, in the above equation is an area averaged or a~parent thermal conductivity of the porous material. The rate of heat transfer out of the control volume in the x-direction is given by:! (10.28) Hence, the difference between the rate at which heat is transferred into the control volume and the rate at which it is transferred out of the control volume in the x-direction is given by: . a Qx - Qx+dx = - ax(Qx)dx = a + a x [ ka a T] d xdydz ax (10.29) / P CHAPI'ER 10: Convective Heat Transfer Through Porous Media 497 I f the apparent thennal conductivity is assumed to be constant, this gives: a2T' Qx - Qx+dx = ka a x2 d xdydz (10.30) Similarly, in the y- and z-directions, the differences between the heat transfer rate into the control volume and the heat transfer rate out of the control volume are· given by: . • (10.31) and: Qz - Qz+dz " = ka a2T z a 2 d xdydz (10.32) Hence, the net difference between the rate that heat is transferred into the control volume and the rate at which it is transferred out of the control volume is gi,ven by: a2T ka [ a x2 + a2T' a2T] ay2 + ay2 d xdydz (10.33) Substituting Eqs.(10.26) and (10.33) into Eq. (10.20) and canceling ( dxdydz) . gives, after rearrangement: u ax aT + v ay + W aT aT ( ka ) (a2T az = p fcpfax2 + a2T ay2 +. az2 a2T) (10.34) The quantity ( ka/Pfcpf) is tenned the "apparent thennal diffusivity" and will· here be given the symbol i ra. I t is not the thennal diffusivityof the fluid since it is the .ratio of the apparent conductivity of the porous medium to the product of the density arid specific heat at constant pressure of the fluid. Eq. (10.34) together with Eqs. (10.11) to (10.14) constitutes the set of equations governing forced convective flow through a porous medium. As discussed in the previous section, the distribution of the velocity components is the saine as would exist with potential flow in the' same geometrical situatio:n. This potential flow solution gives the values of u, v, and W which can then be used in Eq. (10.34) to give the temperature distribution. The apparent conductivity that occurs in the energy equation i sa result of conduction in the solid material and in the fluid. I f a simple "parallel" path model is . assumed, ka will be given by: (10.35) k f and ks being the conductivities of the fluid and solid material respectively andcf> is the porosity as previously defined. Other more complex models that relate, ka t o cf>, kf ,and ks usually must be used or ka haS to be measured experimentally. / 498 Introduction to Convective Heat Transfer Analysis 10.5 BOUNDARY LAYER SOLUTIONS FOR TWO·DIMENSIONAL FORCED CONVECTIVE HEAT TRANSFER I f the Darcy assumptions are used then with forced convective flow over a surface i n a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there i s therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which.there are significant temperature changes in the direction normal to the surface. U nder' many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, t he ratio o f the boundary layer thickness, 8T , to the size o f the body as measured by some dimension, L, being small [15],[16]. The velocity component in the x-direction shown i n Fig. 10.9 can, because the boundary layer is assumed to be thin, be taken as equal to the velocity at the surface, i.e., as equal to the velocity that would exist at the surface at the value of x considered i n inviscid flow over the surface (see discussion in Section 10.3 above). The boundary layer form o f the full energy equation for porous media flowis derived using the same procedure as used in dealing with pure fluid flows, this procedure having been discussed in Chapter 2. Attention will be restricted to two-dimensional flow. Now the continuity equation gives: - av ay =- au ax Therefore, i f U<x> i s a\ measure o f the value of u equal, for example, to the velocity in the free-stream ahead o f the body and i f L is some measure o f the size of the body in the x-direction then because y is o f the order o f 8T i n the boundary layer, the above equation shows that: (10.36) / F IGURE 10.9 ThennaI boundary layer in a porous medium. CHAPTER 10: Convective Heat Transfer Through Porous Media 499 Defining the following for convenience: X = ~' Y = Z. , u = ~ , V = ~ , (J = L uT ~ Ur» Ur» TT r- - .Tr»r» T. w (10.37) where Tr» is the temperature in the free-stream temperature outside the boundary layer and Tw r is a measure of the wall temperature. X, Y, and (J will all be of order 1 in the boundary layer. The energy equation for the tWo-dimensional flow which is being considered here can be written as: aT aT u -+v-= ax (pfcpf a-T2+aka )(a2 · ~2T) a y· x y2 (10.38) I n terms of the variables defined above, this equation gives: U ;; + v;~ (7) = (:.~}[;~ + (8T~L)2 :~] (10.39) The left-hand. side of this equation is of order 1. Considering the right-hand side, 8 TIL is by assumption small so, since a2(JlaX2 and a2(Jlay2 are both of order 1, the first term on the right-hand side of the above equation will be much less than the second and the first term is, therefore, negligible compared to the second. Hence, the following form of the energy equation applies in two-dimensional boundary layer flow in a porous medium: .. U aT ax aT + v ay = (Pk fcpf a ) a2T ay2 (10.40) Further, since the left-hand side of Eq. (10.39) has order of magnitude 1, the right-hand side of this equation must also be of order 1. Since a2(Jlay2 is of order 1, this requires that: 1 (uaa) (8TIL)2 = 0(1) lL i.e.: (lOA1) where: (10.42) s the Peelet number. For the boundary layer assumptions to apply, i.e., for 8 TIL to Ie small,it is necessary, therefore, that the Peclet number be large. . I n order to illustrate the use of the boundary layer equations, consider, first, twoimensionalforced convection flow over a flat plate that is buried in a porous material 1 such a way that it is aligned With the fluid flow. The situation being considered is lUSas shown in Fig. 10.10. . 500 Introduction to Convective Heat Transfer Analysis FIGURE 10.10 Flow over a flat plate in a porous medium. ' Because the plate is aligned with the flow and because the effects o f viscosity are being ignored, the plate does not disturb the flow, and the velocity distribution is given by: U = Ur», V = ' 0 everywhere, this being the solution for inviscid flow i n t he situation. With this solution for the velocity components, the energy equation gives i f a boundary layer-type flow is assumed: (10.43) I t will first be assumed that the plate is at a uniform temperature, Tw. T he fol-, lowing is then defined for this case: o= ao Tw - T Tw - Tr» (10.44) T he governing equation, Eq. (10.43), then becomes: a20 Ua> a x = l ¥a ay2 (1O.45) N ow i t seems reasonable to assume that the temperature profiles i n t he boundary l ayer at all VJlhieS o f x will be similar, i.e., that: (10.46) where the function f does not vary with x. This is illustrated i n Fig. 10.11. / FIGURE 10.11 Similar temperature profiles i n boundary layer. CHAPI'ER 10: Convective Heat Transfer Through PorouS Media 501 Now 8T is basically not a clearly defined quantity. Hence, in view ofEq. (10.41) and noting that 8T is here the bo~ layer thickness at any value of x, 8T is .assumed to be proportional to x l Juooxla a • Eq; (10.46) is, therefore, written as: 8 = funCtiOn( y ) = 8("1) x l JUooxla a . (10.47) where "1 is here given by: l12 '." = -l. JUoox = l.Pex , x aa x (10.48) Here, x is the Peelet number based on x. Substituting Eq. (10.47) into Eq. (10.45) and noting that, by assumption, 8 does not depend on x, gives the following: - Pe u 8' a"1' = aa 8" (a"1)2 00 ax ay (10.49) where the primes denote differentiation with respect to "1, i.e: 8' = d"1' d 8. 8 " d: . d"12 d 28 Using Eq. (10.48), Eq. (10.49) becomes i.e.: _(Uoo) "18 ' aa 2 x i.e.: = 8" Pe2x x i.e.: 8 " + "18 ' = 0 -2 (10.50) The fact that an ordinary differential equation 'has been obtained indicates that the assumption of similar temperature profiles is valid. Now, the boundary conditions on temperature are: y = 0, T = Tw y large, T - + Too y large again being meant to indicate y values outside the boundary layer. In terms of 8 and "1, these boundary conditions are: "1 = 0 ,8 = 0 11 large, 8 - + 1 / / (10.51) (10:52) S02 Introduction to Convective Heat Transfer Analysis The temperature distribution is therefore given by solving Eq. (10.50) subject to the boundary conditions given in Eq. (10.52). Now Eq. qO.50) cari be written as: -=- ( 8')' 11 8' 2 (10.53) Integrating this equation once gives: 9' = and then integrating again gives: Clex+~) + C. 9 = Cl f: ex+ ~}1/ where Cl and C2 are constants of integration. Applying the boundary conditions . gives: C2 =0 ,r 1 Cl = - --.,.--..,...-- exp (- ~)d1/ Hence the solution can be written as: (10.54) This can be written as: 8 = e rf(i) (10.55) where erf is the so-called error function. The form o f this solution is shown in Fig. 10.12. The local heat transfer rate at any value of x is given by: \ .qw a ' = - ka T TI = +ka(Tw - u y y=o . Pe l12 T<¥l)-X-8 ,0 x. where: 8' o - dl1 _d~I'11=0 / - This equation can be rearranged t o give: _ 8 ' P 1 12· l\T qw x ka(Tw - T<¥l) - 0 e x , I .e., ./.'lUx - 8' P 112 0 ex (10.56) CHAPTER 10: Convective Heat Transfer Through Porous Media 503 5~--------~~--------~ 4 3 11 2 1 F IGURE 10.12 0.5 () 1.0 Variation of (J with 7J in boundary layer on an isotheimal t:Iat plate. . . Now the error function is such that: 8'0 = _ 1_ = 0.564 1T112 Hence, Eq. (10.56) gives: Nux ,,; 0.564Pe~ (10.57) This equation allows the determ.in.ation of the variation of the local heat transfer rate along the plate. The mean heat transfer rate is given, as before, b y:' . qw = 1r IJo qwdx L i.e., using Eq. (10.56) qw = ka{Tw - Ta)8'o(uoo)ll2 fL aa Jo x ~~ = ka{Tw - ia)8'o(~~)112 2L112 aa (10.S8) which, using Eq.(lO.57), can be rearranged to give: where NUL i s the mean Nusselt number for the plate, i e., h Uka, h being the mean heat transfer coefficient. Similarity solutions to the boundary layer equations for certain other thermal boundary conditions at the surface of the plate. can be obtained, e.g., such a solution can be obtained for a plate with a uniform heat flux at the surface. A heat pump system rejects heat to a water flow thattrickles through a b ed of small pebbles i n the ground. The water velocity in the bed is estimated.to be 1 cm/s. The heat exchanger can be approximately modeled as a. series of 10-cm wide flat plates that are aligned with the water flow. I f the plates are at a temPerature of 20°C and i f the ground water has a temperature of 10°C, find the heat transfer rate from the plates to the water per meter length of plate considering both sides of the plate. The pebble bed has a porosity ofO.3S. E XAMPLE 1 0.2. / 504 Introduction to Convective Heat Thansfer Analysis FIGURE EIO.2 Solution. The water properties will,be evaluated at the mean temperature of the water in the boundary layer, i.e., at (20 + 10)/2 = 15°C. At this temperature, for water: Pf . c pf = 999.1 kglm3 = 4186 Jlkg K = 0.59 W /mK = 1.8W/mK kf The thermal conductivity of the pebbles will be assumed to be given by: ks In the absence of other information, the thermal conductivity of the porous material will be assumed to be given by: fea = cfJkf + (1 - cfJ)ks = 0.35 x 0.59 + 0.65 x 1.8 = 1.38 W/m K Hence: ~a = -Pf-Cp-f ks = 1.38 _ 0 33 999.1 x 4186 - '. X 1 0-6 21 ms The Peelet number is therefore given by: Pe L = 0.01 x 0.1 0.33 x 10-6 = 3030 i Eq. (10.58) then gives: Ii ;j~l = 1.128 x Hence: Therefore, considering both sides of the plate: (3030)112 / Q = 2 X 857 x 0.1 x 1 x (20 - 10) = 1.71 kW Hence, the heat transfer rate from the two sides of the plate per meter 18R_ is 1.71 kW. CHAPrER 10: Convective Heat Transfer Through Porous. Media 50S FIGURE 10.13 Flow in the region of a stagnation point. As another example o f a situation i n which a similarity-type solution can b e obtained, consider flow i n the region o f a stagnation point o f an isothermal body as shown i n Fig. 10.13. The inviscid flow solution gives for this region: Ul = 4Uoo(xID) (10.59) .whe~e D i s twice the radius o f curvature o f t he leading edge. The velocity given by this equation will be taken as the x-component o f the ·velocity in the boundary layer i n the stagnation point region. The y-component o f velocity in the boundary· layer is then given b y using the continuity equation to give: - av au =ay ax i.e., because u is assumed to b e equal to.Ul as given by Eq. (10.59) i t follows that: av = _4U<Xl ay D Because v is zero' at the wall, i.e., at y = 0, this can be integrated to give: v = -4Uoo(ylD ) (10.60) The boundary layer energy equation therefore gives for the stagnation point region: . aT aT 4u<Xl(xID) - a - 4u<Xl(yID)-a = x . Y Again defining: (ka ) a2 T -a 2 P fcpf. y (10.61) and assuming that: 8 = 8(11) (10.62) where, because i t is to be expected from stagnation region solutions for pure fluids that the bOQ.ndary layer thickness will b e constant in this stagnation point region, 11 is .defined by: '. (10.63) 506 Introduction to Convective Heat Transfer Analysis Because this quantity is independent of x, the use of Eq. (10.62) implies that in the stagnation point region, a81ax = O. SubstitutingEq. (10.62) into Eq~ (10.61) therefore gives: 4 - ( y) a8 dT/ ~rx> D a'YJ d y = 28 (ka ) aT/2 [dT/]2 p jcpf a dy Le., because dT/ldy = Le.: Pe112ID: -4T/8',i::: 8 " 8" y + 4T/8' =0 , (10.64) The boundary conditions on temperature are:, = 0, T = Tw y large, T - + Trx> (10.65) y large again being meant to indicate y values outside the boundary layer. In terms of 8 and T/, these boundary conditions are: T/ = 0 ,8 = '0 T/ large, 8 - + 1 (10.66) The temperature distribution is therefore given by solving Eq. (10.64) subject to the bopndary conditions given in Eq. (10.66). Now Eq. (10.64) can be written as: 6 ' = -4T/ Integrating this equation once gives: 8 ' = Cl exp (-2T/ 2) and then integrating again gives: 8 = ClIo" exp (--'2T/ 2)dT/ + C2 where Cl and C2 are constants of integration. ~~: ( 8')' (10.67) (10.68) the boundary conditions , A~plying " , C2 = 0, Cl = L 1' - r x > - '- - - - (10.69) exp(-2T/2)dT/ Hence, the solution can be written as: (10.70) CHAPrER 10: Convective Heat Transfer Through Porous Media 507 4~------~~--------~~ 3 1'/ 2 1 o I...oo:::::-_ _ _. . i -_ _ _ _.....J 0.0 0.5 1.0 () FIGURE 10.14 Variation of 8 with TJ in boundary layer in stagnation point region. This can be written as: (J = erf(211271) (10.71) The fOrIn of this solu,tion is shown in Fig. 10.14. The heat transfer rate is givep. by: " qw = a" T e l12 - ka -- I " +ka(Tw - Too) PDD (J'o a "= y y=o (10.72) where: (J'o = d(J d71 I (10.73) '11=0 Eq. (10.72) can be rearranged to give: qw D /(a(Tw - Too) 1 = (J' 0 P e1 12·I .e., n UD = (J' 0 P eD12 D' AT (10.74) Now the error function is such that: (J'O - = - = 1.596 '1T112 = 1 .596Pelf 2312 (10.75) Hence, Eq. (10.74) gives: N UD (10.76) There are many situations in which similarity-type solutions to the boundaly layer equations cannot b e obtained. Numerical solutions to these equations can be obtained in such; cases. In general, such solutions first involve numerically solving for the surface velocity distribution and then. using the energy equation to obtain the temperature distribution. Here, in order to illustrate-how the energy equation can be numerically solved, it will be assumed that the variation ~f the surface velocity With 508 Introduction to Convective Heat Transfer Analysis distance x about the surface is known, i.e., it will be assumed that: (10.77) where Uoo is some characteristic velocity such as the free-stream velocity ahead of the body and D is some characteristic dimension o f the body involved. The function, F , is assumed to b e known. T he continuity equation is again used to obtain an expression for the velocity component i n the y-direction in the boundary layer, i.e., using: av = -au ay ax and noting that Eq. (10.77) gives: du dx w here F ' = = Uoo F ' D dFld(xID), i t follows that because v is zero at the wall, i.e., at y v = 0: = -uoo(yID)F ' (10.78) B ecause F is assumed to b e a known function, this equation allows the value of v at any value of y i n the boundary layer to b e found. Attention will be given to flow over a surface with a specified temperature and the following dimensionless variables are introduced for convenience: X = ..:. ' y D = 1.Pe1/2 U = !:!-. , V = ~ PeD/2' () = TT - _ Too 1 D D' ' T' U oo Uoo wr .1 (10.79) 00 where Too is the temperature in the free-stream, T w r is a measure o f the wall temperature, and P eD is the Pec1et number based on Uoo and D. Eqs. (10.77) and (10.78) then give: U = F (X), V =- y ~~ (10.80) I n terms o f the above dimensionless variables, the energy equation for the boundary layer becomes: (10.81) The boundary conditions on the solution are: Y = 0 : () = (}w(X), Y l arge: () ~ 0 (10.82) A forward-marching implicit finite-difference solution o f t he energy equation will again be considered. In order to obtain this solution, a series o f nodal lines running parallel to t he x - and y-axes are again introduced as shown i n Fig. 10.15. A uniform grid spacing in the Y -direction will b e u sed here. Consider, as was done i n C hapter 3, the nodal points shown i n Fig. 10.16. CHAPTER 10: Convective Heat Transfer Through Porous Media 509 F IGURE 10.15 Nodal lines used in numerically solving the energy equation in the boundary layer. F IGURE 10.16 Nodal lines used in deriving finite-difference·· approximations. A t any stage of the solution, fJ is known at aU points on the ( i - 1) line and must be determined at all points on the i-line. Because U and V are known at all points in the boundary layer and because U is assumed to depend only on X, the following finite-difference approximations are introduced: afJ)l. (u ax .. I.J ~J = U.(fJi.j I fJi-l.j) ! U{ (10.83) (10.84) a (V afJ)1 .. = V,... y 'J ((Ji.j+1 - 2AY . fJi.j-l) (10.85)· i .j Substituting these into the energy equation then gives, on rearrangement, an equation that has the form: . .. E·fJ·I,J. + F·(J·I,J'+1 + G·fJ·I,J' -1 J J J where the coefficients in this equation are given by: EJ = = HJ .· . (10.86) (~)+ C~~) (10.87) (10.88) F j - AY· - _ (Vi,j) (Ay2 1.) 510 Introduction to Convective Heat Transfer Analysis Gj = J. (~ii )- C~~ AX ) (10.89) (10.90) H . = ( Ui(Ji-l,j) Because the boundary conditions give 8 i,l and (J i ,N, t he outermost g tid point being chosen to lie outside both -the boundary layers, the ~pplication o f Eq.(10.86) to each o f the internal points on the i-line, i.e., j = 2, 3, 4, . .. , N - 2, N - 1 again gives a set o f N equations in the N unknown values o f (J. B ecause (Ji,N is 0, this set o f equations has the following for.n1: (Jj,1 = (Jw E2(Ji,2 E3(Ji,3 + F2(Ji,3 + G2(Ji,1 + F3(Ji,4 + G3(Ji,2 = H2 = H3 (10.91) E N-l(Ji,N-l + F N-l(Ji,N + G N-l(Ji,N-2 = (Ji,N H N-l =0 0 0 0 0 0 0 0 0 ( J'1 I, (J., 2 I (J., 3 I (J., 4 I (Jw H2 H3 i.e., has the form: 1 0 E2 0 G2 0 0 F2 G4 0 0 G3 0 E3 . F3 E4 0 0 0 F4 0 0 0 0 - 84 ooooo ooooo i.e., has the form: G N -l E N-I· F N-l 001 ( Ji,N-l ( J'N I, (10.92) where Q r i s a tridiagonal matrix. This equation can b e solved, as discussed i n Chapter 3, using the standard tridiagonal matrix solver al~orithm o ften termed the Thomas algorithm. I t should b e noted that at any stage o f the procedure, i t i s only necessary to know the values o f the variables on two a djacentgrid lines. .. I n the above procedure, i t was assumed that the outermost grid point, i.e:, the Npoint, was always outside the boundary layer. To ensure that this i s always the case, the solution starts with a relatively small number o f j -grid l ines and the boundary layer growth is monitored. When the boundary layer has reached to within a few nodal points o f the outermost point, the number o f j -lines i s increased. Since the additional points so generated initially lie outside the boundary layer, the values of the variables on these points are initially known. The procedure was discussed in Chapter 3. Once t he distribution of() has b een determined using this procedure, any other property o f the flow can be determined. In the present discussion, the heat transfer CHAP'IER 10: Convective Heat Transfer Through Porous Media 511 rate at the waIl, qw, is the most important such property. This is, o f course, given by Fourier's law as: qw = - ka ~ uTI . u y y=o (10.93) In terms o f the dimensionless variables being used here this gives: ( Twr This can be rearranged as; qwD =_ P elf - TriJ)ka uY y=o u81 (10.94) NUD 1) JPeD = - uY y=o 8w u81 I( . (10.95) where NUD is the local Nusselt number and 8 w iS'the local dimensionless wall temperature. Now since point j = 1 lies o nthe wall: ;~ IY~o = (;n.l 8i.2 = 8i,1 + (HI.%) I n order to determine this from the values o f 8 calculated a t t he nodal points, i t is noted that, to the same degree o f approximation as previously used: (;~ l\,1 ~Y + (;~~ ~1 l l:: \ (10.97) B ut the application o f the boundary layer energy equation to conditions at the wall gives: (10.98) Therefore, Eq.(1O.97)can be rearranged to give: uY and Eq.(10.95) then gives: u81 i,1 = (8i.2 - 8i ~Y 'l) w (10.99) NUD = JPeD ({hI ~-Y' 8 8i.2)(~) (10.100) 8 w being, o f course, the same as 8i 1. In order to illustrate the use o f this procedure, consider flow over an isothermal' cylinder o f diameter D. In this case: U Eq. (10.80) then gives: = 2 s4t(2X) (10.101) / v = - 4Ycos(2X) (10.102) 512 Introduction to Convective Heat Transfer Analysis F IGURE 10.17 Boundary layertlrickness variation around cylinder. . Using these equations, the numerical procedure outlined above can b e used to find t he variation o f the local heat transfer rate in the form o f NuvlPe112 around the cylinder. The solution procedure does not really apply near the rear stagnation point (R i n Fig. 10.17) because the effective boundary thickness becomes very large in this a rea as indicated in Fig. 10.17. This is however, o f little practical importance because the heat transfer rate is very low in the region o f the rear stagnation point. O nce the distribution o f t he local heat transfer rate about t he the cylinder has been found, the mean heat transfer can be found using: qw =- ,1 J1T 'TT' 0 qw dX (10.103) i.e., i f N uv is the mean Nusselt number: MiD =.! JPei> \ (1T N uv d X 'TT' Jo (10.104) J Pev A simple computer program, PORCYL, written in F ORTRAN a nd based on the procedure outlined above is available as discussed in the Preface. The variation of N uvlJPev with X given by this program is shown i n Fig. 10.18. . 2r-----~----~-----.-----. o a.....-_.........a._ _- -L._ _- -'-_ _ 0.0 0.4 0.8 X ..l<..I 1.2 1.6 F IGURE 1 0.18 , Variation o f local dimensionless heat transfer rate around cylinder.. CHAPTER 10: Convective Heat Transfer Through Porous Media 513 I t will be seen from Fig. 10.18 t hat in the forward stagnation point region the heat transfer rate obtained is i n a greement with that previously given for the stagnation . point region, i.e., Nun/ PeK2 = 1.596. T he program gives the mean heat transfer rate from the cylinder as: JPeD E XAMPLE 1 0.3. NuD = 1.015 (10.105) Consider the same situation as dealt with in Example 10.2, i.e., in which a heat pump system rejects heat to a water· flow that trickles through a bed of small pebbles in the ground, the water velocity in the bed being estimated to b e 1 cm/s. I n the present case, the heat exchanger consists o f a series o f long 2.5-cm diameter pipes that are arranged with their axis normal to the water flow. I f the pipes are again at a temperature o f 2 0°C and i f the ground water again has a temperature of 10°C, find the heat transfer from the pipes to the water per meter length of pipe. As in Example 10.2, assume that the pebble bed has a porosity o f 0.35 .. Solution. As discussed in Example 10.2, the water properties are evaluated at the mean temperature o f the water in the boundary layer, i.e., at 15°C. A t this temperature, for water: Pf = 999.1 k g/m3 c pf = 4186 J/kg K k f = 0.59 W /m K As was also discussed in Example 10.2, the thermal conductivity o f the porous material will be assumed to be given by: ka = 4>kf + (1 - 4»ks = 1 .38 W /m K Hence: aa = -Pf-C-p-f = 999.1 ks 1.38 x 4186 =. 0 33 x 10-6 ms 21 T he Peelet number based on the pipe diameter is therefore given by: PeD = 0.01 x 0 .025 0.33 x 1 0- 6 = 7 576 • FIGURE EIO.3 514 Introduction to Convective Heat Transfer Analysis This is assumed to be large enough for the boundary layer assumptions to apply. It is also assumed that the pipes are far enough apart to avoid interaction between the flows over the pipes. Eq. (10.105) is therefore used and gives: Ii ~.~~25 = 1.015 x (757.6)112 Therefore: Ii = 1542 W/m2 Considering a I-m length of 1 pipe: Q = 1542 X 71' X 0.025 X 1 X (20 - 10) = 1.21 kW Hence, the heat transfer rate per meter length of the pipes is 1.21 kW. T he approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed i n Chapters 2 and 3, this integral equation method has largely.been superceded by purely numerical methods o f the type discussed above. However, integral equation methods are still sometimes used a nd.it therefore appears to be appropriate to briefly discuss the use o f t he method here. Attention will continue to, b e restricted to two-dimensional constant fluid property forced flow. As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body i n a porous medium, i f t he Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given b y the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be: d d x [r8T u(T - Jo Too)d Y] = qw P fcpf (10.106) where S r i s the boundary layer thickness as shown in Fig. 10.19. Temperature Variation i n Boundary Layer Saturated n nn/Hlnt Layer F IGURE 10.19 Boundary layer on a surface in a porous medium. CHAPI'ER 10: Convective Heat Transfer Through Porous Media 515 B ecause BT is assumed to be small, the velocity u can, as discussed before,' b e assumed to b e i ndependent o f y a nd equal to the value o f u a t the sUrface a t the particular value o f x being considered that is given by the potential flow solution, this value here being designated by U t(x). Eq. (10.106) c an therefore be written as: -d [ U t' d x i8r (T - Too)dy.] ::;: Pfcpf qw 0 (10.107) I n t he integral method, the form o f the temperature profile is assumed. For example, i f a situation in which the wall temperature variation is specified is considered then i t is assumed that: T - Too - --::;:f- Tw -Too (y) BT (10.108) 'where the form o f the function f is assumed, any coefficients in this function being detennined b y boundary conditions on temperature. Using Eq. (10.108), th~ integral equation as given in Eq. (10.107) c an be written as: [io 1 fd T" UT (Y)~'d" - d [ul(Tw X Too)BT] ::;:. qw P fcpf (10.109) T he h eat transfer rate at the wall is given by Fourier's Law as: . _ - k a TI _ - ka(Tw - Too)' d f qw aa y y=O BT d(yIBT) I - _-ka(Tw - Too)." BT JO ylST = 0 (10.110) wherekais, as before, the apparent thermal conductivity o f the porous medium. Substituting this ill-to Eq. (10.109) then gives: .[Io' / d(;')] : x [ul(Tw - T.)8rj ., _a.<Tw ~ T.) / . (1O.111) wh~re a a is, as~before, the app~ent thermal diffusivity, i.e., kal P fCpf. I n o rder to illustrate the procedure, the temperature profile will here be assumed to b e d escribed b y a second-order' polynomial, i.e., i t will b e assumed that i n t he boundary layer: T ::;: a + b y + cy2 (10.112) / ' The coefficients in this equation, i.e., a, b, a nd c, are determined by applying the b oundaiy conditions on temperature at the inner and outer edges o f t he thermal boundary layer. Because the case where the wall temperature variation is specified/ is being considered, these boundary conditions are: Aty::;: 0: T = Tw A t y = BT: T = TfYJ A ty =,BT: a Ttay = 0 (10.113) / 516 Introduction to Convective Heat Transfer Analysis The first of these conditions follows from the requirement that the fluid in contact with the wall must attain the same temperature as the wall. The other two conditions follow from the requirement that the boundary layer temperature profile must blend smoothly into the free-stream temperature distribution at the outer edge of the boundary layer. Applying the boundary conditions given in Eqs. (10.113) to Eq. (10.112) then gives: Tw = a Tl = a + b8T + c8'f 0 = b + 2c8T (10:114) This is a set of three equations in the three unknown coefficients. Solving between them then gives: a = Tw, b = - 2(Tw - T 1)l8T, c , = (Tw - Tl)18 T . 2 (10.115) Substituting these values into Eq. (10.112) and rearranging gives the temperature profile as: Tw T- Too _ 1 _ 2(L.)+ (L.)2 Too 8 8 - T T (10.116) Comparing Eq. (10.116) with Eq. (10.108) shows that: f= 1-2(Ir)+(Ir)' = (10.117) Hence: Jefd(L.) = ~,fo o 8T 3 d[ d(yI8 T ) df I = -2 . (10.U8) ylBr = 0 Substituting these values into Eq. (10.111) gives: d x u l(Tw -Too )8T = 6a a ] ( Tw 8T - Too) (10.119) I f attention is restricted to a surface that has a uniform temperature, i.e., for which Tw - Too i s constant, Eq. (10.119) becomes: (10.120) ( Ul Consider, first, the case of flow over a flat plate for which, as discussed before, is a constant, equal say to uoo . In this case, 'Eq. (10.120) gives: (10.121) This equation can be directly integrated, because 8T is by assumption 0 at x = 0, (see Fig. 10.20), to give: (10.122) CHAPTER 10: Convective Heat Transfer Through Porous Media 517 FIGURE 10.20 Boundary layer growth on a flat plate in a porous medium. i.e.: x 3.464 • = PeO5 = PeO.5 (1.0.123) x x where Pex is, as before, the Peelet number based on x. Now Eq. (10.110) gives usiQ.g Eq. (10.118): , _ 2 k (Tw ~ TCIJ) qw a aT (10.124) which when combined with Eq. (10.123) gives for flow over a flat plate in a porous medium: k ( qw x a T w-TCIJ ) 2 = 3.464' Pe~·5, i.e.,· Nux = 0.577, Pe~·5 (10.125) This equation gives the variation of the local heat transfer rate along the plate. The mean heat transfer rate is given by: 1 qw,= L Jo (L qw dx i.e., using Eq. (10.125): which can be rearranged to give: N UL = , 1.154Pet2 (10.126) where N UL is the mean Nusselt number for the plate, i.e., Tiu ka, Ti being the mean heat transfer coefficient. The results obtained using the integrai equation method as given in Eqs. (10.125) and (10.126) agree to within about 2% with the exact result detj.ved earlier. E XAMPLE 1 0.4. Consider forced convective boundary layer flow over a flat plate in a porous medium, the plate being aligned with the forced flow. The surface temperature .of the plate is given by Tw - T = Cxm ~ Use the integral equation method to determine an expression for the variation of the local heat transfer rate along the plate. ,' tX> / 518 Introduction to Convective Heat Transfer Analysis Solution. The following applies to flow over a flat plate whatever the form. of the wall temperature variation: . which can be written as: Substituting: Tw - T«> = ex'" 6a a f into this equation gives: .d 8t. -+-=-dx x u",8 m8T Le.: (i) An inspection of the form. of this equation shows that the solution for 8T must have the form: 8T = A x1n 6a ( ii) where A is a constant. Substituting this into Eq. (i) then gives: A. _2 +mA2 2 = _a ~ Le.: A2 = 12a a ~(l + 2m) 1 SubstitUting this into Eq. ( ii) then gives: x = .y"f+2m Peo.s x But the heat transfer rate is given by: q. 8T ru ( iii) w =. 2ka . (Tw - T«».· 8T · .ux ' I.e., N = ~I ~x 2 Substituting Eq. ( iii) into this equation then gives: Nux .~. [ 2(1 ~m)] P~s ~ 0.577(1 + 2m)Pe~s As another example o f the use of the energy integral· equation, conside~ again the flow i n the region of a· stagnation point on an isothermal body as shown in Fig. 10.21. . . CHAPTER 10: Convective Heat Transfer Through Porous Mecija 519 F IGURE 10.21 Flow considered i n integral equation solution· for stagnation region. In this case, as discussed before, for small x: (10.127) where D is twice the radius of curvature of the leading edge and Ucc is the velocity in the free-stream ahead of the body. Substituting this expression for u into Eq. (10.120) gives: 4Ucc d ( ~ ) _ 6 a a - -XUT - D dx ~T i.e.: ~T -(X~T) d dx = 1.5aaD Ucc 1.5aa D Ucc i.e.: 2 d~T ~T+X~T- dx = (10.128) Because the right-hand side of this equation'is independent of x, it follows that in the stagnation point region d~Tldx = 0 and in thisregion therefore: ~f. = 1.5a a D (10.129) Ucc , SubstitUting this result into Eq~ (10.124) gives: 0.5 2 qw . ( Ucc ) ka(Tw - T~) = 1.5 112 a aD i.e.: N UD = 1.633Pe~5 (10. 1 30( Comparing this resUlt with the exact result given in Eq. (10.76) shows that the integral equation method agrees in this case to within 3% of the exact solution. As a last example of the use of the integral equation method consider again twodimensional flow about an isothermal cylinder in aporolls medium. The situation considered is shown in Fig. 10.22. ~ 520 Introduction to Convective Heat Transfer Analysis F IGURE 1 0.22 Flow considered i n integral equation solution for flow over a cylinder. I n this case, as previously discussed: u= 2Uoosm(~) 8T Uw J I0.131) Eq. (10.120) therefore gives for this case: 4COS(2X)(8T ) + 2 sin (2X dd8xT ) = DD D T he following are then defined for convenience: X 6a a (10.132) = -x 1:1 = D' 8T 112 - Pe D PeD D' ~D = -a a (10.133) I n t erms o f these variables, Eq. ( 10.132) gives: 21:1 2 cos (2X) + sin (2X)A ~~ =3 (10.134) F or small values o f X, i.e., i n t he r egion o f the forward s tagnation point, 1:1 will, as d iscussed before, be a constant a nd i s given by Eq. ( 10.134) a s: A5 = ~, i.e., 1:10 = 1.225 (10.135) Starting with this value o f 1:1 a tX = 0, Eq. (10.134) c an b e i ntegrated to give the variation o f 1:1 with X around t he cylinder. For t his purpose, Eq. ( 10.134) is written as: dl:1 3 - 21:12 cos (2X) dX 1:1 sin (2X) (10.136) T he variation o f A with X o btained b y t he integration o f t his equ~tion is shown in Fig. 10.23. Once t he variation o f A h as b een o btained i n this way, t he l ocal h eat t ransfer rate distribution can b e obtained b y u sing Eq. (10.124) w hich c an b e r earranged to give: qwD 2Pe~5. k iTw - T < ¥) - Ll NUD 2 , I.e., Pe~5 - 1:1 (10.137) CHAPl'ER 10: Convective Heat Transfer Through Porous Media 521 8~----~----~----~~--~ 6 4 2 o L -.._ _l ...--_---'_ _- -L_-.-::::::-.I 0.0 0.4 0.8 1.2 1.6 x FIGURE 10.23 Variation of ~. and NuDIPe~5 with X around cylinder. U sing t he calculated variation o f !1 with X, this equation allows the variation ofNuDIPe~5 with X to be determined. The variation so obtained is also shown in Fig. 10.23. With the variation o f the local dimensionless heat transfer rate obtained in this way, the mean heat transfer rate for the cylinder can b e obtained by using: qw -- = 1 (1T/2 . '1T12 Jo qw d X . /2 i.e., i f NUD i s the mean Nusselt number: NUD _ 1 Pe~5 - '1T12 Jo (11: NUD d X Pe~5 (10.138) Using this with the cal~ulated variation o f NUDI Pe~5 with X gives: NUD = 1.039 Pe~5 (10.139) T his value agrees to within 3% with that obtained by numerically solving the boundary layer energy equation. . 10.6 FULLY D EVELOPED DUCT F LOW While flow i n a duct containing a porous medium [19] does not occur extensively in ... practice there are a few important situations that can be represented by this type o f flow. F or example, it has been suggested that filling a duct with a porous medJum can b e u sed as a method o f enhancing the· heat transfer rate from the walls· o f the duct. T he disadvantage of this is, o f course, that a much higher pressure drop will be incurred. S ome geological flows can also be adequately modeled as flow ~ugha porous medium~filled duct with heated or cooled walls. / 5 22 Introduction to Convective Heat Transfer Analysis F IGURE 1 0.24 Flow situation considered. In order to illustrate how fully developed flow through a duct filled with a porous medium can be analyzed, consider flow through a wide duct with plane walls, ie., flow between parallel plates, with a unifonn heat flux at the wall. The flow situation .is thus as shown in Fig. 10.24. Because in the porous media flow model being used, the effects of viscosity are assumed to be negligible, the velocity will be unifonn across the duct, i.e., the velocity will be equal to Urn everywhere and the cross-stream velocity compone.nt, v, will, therefore, be zero everywhere. I t will further be assumed that the temperatUre gradients across the flow, i.e., in the y-direction, will be much greater than those in the z-direction because W < L, L being the length of the duct. With these assumptions, the governing equation, i.e., Eq. (10.34), reduces to: Urn aT az = aa a2T ay2 (10.140) Now it is being assumed that the flow is fully developed, i.e., that i f Tc is the centerline temperature: Tw - T = f (Y). Tw - Tc W the function f not being dependent on z, i.e.: (10.141) ~(Tw-T)=O dZ Tw-Tc (10.142) From this i t follows that: T)(Tw T T (dTw _ aZ Tw -- Tc )(dTw _ ddzc) = 0 dz d d z· But the heat transfer rate at the wall is given by: (10.143) qw = aTI +ka~ . vy (10.144) y =w the ( + ) sign OCCurring because the heat flux is taken as positive from the wall to the fluid, i.e., in the negative y-direction. Substituting Eq. (10.141) into Eq. (10.144) then gives: . (10.145) CHAPI'ER 10: Convective Heat Transfer Through Porous Media 523 Because qw is constant and because f does not depend on z, this equation shows that (Tw - Tc) = constant, i.e., that d Tw d Tc -=dz dz (10.146) Substituting this into Eq. (10.143) then gives: d Tw aT dz = az (10.147) Hence, Eq. (10.140), the governing equation, can be written as: d Tw a2T Um d z = a aay2 (10.148) Integrating this once and noting that the left-hand side does not depend on y gives: aT U m d Tw C a y = aa d z y + 1 (10;149) where Cl is a constant of integration. But, with the y-coordinate defined as in Fig. 10.24, the following boundary condition applies because the temperature profile is symmetrical about the center line: . aT y = 0: a y =0 (10.150) This gives Cl = 0 ID.tegrating again then gives: T - Um d Twy2 C - - - d - 2+ 2 aa z (10.151) where C2 is a second constant of integration. But the other boundary condition is: y = w ( = W/2): T = T w (10.152) Hence: (10.153) i.e.: C2 = Tw _ d Tw aa d z 2 Um w2 (10.154) Substituting this back into Eq. (10.151) then gives: T = Tw _ Um d Tw aa d z (w2 _ y2). 2 2 (l0.155) S24 Introduction to Convective Heat Transfer Analysis T he wall temperature gradient is now related to the specified uniform wall heat flux qw b y substituting Eq. (10.155) into Eq. (10.144). This gives: . Urn d Tw .qw = k a---w aa dz i.e.: Urn d Tw qw aa dz = ka w Substituting this back into Eq. (10.155) gives the temperature profile as: T _T = w (10.156) (k~)(w2 a 2 w _y22) (10.157) From this it follows that: (10.158) Hence: f = Tw - T = 1 _ Tw - Tc (y)2 W (10.159) Now i n duct flow, the Nusselt number is defined i n t erms o f the difference between the wall temperature and the mean fluid temperature, this mean temperature being defined by: Trn " = - 1 .W u Tdy = -1 Urn W 0 i W0 i W Tdy (10.160) because u = Urn a t all y. Using Eq. (10.157), this gives: Trn = i.e.: ! W [TwW _ (~)(w3 w3)~ = kaw 2 _ 6~ Tw _ qw W ka 3 qwW . = 3 . (Tw - Trn)ka (10.161) I But the mean Nusselt number is conventionally defined i n terms o f the full duct width W( = 2w) so Eq. (10.161) is written as: i.e.: Nu = 6 (10.162) / Therefore, with fully developed flow i n the gap between parallel plate~, when this gap is filled with a porous medium and when there is· a uniform heat flux at the CHAPTER 10: Convective Heat Transfer Through Porous Media 525 plates, the Nusselt number is 6. Using the same approach it can b e shown that, i f the wall temperature is constant, Nu = 5. For flow through a pipe filled with a porous medium i t can be shown that N u = 8 for the uniform heat flux case and N u = 5.78 for the uniform temperature case. The Nusselt number, Nu, is here based on the pipe diameter, D. Notice that these Nusselt numbers are all much higher than those that would exist with the flow o f a p ure fluid in the same geometrical situation and that they are based on the effective conductivity. This is basically the reason that has led to the suggestion that filling a channel with a porous medium b e u sed as a means of increasing, i.e., enhancing, the heat transfer rate. E XAMPLE 1 0.5. In a heat exchanger, air flows between parallel plates that are separated b y 3 mm. The plates are effectively kept at a uniform temperature o f 40°C and the air is heated from 10°C to 20°C as it passes through the heat exchanger. T he air velocity between the plates is 1 mls. I t has been proposed that the size o f t he heat exchanger can b e reduced by incorporating a loosely packed porous medium into the gap between the plates. Evaluate this proposal by finding the plate lengths with and without the porous medium and by finding the pressure drop across the heat exchanger with and without the porous medium. Neglect any entry region effects. Solution. First consider the flow without the porous medium. A t the mean temperature o f ISoC, air has the following properties: p k = 1.22 kg/m3, cp = 1007 J/kg K 18 X 1 0-6 kg/ms = 0.026 W /m K, X J.t = The Reynolds number based on the gap between the plates is therefore: Re = p uW/ J.t = 1.22 1 X 0,003/18 X 1 0-6 = 2 03 T he flow is therefore laminar and as a consequence, as shown in Chapter 4, Nu = 4.12. Hence: h = NuklW = 4.12 X 0.026/0.003 = 3S.7 W /m2 K Tin) Consider a unit width o f a passage. Q = mcp(Tout - = puAcp(Tout - Tin) = 1.22 X But: 1 X (0.03 X 1) X 1007 X (20 - 10) = 3 68.6 W Q = hAwall(Tw - Tair ) = h(2 X L X 1)(40 - IS) w here L is the length o f the channels. I t has been recalled that there i s h eat transfer from the top and bottom walls. Using t he above results then gives: L = Q/(SOh) = 368.6/(SO X 3S.7) = 0.207 m Therefore, the length o f t he heat exchanger is 207 mm. The pressure drop through one channel and therefore across the heat exchanger is given by: t::.. p = 12 X JLuL W2 = 12 18 X . 1 0- 6 X 1 X 0.207 0.0032 = 4 .97 P a 526 Introduction to Convective Heat Transfer Analysis Consideration will next be given to the case where the porous medium is incorporated into the system. I t will be assumed that: K = 1 0-9 m2 , ka = 0.04 W /mK In this case, Nu = 5. Hence: h = N ukIW = 5 X 0.026/0.04 = 81.3 W /m2 K I n this case then: L = QI(50h) = 368.6/(50 X 81.3) = 0.091 m Thus, the required length of the heat exchanger is reduced by more than 50%. The pressure drop through one channel and therefore across the heat exchanger when the porous medium is in the channel is given by: Il. _ . p.uL _ 18 X 1 0-6 X 1 X 0.0,91 P- K 10-9 = 1638 P a While this is much higher than the pressure drop when there is no porous medium, it may be acceptable under many circumstances. 10.7 NATURALCONVEC~BOUNDARYLAYERFLOWS Natural convective flows i n porous media occur in a number o f important practical situations, e.g., in air-saturated fibrous insulation material surrounding a heated body and about pipes buried i n water-saturated soils. To Illustrate how such flows can be analyzed; e.g., see [20] to [22], attention will be given i n this section to flow over the outer surface o f a body in a porous medium, the flow being caused purely by the buoyancy forces resulting from the temperature differences in the flow. The simplest such situation is two-dimensional flow over an isotherinal vertical flat surface imbedded in a porous medium, this situation being shown schematically in Fig. 10.25. ' F IGURE 10.25 Natural convective Bow over a v~rtical plate. CHAPI'ER 10: Convective Heat Transfer Through Porous Media 527 . .It is assumed that the thickness of the layer, i.e., the boundary layer,' in which there are significant temperature differences. and velocities parallel to. the· surface is small, t e., that 81L « 1. In such a flow, as discussed previously, the pressure changes across the boundary layer are negligible and the momentum equation for the x-direction, i.e., the vertical direction, is (10.163) T~ being the temperature far from the plate, i.e., outside the boundary layer. Eq. (10.163) can be written as: .u ' ( 3gKpfrr - TIXJ ) I Lf . (10.164) = Because a boundary layer type flow. is being assumed, the gradients in the,ydirection are much greater than those in the x-direction and the energy equation can, therefore, as before, be approximated' by: il- aT ax + v - .y a y = CX a - 2 u aT a2T ' (10.165) It is also noted that the continuity equation gives:' ax au + av ay = 0(10.166) , Eg. (10.166) is, as previously discussed, satisfied by a stream furiction, t/!, which. is defined, as before, by: at/! ' . at/! u= v =-ay' . ax . ," " (10.167) ' Eqs. (10.164) and (10.165) can be written in terms of the stream function as follows: . . . ", , .' . at/!. = (3gKpt<T - TIXJ) ay I Lf - - - - - . = CX a - (l0.-168) (10.169) at/! a T ay a x at/! a T a x ay a2T iJy2 The first of these equations can be written as: iJ2f/J _ f 3gKpf a T iJy2 ILf iJy (10.170) The order of magnitude of t/! in the boundary layer. must, therefore, be such that:· " !l!... where 82 = o({3 gKPf a8T) I Lf (10.171) aT is a measure of the temperature change across the boundary layer. Hence: r/J= o~g~~TIl) (10.172) 528 Introduction to Convective Heat Transfer Analysis Similarly, i f the first and last terms in Eq. (10.169) are considered, the orders of magnitude are such that: (10.173) i.e.: (10.174) Eqs. (10.172) and (10.174) together give: aa X = o(~gKPfAT8) I Lf 8 i.e.: ~: ;. O~g;;:'iTX) i.e., defining the following as the Darcy-modified Rayleigh number: (10.175) Ra = ~gKpfATx = ~gKATx a aILf a avf (10.176) it follows fromEq. (10.175) that: (10.177) and substituting this result into Eq. (10.174) gives: " , = o(aaRao. s ) (10.178) Now it is to be expected that the stream function and temperature profiles in the boundary layer will be similar, Le.,that in view of the results given in Eqs. (10.177) and (10.178) that: (10.179) and f:laRa o.s ", - 1(71) - (10.180) I where·Tw is the wall temperature which is assumed to be uniform and J1 is the similarity variable which is given by: x and the Darcy-modified Rayleigh number is given as before by: Ra = ~gK(Tw'- T«»x a avf 71 = ~Rao.s (10.181) CHAPI'ER 10: Convective Heat Transfer Through Porous Media 529 I n terms. o f the variables introduced i n Eqs. (10.179) and (10.180), Eqs. (10.170) a nd (10.169) become, i.e.: I" and = 0' (10.182) 0" =- 10' 2 (10.183) respectively. T he boundary conditions on the solution are: y = 0, v = 0, T = Tw, 0 0, i.e., T/ = 0, I = 0, 0 = 1 --+ (10.184) y large, u --+ 0, T --+ i.e., T/ l arge!, 0 ,0 --+ 0 Eqs. (10.182) and (10.183), whi~h are a pair o f simultaneous ordinary differential equations, can b e integrated simultaneously to give the variations o f f a nd 0 with T/. F or this purpose it is convenient to note that Eq. (10.183) can be written as: Integratin~ 0" 0' =- I 2 (10.185) this gives: 0: 00 = exp [ _ Jo (11 I dT/] 2 (10.186) is the value o f 0 ' a t T/ = O. where Integrating this equation again then gives: Oil = 80Ll1exp [ - Ll1 8b { dT/ ] dT/ + 1 ~(10.187) the fact that 8 = 1 a t T/ = 0 having been used. I f the fact that 0 tends to zero a t large T/ is next used, the above equation gives: 80 = fo ~ [-; exp - fo' { d T/fT/ 1 (10.188) Substituting this back into the original equation then gives: .. o= 1- Jo (11 exp [ _ (11 { dT/]dT/ Jo . (10.189) fo exp [ - fOl1 { dT/] dT/ oo Integrating Eq. (10.182) gives: . '.' I' - fcr = J(o . 11 0' dT/ = 8 - 1 (10.1.90) / .530 Introduction to Convective Heat Transfer Analysis where 10 is the value of I ' at 71 infinity. I t follows therefore that fo = = O. But I ' and 8 both tend to 0 as 71 tends to 1 so the above equation gives: I' =8 (10.191) (10.192) Integrating this and recalling that I = 0 at 71 = 0 then gives: 1 = fo'" 8 d71 These equations are easily solved to give the variations of I and 8 w~th 71. The variations so obtained are shown in Fig. 10.26, . The heat transfer rate at the wall is given by: qw = - ka i.e.~ aTI,y=o ay (10.193) terms of the dimensiOliless temperature. qw x R 0.58'1 ka(Tw - T«» = - a 0 But the calculated variation of 8 with 71 gives: 8'10 so Eq. (10.193) gives: = 0.444 . Nux = 0.444Rao. 5 (10.194) where Nux is the local Nusselt number based on x. This gives the variation of the local heat transfer rate with x. The mean heat transfer rate is then given by using: 2~ _ _- ---____p -_ _ _ _ _ _ _ _ _ _ ~ 1 5 11 F IGURE 10.26 1 0' Variations o f dimensionless stream function. and temperature with similarity variable, ",. CHAP1ER 10: Convective Heat Transfer Through Porous Media 531 Using Eq. (10.194) in this equation and integrating gives: NUL· = 0.888Ra2 5 (10.195) where NUL is the mean Nusselt number based on the plate length L and RaLis .the Darcy-modified Rayleigh number based on L. Natural convective boundary layer-type solutions have been obtained for a number o f other geometrical configurations. A number o f studies o f m ixed convective flows in porous media are also available, e.g., [23] to [35]. E XAMPLE 1 0.6. I n order to k eep it cool during ajourney, a bottle o f a medication is kept . i n t he center o f a large box containing a loosely packed low quality granular insulating material that has a permeability o f 1 0-7 m2 • T he bottle can be approximately modeled as a 2 0-cm high by lO-cm wide flat plate placed vertically in the insulating material. T he initial temperature of the medication is 5°C and the temperature o f t he insulation is equal to the ambient temperature which is 30°C. Find the initial rate at which heat is . transferred to the medication. Solution. I t will be assumed that the insulation material and air have the following properties: K = 1 0-7 m2, aa = 37 X 1 0-6 m2 /s, ka = 0.046 W /m K v f = 14.8 X 1 0- 6 m 2 /s, f3 = 3.5 X 10- 3 K - 1 U sing these values the Darcy-modified Rayleigh number based on the height o f the bottle is: Ra = f 3gKATx a avf = 0.0035 x 9.81 x 0.0000001 x 0.2 x (30 0.000037 5) = 31.3 x 0.0000148· W hile this is rather small to apply the boundary layer assumptions i t will, nevertheless, b e assumed that the mean Nusselt number is given by: NUL =O.888Ra~5 = 0.888 x (31.3)°·5 = 4.97 T he mean heat transfer coefficient is therefore given by: h = NUt ka = 4 .97 ;'~.046 = 1.14 W /m2-K T he initial rate o f heat transfer from the bottle is therefore given, since there are two sides to the bottle, by: Q = hA(Tw - T«» = 1.14 x (2 x 0.2 x 0.1) X (5 - 30) = - 1.14W T he negative sign indicates that the heat transfer is to the bottle. 10.8 NATURAL CONVECTION IN POROUS MEDIA·FILLED ENCLOSURES H eat transfer by natural convection across porous media-filled enclosures occur$ in a number o f practical situations and will b e considered in this section, [36] to [51]. 5 32 Introduction to Convective Heat Transfer Analysis Situations o f this type arise for example in some building applications in which heat is transferred across an insulation-filled enclosure. Attention will here b e restricted to two-dimensional steady flow in a rectangular porous medium-filled enclosure which is, in general, inclined at an angle to the vertical. O ne wall o f the enclosure is kept at a uniform high temperature and the opposite wall is kept at a uniform low temperature. The other two walls o f the enclosure are assumed to be adiabatic, i.e., it is assumed that no heat is transferred into or out o f these walls. The situation is, therefore, as shown in Fig. 10.27. Using the Darcy flow model and the Boussinesq approximation, the governing . equations are: oy ox + OV oy = 0 (10.196) (10.197) (10;198) (10.199) K JLfu JLiV = ~ OX op + f 3gPf(T , - Tc) cos c/> Tc)sincf> =- ~~ + f 3gPf(T =( U oT ox + v oT .a y· ) (a2T P fcpf o x2 ka + o 2T) oy2 T he first o f these equations expresses, as before, conservation o f mass, the second and third express conservation o f momentum in the x - and y-directions respectively while the last expresses conservation o f energy. T he c old wall temperature, T c , has been taken as the reference temperature. . T he solution will, here, b e obtained in terms o f t he stream function. I t must be stressed that this is not necessary, solutions in terms o f the so-called primitive variables, i.e., u, v, and p , being in fact more widely used. T he stream function is, as before, defined by: U=-,V.=-- aI/J oy . oI/J ox (10.200) c B D Horizontal F IGURE 10.27 A Situation being .analyzed. CHAPtER 10: Convective Heat Transfer Through Porous Media 533' The stream function, so defined~ satisfies the continuity equation. Now, i f the x-derivatiye is taken of Eq. (10.197) and the y-derivative is taken ofEq. (10.198) and the two results are subtracted, the pressure is eliminated and the following is obtained: ILf a - , (au - - v) = K ay ax f3gPf (aT' ' - '-SIDcf> - coscfJ aT.) ay ax (10.201) i.e., using the definition of the str~am function: • a2 + a2f/! f/! ax2 ay2 = f3gpfK(ar coscf> _ aT Sinet» ILf ay, , ax ' ( ka )(a2 a2T) T \PfCp/ ·ax2 + ay2 , '(10.202) In t ennsofthe stream function, the energy equation, i.e., Eq. (10.199), is: af/! aT af/! aT ay ax - ax ay, = (10.203) Before discussing the solution to the above pair of equations, they will be written in diu,.ensionless form. For this purpose, the following dimensionless variables are ' defined: x= "I' = f/!Iaa, (J ~~~ xlW, Y = ylW = (T - Tc)/(TH - Tc ) , (10.204) where, as before, the app~nt thermal diffusivity of the porous material, aa, is equal In terms of the variables defined in Eq. (10.204), Eqs. (10.202) ,and (10.203) become: a2 '\}f a2 '\}f aX2 + ay2 = Raw ay cos 4> - ax S lll!/J = (a(J a (J.) (10.205) (10.206) a2 (J a2(J aX2 +ay2 a"l' a(J a'\}f a(J ay ax - ax ay where Raw is the Darcy-modified Rayleigh number based on the enclosure width , W, i.e.: (1Q.207) Now, the boundary conditions on the solution for flow in the enclosure are: Velocity component normal,to wall = 0 on all walls T aT = TH at x ':;: 0 (10.208) T = Tc atx::;:: W - y = O aty = Oandy = H a , 534 Introduction to Convective Heat Transfer Analysis I n terms o f the stream function, this means that: a", ay = 0 at x = 0 and x =W (10.209) ~~ = O aty = O andy = W B ut the absolute value o f the stream function is quite arbitrary because it is only its derivatives that occur in the governing equations. Hence, i t will arbitrarily be assumed t hat the stream function has a value o f 0 at the point A shown in Fig. 10.27. B ut the boundary conditions indicate that a",lay is 0 along AB. Hence, s ince", is o at point A, it is zero everywhere along AB. Along B C the boundary conditions indicate that a ",lax = O. Hence, s ince", is zero at B, it is 0 everwhere along Be. In a similar way, it can be deduced t hat'" is 0 everywhere along C D and DA. Hence, on all walls, i.e., along all o f ABCD, ' " is 0 i f it i s arbitrarily taken as 0 at point A. Hence, the boundary condition on the stream function is: All wall surfaces: ' " = 0 (10.210) I n terms o f the dimensionless variables defined in Eq. (10.204), the boundary conditions are: O n all walls: X = 0: X = 1: 8 "It =0 (10.211) Y = 0: Y ay =1 B=O aB = 0 = A: aB ar = 0 Here A = H I W i s the so-called aspect ratio o f the enclosure. A number o f approximate analytical solutions to Eqs. (10.205) and (10.206) have been obtained. However, those solutions are o f l imited applicability and it is now usual to obtain a numerical solution to the equations. A very simple finitedifference numerical procedure, basically identical to that used before to solve for the flow i n a fluid-filled enclosure, will b e discussed here. I f a uniformly spaced grid is used and i f attention is directed to. the grid points shown in Fig. 10.28, the following finite-difference form o fEq. (10.206) is obtained: I '-1 . I ,} I, I [ 8'+1 }. + 8I , }' - 28·.}.. ] +. [ 8"+1 AJ(2 + 8,I .}' -I Ay2 - 28,,} I 'J = ['lti,j+~~y'lti,j-l ] [8i+l'~~:i-l.j ] _ ['lti+1.~~'lti-l,j ] [8i,j+12~ 8i,j-l ] (10.212) / A n iterative procedure is actually used in which the values o f the variables at all nodal points are first guessed. Updated values are then obtained by applying tQ~ CHA.P1'ER 10: Convective Heat Transfer T hrough Porous Media 535 F IGURE 10.28 Nodal points used. governing equations and the process is repeated until convergence is attained. For this reason, Eq. (10.212) is written as: 6" = WI+ ~~i~ ~j ) + e,j+ ~;:i,j"1:- Cfr"i+~~:'J"1+1'~::'-l,j) t ) 1 )]/ + Cfri+1.i~'frH'i )(9'i+';;.:'i8'L = 89. iJ ' ,J (~ + d;') (10.213) The right-hand side of this eqqation is calculated usiI).g the "most recent" values of the variables. Under-relaxation is actually used so the updated value of 8 is given by: + r(8 9,J I * ~ 89 .). I,J . (10.214) whe~e 8~1c is .the value given by Eq. (10.213) and 82 j is the value of 8i,j at the prevIous Iteration. Eq. (10.213) is applied at all'~intemal" nodal points, i.e., at all points within the enclosure. The boundary conditions determine the dimensionless temperatures on the walls. These give:' .(10.215) = 1, N: 8·I,J.·= 1.0, 8 M' = 0 . ,J there being N nodal points mhe Y -direction and M in the X-dir~ctiOIi. t " On the other two walls, since the gradient in the Y-direction is zero, to .first order J a~curacy: i = i, M : 8i,1 = 8i,2,8i,N = 8 i,N-l (10.216) I The stream function equation, i.e., Eq. (10.205), is treat~d i n the same 'wayas the energy equation. The following finite-difference form o f Eq. (10.2Q5) is-, therefore, obtained: .' ( Vi+l,j . + ' l'i-l,j ~ 2 'l!i,j) + ('I'i,j+l + ' l'i,j-l t:.x2 ~y2 - 2'f!i,j) = Raw [(9 i,i+';;.:',J-l )COS4> - (9i+1,;;;;i-l,J )sin 4>] (10,217) / 5 36 Introduction to Convective Heat Transfer Analysis T his can be rearranged to give: '\}I'. . = { ('I'i+l'i ',J i lX2 + ' l'i-l,i) + (qri'i+l ilYZ + ' l'i,i- l ) - Raw [ei.j+~~:i.j-l )cos~ - (81+1.~I-LJ )sin~]}! (~ + a;') (10.218) T he right-hand side o f this equation is again calculated using the "most" recent values o f the variables. Under-relaxation is also again used so the updated value o f 'l'i.i i s actually given by: '\}t.l. = '\}t!>. ',J ',J + r('\}t~l!lc ',J "It!>.) ',J (10.219) . w here 'l'i~fc is the value given by.Eq. (10.218) and'l'i?i is the value o f'l'i,i a t the previous iteration. r ( < 1) is again the under-relaxation p arameter.Eq. (10.218) is applied at all "internal" nodal points. The boundary conditions give the value o f ' I' = 0 on all boundary points, i.e.: . j = 1, N: 'l'i,i = 0, qrM,i = 0 i = 1, M: q ri,i= 0, qri,N = 0 T he above procedure is actually implemented in the following way: (10.220) 1. T he values Ofqri,i and Bi,i a t all nodal points are first set equal to arbitrary initial values. Typically, the following are used: '\}I'. . = ',J 0 BI·· = 1•O -X",J ,J ' (10.221) 2. 3. . 4. 5. T he assumed B distribution is that which would exist i f there was no connective motion, i.e., i f conduction alone existed. Its use is consistent with the assumed distribution o f qr which implies that there is no flow in the enclosure. Eq. (10.218) i n conjunction with Eq; (10.219) is used to obtain updated values o f ' l'i,i' B ecause iteration i s being used, this process should really b e repeated over and over until the values Of'l'i,i corresponding to the initially assumed distributionof B are obtained. Experience suggests, however, that i t is quite adequate to undertake this step j ust twice. Eq. (10.213) i n conjunction with Eq. (10.214) is used to obtain updated values o f Bi,i' T his step·is also undertaken twice . S teps (2) and ( 3) are repeated over .and over until convergence is obtained to a . specified degree. . The heat transfer rate distribution is obtained by applying Fourier's law at the heated and cooled walls, i.e.: (10;222) CHAPrER 10: Convective Heat Transfer Through, Porous Media 537 I n terms o f the dimensionless variables, this equation becomes: . 1 ~(h,j N UI,j= AY , lI.T· 8M - H UM,j = I AY' ,} (lQ.223). where N u is the local Nusselt number based on W a nd where it has been rioted that 8 i,j = 1.0 a nd 8 M,j = O . . Because uniform· grid spacing· has been used, the mean· Nusselt numbers are . given by: N UB = ~ A NUll' .' 2 AY (2 + NUI,2 + N." + ... + NUM,N-I + N Ul N ) UI,3 . :. .. .' (10.224) -' ~uc = A N M. . AY (2 UNUM,2i + NUM,3 + ... ~NUM'N-I + N UMN) . + i' Because a steady-state situatipn is being considered, N UB a nd N uc will be equal. Numerically, slightly different values will normally b e obtained because o f the finite degree o f convergence used. A computer program, ENCLPOR, written in FORTRAN that is based on this procedure is available as discussed in the Preface. This program. applies to a ''vertical" enclosure, i.e., to the case cf> = 90°. ' T he solution discussed above h as,as parameters, R aw, the Darcy...;modified Rayleigh number based on the enclosure width, and A, the enclosure aspect ratio 11.IW. S ome typical results computed with the above program are shown i n Figs. l O.i9and 10.30. l t will be seen from Fig. 1 0.29 that for small Raw, when the convective motion is wealc,Ute Nusselt number tends to the conduction limit value o f 1. However, when R aw is large, the flpw will consist mainly of a boundary layer flow up the hot wall and a ~oundary l aye, flow down the cold wall. I t is to b e expec~d, therefore, that i n 3 .~----------~----------~ Nu 2 11 10 Raw Variation of mean Nusselt number with R aw for various A. 538 Introduction to Convective Heat Transfer Analysis CONTOUR VALUES CONTOUR VALUES· -0.500 -1.000 -1.500 -2.000 -2.500 -3.000 -3.500 -4.000 -4.500 0.100 0.300 0;400 0.500 0.600 0.900 0.200 0.700 0.800 F IGURE 10.30 'JYpical streamline and isotherm patterns (Raw = 50, A = 1). view of the form of the boundary layer solution obtained earlier, i.e: q ;H' (TH - Tc)ka ocR 112 aH .. that: Nu oc R aW/A 112 (10.225) and, indeed, it will be seen from the results that at large values of Raw: . -_ Nu - 0.508 RaW A II2 (10.226) This.equation will apply as long as R aW/A 112 is large, i.~., as long.as A « Raw. The wall o f an insulated shipping container can be modeled as a 10 c m thick b y 5 0 c m high vertical enclosure. The inner and outer surfaces caI\ b e assumed t o b e a t 5 0°C and 10°C respectively. The enclosure is filled with an air-saturated loosely packed granular insulating material that has a permeability o f2 X 1 0-7 m 2 • Estimate the heat transfer rate across the wall per m width. E XAMPLE 10.7. Solutio". It will b e assumed that the insulation material and air have the following properties: . K = 2 X 1 0-7 m2, a a = 40 X 1 0-6 m2 /s, ka = 0.048 W /m K vf = 15 X 1 0-6 m 2/s, /3 = 3.3 X 1O-~ K - I Using these values the Darcy-modified Rayleigh number based on the width o f the enclosure is: Raw = /3gK(TH - Tc)W = 0.0033 a avf X 9.81 X 0.0000002 X (50 - 10) X 0.1 = 43 0.00004 X 0.000015 . T he aspect ratio of the enclosure is 50110 = 5. Hence Raw = 43 and A = 5. For these values o f Raw and A, Fig. 10.29 indicates that because RaW-IAI12 = 2.9, . Eq. (10.226) applies, i.e.: _ Rat12 43 112 Nu = 0.508 A 1~ = 0.508 5112 = 1.49 CHAPTER 10: Convective Heat Transfer Through Porous Media 539 The mean heat transfer coefficient is therefore given by: Therefore the rate of heat transfer across the enclosure per unit length is given by: Q = hA(TH - Tc) = 0.72 x (0.5 x 1)X (50 - 10) = 14.3 W T he computer program for the calculation o f flow i n an enclosure discussed above is easily modified to deal with nonverticalenclosures. Some results obtained using such a modified program are shown in Fig. 10.31. These result~are for a square enclosure, i.e., for an enclosure with an aspect ratio o f 1 a nd a Darcy-modified Rayleigh number o f 100. This figure shows the variations o f m ean Nusselt number and o f the stream function at the center o f the enclosure, i.e., 'ftcenter, with angle o f inclination. The magnitude o f ' ftcenter is a measure o f the strength o f the fluid motion i n the enclosure. I t will b e seen that the heat transfer rate is a maximum when .c:f>. is about 5 00 • A t this angle, the flow along both the heated wall and the adiabatic "top" wall is acted upon b y buoyancy forces that are near parallel to the wall as shown in Fig. 10.32. I t will also b e noted that the highest magnitude o f 'ftcenter occurs near this value o f c:f> • . W hen c:f> = 1800 , the hot wall is horizontal and at the top there is no fluid motion and the Nusselt number has its pure conduction value. When c:f> = 0 0 , the hot wall is horizontal and a t the bottom and the flow is unstable. However~ t he Rayleigh number for the aspect ratio considered is too low for. this to occur and the conduc.tion value for the Nusselt number is obtained at c:f> = 00 • However, inclining the enclosure b y a small amount provides this "trigger" leading to a value o f Nusselt number that is much higher than the conduction value, e.g., Nu is calculated to b e 1 when c:f> = 00 and to be approximately 2.6 when w hen c:f> = 10. 4 2 o -2 Results for iP = 0 0 -4 -6 -8~--~--~--~--~----L-~ F IGURE 10.31 o 30 60 90 120 iP-degrees 150 180 Variation of mean Nusselt number and dimensionless center-stream function with angle of inclination for an enclosure . with an aspect ratio of 1. 540 Introduction to Convective Heat Transfer Analysis Indicates a Component of the Buoyancy Force. F IGURE 10.32 Buoyancy force components in an inclined enclosure. 10.9 STABILITY O F HORIZONTAL POROUS LAYERS HEATED F ROM B ELOW Consider an essentially infinite horizontal layer o f saturated porous medium that is heated from below and cooled from the top. The situation is shownin'Fig. 10.33. I f the temperature difference i s very small no fluid motion will occur and ,the heat transfer from the hot surface to the cold surface will be by conduction alone. In this case, the temperature varies linearly through the material and the temperature distribution is given by: (10.227) .F IGURE 10.33 Porous·layer heated from below. CHAPfER 10: Convective Heat Transfer Through Porous,Media 541 where A T = TH - Tc is the temperature difference across the layer. As the tem..: perature difference is increased, a condition will b e reached a t which' fluid motion begins, i.e., the fluid layer becomes ''unstable'',e.g., see [12], [52] to [56]. The fluid motion is, o f cO'ilrSe, caused by the buoyancy forces, the least dense fluid being adjacent t o the lower hot surface which tends to rise while the fluid with the highest density is adjacent to the upper cold surface and tends to fall. Here, an attempt will b e made to determine the conditions under which the instability develops, i.e., under what cOllditions fluid motion first occurs. The equations governing the fluid motion are: au ax + av ay =0 . (10.228) (10.229) K ap u=--ILf ax , ~ = _~ a p I Li a y aT aT + ~gPfK(T aT Tc) . ILf' (10.230) U a t + u ax + v ay ( a 2T = aa dX2 + a2T) ay2 (10.231) B ecause the growth o f the disturbances must b e considered, the unsteady form o f t he governing equations has been used. The term u is given by: s u - - - - ....:...;...::......::-'-----'---'-P fcf P fcf _ pc _ cfJPfcf + (1 - c/J)psc (10.232) the subscripts f a nd s referring to the fluid and the solid matrix respectively, c/J being the porosity. This term arises because the energy required to produce a given change i n temperature will depend on p c whereas the enthalpy changes will depend. on P fcf. I t i s again convenient to eliminate the pressure, p, b y differentiating Eq. (10.229) with respect to y a nd Eq. (10.230) with respect to x a nd subtracting the two results. This gives: au ay _av = _~Pfg~ a T ax I Lf ax (10.233) I f t he flow i s stable, there is no fluid motion and under these circumstances: u = 'v = 0, T TH -(;)aT = ( 10.234) T he interest here is with the conditions under which the flow j ust begins, i.e., under which the flow deviates only slightly from'that existing under stable conditions. 542 Introduction to Convective Heat Transfer Analysis H ence,the following solution is assumed: T= (TH '- ~4T)+T' u = u' v = (10.235) (10.236) (10.237) v' where u', v', and T ' are functions o f x, y, a nd t and are small. In terms o f these variables the governing equations become: iJu' iJv' -'+-=0 iJx iJy , iJu" IJv' K iJT' ,_ _ _ = (3gp1 _ iJy U - -;ut (10.238) (10.239) iJx IL I iJx' i JT', iJT' _ (2 i 2T')' + u -u X + v , [ iJT'"- A T] - , uiJ T' + -J:;-r :;-u y :;H aa ~ X. uy (10.240) Now u', v', a ndT' are, by assumption,·small so products o f s uch terms will be 'Very srhall and will be neglected i n t he present analysis. As a result, Eq. (10.240) can be written: (10.241) 'Because this equation does not contain u', it is convenient to eliminate u' between Eqs. (10.238) and (10.239) by differentiating Eq. (10.238) with respect to y and E q.. (10~239) with respect to x and subtracting the two results. This gives: -2 iJ2 v' i Jx' + -2 2 (3gK a T' iJ2 v' =-2 . iJy i ii iJx ,(10.242) I t i s conveni~ntto w riteEqs. (10.241) and (10.242) in dimensionless form by introducing t he following dimensionless variables: x V ' = v' H aa y T a at = H2o: AT . X = H' Y = H' (10.243) 0' = T' .' :. . .' . I n terms o f these variables, Eqs. (10.241) and (10.242) become: iJO' 2 0' J20' t f'_ iax2 + iiJJy2 aT - t - (10.244) and (10.245) CHAPTER 10: Convective Heat Transfer Through Porous Media 543 where as before R a = f3gKp f ll.TH a a/J-f (10.246) is the Darcy-modified Rayleigh numberpased on H. The boundary conditions on the above pair of equations are: Y = 0 and Y = 1: V ' = 0, T ' = 0 (10.247) Now the conditions under which the disturbances grow are being sought and physical observation indicates that the disturbances grow periodically with X. It will, therefore, be assumed that: (J I = A(y)eTT+iaX (10.248) (10.249) V' = B(Y)eT'T+iaX Substituting these into Eqs. (10.244) and (10.245) gives: rA - B = - a 2 A + d2 A dy2 (10.250) and (10.251) the boundary conditions being: Y = 0 and Y = 1: A = 0, B = 0 (10.252) Now, consideration ofEqs. (10.248) and (10.249) indicates t hatifr < 0, the disturbance will decrease with time, i.e., will die out. However, i f r > 0, the disturbance will increase with time arid flow will develop. The condition r = 0 represents· neutral stability, the disturbances, therefore, neither increasing or decreasing with time under these conditions. This neutral stability condition will determlne the conditions beyond which inst~bility develops. I f r = 0, Eq. (10.250) gives (10.253) Using this. value of B to evaluate the left-hand sid~ ofEq. (10.251) then gives: - a 4 A + a 2 d 2 A + a 2 d 2 A _ d 4 A = - a 2RaA d y2 d y2 d y4 Le.: d d4A y4 - 2a d 2 d 2A y2 +a 4 A - a R aA _ 2 (10.254) / Now consider the following solution to this equation: A = C sin(mrY) (10.255) 544 Introduction to Convective Heat Transfer Analysis Substituting this into Eq. (10.254) g ives Cn4'TT'4 s in(mrY) + 2a 2Cn2'TT'2 sin(n'TT'Y) + a 4C sin(n'TT'Y) = a 2RaC sin(n'TT'Y) (10.256) In order for the boundary conditions to be satisfied, n m ust b e a n integer. Eq. (10.256) t hen gives: (10.257) i.e.: For any value o f n, this allows the value o f RA at which instability starts to develop to be determined as a function o f a . Some 'typical results are shown i n Fig. 10.34. I t w ill b e seen that for each value o f n there is a m inimum Rayleigh number i n t he variation with a . I t will further b e seen that the lowest o f t hese minimum Rayleigh numbers occurs when n = 1. This minimum Rayleigh number with n = 1 is, therefore, the lowest Rayleigh number at which instability could b e expected to occur. Now, when n = 1, Eq. (10.257) gives: Ra = ~--:,----:­ ('1T2 + ( 2)2 a2 (10.258) T he m inimum i n the curve will occur when: d Ra = 4('TT'2 + ( 2)a a2 _ 2('TT'2 + d 2)2 = 0 da a3 1~~~--------~-------------' 1000 Ra 100 10~----------~~------~--~ 1 10 100 FIGURE 10.34 Variationofrrrinhnum Rayleigh n~ber for ,stability. a CHAPTER 10: Convective Heat Transfer Through Porous Media 545 4 ~~------~--~--------~ 3 Nu • .2 I -~-:--~l·~·· Pure Predi dR . • • •• ••J • Conduction 100 Ra F lGURE 10.35 . .. Heat transfer rate across a horizontal . porous layer. i.e., when: Le., when: ' . a 2 = 'IT'?- (10.259) . . Substituting this result back into Eq. (10.258) then gives: . . i . . . (10.260) Hence, t his analysis indicates that instability will first occur, i.e., convective - motions will first occur, when the Darcy-modified Rayleigh number based on the thickness o f the layer l'eaches 39.5. I f Ra is grea~r than this value, convective motion will occur. This value for Ramin is i n good agreement with experiment,some experimental results being shown in Fig. 10.35. 10.10 NON-DARCY AND OTHER E FFECTS T he Darcy model is based, as discussed earlier, on the assumptions that: 1. T he inertia terms in the governing· equations are negligible. 2. The viscous terms in the. governing equations' are negligible. 3. T he losses associated with the flow over the particles o f the porous Jl!.aterialare proportional to the volume-averaged velocity. The use o f these assumptions is, however, not always justified, e.g., see [57] to [71]. I f the "pore" Reynolds number,.i.e., the Reynolds number based on the local velocity vector and.on K I I2, i.e.: ( u 2 + v 2 + w 2 )112 K ll2: , Rep = . ..:..-------:--'II (10.261) 546 Introduction to Convective Heat Transfer Analysis is greater than 1, t he drag on the particles associated with the square o f the velocity starts to become important, i.e., Eq. (10.7) a nd its equivalent i n the other coordinate directions must b e written as: _ a p = I Ll u + bp/lulu ax K - ap = JLI v + bP/lvlv ay K .. - (10.262) (10.263) (10.264) ~~ = 7 w + bP/lwlw, The last term, which is written as IVlv to ensure that the direction o f the loss is correctly specified, is termed the Forchheimer extension o f t he Darcy model. The factor b is a constant that depends on the characteri~tics o f t he porous material. I f the material is assumed to b e m ade up o f tightly packed spheres, i t c an b e shown that b = 1.75(1 - cf» cf>3 d (10.265) where d is again the diameter o f the particles. The above equations apply to forced convection. Their extension and theextension o f the rest o f the equations given in this\~ection to natural convective flows is straight-forward and will not be discussed here. : I f the permeability o f the material is very high, i t m ay not b e possible to ignore the viscous forces in setting up the momentum equation. In this case, i f the effective viscosity is taken to b e the viscosity o f the fluid, the Darcy equations, i.e., Eqs. (10.12) to (10.14) become:. K u ;:: - ILl K v = - ILl ap (a 2u a2u a2u) ax + K ax2 + ay2 + az2 ap (a 2 a2 a2 v v v) ay + K ax2 + ay2 + az2 ap (a 2 a2 iJ2 w w w) az + K ax2 + ay~ + az2 (10.266) (10.267) K w = - ILl (10.268) These equations repre,sent the Brinkman extension o f t he Darcy model. The Forchheimer and Brinkman extensions o f the basic Darcy model often must be simultaneously used. While i t is easy to extend the governing equations to include the inertia terms, these terms are seldom important i n studying the flow through a porous medium. " I n addition to the modifications to the Darcy flow model, i t is also sometimes necessary to note that the apparent thermal conductivity, ka, as used above arose from conduction through the fluid and solid media. This was the basis for Eq. (10.36). However, i Ii fact, as the fluid flows through the tortuous paths between the solid CHAPTER 10: Convective, Heat Transfer Through Porous M ediaS47 KHigherIn 'Fhis Region Approximately Spherical Particles F IGURE 10.36 Conditions near a wall. matrix, additional mixing occurs which effectively causes an'increase ln'the thermal conductivity. This effect is referred to as "thermal dispersion". The effective thermal conductivity is thus higher than the value calculated by only accounting for the molecular level conduction in the fluid and in the solid matrix. , Near a wall it is possible for K to vary' with distance from the wall. 'This can cause so-callcxl, tunnelling near the, wall and so affect the heat transfer rate a t the wall (see Fig. 10.36). " ,, 10.11 CONCLUDING REMARKS Flow through a solid matrix which is saturated with a f ttiid and through which the fluid is flowing occurs in many practical situations. I n many such cases, temperature differences exist and heat transfer, there'ioie, occurs. The extension of the methods of analyzing convective heat transfer rates that were discussed in the earlier chapters of this book to deal with heat transfer iit porous media flows have been Qiscussed in this chapter. Both forced and natural convective flows ~ave been discussed. ' PROBLEMS 10.1. Unidirectional flow through a porous medium can often be approximately modeled as flow through a series of parallel plane channels as shown in Fig. P10.1. Using this model derive an expression for the permeability, K, in terms of the channel size, W, and the porosity, cf>. ' 10.2. Show that a similarity-type solution can be obtained for the case of two-dimensional flow over a flat plate in a porous medium. the plate being aligned with a forced flow I 548 Introduction to Convective Heat Transfer Analysis FIGURE P10.1 through the porous medium, for the case where there is a unifonn heat flux at the surface of the plate. In treating this problem define: () = T -Too qwx1ka. where qw is the unifonn heat flux at the surface o f th,e plate. 10.'3. A wide flat plate with a length of 20 c m kept at a unifonn surface temperature of 30°C is buried in a bed of approximately spherical pebbles with a mean diameter o f 7 mm with a porosity of 0.35. Water trickles through the bed at a velocity o f 0.03 mfs, the plate being aligned with the water flow. I f the water has a temperature of 20°C, find the variation of the heat transfer rate along the plate. 10.4. I n Chapter 3 it was shown that similarity-type solutions could be found for fluid flow over is~thennal surfaces whose shape is such that the the velocity distribution outside the boundary is described b y U l = Axm. Investigate whether this is also true when . such bodies are placed in a flQwthrough a porous medium. 10.5. A long 5-cm diameter p ipe kept at a surface temperature o f 15°C i s buried in a bed . o f approximately spherical pebbles with a mean diameter of 5 mm with a porosity of 0.4. Water trickles through the bed at a mean velocity of 0.015 mfs, the axis o f the pipe being normal to the water flow. I f the water has a temperature of 10°C,. find the heat transfer from the pipe to the water per meter length o f pipe. 10.6. A cylindrical container kept at a surface temperature o f 40°C is buried in a bed o f approximately spherical pebbles with a m ean diameter o f 5 m m with a porosity o f 0.3; Water at a temperature o f 1 5°C trickles through the bed at a mean velocity ofO.Ol mfs, the axis of the cylinder being normal to the water fl()w. Find the distribution of the local heat transfer rate around the container assuming two~dimensional flow. 10.7. U se the integral equation method to derive an expression for the variation o f the local heat transfer rate along a wide flat plate buried in a porous medium through which C HAPTER 10: Convective Heat Transfer Through Porous Media 549 there is a forced flow parallel to the plate surface. The surface temperature of the plate is given by Too + A(xIL)0.5, L being the length of the plate in the flow direction. 10.S. Derive the value for the Nusselt number for fully developed flow through a porous medium-filled pipe with a uniform heat flux at the wall. 10.9. A heat pump system utilizes a heat exchanger buried i n water-saturated s oilas a heat source. The heat exchanger basically consists o f a series o f vertical plates with height o f30 c m and a width o f 10 cm. These plates, are effectively at a uniform temperature o f 5°C. The soil can be assumed to have a permeability o f 1 0- 10 m 2 and an apparent thermal conductivity o f 0.1 W/m-K. The temperature o f the saturated soil far from th,e heat exchang~r is 30°C. Assuming natural convective flow and that ,there is no interference between the flows over the individual plates, find the mean heat transfer ' rate to a plate. 10.10. Discuss how the analysis o f natural convective flow over a vertical flat plate in a saturated porous medium must be modified i f there is a uniform heat flux r~ther than a uniform temperature a t the surface. 10.11. Explain how the computer program for the calculation o f flow i n a porous medium" filled vertical enclosure given in this chapter must b e modified to deal with the case where the heat flux rather than the temperature at the hot wall is specified. For this purpose, define the following dimensionless temperature: () = T - Tc qwWlka where qw is the uniform heat flux at the hot wall. Also introduce a suitably defined heat flux Rayleigh number. Use the modified program to determine the variation of , Nu with Rayleigh number for the particular case o f a "vertical" enclosure with A = 1. 10.12~ The wall of an insulated container can be modeled as a, 5 -cm wide vertical enclosure with an aspect ratio o f 5. The enclosure is filled with a low grade insulating material that has a permeability of 1 0- 8 m2 and an apparent thermal conductivity of 0.04 W/m-K. The inner wall o f the enclosure is at 5°C. Plot a curve showing 'how the heat transfer rate across the enclosure varies with outer wall temperature.' 10.13. An approximate model o f the flow in a vertical porous medium-filled enclosure assumes that the flow consists of boundary layers on the hot and cold walls with a stagnant layer between the two boundary layers, this layer being at a temperature that is the average o f the hot and cold wall temperatures. Use this model to find an expression for the heat transfer rate across the enclosure and discuss the conditions under which this model is likely to be applicable. 10.14. Explain how the computer program for the calculation o f flow in an enclosure discussed in this chapter must be modified to deal with nonisothermal walls. 10.15. Using the procedure outlined in this chapter for using the boundary layer equations to find the forced convective heat transfer rate from a circular cylinder buried in it saturated porous medium, investigate the heat transfer rate from cylinders w ith an elliptical cross-section with their major axes aligned with the forced flow. The surface velocity distribution should be obtained from a suitable book on fluid mechanics. 550 Introduction to Convective Heat Transfer Analysis 10.16. An insulated wall can be modeled as a 6-cm wide vertical enclosure with an aspect ratio o f 3. The inner and outer surfaces are at temperatures o f 50°C and 10°C respectively. This wall is divided into two 3-cm wide sections by a vertical impermeable barrier. The inner section is filled with an insulating material that has a permeability o f t o- 10 m2 and an apparent thermal conductivity o f 0.02 W /m-K while the outer section is filled with an insulating material that has a permeability o f 1 0-8 m 2 and an apparent thermal conductivity o f 0.05 .WIm- K. Assuming that the barrier between the two sections is at a uniform temperature and that the flow i n the insulation is two-dimesional, find the mean heat transfer rate acrosS the wall p er m2 o f wall area. 10.17. Consider air flow through a 1000-mm long, 5-mm diameter pipe whose wall is kept at a uniform temperature o f 50°C. T he air is heated from 20°C to 30°C as it passes through the pipe and the mean air velocity i n the pipe is 0.6 mls. Using results given in Chapter 4 ,find the mean wall heat flux assuming full':'developed flow, i.e., neglect any entry region effects. Also find the mean heat transfer rate i f the pipe is filled with a loosely packed porous medium which has a permeability o f 1 0-9 m2 and an apparent thermal conductivity o f 0.04 W/m-K. A gain assume fully developed flow. Compare and discuss the values o f the mean heat transfer rates obtained for the two cases. As p art o f this discussion, the pressure drop across the pipe in the two cases should be calculated. .REFERENCES 1. Kaviany, M., Principles o fHeat Transfer in Porous Media, Springer-Verlag, New York, 1991. 2. Scheidegger, A.E., The Physics o f Flow Through Porous Media, 3rd ed., University o f , Toronto Press, Toronto, 1974. 3. Bear, J., Dynamics o f Fluids in Porous Media, American Elsevier, New York, 1972.. .4. Bear, J. and Bachmat, Y., Introduction to Modeling o f Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, 1991. 5. 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