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5 INTERNAL FORCED CONVECTIVE HEAT AND MASS TRANSFER 5.1 Introduction Internal heat and mass transfer have significant applications in a variety of technologies, including heat exchangers and electronic cooling. Internal convective heat and mass transfer can be classified as either forced or natural convection. An initial simple approach to internal convective heat transfer is to utilize the dimensional analysis presented in Chapter 1 to obtain important parameters and dimensionless numbers for the steady laminar flow of an incompressible fluid in a convectional tube, i.e., h = f (k , μ , c p , ρ , u, D, x , ΔT ) (5.1) The local heat transfer coefficient is a function of the fluid properties (viscosity, μ; thermal conductivity, k; density, ρ; specific heat, cp), geometry (D), temperature (ΔT), and flow velocity (u). In dimensionless form, as shown in Chapter 1, Nu = g(Re, Pr, x / D ) (5.2) The above relation indicates that the local Nusselt number for flow in a circular tube is a function of the Reynolds number, Prandtl number, and x/D. The goal of this chapter is to develop the heat and mass transfer coefficients for various internal flow configurations under different operating conditions. The objective of this chapter is to present fundamental models, and analytical and numerical solutions of both laminar and turbulent internal forced convections. Section 5.2 introduces the basic definitions, terminologies, and governing equations for internal flow; followed by discussions on uncoupled fully developed laminar flow and the thermal entry effects in Section 5.3 and 5.4. The fully developed laminar flow with coupled thermal and concentration entry effects is taken up in Section 5.5. While the flow in Sections 5.3 – 5.5 is assumed to be fully developed, the combined hydrodynamic, thermal, and concentration entry effects are discussed in Section 5.6. The full numerical solution of internal forced convection problem based on full Navier-Stokes equations using the finite volume method is discussed in Section 5.7; this is followed by a discussion on forced convection in microchannels. This chapter is closed by a very detailed discussion on the turbulence for internal forced convection in Section 5.9. 438 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.2 Basic Definitions, Terminology and Governing Equations It is important to clarify some basic definitions, terminologies and criteria that are often used in internal convective heat and mass transfer. These include: 1. Mean velocity, temperature, and concentration 2. Fully developed flow, temperature, and concentration profiles 3. Hydrodynamic, thermal, and concentration entrance lengths Figure 5.1(a) shows the development of a velocity profile inside a duct or tube with uniform inlet velocity for laminar flow of an incompressible Newtonian fluid. The velocity profile at some distance away from the tube’s inlet no longer changes along the flow direction, where it is referred to as the fully developed flow condition. The fully developed condition is often met at some distance away from the inlet. However, there are also applications in which fully developed flow is never reached. Momentum, thermal, and concentration boundary layers form on the inside surface of the tube. The thickness of the layers increases in a similar manner as boundary layer flow over a flat plate (which was presented in detail in Chapter 4). Figure 5.1(a) shows how the momentum boundary layer builds up in a pipe along the flow direction. At some distance away from the inlet, the boundary layer fills the flow area. The flow downstream from this point is referred to as Hyrdrodynamic entrance length Fully developed flow uin u δ r δ (a) Velocity profile ro friction factor (b) Friction x Figure 5.1 Velocity profiles and friction factor variation in laminar flow in a circular tube. Chapter 5 Internal Forced Convective Heat and Mass Transfer 439 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press fully developed flow since the velocity slope does not change after this point. The distance downstream from the inlet to where the flow becomes fully developed is called the hydrodynamic entrance length. If the flow is laminar (Re < 2300 for flow inside circular tubes), the fully developed velocity is a parabolic shape. It should be noted that the fluid velocity outside the boundary layer increases with x, which is required to satisfy the conservation of mass (or continuity) equation. The center line velocity finally reaches a value two times the inlet velocity, uin , for fully developed, steady, incompressible, laminar flow inside tubes. It should be noted that the hydrodynamic entrance length for fully developed flow does not start from the point where the friction coefficient, τw cf = (5.3) 2 ρ uin / 2 does not change along the flow. The friction coefficient variation for laminar flow inside a circular tube with uniform inlet velocity is shown in Fig. 5.1(b). The friction coefficient is highest at the entrance and then decreases smoothly to a constant value, corresponding to fully developed flow. Two factors cause the friction coefficient to be higher in the entrance region of tubes than in the fully developed region. The first factor is the larger velocity gradient at the entrance on the wall. The gradient decreases along the pipe and becomes constant before the velocity becomes fully developed. The second factor is the velocity outside the boundary layer, which must increase to satisfy the conservation of mass or continuity equation. Accelerating velocity in the core produces an additional drag force when its effect is considered in the friction coefficient. The turbulent velocity profile and friction coefficient variation for a circular pipe are shown in Fig. 5.2. Even for a very high inlet velocity, the boundary layer will be laminar over a part of the entrance. This transition from laminar to turbulent is clearly shown by the sudden increase in momentum boundary layer thickness as shown in Fig. 5.2(a). The friction coefficient variation for turbulent flow in a pipe entrance is shown in Fig. 5.2(b). The hydrodynamic entry length required for fully developed flow should be obtained by a complete solution of the flow and thermal field in the entrance region. A rule of thumb to judge whether or not the flow is fully developed for circular pipes is LH ≥ 0.05 Re D LH ≥ 0.625 Re0.25 D for laminar flow for turbulent flow (5.4) (5.5) where LH is the hydrodynamic length and the Reynolds number is defined by Re = um D ν 440 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Hyrdrodynamic entrance length δ Fully developed flow u r0 δ (a) Velocity profile r Laminar boundary layer friction factor Turbulent layer boundary Fully developed turbulent (b) Friction factor Figure 5.2 Velocity profiles and friction factor in turbulent flow in a circular tube. x Uniform inlet temperature, Tin Tin-Tw δT r δT ro Figure 5.3 Temperature development along the flow in a circular tube. A similar behavior is expected for thermal cases with the thermal boundary layer growth at the entrance of a tube as shown in Figure 5.3, which corresponds to a case where there may be an unheated length in which the velocity is fully developed before heating starts. One expects that the thermal boundary layer increases in the thermal entry region before the heat transfer coefficient becomes constant. It should be noted that the requirement for a fully developed thermal region is that the dimensionless temperature, θ , Chapter 5 Internal Forced Convective Heat and Mass Transfer 441 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press θ= Tw − T Tw − Tm or Tw − T Tw − Tc does not change with distance along the flow direction. The mean and centerline temperatures are Tm and Tc, respectively. Similar requirements exist for the fully developed concentration profile where θ is replaced with φ , φ= cw − c c w − cm or cw − c c w − cc where the mean and centerline concentrations (or mass fractions) are cm and cc, respectively. In the subsequent sections, we use the following definitions to mathematically define the fully developed flow, temperature, and concentration profiles: Fully developed flow profile u uc or r u = f  um  ro  (5.6) Fully developed temperature profile θ= Tw − T Tw − Tm cw − c c w − cm or r Tw − T = g  Tw − Tc  ro  r cw − c = h  c w − cc  ro  (5.7) Fully developed concentration profile φ= or (5.8) We can now define the local heat and mass transfer coefficients (h and hm) based on the mean temperature or concentration. ′′ qw = h ( Tw − Tm ) = − k ∂T ∂r (5.9) r = ro  ′′ mw = hm (ωw − ωm ) = − ρ D ∂ω ∂r (5.10) r = ro where D is mass diffusivity. Since we define the fully developed temperature profile as when the nondimensional temperature profile (Tw − T ) / (Tw − Tm ) is invariant in the flow direction (x-direction), we can write the following equation: ∂T − ∂r r = ro h  Tw − T  ∂ = constant = = = constant   ∂r  Tw − Tm  r = r Tw − Tm k o (5.11) The above conclusion, that the local heat transfer coefficient is constant along the flow direction for a fully developed temperature profile, is only valid for constant wall heat flux or constant wall temperature conditions. The requirement for the dimensionless temperature to be invariant for a fully developed temperature profile can also be presented as 442 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ∂  Tw − T   =0 ∂x  Tw − Tm  (5.12)  dTm   dx Differentiating the above equation yields ∂T dTw  Tw − T = − ∂x dx  Tw − Tm  dTw  Tw − T +   dx  Tw − Tm (5.13) In external flow, the heat and mass transfer coefficients are usually defined by a driving differential (Tw - T∞) or (ωw – ω∞) where T∞ and ω∞ are the temperature and mass fraction of the fluid in the free stream (far away from the wall). In most cases, T∞ and ω∞ are known and constant for external flows. However, in internal flow configurations, there is not usually a well-defined temperature or concentration (mass fraction), except at the inlet and/or the boundaries. In internal flow, the temperature and concentrations may change both in the axial direction and perpendicular to the flow direction. Therefore, there are several choices available for the driving differential for temperature and concentration in internal flow. The most common choice for defining the driving temperature or concentration is based on mean temperature or concentration (mass fraction or mass density). The mixed mean fluid temperature or concentration is defined at a given local axial location based on the convective thermal energy or mass balance, i.e., Tm = 1 Aum ρm c p , m 1 Aum  A A uT ρ c p dA (5.14) (5.15) ρ A ,m =  u ρ A dA where ρ A ,m is the mean mass density for a given component A, and ρm is the mean density for the fluid where the mean velocity is defined as um = 1 A ρm  A u ρ dA (5.16) Assuming constant properties for mean velocity, temperature and mass density in the above equation, we obtain 1 udA A A 1 Tm = uTdA Aum  A um = (5.17) (5.18) (5.19) ρ A,m = 1 Aum  A ρ A udA We will now focus our attention on two special conventional boundary conditions; constant wall heat flux and constant surface temperature. First, consider the constant heat flux or heat rate at the wall, which occurs in many applications such as electronic cooling, electric resistance heating, and radiant Chapter 5 Internal Forced Convective Heat and Mass Transfer 443 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ′′ heating. From eq. (5.9), since h and qw are constant, we can conclude Tw – Tm = constant. Differentiating leads to dTw dTm = dx dx Substituting into eq. (5.13) gives us ∂T dTw dTm = = ∂x dx dx (5.20) Now consider the case of constant surface or wall temperature, which also occurs in many applications including condensers, evaporators and any heat exchange surface where the heat transfer coefficient is extremely high. Using eq. (5.13) and the fact that dTw /dx = 0 for constant surface temperature, we get ∂T  Tw − T  dTm =  ∂x  Tw − Tm  dx (5.21) It should be emphasized that eqs. (5.20) and (5.21) are only applicable when the temperature profile is fully developed. The variations of wall and mean temperature for the fully developed temperature profile along the flow for constant heat rate or surface temperature are shown in Fig. 5.4. Finally, to obtain the convective heat and/or mass transfer coefficients, one needs to solve the continuity, mass, momentum, energy and appropriate species equations. In convective heat and mass transfer problems, it is important to obtain information about the flow by solving the continuity and momentum equations, in addition to the energy and species equations. These conservation equations are mostly decoupled, except for circumstances such as a variable property, or coupled governing equations or boundary conditions due to physical circumstances (which happens in applications such as natural convection, absorption, sublimation, evaporation and condensation problems). T Tw Tm Tm T Tw x (a) Constant heat flux at wall (b) Constant wall temperature x Figure 5.4 Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile. 444 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press It is obviously more accurate to solve the complete transport conservation equations (elliptic form) for internal flow without making boundary layer assumptions (parabolic form) as discussed in Chapter 4. However, in most cases it is not practical due to complexity of the geometry and/or solution techniquesm, as well as the requirement of additional boundary conditions in both analytical or numerical methods. For the case of two-dimensional fully developed steady laminar flow with constant properties, the momentum equation in a circular tube, including boundary conditions, as shown in Chapter 2 is: dp μ d  du  (5.22) = r  dx u=0 r dr  dr  at r = ro du =0 dr at r=0 (5.23) Integrating the above equation twice and using the boundary conditions yield a parabolic velocity profile: u= −ro2  dp   r 2    1 −  4 μ  dx   ro2  (5.24) Using the definition of mean velocity, um , for constant properties and the above equation, we obtain π ro2 Equation (5.24) in terms of mean velocity is A  r2  u = 2um  1 − 2   ro  um  = A udA  = ro 0 2π rudr =− ro2 dp 8μ dx (5.25) (5.26) The shear stress at the wall can be calculated from the velocity gradient at the wall. 4u μ r dp ∂u (5.27) = m =− o τw = μ ∂r r = ro ro 2 dx The above result can be presented in terms of the friction coefficient, cf. τw 8μ 16 cf = = = (5.28) 2 ρ um / 2 ro ρ um Re In addition to the above friction coefficient, the following friction factor is also widely used: f= −(dp / dx ) D 2 ρ um / 2 (5.29) It follows from eq. (5.27) that 4τ w 64 f= = 4c f = 2 ρ um / 2 Re (5.30) Chapter 5 Internal Forced Convective Heat and Mass Transfer 445 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Let’s make an analysis of the energy equation to get a feeling about the importance of various terms, since determination of the temperature field in the fluid is required for the heat transfer coefficients. To simplify the analysis, lets consider a two-dimensional cylindrical geometry with the following assumptions: 1. Steady laminar flow 2. Constant properties 3. Fully developed flow 4. Newtonian incompressible fluid The energy equation under the above assumptions is 2  1 ∂ ( r ∂T / ∂r ) ∂ 2T  ∂T μ  ∂u  =α  + 2 + (5.31) u  ∂x ∂r ∂x  ρ c p  ∂r  r The above equation is non-dimensionalized using the following variables to show the effect of axial conduction and viscous dissipation: u+ = T − Tr θ= Tin − Tr u um , x+ = 2( x / D) Re Pr , E= + 2 um c p ΔT , ΔT = Tin − Tr r ,r= ro (5.32) , Pe = Re Pr where Tr is a reference temperature and E and Pe are Eckert and Peclet numbers, respectively. The resulting dimensionless energy equation is  ∂u +  u + ∂θ 1 ∂  ∂θ  1 ∂ 2θ = + +  r+ +  + + E Pr  +  2 ∂x + r ∂r  ∂r  2 Pe 2 ∂x + 2  ∂r  2 (5.33) The second term on the right hand side of the above equation is due to axial heat conduction, and the last term is due to the viscous dissipation effect. If E Pr is small, viscous dissipation can be neglected. This is true for flow with a low velocity and low Prandtl number. The second term on the right hand side (axial heat conduction) is neglected when the Peclet number, Pe, is greater than 100. Axial heat conduction should be accounted for when the Peclet number is small, in the case of liquid metals. Example 5.1. Estimate the hydrodynamic entry length, LH, using Blasius’ result for the momentum boundary layer thickness. Solution: From Blasius’ solution δ 5 x ≈ Re x1/ 2 (5.34) For fully developed flow conditions, δ = D / 2, and therefore D/2 5 = LH Re x1/ 2 LH Re x1/ 2 = = 0.1Re x1/ 2 10 D (5.35) 446 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.3 Hydrodynamically and Thermally Fully Developed Laminar Flow Constant Wall Heat Flux In this section, we consider the case of fully developed laminar flow and constant properties in a circular tube with a fully developed temperature and concentration profiles. We first consider the case of constant heat rate per unit surface area for steady, laminar, fully developed flow. The energy equation in a circular tube, by neglecting axial heat conduction and viscous dissipation terms, is: u ∂T  1 ∂  ∂T =α  r ∂x  r ∂r  ∂r    (5.36) For a fully developed flow with constant wall heat flux, eq. (5.20) can be substituted into eq. (5.36) to obtain u dTm  1 ∂  ∂T =α  r dx  r ∂r  ∂r    (5.37) The boundary conditions are −k ∂T ′′ = qw ∂r at r = ro at r = 0 ∂T =0 ∂r (5.38) Integrating eq. (5.37) twice and applying the boundary conditions in eq. (5.38) to get the temperature distribution gives us T = Tw − 2um dTm  3ro2 r 2 r4  −+   α dx  16 4 16ro2  (5.39) Using the definition of mean temperature presented in the last section with the above profile for temperature, and assuming constant properties: Tm  uTdA = 2 π ruTdr = πr u  udA A 0 A 2 o m ro Substituting eq. (5.39) into the above expression yields: Tm = Tw − 11  2um  dTm 2 ro 96  α  dx   (5.40) 2  ro  The heat flux at the wall can be obtained using the above relation for Tm  11   2u  dT qw′′ = h ( Tw − Tm ) = h   m  m  96  α  dx (5.41) The heat flux at the wall can also be calculated using eq. (5.39) for the temperature profile and Fourier’s law of heat conduction ′′ qw = −k ∂T ∂r r = ro  u   dT  = ρ c p ro  m   m   2   dx  (5.42) Combining eqs. (5.41) and (5.42) and solving for the heat transfer coefficient, h, yields Chapter 5 Internal Forced Convective Heat and Mass Transfer 447 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press h = 4.364 k / D or in terms of the Nusselt number, Nu = 4.364 (5.43) Example 5.2. Make an energy balance on a control volume element in a circular tube with constant wall heat flux for the fully developed flow and temperature profile region to prove eq. (5.42). Solution: Applying the conservation of energy for the control volume element shown in Fig. 5.5 yields ′′ qw ( 2π ro ) = ( ρ c p ) (π ro2 ) um dTm dx Solving for heat flux at the wall, ′′ qw = ρ c p r0 dT um m 2 dx ′′ qw (2π ro )Δx ρ c p (π ro2 ) umTm x r r0 Δx ρ c p (π ro2 ) um  Tm +   dTm  Δx  dx  Figure 5.5 Energy balance on a control volume element for fully developed flow and temperature profile in a circular tube with constant wall heat flux Constant Surface Temperature We begin with the energy eq. (5.31) and neglect the effects of axial heat conduction and viscous dissipation. We already showed that for a fully developed flow and temperature profile with constant surface temperature T − T dTm ∂T =w ∂x Tw − Tm dx The energy equation and boundary conditions, using the fully developed velocity profile eq. (5.26) and the above equation, are  r 2   T − T  dTm  1 ∂  ∂T   2um 1 − 2   w =α   r   r ∂r  ∂r    ro   Tw − Tm  dx r = ro , T = Tw r = 0, ∂T = 0 or T = finite ∂r (5.44) (5.45) 448 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The above equation and boundary conditions have been solved by various techniques in literatures, including separation of variables and infinite series. For additional detailed information the reader should refer to Burmeister (1993), Kakac, et al. (1987), Kays et al. (2005), and Bejan (2004). The solution for eqs. (5.44) and (5.45) in the form of an infinite series for the temperature and Nusselt number is (Kakac et al. 1987): ∞ r T − Tw =  C 2m   Tin − Tw m = 0  ro  2m (5.46) λ02 where co = 1, 1 c2 = − λ02 = −1.828397, 4 λ0 = 2.704364 Nu = 12 λ = 3.657 2 c2 m = ( 2m ) 2 ( c2 m − 4 − c2 m − 2 ) The Nusselt number corresponding to the above temperature distribution is (5.47) The temperature slope for constant wall heat flux at the wall is higher than the temperature slope for constant surface temperature. This effect has resulted in a 16 percent increase in Nusselt number for constant wall flux versus constant wall temperature for a fully developed flow and temperature profile. The two cases of constant wall temperature and constant wall heat flux are special cases of a more general exponential heat flux boundary condition: 1  ′′ qw = A exp  nx +  2  (5.48) where A and n are both constants and n can be assumed to be either a positive or negative value, and x + = x / ro . n = 0 corresponds to constant heat flux at the Re Pr wall and n = –14.63 corresponds to constant wall temperature. Shah and London (1978) developed the following correlation, which fits the exact solution of eq. (5.48) within 3 percent for –51.36 < n < 100. Nu = 4.3573 + 0.0424n − 2.8368 × 10−4 n 2 + 3.6250 × 10 −6 n 3 − 7.6497 × 10−8 n 4 + 9.1222 × 10−10 n 5 − 3.8446 × 10−12 n 6 (5.49) Example 5.3. When Pr = ν α  1 , it is possible to assume a uniform velocity in the radial direction within a short distance from the inlet, since the diffusion of momentum is much less than the diffusion of heat. This is referred to in literature as a “slug flow” condition. Determine the Nusselt number for “slug flow” with a fully developed temperature profile in a circular pipe and uniform heat flux at the wall. Solution. The energy equation and boundary conditions for this case can be found by assuming u = um = constant and ∂T/∂x = dTw/dx = dTm/dx = constant. Chapter 5 Internal Forced Convective Heat and Mass Transfer 449 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press um dTw  1 ∂  ∂T   =α  r  dx  r ∂r  ∂r   ∂T r = 0, =0 ∂r r = ro , T = Tw (5.50) Integrating eq. (5.50) twice with respect to η and using the above boundary conditions yields T = Tw − 2 um ro2 dTw   r   1 −    α 4 dx   ro     (5.51) Both the heat flux at the wall and the mean temperature are obtained from the above equation ′′ qw = −k dT dr =k r = r0 um ro dTw 2α dx (5.52) q′′ r (5.53) = = Tw − w o π ro2 um 4k ro2 Using the above equation and the definition of the Nusselt number, Tm 0 0  = r0 2π rum Tdr 2 rTdr r0 Nu = qw′′ D hD = =8 k (Tw − Tm ) k (5.54) Obviously, as one expects, the Nusselt number for “slug flow” is higher than when assuming a parabolic velocity profile. The velocity is higher near the wall assuming “slug flow” compared to the parabolic profile. The problem of fluid flow and heat transfer in an annulus (see Fig. 5.6) is also of considerable interest in various applications, including heat exchangers and heat pipes, due to an increase and flexibility in the heating and cooling of surface area. We present the result below for the case in which both the inner wall (radius ri) and outer wall (radius ro) are kept at a constant heat flux for various values of K = ri/ro. K = 0 corresponds to conventional circular tubes, and K =1 corresponds to flow between two parallel planes. ′′ qo ro ri r qi′′ ′′ qo Figure 5.6 Flow and heat transfer in an annulus 450 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The momentum and energy equations, as well as the boundary conditions for steady, laminar, fully developed flow and temperature profile, by neglecting radial conduction of the wall, axial heat conduction in the fluid, and viscous dissipation, and assuming constant properties, are dp μ d  du  (5.55) = r  dx r dr  dr  dTm  1  ∂ ∂T   =α    dx  r  ∂r ∂r   ∂T r = ri , u = 0, −k = qi′′ ∂r r =ri u r = ro , u = 0, k ∂T ∂r = qo′′ r = ro (5.56) (5.57) (5.58) where qi′′ and qo′′ are the inner and outer wall heat fluxes, respectively. The velocity profile can be obtained by integrating eq. (5.55) twice and applying no slip boundary conditions at both the inner and outer walls. 2 u 2 r r = 1 −   + B ln  um A   ro  ro    (5.59) where A = 1 + K − B, 2 K 2 −1 B= , ln K um = (r  ro ri 2 o urdr − ri 2 ) The local heat transfer coefficients and Nusselt numbers are hi = qi′′ qo′′ , ho = Ti − Tm To − Tm hi Dh hD , Nuo = o h k k Nui = where the hydraulic diameter, Dh = 2(ro – ri). The energy eq. (5.56) can also be solved using the velocity profile given by eq. (5.59) in a similar manner as presented for a circular tube, except that it can involve lengthy algebra. Since the energy eq. (5.56) is linear and homogeneous, the principle of superposition was used to obtain the solution of the above problem as a sum of two problems. One problem is the outer wall heated uniformly and the inner wall insulated. The second problem is the inner wall heated uniformly and the outer wall insulated. The principle of superposition can be applied to linear homogeneous differential equations as long as the summation of the governing equations and boundary conditions for each subset problem will add to the original problem. Kakaç and Yucel (1974) performed a numerical solution for fully developed velocity and temperature profile. Table 5.1 shows the inner and outer wall Nusselt number for various K values obtained by Kakaç and Yucel (1974). Chapter 5 Internal Forced Convective Heat and Mass Transfer 451 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The inner and outer wall Nusselt numbers, based on the average wall heat flux ratio, can be calculated from the results in Table 5.1 using the following equations: Nui = Nuo = Nuii ′′ 1 − ( qo / qi′′) θ i ∗ Nuoo ′′ 1 − ( qi′′ / qo ) θ o∗ (5.60) (5.61) where Nuii is defined as the inner wall Nusselt number when the inner tube is heated and outer wall is insulated. Similarly, Nuoo is the outer tube Nusselt number when the outer wall is heated and inner tube is insulated. θ i∗ and θ o∗ are defined as influence coefficients and are a function of K only for laminar flow. Table 5.1 Nusselt number for fully developed flow and temperature profile with constant wall heat flux in an annulus, K = ri /ro (Kakaç and Yucel, 1974) K 0 0.10 0.25 0.50 1.00 Nuii ∞ 11.900 7.735 6.181 5.384 Nuoo 4.364 4.834 4.904 5.036 5.384 θi* ∞ 1.3835 0.7932 0.5288 0.3460 θo* 0 0.0562 0.1250 0.2160 0.3460 Example 5.4. Determine the Nusselt number for a fully developed flow and temperature profile between two parallel planes when both surfaces are heated at constant identical heat fluxes using the results presented in Table 5.1. Solution: As indicated before, flow and heat transfer between parallel planes is the limiting case of an annulus where K = 1. In this case, eqs. (5.60) and (5.61) are identical. Nui = Nuo = Nuoo 5.385 = = 8.234 ∗ 1 − θo 1 − 0.346 The fluid and heat transfer solution for tubes of noncircular cross section with a fully developed flow and temperature profile can be obtained by solving the energy equation for the particular geometry. The results for some of the more conventional geometries obtained by Shah and London (1974) are presented in Table 5.2. There are three different boundary conditions in this table; “H1” which refers to the circumferentially constant wall temperature and the axial constant wall heat flux boundary condition, “H2” is for both axially and circumferentially constant heat flux at the wall, and “T” represents a constant wall temperature boundary condition. For the case of a symmetrically heated straight duct having no corners and a constant peripheral curvature, e.g. parallel planes and circular pipe, H1 and H2 boundary conditions are the same. 452 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table 5.2 Solutions for Nusselt Number and Friction Coefficient for fully developed laminar flow and temperature profile for various geometries (Shah and London 1974) Geometry (L/Dh > 100) NuH1† 3.014 NuH2† 1.474 NuT 2.39* cfRe 12.630 b a a b b a b 3 = a 2 60° b 3 = a 2 b =1 a 3.111 1.892 2.47 13.333 3.608 3.091 2.976 14.227 4.002 b1 = a2 3.862 3.34* 15.054 b 4.123 3.017 3.391 15.548 a 4.364 4.364 3.657 16.000 b a b a b a b = 0.9 a 5.099 4.350* 3.66 18.700 b1 = a4 5.331 2.930 4.439 18.233 b1 = a8 b =0 a 6.490 8.235 2.904 8.235 5.597 7.541 20.585 24.00 * Interpolated values. † Nusselt numbers are averaged with respect to tube periphery 5.4 Hydrodynamically Fully Developed and Thermally Developing Laminar Flow In the previous section, we considered problems where the velocity and temperature profile were fully developed, so that the heat transfer coefficient was constant with distance along the pipe. In this section, we consider problems in which only velocity is fully developed at the point where the heat transfer starts. Furthermore, as before, we consider two cases of constant wall temperature and Chapter 5 Internal Forced Convective Heat and Mass Transfer 453 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press wall heat flux, both by assuming uniform temperature at the inlet. Under these conditions, the heat transfer coefficient is not constant but varies along the tube. Whiteman and Drake (1980), Lyche and Bird (1956) and Blackwell (1985) studied the case of fully developed flow with thermal entry effects for nonNewtonian fluids. Sellars et al. (1956) obtained thermal entry length solutions for the case of a Newtonian fluid with constant wall temperature and fully developed flow, which are presented below. The following assumptions are made in order to obtain a closed form solution for heat transfer analysis for fully developed flow and developing temperature profile in a circular tube: 1. Incompressible Newtonian fluid 2. Laminar flow 3. Two-dimensional steady state 4. Axial heat conduction and viscous dissipation are neglected 5. Constant properties This does not mean that one cannot obtain analytical solutions when one or more of the above assumptions is valid, but the solution will be much easier by making the above assumptions. Since the fully developed velocity was already obtained in Section 5.2, we will focus on the solution of the energy equation and boundary conditions for a developing temperature profile. 5.4.1 Constant Wall Temperature The dimensionless energy eq. (5.33) and boundary conditions using the above assumptions for the case of constant wall temperature are reduced to u + ∂θ 1  ∂  ∂θ   (5.62) = +  +  r + +  + 2 ∂x + θ (1, x + ) = 0 θ ( 0, x + ) = finite or ∂θ ( 0, x + ) = 0 ∂r + θ (r , 0) = 1 r  ∂r  ∂r   (5.63) where r+ = T − Tw x / r0 r u , θ= , u+ = , x+ = ro Tin − Tw um Re Pr For a fully developed laminar flow, the parabolic velocity profile previously developed is applicable, i.e.,  r2  2 u = 2um  1 − 2  or u + = 2 1 − r +  ro  ( ) (5.64) Substituting the above equation into the energy eq. (5.62), we get ∂θ ∂ 2θ 1 ∂θ + (1 − r ) ∂x 2 + = ∂r + 2 + r + ∂r + 454 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Since the above partial differential equation is linear and homogeneous, one can apply the method of separation of variables. The separation of variables solution is assumed of the form (5.65) θ (r + , x + ) = R (r + ) X ( x + ) The substitution of the above equation into eq. (5.64) yields two ordinary differential equations X′ + λ2X = 0 (5.66) R′′ + 2 1 R′ + λ 2 R 1 − r + = 0 r+ ( ) (5.67) where X′ = 2 dX , dx + X ′′ = d2 X dx + 2 , R′ = dR d2 R , R′′ = +2 + dr dr 2+ and – λ is the separation constant or eigenvalue. The solution for eq. (5.66) is a simple exponential function of the form e − λ x while the solution of eq. (5.67) is of infinite series referred to by the SturmLiouville theory. The solution is of the form θ ( r + , x + ) =  cn Rn ( r + ) exp ( −λn 2 x + ) n =0 ∞ (5.68) where λn are the eigenvalues, Rn are the eigenfunctions corresponding to eq. (5.67), and cn are constants. The local heat flux, dimensionless mean temperature, local Nusselt number and mean Nusselt number can be obtained from the following equations, using the above temperature distribution (Tw − Tin ) ∂θ ∂T ′′ qw = −k ∂r = −k r = ro ro ∂r + r + =1 2k = − ( Tw − Tin )  Gn exp ( −λn 2 x + ) ro n =0 ∞  exp ( −λn 2 x + )  Tm − Tw  θm = = 8 Gn  λn 2 Tin − Tw   n =0   ∞ (5.69) (5.70) exp ( −λn 2 x + ) Nux = hx ( 2ro ) k = (Tw − Tin ) kθ m 1 =+ x −qw′′ ( 2ro ) −2 ∂θ = θ m ∂r + = r + =1 G ∞ n =0 n =0 ∞ n 2 Gn exp ( −λn 2 x + ) / λn 2 (5.71) Nu m = hx ( 2ro ) k  x+ 0  ∞ Gn exp ( −λn 2 x + )  1  Nux dx = − + ln 8 2x λn 2  n =0    + (5.72) where 1 ′ Gn = − cn Rn (1) 2 Chapter 5 Internal Forced Convective Heat and Mass Transfer 455 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The first five terms in eqs. (5.69) – (5.72) are sufficient to provide accurate solutions to the above infinite series. The eigenvalues, λn and Gn, used to calculate qw′′ , θm, Nux and Num for the above problem are presented in Table 5.3. Table 5.3 Eigenvalues and Eigenfunctions of a Circular Duct; Thermal Entry Effect with Fully Developed Laminar Flow and Constant Wall Temperature (Blackwell 1985) n 0 1 2 3 4 λn2/2 3.656 22.31 56.9 107.6 174.25 Gn 0.749 0.544 0.463 0.414 0.383 Table 5.4 Nusselt Solution for Thermal Entry Effect of a Circular Tube for Fully Developed Laminar Flow and Constant Wall Temperature (Blackwell 1985) x+ 0 0.001 0.004 0.01 0.04 0.08 0.1 0.2 ∞ Nu x ∞ 10.1 8.06 6.00 4.17 3.79 3.71 3.658 3.657 Nu m ∞ 15.4 12.2 8.94 5.82 4.89 4.64 4.16 3.657 θm 1 0.940 0.907 0.836 0.628 0.457 0.395 0.190 0 Table 5.4 provides the variations of Nux, Num and θm with distance along the tube. It can be easily observed from Table 5.4 that the fully developed temperature profile starts at approximately: x+ = x / r0 = 0.1 Re Pr (5.73) Therefore, (LT,T / D) = 0.05RePr where LT,T is the thermal entrance length for constant wall temperature The thermal entry length increases as both the Reynolds number and Prandtl number increase. A very long thermal entry length is needed for fluids with a high Prandtl number, such as oil. Therefore, care should be taken to make a fully developed temperature profile assumption for fluids with a high Prandtl number. 5.4.2 Constant Heat Flux at the Wall The laminar fully developed flow with thermal entry length effects (developing temperature profile) for constant wall heat flux is very similar to that 456 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press in the case of constant wall temperature, except that the dimensionless temperature and boundary conditions are defined as θ= ′′ qw = −k Tin − T ′′ qw D / k = constant r = ro (5.74) (5.75) ∂T ∂r Siegel et al. (1958) solved the above problem for laminar fully developed flow using separation of variables and the Strum-Liouville theory, of which the result is presented below: θ = θ * (r + , x + ) + 4x + + r + − 2 where θ* is given below ∞ n =1 7 r+ − 4 24 2 (5.76) (5.77) θ * ( r + , x + ) =  cn Rn exp ( −λn 2 x + ) The eigenvalues, λn, eigenfunctions, Rn, and constants, cn, are presented in Table 5.5. Table 5.5 Eigenvalues and Eigenfunctions for Thermal Entry Effect of a Circular Tube for Fully Developed Laminar Flow and Constant Wall Heat Flux (Siegel et al. 1958) n 2 λn Rn(1) cn 1 2 3 4 5 6 7 25.6796 83.8618 174.167 296.536 450.947 637.387 855.850 -0.492517 0.395508 -0.345872 0.314047 -0.291252 0.273808 -0.259852 0.403483 -0.175111 0.105594 -0.0732804 0.0550357 -0.043483 0.035597 Table 5.6 Local Nusselt Number for Thermal Entry Effect with Fully Developed Flow of a Circular Tube with Constant Wall Heat Flux (Siegel et al. 1958) x+ 0 0.0025 0.005 0.01 0.02 0.05 0.1 0.2 ∞ Nu x ∞ 11.5 9.0 7.5 6.1 5.0 4.5 4.364 4.364 Chapter 5 Internal Forced Convective Heat and Mass Transfer 457 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The local Nusselt number, based on the above solution, is given below and numerical values are presented in Table 5.6 Nux = 1 + ( 24 / 11)  cn exp ( −λn 2 x + ) Rn (1) n=0 ∞ ( 48 / 11) (5.78) The thermal entrance for constant wall heat flux based on the numerical results presented in Table 5.6 is x + ≈ 0.05 or (5.79) ( LT , H ) = 0.05 ( Re Pr )( D ) where LT, H is the thermal entry length for fully developed flow with constant wall heat flux. The mean temperature variation can be obtained from the Nusselt number, eq. (5.78), using the following equation: Tw − Tm = qw′′ qw′′ D = hx Nux k (5.80) The thermal entry length solutions presented above for hydrodynamically fully developed flows are based only on either constant wall temperature or constant heat flux. Although the wall temperature or heat flux may be assumed constant along the tube length in the entrance region for certain internal convection problems, there are cases where the wall temperature or heat flux varies considerably with tube length. In these cases, a decision must be made whether to assume constant wall temperature or constant heat flux, use a mean overall wall temperature or heat flux with respect to length, or attempt to take into account the effect of varying wall temperature or heat flux. If the latter is desirable, the solution can be obtained by superposing the thermal entry length solutions at infinitesimal and finite surface-temperature steps (Kays et al., 2005). 5.5 Hydrodynamically Fully Developed Flow with Coupled Thermal and Concentration Entry Effects There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase. In this section, coupled forced internal convection in a circular tube will be presented for both adiabatic and constant wall heat flux. 5.5.1 Sublimation inside an Adiabatic Tube In addition to the external sublimation discussed in subsection 5.6.2, internal sublimation is also very important. Sublimation inside an adiabatic and externally 458 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press heated tube will be analyzed in the current and the following subsections. The physical model of the problem under consideration is shown in Fig. 5.7 (Zhang and Chen, 1990). The inner surface of a circular tube with radius ro is coated with a layer of sublimable material which will sublime when gas flows through the tube. The fully-developed gas enters the tube with a uniform inlet mass fraction of the sublimable substance, ω0, and a uniform inlet temperature, T0. Since the outer wall surface is adiabatic, the latent heat of sublimation is supplied by the gas flow inside the tube; this in turn causes the change in gas temperature inside the tube. It is assumed that the flow inside the tube is incompressible laminar flow with constant properties. In order to solve the problem analytically, the following assumptions are made: 1. The entrance mass fraction, ω0, is assumed to be equal to the saturation mass fraction at the entry temperature, T0. 2. The saturation mass fraction can be expressed as a linear function of the corresponding temperature. 3. The mass transfer rate is small enough that the transverse velocity components can be neglected. The fully developed velocity profile in the tube is   r 2  u = 2um 1 −      ro     (5.81) where um is the mean velocity of the gas flow inside the tube. Neglecting axial conduction and diffusion, the energy and mass transfer equations are ∂T ∂  ∂T  = α r  ∂x ∂r  ∂r  ∂ω ∂  ∂ω  ur = D r  ∂x ∂r  ∂r  ur (5.82) (5.83) where D is mass diffusivity. Equations (5.82) and (5.83) are subjected to the following boundary conditions: T = T0 , x = 0 (5.84) ω = ω0 , x = 0 (5.85) Figure 5.7 Sublimation in an adiabatic tube. Chapter 5 Internal Forced Convective Heat and Mass Transfer 459 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ∂T ∂ω = = 0, r = 0 ∂r ∂r ∂T ∂ω −k = ρ Dhsv , r = ro ∂r ∂r (5.86) (5.87) Equation (5.87) implies that the latent heat of sublimation is supplied as the gas flows inside the tube. Another boundary condition at the tube wall is obtained by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature (Kurosaki, 1973). According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship: ω = aT + b , r = ro (5.88) where a and b are constants. The following non-dimensional variables are then introduced: 2u r r x α , Le = , Re = m o , η= , ξ= ro r0 Pe D ν (5.89) ω −ωf T − Tf 2u r Pe = m 0 , θ = , ϕ= T0 − T f α ω0 − ω f where Tf and ωf are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations (5.82) – (5.88) then become ∂θ ∂  ∂θ  = (5.90) η (1 − η 2 ) η  ∂ξ ∂η  ∂η  η (1 − η 2 ) ∂ϕ 1 ∂  ∂ϕ  = η  ∂ξ Le ∂η  ∂η  (5.91) (5.92) (5.93) (5.94) (5.95) θ = ϕ = 1, ξ = 0 ∂θ ∂ϕ = = 0, η = 0 ∂η ∂η ∂θ 1 ∂ϕ − = , η =1 ∂η Le ∂η ϕ =  ahsv c p  θ , η = 1   The heat and mass transfer eqs. (5.90) and (5.91) are independent, but their boundary conditions are coupled by eqs. (5.94) and (5.95). The solution of eqs. (5.90) and (5.91) can be obtained via separation of variables. It is assumed that the solution of θ can be expressed as a product of the function of η and a function of ξ, i.e., θ = Θ(η )Γ(ξ ) (5.96) Substituting eq. (5.96) into eq. (5.90), the energy equation becomes d  dΘ    Γ′ dη  dη  = = −β 2 Γ η (1 − η 2 )Θ (5.97) 460 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press where β is the eigenvalue for the energy equation. Equation (5.97) can be rewritten as two ordinary differential equations: (5.98) Γ′ + β 2 Γ = 0 d  dΘ  2 2   + β η (1 − η )Θ = 0 dη  dη  (5.99) (5.100) (5.101) The solution of eq. (5.98) is Γ = C1e − β 2 ξ The boundary condition of eq. (5.99) at η = 0 is Θ′(0) = 0 2 The dimensionless temperature is then θ = C1Θ(η )e − β ξ (5.102) Similarly, the dimensionless mass fraction is ϕ = C 2 Φ (η )e −γ ξ (5.103) where γ is the eigenvalue for the conservation of species equation, and Φ (η ) satisfies 2 d  dΦ  2 2   + Leγ η (1 − η )Φ = 0 dη  dη  (5.104) and the boundary condition of eq. (5.104) at η = 0 is (5.105) Substituting eqs. (5.102) – (5.103) into eqs. (5.94) – (5.95), one obtains β =γ (5.106)  ah −  sv c p  Θ(1) Θ′(1) = Le   Φ (1) Φ ′(1)  Φ ′(0) = 0 (5.107) To solve eqs. (5.99) and (5.104) using the Runge-Kutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (5.101) and (5.105), respectively. Since both eqs. (5.99) and (5.104) are homogeneous, one can assume that the other boundary conditions are Θ(0) = Φ(0) = 1 and the solve eqs. (5.99) and (5.104) numerically. It is necessary to point out that the eigenvalue, β, is still unknown at this point and must be obtained by eq. (5.107). There will be a series of β which satisfy eq. (5.107), and for each value of βn there is one set of corresponding Θn and Φn functions (n = 1, 2,3,) . If we use any one of the eigenvalues, βn, and corresponding eigenfunctions, Θn and Φn, in eqs. (5.102) and (5.103), the solutions of eq. (5.90) and (5.91) become θ = C1Θn (η )e − β ξ (5.108) n 2 ϕ = C 2 Φ n (η )e − β ξ 2 n (5.109) Chapter 5 Internal Forced Convective Heat and Mass Transfer 461 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press which satisfy all boundary conditions except those at ξ = 0 . In order to satisfy boundary conditions at ξ = 0 , one can assume that the final solutions of eqs. (5.90) and (5.91) are θ =  Gn Θn (η )e − β n =1 ∞ ∞ n 2 ξ (5.110) (5.111) ϕ =  H n Φ n (η )e − β ξ 2 n n =1 where Gn and H n can be obtained by substituting eqs. (5.110) and (5.111) into eq. (5.92), i.e., 1 =  Gn Θn (η ) n =1 ∞ ∞ (5.112) (5.113) 1 =  H n Φ n (η ) n =1 Due to the orthogonal nature of the eigenfunctions Θn and Φ n , expressions of Gn and H n can be obtained by  Θ (1)  1 )Θn (η )dη +  n   η (1 − η 2 )Φ n (η )dη  Φ n (1)  0 Gn = 2  1  Θ (1)  2  2  η (1 − η 2 ) Θn (η ) + ( Ahsv / c p )  n  Φ n (η )  dη 0 Φ n (1)       Ahsv Θn (1) Hn = Gn c p Φ n (1)  η (1 − η 0 1 2 (5.114) (5.115) The Nusselt number due to convection and the Sherwood number due to diffusion are −k Nu = ∂T ∂r r = ro Tm − Tw 2ro 2 =− θm − θ w k  G e β ξ Θ′ (1) n − 2 n ∞ n (5.116) n =1 −D Sh = (5.117) ωm − ωw n =1 where Tm and ωm are mean temperature and mean mass fraction in the tube. Figure 5.8 shows heat and mass transfer performance during sublimation inside an adiabatic tube. For all cases, both Nusselt and Sherwood numbers become constant when ξ is greater than a certain number, thus indicating that heat and mass transfer in the tube have become fully developed. The length of the entrance flow increases with an increasing Lewis number. While the fully developed Nusselt number increases with an increasing Lewis number, the Sherwood number decreases with an increasing Lewis number, because a larger Lewis number indicates larger thermal diffusivity or low mass diffusivity. The effect of (ahsv / c p ) on the Nusselt and Sherwood numbers is relatively n ∂ω ∂r r = ro 2ro 2 =− ϕm − ϕw D H ∞ e − βn ξ Φ ′ (1) n 2 462 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ahsv cp ahsv cp Figure 5.8 Nusselt and Sherwood numbers for sublimation inside an adiabatic tube (Zhang and Chen, 1990). insignificant: both the Nusselt and Sherwood numbers increase with increasing (ahsv / c p ) for Le < 1, but increasing (ahsv / c p ) for Le > 1 results in decreasing Nusselt and Sherwood numbers. 5.5.2 Sublimation inside a Tube Subjected to External Heating When the inner wall of a tube with a sublimable-material-coated outer wall is heated by a uniform heat flux, q′′ (see Fig. 5.9), the latent heat will be supplied by part of the heat flux at the wall. The remaining part of the heat flux will be used to heat the gas flowing through the tube. The problem can be described by eqs. (5.81) – (5.88), except that the boundary condition at the inner wall of the tube is replaced by ∂ω ∂T ρ hsv D +k = q′′ at r = ro (5.118) ∂r ∂r where the thermal resistance of the tube wall is neglected because the tube wall and the coated layer are very thin. ro Figure 5.9 Sublimation in a tube heated by a uniform heat flux. Chapter 5 Internal Forced Convective Heat and Mass Transfer 463 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The governing equations for sublimation inside a tube heated by a uniform heat flux can be non-dimensionalized by using the dimensionless variables defined in eq. (5.89), except the following: h (ω − ωsat ,0 ) k (T − T0 ) θ= , ϕ = sv (5.119) q′′ro c p q′′ro where ωsat ,0 is the saturation mass fraction corresponding to the inlet temperature T0. The resulting dimensionless governing equations and boundary conditions are ∂θ ∂  ∂θ  (5.120) η (1 − η 2 ) = η  ∂ξ ∂η  ∂η  η (1 − η 2 ) ∂ϕ 1 ∂  ∂ϕ  = η  ∂ξ Le ∂η  ∂η  (5.121) (5.122) (5.123) (5.124) (5.125) (5.126) θ = 0, ξ = 0 ϕ = ϕ0 , ξ = 0 ∂θ ∂ϕ = = 0, η = 0 ∂η ∂η ∂θ 1 ∂ϕ + = 1, η = 1 ∂η Le ∂η  ahsv  θ , η =1 c    p where ϕ0 = khsv (ω − ωsat ,0 ) / (c p q′′ro ) in eq. (5.123). ϕ = The sublimation problem under consideration is not homogeneous, because eq. (5.125) is a nonhomogeneous boundary condition. The solution of the problem is consistent with its particular (fully developed) solution as well as the solution of the corresponding homogeneous problem (Zhang and Chen, 1992): θ (ξ ,η ) = θ1 (ξ ,η ) + θ 2 (ξ ,η ) (5.127) ϕ (ξ ,η ) = ϕ1 (ξ ,η ) + ϕ 2 (ξ ,η ) (5.128) While the fully developed solutions of temperature and mass fraction, θ1 (ξ ,η ) and ϕ1 (ξ ,η ) , respectively, must satisfy eqs. (5.120) – (5.121) and (5.124) – (5.126), the corresponding homogeneous solutions of the temperature and mass fraction, θ 2 (ξ ,η ) and ϕ2 (ξ ,η ) , must satisfy eqs. (5.120), (5.121), (5.124), and (5.126), as well as the following conditions: θ 2 = −θ1 (ξ ,η ) , ξ = 0 (5.129) ϕ2 = ϕ0 − ϕ1 (ξ ,η ) , ξ = 0 (5.130) ∂θ 2 1 ∂ϕ 2 + = 0, η = 1 (5.131) ∂η Le ∂η The fully developed profiles of the temperature and mass fraction are 464 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press θ1 = 1  1 4ξ + η 2 1 − η 2  + ϕ0 1 + ahsv / c p  4  11Le ahsv / c p − 18ahsv / c p − 7  +  24(1 + ahsv / c p )   ahsv / c p  1 4ξ + Le η 2 1 − η 2  + ϕ0 ϕ1 = 1 + ahsv / c p  4  (5.132) 7 Le ahsv / c p + 18 Le − 11  −  24(1 + ahsv / c p )   (5.133) The solution of the corresponding homogeneous problem can be obtained by separation of variables: θ 2 =  Gn Θn (η )e − β n =1 ∞ n 2 ξ (5.134) (5.135) ϕ2 =  H n Φ n (η )e − β ξ 2 n ∞ n =1 where Gn =  η (1 − η 0 1  Θ (1)  1 )θ 2 (0,η )Θn (η )dη +  n   η (1 − η 2 )ϕ 2 (0,η )Φ n (η )dη  Φ n (1)  0 2   1  Θn (1)  2  2 2 0 η (1 − η ) Θn (η ) + ( ahsv / c p )  Φ n (1)  Φ n (η )  dη       ahsv Θn (1) Hn = Gn c p Φ n (1) 2 (5.136) (5.137) and β n is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is Nu = = 2q′′r0 2 = k (Tw − Tm ) θ w − θ m 2(1 + Ahsv / c p ) 11  ahsv + 1 + 24  cp  ∞ − β 2ξ   Gn e n  n =1    4 Θn (1) + 2 Θ′ (1)  βn n   (5.138) where θ w and θ m are dimensionless wall and mean temperatures, respectively. The Nusselt number based on the convective heat transfer coefficient is 2h r 2ro  ∂T  2  ∂θ  = Nu * = x o =     k Tw − Tm  ∂r r = r θ w − θ m  ∂η η =1 o = 2 + 2(1 + ahsv / c p ) Gn e − βn ξ Θ′ (1) n 2 ∞ (5.139) n =1 11  ahsv + 1 + 24  cp  ∞ − β 2ξ   Gn e n  n =1    4 Θn (1) + 2 Θ′ (1)  n βn   Chapter 5 Internal Forced Convective Heat and Mass Transfer 465 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The Sherwood number is Sh = 2hm , x r0 D = 2r0 ∂ω ωw − ωm ∂r = r = ro 2 ∂ϕ ϕw − ϕm ∂η η =1 2Le = 2 ahsv ah ∞ + 2(1 + sv ) H n e − βn ξ Φ ′ (1) n cp c p n =1 11 ahsv  ahsv Le + 1 +  24 cp cp  ∞  4 − β 2ξ    Gn e n  Φ n (1) + 2 Φ ′ (1)  n  n =1 β n Le    (5.140) When the heat and mass transfer are fully developed, eqs. (5.138) – (5.140) reduce to  ah Nu = 1 + sv  cp   48   11  (5.141) 48 (5.142) 11 48 Sh = (5.143) 11 The variations of the local Nusselt number based on total heat flux along the dimensionless location ξ are shown in Fig. 5.10. It is evident from Fig. 5.10(a) that Nu increases significantly with increasing (ahsv / c p ) . The Lewis number has very little effect on Nux when (ahsv / c p ) = 0.1, but its effects become obvious in the region near the entrance when (ahsv / c p ) = 1.0 and gradually diminishes in the region near the exit. ϕ0 has almost no influence on Nu in almost the entire region when (ahsv / c p ) = 1.0, as seen in Fig. 5.10(b). When (ahsv / c p ) = 0.1, Nux increases slightly when ξ is small. The variation of the local Nusselt number based on convective heat flux, Nu*, is shown in Fig. 5.11(a). Only a single curve is obtained, which implies that Nu* remains unchanged when the mass transfer parameters are varied. The value of Nu* is exactly the same as for the process without sublimation. Figure 5.11(b) Nu* = (a) ϕ0 = 0 Figure 5.10 Nusselt number based on total heat flux. (b) Le=3.5 466 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press (a) Nusselt number based on convection (b) Sherwood number Figure 5.11 Nusselt number based on convective heat flux and Sherwood number. shows the Sherwood number for various parameters. It is evident that (ahsv / c p ) and ϕ0 have no effect on Shx, and Le has an insignificant effect on Shx in the entry region. Example 5.5 Air flows through a circular tube that has a radius of ro and is heated by external convection. The external convective heat transfer coefficient and fluid temperature are he and Te, respectively. The inner surface of the tube is coated by a layer of sublimable material. The fluid, with a mass fraction of sublimable substance ω0 and a temperature T0, enters the tube with a velocity U. For the sake of simplicity, the flow inside the tube is assumed to be slug flow (uniform velocity). The heat and mass transfer inside the tube are assumed to be developing. Find the Nusselt number based on total heat transfer and convective heat transfer, as well as the Sherwood number. The thermal diffusivity and mass diffusivity are assumed to be the same, i.e., Le = 1. Solution: The physical model of the problem is shown in Fig. 5.12. The conservations of energy and species equations are ∂T ∂  ∂T  = α r  ∂x ∂r  ∂r  ∂ω ∂  ∂ω  Ur = D r  ∂x ∂r  ∂r  Ur (5.144) (5.145) (5.146) (5.147) (5.148) (5.149) with the following boundary conditions: T = T0 , x = 0 = 0, r = 0 ∂r ∂T ∂ω k + ρ Dhsv = he (Te − T ) , r = ro ∂r ∂r ∂r ω = ω0 , x = 0 ∂T ∂ω = where D is mass diffusivity. Chapter 5 Internal Forced Convective Heat and Mass Transfer 467 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 5.12 Sublimation in a tube heated by external convection. Introducing the following non-dimensional variables, η= (5.150) ωe − ω ϕ= , ωe = aTe + b ωe − ω0 where ωe is the saturation mass fraction corresponding to Te, the governing equations become η ∂θ ∂  ∂θ  (5.151) = η  2 ∂ξ ∂η  ∂η  T −T θ= e , Te − T0 2Uro hr r x , ξ= , Pe = , Bi = e o , α ro ro Pe k η ∂ϕ ∂  ∂ϕ  = η  2 ∂ξ ∂η  ∂η  θ = ϕ = 1, ξ = 0 ∂θ ∂ϕ = = 0, η = 0 ∂η ∂η ahsv ∂θ ∂ϕ + = − Biθ w , η = 1 c p ∂η ∂η (5.152) (5.153) (5.154) (5.155) ϕw = θ w , η = 1 (5.156) Equations (5.151) and (5.152) can be solved using separation of variables, and the resulting temperature and mass fraction distributions are (Zhang, 2002) ∞ 2 J1 ( β n ) J 0 ( β nη ) e−β ξ (5.157) θ =ϕ =  2 2 n =1 β n  J 0 ( β n ) + J1 ( β n )    2 n where J0 and J1 are the zeroth and first order Bessel functions. The Nusselt number based on the total heat supplied by the external fluid is 2r h (T − Tw ) 2 Biθ w Nu = o e e = (5.158) k (Tw − Tm ) θ m − θ w 468 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The Nusselt number based on the heat transferred to the fluid inside the tube is 2ro 2  ∂θ   ∂T  Nu * = − = (5.159)     (Tw − Tm )  ∂r r =r θ m − θ w  ∂η η =1 o The Sherwood number is Sh = − 2ro  ∂ω  2 = ωw − ωm  ∂r r =r ϕm − ϕ w   o  ∂ϕ     ∂η η =1 (5.160) The Nusselt number based on the heat transferred to the fluid inside the tube and the Sherwood number are identical, since θ = ϕ as indicated by eq. (5.157). Fig. 5.13 shows the variation of the local Nusselt number based on the total heat supplied from the fluid outside the tube. The dimensionless lengths of the entrance slightly increase with decreasing Biot number and are approximately equal to 0.1. Nusselt numbers become constants after ξ becomes greater than 0.1. The fully developed ahsv = 1.0 cp Figure 5.13 Effect of Biot number on Nu (ahsv / c p = 1) . ahsv = 1.0 cp Figure 5.14 Effect of Biot number on Nu* or Sh (ahsv / c p = 1) . Chapter 5 Internal Forced Convective Heat and Mass Transfer 469 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Nusselt number increases with a decreasing Biot number. Fig. 5.14 shows the variation of the local Nusselt number based on heat transferred to the fluid inside the tube, or the local Sherwood number. The variations of Nu* and Sh are similar to that of Nu in Fig. 5.13. 5.6 Developing Flow, Thermal and Concentration Effects All of the forced convective heat and mass transfer problems considered so far assumed that the flow is fully developed, which, as previously shown, occurs at x/D approximately equal to 0.05Re for a circular tube. For forced convective heat and mass transfer with constant properties, the hydrodynamic entrance length is independent of Pr or Sc. It was also shown that when assuming fully developed flow, the point at which the temperature profile becomes fully developed for forced convection in tubes is linearly proportional to RePr. Analysis of these criteria for a fully developed flow and temperature profile shows that when Pr  1, as is the case with fluids with high viscosities such as oils, the temperature profile takes a longer distance to completely develop. In these circumstances (Pr  1), it makes sense to assume fully developed velocity since the thermal entrance is much longer than the hydrodynamic entrance. Obviously, from the definition of Prandtl number and the above criteria, one expects that when Pr ≈ 1 for fluids such as gases, the temperature and velocity develop at the same rate. When Pr  1, as in the case of liquid metals, the temperature profile will develop much faster than the velocity profile, and therefore a uniform velocity assumption (slug flow) is appropriate. Similar analysis and conclusions can be made with the Schmidt number, Sc, relative to mass transfer problems concerning the entrance effects due to mass diffusion. If one needs to get detailed information concerning the hydrodynamic, thermal or concentration entrance effects, the conservation equations should be solved without a fully developed velocity, concentration, or temperature profile. Consider laminar forced convective heat and mass transfer in a circular tube for the case of steady two-dimensional constant properties. The inlet velocity, temperature and concentration are uniform at the entrance with the possibility of mass transfer between the wall and fluid, as shown in Fig. 5.15. r x ro L Figure 5.15 Geometry and coordinate system flow for forced convective heat and mass transfer in a circular tube. ωin Tin Uin 470 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The conservation equations with the above assumptions, as well as neglecting the viscous dissipation and assuming an incompressible Newtonian fluid, are continuity x - momentum u r - momentum u energy u species u ∂u 1 ∂ + ( rv ) = 0 ∂x r ∂r (5.161) (5.162) (5.163) (5.164) (5.165)  1 ∂  ∂u  ∂ 2 u  ∂u ∂u 1 ∂p +v =− +ν  r  + 2  ∂x ∂r ρ ∂x  r ∂r  ∂r  ∂x   1 ∂  ∂v  ∂ 2 v  ∂v ∂v 1 ∂p +v =− +ν  r  + 2  ∂x ∂r ρ ∂r  r ∂r  ∂r  ∂x   1 ∂  ∂T ∂T ∂T +v =α  r ∂x ∂r  r ∂r  ∂r 2  ∂ T + 2   ∂x   1 ∂  ∂ω1  ∂ 2ω1  ∂ω1 ∂ω +v 1 = D r + 2 ∂x ∂r  r ∂r  ∂r  ∂x  no slip boundary condition Typical boundary conditions are: Axial velocity at wall u ( x , ro ) = 0 vw = 0 impermeable wall vw > 0 injection and vw < 0 suction   Radial velocity at wall v ( x , ro ) = mw′′ = mass flux due to diffusion    ∂ω = ρ ω1, w vw − D12 1   ∂r r = ro       Tw = const.  ∂T = const. or qw′′ = −k  ∂r r = ro  Tw = f ( x ) or   ′′ qw = g ( x ) T = Tin  ω1 = ω1, in u = uin  T = ? ω1 = ? u = ?  P = ? Thermal condition on wall at ( r = ro ) Inlet condition at x = 0 Outlet condition at x = L Clearly there are five partial differential equations and five unknowns (u, v, P, T, ω1). All equations are of elliptic nature (Chapter 2) and one can neglect the  ∂ 2 u ∂ 2 v ∂ 2 T ∂ 2ω  axial diffusion terms,  2 , 2 , 2 , 21  , under some circumstances in order  ∂x ∂x ∂x ∂x  to make the conservation equations of parabolic nature. These axial diffusion terms can also be neglected under boundary layer assumptions. Chapter 5 Internal Forced Convective Heat and Mass Transfer 471 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Making boundary layer assumptions makes the result invalid very close to the tube entrance where the Reynolds number is very small. Shah and London (1978) showed that the momentum boundary layer assumption will lead to error if Re < 400 and LH / D < 0.005Re. In these circumstances, the full Navier Stokes equation should be solved. It was also shown in Section 5.2 that there are circumstances other than boundary layer assumptions where axial diffusion terms, such as the axial conduction term, can be neglected. However, as we showed in the case of the energy equation, one cannot neglect axial conduction for a very low Prandtl number despite the thermal boundary layer assumption. 40 35 30 25 Nux 20 15 10 5 0 0.0001 0.001 x/(D Pe) 0.01 0.1 3.66 ReD=100 Pr=0.7 Pr=2 Pr=5 (a) Local Nusselt number 100 90 80 70 60 Num 50 40 30 20 10 0 0.0001 0.0010 0.0100 x/(D Pe) 0.1000 3.66 ReD=100 Pr=0.7 Pr=2 Pr=5 (b) Average Nusselt number Figure 5.16 Local and average Nusselt numbers for the entrance region of a circular tube with constant wall temperature. 472 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table 5.7 Local and average Nusselt number for the entrance region of a circular tube with constant wall temperature Nux Num Pr = 0.7 Pr = 2 Pr = 5 Pr = 0.7 Pr = 2 Pr = 5 x+ 0.001 17.0 12.5 10.6 60.7 34.9 24.9 0.002 11.6 8.86 8.04 37.3 22.6 17.0 0.004 8.29 6.64 6.30 23.3 15.1 12.0 0.008 6.29 5.23 5.09 14.9 10.4 8.80 0.01 5.81 4.89 4.80 13.1 9.36 8.03 0.02 4.69 4.12 4.12 8.9 6.89 6.21 0.04 4.01 3.75 3.76 6.46 5.39 5.06 0.06 3.79 3.67 3.68 5.56 4.83 4.61 0.08 3.71 3.66 3.66 5.09 4.54 4.37 0.1 3.68 3.66 3.66 4.81 4.36 4.23 0.12 3.66 3.66 3.66 4.62 4.24 4.14 3.66 3.66 3.66 3.66 3.66 3.66 ∞ 60 50 40 Nux 30 20 10 0 0.0001 0.001 x/(D Pe) 0.01 0.1 ReD=100 Pr=0.7 Pr=2 Pr=5 4.36 (a) Local Nusselt number 140 120 100 80 Num 60 40 20 0 0.0001 0.001 x/(D Pe) 0.01 0.1 4.36 ReD=100 Pr=0.7 Pr=2 Pr=5 (b) Average Nusselt number Figure 5.17 Local and average Nusselt numbers for the entrance region of a circular tube with constant heat flux. Chapter 5 Internal Forced Convective Heat and Mass Transfer 473 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table 5.8 Local and mean Nusselt number for the entrance region of a circular tube with constant wall heat flux Nux Nu m x+ 0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.06 0.08 0.1 0.12 0.16 ∞ Pr = 0.7 23.0 15.6 10.9 7.92 7.24 5.69 4.81 4.53 4.43 4.39 4.37 4.36 4.36 Pr = 2 17.8 12.5 9.16 7.01 6.49 5.30 4.64 4.46 4.39 4.37 4.36 4.36 4.36 Pr = 5 15.0 11.1 8.44 6.65 6.22 5.21 4.62 4.45 4.39 4.37 4.36 4.36 4.36 Pr = 0.7 61.0 39.8 26.3 17.7 15.7 11.0 8.09 6.94 6.33 5.95 5.69 5.36 4.36 Pr = 2 43.4 29.0 19.8 13.8 12.4 9.11 7.00 6.18 5.74 5.47 5.29 5.23 4.36 Pr = 5 34.2 23.5 16.5 11.9 10.8 8.23 6.54 5.87 5.51 5.29 5.13 4.94 4.36 Table 5.9 Local Nusselt number for the entrance region of a group of circular Tube Annulus with Constant Wall Heat Flux (Heaton et al., 1964; Reproduced with permission from Elsevier) Parallel planes Circular-tube annulus K= 0.50 Pr Nu11 θ1* Nuii Nuoo θ i* θ i* 24.2 0.048 -24.2 -0.0322 11.7 0.117 -11.8 -0.0786 8.8 0.176 9.43 8.9 0.252 0.118 0.01 5.77 0.378 6.4 5.88 0.525 0.231 5.53 0.376 6.22 5.6 0.532 0.238 5.39 0.346 6.18 5.04 0.528 0.216 18.5 0.037 19.22 18.3 0.0513 0.0243 9.62 0.096 10.47 9.45 0.139 0.063 7.68 0.154 8.52 7.5 0.228 0.0998 0.7 5.55 0.327 6.35 5.27 0.498 0.207 5.4 0.345 6.19 5.06 0.527 0.215 5.39 0.346 6.18 5.04 0.528 0.216 15.6 0.0311 16.86 15.14 0.045 0.0201 9.2 0.092 10.2 8.75 0.136 0.0583 7.49 0.149 8.43 7.09 0.224 0.0943 10 5.55 0.327 6.35 5.2 0.498 0.204 5.4 0.345 6.19 5.05 0.527 0.215 5.39 0.346 6.18 5.04 0.528 0.216 In general, elliptic equations are more complex to solve analytically or numerically than parabolic equations. Furthermore, to solve the equations as elliptic you need pertinent information at the outlet as well, which in some cases is unknown. The momentum equation is nonlinear while the energy equation is linear under the constant property assumption. In most cases, the momentum, energy, and species equations are uncoupled, except under the following circumstances which make the equations coupled. 1. Variable properties, such as density variation as a function of temperature in natural convection problems. 474 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 2. Coupled governing equations and/or boundary conditions in phase change problems, such as absorption or dissolution problems. 3. Existence of a source term in one conservation equation that is a function of the dependent variable in another conservation equation. Langhaar (1942) and Hornbeck (1965) obtained approximate solutions for the momentum equation for circular tubes by solving the linearized momentum equation. Hornbeck (1965) solved the momentum equation numerically by making boundary layer assumptions (parabolic form). Several investigators solved the energy equation either using Langhaar’s approximate velocity profile, or solving the momentum and energy equations numerically for both constant wall temperature and constant wall heat flux in circular tubes. Heat transfer in hydrodynamic and thermal entrance region has been solved numerically based on full elliptic governing equations (Bahrami, 2009). Variations of local and average Nusselt numbers for different Prandtl numbers under constant wall temperature and constant heat flux using full elliptic governing equations are shown in Figs. 5. 16 and 5.17, respectively. The local and average Nusselt numbers for different Prandtl numbers and boundary conditions are also presented in Tables 5.7 and 5.8. Heaton et al. (1964) approximated the result for linearized momentum and energy equations using the energy equation for constant wall heat flux for a group of circular tube annulus for several Prandtl numbers. Table 5.9 summarizes the results for parallel plates and circular annulus. 5.7 Full Numerical Solutions The algorithms to solve the convection-diffusion equation and flow field that were addressed in Section 4.8 are still applicable to the internal convection problems. However, special attention must be paid to proper treatment of the boundary conditions. For an internal flow and heat transfer problem, there are several different types of boundary conditions (see Fig. 5.18): (1) inflow condition, (2) axisymmetric condition, (3) impermeable solid surface, and (4) outflow condition. The inflow conditions are normally specified by a given distribution of the inlet velocity and the general variable, φ , at the inlet ( x = 0 ). The axisymmetric condition can be implemented by setting the gradient of the velocity component in the y-direction, ∂u / ∂y , and the gradient of the general variable, ∂φ / ∂y , equal to zero along the axisymmetric line ( y = 0 ). In addition, the velocity component in the y-direction, v, at the axisymmetric boundary should also be zero. For the impermeable solid surface, all three kinds of boundary conditions for the general variables are possible: specified φ (the first kind), specified gradient of φ in the direction perpendicular to the impermeable surface, ∂φ / ∂y , (the second kind), or specified relation between φ and ∂φ / ∂y (the third kind). The velocity components in both the tangential and normal direction of the impermeable Chapter 5 Internal Forced Convective Heat and Mass Transfer 475 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Impermeable solid (y = H) Inflow (x = 0) Outflow (x = L) x, u y, v 0 axisymmetric (y = 0) Figure 5.18 Typical boundary conditions for internal convection. surface are zero, i.e., u = v = 0 , due to no-slip and impermeable conditions. No special treatment for the boundary conditions at inflow, axisymmetric, and impermeable surfaces is required. On the contrary, the outflow boundary condition requires special treatment as outlined below. Since the momentum equation in the internal flow direction (the x-direction in Fig. 5.18) and the conservation equation for the general variable (see eq. (4.200)) are elliptic, the second order derivatives with respect to x appeared in the partial difference equation. Mathematically, boundary conditions at both the inflow and outflow boundary need to be specified. While the inflow boundary conditions are always known, the outflow boundary conditions (at y = L in Fig. 5.16) are usually unknown. Unless the experimentally measured distribution at the outflow boundary is available, we cannot directly use the algorithms introduced in the previous chapter to solve the internal convection problem. A coordinate (such as x in Fig. 5.18, or time for a transient problem) can be either two-way or one-way depending on the nature of the problem (Patankar, 1980, 1991). If the condition at a given location, x, is influenced by changes of conditions at either side of the given location, the coordinate is said to be twoway. On the other hand, if the conditions at the given location, x, are influenced by changes of conditions from only one side, the coordinate is said to be oneway. Mathematically, if the second order derivation with respect to x appeared in the partial differential equation, the coordinate in the x-direction is two-way. For example, the spatial coordinate in the one-dimensional steady-state heat conduction in a fin (see Section 3.2.1) is two-way because the temperature at any point is influenced by the temperatures at either side, and the second order derivative appears in the energy equation that describes heat conduction in the fin. On the other hand, time is a one way coordinate because the conditions at a given time are only influenced by what happened before that time, and what happen afterward will not affect the conditions at the current time. While the space coordinate in heat and mass transfer is normally always twoway, it can become one-way for some special cases. For internal flow, the change of condition at a given point will have a more profound effect on the conditions of the points downstream, while its influence on the conditions of the points upstream will be relatively weak. For the case that convection overpowers 476 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press N Outflow boundary W P E S Figure 5.19 Control volume for continuity equation at boundary diffusion, one can assume that the condition changes at one point can only propagate downstream and the coordinate becomes one-way. The most commonly used approach to handle the outflow boundary condition is to assume the coordinate at the outflow boundary is locally one-way. Thus, the value of the general variable, φ , at the inner point, P, shown in Fig. 5.19 is not affected by the value of φ at the outflow boundary grid point, E, which is located downstream. Computationally, one can set aE = 0 in the discretized equation for the point P. This simple treatment effectively avoids the problem associated with an unknown value of φ at the outflow boundary. It follows from Section 4.8 that this treatment will be more accurate for cases with a high Peclet number. Another situation that can be handled by a similar approach is the case that the unknown variable is fully developed at the outflow boundary, in which case the outflow boundary condition becomes ∂φ / ∂x = 0 . The implication of the fully developed condition at the outflow is exactly the same as the local one-way behavior discussed above, and can be treated using the same approach. However, it should be pointed out that for the case of fully developed heat transfer, one must not use ∂T / ∂x = 0 , and the correct condition for fully developed heat transfer should be ∂[(T − Tm ) / (Tw − Tm )] / ∂x = 0 (see Section 5.3). Many practical problems are not similar to the cases corresponding to simple internal channel flow presented in Sections 5.1-5.6. In many practical applications for internal flow, one needs to deal with conjugate effects, compressibility, multi-domain, and porous media, as well as non-conventional boundary conditions. To show the power of numerical simulation, the modeling of a conventional wicked heat pipe with variable heat flux (see Fig. 5.20) will be presented here, compared to the non-conventional internal flow case presented in previous sections. The problem is modeled as a two-dimensional conjugate, with compressible flow including the effect of porous media and coupling between various regions. Chapter 5 Internal Forced Convective Heat and Mass Transfer 477 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Heater Heat pipe screen wick Heat pipe wall r z Ri R0 Rv v w Vapor Multiple evaporator Figure 5.20 Heat pipe configuration. Adiabatic section Condenser section Since the heat pipe in Fig. 5.20 is closed at both ends, it is required that the vapor which flows out of the evaporator segment enters into the condenser section. The conservation of mass, momentum, and energy equations for the compressible vapor flow region including viscous dissipation are 1∂ ∂ ( ρ v rvv ) + ( ρ v wv ) = 0 ∂z r ∂r ∂wv ∂wv  ∂pv  ρ v  wv + vv =− ∂z ∂r  ∂z   2  4 ∂ wv 1 ∂  ∂wv  1 ∂  ∂vv  2 ∂  1 ∂  + μv  +  r ∂r  + r ∂r  r ∂z  − 3 ∂z  r ∂r (rvv )   2 r ∂r        3 ∂z ∂vv ∂v  + vv v  ∂z ∂r   2 ∂ v ∂p 4 ∂  ∂vv  4 vv 1 ∂ 2 wv  = − v + μv  2v + +  r − ∂r 3r ∂r  ∂r  3 r 2 3 ∂z ∂r   ∂z (5.166) (5.167) ρ v  wv  (5.168)   ∂  ∂Tv k = v  r r  ∂r  ∂r ρ v c pv  wv  ∂Tv ∂T  + vv v  ∂z ∂r  ∂ 2T  ∂p ∂p  + r 2v  + vv v + wv v + μv Φ  ∂z  ∂r ∂z  (5.169) 478 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press where the viscous dissipation is  ∂v 2  v 2  ∂wv 2 1  ∂vv ∂wv 2 1  2 Φ = 2  v  +  v  +   + 2  ∂z + ∂r  − 3 (∇ ⋅ Vv )     ∂r   r   ∂z     (5.170) and ∇ ⋅ Vv is given by ∂wv 1∂ (rvv ) + (5.171) ∂z r ∂r The ideal gas law ( p = ρ v RgTv ) is employed to account for the ∇ ⋅ Vv = compressibility of the vapor. The use of liquid capillary action is a unique feature of the heat pipe. From a fundamental point of view, the liquid capillary flow in heat pipes with screen wicks should be modeled as a flow through a porous media. It is assumed that the wicking material is isotropic and of constant thickness. In addition, the wick is saturated with liquid, and the vapor condenses and the liquid evaporates at the liquid-vapor interface. The averaging technique has been applied by many investigators to obtain the general equation which describes the conservation of momentum in a porous structure. Since the development of Darcy’s semi-empirical relation, which characterizes the fluid motion under certain conditions, many researchers have tried to develop and extend Darcy’s law in order to see the effect of the inertia terms. In this respect, those who have tried to model the flow with the NavierStokes equations were the most successful (Chapter 2). The general equations of continuity, momentum and energy for steady state laminar incompressible liquid flow in porous media in terms of the volume-averaged velocities as presented in Chapter 2 are: ∂ 1∂ ( ρ  w ) + ( ρ rv ) = 0 ∂z r ∂r ∂w ∂w  1 1 ∂p ν  w ν   1 ∂  ∂w + v w  = − ρ ∂z − K + ε  r ∂r  r ∂z 2 ∂z ∂r  ε    (5.172) 2  ∂ wv  +  2  ∂z  (5.173) (5.174) (5.175) ∂v ∂v  1 1 ∂p ν  v ν   1 ∂  ∂v  v ∂ 2 vv  − + −+ w  + v   = − r  2 ∂z ∂r  K ε ρ ∂r ε  r ∂r  ∂z  r 2 ∂z 2    ∂T ∂T  1 ∂  ∂T  ∂  ∂T   ρ c p  w  + v   = rkeff   +  keff   ∂z ∂r  r ∂r  ∂r  ∂z  ∂z    where ε is the volume fraction or porosity of the wick, and K is the permeability of the wick structure. The effective thermal conductivity of the wick, keff, is related to the thermal conductivity of the solid and liquid phases k [(k + ks ) − (1 − ε )(k − ks )] (5.176) keff =   [(k + ks ) + (1 − ε )(k − ks )] The steady state energy equation that describes the temperature in the heat pipe wall is Chapter 5 Internal Forced Convective Heat and Mass Transfer 479 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ∂Tw  ∂  ∂Tw  1 ∂   kw ∂z  + r ∂r  kw r ∂r  = 0 ∂z     (5.177) where kw is the local thermal conductivity of the heat pipe wall. The boundary conditions are w(0, r ) = v(0, r ) = 0 (5.178) w( Lt , r ) = v( Lt , r ) = 0 (5.179) w( z , Rv ) = 0 (5.180) w( z , Ri ) = 0 (5.181) ∂w ∂v ( z , 0) = ( z , 0) = 0 ∂r ∂r ∂T ∂T 1 v( z , Rv ) = + kv v  −k ρ v hv  eff ∂r r = Rv ∂r  v( z , Ri ) = 0 ∂T ∂T (0, r ) = ( Lt , r ) = 0 ∂z ∂z  1 Rg  pv   ln    T ( z , Rv ) =  −  T0 hv  p0     ∂T Q( z ) kw w ( z , R0 ) = ± ∂r A −1 (5.182)     (5.183) (5.184) (5.185) (5.186) (5.187) r = Rv In the boundary conditions specified by eqs. (5.178)-(5.187), the heat flux (Q(z)/A) was uniform and positive at the active evaporators, and zero in all adiabatic and transport sections as well as in the inactive evaporators. In the condenser section, the heat flux was assumed to be uniform and negative based on the total heat input through the evaporators. The temperature along the liquidvapor interface was taken as the equilibrium saturation temperature corresponding to its equilibrium pressure condition. Through the above procedure, both the effects of energy conduction in the heat pipe wall and the liquid flow in the wick are taken into account. For simplicity and generality, the problem should be solved as a single domain problem using conjugate heat transfer analysis. To achieve this, the conservation equations for mass, momentum, and energy are generalized such that the energy equations with temperature as a dependent variable in the three regions (i.e., wall, wick, and vapor) have the same source term. In addition, the continuity of temperature, heat flux, and mass flux, as well as some special boundary conditions such as thermodynamic equilibrium, should be satisfied at each of the interfaces. The energy equation can be written in terms of temperature as ρ DT k s = ∇ 2T + Dt c p cp (5.188) where s is the source term which includes viscous dissipation and pressure work. It should be noted that solving the problem in terms of enthalpy does not 480 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press preserve the condition of temperature continuity at an interface when harmonic averaging is employed, and will lead to an incorrect calculation of diffusion. For the vapor region, the energy equation is ρv c p , / c p , v results in k DT s = v ∇ 2T + Dt c p , v c p,v (5.189) Multiplying both sides of the energy equation for the liquid-wick region by ρl c p ,l DT k s = l ∇ 2T + c p , v Dt c p , v c p,v c p , w DT kw 2 s = ∇T+ c p , v Dt c p , v c p,v (5.190) Similarly, the energy equation for the heat pipe wall is ρw (5.191) From this transformation, the following can be observed: 1. The source term for the energy equation is divided by cp,v for all three regions. 2. The density is equal to the modified density, ρ*, for different regions, i.e., ρv for vapor, ρ c p,  / c p, v for liquid, and ρwcp,w/cp,v for solid. 3. The diffusion coefficients contain only one cp, namely, cp,v. The transformation makes possible an exact representation of diffusion across an interface when the harmonic average is used for the diffusion coefficient at the interfaces. The momentum equation for the vapor flow does not need any special treatment since ρ*= ρv in the vapor region. In the wick region, the momentum equation can be transformed in the same manner: v ρ *ε DV ρ* V = −∇p * + vl ρ *V − l (5.192) Dt K It should be noted that: 1. A source term was added in the momentum equation for the porosity effect in the wick region. 2. The pressure solved for is the modified pressure, p*, which is proportional to the actual pressure. The actual pressure drop between two points in the flow fluid can be calculated as Δp = c p , v / c p,l Δp * . The transformed continuity equation can be written as ∇ ⋅ ( ρV ) = 0 (5.193) The finite volume method discussed above was employed by Faghri and Buchko (1991) in the solution of the elliptic conservation eqs. (5.166)-(5.169) and (5.177) subject to the boundary conditions (5.178)-(5.187). The source terms due to viscous dissipation and pressure work in the energy equation, and the source term due to the porous matrix in the momentum equation are linearized, and the SIMPLEST algorithm was employed for the momentum equations. The solution procedure is based on a line-by-line iteration method in the axial direction and the Jacobi point-by-point procedure in the radial direction. Chapter 5 Internal Forced Convective Heat and Mass Transfer 481 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press T(ºC) 95 Experimental vapor temp. 90 85 80 75 70 65 60 0 0.2 0.4 0.6 0.8 1 Figure 5.21 Heat pipe wall and vapor temperature versus axial location (Faghri and Buchko, 1991). Experimental wall temp. Numerical wall temp. Numerical vapor temp. Evaporator 1=97 (W) Evaporator 2=0 (W) Evaporator 3=0 (W) Evaporator 4=0 (W) The energy equation is not continuous at the liquid-vapor interface due to the latent heat of evaporation and condensation. The term hv must be added as a source term in the energy equation at the liquid-vapor interface. For the first few iterations, it was assumed the heat flux at the liquid-vapor interface is equal to the outer wall heat flux. An exact energy balance given by eq. (5.183) is satisfied after these initial iterations. Numerical results based on the mathematical modeling presented above are shown by Faghri and Buchko (1991) in Fig. 5.21, which show excellent agreement with experimental data for the wall and vapor temperatures along the heat pipe. In the condenser section, the experimental and numerical values of the outer wall temperature do not coincide. This is due to the assumption of a constant heat flux along the condenser section at the outer wall, which does not precisely correspond to the experimental specifications of a cooling jacket around the condenser sections. 5.8 Forced Convection in Microchannels 5.8.1 Introduction Due to recent advances in micro fabrication and manufacturing, various devices having the order of microns, such as micro pumps, micro-heat sinks, microbiochips, micro-reactors, micro-motors, micro-valves, and micro-fuel cells, have 482 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press been developed. These micro-devices found their applications in microelectronics, microscale sensing and measurement, power systems, spacecraft thermal control, biotechnology, and microelectromechanical systems. For example, microchannel heat sinks are one of the ultimate heat removal solutions in most microscale devices. The need for efficient and effective heat transfer techniques in miniature devices has, in recent years, fostered extensive research interest in microscale heat transfer with an emphasis on microchannels with both circular and rectangular cross-sections (Karniadakis et al., 2005; Kandlikar et al., 2006). Several investigators have reported significant deviation from classical theory (e.g. with respect to the solution of the Navier-Stokes and energy equations with simplifying assumptions like no-slip boundary conditions for velocity and temperature) used in macroscale applications, while others have reported general agreement, especially in the laminar region. The common channel flow classification based on the hydrologic diameter divides the range from 1 to 100 µm as microchannels, 100 µm to 1 mm as mesochannels, 1 mm to 6 mm as miniature channels, and greater than 6 mm as conventional channels. It is convenient to differentiate the flow regimes for experimental and theoretical predictions as a function of Knudsen number (Kn). Kn is a parameter that physically indicates the relative importance of rarefaction or non-continuum effects. It is the ratio of the flux gas mean free path, λ, to the characteristic dimension of flow field, D. The following classification is commonly accepted, as described in Chapter 1: • For Kn < 10-3, the flow is a continuum flow and is accurately modeled by the Navier-Stokes and energy equations with classical no-slip boundary conditions for velocity and temperature. • For 10-3 < Kn < 10-1, the flow is a slip flow and the Navier-Stokes and energy equations remain applicable, provided a first order velocity slip and a temperature jump are taken into account at the walls. These new boundary conditions indicate the rarefaction effects at the walls. • For 10-1 < Kn < 10, the flow is a transition flow and the continuum approach of the Navier-Stokes equations is no longer valid. However, the intermolecular effects are not yet negligible and should be taken into account. • For Kn > 10, the flow is a free molecular flow and the occurrence of intermolecular collisions is negligible compared with the collisions between the gas molecules and the walls. As noted above, some special effects or conditions that are typically neglected at the macroscale should be included at the microscale. One such condition is slip flow when the fluid is rarefied or the geometry is at the microscale level. In contrast to continuum flow phenomena, the fluid no longer reaches the surface velocity or temperature. Two major characteristics of slip flow are velocity slip and temperature jump at the surface. These can be Chapter 5 Internal Forced Convective Heat and Mass Transfer 483 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press us r, y uc 0 x Figure 5.22 Configuration for the slip boundary condition in a microchannel. determined using the kinetic theory of gases. For a cylindrical microchannel, the velocity slip condition is: uS = − 2 − F ∂u λ F ∂r r =ro r0 , H (5.194) where us is the slip velocity, as shown in Fig. 5.22, λ is the molecular mean free path, and F is the tangential momentum accommodation coefficient; and the temperature jump is 2 − Ft 2γ λ ∂T TS − Tw = − (5.195) Ft γ + 1 Pr ∂r r =r where Ts is the temperature of the fluid at the wall, Tw is the wall temperature, and Ft is the thermal accommodation coefficient. To make the analysis simpler, F and Ft are usually denoted by F and assumed to be 1. The above relations clearly indicate that with an increase in the mean free path value, the slip velocity condition at the walls, as well as the temperature jump, increases. Detailed investigations were made to compare the conventional (continuum) theories with the experimental data in microscale flows and heat transfer problems. Hetsroni, et al. (2005) compared the experimental data from the literature with small Knudsen numbers (0.001 – 0.4) and Mach numbers (0.07 – 0.84) that correspond to continuum models in circular, rectangular, triangular, and trapezoidal microchannels with hydraulic diameters ranging from 1.01 μm to 4010 μm and noted, in general, a good agreement with the conventional theories. The no-slip boundary conditions are used for these models for both velocity and temperature. Hetsroni et al. (2005) concluded that the existing experimental friction factor in the literature agrees quite well with that of the conventional continuum for fully developed laminar gas flow for 0.001 ≤ Kn ≤ 0.38 . There are, however, contradictory results despite the existence of significant experimental and theoretical investigations in microchannels. For the microchannels with small Knudsen numbers which lie within the noslip flow region, the conventional solutions apply with high accuracy. The o 484 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press solutions for the circular tube fully developed flow and temperature profile are thus Nu = 3.66 for the constant wall temperature and Nu = 4.36 for the constant wall heat flux boundary conditions. The Nusselt number for a rectangular microchannel has a dependence on the channel aspect ratio. Myong et al. (2006) adopted Langmuir’s slip model to characterize the slip boundary conditions. They developed a physical approach to account for the interfacial interaction between the gas molecules and surface molecules. In this approach, the gas molecules are assumed to interact with the surface of the solid via long range attractive forces. Consequently, the gas molecules can be adsorbed into the surface and then desorbed after some time lag. They found that, for most physical applications, this model always predicts the reduction of heat transfer with increasing gas rarefaction. Several studies have also examined the transition point from laminar to turbulent flow in microchannel passages and found that the critical Reynolds number is still approximately 2300. The classical flow and thermal regions such as: fully developed flow and temperature profiles, fully developed flow profile but developing thermal profile, fully developed thermal profile but developing flow profile, and simultaneously developing flow and temperature profiles, are also applicable to the analysis of microchannels. For the microchannels with the Knudsen numbers within the slip flow condition, the flow and heat transfer is characterized by Knudsen number (Kn), Peclet number (Pe), and Brinkman number (Br). 5.8.2 Fully Developed Laminar Flow and Temperature Profile Fully Developed Velocity Distribution The velocity distribution is derived from the continuity and momentum equations along with the slip condition. The velocity profile is expressed in terms of Knudsen number. The model for the analysis of velocity distribution can be considered as the flow of a fluid in a circular tube of radius ro, as shown in Fig. 5.22. The continuity equation in cylindrical coordinates is given as: ∂ρ 1 ∂ ∂ + (5.196) ( ρ rv ) + ( ρ u ) = 0 ∂t r ∂r ∂x For steady and fully developed flow of an incompressible fluid, it becomes: ∂u =0 ∂x (5.197) Therefore, rv = constant. Since the radial velocity at the wall is zero (impermeability condition), it can be concluded that v = 0 everywhere in the flow field. The steady parabolic momentum equation in the x-direction can be written in cylindrical coordinates as: Chapter 5 Internal Forced Convective Heat and Mass Transfer 485 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ρu ∂u ∂u μ ∂  ∂u  ∂p + ρv = r  − ∂x ∂r r ∂r  ∂r  ∂x (5.198) For fully developed, steady flow of incompressible fluid, it reduces to: dp μ d  du  − + (5.199) r =0 dx r dr  dr  1 dp 4 μ dx As the velocity is independent of x, a constant C1 can be defined as: C1 = − (5.200) Therefore, the momentum equation reduces to: 4C1 + 1 d  du  r =0 r dr  dr  (5.201) The above equation is integrated twice to yield: (5.202) Since the velocity, u, is finite at the center of the tube (r = 0), we conclude C2=0. At the centerline of the tube (r = 0), the velocity is equal to uc, therefore, u(0) = uc = C3. At the inner surface of the tube wall (r = ro), the velocity is not zero due to the slip flow condition, but is equal to a finite velocity us: u(ro ) = us = uc − C1ro2 (5.203) Rearranging and solving for C1 we have: u = C 2 ln r + C 3 − C1r 2 C1 = uc − us 1 dp =− 2 4 μ dx r0 (5.204) The velocity profile is obtained as follows: (5.205) The mean velocity in the tube, um , is now evaluated. The volumetric flow rate is: u = uc − (uc − us )(r / ro ) 2 = uc [1 − (r / ro )2 ] + us (r / ro )2 π r02 um =  2π rudr 0 r0 (5.206) (5.207) therefore, um = 2[uc / 2 − (uc − us ) / 4] = (uc + us ) / 2 The centerline velocity can be written as: uc = 2um − us (5.208) Using the above relation, we get the following velocity profile in a microchannel: u = 2(um − us )(1 − r +2 ) + us (5.209) + where r =r/ro. If the slip velocity is zero, the velocity distribution reduces to the Poiseuille flow distribution for conventional channels. The velocity slip is given by the following condition [see eq. (5.194)]: us = − 2 − F du λ F dr r =ro (5.210) For most applications, F has values near unity. Therefore, us = −λ du dr r = ro (5.211) 486 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 2 1.6 1.2 u/um 0.8 Kn=0.00 =0.02 =0.04 =0.06 =0.08 =0.10 =0.12 Kn 0.4 0 0 0.2 0.4 r+ Figure 5.23 Non-dimensional, fully developed velocity profile in a microchannel as a function of Knudsen number. 0.6 0.8 1 The slip velocity is derived using eq. (5.209) for velocity profile: 4λ (um − us ) 8λ (um − us ) λ du us = − ro dr + = r + =1 ro = D (5.212) The Knudsen number, Kn = λ / D , is introduced in the above relation to obtain the following equation for the slip velocity: us 8Kn = um 1 + 8Kn (5.213) Substituting the above into eq. (5.209), the non-dimensional expression for the velocity profile is obtained: u 2(1 − r +2 ) + 8Kn = um 1 + 8Kn (5.214) The velocity distribution is shown in Fig. 5.23 for different Kn numbers. The slip velocity at the wall increases with an increasing Kn number. Fully Developed Heat Transfer Coefficient in Microchannel The flow is considered to be steady, laminar, and fully developed both hydrodynamically and thermally. The conventional continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump, as noted above. The energy equation including the viscous dissipation term for steady, fully developed flow in a pipe and neglecting axial conduction is: Chapter 5 Internal Forced Convective Heat and Mass Transfer 487 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press u ∂T α ∂  ∂T  ν  ∂u  = r +   ∂x r ∂r  ∂r  c p  ∂r  2 (5.215) The variation of temperature in the r-direction at the center is zero due to axisymmetric conditions: ∂T ∂r =0 r =0 (5.216) Also, the temperature jump condition for the fluid at the wall is written as follows: 2 − Ft 4γ Kn ∂T Ts − Tw = − (5.217) Ft 1 + γ Pr ∂r r = r Both conventional boundary conditions of constant heat flux and constant temperature at the wall are solved analytically by Aydin and Avci (2006) for microchannel with circular cross-section, which are presented below. o • Constant Heat Flux at the Wall The constant heat flux at the wall is described by: k ∂T ∂r ′′ = qw r = r0 (5.218) " where qw is positive when its direction is to the fluid (the hot wall) and negative when its direction is from the fluid (the cold wall). For the constant wall heat flux, the following equation, similar to the analysis presented for conventional heat pipes, is applicable: ∂T dTw dTs = = ∂x dx dx (5.219) Substituting eq. (5.214) into eq. (5.215) and non-dimensionalizing the resultant equation yield: 1 d  + dθ q  r+ + (5.220) r  = β1 1 − r + 4Kn + 32 Brq + + + 2 r dr  Brq = dr  ( 2 ) 2 (1 + 8Kn ) where 2 μ um Dq " w , θq = 2 ρ c p um ro dTs T − Ts , β1 = − ′′ ′′ qw r0 / k (1 + 8Kn ) qw dx (5.221) Since β1 is unknown, three non-dimensional boundary conditions are required for the second order ODE, eq. (5.220): ∂θ q + ∂r + = 0 at r = 0 ∂θ q ∂r + (5.222) + θ q = 0 and θ q ( r + ) = β1  2 = −1 at r = 1   r +4  + β2    16    − β3  The solution of eq. (5.220) using the above boundary conditions is:  r + r +4 2 − + Knr + 4 16  (5.223) 488 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press where β2 = 32 Brq (1 + 8Kn ) 2 , β1 = − β2 + 4 1 + 8Kn , β3 = β 2 + β1 ( 3 + 16Kn ) 16 (5.224) The local heat transfer coefficient, h, is: h= k ∂T Tw − Tm ∂r (5.225) r = ro The Nusselt number, based on θ q is: Nu = hD 2 = θq ,m − θq , w k (5.226) where the non-dimensional mean temperature ( θ q , m ) and slip temperature at the wall ( θ q , w ) are given as follows: θq ,m = + 1 5 + β 2 + Kn ( 32 + 6β 2 ) 2 4 24 (1 + 8Kn ) 2-Ft 4γ Kn Ft 1 + γ Pr (5.227) (5.228) θ q , w =- Aydin and Avci (2006) investigated the effects of the Brinkman number and Knudsen number for both fully developed flow and temperature profile in a microchannel with circular cross-section using the above analytical technique. Kn = 0 represents the macroscale case, while Kn > 0 holds for the microscale case, and Brq = 0 represents the case without the effect of the viscous dissipation. 6.5 6 5.5 5 Wall Cooling Brq=-0.10 =-0.05 =0.00 =+0.05 =+0.10 Nu 4.5 4 3.5 3 2.5 2 0 0.02 0.04 0.06 Kn 0.08 Wall Heating 0.1 0.12 Figure 5.24 The variation of the Nusselt number for microchannel, with the Knudsen number, for different values of the Brinkman number for constant heat flux at the wall. Chapter 5 Internal Forced Convective Heat and Mass Transfer 489 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 5.24 shows the variation of Nusselt number with the Knudsen number for different Brinkman numbers for the case of constant wall heat flux (Bahrami, 2009). For Brq = 0, an increase at Kn decreases Nu due to the temperature slip at the wall. Viscous dissipation significantly affects Nu. Positive values of Brq ′′ correspond to wall heating ( qw > 0 ), while the opposite is true for negative values of Brq. With no viscous dissipation, the solution is independent of whether there is wall heating or cooling. Nu decreases with increasing Brq for the wall heating case. Increasing Brq in the negative direction increases Nu. The trend followed by Nu versus Kn for lower values of the Brinkman number, either in the case of wall heating (Brq = 0.05) or in the case of wall cooling (Brq = – 0.05) is very similar to that of Brq = 0. For the wall cooling case, at Brq = –0.1, the decreasing effect of Kn on Nu is more significant. At Brq = 0.1, increasing Kn increases Nu up to Kn ≈ 0.01 where a maximum occurs, after which Nu decreases with increasing Kn. Constant Wall Temperature Aydin and Avci (2006) investigated the effects of the Brinkman and Knudsen numbers for both fully developed flow and temperature profile in a microchannel with circular cross-section tube subjected to constant wall temperature. They assumed that the fluid temperature at the wall does not change along the tube length, i.e. dTs / dx = 0 . However, their analysis for the viscous dissipation effect on Nusselt number was found to be inconsistent with other researchers’ results (Hooman, 2008). In the following the effect of the Knudsen number for both fully developed flow and temperature profile in a microchannel subjected to constant wall temperature considering variation of fluid temperature at the wall (i.e. dTs / dx ≠ 0 ) is presented (Bahrami, 2009). The non-dimensional temperature profile is defined as: θ= Ts − T Ts − Tc • (5.229) where Tc is the fluid temperature at the centerline. Substituting eq. (5.214) into eq. (5.215), neglecting the viscous dissipation, and non-dimensionalizing the resultant equation give us: 1 d  + dθ  r = ( β1θ + β 2 (1 − θ ) ) 1 − r + + 4Kn (5.230) + + + r dr  dr  ( 2 ) where: β1 = β2 = 2um ro2 dTc (1 + 8Kn ) α (Tc − Ts ) dx 2um ro2 dTs (1 + 8Kn ) α (Tc − Ts ) dx (5.231) (5.232) Equation (5.230) is subject to the following boundary conditions: 490 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The Nusselt number based on θ is: ∂θ = 0 and θ = 1 at r + = 0 ∂r + θ = 0 at r + = 1 (5.233)  ∂θ  −2  +  hD  ∂r r + =1 Nu = = θm − θw k (5.234) where θ m is the non-dimensional mean temperature, and θ w is the slip temperature at the wall. The relation between β1 and β 2 can be obtained by taking the derivative of slip temperature boundary conditions along the xdirection, and the results are given bellow: dTc 2 − Ft 4γ Kn ξ dTs = , ξ =− Nu (θ m − θ w ) dx ξ + 1 dx Ft 1 + γ Pr A closed form solution for Nu cannot be obtained for this case. However, the solution of θ can be obtained by using an iterative procedure. The temperature profile for the constant heat flux at the wall can be used as the first approximation, and eq. (5.230) is then integrated to obtain θ. This iterative procedure is repeated until an acceptable convergence is obtained. However, in order to get a very good accuracy, a forth-order Rung-Kutta procedure is employed to solve eq. (5.230). Figure 5.25 presents Nu versus Kn for the case of constant wall temperature (Bahrami, 2009). Similar trends to those obtained for the case of constant heat flux at the wall are observed. Nu values for the case of constant wall temperature are, for the same Kn, lower than those Nu values for the case of constant heat flux at the wall. 4 3.8 3.6 3.4 3.2 Nu 3 2.8 2.6 2.4 2.2 2 0 0.02 0.04 0.06 Kn 0.08 0.1 0.12 Figure 5.25 Variation of the fully developed Nusselt number for microchannel with the Knudsen number for constant wall temperature and neglecting viscous dissipation effects. Chapter 5 Internal Forced Convective Heat and Mass Transfer 491 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table 5.10 Fully developed Nusselt numbers for microchannels with Br = 0, and constant temperature at the wall Pr=0.6 Pr=0.65 Pr=0.7 Pr=0.75 Pr=0.8 Pr=0.85 Pr=0.9 Pr=0.95 Pr=1.0 Kn=0 Kn=0.02 Kn=0.04 Kn=0.06 Kn=0.08 Kn=0.1 Kn=0.12 3.657 3.432 3.191 2.952 2.728 2.522 2.337 3.657 3.462 3.245 3.024 2.812 2.614 2.433 3.657 3.488 3.292 3.088 2.887 2.697 2.521 3.657 3.512 3.334 3.145 2.955 2.773 2.603 3.657 3.532 3.372 3.196 3.017 2.843 2.678 3.657 3.550 3.405 3.243 3.074 2.907 2.747 3.657 3.566 3.436 3.285 3.126 2.967 2.812 3.657 3.580 3.463 3.323 3.173 3.022 2.872 3.657 3.593 3.488 3.359 3.217 3.073 2.929 Table 5.11 Fully developed Nusselt numbers for microchannels with Brq = 0, and constant heat flux at the wall Kn=0.00 Kn=0.02 Kn=0.04 Kn=0.06 Kn=0.08 Kn=0.10 Kn=0.12 Pr=0.6 4.364 3.981 3.599 3.252 2.949 2.687 2.461 Pr=0.65 4.364 4.029 3.678 3.350 3.057 2.799 2.575 Pr=0.7 4.364 4.071 3.749 3.439 3.156 2.904 2.681 Pr=0.75 4.364 4.108 3.812 3.519 3.247 3.000 2.781 Pr=0.8 4.364 4.141 3.870 3.593 3.331 3.091 2.874 Pr=0.85 4.364 4.171 3.922 3.661 3.409 3.175 2.962 Pr=0.9 4.364 4.197 3.969 3.723 3.481 3.254 3.044 Pr=0.95 4.364 4.221 4.013 3.781 3.549 3.327 3.122 Pr=1 4.364 4.243 4.053 3.834 3.612 3.397 3.195 The fully developed Nusselt numbers for 0 < Kn < 0.12 and no viscous dissipation are shown in Tables 5.10 and 5.11 for constant wall temperature and heat flux, respectively (Bahrami, 2009). The fully developed Nusselt number decreases as Kn increases. The effect of temperature jump on the Nusselt number is shown in Figs. 5.26 and 5.27. The solid and dashed lines represent the results obtained for considering the temperature jump and neglecting the temperature jump conditions, respectively. When the temperature jump condition is not accounted 7 Neglecting the Temperature Jump 6 Nu 5 Brq=-0.10 =0.00 =+0.10 Considering the Temperature Jump 4 3 2 0 0.02 0.04 0.06 Kn 0.08 0.1 0.12 Figure 5.26 Effect of temperature jump on fully developed Nu in microchannel for different Brq when the wall is subjected to constant heat flux. 492 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.5 Considering temperature jump 5 4.5 4 Neglecting temperature jump Nu 3.5 3 2.5 2 0 0.02 0.04 0.06 Kn 0.08 0.1 0.12 Figure 5.27 The effect of temperature jump on fully developed Nu for microchannel when the wall is subjected to constant temperature. for, i.e. only the velocity slip condition is taken into consideration, the Nusselt number increases with increasing Kn, which indicates that the velocity slip and temperature jump have opposite effects on the Nusselt number. 5.8.3 Fully Developed Flow with Developing Temperature Profile Convective heat transfer for steady state, laminar, hydrodynamically developed flow and developing temperature profile in microchannels with both uniform temperature and uniform heat flux boundary conditions were solved by Tunc and Bayazitoglu (2001, 2002) using the integral transform technique that is presented below. Uniform Temperature The energy equation assuming fully developed flow, including viscous dissipation and neglecting axial conduction, is the same as eq. (5.215). The boundary and inlet conditions for constant wall temperature are: T = Ts at r = ro (5.235) ∂T = 0 at r = 0 ∂r T = T0 at x = 0 (5.236) (5.237) The fully developed velocity profile with slip boundary condition given by eq. (5.214) is used. The slip boundary condition given by eq. (5.195) is also used to express the wall temperature jump. The following non-dimensional variables are introduced: for temperature (θ), radial coordinate (r+), axial coordinate (x+), and velocity (u+): θ= T − Ts r x u , r + = , x + = , u+ = T0 − Ts ro L um (5.238) Chapter 5 Internal Forced Convective Heat and Mass Transfer 493 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The non-dimensional energy equation and boundary conditions are obtained through use of the above variables: Gz (1 − r + 2 + 4Kn ) ∂θ 1 ∂  ∂θ  16Br = + + r+ +  + r +2 + 2 (1 + 8Kn ) ∂x r ∂r  ∂r  (1 + 8Kn )2 (5.239) θ = 0 at r + = 1 (5.240) ∂θ = 0 at r + = 0 (5.241) ∂r + θ = 1 at x + = 0 (5.242) where the Graetz number (Gz) and the Brinkman number (Br) are defined as: 2 μ um RePrD Gz = and Br = (5.243) L k ΔT where ΔT is the difference between the temperature of the fluid at the wall, Ts, and at the tube entrance, T0, i.e. ΔT = T0 − Ts . Uniform Heat Flux For the case of constant heat flux at the wall, the following non-dimensional variables are used: 2 T − T0 μ um (5.244) θ= and Br = ′′ qw r0 / k ′′ qw D Upon use of the above non-dimensional variables, the non-dimensional energy equation for constant wall heat flux can be obtained: Gz (1 − r +2 + 4Kn ) ∂θ 1 ∂  ∂θ = + +  r+ + + 2 (1 + 8Kn ) ∂x r ∂r  ∂r 16Br  r +2 + 2  (1 + 8Kn ) (5.245) where the centerline symmetric and uniform inlet temperature conditions are the same as eqs. (5.241) and (5.242), respectively. However, the boundary condition at the wall is given by ∂θ / ∂r + = 1 . An integral transform technique based on separation of variables was used by Tunc and Bayazitoglu (2001) to solve this problem. An appropriate integral transform pair was developed. Under the transformation, the variable x+ was eliminated from the partial differential governing equation, which transformed the governing equation into an ordinary differential equation. The effect of viscous heating is presented in Fig. 5.28 for the constant wall temperature case, where Kn = 0.04 and Pr = 0.7. The inclusion of viscous dissipation causes an increase in Nu. The Nusselt number first reaches the fully developed condition as if there was no viscous dissipation, and then makes a jump to its final value for a given Br. Figure 5.29 shows the effect of viscous heating on the Nusselt number for a uniform wall heat flux. Since the definition of the Brinkman number is different for the uniform wall heat flux boundary condition case, a positive Br means that the heat is being transferred to the fluid from the wall, as opposed to the uniform wall temperature case. For constant heat flux at the wall, the Nusselt number decreases as Br (> 0) increases. 494 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 6.5 6.5 6 Kn=0.04 Pr=0.7 5.5 Nusselt Number Br=0.01 5 Br=0.015 Br=0.006 4.5 Br=0.003 Br=0.001 4 Br=0.0 3.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Non-dimensional axial length Figure 5.28: The effect of viscous dissipation on Nusselt number for constant wall temperature (Tunc and Bayazitoglu, 2001). 4.1 4.1 Kn=0.04 4.0 Pr=0.7 3.9 Nusselt Number 3.8 Br=0.0 3.7 Br=0.003 Br=0.006 Br=0.01 3.6 Br=0.015 3.5 3.4 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Non-dimensional axial length Figure 5.29 The effect of viscous dissipation on Nusselt number for uniform wall heat flux (Tunc and Bayazitoglu, 2001). Jeong and Jeong (2006) extended the analysis for application to microchannels of rectangular cross-section, including axial conduction and viscous dissipation. The configuration for developed flow with developing temperature profile for rectangular microchannels is similar to Fig. 5.22. The fluid temperature changes from the value T0 at the entrance, to the value Ts on the walls. The Chapter 5 Internal Forced Convective Heat and Mass Transfer 495 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press governing energy equation and boundary conditions, including axial conduction and viscous dissipation, for laminar flow are: ρcpu  ∂ 2T ∂ 2T   ∂u  ∂T = k 2 + 2 + μ  ∂x ∂x ∂y   ∂y   T = T0 at x = 0 2 (5.246) (5.247) (5.248) (5.249) T − Tw =  2 − Ft 2γ λ ∂T at y = ± H Ft γ + 1 Pr ∂y ∂T = 0 at y = 0 ∂y where H is half the microchannel height, and the wall length in the x-direction is L. The fully developed velocity profile in the rectangular microchannel is: u ( y) = − 2  H 2 dP   y  2− F 1−   + 8 Kn   2 μ dx   H  F    (5.250) which satisfies the slip boundary condition: us =  2 − F  ∂u  λ   at y = ± H F  ∂y  , Kn = (5.251) λ Dh Defining the following dimensionless variables: 2 T − Tw μ um x y θ= , x+ = , y + = , Br = T0 − Tw RePrH H k ( T0 − Tw ) 2 (5.252) Equations (5.250) and (5.254) are respectively non-dimensionalized as:  ∂u +  1 + ∂θ 1 ∂ 2θ ∂ 2θ u = 2 + 2 + + 2 + Br  +  4 ∂x + Pe ∂x ∂y  ∂y  (5.253) and 2−F   1 − y+ 2 + 8 Kn  u 3 F   u= = um 2 C1 + (5.254) where C1 = 1 + 12 2−F Kn F (5.255) (5.256) (5.257) (5.258) The non-dimensional boundary conditions are: θ = 1 at x + = 0 ∂θ at y + = 1 θ = −4C 2 ∂y ∂θ = 0 at y + = 0 ∂y where 496 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 20 16 Pe=106 Br=0 12 Kn=0 Kn=0.04 Kn=0.08 Nu 8 7.54 4 0 0 0.25 0.5 0.75 1 x/(Pe·H) x/(PeH) Figure 5.30 Nusselt number distribution for a constant temperature boundary condition for rectangular microchannels. (Jeong and Jeong, 2006). C2 = 2 − Ft 2γ Kn Ft γ + 1 Pr The effects of the Knudsen number on the Nusselt number variation along a rectangular microchannel neglecting axial conduction and viscous dissipation are shown in Fig. 5.30. For Kn = 0, the fully developed Nusselt number is approximately 7.54, which is the result for a pipe of conventional size (classic Graetz problem). The Nusselt number decreases as Kn increases due to the temperature jump at the wall. The effect of the Knudsen number on the Nusselt number distribution in a rectangular microchannel with constant wall heat flux is presented in Fig. 5.31. When the channel is subjected to a constant wall temperature, as x + → ∞ , the Nusselt number for Br ≠ 0 is independent of Br and different from that for Br = 0 . The thermally fully developed Nusselt number was obtained from the fully developed temperature profile for constant wall temperature and heat flux by Jeong and Jeong (2006). For constant wall temperature the Nusselt number as x + → ∞ ( Br ≠ 0 ) is Nu ∞ = 140C1 1 + 7C1 + 140C1C 2 (5.259) And for constant heat flux Nu ∞ = C 2 1 ( 35C 2 1 + 14C1 + 2 + 420C12 C 2 ) + Br ( 42C12 + 33C1 + 6 ) 420C14 (5.260) Chapter 5 Internal Forced Convective Heat and Mass Transfer 497 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 20 18 16 14 Pe=106 Br=0 Kn=0 Kn=0.04 Kn=0.08 Nu 12 10 8 6 4 2 0 0.2 0.4 0.6 x/x/(PeH) (Pe H) 0.8 1 8.24 5.72 4.26 Figure 5.31 Nusselt number distribution for a constant heat flux boundary condition for rectangular microchannels. (Jeong and Jeong, 2006). where  2−F  C1 = 1 + 12   Kn F 2 − Ft 2γ Kn C2 = . Ft γ + 1 Pr Unless Br is a large negative number, Nu, is always positive. Example 5.6: Develop the analytical expression of the Nusselt number for constant wall temperature for rectangular microchannels as x + → ∞ , including viscous dissipation, given by eq. (5.253). Solution: As x + → ∞ and ∂θ / ∂x + → 0 , eq. (5.253) becomes:  ∂u +  ∂ 2θ = −Br  +  ∂y +2  ∂y  2 (5.261) The non-dimensional temperature profile, θ ∞ , can be derived by integrating both sides of eq. (5.261) and applying the boundary conditions given by eqs. (5.257) and (5.258): θ∞ ( y ) = Br C12  3 +4   − 4 ( y − 1) + 12C 2    (5.262) The non-dimensional mean temperature and Nusselt number can be obtained using the following equations: θ m =  u +θ dy + 0 1 (5.263) 498 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Nu = hDh h ( 4 H ) 4 ∂θ = =− θ m ∂y + k k (5.264) y + =1 Through the use of the temperature profile given by eq. (5.262), one obtains the full relation for the fully developed Nusselt number: Nu ∞ = 140C1 for Br ≠ 0 1 + 7C1 + 140C1C 2 (5.265) 5.9 Turbulence 5.9.1 Time-Averaged Governing Equations The generalized governing equations for three-dimensional turbulent flow have been presented in Chapter 2 (see Section 2.5). For two-dimensional steady-state turbulent flow inside a cylindrical coordinate system (see Fig. 5.2), the governing equations are: ∂u 1 ∂ (rv ) + =0 ∂x r ∂r ∂u ∂u 1 dp 1 ∂  ∂u  u +v =− + r (ν + ε M ) ∂r  ρ dx r ∂r  ∂x ∂r  u ∂T ∂T 1 ∂  ∂T  +v = r (α + ε H )  ∂x ∂r r ∂r  ∂r  (5.266) (5.267) (5.268) which can be obtained by using the boundary layer theory similar to Chapter 4. The second order derivatives of u and T in the x-direction have been dropped based on the similar arguments for laminar flow in a duct. The time-averaged pressure is not a function of r, but is a function of x only. It can also be observed from eqs. (5.267) and (5.268) that the both momentum and energy diffusions are governed by molecular and eddy diffusions. Similar to the cases of external turbulent boundary layers, the momentum and thermal eddy diffusivities, ε M and ε H , are defined as: ∂u ∂r ∂T − ρ c p v′T ′ = ρ c p ε H ∂r − ρ u′v′ = ρε M (5.269) (5.270) where u′, v′ and T ′ are the fluctuations of axial velocity, radial velocity, and temperature, respectively. Appropriate turbulent models in either algebraic or differential equation forms must be employed to obtain the eddy diffusivities. Chapter 5 Internal Forced Convective Heat and Mass Transfer 499 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.9.2 Velocity Profile and Friction Coefficient for Fully Developed Flow Since the turbulent boundary layer grows much faster than the laminar boundary layer, the lengths of hydrodynamic and thermal entrances for turbulent internal flow are also much shorter than those for laminar flow. When the Prandtl number of the fluid is on the order of 1 (e.g., air or water), the lengths of the hydrodynamic and thermal entrances are about 10 times of the diameter of the tube, i.e., LH LT   10 D D (5.271) The internal turbulent flow becomes fully developed after x > LH or LT . Similar to the laminar internal flow, we have v = 0, and ∂u / ∂x = 0 for fully developed flow, and the momentum eq. (5.267) becomes dp 1 ∂ (rτ app ) − = (5.272) dx r ∂r where τ app = ρ (ν + ε M ) ∂u ∂y (5.273) is the apparent or total shear stress and y is the distance measured from the wall ( y = ro − r ). The apparent shear stress is equal to τ w at the wall and zero at the centerline. Integrating eq. (5.272) from the centerline to the wall yields, τw dp − dx =2 ro which can be substituted into eq. (5.272) to yield τ w 1 ∂ (rτ app ) −2 r0 = r ∂r (5.274) Integrating eq. (5.274) in the interval of (0, r), one obtains:   y (5.275)  = τ w ⋅ 1 −  ro    where y is measured from the tube wall ( y = r0 − r ). Equation (5.275) shows that τ app = τ w ⋅  r  ro the shear stress is a linear function for internal turbulent flow. Close to the wall where r is near r0 (or y is near 0), the apparent shear stress is nearly a constant, i.e., τ app ≈ τ w . The law of the wall resulting from the two-layer turbulent model (see Section 4.11.2) can be applied near the wall to yield: u + = 2.5ln y + + 5.5 (5.276) which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in Section 4.11.2 and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as u+ = yu u , y+ = τ uτ ν (5.277) where 500 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press τw (5.278) ρ It should be pointed out that the Nikuradse equation (5.276) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (5.276) is a finite value, not zero as it should be. In addition, eq. (5.276) also implies that ε M / ν = 0 at the centerline, which is also not true because the centerline is also in the fully turbulent region. Reichardt (1951) suggested the following empirical correlation for the eddy diffusivity: uτ = r ε M κ y+  r   =  1 +  1 + 2   6  ro   ν  ro   2     (5.279) which becomes ε M / ν = κ y + near the wall – a result that coincides with the mixing length theory. Equation (5.279) produces finite eddy diffusivity at the centerline. Assuming ε M  ν , eq. (5.273) becomes τ app = ρε M ∂u ∂y (5.280) Substituting eqs. (5.275) and (5.279), and integrating the resultant equation, the following velocity profile is obtained  3(1 + r / ro )  u + = 2.5ln  y +  + 5.5 2  2[1 + 2(r / ro ) ]  (5.281) which becomes identical to eq. (5.276) near the wall and produces zero slope at the centerline. The friction factor for internal turbulent flow is defined as τw cf = (5.282) 2 ρ um / 2 where um is the mean velocity over the cross-section of the duct. For axisymmetric flow in a circular tube, it is obtained by um = 1/ 2 2 ro2  ro 0 urdr (5.283) 1/ 2 The definition of the friction factor, eq. (5.282), can be rewritten as τw   ρ  cf  = um   2 (5.284) For moderate Reynolds number, the velocity profile in the entire tube can be approximated as (Kays et al., 2005) u + = 8.6( y + )1/ 7 (5.285) At the centerline where u = uc and y = r0 , the centerline velocity satisfies: r τ / ρ = 8.6  o w  ν τw / ρ  uc     1/ 7 (5.286) The velocity at any radius is related to the centerline velocity by Chapter 5 Internal Forced Convective Heat and Mass Transfer 501 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press u y =  uc  ro  1/ 7 (5.287) Substituting eq. (5.287) into eq. (5.283), a relationship between the mean velocity and centerline velocity is obtained: um = 0.817uc (5.288) Substituting eq. (5.284) and (5.288) into eq. (5.286) and considering the definition of Reynolds number Re D = um D / ν , the friction coefficient can be obtained as c f = 0.078 Re −1/ 4 (5.289) D which agreed with the experimental data very well up to Re D = 5 ×104 . For even higher Reynolds number, the following empirical correlation works better for smooth tubes (see Fig. 5.32): (5.290) c f = 0.046 Re −1/5 , for 3 × 104 < Re < 106 D Instead of the one-seventh law, eq. (5.285), the law of the wall, eq. (5.276), can be used to obtain the following correlation: c −1/ 2 = 1.737 ln(c1/ 2 Re D ) − 0.396 (5.291) f f which is referred to in the literature as the Kármán-Nikuradse relation. Equation (5.291) is valid up to ReD = 106. For fully-developed turbulent flow in a non-circular tube, eq. (5.291) is still applicable provided the hydraulic diameter is used in the definition of the Reynolds number. In this case, the friction coefficient, cf, is defined based on the perimeter-averaged wall shear stress because the shear stress is no longer uniform around the periphery of the cross-section. Figure 5.32 Friction factor for duct flow. 502 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press When the inner surface of the tube is not smooth, the friction will significantly increase with roughness (see Fig. 5.32). The effect of the surface roughness can be measured by the roughness Reynolds number defined as: ku k (τ / ρ )1/ 2 (5.292) Rek = s τ = s w ν ν where ks is the roughness. When the roughness Reynolds number is greater than 70, the friction coefficient is no longer a strong function of the Reynolds number and becomes a constant, which is referred to as a fully rough surface. In the fully rough surface regime, the roughness size excees the order of the magnitude of what would have been the thickness of the viscous sublayer for a smooth surface. The friction coefficient for the fully rough surface regime can be obtained from the following empirical correlation:   D c f   1.74 ln + 2.28  ks   −2 (5.293) 5.9.3 Heat Transfer in Fully Developed Turbulent Flow Heat transfer in fully-developed turbulent flow in a circular tube subject to ′′ constant heat flux ( qw = const) will be considered in this subsection (Oosthuizen and Naylor, 1999). When the turbulent flow in the tube is fully developed, we have v = 0 and the energy eq. (5.268) becomes u ∂T 1 ∂  ∂T  = r (α + ε H )  ∂x r ∂r  ∂r  (5.294) After the turbulent flow is hydrodynamically and thermally fully developed, the time-averaged temperature profile is no longer a function of axial distance from the inlet, i.e., ∂  Tw − T  ∂x  Tw − Tc  =0  (5.295) where Tc is the time-averaged temperature at the centerline of the tube, and Tw is the wall temperature. Thus, (Tw − T ) / (Tw − Tc ) is a function of r only, i.e., Tw − T = f (r ) Tw − Tc (5.296) where f is independent from x. Differentiating (5.295) yields ∂T dTw  Tw − T   dTw dTc  = − −   ∂x dx  Tw − Tc   dx dx  (5.297) At the wall, the contribution of eddy diffusivity on the heat transfer is negligible, and the heat flux at the wall becomes ′′ qw = k ∂T ∂r (5.298) r = r0 Chapter 5 Internal Forced Convective Heat and Mass Transfer 503 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Substituting eq. (5.296) into eq. (5.298), one obtains: ′′ qw = −k (Tw − Tc ) f ′(r0 ) (5.299) ′′ Since the heat flux is constant, qw = const, it follows that (Tw − Tc ) = const., i.e., dTw dTc = (5.300) dx dx Therefore, eq. (5.297) becomes: ∂T dTw = dx ∂x (5.301) For fully developed flow, the local heat transfer coefficient is: hx = ′′ qw = const Tw − Tm 2 r02 (5.302) where Tm is the time-averaged mean temperature defined as: Tm =  r0 0 uTrdr (5.303) ′′ Since qw = const, it follows from eq. (5.302) that (Tw − Tm ) = const., i.e., dTw dTm = dx dx (5.304) Combining eqs. (5.300), (5.301) and (5.304), the following relationships are obtained: ∂T dTw dTc dTm (5.305) = = = dx dx dx ∂x The time-averaged mean temperature, Tm , changes with x as the result of heat transfer from the tube wall. By following the same procedure as that in Example 5.2, the rate of mean temperature change can be obtained as follows: ′′ dTm 4π qw (5.306) = ρ c p Dum dx Substituting eq. (5.305) into eq. (5.294), the energy equation becomes: u dTm 1 ∂ ∂T  = (r0 − y)(α + ε H )  ∂y  dx r0 − y ∂y  (5.307) where y = r0 − r is the distance measured from the tube wall. Equation (5.307) is subject to the following two boundary conditions: ∂T = 0, y = r0 (axisymmetric condition) ∂y (5.308) (5.309) Integrating eq. (5.307) in the interval of (r0, r) and considering eq. (5.308), we have: T = Tw (unknown), y = 0 (r0 − y)(α + ε H ) ∂T dTm = dx ∂y  y r0 (r0 − y)udy (5.310) which can be rearranged to 504 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press dTm I ( y) ∂T = ∂y (r0 − y)(α + ε H ) dx (5.311) where I ( y) =  (ro − y)udy ro y (5.312) Integrating eq. (5.311) in the interval of (0, y) and considering eq. (5.309), one obtains: T − Tw = dTm dx  y 0 I ( y) dy (ro − y)(α + ε H ) (5.313) If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (5.313) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, u , in eq. (5.312) can be replaced by um , and I(y) becomes: I ( y)  − 2 um (ro − y) 2 2 (5.314) Substituting eqs. (5.314) and (5.306) into eq. (5.313) yields: T − Tw = − ′′ qw ρcp +  y 0 (1 − y / ro ) dy (α + ε H ) (5.315) which can be rewritten in terms of wall coordinate q′′ (1 − y / ro ) ρy + T − Tw = − w 0 [1 / Pr + (ε M / ν ) / Prt ] dy ρcp τ w (5.316) where y+ is defined in eq. (5.277). To consider heat transfer in an internal turbulent flow, the entire turbulent boundary layer is divided into three regions: (1) inner region ( y + < 5 ), (2) buffer region ( 5 ≤ y + ≤ 30 ), and (3) outer region ( y + > 30 ). In the inner region ε M = ε H = 0 and eq. (5.316) becomes T − Tw = − y+ ′′ qw ρ Pr  (1 − y / ro )dy + 0 ρcp τ w (5.317) Since the inner region is very thin, y / r0  1 and 1 − y / r0 is effectively equal to 1. Therefore, the temperature profile in the inner region becomes:  q′′  ρ T − Tw = −  w  Pr y + (5.318)  ρc  τ  p w The temperature at the boundary between the inner and buffer regions ( y + = 5 ), Ts , can be obtained from eq. (5.318) as  q′′ Ts − Tw = −5  w  ρc p ρ Pr  τ w (5.319) Chapter 5 Internal Forced Convective Heat and Mass Transfer 505 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press In the buffer region where 5 ≤ y + ≤ 30 , the eddy diffusivity in the buffer region is: ε M y+ = −1 (5.320) 5 ν Substituting eq. (5.320) into eq. (5.316) and assuming the turbulent Prandtl number Pr t = 1 , the following expression is obtained: q′′ (1 − y / ro ) ρy + T − Ts = − w (5.321) 5 [1 / Pr + y+ / 5 − 1] dy ρcp τ w Since the buffer region is also very thin, 1 − y / r0 in eq. (5.321) is effectively + equal to 1. Defining T + = (T − Tw )  −  yields  ′′ qw ρ  and Integrating eq. (5.321)   ρcp τ w   i.e., T+ 5 Pr dT + =  y+ 5 dy + 1 / Pr + ( y + − 5) / (5 Pr t ) (5.322)  ρ  y+  ln  Pr − Pr + 1 , 5 < y + < 30 (5.323)  τ 5  w The temperature at the top of the buffer region where y + = 30 , Tb , becomes  q′′  ρ Tb − Ts = −5  w  ln(5 Pr + 1) (5.324)  ρc  τ  p w For the outer region where ε M  ν and ε H  α , eq. (5.316) becomes q′′ ρ y+ (1 − y / ro ) + T − Tb = − w dy (5.325) ρ c p τ w 30 ε M / ν  q′′ T − Ts = −5  w  ρc p where the turbulent Prandtl number is assumed to be equal to 1. It is assumed that the Nikuradse equation (5.276) is valid in the outer region and the velocity gradient in this region becomes: ∂u + 2.5 = ∂y + y + (5.326) The expression of apparent shear stress in this region, eq. (5.280) , can be nondimensionalized using eqs. (5.277) and (5.278) as: τ app ε M ∂u + (5.327) = τw ν ∂y + Substituting eqs. (5.275) and (5.326) into eq. (5.327), the eddy diffusivity in the outer region is obtained as: ε M  y  y+ = 1 −  (5.328) ν  ro  2.5 Substituting eq. (5.328) into eq. (5.325), the temperature distribution in this region becomes: 506 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press νρ The temperature at the center of the tube, Tc , can be obtained by letting y + = yc+ in eq. (5.329), i.e.  q′′ Tc − Tb = −2.5  w  ρc p  ρ  r0 ln    τ  w  30ν  q′′  ρ y+ 1  q′′  ρ  y +  T − Tb = −2.5  w  dy + = −2.5  w  ln   (5.329)  ρ c  τ 30 y +  ρc  τ  p w  p  w  30  which is valid from y + = 30 to the center of the tube where yc = r0 or r τw + (5.330) yc = 0 τw ρ     (5.331) The overall temperature change from the wall to the center of the tube can be obtained by adding eqs. (5.319), (5.324) and (5.331):  q′′ Tw − Tc =  w   ρcp ρ    τw  r  2.5ln  0    30ν  2 τ w = c f ρ um τw ρ    + 5ln(5 Pr + 1) + 5 Pr      (5.332) It follows from the definition of friction factor, eq. (5.282), that 1 2 (5.333) Substituting eq. (5.333) into eq. (5.332) and considering the definition of Reynolds number, Re D = um D / ν , eq. (5.332) becomes:  cf   + 5ln(5 Pr + 1) + 5 Pr  (5.334) 2    ′′ In order to obtain the heat transfer coefficient, h = qw / (Tw − Tm ) , the temperature  qw ′′ Tw − Tc =   ρc u  pm  Re  2  2.5ln  D  c  60  f  difference Tw − Tm must be obtained. If the velocity profile can be approximated by eq. (5.287), and the temperature and velocity can also be approximated by the one-seventh law, i.e., Tw − T  y  =  Tw − Tc  ro  1/ 7 , u y =  uc  ro  1/ 7 (5.335) it follows that Tw − Tm  = ro 0 u (Tw − T )2π rdr  ro 0 u 2π rdr = 5 (Tw − Tc ) 6 (5.336) Substituting eq. (5.334) into eq. (5.336) results in: ′′ 5  qw Tw − Tm =   ρc u 6 p m  Re  2  2.5ln  D  c  60  f   cf   + 5ln(5 Pr + 1) + 5 Pr  2    (5.337) which can be rearranged to the following empirical correlation Chapter 5 Internal Forced Convective Heat and Mass Transfer 507 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Nu D = Re D Pr cf  Re 5  2.5ln  D  60 6   2   cf  + 5ln(5 Pr + 1) + 5 Pr  2    (5.338) which can be used together with appropriate friction coefficient discussed in the previous subsection to obtain the Nusselt number. Example 5.7 Water is to be heated from 30 °C to 70 °C by flowing through a smooth tube with a diameter of 50 mm and heated at constant heat flux. If the mass flow rate of water is 2.5 kg/s, what is the heat transfer coefficient? Solution: The properties of water can be determined at the average temperature of the water at Tm = (Tm ,i + Tm , e ) = (30 + 70) / 2 = 50 °C. From Table C.8, we have ρ = 988.1kg/m3 , c p = 4180J/kg- o C , ν = 0.554 × 10−6 m 2 /s , k = 0.64W/m- o C , and Pr = 3.57 . The mean velocity of water is um =  4m 4 × 2.5 = = 1.288m/s ρπ D 2 988.1× π × 0.052 um D = 1.288 × 0.05 = 1.16 × 105 −6 0.554 × 10 The Reynolds number is ν The friction coefficient under this Reynolds number can be found from eq. (5.290), i.e., c f = 0.046 Re −1/5 = 0.046 × (1.16 × 105 ) −1/5 = 4.46 × 10 −3 D Re D = The Nusselt number can be obtained from eq. (5.338), i.e. Nu D = Re D Pr  Re 5  2.5ln  D  60 6   cf 2   cf  + 5ln(5 Pr + 1) + 5 Pr  2    4.46 × 10−3 2 The heat transfer coefficient is therefore: h= Nu D k 535.56 × 0.64 = = 6855.17W/m 2 - o C D 0.05  1.16 × 105 5  2.5ln   6 60   = 535.56 = 1.16 × 105 × 3.57 × 4.46 × 10−3 2    + 5ln(5 × 3.57 + 1) + 5 × 3.57      508 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press References Aydin, O., M. Avci, 2006, “Heat and Fluid Flow Characteristics of Gases in Micropipes,” Int. J. Heat Mass Transfer, Vol. 49, pp. 1723-1730. Bahrami, H., 2009, Personal Communication, Storrs, CT. Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ. Blackwell, B. F., 1985, “Numerical Solution of the Graetz Problem for a Bingham Plastic in Laminar Tube Flow with Constant Wall temperature,” ASME J. Heat Transfer, Vol. 107, pp. 466-468. Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ. Faghri, A., and Buchko, B., 1991, “Experimental and Numerical Analysis of Low Temperature Heat Pipes with Multiple Heat Sources”, ASME J. Heat Transfer, Vol. 113, pp. 728-734. Heaton, H. S., Reynolds, W. C. and Kays, W. M., 1964, “Heat Transfer in Annular Passages: Simultaneous Development of Velocity and Temperature Fields in Laminar Flow”, Int. J. Heat Mass Transfer, Vol. 7, pp. 763-781. Hetsroni, G., Mosyak, A., Pogrebnyak, E., Yarin, L.P., 2005, “Fluid Flow in Micro-channels,” Int. J. Heat Mass Transfer, Vol. 48, pp. 1982-1998. Hooman, K., 2008, “Comments on ‘Viscous-dissipation effects on the heat transfer in a Poiseuille flow’ by O. Aydin and M. Avci”, Applied Energy, Vol. 85, pp. 70-72 Hornbeck, R.W., 1965, “An All-numerical Method for Heat Transfer in the Inlet of a Tube,” ASME Paper No. 65-WA HT-36. Jeong, H.E., Jeong, J.T., 2006, “Extended Graetz Problem including Streamwise Conduction and Viscous Dissipation in Microchannel,” Int. J. Heat Mass Transfer, Vol. 49, pp. 2151-2157. Kakaç, S., Shah, R., Aung, W., 1987, Handbook of Single-Phase Convective Heat Transfer, John Wiley, New York. Kakaç, S., and Yucel, O., 1974, Laminar Flow Heat Transfer in an Annulus with Simultaneous Development of Velocity and Temperature Fields, Technical and Scientific Council of Turkey, TUBITAK, ISITEK No. 19, Ankara, Turkey. Kandlikar, S.G., Garimella, S., Li, D., Colin, S. and King, M.R. (2006), Heat Transfer and Fluid Flow in Minichannels and Microchannels, Elsevier, San Diego, CA, USA Karniadakis, G., Beskok, A., and Narayan, A., 2005, Microflows and Nanoflows, Springer Verlag, Berlin. Chapter 5 Internal Forced Convective Heat and Mass Transfer 509 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY Kurosaki, Y., 1973, “Coupled Heat and Mass Transfer in a Flow between Parallel Flat Plate (Uniform Heat Flux),” Journal of the Japan Society of Mechanical Engineers, Part B, Vol. 39, pp. 2512-2521 (in Japanese). Langhaar, H.L., 1942, Steady Flow in the Transition Length of a Straight Tube,” J. Appl. Mech., Vol. 9, pp. A55-A58. Lyche, B. C., and Bird, R. B., 1956, “Graetz-Nusselt Problem for a Power-Law Non-Newtonian Fluid,” Chem. Eng. Sci., Vol. 6, pp. 35-41. Moody, L. F., 1944, “Friction Factors for Pipe Flow,” Trans. ASME, Vol. 66, pp. 671-684. Myong, R.S., Lockerby, D.A., Reese, J.M., 2006, “The Effect of Gaseous Slip on Microscale Heat Transfer: An Extended Graetz Problem,” Int. J. Heat Mass Transfer, Vol. 49, pp. 2502-2513. Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York. Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC. Patankar, S.V., 1991, Computation of Conduction and Duct Flow Heat Transfer, Innovative Research. Reichardt, H., 1951, “Die Grundlagen des turbulenten Wärmeüberganges,” Arch. Gesamte Waermetech, Vol. 2, pp. 129-142. Sellars, J.R., Tribus, M., and Klein, J.S., 1956, “Heat Transfer to Laminar Flow in a Flat Conduit –The Graetz Problem Extended,” Trans. ASME, Vol. 78, pp. 441-448. Shah, R. K.; London, A. L., 1974, “Thermal Boundary Conditions and Some Solutions for Laminar Duct Flow Forced Convection,” ASME J. Heat Transfer Vol. 96, pp. 159-165. Shah, R. K.; London, A. L., 1978, “Laminar Flow Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Irvine, T. F. and Harnett, J.P., Eds., Academic Press, San Diego, CA. Siegel, R., Sparrow, E.M., and Hallman, T.M., 1958, “Steady Laminar Heat Transfer in a Circular Tube with Prescribed Wall Heat Flux,” Appl. Sci. Res., Ser. A, Vol. 7, pp. 386-392. Tunc, G., and Bayazitoglu, Y., 2001, “Heat Transfer in Microtubes with Viscous Dissipation,” Int. J. Heat Mass Transfer, Vol. 44, pp. 2395-2403. Tunc, G., and Bayazitoglu, Y., 2002, “Heat Transfer in Rectangular Microchannels,” Int. J. Heat Mass Transfer, Vol. 45, pp. 765-773. 510 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Whiteman, I. R., and Drake, W. B., 1980, Trans. ASME, Vol. 80, pp. 728-732. Zhang, Y., 2002, “Coupled Forced Convective Heat and Mass Transfer in a Circular Tube with External Convective Heating,” Progress of Computational Fluid Dynamics Journal, Vol. 2, pp. 90-96. Zhang, Y., and Chen, Z.Q., 1990, “Analytical Solution of Coupled Laminar Heat-Mass Transfer inside a Tube with Adiabatic External Wall,” Proceedings of the 3rd National Interuniversity Conference on Engineering Thermophysics, Xi’an Jiaotong University Press, Xi’an, China, pp. 341-345. Zhang, Y., and Chen, Z.Q., 1992, “Analytical Solution of Coupled Laminar Heat- Mass Transfer in a Tube with Uniform Heat Flux,” Journal of Thermal Science, Vol. 1, No. 3, pp. 184-188. Problems 5.1. Estimate the hydrodynamic entry length for laminar flow with constant properties inside ducts using the integral methods developed in Chapter 5 for flow over a flat plate. Obtain the Nusselt number for laminar flow in a circular pipe assuming “slug flow” and a fully developed temperature profile with constant wall temperature. Develop both the temperature distribution and Nusselt number for laminar flow as a function of r and x for a developing temperature profile with constant wall temperature by assuming “slug flow” and using the separation of variables technique. What is the limit of Nusselt number as x → ∞ ? Is there a physical significance for the results as x → ∞ ? Repeat Problem 5.3 for the case of constant heat flux at the wall. Consider the flow and heat transfer between two infinite parallel plates with uniform inlet temperature and by assuming “slug flow” and the same constant wall temperature on both walls. Determine dimensionless temperature distribution, mean temperature and Nusselt number for the developing temperature region. Repeat Problem 5.5 for the case of the same constant heat flux in each wall. Discuss the physical significance of the result for Example 5.4 if qo′′ qi′′ = 1 rather than 1. 0.346 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. Chapter 5 Internal Forced Convective Heat and Mass Transfer 511 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.8. Develop the governing equations and boundary conditions for laminar, steady and two-dimensional forced convective heat transfer in a plane duct with developing velocity and temperature profiles for an incompressible Newtonian fluid with constant wall temperature using the parabolic governing equation or boundary layer assumption (see Fig. P5.1). Nondimensionalize the conservation governing equations and boundary conditions so that the only dimensionless parameter in the governing equations and boundary conditions is Prandtl number. x, u y, v Fig. P5.1 2H 5.9. Calculate the heat transfer coefficient and heat transfer rate for laminar fully developed flow and temperature profile of water in a circular pipe of ¼” diameter with constant wall temperature of 80ºC. Assume an inlet temperature of 50ºC. 5.10. Repeat Problem 5.9 for air as the working fluid instead of water. 5.11. Repeat Problem 5.9 for oil as the working fluid instead of water. 5.12. Air at 300ºK enters a 1.5cm ID tube, 30cm long with constant wall temperature of 340ºK and mean inlet velocity of 0.6m/s. Determine the pressure drop, drag force, heat transfer coefficient and outlet temperature by assuming fully developed flow and temperature profile. Discuss the appropriateness of assuming a fully developed flow and temperature profile. 5.13. Repeat Problem 5.12 for oil as the working fluid instead of air. 5.14. Determine the Nusselt number for fully developed flow and temperature profile in a plane duct (i.e. between two large parallel plates; see Fig. P5.2) for the case of the same constant wall heat flux in both walls. 5.15. Repeat Problem 5.14 with one wall kept at uniform flux and the other wall insulated. Tw x, u y, v H Tw Fig. P6.2 512 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.16. It is common practice (but not acceptable) that when dealing with flow and heat transfer in non-circular tubes, one can use the results of flow in circular tubes based on hydraulic diameter (4 × area/wetted perimeter). Use the results of Example 5.4 to show that this gives significant error for the case of flow between parallel plates. 5.17. Couette flow consists of a fluid contained between two parallel plates where one moves with constant velocity U and the other one is stationary (see Fig. P5.3). Furthermore, assume the two plates are porous and fluid enters the space between the plates through both plates with constant velocity vw. Assume steady and laminar flow with no axial pressure gradient, as well as no end effects. The upper and lower walls are kept at uniform temperature T1 and T2, respectively. Determine the temperature distribution within the fluid, and the shear stress and heat flux at the walls. U H y x Fig. P5.3 5.18. For sublimation inside a circular tube subject to constant heat flux heating (see Section 5.5), show that the dimensionless mean temperature and concentration are related by θ m + ϕm − ϕ0 = 4ξ . 5.19. Show that the fully developed dimensionless temperature and mass fraction distributions for sublimation inside a circular tube subject to constant heat flux heating discussed in Section 5.5 are eqs. (5.132) and (5.133). 5.20. The inner surface of a circular tube with radius R is coated with a layer of sublimable material, and the outer wall of the tube is kept at a constant temperature Tw . The fully developed gas enters the tube with a uniform inlet mass fraction of the sublimable substance ω0 that equals the saturation mass fraction corresponding to the inlet temperature T0. The thermal and mass diffusivities are assumed to be the same, i.e., Le = 1. Find the local Nusselt number based on convective heat flux and the total heat flux at the wall, and the local Sherwood number. 5.21. Obtain the fully developed Nusselt number based on convective heat flux and the total heat flux at the wall, and the local Sherwood number for the sublimation problem discussed in Example 5.5. 5.22. Develop an analytical solution, which shows that the fully developed Nusselt number for constant wall heat flux in rectangular microchannels is given by eq. (5.260). Chapter 5 Internal Forced Convective Heat and Mass Transfer 513 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 5.23. If the tube in Example 5.7 is heated at constant wall temperature at 100 °C and eq. (5.338) is assumed to be valid under constant wall temperature, what is the length of the tube? 5.24. Obtain the Nusselt number for fully developed internal turbulent flow based on analogy between momentum and heat transfer. The Prandtl number and turbulent Prandtl number can be assumed to be equal to 1. 514 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press