Heat Transfer Handbook part 55 ppt

BOOKCOMP, Inc. — John Wiley & Sons / Page 532 / 2nd Proofs / Heat Transfer Handbook / Bejan 532 NATURAL CONVECTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [532], (8) Lines: 216 to 254 ——— 2.85013pt PgVar ——— Long Page * PgEnds: Eject [532], (8) δ T δ = O  1 Pr 1/2  (7.13) where Gr is the Grashof number based on a characteristic length L and Pr is the Prandtl number. These are defined as Gr = gβL 3 (T w − T ∞ ) ν 2 Pr = µc p k = ν α (7.14) where ν is the kinematic viscosity and α the thermal diffusivity of the fluid. These dimensionless parameters are important in characterizing the flow, as discussed in the next section. The resulting boundary layer equations for a two-dimensional vertical flow, with variable fluid properties except density, for which the Boussinesq approximations are used, are then written as (Jaluria, 1980; Gebhart et al., 1988) ∂u ∂x + ∂v ∂y = 0 (7.15) u ∂u ∂x + v ∂u ∂y = gβ(T − T ∞ ) + 1 ρ ∂ ∂y  µ ∂u ∂y  (7.16) ρc p  u ∂T ∂x + v ∂T ∂y  = ∂ ∂y  k ∂T ∂y  + q  + βTu ∂p a ∂x + µ  ∂u ∂y  2 (7.17) where the last two terms in the energy equation are the dominant terms from pressure work and viscous dissipation effects. Here u and v are the velocity components in the x and y directions, respectively. Although these equations are written for a vertical, two-dimensional flow, similar approximations can be employed for many other flow circumstances, such as axisymmetric flow over a vertical cylinder and the wake above a concentrated heat source. There are several other approximations that are commonly employed in the analysis of natural convection flows. The fluid properties, except density, for which the Boussinesq approximations are generally employed, are often taken as constant. The viscous dissipation and pressure work terms are generally small and can be neglected. However, the importance of various terms can be best considered by nondimension- alizing the governing equations and the boundary conditions, as outlined next. 7.2.3 Dimensionless Parameters To generalize the natural convection transport processes, a study of the basic nondi- mensional parameters must be carried out. These parameters are important not only in simplifying the governing equations and the analysis, but also in guiding experiments that may be carried out to obtain desired information on the process and in the presentation of the data for use in simulation, modeling, and design. In natural convection, there is no free stream velocity, and a convection velocity V c is employed for the nondimensionalization of the velocity V, where V c is given by BOOKCOMP, Inc. — John Wiley & Sons / Page 533 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 533 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [533], (9) Lines: 254 to 291 ——— 2.61385pt PgVar ——— Long Page PgEnds: T E X [533], (9) V c = [ gβ(T w − T ∞ ) ] 1/2 (7.18) The governing equations may be nondimensionalized by employing the following dimensionless variables (indicated by primes): V  = V V c p  = p ρV 2 c θ  = T − T ∞ T w − T ∞ Φ  v = Φ v L 2 V 2 c t  = t t c ∇  = L∇ (∇  ) 2 = L 2 ∇ 2 (7.19) where t c is a characteristic time scale. The dimensionless equations are obtained as ∇  · V  = 0 (7.20) Sr  ∂v  ∂t  + V  ·∇  v   =−eθ  −∇  p  d + 1 √ Gr (∇  ) 2 V  (7.21) Sr  ∂θ  ∂t  + V  ·∇  θ   = 1 Pr √ Gr (∇  ) 2 θ  + (q  )  + βT gβL c p  Sr ∂p  ∂t  + V  ·∇  p   + gβL c p 1 √ Gr Φ  v (7.22) where e is the unit vector in the direction of the gravitational force. Here Sr = L/V c t c is the Strouhal number and q  is nondimensionalized with ρc p (T w −T ∞ )V c /L to yield the dimensionless value (q  )  . It is clear from the equations above that √ Gr replaces Re, which arises as the main dimensionless parameter in forced convection. Similarly, the Eckert number is replaced by gβL/c p , which now determines the importance of the pressure and viscous dissipation terms. The Grashof number indicates the relative importance of the buoyancy term compared to the viscous term. A large value of Gr, therefore, indicates small viscous effects in the momentum equation, similar to the physical significance of Re in forced flow. The Prandtl number Pr represents a comparison between momentum and thermal diffusion. Thus, the Nusselt number may be expressed as a function of the Grashof and Prandtl numbers for steady flows if pressure work and viscous dissipation are neglected. The primes used for denoting dimensionless variables are dropped for convenience in the following sections. 7.3 LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 7.3.1 Vertical Surfaces The classical problem of natural-convection heat transfer from an isothermal heated vertical surface, shown in Fig. 7.1, with the flow assumed to be steady and laminar and the fluid properties (except density) taken as constant, has been of interest to investigators for a very long time. Viscous dissipation effects are neglected, and no BOOKCOMP, Inc. — John Wiley & Sons / Page 534 / 2nd Proofs / Heat Transfer Handbook / Bejan 534 NATURAL CONVECTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [534], (10) Lines: 291 to 352 ——— 0.02415pt PgVar ——— Custom Page (1.0pt) PgEnds: T E X [534], (10) heat source is considered within the flow. Therefore, the problem is considerably sim- plified, although the complications due to the coupled partial differential equations remain. The governing differential equations may be obtained from eqs. (7.15)–(7.17) by using these simplifications. An important method for solving the boundary layer flow over a heated vertical surface is the similarity variable method. A stream function ψ(x,y) is first defined so that it satisfies the continuity equation. Thus, we define ψ by the equations u = ∂ψ ∂y v =− ∂ψ ∂x (7.23) Then the similarity variable η, the dimensionless stream function f , and the temperature θ are defined so as to convert the governing partial differential equations into ordinary differential equations. Gebhart et al. (1988) have presented a general approach to determine the conditions for similarity in a variety of flow circumstances. For flow over a vertical isothermal surface, the similarity variables which have been used in the literature and which may also be derived from this general approach may be written as η = y x  Gr x 4  1/4 ψ = 4νf(η)  Gr x 4  1/4 θ = T − T ∞ T w − T ∞ (7.24) where Gr x = gβx 3 (T w − T ∞ ) ν 2 (7.25) The boundary conditions are: at y = 0: u = v = 0,T = T w ; as y →∞: u → 0,T → T ∞ (7.26) These must also be written in terms of the similarity variables in order to obtain the solution. Note that the velocity component v for y →∞is not specified as zero in order to account for the ambient fluid entrainment into the boundary layer. The governing equations are obtained from the preceding similarity transforma- tions as f  + 3ff  − 2(f  ) 2 + θ = 0 (7.27) θ  Pr + 3f θ  = 0 (7.28) where the primes here indicate differentiation of f(η) and θ(η) with respect to the similarity variable η, one prime representing the first derivative, two primes the second derivatives, and three primes the third derivative. The corresponding boundary conditions are at η = 0: f = f  = 1 − θ = 0; as η →∞: f  → 0, θ → 0 BOOKCOMP, Inc. — John Wiley & Sons / Page 535 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 535 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [535], (11) Lines: 352 to 396 ——— 2.09578pt PgVar ——— Custom Page (1.0pt) * PgEnds: Eject [535], (11) which may be written more concisely as f(0) = f  (0) = 1 − θ(0) = f  (∞) = θ(∞) = 0 (7.29) where the quantity in parentheses indicates the location where the condition is applied. The solution of these equations has been considered by several investigators. Schuh (1948) gave results for various values of the Prandtl number Pr, employing approximate methods. Ostrach (1953) numerically obtained the solution for the Pr range 0.01 to 1000. The velocity and temperature profiles thus obtained are shown in Figs. 7.3 and 7.4. An increase in Pr is found to cause a decrease in the thermal boundary layer thickness and an increase in the absolute value of the temperature gradient at the surface. This is expected from the physical nature of the Prandtl number, which represents the comparison between momentum and thermal diffusion. An increasing value of Pr indicates increasing viscous effects. The dimensionless maximum velocity is also found to decrease and the velocity gradient at the surface to decrease with increasing Pr, indicating the effect of greater viscous forces. The location of this maximum value is found to shift to higher η as Pr is decreased. The velocity boundary layer thickness is also found to increase as Pr is decreased to low values. These trends are expected from the physical mechanisms that govern this boundary layer flow, as dicussed earlier. It is also worth noting that the results indicate the coupling between the velocity and temperature fields, as evidenced by the presence of flow wherever a temperature difference exists, such as the profiles at low Pr. Additional results and discussion on the flow are given in several books; see, for instance, the books by Kaviany (1994), Bejan (1995), and Oosthuizen and Naylor (1999). The heat transfer from the heated surface may be obtained as q  x =−k  ∂T ∂y  0 =−k(T w − T ∞ ) 1 x  Gr x 4  1/4  ∂θ ∂η  0 =  −θ  (0)  k(T w − T ∞ ) x  Gr x 4  1/4 (7.30) The local Nusselt number Nu x is given by Nu x = h x x k = q  x T w − T ∞ x k We have for an isothermal surface Nu x =  −θ  (0)   Gr x 4  1/4 = −θ  (0) √ 2 Gr 1/4 x = φ(Pr)Gr 1/4 x (7.31) where φ(Pr) =  −θ  (0)  / √ 2. Therefore, the local surface heat transfer coefficient h x varies as h x = Bx −1/4 where B = k[−θ  (0)] √ 2  gβ(T w − T ∞ ) ν 2  1/4 BOOKCOMP, Inc. — John Wiley & Sons / Page 536 / 2nd Proofs / Heat Transfer Handbook / Bejan 536 NATURAL CONVECTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [536], (12) Lines: 396 to 402 ——— * 58.25099pt PgVar ——— Normal Page * PgEnds: PageBreak [536], (12) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 123456 7 ␩ y x = Gr 4 x 1/4 ( ( f ux Ј␩ ͌ ()= 2Grν x Pr = 0.01 Pr = 0.01 1 0.72 2 10 100 1000 0 0.1 0.2 0.3 6 8 10 12 14 16 18 20 22 24 f Ј␩() Figure 7.3 Calculated velocity distributions in the boundary layer for flow over an isothermal vertical surface. (From Ostrach, 1953.) BOOKCOMP, Inc. — John Wiley & Sons / Page 537 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 537 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [537], (13) Lines: 402 to 419 ——— 0.71503pt PgVar ——— Normal Page * PgEnds: Eject [537], (13) 0 0.2 0.8 0.4 1.0 0.6 123456 ␩ y x = Gr 4 x 1/4 ( ( ␪␩ Ϫ ()= TT ϱ TT w Ϫ ϱ Pr = 0.01 Pr = 0.01 1 0.72 2 10 100 1000 0 0.2 0.4 0.6 6 8 10 12 14 16 18 20 22 24 ␪␩() Figure 7.4 Calculated temperature distributions in the boundary layer for flow over an isothermal vertical surface. (From Ostrach, 1953.) The average value of the heat transfer coefficient ¯ h may be obtained by averaging the heat transfer over the entire length of the vertical surface, to yield ¯ h = 1 L  L 0 h x dx = 4 3 B L L 3/4 Therefore, BOOKCOMP, Inc. — John Wiley & Sons / Page 538 / 2nd Proofs / Heat Transfer Handbook / Bejan 538 NATURAL CONVECTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [538], (14) Lines: 419 to 472 ——— 1.31822pt PgVar ——— Normal Page PgEnds: T E X [538], (14) Nu = 4 3  −θ  (0)  √ 2 Gr 1/4 = 4 3 φ(Pr) Gr 1/4 = 4 3 Nu L (7.32) The values of φ(Pr) can be obtained from a numerical solution of the governing differential equations. Values obtained at various Pr are listed in Table 7.1. The significance of n and the uniform heat flux data in the table is discussed later. An approximate curve fit to the numerical results for φ(Pr) has been given by Oosthuizen and Naylor (1999) as φ(Pr) =  0.316Pr 5/4 2.44 + 4.88Pr 1/2 + 4.95Pr  1/4 (7.33) It must be mentioned that these results can be used for both heated and cooled surfaces (i.e., T w > or <T ∞ ), yielding respectively a positive q  value for heat transfer from the surface and a negative value for heat transfer to the surface. In several problems of practical interest, the surface from which heat transfer occurs is nonisothermal. The two families of surface temperature variation that give rise to similarity in the governing laminar boundary layer equations have been shown by Sparrow and Gregg (1958) to be the power law and exponential distributions, given as T w − T ∞ = Nx n and T w − T ∞ = Me mx (7.34) TABLE 7.1 Computed Values of the Parameter φ(Pr) for a Vertical Heated Surface φ(Pr) φ  Pr, 1 5  (Isothermal), (Uniform Heat Flux), Pr n = 0 n = 1 5 0 0.600Pr 1/2 0.711Pr 1/2 0.01 0.0570 0.0669 0.72 0.357 — 0.733 — 0.410 1.0 0.401 — 2.0 0.507 — 2.5 — 0.616 5.0 0.675 — 6.7 — 0.829 7.0 0.754 — 10 0.826 0.931 10 2 1.55 1.74 10 3 2.80 — 10 4 5.01 — ∞ 0.503Pr 1/4 0.563Pr 1/4 Source: Gebhart (1973). BOOKCOMP, Inc. — John Wiley & Sons / Page 539 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 539 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [539], (15) Lines: 472 to 517 ——— 0.53522pt PgVar ——— Normal Page PgEnds: T E X [539], (15) where N,M,n, and m are constants. The power law distribution is of particular interest, since it represents many practical circumstances. The isothermal surface is obtained for n = 0. From the expression for q  x , eq. (7.30), it can be shown that q  x varies with x as x (5n−1)/4 . Therefore, a uniform heat flux condition, q  x = constant, arises for n = 1 5 . It can also be shown that physically realistic solutions are obtained for − 3 5 ≤ n<1 (Sparrow and Gregg, 1958; Jaluria, 1980). The governing equations are obtained for the power law case as f  + (n + 3)ff  − 2(n + 1)(f  ) 2 + θ = 0 (7.35) θ  Pr + (n + 3)f θ  − 4nf  θ = 0 (7.36) The local Nusselt number Nu x is obtained as Nu x Gr 1/4 x = −θ  (0) √ 2 = φ(Pr,n) (7.37) The function Nu x /Gr 1/4 x is plotted against n in Fig. 7.5. For n<− 3 5 , the function is found to be negative, indicating the physically unrealistic circumstance of heat transfer to the surface for T w >T ∞ . The surface is adiabatic for n =− 3 5 , which thus represents the case of a line source at the leading edge of a vertical adiabatic surface, so that no energy transfer occurs at the surface for x>0. For the case of uniform heat flux, n = 1 5 and q  x = q  , a constant. Therefore, from eq. (7.30), q  = k  −θ  (0)  N  gβN 4ν 2  1/4 which gives N =  q  k[−θ  (0)]  4/5  4ν 2 gβ  1/5 (7.38) Therefore, for a given heat flux q  , which may be known, for example, from the electrical input into the surface, the temperature of the surface varies as x 1/5 and its magnitude may be determined as a function of the heat flux and fluid properties from eq. (7.38). The parameter −θ  (0) is obtained from a numerical solution of the governing equations for n = 1 5 at the given value of Pr. Some results obtained from Gebhart (1973) are shown in Table 7.1 as φ  Pr, 1 5  . 7.3.2 Inclined and Horizontal Surfaces In many natural convection flows, the thermal input occurs at a surface that is itself curved or inclined with respect to the direction of the gravity field. Consider, first, a flat surface at a small inclination γ from the vertical. Boundary layer approximations, BOOKCOMP, Inc. — John Wiley & Sons / Page 540 / 2nd Proofs / Heat Transfer Handbook / Bejan 540 NATURAL CONVECTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [540], (16) Lines: 517 to 522 ——— 0.09702pt PgVar ——— Normal Page * PgEnds: Eject [540], (16) Ϫ1.0 Ϫ1.0 Ϫ0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 Ϫ0.8 Ϫ0.6 Ϫ0.4 Ϫ0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 Pr 1.0= Pr 0.7= Nu /(Gr /4) xx 1/4 n Figure 7.5 Dependence of the local Nusselt number on the value of n for a power law surface temperature distribution. (From Sparrow and Gregg, 1958.) similar to those for a vertical surface, may be made for this flow. It can be shown that if x is taken along the surface and y normal to it, the continuity and energy equations, eqs. (7.15) and (7.17), respectively, remain unchanged and the x-direction momentum equation becomes u ∂u ∂x + v ∂u ∂y = gβ(T − T ∞ ) cos γ + 1 ρ ∂ ∂y  µ ∂u ∂y  (7.39) BOOKCOMP, Inc. — John Wiley & Sons / Page 541 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES 541 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [541], (17) Lines: 522 to 553 ——— 4.21005pt PgVar ——— Normal Page PgEnds: T E X [541], (17) Therefore, the problem is identical to that for flow over a vertical surface except that g is replaced by g cos γ in the buoyancy term. Therefore, a replacement of g by g cos γ in all the expressions derived earlier for a vertical surface would yield the corresponding results for an inclined surface. This implies using Gr x cos γ for Gr x and assuming equal rates of heat transfer on the two sides of the surface. This is strictly not the case since the buoyancy force is directed away from the surface at the top and toward the surface at the bottom, resulting in differences in boundary layer thicknesses and heat transfer rates. However, this difference is neglected in this approximation. The preceding procedure for obtaining the heat transfer rate from an inclined surface was first suggested theoretically by Rich (1953), and his data are in good agreement with the values predicted. The data obtained by Vliet (1969) for a uniform- flux heated surface in air and in water indicate the validity of this procedure up to inclination angles as large as 60°. Additional experiments have confirmed that the replacement of g by g cos γ in the Grashof number is appropriate for inclination angles up to around 45° and, to a close approximation, up to a maximum angle of 60°. Detailed experimental results on this problem were obtained by Fujii and Imura (1972). They also discuss the separation of the boundary layer for the inclined surface facing upward. The natural convection flow over horizontal surfaces is of considerable importance in a variety of applications, for instance, in the cooling of electronic systems and in flows over the ground and water surfaces. Rotem and Claassen (1969) obtained solutions to the boundary layer equations for flow over a semi-infinite isothermal horizontal surface. Various values of Pr, including the extreme cases of very large and small Pr, were treated. Experimental results indicated the existence of a boundary layer near the leading edge on the upper side of a heated horizontal surface. These boundary layer flows merge near the middle of the surface to generate a wake or plume that rises above the surface. Equations were presented for the power law case, T w −T ∞ = Nx n , and solved for the isothermal case, n = 0. Pera and Gebhart (1972) have considered flow over surfaces slightly inclined from the horizontal. For a semi-infinite horizontal surface with a single leading edge, as shown in Fig. 7.6, the dynamic or motion pressure p d drives the flow. Physically, the upper side of a heated surface heats up the fluid adjacent to it. This fluid becomes lighter than the ambient, if it expands on heating, and rises. This results in a pressure difference, which causes a boundary layer flow over the surface near the leading edge. Similar consid- erations apply for the lower side of a cooled surface. The governing equations are the continuity and energy equations (7.15) and (7.17) and the momentum equations u ∂u ∂x + v ∂u ∂y = 1 ρ ∂ ∂y  µ ∂u ∂y  − 1 ρ ∂p d ∂x (7.40) gβ(T − T ∞ ) = 1 ρ ∂p d ∂y (7.41) This problem may be solved by similarity analysis, as discussed earlier for vertical surfaces. The similarity variables, given by Pera and Gebhart (1972), are . surface heat transfer coefficient h x varies as h x = Bx −1/4 where B = k[−θ  (0)] √ 2  gβ(T w − T ∞ ) ν 2  1/4 BOOKCOMP, Inc. — John Wiley & Sons / Page 536 / 2nd Proofs / Heat Transfer Handbook. used for both heated and cooled surfaces (i.e., T w > or <T ∞ ), yielding respectively a positive q  value for heat transfer from the surface and a negative value for heat transfer to the. vertical surface. (From Ostrach, 1953.) The average value of the heat transfer coefficient ¯ h may be obtained by averaging the heat transfer over the entire length of the vertical surface, to yield ¯ h