BOOKCOMP, Inc. — John Wiley & Sons / Page 542 / 2nd Proofs / Heat Transfer Handbook / Bejan 542 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 [542], (18) Lines: 553 to 579 ——— 0.4231pt PgVar ——— Normal Page PgEnds: T E X [542], (18) Figure 7.6 Natural convection boundary layer flow over a semi-infinite horizontal surface, with the heated surface facing upward. η = y x Gr x 5 1/5 ψ = 5νf(η) Gr x 5 1/5 (7.42) Figure 7.7 shows the computed velocity and temperature profiles for flow over a heated horizontal surface facing upward or a cooled surface facing downward. For a heated surface facing downward or a cooled surface facing upward, a boundary layer type of flow is not obtained for a fluid that expands on heating. This is because the fluid does not flow away from the surface due to buoyancy. The local Nusselt number for horizontal surfaces is given by Pera and Gebhart (1972) for both the isothermal and the uniform-heat-flux surface conditions. The Nusselt number was found to be approximately proportional to Pr 1/4 over the Pr range 0.1 to 100. The expression given for an isothermal surface is Nu x = h x x k = 0.394Gr 1/5 x · Pr 1/4 (7.43) and that for a uniform-flux surface, Nu x,q ,is Nu x,q = h x x k = 0.501Gr 1/5 x · Pr 1/4 (7.44) It can be shown by integrating over the surface that for the isothermal surface, the average Nusselt number is 5 3 times the value of the local Nusselt number at x = L. Therefore, the natural convection heat transfer from inclined surfaces can be treated in terms of small inclinations from the vertical and horizontal positions, de- tailed results on which are available. For intermediate values of γ, an interpolation between these two regimes may be used to determine the resulting heat transfer rate. Numerical methods, such as the finite difference method, can also be used to solve the governing equations to obtain the flow and temperature distributions and the heat transfer rate. This regime has not received as much attention as the horizontal and BOOKCOMP, Inc. — John Wiley & Sons / Page 543 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 543 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 [543], (19) Lines: 579 to 593 ——— 0.12099pt PgVar ——— Normal Page PgEnds: T E X [543], (19) Figure 7.7 Calculated (a) velocity and (b) temperature distribution in natural convection boundary layer flow over a horizontal surface with a uniform heat flux. (From Pera and Gebhart, 1972.) vertical surfaces, although some numerical and experimental results are available such as those of Fujii and Imura (1972). 7.4 EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 7.4.1 Horizontal Cylinder and Sphere Much of the information on laminar natural convection over heated surfaces, dis- cussed in Section 7.3, has been obtained through similarity analysis. However, neither the horizontal cylindrical nor the spherical configuration gives rise to similarity, and therefore several other methods have been employed for obtaining a solution to the governing equations. Among the earliest detailed studies was that by Merk and Prins (1953–54), who employed integral methods with the velocity and thermal boundary layer thicknesses assumed to be equal. The variation of the local Nusselt number with φ, the angular position from the lower stagnation point φ = 0°, is shown in Fig. 7.8 for a horizontal cylinder and also for a sphere. The local Nusselt number Nu φ decreases downstream due to the increase in the boundary layer thickness, which is predicted to be infinite at φ = 180°, resulting in a zero value for Nu φ there. However, Merk and Prins (1953–54) indicated the inapplicability of the analysis for φ ≥ 165° due to boundary layer separation and realignment into a plume flow near the top. BOOKCOMP, Inc. — John Wiley & Sons / Page 544 / 2nd Proofs / Heat Transfer Handbook / Bejan 544 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 [544], (20) Lines: 593 to 605 ——— 0.14706pt PgVar ——— Long Page PgEnds: T E X [544], (20) 0 0 30 30 60 60 90 90 120 120 150 150 180 180 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 (deg) (deg) ()a ()b Nu /(Gr Pr) 1/4 Nu /(Gr Pr) 1/4 Pr = ϱ Pr = ϱ 10 10 1.0 1.0 0.7 0.7 Figure 7.8 Variation of the local Nusselt number with downstream angular position φ for (a) a horizontal cylinder and (b) a sphere. (From Merk and Prins, 1953–54.) The mean value of the Nusselt number Nu is given by Merk and Prins (1953–54) for a horizontal isothermal cylinder as Nu = ¯ hD k = C(Pr)(Gr ·Pr) 1/4 (7.45) where Nu and Gr are based on the diameter D. The constant C(Pr) was calculated as 0.436, 0.456, 0.520, 0.523, and 0.523 for Pr values of 0.7, 1.0, 10.0, 100.0, and ∞, respectively. BOOKCOMP, Inc. — John Wiley & Sons / Page 545 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 545 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 [545], (21) Lines: 605 to 630 ——— 0.15605pt PgVar ——— Long Page * PgEnds: Eject [545], (21) The preceding expression is also suggested for spheres by Merk and Prins (1953– 54), with C(Pr) given for the Pr values of 0.7, 1.0, 10.0, 100.0, and ∞ as 0.474, 0.497, 0.576, 0.592, and 0.595, respectively. There are many other analytical and experimental studies of the natural convection flow over spheres. Since this configu- ration is of particular interest in chemical processes, it has also been studied in detail for mass transfer. Chiang et al. (1964) solved the governing equations, using a series method, and presented heat transfer results. Trends similar to those discussed earlier were obtained. A considerable amount of experimental work has been done on the heat transfer from spheres. Amato and Tien (1972) have discussed such studies and have given the heat transfer correlation as Nu = 2 +0.5(Gr · Pr) 1/4 (7.46) where the constant 2 in the expression can be shown analytically to apply for pure con- duction. Additional correlations for transport from horizontal cylinders and spheres are given later. Work has also been done on the separation of the flow to form a wake near the top of the body. This realignment of the flow can significantly affect the heat transfer rate in the vicinity of the top of a cylinder or a sphere (Jaluria and Gebhart, 1975). 7.4.2 Vertical Cylinder Natural convection flow over vertical cylinders is another important problem, being relevant to many practical applications, such as flow over tubes and rods (as in nuclear reactors), over cylindrical heating elements, and over various closed bodies (including the human body) that can be approximated as a vertical cylinder. For large values of D/L, where D is the diameter of the cylinder and L its height, the flow can be approximated as that over a flat plate, since the boundary layer thickness is small compared to the diameter of the cylinder. As a result, the governing equations are the same as those for a flat plate. However, since this result is based on the boundary layer thickness, which in turn depends on the Grashof number, the deviation of the results obtained for a vertical cylinder from those for a flat plate must be given in terms of D/L and the Grashof number. Sparrow and Gregg (1956) obtained the following criterion for a difference in heat transfer from a vertical cylinder of less than 5% from the flat plate solution, for Pr values of 0.72 and 1.0: D L ≥ 35 Gr 1/4 (7.47) where Gr is the Grashof number based on L. When D/L is not large enough to ignore the effects of curvature, the relevant governing equations must be solved. Sparrow and Gregg (1956) employed similarity methods for obtaining a solution to these equations. Minkowycz and Sparrow (1974) obtained the solution using the local nonsimilarity method. Cebeci (1974) gave results on vertical slender cylinders. LeFevre and Ede (1956) employed an integral method to solve the governing equations and gave the following expression for the Nusselt number Nu: BOOKCOMP, Inc. — John Wiley & Sons / Page 546 / 2nd Proofs / Heat Transfer Handbook / Bejan 546 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 [546], (22) Lines: 630 to 654 ——— -0.12282pt PgVar ——— Normal Page PgEnds: T E X [546], (22) Nu = ¯ hL k = 4 3 7Gr ·Pr 2 5(20 +21Pr) 1/4 + 4(272 +315Pr)L 35(64 +63Pr)D (7.48) where both Nu and Gr are based on the height L of the cylinder. Other studies on vertical axisymmetric bodies are reviewed by Gebhart et al. (1988). 7.4.3 Transients We have so far considered steady natural convection flows in which the velocity and temperature fields do not vary with time. However, time dependence is important in many practical circumstances (Jaluria, 1998). For instance, the change in the thermal condition that generates the natural convection flow could be a sudden or a periodic one, leading to a time-dependent variation in the flow. The startup and shutdown of thermal systems, such as furnaces, ovens, and nuclear reactors, involves a consideration of time-dependent or unsteady natural convection if buoyancy effects are significant. If the heat input at a surface is suddenly changed from zero to a specific value, the steady natural convection flow is eventually obtained following a transient process. As soon as the heat is turned on, the surface starts heating up, this change being essentially a step variation if the thermal capacity of the body is very small. In re- sponse to this sudden change, the fluid adjacent to the surface gets heated, becomes buoyant, and rises, if the fluid expands on heating. However, the flow at a given loca- tion is initially unaffected by flow at other portions of the surface. This implies that the fluid element behaves as isolated, and the heat transfer mechanisms are initially not influenced by the fluid motion. Consequently, the initial transport mechanism is predominantly conduction and can be approximated as a one-dimensional conduction problem up to the leading edge effect, which results from flow originating at the lead- ing edge and which propagates downstream along the flow, is felt at a given location x. The heat transfer rates due to pure conduction being much smaller than those due to convection, it is to be expected that for a step change in the heat flux input, there may initially be an overshoot in the temperature above the steady-state value. Similarly, for a step change in temperature, a lower heat flux is expected initially, ultimately approaching the steady-state value, as the flow itself progresses through a transient regime to steady-state conditions. The preceding discussion implies that at the initial stages of the transient, the solution for a step change in the surface temperature, or in the heat flux, is independent of the vertical location and is of the form obtained for semi-infinite conduction solutions. Employing Laplace transforms for a step change in the heat flux, the solution is obtained as (Ozisik, 1993) ˜ θ = 2q √ αt k exp(−η 2 ) √ π − η erfc(η) (7.49) where η = y/ √ αt,α being the thermal diffusivity of the fluid. Here, erfc(η) is the conjugate of the error function and q is the constant heat flux input imposed at BOOKCOMP, Inc. — John Wiley & Sons / Page 547 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 547 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 [547], (23) Lines: 654 to 679 ——— 0.01164pt PgVar ——— Normal Page PgEnds: T E X [547], (23) time t = 0, starting from a no-flow, zero-heat-input condition. The temperature ˜ θ is simply the physical temperature excess over the initial temperature T ∞ . The heat transfer coefficient h is obtained from the preceding temperature expression, by using Fourier’s law, as h = q [ ˜ θ] 0 = k 2 π αt (7.50) Similarly, for a step change in the surface temperature, the solution is (Ozisik, 1993) T − T ∞ T w − T ∞ = θ = erf(η) (7.51) The velocity profile is obtained by substituting the preceding temperature solution into the momentum equation and solving the resulting equation by Laplace trans- forms to obtain u(y). Numerical solutions to the governing time-dependent boundary layer equations have been obtained by Hellums and Churchill (1962) for a vertical surface subjected to a step change in the surface temperature. The results converge to the steady-state solution at large time and show a minimum in the local Nusselt number during the transient, as shown in Fig. 7.9. An integral method for analyzing unsteady natural convection has also been developed for time-dependent heat input and for finite ther- mal capacity of the surface element. This work has been summarized by Gebhart (1973, 1988) and is based on the analytical and experimental work of Gebhart and co-workers. This analysis is particularly suited to practical problems, since it con- siders the thermal capacity of the bounding material and determines the temperature variation with time over the entire transient regime. 01 2 3 4 5 0 0.2 0.6 0.4 0.8 1.0 νt x 2 ͌Gr x Nu /Gr xx 1/4 Steady state Conduction solution Figure 7.9 Variation of the heat transfer rate with time for a step change in the surface temperature of a vertical plate. (From Hellums and Churchill, 1962.) BOOKCOMP, Inc. — John Wiley & Sons / Page 548 / 2nd Proofs / Heat Transfer Handbook / Bejan 548 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 [548], (24) Lines: 679 to 717 ——— 0.81311pt PgVar ——— Custom Page (6.0pt) PgEnds: T E X [548], (24) Churchill (1975) has given a correlation for transient natural convection from a heated vertical plate, subjected to a step change in the heat flux. The thermal capacity of the plate is taken as negligible, and the local Nusselt number is given as Nu x,q n = πx 2 4αt n/2 + Ra x /10 1 +(0.437/Pr) 9/16 16/9 n/4 (7.52) where Ra x = gβ(T w − T ∞ )x 3 να (7.53) Employing the available experimental information, the appropriate value of n is given as 6. With this value of n, the preceding correlation was found to give Nusselt number values quite close to the experimental results. A temperature overshoot was not considered, since the experimental studies of Gebhart (1973) showed no significant overshoot. For a step change in surface temperature, Churchill and Usagi (1974) have obtained an empirical correlation approximating the entire transient domain. 7.4.4 Plumes, Wakes, and Other Free Boundary Flows In the preceding sections we have considered external natural convection adjacent to heated or cooled surfaces. However, there are many important natural convection flows that, although generated by a heated or cooled surface, move beyond the buoy- ancy input so that they occur without the presence of a solid boundary. Figure 7.10 shows the sketches of a few common flows, which are often termed free boundary flows. Many of these flows are of interest in nature and in pollution and are usually turbulent (Gebhart et al., 1984). Thermal plumes which are assumed to arise from heat input at point or horizontal line sources represent the wakes above heated bodies. The former circumstance is an axisymmetric flow and is generated by a heated body such as a sphere, whereas the latter case is a two-dimensional flow generated by a long, thin, heat source such as an electric heater. Figure 7.11 shows a sketch of the flow in a two-dimensional plume, ( ) Plumea ( ) Thermalb ( ) Starting plumec ( ) Jetd Nozzle Figure 7.10 Common free boundary flows. BOOKCOMP, Inc. — John Wiley & Sons / Page 549 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES 549 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 [549], (25) Lines: 717 to 725 ——— 4.77504pt PgVar ——— Custom Page (6.0pt) * PgEnds: Eject [549], (25) Figure 7.11 (a) Sketch and coordinate system for a two-dimensional thermal plume arising from a horizontal line source. Also shown are the calculated (b) velocity and (c) temperature distributions. (From Gebhart et al., 1970.) along with a coordinate system which is similar to that for flow over a heated vertical surface. Using the nomenclature and analysis given earlier for a vertical surface, it can easily be seen that n = 3 5 because the centerline, y = 0, is adiabatic due to symmetry. Also, the vertical velocity is not zero there but a maximum, since a no- shear condition applies there rather than the no-slip condition. Then the similarity variables given earlier in eq. (7.24) may be used with T w −T ∞ = Nx −3/5 , where T w is the centerline temperature. The governing equations are then obtained from eqs. (7.35) and (7.36) as f + 12 5 ff − 4 5 (f ) 2 + θ = 0 (7.54) θ Pr + 12 5 (f θ + f θ) = 0 (7.55) BOOKCOMP, Inc. — John Wiley & Sons / Page 550 / 2nd Proofs / Heat Transfer Handbook / Bejan 550 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 [550], (26) Lines: 725 to 769 ——— 5.31909pt PgVar ——— Normal Page PgEnds: T E X [550], (26) The boundary conditions for a two-dimensional plume are θ (0) = f(0) = f (0) = 1 −θ(0) = f (∞) = 0 (7.56) The first and third conditions arise from symmetry at y = 0, or η = 0. The overall energy balance can also be written in terms of the total convected energy in the boundary layer q c as q c = ∞ −∞ ρc p u(T − T ∞ )dy = 4µc p N gβN 4ν 2 1/4 x (5n+3)/4 ∞ −∞ f (η)θ(η)dη (7.57) Therefore, the x dependence drops out for n =− 3 5 and q c represents the total energy input Q per unit length of the line source. Then the constant N in the centerline temperature distribution, T w − T ∞ = Nx −3/5 , is obtained from eq. (7.57) as N = Q 4 64gβρ 2 µ 2 c 4 p I 4 1/5 (7.58) where I is the integral I = ∞ −∞ f (η)θ(η)dη (7.59) This integral I can be determined numerically by solving the governing similarity equations and then evaluating the integral. Values of I at several Prandtl numbers are given by Gebhart et al. (1970). For instance, the values of I calculated at Pr = 0.7, 1.0, 6.7, and 10.0 are given as 1.245, 1.053, 0.407, and 0.328, respectively. Figure 7.11 also presents some calculated velocity and temperature profiles in a two-dimensional plume from Gebhart et al. (1970). These results can be used to calculate the velocity and thermal boundary layer thicknesses, which can be shown to vary as x 2/5 , and the centerline velocity, which can be shown to increase with x as x 1/5 . The centerline temperature, which decays with x as x −3/5 , can be calculated by obtaining the value of N from eq. (7.58) for a given Pr and heat input Q. Note that this analysis applies for a line source on a vertical adiabatic surface as well, since q c is constant in this case, too, resulting in n =− 3 5 (Jaluria and Gebhart, 1977). Therefore, the governing equations are eqs. (7.54) and (7.55). However, the boundary condition f (0) = 0 is replaced by f (0) = 0 because of the no-slip condition at the wall. Similarly, a laminar axisymmetric plume can be analyzed to yield the temperature and velocity distributions (Jaluria, 1985a). The wake rising above a finite heated body is expected finally to approach the conditions of an axisymmetric plume far downstream of the heat input as the effect of the size of the source diminishes. However, as mentioned earlier, most of these flows are turbulent in nature and in most practical applications. Simple integral analyses have been carried out, along BOOKCOMP, Inc. — John Wiley & Sons / Page 551 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 551 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 [551], (27) Lines: 769 to 795 ——— -3.41595pt PgVar ——— Normal Page PgEnds: T E X [551], (27) with appropriate experimentation, to understand and characterize these flows (Turner, 1973). Detailed numerical studies have also been carried out on a variety of free boundary flows to provide results that are of particular interest in pollution, fires, and environmental processes. 7.5 INTERNAL NATURAL CONVECTION In the preceding sections we have considered largely external natural convection in which the ambient medium away from the flow is extensive and stationary. However, there are many natural convection flows that occur within enclosed regions, such as flows in rooms and buildings, cooling towers, solar ponds, and furnaces. The flow domain may be completely enclosed by solid boundaries or may be a partial enclosure with openings through which exchange with the ambient occurs. There has been growing interest and research activity in buoyancy-induced flows arising in partial or complete enclosures. Much of this interest has arisen because of applications such as cooling of electronic circuitry (Jaluria, 1985b; Incropera, 1999), building fires (Emmons, 1978, 1980), materials processing (Jaluria, 2001), geothermal energy extraction (Torrance, 1979), and environmental processes. The basic mechanisms and heat transfer results in internal natural convection have been reviewed by several researchers, such as Yang (1987) and Ostrach (1988). Some of the important basic considerations are presented here. 7.5.1 Rectangular Enclosures The two-dimensional natural convection flow in a rectangular enclosure, with the two vertical walls at different temperatures and the horizontal boundaries taken as adiabatic or at a temperature varying linearly between those of the vertical boundaries, has been thoroughly investigated over the past three decades. Figure 7.12a shows a typical vertical enclosure with the two vertical walls at temperatures T h and T c and the horizontal surfaces being taken as insulated. The dimensionless governing equations may be written as 1 Pr V ·∇ω =∇ 2 ω −Ra ∂θ ∂Y (7.60) V ·∇θ =∇ 2 θ (7.61) where the vorticity ω =−∇ 2 ψ, θ = (T − T c )/(T h − T c ), Y = y/d, and the Rayleigh number Ra = Gr · Pr. The width d of the enclosure and the temperature difference T h − T c are taken as characteristic quantities for nondimensionalization of the variables. The velocity is nondimensionalized by α/d here. This problem has been investigated numerically and experimentally for a wide range of Rayleigh and Prandtl numbers and of the aspect ratio A = H/d. Figure 7.12b shows the calculated isotherms at a moderate value of Pr. A recirculating flow arises and distorts the temperature field resulting from pure conduction. . resulting heat transfer rate. Numerical methods, such as the finite difference method, can also be used to solve the governing equations to obtain the flow and temperature distributions and the heat transfer. as 0.436, 0. 456, 0.520, 0.523, and 0.523 for Pr values of 0.7, 1.0, 10.0, 100.0, and ∞, respectively. BOOKCOMP, Inc. — John Wiley & Sons / Page 545 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL. the error function and q is the constant heat flux input imposed at BOOKCOMP, Inc. — John Wiley & Sons / Page 547 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTERNAL LAMINAR NATURAL CONVECTION