Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 82
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An important condition for the validity of these approximations is that β(T − T∞ ) 1(Jaluria, 1980). Therefore, the approximations are valid for small temperature differences if β is essentially unchanged. However, they are not valid near the densitymaximum of water at 4°C, where β is zero and changes sign as the temperature variesacross this value (Gebhart, 1979). Similarly, for large temperature differences encountered in fire and combustion systems, these approximations are generally notapplicable.Another approximation made in the governing equations is the extensively employed boundary layer assumption. The basic concepts involved in using the boundary layer approximation in natural convection flows are very similar to those in forcedflow.
The main difference lies in the fact that the pressure in the region outside theboundary layer is hydrostatic instead of being the externally imposed pressure, as isthe case in forced convection. The velocity outside the layer is only the entrainmentvelocity due to the motion pressure and is not an imposed free stream velocity. However, the basic treatment and analysis are quite similar. It is assumed that the flowand the energy, or mass, transfer, from which it arises, are restricted predominantlyto a thin region close to the surface.
Several experimental studies have corroboratedthis assumption. As a consequence, the gradients along the surface are assumed to bemuch smaller than those normal to it.The main consequences of the boundary layer approximations are that the downstream diffusion terms in the momentum and energy equations are neglected in comparison with the normal diffusion terms. The normal momentum balance is neglectedsince it is found to be of negligible importance compared to the downstream balance.Also, the velocity and thermal boundary layer thicknesses, δ and δT , respectively, aregiven by the order-of-magnitude expressionsδ1=OLGr 1/4BOOKCOMP, Inc. — John Wiley & Sons / Page 531 / 2nd Proofs / Heat Transfer Handbook / Bejan[531], (7)(7.12)Lines: 188 to 216———3.00002pt PgVar———Long PagePgEnds: TEX[531], (7)532123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION1δT=OδPr 1/2(7.13)where Gr is the Grashof number based on a characteristic length L and Pr is thePrandtl number.
These are defined asGr =gβL3 (Tw − T∞ )ν2Pr =µcpν=kα(7.14)where ν is the kinematic viscosity and α the thermal diffusivity of the fluid. Thesedimensionless parameters are important in characterizing the flow, as discussed in thenext section.The resulting boundary layer equations for a two-dimensional vertical flow, withvariable fluid properties except density, for which the Boussinesq approximations areused, are then written as (Jaluria, 1980; Gebhart et al., 1988)∂u ∂v+=0∂x∂y∂u1 ∂∂u∂uu+v= gβ(T − T∞ ) +µ∂x∂yρ ∂y∂y 2∂T∂pa∂T∂∂T∂uρcp u+v=k+ q + βT u+µ∂x∂y∂y∂y∂x∂y(7.15)Lines: 216 to 254(7.16)2.85013pt PgVar——————Long Page(7.17) * PgEnds: Ejectwhere the last two terms in the energy equation are the dominant terms from pressurework and viscous dissipation effects. Here u and v are the velocity components in thex and y directions, respectively.
Although these equations are written for a vertical,two-dimensional flow, similar approximations can be employed for many other flowcircumstances, such as axisymmetric flow over a vertical cylinder and the wake abovea 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 theBoussinesq approximations are generally employed, are often taken as constant.
Theviscous dissipation and pressure work terms are generally small and can be neglected.However, the importance of various terms can be best considered by nondimensionalizing the governing equations and the boundary conditions, as outlined next.7.2.3Dimensionless ParametersTo generalize the natural convection transport processes, a study of the basic nondimensional parameters must be carried out. These parameters are important not onlyin 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 thepresentation of the data for use in simulation, modeling, and design.In natural convection, there is no free stream velocity, and a convection velocityVc is employed for the nondimensionalization of the velocity V, where Vc is given byBOOKCOMP, Inc.
— John Wiley & Sons / Page 532 / 2nd Proofs / Heat Transfer Handbook / Bejan[532], (8)[532], (8)LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445Vc = [gβ(Tw − T∞ )]1/2533(7.18)The governing equations may be nondimensionalized by employing the followingdimensionless variables (indicated by primes):V =Φv = ΦvVVcL2Vc2p =t =pρVc2ttcθ =∇ = L∇T − T∞Tw − T ∞(∇ )2 = L2 ∇ 2(7.19)where tc is a characteristic time scale. The dimensionless equations are obtained as∇ · V = 0(7.20) ∂v1Sr+ V · ∇ v = − eθ − ∇ pd + √ (∇ )2 V(7.21)∂tGr ∂θ1∂p gβL 2 SrSr + V · ∇ p+ V · ∇ θ = √ (∇ ) θ + (q ) + βT∂t cp∂tPr Gr+gβL 1√ ΦvcpGr(7.22)where e is the unit vector in the direction of the gravitational force.Here Sr = L/Vc tc is the Strouhal number and q is nondimensionalized with/L to yield the dimensionless value (q ) .
It is clear from the equaρcp (Tw − T∞ )Vc√tions above that Gr replaces Re, which arises as the main dimensionless parameterin forced convection. Similarly, the Eckert number is replaced by gβL/cp , whichnow determines the importance of the pressure and viscous dissipation terms. TheGrashof number indicates the relative importance of the buoyancy term compared tothe viscous term. A large value of Gr, therefore, indicates small viscous effects inthe momentum equation, similar to the physical significance of Re in forced flow.The Prandtl number Pr represents a comparison between momentum and thermaldiffusion. Thus, the Nusselt number may be expressed as a function of the Grashofand Prandtl numbers for steady flows if pressure work and viscous dissipation areneglected.
The primes used for denoting dimensionless variables are dropped forconvenience in the following sections.7.3LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES7.3.1 Vertical SurfacesThe classical problem of natural-convection heat transfer from an isothermal heatedvertical surface, shown in Fig. 7.1, with the flow assumed to be steady and laminarand the fluid properties (except density) taken as constant, has been of interest toinvestigators for a very long time.
Viscous dissipation effects are neglected, and noBOOKCOMP, Inc. — John Wiley & Sons / Page 533 / 2nd Proofs / Heat Transfer Handbook / Bejan[533], (9)Lines: 254 to 291———2.61385pt PgVar———Long PagePgEnds: TEX[533], (9)534123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONheat source is considered within the flow.
Therefore, the problem is considerably simplified, although the complications due to the coupled partial differential equationsremain. 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 verticalsurface is the similarity variable method. A stream function ψ(x,y) is first defined sothat it satisfies the continuity equation. Thus, we define ψ by the equationsu=∂ψ∂yv=−∂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 intoordinary 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 beenused in the literature and which may also be derived from this general approach maybe written asGr x 1/4T − T∞y Gr x 1/4(7.24)ψ = 4νf (η)θ=η=x44Tw − T ∞whereGr x =gβx 3 (Tw − T∞ )ν2at y = 0: u = v = 0, T = Tw ;as y → ∞:(7.25)The boundary conditions are:u → 0, T → T∞(7.26)These must also be written in terms of the similarity variables in order to obtain thesolution.
Note that the velocity component v for y → ∞ is not specified as zero inorder to account for the ambient fluid entrainment into the boundary layer.The governing equations are obtained from the preceding similarity transformations asf + 3ff − 2(f )2 + θ = 0(7.27)θ+ 3f θ = 0Pr(7.28)where the primes here indicate differentiation of f (η) and θ(η) with respect to thesimilarity variable η, one prime representing the first derivative, two primes thesecond derivatives, and three primes the third derivative.
The corresponding boundaryconditions areat η = 0:f = f = 1 − θ = 0;BOOKCOMP, Inc. — John Wiley & Sons / Page 534 / 2nd Proofs / Heat Transfer Handbook / Bejanas η → ∞:f → 0, θ → 0[534], (10)Lines: 291 to 352———0.02415pt PgVar———Custom Page (1.0pt)PgEnds: TEX[534], (10)LAMINAR NATURAL CONVECTION FLOW OVER FLAT SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445535which may be written more concisely asf (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, employingapproximate methods.
Ostrach (1953) numerically obtained the solution for the Prrange 0.01 to 1000. The velocity and temperature profiles thus obtained are shown inFigs. 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 gradientat the surface. This is expected from the physical nature of the Prandtl number, whichrepresents the comparison between momentum and thermal diffusion. An increasingvalue of Pr indicates increasing viscous effects. The dimensionless maximum ve[535], (11)locity is also found to decrease and the velocity gradient at the surface to decreasewith increasing Pr, indicating the effect of greater viscous forces.
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