Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 86
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(From Gebhart et al., 1970.)along with a coordinate system which is similar to that for flow over a heated verticalsurface. Using the nomenclature and analysis given earlier for a vertical surface, itcan easily be seen that n = 35 because the centerline, y = 0, is adiabatic due tosymmetry. Also, the vertical velocity is not zero there but a maximum, since a noshear condition applies there rather than the no-slip condition. Then the similarityvariables given earlier in eq. (7.24) may be used with Tw − T∞ = N x −3/5 , where Twis the centerline temperature. The governing equations are then obtained from eqs.(7.35) and (7.36) asf +12 4 2ff − (f ) + θ = 055(7.54)θ12+ (f θ + f θ) = 0Pr5(7.55)BOOKCOMP, Inc. — John Wiley & Sons / Page 549 / 2nd Proofs / Heat Transfer Handbook / Bejan550123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONThe boundary conditions for a two-dimensional plume areθ (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 overallenergy balance can also be written in terms of the total convected energy in theboundary layer qc as ∞qc =ρcp u(T − T∞ )dy−∞= 4µcp NgβN4ν21/4x (5n+3)/4∞−∞f (η)θ(η)dη(7.57)Therefore, the x dependence drops out for n = − 35 and qc represents the total energyinput Q per unit length of the line source. Then the constant N in the centerlinetemperature distribution, Tw − T∞ = Nx −3/5 , is obtained from eq. (7.57) as1/5Q4N=(7.58)64gβρ2 µ2 cp4 I 4where I is the integralI=∞−∞f (η)θ(η) dηLines: 725 to 769———5.31909pt PgVar———Normal PagePgEnds: TEX(7.59)This integral I can be determined numerically by solving the governing similarityequations and then evaluating the integral. Values of I at several Prandtl numbersare 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 ina two-dimensional plume from Gebhart et al. (1970). These results can be used tocalculate the velocity and thermal boundary layer thicknesses, which can be shownto vary as x 2/5 , and the centerline velocity, which can be shown to increase with x asx 1/5 . The centerline temperature, which decays with x as x −3/5 , can be calculated byobtaining the value of N from eq. (7.58) for a given Pr and heat input Q.
Note thatthis analysis applies for a line source on a vertical adiabatic surface as well, since qc isconstant in this case, too, resulting in n = − 35 (Jaluria and Gebhart, 1977). Therefore,the governing equations are eqs. (7.54) and (7.55).
However, the boundary conditionf (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 temperatureand velocity distributions (Jaluria, 1985a). The wake rising above a finite heatedbody is expected finally to approach the conditions of an axisymmetric plume fardownstream 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 inmost practical applications. Simple integral analyses have been carried out, alongBOOKCOMP, Inc. — John Wiley & Sons / Page 550 / 2nd Proofs / Heat Transfer Handbook / Bejan[550], (26)[550], (26)INTERNAL NATURAL CONVECTION123456789101112131415161718192021222324252627282930313233343536373839404142434445551with appropriate experimentation, to understand and characterize these flows (Turner,1973). Detailed numerical studies have also been carried out on a variety of freeboundary flows to provide results that are of particular interest in pollution, fires, andenvironmental processes.7.5INTERNAL NATURAL CONVECTIONIn the preceding sections we have considered largely external natural convection inwhich the ambient medium away from the flow is extensive and stationary.
However,there are many natural convection flows that occur within enclosed regions, such asflows in rooms and buildings, cooling towers, solar ponds, and furnaces. The flowdomain may be completely enclosed by solid boundaries or may be a partial enclosurewith openings through which exchange with the ambient occurs. There has beengrowing interest and research activity in buoyancy-induced flows arising in partialor complete enclosures.
Much of this interest has arisen because of applicationssuch as cooling of electronic circuitry (Jaluria, 1985b; Incropera, 1999), buildingfires (Emmons, 1978, 1980), materials processing (Jaluria, 2001), geothermal energyextraction (Torrance, 1979), and environmental processes. The basic mechanisms andheat transfer results in internal natural convection have been reviewed by severalresearchers, such as Yang (1987) and Ostrach (1988).
Some of the important basicconsiderations are presented here.7.5.1Rectangular EnclosuresThe two-dimensional natural convection flow in a rectangular enclosure, with thetwo vertical walls at different temperatures and the horizontal boundaries taken asadiabatic 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 atypical vertical enclosure with the two vertical walls at temperatures Th and Tc and thehorizontal surfaces being taken as insulated. The dimensionless governing equationsmay be written as1∂θV · ∇ω = ∇ 2 ω − RaPr∂YV · ∇θ = ∇ 2 θ(7.60)(7.61)where the vorticity ω = −∇ 2 ψ, θ = (T − Tc )/(Th − Tc ), Y = y/d, and theRayleigh number Ra = Gr · Pr. The width d of the enclosure and the temperaturedifference Th − Tc are taken as characteristic quantities for nondimensionalizationof the variables. The velocity is nondimensionalized by α/d here.
This problem hasbeen investigated numerically and experimentally for a wide range of Rayleigh andPrandtl numbers and of the aspect ratio A = H /d. Figure 7.12b shows the calculatedisotherms at a moderate value of Pr. A recirculating flow arises and distorts thetemperature field resulting from pure conduction.BOOKCOMP, Inc. — John Wiley & Sons / Page 551 / 2nd Proofs / Heat Transfer Handbook / Bejan[551], (27)Lines: 769 to 795———-3.41595pt PgVar———Normal PagePgEnds: TEX[551], (27)552123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION[552], (28)Lines: 795 to 803Figure 7.12 (a) Typical vertical rectangular enclosure; (b) calculated isotherms for Pr = 1.0at Ra = 2 × 104. (From Elder, 1966.)At small values of Ra, Ra ≤ 1000, there is little increase in the heat transferover that due to conduction alone, for which the Nusselt number Nu = hd/k = 1.However, as Ra increases, several flow regimes have been found to occur, resultingin a significant increase in the Nusselt number.
In laminar flow, these regimes includethe conduction, transition, and boundary layer regimes. The conduction regime ischaracterized by a linear temperature variation in the central region of the enclosure.In the boundary layer regime, thin boundary layers appear along the vertical walls,with horizontal temperature uniformity between the two layers. In the transitionregime, the two boundary regions are thicker and the isothermal interior region doesnot appear.
The characteristics are seen clearly in Figs. 7.13 and 7.14. As Ra increasesfurther, secondary flows appear, as characterized by additional cells in the flow, andtransition to turbulence occurs at still higher Ra. Detailed studies of the different flowregimes and the corresponding heat transfer have been carried out, along with threedimensional transport, for wide ranges of the governing parameters, as reviewed byGebhart et al.
(1988).An interesting solution was obtained by Batchelor (1954) for large aspect ratios,H /d → ∞. The flow can then be assumed to be fully developed, with the velocityand temperature as functions only of the horizontal coordinate y. This problem canbe solved analytically to yield the velocity U and temperature θ distributions asU=RaY (1 − Y )(1 − 2Y ) and12BOOKCOMP, Inc. — John Wiley & Sons / Page 552 / 2nd Proofs / Heat Transfer Handbook / Bejanθ=1−Y(7.62)———12.69101pt PgVar———Normal Page* PgEnds: Eject[552], (28)123456789101112131415161718192021222324252627282930313233343536373839404142434445BOOKCOMP, Inc. — John Wiley & Sons / Page 553 / 2nd Proofs / Heat Transfer Handbook / Bejan20x = 34.2 cm33.030.533.01020.315.210.02030.54025.42020.315.210.00605.12.54402.541.27201.275.1100T ⫺ Tc (°C)T ⫺ Tc (°C)25.40714y ⫻ 103 (m)(a)031.729.224.119.04013.96308.93.8200714y ⫻ 103 (m)(b)4030201001.271000x = 34.2 cm60T ⫺ Tc (°C)x = 34.2 cm01428y ⫻ 103 (m)(c)Figure 7.13 Measured temperature distributions in air in a vertical rectangular enclosure: (a) conduction regime; (b) transition regime; (c) boundarylayer regime.
(From Eckert and Carlson, 1961.)553[553], (29)Lines: 803 to 812———* 528.0pt PgVar———Normal Page* PgEnds: PageBreak[553], (29)554123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION[554], (30)Lines: 812 to 827———0.25099pt PgVar———Normal PagePgEnds: TEX[554], (30)Figure 7.14 Measured velocity distributions at midheight in a rectangular enclosure at various values of Ra. (From Elder, 1965.)where the nondimensionalization given earlier for eqs. (7.60) and (7.61) is used.Therefore, the temperature distribution is independent of the flow and the Nusseltnumber Nu is 1.0.7.5.2Other ConfigurationsMany other flow configurations in internal natural convection, besides the rectangular enclosure, have been studied because of both fundamental and applied interest.Inclined and horizontal enclosures have been of interest in solar energy utilizationand have received a lot of attention. Horizontal layers, with heating from below,provide the classical Benard problem whose instability has been of interest to manyresearchers over many decades.
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