Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 85
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This realignment of the flow can significantly affectthe heat transfer rate in the vicinity of the top of a cylinder or a sphere (Jaluria andGebhart, 1975).[545], (21)Lines: 605 to 630———0.15605pt PgVar———Long PageNatural convection flow over vertical cylinders is another important problem, being * PgEnds: Ejectrelevant to many practical applications, such as flow over tubes and rods (as in nuclearreactors), over cylindrical heating elements, and over various closed bodies (including[545], (21)the human body) that can be approximated as a vertical cylinder.
For large valuesof D/L, where D is the diameter of the cylinder and L its height, the flow can beapproximated as that over a flat plate, since the boundary layer thickness is smallcompared to the diameter of the cylinder. As a result, the governing equations are thesame as those for a flat plate. However, since this result is based on the boundary layerthickness, which in turn depends on the Grashof number, the deviation of the resultsobtained for a vertical cylinder from those for a flat plate must be given in termsof D/L and the Grashof number. Sparrow and Gregg (1956) obtained the followingcriterion for a difference in heat transfer from a vertical cylinder of less than 5% fromthe flat plate solution, for Pr values of 0.72 and 1.0:7.4.2 Vertical CylinderD35≥ 1/4LGr(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 relevantgoverning equations must be solved.
Sparrow and Gregg (1956) employed similaritymethods for obtaining a solution to these equations. Minkowycz and Sparrow (1974)obtained the solution using the local nonsimilarity method. Cebeci (1974) gave resultson vertical slender cylinders. LeFevre and Ede (1956) employed an integral methodto solve the governing equations and gave the following expression for the Nusseltnumber Nu:BOOKCOMP, Inc.
— John Wiley & Sons / Page 545 / 2nd Proofs / Heat Transfer Handbook / Bejan546123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONNu =1/4h̄L4(272 + 315Pr)L47Gr · Pr 2+=k3 5(20 + 21Pr)35(64 + 63Pr)D(7.48)where both Nu and Gr are based on the height L of the cylinder. Other studies onvertical axisymmetric bodies are reviewed by Gebhart et al. (1988).7.4.3 TransientsWe have so far considered steady natural convection flows in which the velocity andtemperature fields do not vary with time.
However, time dependence is importantin many practical circumstances (Jaluria, 1998). For instance, the change in thethermal condition that generates the natural convection flow could be a sudden ora periodic one, leading to a time-dependent variation in the flow. The startup andshutdown of thermal systems, such as furnaces, ovens, and nuclear reactors, involvesa consideration of time-dependent or unsteady natural convection if buoyancy effectsare significant.If the heat input at a surface is suddenly changed from zero to a specific value, thesteady 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 beingessentially a step variation if the thermal capacity of the body is very small.
In response to this sudden change, the fluid adjacent to the surface gets heated, becomesbuoyant, and rises, if the fluid expands on heating. However, the flow at a given location is initially unaffected by flow at other portions of the surface. This implies thatthe fluid element behaves as isolated, and the heat transfer mechanisms are initiallynot influenced by the fluid motion. Consequently, the initial transport mechanism ispredominantly conduction and can be approximated as a one-dimensional conductionproblem up to the leading edge effect, which results from flow originating at the leading edge and which propagates downstream along the flow, is felt at a given locationx. The heat transfer rates due to pure conduction being much smaller than those due toconvection, it is to be expected that for a step change in the heat flux input, there mayinitially 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, ultimatelyapproaching the steady-state value, as the flow itself progresses through a transientregime to steady-state conditions.The preceding discussion implies that at the initial stages of the transient, thesolution for a step change in the surface temperature, or in the heat flux, is independentof the vertical location and is of the form obtained for semi-infinite conductionsolutions. Employing Laplace transforms for a step change in the heat flux, thesolution is obtained as (Ozisik, 1993)√ 2q αt exp(−η2 )θ̃ =− η erfc(η)(7.49)√kπ√where η = y/ αt, α being the thermal diffusivity of the fluid. Here, erfc(η) is theconjugate of the error function and q is the constant heat flux input imposed atBOOKCOMP, Inc. — John Wiley & Sons / Page 546 / 2nd Proofs / Heat Transfer Handbook / Bejan[546], (22)Lines: 630 to 654———-0.12282pt PgVar———Normal PagePgEnds: TEX[546], (22)EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCEStime t = 0, starting from a no-flow, zero-heat-input condition.
The temperature θ̃is simply the physical temperature excess over the initial temperature T∞ . The heattransfer coefficient h is obtained from the preceding temperature expression, by usingFourier’s law, ask πq=h=(7.50)2 αt[θ̃]0Similarly, for a step change in the surface temperature, the solution is (Ozisik, 1993)T − T∞= θ = erf(η)Tw − T ∞(7.51)The velocity profile is obtained by substituting the preceding temperature solutioninto the momentum equation and solving the resulting equation by Laplace transforms to obtain u(y).Numerical solutions to the governing time-dependent boundary layer equationshave been obtained by Hellums and Churchill (1962) for a vertical surface subjectedto a step change in the surface temperature.
The results converge to the steady-statesolution at large time and show a minimum in the local Nusselt number during thetransient, as shown in Fig. 7.9. An integral method for analyzing unsteady naturalconvection has also been developed for time-dependent heat input and for finite thermal 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 andco-workers. This analysis is particularly suited to practical problems, since it considers the thermal capacity of the bounding material and determines the temperaturevariation with time over the entire transient regime.1.00.8Nu x /Grx1/41234567891011121314151617181920212223242526272829303132333435363738394041424344455470.6Steady state0.4Conduction solution0.200123νt 公Grxx245Figure 7.9 Variation of the heat transfer rate with time for a step change in the surfacetemperature of a vertical plate.
(From Hellums and Churchill, 1962.)BOOKCOMP, Inc. — John Wiley & Sons / Page 547 / 2nd Proofs / Heat Transfer Handbook / Bejan[547], (23)Lines: 654 to 679———0.01164pt PgVar———Normal PagePgEnds: TEX[547], (23)548123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONChurchill (1975) has given a correlation for transient natural convection from aheated vertical plate, subjected to a step change in the heat flux.
The thermal capacityof the plate is taken as negligible, and the local Nusselt number is given asn/4 2 n/2 nπxRax /10+ (7.52)Nux,q =16/94αt1 + (0.437/Pr)9/16whereRax =gβ(Tw − T∞ )x 3να(7.53)Employing the available experimental information, the appropriate value of n is givenas 6. With this value of n, the preceding correlation was found to give Nusselt numbervalues quite close to the experimental results. A temperature overshoot was notconsidered, since the experimental studies of Gebhart (1973) showed no significantovershoot.
For a step change in surface temperature, Churchill and Usagi (1974) haveobtained an empirical correlation approximating the entire transient domain.[548], (24)Lines: 679 to 717———7.4.40.81311pt PgVarPlumes, Wakes, and Other Free Boundary FlowsIn the preceding sections we have considered external natural convection adjacentto heated or cooled surfaces. However, there are many important natural convectionflows that, although generated by a heated or cooled surface, move beyond the buoyancy input so that they occur without the presence of a solid boundary. Figure 7.10shows the sketches of a few common flows, which are often termed free boundaryflows.
Many of these flows are of interest in nature and in pollution and are usuallyturbulent (Gebhart et al., 1984).Thermal plumes which are assumed to arise from heat input at point or horizontalline sources represent the wakes above heated bodies. The former circumstance is anaxisymmetric flow and is generated by a heated body such as a sphere, whereas thelatter case is a two-dimensional flow generated by a long, thin, heat source such as anelectric heater.
Figure 7.11 shows a sketch of the flow in a two-dimensional plume,Nozzle(a) Plume(b) Thermal(c) Starting plumeFigure 7.10 Common free boundary flows.BOOKCOMP, Inc. — John Wiley & Sons / Page 548 / 2nd Proofs / Heat Transfer Handbook / Bejan(d ) Jet———Custom Page (6.0pt)PgEnds: TEX[548], (24)EXTERNAL LAMINAR NATURAL CONVECTION FLOW IN OTHER CIRCUMSTANCES123456789101112131415161718192021222324252627282930313233343536373839404142434445549[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 arisingfrom a horizontal line source. Also shown are the calculated (b) velocity and (c) temperaturedistributions.
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