Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 67
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The relationship between x ÷ and x* is simply given byx* = x+/Pr(5.14)FORCED CONVECTION,INTERNALFLOWIN DUCTS5.5Corresponding to the hydrodynamic entrance length, the thermal entrance length Lth isdefined as the axial distance needed to achieve a value of the local Nusselt number Nux, whichis 1.05 times the fully developed Nusselt number value. The dimensionless thermal entrancelength is expressed as Lth = Lth/(Dh Pe).Thermal Boundary ConditionsThe following thermal boundary conditions are the most important and frequently encountered in practical use:1. Uniform wall temperature, denoted by 03.
This boundary condition is present when theduct has a constant wall temperature in both the circumferential and axial directions. Uniform wall temperature has frequent application in condensers, evaporators, and automotive radiators with high flow rates.2. Convective boundary condition, denoted by (~.
The convection between the duct wall andthe environment is taken into consideration in the (~ boundary condition. The duct walltemperature is considered to be uniform in the axial direction. The practical applicationsare the same as those in the ~ boundary condition, except for the case of finite thermalresistance in the wall.3. Radiative boundary condition, denoted by @.
This boundary condition involves radiativeheat transfer from the duct to the environment. The wall heat flux is proportional to thefourth power of the absolute wall temperature, and the environment temperature is uniform in the axial direction. This boundary condition can be found in high-temperature systems such as space radiators, liquid-metal exchangers, and heat exchangers involvingheat-radiating gases.4.
Uniform wall heat flux axially, but uniform wall temperature circumferentially, denotedby(~. This boundary condition is found in electric resistance heating, nuclear heating, andcounterflow heat exchangers, in which two fluids have nearly the same fluid capacity ratesand the wall is highly conductive.5. Uniform wall heat flux axially and circumferentially, denoted by @.
This boundary condition has applications that are similar to those listed for the (~ boundary condition exceptthat the thermal conductivity of the wall material is low and the wall thickness is uniform.6. Conductive boundary condition, denoted by @. This boundary condition refers to the axialuniform wall heat flux and finite heat condition along the wall circumference. The sameapplications can be found as those for the (~ boundary condition except for the existenceof heat conduction in the circumferential direction.7. Exponential wall heat flux, denoted by @. This boundary condition represents a duct withcircumferentially constant wall temperature and exponentially varying wall heat flux alongthe axial direction.
Exponential wall heat flux can be seen in parallel and counterflow heatexchangers with an appropriate value of m.The preceding boundary conditions are applicable to both singly connected and doubly connected ducts. Detailed descriptions of the various boundary conditions are available in Shahand London [1] and Shah and Bhatti [2].CIRCULAR DUCTSIn industry, circular ducts are widely used in various applications. Fluid and heat transferinside circular ducts have been analyzed in great detail.
A discussion of laminar and turbulentflows and heat transfer in circular ducts is presented in the following sections.5.6CHAPTER FIVELaminar FlowFor a circular duct with a diameter of 2a, the characteristics of laminar flow and heat transferfor four kinds of flows, namely, fully developed, hydrodynamically developing, thermallydeveloping, and simultaneously developing, are outlined in the following sections.Fully Developed FlowVelocity Profile and Friction Factor.
The velocity profile of fully developed laminar flowof a constant-property fluid in a circular duct with an origin at the duct axis is given by theHagen-Poiseuille parabolic profile, as follows:[u _ 2 1Urn(5.15)where um can be obtained by the following equation:a( x)um = - ~a2(5.16)The product of the friction factor and the Reynolds number for fully developed flow in acircular duct is found to be constant. This is obtained from Eq. 5.15 as:f R e =16(5.17)Heat Transfer on Walls With Uniform Temperature. For this boundary condition, denotedas (t), temperature distribution in a circular duct for fully developed laminar flow in theabsence of flow work, thermal energy sources, and fluid axial conduction has been solved byBhatti [3] and presented by Shah and Bhatti [2], as follows:T,-T-( r ) 2~Tw- T,~ =,~_-oC2, a(5.18)where the coefficients C2, are given byCo = 1x~0C2 = - -~-=-1.828397~2 z (C2, -4 - C2,,_2)Cz. = (2n)k0= 2.7043644199(5.19)The bulk mean temperature of the fluid can be obtained by:r~-rm= 0.819048 exp(-2~,~x ÷)(5.20)It should be noted that Eq.
5.20 is valid for x* > 0.0335 [3].The Nusselt number corresponding to Eq. 5.18 is as follows:~0Nut = T = 3.657(5.21)The P6clet number is introduced to consider the effect of fluid axial conduction. It hasbeen found that the axial conduction of fluid can be ignored when the P6clet number isF O R C E D CONVECTION, I N T E R N A L FLOW IN DUCTS5.7greater than 10. For a small P6clet number, the following formulas by Michelsen and Villadsen [4] are recommended:Nu -I- 0 04 9 e +[3.6568(1 + 1.227/pe2 +iii))Pe < 1.5Pe > 5(5.22)The Brinkmann number Br = (~tUZm)/[k(Tw.,, - Te)] is introduced to account for the influence of viscous dissipation, such as heating or cooling of the fluid due to internal friction inhigh-velocity flow, highly viscous fluid, or in cases in which viscous dissipation cannot beignored.
When viscous dissipation is considered, the asymptotic Nusselt number in a verylong pipe, found by Ou and Cheng [5], is 9.6 and independent of the Brinkmann number.Heat Transfer on Walls With Uniform Heat Flux. For circular ducts with symmetricalheating, the same heat transfer results for fully developed flow and developing flow areobtained for boundary conditions ~ through ~ .
Therefore, the uniform wall heat flux boundary conditions are simply designated as the @ boundary condition. Shah and Bhatti [2] havederived the temperature distribution and Nusselt number by recasting the results reported byTyagi [6] for heat transfer in circular ducts.
These follow:T~ - Tm = 6 1 -44 + 37q'~ Dh192Nun = k(Tw - Tm) - 44 + 37T= S* + 64 Br'(5.23)(5.24)where S* is the dimensionless thermal energy source number (S* = SDh/q~) and Br' is thedimensionless Brinkmann number, defined as Br' = ~tu 2m/qwDh,,,for the uniform wall heat fluxboundary condition. For the case of negligible viscous dissipation and no thermal energysources (7 = 0), the Nusselt number can be obtained from Eq. 5.24:NUll = 4.364(5.25)Eq. 5.24 can also be applied in the case of finite axial fluid conduction.Heat Transfer on Convection Duct Walls. For this boundary condition, denoted as ~ , thewall temperature is considered to be constant in the axial direction, and the duct has convection with the environment.
An external heat transfer coefficient is incorporated to representthis case. The dimensionless Biot number, defined as Bi = heDh/kw, reflects the effect of thewall thermal resistance, induced by external convection.For cases in which the external heat transfer coefficient he is constant, Hickman [7] developed the following formula to calculate the Nusselt number:4.3636 + BiNuT3 = 1 + 0.2682Bi(5.26)From the preceding equation, the uniform wall heat flux and uniform wall temperature can beobtained when Bi = 0 and Bi = o% respectively.Heat Transfer on Radiative Duct Walls. Heat transfer on radiative duct walls is of theboundary condition type. Kadaner et al.
[8] obtained the following equation for the fullydeveloped Nusselt number under the @ boundary condition:NUT4 =8.728 + 3.66Sk (Ta/Te) 32 + Sk (Ta/Te)3(5.27)where Sk is the dimensionless Stark number, defined as Sk =Ew(yT3eDh/k, and Ta and Te are theabsolute temperatures of the external environment and the internal fluid at the location ofCHAPTERFIVE5.8the impingement of the radiation flux, respectively.
In Eq. 5.27, when Sk = 0 and Sk = 0% NUT4reduces to NUll and Nux.Heat Transfer on the Walls With Exponentially Varying Heat Flux. Exponentially varyingwall heat flux is delineated by the @ boundary condition and represented by q'w'= q~' exp(mx*),where the exponent m can have both positive and negative values corresponding to the exponential growth or decay of the wall heat flux.Shah and L o n d o n [1] have obtained the Nusselt n u m b e r for -51.36 < m < 100 as the following correlation with a m a x i m u m error of 3 percent:NUll5 = 4.3573 + 0.0424m - 2.8368 x 10-8m 2 + 3.6250 x 10-6m 3- 7.6497 x 10-8m 4 + 9.1222 x 10-4m 5 - 3.8446 x 10-12m6(5.28)W h e n m = 0 and m = -14.627, the @ boundary condition corresponds to uniform wall heatflux and uniform wall temperature, respectively. Recently, Piva [9] obtained the fully developed Nusselt n u m b e r for the @ boundary condition based on confluent hypergeometric functions.
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