Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 66
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J. HeatMass Transfer (38/18): 3329-3339, 1995.302. B. Gebhart, Y. Jaluria, R. Mahajan, and R. Sammakia, Bouyancy-lnduced Flows and Transport,Hemisphere, 1998.CHAPTER 5FORCED CONVECTION,IN'rl RNAL FLOW IN DUffl'SM. A. Ebadian and Z. F. DongFlorida International UniversityINTRODUCTIONScope of the ChapterThis chapter deals with internal flow and heat transfer in ducts such as circular pipes, rectangular pipes, and other pipes with irregular cross sections. The scope of the chapter isrestricted to the study of the steady, incompressible flow of newtonian fluids.
The effects ofnatural convection, phase change, mass transfer, and chemical reactions have been ignored.This chapter is organized according to duct geometry. Hydrodynamics and heat transfer characteristics will be presented for each duct in terms of mathematical expressions, tables, andgraphs in the corresponding sections. To the authors' knowledge, the most accurate andupdated correlations and data for the friction factor and Nusselt number are provided for usein practical calculations.Characteristics of Laminar Flow in DuctsAs a result of the development of the hydrodynamic and thermal boundary layers, four typesof laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing).
In this chapter, the termfully developed flow refers to fluid flow in which both the velocity profile and temperatureprofile are fully developed (i.e., hydrodynamically and thermally developed flow). In suchcases, the velocity profile and dimensionless temperature profile are constant along the flowdirection. The friction factor and Nusselt number are also constant.Hydrodynamically developing flow is isothermal fluid flow in which the velocity profilevaries in the flow direction. Fluid flow from the entrance of the duct to the location at whichthe fully developed velocity profile forms is referred to as hydrodynamically developing flow.The distance over which the velocity distribution changes and the hydrodynamic boundarylayer develops is referred to as the hydrodynamic entrance length.
The friction factor in thehydrodynamic entrance is a function of the axial location.5.15.2CHAPTERFIVEThe term thermally developing flow refers to fluid flow in which the temperature profile isdeveloping and the velocity profile has already developed (i.e., the velocity distribution isinvariant with axial length, and the nondimensional temperature profile changes with axiallength). In other words, the hydrodynamic boundary layer is already developed while thethermal boundary is developing. This kind of flow is alternately termed thermal entrance flow.The distance over which the nondimensional temperature distribution changes or the thermalboundary layer develops is termed thermal entrance length, corresponding to hydrodynamicentrance length in hydrodynamically developing flow.
The Nusselt number for thermallydeveloping flow changes with axial location.Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developingin the entrance region of the duct. Both the friction factor and Nusselt number vary in theflow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1]and Shah and Bhatti [2].Characteristics of Turbulent Flow in DuctsIn turbulent flow, the fluid particles do not travel in a well-ordered pattern. These particlespossess velocities with macroscopic fluctuations at any point in the flow field.
Even in steadyturbulent flow, the local velocity components transverse to the main flow direction change inmagnitude with respect to time. Instantaneous velocity consists of time-average velocity andits fluctuating component. When heat transfer is involved in turbulent flow, the instantaneous temperature is composed of the time-average temperature and its fluctuating components.Similar to laminar flow in ducts, turbulent flow can be divided into four typesmfully developed, hydrodynamically developing, thermally developing, and simultaneously developing.Nevertheless, the hydrodynamic and thermal entrance lengths in turbulent duct flow are characteristically much shorter than their corresponding lengths in laminar duct flow. Consequently, the results of fully developed turbulent flow and heat transfer are frequently used indesign calculations without reference to the hydrodynamic and thermal entrance regions.However, caution must be taken in using the fully developed results for low Prandtl numberfluids such as liquid metals inasmuch as entrance effects are very important for these fluids.Table 5.1 illustrates the relationships between the types of flow, boundary layers, velocity andtemperature distributions, the friction factor, and the Nusselt number.TABLE 5.1 Terminologyfor Flow TypesFlow typeFully developed flowHydrodynamicallydeveloping flowThermallydeveloping flowSimultaneouslydeveloping flowHydrodynamicboundary layerVelocitydistributionin the f l o wdirectionDevelopedDimensionlesstemperaturedistributioninthe flowNusseltnumberInvariantConstantFrictionfactorThermalboundarylayerInvariantConstantDevelopedDevelopingVariantVariantDevelopedInvariantConstantDevelopingVariantVariantDevelopingVariantVariantDevelopingVariantVariantnnF O R C E D C O N V E C T I O N , I N T E R N A L F L O W IN D U C T S5.3Hydraulic DiameterFor fluid flow and heat transfer inside a duct, various dimensionless parameters are used.
Inthese parameters, a characteristic length of the duct is commonly involved. The hydraulicdiameter Dh of the duct serves this purpose. It is defined as follows:Dh=4Ac/P(5.1)where Ac is the flow cross-sectional area and P is the wetted perimeter of the duct.For a circular duct, the hydraulic diameter is equal to its physical diameter. For a noncircular duct, it is convenient to use the hydraulic diameter to substitute for the characteristicphysical dimension.
However, for ducts with very sharp corners (e.g., triangular and cuspedducts), the use of the hydraulic diameter results in unacceptably large errors in the turbulentflow friction and heat transfer coefficients determined from the circular duct correlation.Other dimensions have been proposed as substitutes for hydraulic diameter. These equivalent diameters, provided for specific ducts only, will be presented elsewhere in this chapter.Fluid Flow ParametersOne of the flow parameters most commonly used in practice is the friction factor, alsoreferred to as the Fanningfriction factor f, which is defined as follows:'~wf= pu2 /2(5.2)where xw is wall shear stress, Umis the mean velocity, and p is the density of fluid.The Reynolds number Re, the parameter that represents the status of the flow, is definedasRe -UmDh(5.3)vThe hydrodynamic entrance length Lhy is defined as the axial distance required to attain99 percent of the ultimate fully developed maximum velocity when the entering flow is uniform.
The dimensionless hydrodynamic entrance length is expressed as L~y = Lhy]Oh Re.In this hydrodynamic entrance region, the apparent friction factor fapp is employed toincorporate the combined effects of wall shear and the change in momentum flow rate dueto the developing velocity profile. Based on the total axial pressure drop from the duct inlet(x = 0) to the point of interest, the apparent friction factor is defined as follows:Ap* - Po - P = lapp 2Xpu~/2The incremental pressure drop numberexpressed as(5.4)DhK(x) in the hydrodynamic entrance region is2xg(x)-- ( L p p- hd) Oh(5.5)where Jyd is the friction factor for fully developed laminar flow.
K(x) is sometimes referred toas the incremental pressure defect. It increases from a value of zero at x = 0 to a constant valueK(oo) in the hydrodynamically developed region at x > Lhy.5.4CHAFFERFIVEThe relationship between the friction factor, axial pressure drop, and incremental pressuredrop number is the following:Ap* = (fapp Re)(4x ÷) = K(x) + ( f R e ) ( 4 x ÷)(5.6)where x + is the dimensionless axial length, defined as x/(Dh Re).Heat Transfer ParametersThe most useful parameters for heat transfer are the fluid bulk mean temperature and theheat transfer coefficient. The fluid bulk mean temperature Tm, also known as the mixing cupor flow average temperature, is defined asTm - AcumafAuT dAc(5.7)cThe circumferentially averaged but axial local heat transfer coefficient hx is defined byq~'= hx(Tw, m - Tin)(5.8)where Tw, m is the wall mean temperature and Tm is the fluid bulk mean temperature given byEq.
5.7. In Eq. 5.8, the heat flux qx' and the temperature difference Tw, m - Tm are vector quantities. The direction in which the heat is transferred is from the wall to the fluid, and the temperature consistently drops from the wall to the fluid.The average circumferential and axial heat transfer coefficient can be obtained by meansof the following expression:hm ---1 Ix hx dx(5.9)x JoCorrespondingly, the circumferentially averaged but axially local Nusselt number isdefined asNux -hxDhk(5.10)The mean Nusselt number based on hm in the thermal entrance region is defined asNum =--1 tx Nuxdx- hmDhxk(5.11)Two dimensionless axial distances will be used in this chapter. The term x +, which denoteshydrodynamically developing flow, is defined asx÷-X/DhRe(5.12)The term x*, denoting thermally and simultaneously developing flows, is expressed as:x* -X/DhX/DhPeRe Pr(5.13)where Pe is the P6clet number, defined as Re Pr.
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