Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 22
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A. L. Harvath, Physical Properties of Inorganic Compounds SI Units, Crane, Russak & Co., NewYork, 1975.11. C. L. Yaws, Physical Properties, McGraw-Hill, New York, 1972.CHAPTER 3CONDUCTION AND THERMALCONTACT RESISTANCES(CONDUCTANCES)M. M. YovanovichUniversity of WaterlooINTRODUCTIONWhen steady-state conduction occurs within and outside solids, or between two contactingsolids, it is frequently handled by means of conduction shape factors and thermal contact conductances (or contact resistances), respectively. This chapter covers the basic equations, definitions, and relationships that define shape factors and the thermal contact, gap, and jointconductances for conforming, rough surfaces, and nonconforming, smooth surfaces.Shape factors for two- and three-dimensional systems are presented.
General expressionsformulated in orthogonal curvilinear coordinates are developed. The general expression isused to develop numerous general expressions in several important coordinate systems suchas (1) circular, elliptical, and bicylinder coordinates and (2) spheroidal coordinates (spherical,oblate spheroidal, and prolate spheroidal).
The integral form of the shape factor for an ellipsoid is presented and then used to obtain analytical expressions and numerical values for theshape factors of several isothermal geometries (spheres, oblate and prolate spheroids, circular and elliptical disks). It is demonstrated that the dimensionless shape factor is a weak function of the geometry (shape and aspect ratio) provided that the square root of the total activesurface area is selected as the characteristic body length. A general dimensionless expressionis proposed for accurate estimations of shape factors of three-dimensional bodies such ascuboids. Shape factor expressions are presented for two-dimensional systems bounded byisothermal coaxial (1) regular polygons, (2) internal circles and outer regular polygons, and(3) internal regular polygons and outer circles.
A method is given for estimating the shape factors of systems bounded by two isothermal cubes and other combinations of internal andexternal geometries. The shape factor results of this chapter are used in the chapter on natural convection to model heat transfer from isothermal bodies of arbitrary shape.Transient conduction within solids and into full and half-spaces is presented for a widerange of two- and three-dimensional geometries.Steady-state and transient constriction (spreading) resistances for a range of geometriesfor isothermal and isoflux boundary conditions are given.
Analytical solutions for half-spacesand heat flux tubes and channels are reported.Elastoconstriction resistance and gap and joint resistances for line and point contacts arepresented. Contact conductances of conforming rough surfaces that undergo (1) elastic, (2)3.13.2CHAPTER THREEplastic, and (3) elastoplastic deformation are reported. The gap conductance integral is presented. The overall joint conductance is considered.Analytical solutions and correlation equations are presented rather than graphic results.The availability of many computer algebra systems such as Macsyma, MathCad, Maple, MATLAB, and Mathematica, as well as spreadsheets such as Excel and Quattro Pro that providesymbolic, numerical, and plotting capabilities, makes the analytical solutions amenable toquick, accurate computations.
All equations and correlations reported in this chapter havebeen verified in Maple worksheets and Mathematica notebooks. These worksheets and notebooks will be available on my home page on the Internet. Some spreadsheet solutions willalso be developed and made available on the Internet.*BASIC EQUATIONS, DEFINITIONS, AND RELATIONSHIPSShape factors of isothermal, three-dimensional convex bodies having complex shapes andsmall to large aspect ratios are of considerable interest for applications in the nuclear,aerospace, microelectronic, and telecommunication industries.
The shape factor S also hasapplications in such diverse areas as antenna design, electron optics, electrostatics, fluidmechanics, and plasma dynamics [27].In electrostatics, for example, the capacitance C is the total charge Oe required to raise thepotential ~e of an isolated body to the electrical potential Ve, and the relationship betweenthem is (e.g., Greenspan [27], Morse and Feshbach [68, 69], Smythe [98], and Stratton [111])Qe ffm-e. -~nO~e dAC=--~e =where e is the permittivity of the surrounding space, (~eis the nondimensional electric potential,n is the outward-directed normal on the surface, and A is the total surface area of the body.Mathematicians prefer to deal with the capacity C* of a body, which they [81,113] define as1~ ffa --~na~e dAC*=T~-ShapeFactor, Thermal Resistance, and Diffusion Length.
The shape factor S, the thermal resistance R, and the thermal diffusion length A are three useful and related thermal parametersThey are defined by the following relationships:1AQS - k---R- A - k(T0- T~)_ ff- JJA~)~-~n dZ(3.1)where k is the thermal conductivity, To is the temperature of the isothermal body, T. isthe temperature of points remote from the body, and ~ is the dimensionless temperature( T(7.) - T~)/(To - T~).The relationships between the shape factor S, the capacitance C, and the capacity C* areS = __C= 4nC*(3.2)EThe three parameters have units of length.Analytical solutions are available for a small number of geometries such as the family ofgeometries related to the ellipsoid (e.g., sphere, oblate and prolate spheroids, elliptical andcircular disks).
Precise numerical values of S for other axisymmetric convex bodies have beenobtained by various numerical methods such as that proposed by Greenspan [27] and thatproposed by Wang and Yovanovich [123].* The Internet address is mmyovemhtl.uwaterloo.ca.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.3Chow and Yovanovich [15] showed, by analytical and numerical methods, that the capacitance is a slowly changing function of the conductor shape and aspect ratio provided the totalarea of the conductor is held constant.Wang and Yovanovich [123] showed that the dimensionless shape factorS~A-- -S,:~,:~ ffA -~n--3(~)dZ-A - Z(3.3)where the characteristic scale length ~ was chosen to be ~ as recommended by Yovanovich[133], Yovanovich and Burde [134], and Yovanovich [144-146], when applied to a range ofaxisymmetric, convex bodies is a weak function of the body shape and its aspect ratios.This chapter reports and demonstrates, through inclusion of additional accurate numericalresults of Greenspan [27] for complex body shapes such as cubes, ellipsoids, and circular andelliptical toroids, a lens that is formed by the intersection of two spheres such that S~A is a relatively weak function of the body shape and its aspect ratios.This chapter also introduces the geometric length A, called the diffusion length, and showsthat this physical length scale is closely related to the square root of the total body surfacearea when the body is convex.The dimensionless geometric parameter X/~/A is proposed as an alternate parameter fordetermination of shape factors of complex convex bodies.Shape FactorsFormulation of the Problem in General Coordinates.
Consider the steady flow of heat Qfrom an isothermal surface A1 at temperature T1 through a homogeneous medium of thermalconductivity k to a second isothermal surface A2 at temperature T2(T1 > T2). The stationarytemperature field depends on the geometry of the isothermal boundary surfaces. When theseisothermal surfaces can be made coincident with a coordinate surface by a judicious choice ofcoordinates, then the temperature field will be one-dimensional in that coordinate system. Inother words, heat conduction occurs across two surfaces of an orthogonal curvilinear parallelepiped (Fig. 3.1a), and the remaining four coordinate surfaces are adiabatic.Let the general coordinates ul, U2, /'/3 be so chosen that T = T(Ul) and, therefore, t)T[~u2 =OT/Ou3 = 0.
Under these conditions, the heat flux vector will have one component in the Uldirection:q, =-k(dT/ds) =-k(dT/V~g~ du,)where ~ is the metric coefficient in the Ul direction. The metric coefficients are defined bythe general line element ds expressed in terms of the differentials of arc lengths on the coordinate lines [67](ds) 2= gl(dUl) 2 + gz(du2) 2 + g3(du3) 2uaThe product terms such as dui duj (i :~ j) do not appearbecause of the orthogonality property of the chosen coordinate system.
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