Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 24
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Let Ul = 0; therefore, u2 = ~, u3 = 1], andglV~1a s i n h r l sin 0Limits of integration were given previously.S-l= R k , -- In [tan (132/2)] - I n [tan (131/2)]a ~ c o s h r12- cosh rll)(3.27)direction. Let Ul = ~; therefore, u2 = rl, u3 = 0, andg~V~sinh 1"1sin 0a(sinh 2 1"1+ sin 2 0)Limits of integration were given previously.S -l= ek a =(3.28)'~a[(sinh 2 1"1- sin 2 0)/sinh I"1 sin 0] dO drlnl1The basic relations given above for several curvilinear coordinates can be used to obtainexpressions for the shape factor for many problems of interest to thermal analysts. Severaltypical two- and three-dimensional examples are presented in Fig.
3.2. The material in the following section provides shape factors for three-dimensional isothermal bodies in full space.!¸ii~:~ % ~'YliiI!,(a)(b)FIGURE 3.2 (a) circular pipe wall; (b) single spherical shell.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)=b=iI - - - - b2~i3.11............I!|I(c)j(d)T2I(e)T2,"insulatedinsulated:i#T~Ii(f)O0(g)(h)FIGURE 3.2 (Continued) (c) strip in elliptical cylinder; (d) pipe in eccentric circular cylinder; (e) confocal elliptical cylinders;(f) sphere embedded in a half-space; (g) circular pipe embedded in a half-space; (h) hemispherical cavity in a half-space.Shape Factors for Ellipsoids: Integral Form for Numerical CalculationsThe capacity and/or the capacitance of an isopotential ellipsoid are presented in several textsand handbooks such as those by Flugge [22], Jeans [40], Kellogg [43], Mason and Weaver [62],Morse and Feshbach [68, 69], Smythe [98], and Stratton [111].
The results presented in thesetexts are used to develop expressions for the shape factors of several bodies: spheres, oblateand prolate spheroids (see Fig. 3.3), circular and elliptical disks, and ellipsoids. The shape factor for the ellipsoid is general; it reduces to the shape factor for the other bodies.The capacity of an isopotential ellipsoid having semiaxes a > b ___c was given in the integralform [113]:11 c~dvC* - 2 Jo V~(a 2 + v)(b 2 + v)(c 2 + v)(3.29)where v is a dummy variable. This expression is used to develop the expression for the dimensionless shape factor of isothermal ellipsoids.
Since S = 4rtC*, one can set the space variablev = a2t, where t is a dimensionless variable. Next we normalize the two smaller axes b, c of theellipsoid with respect to the largest semiaxis a such that 13= b/a and 7 = c/a. This leads to thefollowing dimensionless integral [150]:8rtaI~dtS - 1(13, 7) =M'(1 + t)(f~ 2 + 0(72+ t)'0 _<7_< 13< 1(3.30)3.12CHAPTER THREEOBLATE SPHEROID(AR--.O.5)SPHERE(AR=I)PROLATE SPHEROID(AR=1.93)PROLATE SPHEROID(AR=10)CIRCULAR DISK(AR=0)RECTANGULAR STRIP(AR--O)SQUARE DISK(AR--0.1)CUBE(AR=I)TALL CUBOID(AR=2)TALL CUBOID(AR=10)FIGURE 3.3 Three-dimensional bodies.The ellipsoidal integral can be expressed in terms of the incomplete elliptical integral ofthe first kind F0c, ~) [10, 60]:2(I(13,1,) = V'I - T-----F~ sin -~ V'I - T2,J )11-- 72132(3.31)where ~: and ~ are the modulus and amplitude angle, respectively.
Computer algebra systemscan be used to evaluate the above special function accurately and quickly.The ellipsoidal integral reduces to several special cases, which are presented next.Sphere.a = b = c; ~5= y = 1.dtI(1, 1)=3'---------~(1 + t) = 2(3.32)which gives S = 4~z, a well-known result.Circular disk. a = b, c = 0; 15= 1, T = 0.I(1, 0 ) =f~which gives S = 8a, also a well-known result.dt(1 + t)X/t = r~(3.33)CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)Elliptical disk.3.13a > b, c = O; 0 < 13 < 1, 7 = O.f~1([3, 0) =dt%/(1 + 0(132 + t)t = 2 K ( V ' I - 132)(3.34)w h e r e K ( n ) is the c o m p l e t e elliptical integral of the first kind of m o d u l u s n = V'I - 132.
T h e r eare several m e t h o d s available to c o m p u t e accurately the c o m p l e t e elliptical integral [1]. H e r eare two simple approximations:K(V'I-[~2) m(12/1:"~/-o'\2 '+ vp)/4\K ( V ' I - 132) = In t-~),andOblate spheroid,0.2 < 13 < 1(3.35)0<[3<0.2(3.36)a = b > c; ~ = 1, 0 > 7 < 1.ffI(1, 7) =Prolate spheroid,dt(1 + t ) ~2= V/1 - y2 cos-17(3.37)1(3.38)a > b = c; ~ = 7 <- 1.I(7, 7) =s: (72 + t)V/i + t --V'I- - - - - ~7 In 1 - V'I -The above results c o r r e s p o n d to an i m p o r t a n t family of axisymmetric, convex geometries.The results p r e s e n t e d in dimensional form or in n o n d i m e n s i o n a l form as given above do notreveal an i m p o r t a n t p r o p e r t y p r e s e n t e d by this family of geometries and o t h e r geometries w h e nthe appropriate physical characteristic scale length is used for the nondimensionalization.The n u m e r i c a l values S* for oblate spheroids ([3 = 1, 0 <_ 7-< 1), prolate spheroids (13 = 7,1 >__7 _> 10), and elliptical disks (0 _< 13 --- 1, 7 = 0) are p r e s e n t e d in Tables 3.1-3.3.TABLE3.1Shape Factors and Diffusion Lengths for Oblate SpheroidsacS*VqA123456712.566410.39239.624769.230858.990908.829328.713083.544913.529033.493923.459393.429943.405533.38530-TABLE 3.2891010210 310 4S*8.625468.557008.502068.050858.005098.00051V~A3.368413.354133.341943.210983.193563.19174Shape Factors and Diffusion Lengths for Prolate Spheroidsacs*1234512.56648.263596.721155.896645.37092-ac-V?A3.544913.566133.627693.706383.79053acS*6789105.000474.722054.503194.325394.17723-V~A3.875333.958784.040054.118834.195083 .
1 4CHAPTERTHREETABLE 3.3Shape Factors and Diffusion Lengths for Elliptical Disksa-b12345V~S*8.0(0)005.827164.969644.486064.16641aSa*-A3.191543.287633.433973.579363.71670b6789103.935113.757633.615763.498883.40033V~A3.845413.966184.079954.187554.28974The range of dimensionless shape factor for the oblate spheroids is 8 < S* < 4n.
The highest and lowest values correspond to the sphere and the circular disk, respectively. The radii ofthe disk and sphere are set to one unit.The dimensionless shape factor range for the prolate spheroids is approximately 4.177 <_S*_< 4n for the aspect ratio range 1 < a/b <_10, and the major axis is set to 2a = 2.The dimensionless shape factor range for the elliptical disks is approximately 3.4 < S*_< 8.The highest value corresponds to a circular disk of unit radius, and the lowest value corresponds to an elliptical disk with a 10 to 1 aspect ratio.The range of all values of S* presented in the three tables is quite large. The ratio of thelargest and smallest values is approximately 3.7.
These values correspond to the sphere andthe elliptical disks of large aspect.In the next section the range of the dimensionless shape factors will be reduced significantly by the introduction of a scale length based on the square root of the total surface area.Surface Area of Ellipsoids.The expression for the total surface area of an ellipsoid is writ-ten as [150]:A _ ~¢2 -k- ~2ha 2sinwith[~2F((~, K~) -k- (1 - y2)E(~, ~¢)] = ~(13, y)~ = cos-1 7and~¢=(3.39)1 - (~¢/1~) 2 )1/21 -- ~¢2The total surface area related to the semimajor axis is a function of the two aspect ratios 13andy.
The special functions F(¢~, n) and E(~, ~¢) are incomplete elliptical integrals of the first andsecond kind, respectively. They depend on the amplitude angle ~ and the modulus ~¢. Thesespecial functions can be computed quickly and accurately by means of computer algebra systems such as Mathematica [153]. Their properties are given in Abramowitz and Stegun [1].The relationship between the square root of the total surface area and the semimajor axisis [150]:v~a- V'2n~/(13, ~,)(3.40)Dimensionless Shape Factor and Diffusion Length of Ellipsoids.The dimensionless shapefactor S ~ and the proposed dimensionless diffusion length V~-/A for isothermal ellipsoidscan be obtained from the shape factor integral 1(13, 7) and the relationship ~/-A/a given previously.
The equation isS~-k- 1(13,~,)k/'~(13, 3,)(3.41)The functions that appear in this expression were computed quickly and accurately usingMathematica. The numerical values for oblate spheroids are presented in the third column ofCONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.15Table 3.1. The range of values has been significantly reduced. The ratio of the values for thesphere a/c = 1 and the circular disk a/c ~ oo has been reduced from 1.57 to 1.11.The numerical values for prolate spheroids are presented in the third column of Table 3.2.Here the reduction in the range is much greater.
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