Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 26
Текст из файла (страница 26)
The dimensionless shape factor of cuboids with side dimensions L1 > L2 > L3 isapproximated by the expression [150](1 +S~AA = [S~A]rect0.8688(L1/DGM)°76)V/1 + 2L1/DGM(3.56)where the equivalent circular cylinder diameter is based on the cuboid lengths L2 and L3:DGM = ~ 2 (L2 + L3)X/L[ + L~The values for [S{/~A]r~ctare obtained from the expressions for the rectangular plates givenabove. This parameter depends on the cuboid lengths L2 and L3.
The proposed expression forcuboids predicts the dimensionless shape factor to within _+5 percent.Three-Dimensional Bodies With Layers: Langmuir MethodShape factors for three-dimensional systems such as regions bounded by isothermal concentric spheres or concentric cubes; inner sphere and outer cube; and inner cube and outersphere are presented in this section. The systems fall into two categories: (1) uniform thickness layers, and (2) nonuniform thickness layers. Warrington et al. [124] reported in graphicform numerical results for the cube-in-cube, sphere-in-cube, and cube-in-sphere systems. Thenumerical results for the shape factors were normalized with respect to the classical spherein-sphere solution and plotted against the ratio Di/Do, where D / a n d Do are the inner andouter equivalent diameters, respectively.
Hassani and Hollands [33] proposed an approximatemethod for calculating shape factors for a region of uniform thickness surrounding an isothermal convex body of arbitrary shape. The proposed method is based on the asymptotic resultscorresponding to very thin layers where the shape factor is given byS0(A) -AiA(3.57)3.20CHAPTER THREEwhere Ai and A are the surface area of the inner body and the layer thickness, respectively,and the result corresponding to infinitely thick layerss . = c.v,(3.58)where C= is a constant that is close to the value 2V~n for many body shapes [15] as shown inthe sections on transient one-dimensional conduction in half-spaces and external transientconduction from long cylinders.
Hassani and Hollands [33] set C~ = 3.51 for all bodies andproposed the following equation, which accurately interpolates between the two asymptoticsolutions:Sn(A) = [S¢ + S,"]TM(3.59)where n is a constant that is a function of the body shape. By trial and error, Hassani and Hollands found the following empirical formula:n = 1.26 - 9V'l.0 - 4.79V~ 3/Ai ' 1.0(3.60)maxwhere Vi and Ai are the volume and surface area of the inner body, and Ls is the longeststraight line passing through the inner body.
They gave the following rule: Y = Ix1, X2]maxmeansthat Y = xl if xl > x2 and Y = x2 if x2 > x~. The shape parameter n was found to lie in the range1 < n < 1.2 for a very wide range of body shapes. The results obtained through this methodshow agreement to within about 5 percent with those obtained from numerical or existinganalytical techniques.Shape Factors for Two-Dimensional SystemsThe shape factors for two-dimensional systems are available for (1) long cylinders boundedby homologous, regular polygons having N sides (Fig. 3.5), (2) long cylinders bounded internally by circles and bounded externally by regular polygons (Fig. 3.6), and (3) long cylindersbounded internally by regular polygons and bounded externally by circles (Fig.
3.7). In allthree cases, as the number of sides N of the regular polygon becomes large (N > 10), the shapefactor approaches the shape factor for the system bounded by two coaxial circular cylinders.N=3N=4N=5~iliiN=6N=3N=4N=5i!N~ooFIGURE 3.5 Regions bounded by regular polygons.N=6N~ooFIGURE 3.6 Regions bounded by inner circles and outerregular polygons.CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)N=3N=43.21N=5N=6N~ooFIGURE 3.7 Regions bounded by inner regular polygonsand outer circles.Regular Polygon Inside a Regular Polygon. The shape factor of two-dimensional regionsbounded internally and externally by isothermal regular polygons of sides N > 3 (Fig. 3.5) isobtained by means of the general expressionSL-4nN>3[(D/d)2- I ] 'In 1 + (N/n) tan (n/N)(3.61)-where d and D are the diameters of the inscribed circles of the inner and outer polygons,respectively.
For the region bounded by two squares (N = 4), the general expression reduces toSL - In [1 +4n(n/4)[(D/d) 2- 1]](3.62)The square/square problem has a complex analytical solution that requires the numericalcomputation of complete elliptical integrals of the first kind (see Ref. 8). For a large range ofthe parameter d/D, the analytical solution provides an accurate asymptotic expression:S[1 + (d/D)]8 In 2,L - 4 [1 - (d/D)] - n-0.3 <-d/O < 1-(3.63)Two correlation equations based on electrical measurements were reported by Smith et al.
[93]:S2nL - 0.93 In [0.947(D/d)] 'andS2nL - 0.785 In (D/d)'D~d > 1.4D~ < 1.4(3.64)(3.65)Circle Inside a Regular Polygon. Several expressions have been developed for this system(Fig. 3.6). Two relationships that give results to within a fraction of 1 percent are given. Thefirst is [52]S2n- ~LIn [As[3](3.66)3.22CHAPTERTHREEwhere 13= b/a > 1 is the ratio of the radius of the inner circle to the radius of the inscribed circle.
The parameter As is obtained by means of a numerical integration of the integralAs =iSo1(1 + uN) -2INdu,N >3(3.67)Laura and Susemihl [52] gave several values of the parameter As for several values of N. Thealternate relationship [91] isS4N[V'A2+Irt]L - A V ' A z + 1 tan-1 ~tan -~ ,N>3(3.68)where the parameter A = V'2 In 13. The second relationship does not require a numerical integration.Regular Polygon Inside a Circle. Numerous analytical, numerical, and experimental studies have produced results for shape factors and resistances for regions bounded internally byisothermal regular polygons of N sides where N > 3 and externally by an isothermal circle(Fig. 3.7).
Lewis [55] gave the following general analytical resultS2nL In [AN(D/d)](3.69)where the coefficients are given byAN = [(N/N- ~ / ' N - 2)]2/N(N- 2) ~/N,N >_3(3.70)The above relationship is limited to the range 0 < d/D < cos n/N, where d and D are the diameters of the inscribed circle and the outer circle, respectively. The relationship gives accuratevalues of S for small values of d/D and for all values of N.
The accuracy decreases for values ofd/D ~ cos n/N for small values of N. Ramachandra Murthy and Ramachandran [82] obtainedtwo empirical correlation equations for regions bounded by squares and hexagons Their correlation equations were developed from electrical measurements and have been shown to bein good agreement with the above relationship for a limited range of values of d/D.Polygons With Layers. Hassani et al. [34] presented a procedure for obtaining a close upperbound for shape factors for a uniform thickness two-dimensional layer on cylinders havingcross sections of the following forms: an equilateral triangle, a square, a rhombus, and a rectangle (Fig. 3.8). The shape factor per unit length of the inner cylinder is obtained fromS2nL - In [1 + (2nB/Pi)](3.71)B°(a)(b)FIGURE 3.8 Polygonswith uniform thickness layers.(c)(d)CONDUCTION AND THERMAL CONTACTRESISTANCES(CONDUCTANCES)3.23where B is the layer thickness and P; is the perimeter of the inner boundary. The accuracy ofthe above relationship was verified by comparison of the predicted values against numericalvalues for the layer thickness-to-side dimension range 0.10 < B/L < 3.00.
The proposed relationship overpredicts the shape factor by approximately 1-3 percent.TRANSIENT CONDUCTIONIntroductionTransient conduction internal and external to various bodies subjected to the boundary conditions of the (1) first kind (Dirichlet), (2) second kind (Neumann), and (3) third kind(Robin) are presented in this section. Analytical solutions are presented in the form of seriesor integrals.
Since these analytical solutions can be computed quickly and accurately usingcomputer algebra systems, the solutions are not presented in graphic form.Internal Transient ConductionInternal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature ¢ = 0/0i is afunction of three dimensionless parameters: (1) dimensionless position ~ = x/Y, (2) dimensionless time Fo = ott/~g2, and (3) the Biot number Bi = h~/k, which depends on the convectiveboundary condition.
The characteristic length ~ is the half-thickness L of the plate and theradius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant.The basic solutions for the plate and the cylinder can be used to obtain solutions withinrectangular plates, cuboids, and finite circular cylinders.
The equations and the initial andboundary conditions are well known [4, 11, 23, 28, 29, 38, 49, 56, 80, 87]. The solutions presented below follow the recent review of Yovanovich [151]. The Heisler [36] cooling charts fordimensionless temperature are obtained from the series solution:~o¢ = ~" A, exp(-5 ] Fo)S,(5,~)(3.72)n=lwith the temperature Fourier coefficients A, for the plate, cylinder, and sphere, respectively,given in Table 3.5. The spatial functions for the three basic geometries are given in Table 3.6.The eigenvalues 5, are the positive roots of the characteristic equations found in Table 3.6where Bi, the physical parameter, ranges between 0 and oo.TABLE 3.5Fourier Coefficients for Temperature and Heat LossA.B.2Bi 2Plate2 sin 5.5, + sin 5, cos 5,52(Bi 2+ Bi + 52)Cylinder2./,(5.)5.[./2(5.) + J2(5.) l4Bi 252(52 + Bi 2)GeometrySphere(_1) ,,+,2Bi [52 + (Bi- 1)21~a(5.2+ Bi2-Bi)6Bi22 2+Bi 2_ Bi)5.(5.3.24CHAPTERTHREETABLE 3.6 Space Functions and Characteristic EquationsiGeometryS.Characteristic equationPlateCylindercos (~5.~)J0(~i.~)x sin x = Bi cos xxJl(x) = Bi Jo(x)Spheresin (~.~)(~.~)(1 - Bi) sin x = x cos xThe Grober charts [29] for the heat loss fraction Q/Qi, where Qi =internal energy, are obtained from the series solution:pcpVOiis the initial total,,oQ - 1 - ~ ' B, exp(-~5 ] Fo)ai(3.73)n=1The Fourier coefficients B, are given in Table 3.5 for the three geometries.
These coefficientsdepend on the Biot number.Lumped Capacitance ModelWhen the Biot number is sufficiently small (Bi < 0.2), the series solutions converge to the firstterm for all values of Fo > 0. The values of the Fourier coefficients A1 and B1 approach 1, andthe dimensionless temperature and the heat loss fraction approach the general lumped capacitance solutions(~ = e-(hS/pcpv) tandQ - 1 - e -(hs/pcpv)taiwhere S and V are the total active surface area and the volume. The lumped capacitance solutions for the three geometries are given in Table 3.7.TABLE 3.7Lumped Capacitance Solutions Bi < 0.2Q/QiGeometry~PlateCylinderSpheree - B i Vo1-e-2Bi ro1 -- e -2Bi Foe-3Bi Vo1 -- e -3Bi voe -Bi FoHeisler and Grober Charts--Single-Term ApproximationsThe Heisler [36] cooling charts and the Grober [29] heat loss fraction charts for the threegeometries can be calculated accurately by the single-term approximations [28, 56]0--exp(-512= A10iandOQi- 1-O 1Fo)Sl(~l~)exp(-5 2 Fo)CONDUCTION AND THERMAL CONTACTRESISTANCES (CONDUCTANCES)3.25Asymptotic Values of First Eigenvalues,Correlation Parameter n, and Critical Fourier NumberTABLE 3.8GeometryBi ~ 0PlateCylinderSphere81--->~81--->~81--->3 V ~Bi --->oonFoc81 --->re/22.139 0.2481--->2.4048255 2.238 0.2181--->rt2.314 0.18for all values of the Biot number, provided Fo > Foc.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
