Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 23
Текст из файла (страница 23)
These metric coefficients can also be generatedby means of the following formula [67]:ds au2ds2U1FIGURE 3.1aOrthogonal curvilinear parallelepiped.g~ - (axlaui) 2 + (ay/aui) 2 + (azlaui) 2i=1,2,3provided that the Cartesian coordinates x, y, z can beexpressed in terms of the new coordinates ul, u2, u3 by theequationsX = X(Ul, U:, U3),y -- y(Ul, U2, U3),Z = Z(Ul, U2, U3)3.4CHAPTERTHREEThe elemental coordinate surface located at u~, which is orthogonal to thethereforeU 1 direction,isdA1 = ds2 ds3 = V~zg3 du2 du3and the heat flow per unit time through this surface into the volume element is, according toFourier's law of conduction,Q1 = - k da,(dT/dsl) =-k(V/-g/g~)(dT/dul) du2 du3where g - glgzg3 [67]. The heat flow rate out of the volume element isQ~ + (dQ~/ds~) ds~ = Q~ + (dQ~/du~) dulneglecting the higher-order terms of the Taylor series expansion of Q1 about Ul.The net rate of heat conduction out of the volume element in the Ul direction is(d/du~)[k(X/-g/gl)(dr/dUl)] dUl duz du3For steady-state conditions with no heat sources within the volume element, the Laplaceequation in general coordinates is obtained by dividing by the elemental volume V~ dUl du2du3 and equating to zero.
Therefore,(1/X/-g)(d/dul )[k(X/-g/g~ )(dT/du~ ) ] = 0The above equation is the governing differential equation and it is nonlinear when k is a function of temperature.The isothermal temperature boundary conditions areul =a,Ul =b,T= T1T= T2The above equation can be reduced to a linear differential equation by the introduction of anew temperature 0 related to the temperature Tby the Kirchhoff transformation [4, 11]:0 = (l/k0) I, T° k d rwhere k0 denotes the value of the thermal conductivity at some convenient reference temperature, say T = O.
It follows thatdO/dUl -- (k/ko)(dT/du,)and, therefore, we have(d/dul)[(N/ g/gl)(dO/dUl) ] = 0after multiplying through by ~/-g/ko. The boundary conditions becomeU 1 = a,0 : 01 : ( 1 / k o ) fo T1 k d TUl - b ~0 = 02 = (l/k0) foT2k d rThe solution of the linear equation is0 = C1(gl/V~) dul + C2CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)3.5where the constants C 1 and C2 are obtained fromandC 2 = 01 - C 1fOa (gl/X/g) dUlTemperature Distribution in Orthogonal Curvilinear Coordinates.The temperature dis-tribution in orthogonal curvilinear coordinates isf;l (gl/'~/-g)01 -- 0d/'il-o,- o~,a < U1 < b(3.4)f~ (g, lXl~) du,The local heat flux in the Ul direction is-k0 dOk0(01 - 02)_ql- X/~l dUlThe heat flow rate through the elemental surface dA1qldZl(3.5)gv~2g3f~ (gl/X/-g) duaisf k0(01(gl/X/g)02)du1duzdu3(3.6)The total heat flow rate through the thermal system can be obtained by integration betweenthe appropriate limits.
Therefore,Q= k°(Oa-Oz) f2 fa~(ga/X/g)dU2dU3dua(3.7)An examination of the above equation shows thatk0(01- 02) =k dT= ka(Zl -- Z2)(3.8)where ka, the average value of the thermal conductivity, is given byk~ = ko[1 + (1~/2)(T1 + 7"2)](3.9)if k = k0(1 + 13T).Shape Factor and Thermal Resistance in Orthogonal Curvilinear Coordinates.The definition of thermal resistance of a system (total temperature drop across the system divided bythe total heat flow rate) yields the following general expression for the thermal resistance Rand the conduction shape factor S:S_(Rka)_X__ff f~ du~du~u~ .~(3.10)(gllW-~) du,The right side of the previous equation has units of length and depends on the geometry ofthe body only.
There are several very important and useful coordinate systems that can beused to solve many seemingly complex conduction problems. Since each coordinate system3.6CHAPTERTHREEhas three principal directions, there are three sets of shapefactors corresponding to each of these directions. The conduction shape factors for several coordinate systems aregiven in the following section. This section by no means represents the total n u m b e r of coordinate systems that areamenable to this type of analysis. It does, however, containthe most frequently used coordinate systems.rd2II,~',z)x¥=0General Expressions f o r Conduction Shape FactorsiiCircular Cylinder Coordinates (r, ~, z): Fig.
3.1br direction. Let Ul = r; therefore, u2 = ~, u3 = z, and gl/X/g =1/r. IfFIGURE 3.1b Circular cylinder coordinates.a<r<b0<~<[~0<z<L~max = 2XS -1= Rka = ~ l n (b/a)then(3.11)~Ldirection. Letjust previously.U 1 : ~;therefore,U 2 = Z, U3 = F~ andS - l = Rka =z direction.
LetU 1 "--Z; therefore,gl/X/-g = r. Limits of integration are given13(3.12)L In (b/a)U 2 = F~ U 3 = llJJ, andgl]X/g = 1]r. Limits of integration weregiven previously.2LS -1=Rka = [3(b2 _ a2)(3.13)Spherical Coordinates (r, 0, ~¢): Fig. 3.1c(ds)2 = ( d r ) 2 + r2(dO) 2 + r ~ sin 2 0 ( d r ) 2gr = 1,go = r 2,rgv = r2 sin2 0,direction.Letu 1 = r;X/g = r 2 sin 0therefore,U2 = 0, U3 = ~,andgl/Vg =1/(r 2 sin 0).
If0=0ia<r<bI'[~1 < 0 < [~2~min-0<~ <77max= 2no0,V)~/2thenS-'=Rka =0and~max =[(l/a) - (l/b)]~(cos ~ , - cos ~)(3.14)0 direction. Let u 1 = 0; therefore, u 2 = ~l/, u 3 = F~ and gl]Vg =I!1/(sin 0). Limits of integration are given just previously.!S -l= Rka = In [tan (132/2)] -In [tan (~1/2)]FIGURE 3.1c Spherical coordinates.~b-a)(3.15)CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)rd2direction.
L e t u I = I1/; t h e r e f o r e , U 2 = F~ U 3 "-- O, andsin O. L i m i t s of i n t e g r a t i o n w e r e g i v e n previously.7S-1= Rk~ = (b - a){ln [tan (132/2)] - In [tan (131/2)]}3.'1gl[Vg-(3.16)Elliptical Cylinder Coordinates (1], ~/, z): Fig. 3.1d~=0(ds)2 = a2(cosh 2 1] - cos 2 ~ ) [ ( d r l ) 2 + ( d ~ ) 2] + ( d z ) 2gn = gv = a2( cOsh2 I"1 - c0S2 II/),gz = 1,= a2(cosh 21] - cos 2 ~1/)11 direction. L e t Ul = 1]; t h e r e f o r e , u2 = ~ , u3 = z, a n d gl/X/g =1. IfFIGURE 3.1d Elliptical cylinder coordinates.S -1=thenwhere1111 = ~ l n111 < 11 < 1"12Tlmin = 0 ,0 < ti/< ~0<z<L~max = 2/I;Rk~ = (112-111)(3.17)L~[bl+Cl]bl-cland~max ~ oo1 E2+c2]'112 = -~ Inb2 - c2a = X/b 2 - c21 = V'b22 - c 2direction.
L e t ul = ~; t h e r e f o r e , u: = z, u3 = 11, a n d g l / V g = 1. L i m i t s o f i n t e g r a t i o n w e r egiven previously.S-1= Rka =(3.18)L(112 - rll)z direction. L e t Ul = z; t h e r e f o r e , u2 = r, u3 = q~, a n dgl1X/ga2(cosh 21] - cos 2 ~/)L i m i t s of i n t e g r a t i o n w e r e g i v e n p r e v i o u s l y .LS-l=Rk~=,n2a f f.1"11(3.19)(cosh21]-cos2qt) drldq/1Bicylinder Coordinates (11, ~, z): Fig.
3.1ea2(ds)2 = ( c o s h 1"1- c o s V) 2 [(drl)2 + ( d r ) : ] + (dz) 2a2gn = gv = ( c o s h 11 - cos ~)2,a2gz = 1,~= ( c o s h 11 - cos ~)21] direction. L e t ul = 1]" t h e r e f o r e , u: = ~, u3 = z, a n d g l / V g = 1. If111 > 11 > 112~min = ---o%O<q/<~0<z<L~max -- 2~'~max "--3.8CHAPTERTHREEY~'kT:O,..._oF I G U R E 3.1eBicylinder coordinates.thenwhereS-~=]TIllRka == s i n h - ' / ( w ~~] 2- 1~ \ r~ /012--1"]1)~(3.20)L~and11121 = sinh -~direction. L e t u~ = ~ ; t h e r e f o r e , u2 = z, u3 = rl, a n dgiven previously.g~/X/g =J(w212~~\r2 /- 11.
L i m i t s o f i n t e g r a t i o n w e r e(3.21)S -1 = R k a =L ( T I 2 - rll)z direction.L e t ul =z; t h e r e f o r e ,u2 = rl, u3 = ~ , a n dgl m ( c o s h 1"1- c o s ~)2V~a2Limits of integration were given previously.S-i=Rk,L=~ ,1(3.22)a2 fo fn [drld~/(cosh rl-COS ~) 2]2wherea = V/w 2 - ~ = V ' w ~ -Oblate Spheroidal Coordinates (1"1,O, ~t): Fig. 3.1f(ds) 2 = a2(cosh 2 rl - sin 2 0)[(drl) 2 + ( d 0 ) 2] + a 2 c o s h 2 ri sin 2 0 ( d r ) 2gn = go = a2(cosh 2 I"1 -sin 2 0)gv = a2 c°sh21"1 sin 2 0V g = a3(cosh 2 q - sin 2 0) c o s h rl sin 0direction.L e t ul = rl; t h e r e f o r e , u2 = 0, u3 = ~ , a n dglV~m1a coshTI sin 0CONDUCTION AND THERMAL CONTACT RESISTANCES (CONDUCTANCES)IfT~I < T I < 1"12T~min "- 0 ,Tlma x _-- oo[~1 <( 0 < [~2~min : 0 ,~max :0 < ~g < 77max = 2ntan -1 (sinh r12) - tan -1 (sinhS-1= Rk,, =thenwhere'1~1)3.9(3.23)ay[cos 151- cos 132]a = V'b21 - c 2 = V'/:P2- caandrll = t a n h -1 (Cl/b~),I"12= t a n h -~ (c2/b2)0 direction.
L e t Ul = 0; t h e r e f o r e , u2 = ~, u3 = 1"1,andg~X/g1Ia cosh 11 sin 0Limits of i n t e g r a t i o n w e r e given previously:S -1= Rko =In [tan (132/2)]- In [tan (131/2)](3.24)ay(sinh 112- sinh 1"11)~g direction. L e t ul = ~g; t h e r e f o r e , u2 = rl, u3 = 0, andglV~Ecosh rl sin 0a ( c o s h 211 - sin 2 0)Limits of i n t e g r a t i o n w e r e given previously.S -1 -~Rk,, =7a(3.25)[(cosh 2 ~ - sin 20)/cosh 11 sin O] dO dTI1111Prolate Spheroidal Coordinates (1"1,O, ~): Fig. 3.1g(ds) 2 = aa(sinh 211 + sin 20)[(dTI) 2 + (dO) a] + a 2 sinh 211 sin 2 0 ( d ~ ) 2g,1 = go - a2(sinh 211 + sin 2 O)gv = a2 sinh211 sin 2 0V ~ - aB(sinh 21"1 + sin 2 O) sinh 1"1sin 00=00=0I1!•°ri=0rd2FIGURE 3.1f Oblate spheroidal coordinates•..
n/2FIGURE 3.1g Prolate spheroidal coordinates.3.10CHAPTER THREE1"1direction. Let//1 = ~ ; therefore, u2 = O,//3 = ~ , andglV~Ifthen1a sinhl"l sin 0q l < 1] < q2]]min -" O,T~max -- oo1~1 < 0 < 1~20 < lq/< ~]~min -- O,~max= 2rt]~max "- ~S-~= R k , = In [tanh(n2/2)] - I nay(coswhereand[tanh (n~/2)](3.26)1~1- COS ~2)a = V'b] - c] = V'b~- c221 i +c I111 = ~- Inbl - Cl 'I"12= ~- In b2 - c20 direction.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
