Pandey, R. N.

Shukla, K. N.

Tripathi, G.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

GENERAL SOLUTION TO A CLASS OF UNSTEADY HEAT CONDUCTION PROBLEMS IN A SOLID CYLINDER WITH THE CONVECTIVE TYPE OF TIME- DEPENDENT BOUNDARY CONDITIONS By G. TRIPATHI, K. N. SHUKLA and R. N. PANDEY Institute of Technology, Banaras Hindu University Received July 6, 1972 m,n X, cp, Z x x=R; z ZZ=R WR 2 Po(x, cp, z, Fo) -,-j,Ta at Fo = R2 Ta 8(x,cp, z, Fo) = 0 = T Notations Constant boundary coefficit'nts on surface. Coefficient defined by (7) Coefficient defined by (10) Coefficient defined by (11) Source function prescribed on surface ::\Iodified Bessel Function of the first kind of order k and argument x Bessel Function of the first kind of order k on argument % 0, L 2, .. . 1,2, .. . Cylindrical co-ordinate Dimensionless co-ordinate Pomerantesev criterion Fourier number Dimensionless temperature distribution Oo/x, cp,:;, Fo) W To = BOj Dimensionless temperature distribution T a Ta R ?OJ' oij /'lm1ll 11lim lP/m1ll (x, 0 z) ( ) (~) (-) Zn(:;) , Cin /'n=---p-Volume- heat source Temperature distribution Thermal diffusivitv Ambient temperature Radius Kronecker delta Eigenvalues in space x, cp, ::; defined by Equ. (7) Eigenfunction defined in space x, cp, Z (a) finite cosine transform of ( ) defined by Eq. (15) (b) finite trigonometrical transform of (-) defined by Eq. (16) (c) finite Hankel transform of () defined by Eq. (17) defined by Eq. (10) defined by Eq. (16). Introduction OLCER [1] studied convective heat transfer in finite region under the bllundary conditions of third kind and later on he [2] extended his result for a three-dimensional rectangular region under the influence of an arbitrarily 72 G. TRIPATHI et al. internal heat source and 'with arbitrary homogeneous boundary conditions of convective type. In this paper we consider a more general and complete problem for a finite solid cylinder is subjected on its entire surface to boundary conditions of the third kind. The solution of the problem is obtained in quasi-steady and transient terms. A number of deductions have been obtained which are avail-able in the literature. Statement of the prohlem Consider a three-dimensional problem of unsteady temperature distri-bution in a right circular cylinder of finite length 2b. The entire surface is subjected to boundary conditions of the third kind. Using cylindrical polar coordinates x, cp, z and choosing the z cordinate along the geometrical axis of the cylinder, the flow of heat conduction in dimensionless form can be "n·it-ten as 1 80 1 8 20 820 80 ---+ --+ Po(x, cp, z, Fo) =--x 2 8([>2 8z 2 8 Fo x 8x Fo > 0, 0 <. x < 1, 0 <. cp <. 2x, I z I < b (1) wbere e = e(x, cp, z, Fo) is the unsteady temperature distribution. Associated with (1), the boundary conditions are 8e Al --+ B l0 = fl(X, Cf, Fo) 0 <. x < 1, 0 <. cp <. 2x, z = - b, Fo > 0 (2a) 8z 80 A z 8z + B 20 = fz(x, cp, F 0) 0 <. x < 1, 0 <. rp <. 2x, z = b, Fo > 0 (2b) 8e A3 --+ B3e = f3(z, Cf, Fo) x = 1,0 <. rp <. 2x, z = b, Fo > 0 (2c) 8x where Bi(i = 1,2,3) are nonzero boundary coefficients and Ji (i = 1,2,3) are in tegrable functions prescribed on the bases and curved surface. The statement of the problem is completed by introducing the initial condition (e) Fo = 0 = e(x, rp, z, 0) = F(x, rp, z). (2d) Solution of the Prohlem The eigenvalue problem corresponding to Eq. (1) is 8 21jJ 1 811) 1 8 21jJ 821jJ __ ...L __ ...L ___ ...L -- ...L}/ 1/l = 0 (3) 8 ? I 8 I 2 8 ? I 8? I • rmn T x- X X X cp- z-o <. rp <. 2x, 0 <. x < 1, I z I < b and uNSTEADY HEAT CONDUCTION PROBLEMS Al~ BOb I1p = , 0 < x < I, 0 < rp < 2x, Z = -8z A 81p I B,"-O ?-, ?T-- 8z -0< x < I, 0 < rp < 2x, z = b A3 alP + B31p = 0 X = I, 0 < rp < 2x, I z I < b 8z 73 (4) The eigenfunetion of the differential equation (3) well behaved at the origin is where 1J!"mn(x, Cf, z) = J k (flkm x) {c~s kr} Zn (z) SIll krp (k = 0,1,2, ... , 00 m, n = 1,2,3, ... , '"Xl) Zn(z) = cos ttn (1+ ~) + ~lsin ~n (I + ~) The eigenvalue I'kmn is given by '2 2 -L ttn /'kmn = flkm I --b2 where ,Ukm is the root of (k fl) J,,(flkm) = flkmJ,,+ l(flkm) B fl = :4>0 and ~n is the nth root of where (J'\l11 + M 2) ~n cos 2ttn = (tt~ - M1 M 2) sin 2xn Zn (- b) = I, Zn (b) = tt~ + Mr cos 2xn tt~ - M11VI2 (5) (6) (7) (7a) (7h) (7c) Taking 1p(l.kmnX) as kernel, a three-dimensional finite integral transform of 8(x, rp, z, Fo) is defined as 2.1. b 1 G(k, m, n Fo) = S S S 1p(}'kmn x) G(x, rp, z, Fo)x dxdrdz (8) o -b 0 with inversion = G(x, r, z, Fo) = S S S Ckmn 1J!(). kmn x)G(k, m, n, Fo) (9) le m n 74 G. TRIPATHI et al. Ckmn is defined as 1 1 1 b where 1 J zg (z)dx -b b(xn + MD (Xn M~) + -+ b(M] + NI2) (x; + Ji1NI2) = 2b I 2;. 7l (~ = -- Vam .) flom X~ (x; + MoD n = 1,2 Xo = 0 k = 1,2, ... = (10) (11) It is difficult to show directly from (9) that it satisfies the boundary conditions (2). Such type of difficulty arises in dealing with the convergence of the series form of the solution. To get rid of this, it is essential to obtain from (9) an alternate form of solution composed of quasisteady state and transient parts. Thus the solution to the system of Eqs (1) and (2) can now be written down directly from the general expression (16) giwn in [1]. The result IS 3 G(x, (r, z, Fo) = .2'GoAx, rp, z, Fo) + Z Z Z C"m',]k(,Ukmx) X ,=0 le m 11 k=m=Il#O Zn(z)e-i.~,""FO X If .\2 .( J k(pk71x)Zn(z) [ F(x, rp, z) - t GOj(x, rp, z, Fo) xdxdrpdz o -b 0 J=o 3 Fo 2}. 11 1 - Z r eXi:mnFO J J J Jk(f-lkmX)Zn(z) X j=O iJ 0 -00 X SGoj (x,rp,z, FO)XdXdrpdZdFO'] SFo The functions 8oj{x, rp, z, Fo) are defined by 1 SGOj 1 1 S2GOj SGOj ~ ( F ) 0 - -- T --+ --+ UOj Po x, (r, z, 0 = x Sx x 2 Srp2 SZ2 o <: x < 1, I z I < b, 0 <; er <; 2). (12) (13) UNSTEADY HEAT COSDUCTIOZY PROBLKUS 75 and o < x < 1, z = - b, ° < cp < 271: _,in 8Go j BG £.f'( F) ./i., 2-0j=U2jJ2 X ,V, 0 - 8z ° < X < 1, ° < er < 271:, z = b (14) x = 1, ° « er < 271: i z I < b Determination of functions: Go j(x, v' z, Fo) The so-called pseudo-steady functIOns Aoj(x, (P, z, Fo), (j = 0,1,2,3) are yet to be determined. We first define three finite transforms as follows: (a) Finite cosine transform with respect to (r 2;. eOj (x, k, rI, z, Fo) = J GO j (x, (r, z, Fo) cos k(cp - r')dcp (15) o with its inversion 1 - 1 =-80j (x, r, z, Fo) = -;;-:- GO j (x, 0, crI, z, Fo) + -:- ::E GO j (x, k, cpl, z, Fo) (16) -I. I. k=p (b) Finite trigonometrical transform with respect to z b GO j (x, n, k, (p', Fo) = J GOj tx, k, (PI, z, Fo) ZIl(z)dz -b 'with its inversion (17) GO j (x, k, cr', z, Fo) = J: An eOj (x, n, k, cp', Fo) Zn(z). (18) 11 c) Finite Hankel transform with respect to x ~ 1 GOj (k, m, (r', z, Fo) = I' Goj(x, k, V', z, Fo) X JkCUkmX)dx (19) o ·with its InverSIOn 80j (x, k, cp', z, Fo) = i: D km Jk(PkmX)00j(m, k, r', z, Fo) (20) m=l Determination of function: Goo(x, cp, z, Fo) From (13) with j = 0, the differential equation and the boundary con-ditions defining 8oo(x, 0, z, Fo) are 76 828 __ 0_0 8x2 1 8800 ---x 8x G. TRIPATHI et al. 1 828 828 ___ 0_0 .l __ 0_0 .l Po(x rr, z, Fo) = 0 , , I 8' I "n x- 8rp- z-o <; X < 1, 0 <; q: <: 2)., I z I < b (21a) and 8800 - A 1-- + B 18 00 = 0 8z a800 D A,-- + B,ooo = 0 - 8z -A 8800. BD 3 8x 3°00 = 0 o <; x < 1, z = - b, 0 <; cp <; 2n o <; x < 1, z = b, 0 <; cp <; 2n (21b) x = 1, i z I < b, 0 <; rp <; 2n Under the transform (a) and (b) the set of Eqs (21) takes the form 1 a (k2 ) J = = - + - -2 -;- ).~ 800 + Po(x, k, n, rp', Fo) = 0 x 8x x and (22) where __ 2:r b Po (x, k, n, rp', Fo) = S J Po(x, rp, z, Fo) cos k(rp - rp')Zn(z)drpdz ° -b The solution of the system (22) well behaved at x = 0 is - % -eoo(x, k, n, (P', Fo) = .r {Gkn(e, x) + H'm(Q, x)} Po(x, k, 11, rp', Fo)· 12 de + o ~ = I (23) + .1 {Gkn(x, 12) + H kn (x, Q)} Po(x, k, n, rp , Fo) 12 de 1 where GIm (e, x) = )'n h P'n x) A3 [Kk P.n 12) I~ (J'Jl) - Ik P'Jl e) K~ P.n)] A3 )'n I~ (J'n) + B3 Ik (}.n) (24a) and H'm (e, x) = ,~3,Ik (:.n x) [Ik P'n) Kk (}.n e) - Kk(}.n) Ik (J'n e)] A3 in Ik (i.n) I B3 Ik (}.n) The inverted transform is 1 = Boo (x, rp, z, Fo) = - ~ AAZn (z) X 2n Jl=l (24b) UNSTEADY HEAT CONDUCTION PROBLEMS 77 ~ 2:r b S S S {Gon (12, x) + Hon (12, x)} Po(x, q:, z, Fo)Zn (z)e de dq: dz+ o 0 -b % 2~ b J .\ .\ {Gon (x, e) + Hon (x, en Po(x, q:, z, Fo)Z (z) ed ed q: dz+ 1 0 -b X 2:t' b J S S {Gkn (12, x) + H'm (I?, x)} Po(x, q:, z, Fo) cosk (q: - q:')Zn (z)ededg;dz o b -b % 2:r b + S j' S {Gkn (x, e) + Him (x, e)} Po(x, q:, z, Fo) cosk (rp - q:') Z" (z) ededq:dz 1 0 -b (25) It is to be noted here that in determining an alternate expression for 80 0 , a finite Hankel transform can be applied instead of the finite trigono-metrical transform. Thus under the transform (a) and (c) Eq. (21) for {Joo reduces to Po* (0 < x -..:::: 1, 0 < rp < 27[, I z I < b) and (0 < x < 1, 0 < rp < 27[, z = - b) (26) (0 -< x < 1, 0 < q: < 27[, z = b) where 2:t 1 Po* = S j' Po (x, q:, z, Fo) cosk (q: - q:')x Jk(flkm x) dx dg; o 0 The solution of the system (26) is eoo (k, m, rpl, z, Fo) = b [(Alflkm cosh flkm b + B1sinh,Llkm b) S Po* {A2fLI:m cosh,Llkm (z' - b) -o - B2 sinh ,Ukm(Z' - b)} dz' + (Az flkm cosh ,ukm b + Bz sinh ,ukm b) . o . \ Po* {Al,Llkm 'coshflkm (z - b) Bl sinh ,ukm (z' - b)} dz'] cosh flkm Z -':b 78 G. TRIPATHI et al. b + (- (A1fJlan sinh fJlcm b + Bl COShfJkm b) ,I' Po* {AzfJkmCosh,Llkm (z' - b) -o - B z sinh fJkm (z' - b)} dz' + (A2 fJkm sinh Pkm b B2 cosh fJkm b) • o J Po* {Alfl COShfJkm (z' - b) + BI sinh Pkm (z' - b)} dz'] sinh fLkm z) -b z f Po* . h (' ) d ' -- -- SIn fJkm Z - Z Z Pkm (27) With its inversion ~ ..... ----2n :r ,Llom (BIBz -------~-~~~~------------ X ([ - (AIfJom cosh Pom b + BI sinh Pom b)· b 2" 1 .\' .\' J Po(x, er" z, Fo) x 10 (p om x) {A2 Pom cosh Pom (z' - B) -o 0 0 - B2 sinh Pom (z' b)} dx der dz' - (A2 Pom cosh Pom b + B2 sinh Pom b) o 2:r 1 .f J .\' Po (X, rr, z, Fo) x10 (Pom X) {AI Pom cosh ,Llom (z' - b) + -b 0 0 BI sinh Pom (z' - b) dx der dz'] cosh flom z + [- (AIPom sinh Pomb BI cosh Pomb) b 2" 1 J J J PO(x, C(, z, Fo) X 10 (Pom x) {Az Pom cosh Pom (z' - b) -o 0 0 - B2 sinh .uom(z' - b)} clx dC( dz' (A2Pom sinh Pom b + B2 cosh Pomb) o 2" 1 J .1" .\' PO(X, c(, z, Fo) x10 CUomx) {AI,u om cosh Pam (z' - b) + -h 0 0 + BI sinh Pom (z' - b)} dx drr dz'] sinh Pom z) Z2:-rI _1_ i f J"J~ .-!~ (x, er, z', Fo) sinh .uom (z' - z) Dom10 (.uom x) xdxdcpdz' • 2n m=l. ,Llom o 0 [;l\STEADY HEAT CONDUCTION PROBLEZ>rS b 2:I 1 J .\' ,\, PO(X, CF, Z', Fo) cosk (cp - cp')Jk (Pkm X) {A2flkm cosh f1km (z' - b) -o 0 0 - Bz sinh f1km (z' - b)xdxdcpdz' - (A z f1km cosh Pkm b B2 sinh f1km b) • o 2:r 1 . J J J Po(x, If, z, Fo) cosk (cp - cp') J k (Pkm X) {AIJ1km cosh f1km (z' - b) + -b 0 0 Bl sinh Pkm (z' - b)} xdxdcpdz'] cosh flkm Z + [- (A1Pkm sinh Pkm b + Bl cosh flkm b) b 2" 1 .1' .1' .\' Po(x, r, z', Fo) cosk (If - cp') J k (P"'m X) {A2 flkm cosh ,ukm(Z' - b)+ o 0 0 + Bz sinh ,ukm (z' - b)} xdxdcpdz' + (A2 ,ukm sinh ,ukm B z cosh flkm b) o 2:-: 1 .r J .1' Po(X, cr, z, Fo) cosh (cp - cp') J k (Pkm x) {A1,ukm cosh ,ukm(Z' - b) bOO + Bl sinh ,ukm (z' - b)} xd.xdlfdz'] sinh flkm z) Z 2:t 1 I '" "'D 0 1 = = • 'Of P :r 1I~1 "=1 J J Pkm ,- /.,.;;;;;. km (x, If, z', Fo) sinh flkm (z' - z)xJ[( (Pkm x), o 0 79 . cosk (cr - rr')dxdrdz') (28) Determination of function 6 l (x, cp, z, Fo) From systems (13) and (14) with j = 1 the differential equation and boundary conditions defining 6 0l(x, cp, z, Fo) are and --+--+- --+ -- BOl(x,cp,z, Fo) = 0 ( 82 1 8 1 82 82 ) 8x 2 X 8x x 2 8cp2 8z2 o <; x < 1, 0 <; rp <; 2:r, Izl < b (29) A e601 D - 1-- + Bl 1701 =/1 (x, p, Fo) 8z o <; x < 1, 0 <; rp <; 2n, z = - b 80 G. TRIPATHI et al. BeOl a A? -- + B? (7101 = 0 - Bz -BeOl a Aa -- + Ba (7101 = 0 Bx o <: x < 1, 0 <; rp <; 2JT z = b (30) x = 1, 0 <: rp <; 2JT, Izj < b Under the transform (a) and (b) the sets of equations (29) and (30) yield for the expression e01(x, k, rpl' n, Fo) as % I ~l I J{Glm (e, x) + Him (e, x)} 11 QdQ + f {Gkn (x, e) H"n (x, e)} 11 ede] o ~ where (31) 2:r 11 = J flex, rp, Fo) cosh k(rp - rp')der' o with its inversion 1 = ,2cr% ---~A~Zn(Z)ll' f{Gon(Q,x) + H on (e,x)}f1(x,rp, Fo) ededrp 2:-rA l n=1 b 0 r !. {Gon (x, p) + Hon (x, Q)} f1(x, er, Fo) ed eder J __ 1_~ ~A~Zn(Zrl y'.f {G,m (e,x)+Hkn (Q,x)) f1(x,rp, Fo) cosk (rp-r') ededtp-I :rAl n=1 k=1 0 0 2:r 1 I J' J {Gkn (x, e)+Hkn (x, e)} fl(X'tp, Fo) cosk (r - cr') QdQd(p - 0~ An alternate expression for eOl(x,p,z, Fo) under the transform (a) and (c) is with its inversion L'"STEADY HEAT COSD(;CTJOS PROBLEJlS 81 I 2:'t J J fleX, rr, Fo)Jo(Pomx)xdxdf[ o 0 + 1 Y :Y _DlmlJd,uI'!..n:0J~2fll;m_Cosh ,ul:m(b _ zL_ l]2sinh ,ukm(~ __ ~) 1 ::T t=i m--:-; (,u~11I A1A2 B 1B2) sinh '1.,uI'l1I b ,ukmV11B2 + BzA1) cosh 2Pkmb 1 2:r I'r fl(x,r[, Fo)Jl{pJ;l7Jx) cosh (rr q')xdxdq iJ 0 (34) Determination for function e02(x, e, z, Fa): The expression fo1' e Il2(x,q,z,Fo) und('r the transform (a) and (b) is x It '~"{~''''('' ,.) + H,,,, (c, x)} ,j,(x, r, Fo) ,oek ('f - 'f ') a,aql (35) .1 .1 {G',n (x, (2)"':'" Him (x, Q)} h(x, q, Fa) cosk (r[ - q) dQdr[ _ 0 0 with its inversion X (An cos x" (1 + ~) + sin x" (1 ~ 1 b Xn b I 2:-r y. >< J J {Gon(Q,x) Hon(Q,x)}h(x,f[, FO)QdQdr[+ o 0 2::7 % \ r {Gon(X, (2) +Hon (x, Q)} f2(X,Q, Fo) + o i + 2' 2'- cos 2Xn = = 1 ( 11=11.:=1 A2 l,\fl_ sinh 9x ) A fCOSk x (1..L ~ ~n n nib Xn NI1 + (I Xn I. :Jl It' i( {G~n (x, (2) Him (x, Q)} hex, f[, Fa) cosk (q - f[)QdQdrr ] 2:r :-! J J {Gkn (12, x) Him (12, x)} f2(x, f[, Fa) cosk (fp-f[) edQdrr sin Xn o 0 (36) 6 Periodica Polytechnica ~L 18/1 82 G. TRIPATll I d "I. An alternate cxpres~ion for e,)2CI:,q;,z,Fo) under the transform (a) and (c) lS obtained as 2:7 b r \. f2(X, q, Fo) cosk (rr - q )J"Cu/unx)xdxdq o '-b with its inversion 2:r b (37) [LlamAl cosh ,uom(b + z) + Bl sinhpcn,{b --'- z)] \ \ fkr, rf, Fo)JoCuc.",x)xdxdrr o '-b ~:r b [,u/m;A 1 cosh ,ulcm( b-i-z) B 1 sinh ,ut,m( b--'- z)] \' \J;(x,,/, Fo) cosk(q-q )xJ,Jfliu71x)dxdr( il '-b (38) The ('xpre""ioIl for eu3(x,Q),z,Fo) under the trall~forlll (<1) and (b) is with its inver"ion 2:r b IOU/l x) j' .\' 11(z, r(, Fo)Zll (z)dzdq; o -0 An alternate expression for erdx, fr. z. Fo) function is l·.vSTEADY HEAT COSDCCTlUS I'lW13LE.l1S 1 b 2:< (- [(A 1,uk/11 cosh /ll:1n b -'- Bl sinh filml b) ,,'.\' f3(z, 'I, Fa)]" (P"m) cosk (Ir - q ') , o 0 B2 sinh ,u':m (z' b)f dz'd,! o 2:7 (Az,uli'" cosh ,llkm b -;- B2 sinh ,u':m b) ,\' J f3(::;' 'I, Fa)]" Cu/:m) cosk ('I -. '(') -bO b) - Bl sinh 11km (z' - b)} dz'd,{ cosh Plan Z b 2;'1 [-(Al,ukmsinh,u'UlIb--BlC05h,UI:mb) I' \' j~(z,fj, Fa)]/:(,ul;m)co5k(q '1'\ o 0 b)} d::;'cZq o 2:r + (A2Plunsinh,u'allb Bzcosh,u,u"b)J J f3(Z,Q, Fa)]/:(,Ulun)cosk('1 ._,/') -bO b)} dz'dq] sinh P!m1 ::;) Z f - S _3_ (::;' 'f, Fa)]" (,u)",,) sinh ,ukm (z' -- ::;) ('osk (q- If') clz' dq (41) Pl(Tn with its inversion b 2:r (--[(Al,uom cosh /1 0mb 13 1 sinh ,uomb) J ,I' j~(z, If, FO)]o(Pul1l) o 0 b) B2 sinh ,uom(z' - b)} dz'dr( o 2:r B2 sinh ,uomb).\' J f3(Z, Cf, Fa )]o(,uOIl1) -bO b) 6* 1 C. TB/PATH! et al. b 2:7 [ - (Alllom sinh f/omb -1- Bl cosh .£10mb) J J 13(z,r(, Fo)Jo CU om ) o 0 A 2 flom cosh f/0m (z' - b) - B 2sinhpom{(z' - b)}dz'dr( o 2:'1 -1- (A2 f/Ol7l sinh f/om b + B2 cosh ,£lam b) .I' J 13(z, r, Fo)JoCuom ) -b 0 {AI,uom cosh ,uom(z' b) + Bl sinh ,uom(z' - b)} dz'dr] sinh .£I omz) 2" Z i; D om J o CL/om x) j~ r 13 - (z', rf, Fa )Jo (Pom) sinh [lonJZ' - .:;) dz'drf 2:T m~l J flom o 0 b " ( [(AI fll:m cosh ,ulun b -1- Bl sinh filun b) I' \"13(rf,':;, Fo)Jk (Ukm) cosk (7 - r) iJ 0 {Az,ukm cosh Pkm (z' - b) - Bz sinh ,ukm (.:;' - b)} d.:;'dr o 2:r + (A2,uJim cosh Piun b -1- B2 sinh PUn b) I' I" 13(Q, z, Fo)Jk (Pkm) cosk(rr - rr') '-:'b 0 {AI ,ukm cosh flkm (z' b) -1- BI sinh Pkm (.:;' - b)} d.:;'dq] cosh ,ukm z b 2:7 (AI flrun sinh [lkm b --'-- Bl cosh ,ukm b) J J 13(q, Z, Fo)Jk eUlmt) cosk (rr rp') -b 0 {A2 ,Illun cosh ,ukm (z' - b) - B2 sinh ,ulun (z' b)} dz'dq o 2:r (A2 flkm sinh flkm b Bz cosh ,ukmb) J JJ3(Q, Z, Fo)Jk C1I km) cosk (rr - rr') -0 0 {Al,ukm cosh f/km (z' - b) + Bl sinh flkm (z' - b)} dz'dr] sinh ,u km z) Z 2;-r 1 .i i Diem J k CUkm x) J~j' 13(Z, q, Fa) J k (fllu,,) sinh flkm (.:;' - z) :T m~l k~l o 0 cosk (rr - rI") dz'dr{ (42) The boundary conditions given in (2) cover a wide variety of cases occurring in technological applications. Specially, conditions of prescribed surface temperature can be obtained by putting Bi = 0, i = 1,2,3) and New-r;.VSTEADY HEAT COSDUCTIOS PIWBLKiIS 85 tonian boundary conditions by putting fi = Bi ai or any comhination of these can be obtained by assigning appropriate yalues to Ai' Bi and fJi = = 1,2, 3). The general problem treated by OJALYO [3] happens to be a partic-ular case of the prohlem trpated here. Summary General expressions have been derived for the unsteady heating of a finite solid cylinder nnder the influence of an arbitrary yolume heat source and an arbitrary initial temperature distribution when conYective type of time dependent boundary conditions are prescribed on the bases and cnrved snrface of the cylinder. By using finite integral transform techniques, expressions for temperature distributions are obtained in various forms and contain valuable results of technological importance. References 1. OLCER, ::\l:RETTIN Y. Conductive heat transfer in finite regions. Int. J. Heat &. ::\lass Transfer 7 196,1" (307-313) 2.0LCER, ::\URETTI:'i" Y.: Solution to a class of unsteady heat-conduction problem. lnt. J. Heat 8; ::\lass Transfer 12 1969. (393-4.11). 3. OJALVO, 1. n.: Conduction with time dependent heat sources and boundary conditions. Int. J. Heat &. ::\1ass Transfer 1962. (1105-1109). 4. CARSLAW, H. S. & JAEGER, J. C.: Conduction of heat in solids, Ind. Ed, p. 10, Clarendon Press, Oxford 1959. 5. LLIKOV, A. V.: Analytical Heat Diffusion Theory, Academic Press (1968). G. TRIPATHI 1 K. N. SHUKLA J R. N. PANDEY Institute of Technology,. Banaras Hindu UniYersity Varanasi- India 

GENERAL SOLUTION TO A CLASS OF UNSTEADY HEAT CONDUCTION PROBLEMS IN A SOLID CYLINDER WITH THE CONVECTIVE TYPE OF TIME-DEPENDENT BOUNDARY CONDITIONS

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GENERAL SOLUTION TO A CLASS OF UNSTEADY HEAT CONDUCTION PROBLEMS IN A SOLID CYLINDER WITH THE CONVECTIVE TYPE OF TIME-DEPENDENT BOUNDARY CONDITIONS

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