Jounal of Engineeing Physics and Themophysics. Vol. 73. No. 5. 2 METHOD OF PAIED INTEGAL EQUATIONS WITH L-PAAMETE IN POBLEMS OF NONSTATIONAY HEAT CONDUCTION WITH MIXED BOUNDAY CONDITIONS FO AN INFINITE PLATE P. A. Mandik UDC 517.968,536.24 Fo solving a nonstationay equation of heat-conduction in cylindical coodinates with mixed discontinuous bounday conditions pescibed on one of the sufaces (z = O) of an infinite plate, a method oj paied integal equations with L-paamete is used. Moeove, on the othe suface (z = h) of the plate the bounday conditions ae pescibed unmixed. It is equied to solve the heat-condition equation fo the excess-tempeatue function: O (, z, T) + 1 O (, Z, "C) + _. (, z, x) = 1 O (, Z, z), >, x>, <z<h, (1) and z ae the cylindical coodinates, ~ is the time, a > is the themal diffusivity coefficient, (, z, ~) = T(, z, x) - To, and To = const is the initial tempeatue of the plate. Pescibed ae the initial condition a homogeneous bounday condition at z = h and mixed bounday conditions at z = (, z, ) = T (, z., ) - T O =, (2) (, h, x) = (3) q* (, x) -O:(,O,'c)-----~=q(,z), <<, (4) (,,'=, <<~o, (5) )v > is the themal conductivity coefficient. Applying the Hankel-Laplace tansfomation to poblem (1)-(5) of the fom <a co. Co, s) =J" j' (, exp (- sx) Jo (p) dd'c, e s >, (6) Jo(p) is the Bessel function of the fist kind and zeo ode, and taking into account the boundedness of the tempeatue (, z, "c) on the axis = and fo ----> ~o, the solution of the heat-conduction equation (1) in the egion of L-tansfoms can be witten [ 1] in the fom Belausian State Univesity, Minsk, Belaus. Tanslated fom Inzhenemo-Fizicheskii Zhunal, Vol. 73, No. 5, pp. 92-96, Septembe-Octobe, 2. Oiginal aticle submitted Febuay 1, 2. 884 162-125//735-884525. 2 Kluwe Academic/Plenum Publishes
sinh[(h-z)~(p2+s 1 (,z,s)= T(, z,s)- T = fa (p,s) Jo (p)dp, (7) m A(p, s) is the unknown analytical function. Hee and hencefoth it is assumed that e s >, and, fo bevity, this is omitted in the epesentation. Afte application of the L-tansfomatioh to conditions (4) and (5) the mixed bounday conditions fo z = will take the fom -O:(,O,s)=q(,s), <<, (8) (,, s) =, <<~o. (9) Assuming in fomula (7) z = and taking into account conditions (8) and (9), it is possible to pass to paied integal equations with the L-paamete: f ~- (p, s) ~J/p2 + S)cot 2+ jo(p)dp=q(,s), <F<, (1) fa(p,s)jo(p)dp=o, <<o~, (11) fom which it is necessay to detemine the analytical function A(p, s). To solve the paied integal equations (1) and (l l), we intoduce the new analytical function q(t, s) with the aid of the elation [ 1 ] A (17, s) = /9 ~ ~ (t, s) sin t 2 + dt. (12) 2+ a s) On substitution of (12) into the second paied equation (11), we can easily veify that Eq. (11) is eadily satisfied accoding to the value of the discontinuous integal PJo(P) sin t 2+ dp=. ~ lj2 / <t<, (13) if-f- ' <<t. Substituting (12) into the fist paied equation (1), we aive at the integal equation with L-paamete fo detemining the unknown t(, s): 885
i~exp 41s(2-t2)))dt-! ~/t 2 _ 2 sin (t 2 - ) dt+ + I (t, s) sin t at oo exp l-- h 4(p2 + s)/ sinlt41p2+s)l J` (p) dp = S q (p, s) pd p, O<<. (14) Equation (14) is the analytical basis fo finding the unknown function q)(, s), but it is not vey convenient fo solution. Theefoe, we educe (14) to an integal equation simila to the Fedholm equation, but with the L-paamete. Fo this pupose, eplacing by ~t in Eq. (14), we multiply the left- and ight-hand sides of the equation by the integating facto cos ('~s--(2- a la 2))21a/~ tion fo la going fom zeo to. As a esult we have o la 2) and then integate the esulting equa- <<, (15) K (, p, s) - - - p- p+ h exp (- x--) Icos/~-~ x)- cos (-~--5 x)] dx sinh (x) [ (./ 1T,~, ~/ 2 - ~t2 d~t. We note that fo h --~ oo we diectly obtain a solution in the egion of L-tansfoms fom fomula (7), the paied integal equations with the L-paamete fom fomulas (1) and (11), and the coesponding integal equations fo ~(t, s) fom Eqs. (14) and (15) fo an isotopic semispace with the mixed bounday conditions (8) and (9) at z =. Thus, the poblem set is actually solved. The main difficulty of calculating the coesponding tempeatue fields in an infinite plate with the mixed bounday conditions (4) and (5) and unmixed condition (3) consists of the detemination [2] of the analytical function ~p(, s) fom Eq. (15). In local means of heating the body suface though a cicula egi_on < <, z = the heat-flux density q*(, s) = L[q*(, "~)] can be epesented in the fom of the poduct q*(, s) = W(s)q(), W(s) = 886
L[W(x)] is the tansfom of the specific powe of the heat flux that fo the oiginal function W(x) depends only on time, and q() is the distibution of the dimensionless heat flux along the cylindical coodinate in the cicula egion < <. Then Eq. (15) can be witten in the fom 14(s (2 - B2))IB q (B) COS d-~* (. s) - I f "~* (B.s) K (. B. s) d,: _~ ~ /~ASO 4 2 - B 2 db,o<<, (16) g tp (,s) -Y('s), m esw(s)>o. (17) sw (s) Next, we epesent the analytical function tp*(, s) in the fom of a seies [3]: * (, s) = exp tp n () (~S) n-2", (18) we substitute this expession into Eq. (16), expand the kenel K(, pt, s) and the well-known function on the ight-hand side of Eq. (16) into the coesponding seies, and pefom opeations of multiplication of the seies obtained. As a esult, we come to the following equation: n= oo n=o n = 2_2_ Z Z (~s) Ann, q(b) n-- m-- n ± Z Y ( )" - --g n=o m=o 1 ~ ~ (~s-s 16h [md.,.,(b,)-e,.j(~t,)]tpn ( ) -- -"--q---z--'-'z--------4-g--... dla,o < <. h n= m=l k---o I--- [4h'm- + (B + )-] [4h-m- + (B - )'] (19) m! (n - m)! (~--)" ' Fk-I - (k - l )! C m(p, ) m! ('~a) m [(g + )m-i (g- )m-l] ' Dm,/(B, ) = cos 1 -~ { [4h2m2 + (P- + ) 2] 4h 2 l! (~aa)t 887
X (IX -- ) I - [4h2m 2 + (IX - )21 (It + ) l} ; Ea, 1 (IX, ) = {[4hZm 2 + (IX - ) 2] 8h 31! ({-ffa)~ x (Ix + ) 1+1 - [4h2m 2 + (IX + ) 2] (ltl - )l+l} Thus, at n =, fom Eq. (19) we can wite the l~edholm integal equation of the second kind to detemine (): P () = 2_2 i q (gix) ~ dix _ co 16h4m x E m=l [4h2m2 + (Ix + )2] [4h2m2 + (Ix - )2] dix. The emaining necessay values of q)i(), i = 1, 2, ae also detemined fom fomula (19) on equating the tems at the same powes of ~ on the left-hand and ight-hand sides. Substituting q)n() in&o seies (18), we find_the values of q)*(, s) and, consequently, fom fomula (17) we detemine the values of cp(, s). Futhe, using q)(, s) in fomula (12), we find the value of A(p, s) and then also the tansfom of the unknown tempeatue fom fomula (7). Finally, applying the Laplace integal invesion fomula, we can find diectly the oiginal function of the excess tempeatue (, z, x) = L-l[(, z, s)]. EFEENCES 1. 2. 3. V. P. Kozlov, P. A. Mandik, and N. I. Yuchuk, Inzh.-Fiz. Zh., 72, No. 3,. 555-571 (1999). V. P. Kozlov, P. A. Mandik, and N. I. Yuchuk, Vesm. Belous. Univ., Se. 1, No. 2, 37-42 (1999). V. P. Kozlov, P. A. Mandik, and N. I. Yuchuk, Inzh.-Fiz. Zh., 71, No. 4, 734-743 (1998). 888