ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES

Transcription

1 R 888 Philips Res. Repts 30, 31*-39*, 1975 Issue in honour of c. J. Bouwkamp Abstract ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES by Murray S. KLAMKIN University of Waterloo Waterloo, Ontario, Canada (Received July 31, 1974) It is physically intuitive that the dominant term in the asymptotic temperature distribution of an arbitrary body with constant thermal properties and subject to a uniform heat flux at its surface can be gotten by assuming it will be the same for all points of the body. Whence, T...,at. This result is shown to hold even if the surface flux varies with position. In addition, the rest of the quasi-steady-state solution is shown to depend only on position and is given by the solution of a corresponding Neumann problem. The results are then extended to periodic heat-flux conditions and to composite bodies. 1. Introduetion There are not too many exact solutions known for heat-conduction problems in which the geometry and/or boundary conditions are not simple. However, if one is mainly interested in "large"-time solutions, one can still, exactly, obtain the dominant term in the asymptotic expansion. Here, we determine the asymptotic temperature distribution for larbitrary homogeneous bodies and then for homogeneous composite bodies with constant thermai properties and subject to two kinds of non-constant boundary conditions. To illustrate the method, we first start off with the simplest of this class of problems. 2. Here, we determine the asymptotic temperature distribution for an arbitrary smooth homogeneous body with constant thermal properties and which is subject to a space-varying heat flux over its surface (fig. I). Consequently, we wish to solve the B.V.P. (boundary-value problem) 'è)t -= ct \12 T 'è)t 'è)t -k-= Q(x,y,z) 'è)n (in V), (on 8), (1) (2) T(x, y, z, 0) - To(x, y, z), for "large" values of the time. As usual, k denotes the thermal conductivity and ct denotes the thermal diffusivity k/c(! (c is the heat capacity, (! the density). (3)

2 32* MURRAY S. KLAM KIN Fig. I. Since the total amount of heat which has flown into the body at time t is t f f Q ds, we expect intuitively that the leading term of the asymptotic temperature distribution will be proportional to t. This is indeed valid. To prove it, we let where T(x, y, z, t) = a IX t + 4>(x,y, z) + T(x, y, z, t), (4) (in V), "ö4> -k-=q "ön (6) (a is a constant to be determined). These conditions on 4> were set to insure that T also satisfies (1) subject to an insulated boundary condition and consequently T will approach a constant asymptotically. From (5) and (6), it follows by the divergence theorem that (5)!!QdS=k!! s v!\12cpdv=kav=qs, where Q denotes the average flux over the surface. Consequently, in order that (5) and (6) have a solution it is necessary that QS a=--. kv (7) Furthermore, that the solution of 4> is unique, up to an arbitrary constant, follows from the known uniqueness of the solution of \12 Ä = 0 (in V) subject to "öä/"ön = 0 aside from an arbitrary constant. To fix this arbitrary

3 ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES 33* constant, we also impose the condition f f f [To(x, y, z) - cp(x, y, z)] d V = O. v This will insure that T(x, y, z, t) will approach zero as t co instead of some non-zero constant. Equations (5) and (6) can be easily transformed into a classical Neumann problem by letting where r 2 = x 2 + y2 + Z2. Whence, <:rx=o cp(x, y, z) = a r 2 /6 + x(x, y, z), ()X k a ()r~ -k-=q+-- ()n 6 ()n (in V), f f f [To - X - a r 2 /6] dv = O. v This Neumarm problem has a unique solution for smooth S since ff( ka ()r 2 ) Q+--,- ds=o s 6 (jn 1 by virtue of a = QS/kV. From the definition of cp, it follows that (as was planned) ()T - = C( \12 T (in V), ()t ()T -=0 ()n 'Ï' (x, y, z, 0) = To(x, y, z) - cp(x, y, z). Consequently, 'Ï' is the solution of the heat-flow problem of a body which is insulated and whose.average initial temperature is zero. Whence T 0 as t We have thus established that the quasi-steady-state temperature is exactly given by a «t + cp(x, y, z). For simple geometries (i.e., one-dimensional flow in slabs, cylinders and spheres), cp can be determined explicitly. For more complicated bodies, cp can be determined numerically.

4 34* MURRAY S. KLAM KIN 3. Here, we change the problem in the preceding section by considering a periodic (in time) heat-flux boundary condition. Consequently, (2) is changed to 'öt -k - = Ql(X, y, z) sin (wt) + Q2(X, y, z) cos (wt) (2') cm while (1) and (3) remain the same. Since we physically expect the temperature to asymptotically have a form similar to the r.h.s. of (2'), we let T(x, y, z, t) = 4>(x, y, z) sin (wt) + X(x, y, z) cos (wt) + T(x, y, z, t), (8) where T is to satisfy (1) and should approach a constant value. Whence, [4> cos (wt) - X sin (wt)] co]«= sin (wt) \12 4> + cos (wt) \12 X (in V), and ( 'ö4> 'öx) -k sin (wt) - + cos (wt) - = Ql sin (wt) + Q2 cos (wt) (on S). Consequently, a \12 4> = -w X, a \12 X = w 4> (in V), (9) subject to the B.C.'s (on S). (10) We first show that 4> and X are unique up to arbitrary constants. Then by a known theorem, the uniqueness implies the existence of 4> and X *). Assume that there exist two different solutions 4>1' Xl and 4>2' X2' and form their differences Whence, - - a \12 4> = -w X, - - a \12 X = t» 4> (in V), (11) 'ö'f 'öx -=-=0 (on S). (12) From (10) it follows that f f f (~\J2~+X\12X)dV=0. (13) v Since.) I am indebted to L. Nirenberg (Courant Institute) for this argument.

5 ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES 35* (13) can be rewritten (using the divergence theorem) as By virtue of the B.C.'s (11), we must then have - - (\7 cp)2 + ('v X)2 = 0 - and thus cp and X must be constants. Explicit solutions for cp and X can be obtained for one-dimensional flow in slabs, cylinders and spheres 1). For more complicated geometry, we can again resort to numerical methods. It follows from (8), (11) and (12) that ()T - = a \12 T ()t ()T -k-=o ()n T(x, y, z, 0) = To(x, y, z) - (on V), x(x, y, z). Consequently, lirn T(x, y, z, t) = - f f f (To - 1 t-+co V y X) d V = To (constant) and the asymptotic temperature distribution is given by Incidentally, T(x, y, z, t) '" To + cp(x,y, z) sin (cot) + x(x, y, z) cos (cot). (15) f f f cp dv = - kacof f Q2 ds, Y S fffxdv= y 4. Here, we generalize the problem in sec. 2 to arbitrary composite bodies, fig. 2. However, for simplicity it suffices to just consider a composite of two different materials, see fig. 3. Our B.V.P. is now

6 36* MURRAY S.KLAMKIN Fig.3. bt -=a \J2T (in V), (16) bt bti -= al \J2 TI bt (in VI), (17) "öt -k-= Q(x,y,z) (18) "ön We Iet T=TI "öt "öti (on SI), (19) k-=klbn ) "ön T(x, y, z, 0) = To, TI(x, y, z, 0) = Tal' (20) T = a a t + cp (x, y, z) + l'(x, y, z, t), TI = al al t + cp I(x, y, z) + 1'1(X, y, z, t),

7 ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES 37* where l' and T, satisfy (16) and (17), respectively. Consequently, \12 cp = a (in V), (21) In order to make l' and T, approach constants as t , we impose the B.C.'s: 'öcp -k-= Q(x,y,z) (22) (23) a a = a,a" (24) cp = cp" ) 'öcp 'öcpi (on S,), (25) k-=k,- In order for the latter Neumann type of problem to have a solution, the heat input per unit time via the flux Q must be exactly picked up by the distributed heat sinks corresponding to a and a, in the Poisson equations (21) and (22). Consequently, as in sec. 2, we must have by the divergence theorem that or Also, Whence, and 111 \12 1> dv = - 11 :~ ds + 11 :~ as, v s ~ Q S k, IJ 'öcpi av=-+- -ds,. k k s, - 11:~' ds, = J 11 \121>ldVI=a, v" s, v, Q S = k a V + k, a, V, a= a,k V+ aki V, It follows now that cp and CPIare determined uniquely up to the same arbitrary constant. As in sec. 2, we fix this constant by also requiring that 111(To - 1» dv+ J 11 (Tol- 1>,) dv,= O. V v,

8 38* MURRAY S. KLAMKIN Since the B.V.P. for T and T, is è:>t - = a \12 T, è:>t è:>t -k-=o è:>n è:>t, - = a, \12 T" è:>t T=T, k ot ~k:t, 1 è:>n <)n (on S,), f f f T d V + f f f T, d v, = 0 v V, (att=o); T and T, -+ 0 as t The last problem to be considered is the same as the one in sec. 4 with the exception that boundary condition (18) is replaced by We now let è:>t -k - = Ql(X, y, z) sin rot + Q2(X, y, z) cos rot. (18') bn T = cp (x, y, z) sin rot + X (x, y, z) cos rot + T(x, y, z, t), T, = cp,(x, y, z) sin rot + X,(x, y, z) cos rot + T,(x, y, z, t), where T and T, are to also satisfy (16) and (17), respectively, and approach constant values as t This requires (as in sec. 3) that a \12 cp = -ro X, a \12 X = to cp (in V), a, \12 CPI= -ro X" a, \12 XI = ro CPI (in V,), (26) (27) è:>cp è:>x -k-= Ql' -k-= Q2 bn è:>n cp = cp" X=XI bcp è:>cpi k-=k,-, k Ox ~ k~x, è:>n bn <)n bn I (on S,), (28) (29)

9 ASYMPTOTIC HEAT CONDUCTION IN ARBITRARY BODIES 39* By a similar argument to that used in sec. 3, we can establish uniqueness up to arbitrary constants for ~, X, ~" XI since we can obtain where ~ = ~1 - ~2, etc. Explici~ solutions for ~, X, ~I and XI can be obtained fairly simply in this manner for composite slabs, cylinders and spheres and will be given in a subsequent paper. REFERENCE 1) H. S. Carsla wand J. C. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, 1959.