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HW#8: March 16, 2000 Dianbo Li 5-1: A slab, 0 x L is initially at zero temperature. For times t > 0, the boundary surface at x = 0 is subjected to a time-varying temperature f(t) = b +ct, while the boundary surface at x = L is kept at zero temperature. Using Duhamel’s theorem, develop an expression for the temperature distribution T(x, t) in the slab for times t > 0. Solution: The mathematical formulations are: 2T ( x, t ) 1 T ( x, t ) in 0 < x < L, t > 0; t x 2 T ( x,t ) = f(t) = b+ ct at x = 0, t > 0; T=0 at x = L, t > 0; T=0 at t = 0, in 0 x L The corresponding auxiliary problems becomes: 2 ( x, t ) 1 ( x, t ) in 0 < x < L, t > 0; t x 2 ( x,t ) = 1 (x,t) = 0 (x,t) = 0 at x = 0, t > 0; at x = L, t > 0; at t = 0, in 0 x L m x 2 m2 ( t ) 1 the solution is ( x, t ) e sin( mx / L) L L m1 m / L here I use t - replace , and m = m/L t df ( ) Since T ( x, t ) ( x, t ) and df()/d = c d d 0 so t x 2 m2 ( t ) 1m T ( x, t ) e sin( mx / L) cd L m 1 m / L 0 L final solution is : 2 1 sin( mx / L) x 2 T ( x, t ) c t c (1 e m (t ) ) L L m 1 (m / L) 3 m 7-4:A slab, , 0 x L, is initially at uniform temperature To. For time t > 0, the boundary surface at x = 0 is kept insulated and the boundary surface at x = L is kept at zero temperature. Obtain an expression for the temperature distribution T(x,t) in the slab valid for very small times. Solution: the mathematical formulation of this problem is given as: 2T ( x, t ) 1 T ( x, t ) in 0 < x t > 0; t x 2 T 0 at x = 0, t > 0; x T=0 at x = L, t > 0; T = To at t = 0, in 0 x L The Laplace transform of these equations are _ T d 2 T ( x, s ) s _ T ( x, s ) o 2 dx in 0 x L _ dT 0 dx at x = 0; _ T 0 at x = L _ T ( x, s ) 1 cosh( x s / ) To s s cosh( L s / ) simplify the equation and use binomial method (P278) so the solution is: _ T ( x, s ) 1 (1 (1) n e [ L (1 2 n ) x ] To s n 0 s / (1) n e [ L (1 2 n ) x ] s / ) n 0 From Table 7-1, inverting term by term: L(1 2n) x L(1 2n) x T ( x, t ) 1 (1) n erfc (1) n erfc T0 4t 4t n 0 n 0 9-2. A semiinfinite medium x >0 is intially at a uniform temperature Ti. For times t > 0 , the boundary surface at x=0 is subjected to a prescribed heat flux, that is k(T/x) = f(t) at x = 0, where f(t) varies with time. Obtain an expression for the temperature distribution T(x,t) in the medium using the integral method and a cubic polynomial representation for T(x,t). Solution: the mathematical formulation of this problem is given as: 2T 1 T x 2 t T f (t ) x T = Fi Therefore, in 0 < x t > 0; at x > 0 , t > 0; at t = 0, in x 0 T x x T x x 0 1 d Tdx T dt 0 x d dt In view of the conditions: T T T x Ti x 0 x 0 f (t ) x x then d f (t ) Tdx Ti ------------------ ( 1 ) dt x 0 Now apply a cubic polynomial representation for T(x,t), 2T x 2 x T(x,t) = a1 + a2x + a3x^2 + a4x^3 T ( x, t ) Ti x (1 ) 3 T (t ) Ti it turns out that 2 4 d T Ti f (t ) 3 dt f (t ) Using equation ( 1 ) & ( 2 ): or f ' (t )T Ti 4 3 f (t ) 3 2 ----------------------- ( 2 ) f or t > 0, 0