D,4 • SLAB OF FINITE THICKNESS 



In accordance with the remarks in Art. 2 the boundary conditions (see 

 Fig. D,4) are: 



^^ = h{T, - T) 



k 



dx 



= 



The initial condition is T = at f = 0. 



Introducing a transformation to nondimensional variables : 



T X kt 



Kt 



(4-2) 



p =x 



O =3 X 



I 

 I 



at ^ = 



where k is the thermal diffusivity of the material, 

 K = k/pc, we obtain the differential equation and 

 boundary conditions in nondimensional form: 



a^e _ ae 



W^ ~ dr 



d^~ k 



= ) at 1=1 



with the initial condition 



= -1 at T = 



When the system is solved by separation of 

 variables as discussed in Art. 2, we obtain the 

 following results: The eigenf unctions are 



<Pni^) = COS M7^(l - (n = 1,2, . . .) 



(4-3) 



(4-4) 



(4-5) 



3- 



■H 



Fig. D,4. 



where the eigenvalues fin are the roots ^ of the eigenvalue equation 



hd 

 n tan jLt = -7- 



(4-6) 



The complete solution is 



ei^, r) = ^ = 1 + e(^, r) = 1 -^ £ A„ cos M»(l " Oe-<^ (4-7) 



«.= 1 



where the amplitudes, determined from the initial condition 6(^, 0) 

 are 



An — 



4 sin iin 



- j^ cos m™(1 - ^)d^ 



j^ cos^ m™(i - m^ ~2i^^:T^^^%u 



1, 



(4-8) 



It is seen from the nondimensional formulation (Eq. 4-3, 4-4, and 4-5) 



1 The first six roots of this transcendental equation, for ^ hd/k ^ 00 , are given 

 in [6 (Appendix IV, Table 1)]. 



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