Page 463 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 443

10.2.1.2 Solution for short times

Let s be the distance measured from the surface of the slab, i.e.,
s=L-r (10.2-40)

so that Eq. (10.2-7) reduces to

(10.2-41)

At small values of time, the heat does not penetrate very far into the slab. Under
these circumstances, it is possible to consider the slab as a semi-infinite medium in
the s-direction. The initial and boundary conditions associated with Eq. (10.2-41)
become
at t=O T = To
at s=O T=Tl (10.2-42)
at s=oo T=T,
Introduction of the dimensionless temperature

&I=- T - To (10.2-43)
Tl - To
reduces Eqs. (10.2-41) and (10.2-42) to

(10.2-44)

at t=O &I=O
at s=O +=l (10.2-45)
at s=oo +=O
Since there is no length scale in the problem, this parabolic partial differential
equation can be solved by the similarity solution as explained in Section B.6.2 in
Appendix B. The solution is sought in the form
4 = f (4 (10.2-46)

where
(10.2-47)

The chain rule of differentiation gives

(10.2-48)