156 Chapter 2 The Heat Equation
5.g(x)=T 0 (constant).
6.g(x)=βx(βis constant).
7.g(x)=β(a−x)(βis constant).8.g(x)=
2 T 0 x
a ,^0 <x<a
2 ,
2 T 0 (a−x)
a ,a
2 <x<a.
9.A.N. Virkar, T.B. Jackson, and R.A. Cutler [Thermodynamic and kinetic ef-
fects of oxygen removal on the thermal conductivity of aluminum nitride,
Journal of the American Ceramic Society, 72 (1989): 2031–2042] use the fol-
lowing boundary value problem to study the kinetics of oxygen removal
from a grain of aluminum nitride by diffusion:∂C
∂t =D∂^2 C
∂x^2 ,^0 <x<a,^0 <t,
C( 0 ,t)=C 1 , C(a,t)=C 1 , 0 <t,
C(x, 0 )=C 0 , 0 <x<a.In these equations,Cis the oxygen concentration,Dis the diffusion con-
stant,ais the thickness of a grain,C 0 andC 1 are known concentrations.
a. Find the steady-state solution,v(x).
b. State the problem (partial differential equation, boundary conditions
and initial condition) for the transient,w(x,t)=C(x,t)−v(x).
c. Solve the problem forw(x,t), and write out the complete solution
C(x,t).
d. The concentration in the center of the grain,C(a/ 2 ,t),variesfromC 0
at timet=0towardC 1 astincreases. Suppose we want to find out how
long it takes for this concentration to complete 90% of the change it will
make fromC 0 toC 1 ; that is, we want to solve this equation fort:C(a
2,t)
−C 0 = 0. 9 (C 1 −C 0 ).
Show that this equation is equivalent to the equationw(a
2 ,t)
=− 0. 1 (C 1 −C 0 ).
Find an approximate formula for the solution by using just the first term
of the series forw(x,t).