1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

170 Chapter 2 The Heat Equation

In these equations,Dis the diffusion constant,Lis the length of the cylin-

der, andC 0 is the saturation concentration, which depends on the porosity

of the material. FindC(x,t).

15.Use the solution of Exercise 14 to find an expression for the total weight

of water absorbed by the rod,

W(t)=A

∫L

0

C(x,t)dx.

16.A plot ofW(t)/C 0 as a function ofs=

√

Dt/L^2 over the range 0 to 2

resembles a slanted line segment joined by a curve to a horizontal line

segment. The slope of the slanted line segment in this graph is approx-

imately 1. Experimenters plot measured values ofW(t)/C 0 vs

√

tto get

a similar graph. Then they use the slope of the slanted line segment to

findD.Explainhow.

2.6 Example: Convection

We have seen three examples in which boundary conditions specified either
uor∂u/∂x. Now we shall study a case where a condition of the third kind
is involved. The physical model is conduction of heat in a rod with insulated
lateral surface whose left end is held at constant temperature and whose right
end is exposed to convective heat transfer. The boundary value–initial value
problem satisfied by the temperature in the rod is

∂^2 u

∂x^2

=^1

k

∂u

∂t

, 0 <x<a, 0 <t, (1)

u( 0 ,t)=T 0 , 0 <t, (2)

−κ∂u

∂x

(a,t)=h

(

u(a,t)−T 1

)

, 0 <t, (3)

u(x, 0 )=f(x), 0 <x<a. (4)

We found in Section 2 that the steady-state solution of this problem is

v(x)=T 0 +xh(κT+^1 −haT^0 ). (5)

Now, since the original boundary conditions were nonhomogeneous, we
form the problem for the transient solutionw(x,t)=u(x,t)−v(x).Bydirect
substitution it is found that

∂^2 w

∂x^2 =

1

k

∂w

∂t,^0 <x<a,^0 <t, (6)