Answer 6.1

(a) Let us derive
Substituting and in Fourier's Law of Heat Conduction , we get
 
or

(b) Use the above result to rewrite the equation (6.2):
 
 
Using equation (6.1) to substitute for k, we obtain the differential equation as follows.
Since
then
 
Let
Then

NOTE:      Equation (6.3), together with the boundary temperatures Ta and Tb at r = a and r = b, respectively, define the boundary value problem

(BVP)

in which the constants , a, b, Ta, and Tb are given and the constant p is to be determined along with the solution T = T(r). The BVP above is solved in Problems 6.2, 6.3, and 6.4.

Generally, two boundary conditions arise in conjunction with a second order ordinary differential equation (ODE). But equation (6.3) is a first order ODE which usually has a single associated boundary condition to fix the arbitrary constant of integration.

The additional boundary condition is needed to determine the value of p.

 

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