Answer 6.1
Substituting and in Fourier's Law of Heat Conduction , we get
(a) Let us derive
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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