Understanding a Thermo Diffusion Equation Problem: A Comparison of Solutions

[Quadruple Bypass]

im given the equation:

k d^{2}T/dx^{2} = 0 over 0 < x < L = 0.4

with T(0)=T and -k dT(L)/dx = h(T(L)-Tinf)



i tried to solve it and i got

T(x) = (-h(T(L)-Tinf)x)*(1/k)

the book gives

T(x) = ( (k+h(L-x))T+h*x*Tinf)*(1/ (k+h*L) )

i don't understand what went wrong... help!

 

[Mute]

If you plug x = 0 into your expression, you'll find you get T(0) = 0, which is not the boundary condition you're given. You want T(0) = T.

Solving the ODE gives you T(x) = Ax + B. In your solution you appear to have dropped the B term and only solved for A. Keeping B and solving for it should fix the problem.

 

[Quadruple Bypass]

thanks i don't know why that happened. that just adds a +T after my original solution. i still don't understand where they got the '(k+h*L)' from...

 

[Mute]

Okay, I see what the other step they did is. Your condition on the derivate of T is defined in terms of T(L), so you get as your solution

T(x) = -\frac{h}{k}(T(L) - T_{\infty})x + T

So, at x = L, you have

T(L) = -\frac{h}{k}(T(L) - T_{\infty})L + T

so you need to solve for T(L). Plugging that back into your original expression should (hopefully) get you the book's solution.