- #1

Vector1962

- 61

- 8

## Homework Statement

Let a slab [itex]0 \le x \le c [/itex] be subject to surface heat transfer, according to Newtons's law of cooling, at its faces [itex] x = 0 [/itex] and [itex] x = c [/itex], the furface conductance H being the same on each face. Show that if the medium [itex] x\le0[/itex] has temperature zero and medium [itex]x=c[/itex] has the constant temperature T then the boundary value problem for steady-state temperatures in the slab is [tex]u''(x)=0[/tex] [tex]Ku'(0)=Hu(0)[/tex] [tex]Ku'(c)=H[T-u(c)][/tex] where K is the thermal conductivity of the material in the slab, write [itex]h=\frac{H}{K}[/itex] and derive the expression [tex]u(x)=\frac{T}{ch+2}(hx+1)[/tex]

## Homework Equations

[itex]U_t=K\nabla^2U[/itex]

## The Attempt at a Solution

.[/B]I have the first part of the question complete. I'm not sure how to apply the boundary conditions? I've solved similar problems with boundary's [itex]u(0)=0[/itex] and [itex]u(c) = T[/itex] and I've solved various problems with heat flux boundary's. For some reason, applying the Newton boundary conditions is messing me up.

I can get to [itex]u(x)=c_1x+c_2[/itex] comfortably which leads to [tex]u(x)=u_x(x)x+u(0) [/tex] [tex]u(c)=u_x(c)c+u(0) [/tex] but from there I'm lost.