What would be my boundary conditions? Heat Equation

[punkstart]

1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celsius. Find the boundary conditions at this x=4 end of the rod to be used in the one dimensional heat equation.



2. Heat flows out of rod at -$$\lambda A u_{x}(a,t)$$




3. with my substitutions i get -$$u_{x}(4,t)=2(u(4,t)-23)$$
So that my first boundary condition is the above. But i think " they " want another boundary condition here at x=4,but what ? I was thinking something like $$u(4,t)=u(4,0)-tu_{t}(4,t)$$ in other words temperature now = initial temperature - time*rate of temperature loss. i am not very experienced with these problems,can someone please point me in the right direction ?

 

[lanedance]

so based on the boundary condition the you can calculate the heat moving out the end of the rod
$$Q = 2(u(4,t) - 23))$$

Assuming the heat flow is reasonably continuous which is a good assumption you should be able to relate the heat flow to the thermal gradient at the end of the rod $$u_x(4,t), using the thermal conductivity$$