I have a bar which is made of 2 different materials. The two materials have the heat conductivity $k_l$ and $k_r$ and the single pieces have the length $l_1$ and $l_2$.
What's the heat flow if there are the two temperatures $T_l$ and $T_r$ at the edges (as a function of the difference $T_l - T_r$)?
There are several conditions of linking the two materials, aren't there?
Thanks for any advice!
When the dynamic equilibrium is reached, the energy that goes in on the left exits on the right. The heat flow in every section of the bar is constant. Also, there is an intermediate temperature $T_i$ at the point of contact of the two materials. So if we write the heat flow $\frac{dQ}{dt}$ for each of the two materials, we have a system of two equations with two unknowns: $$\frac{dQ}{dt}=-k_1\frac{T_l-T_i}{l_1}\\\frac{dQ}{dt}=-k_2\frac{T_i-T_r}{l_2}$$ First, get the expression for $T_i$, then plug it into any of the above equations to find $\frac{dQ}{dt}$.