In this screencast, we're going to find temperatures in a composite wall using the thermal circuit
concept. A composite wall is made up of different materials, therefore they have different thermal
conductivities. If we recall, q - our heat transfer rate - is delta T (or the change in temperature)
divided by the total resistances between those two temperatures. So to start with, we're
going to find the difference in temperature between T infinity 1, or the surrounding temperature,
minus T infinity 2, which is the surrounding temperature on the other side of the wall,
divided by the total resistances. And in this case, it's going to be all the resistances
because we are going from the outside temperature to the inside temperature. So let's look at
how we find our total resistances. So the first resistance that we have - and let's
go up and look at this picture again - is that convective resistance. So that's 1 over
h1 A, and we add to that the first conductive resistance, which is L1 over k1 A, our second
conductive resistance L2 over k2 A, our third conductive resistance L3 over k3 A. And we
have three conductive resistances because we have three different materials with three
different thermal conductivities. And finally, we have our second convective resistance between
the wall and the outside temperature. I'll demonstrate how to find some of these resistances.
Let's start with the first one, which is 1 over h1 (the heat coefficient, or the convective
heat transfer coefficient) times the area that the heat transfer goes through, which
equals 1 divided by 50 watts per meter squared K, times the area, which is 0.01 meters squared.
This equals 2 K (which is kelvin) per watt. This also, because our h's are the same, is
going to equal 1 over h2 A. Now let's demonstrate how to do a conductive resistance. So that's
L1 over k1 times the area, which is 1 meter divided by 30 watts per meter K times that
area, which is 0.01 meters squared. And this is going to equal 3.3 kelvin over watts. So
when we do this for all of the resistances, and it might be a good practice to figure
them out yourself, the first one is 2, plus 3.3, plus 2, plus 5, plus 2, all in kelvin
divided by watts. And that equals 14.3 kelvin divided by watts. Now, if we want to find
our q, we look at the two temperatures that these resistances are between, which is 800 K minus
200 K, all divided by our total resistance, which is 14.3 K per W, so our q is 42 watts.
One way to look at this is as a complete thermal circuit, with the temperatures separated by
their respective resistances. So for example, let's say we want to find Ts3. So we use that
q equals our delta T, which is T infinity 1 minus Ts3, over the total resistances. But
remember those total resistances are the resistances between the two temperatures. So our total
resistances here are 1 over h1 A, plus L1 over k1 A, plus L2 over k2 A. Those are the
resistances between T infinity 1 and Ts3. So now we can rewrite it such that Ts3 equals
T infinity 1 minus q times these total resistances. Knowing our q equals 42 watts, and summing
our total resistances that we calculated before (which is 2, plus 3.3, plus 2, all in K divided
by W, which equals 7.3 K divided by W) we can now find Ts3, which equals 800 K minus
42 watts times 7.3 K over W, which equals 493.4 K. And we can do this to find any one
of the temperatures in our circuit by subtracting that temperature which we want to find from
a known temperature, and dividing that by the resistances between them, and setting
it equal to k.