 Alright, what we've been doing thus far in the course, we've been looking at the different modes of heat transfer. We looked at conduction, convection, and radiation, and then we looked at an approach called the surface energy balance, which enables you to formulate an equation when you have these different modes of heat transfer in a particular system. So what we're now going to do, we're going to work an example problem, and this example problem is kind of going to unite all of the three different forms of heat transfer that we've looked at thus far. So this will be combined modes of heat transfer. Okay, so there's our problem kind of a long statement, but what we have is we have a furnace, and within the furnace are combustion gases, hot combustion gases, and there is a brick wall that forms the wall of the furnace, and what we are told is that the ambient air temperature is 25 degrees C. The brick wall is 0.15 meters thick. We're told the thermal conductivity of the brick, it's 1.2 watts per meter Kelvin, and the surface emissivity of the brick is 0.8. And we're told that under steady state conditions, the outer temperature of the brick wall is measured to be 100 degrees C, and we know that we have free convection, so that is convection without any kind of pumping source, so it's just due to the buoyancy of the air. On the outside of the brick wall has a convective heat transfer coefficient of 20 watts per meter square degree C, I should say. And we're then asked to determine what is the inner wall temperature of the furnace on the inside of the brick wall. So here we have a problem on just analyzing it quickly. Given that we have thermal conductivity, that means we have conduction going through the brick wall. We're given a convective heat transfer coefficient, so we have to have convective heat transfer, and they give us emissivity, and that means we have radiative heat transfer as well. So what we're going to do, we're going to use the surface energy balance, that's a concept that we looked at in the last segment, and we're going to use it to solve the problems. So just like all the other problems, we begin by writing out what we know, and then what we're trying to find, and then we'll write out a schematic. Okay, so that's what we know. Now let me check here. Yeah, 25 degrees C, so the ambient was at 25 degrees C, that's how we get the surrounding temperature. Now what are we after? We're trying to find the temperature on the inside of this brick wall, which we'll draw a schematic in a moment, and things will make a little bit more sense. Okay, so we're going to call that T1 the inner wall temperature. Now what I'm going to do, let me go to a new slide because the schematic is kind of large. So the schematic that we're looking at, and so here we have our brick wall, and this is very similar to the energy balance that we looked at in the previous segment, but over here we have combustion gases. Now in reality, what's going to happen is there'll be convective heat transfer on the inside as well here, and probably also radiation because you have a flame. We're neglecting those and we're just saying that we know this temperature, that's what we're trying to solve for, T1. So let me sketch out the rest of it and then we'll come back and work through the problem. Okay, let's see here. I'm going to put a surface temperature here, T surface. All right, so what we're going to do, we are after this here, we're looking for this temperature, but when approaching this type of problem, what we need to do, we need to come up with an equation where we have all of the different components involved, and in order to do that, we're going to do this surface energy balance. So let me draw that in here because that is where conduction is going to go into both convection and radiative heat transfer. So that will be our surface, the surface where we're going to apply the surface energy balance. Now a couple of assumptions that we have here, steady state, 1D conduction through the wall. So steady state, that means that you're not changing any of the conditions. Second one is 1D conduction through the brick wall, and a third thing is surroundings are large compared to the brick wall. Another one that I should add here is that we're dealing with a gray surface. So with that, that means that the emissivity and the absorptivity are equal to one another. All right, so what we have, we have all the different components here. We're now going to perform a surface energy balance on the outer wall, and that will enable us to come up with our equations. So just like before, what we were going to have, we're going to have what's coming in is going to be that, and then what is going out is going to be these two here. So let's put that together. Okay, so we have conduction, conduction, radiation, and let's plug in all of the different terms. So we have Fourier's law and Newton's law of cooling, and then our radiative heat transfer equation. Okay, now what we can do immediately, we can cancel out the area because that's common to all of them. So area disappears. I'm going to expand the temperature gradient term. So let's do that, and I'll put the other two terms onto the right hand side of the equation. Now note the way that I'm doing this. I'm putting Ts minus T1, and so Ts is the value of the temperature. This would be Ts is here, so I put Ts surface. Actually, let me correct that. I'm going to make that Ts because that is what we have, and then we have Ts surrounding. So that is the surface temperature Ts is for this surface right here. So when I do conduction, I'm taking Ts, and that is going to be at x equals L, and then we will have T1, and that is at x equals zero. And we then are able to evaluate the temperature gradient term as we have there. Now on the right hand side, what we'll do, we'll take conduction or convection and radiation and bring it over to the right hand side. So we have this equation here, and at this point it's good to reflect where we're going in terms of solving the problem. So let's come back and look at our schematic. First of all, what are we trying to find? We're trying to find the inner wall temperature T1. So let's go back to our schematic. This is the thing that we're after. Consequently, with our energy balance equation, what we really should be doing is we should be trying to isolate for T1. So let's try to rearrange this equation and isolate for T1. Okay, so we get this equation here. We can now go ahead and plug in the values because we know everything in this equation with the exception of T1. Okay, so we get that and be certain to keep the temperature in degree Kelvin when you're using the radiative heat transfer equation, which we've done. For the other ones, you can leave it in degree C because they're not being raised to the power 4. But when you have it in the radiative heat transfer equation, that has to be in degrees Kelvin. So once we plug those in, let's see what we get. Okay, so we get that for our final answer. Temperature is 352.5 degrees Celsius. And so that would be the inner wall temperature of our brick wall. So we have this brick wall. We have our furnace. We have natural convection taking place out here. And then we have some big surroundings where we have radiative heat transfer. But what we've been able to find is that this is 352.5 degrees C, this inner wall temperature. Now, this was fairly easy given that a lot of the information was given to us. Had I asked you to solve for the surface temperature, that would have been a little bit more challenging because you would have had a fourth order equation that you would have to solve. But this was a little bit more direct the way that we had it trying to solve for T1. And that is then an example of combined modes of heat transfer. So here we just solved a problem involving conduction, Q conduction. We had convection. And we had radiation. So all three of them put together enabling us to solve something in terms of an engineering application. Let me get rid of that because that's just a hypothetical. But that is an example of putting things together. And you can see using this control balance energy surface or control surface, I should say, enables you to do quite a bit. And we're going to use that over and over and over again throughout our calculations and heat transfer. Whenever you have conduction on the inside of then you're going to other modes on the outside. It's a very, very handy way of being able to handle things. So that concludes this segment looking at an example problem of combined modes of heat transfer.