00:02
All right, so for this question, we're going to consider a hollow cylinder, a hollow cylinder for heat transfer medium, with the inner and outer radine of r and r sub 0, respectively corresponding surface temperatures t and t sub 0.
00:27
So if the thermal conductivity variation can be described as a linear function of temperature, according to the temperature, to the equation, i'm going to pretty much show you this equation here, and we're just giving it a question.
00:43
The equation is like this, and we'll kind of show you the cellular two.
00:50
So the question here is that we should calculate the steady state of heat transfarrow ring in the radial direction using this equation here.
01:05
Using this relationship this model and we after we do that we have to compare the result with the huge transfer rate calculated using a k value calculated at the apripetic mean temperature all right that's a kind of amountful but in summary what we have to do here this question is to for this hollow we're a hollow we're so again, mean that we're not going to have the tops right here with a hollow cylinder.
01:47
We're to just calculate the steady state heat transfer rate in radial direction, so we're going from in here, going outward, you know.
02:00
And compare that with ewat determined using an average thermal conductivity.
02:06
That's really what the question is saying to do.
02:09
Okay, so let's get -go.
02:17
It's usually we hit transfer.
02:19
One of the places we can start is coming up with some, you know, one of the laws.
02:26
And a good goal to think about here would be failure as long.
02:31
And by the way, i've already worked out the problem here, i'm just going to explain how i get through the writing here also involved some latex, so you know, using latex i was able to represent this so i give them and be better than having to write it out but enjoy so again the if we start with fourier's law and so this is what fourier's law looks like 4reras law is well q equals to negative k a delta t so d t d t r where is what i explained if you're not familiar with this, although you should be familiar with free res.
03:26
Low by now, if you're doing a heat transfer class.
03:31
Q is the heat transfer rate, and k is a thermal conductivity.
03:39
The dtdr would be temperature gradient in the radio direction, basically.
03:45
The further out or in, basically your position along a radio line, a given radio line, is variant based on where you are, so the temperature at that point to be very based on where you are along that radio line.
04:03
And so this is dtdr, it's just the radiation is the rate of change of the radiation.
04:11
And a here will be the surface area that has to be normal to the direction of the heat flow.
04:17
So in the case of this cylinder, imagine you unwrapped the cylinder, you're going to get a rectangle.
04:24
Triangle essentially right or a square but square is a little tangle basically i'm not the other way around but again i want to talk to geometry here or just i'll just say you wrap this cylinder you should get some surface area where the heat is going to be transferred through and the area of this of this southern surface area this is an i will be 2 irm.
05:02
So what we can do since we've laid out a couple of things here is i can start thinking about making some substitutions here.
05:11
So for example, remember that we have k here represented by that model.
05:16
We talked about the linear model.
05:19
And then we also have this heat transfer, sorry about the heat transfer here, surface area, like this.
05:26
So we plop all of those two things.
05:30
Into this from your low equation, we should get something that looks like this.
05:39
And again, the k here, the k here is just this.
05:44
This is all just a k here.
05:47
And 2xrl is just from a.
05:50
Dt, dr, we don't touch.
05:55
Now, if you, this is a differential equation.
06:02
So one of the techniques of differential equations would be to do the one i can look that i use here is separation of variables to do that you multiply both sides by dr and then you have dr the dt at this side and that means you have to take the integral of both sides with respect to the error variables we would not only multiply both sides by dr will also divide both sides by 2xr r l or divide both sides by r rather than divide both sides by r rather than divide both sides of the r, you would get 1 over r times the r times q and then get this other stuff here that you get d t is here.
06:51
So take the integral of both sides, the r you're integrated with respect to r which you're different integrate with respect to t here and you're going to get this.
07:02
Okay.
07:03
Now you know what i'm actually just show you the whole thing.
07:07
Okay before we get to this part about factoring.
07:11
Actually, i'll just show you this first.
07:15
So essentially you are going to now, when you're applying the integrating, you have to consider limits of integration...