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0:14
>> Let's do a quick review of last time where we left off pretty much on the right side of the board here is what we covered last time.
0:27
Put this over here.
0:33
Number 1, heat conduction. We're talking about three modes of heat transfer. The first one was heat conduction is governed by Fourier's equation. The general one-dimensional form is qx double prime, double prime means per unit area, surface area minus K dT, dx, K is a property of the material called the thermoconductivity. If it's one-dimensional heat transfer, the equation becomes a bit simpler because you can replace the differential dT, dX by Delta T over Delta X, so the equation becomes
1:14
Q double prime equal K Delta T over L. L is the thickness of the material. Now, number 2, convection heat transfer called Newton's Law of Cooling. Q double prime again is equal to h T_S minus T_infinity. H is not a property, h depends on many things, it's called the convection heat transfer coefficient. You have to tell me first of all, what's the fluid? What it's temperature? What it's pressure for gas? You have to tell me, what's a geometry? Is it spherical, cylindrical,
1:56
a plain wall, a flat surface? You have to tell me the floor regime. Is it laminar? Is it turbulent? Is it max floor? What's going on with floor? What's the velocity and so on. So it's very complicated. You won't find h in the back of the book in a table, you've got to calculate h and that's what chapter 6, 7, 8, and 9 talk about is how to calculate h. Now, number 3, radiation heat transfer, thermal radiation. The basic law about the emission of energy from the surface
2:32
is the Stefan-Boltzmann Law. I think in class last time if you check your notes, I might have put a q here, that's not right, it's capital E, so if you can change your handwritten notes on your laptop, whatever, that should be capital E. Capital E is called the emissive power. I'll put that down here.
2:59
Let's put it right here.
3:09
It's in watts per square meter, but it's called the emissive power. For a black body, which is a perfect emitter for a thermal radiation, it's Sigma T_S_4. Sigma is a constant, the Stefan-Boltzmann constant, it's in your notes. If it's not a perfect emitter of radiation, then we have just E without the subscript B. The subscript B means black body. So E equals Sigma, now we add something called the emissivity Epsilon, T_S_4. The emissivity can vary from 0-1. If the emissivity is one,
3:51
the body behaves as if it's a black body.
3:56
Now, everywhere in radiation, you have to put the temperature in absolute, got to be, don't use degree c, use degree k. If it's conduction, you can use either one, just use all degree c, use all degree k, the answer is going to be the same. If it's convection, you can use either all degree c or all degrees k, the answer is their same. When you get to radiation, you have to use absolute degree k. Now, we said last time, if you have a small object in a large enclosure and you
4:35
want to know the exchange of thermal energy between those two objects, here's the equation, q in watts equal emissivity of what? The small object? A of what? The small object times Sigma, times T surface, of what? The small object minus T surroundings. What is that? The temperature of the surrounding walls in a room or if you put a dime in a basketball, that's a small object, a dime in a large enclosure, a basketball. Now, we didn't discuss this last time but I added one more thing.
5:12
One more, in chapter 1, we also talk about absorbed incident radiation. First of all, the symbol capital G, it's called the irradiation. It's how much energy comes into a surface from its surroundings. The thermal radiation coming into surface from it's surroundings. This is the amount absorbed, G_abs, absorb. This is a constant for the surface in our case, Alpha. Alpha is the absorptivity, it varies between zero and one. So now, we have two properties over here,
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emissivity Epsilon, absorptivity Alpha. We have one capital letter E, called the emissive power. We have one capital G called the irradiation. They're both in watts per square meter.
6:10
We also have a q, the q is in watts. We also have a q prime, watts per square meter. So we're going to use those three modes of heat transfer in examples today. But before we do that, just to show you how they're related to real life. Here's my coffee cup. I heat it before my eight o'clock class and by 8:50, it was getting cool, so I went down the breeze way and heat it with microwave, now it's backup again, the good temperature.
6:49
It's pretty good. Yeah. By 9:50, 50 minutes from now, these [inaudible] are going to be pretty cool again. I'll probably go down and heat it again in the microwave for the 10 o'clock class. Say, gosh isn't there, this is styrofoam, okay? It's supposed to insulate the hot coffee from cooler room air. It's doing an okay job if I drink fast, but the temperature of the coffee is going to go down with time. So I think, okay, let's see. I've had heat transfer, so why should people solve this problem.
7:30
I want to put a lid on it. Why it's hard to drink, I don't like to drink on it without lid. Well, heat is lost at the top two ways, if it's steaming coffee, the steam is taking heat out. If I have this coffee cup outside and there's air blowing over,
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I've got convection of the top surface. I've got conduction through the styrofoam walls of the coffee cup. Doing this prevents some convection. There's still convection on the outside but not off the top. I think yeah, I know. 9:50, I don't know, I don't trust it. I want to keep it really warm, well, it's kind of doomed. I'm going to put two of these guys together. Double insulated. Yeah. Okay, good. Now, I've got two times the thickness which means what? Q goes down. Yeah smart.
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Double insulated, and matter of fact by doing that, I trap air in there. When you trap air in something, wow, is that good insulation? That's about the best you can get except for a vacuum. Trap air so it doesn't move. This coffee will last longer, maybe until 10:50.
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So this funny, I don't like the way that looks. I know it's good, but I'm going to go out and spend some money and get something really nice. Now, that's the same thing as a styrofoam. There's two aluminum layers in there, but there's air between inside, outside. It's a gift of the army department of faculty. Thank you very much. I don't like to use it though. Nice lid, nice pops and they're really good and solid, oh yeah, it's great. Between these inner and outer layers air,
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air is the great insulator. You want to spend more money? Take the air out of there. Make a vacuum inside of there between those two layers, that's great. Now, you've eliminated your convection because there's no air in there, it's going to cost you more, depends how much you want to spend. But yeah, and also this isn't painted black on both sides, that be bad radiation, you want to be reflected and not to be absorbed. This is combination of radiation, convection, and conduction,
10:06
and something like this. So yeah, everything in your real-life, whether it's in your kitchen, or your car or whatever impacts. Your engine block is really hot, don't touch it after you've been driving for awhile. Freeway speeds it. Is it radiating heat? Anything, oh yeah, to the inside of your hood and inside of your engine compartment. So yeah, there's definitely radiation in there. There's convection in there for sure. There's conduction through that solid engine block, so you take the heat
10:44
out of the water by what? Tubes, by conduction. Yeah. You take the heat out by convection, yeah, the tubes in there and the fins. Oh yeah, complicated, but they all interact. Those three modes of heat transfer can be found all over our real life like that, so they're everywhere. Now, Just to remind you what Q double prime means and what Q means, Q double prime equal Q over A. This is called the heat flux, this is called the heat rate, sometimes called the heat transfer,
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heat flux, heat rate or heat transfer. So let's take conduction first, our problem is a cubicle furness compartment, two meters on aside, insulated, it's the wall thickness, so here's a k of, let's see, I think it's made out of the installation, yeah, styrofoam.
11:50
>> The value of K is given,
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0.030 watts per meter K. That came from the back at the book, but in Chapter 1, they give the value of K. In Chapter 2, they're going to ask you to go find K in the back of the book. We're also given that the heat load on this must be less than 500 watts.
12:34
We're also given that the temperature of inside the freezer, we're going to assume that the temperature inside their freezer is same as at wall temperature of the styrofoam inside the freezer. It's minus 10 degrees C, and the outside of the freezer compartment is at a temperature of 35 degrees C. Then we're asked to find the insulation thickness.
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So that's the given and what we're asked to find.
13:28
So we go to our library of possible solutions and we say, let's see, this is definitely a conduction problem. You've got that right. Is It 1d? Yeah, we're going to assume 1d. It's 1d. As I said before, the equation you use in Chapter 1 for conduction pretty much is the upper right hand equation. There it is, 1d.
13:59
Do I want q double prime or do I want to use use q? Let's go back to the units again. Q double prime, watts per square meter. Q watts, look at what you're given. I'm not given watts per square meter, I'm given watts. You can guess which one I'm going to use. I'm going to use q. So we take our equation, over there, just like here. Q equal q double prime times a. Q double prime times a. KA Delta T over L. L is the thickness of the insulation.
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So I want to solve for that, it's how the picture would be. So L equal KA Delta T over q. K was given.
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Area. I didn't mention that.
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Bottom is perfectly insulated.
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It's assumed to be on the ground, and for that, we're going to assume that there's no real heat gained from the ground for this problem. So where does the heat gain come from? Well, the top and the four sidewalls. Top and the four sidewalls. Each one is 2 by 2 meters. How many are those squares are there? 5, times 1 square 2 by 2. 5 meters,
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2 meters, 2 meters.
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Hot temperature, 35.
16:28
Minus cold temperature, minus 10, degree C, divided by q. Now, what's q? Q is called a heat load. What does a heat load mean?
16:44
Let's take this room and I block off the HVAC duct, so no air is coming in here. The air in here is stagnant. It's going to get warmer with time. The sun is hitting east facing-wall, the bricks on the outside. Heat's coming through that wall. The lights generate heat. Your bodies generate heat. By 2:00 o'clock today, it's going to be pretty warm in here. Well, how do we prevent that happening? We turn on the air-conditioning, for instance, maybe.
17:23
How much heat has to taken up by the AC unit? That's called the heat load. Heat load. So in order to keep this stuff inside at minus 10 degrees C, because right now, the outside's warm, the inside's cool, 500 watts are coming in, so you've got take that heat out somehow. That's what heat load means, 500 Watts. There. You see, that last thing drives some people crazy. They say, hold on there, you can't cancel those out. That's degree K, that's degree C. Illegal. I say, no, not illegal.
18:18
Sorry. I don't care. If don't like this right here, you take your finger, you erase the K, and you put degree C. That's the rules. Drives people crazy sometimes. No, you've got to add 273. No, you don't because here's the magic word, P-E-R, per. This is watts per degrees C. You say, well, I'm going to leave that K, but I'm going to change these guys to K. Okay, let's do that. 35 plus 273 minus, minus 10 plus 273. Degree K, degree K, equal, plus 273 minus 273, 35 minus minus 10,
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degree K. They look the same. Of course they do. Drive some people crazy. They say, wow, I don't get that. Let's take freezing water, 0 degrees C, 100 degrees C. Freezing 273, 373. The difference here, 100 degree C. The difference here, 100 degree K. No, it doesn't matter, as long as you put the minus sign there, the minus sign. I'm not saying that you can't just say 100 degrees C is 100 degrees K, illegal. 100 degrees C is 100 plus 273 degree K. But when I take the difference, the minus sign,
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or I say the word per, 0.030 watts per K, it can also be 0.030 watts per degrees C. Just so that won't drive you crazy.
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The answer is 0.054 meters, 5.4 centimeters.
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About two inches. So that freezer compartment, there's two inches of insulation around it. There, that much.
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That's a pretty simple and straightforward problem. Now, let's do another one. >> Professor, on problems that we're given, is it stated that the bottom is [inaudible]? >> Yeah, it did. >> Okay. >> It did in problem stated. Yeah. I just didn't throw in there first. Yeah. >> Professor, just to reiterate, the heat load is just how much heat is being taken out of the room? >> To keep the temperature at what it was previously. >> Okay. >> Yeah. For instance, you put this as a computer lab,
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a lot more heat generated. You need more AC to keep the room cool. >> Okay. >> Yeah. Yes, sir. >> So you put it in to per time as well if you make heat 500 through one hour or something? >> Yeah. Watt hour multiplied by time gives you what? What's a watt? Energy per time. Energy per time times a time gives you what? Energy. That's right. There's a difference in energy and power. Energy is joules, power is watts. One joule per second. Energy per time. One joule per second equal one watt.
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What's one watt? Power. Keep that in mind too. >> Does energy alone exist? >> Pardon me? >> Does energy alone exist? >> Yeah. Should be. Yeah. So we just worked one in conduction, let's work one in convection. Example. Convection. We have electric resistance heater
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embedded in a long cylinder
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of diameter 30 millimeters.
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Air blows over the cylinder. Air temperature is 23 degrees C.
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Surface temperature to cylinder is, I think it was 90 degrees C. Heater puts out
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400 watts per meter.
24:41
I think that's all we need. What is the value of the convection coefficient?
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So here's our picture.
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Cylinder heater
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air blowing over it.
25:21
Convection coefficient on the outside, T surface on the outside, air temperature, T infinity. So that's what's going on.
25:35
Here's what they do. You can buy these things. They're electric resistance heaters. They look like this. Maybe they're called cartridge heaters. They look like a cylinder like this. You might want to drill a hole in an aluminum cylinder. Drill a hole in the middle, aluminum cylinder, stick this guy in the hole in the cylinder. My hand's a cylinder, this is the heater. So this is the outside surface temperature of this. Put this guy across a power supply, it generates power.
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How much? Four hundred watts per meter. Where's that heat go? Out to the air through the outside surface of the cylinder. Chapter 1, we have no choice. Go to number 2, convection heat transfer. I wonder what equation I should use. I don't have any choice, there's number 2. Number 2. So over here,
26:36
we have q double prime. Now, let's look at this and get this right.
26:47
We have to answer the question, which q do we want? Do we want q, q prime, or q double prime? So we have q in watts, q prime in watts per meter, q double prime in watts per meter square. There's a big giveaway. He gave me the heater output, 400 watts per meter. I want that guy, q prime. But I'm going to start with q because none of my equations have q prime in them yet. So I'll start with q. Convection, q equal hA, T_s minus T_infinity. Solve for h. h equal q. No, we don't want to solve for it.
27:51
q prime. Yeah. Our q prime. Got it. So h, and we have our area, T_s minus T_infinity. It's q, not a prime.
28:09
So we've got that. Now, I'm going to do it this way. It might be easier to see. What is h? h is equal to A. What is the area of that cylinder? Pi DL. Pi DL T_s minus T_infinity. This makes better sense to work it that way. We'll come back to that. I don't want q because I'm given q prime. What's q prime? I'll put it here.
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q over L. So I want q prime. h Pi DL T_s minus T_infinity divided by L. Cancel, cancel. Now I'll solve for h this way. So h is equal to q prime over Pi D T_s minus T_infinity.
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So we have 400 watts per meter divided by Pi, diameter in meters. Temperature, 90 minus 25, 65.
30:00
>> Chapter 1, pretty simplistic problems of course, but that's all right. Yes ma'am? >> I don't agree with that >> Pardon me? [inaudible]. >> Yeah. I'm looking back, I think that problem was 35. This guy watt 23. Thank you. Yes. >> I believe the diameter should be 30. >> Did I get the diameter wrong too? Am I taking 100? I know. Yeah, 35 you said? Right 30. I'm doing the next problem, I'm ahead of myself. The next problem is a 100. Thank you, though. Yes, ma'am.
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>> Sir, if you use heat transfer to Q double prime that is heat flux, what do you call Q prime? >> Okay, good point. Yeah, okay. So you call this guy the heat rate or just the heat transfers, okay? This guy you call the heat flux, this guy you don't call him anything. He has no name. It's just watts per meter. He's not given a special name.
31:12
Okay. So what you do in a problem, you look for keys. One key is in this problem, the key is there, watts per meter. This problem, the key is there, watts. That gives you a hint of where you want to go. When you start with your equation where the equation is Q to over prime, I want Q prime, how do I get there? I convert Q double prime to Q and then Q to Q prime. Otherwise, you can get messed up with those three Q terms there. Okay so now that take care too we did one with conduction,
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one with convection, now we're going to do one with radiation. Okay. So let's look at the radiation one. This is an instrumentation package, spherical instrumentation package.
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Okay, got it. There's my 100 millimeters,
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with an emissivity 0.23.
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Got it. It's in a large space simulation chamber
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whose walls are at 77 K.
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The outside of the package is at 40 degrees C.
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Okay. I think we got what we need. Yeah. Okay. We're asked to find the power being dissipated by the package.
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So let's take a look at that. So this is some spherical instrumentation package generating heat, is placed in a large simulation chamber, simulating black space for instance. The wall to that chamber are very low, 77 degree K, simulating a black space environment. The packages itself has an outside surface temperature of 40 degrees C. Okay. How much power is being dissipated? Now he said power okay, big giveaway. Power is only one thing, watts. I want Q. He said the magic word.
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The magic word large. Thank you.
35:40
Because only if I see the word large, can I use this equation. Large enclosure. Okay. He's legal. There is my Q in watts. So Q is equal to Epsilon Sigma A T_s to the 4th minus T surroundings to the 4th.
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Epsilon is Epsilon of the small object, 0.23.
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Sigma is the constant 5.67 ten to the minus eight. Area of a sphere, surface area, pi D squared,
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multiplied by surface temperature. You better be careful, it's not degree K. I told you over there, everything in radiation has to be degrees K. 40 plus 273 to the 4th minus 77 to the 4th.
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Power dissipated, 4.3 watts.
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>> Officially you're not supposed to put the degree with K when you write a technical paper. You don't put degrees with K in SI.
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With C, you put degrees, you do with Fahrenheit, you do with Rankine, degree, degree, degree, but K, no, they don't do it. I'll probably do it in class because I don't want you to get that K confused with that K up there. So I'm going to put degree with it just so that you see it clearly.
38:16
So that takes care of the radiation problem. Now, the last thing in Chapter 1. There are two important energy balance equations. So we'll first do an energy balance on a control volume. So here's our control volume. The control surface surrounds the control volume, the control surface. Energy can enter the control volume. E stands for energy. It can cross the control surface going in, it can cross the control surface going out.
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There may be some storage of energy inside, there may be some generation of energy inside, E dot in. Let's put down the official words on these guys so I get them all right here. We'll call it the rate at which thermal energy enters.
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E dot going out, the rate at which thermal energy leaves the control volume. E dot storage equal the rate at which energy is stored.
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Finally, E dot g,
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rate at which energy is generated.
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If you have more energy coming in than going out, it is going to be stored. But what does generation mean? Well, the classic example is a wire carrying a current. As the wire is carrying the current, there's energy generated in that wire. It starts to glow dull-red maybe.
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Nuclear fuel at assembly is internally generated then for sure. That's the whole purpose of it. So that's the case of energy generation. So when we do this now, we write and we know what's going on.
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So energy balance is what comes in minus what goes out plus the amount of energy generated internally in it, must go to a change in the storage of energy. So this is the basic equation we use throughout the semester.
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There and again, what is E? E is energy. What is energy? Joules. What does the dot mean above it? Per time. So E dot is energy per time, joules per second, Watts. Those guys are Watts.
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Now, the second equation is we consider a surface of the solid.
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So here is our material. The dashed lines are our control surface.
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What's inside of there is the control volume and the dashed lines, control surface.
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Apply the energy balance equation for the control volume for our surface. E dot in minus E dot out plus E dot gen equal E dot storage.
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So a trick question if I can ask somebody. Ask him, "How thick is the surface of that wall? How thick is it?" The guy says, "How thick? I don't know. Thousands of an inch, a tenth of an inch. I don't know. That's a surface." The answer is now that the surface volume approaches zero, the thickness approaches zero. That's what a surface is. Surface means not inside. It's on the surface. How thick is a surface? Negligibly thick, approaches zero. For a surface, the volume approaches zero.
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What is mass? Density times volume. Mass is density times volume. If the volume goes to zero,
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the mass is going to approach zero. If there is no mass, how can it store energy? Don't forget what the storage in simple terms is, MC_p Delta T, MC_p Delta T storage. If mass is zero, guess what storage can be, zero.
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If there is no mass, how can it generate energy? You need mass to generate energy, true. If that g is zero,
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so we're left with what comes in must equal what goes out energy-wise.
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So that's the equation for the energy balance on a surface of an object. What comes in must go out because the volume and the mass approach zero.
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So we're going to work a problem involving these two equations, but we'll wait till next time to do that. So then we'll see you on Monday.
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