 Hi, I'm Zor. Welcome to Unizor education. I would like to talk about heat transfer. In particular, one specific kind of transfer called conduction or conductivity. Now this lecture is part of the course called Physics 14 presented on Unizor.com. I do suggest you to listen to this lecture, which and listen to this lecture from the website Unizor.com. Because if you found it somewhere else, let's say on YouTube, for instance, where it's actually physically stored, you will have only this particular lecture. The website contains basically the course, which means many different lectures logically connected to each other with the textual explanation of each lecture, which basically like a textbook with problems solving and exercises and even exams for those who would like to challenge themselves. And the site is completely free, there are no ads, so it's better to go to the website. Now, speaking about heat transfer, well, first of all, let's recall that heat is basically a molecular movement, actually the intensity of the molecular movement. So what does it mean that the heat is transferred from one object to another or from one part of the object to another? Well, basically it means that intensity of the movement of the molecules is changing. Now why? Well, it's actually very easy to imagine. For example, if you have a certain object and this thing is hot, let's say we are putting some kind of a source of heat to this particular site. Now, here we will measure the temperature. Now, initially, for instance, the temperature is different from this one. Let's say it's colder. Now, if we will measure the temperature of this site, eventually it will heat up basically to the same temperature as this one. Now, why is it happening? Well, it's a simple explanation because if these molecules are moving very fast, they are heating the neighbors, which basically forces those neighbors to move a little bit faster. Those are heating their neighbors, etc. That's how gradually the intensity of this movement causes more intense movement of this. If this is maintained at constant temperature, let's say we are putting some kind of a heat source, like a flame, for instance, through this end. So this end will always have the same intensity. Eventually, these movements will force the movements of all molecules of the entire object to move as intensely as this one, which means it will reach exactly the same temperature because the intensity of the movement is the kinetic energy of the molecules. The temperature is average kinetic energy, basically, measurement of average. So that's how the heat is transferred in this particular case. Now, in some cases, that's the only way how it's transferred. Now, if it's a solid, for instance, where the molecules take relatively stable position, they're just oscillating around this particular position. This oscillation is actually transferred from this place to this, and that is what conductivity is about. Now, there are other ways to transfer the energy. For instance, if you are boiling a pot of water, the hot molecules of the pot itself, the metal, for instance, they are forcing the neighboring molecules of water to move faster, and these faster moving molecules are actually rising up, and it's not just the movements is transferred, the whole molecules which are moving fast are transferred to the upper layers. So this is a different, this is a conduction, which we will address in a later lecture. So today we'll talk about conduction only. And this is related to, again, let's assume that this is oscillation of the molecules around there, some kind of a midpoint, for each molecule, there is such a midpoint, and the distance between molecules, again, it's changing, it's oscillating, but within certain limits. So this molecule is moving around its midpoint, and it forces this molecule also to move around this midpoint, and if this one is hotter, moves much more intensely, then this one will also be moving a little bit more intensely. And then the neighbors, and then the neighbors, and that's how the heat is transferred. So basically, this is the, I would say, qualitative picture of how the transfer of the energy is occurring. So how can we measure the conductivity? Well, basically the same way as I was saying, I was presenting before in this picture, we can hit one end of the object and measure the temperature of another at certain time intervals. Now that basically kind of gives you a certain information about conductivity because if my other end of the object is heated faster, that means the conductivity is better in this particular material. Now obviously everybody knows that if you put a, let's say, silver spoon into a hot tea, it will heat up much faster than, let's say, steel spoon. Well, if you don't know, try it. All you need is just a silver spoon and a steel spoon. And the steel spoon, in turn, will much faster heat up than, let's say, plastic or wooden spoon. You might even not feel the heat if it's a wooden spoon for a very, very long time. So that means that the conductivity of silver is greater than the heat conductivity of steel, which in turn is significantly greater than the heat conductivity of the wood. That's why the handle for the kettle, let's say, is made of wood or plastic and not the metal. I mean, if it's metal, it means it will really heat up very, very fast and you won't be able to hold it without some kind of glove or whatever. So that's basically how we can measure the heat conductivity. By the way, the highest heat conductivity is that of diamonds. Diamonds are extremely fast transferring the heat from one place to another. Okay, now what's next? Next, we will talk about different measures of conductivity and we would like to introduce some, not only qualitative, but quantitative measure. How can we actually measure it? Okay, let's assume that we have a situation where both ends of the object are maintained at certain temperatures. One is warm, another is cold. An example is if you have, let's say, the building wall. So this is the wall. This is outside and there is a temperature of air, which is basically maintained on this surface. And this is inside. This is the room. And it has a temperature of room, which is warmer. So if this is the x direction, I can say that the temperature of, well, this is zero. And this is, let's say, this length is L. So the temperature of zero is equal to T air, some kind of a constant. And temperature at L is equal to T room. And temperature in between, obviously, is changing from a lower to the higher. Now, what happens with the heat? Well, obviously, heat comes from the warmer to the colder side of the wall. And it comes with some kind of a constant flow of heat. That's why we have to really heat the room in winter, when it's cold, at least in the northern hemisphere. So we have to constantly heat the room because the heat is always going away through the walls and windows, etc. We're talking about the walls right now. And for simplicity, let's consider that the wall is very much very much evenly made for the same material. So we don't really kind of have any complications of having one kind of conductivity in one place of the wall and another in another. So the heat is flowing constantly from this side to this side. Now, let's consider the rate of heat flowing through this wall. Well, first of all, we obviously have to, whenever we are measuring something, we have to establish some kind of units, criteria, etc. So our criteria is how the heat is going through one particular area of unit of area, let's say one square meter or one square whatever. And we have to measure amount of heat which goes through the sink during certain unit of time. So if we will be able to find out how much heat goes through the unit of area, this is our heat goes through the unit of area during the unit of time, that would be a good measurement of how conductive this particular material is for the heat, right? So let's just think about, I would say, some reasonable assumptions. Now, they are not only kind of obvious from the logical standpoint that they are also experimentally confirmed. Now, what if the difference between these two temperatures is greater? Do you think that the flow will be more intense? Well, obviously, because during the same distance we have to lose more temperature, which means more heat should actually go. So there is a reasonable assumption that the flow of the heat will depend on the difference between the temperatures. So in this particular case, if I will subtract getting the difference between the temperature from the T room to T air, probably the greater this difference is, the greater the flow of heat will be. That's number one. Now, number two, we assume and again it's experimentally confirmed that this flow of heat is proportional to this difference, which means if this difference doubles, then the amount of heat which goes through this unit of air will also double. It's a reasonable assumption, experimentally confirmed. Now, on the other side, the thicker the wall, the longer it will take for the heat to go from the warm to the cold side, which means that the flow will be slower, which means it's inversely proportional to the thickness of the wall. So this is again a reasonable assumption that our heat flow will be proportional to the difference between the temperatures and inversely proportional to the thickness of the wall. Now, obviously, we need some kind of a coefficient of proportionality. Now, we also have to put the minus sign because the heat goes opposite to the x and that might actually be something which we can call the rate of the heat flow or there is a good word called flux, heat flux. This is the flow of the heat which goes through the unit of air during the unit of time and it's reasonable to assume this particular dependency on the difference between the temperature and the lengths and the widths of the wall. Now, this actually gives you the whole wall, basically, conductivity. Now, if the wall is made of the same material, then obviously it should be the same on each slice of the wall because the heat is going through slices of the wall from this place to this place. So first it goes to a thin layer which is close to this one, then the next thin layer, then the next thin layer, etc, etc. So if we will break this wall into slices like this, for each slice, let's say from x to x plus delta x, I will probably have to have x plus delta x minus x and the widths, the thickness of the slice is delta x and that would be my, not my average, but exact flux, heat flux at slice at the distance x from the left side. Why is it necessary? Well, because the wall might not be of the same material. So in some places it will be a little bit more conductive, let's say, cement is more conductive than brick or something like this. I don't know. But anyway, this gives you exact heat flux at any position within the wall. So if you have some kind of a material which separates one temperature from another temperature, then based on your position, you can find what's the heat flux at this point, at this point, at this point, as long as you have temperature at every point. So as the temperature goes, you can calculate, so all you need is basically the function T of x, temperature at the slice on a distance x from the left side. Now, to make it more palatable, let me go to an example, completely unrelated to this, but related logically. If you will consider a river, which goes along a straight line, along a a homogenous slope down. Well, obviously, if you have the level at one point, this is the level of the river at one point, and this is the level over the sea level, let's say, of another point. So there is a difference here, right? This difference in the position, geographical position of the river relative to the sea level is basically the reason why the river flows down the slope, right? So the difference in the levels above the sea is the reason why the river flows. Now, that's exactly similar to the difference between the temperatures. So I would like to say that the height of this point above the sea level minus height of this is actually similar to the difference between the temperature, which we have. Now, if the same difference is on a shorter distance, let's say, now, which river goes faster, this one or this one? Well, obviously, this one. Now, if this one is very far, then the river will be very, very quietly going. But if this is the close, then the difference between the heights above the sea level will be on a shorter distance and the water will almost fall down, right? So that actually is why we have this in the denominator. So this distance is definitely affects the flow of the river, the greater the distance, the slower the river. So the situation with temperature and the distance here is very much similar to this situation with river flows. Now, what if it's not really a constant slope with the same angle? What if the river goes like this and then like this? Now, this will be faster and this will be slower, right? So it all depends on the function, which is if the height is a function of this x, this is x and this is height. Then obviously, it's the derivative of the function h, which is tangential line, right? Which basically determines the speed of the river, the flow, the intensity of the flow. So it's very much like this case with the temperature. So the different levels of the water in the river is analogous to the different temperatures and the distance on which this difference is stretched is actually analogous to the widths of the wall. So everything is the same. Now, if the material of the wall is different, our temperature will also be not, okay, let me just go one more step here. This is D, one more thing. This is equals to minus K, D, okay. Why did they do this? Well, obviously, I have to, if I would like to measure exactly at certain position, I have to make my slice thinner and thinner. So that's why we have to go through the limit. Well, I shouldn't really put equal sign. It's actually an arrow here. As delta x goes to zero, obviously. So it goes to a derivative at point x and the point x is where we are measuring our flux, heat flux. Same thing here. It all depends on the derivative of h of x by x, where h is the height of the river or river bottom, whatever, relative to the sea level. Okay, so what have we actually determined? We have determined the dependency of the heat flux from the function which is a dependency of the temperature on the distance from the beginning from the source of energy or the recipient of energy or whatever. And this K what is this K? Well, it's a characteristic of the material. The wall obviously has one conductivity and if it's some other, if it's a metal, for instance, if obviously would be faster, why, for instance, we don't really do the wall made of metal? Why do we have to insulate it? Well, so because the metal will just transfer the energy from room to outside cold air much faster and we don't really need this, right? If we will insulate it, then it will be much slower. So what if we will insulate it and we will put something like this, this and something in between. So this will be higher conductivity, but this will be very lower conductivity. How the temperature would change? Well, from a high temperature, it will relatively fast go through this high conductivity part of the wall, then it will go much slower because this coefficient will be significantly lower and then again faster. So it's again like the river going fast, then maybe slow and then fast again. Exactly the same thing. So you can really view this river model as a very good model actually for the flow of the heat from the warm to the cold place. And finally what I can say is that this is called a Fourier formula for thermal conductivity and this is the definition of the conductivity coefficient of different materials. So we can, if we measure the conductivity of each material, then knowing how the temperature is going within this material, then you will get your heat flux. Now obviously if T of x is gradually changing from air to room, so if it's like a linear function of x, then the derivative will be constant, right? If this is a linear function of x, the derivative will be constant, which means that the flux will be constant. But if you have a situation like this with insulation in between, then obviously it will be constant during this, then it will be constant but different during the next layers and then another one at the end. So it all depends on how this wall is constructed. Now the coefficients of conductivity are obviously known, they're experimentally obtained and it's all available in the books. Now what if I would like to know how much heat actually I lose in the room? Well I have to really have the area of this wall obviously and multiply q of x times a and that would be my amount of heat energy which is going through the wall, through the entire wall, a is the area of the wall, right? During the unit of time and again if you need it during the certain time interval you have to multiply it by time. But this is how you can calculate how much energy for instance you need to heat up the room because you know the temperature of the room, you know the temperature of outside air, you know the area of the wall which goes from the room to the outside air and you know the material, you know the widths of the wall. I mean it all can be calculated using the only thing which you really need is material and its conductivity from which the wall is made. All right now this is the lecture which I have also complemented with rather long textual explanation. Almost everything but whatever I'm just saying is exactly reflected in the text for this lecture. So I do recommend you to read it as a text. It's probably smoother than when I'm talking about this but in any case it's very important to understand the concept of heat flux or thermal flux or flow rate of the heat. And again this analogy with the water flowing from the higher level to the lower level is very useful in this case for understanding of how the whole thing actually is made. And again this is the four years law of thermal conductivity is very important for calculations of different heat flows whenever we need it for construction or whatever. Okay that's it for today. Thank you very much and good luck.