Heat Transfer - Online Tutor, Practice Problems & Exam Prep

Heat transfer occurs through conduction, convection, and radiation. $j^4=\epsilon \sigma T^4$ describes thermal radiation from black bodies, which emit electromagnetic waves. Conduction involves direct contact, while convection transfers heat through fluids. The Stefan-Boltzmann law quantifies radiance, and Wien's displacement law relates temperature to emitted color. Understanding these concepts is crucial for grasping thermodynamics and energy transfer in various systems.

Overview of Heat Transfer

Hey folks. We talked a lot about heat up until now. We found that it can change an object's temperature, its phase, or even its physical size. Now, what I want to do in this video is focus a bit more on the nitty-gritty of how exactly heat gets transferred. Right? Remember that heat is just a transfer of thermal energy from one substance to another. What I want to show you is just a very brief overview of the 3 ways or 3 methods that heat gets transferred, and then later on, we'll go into a bit more detail. Alright? Let's check this out.

We're gonna start things off by talking about conduction, which is probably the way that you're most familiar with in terms of heat. Conduction is just a direct transfer of heat from one substance to another at different temperatures, and it's basically via direct contact. So, in other words, when 2 things are touching, right? So when you have two things that are touching at different temperatures, they're gonna exchange thermal energy. The classic example is when you're touching a boiling pot of water. Basically, what happens here is if the temperature of one substance is much greater than another, remember that the temperature is a measure of the average kinetic energy of particles. So, in this one in substance A, you're gonna have these particles that are bouncing around all over the place, and they're very energetic. Now, if temperature B is going to be lower, then it means that its particles are a bit less sort of energetic. They don't move as fast around like this. Now, when two things are in direct contact with each other when these things are touching, then basically what happens is at the barrier where these two things are kind of touching each other, you're gonna have one particle that sort of ricochets off another and it's gonna carry off some kinetic energy. And basically, that's all that's happening here. Right? You have all these energetic particles, they're meeting a bunch of somewhat slower particles and they overall just give off some energy this way. Alright? They're never really gonna give energy that way because these are less energetic and so that transfer of thermal energy is called heat. Alright. This also is how heat is transferred in your calorimetry problems. When we had two liquids that were mixing together in a cup or something like that, this is basically what was happening. You had all this transfer of thermal energy. Alright. So it's just direct transfer of heat by actual contact. That's conduction. Let's move on because that was pretty straightforward.

Convection is going to be a little bit different whereas conduction was a direct transfer of heat, convection is an indirect transfer of heat. So it's an indirect transfer of heat and the way that it does this is by heating a fluid. Remember a fluid can either be a liquid or air or something like that that surrounds the hot substance and a classic example of this is when you hold your hand above a flame. You're not in direct contact with the flame, so it's not conduction, but, basically, what happens is that the hot flame actually heats up the air that's around the flame. Remember, air is a fluid. So what happens here is it basically just excites all the air molecules around and that heated air ends up rising up, and so that heated fluid rises due to increased buoyancy and it carries all of that thermal energy up with it. So basically, what happens is if this is air like this, a flame will make that air hotter, and then that hotter air rises towards your hand and that's what you feel. So overall, it is an indirect transfer of thermal energy, but it's not through direct contact. Alright? Now, the convection is governed by a lot of complicated equations. A lot of this stuff is still actively researched in physics. You're not going to need to know this. This is mostly just conceptual. Let's move on to the last one here which is called radiation. Alright. So conduction is direct, convection was indirect, radiation is also going to be indirect. Alright. So this is also an indirect transfer of heat, and it does this by emitting electromagnetic waves. So you're emitting electromagnetic waves, which, by the way, is also known as just light. So objects will emit light. The classic example of this is either the sun or the glowing filament in a light bulb. Right? So this glowing filament is usually made out of a material like tungsten. And when you heat it up really, really, hot, it glows like this yellow or white-hot sort of color. What happens is if you hold your hand near the light bulb, you're not touching it. It's not conduction, and it's not convection. Basically, what's going on here is that certain materials can actually emit radiation, that's infrared radiation. That stuff touches your skin and that radiation warms up your skin, and that's what you perceive as heat. Right? So these things basically are also a transfer of heat, but it's not through direct contact. It's basically just because it's emitting infrared light. Now, these waves carry energy equal to the heat that it's lost by whatever substance it is. Right? So again, an example is like the glowing filament of a light or this is also how the sun's light works. Right? The sun's light comes from like 150 million kilometers away. It hits your skin, and it's like infrared radiation, and basically, your skin absorbs it, and it warms up. Right? And that's what you perceive as heat. Alright, folks. So that's sort of like a general overview of these different types of processes. Let's move on.

Conduction

Hey guys. In this video, we're going to talk about conduction in more detail. While we've talked about conduction in the qualitative sense, the conceptual sense, we haven't used any equations to describe conduction. Specifically, how quickly can heat be conducted from one object to another. Alright, that's what we're going to focus on in this video. Let's get to it. Remember that conduction is the transfer of heat through direct contact. Okay? Conduction is the most common type of heat transfer you're going to encounter in your studies, in your introductory physics courses. That's why conduction was basically the only type of heat transfer that we've seen up to this point. Okay? When studying calorimetry, all heat transfers were via conduction. Okay, and that was another point that I made. When you put two objects in thermal isolation together in contact the heat transfer is always going to be conduction. Okay? What we're interested in is how rapidly heat can be conducted from a hot substance to a cold substance, right? It always goes from hot to cold, and we're going to get to that later on when we cover the second law of thermodynamics. But we want to know how quickly this happens, how long it takes to happen. Okay? Materials have a natural allowance for heat flow known as the thermal conductivity, given by k. Okay, it's how easily they allow heat to be transferred quickly through them. Okay, the larger the thermal conductivity, the faster heat is conducted. Okay, so materials with a high thermal conductivity are called thermal conductors, and materials with low thermal conductivity are called thermal insulators. Alright.

Now when dealing with heat, we talk often about a heat current. Okay? The current for the heat is just how rapidly the heat is moving per second. Okay? So it's just q ∕ Δ t. We've seen problems before that says heat was entering at 95 joules per second. That was the heat current. Okay? How much energy per second. Okay? So the conduction current is the heat current for conduction. Alright, let me minimize myself. We have two substances here. We have one at a hot temperature, one at a high temperature which we just call hot, and one at a low temperature which we just call cold, and described by three things. It's got a cross-sectional area, it's got a length, and not written here, it has a conductivity. Those are the three aspects that describe the conductor. Besides that, you also have the temperature of the hot substance and the temperature of the cold substance which have nothing to do with the conductor. Those are the conductor. Given by k ⋅ a ⋅ (hot temperature − cold temperature) ÷ l. Okay, and this is a very important equation, and the units are going to be joules per second because it's just the amount of heat transferred per second. Alright? There are a few important consequences of this equation. First, the conduction current, like I said, is the rate at which heat is conducted through the surface. There was a substance. Okay. I explained that. Let's move past that. The heat conducted would then just be given by h ⋅ Δ t. Okay, as long as h is a constant. If h is not a constant, then you couldn't just multiply it by the amount of time because h might change as that time goes on. If you knew the average conduction current, you could multiply it by the amount of time and find the total heat transfer, but this equation right here typically only works if h is a constant. Okay? Now notice h should not be a constant. Okay. The conduction current should absolutely change as the hot substance became colder because it's releasing heat, and the cold substance becomes hotter. So naturally, from the hot substance to the cold substance. So h should not be a constant. The conduction current will be constant if the hot and cold substances are what we call reservoirs. Like a reservoir of water. A reservoir of water is a giant source of water. Okay? What a reservoir is for anything, and we use it a lot in thermodynamics, is a reservoir is an infinite source or sink of heat. That means it can absorb and release an infinite amount of heat without changing its temperature one big. Okay? That's what it means to be a reservoir. So if we look at our conduction current equation, imagine now that the hot objects and the cold objects were reservoirs, and the conductor and the conductor were connected between the two reservoirs, then no matter how much heat went through the conductor the temperature of the reservoirs would never change. That's the point of being a reservoir. It's an infinite source so it can produce as much heat as it wants. It's an infinite sink so it can absorb as much heat as it wants, all without leading to any change in temperature. So if this substance and this substance here were reservoirs, then the conduction current through the conductor would, in fact, be a constant. Okay? And that's an important point to make because you'll probably see reservoirs quite a bit in thermodynamics. Alright, let's do an example. A hot reservoir at 100 degrees Celsius is connected to a cold reservoir at 0 degrees Celsius by a 15 centimeter piece of iron in 5 seconds? And then it gives us the thermal conductivity of iron. Okay, so we're talking about how much heat in some amount of time. So we know that we need to use q = h ⋅ Δ t. Okay? And we know that h, the conduction current, is k ⋅ a ⋅ 100 − 0 ÷ l. We're told that the hot source and the cold source are actually reservoirs in this problem. A hot reservoir and a cold reservoir. So the conduction current is going to be constant. Let's calculate that. The thermal conductivity of iron is 79.5, and the units of watts per meter-kelvin are SI units. The cross-sectional area is 0.05. The hot reservoir is 100 degrees Celsius minus 0 degrees Celsius. Okay? Now, because this is a change in temperature, this is a difference in temperature, right? You have a hot minus a cold. Even though there's no delta there, because there's a change in temperature, we can simply leave this in degrees Celsius because that change in Celsius is equivalent to a change in Kelvin. And we do need Kelvin because if you notice the SI unit right here is Kelvin, okay, divided by the length, and we're told that it's a 15-centimeter long piece of iron. So this is 0.15 meters. Okay? Plugging all of that in, the heat current is 2650 watts. Okay, joules per second is what I gave it as the units for conduction current, but a joule per second is just a watt, so most of the time, conduction current is given in watts. Okay, now we can find the total heat transferred, and we're perfectly allowed to use this equation because since the hot source and the cold source are reservoirs, their temperatures don't change and therefore for h doesn't change. So this is 2650 times, we were asked for it in 5 seconds, And so this is 13,250 joules or 13.3 kilojoules. Like I said, typically like to give these units in kilojoules because most of these heats are large enough to be represented as kilojoules and too large to be represented as joules. Alright, that wraps up our discussion on the conduction current and conduction in specific. Alright? Thanks for watching, guys.

A cubic Styrofoam cooler containing ice on a hot day is shown in the following figure. The thickness of each wall of the cooler is 15 mm, with a side length of 1 m. If it is 40°C outside, how long will 2 kg of ice last in the cooler? Assume that during the melting process, the temperature inside the cooler remains at 0°C and that no heat enters from the bottom of the cooler. Note that the latent heat of fusion for water is 334 kJ/kg and the thermal conductivity of Styrofoam is 0.033 W/mK.

• A. 8.4 hrs
• B. 4.2 hours
• C. 2.1 hours
• D. 0.42 hours

Radiation

Hey guys. In this video, we want to talk more in detail about radiation as a method of heat transfer. Alright? Let's get to it. Remember that certain hot objects can expend heat. They can emit heat in the form of electromagnetic radiation, which is another word for electromagnetic waves. Okay? These substances that can do it are known as black bodies or blackbody-like. Okay? A blackbody is an object that can emit the maximum amount of thermal radiation at a given temperature. A blackbody-like object will always emit less energy in the form of thermal radiation than a true blackbody. Okay?

As with all waves, a particular wave or a particular electromagnetic wave in this case is defined by its frequency. Okay? It can also be defined by its wavelength but the frequency remains constant no matter what medium the wave is in whereas the wavelength changes between media so it's better to define it by frequency. Electromagnetic waves, as I said, are just a fancy way of saying light. So for light a particular frequency will be referred to as its color. Visible light, which is what we can see, only occupies a very very small amount of the electromagnetic waves. There are other types, radio waves, x-rays, gamma rays, microwaves, etc. Okay. But for visible light, the color is absolutely dependent on the frequency. And so we just extend that convention to everything. Even talking about x-rays we'll talk about the color of an x-ray as the frequency of the light. Okay? Black bodies do not emit light at a single color. Okay. This remember this verbiage, this wordage that I'm using, it doesn't have to be visible light for me to say it's a single color. It could be entirely x-rays and all that means is it just doesn't emit x-rays at a single frequency. Okay. Black bodies don't emit light at a single color. They emit light across a spectrum of colors. A spectrum is just a whole bunch of different colors each coming at a different probability. So let me minimize myself. This picture is a spectrum, a black body spectrum, of light. We have in the vertical axis the brightness of the light, and in the horizontal axis the frequency, so the color of the light. And as you can see at low temperature, the most probable light, sorry, the brightest light is at a lower frequency than at a higher temperature. Okay? In the visible light spectrum and light that we can see, low frequency light is red, moderate frequency light is yellow, and high frequency light is blue. That's why I showed the hotter black body as having a blue curve because the color of light is going to be closer to blue and the cold black body emitting light as a red curve because its brightest color is gonna be closer to red. Okay? The particular shape of the spectrum, what the brightest color is, how wide it is, etcetera, is going to be determined by the temperature, okay, of the black body. The color of the light that is seen, what you will actually see is going to be the brightest color. That's going to be the one that survives and that's going to be the one that you can see. Okay? As temperature increases, the light shifts from red to blue. Okay? So maybe you've heard that blue flames are hotter than red flames. Okay? That's typically true because for black bodies when you're emitting blue light it's because they're at a higher temperature than a black body that emits red light okay. But there could also be a chemical process going on where the chemical that you're heating up specifically emits blue light or red light and that has nothing to do with black body radiation. Now at very, very high temperatures this spectrum shifts away from the visible light. Now it's so high in frequency it's no longer visible. What ends up happening is the colors that you see are only the tail end of this right here. This tail end that happens to be in the visible range and the combination of the colors you see is white light. So at very very high temperatures when this spectrum shifts out of the visible range this light shifts from blue, which was hot black bodies, to white, which are black bodies that are so hot that they're emitting like ultraviolet light or x-rays even, low energy x-rays, so that all that you can see, because all we can see is visible light, is the tail end of this spectrum right here. And all of those lights are emitted at very similar brightnesses, brightnesses. There's no clear peak brightness. And a combination of colors produces white light. Okay so that's why really really hot metals glow white. Okay? Like we saw in the blacksmith picture when I introduced heat transfer. Alright?

Now, the radiance, which is something I'll talk about in a second, of thermal radiation emitted by a blackbody-like object is given by the Stefan-Boltzmann law. And the Stefan-Boltzmann law, the radiance is given by j. The Stefan Boltzmann law says it's equal to εσT4. Okay? ε is known as the emissivity. It's how closely to a true blackbody a blackbody-like object is. A true blackbody has an emissivity of 1. All blackbody-like objects have to have an emissivity less than 1 because they emit less light, less thermal radiation, than true black bodies for a given temperature. Okay? σ is known as the Stefan-Boltzmann constant and it has some value object emitting the power per unit surface area of the object emitting the thermal radiation. Okay? Radiance is very very similar in its definition and has identical units to intensity but it's different than intensity. Okay. They both have the same units, watts per meter squared. And the best way to explain the difference is like this. Imagine the sun. Okay? The sun is emitting light. Okay? We're over here on earth and some of that light travels all the way to earth to reach us. Okay? What can we measure? Okay. We always measure intensities of light. The watts per meter squared. What is the sun actually emitting that's inherent to the sun? It's emitting power which is in watts. Okay radiance is not power. Intensity is power per unit area. This light is being emitted what's called isotropically, the same in all directions. So the light creates a sphere of some radius r where r is the distance between the sun and the earth. That's how big the sphere is where all the light passes through. So the intensity that we measure is the power of emitted light over the surface area of that sphere which is 4pi r squared, right, where r is the distance between the earth and the sun. Now what's the radiance? The radiance is the power emitted by the sun which is remember a unique quality of the sun. The sun emits power. That's determined by internal things about the sun whereas the intensity is determined by how far away from the sun you're measuring. Okay? What the radiance is is it's the intensity at the surface of the sun. Okay? It is the power per unit surface area of the object emitting the light. So it's however much power that object is emitting divided by the surface area of that object. So it's that same power but this time it's divided by the surface area of the sun. And the radiance doesn't change with distance because the radiance is only measured at one point. It's only measured at the surface of the object emitting the light. Intensity could be measured anywhere, but radiance is always measured at the surface. Okay?

Now, the brightest color in the emission spectrum of black body radiation or thermal radiation is given by Wien's displacement law, and it's just b/T. Where b is Wien's displacement constant and it's some value. Okay? That'll be the color that you see if it's in the visible light region. If it's past the visible light region you're gonna see white light instead. Okay? Let's do a problem. A spherical object of 0.01 meter radius with an emissivity of 0.8 is heated to a temperature of 1000 kelvin. How much heat is radiated by this object in 5 milliseconds? What is the brightest color of this emission? So I'm gonna call this a and I'm gonna call this b. Okay? So part a. We're talking about thermal emission so we're going to have to use the Stefan-Boltzmann law first. First. So the Stefan-Boltzmann law is the radiance equals the emissivity times the Stefan-Boltzmann constant times the temperature to the 4th power. Okay the emissivity is 0.8, the temperature is 1000 kelvin and the Stefan-Boltzmann constant is just a constant. So it's 0.8. The Stefan-Boltzmann constant is 5.67 times 10 to the negative 8 and the temperature is 1000 kelvin. If it was given in Celsius you'd have to convert it to kelvin. This is an absolute temperature, this is not a difference in temperatures. So degrees Celsius and kelvin not the same unit. And this is to the 4th power. Okay? So the radiance is going to be 45,360 watts per square meter. Okay? Now we want to know how much heat is radiated by this object in 5 meters per second. Well what does the radiance tell us? The radiance tells us the power emitted by the object at the surface of the object. Okay? So the power is gonna be the radiance sorry. It's gonna be the radiance times the surface area of the object. Okay? Remember the power is the radiance at the surface of the object. So we already have the radiance 45,360 in our SI units. This is a spherical object so the surface area of the sphere is 4pi r squared. The radius is 0.01 meters squared, so the power is 57 watts. Now what we want to know is how much heat is radiated in 5 milliseconds. Well now that we know the power, which is the amount of heat per second, we can simply say that the heat is the power times the amount of time, which is 57 watts times 0.005, that's 5 milliseconds, and that is going to be 0.285 joules. Okay that wraps up our discussion on thermal emission and radiation as a form of heat transfer. Thanks for watching guys.

If the intensity of sunlight measured at the Earth's surface is 1400 W/m² , what is the surface temperature of the Sun? Treat the Sun like a true blackbody. Note that the distance from the Earth to the Sun is 1.5 x 10¹¹ m and the radius of the Sun is 696 million meters.

• A. 3686 K
• B. 5819 K
• C. 5777 K
• D. 6.5 × 10⁷ K

Here’s what students ask on this topic:

What are the three main methods of heat transfer?

The three main methods of heat transfer are conduction, convection, and radiation. Conduction involves the direct transfer of heat through contact between substances at different temperatures. Convection is the transfer of heat through fluids (liquids or gases) by the movement of the fluid itself. Radiation involves the transfer of heat through electromagnetic waves, such as infrared radiation, without the need for a medium. Understanding these methods is crucial for studying thermodynamics and energy transfer in various systems.

How does conduction transfer heat?

Conduction transfers heat through direct contact between substances at different temperatures. When two objects touch, the faster-moving particles in the hotter object collide with the slower-moving particles in the cooler object, transferring kinetic energy. This process continues until thermal equilibrium is reached. The rate of heat transfer by conduction depends on the thermal conductivity of the materials, the temperature difference, the cross-sectional area, and the length of the path through which heat is conducted.

What is the Stefan-Boltzmann law?

The Stefan-Boltzmann law describes the power radiated from a black body in terms of its temperature. The law is given by the equation:

$j=\epsilon \sigma T^4$

where $j$ is the radiance, $\epsilon$ is the emissivity of the material, $\sigma$ is the Stefan-Boltzmann constant (approximately $5.67\times {10}^{-8}\text{W}·{\text{m}}^{-2}·{\text{K}}^{-4}$), and $T$ is the absolute temperature in kelvin. This law is essential for understanding thermal radiation and energy transfer in various systems.

What is Wien's displacement law?

Wien's displacement law relates the temperature of a black body to the wavelength at which it emits the most radiation. The law is given by the equation:

$\lambda {\max }_{}=\frac{b}{T}$

where $\lambda {\max }_{}$ is the peak wavelength, $b$ is Wien's displacement constant (approximately $2.898\times {10}^{-3}\text{m}·\text{K}$), and $T$ is the absolute temperature in kelvin. This law helps determine the color of the radiation emitted by a black body and is crucial for understanding thermal emission spectra.

How does convection transfer heat?

Convection transfers heat through the movement of fluids (liquids or gases). When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks. This creates a circulation pattern that transfers heat. Convection can be natural, driven by buoyancy forces due to temperature differences, or forced, driven by external means like fans or pumps. This method of heat transfer is essential in many natural and engineered systems, such as weather patterns, ocean currents, and heating systems.