## Heat transfer

12-6-99

Sections 14.7-14.9

### Heat transfer

There are three basic ways in which heat is transferred. In fluids, heat is often transferred by convection, in which the motion of the fluid itself carries heat from one place to another. Another way to transfer heat is by conduction, which does not involve any motion of a substance, but rather is a transfer of energy within a substance (or between substances in contact). The third way to transfer energy is by radiation, which involves absorbing or giving off electromagnetic waves.

### Convection

Heat transfer in fluids generally takes place via convection. Convection currents are set up in the fluid because the hotter part of the fluid is not as dense as the cooler part, so there is an upward buoyant force on the hotter fluid, making it rise while the cooler, denser, fluid sinks. Birds and gliders make use of upward convection currents to rise, and we also rely on convection to remove ground-level pollution.

Forced convection, where the fluid does not flow of its own accord but is pushed, is often used for heating (e.g., forced-air furnaces) or cooling (e.g., fans, automobile cooling systems).

### Conduction

When heat is transferred via conduction, the substance itself does not flow; rather, heat is transferred internally, by vibrations of atoms and molecules. Electrons can also carry heat, which is the reason metals are generally very good conductors of heat. Metals have many free electrons, which move around randomly; these can transfer heat from one part of the metal to another.

The equation governing heat conduction along something of length (or thickness) L and cross-sectional area A, in a time t is:

k is the thermal conductivity, a constant depending only on the material, and having units of J / (s m °C).

Copper, a good thermal conductor, which is why some pots and pans have copper bases, has a thermal conductivity of 390 J / (s m °C). Styrofoam, on the other hand, a good insulator, has a thermal conductivity of 0.01 J / (s m °C).

### Example

Consider what happens when a layer of ice builds up in a freezer. When this happens, the freezer is much less efficient at keeping food frozen. Under normal operation, a freezer keeps food frozen by transferring heat through the aluminum walls of the freezer. The inside of the freezer is kept at -10 °C; this temperature is maintained by having the other side of the aluminum at a temperature of -25 °C.

The aluminum is 1.5 mm thick. Let's take the thermal conductivity of aluminum to be 240 J / (s m °C). With a temperature difference of 15°, the amount of heat conducted through the aluminum per second per square meter can be calculated from the conductivity equation:

This is quite a large heat-transfer rate. What happens if 5 mm of ice builds up inside the freezer, however? Now the heat must be transferred from the freezer, at -10 °C, through 5 mm of ice, then through 1.5 mm of aluminum, to the outside of the aluminum at -25 °C. The rate of heat transfer must be the same through the ice and the aluminum; this allows the temperature at the ice-aluminum interface to be calculated.

Setting the heat-transfer rates equal gives:

The thermal conductivity of ice is 2.2 J / (s m °C). Solving for T gives:

Now, instead of heat being transferred through the aluminum with a temperature difference of 15°, the difference is only 0.041°. This gives a heat transfer rate of:

With a layer of ice covering the walls, the rate of heat transfer is reduced by a factor of more than 300! It's no wonder the freezer has to work much harder to keep the food cold.

The third way to transfer heat, in addition to convection and conduction, is by radiation, in which energy is transferred in the form of electromagnetic waves. We'll talk about electromagnetic waves in a lot more detail in PY106; an electromagnetic wave is basically an oscillating electric and magnetic field traveling through space at the speed of light. Don't worry if that definition goes over your head, because you're already familiar with many kinds of electromagnetic waves, such as radio waves, microwaves, the light we see, X-rays, and ultraviolet rays. The only difference between the different kinds is the frequency and wavelength of the wave.

Note that the radiation we're talking about here, in regard to heat transfer, is not the same thing as the dangerous radiation associated with nuclear bombs, etc. That radiation comes in the form of very high energy electromagnetic waves, as well as nuclear particles. The radiation associated with heat transfer is entirely electromagnetic waves, with a relatively low (and therefore relatively safe) energy.

Everything around us takes in energy from radiation, and gives it off in the form of radiation. When everything is at the same temperature, the amount of energy received is equal to the amount given off. Because there is no net change in energy, no temperature changes occur. When things are at different temperatures, however, the hotter objects give off more energy in the form of radiation than they take in; the reverse is true for the colder objects.

The amount of energy an object radiates depends strongly on temperature. For an object with a temperature T (in Kelvin) and a surface area A, the energy radiated in a time t is given by the Stefan-Boltzmann law of radiation:

The constant e is known as the emissivity, and it's a measure of the fraction of incident radiation energy is absorbed and radiated by the object. This depends to a large extent on how shiny it is. If an object reflects a lot of energy, it will absorb (and radiate) very little; if it reflects very little energy, it will absorb and radiate quite efficiently. Black objects, for example, generally absorb radiation very well, and would have emissivities close to 1. This is the largest possible value for the emissivity, and an object with e = 1 is called a perfect blackbody,

Note that the emissivity of an object depends on the wavelength of radiation. A shiny object may reflect a great deal of visible light, but it may be a good absorber(and therefore emitter) of radiation of a different wavelength, such as ultraviolet or infrared light.

Note that the emissivity of an object is a measure of not just how well it absorbs radiation, but also of how well it radiates the energy. This means a black object that absorbs most of the radiation it is exposed to will also radiate energy away at a higher rate than a shiny object with a low emissivity.

The Stefan-Boltzmann law tells you how much energy is radiated from an object at temperature T. It can also be used to calculate how much energy is absorbed by an object in an environment where everything around it is at a particular temperature :

The net energy change is simply the difference between the radiated energy and the absorbed energy. This can be expressed as a power by dividing the energy by the time. The net power output of an object of temperature T is thus:

### Heat transfer in general

We've looked at the three types of heat transfer. Conduction and convection rely on temperature differences; radiation does, too, but with radiation the absolute temperature is important. In some cases one method of heat transfer may dominate over the other two, but often heat transfer occurs via two, or even all three, processes simultaneously.

A stove and oven are perfect examples of the different kinds of heat transfer. If you boil water in a pot on the stove, heat is conducted from the hot burner through the base of the pot to the water. Heat can also be conducted along the handle of the pot, which is why you need to be careful picking the pot up, and why most pots don't have metal handles. In the water in the pot, convection currents are set up, helping to heat the water uniformly. If you cook something in the oven, on the other hand, heat is transferred from the glowing elements in the oven to the food via radiation.