# Heat transfer

#### Prerequisites

The basic idea of temperature is that objects share thermal energy. A hot object placed next to a cold object gets colder, and the colder object gets warmer. The analysis of the energy of molecule motion in kinetic theory shows a bit of how this works. Temperature is a measure of the average energy of random motion of molecules, and in hotter objects the molecules are moving faster. Collisions at the interface of hot and cold objects tend to speed up the cold molecules and slow down the hot ones, sharing the energy. Exactly how this happens is governed by the number of ways energy can be shared, as we will see in our discussion of the second law of thermodynamics and entropy.

What matters is not only the fact that temperatures tend to equalize. Organisms live in time and it matters how fast this equalization process takes place. There is a delicate (and critical) balance between the rates at which organisms internally generate and exchange thermal energy with their environments. What is critical for an organism is the rate at which thermal energy is exchanged.

At the macroscopic level, the simple model of direct contact described in the first paragraph is only one mechanism by which energy is shared and temperatures tend to equalize. There are two other important mechanisms, and the rates of energy transfer depend on which mechanism is in play. The three macroscopic mechanisms are:

• conduction
• convection

1) Conduction  This is the mechanism we described in the first paragraph when we were building the basic concept of temperature: exchange of energy through contact at surfaces. Since biological organisms are rarely in a vacuum, they are be in constant contact with molecules of the surrounding medium (typically air or water). Thermal energy is readily transferred to or from this surrounding medium through collisions of the molecules on either side of the surface. If the object is warmer than the surrounding medium, its molecules will give up some of their energy to the surroundings. Conduction assumes that the surroundings are stable and do not move — at least on the time scales we are considering.

The rate at which heat is transferred between two surfaces is dependent on the molecules randomly bouncing off the surface from both sides. The simplest example to think about is heat flowing within a single material that is not at a constant temperature.

This process is similar to the random motion that causes molecules to diffuse from a non-uniform concentration. It results in an equation that is very similar to Fick’s law of diffusion ($J = -D dn/dx$ where $J$ is the flux of molecules and $n$ is their concentration). The analogous equation for heat is called Fourier's law:

$$J_{conduction} = - \kappa \frac{dT}{dx}$$

Here $J_{conduction}$ is the heat flux density — the amount of energy crossing a surface per unit area, per unit time. An appropriate unit is Joules/m2/s or Watt/m2.

The equation shows that the variable that drives thermal energy flow is a temperature gradient. The constant κ is called the thermal conductivity of the medium. Since the thermal energy flux is given in W/m2, and the temperature gradient is K/m, the units of thermal conductivity must be W/m⋅K.  Thermal conductivity tends to be higher for denser media where the molecules are closer together and so more readily bounce into each other to transfer the kinetic energy away from the warm part of the object. (Though that isn't the whole story.)

When two different items are in contact, the surfaces will come to a common value and the rate at which the energy is flowing through the surface will depend on the thermal conductivity on both sides. The value of $κ$ for some materials are given in the table below.

 Material $\kappa$ (W/m oK) water 0.60 air 0.026 muscle 0.49 fat 0.21

Thermal conductivity is 23 times greater in water than in air. This makes air a useful insulator. For a more complete thermal analysis of the properties of body tissues, check the IS'IT website.

These values have a lot of real-world implications. They are the reason that animals that live in cold water (seals, polar bears, whales) have a layer of fat (blubber) right under the skin. Fat has a thermal conductivity of about 0.20 W/m⋅K, the lowest of any body tissue, so it slows the loss of heat through the skin. The low value of thermal conductivity for air is the reason why the best insulating clothing (and insulation in houses) contains lots of tiny air pockets.

For heat flow for conduction, the equation shows us that energy will flow given proportional to the amount of surface area, and will be proportional to the temperature gradient, larger differentials driving faster flow.

Note that Fourier's law is one more example of gradient-driven flow — like not only Fick's law, but like the Hagen-Pouiseuille equation describing fluid flow, and Ohm's law describing electric currents. Like all of these laws, the proper form of Fourier's law in 3D would be to use a vector derivative — the gradient — since in 2D or 3D the heat will flow in the direct of the most rapid temperature change — pointing "down the hill."

2)  Convection:  In our discussion of heat flow in conduction, we assumed that the media conducting the thermal energy were stationary — only the energy was flowing; matter was not. We know from our considerations of diffusion (and, say, a perfume bottle opened in a room) that when we are talking about diffusion in fluids, the motion of the fluid carrying the diffusing molecules can be equally or more important than diffusion (depending on the scale we are considering).

The same thing is true of heat flow. If the surrounding medium is in motion, then thermal energy can be carried much faster than if it were moving only by conduction. The Gulf Stream brings warm water from the Caribbean up to the North Atlantic at a much faster rate than the thermal energy can diffuse out into the surrounding water. As a result, the Gulf Stream brings warmth to Northern Europe, making it much warmer in winter than corresponding latitudes in North America.

Since many organisms live in fluids, and since fluids of different temperatures have different densities, warming the medium the organism is in may result in the layer of medium next to an object rising and continually being replaced.  Therefore, the air next to an object never heats up as that layer is continually mixing in with the rest of the surroundings. As a result, the temperature gradient from the object to the surroundings remains larger than it would for stagnant air and more heat is transferred more quickly from the object to the air. This is why you don't feel so cold on a cold day with still (and dry) air. (Dry air has a lower heat conductivity than moist air.) But when the wind picks up, the air near your surface doesn't form a layer of warmed air around you.

3) Radiation: In our discussion of temperature (and kinetic theory) we only have discussed the energy stored in the kinetic and potential energy of matter — molecules. But light also carries energy. Light is electromagnetic radiation of particular wavelengths, ranging from about 300-700 nm.

Interestingly, vibrating matter is continually emitting and absorbing electromagnetic radiation and so goes to a thermal equilibrium that includes the energy of the radiation. This means that "light can have a temperature." Hotter light has shorter wavelengths, cooler light has longer wavelengths. The temperature of the surface of the sun is nearly 6000K and this is why the light from the sun looks the way it does. The most likely wavelength at this temperature is about 600 nm and much of the electromagnetic radiation coming from the sun is in the visible. (Of course this is not an accident! Evolution occurs in the context of organisms being bathed in radiation of these wavelengths.) Objects at temperatures of 30°C and below radiate in the infrared — 10,000 nm (10 microns) and longer.

As of this writing (with LED bulbs replacing compact fluorescents, and incandescents are being totally phased out), bulbs are being sold by the color — and that color is identified by the temperature of the the electromagnetic radiation that it best matches. A "daylight" LED bulb is sold as "color = 5000K" (see the chart at the right). This does not mean that the bulb gets that hot! (Good thing.)

All objects that are at a different temperature than their environment will gain or lose energy through absorbing or emitting electromagnetic radiation. The rate at which they do this depends on their temperature and on their surface. A perfect absorber is called a "black body" because it absorbs everything that falls on it.* But a perfect absorber is also a perfect radiator. Wearing white clothing in a hot climate is good because it reflects the radiation from the sun but it also slows the radiation from getting out.

The rate at which a perfect radiator (black body) emits thermal energy proportional to the fourth power of the Kelvin temperature. This is given by the Stefan-Boltzmann equation:

$$J_{radiation} = - \sigma T^4$$

In contrast to our previous equations for heat flow, the energy carried away by radiation does not depend on a temperature difference but on an absolute temperature. This is because electromagnetic energy, like light, can propagate through empty space — and through itself. So every object just pours out electromagnetic energy, whatever environment it is in. (Of course it might be absorbing electromagnetic energy from its environment as well.)

The negative sign here is to remind us that the energy is flowing out of the object. With equations of heat flow, we often suppress the sign because we know which way the heat is going. If an object radiated electromagnetic radiation away and absorbed none, for example, we know that it would cool down.

Here, the energy flux is the energy per area per time, just as before. Appropriate units are J/s⋅m2 or W/m2.  $T$ is the temperature in Kelvin degrees, and $σ$ is the Stefan-Boltzmann constant (5.67 x 10-8 W/m2K4), a property of radiation, independent of the material of which the black body is made. Of course imperfect radiators will give off less heat than this and so a blackbody is a good upper limit to the amount of heat an object will radiate. The total radiation emitted will be proportional to the surface area emitting.

Which of these three mechanisms dominate will depend on the specific circumstances. See the following examples for two specific calculations.

• The reason a perfect radiator/absorber is called a "black-body" is because the perfect absorber is just a hole. If you have a closed container, radiation hitting the hole will pass through it without resistance. Such a hole looks black because it does not reflect back any light.

Julia Gouvea, Wolfgang Losert, and Joe Redish 11/24/14

Article 436