Basics of Heat Transfer by Conduction

Who Cares About Heat Transfer?

Somebody made a fortune by selling small cardboard rings to cafes that sell hot coffee in paper cups. This person found a simple solution to a basic heat transfer problem: how to keep customers fingers from getting burned from hot coffee in a paper cup. They realized that if you put some cardboard between the cup and your fingers, you can safely hold the cup. Why? Because the heat of the coffee is not transferred efficiently through the cardboard -- the cardboard does not conduct heat well, but the paper of the cup does. But why does one material conduct heat better than the other? That is a basic question we wish to answer in this module.

You are probably familiar with other examples of heat transfer from everyday life or your engineering classes. Some other examples include

Macroscopic View of Heat Transfer

Here we will only consider conduction.

Imagine a solid system, such as a sheet of metal, where we can bring the two large faces of the solid in contact with fluids of arbitrary temperatures, keeping the thin sides of the sheet insulated. If the two large faces are at different temperatures, the solid system will be perturbed from equilibrium by a temperature gradient dT / dx. A flux of heat will arise, sending thermal energy in the direction of lower temperature, which works to eliminate the gradient and bring the system to equilibrium. This flow of heat is known as thermal conduction. The coefficient of thermal conductivity k is then defined as the proportionality constant between the flux J and the temperature gradient. In one dimension we write

This equation is called Fourier's law. The negative sign indicates that the flux is down the gradient, and it can be shown from irreversible thermodynamics that the coefficient is always positive.

The thought experiment in the previous paragraph was fairly straightforward for detemining k for solids. Now imagine applying a temperature gradient to a fluid, either gas or liquid, initially at rest. In this case, the temperature gradient will give rise to a density gradient. This will create a bouyancy force that will be opposed by viscous resistance of the fluid, and mechanical non-equilibrium will result. This mechanical instability will cause bulk convective motion, i.e. the bodily motion of whole portions of the fluid. This makes experimental measurements of thermal conductivity for gases and liquids more complicated, although it is certainly possible. If you are interested in learning more about how these experiments are performed, McLaughlin (1964) provides a good discussion.

The table below gives thermal conductivities for a variety of solids, liquids, and gases (Bird et al., 2002). You can see that generally speaking, thermal conductivities increase in the order gases < liquids < solids. For gases, k depends rather strongly on temperature. For many simple organic liquids, the thermal conductivities are 10 to 100 times larger than those of the corresponding gases at the same temperature, with little effect of pressure in the liquid state. Values of k for most common organic liquids range between 0.1 and 0.17 W / (m⋅K) at temperatures below the normal boiling point, but water, ammonia, and other highly polar molecules have values several times as large (Reid et al., 1987).

There are many systems where experimentally determined thermal conductivities are not available. This lack of data often makes it necessary to estimate k for a given substance. In addition, even if an experimentally determined k can be found for the material of interest, it may have been measured at a different temperature or pressure than we want. Thus, engineers are interested in mathematical expressions for how k varies with temperature and pressure. In the following two paragraphs, we give a short qualitative description of classical theories for estimating thermal conductivities and their temperature and pressure dependences. Further details and procedures for estimating these properties can be found in the texts by Bird et al. (2002) and Reid et al. (1987)

Thermal conduction in gases can be analyzed within the framework of kinetic gas theory, which you probably learned about in your freshmen or physical chemistry classes. In this theory, we assume all molecules to be identical, non-interacting, rigid spheres of given diameter and mass and moving randomly with a mean velocity v. The molecules move about in the gas and collide (like in the simulation applet). They may transfer energy between different regions of the gas if there are temperature gradients. Consider mentally dividing the gas into layers. The net flux of energy between two adjacent layers is assumed to be proportional to the energy gradient

Kinetic theory also provides similar expressions for the diffusivity and viscosity (the other common transport coefficients), which give reasonable results compared to experimental data. The above theory is pretty accurate in prediciting thermal conductivity of low density gas, however, it is not sufficient to describe high density gases or liquids.

Bird et al. (2002) describe a simple theory for estimating k for dense liquids. It pictures each molecule oscillating in a "cage" formed by its nearest neighbors. In this framework, the thermal conductivity ends up being related to the speed of sound through the liquid. A detailed discussion of thermal conduction in solids is given by Jakob (1949). It can depend on things like the degree of molecular orientation, the phase and crystallite size (for crystalline materials), and the void fraction (for porous solids). In general, the thermal conductivies of solids show the following behavior: metals conduct heat better than nonmetals (rememeber, metal is able to conduct heat by both lattice vibration and particle collision), and crystalline materials conduct heat better than amorphous ones (crystalline materials have better defined lattice than amorphous ones). Dry porous solids make good insulation. It should also be noted that thermal and electrical conductivity go hand in hand. Free electrons are major heat carriers in metals. The Wiedeman, Franz, Lorenz equation relates thermal conductivity and electrical conductivity for pure metals (Bird et al., 2002). The equation is not suitable for nonmetals, in which the concentration of free electrons is so low that energy transmission by molecular motion predominates.

So, while a simple theory exists for dilute monatomic gases, it does not work for high density gases. And more complicated systems like dense polar gases, or liquids, or solids also cannot be treated with simple pencil-and-paper theories. Increasingly, therefore, engineers are turning to molecular simulation as a means to predict such quantities. To understand how this is done, first we'll need some background on molecular simulation in general.