Thermal Modality Handbook

Convection is the heat transfer caused by the macroscopic movement of matter. Just as molecules of high temperature possess more kinetic energy, fluids of a high temperature are less dense than of low temperature. The difference in the density (and temperatures) within the fluid create the macroscopic convection currents. As a first example consider a pocket of air. The hot air (less-dense) rises due to buoyant forces and the colder air (more-dense) falls.

To be explicit, convection is really the combined heat transfer due to the macroscopic bulk movement and the random molecular motion. This can be demonstrated considering the example of a fluid in motion and its barrier, both at different temperatures. At the surface boundary, the velocity of the fluid is zero, and thus all of the heat transfer is due solely to the random molecular motion. As you go further from the boundary, the velocity of the fluid increases to some finite value and additionally, the temperature decreases.

The macroscopic heat transfer is due to the fact that the thermal boundary layer grows as the flow progresses in the x direction and heat in this layer is swept downstream and eventually to the fluid outside the boundary layer.

There are essentially two types of convection: forced and natural. The hot air example above is an example of natural convection. Air blowing over the earth as a result of atmospheric wind is an example of forced convection. However, both obey the same rate equation given below.

\begin{equation} q_\mathrm{conv} = h(T_a - T_b) \end{equation}

where h is the convection heat transfer coefficient and T_a and T_b represent the temperature gradient.

Conduction is the mechanism described by heat transfer through a medium in direct contact with surfaces of different temperatures. The first requirement for conduction is that the medium is not moving. Thus the medium must be a rigid solid, or if fluid, it must have no circulating currents (note that circulation is sometimes referred to as the fourth heat transfer mechanism).

If there is no large-scale movement then the heat transfer must be at the molecular and atomic level. Recall that molecules and atoms at any temperature above absolute zero are in motion. The motion is measured by the kinetic energy of the atoms and molecules. Temperature is by definition the measure of the average kinetic energy. This kinetic energy is due to the motion of the atoms and molecules globally, as well as the internal molecular motion; rotational, vibrational, etc. When atoms or molecules collide, energy is transferred. In terms of temperature, hotter (faster) atoms lose some energy, while the (slower) cooler atoms gain energy. Conduction is defined as the transfer of the energy in this manner.

Consider a group of atoms with high kinetic energy isolated from a group of atoms at a lower temperature via an ideal thermal insulator. At this point nothing interesting happens. If however the thermal insulator is removed, the atoms begin colliding randomly with each other, and at some time t1 > t0, the atoms will reach an equilibrium temperature. Now consider replacing the thermal insulator with a material of some given thermal conductivity. Thermal conductivity is a material property which describes the rate at which the material conducts heat. All materials have some thermal conductivity as a perfect vacuum is the only ideal insulator.

We can now quantify the heat transfer rate in terms of the thermal conductivity, k, temperature difference, dT, and the direction of energy transfer dx.

\begin{equation} q_\mathrm{cond} = -k\frac{dT}{dx} \end{equation}

Radiation is the mechanism of heat transfer where electromagnetic energy is either absorbed from an external source by the object or emitted by the object into the environment. Remember that any object at a temperature above absolute zero will radiate energy. A perfect radiator, also known as a blackbody, has the characteristic that it is a perfect absorber and in turn a perfect emitter. Planck derived the equation of spectral radiant exitance from a blackbody. It is given by Planck’s equation below.

\begin{equation} M_{\lambda BB}= 2 \pi h c^2 \lambda^5 (e^\frac{hc}{\lambda k T} - 1)^{-1} \end{equation}

Integrating Planck’s Equation yields the total exitance from a blackbody. This is knows as the Stefan-Boltzmann equation. Note that the total energy is proportional to the Temperature of the blackbody raised to the fourth power.

\begin{equation} M = \sigma T^4 \end{equation}

where sigma is the Stefan-Boltzmann constant.

In reality real objects are not perfect absorbers or emitters. We introduce emissivity as the ratio of the object’s spectral exitance to that of an ideal blackbody at the same temperature. From the equation it is obvious that the emissivity is a number between 0 and 1 and it can vary spectrally (as a function of wavelength).

An object is simultaneously absorbing and emitting with surrounding surfaces. Determination of the overall rate of heat transfer becomes complicated very quickly. A common example used in heat transfer is a small object completely surrounded by a larger surface (picture a ball hovering within a larger sphere). Whereas convection and conduction rely on a medium for heat transfer, radiation can occur in a vacuum. The heat transfer of the surface and surroundings is given in the following equation.

\begin{equation} q_\mathrm{rad} = \varepsilon \sigma {(T_o^4 - T_s^4)} \end{equation}

where e is the spectral emissivity, T_o is the temperature of the object and T_s is the temperature of the surroundings.

The convection heat transfer coefficient h is difficult to parameterize in thermal models because it can depend on many factors. Surface roughness, fluid properties such as viscosity, stability (vertical temperature change), and humidity (of air) can affect the convection rate. This coefficient tends to be the largest source of uncertainty in temperature prediction models. Some models, such as MuSES allow the user to explicitly define h, while some models such as THERM calculate h at runtime based on a parameterization that depends on the problem conditions (e.g. Wind Speed).