7.2. Technical Description
Heat transfer is the energy transferred between an object and its environment due to a temperature difference between the two. There are three basic mechanisms for heat transfer: convection, conduction and radiation. They are described individually for clarity, however in reality a given system will typically exhibit all three processes simultaneously. The following three sections give a brief description of the theory and relevant parameters for the three mechanisms. More detailed explanation of the theory can also be found in a variety of physics or heat transfer texts, such as [Keller 1993] or [Incropera 1990].
7.2.1. Conduction
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 [Keller 1993].
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.
Thermal conductivity has units of [L-cm / hr /C]
7.2.2. Convection
Convection is the heat transfer caused by the macroscopic movement of matter. Just as molecules of high temperature possess more kinetic energy, areas of a high temperature (fluid) are less dense than areas 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 [Keller 1993]. 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.
h is the convection heat transfer coefficient and is in units of [W/m^2 K].
7.2.3. Radiation
Radiation is the mechanism of heat transfer where energy is emitted due to an object's temperature. Therefore 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.
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.
In reality however, real objects are not perfect absorbers or emitters. We introduce the emissivity as a means to modulate the spectral exitance. The emissivity is calculated as the ratio of the object's spectral exitance to that of the blackbody at the same temperature. From the equation it is obvious that the emissivity is a number between 0 and 1.
Combining the Stefan-Boltzmann result with the concept of emissivity yields the heat flux emitted by a real surface.
So far we have only described how a surface emits energy via radiation. We need to describe how this radiant energy affects a receiving surface. The surface will absorb some amount of the incoming radiation. Again, real objects are not perfect blackbodies, and therefore will not absorb 100% of this energy. Similar to the emissivity is a property which describes the rate at which a surface absorbs, the absorptivity. The absorbed energy is expressed as:
One final note is that in reality, a surface is simultaneously absorbing and emitting with surrounding surfaces. Determination of the overall rate of heat transfer becomes complicated very quickly. One example that [Keller 1993] explains is the case of heat transfer between a large surface that completely surrounds smaller a surface. Whereas convection and conduction rely on a medium for heat transfer, radiation occurs most efficiently in a vacuum. Therefore, the effect due to the surrounding air has virtually no effect. The heat transfer of the surface and surround is given in the following equation.
7.2.4. Enhancements and Limitations
7.2.4.1. Sun/Shadow History
A vast improvement of THERM's process of predicting object temperature involved the incorporation of a facet's sun/shadow history, i.e., the inclusion of a record chronicling when a particular object facet (or pixel) is in shadow and when it is in sun. Supplied with the proper ephemeris data, it is a fairly straight forward process for the ray-tracer sub model to determine the obscuration of a given scene point from the sun due to adjacent scene objects. This information allows the direct insolation terms in the weather data in THERM to be toggled on or off accordingly or to be modified by cloud transmission values. This represents a more accurate indicator of each object's resulting thermal history.
The sun/shadow history is determined based on pre-computed sun positions generated as a function of time by THERM. The sun position from a specific geographical coordinate is represented by a declination angle measured from the ground normal and an azimuth angle measured from the north direction in a clockwise sense. From these two angles, a vector directed at the sun's position can be computed. A family of vectors are created pointing to the diurnal sun positions. Rays are cast along each vector to determine if an object is hit or if the sun would be hit. This process is used to determine whether a given point on a facet is experiencing both direct and indirect insolation (1=sun) or simply indirect insolation (0=shadow) or a fractional value due to a partially transmitting cloud. This history is represented as a data vector. The position of each entry corresponds to the position vector of the sun at a given time and the value of the entry represents the condition fraction of sunlight passing through any obscuring object (e.g., sunlit(1), shadowed (0), thin cloud (0.9), thick cloud (0.1)).
7.2.4.2. Lateral Conduction
THERM has been shown to produce accurate temporal predictions of temperatures of passive real world objects. However, THERM does not compute the conduction between adjacent object facets and therefore does not have all of the functionality of fully conducting model which would require the use of finite element analysis. Such models are complex and require substantial amounts of computing time to add facet-to-facet conduction. THERM does allow for facets with self-generated (internal) power which can, when properly implemented, be used to overcome some of the limitations of non-conduction between facets.
7.2.5. Adding thermal properties to materials
The thermal properties for each material are defined in the Material Database File. They can be edited using the material editor (mat_edit). Directions on using the material editor to edit these properties can be found in the Materials chapter. That section also shows some simple examples of how the diurnal cycle of a material is modeled and changes based on the thermal properties assigned.