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Heat:
Heat is particularly important for environmental issues, because temperature is a main characteristic of our climate system and controls the rates of many chemical reactions. First, a few definitions:
Thermal energy: kinetic energy associated with random motion of particles. Informally, "heat" is often used as a synonym for thermal energy, although strictly speaking, heat really describes the transfer of thermal energy from hot objects to cold ones. For simplicity, I will use the two terms as synonyms. Temperature is a measure of the average thermal energy of a system of particles.
Conduction: one of two main means of transporting heat. During conduction, the motion of hot particles causes nearby cold particles to move faster, thereby causing the nearby temperature to rise. The greater the difference in temperature between the particles, the greater the rate of heat transport. Formally, this can be expressed as flux~Δtemperature/Δdistance. The rate of heat transfer also depends on the thermal properties of the conducting materials, expressed as thermal diffusivity K (for rocks, K is ~1 mm²/s).
An example of conduction is the heating of a metal pot through contact with a hot electric burner. Conduction also transmits this heat from the pot base to the handle, making the pot hot. Metals tend to be good conductors, which is why we construct pots out of metal. Plastic is not, which explains why pot handles are often made of plastic. Air is also a notably poor conductor. For conduction,
Δtemperature/Δtime=-K*Δtemperature gradient/Δdistance
Advection: transport of energy through the motion of heat by a moving medium, such as a fluid. The rate that energy is transported depends on the temperature of the material being moved and the velocity at which it is moving.
An example of advection is the cooling of a hot beverage by blowing on it, thereby advecting thermal energy away from the beverage. Convection (circular motion of a fluid that is being heated from below) is a special type of advection. In general, advection tends to be much more efficient at transporting heat, which explains why we blow on hot things to cool them off and why we try to prevent air currents from cooling off our houses in the winter.
Geothermal heat: Earth's underground temperatures provide a good example of how to apply the different processes of heat transport to a steady-state system. Earth's interior is much hotter than Earth's surface, a result of remnant heat released during the accretion of Earth at the birth of our solar system 4.5 billion years ago and of heat released during radioactive decay of isotopes since Earth's formation. The rate that temperature increases with depth is known as the geothermal gradient, and is typically ~20-25 °C/km (i.e., 1 km below ground is ~25 °C warmer than Earth's surface). This gradient is relatively constant because it is this temperature gradient that dictates the rate of heat conduction from Earth's interior to the surface. Since deep temperatures are relatively stable, local areas must constantly be in a state in which energy flowing into an area from below is balanced by the amount of energy flowing upward toward the surface. This upward heat flow provides an excellent source of heat for buildings in the winter. In the summer, underground regions are cooler than the surface and can therefore be used to cool buildings.
The geothermal gradient is only applicable for Earth's lithosphere, however. Below the lithosphere, the asthenosphere experiences convection (the motion that controls plate tectonics!), making temperatures in the asthenosphere much more uniform.
Instantaneous heating and cooling: how temperatures change in non-steady systems is complex and general equations can only be determined for very special cases. One famous case is the instantaneous heating or cooling of an infinite half-space. Assumptions are:
1. The region being heating (or cooled) is an infinite half-space, extending from a surface upward and outward (but not downward) toward infinity. This also works upside-down, with the half-space extending downward (e.g., underground regions extend from Earth's surface downward and outward for great distances).
2. Temperature of the region being heated (or cooled) is uniform to start, with a temperature T0.
3. Temperature change at the surface is instantaneous to a new temperature T1.
4. Heat is transported only through conduction.
Given these conditions and a material with a thermal diffusivity K, temperature T as a function of time t and distance from the surface z can be expressed as
(T-T0)/(T1-T0)=erfc(z/(2(Kt)½))
Example 1: What is the age of Earth?
Lord Kelvin used the above equation and Earth's geothermal gradient to calculate its age in the late 1800s at a time when the age of Earth was particularly critical to theories like evolution. Using a geothermal gradient of 25 °C/km, K of 1 mm²/s, T0 of 1300 °C, and T1 of 0 °C, you can determine that Earth is ~25 million years old. This is well below Earth's true age. The failure of this calculation indicates that one important process unknown to Kelvin, radioactivity, is a very significant source of heat, effectively keeping Earth much hotter than you would predict given the planet's age.
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