What basic physics is needed?
What determines the temperature of a single body?
The temperature of a single body is determined by the flow of heat into and out of the body. If it receives energy at a faster rate than it disposes of it, it will heat up. Conversely if it looses heat faster than it gains heat its temperature will fall. These transient changes in temperature will continue until the 'in' and 'out' heat flows are in balance after which the temperatures remains steady.
What determines the temperature of two bodies in a thermal system?
A hotter body looses energy to a cooler body by one of the three mechanisms above thereby loosing some of its internal energy. If the body cannot replenish its internal energy with a heat source the temperature of the body will therefore decrease. This continues until the temperature of the two bodies is the same and there is then no net flow of energy between them. The components are then said to be in thermal equilibrium. Clearly the Universe is not in a state of thermal equilibrium because we only 'see' the Universe by virtue of an unequal exchange of energy. To maintain a steady flow of heat between two bodies (i.e. to maintain the temperature difference (gradient)) therefore requires the hotter body to have an internal source of energy; i.e. it has to have some internal heat dissipation. In this situation we speak about a state of dynamic thermal equilibrium, or simply thermal steady state, where the net heat flows are steady and the consequent temperatures are different to each other though unchanging. A different insight into these concepts can be obtained from consideration of water in a bath. If you would like to explore this further please click here.
How are temperature gradients calculated?
The rate at which heat flows between one body and another (i.e. the power) depends on that property which is supporting the heat flow and that property which is opposing the heat flow. The first is clearly related to the temperature gradient because without a gradient there is no heat flow. The second relates to the thermal resistance between the bodies; if the resistance is infinite then again there will be no heat flow. An interesting analogy is the flow of electricity through a wire. To explore this further click here. In general terms we can see therefore that the temperature gradients are in some way proportional to the power and in some way inversely proportional to the resistance. We can expect the functional relationships to be different for the three different transfer mechanisms. As an example consider the human body. We have an internal supply of energy which we maintain through eating. Notwithstanding the various mechanisms the body has for maintaining our body temperature nearly constant, when we feel cold we put on more clothing which resists the loss of heat flow; the clothing acts as a thermal insulator and consequently we feel warmer.
