Thermal Systems Background

Animation by Ames Bielenberg   (Note)

The animation shows two thermal capacitances (C1 and C2). C1 is separated from C2 by a thermal resistance R1. C2 is separated from ambient by R2. Heat (qin) flows into C1, increasing its temperature. Heat then flows from C1 to C2, increasing the temperature of C2. Finally heat flows from C2 to ambient.


Thermal systems are those that involve the storage and transfer of heat.  Heat stored in a material object is manifested as a higher temperature.  For example, a hot block of metal has more heat stored in it than an equivalent cool block.  Heat flows between objects by one of three mechanisms: conduction, convection (or mass transfer), and radiation.  Conductive heat transfer occurs when a temperature difference exists across an object.  An example is the flow of heat that occurs through the wall of a building if the temperature inside is higher (or lower) than the temperatures outside.  Convective heat transfer involves the flow of heat in a liquid or gas, as when a fan blows cool air across a hot object; the air carries away some of the heat of the object.   Radiative heat transfer, like conductive transfer, is caused by a temperature difference between objects, does not require a physical medium for heat flow (i.e., radiative heat can flow through a vacuum).  It is exemplified by the heat transfer from sun to earth, but it is highly nonlinear (it depends on the fourth power of the temperature difference) and will not be discussed here.  Our discussion will also be limited in several other ways listed, and briefly discussed, here.

A list of the fundamental units of interest is listed below.  The next tab (system elements) gives a description of the building blocks of these system (thermal resistance, capacitance and fluid flow). This is followed by a description of  methods to go from a drawing of a system to a mathematical model of a system in the form of differential equations.  Methods for solving the equation are given elsewhere.  The last section discusses topics relevant to energy storage and dissipation in these systems.

This page does not discuss the solution of these equations, only the development of the equations through a physical model of the system.

Table of units

The table below lists commonly used units of measure for translating mechanical systems in SI units.  More complete tables are available.

Fundamental Quantities   SI unit
Time - t second (s)
Energy - w Joule (J)
Power (or heat flow) - q Watt (J/s)
Temperature - θ K
(note we will generally be interested in temperature differences. 
Since temperature differences are equal on Kelvin and Celsius
 scales, we will generally use °C instead of K)
Thermal Resistance - R K/W
Thermal Conductance - KT W/K
Thermal Capacitance - C J/K
Mass flow rate - G kilogram/sec (kg/s)
Specific heat - cp  J/(kg-K)

Energy is sometimes measure in calories rather than Joules.  The conversion is 1 J = 0.239 calories.