Download ME 3210 Mechatronics – Thermal Systems

ME 3210 Mechatronics – Thermal Systems
1-D Thermal Systems
Thermal systems are the last system that will be modeled in this class. Only very simple
one dimensional systems will be modeled. Courses in heat transfer and thermodynamics
will present more realistic and complex modelling. The approach will be like the other
systems, but with some significant differences. The variables that describe the energy are
temperature, the across variable, and heat flow, the through variable. Unlike the other
systems only two elements will be used to model thermal systems, one storage and one
dissipative.
Variables
Temperature, T, is the across variable as it is the variable that is different across elements.
Heat flow, q, flows through thermal systems.
Elements
There are only two elements with thermal systems, thermal capacitance, and thermal
resistance. Thermal capacitors store temperature and resistors dissipate energy. There is
no element that can store heat flow. There is no such thing as thermal inertia. The
consequence of this is that there are only first order differential equations to describe
thermal systems. There may be multiple first order parts but never a second order, underdamped part.
Thermal Capacitors
An example of a thermal capacitor is a mass with a specific heat. When a mass has a
temperature it keeps that temperature until the heat flow goes to a lower temperature. The
governing elemental equation for a thermal capacitor is given by equation (1). Cp is the
specific heat of the material and m is the mass.
q = mcp
dT
dt
(1)
Thermal Resistance
Thermal resistance is the major point of courses in heat transfer and thermodynamics. For
the purposes here, Fourier's law will be used. While this is not a very good approximation
for the heat flow in many systems, it is the only model that results in linear differential
equations. The resistance for a thermal system is given by equation (2).
q=
kA
(T)
L
(2)
A is the area of the thermal resistance, L is the length, and k is the coefficient of thermal
resistance.
Sources
There are sources which can provide a defined temperature or a defined heat flow.
Determining the proper directions for the sources is much like that for electrical systems. A
temperature source can provide a higher temperature to system. This is like connecting the
positive voltage to the system. If a lower temperature is applied to the system, this is like
connecting the negative side of a voltage source. To put the proper direction of a voltage
sources in a linear graph, simply put an arrow from plus to minus as is done with electrical
systems. The minus side (low temperature) is often ground.
Heat flow sources are similar to electrical systems. The schematic of a flow source defines
the direction of the flow.
Power
The power for a thermal system is defined differently than that for a mechanical, an
electrical system, or hydraulic system. Heat flow is the power, or:
Power =q