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Transcript
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