Thermofluid effects in dynamic systems

Transcription

1 Thermofluid effects in dynamic systems

2 Use of entropy (from BP)

3 From BP: Equilibrium and states When we make a thermodynamic assumption, we assume a homogeneous substance has near uniform properties of temperature and pressure. These are intensive variables. Extensive variables distribute among the parts. Intensive are common over a substance, while extensive sum over the substance.

4 U SVN Gibbs internal energy The state of a simple thermodynamic system can be quantified by, = U ( S, V, N ) = Gibbs (internal) energy i S = entropy, measures dispersal of system energy V N i = volume, measures size = moles, measures amount or matter content Now we quantify power flow into the thermodynamic system, P P = power du SVN U SVN U SVN U SVN = = Sɺ + Vɺ + Nɺ dt S V N i T P µ i i

5 Thermodynamic C Now we can represent this storage of energy using a multiport-c element:

6 U is a homogeneous 1 st order function Example: Basic Gibbs equation for pure substance: du = Tds Pdv

7 Energy stored in a solid 10 T C This is an equation of state; i.e., a constitutive relation for the C S

8 For ideal gas see BP Where: Pv = RT and du V dt = c (constant volume specific heat)

9 See Example 7-2 from BP T S ɺ = F 0 pɺ = mg + A( P P ) Vɺ Sɺ A = p m = 0 (adiabatic) Vo P = Po V Vo T = To V γ γ 1 A

10 Electro-thermal-mechanical system Consider the electro-thermo-mechanical system shown below. A piston is forced to move by the expansion of air in the cylinder. The cylinder and piston are made of steel. The piston is h thick and the cylinder walls are t w with the inner radius of the cylinder being r c. The height of the cylinder is L c. The ambient temperature of T a is fixed and known. The heater coil has known electrical resistance R and the voltage input is AC at 60 Hz. a. Develop a bond graph model of this system. b. Develop state equations for this system. c. Starting with the air at 25 deg C and compressed enough to balance the piston, the heater is turned on. Perform a simulation of the system. What are the steady state values of critical variables? Need to model the heat generated by the resistor as well as heat transfer. Let s look at how to model these elements.

11 Basic conduction R Other modes of heat transfer have same bond graph form, just the constitutive relation changes.

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13 P F Sɺ T P 2 v = fs = T R( T ) S S o = To exp cv S S o = Po exp cv

14 P F You now have enough to model a piston compressing air in a cylinder or chamber.

15 KMR Note: KMR reverse the sign and bond on the pressure port equivalent formulation.

16 KMR

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18 Thermal effects in a PMDC motor

19 You can derive a fairly good estimate of the thermal limitations on PMDC motors with this basic model. Here, it is critical that the rotor/windings temperature not exceed a specified limit. In turn, this limits the torque capacity of the motor.

20 Summary We can incorporate thermal effects into our system models using basic elements that represent the generation, storage and transfer of thermal energy/power. Up to this point, only systems with a fixed amount of matter (closed) have been considered. In some practical systems, it may be necessary to keep track of how much matter enters and/or leaves the system, and for those cases we need to track moles or mass and the conveyance of energy as well. This can be done with an extension to the methods we ve already described. References [1] J.J. Beaman and H.M. Paynter,, notes for ME 383Q, UT-Austin, Chapter 7. [2] D.C. Karnopp, et al, System Dynamics, Wiley (any edition). Chapter 12.

21 These models don t tell us anything about the internal energy storage.

22 Example: Air-spring suspension (KMR, P12-12)