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thermal_damping

# thermal_damping - EXAMPLE THERMAL DAMPING work in air...

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EXAMPLE: THERMAL DAMPING air sealed outlet work in A BICYCLE PUMP WITH THE OUTLET SEALED . When the piston is depressed, a fixed mass of air is compressed. —mechanical work is done. The mechanical work done on the air is converted to heat. —the air temperature rises A temperature difference between the air and its surroundings induces heat flow. —entropy is produced The original work done is not recovered when the piston is withdrawn to the original piston. —available energy is lost Mod. Sim. Dyn. Syst. Thermal damping example page 1

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MODEL THIS SYSTEM G OAL : the simplest model that can describe thermal damping (the loss of available energy) E LEMENTS : T WO KEY PHENOMENA work-to-heat transduction a two port capacitor represents thermo-mechanical transduction entropy production a two port resistor represents heat transfer and entropy production B OUNDARY CONDITIONS : For simplicity assume a flow source on the (fluid-)mechanical side a constant temperature heat sink on the thermal side Mod. Sim. Dyn. Syst. Thermal damping example page 2
A BOND GRAPH IS AS SHOWN . R T gas T o dS o /dt C P –dV/dt dS gas /dt S f S e 0 :T o Q(t): (fluid) mechanical domain thermal domain C AUSAL ANALYSIS : The integral causal form for the two-port capacitor (pressure and temperature outputs) is consistent with the boundary conditions and with the preferred causal form for the resistor Mod. Sim. Dyn. Syst. Thermal damping example page 3

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C ONSTITUTIVE EQUATIONS : Assume air is an ideal gas and use the constitutive equations derived above. T T o = V V o R c v exp S – S o mc v P P o = V V o R c v + 1 exp S – S o mc v Assume Fourier’s law describes the heat transfer process. ˙ Q = kA l (T 1 - T 2 ) Mod. Sim. Dyn. Syst. Thermal damping example page 4
ANALYSIS: For simplicity, linearize the capacitor equations about a nominal operating point defined by S o and V o Define δ S = S - S o δ V = V - V o δ T = T - T o (S o ,V o ) δ P = P - P o (S o ,V o ) Taylor series expansion:

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thermal_damping - EXAMPLE THERMAL DAMPING work in air...

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