EXAMPLE THERMAL DAMPING work in air sealed outlet 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and its surroundings induces heat flow entropy is produced The original work done is not recovered when the piston is withdrawn to the original piston available energy is lost Mod Sim Dyn Syst Thermal damping example page 1 MODEL THIS SYSTEM GOAL the simplest model that can describe thermal damping the loss of available energy ELEMENTS TWO 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 BOUNDARY 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 Q t Sf P dV dt fluid mechanical domain C Tgas dSgas dt 0 R To dSo dt Se To thermal domain CAUSAL 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 CONSTITUTIVE EQUATIONS Assume air is an ideal gas and use the constitutive equations derived above T T o P P o R V c V v o R V c V v o S S o exp mc v 1 S S o exp mc v Assume Fourier s law describes the heat transfer process kA Q l T1 T2 Mod Sim Dyn Syst Thermal damping example page 4 ANALYSIS For simplicity linearize the capacitor equations about a nominal operating point defined by So and Vo T T o S o mcv T TR o V o Vo c v P P o S o mc v P P R o 1 V o Vo c v Po To mc mc v v 1 Inverse capacitance C P P R o o 1 mc v Vo c v equality of the off diagonal terms the crossed partial derivatives is established using Po Vo mRTo Linearized constitutive equations To T mc v Po P mc v S Po R 1 V Vo c v Po mcv where S S So V V Vo T T To So Vo P P Po So Vo Mod Sim Dyn Syst Thermal damping example page 5 NETWORK REPRESENTATION The linearized model may be represented using the following bond graph T TF 1 P V To Po C Vo Po 0 S C mcv To This representation shows that in the isothermal case T 0 the fluid capacitance is Cfluid Vo Po 0 the thermal in the constant volume case V capacitance is C thermal mc v To the strength of thermo fluid coupling is To Po This uses the convention that the transformer coefficient is for the flow equation with output flow on the output power bond To S and hence T To P V Po Po though causal considerations may require the inverse equations Mod Sim Dyn Syst Thermal damping example page 6 ALTERNATIVELY It may be useful to express the parameters in term of easily measured reference variables To and Vo as follows T P V TF 1 Vo mR C Vo2 mRTo 0 S C mcv To This representation shows that the strength of the coupling is Vo mR proportional to the nominal gas volume inversely proportional to the mass of gas Vo S and T Vo P V mR mR Mod Sim Dyn Syst Thermal damping example page 7 RESISTOR EQUATIONS The two port resistor constitutive equations are S1 Q kA T1 T2 T1 l T1 S2 Q kA T1 T2 T2 l T2 Linearize the resistor constitutive equations about a nominal operating point defined by T1o and T2o T2o S 1 2 kA T1o l 1 S 2 T 2o 1 T T1o 1 T1o 2 T2 T2o This is in conductance form f Ge Note that this conductance matrix is singular G T1oT2o 1 0 T12oT22o T1o T2o this is because both entropy flows are associated with the same heat flow Mod Sim Dyn Syst Thermal damping example page 8 LINEARIZE ABOUT ZERO HEAT FLOW If the two nominal operating temperatures are equal T1o T2o To the linearized constitutive equations are 1 S 1 kA To l 1 S 2 T o 1 T To 1 1 T2 To This simple form can be represented by an equally simple bond graph T1 S1 1 T2 S2 R Tol kA This follows the usual convention of writing the resistor parameter in resistance form Mod Sim Dyn Syst Thermal damping example page 9 ASSEMBLE THE PIECES LINEARIZED BOND GRAPH TF 1 So 0 1 Tgas Sf Pgas Vgas C R mcv To Tol kA Se To C Vo Po Po To Note the sign change on the capacitor thermal port to avoid a superfluous 0 junction Causal assignment indicates a first order system Time constant is determined by thermal conduction resistance and thermal capacitance Gas pressure is determined by fluid capacitance and reflected thermal capacitance and resistance Mod Sim Dyn Syst Thermal damping example page 10 INCLUDE PISTON INERTIA BOND GRAPH 1 mcv To Tol kA C R C Vo Po Se Tambient To Po 0 TF mpiston 1 Apiston 1 Sout of gas Tgas TF I Vgas Causal analysis indicates a third order system capable of resonant oscillation In this model the only damping is in the thermal domain heat transfer entropy flow Mod Sim Dyn Syst Thermal damping example page 11 SUMMARIZING THE GAS STORES ENERGY It also acts as a transducer because there are two ways to store or retrieve this energy two interaction ports energy can be added or removed as work or heat The energy storing transducer behavior is modeled as a two port capacitor just like the energy storing transducers we examined earlier Mod Sim Dyn Syst Thermal damping example page 12 IF POWER FLOWS VIA THE THERMAL PORT AVAILABLE ENERGY IS REDUCED the system also behaves as a dissipator The dissipative behavior is due to heat transfer Gas temperature change due to compression and expansion does not dissipate available energy If the walls were perfectly insulated no available energy would be lost but then no heat would flow either Without perfect insulation temperature gradients induce heat flow Heat flow results in entropy generation Entropy generation means a loss of available energy THE SECOND LAW Mod Sim Dyn Syst Thermal damping example page 13 DISCUSSION ALL MODELS ARE FALSE It is essential to understand what errors our models make and when the errors should not be ignored It is commonly assumed that modeling errors become significant at higher frequencies not so Compression and expansion of gases is common in mechanical systems Hydraulic systems typically include accumulators to prevent over pressure during flow transients The most common design uses a compressible gas Compression and expansion of the gas can dissipate available energy Mod …