ideal_gas - EXAMPLE IDEAL GAS MANY LOW-DENSITY GASES AT...

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EXAMPLE: IDEAL GAS M ANY LOW - DENSITY GASES AT MODERATE PRESSURES MAY BE ADEQUATELY MODELED AS IDEAL GASES . Are the ideal gas model equations compatible with models of dynamics in other domains? A N IDEAL GAS IS OFTEN CHARACTERIZED BY THE RELATION PV = mRT P: (absolute) pressure V: volume m: mass R: gas constant T: absolute temperature Mod. Sim. Dyn. Syst. Ideal gas example page 1
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I N THE FORM PV = mRT THE IDEAL GAS EQUATION IS A RELATION BETWEEN VARIABLES WITH NO CAUSAL MEANING . Used as an assignment operator relating input to output PV := mRT it would imply that both effort, P, and displacement, V, on the mechanical power port are outputs. That is physically meaningless. Similarly, the form T := PV/mR would imply that both effort, P, and displacement, V, on the mechanical power port are inputs. That is also physically meaningless. Mod. Sim. Dyn. Syst. Ideal gas example page 2
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T HE IDEAL GAS EQUATION MAY BE RE - ARRANGED INTO TWO FORMS THAT ADMIT A MEANINGFUL CAUSAL INTERPRETATION . O NE FORM IS COMPATIBLE WITH THE CAUSAL ASSIGNMENT ASSOCIATED WITH THE H ELMHOLTZ FUNCTION . the “Helmholtz form” P := mRT/V T HE OTHER FORM IS COMPATIBLE WITH THE CAUSAL ASSIGNMENT ASSOCIATED WITH THE G IBBS FUNCTION . the “Gibbs form” V := mRT/P Mod. Sim. Dyn. Syst. Ideal gas example page 3
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AN IMPORTANT POINT T HE RELATION PV = M RT DOES NOT COMPLETELY CHARACTERIZE THE GAS . We model the interacting thermal and mechanical effects using a two-port capacitor. A two-port capacitor needs two constitutive equations. T HE SECOND IS USUALLY OBTAINED BY ASSUMING A PARTICULAR RELATION BETWEEN INTERNAL ENERGY AND TEMPERATURE . Common practice: assume c v is constant. c v : specific heat at constant volume W HAT S SPECIFIC HEAT ”? A ND HOW DOES IT DETERMINE THE RELATION BETWEEN INTERNAL ENERGY AND TEMPERATURE ? “S PECIFIC IN THIS CONTEXT MEANS PER UNIT MASS ”. Mod. Sim. Dyn. Syst. Ideal gas example page 4
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E XTENSIVE VARIABLES : A LL QUANTITIES THAT VARY WITH THE AMOUNT ( OR EXTENT ) OF A SUBSTANCE ( ALL OTHER FACTORS BEING EQUAL ) ARE EXTENSIVE VARIABLES ( OR PROPERTIES ). mass volume (total) entropy (total) internal energy (total) enthalpy etc. Mod. Sim. Dyn. Syst. Ideal gas example page 5
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E VERY EXTENSIVE VARIABLE ( PROPERTY ) HAS A SPECIFIC COUNTERPART . specific internal energy, u u = U/m internal energy per unit mass specific entropy, s s = S/m entropy per unit mass specific volume, v v = V/m volume per unit mass the inverse of density, ρ v = 1/ ρ (Note: “specific mass” -- mass per unit mass -- has dubious value and usually is not defined.) Mod. Sim. Dyn. Syst. Ideal gas example page 6
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EXTENSIVE VS. INTENSIVE I NTENSIVE VARIABLES : A LL QUANTITIES THAT DO NOT VARY WITH THE AMOUNT ( OR EXTENT ) OF A SUBSTANCE ( ALL OTHER FACTORS BEING EQUAL ) ARE INTENSIVE VARIABLES ( OR PROPERTIES ). pressure
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ideal_gas - EXAMPLE IDEAL GAS MANY LOW-DENSITY GASES AT...

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