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Non-tabular approaches to calculating properties of real gases

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The critical state At the critical state (T c, P c ), properties of saturated liquid and saturated vapor are identical if a gas can be liquefied at constant T by application of pressure, T·T c. if a gas can be liquefied at constant P by reduction of T, then P·P c. the vapor phase is indistinguishable from liquid phase

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Properties of the critical isotherm The SLL and SVL intersect on a P-v diagram to form a maxima at the critical point. On a P-v diagram, the critical isotherm has a horizontal point of inflexion. –

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Departures from ideal gas and the compressibility factor For an ideal gas One way of quantifying departure from ideal gas behavior to evaluate the compressibility factor (Z) for a true gas: Both Z 1 is possible for true gases

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The critical state and ideal gas behavior At the critical state, the gas is about to liquefy, and has a small specific volume. is very large Z factor can depart significantly from 1. Whether a gas follows ideal gas is closely related to how far its state (P,T) departs from the critical state (P c,,T c ).

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Critical properties of a few engineering fluids Water/steam (power plants): –CP: 374 o C, 22 MPa –BP: 100 o C, 100 kPa (1 atm) R134a or 1,1,1,2-Tetrafluoroethane ( refrigerant ) : – CP: 101 o C, 4 MPa – BP: -26 o C, 100 kPa (1 atm) Nitrogen/air (everyday, cryogenics): –CP: -147 o C, 3.4 MPa –BP: -196 o C, 100 kPa (1 atm)

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Principle of corresponding states (van der Waal, 1880) Reduced temperature: T r =T/T cr Reduced pressure: P r =P/P cr Compressibility factor: Principle of corresponding states: All fluids when compared at the same T r and P r have the same Z and all deviate from the ideal gas behavior to about the same degree.

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Generalized compressibility chart 1949 Fits experimental data for various gases

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Use of pseudo-reduced specific volume to calculate p(v,T), T(v,p) using GCC Z

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Nelson-Obert generalized compressibility chart 1954 Based on curve- fitting experimental data

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Equations of state

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Some desirable characteristics of equations of state Adjustments to ideal gas behavior shoujd have a molecular basis (consistency with kinetic theory and statistical mechanics). Pressure increase leads to compression at constant temperature Critical isotherm has a horizontal point of inflection: Compressibility factor (esp. at critical state consistent with experiments on real gases.)

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Some equation of states Two-parameter equations of state Virial equation of states Z=1+A(T)/v+B(T)/v 2 +…. (coefficients can be determined from statistical mechanics) Multi-parameter equations of state with empirically determined coefficients: –Beattie-Bridgeman – Benedict-Webb-Rubin Equation of State Often based on theory

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Two-parameter equations of states Examples: –Van der waals – Dieterici –Redlich Kwong Parameters (a, b) can be evaluated from critical point data using Van der Waals:

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Critical compressibility of real gases

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First law in differential form, thermodynamic definition of specific heats

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