Departure function

In thermodynamics, a departure function is defined for any thermodynamic property as the difference between the property as computed for an ideal gas and the property of the species as it exists in the real world, for a specified temperature T and pressure P. Common departure functions include those for enthalpy, entropy, and internal energy. Departure functions are often used to "transform" hard problems involving nonideal fluids into easier problems involving state changes of ideal gases. The "easier" problems can then be solved directly, and the departure function allows one to transform back to the nonideal fluid and calculate the properties of the fluid.

Departure functions can be computed via calculus from an equation of state.

Departure functions for Peng-Robinson equation of state

The Peng-Robinson equation of state relates the three interdependent state properties pressure P, temperature T, and molar volume V_m. From the state properties (P, V_m, T), one may compute the departure function for enthalpy per mole (denoted H) and entropy per mole (S)^[1]:

$H_{T,P}-H_{T,P}^{\mathrm {ideal} }=RT_{C}\left[2.078(1+\kappa ){\sqrt {\alpha }}ln\left({\frac {z+2.414B}{z-0.414B}}\right)-T_{r}(z-1)\right]$

$S_{T,P}-S_{T,P}^{\mathrm {ideal} }=R\left[2.078\kappa \left({\frac {1+\kappa }{\sqrt {T_{r}}}}-\kappa \right)ln\left({\frac {z+2.414B}{z-0.414B}}\right)-ln(z-B)\right]$

Where $\alpha$ is defined in the Peng-Robinson equation of state, T_r is the reduced temperature, P_r is the reduced pressure, z is the compressibility factor, and

$\kappa =0.37464+1.54226\omega -0.26992\omega ^{2}$

$B=0.07780{\frac {P_{r}}{T_{r}}}$

Typically, one knows two of the three state properties (P, V_m, T), and must compute the third directly from the Peng-Robinson equation of state. To calculate the third state property, it is necessary to know three constants for the species at hand: the critical temperature T_c, critical pressure P_c, and the acentric factor ω. But once these constants are known, it is possible to evaluate all of the above expressions and hence determine the enthalpy and entropy departures.

References

• ^ Kyle, B.G. Chemical and Process Thermodynamics, 3rd Ed. Prentice Hall PTR, 1999. p. 118-123.