Departure functions

Thermodynamic paths are necessary to evaluate the enthalpy (or internal energy) of the fluid phase and the internal energy of the stationary phase. For gas-phase processes at low and modest pressures, the enthalpy departure function for pressure changes can be ignored and a reference state for each pure component chosen to be ideal gas at temperature and a reference state for the stationarv phase (adsorbent plus adsorbate) chosen to be adsorbate-free solid at. Thus, for the gas phase we have... 

To calculate the enthalpy of liquid or gas at temperature T and pressure P, the enthalpy departure function (Equation 4.78) is evaluated from an equation of state2. The ideal gas enthalpy is calculated at temperature T from Equation 4.81. The enthalpy departure is then added to the ideal gas enthalpy to obtain the required enthalpy. Note that the enthalpy departure function calculated from Equation 4.78 will have a negative value. This is illustrated in Figure 4.9. The calculations are complex and usually carried out using physical property or simulation software packages. However, it is important to understand the basis of the calculations and their limitations. 

The calculation of entropy is required for compression and expansion calculations. Isentropic compression and expansion is often used as a reference for real compression and expansion processes. The calculation of entropy might also be required in order to calculate other derived thermodynamic properties. Like enthalpy, entropy can also be calculated from a departure function ... 

The integral in Equation 4.84 can be evaluated from an equation of state3. However, before this entropy departure function can be applied to calculate entropy, the reference state must be defined. Unlike enthalpy, the reference state cannot be defined at zero pressure, as the entropy of a gas is infinite at zero pressure. To avoid this difficulty, the standard state can be defined as a reference state at low pressure P0 (usually chosen to be 1 bar or 1 atm) and at the temperature under consideration. Thus,... 

Both enthalpy and entropy can be calculated from an equation of state to predict the deviation from ideal gas behavior. Having calculated the ideal gas enthalpy or entropy from experimentally correlated data, the enthalpy or entropy departure function from the reference state can then be calculated from an equation of state. 

Simple vapor and liquid enthalpy models may be developed by recognizing that, on a mass basis, vaporization enthalpies and vapor and liquid heat capacities do not vary widely from component to component, and the latter are relatively independent of temperature. Enthalpy departure functions in mass units are first defined as follows ... 

The enthalpy models are actually for the enthalpy departure function. Normally, the molar enthalpy of a phase is found from the ideal gas enthalpy and the enthalpy departure... 

The ideal gas law is used to calculate the enthalpy and entropy of fluids at low pressure. The principles of thermodynamics can be used to extend these to higher pressure. This is done through the so-called departure functions. For the enthalpy, the departure function is given as follows ... 

Derivation of Helmholtz and Gibbs Free-Energy Departure Functions from the Peng-Robinson Equation of State at Infinite Pressure... 

HE Helmholtz tree-energy departure function (from ideal gas behavior) for the Peng-Robinson equation at a given temperature, pressure, and composition is (Wong and Sandler 1992)... 

Symbolic determination of enthalpy departure function for the Clausius equation of state... 

The proposed approximation amounts to neglecting the partial derivatives of the enthalpy departure functions, the Q s, with respect to the component-flow rates. As shown in Chap. 14, Q appears in the definition of the virtual value of the partial molar enthalpy. For example, for any component i in the liquid phase on plate j, the virtual value of the partial molar enthalpy is given by... 

Most approximations of this class involve the relative magnitudes of the partial derivatives of the activity coefficients, fugacities, and the departure function Q with respect to temperature. If, for example, the Q is independent of temperature or its variation with temperature is small, then the approximation dQ/dT = 0 may be made. 

The vapor enthalpies were expressed in terms of the enthalpy departure function Q (see Chap. 14) as follows... 

The equality given by this equation is readily established by beginning with the fact that the enthalpy of one mole of any mixture may be expressed in terms of h° and the departure function Cl as follows... 

The departure function Q (which is needed in the evaluation of the virtual values of the partial molar enthalpies [Eq. (14-65)]) may be evaluated through the use of an equation of state for the mixture. 

The existence of an accurate corresponding-states relationship of the type Z = Z(Tr, Pr) (or perhaps a whole family of such relationships for different values of Zc or ) allows one to also develop corresponding-states correlations for that contribution to the thermodynamic properties of the fluid that results from molecular interactions, or nonideal behavior, that is, the departure functions of Sec. 6.4. For example, starting with Eq. 6.4-22, we have... 

Although the discussion of the previous section focused on the van der Waals equation and corresponding-states charts for both the compressibility factor Z and the thermodynamic departure functions, the modem application of the corresponding states idea is to use generalized equations of state. The concept is most easily demonstrated by again using the van der Waals equation of state. From Eqs. 6.2-38,... 

This generalized form of the Peng-Robinson equation of state (or other equations of state) can be used to compute not only the compressibility, but also the departure functions for the other thermodynamic properties. This is done using Eqs. 6.4-29 and... 

This example demonstrates that reliable PVT correlation and constant-pressure heat capacity of an ideal gas are sufficient to determine a variety of thermodynamic properties, as enthalpy, entropy, Gibbs free energy, etc., and built comprehensive charts. This approach will be extended by means of departure functions. 

At this point we may introduce an important class of thermodynamic functions called residual or departure functions. In this case, the residual Gibbs free energy may be defined as ... 

Hence, the enthalpy change between T, and Tj. Pi niay be computed from the variation for an ideal gas plus the variation of the departure function, which accounts for non-ideality. The big advantage of the departure functions is that they can be evaluated with z PVT relationship, including the corresponding states principle. Moreover, the use of departure functions leads to a unified framework of computational methods, both for thermodynamic properties and phase equilibrium. 

Note that the difference between residual and departure functions comes only from the quantity M (P,T) M P, T). Thus, residual and departure functions are identical for U and H, but slightly different for S and G. Hence, for the last function the difference is G" = (G -G )- ln(/ / P ), where P is the reference pressure. 

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