Pressure-Explicit Equations of State

Direct application of these results is possible only to equations of state explicit in volume. For pressure-explicit equations of state, alternative recipes are required. The basis is Eq. (4-82), which iu view of Eq. (4-157) may be written... 

When a pressure-explicit equation of state for a liquid mixture is substituted into Eq. (54), we obtain an expression of the form... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

In classical thermodynamics, there are many ways to express the criteria of a critical phase. (Reid and Beegle (11) have discussed the relationships between many of the various equations which can be used.) There have been three recent studies in which the critical points of multicomponent mixtures described by pressure-explicit equations of state have been calculated. (Peng and Robinson (1 2), Baker and Luks (13), Heidemann and Khalil (14)) In each study, a different statement of the critical criteria and... 

Equation (1.3-11) should not be used for densities greater then about half die critical value, aed Eq. (1.3-12) should not ha used for densities exceeding about three-quarters of the critical value. Note that Eq. (1.3-11) can be considered either a volume-explicit or a pressure-explicit equation of state, whereas Eq. (1.3-12) is pressure explicit. 

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only Tc, Pc, and o> were required for the calculation. The proposed equation successfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. 

In conclusion, a new pressure-explicit equation of state has been successfully developed as intended. It is suitable for representing PVT behavior of liquid and gas phases over a wide range of temperature and pressure for pure, nonpolar compounds. Furthermore, the parameters of the proposed equation are generalized in terms of the critical properties and the acentric factor. 

Thus, pressure-explicit equations of state for pure substance 1 (for the first integral) and for the gas mixture (the second integral) are required. Five different equations of state have been used in the analysis of this system (1) the five-constant Beattie-Bridgeman equation (2) the eight-constant Benedict-Webb-Rubin equation (3) the twelve-constant modified Martin-Hou equation and (4) and (5), the virial equation using two sets of virial coefficients. The first of these uses pure-substance second and third virial coefficients calculated from the Lennard-Jones 6-12 potential with interaction coefficients determined by the method of Ewald [ ]. The second set differs only in the second virial coefficients and interaction coefficient, these being found using the Kihara potential Solutions of the theoretical equa-... 

All pressure-explicit equations of state should satisfy this limit. 

The choice hinges on whether the independent variable (P or v) in our equation of state is appropriate for the conceptual whose value we need to compute. Recall from Chapter 3 that the fundamental equations for u and a have v as the independent variable, while those for h and g have P. Consequently, if we need to compute Ah or Aa, then we prefer to use a pressure-explicit equation of state, P(T, v, x ), but if we need to compute Ah or Ag, then we prefer to use a volume-explicit equation, v T, P, x ). Note that if we need As, /), or cp, then little advantage is offered by one kind of equation over the other both kinds involve about the same computational effort. These possibilities summarize the Ihs of the diagram in Figure 4.7. 

In this subsection we consider those situations in which our mixture is described by a pressure-explicit equation of state, P = P(T, v, x ). Our objective is still to relate excess properties to residual properties and to the equation of state, but with v as an independent variable, we would prefer to express those relations in terms of the isometric residual properties, rather than the isobaric ones. However, the development is not as simple as what we did in the previous section because now we have an inconsistency the equation of state and the isometric residual properties use (T, v, x ) as the independent variables, but the excess properties defined by (5.2.1) use (T, P, x)). 

From a pressure-explicit equation of state, the internal energy is the appropriate first-law conceptual to evaluate. Since isometric residual internal energies are identical to isobaric ones (4.2.24), we can immediately write (5.3.4) as... 

Conceptually, the simplest method for solving phase-equilibrium problems is the phi-phi method, but computationally it is usually more complicated than other methods. The conceptual simplifications arise in part because no decisions need to be made about reference states the reference state is the ideal gas and the choice of the ideal-gas reference is implicit in choosing to work with fugacity coefficients. Usually, the same pressure-explicit equation of state is used for all components in all phases, for this produces consistency in the results and helps in organizing the calculations. (The same calculations are to be done for all components in all phases, and therefore computer programs can be structured in obvious modular forms.) However, this need not be done different equations of states can be used for different phases. 

The residual functions as a function of volume can be derived in a similar way as needed for a pressure-explicit equation of state. Table 22 summarizes the different residual functions for pressure- and volume-explicit EOS. 

Derive an expression for the fugacity coefficient from a pressure-explicit equation of state. 

This equation can be used to calculate the fugacity coefficient with the help of a pressure-explicit equation of state. As RTln (p is by definition equivalent to the residual Gibbs energy (g — g )r.p, the above equation is equivalent to the corresponding equation in Table 2.2. 

For P, a pressure-explicit equation of state can be inserted, for example, the Peng-Robinson equation (Eq. (2.166)) ... 

Dl. Fugacity Coefficient for a Pressure-Explicit Equation of State... 

For a pressure-explicit equation of state, the term (3v/3T)p is not very useful. Setting up the total differential of the pressure... 

The above equations are not very practical to use How can we keep f.ii constant when a system is undergoing a change An alternative procedure may be more convenient for the calculation of the critical point of mixtures with three or more components. Since we often use a pressure-explicit equation of state, P — P V, T, A), the following derivations are aimed towards the use of such equations. The first equation for criticality can be written as... 

Although modern empirical equations of state are usually formulated in terms of the reduced Helmholtz energy, the most common form used in the 20 century for multi-parameter equations of state in technical applications was the pressure-explicit form. The discussion that follows highlights several practical forms of multi-parameter pressure-explicit equations of state and indicates applications for each. The virial equation of state is not detailed in this work, but can be found in Chapter 3 of this book. 

The simplest form of the pressure explicit equation of state (with independent variables of density and temperature) are those based on a cubic expression of the fluid volume such as the Peng-Robinson and Soave-Redlich-Kwong and these equations are discussed in Chapter 4. When temperature and pressure inputs are available, the cubic equation can be solved non-iteratively for density. Thus, the calculation speed of cubic equations of state is rapid when compared to other methods explained that are provided below, and the use of these equations is quite popular in many industrial applications. Unfortunately, the advantage of speed of calculation is offset by the disadvantage of higher uncertainties. 

The values of entropy S, enthalpy J/m, internal energy C/m, and heat capacity (Cpm, Cj/.m. or Csat) at various state points are calculated with the pressure explicit equation of state and an ancillary equation that represents the ideal gas heat capacity. For a pure fluid, the equations that represent the vapour pressure and melting curve are used to identify the temperatures of the phase changes from liquid to vapour and solid to liquid, respectively. Properties are evaluated through... 

First, if we have a pressure explicit equation of state, P =f T, v), we can analytically assess the partial derivative [dP/dT) . A volume explicit equation of state, v = f T, P), allows us to obtain (dvl8T)p. Second, we may use the thermal expansion coefficient, )8, and the isothermal compressibility, k as an alternative to equations of state to assess dP/dT) and dv/dT)p. Both alternatives are indicated in Figure 5.3. These properties were defined in Section 4.3 as ... 

It is instructive to compare Equation (5.31) to Equation (5.33). Equation (5.31) was developed using T and v as the independent properties, while Equation (5.33) used T and P. We see that in the former case we get a partial derivative in P. Therefore, this form is amenable to a pressure-explicit equation of state. Conversely, Equation (5.33) gives a partial derivative for v and is more amenable to a volume-explicit equation of state. This observation holds generally that is, T and v are convenient independent properties when we have a pressure-explicit equation of state, while T and P are convenient for a volume-explicit equation. This rule of thumb is reinforced in Example 5.4, where the choices of independent properties T,v) and (T,P) are compared for a calculation using the pressure-explicit Redlich—Kwong equation of state. 

If we integrate Equation (E5.4L), we get a result identical to Equation (E5.4F), so the rest of the problem is equivalent to the part above where we used T and v as independent properties. We come up with the same result applying the thermodynamic web to each path the form h = h T, v) yields an equivalent result to h = h T, P). However, the first choice of independent properties made the math easier. This result is not surprising in light of the discussion after Equation (5.33) that is, T and v are the convenient independent properties when we have a pressure-explicit equation of state. 

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