Cubic Equations-of-State

Equations of state that are cubic in volume are often employed, since they, at least qualitatively, reproduce the dependence of the compressibility factor on p and T. Four commonly used cubic equations of state are the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson. All four can be expressed in a reduced form that eliminates the constants a and b. However, the reduced equations for the last two still include the acentric factor u that is specific for the substance. In writing the reduced equations, coefficients can be combined to simplify the expression. For example, the reduced form of the Redlich-Kwong equation is 

The four cubic equations of state are summarized in Table A3.1. 

A broad range of cubic equation of state models (EOS) are successfully used today. The EOS range from the standard Soave-Redlich-Kwong and Peng-Robinson, which is widely used in the hydrocarbon processing and related industries (oil gas and petrochemicals), to a new class of models that extend the range of applications to chemicals. New models are continually being developed and are too numerous to cite. 

As shown in Problems F.3, F.4, and F.5, it is common to rewrite all the cubic EOSs in z form, and then to redefine the constants, as shown below. For the SRK EOS the forms [1] are 

There is only little interest in computing pme species 

4 filPyt FOR INDIVIDUAL SPECIES IN MIXTURES, BASED ON PRESSURE-EXPLICIT EOSs 

In principle, we should be able to begin with Eq. F.6, take the partial derivatives with respect to yi, and use the method of tangent intercepts to find fi/ytP. In practice, this is very difficult, because it is very difficult to write out the derivatives of z from a pressure-explicit EOS. Instead, we begin with Eq. 7.15, rewritten as 

The dP/drii) is a strange-looking derivative. Remember that we have taken nj-v = F as one of the independent variables, so this is the increase in P when we add 1 mol of i at constant V, T, and Uj. 

The above equations describe a three-dimensional T-P-v surface which is a representation of all the experimental or statistical mechanical information concerning the EoS of the system. It should also be noted that it is observed experimentally that 

The quantity Pv/RT is called compressibility factor. The PvT or volumetric behavior of a system depends on the intermolecular forces. Sizes, shapes and structures of molecules determine the forces between them (Tassios, 1993). 

Volumetric data are needed to calculate thermodynamic properties (enthalpy, entropy). They are also used for the metering of fluids, the sizing of vessels and in natural gas and oil reservoir calculations. 

In the above equations, co is the acentric factor, and Tc, Pc are the critical temperature and pressure respectively. These quantities are readily available for most components. 

The Trebb/e-Bishnoi EoS is a cubic equation that may utilize up to four binary interaction parameters, k=[ka, kb, kc, k i]r. This equation with its quadratic combining rules is presented next (Trebble and Bishnoi, 1987 1988). 

Fluid-phase equilibria v or-liquid, gas-liquid, and liquid-liquid, represent a most important application of chemical engineering thermodynamics in the petroleum and chemical industries. Quantitative description and/or prediction of such equilibria is essential in the design of distillation, extraction, and gas absorption units. 

Cubic Equations of State (EoS) are progressively becoming the main tool for phase equilibria calculations and, even though they are - so far -successful for nonpolar/weakly polar systems only, it will not be long before they can handle polar systems as well. The Soave-Redlich-Kwong (SRK, Soave, 1972) and the PR (Peng and Robinson, 1976) EoS - modifications of the first EoS proposed, that of van der Waals (vdW) - are the most commonly used among them. 

For the successful description of phase equilibria EoS must meet the following requirements  

Provide reliable prediction of vapor pressures of pure compounds over a wide temperature range  

Reflect a three-parameter corresponding states principle, i.e. predict successfully vapor pressures for nonpolar/weakly polar compounds where Jp, and k values are available  

For molecules more complex than the noble gases, a third parameter is introduced (Pitzer et al. 1955 Pitzer 1955) to form a three-parameter theory of corresponding states. This is the acentric factor a, which is a function of the acentricity or noncentral nature of the intermolecular forces. It is completely empirical. For simple fluids, it was observed that at a reduced temperature of 0.7, the saturation pressure (i.e., the vapor pressure of the liquid-gas equilibrium) divided by the critical pressure is very close to 1/10, or 

It is important to realize that however valuable PVT information is (and it is extremely valuable in the chemical industries), it is not thermodynamically complete information. You cannot calculate a heat capacity from PVT data, and this means that you cannot calculate the temperature variation of the Gibbs energy, enthalpy, or entropy. You can calculate the pressure variation of these functions, but you need to start with a baseline showing the variation with T at some pressure. There is a second important class of EoS, sometimes called thermal EoS, which do provide complete information, and these are equations based on G T, P) or A T, V). We will look first at PVT equations of state. 

Apart from the ideal gas equation, equations relating P, V, and T are usually cubic equations. 

The ideal gas law does not work well for real gases at even moderate pressures. Two of its main problems were recognized by van der Waals, and appropriate corrections were incorporated into his famous equation of state of 1873  

The concentration of the outermost zone of molecules is proportional to the gas density 1/V, as is the concentration of molecules in the next inward zone (presuming concentrations in both zones are the same - a potential source of error). The inward attraction is proportional to the number of molecules in both zones or to l/V. Introducing the proportionality constant a, this inward pull should modify the outward force or pressure by —a/V.  

If an equation of state is to represent the PVT behavior of both liquids and vapors, it must encompass a wide range of temperatures and pressures. Yet it must not be so complex as to present excessive numerical or analytical difficultiesin application. Polynomial equations tliat are cubic in molar voliune offer a compromise between generality and simplicity tliat is suitable to many purposes. Cubic equations are in fact tlie simplest equations capable of representing botli liquid and vapor beliavior. 

The first practical cubic equation of state was proposed by J. D. van der Waals in 1873  

a and b are positive constants when tliey are zero, tlie ideal-gas equationis recovered. 

Experimental isothenns do not exliibit tliis smooth transition from saturated liquid to saturated vapor rather, tliey contain a horizontal segment within tlie two-phase region where saturated liquid and saturated vapor coexist in varying proportions at the saturation or vapor 

Cubic equations of state have three volume roots, of which two may be complex. Physically meaningful values of V are always real, positive, and greater than constant b. For an isotherm at T Tc, reference to Fig. 3.12 shows that solutionfor V at ary positive value of P yields only one such root. For the critical isotherm (T = Tc), this is also true, except at the critical pressure, where there are three roots, all equal to Vc- For isotherms at T Tc, the equation may exhibit one or three real roots, depending on the pressure. Although these roots are real and positive, they are not physically stable states for the portion of an isothenii lying between saturated hquid and saturated vapor (under the dome ). Only the roots for P = P namely V (liq) and V (vap), are stable states, coimectedby the horizontal portion of the true isotherm. For other pressures (as indicated by the horizontal lines shown on Fig. 3.12 above 

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The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

AppHcation of equation 226 requires the availabiHty of a single equation of state suitable for both vapor and Hquid mixtures. Cubic equations of state are widely used for VLE calculations. 

Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate Redlich-Kwong-type equations such as the Soave and Peng-Robinson equations are often used. Both require only T, and (0 as inputs. For organic compounds, the Lee-Erbar-EdmisteF" equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. 

Cubic Equations of State The simplest expressions that can (in... 

The modern development of cubic equations of state started in 1949 with publication of the Redlich/Kwong equation (Redhch and Kwong, Chem. Rev., 44, pp. 233-244 [1949]) ... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

The above constrained parameter estimation problem becomes much more challenging if the location where the constraint must be satisfied, (xo,yo), is not known a priori. This situation arises naturally in the estimation of binary interaction parameters in cubic equations of state (see Chapter 14). Furthermore, the above development can be readily extended to several constraints by introducing an equal number of Lagrange multipliers. 

Based on the above, we can develop an "adaptive" Gauss-Newton method for parameter estimation with equality constraints whereby the set of active constraints (which are all equalities) is updated at each iteration. An example is provided in Chapter 14 where we examine the estimation of binary interactions parameters in cubic equations of state subject to predicting the correct phase behavior (i.e., avoiding erroneous two-phase split predictions under certain conditions). 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

Any cubic equation of state can give an expression for the fugacity of species i in a gaseous or in liquid mixture. For example, the expression for the... 

It should be kept in mind that an objective function which does not require any phase equilibrium calculations during each minimization step is the basis for a robust and efficient estimation method. The development of implicit objective functions is based on the phase equilibrium criteria (Englezos et al. 1990a). Finally, it should be noted that one important underlying assumption in applying ML estimation is that the model is capable of representing the data without any systematic deviation. Cubic equations of state compute equilibrium properties of fluid mixtures with a variable degree of success and hence the ML method should be used with caution. 

It is well known that cubic equations of state have inherent limitations in describing accurately the fluid phase behavior. Thus our objective is often restricted to the determination of a set of interaction parameters that will yield an "acceptable fit" of the binary VLE data. The following implicit least squares objective function is suitable for this purpose... 

It is well known that cubic equations of state may predict erroneous binary vapor liquid equilibria when using interaction parameter estimates from an unconstrained regression of binary VLE data (Schwartzentruber et al.. 1987 Englezos et al. 1989). In other words, the liquid phase stability criterion is violated. Modell and Reid (1983) discuss extensively the phase stability criteria. A general method to alleviate the problem is to perform the least squares estimation subject to satisfying the liquid phase stability criterion. In other... 

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