Cubic state equation

Cubic state equations are third degree equations with regards to volume. The most well-known of these equations is the Van der Waals... 

Coefficients B2, B, etc. are called the second, third, etc. coefficient of the virial. We can calculate the first coefficients of the virial using the cubic state equations as we did for the Dietrerici equation, but it does not provide any additional information. For example, for the Van der Waals equation we obtain, respectively ... 

Some authors inspired by the fact that cubic state equations, such as the van der Waals equation, account for the change in state from vapor-liquid and the critical point, and they have proposed to describe liquid and gas phases using the same model. These methods are called state equation methods. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equiUbrium, or the three roots (vapor, Hquid, sohd) characteristic of the triple point. 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

In this model (Table 3), substrate, A, is transformed to product, B, by an enzyme, E. The supply of A is large, ensuring far-from-equilibrium conditions. An intermediate, X, is produced autocatalytically, and degraded by the enzyme. (This feature of the model makes it unrealistic, as few autocatalytic processes arise this way.) The steady-state equation for X is cubic, and has three roots, or solutions for certain values of the parameters. One of the solutions is unstable a real system cannot maintain a steady-state concentration, [X]ss, with a value corresponding to this solution. Therefore, before [X]ss approaches such a value too closely, it jumps to a different value, corresponding to one of the stable solutions. This behavior leads, to hysteresis, as shown in Fig. 1. 

Tochigi, K., Kurita, S., and Matsumoto, T, Prediction of PVT and VLE in polymer solutions using a cubic-perturbed equation of state, Fluid Phase Equilibria, 158-160, 313, 1999. 

Up to the present time, many versions of state equations [8, 9] have been suggested. The majority of them aim at the description of behavior of hydrocarbon systems. These equations can be conventionally divided into two basic types the multi-coefficient- and cubic equations of state. 

Kiselev, S.B. (1998) Cubic crossover equation of state, Fluid Phase Equilibria 147, 7-23. 

Aris et al. have primarily analyzed whether the steady-state multiplicity features in a CSTR arising from a cubic rate law also can arise for a series of successive bimolecular reactions [26]. Aris et al. have showed that the steady-state equations for a CSTR with bimolecular reactions scheme reduces to that with a cubic reaction scheme when two parameters e(=k,Cg/k j) and K(=kjC /k j) arising in system equations for the bimolecular reactions tend to zero. Aris et al. have shown that the general multiplicity feature of the CSTR for bimolecular reactions is similar to that of the molecular reactions only at smaller value of e and K. The behavior is considerably different at larger values of e and K. Chidambaram has evaluated the effect of these two parameters (e and K) on the periodic operation of an isothermal plug flow reactor [18]. 

In Eq. (43) the energy halfway between the 6sl8s 5o and states has been taken as zero. A px = (5d7d Po, / = 0) - (6sl8s S, 7 = 0) denotes the perturber-triplet separation (cf. Figure 34). The present example requires the solution of a cubic secular equation, in contrast to the unperturbed case [cf. Eq. (24)]. Although general analytic expressions can be derived... 

The representation of pVTx properties of mixtures by using the cubic EOS is still a subject of active research. Kiselev (1998), Kiselev and Friend (1999), and Kiselev and Ely (2003) developed a cubic crossover equation of state for fluids and fluid mixtures, which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original classical cubic equation of state far away from the critical point. Anderko (2000) and Wei and Sadus (2000) reported comprehensive review of the cubic and generalized van der Waals equations of state and their applicability for modeling of the properties of multicomponent mixtures. 

Fig. 11.1. Typical hysteresis loop for a one-variable system with a cubic kinetic equation plot of concentration c vs. influx coefficient. Solid lines, stable stationary states (nodes) broken line, unstable stationary state. For a discussion of lines A and B and numbers, see the text...

These results were used in sections 4.2.3 and 4.2.4. Note, however, that these relatively simple results are valid only for two-state units. The case of three-state units already involves a cubic secular equation, instead of (4.3.24), which is much more difficult to solve for the largest eigenvalue. 

Besides cubic equations, the number of state equations is constantly increasing of the different general laws proposed, we will only mention the most widespread. 

Another form of the state equation with two parameters, but which is no longer a cubic equation, proposed by Dieterici, is written ... 

We obtain the same type of result every time the state equation is a cubic equation, whose two parameters can be calculated from the two definition relations of the critical point, namely the cancehng out of both derivatives... 

The Yoimg s modulus obviously depends on the temperature, but these data can sometimes be replaced by a state equation in the form F(F,1,T) = 0, such as that given for rubber, which links the temperature T, the length / of a cylinder of mbber with section s, to the force of traction F exerted upon it in the elastic domain by way of two constants, B and its cubic expansion coefficient which is of the form ... 

For polyatomics, ordinarily only the last two tenns of equation (C3.5.6), the cubic and quartic anlrannonic tenns, need be considered [34]. In a cubic anlrannonic process, excited vibration D relaxes by interacting with two other states, say airother vibration cr aird one phonon (or alternatively two phonons). In the quartic process, D relaxes by interacting with tlrree other states, say two vibrations aird one phonon. The total rate constairt for energy loss from Q for cubic... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

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). 

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... 

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... 

Cubic equations, although simple and able to provide semiquantitative descriptions of real fluid behavior, are not generally useful for accurate representation of volumetric data over wide ranges of T and P. For such appHcations, more comprehensive expressions with large numbers of adjustable parameters are needed. 7h.e simplest of these are the extended virial equations, exemplified by the eight-constant Benedict-Webb-Rubin (BWR) equation of state (13) ... 

Mixing mles for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 9 for mixtures are usually given by the following, where, as for the second virial coefficient, = 0-. 

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. 

Reduced Equations of State. A simple modification to the cubic van der Waals equation, developed in 1946 (72), uses a term called the ideal or pseudocritical volume, to avoid the uncertainty in the measurement of volume at the critical point. 

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. 

---