Non-cubic equations of state

The non-cubic equations of state are characterized by the use of a repulsive term that is based on the Camahan-Starling or on the HCB expressions already reported in Table 3. The attractive part is generally based on that derived from the perturbed hard chain theory (PHCT) [49], or from the statistical associating fluid theory SAFT [50, 51]. These approaches were the precursors of many theoretical attractive terms and consequently of different equations of... 

The phase behaviour of binary polymer - supercritical fluid systems can be modelled with an equation of state model. In general, non-cubic equations of state are used, mainly from the PHCT and SAFT families. Lattice-fluid equations of state are also commonly used for the... 

All of the applications of cubic and non-cubic equations of state presented so far refer to equilibrium thermodynamics. Cubic equations of state have been also used for the calculation of transport properties of pure components and mixtures, including viscosity, diffusion coefficient and thermal conductivity. Some recent viscosity calculations will be presented here. 

Following the principle of corresponding states, the equation of state should be able to provide reasonably good results for nonpolar substances with two, and still better, with three adjustable parameters. Indeed, the typical cubic EoS contain two or three parameters. For very accurate results, however, non-cubic equations of state with a large number of parameters are used, such as the Benedict-Webb-Rubin one. 

One of the oldest, but still used, non-cubic equations of state is that proposed by Benedict, Webb and Rubin (1940 1942 1951). It contains eight parameters that are determined from experimental data. Parameter values for several compounds are given by Reid et al. 

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. 

However, it was Soave s modification [30] of the temperature dependence of the a parameter, which resulted in accurate vapour pressure predictions (especially above 1 bar) for light hydrocarbons, which led to cubic equations of state becoming important tools for the prediction of vapour-liquid equilibria at moderate and high pressures for non-polar fluids. 

Several cubic equations of state such as Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson have been used to calculate vapor liquid equilibria of fatty acid esters in supercritical fluids. Comparisons are made with experimental data on n-butanol, n-octane, methyl oleate, and methyl linoleate in carbon dioxide and methyl oleate in ethane. Two cubic equations of state with a non quadratic mixing rule were successful in modeling the experimental data. 

The second example shows the solubility of methane in water. In general, cubic equations of state do quite poorly with aqueous solutions, since most are tuned to non-polar compositions such as hydrocarbon mixtures. The HCToolkit allows one to quickly compare the abilities of... 

A cubic equation of state model cannot predict all the properties with equal accuracy. Usually there is a non-negligible error in estimating liquid volume, which produces also errors in computing enthalpies, frequently underestimated. More accurate methods for enthalpy and entropy are based on corresponding states correlation (Lee-Kesler). 

This equation improves the liquid density prediction, but still cannol describe volumetric behavior around the critical point because of fundamental reason that will be discussed later. There are thousands of cubic equations of states, and many noncubic equations. The non cubic equations such as the Benedict-Webb-Rubin equation (1942) ant its modification by Starling (1973) have a large number of constants they describe accurately the volumetric behavior of pure substances But for hydrocarbon mixtures and crude oils, because of mixing rub complexities, they may not be suitable (Katz and Firoozabadi, 1978) Cubic equations with more than two constants also may not improv the volumetric behavior prediction of complex reservoir fluids. In fact most of the cubic equations have the same accuracy for phase-behavio prediction of complex hydrocarbon systems the simpler equation often do better. Therefore, the discussion will be limited to the Peng... 

In all of the equations of state discussed so far, the repulsive term has remained unchanged and equal to that proposed by van der Waals and given by the first term on the r.h.s. of eq 4.1. Thanks to the development of molecular simulation methods starting in the 1960s, we are able today to quantify the effects of different interactions on the thermodynamic properties of a fluid. A simple comparison of the van der Waals repulsive term against molecular simulation data for hard spheres reveals the inaccuracy of the former. A more accurate simple repulsive term was proposed by Elliott et a/. and was incorporated into a cubic equation of state that also accounts for the shape (non-sphericity) of the molecules. The Elliott-Suresh-Donohue equation of state is given by ... 

The modern cubic equations of state provide reliable predictions for pure-component thermodynamic properties at conditions where the substance is a gas, liquid or supercritical. Walas and Valderrama provided a thorough evaluation and recommendations on the use of cubic equation of state for primary and derivative properties. Vapour pressures for non-polar and slightly polar fluids can be calculated precisely from any of the modem cubic equations of state presented above (Soave-Redlich-Kwong, Peng-Robinson or Patel-Teja). The use of a complex funetion for a (such as those proposed by Twu and co-workers ) results in a significant improvement in uncertainty of the predicted values. For associating fluids (such as water and alcohols), a higher-order equation of state with explicit account for association, such as either the Elliott-Suresh-Donohue or CPA equations of state, are preferred. For saturated liquid volumes, a three-parameter cubic equation of state (such as Patel-Teja) should be used, whereas for saturated vapour volumes any modern cubic equation of state can be used. 

The description of hydrocarbon mixture VLB at low and high pressure is of major importance to the oil industry. For such mixtures, any of the modern cubic equations of state (such as Redlich-Kwong, Peng-Robinson or Patel-Teja) provide precise predictions when used with a temperature-independent binary interaction parameter of relatively small value (in most cases in between — 0.1 and 0.1). For the case of non-polar hydrocarbon mixtures of similar size, even kij = Q.Q results in excellent prediction of VLB. 

Although originally developed to model thermodynamic properties of volatile fluids, cubic equations of state have also been applied to polymer mixtures. Sako, Wu and Prausnitz proposed a cubic equation of state for polymers based on the Soave-Redlich-Kwong equation of state with the Prigogine s parameter c to account for the non-sphericity of chain molecules. The Sako-Wu-Prausnitz... 

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. 

One drawback of the MF1V2 model is the inability of UNIFAC to predict (vapour + liquid) equilibria (VLB) and (liquid + liquid) equilibria (LLE) conditions using the same set of group-interaction parameters. In general, cubic equations of state do not provide precise predictions of the phase equilibria when the mixture is asymmetric in size that is attributed to the large differences in the pure-component co-volumes. The Carnahan -Starling equation for hard spheres is a more realistic model for the repulsive contribution than that proposed by van der Waals. Mansoori et al. proposed an equation for mixtures of hard spheres that has been found to correlate the phase behaviour of non-polar mixtures with large molecular size differences. 

For the simultaneous solution of phase and chemical equilibria special emphasis has to be given to the development of algorithms that solve the complex non-linear problem. Frequently, traditional cubic equations of state with classical mixing rules have been applied to highly non-ideal mixtures this invalidates the precise numerical solutions obtained. 

Trebble, M.A. Bishnoi, PR. (1986). Accuracy and consistency comparisons of ten cubic equations of state for polar and non- lar compounds. Fluid Phase Equil, 29,465-474. 

Cubic equations of state have become the main tool for high pressure VLE calculations. They combine simplicity with accuracy comparable to -or better than - that of other methods, including non-cubic EoS. For a comparison of the EoS approach with the Chao-Seader method, see Maddox and Erbar (1981). 

At 10 MPa and 35 °C, C02 has a density of approximately 700kg/m3. Under these conditions, a cubic meter of sandstone with 10% porosity contains approximately 70 kg of C02 if the pore space is completely filled by C02. However, saturation of C02 is not complete, and some brine remains in the invaded pore spaces (Saripalli McGrail 2002 Pruess et al. 2003). In addition, non-uniform flow of C02 bypasses parts of the aquifer entirely. Darcy-flow based analytical and numerical solutions are used to evaluate some of these effects by simulating the advance of the C02 front over time-scales of decades to hundreds of years and over lateral distances of tens to hundreds of kilometers. To account for the extreme changes in density and viscosity of C02 with pressure and temperature, these models must incorporate experimentally constrained equations of state (Adams Bachu 2002). 

Experimental results are presented for high pressure phase equilibria in the binary systems carbon dioxide - acetone and carbon dioxide - ethanol and the ternary system carbon dioxide - acetone - water at 313 and 333 K and pressures between 20 and 150 bar. A high pressure optical cell with external recirculation and sampling of all phases was used for the experimental measurements. The ternary system exhibits an extensive three-phase equilibrium region with an upper and lower critical solution pressure at both temperatures. A modified cubic equation of a state with a non-quadratic mixing rule was successfully used to model the experimental data. The phase equilibrium behavior of the system is favorable for extraction of acetone from dilute aqueous solutions using supercritical carbon dioxide. 

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