Equation pressure-explicit

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... 

If we can write an equation of state for liquid mixtures, we can then calculate partial molar volumes directly by differentiation. For a pressure-explicit equation, the most convenient procedure is to use the exact relation... 

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). 

Since most equations of state are pressure-explicit, Eqs. (6) and (99) are often more convenient than Eqs. (5) and (98). With these equations, basing his calculations on van der Waals equation of state, Temkin (Tl) showed that gas-gas immiscibility may occur if the van der Waals constants a and b... 

The effect of tearing is to delete the tear variables and tear equations from the original set and to solve them iteratively external to the remaining set of equations and variables. In order for tearing to be a viable strategy, the number of tear variables required must be small and the tear equations must not be too difficult to solve. In this example, after tearing the iteration will involve only one equation, assuming the model equations are pressure explicit. 

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... 

It will be noted that the universal isotherm equation as written here has formal similarity to pressure explicit forms of Langmuir, Langmuir-Freundlich and LRC models. One key advantage of the universal form is that the heat of adsorption and the adsorption equilibrium are bound to be self-consistent. 

Most of the equations of state are pressure explicit, and Eq. (10.6) can be used for equilibrium calculations. As the integration is made from F to oo the EOS has to be valid in the density range from zero to the actual density. 

In the isothermal case, any appropriate PVT equation of state may be used to eliminate either P or V from this equation and thus permit integration. Since most of the useful equations of state are pressure-explicit, it is simpler to eliminate P. Take the example of one of the simplest of the non-ideal equations, that of van der Waals... 

The Vina/Expansion. Many equations of state have been proposed for gases, but the virial equation is the only one having a firm basis in theory (1,3). The pressure-explicit form of the virial expansion is... 

Equation 4.11 does not contain pressure explicitly. If Equation 4.11 is recast as ... 

This equation is explicit in pressure, but cubic in volume. Solution for V is usually done by an iterative scheme with a calculator. 

Alternative Property Formulations Direct application of Eqs. (4-159) and (4-161) can be made only to equations of state that are solvable for volume, that is, that are volume explicit. Most equations of state are in fact pressure explicit, and alternative equations are required. 

The tliree-term viriaf equation, Eq. (3.39) is tlie simplest pressure-explicit equation of... 

The calculation of the velocities and temperatures requires the solution of the continuity, momentum and energy equations. An explicit solution of the system of equations is not possible. A numerical solution has been communicated by Stephan [3.24] and later by several other authors see [3.26], As a result the pressure drop Ap = pi — p between the pressure pi at the inlet and the pressure p at any tube cross section is obtained. It can be approximated by an empirical correlation [3.30]... 

In this equation V is total volume, and Z = PV/RT is the compressibility factor computed from an equation of state, and V is the molar volume of the mixture. Most equations of state used in engineering are pressure explicit, that is, they arc in a form in which the pressure is explicit and the volume dependence is more complicated. One such example is the virial equation... 

Substituting the Maxwell relation (35/37)7 = (dP/3T y results in the second form of Equation 1.39, which would be used if the equation of state is pressure-explicit. 

The above discussion presumes the availability of a volume-explicit equation of state. For applications to gases at moderate to high pressures or densities or to vapors and liquids, realistic equations of state are not volume explicit but are instead pressure explicit. That is, Z is expressed as a function of T. v. and x or. equivalently, of T, p (molar density e o->), and Jt ... 

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. 

To model the measured transient foam displacements, equations 2 through 12 are rewritten in standard implicit-pressure, explicit-saturation (IMPES) finite difference form, with upstream weighting of the phase mobilities following standard reservoir simulation practice (10). Iteration of the nonlinear algebraic equations is by Newton s method. The three primitive unknowns are pressure, gas-phase saturation, and bubble density. Four boundary conditions are necessary because the differential mass balances are second order in pressure and first order in saturation and bubble concentration. The outlet pressure and the inlet superficial velocities of gas and liquid are fixed. No foam is injected, so Qh is set to zero in equation... 

The simplest useful polynomial equation of state is one that is cubic in molar volume, for such an expression is capable of yielding the ideal gas equation in the limit as V ->> oo, and of representing both liquid-and vapor-like volumes for sufficiently low temperatures. If we require that the equation be explicit in pressure, then algebraic arguments lead us to a five-parameter expression of the form (I)... 

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. 

The origin of the cubic equations of states goes back in history to the famous van der Waals equation (VdW EOS), which corrects the ideal gas law by an attraction term on pressure and a repulsion term on volume. Van der Waals equation is explicit in pressure and implicit (cubic) in volume. It contains two parameters, a and b, which can be expressed as function of Tc and Pc- We write it here again as ... 

Abstract A newly developed numerical simulator of two-phase flow using three-dimensional finite element method is presented in this paper. It is described that the fundamental simultaneous equations, the deduction to implicit pressure explicit saturation formulation and their finite element discretization method. Furthermore, its practical application to the numerical simulation project of predicting Horonobe natural gas product is also introduced. 

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-... 

Volumetric equations of state (3.5.1) typically take one of two forms, either a pressure-explicit form. 

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

A volumetric equation of state takes one of two forms. A volume-explicit equation has the form v = v T, P, x ), while a pressure-explicit equation has the form P = P(T, v, x ). Therefore, our expressions for conceptuals divide into two classes, depending on whether P ( 4.4.1) or v ( 4.4.2) is independent. In a particular problem, calculations are often simplified by using one set of independent variables rather than the other. To choose between the two sets, we follow the steps given in Figure 4.7. 

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. 

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