Compressibility equation

Comparing the two results and substituting the relation of the mean square number fluctuations to isothennal compressibility, equation (A2.2.128) one has... 

The first tenn in the compressibility equation is the ideal gas temi and the second temi, the integral of g r)- ... 

The compressibility equation can also be written in tenns of the direct correlation fiinction. Taking the Fourier transfomi of the Omstein-Zemike equation... 

The CS equation for the pressure is found to be the weighted mean of the pressure calculated from the virial and compressibility equations ... 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

This equation is analogous to the compressibility equation for fluids and diverges with the same exponent y as the critical temperaUire is approached from above ... 

Other volume-explicit equations of state are sometimes required, such as the compressibility equation V = zRT/P or the truncated virial equation V= (1 -i- B P)RT/P. The quantities z a.ndB are not constants, so some land of averaging will be required. More accurate equations of state are even more difficult to use but are not often justified for kinetic work. 

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. 

Fig. 10(b)). One of the reasons for the differences between both theories is a different form of a hard sphere part of the free energy functional. Segura et al. have used the expression resulting from the Carnahan-Starhng equation of state, whereas the Meister-Kroll-Groot approach requires the application of the PY compressibility equation of state, which produces higher oscillations. 

The former of these two relations is the compressibility equation for the fluid in the matrix. 

Finally, in this part of the work we would like to discuss to some extent practical tools to obtain thermodynamic properties of adsorbed fluids. We have mentioned above that the compressibility equation is the only simple recipe, for the moment, to obtain the thermodynamics of partly quenched simple fluids. The reason is that the virial equation is difficult to implement it has not been tested for partly quenched systems. Nevertheless, for the sake of completeness, we present the virial equation in the form [22,25]... 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

The adiabatic temperature increase for an ideal gas is computed from the thermodynamic adiabatic compression equation ... 

Equation (7.27) will hold for incompressible fluids and for compressible fluids with small values of AP. If the pressure gradient across the bed is large and the fluid is compressible, equation (7.27) takes the form... 

Researchers have proposed hundreds of equations of state for real gases. We will consider first the compressibility equation of state. This equation of state is the one used most commonly in the petroleum industry. This equation does have some limitations therefore, we will examine later several other equations of state which are used to a lesser extent by petroleum engineers. 

The accuracy of the compressibility equation bf state is not any better than the accuracy of the values of the z-factors used in the calculations. The accuracy of Figures 3-7 and 3-8 was tested with data from 634 natural gas samples of known composition.8 Experimentally determined z-factors of these gases were compared with z-factors obtained from the charts using Kay s rules for calculating the pseudocritical properties and Figure 3-10 for properties of heptanes plus. 

One of the limitations in the use of the compressibility equation of state to describe the behavior of gases is that the compressibility factor is not constant. Therefore, mathematical manipulations cannot be made directly but must be accomplished through graphical or numerical techniques. Most of the other commonly used equations of state were devised so that the coefficients which correct the ideal gas law for nonideality may be assumed constant. This permits the equations to be used in mathematical calculations involving differentiation or integration. 

The volume of n moles of a gas at reservoir conditions may be obtained with the compressibility equation of state. 

This coefficient normally is referred to simply as compressibility or gas compressibility. You must understand that the term compressibility is used to designate the coefficient of isothermal compressibility whereas, the term compressibility factor refers to z-factor, the coefficient in the compressibility equation of state. Although both are related to the effect of pressure on the volume of a gas, the two are distinctly not equivalent. 

The compressibility equation is the most commonly used equation of state in the petroleum industry. We will combine this equation with the equation which defines the coefficient of isothermal compressibility. Since z-factor changes as pressure changes, it must be considered to be a variable. 

The volume of the gas in the reservoir may be calculated using the compressibility equation of state. This calculation is based on 1 lb mole of gas using Equation 3-39. The composition of the gas in the reservoir calculated by the recombination method can be used to compute the pseudocritical properties so that the compressibility factor may be obtained in the same manner as illustrated in Examples 3-10 and 3-12. 

The complete graph of compressibility as a function of reservoir pressure is given in Figure 8-7. There is a discontinuity at the bubble point. The evolution of the first bubble of gas causes a large shift in the value of compressibility. Equation 8-7 applies at pressures above the bubble point, and Equation 8-24 applies at pressures below the bubble point. 

The Compressibility Equation of State —The Law of Corresponding States— The Compressibility Equation of State for Gas Mixtures. 

---