Critical pressure compression factor

In above equations Tc, Pa >, MW, ppp, Zpg, vpp are critical Temperature, critical pressure, acentric factor, molecular weight, Uquid density calculated by PR EOS, compressibility factor calculated by PR EOS, and liquid volume calculated by PR EOS respectively. Now, the error of calculating liquid density by PR EOS can be obtained as follows ... 

The critical pressure, critical molar volume, and critical temperature are the values of the pressure, molar volume, and thermodynamic temperature at which the densities of coexisting liquid and gaseous phases just become identical. At this critical point, the critical compressibility factor, Z, is ... 

For pure organic vapors, the Lydersen et al. corresponding states method is the most accurate technique for predicting compressibility factors and, hence, vapor densities. Critical temperature, critical pressure, and critical compressibility factor defined by Eq. (2-21) are used as input parameters. Figure 2-37 is used to predict the compressibihty factor at = 0.27, and the result is corrected to the Z of the desired fluid using Eq. (2-83). 

An analytical method for the prediction of compressed liquid densities was proposed by Thomson et al. " The method requires the saturated liquid density at the temperature of interest, the critical temperature, the critical pressure, an acentric factor (preferably the one optimized for vapor pressure data), and the vapor pressure at the temperature of interest. All properties not known experimentally maybe estimated. Errors range from about 1 percent for hydrocarbons to 2 percent for nonhydrocarbons. 

W = gas rate, Ib/hr Z = gas compressibility factor Pc = Pcriv critical pressure, psia... 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

These equations will be sufficiently accurate up to moderate pressures, in circumstances where the value is not critical. If greater accuracy is needed, the simplest method is to modify equation 8.3 by including the compressibility factor z ... 

Fig. 3.2 Plot of the compressibility factor as a function of reduced pressure and parameterized by the reduced temperature. The reduced values are normalized by their corresponding values at the critical point. This plot is adapted from one originally prepared by Nelson and Obert [295,332],...

Table 2.4 displays critical constants Tc, Pc, Vc and critical compressibility factor Zc for a number of common gases. (Accurate determination of the critical point is experimentally challenging, and quoted values are generally uncertain in the final decimal.) One can see from the table that many common gases (including N2, 02, and CH4) are actually supercritical fluids ( permanent gases ) under ambient temperature conditions, incapable of liquefaction by any applied pressure whatsoever. (Aspects of cryogenic gas-liquefaction techniques are discussed in Section 3.6.3.)... 

The above equations were obtained from twenty non-polar gases including inert gases, hydrocarbons and carbon dioxide (but not hydrogen and helium). Hence, possible errors can be as large as 20%. The maximum pressure corresponds to a reduced density of 2.8. In the above equations, Zc represents the critical compressibility factor. The value of gamma is calculated using Eqn. (3.4-26). 

The critical pressure P = PoAso/q2 is more difficult to evaluate. In the earlier literature there is a large spread of values [17]. The recent MC simulations of Orkoulas and Panagiotopoulos [52] yield P c = 8 x 10-5 near the lower limit of earlier estimates, along with a critical compressibility factor of Zc — PJ(pcTc) = 0.024 which is one order of magnitude lower than observed for nonionic fluids (e.g., Zc = 3/8 = 0.375 for the van der Waals fluid). 

This equation is useful for gases above the critical point. Only reduced pressure, /J, and reduced temperature, T, are needed. In the form represented by equation 53, iteration quickly gives accurate values for the compressibility factor, Z. However, this two-parameter equation only gives accurate values for simple and nonpolar fluids. Unless the Redlich-Kwong equation (eq. 53) is explicidy solved for pressure in nonreduced variables, it does not give accurate liquid volumes. 

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

Equation (6-17) can be easily adapted for the critical state, if p is substituted with the critical pressure, pc, obtained with Eq. (6-12) and if W/ in the exponent is substituted with w, e. This takes into account the absence of a pure translational energy contribution in the critical state. On the contrary, an additional negative term, the critical compression factor Zc = -w,lw, is introduced in the exponent, taking into account the decrease in diffusion velocity caused by attraction between the particles. As a result the following equation gives the coefficient of self-diffusion in the critical state ... 

Critical properties of gaseous compounds are useful in determining the P-V-T Pressure-Volume-Temperature) properties at nonideal conditions. The compressibility factor Z is defined by the following relationship ... 

Table 2 illustrates the weighting method that can be used. The required input of parameters to perform computations are the number of components, the mole fraction of ith component, the critical pressure of ith component in units of atm, the critical temperature of the ith component (°K), the molecular weight of the ith component, the pressure of the mixture (atm), the temperature of the mixture (°F), and the compressibility factor of the mixture, Zm. 

For use of the generalized Redlich/Kwong equation one needs only the critical temperature and critical pressure of the gas. This is the basis for the two-parameter theorem of corresponding states All gases, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor, and all deviate from ideal-gas behavior to about the same degree. 

To compute u at a given reservoir temperature and pressure the value of the compressibility factor under these conditions must be known. If an experimental value of Z is not available and it is necessary to estimate a value for a reservoir gas one has recourse to the methods described in Chapter 2. If the composition of the gas is known a pseudo-reduced temperature and pressure may be calculated and the compressibility factor obtained from Figure 10. If, on the other hand, the composition is not known but a value of the gas gravity is available, it is still possible to evaluate the pseudo-critical temperature and pressure from Figures 11 and 12. With these pseudo-critieals and the values of the reservoir temperatiue and pressure, the pseudo-reduced temperature and pressure can be computed and the compressibility factor obtained from Figure 10 as before. 

Fluid Molecular weight, (gm/mol) Density of Liquid (gm/mol) Temp, of liquid ( K) Dipole moment (debyes) Critical Temp. Factor, (Tytc) C>K/ C) Critical Pressure (F e) (bar) (atm) (psi) Critical Volume, (cmVmol) Critical Compress. (Z)c ... 

Explain in your own words and without the use of jargon (a) the three ways of obtaining values of physical properties (b) why some fluids are referred to as incompressible (c) the liquid volume additivity assumption and the species for which it is most likely to be valid (d) the term equation of state (e) what it means to assume ideal gas behavior (f) what it means to say that the specific volume of an ideal gas at standard temperature and pressure is 22.4 L/mol (g) the meaning of partial pressure (h) why volume fraction and mole fraction for ideal gases are identical (i) what the compressibility factor, z, represents, and what its value indicates about the validity of the ideal gas equation of state (j) why certain equations of state are referred to as cubic and (k) the physical meaning of critical temperature and pressure (explain them in terms of what happens when a vapor either below or above its critical temperature is compressed). 

The basis for estimating z in this manner is the empirical law of corresponding states, which holds that the values of certain physical properties of a gas—such as the compressibility factor— depend to great extent on the proximity of the gas to its critical state. The reduced temperature and pressure provide a measure of this proximity the closer Tx and r are to 1, the closer the gas is to its critical state. This observation suggests that a plot of z versus Tx and Px should be approximately the same for all substances, which proves to be the case. Such a plot is called the generalized compressibility chartJ... 

Figure 5.4-1 shows a generalized compressibility chart for those fluids having a critical compressibility factor of 0.27. Conditions for both gases and liquids are illustrated, although in our discussions here we only consider estimation of z for gases. Note the increasing deviations from ideal gas behavior as pressures approach Pc O-e-, when Pr 1). 

To perform PVT calculations for nonideal gas mixtures, you may use Kay s rule. Determine pseudocritical constants (temperature and pressure) by weighting the critical constants for each mixture component by the mole fraction of that component in the mixture then calculate the reduced temperature and pressure and the compressibility factor as before. 

The compilations of CRC (1-2), Daubert and Danner (3), Dechema (15), TRC (13-14), Vargaftik (18), and Yaws (19-36) were used extensively for critical properties. Estimates of critical temperature, pressure, and volume were primarily based on the Joback method (10-12) and proprietary techniques of the author. Critical density was determined from dividing molecular weight by critical volume. Critical compressibility factor was ascertained from application of the gas law at the critical point. Estimates for acentric factor were primarily made by using the Antoine equation for vapor pressure (11-12). 

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