Critical compressibility factor, defined

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). 

Hence there must be one relation involving pc, Tc and Vc which is independent of the parameters a and b. This relation defines the critical compressibility factor Zc ... 

Once this function is determined, it could be applied to any substance, provided its critical constants Pc, T, and V are known. One way of applying this principle is to choose a reference substance for which accurate PVT data are available. The properties of other substances are then related to it, based on the assumption of comparable reduced properties. This straightforward application of the principle is valid for components having similar chemical structure. In order to broaden its applicability to disparate substances, additional characterizing parameters have been introduced, such as shape factors, the acentric factor, and the critical compressibility factor. Another difficulty that must be overcome before the principle of corresponding states can successfully be applied to real fluids is the handling of mixtures. The problem concerns the definitions of Pq P(> and Vc for a mixture. It is evident that mixing rules of some sort need to be formulated. One method that is commonly used follows the Kay s rules (Kay, 1936), which define mixture pseudocritical constants in terms of constituent component critical constants ... 

Parameter fc is an apparent critical compressibility factor. We give it a special symbol to preclude its general identification with the experimental critical compressibility factor Zc, to which, in practical applications of cubic equations of state, it is often assumed not equal. In such cases, the reduced volume Vr is defined always with respect to a (possibly hypothetical) critical volume Vc defined in terms of Pc, Tc, and fc via Equation 5. 

This procedure would define a pseudo-critical volume and temperature. However, it is more convenient to change the pseudo-critical volume variable to a pseudo-critical pressure by defining an empirical pseudo-critical compressibility factor for the mixture ... 

In selecting this pure substance, an additional parameter accounting for the shape of the molecules should be used. The average critical compressibility factor Zg as defined by (33) is often used as an empirical shape factor parameter. However, when e molecular weights of the gases become small or at low temperatures, the pseudo-mass as calculated in (40) becomes important in addition to shape factors. If a reference substance with known properties is selected in this manner, it is convenient to define an equivalent temperature and pressure, 7 and P , as ... 

Three such parameters have been proposed through the years the critical compressibility factor the Riedel factor and the acentric factor omega ( ). The last one, proposed by Pitzer in 1955, has found the widest application and is defined as follows ... 

No specific mixing rules have been tested for predicting compressibility factors for denned organie mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibihty faclor and hence the vapor density. 

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. 

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 ... 

Saturated Vapor Densities. We formulate the compressibility factor for saturated vapor as a function of temperature by using the vapor-pressure equation. Subscripts are omitted because we refer always to saturated vapor and to the vapor pressure. We define the constant, A0 Zc — 1, where Zc is the value of the compressibility factor at the critical point, and the variables... 

The principle of corresponding states can be used to express the pressure, tan-perature, and specific volume in terms of reduced variables. Experimental observations reveal that the compressibility factor, Z Equation (2.24), for different fluids exhibits similar behavior when correlated as a function of reduced temperature, T, and reduced pressure, The reduced variables may be defined with respect to some characteristic quantity. For example, they can be defined as follows with respect to critical temperature and critical pressure ... 

The van der Waals equation predicts that the value of the compression factor at the critical point is equal to 0.375 for all substances. There is even a greater degree of generality, expressed by an empirical law called the law of corresponding states All substances obey the same equation of state in terms of reduced variables. The reduced variables are dimensionless variables defined as follows The reduced volume is the ratio of the molar volume to the critical molar volume ... 

Microindentation hardness is currently measured by static penetration of the specimen with a standard indenter at a known force. After loading with a sharp indenter a residual surface impression is left on the flat test specimen. An adequate measure of the material hardness may be computed by dividing the peak contact load, P, by the projected area of impression (Tabor, 1951). The microhardness, so defined, may be considered as an indicator of the irreversible deformation processes which characterize the material. The strain boundaries for plastic deformation below the indenter are critically dependent, as we shall show in the next chapter, on microstructural factors (crystal size and perfection, degree of crystallinity, etc.). Indentation during a microhardness test permanently deforms only a small volume element of the specimen (V 10 -10 nm ) (non-destructive test). The rest of the specimen acts as a constraint. Thus the contact stress between the indenter and the specimen is much larger than the compressive yield stress of the specimen (about a factor of 3 higher). 

For AS ME Code vessels the allowable compressive stress is Factor B. The ASME Code, factor B. considers radius and length but does not consider length unless external pressure is involved. This procedure illustrates other methods of defining critical stress and the allowable buckling stress for vessels during transport and erection as well as equipment not designed to the ASME Code. For example, shell compressive stresses are developed in tall silos and bins due to the side wall friction of the contents on the bin wall. 

From the above derivation, it is important to note that the effect of constraint increases the stiffness of the component by the factor (1 +a l2h ). As seen in Fig. 15, for large thin blocks of rubber, the effective modulus approaches that of the bulk modulus, B, for compression when a/h is greater than about 10. Thornton et al. analyzed the interfacial stresses for the intermediate case when some amount of friction or adhesion occurs between the rubber disk and the rigid blocks [8]. A new variable, ii, defines the static coefficient of friction for which a critical radius exists at which the shear stresses cannot overcome the coefficient... 

There are statistical, and perhaps systematic, uncertainties associated with the primary measured quantities used to define the thermodynamic state these should be considered in the reporting of experimental uncertainties of the measured quantity and accounted for in the regressions used to determine a correlation. All the uncertainties are likely to have state point dependence, so that the assessment in the dilute gas regime will differ markedly firom that in the critical region or in the compressed liquid (Perkins et al. 1991a). The uncertainty reported for a correlation should be a function of the fluid state point. The coverage factor (based on, perhaps, two standard deviations in a normal distribution) should be applied to the appropriate distribution associated with the combined standard uncertainty of the correlation. 

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