## Ycc

Substituting into the equation of equilibrium and noting that x + x2 1 and x + x'i 1, we obtain exp M1 tX )2 x' exp M1-tX' 2 (8-13.7) and (1 - x ) exp (1 - x' ) exp (8-13.8) Equations (8-13.7) and (8-13.8) contain two unknowns (x and x), which can be found by iteration. Mathematically, several solutions of these two equations can be obtained. However, to be physically meaningful, it is necessary that 0 < x < 1 and 0 < x < 1. Similar calculations can be performed for ternary (or higher)...

## Info

SPECIAL CORRECTIONS FOR FIRST FEW (-CH2J IN NORMAL SERIES For the first three -CH2- in normal paraffins first -CH2_ 1-45 1.47 1.03 -6.7303 first and_ second -CH2- -0.23 2.72 -1.01 -2.4878 first ana second and third -CH2- 0.11 2.60 -0.59 -1.3214 For the first three (-CH2-) in normal al kyl benzenes first -CH2- 0.42 1.76 -0.08 -2.6088 first and second -CH2- -0.92 3.02 -1.77 3.8192 first ano second and third -CH2- -1.93 2.89 -2.71 4.9593 For the first two (-CH2-) in normal monoolefins first -CH2-...

## Przezdziecki and Sridhar method [160

In this technique the authors propose using the Hildebrand-modified Bat-schinski equation 15, 99, 217 . and the parameters E and V0 are defined below. 12.94 + 0.10M - 0.23pc + 0.0424T, - U.58(Tf Tc) (9-11.10) v0 0.0085cotc - 2.02 + q + q (9-11.11) where Tc critical temperature, K Pc critical pressure, bar Vc critical volume, cm3 mol M molecular weight, g mol 7) freezing point, K to acentric factor Vm liquid molar volume at 7), cm3 mol Thus, to use Eq. (9-11.9), one must have values for Tc, Pc,...

## 1

Some values of the binary interaction coefficient k j for the Soave and Peng-Robinson equations are given in Table 4-2. A more extensive tabulation is given in Ref. 28. Values for ky for specific systems and as a function of temperature are given in 9, 16, 23, 25, 33 . For hydrocarbon pairs, ky is usually taken as zero. If all k J are zero, Eq. (4-5.1) reduces to H can be shown that the second virial coefficient B is given by This provides a relation between ky...

## Rt Conductivity

In these equations x, is the mole fraction of component i and the summations in Eqs. (8-10.50) and (8-10.51) are over all components, including component i, 6, is the area fraction, and < i> , is the segment fraction, which is similar to the volume fraction. Pure component parameters r, and q,-are, respectively, measures of molecular van der Waals volumes and molecular surface areas. In UNIQUAC, the two adjustable binary parameters ry and r i appearing in Eq. (8-10.51) must be evaluated...

## Yivi

Xi is the mole fraction of component i, and lt t gt , is the superficial volume fraction of i. V, is the molar volume of the pure liquid. For a binary system of 1 and 2, Eq. 10-12.17 becomes Xm 0 X, 2 lt t gt lt t gt 2 2 lt t gt 22X2 10-12.20 The harmonic mean approximation for X1 was chosen over a geometric or arithmetic mean after extensive testing and comparison of calculated and experimental values of Xm. Also, it was found that the V, terms in Eq. 10-12.19 could be replaced by critical...

## Estimation of Liquid Viscosity at High Temperatures

Low-temperature viscosity correlations as covered in Sec. 9-10 usually assume that In rn is a linear function of reciprocal absolute temperature. Above a reduced temperature of about 0.7, this relation is no longer valid, as illustrated in Fig. 9-10. In the region from about Tr 0.7 to near the critical point, many estimation methods are of a corresponding states type that resemble or are identical with those used in the first sections of this chapter to treat gases. For example, Letsou and...

## References

Di Filippo, and G. Latini Comparison between Predicted and Experimental Thermal Conductivity Values for the Liquid Substances and the Liquid Mixtures at Different Temperatures and Pressures, paper presented at the Workshop on Thermal Conductivity Measurement, IMEKO, Budapest, March 14-16, 1983. 2. Baroncini, C., P. Di Filippo, and G. Latini Intern. J. Refrig., 6 1 60 1983 . 3. Baroncini, C., P. Di Filippo, G. Latini, and M. Pacetti High Temp. High Press., 11 581 1979 . 4....

## Index

Table, 656-732 Activity coefficients from ASOG group contribution method, 313-314 from azeotropic data, 307-309 correlations for, 254-257 definition of, 248 estimation of, 283-332 at infinite dilution, 290-307 from mutual solubilities, 309-311 one-parameter correlations for, 255-259 from UNIFAC group contribution method, 314-332 Ambrose estimation method for critical properties, 12-14 Amoco Redlich-Kwong equation of state, 44, 346 Andrade'correlation for liquid viscosity, 439 Antoine vapor...

## Low pressure

This region extends from approximately 10 3 to 10 bar and includes the domain discussed in Sees. 10-3 and 10-4. The thermal conductivity increases about 1 percent or less per bar 68,174,175,176 . Such increases are often ignored in the literature, and either the 1-bar value or the zero-pressure extrapolated value may be referred to as the low-pressure conductivity. TABLE 10-3 Thermal Conductivities of Some Gases at About 1 Bar1 X A BT CT- DT3 X in W m K and T in keivins TABLE 10-3 Thermal...

## Recommendations for estimating lowtemperature liquid viscosities

Three estimation methods have been discussed. In Table 9-11 calculated liquid viscosities are compared with experimental values for 35 different liquids usually of simple structure . Large errors may result, as illustrated for all methods. The method of van Velzen et al. is not recommended for first members of a homologous series, and the method of Przezdziecki and Sridhar should not be used for alcohols. The method of van Velzen et al. assumes that log in is linear in T l, whereas the Orrick...

## A

Thus, all the A terms in Eqs. 5-6.31 and 5-6.32 are second derivatives of the total Helmholtz energy A with respect to moles at constant temperature and total volume V. The determinants expressed by Eqs. 5-6.31 and 5-6.32 are solved simultaneously for the critical volume and critical temperature. The critical pressure is then found from the original equation of state. Peng and Robinson 61 used their equation of state to calculate mixture critical points later Heidemann and Khalil 32 ,...

## Notation

A group contribution sum Eq. 9-4.21 b0 excluded volume, 2 3 7rN0 lt T3 B viscosity parameter in Eq. 9-11.2 C heat capacity at constant volume, J mol-K C,-, structural contribu D diffusion coefficient, cm2 s or m2 s Fc shape and polarity factor in Eq. 9-4.10 FP, low-pressure polar cor rection factor in Eq. 9-4.17 Fq, low-pressure quantum correction factor in Eq. 9-4.18 FP, high-pressure polar correction factor in Eq. 9-6.8 Fq, high-pressure quantum correction factor in Eq. 9-6.9 Glt Go...

## T

lt r a i 0.809 V f 9-5.32 9-5-35 con u gt , 9-5.36 and fy are binary interaction parameters which are normally set equal to unity. The FCm term in Eq. 9-5.24 is defined as in Eq. 9-4.10 . FCm 1 - 0.275a m 0.059035M Km 9-5.41 In these equations, Tc is in kelvins, Vc is in cm3 mol and is in debyes. The rules suggested by Chung et al. are illustrated for a binary gas mixture in Example 9-8. As with the Lucas approach, the technique is not interpolative between pure component viscosities. Some...

## Roy and Thodos estimation technique

In the same way that the viscosity was nondimensionalized in Eqs. 9-4.12 and 9-4.13 , a reduced thermal conductivity may be expressed as In SI units, if R 8314 J kmol-K , N0 Avogadro's number 6.023 X 1026 kmol -1, and with Tc in kelvins, M' in kg kmol, and Pc in N m2, T has the units of m K W or inverse thermal conductivity. In more convenient units, where F is the reduced, inverse thermal conductivity, W m-K Tc is in kelvins, M is in g mol, and Pc is in bars. TABLE 10-1 Recommended f Tr...

## Method of Chung et al [27

Chung et al. employed an approach similar to that of Mason and Mon-chick to obtain a relation for X. By using their form and a similar one for low-pressure viscosity Eq. 9-4.9 , one obtains where thermal conductivity, W m K M' - molecular weight, kg mol r low-pressure gas viscosity, N-s m2 C heat capacity at constant volume, J mol-K R gas constant, 8.314 J mol-K 1 a 0.215 0.28288a - 1.061 3 0.26665Z 0.6366 pZ P 0.7862 - 0.7109a 1.3168a 2 Z 2.0 10.5T2 The term is an empirical correlation for t 1...

## Method of Grunberg and Nissan [87

In this procedure, the low-temperature liquid viscosity for mixtures is given as since Gu 0. In Eqs. 9-13.1 and 9-13.2 , x is the liquid mole fraction and Gij is an interaction parameter which is a function of the components i and j as well as the temperature and, in some cases, the composition . This relation has probably been more extensively examined than any other liquid mixture viscosity correlation. Isdale 107 presents the results of a very detailed testing using more than 2000...

## MlJM l

The method of Lucas does not necessarily lead to the pure component viscosity tji when all yj 0 except y, 1. Thus the method is not inter-polative in the same way as are the techniques of Reichenberg, Wilke, and Herning and Zipperer. Nevertheless, as seen in Table 9-4, the method provides reasonable estimates of rim in most test cases. Example 9-7 Estimate the viscosity of a binary mixture of ammonia and hydrogen at 33 C and low pressure by using the Lucas corresponding states method. solution...

## Liquid Mixture Viscosity

Essentially all correlations for liquid mixture viscosity refer to solutions of liquids below or only slightly above their normal boiling points i.e., they are restricted to reduced temperatures of the pure components to values below about 0.7. The bulk of the discussion below is limited to that temperature range. At the end of the section, however, we suggest approximate methods to treat high-pressure, high-temperature liquid mixture viscosity. At temperatures below Tr 0.7, liquid viscosities...

## Phase envelope constructiondew and bubble point calculations

Calculations at low pressures present little difficulty, but high-pressure calculations can be complicated by both trivial root and convergence difficulties. Trivial root problems can be avoided by starting computations at a low pressure and marching toward the critical point in small increments of temperature or pressure. When the initial guess of each calculation is the result of a previous calculation, trivial roots are avoided. Convergence difficulties are avoided if one does dew or bubble...

## Gqh

CpL 0 C 2 CH3 -1- -CH -OH 2X40.0 23.8 33.5 137.2 J mol K The experimental value is 135.8 J mol K 25 . Several corresponding states methods for liquid heat capacity estimation have been cast in the form of Eq. 5-5.4 . For example, with Tables 5-8 and 5-9, one can estimate the heat capacity departure function CPL Cp for liquids as well as for gases. Good results have also been reported by using an analytical form of the Lee-Kesler heat capacity departure function 47 for calculating liquid heat...

## Viscosity of Gas Mixtures at High Pressures

The most convenient method to estimate the viscosity of dense gas mixtures is to combine, where possible, techniques given previously in Sees. 9-5 and 9-6. In the pure dense gas viscosity approach suggested by Lucas, Eqs. 9-6.4 to 9-6.10 were used. To apply this technique to mixtures, rules must be chosen to obtain Tc, Pc, M, and m as functions of composition. For Tc, Pc, and M of the mixture, Eqs. 9-5.18 to 9-5.20 should be used. The polarity and quantum corrections are introduced by using...

## Aay12i f yK yAi

In Eq. 4-8.1 , there are three parameters per binary in Eq. 4-8.2 there are two. Both of these equations make kijt as defined in Eq. 4-5.1 , a linear function of composition. The composition dependence expressed by Eq. 4-8.2 is obtained from Eq. 4-8.1 if is set equal to minus Burcham et al. 7 have examined the case when one of the two d parameters in Eq. 4-8.1 is set equal to zero. Several authors have added a second binary interaction parameter in the b constant 8, 15, 49, 55 , i.e. See also...

## Discussion and recommendations to estimate the lowpressure viscosity of gas mixtures

As is obvious from the estimation methods discussed in this section, the viscosity of a gas mixture can be a complex function of composition. This is evident from Fig. 9-4. There may be a maximum in mixture viscosity in some cases, e.g., system 3, ammonia-hydrogen. No cases of a viscosity minimum have, however, been reported. Behavior similar to that of the ammonia-hydrogen case occurs most often in polar-nonpolar mixtures in which the pure component viscosities are not greatly different 101,...

## Viscosity

The first part of this chapter deals with the viscosity of gases and the second with the viscosity of liquids. In each part, methods are recommended for 1 correlating viscosities with temperature, 2 estimating viscosities when no experimental data are available, 3 estimating the effect of pressure on viscosity, and 4 estimating the viscosities of mixtures. The molecular theory of viscosity is considered briefly. 9-2 Definition and Units of Viscosity If a shearing stress is applied to any...

## In QO p

Again, simple algebraic substitution shows that the departure functions H - H , U - U , and ln P do not depend upon the choice of P or V . 5-4 Evaluation of Departure Functions The departure functions shown in Eqs. 5-3.5 to 5-3.11 or in Eqs. 5-3.12 to 5-3.18 can be evaluated with PVT data and, when necessary, a definition of the reference state. Generally, either an analytical equation of state or some form of the law of corresponding states is used to characterize PVT behavior, although, if...

## Bkm8km 99876543 Isbn 0070517991

The editors for this book were Betty Sun and Galen H. Fleck, the designer was Naomi Auerbach, and the production supervisor was Thomas G. Kowalczyk. It Anas set in Century Schoolbook by University Graphics, Inc. 1. The Estimation of Physical Properties 1 1-1 Introduction 1 1-2 Estimation of Properties 3 1-3 Types of Estimation 4 1-4 Organization of the Book 8 References 10 2-1 Scope 11 2-2 Critical Properties 12 2-3 Acentric Factor 23 2-4 Boiling and Freezing Points 25 2-5 Dipole Moments 26...

## U5115

From Table 3-8, C 14.8, H 3.7, CI 24.6, and the ring -15.0. Therefore, Vb 6 14.8 5 3.7 24.6 15.0 117 cmVmol Tyn and Calus Method. With Eq. 3-10.1 , Vb 0.285 yj0,8 0.285 3081048 115 cmVmol Even if no data are available, there are a number of techniques for estimating pure liquid specific volumes or densities. Three techniques are presented to estimate saturated liquid densities one is presented for compressed liquids. Hankinson-Brobst-Thomson HBT technique Hankinson and Thomson 33...

## Lcd Thermal Conductivity

Figure 7-6 Comparison between calculated and experimental values of AHJ RTC AZv for propane. volumes of nonpolar materials, the following equation presented by Thompson and Sullivan 86 may be used from the triple point to the critical point 1 - Pvpr l - 7-10.9 Vr is the reduced saturated vapor volume values of 3 and X are listed in 86 or may be estimated by X 1 0.085 1 0.19a 8 7-10.11 Equations 7-10.9 to 7-10.11 gave errors which were usually less than 3 percent but could be as high as 10...

## Mek Thermal Conductance

Depend upon the type numbers of the carbon atoms involved. As an example, for 2-methyl-2-butene, we would first synthesize 2-methylbutane as above and insert a double bond of the type 2 3. Corrections are also given in Table 6-2d for adjacent and conjugated double and triple bonds. With steps 1 to 3 we can synthesize hydrocarbons and estimate their ideal-gas properties. For nonhydrocarbons, one must first prepare a suitable hydrocarbon and then insert the desired functional groups by...

## Fluid Phase Equilibria in Multicomponent Systems

In the chemical process industries, fluid mixtures are often separated into their components by diffusional operations such as distillation, absorption, and extraction. Design of such separation operations requires quantitative estimates of the partial equilibrium properties of fluid mixtures. Whenever possible, such estimates should be based on reliable experimental data for the particular mixture at conditions of temperature, pressure, and composition corresponding to those of interest....

## Parameter Margules Equation

A M ln T,Ul 4 ln 72 X2 4 1 85'8 Upon differentiating Eq. 8-5.5 as indicated by Eqs. 8-5.3 and 8-5.4 , we find With these relations we can now calculate activity coefficients 71 and 72 at any desired x even though experimental data were obtained only at one point, namely, X x2 i This simplified example illustrates how the concept of excess function, coupled with the Gibbs-Duhem equation, can be used to interpolate or extrapolate experimental data with respect to composition. Unfortunately, the...

## Discussion and Recommendations Enthalpy of formation

Values estimated from these methods are compared with literature values in Table 6-7. The methods of Benson Sec. 6-6 and Yoneda Sec. 6-4 yield the smallest errors with only a few large deviations. Both techniques allow for effects of next-nearest neighbors and require some patience to master the method of calculation. Thinh et al.'s procedure Sec. 6-5 is also quite accurate but is limited to hydrocarbons. These authors 26, 27 , in a much wider test, found errors...

## Thermodynamic Properties

In this chapter we first develop relations to calculate the Helmholtz and Gibbs energies, enthalpies, entropies, and fugacity coefficients. These relations are then used with equation-of-state correlations from Chap. 3 to develop estimation techniques for enthalpy and entropy departure functions and fugacity-pressure ratios for pure components and mixtures. In Sec. 5-5 methods are presented for determining the heat capacities of real gases. The true critical properties of mixtures are discussed...

## Q5 06 07 08 09

Figure 3-1 Generalized compressibility chart. Vri is V RTJPC . From Ref. 73. Figure 3-2 Generalized compressibility chart. Vri is V RTC Pr . From Ref. 73. Reduced pressure Pr -3 Generalized compressibility chart. Vri is V RTC P, . From Ref. 73. Vn is not defined in the usual manner, that is, V Vc, but instead is an ideal reduced volume given by Equation 3-2.2 is an example of the law of corresponding states. This law, though not exact, suggests that reduced properties of all fluids are...

## Antoine Vapor Pressure Correlation

Antoine 10 proposed a simple modification of Eq. 7-2.3 which has been widely used over limited temperature ranges. When C 0, Eq. 7-3.1 reverts to the Clapeyron equation 7-2.3 . Simple rules have been proposed 30, 89 to relate C to the normal boiling point for certain classes of materials but these rules are not reliable, and the only way to obtain values of the constants is to regress experimental data 15, 46, 58, 63, 74, 88 . Values of A, B, and C are tabulated for a number of materials in...