The Chao-Seader Method

Chao and Seader (1961) describe high pressure vapor-liquid equilibrium using the gamma-phi approach, applied to low pressure mixtures in Section 13.4  

Vapor phase fugacity coefficients are calculated using the Redlich-Kwong equation of state, while the activity coefficients are calculated from the regular solution theory of Hildebrand (Hildebrand and Scott, 1964)  

AHf is the enthalpy of vaporization and V, the liquid molar volume, both at 298 K. The quantity 6 refers to the volumetric average value of the solubility parameter for the components of the mixture  

The standard state fugacity coefficient is calculated from a Pitzer-type corresponding states correlation  

Expressions for log /° and log / are given by the authors along with values for the acentric factor 

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Calculations were made with the Grayson-Streed modification of the Chao-Seader method for K values and the Lee-Kesler method for enthalpy departures. Initial estimates for stage temperatures and flow rates were as follows, where numbers in parentheses are consistent with specifications ... 

The Grayson-Streed method was developed for hydrogen-petroleum systems at high temperatures and pressures, i.e., hydrofining conditions. Thus the constants, including some critical temperatures and pressures, differ slightly from those recommended in the Chao-Seader method. 

Example 6 ReboUed Stripper Compute stage temperatures and interstage vapor and liquid flow rates and compositions and reboiler heat duty for the reboiled stripper shown in Fig. 13-48. Thermodynamic properties may be estimated by using the Grayson-Streed modification of the Chao-Seader method. 

A number of other equations of state have been proposed. Some of these are presented in Table 14-9. Lee et al.38 proposed an equation of state in which a procedure similar to the Chao-Seader method (described below) was used for the calculation of K values. Lee et al. state that the K values obtained by this method are more accurate than those given by the Chao-Seader method,12 particularly at low temperatures. 

A number of methods are presented for the prediction of K values. First, methods are presented for the calculation of K values for mixtures which form ideal solutions in both the vapor and liquid phases. Then the convergence-pressure method, the Chao-Seader, and the Grayson-Streed modification of the Chao-Seader method are presented. 

The Chao-Seader method uses the Redlich-Kwong equation of state for the calculation of Hildebrand s equation for the calculation of the liquid activity coefficient yf, and an extension of Pitzer s modified form of the principle of corresponding states for the calculation of the liquid fugacity ratio [. 

The Grayson-Streed Modification of the Chao-Seader Method for Calculation of K Values25... 

Note that the RKJZ method not only predicts qualitatively the approach to mixture critical conditions it is also quantitatively superior to the Teja and Rawlinson procedure in this instance. The ability of the RKJZ method to sense the approach to mixture critical conditions has been a great advantage in its application, by comparison with the Chao-Seader method, which has a stated limitation of pressure less than 0.8 times the true critical pressure. 

For companies using the Chao-Seader method, where 5 is already available in computer storage, transition to Soave is certainly advantageous. 

Separate treatment of the two phases, mainly the Chao-Seader method 3.Simultaneous description of both phases with an equation of state. 

Within the limits of the Chao-Seader method, especially the restriction to pressures below about 0.8 of the critical value of the mixture, good results are obtained. 

Cubic equations of state have become the main tool for high pressure VLE calculations. They combine simplicity with accuracy comparable to -or better than - that of other methods, including non-cubic EoS. For a comparison of the EoS approach with the Chao-Seader method, see Maddox and Erbar (1981). 

Prepare a flow diagram for a bubble point pressure calculation with the Chao-Seader method. 

Calculate the total pressure and vapor phase composition for the system Ci(l) - C02(2) at 230 K, jcj = 0.170 and = 0.543, using the Chao-Seader method. Compare your findings with the experimental values given in Table 14.P.1. 

The correlation of Chao and Seader has been computerized and has been used extensively in the petroleum industry. It provides a useful method for estimating high-pressure vapor-liquid equilibria in hydrocarbon systems over a wide range of temperature, pressure, and composition, and presents a significant improvement over the previously used A -charts first introduced by W. K. Lewis, B. F. Dodge, G. G. Brown, M. Souders, and others (see D6) almost forty years ago. However, the Chao-Seader correlation is unreliable at conditions approaching the critical. Various extensions have been proposed (G2), especially for application at extreme temperatures. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

The module FLASH simulates the reformer flash separator by using a modified Chao-Seader method. The necessary constants for the Chao-Seader correlation have been developed for the lumped components. The output from this module determines the liquid yield and the composition of recycle gas. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Estimating the unknown but required starting values of conditions and compositions is an important and sensitive part of these calculations. The composition of the feed is always known, as is the composition of one of the two phases in bubble and dew point calculations. With the Chao-Seader, Grayson-Streed, and Lee-Erbar-Edmister methods, it is possible to assume that both phases have the composition of the feed for the first trial. This assumption leads to trouble with the Soave-Redlich-Kwong, the Peng-Robinson and the Lee-Kesler-Ploecker... 

Experimental vapor-liquid equilibrium data obtained by Grayson and Streed25 were used to extend the temperature range of the Chao-Seader X-value method... 

In the case of hydrocarbon mixtures the Chao-Seader (1961) method is often used. The fugacity coefficients may be calculated by a corresponding states formulation, as the sum of two contributions, for spherical molecule and deviation from sphericity ... 

Use the Naphtali-Sandholm SC method with the Chao-Seader correlation for thermodynamic properties to calculate product compositions, stage temperatures, interstage flow rates and compositions, reboiler duty, and condenser duty for the following distillation specifications. 

Derivatives of properties are needed in the Naphtali-Sandholm SC method. For the Chao-Seader correlation, determine analytical derivatives for... 

A saturated liquid feed at 125 psia contains 200 IbmoIe/hr of 5 mole% /C4, 20mole% nCj, 35mole% iCs, and 40mole% nCs- This feed is to be distilled at 125 psia with a column equipped with a total condenser and partial reboiler. The distillate is to contain 95% of the nC4 in the feed, and the bottoms is to contain 95% of the iCs in the feed. Use the Naphtali-Sandholm SC method, with the Chao-Seader correlations for thermodynamic properties, to determine a suitable design. Twice the minimum number of equilibrium stages, as estimated by the Fenske equation in Chapter 12, should provide a reasonable number of equilibrium stages. 

We can now use the Redlich-Kwong equation of state [6] and a liquid-phase correlation (or an equation of state) to obtain expressions for and as functions of temperature, pressure and component critical properties. This is the approach taken by the very popular Chao-Seader [6] and Grayson-Streed [6] methods. The only factor that remains undefined is the liquid activity coefficient. The Chao-Seader and Grayson-Streed methods use the regular solution theory to obtain an expression for as follows ... 

There are many other specific techniques applicable to particular situations, and these should often be investigated to select the method for developing the vapor-liquid relationships most reliable for the system. These are often expressed in calculation terms as the effective K for the components, i, of a system. Frequently used methods are Chao-Seader, Peng-Robinson, Renon, Redlich-Kwong, Soave Redlich-Kwong, Wilson. 

The physical property package to be used must be specified. Chao-Seader usually works well in this type of hydrocarbon system. Click Properties and Specifications. Use the dropdown arrow in the Base method box to select CHAO-SEA as shown in Figure 2.40a and 2.40b. 

Six alternate methods for predicting the thermodynamic properties are included. These are known by the names of the authors of the methods, which are Chao-Seader (2), Grayson-Streed (3), Lee-Erbar-Edmister (4), Soave-Redlich-Kwong (5), Peng-Robinson (6) and Lee-Kesler-Ploecker (7, 12). 

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