Equation Chao-Seader

The Chao-Seader equation gives accurate predictions for light hydrocarbons and hydrogen, but is limited to temperatures below 530 K Chao and Seader (1961). 

Grayson and Stread (1963) extended the Chao-Seader equation for use with hydrogen rich mixtures, and for high pressure and high temperature systems. It can be used up to 200 bar and 4700 K. 

Table VII shows results for the first system, isopropanol -isopropyl ether - water - propylene, in which the experimental compositions in each of the three phases are compared with the values predicted by the method just described. A modified Redlich-Kwong equation of state for vapor fugacity, Chao-Seader equation with adjusted parameters for liquid fugacity, and the Wohl equation for the activity coefficients were used. The predictions were based only on data for binary systems.

The amount of hydrogen gas dissolved in the liquid can also be calculated from experimental data by using the Aspen-Hysys simulator and the vapor-liquid equilibrium at typical operating conditions in the storage container of a hydrotreated liquid product. The Chao Seader equation of state (EOS), which is reported as valid for temperatures below 257°C (Mapiour et al., 2010), can be applied to perform equilibrium calculations. 

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 first three methods use one set of equations for the vapor phase and another for the liquid, in a similar technique. These methods are identified as Chao-Seader (2), Grayson-Streed (3), and Lee-Erbar-Edmister (4). The other three methods employ the same equations for both vapor and liquid phases. They are identified as Soave-Redlich-Kwong (5), Peng-Robinson (6), and Lee-Kesler-Ploecker (7, 12). At this writing, the present... 

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. 

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. 

The Chao-Seader method12 is an example of the use of multiple equations of state for the calculation of K values. The Redlich-Kwong equation of state is used to compute the vapor-phase fugacity coefficient the Hildebrand equation for the calculation of the liquid-phase activity coefficient y/% and an extension of Pitzer s modified form of the principle of corresponding states for the calculation of the liquid-phase fugacity coefficient 4> ... 

Case 3 T > Tci In this case, the liquid phase does not exist at any pressure. Lewis and Kay40 suggested the extrapolation of a plot of log f tPi or log P( vs. /Tr while Souders et al.61 used experimental K values and heats of solution data in their extrapolation procedure. Adler et al.2 suggested the use of the Chao-Seader (or Grayson-Streed) equations or back-calculation from experimental vapor-liquid equilibrium data. 

A first issue is the thermodynamic modelling. For the high-pressure part including the stabiliser, an equation-of-state model is appropriate, as for example Peng-Robinson or Soave-Redlich-Kwong. A specific model for hydrocarbons, as Chao-Seader or Grayson-Streed, may be used equally for the low-pressure part. 

Example 5.3. Solve Example 4.4 using the liquid enthalpy equation of Edmister, Persyn, and Erbar, which is based on the Chao-Seader correlation. 

The Chao-Seader correlation is widely used in the petroleum and natural gas industries. Waterman and Frazier describe its use in the design of a wide variety of distillation separations involving light hydrocarbons. Correlations more sophisticated than the C-S correlation can give more accurate results in certain ranges of conditions. However, Lo showed that computing requirements can become excessive and extrapolation more uncertain when more complex equations are utilized. 

In the Chao-Seader (C-S) correlation, the R-K equation of state (4-72) is used to compute which is close to unity at low pressures. As pressure increases, <, v remains close to one for very volatile components in the mixture. However, for components of low volatility, 4>iv will be much less than one as pressure approaches the convergence pressure of the mixture. 

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. 

A Fortran programme has been elaborated by Williams and Henley [89] for tlie computation of multicomponent vapour-liquid equilibria. To take into account real behaviours a number of subprogrammes are available which enable fugacities to be calculated by means of the virial equation, the Redlich-Kwong relation or according to Chao-Seader. Activity coefficients may be considered following Wilson, van baar or Hildebrand. The state of the art of precalculating vapour-liquid equilibria in multicomponent mixtures was surveyed by Hala [89a]. Lu and Polak [89b] discussed the special requirements for the calculation of phase equilibria at low temperatures (20 K to room temperature). 

Scatchard-Hildebrand Equation, Chao-Seader method Solubility parameters and partial molar volumes are referred to 25 °C, suitable for hydrocarbon systems Chao, K.C., and Seader, J.D., AIChE. J. 7 (1961) 4, 598. 

As pointed out in the preceding paragraph, some equations, by their very nature, complicate matters. As an additional example, consider the specific limitations of the Chao-Seader correlation and others like it, which allow only about 20% methane in the liquid. In a typical flash problem it is conventional for a first set of K constants to be furnished either by the engineer or initialized by the flash program. These K s immediately lead to vapor and liquid compositions. With these compositions the component fugacltles and liquid activity coefficients are calculated, which in turn lead to a seemingly better set of K s. If the first... 

The Chao-Seader correlation and its many modifications, and the Chueh-Prausnitz correlation are examples of this approach. In the Chao-Seader correlation f and ir are combined as a pure liquid fugacity coefficient, v, so that Eq. (5) has three distinct parts —each requiring a unique equation, as just described. 

Chao-Seader Correlation. Reference was made earlier to the well known and much used Chao-Seader correlation for the prediction of vapor-liquid equilibrium for principally hydrogen-hydrocarbon systems with small amounts of CO2, H2S, O2, N2, etc. The heart of the correlation consists of several equations to represent liquid fugacity. The other two constituents, the Scatchard-Hildebrand equation for activity coefficients and the Redllch-Kwong equation for the vapor-phase nonideality, were already well established. 

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 ... 

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

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). 

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