The Pseudo-component Method

Marano and Holder have calculated the VLE of the Fischer-Tropseh system. The pseudo-components were defined with the aid of an analytical molar-mass distribution function (Anderson-Schulz-Flory distribution). The properties of a pseudo-component were based on a hypothetical model component in each carbon-number cut. 

Gonzalez et alP modeled the asphaltenes precipitation in live oils with the (Perturbed Chain-Statistical Associating Fluid Theory (PC-SAFT) see Chapter 8 for additional material. It is not an easy task to apply a complicated model such as PC-SAFT to systems consisting of a very large number of chemical 

Nichita et al calculated the wax precipitation from hydrocarbon mixtures using a cubic equation of state (see Chapter 4) to describe the vapour and the liquid lumping into pseudo-components to simplify the phase equilibrium calculation. However, the information lost in this procedure effected the location of the predicted solid phase transition. This issue was avoided by an inverse lumping procedure, in which the equilibrium constants of the original system are related to some quantities evaluated from lumped fluid flash results. The method was tested for two synthetic and one real mixture yielding good agreement between calculated and experimental results. 

However, the few cases discussed show that pseudo-components can be a successful applied to describe polydispersity and calculate phase equilibrium by 

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However, traditional chemical thermodynamics is based on mole fractions of discrete components. Thus, when it is applied to polydisperse systems it has been usual to spht the continuous distribution function into an arbitrary number of pseudo-components. In many cases dealing, for example, with a solution of a polydisperse homopolymer in a solvent (the pseudobinary mixture), only two pseudo-components were chosen (reproducing number and mass averages of molar mass of the polymer) which, indeed, are able to describe some main features of the liquid-liquid equilibrium in the polydisperse mixture [1-3]. In systems with random copolymers the mass average of the chemical distribution is usually chosen as an additional parameter for the description of the pseudo-components. However, the pseudo-component method is a crude and arbitrary procedure for polydisperse systems. 

The Sako-Wu-Prausnitz equation of state was also applied to high-pressure phase equilibria of polyolefin systems by Tork et alP The calculations were based on the pseudo-component method where the number of pseudo-components used were between 2 and 8. The small number of pseudo-components is a result of the very efficient estimation method used to adjust the pseudo-components to the moments of the distribution function (described in section 9.3.1). In so doing Tork et alP were able to provide a good description of the experimental data and show, perhaps not surprisingly, the agreement between calculated and experimental data improved with increasing number of pseudo-components. 

Nichita et al applied the pseudo-component method to the wax precipitation from hydrocarbon mixtures. To do so a general form of a two-parameter equation of state was used for vapour and liquid phases. The heavy components were assumed to precipitate in a single solid solution. Because lumping in pseudo-components often results in difficulties in solid-liquid equihbrium calculations the authors proposed a delumping procedure (mentioned in section 9.3.1). Lira-Galeana et al calculated wax precipitation in petroleum mixtures by assuming the wax consisted of several solid phases each described as a piue component or pseudo-component immiscible with other solid phases. 

Much effort has been directed at asphaltenes flocculation and precipitation in petroleum crude mixtures that is a significant problem in oil production, transmission and processing facilities. Pressure, temperature, the chemical composition of the oil and the amount of dissolved gases affect this undesired phenomenon. In general, the prediction of asphaltenes precipitation is very difficult and suffers from the definition of an asphaltene. Usually, asphaltenes are defined as the part of the crude oil that is soluble in methylbenzene and benzene but insoluble in pentane or heptane. Asphaltenes consist of many thousands of species, differing in size and chemical structure. The aromatic character of the asphaltenes and their content of heteroatoms influence their solubility in different solvents and the tendency to flocculate. In most of the calculations of the phase equilibria the pseudo-component method has been... 

There are other examples of the calculation of phase equilibrium that are of practical interest and worthy of further discussion. For example, Rodrigues et al studied the phase equihbrium for (rice bran oil + fatty acid-s + ethanol + water + y-oryzanol + tocols). y-oryzanol is a mixture of several ferulic acid [(E)-3-(4-hydroxy-3-methoxy-phenyl)prop-2-enoic acid] esters of sterol and triterpene alcohols. Tocols consist of tocopherols and tocotrienols. y-oryzanol. Tocols are minor components of rice bran oil possessing antioxidant and other beneficial physiological properties. Rodrigues et al applied the pseudo-component method with the NRTL and the UNIQUAC model to calculate the LLE of this compositionally very complex system. For parameter estimation model fatty systems were investigated. 

Usually, product specifications for a crude distillation unit are expressed in terms of the products 15/5 or ASTM distillation curves. The prediction of a product 15/5 distillation is accomplished simply by blending the quantities of the pseudo components in the stream so as to form a true boiling point, 15/5 equivalent, distillation curve. This curve can then be converted to an ASTM type distillation using an empirical method. Figure 5 illustrates how a typical ASTM curve compares to the 15/5 curve for the same material. 

All the above three provide extensive data sources and simulators for estimating various thermodynamic properties of pure compounds, well-defined mixtures and petroleum fluids. The respective manuals describe in detail various methods used for estimating the properties and their limitations while at the same time providing the user with choices. The Aspen Physical Property System and SUPBRTRAPP can also be used to estimate thermodynamic profrerties using the pseudo-component characterization of petroleum fluids. The reader is referred to the respective manuals/sim-ulators for details. 

The reader will note that the pseudo-elastic method is conservative. The stress analysis uses a modulus that is really appropriate only to the most highly strained regions of the design, and applies it to the whole component. Elsewhere, strains are lower, and the creep modulus is greater than that used in the analysis. This results in a small but unavoidable element of over-design a component designed in this way will be somewhat thicker and more complex (e.g. because of ribbing) than strictly it needs to be to meet the specification. 

Feed analyses in terms of component concentrations are usually not available for complex hydrocarbon mixtures with a final normal boihng point above about 38°C (100°F) (/i-pentane). One method of haudhug such a feed is to break it down into pseudo components (narrow-boihng fractions) and then estimate the mole fraction and value for each such component. Edmister [2nd. Eng. Chem., 47,1685 (1955)] and Maxwell (Data Book on Hydrocarbons, Van Nostrand, Princeton, N.J., 1958) give charts that are useful for this estimation. Once values are available, the calculation proceeds as described above for multicomponent mixtures. Another approach to complex mixtures is to obtain an American Society for Testing and Materials (ASTM) or true-boihng point (TBP) cui ve for the mixture and then use empirical correlations to con-strucl the atmospheric-pressure eqiiihbrium-flash cui ve (EF 0, which can then be corrected to the desired operating pressure. A discussion of this method and the necessary charts are presented in a later subsection entitled Tetroleum and Complex-Mixture Distillation. ... 

The second method can be applied to mixtures as well as pure components. In this method the procedure is to find the final temperature by trial, assuming a final temperature and checking by entropy balance (correct when ASp t, = 0). As reduced conditions are required for reading the tables or charts of generalized thermodynamic properties, the pseudo critical temperature and pressure are used for the mixture. Entropy is computed by the relation. See reference 61 for details. ... 

The symbol means that the first nc significant columns of V and T are retained. The number of significant principal components, nc, which is the pseudo-rank of X, is usually unknown. Methods for estimating the number of components are discussed in Section 31.5. 

If the presence of the other components does not significantly affect the volatility of the key components, the keys can be treated as a pseudo-binary pair. The number of stages can then be calculated using a McCabe-Thiele diagram, or the other methods developed for binary systems. This simplification can often be made when the amount of the non-key components is small, or where the components form near-ideal mixtures. 

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

The DCLS method can be applied to simple systems where all of the pure-component spectra can be measured. To construct the DCLS model, the pure-component spectra are measured at unit concentration for each of the analytes in the mixture. Tliese are used to form a matrix of pure spectra (S) and the model is then constructed as the pseudo-inverse of this S matrix. This calibration model is used to predict the concentrations in unknown samples. 

However, Briegleb,6 from X-ray investigations, maintains that in all the allotropic modifications two such pseudo-components do exist, and that these may be separated in some degree by taking advantage of the fact that although their absolute solubilities in carbon disulphide are almost identical, the rates at which they dissolve are different. By spectroscopic methods evidence has been obtained that the two forms exist in equilibrium in this solution and that the equilibrium varies with the temperature. 

Aj k and Bj A are the constants for a pseudo-Clausius-Clapeyron equation for the base component K-value. These constants are unique for every stage and are updated at every trial in the calculation. They are calculated as follows. Using the if-value method chosen for the simu-... 

Calculate the volume using Kay s method. In this method, V is found from the equation V = ZRT/P, where Z, the compressibility factor, is calculated on the basis of pseudocritical constants that are computed as mole-fraction-weighted averages of the critical constants of the pure compounds. Thus, T = Z K, 71, and similarly for Pc and Z, where the subscript c denotes critical, the prime denotes pseudo, the subscript i pertains to the ith component, and Y is mole fraction. Pure-component critical properties can be obtained from handbooks. The calculations can then be set out as a matrix ... 

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