Ind. Eng. Chem. Process Des. Dev. l W 3 , 22, 217-225
t = time At = time increment
Daniel, K. J. “Transient Model of a MovingBed Coal Gasifier”; presented at the AIChE National Meeting, Phlladeiphia, June 1980. Elgln, D. C.; Galland, J. M.; Dlnsmoor, 6.;Tsang, T. “American Coals In Lurgi Pressure-Gaslflcation Plant at WestfieM, Scotland”; Sixth Synthetlc Pipeline Qas Symposium, Chicago, 1974; p 247. Gregory, D. R.; Llttlejohn, R. F. Br. CoaluHI. Res. A s s . W n . Bull. 1085; 29, 173. Hebden. D. ”High Pressure Gasification under Slagglng CondWons”; presented at Seventh Synthetic Plpellne Qas Symposium, Chlcago, 1975. Hoogendorn, J. C. “Gas from Coal with Lurgi Gasification at Sasol”, In “Clean Fuels from Coal”, Institute of Gas Technology, 1973; p 111. Rudolph, P. “The Lurgl Process, The Route to SNG from Coal”, Proceedings of the Fourth Symposium on Synthetlc Pipeline Gas, Chicago, 1972. Yoon. H.; Wel, J.; Denn, M. M. A I C M J . 1978a, 2 4 , 885. Yoon. H.; Wel, J.; Denn, M. M. AIChE J . 1978b. 25, 429.
T = temperature of gas stream { = temperature of solid stream T, = wall temperature U = gas-wall heat transfer coefficient z = distance from bottom of the reactor Greek Symbols a
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number of hydrogen atoms per carbon atom in char
6 = number of oxygen atoms per carbon atom in char
y = stoichiometric coefficient of oxygen in char-oxygen re-
action Literature Cited
Received for review January 27, 1982 Accepted August 30, 1982
Arri. L. E.; Amundson. N. R. A I C N J . 1978, 72. Cho. Y. S. Master of Science Thesis, Washlngton University, St. Louis, MO, 1980. cho. Y. s.;JON*, B. I&. fng. chem. process 08s. Dev. 1081,20,314.
This work was supported by the Department of Energy under Contract No. DE-FG22-80PC30219.
Local Models for Representing Phase Equilibria in Multicomponent, Nonideal Vapor-Liquid and Liquid-Liquid Systems. 1 Thermodynamic Approximation Functions
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Eldred H. Chlmowltz,’ Thomas F. Anderson,’ Sandro Macchletto, and Leroy F. Stutzman Depertment of Chemlcal Englneerlng, Unlverslty of Connectlcut, Stows, Connectlcut 06268
Local models are used to represent equilibrium ratios (K values) essential to the design of separation processes. These simpllfied models are derived from the rigorous thermodynamic relationships that govern phase equilibria. The local models provide an Interface between the design algorithm and real data or an extensive library of thermophysicai property subroutines used for rigorous property evaluations. The models have been developed for both multicomponent vapor-liquid and IiquM-liquid systems. During any computational procedure in which the iocai models are used, parameters In the models are periodically updated by recursive use of the thermophysicai property data base. This ensures that an accurate representation of phase equilibria is maintained at all times at a cost significantly less than by the use of rigorous thermodynamic property evaluations alone.
Introduction Process simulatom are recognized as valuable tools either for new process design or the analysis of existing plants. They are generally classified as either sequential modular or equation oriented, depending on their structure and solution method. Rosen (1979) has reviewed both types and has discussed the advantages afforded by each for process simulation calculations. A more detail discussion of the possibilities inherent in the equation oriented approach is given in a review article by S h a c k et al. (1982). With either approach, considerable time may be spent evaluating thermodynamic properties using rigorous estimation and correlation programs. In addition, since equation-oriented methods often use a Newton-Raphson solution technique, partial derivatives of various thermodynamic properties must also be evaluated. Analytic partial derivatives are usually not available in the physical-properties subroutines and must be generated using perturbation techniques. Derivatives generated in this fashion greatly increase the computational overhead, especially for highly nonideal systems where composition
partial derivatives may be required for convergence of the calculations. The Bmount of time used in evaluating thermodynamic properties and their derivatives may be reduced, and also, the accuracy of derivative information may be increased by using local approximations of the thermodynamic properties. This concept has been investigated by Hutchison and Shewchuk (1974), Shewchuk (1977), Leesley and Heyen (1977), Boston and Britt (1978),Boston (1979), and Barrett and Walsh (1979). Hutchison and Shewchuk proposed a rather interesting approach for representing K values in vapol-liquid systems. The central idea in that work was to develop a linear approximation to the tangent hyperplane at a particular point on the phase equilibrium surface. Following reasoning similar to that used in relating partial molar and total properties in solutions, expressions for mixture equilibrium ratios (K values) were developed. The resultant expressions for the K values are complex functions of the mole fraction and relative volatility of each component in the mixture. However, since the models are only valid in a limited region about the point of interest, they have to be updated at each iteration of the solution procedure. Thus,the linearization approach suffers from the drawback of requiring the use of the rigorous thermophysical property package at each iteration of the calculations. The authors reported that the use of
‘Department of Chemical Engineering,University of Rochester, Rochester, NY 14627. 0196-4305/83/1122-0217$01.50/0
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their composition dependent K-value models had a beneficial effect on the convergence properties of the distillation algorithm described in that paper. Boston and Britt (1978) utilized simple temperature-only thermodynamic models for K values in a novel manner. The model parameters were treated as iteration variables instead of the usual composition and temperature variables which became dependent quantities. The local models presented by Leesley and Heyen were used to represent equilibrium ratios for relatively ideal mixturea. No terms were included in these expressions to explicitly account for the composition dependence of the K values. The resultant expressions are simple functions of temperature and pressure and require only two adjustable parameters per component. These parameters are evaluated from accurate K-value data using standard linear regression methods. These local models were used throughout the simulation with periodic updating of the parameters. Leesley and Heyen reported instances where the local models afforded a savings of up to 80% in computer time on simulation studies. To account for strongly nonideal mixtures, Barrett and Walsh (1979) outlined an approach for explicitly including composition dependence in the local models by using regular solution theory for modeling liquid phase activity coefficients. Their expressions for doing so contain n(n - 1)/2 distinct terms to accommodate composition dependence for each activity coefficient for an n-component mixture. For a large number of components, such expressions would be impractical to use as approximate models. Barrett and Walsh also mentioned the potential benefit of utilizing local analytic models to generate derivatives for Newton-Raphson-based simulation algorithms, and for other design methods requiring derivative information from the describing equations. However, a methodology for generating simple nonlinear local models that accurately represent strongly nonideal mixture K values and the various partial derivatives of these quantities over a region of interest has not been presented in the literature to date. This work develops nonlinear local models for equilibrium ratios (K values) for highly nonideal multicomponent, vapor-liquid systems. In addition, the concept has been extended to include a liquid-liquid system. Some of the models approximate the relative volatilities of the mixture components; others approximate the K values directly. One of the essential ideas has been to treat multicomponent mixtures (over a region of composition and temperature) as pseudo-binary solutions. This has enabled the number of parameters required to model composition dependence of the various activity coefficients to be drastically reduced. The functional form used to model the activity coefficient composition dependence for each pseudo-binary has been derived from basic thermodynamic considerations. The model parameters are obtained by a linear regression fit of the approximate functions to the rigorous K-value data generated by the thermophysical property subroutines. While the reduction in computation time is considered important, the principal goal has been to develop expressions that can be analytically differentiated. In part 2 of this series (Chimowitz et al., 1982), the models developed here have been used to create efficient and robust algorithms for solving a number of fundamental problems encountered in multicomponent, vapor-liquid computations.
Equilibrium Ratios in Vapor-Liquid Systems The fundamental equation describing phase equilibria
between two phases is that which equates the fugacity of component i in one phase to ita fugacity in the other phase. For vapor-liquid equilibria this can be expressed as where f?is the fugacity of component i in the liquid phase and fiv is its fugacity in the vapor phase. By definition, the equilibrium ratio Ki is defiied as yi/xi where yi is the mole fraction of component i in the vapor phase and x i is the mole fraction of component i in the liquid phase. For vapor-liquid equilibria these fugacities can be related to temperature, pressure, and composition by and (3)
where P is the system pressure and for component i, yi is the activity coefficient, f/' is the standard state fugacity, and & is the vapor-phase fugacity coefficient. The equilibrium ratio Ki can be found from eq 1, 2, and 3 as (4)
The equilibrium ratio given by eq 4 is what we will approximate with a local model. Local Models for Ideal Mixtures. A t low pressures eq 4 may be simplified since $q N 1and the standard state fugacity is approximated by the saturation pressure, Pi". The approximate expression for Ki is then given as Tipi"
Ki N P
(5)
For relatively ideal mixtures, yi provides only a small correction to Ki, and to a good approximation, the product yp; can be represented by a vapor pressure type equation (a function of temperature only) i.e., In (yip:) = f ( r ) . Furthermore, if the saturation pressure of any one component in the mixture is known as a function of temperature, it may be chosen as a reference component; the remainder of the components can then be calculated relative to it (i.e., the major temperature effect is accounted for by the reference component). Thus, the product yip: can be approximated by an equation of the form where Pi" is the saturation pressure of the reference component j and Ai,l and Ai, are adjustable parameters to be determined by regression methods using accurate sets of K-value evaluations. Combining eq 5 and 6 leads to a general expression in terms of temperature only for the K values of each component in the mixture. In Ki = Ai,l In Pt(T)+ Ai,2- In P (7)
i = 1, ..., n Leesley and Heyen reported that an expression similar to eq 7 accurately reproduced K values for relatively ideal mixtures over a 50-100 'C temperature range at low pressures (
