2360
Znd. Eng. Chem. Res. 1991,30, 2360-2362
Comments on “Asymptotic Effects Using Semicontinuous vis-&vis Discrete Descriptions in Phase Equilibrium Computations” Sir: A paper by Luks et al. (1990) deserves some comments, since in our opinion the paper creates confusion as to previous concepts involved in the subject, while the assertion that continuous thermodynamics (solely identified as the method of moments) has a serious deficiency in saturation calculations is generalized to the extreme. Even when alternative computational continuous thermodynamics techniques that avoid these shortcomings are currently in use (Le., generalized Gaussian quadrature, etc.), thus declining the generalization, we show in these comments that, by the simple incorporation of a more versatile probability function, very useful results can be obtained in this type of calculation, even when moments are used. Included in the literature on continuous thermodynamics (CT), notably by Cotterman and Prausnitz (Cotterman, 1985; Cotterman and Prausnitz, 1985a,b; Cotterman et al., 1985), there is a consensus about the characteristics of discrete, continuous, and semicontinuous mixtures. According to this classification, a discrete mixture is a mixture having a full composition vector given by a set of identified components. Semicontinuous mixtures are systems in which, apart from having identifiable components, the integral nature of the continuous fractions allows for the use of integration techniques in order to describe the composition of the heavy fractions (cf. Bowman, 1949; Taylor and Edmister, 1971; Khelen and Ratzch, 1985). However, even when integration is used, these systems never lose their polydisperse nature, but, in addition, the type of numerical integration used in the calculations must be mentioned (Prausnitz, 1990). This type of mathematical solution has generally been used to designate all kinds of existing CT methods for phase equilibrium calculations. According to these arguments, the live oil fluid studied by Luks et al., strictly speaking, does not follow a discrete description, as mentioned in the paper (because not all the fluid’s components are identifiable), but due to the use of both discrete compositions and quadrature points, we feel this fluid should be said to follow a semicontinuous quadrature description. The fact that the authors classify this mixture as discrete is not in line with the previous consensus, giving the impression that results are analyzed for a system really made up of identifiable (discrete) components, against an alternative representation for the system using a CT method, which is not the case. Furthermore, the discrete description term was henceforth used to establish a comparison between their discrete approach and the moments method (from which the authors give the impression that this approach constitutes the sole CT method). For example, Cotterman and Prausnitz have developed an alternative method-the quadrature method-and they have stated that both quadrature and the moments approach belong to the general matter of CT. They have also shown that quadrature provides exact solutions to material balances. This characteristic has also been shown in other CT methods developed to date (i.e., Gibbs free energy minimization, orthonormal expansion, etc.). Therefore, we feel that the assertion conservation of mass failure is a recognized and formal shortcoming of CT made in the paper by Luks et al. should be modified, since none of the alternative CT procedures mentioned suffer such deficiencies. 0888-5885191/2630-2360$02.50/0
FEED-I
9 FR FLASH
DRUM
I
I
Figure 1. Distillation column with two feeds and a liquid sidestream.
Finally, to demonstrate that even the predictions made by the method of moments are useful in processes design, in contrast to the points of view of Luks and his coworkers, let us present our experiences in using this method during the hundreds of times it is required for the simulation of separation operations, such as large distillation and absorption columns. Our algorithms are, in fact, the CT analogies of the well-known bubble points method of Wang and Henke (Wang and Henke, 1966) and the Sujata method (Sujata, 19611, respectively. However, our CT separation-column models use a more versatile molecular weight distribution function (cf. Johnson and Kotz, 1970), i.e.
The above distribution is known as the three-parameter gamma distribution (Whitson, 1983; Salazar-Sotelo and Lira-Galeana, 1985; Cotterman, 1985; Cotterman et al., 1985). When a = 1,this distribution becomes the one used by Luks et al. The main feature of our CT separation-column models is the linear dependency that exist in the p parameters of the liquid phase, when the moments equations are written for the continuous fractions a t the system bubble conditions in each plate. Thus, the familiar Thomas algorithm is used to update these parameters along the column during iterations that converge both material and energy balances (Lira-Galeana and Najera-Blanco, 1986a; Najera-Blanc0 and Lira-Galeana, 1986) in a similar way as the composition updating process is employed in discrete columns (Wang and Henke, 1966). For absorption, a similar procedure is used but it varies in the solution of the energy balances. However, only a distillation calculation is illustrated. The detailed results for absorption and pertinent computer programs are available upon request. We fractionate a synthetic paraffinic mixture by distillation. Figure 1 shows all column specifications. Table I shows the types of characterizations used to make the paraffinic mixture. These are (a) a discrete representation, using 14 identifiable compounds in the system, and (b) a semicontinuous representation, using n-heptane as the discrete part of the mixture. The residual fraction is represented by a r distribution, whose characteristic pa1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 10, 1991 2361 CONDENSER
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Figure 2. Calculated internal flow profiles along the column. CONDENSER
I 2
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TEMPERATURE K Figure 3. Calculated internal temperature profiles along the column.
rameters are reported. Calculations are presented from both the true (discrete) multicomponent formalism and the semicontinuous approach using the method of moments. It must be pointed out that both approaches use enthalpies and equilibrium ratios given by a semicontinuous version of the Soave (Soave, 1972) equation of state (Lira-Galeana and Najera-Blanco, 1986b). Figures 2 and 3 show both the temperature and stream flow profiles along the column. A more detailed inspection of the calculation is given in Table 11, where specific
comparisons between our CT approach and the rigorous multicomponent formalism are also given. It is evident that the moments approach (as any other CT formalism) gives improved results when more versatile probabilistic functions are used (see Halpin and Quirke (1990)). In this, as in many other examples, continuous thermodynamics helped us to simulate complex separation operations, both a t a fraction of the computer cost of the multicomponent procedure and without serious damage to the final calculation results. However, Gaussian quadrature, or any other CT formalism, always remains as a useful alternative.
2362 Ind. Eng. Chem. Res., Vol. 30, No. 10, 1991 Table I. Synthetic Fluid Characterizations comDonent mol 7 '0 comDonent mol % (a) Discrete Multicomponent Formalism (DMF) n-heptane 50.000 n-pentadecane 0.340 n-hexadecane 0.124 n-octane 3.710 n-nonane 13.992 n-heptadecane 0.044 n-decane 14.483 n-octadecane 0.015 n-nonadecane 0.05 n-undecane 9.396 n-eicosane 0.020 n-dodecane 4.836 n-tridecane 2.167 av mol wt 122.73 n-tetradecane 0.886 (b) Semicontinuous Approach n-heptane
'Continuous
50.000 3.657 10.436
Yo
107.110
av mol wt
122.73
fraction (paraffinic) distribution parameters.
Table 11. Comparison of the Moments Method to the Discrete Multicomponent Formalism for the Simultation of a Complex Distillation System Distillate, D flow, kg-mol/h mol 70 component DMF this work DMF this work n-heptane 0.363 0.362 90.705 90.568 others 0.037 0.038 9.295 9.432 total 0.400 0.400 100.000 100.000 temp, K 373.77 373.68
component n-heptane others total temp, K
component n-heptane others total temp, K
Liquid Sidestream, SSL flow, kg-mol/h mol % DMF this work DMF this work 0.122 0.128 0.250 398.5
0.122 0.128 0.250 398.5
Bottoms. B flow, kg-mol/h DMF this work 0.015 0.335 0.350 449.38
0.016 0.334 0.350 448.47
48.879 51.121 100.000
48.873 51.127 100.000
~
~~
mol % DMF this work 4.280 95.720 100.000
4.498 95.502 100.000
heat loads, kcal/h DMF this work condenser duty reboiler duty
-1810.0 1992.0
-1895.7 1949.6
preflash sepn at 101.3 kPa. 394.4 K
DMF fractn of feed vaDorized (FFl/F) 0.063 35 re1 comput time of calcnso DMF this work 1 16
this work 0.064 31
"Using a UNIVAC 1100/80computer.
Acknowledgment This work was supported by both the Exploitation Technology and the Project Engineering Branches of the Mexican Petroleum Institute. We are also grateful to the members of the Common Thermodynamics Group, National Autonomous University of Mexico, administered by the University Energy Program for additional support. We thank Professors J. M. Prausnitz and G. A. Mansoori for very stimulating discussions. In addition, the valuable
comments of Candelario Perez-Rosales and Humberto Cortez-Casillas are greatly appreciated.
Literature Cited Bowman, J. M. Distillation of an Indefinite Number of Components. Ind. Eng. Chem. 1949,41, 2004. Cotterman, R. L. Phase Equilibria for Complex Fluid Mixtures at High Pressures. Development and Application of Continuous Thermodynamics. Ph.D. Dissertation, University of California at Berkeley, 1985. Cotterman, R. L.; Prausnitz, J. M. Flash Calculations for Continuous and Semicontinuous Mixtures using an Equation of State. Ind. Eng. Chem. Process Des. Deu. 1985a, 24, 434. Cotterman, R. L.; Prausnitz, J. M. Continuous Thermodynamics for Enhanced Oil Recovery. High Pressure Phase Equilibria for Reservoir Simulation. Presented a t the 3rd European Meeting on Improved Oil Recovery, Rome, Italy, April 1985b. Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Phase Equilibria for Mixtures Containing Very Many Components. Development and Application of Continuous Thermodynamics for Chemical Process Design. Ind. Eng. Chem. Process Des. Dev. 1985,24, 194. Halpin, T. P. J.; Quirke, N. A New Method of Continuous Thermodynamics Applied in an Equation of State. SPE Reservoir Eng. 1990, Nou, 617-622. Johnson, N. L.; Kotz, S. Continuous Uniuariate Distribution-I; Houghton Mifflin: Boston, 1970. Kahlen, H.; Ratzch, M. T.; Bergmann, J. Continuous Thermodynamics of Multicomponent Systems. AIChE J. 1985, 31, 1136. Lira-Galeana, C.; Najera-Blanco, A. "Multistage Column Calculations by Continuous Thermodynamics"; Internal Report DIB86/05, Mexican Petroleum Institute, September 1986a. Lira-Galeana, C.; Najera-Blanco, A. On the Representation of Phase Equilibria Using Continuous Thermodynamics. Abstracts of Papers, 21th Mexican Congress of Pure and Applied Chemistry; J . Mer. Chem. SOC. 1986b, 30 (5), 302. Luks, K. D.; Turek, E. A.; Kragas, T. K. Asymptotic Effects Using Semicontinuous vis-&vis Discrete Descriptions in Phase Equilibrium Computations. Ind. Eng. Chem. Res. 1990,29, 2101-2106. Mansoori, G. A. Private communication, Mexico City, January 1991. Najera-Blanco, A.; Lira-Galeana, C. Application of Continuous Thermodynamics for the Simulation of Equilibrium-Stage Separation Operations. Distillation and Absorption. Presented at the 26th National Meeting of the Mexican Institute of Chemical Engineers, Guanajuato, Gto. Mexico, 1986. Prausnitz, J. M. Private communication, Mexico City, July 1990. Salazar-Sotelo, D.; Lira-Galeana, C. Vapor-Liquid Equilibrium for Heavy Oil Fractions. Presented at the 25th National Meeting of the Mexican Institute of Chemical Engineers, San Luis Potosi, S.L.P., Mexico, 1985. Soave, G. Equilibrium Constants from a Modified Redlich Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. Sujata, A. D. Hydrocarbon Process. Pet. Refin. 1961, 40, 12. Taylor, D. L.; Edmister, W. C. Solutions for Distillation Processes Treating Petroleum Fractions. AIChE J . 1971, 17, 1324. Wang, J. C.; Henke, G. E. Tridiagonal Matrix for Distillation. Hydrocarbon Process. 1966, Aug, 155. Whitson, C. H. Characterizing Hydrocarbon Plus Fractions. SPEJ, SOC.Pet. Eng. J . 1983, 23, 683.
* Author to whom correspondence should be addressed. f
Exploitation Technology Branch. Project Engineering Branch.
C. Lira-Galeana,**tA. Najera-Blanco* Exploitation Technology Branch a n d Project Engineering Branch Mexican Petroleum Institute Eje Central Lazaro Cardenas 152 C P 07730, Mexico, D.F., Mexico
L. Ponce-Ramirez Materials Research Institute National Autonomous University of Mexico P.O. Box 70-360 Coyoacan, Mexico, D.F., Mexico
