Using an Adaptive Parameter Method for Process Simulation of

Apr 12, 2010 - By means of the Aspen PLUS process simulator, a comparison is made between the performances of both the proposed method with ...
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Ind. Eng. Chem. Res. 2010, 49, 4923–4932

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Using an Adaptive Parameter Method for Process Simulation of Nonideal Systems Laura A. Pellegrini,* Simone Gamba, and Stefania Moioli Dipartimento di Chimica, Materiali e Ingegneria Chimica “G. Natta”, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

A thermodynamic model of the φ/φ class of methods that can be easily used with commercial process simulators is proposed. The vapor-liquid equilibrium (VLE) is calculated by means of a Redlich-Kwong-type equation of state (EoS) that uses modified Huron-Vidal mixing rules with activity coefficients: the activity coefficients are derived from the NRTL model whose parameters are evaluated by fitting VLE data of binary mixtures. By means of the Aspen PLUS process simulator, a comparison is made between the performances of both the proposed method with user-created adaptive parameters that can be forced into the simulator database and the Predictive-Soave-Redlich-Kwong (PSRK) method that uses UNIFAC to compute the excess free energy. The ethanol-water separation by extractive distillation is analyzed to point out how an incorrect prediction of the azeotrope can lead to underestimation of the overall process energy requirement. 1. Introduction The choice of the proper thermodynamic package has always been the critical aspect in the use of commercial process simulators. Reid et al. said,1 “Do not expect magic from thermodynamics. If you want reliable results, you need some reliable experimental dataseither from the literature or from your laboratory”. In many cases, the characteristics of the components and the operating conditions do not require a sophisticated thermodynamic modeling for vapor-liquid equilibrium (VLE) calculation. In these cases, it is possible to apply well-known and tested methods such as Soave-Redlich-Kwong (SRK)2 or PengRobinson (PR)3 equations of state (EoSs) in their original form. They are direct methods (φ/φ) based on cubic equations with classical (quadratic) mixing rules which are expressed by a)

∑ ∑xxa

(1)

∑xb

(2)

i j ij

i

b)

j

i i

i

where aij ) aji ) (1 - kij)√aiaj

(3)

and kij is the only adjustable parameter for each component pair. It is common opinion that the classical mixing rules are not valid for systems that contain polar compounds. As a matter of fact, the poor VLE results obtained for systems that contain polar compounds should, instead, be attributed to the inaccuracy in predicting pure-component vapor pressures,4 because of the usual practice of calculating the attractive parameter of the polar components by means of the simplified equation2 R(T) )

a(T) 2 ) [1 + m(ω)(1 - √TR)] a(Tc)

(4)

which is acceptable for nonpolar substances only. The vapor pressures of polar components calculated through eq 4 can be seriously inaccurate, thus affecting the phase equilibria of their * To whom correspondence should be addressed. Tel.: +39 02 2399 3237. Fax: +39 02 7063 8173. E-mail: [email protected].

mixtures as well. There are few reasons to use such an approximated equation, because other expressionssmainly, the three-parameter Mathias-Copeman correlation5sallow the attractive parameter to be directly determined from experimental vapor pressures.6 Poor results from classical mixing rules are restricted mainly to systems that contain associating components, such as water and alkanols. In those cases, and, generally, for better accuracy, more-flexible mixing rules would be required, with more than a single binary interaction parameter for each component pair. When the original classical EoSs are inadequate, the user must face the difficult choice of which thermodynamic package is better to select in the list offered by the simulator. The objective of this paper is to show, based on examples that represent situations of highly nonideal thermodynamic behavior mainly in liquid phase (systems that usually require unconventional separation units: azeotropic distillation, extractive distillation), the best way to attain a proper thermodynamic modeling through the use of a process simulator7 without forcing any user-created routine. The case in which experimental data are missing (predictive method), as well as the one in which experimental data are provided by the user (model with adaptive parameters), are considered and compared. 2. The Utilized Thermodynamic Modeling The thermodynamic model developed in this work calculates the phase equilibrium via the direct method (φ/φ): liquid and vapor fugacities are calculated by means of a modified Redlich-Kwong EoS, using modified Huron-Vidal mixing rules8-10 with activity coefficients derived from the NRTL model.11 2.1. Equation of State. The EoS used for fugacity calculation is an RK-type12 equation (RK, SRK) in which the original alphafunction has been replaced by the Mathias-Copeman alpha function,5 which was developed especially for polar compounds:

√R ) 1 + m1(1 - √TR) + m2(1 - √TR)2 +

10.1021/ie901773q  2010 American Chemical Society Published on Web 04/12/2010

m3(1 - √TR)3

(for TR < 1) (5)

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√R ) 1 + m1(1 - √TR)

(for TR > 1)

(6)

This three-parameter function allows an increased accuracy in the prediction of pure compound vapor pressure mainly at low temperatures for both nonpolar (e.g., hydrocarbons13) and polar substances. The three parameters for the Mathias-Copeman alpha function can be evaluated by fitting pure component vapor-liquid equilibrium data by means of the least-squares method. The objective function that ensures a good fitting over wide temperature ranges is the following: F)

∑ (ln P

sat,calck

k

- ln Psat,expk)2

(7)

Figure 1 reports, as an example, the comparison between the relative errors in the calculation of the vapor pressure of carbon dioxide14,15 by using the Mathias-Copeman parameters from our procedure and by directly using the parameters implemented in Aspen PLUS. The vapor pressures calculated by the simulator present higher relative errors for low temperatures and pressures. For most of the analyzed compounds, average errors, but mainly maximum errors, in pure-component vapor pressure calculations with the optimization procedure are smaller than the errors made using the adjustable parameters m1, m2, and m3 tabulated in Aspen PLUS; the percentage relative errors in the former case are 0.999), as well as that of glycol (∼10-6-10-7). The subsequent operation of solvent recovery by distillation is easy and does not deserve discussion. The comparison between the simulation results obtained using PSRK and the proposed method shows that the latter, which ensures a better fit of experimental VLE data, allows the correct evaluation of duties in the concentration section. The group contribution model is not able to predict the correct azeotropic composition, overestimating the ethanol molar fraction of ∼0.0261 (see Figure 6); as a consequence, the energy consumption/reflux ratio are underestimated. The greatest differences are found for an ethanol composition at the top of the column close to the azeotrope experimental value (x ) 0.88), as reported in Table 5. As a further consequence, the discrepancy from the experimental datum using the predictive method leads to incorrect answers on the ethanol purity required in the concentration section when minimizing the energy objective function for the entire plant. In contrast, the energy consumption in the dehydration section does not present significant differences in duties but does in the value of the minimum glycol flow rate required for a given ethanol purity at the top of the extractive column. The glycol

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Figure 15. VLE for the ethanol-glycol mixture: comparison between proposed and PSRK models at 1 atm.

Figure 16. Concentration section column duties for different ethanol molar fraction in the distillate.

flow rate evaluated by the predictive model is 25% greater. This is probably due to the less accurate prediction of the experimental65 VLE curve for the ethanol-ethylene glycol system:61 the predictive method gives a more difficult separation than it really is (Figure 15). It has already been remarked that the ethanol purity at the outlet of the concentration section is a degree of freedom of the overall process. The objective of the study is to fix this variable to minimize the variable costs of the plant that consist mainly of two terms: the duty at the reboiler of the high-pressure column and the duty at the reboiler of the regeneration column. The analysis has been performed simulating the overall plant for different ethanol purities in the range of 0.74-0.88. The change in the ethanol molar fraction at the top of the distillation column in the concentration section affects both the energy requirement and the feed split factor necessary to ensure the energy coupling between the low-pressure column reboiler and the high-pressure column condenser. A decrease in the ethanol purity requires an increase of the feed fraction to the high-pressure column to maintain the energy coupling between the columns. In the meantime, the operative costs of the two columns decrease to an asymptotic value (because of the increase in the high-pressure column flow rates), as shown in Figure 16. For the dehydration unit, the analysis of the energy consumption for different ethanol molar fractions of the inlet stream requires the analysis of the best condition of the extractive column (for a given number of trays and feed locations, the column presents three degrees of freedom) in terms of glycol

Figure 17. Ethanol molar fraction in the extractive column distillate as a function of the reflux ratio for given glycol molar flow rates (ethanol molar fraction in the feed ) 0.82).

flow rate vs reflux ratio necessary to ensure the final ethanol purity (>99.9% on molar base). The third degree of freedom (i.e., the recovery ratio of ethanol) is fixed to 99.9%. The ethanol purities versus the reflux ratio for different values of glycol flow rate are reported in Figure 17, which shows that, for a given glycol flow rate, there exists a reflux ratio value that ensures the highest purity. For lower values, the reflux ratio is not sufficient to run the column at the best performance, whereas higher refluxes require a higher amount of water to be evaporated. It is possible to note that this optimized reflux ratio presents a minimum when increasing the glycol flow rate. As a matter of fact, at low glycol flow rates, the best operation is ensured by high reflux ratios; the optimal glycol flow rate (which allows a product ethanol molar fraction at least equal to 0.999) corresponds to the minimum reflux ratio; for high values of glycol flow rate the ethanol purity is high enough at any reflux ratio, however the maximum purity is obtained at higher reflux ratios, to avoid the presence of glycol in the top product. By increasing the glycol flow rate, the minimum value sufficient to ensure the desired ethanol purity is reached; it requires a reflux ratio of 1-1.5. For lower values of the ethanol molar fraction in the feed stream of the extractive column than the value x ) 0.82, used for the simulations reported in Figure 17, the range boundary values increase. The reflux ratio has been fixed to the unity since in these conditions the minimum flow rate that ensures the desired ethanol purity is higher of only 4-5% than the absolute minimum flow rate that corresponds to the reflux ratio of 1.5. The operation at a lower reflux ratio allows to diminish by about 20% the energy requirements at reboiler and condenser. For what concerns the solvent recovery column, the duties of reboiler and condenser diminish when increasing the inlet ethanol purity in the extraction column because of the lower water amount to be separated from glycol. In the extractive column the duties do not change. Thus, the optimal ethanol molar fraction at the top of the concentration columns must be determined by minimizing the objective function that takes into account the duties of the highpressure column reboiler in the concentration unit and of the reboiler of the glycol regeneration column. The optimized value for the ethanol molar fraction at the concentration section outlet for the considered case study is 0.82, as shown in Table 6.

Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010 Table 6. Objective Function Values for Several Ethanol Molar Fractions at the Top of the Concentration Columns concentrated ethanol molar fraction

HP column reboiler duty (concentration unit) [kW]

glycol regeneration column reboiler duty [kW]

objective function [kW]

0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74

367.9 281.0 253.8 240.8 234.4 231.7 230.5 230.1

36.0 42.5 49.3 56.4 63.9 71.7 79.8 88.4

403.9 323.5 303.1 297.2 298.3 303.4 310.3 318.5

5. Conclusions The obtained results show the applicability of the proposed way of modeling to a wide range of mixtures (associated compounds, polar substances, polar-nonpolar compound mixtures). The model is mainly capable of describing the equilibrium for solutions of strongly nonideal chemical species with strong inter and intramolecular interactions. In mixtures with strong deviations from ideality, characterized by the high values of the activity coefficients, we have found major discrepancies between the proposed adaptive parameter method and the twin predictive method (PSRK), whose parameters are directly available in a process simulator databank, highlighting where the use of the former method is suggested. The proposed adaptive method ensures satisfactory results in the prediction of pure-compound vapor pressure and, in systems containing azeotropes, temperature, pressure, and composition of the azeotrope are correctly described. Calculating as accurately as possible the azeotropic values for a mixture is crucial when the simulation of a separation process, whose performances are influenced by the presence of physical constraints, must be performed. The value of an accurate description of VLE is revealed in the ethanol-water separation process where authors have experimented how the wrong prediction of azeotrope can heavily affect the process optimization (for a 2% discrepancy of the azeotropic composition, an increased energy consumption in the concentration section of ∼30% for a given ethanol purity is found). The values of the optimized parameters for the MathiasCopeman correlation and for the NRTL correlation are reported for all the components and binary systems analyzed in this paper. Acknowledgment This paper is dedicated to the memory of Prof. Giuseppe Biardi. The authors are grateful to Giorgio Soave for the useful discussions on thermodynamics. Nomenclature a ) energy parameter of the equation of state [J m3/mol2] A ) dimensionless energy parameter of the equation of state; A ) aP/(RT)2 aij ) adaptive parameter in τij expression b ) molar covolume parameter of the equation of state [m3/mol] B ) dimensionless volumetric parameter; B ) bP/(RT) bij ) adaptive parameter in τij expression [K] F ) objective function g ) molar Gibbs energy [J/mol] G ) term contributing to model excess Gibbs energy k ) binary interaction parameter

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m ) alpha function coefficients P ) pressure [Pa] Q ) function of the MHV-1 and PSRK mixing rules R ) universal gas constant [J/(mol K)] T ) temperature [K] x ) liquid molar fraction y ) vapor molar fraction Greek Symbols R ) attractive parameter Rij ) NRTL non-randomness parameter γ ) activity coefficient φ ) fugacity coefficient τij ) NRTL model binary interaction parameter ω ) acentric factor Subscripts C ) critical value calc ) calculated exp ) experimental i, j ) component index ij ) interaction property between components i and j R ) reduced value sat ) saturation Superscript E ) excess property

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ReceiVed for reView November 9, 2009 ReVised manuscript receiVed March 15, 2010 Accepted March 17, 2010 IE901773Q