Reliable initialization strategy for the equal area rule in flash calculations of ternary systems

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1 Instituto Tecnologico de Aguascalientes From the SelectedWorks of Adrian Bonilla-Petriciolet 2007 Reliable initialization strategy for the equal area rule in flash calculations of ternary systems Adrian Bonilla-Petriciolet Available at:

2 AFINIDAD REVISTA DE QUÍMICA TEÓRICA Y APLICADA EDITADA POR LA ASOCIACIÓN DE QUÍMICOS E INGENIEROS DEL INSTITUTO QUÍMICO DE SARRIÁ Reliable Initialization Strategy for the Equal Area Rule in Flash Calculations of Ternary Systems Adrián Bonilla-Petriciolet Instituto Tecnológico de Aguascalientes, Departamento de Ingeniería Química Av. López Mateos CP 20256, Aguascalientes, Aguascalientes, México Estrategia robusta para la inicialización de la regla de igualdad de área en cálculos flash de sistemas ternarios Estratègia robusta per a la inicialització de la regla d igualtat d àrea en càlculs flash de sistemes ternaris Recibido: 5 de agosto de 2006; revisado: 17 de noviembre de 2006; aceptado: 17 de noviembre de 2006 Afinidad (2007), 64 (530),

3 Reliable Initialization Strategy for the Equal Area Rule in Flash Calculations of Ternary Systems Adrián Bonilla-Petriciolet Instituto Tecnológico de Aguascalientes, Departamento de Ingeniería Química Av. López Mateos CP 20256, Aguascalientes, Aguascalientes, México Estrategia robusta para la inicialización de la regla de igualdad de área en cálculos flash de sistemas ternarios Estratègia robusta per a la inicialització de la regla d igualtat d àrea en càlculs flash de sistemes ternaris Recibido: 5 de agosto de 2006; revisado: 17 de noviembre de 2006; aceptado: 17 de noviembre de 2006 RESUMEN En este trabajo se reporta un nuevo procedimiento para la inicialización y aplicación de la Regla de Igualdad de Área (EAR) en cálculos flash de sistemas ternarios. Específicamente, el vector de composición que minimiza globalmente a la Función Distancia del Plano Tangente es utilizado para estimar valores iniciales adecuados para el vector tie-line del algoritmo EAR Esta estrategia ha sido aplicada en cálculos flash de diversos sistemas ternarios incluyendo mezclas de polímeros y los resultados obtenidos muestran que el método propuesto es robusto y eficiente para la inicialización del algoritmo EAR. Palabras clave: Equilibrio de fases. Regla de Igualdad de Área. Flash. Estabilidad de Fases. RESUM En aquest treball, es presenta un nou procediment per a la inicialització i aplicació de la Regla d Igualtat d Àrea (EAR) en càlculs flash de sistemes ternaris. En concret, s utilitza el vector de composició que minimitza globalment la Funció Distància del Pla Tangent per estimar valors inicials adequats per al vector tie-line de l algoritme EAR. Aquesta estratègia s ha aplicat en càlculs flash de diversos sistemes ternaris, incloses barreges de polímers. Els resultats obtinguts mostren que el mètode proposat és robust i eficient per a la inicialització de l algoritme EAR. Mots clau: Equilibri de fases. Regla d Igualtat d Àrea. Flash. Estabilitat de Fases. SUMMARY In this paper, we describe a new approach for the initialization and application of the Equal Area Rule (EAR) algorithm in flash calculations of ternary systems. Specifically, the composition vector that globally minimizes the Tangent Plane Distance Function is used to estimate good initial values for the tie-line vector used in the EAR algorithm. We have applied this strategy in flash calculations of several ternary systems including polymer mixtures and our results show that the proposed approach is robust and efficient for the initialization of EAR algorithm. Key words: Phase equilibria. Equal area rule. Flash. Phase stability. INTRODUCTION The phase equilibria calculation is an important task in the design, synthesis, control and optimization of chemical and petrochemical processes. The aim is to predict the number, identity and composition of phases at equilibrium state for a multicomponent system at fixed pressure, temperature and composition (Ammar and Renon 1987). Until now, several methods have been developed for phase equi- * Author for correspondence: Telephone and fax: (52) ext petriciolet@hotmail.com 529

4 librium calculations (Castillo and Grossman 1981; Michelsen 1982; Ammar and Renon 1987; Eubank et al. 1992; Eubank and Hall 1995; McDonald and Floudas 1995; Lucia et al. 2000; Stateva et al. 2000). Although all proposed approaches try to solve the same problem, they differ in the problem formulation or in the numerical scheme used to reach the solution. According to Ammar and Renon (1987), the methods for phase equilibria calculation can be classified as equation-solving methods or Gibbs free energy minimization methods. The equation-solving methods try to solve a set of nonlinear descriptive equations using an appropiate numerical technique while the other methods try to minimize the Gibbs free energy using local and global optimization strategies. One of the most robust procedures proposed for twophase equilibria calculation is the Equal Area Rule (EAR) algorithm (Eubank and Hall 1995). The EAR is a Gibbs free energy minimization technique that finds the equilibrium compositions using an integrative approach. The EAR is a high sensitive method near to critical points and in the retrograde region. This method has been tested with several binary and ternary systems, including polymer mixtures, with promising results (Shyu et al. 1995; Yu et al. 2000). However, the proposed algorithm for EAR is inefficient and difficult to implement for multicomponent systems. In this paper, we propose a robust procedure for the initialization of EAR. Specifically, the results of phase stability analysis are used to obtain good initial values for the tie-line vector of EAR algorithm. We show that the proposed approach allows the easy application of EAR in flash calculations of ternary systems. EQUAL AREA RULE FOR PHASE EQUILIBRIA CALCULATIONS Eubank and Hall (1995) have showed that an equal rule, similar to that proposed by Maxwell for determining saturation volumes, can be applied in the calculation of phase equilibrium. They showed that the tangent plane criterion also reduces to an equality of areas in the plot of (dg ) versus the mole fraction x 1, where g is the Gibbs free energy of mixing. The equilibrium conditions are the set of compositions that equals the two areas L and U of this plot, see Figure 1. Eubank and Hall (1995) have developed an algorithm for the EAR, which converges in three to five iterations assuming reasonable initial values. For binary systems, the procedure begins assuming an initial value for (dg ) and then the phase compositions are calculated. The calculated compositions are used to adjust (dg ) employing the next equation α β where k is the iteration number, x 1 and x 1 are the mole fractions at the equilibrium phases α and β, respectively. Generally, the initial value for dg is the average value of the stationary points of the plot ( dg ) x 1. The convergence is checked by comparing the difference between (dg ) k and (dg ) k+1, if the difference is less than an acceptable tolerance value, then the calculated compositions are the equilibrium compositions (Eubank and Hall 1995). Application of EAR to multicomponent mixtures (c 3) requires the use of the tie-line vector procedure of Eubank et al. (1992). For ternary systems, the equilibrium compositions of two phases are connected by a straight line which passes through the overall composition of the mixture (z 1, z 2, z 3 ) and satisfies the material balances (Eubank et al. 1992; Shyu et al. 1995). At equilibrium, the relative composition change between two components at the overall composition, along a tie-line, is a constant (Shyu et al. 1995). For a multicomponent mixture, the function of the relative composition change (x ij ) for components (i,j) along a tie-line can be defined as where x α i is the mole fraction of component i at phase α, x β i is the mole fraction of component i at phase β and z i is the mole fraction of component i at the feed, respectively. If the set of x ij and the overall composition are available, x j can be expressed as function of x j using where (1) (2) x j = x ij (x i z i ) + z j j = 1,...,c., for j i (3). This result implies that g depends upon c 1 variables: g = g(x i, x ij ) where j = 1,...c 1 subject to j i. The tie-line functions x ij are unknowns that we need to determine for calculating the equilibrium compositions and they are the key parameters for convergence of EAR algorithm. As shown by Shyu et al. (1995) when the tie-line Figure 1. Equal Area Rule for flash calculations. 530

5 functions are close to its equilibrium values, the function (dg ) exhibits a van der Waals loop behavior. For a system of c components, (dg ) is given by where being µ^i the chemical potential of component i at the mixture. To determine the equilibrium values of functions x ij, the following condition must be satisfied The equilibrium compositions are calculated by searching the tie-line functions x ij to make the Eq. (6) less than a tolerance value. For multicomponent systems, the EAR algorithm involves two stages: an external stage where the functions x ij are searching until satisfy Eq. (6), and an internal stage where the EAR procedure is applied to calculate the equilibrium compositions for the given set of functions x ij. The overall method is outlined as follows for a flash calculation of a multicomponent system 1. Assume initial values for the functions x ij. 2. Calculate the limits (x i min, x i max ), for the proposed values of x ij, by checking the conditions 0 < x 1,...,x c < 1 and c Σ x i = 1. i=1 3. Transform x j (j = 1,...,c j i) to a function of x i and x ij. 4. Do the EAR procedure (internal stage) inside the domain (x i min, x i max ), if van der Waals loop exists for ( dg ) x 1, then check the Eq. (6). 5. If the Eq. (6) is satisfied, then the convergence is achieved. Otherwise adjust the x ij functions and repeat steps 2-5 until satisfy the convergence criterion. 6. Calculate x j (j = 1,...,c., for j i) by the converged x i and x ij. These compositions are thus the equilibrium compositions. (4) (5) (6) For performing flash calculations using this method, it is necessary to locate the intervals for functions x ij where a van der Waals loop behavior exists for dg. The location of those intervals involves a high computational time because it is necessary to perform a full search in the real variable space where x ij (, ). This search is the most critical stage of EAR algorithm. Some researchers have suggested heuristics to estimate initial values for the functions x ij (Shyu et al. 1995; Shyu et al. 1996; Yu et al. 2000). However, there is not a general procedure for the estimation of suitable initial values for x ij in multicomponent systems. Therefore, in this paper a general procedure for solving this problem is proposed and applied to phase equilibrium calculations in ternary systems. In the next section, we will describe our initialization strategy for EAR. NEW PROCEDURE FOR THE INITIALIZATION OF EAR ALGORITHM Our strategy to initialize the EAR algorithm is based on the application of results from phase stability analysis. As indicated by Michelsen (1982), the phase stability of a nonreactive mixture with c components and a global composition z{z 1,...,z c }, at constant pressure and temperature, can be determined through the global minimization of the Tangent Plane Distance Function TPDF = c Σ x i ( µ^i (x) µ^i (z) i=1 ) (7) If TPDF 0 for all compositions x, the mixture is stable otherwise it is unstable and presents phase equilibrium. The composition vector that globally minimizes TPDF can be used to estimate good initial values for x ij employing the next equation TPDF where x i is the mole fraction of component i that corresponds to the global minimum of TPDF. We have performed several calculations employing this approach and the obtained results indicate that the estimations for x ij, using the global minimum of TPDF, are good initial values for performing phase equilibrium calculations with EAR. This is because the composition vector that globally minimizes TPDF, for an unstable mixture, is very near to one equilibrium composition. We perform the stability analysis by minimizing TPDF with respect to all mole numbers of system components. For this purpose, we use the stochastic global optimization method Simulated Annealing (SA) (Rangaiah 2001). However, other global optimization methods as Genetic Algorithm or Tabu Search can be used (Teh and Rangaiah 2003). SA method has been tested with several systems and it has been found that this method is robust for phase stability calculations (Bonilla-Petriciolet et al. 2006). Our initialization procedure for EAR is outlined as follows (see Figure 2): (8) Figure 2. Initialization procedure for the EAR algorithm in flash calculations of multicomponent systems. 531

6 1. Specify temperature, pressure, thermodynamic models and global composition z 2. Do a phase stability analysis by globally minimizing TPDF. 3. If the mixture is unstable, calculate the initial values for the tie-line vector using the global minimum of TPDF and Eq. (8). 4. Perform phase equilibrium calculations using the EAR algorithm for multicomponent mixtures (steps 2 6 of the algorithm described above). This initialization strategy can be applied with any thermodynamic model and phase equilibrium type. It is important to note that the numerical effort involved in the phase stability analysis is lower than searching the intervals for the functions x ij. In the next section, as study case, we present the results obtained from the numerical implementation of this procedure in ternary systems. TEST EXAMPLES AND RESULTS Example 1: N 2 C 1 C 2 We study this ternary system using the SRK EoS with classical mixing rules and the pure component parameters reported by Reid et al. (1987). Three feeds are analyzed at 270 K and 7600 KPa and Table I reports the results of phase stability analysis and the initial and equilibrium values for the function x 12. The equilibrium mole fractions are also reported in this table. Our results indicate that the initial values calculated for x 12 are near to the equilibrium values. There is a mean difference of 5.7% between the initial and equilibrium tie lines. Our calculated values for x 12 improve the numerical behavior of EAR algorithm and allow its easy convergence to the equilibrium compositions. Shyu et al. (1995) have suggested an initial value of x 12 = -1 for systems with small molecules. In this case, this heuristic is not appropriated. Example 2: Hypothetical ternary mixture Our second example is a ternary mixture whose thermodynamic properties are defined by G E = 2.5 x 1 x x 1 x x 2 x 3 (9) RT where G E is the excess Gibbs free energy. Flash calculations are performed for three feeds and the results are reported in Table II. Again, the calculated values for x 12 using the results of phase stability analysis are very close to the equilibrium values. In this case, the mean deviation between initial and equilibrium tie line functions is 3.2%. For these feeds, the heuristic value of Shyu et al. (1995) is a suitable estimation to begin the equilibrium calculations. TABLE I Results of flash calculation for N 2 - C 1 C 2 at 270 K and 7600 KPa. Stability analysis Value of x 12 Equilibrium compositions z x TPDF Initial Equilibrium x α x β (0.14, 0.12, 0.74) (0.3182, , ) (0.1383, , ) (0.3153, , ) TPDF = (0.213, 0.178, 0.609) (0.1269, , ) (0.1311, , ) (0.2834, , ) TPDF = (0.31, 0.165, 0.525) (0.1338, , ) (0.1396, , ) (0.3211, , ) TPDF = TABLE II Results of flash calculation for a hypothetical ternary mixture. Stability analysis Value of x 12 Equilibrium compositions z x TPDF Initial Equilibrium x α x β (0.4, 0.58, 0.02) (0.8578, , ) (0.1536, , ) (0.8248, , ) TPDF = (0.493, 0.411, 0.096) (0.1681, , ) (0.1957, , ) (0.7004, , ) TPDF = (0.25, 0.72, 0.03) (0.8543, , ) (0.1585, , ) (0.8084, , ) TPDF =

7 TABLE III Results of flash calculations for Toluene - Water - Aniline at 1 atm and 298 K. Stability analysis Value of x 12 Equilibrium compositions z x TPDF Initial Equilibrium x α x β ( , , ) (0.0001, , ) (0.0001, , ) (0.3467, , ) TPDF = (0.15, 0.15, 0.7) (0.0001, , ) (0.0001, , ) (0.1512, , ) TPDF = (0.4, 0.4, 0.2) (0.0001, , ) (0.0001, , ) (0.6537, , ) TPDF = Example 3: Toluene Water - Aniline We use a ternary system studied by McDonald and Floudas (1995) at 1 atm and 298 K. At these operation conditions, this mixture presents liquid-liquid equilibrium. The thermodynamic properties are calculated using the NRTL model with the parameters reported by McDonald and Floudas (1995). Three feeds are analyzed and the results of flash calculations appear in Table III. Again, our approach generates good initial values for the tie-line function and there is a mean deviation of only 0.21% between calculated and equilibrium values. In some feeds, the value proposed by Shyu et al. (1995) is far from the equilibrium tie line. Example 4: Polymer 1-Polymer 2-Solvent mixture Our last example is a ternary polymer mixture reported by Yu et al. (2000). The Flory Huggins model for the Gibbs free energy of mixing is used with the parameters: m 1 = 4000, m 2 = 250, m 3 = 1, X 12 = 0.02, X 13 = 0.15 and X 23 = 0.1 where m i is the ratio of the molar volume of component i to that of a reference component and X ij is the Flory interaction parameter, respectively. Three feeds are analyzed and the results of flash calculations are reported in Table IV. Our initial tie lines are very close to the equilibrium values and there is a mean deviation of only 0.77%. In spite of the mole fraction magnitudes of this polymer mixture, our initialization procedure is very reliable. For ternary polymer mixtures, Yu et al. (2000) have proposed that the negative ratio m 1 /m 2 is a good initial value for X 12. In this example, this ratio is equal to 16. However, this value is far from the equilibrium tie line and our initial estimations are more suitable to perform the phase equilibrium calculations. CONCLUSIONS A robust procedure for the initialization of the tie-line functions involved in the Equal Area Rule has been developed. We can calculate good initial values for the functions x ij using the results of phase stability analysis. Our strategy can be applied for any mixture, equilibrium type and thermodynamic model. The proposed approach can be extended for flash calculations in multicomponent mixtures. ACKNOWLEDGEMENTS The author acknowledges the financial support from Instituto Tecnológico de Aguascalientes. TABLE IV Results of flash calculations for Polymer 1 - Polymer 2 - Solvent mixture. Stability analysis Value of x 12 Equilibrium compositions z x TPDF Initial Equilibrium x α x β ( , 0.001, ) ( , , ) ( , , ( , , ) ) TPDF = ( , 0.001, ( , , ( , , ( , , ) ) ) ) TPDF = ( , 0.008, ( , , ( , , ( , , ) ) ) ) TPDF = 2.9E

8 NOMENCLATURE c G E g k m i R T TPDF x i component number excess Gibbs free energy Gibbs free energy of mixing iteration number ratio of the molar volume of component i to that of a reference component gas universal constant temperature Tangent Plane Distance Function mole fraction of component i x min, x max limits for mole fraction x i TPDF mole fraction of component i that corresponds to the global minimum of TPDF x ij relative composition change for components (i,j) along a tie-line z i mole fraction of component i at the feed α, β phases at equilibrium µ^ i X ij chemical potential of component i at the mixture Flory interaction parameter BIBLIOGRAFÍA Ammar M.N. and Renon H.: «The isothermal flash problem: new methods for phase split calculations». AIChE Journal 1987, 33: 926. Bonilla-Petriciolet A., Vázquez-Román R., Iglesias-Silva G.A., Hall K.R.: «Performance of stochastic optimization methods in the calculation of phase stability analyses for nonreactive and reactive mixtures». Industrial Engineering Chemistry Research 2006, 45: Castillo J., Grossmann I.E.: «Computation of phase and chemical equilibria». Computers Chemical Engineering 1981, 5: 99. Eubank P.T., Elhassan A.E., Barrufet M.A., Whiting W.B.: «Area method for prediction of fluid phase equilibria». Industrial Engineering Chemistry Research 1992, 31: 942. Eubank P.T., Hall K.R.: «Equal area rule and algorithm for determining phase compositions». AIChE Journal 1995, 41: 924. Hanif N.S.M., Shyu G.S., Hall K.R., Eubank P.T.: «Calculation of multi-phase equilibria using the equal area rule with application to hydrocarbon/water mixtures». Fluid Phase Equilibria 1996, 126: 53. Lucia A., Padmanabhan L., Venkataraman S.: «Multiphase equilibrium flash calculations». Computers Chemical Engineering 2000, 24: Michelsen M.L.: «The isothermal flash problem. Part I. Stability». Fluid Phase Equilibria 1982, 9: 1. McDonald C.M., Floudas C.A.: «Global optimization for the phase stability problem». AIChE Journal 1995, 41: Rangaiah G.P.: «Evaluation of genetic algorithms and simulated annealing for phase equilibrium and stability problems». Fluid Phase Equilibria 2001, : 83. Reid R.C., Prausnitz J.M., Poling B.E.: «The properties of gases and liquids», 4th ed., McGraw-Hill, New York Shyu G.S., Hanif N.S.M., Alvarado J.F.J., Hall K.R., Eubank P.T.: «Equal area rule methods for ternary systems». Industrial Engineering Chemistry Research 1995, 34: Shyu G.S., Hanif N.S.M., Alvarado J.F.J., Hall K.R., Eubank P.T.: «Maximum partial area rule for phase equilibrium calculations». Industrial Engineering Chemistry Research 1996, 35: Stateva R.P., Cholakov G.S., Galushko A.A., Wakeham W.A.: «A powerful algorithm for liquid-liquid-liquid equilibria predictions and calculations». Chemical Engineering Science 2000, 55: Teh Y.S., Rangaiah G.P.: «Tabu search for global optimization of continuous functions with application to phase equilibrium calculations». Computers Chemical Engineering 2003, 27: Yu M., Sauvé R., Khoshkbarchi M.K., Zhao E.: «A highly convergent algorithm for phase equilibria calculation of ternary polymer/solvent systems using an equal-area rule method». Industrial Engineering Chemistry Research 2000, 39: