Ind. Eng. Chem. Res. 2005, 44, 2845-2847
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PROCESS DESIGN AND CONTROL Heterogeneous Nonequilibrium Mole Fraction Curve Maps L. N. Sridhar, C. Maldonado, Anamaria Garcia, and Jon Irizzarry Chemical Engineering Department, University of Puerto Rico, Mayaguez, Puerto Rico 00681-9046
A strategy for drawing mole fraction curve maps for heterogeneous mixtures, using the nonequilibrium model that involves rigorous mass-transfer calculations, is presented. The calculations involve the use of rigorous stability analysis in the form of the Gibbs tangent plane criterion and nonequilibrium liquid-liquid calculations when the mixture is determined to be unstable. This is followed by a nonequilibrium bubble-point calculation. A comparison of the equilibrium and nonequilibrium mole fraction curve maps for the mixture of ethanol, benzene, and water is presented. Introduction In a recent article,1 a strategy was developed to draw mole fraction curve maps for homogeneous mixtures, using the nonequilibrium model that involves the utilization of mass-transfer models. However, it must be emphasized that these curves were incorrectly referred to as “residue curves”. This error was noted by Dr. Ross Taylor in an E-mail communication, which reads “...the curves that you are calculating are NEQ distillation lines. The mass balance for a packed column at total reflux and CMO reduces to dx/dh ) x - y and that result is independent of the contacting model (it’s just a mass balance). Now, the problem becomes how to relate x to y. There is then no reason you cannot use the model in your paper but you cannot call them NEQ residue curves, because, to my mind, the concept has no meaning...” We will refer to these curves hereafter as mole fraction curves. In this article, we present the extension of the work of Sridhar et al.1 to heterogeneous mixtures. The first attempt to develop residue curve maps for heterogeneous systems involving the equilibrium model was by Pham and Doherty in 1990.2 In this article, they presented a derivation that described the residue curve map equation and developed algorithmic strategy for drawing heterogeneous residue curves. However, this algorithmic strategy lacked a rigorous stability analysis. Subsequently, Wasylkiewicz et al.3 used rigorous stability analysis in the form of the tangent plane criterion to compute all the homogeneous and heterogeneous azeotropes in the mixture. Castillo and Towler4 used Murphree efficiencies to account for departures from equilibrium in the development of distillation line diagrams. Sridhar et al.1 derived the nonequilibrium mole fraction curve map equation (without using efficiencies) for homogeneous systems and showed, numerically and analytically, that the nonequilibrium and equilibrium azeotropes coincide. This approach is more rigorous than simply using an efficiency or any other factor. We first present a derivation for the nonequilibrium mole fraction curve map when the mixture is heterogeneous. We then develop an algorithmic strategy
for drawing these mole fraction curve diagrams and demonstrate this strategy with an example. Derivation of the Nonequilibrium Mole Fraction Curve Equation Consider L moles of a heterogeneous liquid (which is unstable) of overall composition x0i that is being vaporized. The liquid phase that is being unstable will split into two phases, as
Lx0i ) LIxIi + LIIxII i
(1)
A differential portion of this liquid (∆L, which is equal to ∆LI + ∆LII) is vaporized. As a result, the overall composition would change to x0i + ∆x0i . This evaporation can be modified using a mass-transfer approach that is similar to that in Sridhar et al.:1
Lx0i ) NLi 1,VaL1,V + NLi 2,VaL2,V + (L - ∆L)(x0i + ∆x0i ) (2) NL1 1,V and NL1 2,V are the fluxes across the two vapor/ liquid interfaces. The differential amount of vapor formed is given by
Vyi ) NLi 1,VaL1,V + NLi 2,VaL2,V
(3)
Combining eqs 2 and 3, we get
Lx0i ) Vyi + (L - ∆L)(x0i + ∆x0i )
(4)
As in Sridhar et al.,1 we implement the fact that V ) ∆l (by mass balance), and taking the limit ∆L f 0, we get
dxi (dL/L)
) xi - yi
(5)
Setting dL/L ) dt, we obtain the mole fraction curve map equation:
10.1021/ie049470i CCC: $30.25 © 2005 American Chemical Society Published on Web 03/17/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005
dxi ) xi - yi dt
(6)
The vapor mole fraction yi is the vapor phase that we would obtain when we boil a liquid of overall composition x0i . The terms xIi and xII i represent the bulk phase compositions of the two liquid phases, and yi represents that of the vapor phase. These phases are not in equilibrium with each other. The problem that needs to be solved to integrate eq 6 is to find the bulk vapor phase composition yi, given an overall liquid-phase composition x0i , when the liquid phase may split under certain conditions into two liquid phases. The first step required to obtain this result would be a rigorous stability analysis to determine whether the mixture would split into two or more liquid phases. If the mixture is stable, then we follow the procedure outlined in Sridhar et al.,1 calculating the temperature T and the bulk and interface vapor compositions y and yint. (We refer to this calculation as the nonequilibrium BUBL-T calculation.) If the mixture is unstable, then we perform a liquid-liquid nonequilibrium phase split calculation, as described in Lao5 and Sridhar and Torres,6 and obtain two bulk interface compositions (xI and xII) and two liquid interface compositions (xI,int and xII,int). We then perform the nonequilibrium BUBL-T calculation. In the next section, we will describe the stability analysis, the liquid-liquid nonequilibrium phase split calculation, and the procedure for obtaining the vaporphase composition and the temperature. We then will describe the steps of the algorithm used and present an example involving a mixture of ethanol, benzene, and water, where we have compared the equilibrium and nonequilibrium mole fraction curve maps. Results and Discussion Stability Analysis. We used the method described by Michelsen7 to determine whether the mixture is stable. Michelsen7 developed a numerical interpretation of the Gibbs tangent plane criterion (reported previously by Gibbs8 and Baker et al. (in 1981)). Wasylkiewicz et al.9 extended this method to solve problems involving three liquid phases. Briefly, Michelsen7 showed that the tangent plane criterion could be expressed as a set of equations of the form
ln Yi + ln γi - hi ) 0
(7)
where Y represents the vector that demonstrates stability. The parameter γ is the activity coefficient at the composition
xi )
Yi c
(8)
Yi ∑ i)1 and
hi ) ln x0i + ln γi(x0i )
(9)
where x0i represents the overall composition. Michc elsen7 showed that, if ∑i)1 Yi > 1, then the mixture is unstable; otherwise, it is stable. More details can be found in the work by Sridhar and Torres.6 If the mixture is stable, then we perform the nonequilibrium BubL-T
Figure 1. Flowchart for calculating y and T, given composition xo, when the mixture tends to phase-split, using the nonequilibrium model.
calculation that has been described in the work of Sridhar et al.1 If the mixture is unstable, we perform the liquid-liquid phase split calculation (described previously by Lao5 and Sridhar and Torres6), which we briefly discuss in the next subsection. This will be followed by the nonequilibrium BUBL-T calculation. Nonequilibrium Liquid-Liquid-Phase Split Calculation. The nonequilibrium liquid-liquid phase-split calculation described in detail by Lao5 and Sridhar and Torres6 involves the solution of the material balance, the transfer rate, the interface material balance, the interface equilibrium equations, and the composition summation equations. As stated previously by Sridhar and Torres,6 the Michelsen stability test provides excellent starting points for the phase split calculations. We use Newton’s method with finite difference derivatives for the solution of these equations. Details of the variables and equations involved can be found in the work of Lao5 and Sridhar and Torres.6 Nonequilibrium BUBL-T Calculation. The nonequilibrium BUBL-T is described in detail in the work by Sridhar et al.1 and involves the calculation of the vapor-phase composition yi and temperature T, given the liquid-phase composition and pressure. If the mixture is stable, we use x0i ; otherwise, we use xIi , which we obtain as a result of the nonequilibrium liquid-liquidphase split calculation. We solve the transfer rate equations, the energy continuity equation, the interface equilibrium equations, and the summation equations. Details of the variables and equations involved and the solution procedure are discussed in the work by Sridhar et al.1 After the vapor composition y is obtained, eq 6 can be integrated to obtain the mole fraction curve. Algorithm Description. Figure 1 shows a description of the algorithm that we used to obtain y and perform the integration to draw the mole fraction curve map. As mentioned previously, if the mixture is unstable, we perform a liquid-liquid nonequilibrium calculation, followed by a nonequilibrium BUBL-T calculation. If the mixture is stable, we directly perform the nonequilibrium BUBL-T calculation. Pham and Doherty2 have used a similar strategy with the equilibrium model. Figure 2 shows both the equilibrium and
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without the use of efficiencies, has been extended to heterogeneous systems. A comparison between the equilibrium and nonequilibrium mole fraction curve maps for the mixture of ethanol, benzene, and water is presented. It is observed that, in the heterogeneous region, the mole fraction curves are not significantly different from the equilibrium curves. Literature Cited
Figure 2. Comparison of the nonequilibrium and equilibrium mole fraction curve maps for the mixture of ethanol, benzene, and water.
nonequilibrium mole fraction curve maps for the mixture of ethanol, benzene, and water. The liquid-phase diffusivities (which are ∼10-9) are much smaller than the vapor-phase diffusivities (10-4-10-5). Consequently, in liquid-liquid-phase split calculations, the departure from equilibrium is not as pronounced as that in the liquid-vapor calculations. This is the reason, in the liquid-liquid region, the nonequilibrium mole fraction curves do not significantly differ from their equilibrium counterparts. Therefore, it is advisable that, when the mixture is heterogeneous, we can obtain good results by simply drawing equilibrium residue curves. Conclusions The strategy of Sridhar et al.1 for drawing mole fraction curve maps, using the nonequilibrium models
(1) Sridhar, L. N.; Maldonado, C.; Garcia, A. M. Design and Analysis of Nonequilibrium Separation Processes. AIChE J. 2002, 48 (6), 1179. (2) Pham, H. N.; Doherty, M. F. Design and Synthesis of Heterogeneous Azeotropic DistillationssII. Residue Curve Maps. Chem. Eng. Sci. 1990, 45 (7), 1837-1843. (3) Wasylkiewicz, S. K.; Doherty, M. F.; Malone, M. F. Computing All Homogeneous and Heterogeneous Azeotropes in Multicomponent Mixtures. Ind. Eng. Chem. Res. 1999, 38, 4901. (4) Castillo, F. J. L.; Towler, G. P. Influence of Mass Transfer on Homogeneous Azeotropic Distillation. Chem. Eng. Sci. 1998, 53 (5), 963-976. (5) Lao, M. Nonequilibrium Models for Multiphase Separation Processes, Ph.D. Thesis, Clarkson University, Potsdam, NY, 1989. (6) Sridhar, L. N.; Torres, M. Stability Calculations for Nonequilibrium Separation Processes. 1998, 44 (10), 2175. (7) Michelsen, M. L. The Isothermal Flash Problem, Part 1. Stability. Fluid Phase Equilib. 1982, 9, 1. (8) Gibbs, J. W. A Method of Geometric Representation of Thermodynamic Property of Substances by Means of Surfaces. Trans. Conn. Acad. Arts Sci. 1873, 2, 382. (Article 14.) (9) Wasylkiewicz, S. K.; Sridhar, L. N.; Doherty, M. F.; Malone, M. F. Global Stability Analysis and Calculation of Liquid-Liquid Equilibrium in Multicomponent Mixtures. Ind. Eng. Chem. Res. 1996, 35, 1395.
Received for review June 17, 2004 Revised manuscript received January 21, 2005 Accepted February 21, 2005 IE049470I
