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Ind. Eng. Chem. Res. 2001, 40, 3664-3669
A Fast Method to Check Steady-State Input Multiplicities in Vapor-Liquid Flash Separation Rosendo Monroy-Loperena* Simulacio´ n Molecular, Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, 07730 Distrito Federal, Me´ xico
Steady-state input multiplicities in vapor-liquid flash separation processes are common when a product mole fraction is specified and places restrictions on the selection of controlled variables such that more than one set of conditions can produce the same result; if this happens, then potential problems during the process operation can arise. The equations that describe isobaric flash operation with a product composition specified are analyzed to propose a fast method for determining the existence of steady-state input multiplicities. The ability of the method is shown via numerical examples using homogeneous nonazeotropic multicomponent mixtures. The most attractive features of the proposed method are that it is easy to use, has a high reliability in determining the presence of steady-state input multiplicities, and requires only the separation factors of each component at the bubble and dew points at the pressure of the flash separation process. 1. Introduction The possibility of multiple steady states in ternary distillation separations involving homogeneous systems was first conjectured in 1971.1 Doherty and Perkins,2 by studying the stability of the dynamic model equations and restricting their attention to constant molar overflow, showed that multiple steady states cannot occur for binary distillation separations. Sridhar and Lucia3,4 came to the conclusion that the existence of solution uniqueness in those processes is attributed to the homogeneous nature of the mixture (i.e., to the stability of the single vapor and liquid phases) rather than to the assumption of constant molar overflow among stages. Jacobsen and Skogestad5 demonstrated that multiple steady states can arise in binary distillation systems if flow rates are given on a mass or volume basis instead of on a molar basis. They showed that the coupling between molar mass balances and the energy balance also produces multiple steady states for binary distillations. Lucia and Li6 distinguished the cases in which one unique solution and multiple solutions exist for the distillation separation of binary mixtures with nonideal vapor-liquid equilibria, including the energy balance. Gani and Jorgensen7 studied multiplicity in distillation separation processes and showed the effect of overdesign; they concluded that the possibility of multiple solutions increases with increasing number of stages. On the other hand, Lucia8 and Michelsen and Heidemann9 discussed the uniqueness of solutions in isobaric single-stage flash separations and concluded that the solution involving a multicomponent homogeneous mixture is unique. The monotonic behavior of the flash equations was demonstrated for isobaric separations. Recently, Tiscaren˜o et al.10 determined the conditions under which steady-state input multiplicities can be predicted when a product mole fraction is specified in a single-stage flash. * E-mail:
[email protected]. Tel.: +52-5333-8105. Fax: +52-5333-6239.
Figure 1. Input multiplicity.
As illustrated in Figure 1, steady-state input multiplicities in single-stage flash calculations can occur when two or more sets of system input variables (temperature or vaporization) provide the same output condition (outlet composition of component i). Because input variables are those that can be manipulated by controllers, potential problems during process operation can arise. This work concentrates on the development of a fast method for determining steady-state input multiplicities when a product mole fraction is specified in an isobaric flash process. First, a rigorous method for testing for steady-state input multiplicities is proposed. Then, a shortcut procedure for predicting input multiplicities in a faster form is developed. Numerical simulations are also presented to show the application of both the rigorous and simplified methods. 2. Problem Description Consider the flash separation process shown in Figure 2. The typical problem for two-phase vapor-liquid
10.1021/ie000662x CCC: $20.00 © 2001 American Chemical Society Published on Web 07/13/2001
Ind. Eng. Chem. Res., Vol. 40, No. 16, 2001 3665
As suggested by Tiscaren˜o et al.,10 one can develop a method for predicting input multiplicities by finding the shape of the function that defines the problem. Intuitively, if steady-state input multiplicities exist, then eq 4 or eq 5 must be a concave function with a maximum or a convex function with a minimum. To verify this statement, we begin by using eq 4. Assuming that a maximum or a minimum exists, then a stationary point must exist within a real vapor-liquid solution, i.e., 0 e V/F e 1. Thus
dxi d(V/F)
)
-zi[(Ki - 1) + (V/F)* dKi/d(V/F)] [(Ki - 1)(V/F)* + 1]2
) 0 (7)
or Figure 2. Schematic representation of a flash separation.
(V/F)* ) -
equilibria is to solve the system of material balance, equilibrium, and consistency equations given by
[1 - (V/F)]xi + (V/F)yi - zi ) 0
(1)
Kixi - yi ) 0
(2)
C
C
yi - ∑ xi ) 0 ∑ i)1 i)1
(3)
where V/F is the ratio of the total molar flow rate of the vapor to that of the feed stream (usually called flash vaporization); C is the number of species in the mixture; xi, yi, and zi are the mole fractions of component i (i ) 1,... , C) in the liquid phase, vapor phase, and feed stream, respectively; and Ki is the separation factor of component i, which is defined as yi/xi. In general, the Ki’s are obtained from a thermodynamic model and are functions of the temperature, T; pressure, P; liquid mole fractions, xi; and the vapor mole fractions, yi. Equations 1-3 can been rewritten as
xi ) yi ) C
∑ i)1(K
i
zi (Ki - 1)(V/F) + 1 ziKi (Ki - 1)(V/F) + 1 zi(Ki - 1) - 1)(V/F) + 1
)0
(4)
(5)
(Ki - 1)
(8)
dKi/d(V/F)
as both Ki and dKi/d(V/F) are positive in the real solution. Therefore, Ki e 1, so 0 e (V/F)* e 1. Note that, as (V/F)* f 0, eq 7 is positive, and as (V/F)* f 1, eq 7 is negative; therefore, a maximum will exist in the interval 0 e (V/F)* e 1. On the other hand, if we analyze the problem in the temperature domain, to have a real vapor-liquid solution, the temperature must lie between the bubble-point temperature, TB, and the dewpoint temperature, TD. Thus, a stationary point must be stated as
dxi -zi[(Ki - 1) + (V/F)* dKi/dT] ) )0 dT [(K - 1)(V/F)* + 1]2
(9)
i
or
(V/F)* ) -
(Ki - 1) dKi/dT
(10)
as dKi/dT is positive in the real solution. Therefore, as T* f TB, eq 9 is positive, and as T* f TD, eq 9 is negative, so a maximum will exist in the interval TB e T* e TD. For the vapor phase using eq 5, we obtain
dyi
(6)
The last equation is well-known as the Rachford-Rice equation.11 Simple analysis shows that there are C + 2 degrees of freedom, which are usually fixed by specifying C feed mole fractions, zi; the temperature, T; and the pressure, P, of the flash separation process. Assuming that the specifications are chosen below the critical state of a nonazeotropic mixture, one then has a so-called isobaricisothermal problem. Numerical procedures proposed to solve such problem are reviewed by King12 and Henley and Seader.13 Now consider the isobaric-product mole fraction case, where the C + 2 degrees of freedom are fixed by specifying C feed mole fractions, zi; the flash drum pressure, P; and a product composition, say xr or yr. For this case, more than one physical solution might exist.
) d(V/F) -zi[Ki(Ki - 1) + (1 - (V/F)*) dKi/d(V/F)] [(Ki - 1)(V/F)* + 1]2
) 0 (11)
or
(V/F)* ) 1 -
Ki(Ki - 1) dKi/d(V/F)
(12)
as both Ki and dKi/d(V/F) are positive in the real solution. Therefore, Ki g 1, so 0 e (V/F)* e 1. Note that, as (V/F)* f 0, eq 11 is positive, and as, (V/F)* f 1, eq 11 is negative; therefore, a maximum will exist in the interval 0 e (V/F)* e 1. Alternatively, in the temperature domain
dyi -zi[Ki(Ki - 1) + (1 - (V/F)*) dKi/dT] ) 0 (13) ) dT [(K - 1)(V/F)* + 1]2 i
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or
(V/F)* ) 1 -
Ki(Ki - 1) dKi/dT
(14)
as dKi/dT is positive in the real solution. Therefore, as T* f TB, eq 13 is positive, and as T* f TD, eq 13 is negative, so a maximum will exist in the interval TB e T* e TD. It is important to note that the existense of a stationary point is necessary but not sufficient to identify steady-state input multiplicities for component i in either phase. Thus, we need to know the temperature and vaporization range where the stationary point is located. Because a maximum is obtained for both phases, Jensen’s inequality14 for a concave functions holds; that is
f[(1 - λ)w1 + λw2] g (1 - λ)f(w1) + λf(w2) (15) for all 0 e λ e 1 and any two points w1 and w2 in the two-dimensional space of w ) {T,(V/F)}T. Intuitively, the surface f(w) is concave if a line segment joining any two points on the surface lies on or below the surface. A more restrictive concavity definition can be obtained by expanding the left-hand side of inequality 15 using a Taylor expansion to obtain, for small values of λ f[(1 - λ)w1 + λw2] ) f[w1 + λ(w1 - w2)] 2
) f(w1) + λ
∑ i)1
( ) df
dwi
(wi,2 - wi,1)
w1
provided that the second-order terms in the expansion can be neglected. Substituting this approximation into inequality 15 then gives the more restrictive condition of concavity as 2
f(x2) e f(x1) +
∑ i)1
( ) df
dxi
(xi,2 - xi,1)
(16)
x1
The uniquiness analysis of Lucia8 has shown that only one equilibrium temperature exists for an equilibrium vaporization and vice versa. This restricts our analysis to a one-dimensional concave case, either in the vaporization domain or in the temperature domain. Therefore, for the vaporization domain in the vapor phase
yi[(V/F)2] e yi[(V/F)1] +
(
dyi
)
d(V/F)1
(V/F)1
[(V/F)2 - (V/F)1] or
(
dyi
d(V/F)1
)
g
(V/F)1
yi[(V/F)2] - yi[(V/F)1] (V/F)2 - (V/F)1
(17)
for (V/F)2 > (V/F)1. Thus, the slope of yi at the point (V/ F)1 should be greater than the slope of the secant through (V/F)1 and (V/F)2. Alternatively, in the temperature domain
( ) dyi dT1
T1
g
yi(T2) - yi(T1) T2 - T 1
(18)
for T2 > T1, i.e., the slope of yi at the point T1 is greater than the slope of the secant through T1 and T2. The same holds for the liquid phase. From these results, one can conclude that, to identify steady-state input multiplicities, it is only necessary to inspect either the vaporization domain or the temperature domain, because if a component has a maximum with respect to vaporization, then it also has a maximum with respect to temperature. Our rigorous method in the vaporization domain thus consists of the following three steps, for instance, in the vapor phase for component i: (1) Select as many vaporization points as necessary to generate the composition profile of component i by solving eq 6 for temperature. (2) Determine the concave range of the calculated profile, (V/F)min e (V/F) e (V/F)max, using eq 17. (3) If eq 13 evaluted at (V/F)min is positive and at (V/F)max is negative, then component i has steady-state input multiplicity at the specified pressure. 3. Simplified Approach We now develop a shortcut method for determining whether a component shows steady-state input multiplicity either in the liquid phase or in the vapor phase. Consider the worst case in which only the bubble and dew points of the system are known and no thermodynamic method is available to generate intermediate composition points, i.e., eq 4 or/and 5. In the temperature domain, we propose a Clapeyrontype emprirical relation indepedent of composition to interpolate Ki values as a function of temperature between the known bubble and dew points; that is
ln K ˜ i ) ln K0i + (ln K1i - ln K0i )(TD-1 - TB-1)-1 (T-1 - TB-1) (19) where K0i refers to the separation factor of component i at the bubble point, K1i refers to the separation factor of component i at the dew point, and K ˜ i is the estimated separation factor. In the vaporization domain, we propose a linear semilogarithmic dependence of the Ki’s with respect to the vaporization between the known bubble and dew points; that is
ln K ˜ i ) ln K0i + (ln K1i - ln K0i )(V/F)
(20)
This approach is potentially more efficient because it avoids the need for nested iteration loops related to the solution of eq 6. However, the mole fractions calculated by eq 4 and/or eq 5 using eq 20 likely will not sum unity, so they will have to be normalized. Because we are able now to generate intermediate composition points, we propose the following method in the vaporization domain to determine the existence of component steadystate input multiplicities. For component i in the liquid phase: (1) If Ki g 1 at the bubble point and Ki g 1 at the dew point, then component i might not have steady-state input multiplicity. If this is not the case, then go to step 2. (2) Using a direct search method, i.e., Fibonacci search or golden section,15 use eqs 4 and 20 to generate all liquid compositions for each vaporization tested, and normalize these compositions to locate the maximum value of the liquid composition of component i in the interval 0 e (V/F) e 1. (3) If the maximum composition is located
Ind. Eng. Chem. Res., Vol. 40, No. 16, 2001 3667 Table 1. Characteristics of the Study Case at a Pressure of 2068 kPa component
zi
Ki at TB
Ki at TD
C2 C3 iC4 nC4 iC5 nC5 temperature (K)
0.1 0.1 0.1 0.1 0.3 0.3
4.9664 1.6100 0.8586 0.6763 0.3426 0.2870 364.40
6.5124 2.6907 1.6315 1.3550 0.7942 0.6902 414.27
within the interval 0 < (V/F) < 1, then component i has steady-state input multiplicity. For component i in the vapor phase: (1) If Ki e 1 at the bubble point and Ki e 1 at the dew point, then component i might not have steady-state input multiplicity. If this is not the case, then go to step 2. (2) Using a direct search method, use eqs 5 and 20 to generate all vapor compositions for each vaporization tested, and then normalize them to locate the maximum value of the vapor composition of component i in the interval 0 e (V/F) e 1. (3) If the maximum composition is located within the interval 0 < (V/F) < 1, then component i has steady-state input multiplicity. Regarding the proposed method, the following comments are in order: (a) Through knowledge of the separation factors at the bubble and dew points, one can discard those components that will not present steadystate input multiplicities, because they will not generate a maximum in the real solution, i.e., 0 e (V/F) e 1. (b) We suggest that a direct search method be used to find a maximum, rather simply analyzing the sign of eq 7 or eq 11 for (V/F) ) 0 and (V/F) ) 1, because the ˜ i(ln K1i - ln K0i ) always aproximation dK ˜ i/d(V/F) ) K generates concave functions with a maximum in the real solution, i.e., dxi/d(V/F) > 0 for (V/F) ) 0 and dxi/d(V/F) < 0 for (V/F) ) 1. However, the normalized compositions obtained using eq 20 are very similar to the true values and generate concave, convex, and concave-convex functions as expected. (c) We recommend either the Fibonacci search method or the golden section method because, with an accuracy of 10-3, only 16 points are required to obtain the answer. (d) The reliability of the proposed method depends on the fact that eq 20 holds for the mixture under investigation. We have found that, for nonazeotropic mixtures below the critical state and far from the retrograde zone, eq 20 always holds.
Figure 3. Liquid-phase composition behavior as a function of the vaporization.
4. Numerical Example We have conducted numerical simulations on a variety of homogeneous multicomponent mixtures in order to verify the adequacy of the proposed method. We have reworked the example provided by Tiscaren˜o et al.10 The polynomial expressions given by Holland16 were used to estimate the Ki values. Table 1 provides a description of the study case, as well as the separation factors for each component at the bubble point and at the dew point. By making a quick analysis of the values of the separation factors, we can see inmediately that, in the liquid phase, components C2 and C3 will not have steady-state input multiplicities, and in the vapor phase, iC5 and nC5 will not have steady-state input multiplicities. Figures 3-6 were obtained by solving eq 6 for the temperature fixing the vaporization (isothermal-isobaric problem) using the exact separation factors. Figures 3 and 4 show the dependences of the liquid mole
Figure 4. Liquid-phase composition behavior as a function of the temperature.
fraction on the vaporization and the temperature, respectively. We can see that nC4, iC5, and nC5 have concave profiles in range 0 e (V/F) e 1; however, iC4 is only concave in the range 0 e (V/F) e 0.7. Thus, if we only expect that dxi/d(V/F) > 0 for (V/F) ) 0 and dxi/ d(V/F) < 0 for (V/F) ) 1, we will reach an incorrect conclusion. Therefore, it is important to note that, in the rigorous method, one first needs to find the concave range of the function before applying the criterion of change of signs. The components that show a change of sign for the stationary condition in the concave region are only iC4 and nC4, so we can conclude that those components have steady-state input multiplicities in the liquid phase.
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Ind. Eng. Chem. Res., Vol. 40, No. 16, 2001 Table 2. Results with the Rigorous Method liquid phase component C2 C3 iC4 nC4 iC5 nC5
(V/F)*
T*
vapor phase xmax i
(V/F)*
T*
ymax i
out of range 0.127 380.65 0.174 0.035 369.70 0.100 0.374 397.33 0.119 0.109 378.86 0.102 0.531 403.45 0.108 out of range out of range -
Figures 7 and 8 show the estimated dependences on the vaporization of the liquid and vapor mole fractions, respectively. Note that the use of eq 20 reconstructs the composition profiles with a good accuracy. Following the methodology described in section 3 for the liquid phase and using step 1, we discard compo-
Figure 5. Vapor-phase composition behavior as a function of the vaporization.
Figure 7. Estimated liquid-phase composition behavior as a function of the vaporization.
Figure 6. Vapor-phase composition behavior as a function of the temperature.
Figures 5 and 6 show the dependences of the vapor mole fraction on the vaporization and the temperature, respectively. Proceeding with the rigorous method, we find that C3, iC4, nC4, iC5, and nC5 have concave profiles in the range 0 e (V/F) e 1; however, iC5 and nC5 are not considered, because the separation factors might the bubble and dew points are less than unity. Of the remaining species C2, C3, iC4, and nC4, only C3, iC4, and nC4 have steady-state input multiplicities in the vapor phase, because they have changes of sign for the stationary condition when (V/F) ) 0 and (V/F) ) 1. The numerical values obtained from the rigorous method are summarized in Table 2.
Figure 8. Estimated vapor-phase composition behavior as a function of the vaporization.
Ind. Eng. Chem. Res., Vol. 40, No. 16, 2001 3669 Table 3. Results with the Proposed Method liquid phase
vapor phase
component
(V/F)*
xmax i
(V/F)*
ymax i
C2 C3 iC4 nC4 iC5 nC5
0.041 0.130 1.000 1.000
0.100 0.102 0.378 0.435
0.000 0.170 0.428 0.571 -
0.497 0.177 0.121 0.109 -
nents C2 and C3 as not having steady-state input multiplicities, and proceed to find the vaporization that gives the maximum liquid composition for each component. It is found that only iC4and nC4 have maxima located in 0 < (V/F) < 1; therefore, these components have steady-state input multiplicities in the liquid phase. Note that one does not need to first search for the concave region, as the direct search method does this implicity. For the vapor phase, we discard components iC5 and nC5 by applying step 1 of our procedure, because the separation factors might the bubble and dew points are less than unity. Of the remaining components, only C3, iC4, and nC4 have maxima in the range of vaporization with 0 < (V/F) < 1, so these components have steady-state input multiplicity in the vapor phase. The numerical values obtained are summarized in Table 3. Note that these results are in agreement with the results from the rigorous method and with those presented in ref 10. 5. Conclusions Rigorous and simple procedures to identifying steadystate input multiplicities in either vapor or liquid equilibrium phases have been developed. The main advantage of the proposed fast method is that it only requires information on the bubble and dew points and does not require any thermodynamic derivatives to predict steady-state input multiplicities with good accuracy. The importance of having a fast method to test for steady-state input multiplicities arises from the viewpoint of robust control, in that the presence of steadystate input multiplicities creates changes in the process gain, which influences the selection of the control configuration and restricts the range of variables that can be used to infer the state of the process. With no control (manual operation), the operator still requires additional information to determine the operating re-
gion in which the process is located, so that the appropiate changes to the manipulated variables can be made to move the process toward its optimum operating point. Literature Cited (1) Petlyuk, F. B.; Avetyan, V. S. Investigation of three component distillation at infinite reflux. Theor. Found. Chem. Eng. 1971, 5, 499. (2) Doherty, M. F.; Perkins, J. D. On the dynamics of distillation processes: IV. Uniqueness and stability of the steady state in homogeneous continuous distillation. Chem. Eng. Sci. 1982, 37, 381. (3) Sridhar, L. N.; Lucia, A. Analysis and algorithms for multistage separation processes. Ind. Eng. Chem. Res. 1989, 28, 793. (4) Sridhar, L. N.; Lucia, A. Analysis of multicomponent, multistage separation processes: Fixed temperature and pressure profiles. Ind. Eng. Chem. Res. 1990, 29, 1668. (5) Jacobsen, E. W.; Skogestad, S. Multiple steady states in ideal two-product distillation. AIChE J. 1991, 37, 499. (6) Lucia, A.; Li, H. Constrained separations and the analysis of binary homogeneous separators. Ind. Eng. Chem. Res. 1992, 31, 2579. (7) Gani, R.; Jorgensen, S. B. Multiplicity in numerical solution of nonlinear models: Separation processes. Comput. Chem. Eng. 1994, 18, S55. (8) Lucia, A. Uniqueness of solutions to single-staged isobaric flash process involving homogeneous mixtures. AIChE J. 1986, 32, 1761. (9) Michelsen, M. L.; Heidemann, R. A. Uniqueness of solutions to single-stage isobaric flash processes involving homogeneous mixtures. Comments. AIChE J. 1988, 34, 877. (10) Tiscaren˜o, F.; Go´mez, A.; Jime´nez, A.; Cha´vez, R. Multiplicity of solutions of the flash equations. Chem. Eng. Sci. 1998, 53, 671. (11) Rachford, H. H.; Rice, J. D. Procedure for use of electronic digital computers in calculation flash vaporization hydrocarbon equilibrium. J. Petrol. Technol. 1952, 4, 9. (12) King, J. C. Separation Processes, 2nd ed.; McGraw-Hill Book Company: New York, 1980. (13) Henley, E. J.; Seader, J. D. Equilibrium-Stage Separation Operations in Chemical Engineering; Wiley: New York, 1981. (14) Floudas, C. A. Nonlinear and Mixed-Integer Optimization. Fundamentals and Applications; Oxford University Press: New York, 1995. (15) Beveridge, G. D. G.; Schechter, R. S. Optimization: Theory and Practice; McGraw-Hill Book Company: New York, 1970 (16) Holland, C. D. Fundamentals of Multicomponent Distillation; McGraw-Hill Book Company: New York, 1981.
Received for review July 13, 2000 Accepted May 29, 2001 IE000662X
