Ind. Eng. Chem. Res. 1994,33,2631-2636
2631
PROCESS DESIGN AND CONTROL Distillation Column Design Calculations Using a Nonequilibrium Model Sirish Agarwal and Ross Taylor’ Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699-5705
A nonequilibrium (mass-transfer-rate-based) model has been used to determine the minimum reflux ratio and the minimum number of stages in distillation. It was found that, for a column with a n infinite number of stages, Rdn calculated from the rate-based approach was the same a s that computed using a n equilibrium model. A substantial difference in the R- calculated using the different methods was observed for a column with a specified number of stages. Introduction The calculation of the minimum reflux ratio (R,in) is useful in the design of distillation columns. Graphical methods of determining minimum reflux in binary systems may be found in all textbooks on distillation design (see, for example, King, 1980; Henley and Seader (1981)). Other methods are limited to systems of constant relative volatility (Underwood, 1948) or to ternary systems (Levy et al., 1985). Julka and Doherty (1990) have developed a geometric method of determining minimum flows that places no such restrictions on the type of system to which it may be applied. All of the methods cited above are based on the equilibrium stage model of distillation. However, the trays of an actual column are not equilibrium stages. The usual way of dealing with this fact is through the use of an efficiency of some kind. It is common practice t o assume the efficiency is the same for all components on any tray and that the efficiency is the same on all trays. Neither assumption is true in general; for multicomponent systems the individual component efficiencies may take values from --oo to +-oo (see Taylor and Krishna (1993) for a survey of the evidence for this remark). Sunderesan et al. (1987) looked a t the calculation of pinch compositions in countercurrent adsorption and found that finite mass transfer rates have a significant influence on the column profiles. They concluded by saying that “a reexamination of the classical procedures for minimum reflux calculation such as Underwood (1948) method, relaxing the assumption of equilibrium trays and retaining the finite interphase mass transfer rates in the trays, appears warranted.” In this paper we look at the problem of determining the minimum reflux and the minimum number of stages using a mass-transfer-rate-based or nonequilibrium model.
A Nonequilibrium Model for Distillation Column Design In recent years it has become possible to model multicomponentdistillation operations as mass transfer
* To whom correspondence should be addressed. E-mail:
[email protected].
rate processes (see Taylor and Krishna, 1993). The key feature of nonequilibrium models is that the component material balances and energy balances for each phase are solved together with mass and energy transfer rate equations and equilibrium relations for the phase interface to find the actual separation on each stage directly (Krishnamurthy and Taylor, 1985). This makes the calculation of stage efficiencies unnecessary. These nonequilibrium models have been found capable of better predictions of actual column performance than is an efficiency-modified equilibrium stage model (see Taylor and Krishna, 1993). In order to determine the minimum reflux using a nonequilibrium model, it is desirable t o develop an approach that is quite different from that of Krishnamurthy and Taylor (1985). The approach we take here is to develop the material balance equations in a way that allows us to carry out stage-to-stage calculations. In what follows we assume constant molar overflow, fully recognizing that this is not consistent with some of the assumptions underlying nonequilibrium models (Taylor et al., 1994). The assumption of constant molar flows can be relaxed by including the energy balances in the set of model equations (Knight and Doherty, 1986). A material balance around the condenser gives
Dx, =
- Lxi,l
(1)
where the following restriction holds for a total condenser. (2)
y.r,2 = x .r,1 = x .rD For stage j in the rectifying section we have
+
vyiJ+l LXj,-,
= vyw
+ Lxw
(3)
Similarly for the stripping section, a material balance around the reboiler, stage N , gives Bx, = L
x ~-~V-y i~J v
(4)
and the balance for stagej in the stripping section looks like
+
+
Vyjj+l Exjj-l = v-y i j LXjj -
0 1994 American Chemical Society 0888-5885/94/2633-2631$04.50/0
2632 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 The composition of the vapor leaving the froth on tray j can be related to the composition of the vapor below tray j by the following matrix equation of order c - 1:
where (y) is a column matrix (vector) of vapor phase mole fractions of the subscripted stream; (y*) is the vector of mole fractions for a vapor that would be in equilibrium with the liquid. [&I represents the degree of departure from equilibrium. Equation 6 may be rearranged to give
Efficiency and the Structure of [&I. Specifying the elements of [&I is the proper way to fix the efficiencies of the various components. The relationship between the elements of [&I and the component Murphree efficiencies is as follows for a ternary system:
(9)
where Ayz = y*g - yij+i. If the vapor leaving the stage is assumed to be in equilibrium with the leaving liquid, the matrix [&I is null, [&I = [OI, and we see from eqs 8-10 that all of the component efficiencies are unity. [Q] is non-null whenever the tray is not an equilibrium stage (i.e., in all real cases). In cases where the approach to equilibrium of all the components is the same, then [&I is diagonal with all the elements on the main diagonal equal to one another, i.e., [&I = q[Il or Q.. 11 = q Q.. LJ = 0 (i * j = 1, 2, ..., c - l ) , where q is a constant. This is equivalent to assuming the efficiencies of all components are the same with E? = 1 - q. This situation can arise in practice only for mixtures of similar components where the binary pair mass transfer coefficients are equal and the resistance to mass transfer in the liquid phase is entirely negligible. Allowing [&I to be diagonal with unequal diagonal values corresponds to a situation often encountered in practice: where the efficiencies of some components are given different values in order to try and match plant data. [&I will also be diagonal with differing diagonal elements if we use an effective diffusivity model of mass transfer. Again, this is commonly done, even in circumstances where the use of such a model is not justified. If we assume the vapor rises through a well-mixed liquid in plug flow, we find 9
where [Nov]is a matrix of transfer units for multicomponent systems. The calculation of [Nov]from empirical or from fundamental models of mass transfer is discussed in detail by Taylor and Krishna (19931, and in the interests of brevity, we omit further development of methods for calculating [Novl. It must suffice to point out that the elements of this matrix are complicated
functions of the binary mass transfer coefficients of both vapor and liquid phases. [Nov] (and, hence, [&I) will be nondiagonal in general. Other models of flow and mass transfer on a distillation tray lead to different methods for calculating [&I, but the form of eq 6 can be retained. In general, [&I will have nonzero off-diagonal elements (as well as unequal diagonal elements). However, the elements of [&I cannot normally be specified arbitrarily without running the risk of violating the physical constraints of material balance and nonnegativity of the mole fractions. The structure of [&I is determined by the equations that govern mass transfer in multicomponent systems. These restrictions ensure that [Q] will be diagonal (with unequal diagonal elements) in the corners of the ternary diagram, upper or lower triangular along the axes of the diagram, and full in the large central region. The off-diagonal elements of [&I will be small close to the axes of the diagram. While the diagonal elements of [&I must be positive, the off-diagonal elements can take either sign. Stage-to-StageCalculations. The determination of the composition profiles involves a stage-to-stage calculation down from the condenser (or up from the reboiler) until we reach the pinch zone where no further change in composition of the liquid is encountered. The equations to be solved for the rectifying section include the first c - 1material balances, eq 3 combined with eq 7 in the form
For the stripping section the material balances are combined with eq 7 to give
For both sections we need additional equations that allow us to calculate the equilibrium mole fractions
and equations that force the mole fractions t o sum t o 1: C
C
sy,+, = bi j+l - 1 = 0
(16)
i=l C
s,, = cy*il- 1 = 0
(17)
i=l
The unknown variables computed by solving these equations for the rectifying section include the mole fractions of the vapor streamj 1 and liquid stream j , the equilibrium vapor mole fractions, and the stage temperature. Temperature is a variable because the K-values are composition and temperature dependent (the pressure is specified). This set of 3c 1 equations was solved simultaneously using Newton's method. Initial estimates of the unknown variables were the compositions and the temperature of the previous stage for which the above quantities are already known. For the condenser the distillate composition and reflux ratio were specified (hence the flow ratio VIL is known).
+
+
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2633 1 .o
1 .o
0.8
0.8
0.6
0.6
xz
X2
0.4
0.4
0.2
0.2
0.0
0.0
C
x1
Figure 1. Composition profiles in hydrocarbon system for two different reflux ratios.
Figure 2. Composition profiles at two different reflux ratios obtained with nonequilibrium model.
For the stripping section a similar set of equations can be solved, proceeding from the reboiler and calculating upward.
‘.O
Results and Discussion Julka and Doherty (1990) have demonstrated the effect of reflux ratio changes on the distillation mapping. They show that for any specified distillate composition, the location and orientation of the rectifying plane are determined by the reflux ratio. As the reflux ratio is increased, the rectifying plane moves away from the distillate composition and increases in size as well. For the stripping section, the location and orientation of the stripping profile are determined by the reboil ratio. The calculations of Julka and Doherty assume equilibrium on each stage. We wish to see the effect that departures from equilibrium have on the location of the composition profiles. To this end we look a t the composition profiles computed when the elements of [&I are specified. Stage-to-stage calculations were performed for two systems used in the study by Levy et al. (1985). System 1 consists of a mixture of paraffins: n-hexane, nheptane, and n-nonane. Thermodynamic properties were calculated using the SRK equation of state (see Walas, 1985). System 2 is a nonideal mixture of acetone, benzene, and chloroform. Thermodynamic properties for this system were computed using the NRTL model with parameters taken from the DECHEMA data collection (Gmehling and Onken, 1977ff). Figure 1shows the composition profiles for system 1 a t two different values of reflux ratio with the distillate composition fmed a t X U ) = 0.99, x 3 = ~ and xm = 1 - X I D - X ~ D . [&I is null. These profiles agree with those computed by Levy et al. (1985). It can be seen that a reflux ratio of 1.5 does not give us a feasible column because the rectifying and stripping profiles fail to intersect each other. On the other hand, a reflux ratio of 2.0 is more than the minimum reflux ratio required to obtain a feasible column. When the profiles are computed with [&I = 0.5[4 (and all other quantities remaining the same), then the orientation of the profiles changes to that shown in Figure 2. Rectifylng section profiles as a function of q at a reflux ratio of 2 are shown in Figure 3. The variation in the
3
0.8
R = 2 X - 0 9 9 0 0 1 1E-07) [o:ooi , b . 4 i i .os781 = aZ1= 0, = a,? = 0.
x: a,, a,,
0.6 x2
0.4
0.2
0.0 XI
Figure 3. Composition profiles in hydrocarbon system as a function of (Q)a t a reflux ratio of 2.
location of the profiles is seen to be large, though the manifolds are bounded in location by the reflux ratio as expected; i.e., the profiles trace the separatrix but do not cross the boundary defined by a particular reflux ratio and distillate composition. The actual value of the the diagonal elements of [&I determines the orientation of the profile and the number of stages required to achieve a specified separation. As q increases, the number of stages required for a specified separation also increases. Figure 4 shows the profiles for system 1where the diagonal elements are given unequal values. The effect of unequal diagonal elements of [&I is seen to be quite large particularly for the case of a direct split (where x 3 ~ 10-8). This figure provides an indication that there can be large differences between the equilibrium and nonequilibrium models for nonideal systems (where different species may have quite different mass transfer characteristics). Figure 4 includes a profile obtained when the off-diagonal elements of [&I are given small
2634 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 1 .o
1 .o
\
0.8
\
R == 4 (0.0353.4E-04 0.96 X, Y,, = (0.23,7.5E-04,0.77\
0.8
1 Calculated 0 , R = 6.3 2 Levy's method, R = 6.3
0.6
0.6 x2
x2
0.4
0.4
\
0.2
0.2
0.0
0.2
0.6
0.4
0,o 0
0.8
0.2
0.4
0.8
0.6
1
XI
Xl
Figure 4. Composition profiles in hydrocarbon system for (Q) f
m.
Figure 6. Composition profiles for hydrocarbon system determined by Levy's method and the method of this paper. 1 .o
0.8
0.6
xz 0.4
-
0.4
0.2
0.0
0.2
0.6
0.4
0.8
',
I
0.0
1
x1
XI
Figure 5. Rectifying and stripping section profiles for component 1, component 2, component 3 at reflux ratios of 2 and 6.3. (Q) computed from fundamental model.
(but arbitrary) values. Again, we see that there is a significant effect on the location of the composition profile. Minimum Reflux. In our next set of results the matrix [&I is calculated from a fundamental (and quite detailed) model of interphase mass transfer in the froth on a sieve tray. A complete description of the model is beyond the scope of this paper; readers are referred to sections 12.1.7 and 12.2.4of Taylor and Krishna (1993) for a development of the model and example calculations. Figure 5 shows the rectifying and stripping section profiles for system 1 at reflux ratios of 2 and 6.3. At a reflux ratio of 2 the two profiles do not intersect and we do not get a feasible design. If the reflux ratio is increased t o 6.3, we see that the two profiles just about touch each other. Hence, this is close to the minimum reflux ratio for these conditions. In order to determine the minimum reflux ratio for this system, we implemented the method of Levy et al. (1985)which requires the numerical integration of a set
Figure 7. Composition profiles for nonideal system determined by Levy's method and the method of this paper.
of differential equations. While the method of Levy et ai. nas Deen superseaea ~y mat 01 JuiKa ana uonerty (1990), the two methods give identical answers for the systems considered here and the method of Levy et al. was easier for us to implement. To our surprise the method of Levy et al. gave the same minimum reflux as did our calculations where [&I was involved. This is shown for both of our test systems in Figures 6 and 7. The implication is that the minimum reflux ratio computed from the actual tray mass transfer considerations does not differ from the value calculated assuming that equilibrium exists on each tray in the limit of an infinite number of stages. The situation is rather different, however, if we determine the minimum reflux ratio for an existing column, i.e., where the number of trays is specified. The effect of mass transfer on the Rminvalue of an existing column can be determined by computing the rectifying and stripping section profiles starting from the top and the bottom stage respectively and continuing the stageby-stage calculations for a specified number of stages. Figure 8 shows the profiles for system 1 for a column I
,
1
1
1 1
>,
1
l l T
1.
1 - 1
>
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2635 1 .o
1 .o
0.8
0.8
-
R = 7.3 R = 8.7
0.6
0.6
X?
XZ
0.4
0.4
0.2
0.2
0.0 0.0
0.2
0.6
0.4
0.8
1 .o
n -.-n
0.0
0.2
x1
0.8
0.8
0.6
1 .o
x1
Figure 8. Composition profiles for hydrocarbon system for column with 30 stages at two reflux ratios. I
1.0,
0.4
R = 1000
Figure 10. Composition profiles for nonideal system at total reflux.
model. For system 2 the minimum numbers of stages were 16 and 28, respectively. The number of stages required to achieve a specified separation (the final goal in a column design procedure) may now be estimated using Gilliland’s correlation (see King, 1980):
R
0.6
N-Nmin
N+1
x2
x1
Figure 9. Composition profiles for hydrocarbon system a t total reflux.
having 15 stages in each section plus a total condenser and a partial reboiler. If we assume [&I = 0.5[rl,we find that the minimum reflux ratio is about 7.3. However, if [&] = 0.52[11,the profiles shown in Figure 8 do not intersect. IncreasingR to 8.7 helps to close the gap in the profiles, as shown in Figure 8, but is still too low a reflux ratio. A nonequilibrium model can be expected to provide more accurate estimates of the minimum reflux ratio for an existing column. Total Reflux. It is possible to calculate the minimum number of stages a t total reflux using the nonequilibrium model described in this work. The procedure used in all of the above calculations was used with a reflux ratio of 1000. The number of stages required to reach a specified composition can be found for each section of the column after locating the intersection of the two profiles on a ternary diagram. The results for an equilibrium ([&I = [Ol) and the nonequilibrium calculation are shown in Figures 9 and 10. The number of stages at total reflux for system 1 was found to be 10 for the equilibrium model and 18 for the nonequilibrium
= 0.75 - 0.75(
- R,, +
1
)
0.5668
(18)
where Nmh is the minimum number of stages determined from the equilibrium model, N is the number of stages in the final design, R is the operating reflux ratio, and R- is the minimum reflux ratio (which is the same for both models). Using this method we find that system 1 requires 46 stages and system 2 requires 78 stages. The number of actual stages can be determined directly using a nonequilibrium model with the result that system 1requires 39 stages and system 2 requires 56 stages. In both cases the equilibrium-model-based method leads to a substantial overdesign.
Conclusion The true minimum reflux ratio (for a column with an infinite number of stages) is the same whether determined by an equilibrium model or by the nonequilibrium model described in this paper. The results given in this paper suggest that the fears expressed by Sunderesan et al. are unfounded and that existing methods for determining minimum reflux requirements remain useful.
Acknowledgment
S.A. is grateful to Clarkson University for a tuition assistantship for the duration of this work. Nomenclature B = bottoms flow rate c = number of components in the mixture D = distillate flow rate E = equilibrium equation Eov = Murphree efficiency i = component index
2636 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 j = stage index K = equilibrium ratio L = molar flow rate of the liquid phase N = number of stages
[Nov]= matrix of numbers of transfer units [Q]= matrix representing departure from equilibrium S = summation equation T = temperature V = molar flow rate of the vapor phase x: = mole fraction in liquid phase y = mole fraction in vapor phase Matrix Notation [ ] = square matrix of order c - 1 ( ) = column matrix of order c - 1
Literature Cited Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection; DECHEMA: FrankfurVMain, 1977ff. Henley, E. J.; Seader, J. D. Equilibrium Stage Separation Operations i n Chemical Engineering; Wiley: New York, 1981. Julka, V.; Doherty, M. F. Geometric Behavior and Minimum Flows for Nonideal Multicomponent Distillation. Chern. Eng. Sei. 1990,45,1801-1822. King, C. J . Separation Processes; McGraw-Hill Book Company: New York, 1980;pp 428-432. Knight, J. R.; Doherty, M. F. Design and Synthesis of Homogeneous Azeotropic Distillations. 5. Columns with Nonnegligible Heat Effects. Znd. Eng. Chem. Fundam. 1986,25,279-289.
Krishnamurthy, R.; Taylor, R. A Nonequilibrium Stage Model of Multicomponent Separation Processes Part 1: Model Description and Method of Solution. N C h E J . 1985,31, 449-454. Levy, S. G.; Van Dongen, D. B.; Doherty, M. F. Design and Synthesis of Homogeneous Azeotropic Distillations. 2. Minimum Reflux Calculations for Nonideal and Azeotropic Columns. Ind. Eng. Chem. Fundam. 1985,24,463-474. Sunderesan, S.; Wong, J. K.; Jackson, R. Limitations of the Equilibrium Theory of Countercurrent Devices. N C h E J . 1987, 33, 1466-1472. Taylor, R.; Krishna, R. Multicomponent Mass Transfer; Wiley: New York, 1993. Taylor, R., Kooijman, H. A,; Hung, J.-S.A Second Generation Nonequilibrium Model for the Computer Simulation of Multicomponent Separation Processes. Comput. Chem. Eng. 1994, 18, 205-217. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948,44,603-614. Walas, S. M. Phase Equilibria in Chemical Engineering; Buttenvorths: Stoneham, MA, 1985. Received for review February 18, 1994 Revised manuscript received J u n e 28, 1994 Accepted July 19,1994@
@
Abstract published in Advance A C S Abstracts, September
15, 1994.
