Ind. Eng. Chem. Res. 1989,28, 1862-1867
1862
SEPARATIONS Determination of Feasible Reflux Ratios and Minimum Number of Plates Required in Multicomponent Batch Distillation Wen-Hai Wu* and Tzy-Neng Chiou Chung Shan Institute of Science and Technology, P.O. Box 1-4, Lung-Tan, Taiwan, ROC
This work deals with the method for determining reflux ratios and the number of plates in multicomponent batch distillation. From the mass balances, it was found that, under reasonable assumptions, each intermediate state has only 2 degrees of freedom. Analysis on the suitability of the intermediate state shows that the effective concentration in the residue is the determining factor. The effective concentration is thus used as a criterion for precisely determining reflux ratios and the number of plates in the proposed design procedure, in which reflux ratios and the number of plates are determined sequentially from one operation step to the next step rather than simultaneously for all steps as in conventional methods. This considerably simplifies the design. The design procedure is demonstrated by using an example of three-step batch distillation, in which the relative volatilities of the different steps range from 2 to 13. “Multicomponent batch distillation” represents here the batch distillation in which an M-component mixture (M I3) is separated into M products by a sequence of M 1 operation steps. A number of papers dealing with optimal reflux policies have been issued (Converse and Gross, 1963; Coward, 1967; Robinson, 1969; Mayur and Jackson, 1971), but the cases studied were mostly restricted to one-step batch distillations whose relative volatilities range from 1.5 to 3. It is difficult to apply the results in these works to multistep problems or to one-step problems with large relative volatilities (e.g., a > 10). The work of Mayur and Jackson (1971) seems to be the first concerning multistep batch distillation design. This work dealt with the time-optimal problems for a ternary mixture. The authors found that the intermediate state (amount and composition of residue) after the first operation step was restricted on a cylindrical surface and that the terminal state was restricted on a curve in the threedimensional representation. Their approach was to search over all possible points on this surface and thus locate the point for which the sum of the times required for the first and second steps is smallest. They pointed out that the computation time for this method of solution would probably be prohibitive for systems of more than three components. Rose (1985) presented some general aspects for the multicomponent batch distillation design. His method was first to use an order-of-magnitudedesign with a few short-cut calculations and then a detailed design by repeated trials with computer simulation. No detail about how to select reflux ratios and number of plates was given. In this work, we analyzed the restriction of the intermediate state for systems of any number of components. The result shows that, under two reasonable assumptions, each intermediate state has only 2 degrees of freedom. Instead of searching over all possible intermediate states to find an appropriate one, it was found that an intermediate state can be properly determined according to the
* Author to whom correspondence
should be addressed.
relationship between adjacent operation steps. The design procedure proposed in this work considerably reduces the number of calculations and the complexity in multicomponent batch distillation design.
Theoretical Basis of the Design Formulation of the Problem. The batch still consists of a reboiler, a rectifying column surmounting the reboiler, and a total condenser. The rectifying column, together with the reboiler, has a separation efficiency corresponding to N theoretical plates. The feed initially charged into the reboiler is a multicomponent mixture constituted by components 1, 2, ...,M (in order of decreasing volatility). We desired to separate this mixture, in a sequence of M - 1 operation steps, into M - 1 overhead products of for components specified concentrations XD Xb,..., XDM-l 1, 2, ..., M - 1 and into a &a1 residue of specified confor the Mth component. To simplify the centration XBM-l design problem, we assumed that no intermediate cut was taken between each of the two adjacent products. The purity specifications may be described by the following equations:
Xb,= X,,,
r = 1, 2,
..., M - 1
(1)
where D, and B, denote respectively the accumulated overhead product and the residue at the end of the rth operation step and the superscripts identify the components. Degree of Freedom of the Intermediate State. Multicomponent batch distillation may be regarded as a series of operation steps. Each operation step accumulates an overhead product and leaves a residue at the end of the operation. The amounts and compositions of the overhead product and residue after the rth operation step will be called the “rth intermediate state”, and those after the last operation step will be called the “terminal state”. The intermediate or terminal state varies with the reflux policy and the number of plates. In this variation, the variables
08~8-5885/89/2628-1862$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1863 in the intermediate or terminal state do not vary independently but are related by mass balance and summation equations as follows: mass balances (3) Bpi = B, + D, i = 1, 2, ..., M - 1 (4) B,lXkl = B,X& + D,XL, summation equations
xg, + xir+ ... + xg = 1 (5) X b , + xg,+ ... + xg = 1 (6) In the above M + 2 independent equations, the residue
of the (r - 1)th intermediate state is used as the feed for the rth step. When the amount and composition of this feed are known, the above equations contain a total of ,2M + 2 separation variables, namely, B,, D,, Xg,,and Xbr (i = 1, 2, ...,M). These variables constitute the rth intermediate state. Some of these variables have very small values and may be eliminated according to the following two assumptions: (a) The residue does not contain any components that are more volatile than the light key component. (b) The overhead product does not contain any components that are less volatile than the heavy key component. Assumption a indicates that the initial r - 2 components, which are more volatile than the light key component of the (r - 1)th step, are negligible in the residue of the (r 1)th intermediate state, as follows:
xg,, = Xi,,
= ... = xpl = 0
(7)
Since the residue of the (r - 1)th intermediate state is used as the feed of the rth operation step, the initial r - 2 components are also negligible in the rth intermediate state, as follows:
... = xs; = 0 XB, = xg, = ... = xq = 0
Xb, = xg, =
(8)
(9)
Besides eq 8 and 9, from assumption a we have
xg = 0
(10)
and from assumption b we have
XbT2 = Xc3= ... = Xp;', =0
(11)
Equations 8-11 reduce the number of variables from 2M 2 to M - r + 6 for the rth intermediate state ( r > 1)or terminal state. Since the initial r - 2 components are negligible in the feed of the rth step, the equations describing the mass balances of these components may be deleted from eq 4 to reduce the number of equations from M + 2 to M - r 4. The degree of freedom of the rth intermediate state ( r > 1) or terminal state can thus be calculated as (M- r + 6) - (M- r + 4) = 2. For the first intermediate state, the number of variables may be reduced to M + 4 according to eq 11, but the number of equations remains unchanged and equal to M + 2. Hence, the degree of freedom of the first intermediate state also equals 2. Effective Concentration as a Criterion for Determining the Suitability of the Intermediate State. An appropriate intermediate state should fulfill the following two requirements: (1)The overhead product of the state satisfies the product purity specification. (2) The residue of the state, used as the feed of the following operation
+
+
step, has a composition capable of achieving the purity specification of the following step. The first requirement is commonly used as the stopping criterion in the integration in the computer simulation. The fulfillment of the second requirement is judged by the value of the "effective concentration of the light key component in the residuen, defined as the fraction of the light key component in the combined light key component + heavy key component in the residue. The effective concentration of the rth intermediate state, XL,, may be written as
Since the residue of the rth intermediate state is used as the feed of the ( r 1)th operation step and the overhead product accumulated in the ( r + 1)th step is composed mostly of the (r + 1)th component, the concentration of the T h component in this product is approximately equal to XB,, as follows:
+
Xb,, = XB,
(13)
From eq 13, the summation equation for the overhead product of the ( r 1)th intermediate state may be written as
+
1
xb,,+ XDr+, + Xg:, = xi,+ XD*, + xg-,
(14)
Equation 14 shows clearly the influence of the effective concentration of an intermediate state upon the product purity in the following operation step. In view of this equation, the principles for selecting an appropriate effective concentration may be proposed as follows: (1)XB, must be smaller than 1- X D (2) The relative magnitudes of XL,and Xg: depen&n the relative difficulties in the separation of the rth and ( r 1)th steps. For instance, if the separation of the ( r 1)th step is more difficult, X B , should be smaller than Xr,.,". When the purity of the overhead prochct has been specified, the degree of freedom for an intermediate state is reduced from 2 to 1. The other variable that is very suitable to be specified is the effective concentration in the residue since it is the main factor in determining the suitability of an intermediate state and its appropriate values are located in a narrow range (e.g., 0 to 1 - X D , , for XB).
+ +
Design Procedure Combining the above results, the design procedure may be proposed as follows. (1) Properly select the value of the effective concentration for the first intermediate state using the principles stated earlier. (2) Carry out the simulations of the first operation step for various reflux ratios and number of plates and use the selected effective concentration as a criterion to determine the required reflux ratio and number of plates for the first step. Usually, a graph such as Figure 2 is used in this determination. (3) Calculate the first intermediate state by computer simulation using the determined reflux ratio and number of plates. (4) Take the residue of the first intermediate state as the feed of the second operation step and carry out the design for the second step. Proceed in this way until the last step is completed. The simulations in the above design procedure are based on mass balances and an equilibrium relationship and are
1864 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 [Inpiit the initial values) B = Br-l , t = t0 XB
=
i=1,2,..,~
and the specifications of R, N, and V
= xi
Ix;
i=1,2,..,M
r
Figure 1.
performed on the computer. The dynamic mass balances for no hold-up systems (Domenech and Enjalbert, 1981) may be written as
dXi v x$-xi =--
i = 1, 2, ..., M (16) R + l B Assuming constant molal overflow, the equations describing the operating lines and equilibrium curves are operating lines dt
R X$ y; = R + lX;+l + R+1
Table I. Composition and Vapor Pressure Constants of the Initial Charge Antoine constants concn, component molefrac bp, O C ANTA ANTB ANTC -39.94 15.833 2477.07 A 0.361 36 13.558 1789.03 -126.0 B 0.339 108 -38.0 17.776 4514.28 C 0.060 170.5 -52.8 16.965 4276.08 D 0.240 193
lines represents the subroutine for calculating distillate composition Xb contained in eq 16. The formula for revising the estimated distillate composition in the iterative procedure in this subroutine is
. = 1, 2, ..., M
I
p = 1, 2, ..., N (17)
equilibrium curves M
M
yi
(18)
where (Xi)!denotes the calculated reboiler composition at the nth iteration and (Xl,)" denotes the updated distillate composition at the nth iteration.
Figure 1 shows the flow sheet of the multicomponent batch distillation program used in this work. The integration is terminated at the time when the product purity specification is satisfied. The region bound by the dashed
Illustrating Example A four-component mixture is to be separated into its constituents by batch distillation. The composition of this
EX; = i=l
Ei = l Kb
=1
p = 1, 2,
..., N
Table 11. Operation Specifications and Relative Volatilities of Each Operation Step operating re1 volatility pressure, step purity, mol 70 mmHg 50 O C 100 O C 660 13.51 8.003 1 A = 99
B
2 3
80 80
= 97
C = 96 D = 95
12.66 2.218
-.m
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1865
N
X v
G
0.5r
7.483 1.993
R=0.5
I
F!
5
U
\
0.221
b
0.2R= I
0
z
0 0 Y
a
4
0.161
0.121
0 LL
0.11
-
X =0.96 O3
2
-l
0.141
0 6
0. I
U
-
0
\
--
R-2
\
R=3 R=5
1
R=7
0
1
U
m
I-
0.02
R=30 R=25
U
LL 0
1
2
3
4
5
6
7
0
9
1
0
THEORETICAL P L A T E S
mixture and the Antoine constants of each component are shown in Table I. The distillation is carried out by a series of three operation steps, in which products A, B, and C are accumulated respectively in the first, second, and third steps, and product D is left as the final residue at the end of the third step. The purity specifications and operating pressures of each operation step are shown in Table 11. A low pressure is used in the last two steps to avoid thermal decompositionof components B and C. All the simulation results followed are based on 100 mol of the multicomponent mixture as the initial charge. Determination of Reflux Ratios and the Number of Plates. Consider the first operation step. The light key component of this step, A, has a concentration approximately equal to XLl= X&/(Xk:, + Xg,)in the overhead product of the second step. Since the purity specification for the second step is 97 mol % component B, eq 14 may be written as
+
+ XS,
0.97
R=2O
R=17
a
Figure 2. Effective concentration of component A in the residue as a function of the reflux ratio and number of plates for the first step.
1 = Xil 0.97
5
(20)
The initial two steps have close relative volatilities, so according to the mentioned pfinciples, x', may be chosen to be equal to X& and thus XB,= X$,= O.bl5. The ranges of reflux ratios and the number of plates that nearly lead to this effective concentration can be found from Figure 2, which shows the results at the time when the accumulated distillate just meets the purity specification, 99 mol % component A. From Figure 2, R = 5 and N = 7 may be chosen for the first step. The amount and composition of residue obtained by computer simulation with R = 5 and N = 7 are 64 mol and (0.0071, 0.5241, 0.0938, 0.3750) re-
=s
R=15
0.95
-
LL 0
0.941
0 2
0.931
L
// R= I 2
T
/
I
/
0
~ 0 . 8 9 " IO I I
"
I
'
I
"
'
12 13 14 15 16 17 18 19 20 THEORETICAL P L A T E S
Figure 4. Concentration of component D in the residue as a function of the reflux ratio and number of plates for the third step.
spectively. These were used as the data for the feed for the second step. For the second step, eq 14 may be written as 1 i= Xi2+ 0.96 + XB8 (21) Since the relative volatility of the third step is much smaller than that of the second step, X'', should be smaller than Xg , and from eq 21, this may be expressed as X' < 0.02. f i e effects of reflux ratio and the number of platea upon X)B2 are shown in Figure 3. From this figure, the operating condition for the second step may be chosen as R = 5 and N = 7. The values of Xg,and residue composition at this operating condition were calculated to be 0.0143 and (0, 0.0027, 0.1860, 0.8113) respectively. Since the purities of both overhead product and residue have been specified for the third (last) step, the terminal state is fixed. Figure 4 shows the component D concentration in the residue as a function of the reflux ratio and the number of plates at the time when the product purity
1866 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 W
0 W
k 6 1
I-
a N=7
x Z
Y
u u
-0.97 D2
LL
a
t 0.99
I
I
1
I
I
I
I
I
0.98
I
1
1
1
I
Figure 5 . Variation of accumulated distillate composition in the first step.
specification is satisfied. It can be seen that, for each number of plates, only one reflux ratio will lead to the specified residue purity, 95 mol '70 component D. For example, the required reflux ratio equals 15.25 when N = 15 and equals 17.0 when N = 12. A proper choice of R and N for the third step may be obtained by minimizing the sum of the energy cost and column investment. Practically only one column is used in the distillation, so the largest of all the plate numbers selected for each step should be taken as the plate number of the column. In the above procedure, the plate numbers for the initial two steps are both selected as 7, but the plate number determined for the third step is much greater than 7. However, recalculation of the initial steps using the finally determined plate number is unnecessary because the simulation results of these steps remain unchanged when N 2 7, and thus the reflux ratios previously determined for the initial steps remain valid. Practical Reflux Policies. Constant reflux policies determined by the above method are usually too expensive in energy consumption, especially for operation steps of large relative volatility. For such steps, the piecewise constant reflux policy serves as a practical substitute for constant reflux policy. This policy is clearly illustrated in Figure 5 for the first operation step. Each curve in Figures 5-7 represents a constant reflux policy and is terminated at the overhead purity specification. In the initial period, a reflux ratio of 0.3 is high enough to obtain a pure distillate, so it is used up to point A, at which the accumulated distillate concentration reaches a arbitrarily specified criterion approaching 1 (e.g., 0.9999). A larger reflux ratio 0.5 is then used from point A to point B. We proceed in this way up to point D, from which the largest or final reflux ratio 5 is used. This last piece of distillation is terminated a t the overhead purity specification. If the first step is carried out with the piecewise constant policy, as shown in Figure 5 and listed in Table 111, the first intermediate state will be almost the same as that obtained by constant reflux policy with R = 5, but the required total vapor generation is 70.09 mol, only 32.4% of the vapor generation of constant reflux policy with R = 5 (216 mol). For the second step, if we divide the operation into five pieces, as shown in Figure 6, the required total vapor
I
.oo
tt
\
N= I 5
\
\
\
I
I
\
\
I
X zO.96 D3
0.99
t
a 0 0
tt 0.971 0
I
I
I
I
Z
I
AMOUNT
I
I
3
I
I
4
,
I
5
DISTILLED (MOLES1
Figure 7. Variation of accumulated distillate composition in the third step. Table 111. Piecewise Constant Reflux Policy for t h e First Step piece reflux ratio range of amt distilled, mol 0.3 0.5 1 2 5
0-15.16 15.16-24.91 24.91-30.58 30.58-33.28 33.28-36
generation is 58.19 mol, which is only 28.1% of the vapor generation of constant reflux policy with R = 5 (207 mol). The third step has a values much smaller than the initial steps and is the step that determines the required plate number of the column. When the plate number is appropriately selected according to this step, the plot of the variation of distillate composition for different reflux ratios has the common characteristics shown in Figure 7. Consequently, a piecewise constant policy for the third step is more difficult to determine than those illustrated in
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1867 40
El l-
is therefore greatly reduced.
r
30
a
U
X
3
20
J LL W
U
IO
0
0
I
AMOUNT
2
3
DISTILLED
4
5
(MOLES)
Figure 8. Variation of the reflux ratio at constant overhead policy in the third step.
component is present in the distillate before the last piece. Constant overhead is a more suitable reflux policy for the third step. Since the feed of the third step contains a small amount of component B which is present a t a high concentration in the distillate in the initial period; the constant overhead must be specified in terms of the concentration of component C plus component B, which has a fixed value of 97.83% because the terminal state has been fixed by the purity specifications of the overhead product and residue. The variation of the reflux ratio at constant overhead policy is shown in Figure 8. This policy can save about 25% of energy for the third step compared with the constant reflux policy at the same value of N . Conclusions Under reasonable assumptions, the degree of freedom of each intermediate state equals 2. Therefore, when the purity of overhead product is specified, only one variable remains free to be specified for an intermediate state. The effective concentration in the residue, whose appropriate values are located in a narrow range, may be regarded as the variable most suited to be specified. Two principles for properly selecting the effective concentration have been proposed. The main feature of the proposed design procedure is that the effective concentration of the intermediate state is first properly selected, and then the reflux ratio and the number of plates are determined by examining whether they can meet the selected effective concentration. This enables us to determine the reflux ratios and the number of plates separately for each operation step rather than simultaneously for all steps. The complexity of the design
Nomenclature B = moles of material in reboiler B, = moles of residue left in reboiler at the end of the rth step D, = moles of overhead product accumulated in the rth step It = equilibrium ratio of component i on the pth plate Id=number of components in the initial charge N = number of theoretical plates n = number of iterations R = reflux ratio t = time V ,= vaporization rate Xb = mole fraction of component i in the reboiler liquid Xi, = mole fraction of component i in the distillate XbA = mole fraction of component i in the accumulated distillate Xi,= mole fraction of component i in the residue at the end of the rth step Xb,=. effective concentration of the light key component in residue at the end of the rth step XBM-M-l = specified mole fraction of the Mth component in the residue at the end of the final (or (M - 1)th) step X i r = mole fraction of component i in the overhead product accumulated in the rth step XD = specified mole fraction of the light key component in the overhead product accumulated in the rth step Xg = mole fraction of component i in the liquid on the pth plate Yi = mole fraction of component i in the vapor over the pth plate Greek Letters a = relative volatility of the light key component to the heavy
key component el = error tolerance in e2
the termination of the computer program = error tolerance in the distillate composition calculation
Literature Cited Converse, A. 0.;Gross, G. D. Optimal Distillate-Rate Policy in Batch Distillation. Ind. Eng. Chem. Fundam. 1963, 2, 217-221. Coward, I. The Time-Optimal Problem in Binary Batch Distillation. Chem. Eng. Sci. 1967,22,503-515. Domenech, S.; Enjalbert, M. Program for Simulating Batch Rectification as a Unit Operation. Comput. Chem. Eng. 1981, 5, 181-184. Mayur, D. N.; Jackson, R. Time-Optimal Problems in Batch Distillation for Multicomponent Mixtures and for Columns with Holdup. Chem. Eng. J. 1971,2, 150-163. Robinson, E. R. The Optimisation of Batch Distillation Operations. Chem. Eng. Sci. 1969, 24, 1661-1668. Rose, L. M. Batch Distillation. In Distillation design in Practice; Elsevier Science Publishing Company: New York, 1985.
Received for review December 12, 1988 Revised manuscript received June 26, 1989 Accepted August 15, 1989
