Ind. Eng. Chem. Res. 1987,26,983-989
983
Modeling of Reactive Distillation Systems Joel H. Grosser, Michael F. Doherty, and Michael F. Malone* Department of Chemical Engineering, Goessmann Laboratory, University of Massachusetts, Amherst, Massachusetts 01003
A model is proposed for reactive distillation which typically accompanies a continuous polycondensation reactor system, e.g., in Nylon 6,6 production. Extremely high purities are imposed on monomer loss: typically less than 300 ppm monomer concentration in vented vapors. It is shown t h a t such systems are most economical when reactive entrainers are used. T h e latter part of the paper develops guidelines for the use of reactive entrainers t o ease the separation of closely boiling mixtures. We find t h a t there is little economic incentive for the use of reactive entrainers unless the relative volatility of the components t o be separated is less than 1.06. Reactive distillation is a process where fractional distillation is accompanied by a chemical reaction on some or all of the trays in a column. Frequently, this reaction is induced by the deliberate addition of a reactive entrainer. The entrainer selectively reacts with one compound in the column to form products which are removed from the column with relative ease. For example, a difficult binary separation requires an expensive column with many trays and a high reflux ratio. With an appropriate reactive entrainer, the size of the column and its reflux ratio can be significantly reduced to produce the desired separation a t a lower cost. Reactive distillation columns are often used in place of liquid-phase equilibrium reactors. One of the oldest examples of this is a process whereby a column is fed with acetic acid and ethanol to produce ethyl acetate and water by an equilibrium-limited chemical reaction. The column separates the water of reaction and thereby shifts the reaction equilibrium toward the desired product. More recently, a similar process has been commercialized for the manufacture of methyl tert-butyl ether (MTBE), an important antiknock agent to replace tetraethyllead in gasoline. An extensive bibliography on reactive distillation is contained in the paper by Terrill et al. (1985) and will not be repeated here. In this paper, we consider the use of reactive distillation columns for a class of processes which have not been studied previously in the literature. Many important polyamides are formed by equilibrium-limited, liquidphase polycondensation reactions of the form A + B z C
(1)
and A + B e PI+W Pn-k
+ Pk 2 P, + w
(24 (2b)
where A and B are comonomers, C is a salt, and Pi is a polymer with i repeat units. The importance of reaction 1in comparison to 2 depends on temperature, and in what follows, either one may be important depending upon the pressure level in the column. It is desirable to remove the water of reaction, as this will shift the equilibrium toward the polymer product. However, if the reactor is vented, an equilibrium concentration of any volatile reactants with water is removed in the vapor. Since the reactants are valuable and stoichiometric balance is critical to attain high molecular weights, reactants must be separated from the water and returned to the reactor. Moreover, the monomer loss must be both low and constant since a fluctuating monomer loss causes
* Author to whom correspondence should be addressed. 0888-5885 f 87 f 2626-0983$01.50/0
unacceptable variability in the molecular weight of the polymer product. These requirements call for the use of a distillation column to be used in conjunction with the gas/liquid polymer reactor. If both monomers are volatile, reaction 1 or 2a would occur in the liquid phase on each tray of the column with an equilibrium and product distribution dictated by the temperature and compositions. Since the salt and polymer are essentially involatile, this can enhance the separation of the monomers from water. However, when only one of the monomers is volatile (say B), the column must separate only B from water. For reasons mentioned above, it is typical to require no more than a few parts per million of B in the water distillate from the column. Such high-purity separations are quite costly since they require high reflux ratios and a large number of plates. However, if a reactive entrainer stream, rich in A, is fed to the column, then the fast ionic reaction 1will occur on each tray below the entrainer feed point. Such an arrangement allows for very high-purity distillates in relatively small columns. An example we have in mind is a process where A is adipic acid (nonvolatile), B is hexamethylenediamine (volatile) and the polymer is Nylon 6,6. The reactor/ separator is shown in Figure 1 and is described in more detail by Jacobs and Zimmerman (1977) and in US Patent 3 900 450 presented to E. I. du Pont de Nemours & Co. (Aug 19, 1975). In this process, the main salt feed is preheated in the total condenser and fed to the lower part of the column (it could also be fed directly to the heat exchanger adjoining the reactor). The reactive entrainer feed contains adipic acid, salt, water, and traces of diamine. In this paper, we develop a detailed dynamic model for the reactive column shown in Figure 1. Later papers will deal with detailed reactor models and integrated process models. Column Model In this section, we develop a dynamic model suitable for simulating the behavior of reactive columns like the one shown in Figure 1. Reasonable simplifying assumptions are as follows: (i) Components A and C are nonvolatile. (ii) The reaction reaches equilibrium instantaneously on each tray. (iii) The vapor- and liquid-phase equilibrium is instantaneous on each tray. (iv) The solutions are dilute, and hence the temperature change is small and will be ignored. (v) The liquid holdup is constant on each tray. Vapor holdup is assumed to be negligible. (vi) A modified constant molar overflow is assumed (see text for details). (vii) A total condenser provides the reflux. For convenience in deriving the general working equations, the system shown in Figure 2 is considered as a model tray in the column. This column has N equilibrium stages (numbered from bottom), a total condenser, and no 0 1987 American Chemical Society
984 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
tray n. The stoichiometric coefficients are VA = VB = -1 and vc = 1. The three differential equations in eq 3 are not independent. Since the extents of reaction are equal when based on any reactive component, we obtain two independent equations for the conservation of mass
DISTILLATE
TOTAL CONDENSER
SOLUTION
REFLUX
REACTIVE A,B,C A,B,C,WW ENTRAINER FEED
fl,
1'
LIQUID RETUfN LINE
'
tA,n
- tE,n =
(4)
tA,n
- tC,n = 0
(5)
Rearranging eq 3 for .$A,,, 5 leads to
1
I
C O -L U M 'N
[B,,,
and t c , , and using eq 4 and
n = 1, 2, ..., N
(6)
n = 1, 2, ..., N
(7)
VAPOR FEED I
REACTOR
POLYMER PRODUCT
where HEAT EXCHANGER
Figure 1. Reactive distillation column process flow sheet. '-,+I Xn+j
I
hFnn
Vn
-Yn
t
--I
(8)
X2,n
= XA,n + XC,n
(9)
The overall mass balance on tray n is 3
t
I
Xn
-Yn-1
L"
Vn-1
Figure 2. Typical plate.
reboiler. The first stage is fed with vapor from the reactor at some prescribed flow rate and composition. The magnitude of this vapor flow and its composition depend upon the rather complex processes going on in the reactor. In a later paper, we will link the present column model to a detailed reactor model in order to show how the overall polycondensation/reactive distillation system works. We assume that one feed stream Fn,and one vapor and one liquid side stream Pnvand PnL, respectively, may also exist at each stage. This formulation provides the flexibility to investigate alternative entrainer feed points. The transient material balance for component i on plate n is (see Figure 2 for notation)
where
= XA,n - XE,n
and yl,,, z ~ ,and ~ , zZ,, are defined similarly. Equations 6 and 7 incorporate the requirements Y A , ~= Y C , ~= 0. Also, yl,n is equal to - Y B , ~ since Y A , ~= 0; however, we prefer to use yl,n for consistency of notation. Before eq 6 and 7 can be integrated, we need to establish methods for calculating vapor compositions, liquid rates, and vapor rates throughout the column. (a) Liquid and Vapor Rates. We expect the vapor rate throughout the column to be reasonably constant because the diamine composition is very small. Therefore, v, = v 2 = .,. = v, = ... = v, = v (10)
Vn
+ Vn.Yi,n + P n L x i , n + p n v Y i , n ) + viti,n i = A, B, C (3) is the extent of the reaction of component i on (Lnxi,n
Xl,n
+ L n = Vn-1 + L n + l + iC=vl i t i , n + F n
(11)
which, in view of eq 10, reduces to Ln
= Ln+1 +
Cviti,n i=l
+Fn
Equations 10-12 ignore tray hydraulics (but not composition dynamics), in keeping with assumption v. The accuracy of this approximation is well established (see: Levy et al., 1969). If L N +is~ known, then eq 1 2 can be used to determine all the internal liquid flows within the column (see Appendix B for details). Note, if PnLor Pnv# 0, then eq 12 must be modified accordingly. (b) Vapor-Liquid Equilibrium. Recall that we are attempting to separate components B and W by using reactive entrainer A. Components A and B react according to A + B s C , (13) and component W is nonreactive. In addition, components A and C are nonvolatile. This is an example of a perfectly selective reacting entrainer. The entrainer removes B without introducing any volatile componets to contaminate the purified water vapor. The phase equilibria for such a system has already been discussed by Terrill et al. (1985). A modification of their analysis is given in Appendix A. This leads to the following
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 985 equation for y1 as a function of the independent liquidphase compositions, x1 and x2: y1 =
(-& + i)+ i[ +y + SI’”)/ (xl
((a - l){-;(xl
’[
2 (xl -
-
+
a)
a)2 sl‘13 +
+
+ a ( x 2 - 1)) (14)
where K is the equilibrium constant for reaction 13, and a is the relative volatility cywB > 1. Equations 10,12, and 14 enable us to integrate the differential eq 6 and 7. The actual mole fractions of the individual species can be obtained from xl, x2, and y1 as (see Appendix A)
Table I. Feed Mole Fractions in Example 1 feed adioic acid diamine salt 0.0125 0.0542 salt 0.0125 0.0262 0.0384 0.0020 reactive 0.0 3.7 x 10-3 0.0 vapor
water 0.9207 0.9334 1-3.7 x 10-3
Table 11. Dynamic Response of Liquid Flow Rates L,mol/h, at time, h 0.000 0.150 0.300 0.450 0.750 tray 1 0 0 0 0 0 145.05 144.94 145.04 2 145.05 145.05 44.94 44.90 44.91 44.91 44.91 3 44.92 44.94 44.91 44.92 44.92 4 44.93 44.93 44.94 44.94 44.94 5 44.94 44.94 44.94 44.95 44.95 6 44.95 44.95 7 44.94 44.95 44.95 44.95 44.95 44.95 44.95 8 44.94 44.94 44.95 44.95 44.95 44.95 9 43.37 43.37 43.37 43.37 43.37 10 11 43.37 43.37 43.37 43.37 43.37 43.37 43.37 12 43.37 43.37 43.37 4.c
-
=
Qx
3.:
1
2,
z-
3.0
0
Algorithm In order to integrate eq 6 and 7, we need to specify the following: (i) an initial condition, which we take to be pure water on all trays, i.e., xl,,,(t = 0) = x2,,, = ( t = 0) = 0, n = 1 , 2 , ...,N (ii) a phase equilibrium relationship, y1 = y1 (xl, xz), which we take to be eq 14; (iii) the molar liquid holdup on each tray; (iv) the flow rate and composition of all the external flows entering the trays, Le., F,,, z,,, V, yo,LN+l,and x ~ +(v) ~ the ; flow rate of each side stream, PnLand PnV,n = 1, 2, ..., N . We did not consider the use of side-stream take-offs in this work, so all these flows are set to zero. For item iv above, we set xN+l = YN
c 0
2LL
2.5
W
dH
2.c
w
zH
9
0
1.5
w
c
a J 2
1.c
t7
E
0.5
(16)
(i.e., total condenser). In addition, we prefer to set the reflux ratio, R, and then calculate LN+l. These quantities are related via a mass balance around the total condenser V = D + LN+1 = D ( R + 1) (17) Knowing V and R enables us to calculate the distillate rate, D , and LN+1* Finally, knowing LN+lenables us to calculate all the
C
I
I
I
I
I
I
I
I
I
1
01 0.2 0,3 0.4 0.5 0.6 0.7 0.0 0.9 1.0
TIME (hours)
Figure 3. Dynamic response of distillate diamine composition.
A
2.252 x ( O - ~
liquid rates throughout the column from eq 12. Integration of the differential equations is now a simple task, using any of the standard techniques.
Numerical Simulations The simulations can be carried out quite efficiently by using either a fourth-order Runge-Kutta method (st = 1/1000 h) or Gear’s method. Example 1. This example represents a typical 12-tray industrial column; see US Patent (1975) for a discussion of the conditions. The column has a total condenser, two liquid feeds, and a vapor feed from the reactor, exactly as described earlier. The salt feed is a saturated liquid that enters the column on tray 2; a basis of 100 lb mol/h is used. The reactive feed, also a saturated liquid, enters the column on tray 9 at 1.57 lb mol/h. The vapor feed enters the bottom of the column as a saturated vapor a t a rate of 130
RE4CTlVE FEED TR4Y
2.249~10-~
I
I
I
I
I
I
986 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 I0'
10-
REFLUX RATIO = 0.5 SALT FEED: TRAY 2 ADIPIC FEED= TRAY 9 y, IN ENTERING VAPOR= 0.003665
REFLUX RATIO = 0.5 SALT FEED= TRAY 2 ADIPIC FEED= TRAY 5 y, IN ENTERING VAPOR: 0.003665
10-
10-
TALT FEED
10-
10-
10-
10-
10'
1
2
3
4
5
6
7
S
10-
9 10 11 1i
I
TRAY
2
3
4
5
6
7
8
9 10 11 12
TRAY
Figure 5. ndiamineand ydiamine vs. tray: example 1.
Figure 7.
1
ix
xdiamine
and ydiamine vs. tray: example 2
The change in x 2 is small, and x 2 can be considered constant between feed trays. In fact, this is ensured by eq 7 when L is assumed constant. Clearly, the assumption of constant liquid flow rate does not significantly affect the results of this model. This assumption also simplifies the column model, as eq 12 is no longer required. A plot of the liquid- and vapor-phase diamine mole fractions on each tray a t steady state is shown in Figure 5. Trays 6-9 do not effectively separate diamine from the remaining compounds because the diamine mole fraction does not significantly change in that region of the column. Figure 6 shows the steady-state composition of each compound on each tray. Example 2. The specifications for example 2 are exactly the same as those for example 1, except that the reactive feed enters the column on tray 5. This example illustrates the effect of reactive feed location on the purity of the separation. This effect becomes especially evident when Figure 7, a plot of the steady-state liquid- and vapor-phase mole fractions vs. tray, is compared to Figure 5. Here, the diamine mole fractions exhibit a nearly linear semilogarithmic relationship with the tray number. As a result, the distillate diamine mole fraction in example 1is 3.582 X and that of example 2 is 1.083 X Changing the reactive feed point reduces the diamine loss by over 300%. Alternatively, the original diamine loss can be maintained but with a much smaller reflux ratio (and corresponding energy saving).
7 o~w2h 0
YLT
2
4
6
8
1 0 1 2
7
S
9
10
11
~~~~
0
1
2
3
4
5
6
12
TRAY
Figure 6. Steady-state liquid mole fractions vs. tray.
constant on each tray even though no assumption regarding the constancy of the molar liquid flow rate has been made. This is because the flow rate of the reactive feed is small, and the reduction of the liquid rate due to reaction is correspondingly small compared to the reflux rate. Under these conditions, then, the liquid flow rate can be considered constant between feed points. The dynamic response of the distillate diamine composition is given in Figure 3. Calculations show a monotone increasing response. Neither oscillations nor multiple steady states were ever encountered. The calculated distillate diamine mole fraction for a nonconstant liquid When the liquid rate was assumed rate was 3.582 X constant, the distillate diamine mole fraction was 3.593 x Therefore, the assumption of constant liquid molar flow rate does not significantly affect the calculated distillate purity. Figure 4 is a plot of the sum of the mole fraction of nonvolatile compounds (x2) vs. tray number at steady state.
Separating Closely Boiling Mixtures by Reactive Distillation In a previous article, we considered the use of reactive entrainers for separating closely boiling mixtures (Terrill et al., 1985). Industrially important separations in this class include p-xylene + m-xylene (a= 1.02), 1-butene + 1,a-butadiene (a = 1-07), 1,3-dichlorobenzene + 1,4-dichlorobenzene ( a = 1.03), etc. Terrill et al. describe a reactive entrainer method for separating p-xylene from m-xylene. In addition, they discuss a method for recovering the reactive entrainer for recycle.
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 987
t" REACTIVE DISTILLATION L COLUMN
A
1
-
ENTRAINER
BY
L
J
f a
9.6
9.4-
0
$
9.2-
E
2
9.0A B
8
8.4 8.2
t,,,,,,,11111111111
8.0
1.0
1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18
R/Rmin
Figure 9. Relative cost of separation vs. R/Rmi, for various separation parameters.
Optimization of the Reactive Distillation Process The optimization of a reactive distillation process is more difficult than optimizing an ordinary fractional distillation process. In fractional distillation, if the flow rate and the composition of all feed and product streams are specified, then only the ratio R/R- need be optimized to produce the lowest cost, optimal column. In reactive distillation, the reactive entrainer feed rate (a function of x2,the sum of the mole fractions of the nonvolatile compounds) must be optimized, as well. With the optimal values of these variables, quantities such as the number of column trays, heat-exchanger surface areas, and heating and cooling requirements can be calculated, along with their associated costs. Since the entrainer may be expensive, an entrainer recovery system must be included in the reactive distillation process and its cost included in the optimization problem. The detailed design of a recovery system is beyond the scope of this work, but one possible flow sheet is given in Figure 8. The economic tradeoffs of these optimization variables can be intuitively examined. If R/R- is large, the column requires fewer trays for a specified separation but a large diameter, and heating and cooling costs are high. If R/Ris small, heating and cooling requirements are low, but an expensive column of many trays is required. When the entrainer feed rate is low, little or no reaction will occur on each tray, and again, many trays are required. If the entrainer feed rate is high, fewer column trays are needed, but the entrainer recovery costs become excessive. Optimization of R/Rmi, When column parameters such as distillate purity, distillate flow rate, and entrainer feed rate are specified, and when the relative volatility and reaction equilibrium constant are also specified, then changing the value of R/R- influences the size and cost of the column required for the desired separation. There is certainly a different optimum value of R/Rminfor each separation, but if this value is insensitive to the parameters, then a rule of thumb can be developed. Figure 9 is a plot of separation cost vs. R/R- for various parameters, given in Table 111. Clearly, the lowest separation cost occurs near R/Rmin= 1.1in each case. This was also found to be true for wide variations of a, the
Table 111. Parameters for Curves in Figure 9 curve
A B C D
(Y
1.03 1.03 1.03 1.03
K
XZ
ZB
0.345 344.8 344.8 0.690
1X 1 x 10-5 1 x 10-3 1X
0.4 0.4 0.4 0.4
relative volatility, and zB, the feed composition of the volatile reactant. Therefore, it is not necessary to know x 2 (a function of entrainer feed rate) to select an approximately optimal reflux ratio. There is, however, one important exception to this rule. This occurs when ZB is small and x 2 is large. In this case, no reflux is required because sufficient separation occurs by reaction alone.
Optimization of Reactive Entrainer Feed Rate I t is preferred to solve the entrainer feed rate optimization problem in terms of x2, the sum of the mole fractions of nonvolatile compounds, because it is a dimensionless column parameter. The entrainer feed rate equals the product of the column reflux rate and the entrainer mole fraction in the reflux. If x 2 is assumed constant, then the entrainer feed rate = Lxz. While the design of the reactive entrainer recovery system is beyond the scope of this work we can assume values for the total cost of recovery, including capital and operating costs (in $/lb mol of entrainer), and examine the economic incentive for reactive distillation. For a given separation, as the entrainer recovery cost is decreased, the optimal entrainer feed rate increases. The optimal feed rate also increases as the relative volatility approaches unity because then the cost of fractional distillation increases, and we can afford to use more entrainer. These effects are shown in Figure 10 where the ratio of reactive to fractional distillation costs and the optimal value of x 2 are plotted vs. assumed values of entrainer recovery cost for K = 344.8. We see that there is little incentive to use reactive distillation when a > 1.06 because the reactive to fractional distillation cost ratio is close to unity. It is also clear that the optimization of x 2 is more important than reflux ratio optimization because we cannot rely on simple heuristics for guidance. Therefore, the design of the entrainer recovery system is required for the optimal design
988 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
" 8 c
IO-
09I-
m 0
:$
s
08-
2
J
07-
8 0I
06-
w n
*,
05-
5%
04-
U U
03-
PL
02- a = 1.02
FF
aa
I .o
I
W
c
v)
t
"c
5
Fc3
ENTRAINER RECOVERY COST ($/lbmolc)
(b)
$v
8 W U
"I 8 " I-
0
w
u a
W
(C)
L
0
1.o
x2
Figure 11. Calculation of optimal x2. Figure 10. Ratio of reactive to fractional distillation costs and optimal value of x 2 vs. entrainer recovery costs ( x 2 = xA + xJ
of the entire reactive distillation process. However, a preliminary decision to use reactive distillation rather than fractional distillation can be based on the results of Figure 10. For reactive distillation, the following design algorithm is suggested. (a) Assuming no cost for the entrainer recovery, compute the reactive distillation column cost vs. x p as shown in Figure l l a . (b) Design the entrainer recovery system, and compute its cost vs. x 2 as shown in Figure l l b . (c) Plot the sum of the column and entrainer recovery system cost vs. x2. The optimal value of x 2 corresponds to the minimum value of this cost curve, Figure llc.
Conclusions We have developed a model for the reactive separation of volatile monomer from vented water vapor in a typical polycondensation reactor. We predict that the process described in US Patent 3900450 (1975) could be retrofitted by changing the reactive entrainer feed location to attain a significant improvement in product purity. (Compare Figures 5 and 7.) When reactive entrainers are used to separate closely boiling mixtures, there is an optimum reflux ratio and an optimum entrainer feed rate. For the reaction system studied (A + B e C, A and C nonvolatile), the optimal value of R/Rmi, is near 1.1except when the feed mole fraction of the volatile reactant is small and the reactive entrainer feed rate is large. In this case, the desired product purity can be achieved by reaction alone, and no reflux is required. Because the reactive entrainer may be expensive, an entrainer recovery system is required. While this work does not include the detailed design of such a recovery system, we can nevertheless examine the economic incentive for reactive distillation. It was found that there is little incentive to use reactive distillation when the relative volatility is greater than 1.06, and it becomes more attractive as the relative volatility approaches unity.
Acknowledgment We are grateful for the financial and technical support received from E. I. du Pont de Nemours & Co., Textile Fibers Department, Seaford, DE. In particular, we thank Dr. R. Longhi, R. D. Livingston, and Dr. R. Dujari for their help and encouragement.
Nomenclature A = reactive entrainer B = volatile reactant C = salt, reaction product D = distillate rate F = feed rate, mol/h H = liquid holdup, mol K = reaction equilibrium constant L = liquid rate, mol/h N = total number of equilibrium stages PL= liquid product rate, mol/h Pv = vapor product rate, mol/h Pj = polymer containing j repeating units R = reflux ratio V = vapor rate, mol/h W = water, volatile nonreactant x = mole fraction in liquid phase x1 = defined in eq 8 x p = defined in eq 9 y = mole fraction in vapor phase z = mole fraction in the feed Greek Symbols CY
= relative volatility
stoichiometric coefficient extent of reaction Subscripts D = distillate i = component i n = tray n v = =
Appendix A. Phase Equilibrium Equations The binary mixture of water + hexamethylenediamine is decidedly nonideal. However, a constant value of CY correlates the x-y data very well. Of course, a cannot be estimated from the ratio of vapor pressures for the two components. If we assume the vapor-liquid equilibrium
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 989 can still be described by constant a’s when the nonvolatile
components are present, then the reactive-phase equilibrium is governed by YB
=
XB XACYAB+ X B
+ XCCYCB+ XWCYWB
(Ala)
3
Ln
XAXB
+ XB + XC + XW
(A14
YB
= (1 - CY - CYKXJXB
+ a(l - xA)
(A24
= KxAxB
(A2b) XW = 1- XA - X B - K x A x B (A24 Note that eq A1 have 2 degrees of freedom, which we have taken to be XA and xB. Also, for ease of notation, we have taken a = CYWB> 1. With a bit of rearrangement, eq A2 can be written in terms of the independent variables x1 and x p , where XC
x2
XA
- XB
= XA
+ XC
Appendix B. Calculation of Liquid Rates The liquid rates inside the column depend on the extent of reaction for each plate and vary according to eq 12. 3
i=l
032)
= Ln+, - t n + F n
033)
The extent of reaction is found by rearranging any one of eq 4, say i = B
(LnXB,n
+ VYB,n)
(B4)
The derivative on the right-hand side of eq B4 is estimated by backward difference. A forward difference approximation introduces an iteration at each time step owing to the fact that ~ g ( + t 6t) is not known at time t. This backward difference approximation effectively introduces a small artificial time delay into the system since we are essentially replacing tn(t)by tn(t- 6t). Our numerical integrations are carried out with 6t = 1/1000 h, and extensive testing with much smaller values of 6 t demonstrates that our results are not sensitive to this approximation. Literature Cited
The results of this rearrangement are given by eq 14 and 15 in the text.
Ln = L n + 1 + CviEi,n + F n
+ EnCvi + Fn i=l
For our particular application, this reduces to
Noting am = aCB= 0, these equations can be rearranged to give XB
= Ln+l
Ln
K = - XC 1 = XA
components, eq B1 becomes
(Bl)
Since the extent of reaction is the same for all reactive
E. I. du Pont de Nemours & Co. US Patent 3 900 450, Aug 19,1975. Jacobs, D. B.; Zimmerman, J. In Polymerization Processes; Schildknecht, C. E., Ed.; Wiley: New York, 1977; Chapter 12. King, C. J. Separation Processes, 2nd ed.; McGraw Hill: New York, 1980; pp 216-217. Levy,R. E.; Foss, A. S.; Grens, E. A. Ind. Eng. Chem. Fundam. 1969, 8, 765. Terrill, D. L.; Sylvestre, L. F.; Doherty, M. F. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 1062. Received for review March 14, 1985 Revised manuscript received September 29, 1986 Accepted December 4, 1986
