Ind. Eng. Chem. Rod. Res. Dev. 1985, 2 4 , 327-333
327
Modeling of Backmixing in Continuous Polymerization of Caprolactam in VK Column Reactors Alok Gupta and Kandukurl S. Gandhl' Department of Chemlcal Englneerlng, Indlan Institote of Technology, Kanpur 208 0 16, Indh
A stagewise model based on tanks In series approach was proposed to assess the effectsof backmixing in the two-phase zone of VK (Vereinfacht Kontlnuerlich)column reactors. A recycle stream between tanks has been used to represent backmixing due to the vapor stream rising countercurrently to the liquid stream. The model has been used to calculate the conversion, number average degree of polymerization, and the polydispersity index at the end of the reactor. It was found that Increased backmixing increases the waterholding capacity of the stream and therefore increases the conversion at the end of the two-phase zone. As water also lowers the degree of polymerization, it was found that at the end of the reactor where equilibrium is reached, the degree of polymerization decreases with increased backmixing in the top portion of the reactor. The polydispersity index is relatively unaffected as equilibrium is attained at the end of the reactor. If too high temperatures were used at the beginning of the reactor, the streams would contain less water. This would lead to higher degrees of polymerization but to lower rates.
Introduction Hydrolytic polymerization of t-caprolactam is a commercial route of production for Nylon-6. The hydrolytic process consists of polymerizing caprolactam in the presence of water and is carried out either batchwise or continuously. Sittig (1972) has summarized the various reador arrangements that have been proposed. The proposed arrangements vary from batch reactors operating under high pressure to a cascade of stirred tank reactors to simple tubular reactors. The present work focuses on one type of continuous tubular reactor-the VK column. A VK column is essentially a vertical tube or a series of tubes into which an appropriate mixture of water and caprolactam is fed at the top and polymer and unreacted caprolactam are withdrawn at the bottom. The top of the reactor is at atmospheric pressure. The feed temperature varies in the range of 220-240 "C. Although this temperature is considerably higher than the boiling point of water, it still is low for reaction to occur to an appreciable extent. The top zone of the reactor therefore consists of heaters or banks of heaters to bring the reaction mixture up to around 260 "C. It may be noted that 260 "C is very close to the normal boiling point of caprolactam. As the reaction mixture enters and flows down the column, it encounters temperatures well above the boiling point of water. As a result, water vaporizes to a large extent and caprolactam to a small extent. The boiling point of the reaction mixture increases as it flows down the column for two reasons. First, the activity of water in the liquid phase continually decreases due to evaporation as well as to reaction and this pushes the boiling point of the reaction mixture up as water is more volatile than caprolactam. Secondly, the hydrostatic head increases with depth, which also increases the boiling point. Consequently, the vaporization process proceeds only up to a certain depth where, due to a combination of decreased activity of water and increased hydrostatic head, the boiling point of the reaction mixture just equals the local temperature. The reaction mixture is in single phase from that depth onward. The vapor generated in the two-phase zone bubbles through the column and rises to the top. Usually the caprolactam in the vapor is condensed and recycled into the reactor as shown in Figure 1. The VK column therefore can be visualised consisting of two zones. The top zone of the reactor is a two-phase 0196-4321 18511224-0327$01.5010
gas-liquid bubble column reactor while the bottom zone is a single-phase tubular reactor. Usually the bottom zone is modeled as a plug-flow reador (PFR). The mass fraction of vapor phase in the top zone of the reactor, or quality of the mixture in the two-phase flow terminology, is high and hence the countercurrent flow of the vapor causes vigorous mixing action. In view of this, it has been customary to model the two-phase part of the reactor as a single continuous stirred tank reactor (CSTR). In such a scheme the entire VK column is modeled as a CSTR followed by a PFR. Jacobs and Schweigman (1972) and Naudin ten Cate (1973) utilized such models to analyze VK column reactors. A rough estimate made from data in the open literature indicates that the residence time in the two-phase zone could be as much as 30% of the total residence time. Further, as the quality of the mixture decreases with depth in the two-phase zone, the mixing action of the vapor would be less vigorous there. In other words, the reaction mixture throughout the two-phase zone might not be homogeneously mixed. In view of both these factors, the approximation of the entire two-phase zone by a single CSTR is open to criticism. Moreover, as the reaction is carried out essentially to equilibrium in VK column, the water content of the reaction mixture at the end of the two-phase zone essentially determines the nature and properties of the product at the end of the single-phase zone as well. Thus it is desirable to have a more realistic model of the two-phase zone of the reactor. We recently (Gupta and Gandhi, 1984) proposed a model with stagewise backmixing and here the detailed consequences of the model will be evaluated. As data on VK column reactors are not available, it is not possible to verify the model's prediction. However, it is hoped that the model can be tested experimentally in future. Model for t h e Reactor Reaction Mechanism and Kinetics. The hydrolytic polymerization is known to consist of three main reversible reactions: (i) ring opening
0 1985 Amerlcan Chemical Society
328 Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985
(ii) polyaddition F
t
K
Caprolactam + water
Water
Condenser Va pour t I ow
n n
(iii) polycondensation
IIII IIPII 1I1I IIll IIII
81
In addition to these three main reactions, side reactions leading to the formation of cyclic oligomers also occur. However, these have not been considered in this work. The rate expressions have been derived earlier (Gupta and Gandhi, 1982) and are summarized in Table I. These equations are different from those given by Tirrell et al. (1975) but are consistent with those given by Hermans et al. (1958), Hoftyzer et al. (1965),and Reimschuessel(l977). The reaction rate constants for three reactions are usually represented as ki = k?
+IZ~C;
(i = 1, 2, 3)
+
output
Figure 1. VK column reactor.
Table I. Rate Expressions
dM
- = rM = - k , ( M w - s1/K,) - h , ( M c - ( c - s , ) / K , )
ds i
(4)
where the subscripts refer to the reaction number. The above representation reflects the fact that all three reactions are catalyzed by carboxyl end groups. The superscript u refers to the rate constant for the uncatalyzed reaction while c refers to the catalyzed reaction. The rate and equilibrium constants are temperature dependent and an Arrhenius type relationship can be used k i = A i exp(-Ei/RT); 0' = u or c) (5)
(7)
dt
= rsl= h , ( M w
-
sl/K,) - 2 h , ( s , c - w ( c - sl)/ K 2 ) - k , ( M s , - s,/K,)
n- 1
h n
- = rsn = k , C dt
m=l
(8)
m
s,s,.~
+ 2(h,/K,)w
C
Sm -
m=n+l
The equilibrium constant similarly can be written as In this work the values given by Reimschuessel(l977) for these constants have been used and are listed in Table 11. The units of K , and K3 are kg/mol, while K2 is dimensionless. Model with Stagewise Backmixing. The most important feature of the top portion of the reactor is the backmixing caused by the bubbling action of the vapor phase. The effect of mixing in two-phase bubble columns has been evaluated on the basis of axial dispersion models as well as by stagewise models such as tanks in series models etc. In view of the computational simplicity as well as the objective of quantitative but comparative exploration of mixing effects, a stagewise model was proposed earlier (Gupta and Gandhi, 1984) for this purpose. Selection of a stagewise model is also partly justified by the presence of baffles and heating sections in the reactor. The model consists of three tanks in series with recycle liquid streams between them. The recycle stream represents the backmixing caused by the upward moving vapor bubbles. A figurative depiction of the model is displayed in Figure 2. The recycle streams are shown by the dotted lines. Models such as these have been used earlier by Mecklenburgh and Hartland (1975) and Chianese et al. (1981). The number of tanks has been selected to be 3. Chianese et al. (1981) presented a correlation for a , the magnitude of
dz
dt
=
m dsn rt = .Z ( n - 1 ) -d t = h 2 ( c 2- w z / K , ) t
n=l
Table 11. Kinetic Constants of Rate Eauations" i J A,' E,', cal/mol H,, cal/mol 1 u 16.94 X lo5 21040 2114.2 c 41.06 X IO6 18753 22550 -6140.4 2 u 86.87 X lo8 c 23.37 X lo9 20674 3 u 26.2 X lo8 21 269 -4028.3 c 23.72 X lo9 20400
S,, eu -7.87 0.93 -6.95
"A," in kg/(mol h); A,C in kg2/(mo12h).
the fraction of the recycle stream. The values of CY obtained from their correlation were in the range of 0.5 to 0.9. Accordingly a was varied between 0 and 1. The three CSTRs shown in Figure 2 are connected by the vapor streams emerging from the tanks below. Thus the first and second tanks receive a vapor stream from the second and third tanks, respectively. The vapor stream leaving the first tank is sent to a condenser and the caprolactam is recycled to the top of the reactor. The third tank, however, will not receive a vapor stream as the portion below it is a single-phase reactor. The three tanks
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985
Flu1
+ aFiw1- aFlw2 - FZW~- CUFZW~ + (YFZW~+
Flsl,l
+ aFlS1,l - aFlSl,Z - F2s1,2
329
G2~3' - G1~2' + pV2rw,2 = 0 (22)
+ aF2s1,3 +
- ffF2s1,2
PV2rs1,*= 0 (23) F1snpl+ aF1Sn,1 - aF1Sn,2 - F2~n,2- ~ F z s ~+, azF 2 s ~ v 3 + PV2rsn,*= 0 (24)
Two phase reactor model
I
F1
+ G2 - F2 - G1
O.113M2'
:'I/
+ O.OI8W2' + Mz')
wZV(Pm+ pgh,)/(w,'
F?
0
(25)
= 1.0
(26)
= P ~ , Z ' ~ ~ W Z (27) /MO
M2'(Pm + P ~ ~ z ) / ( w + , ' M2') = Pc1,zsatM2/Mo (28) third tank
j u g flow model
VI 4 4
F2M2
(29) F 2 ~ 2+ CYF~W~ - ( Y F ~-wF~3 ~ 3- G2~3' + pV3rW,3= 0 (30)
output
Figure 2. Schematic of reactor model.
have been assumed to be at different temperatures. Thermal equilibrium has been assumed via the well-stirred nature of the tanks. The residence time in each of the tanks has been assumed to be 1 h, giving a total residence time of 3 h in the two-phase region. It is approximately 30% of the total residence time in a typical VK column. The feed flow rate has been arbitrarily selected as 1000 kg/h, a typical production figure. Jacobs and Schweigman (1972) reported thermodynamic equilibrium to exist between the liquid and gas phases as far as mass transport is concerned. Earlier estimates (Gupta and Gandhi, 1984) of mass transfer coefficients supported this. It was therefore assumed that the vapor and liquid phases would be in local thermodynamic equilibrium throughout the two-phase zone of the column. Mathematical Description of the Model. The equations describing the polymerization processes in the model described are given below. The mass balances for monomer, water, aminocaproic acid, and n-mers in the first tank, respectively, are F a 0 - FlMl + aF1M2 - aF1M1 + G1M2' + PVir,, = 0 (13)
FOWO - Fiw1
+ aFlw2 - aFiW1 + GlW2'
+ aF2M2 - aF2M3 - F3M3 - G2M3' + PV3rM,3 = 0
- GOWl'
+
F2~1,2+ aFzsi,z - ~ J ' 2 ~ 1 , 3- F3si,3 + P V ~ ~ = S ,0, ~(31) F2sn92
+ ffF2sn*2- ffF2sn,3- F3sn,3 + Pv3rs,,3 F2 - F3 - G2 = 0
+ pgh,)/(w,' + M3') M3'(pm + pgh,)/(ws' + M3') ~3'(f"
+ aF1~1,2- aFlsl,l + p v ~ r s l =, , 0 -FlSn,l + aFlsn,Z - aF1sn,1 + pV1rsn,,= 0
= P W , ~ ~ ~ ~ W (34) ~/MO
= pc1,3"tM3/M0
(35) (36)
In the above equations, p V i = T ~ Fwhere ; T is the residence time in the CSTR. Generally three properties of the product are of interest: conversion, number average degree of polymerization (DP), and the polydispersity index (PDI), which is the ratio of the weight average degree of polymerization and DP. Noting that m
c =
m
Cs, = po;
z
n=l
+ c = n=l Ens,
= pl
+
it can be seen that DP (= (z c)/c) can be easily calculated if the zeroth and first moments of the molecular weight distribution represented by sn are evaluated. The second moment of the distribution, b2, defined as
".
C n2sn = n=l
(15) (16)
In view of the units being used, i.e., mol/kg, the overall mass balance has to be written separately and is given by
Fo - F1 + G1 - 0.018wlvGo = 0
(32) (33)
0.113M3' 4- 0 . 0 1 8 ~= ~1.0 ~
PVirWl= 0 (14) -FlSi,l
=0
(17)
and the gas-phase compositions have to satisfy (18) 0.113M1' 0 . 0 1 8 ~= ~1 ~ Vapor-liquid equilibrium relations for the first tank give w,'(pm + pgh,)/(w,' + MI') = wlPw,lsat/Mo (19)
+
M,'(Pm + pghJ/(w1' + MI') = MIPcl,lsat/Mo (20) Similarly, the modeling equations for other tanks can also be written. They are: 2nd tank F1M1+ aF1Ml- aFlMz - FZMZ - CXF~MZ + aF2M3 + GzM3' - G1M2' + P V ~ F M = 0, ~(21)
has to be evaluated to get PDI since PDI = p 2 p O / p l 2 . Calculation of Conversion and DP. Equations for c can be obtained for the first tank by summing (15) and (16) for n I2. Similarly, an equation for z + c can be obtained by multiplying (16) by n and summing for n I 2 and adding the sum to (15). The resulting equations are -F1c1 + aF1~2- aF1c1 + pV1rcl = 0 (37)
+ C J + C Y F ~ (+Z ~
-F~(z,
~ 2 -)
+
(YF~+ ( z~~ 1 ) P V I (+~r Z~ J~= 0 (38)
Similarly, the equations for the second and third tanks, respectively, are (1 + a)F1C1 - aF1C2 - ( 1 + C Y ) F ~+CCYFZC~ ~ + pV2rC2= 0 (39)
330
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985
(1 + a)F2c, - C Y F ~-CF~3 ~ 3+ pV3rC3= 0
(41)
1-1/ I \
FZ(1 + ( Y ) ( c ~+ 22) - aFz(c3 + 23) - F 3 ( ~ 3+ 23) + pV3(rC3+ rz3)= 0 (42)
A total of 27 nonlinear algebraic equations, nine for each tank, have to be solved simultaneouslyto obtain conversion and DP. As an example, (13) to (20), (37), and (38) are the equations for the first tank. One more assumption is required to do this as rsl involves s., See (8) of Table 11. It is well-known that assuming s2 = s1 or estimating s2 by Flory’s distribution give results differing only insignificantly. Here it was assumed that s2 = sl. A form of Raoult’s law suggested by Jacobs and Schweigman (1972) on the basis of their data has been used to calculate the vapor-liquid equilibrium relations. This is exemplified in (19), (20), (27), (28), (34), and (35). Tai and Tagawa (1983) quote data of Fukumoto on vaporliquid equilibria of a caprolactam-water-nylon system. However, these data are applicable for caprolactam concentrations of less than 13 wt % or at high conversions. Our attempts to fit data of Giori and Hayes (1970) in a form suitable for the present work were unsuccessful. In view of all this it was decided to stick to the above-mentioned Raoult’s law type of formulation. The vapor pressure of water was calculated from the correlation given by Sherwood et al. (1977) 3814.44 In Pwsat (in atm) = 11.6703 (43) T (K) - 46.13 It might be mentioned here that (43) is very different from that given by Tai et al. (1982). We use (43), as the latter’s equation did not fit the observed vapor pressures of water and gave large deviations starting from 100 “C onward. The vapor pressure data for caprolactam given by Snell and Ettre (1969) have been fitted to Claussius-Clapeyron’s equation resulting in 7024.023 In Pclsat (in atm) = 13.006 (44) T (K) The reaction mixture from the third tank flows to the PFR where boiling does not occur. The equations that describe a PFR are identical with those of a batch reactor which have already been summarized in Table I. The PFR calculations were done at the same temperature as that of the last tank. It is clear that the outlet conditions of the third tank form the inlet conditions of PFR. Calculation of PDI. The second moment of the s, distribution has to be calculated to get the PDI. However, it is well-known that a closure approximation is needed to calculate a finite number of moments in reversible polymerization as the equation for the nth moment would contain the (n 1)st moment as well. In this work the equations for the first three moments were solved using a closure approximation for the fourth moment based on Laguerre polynomials given by us (Gupta and Gandhi, 1982). The equations for the lth moment of the s, distribution in the first tank is obtained by multiplying (16) by n1and summing it from n = 2 to 03 and adding (15) to the sum. The equations for the second and third tank can be similarly obtained by operating on (24) and (23), (32) rnd (311, respectively. These are given by (45) +lPl,l + aF,cc1,2 - (YFlII1,I + PVlrrr,l= 0
+
FlP1,l
+ aFlP1,l - CYFlIL1,2 - F2P1,2 - aFzP1,z + aF2111,3 + P V Z ~= , ~0, , (46) F2P1,2
where
+ aF2Pl,2
- aF2P1,3 - F3P1,3
+ P v 3 r p , , 3 = 0 (47)
The second moment p2 in each tank can be obtained by solving the above equations for 1 = 2 and 3 simultaneously by using the closure approximation for p4 as already mentioned. The equations corresponding to PFR are given by
The second moment of the molecular weight distribution at the end of PFR can be obtained by integrating (48) for 1 = 2 and 3 by using the exit conditions of the third tank as the initial conditions for the PFR. Numerical Computation Procedure. The modeling equations are highly nonlinear algebraic equations. Good initial guesses are required for successful solution of these equations. Brown’s (1973) method was used for solving the equations. Initial guesses were obtained by defining an objective function as the sum of squares of the left-hand side of eq 13, 15, 37, 21, 23, 39, 29, 31, and 41 and minimizing it with respect to the nine variables MI, s ~ ,c l~, M,, , sIp2, c2, M3,~ 1 , 3and , c3. Equations 45 to 47 were solved by employing a similar procedure. Using the above results as feed conditions (i.e., at t = 3 h), eq 7, 8, 11, 12, and 48 were numerically integrated for t > 3 h. It may be noted that the usual water balance equation
w = Wg
+ co - c
eliminates the need for integrating eq 10 in PFR. Runge-Kutta-Gill’s method as given by Ralston and Wilf (1962) was used for integrating the equations. Results and Discussion The modeling equations were solved to obtain the conversion of the monomer, the number average degree of polymerization (DP), and the polydispersity index (PDI). In all calculations the feed was assumed to contain 0.05 mol of water/mol of caprolactam. The three tanks would be at different temperatures. Three sets of temperatures were selected which were in the range of temperatures indicated by Kwatra (1980). For a set of temperatures selected it is not necessary that vapor phase would be generated in all three tanks. Two phases would not be present in a tank if the boiling point of its contents is greater than the temperature specified for it. Thus, once the temperatures of the tanks are specified, the composition of the liquid in the tanks determines the boiling points and hence the presence or absence of two phases. The following procedure was adopted to establish the conditions in the reactor. The composition of the reaction mixture in the three tanks was calculated for a given set of temperatures assuming that only liquid phase is present in all the three tanks. The boiling points of the resulting compositions were computed. If any of the boiling points was less than the temperature specified for that tank, the assumption that only liquid phase is present is not satisfied and hence the solution was rejected. This procedure ruled out the possibility of all three tanks containing only liquid phase for all the temperature sets se-
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985 331
Table 111. Two-Phase Zone Results
"C) stream leaving 3rd tank conv of lactam, % DP 55.6 22.5 33.4 64.7 36.3 67.6 66.3 35.3 62.4 31.4
liq
a
0.0,0.0 0.5,0.5 1.0,1.0 1.0,0.5 1.0,0.0
two phases present in tank no. 1, 2, 3 1, 2, 3 1, 2,3 1, 2,3 1, 2,3
HzO, mol/kg 0.16 0.177 0.181 0.180 0.174
(230-240-253.5
160
-
150-
PDI 1.99 2.12 2.18 2.14 2.04
1LVL
loo -
120
a 0 80'.
Table IV. Two-Phase Zone Results
a
0.0,0.5,0.5 1.0,1.0 1.0,0.5
(240-253.5-265 "C)
liq stream leaving 3rd tank conv of HzO, lactam, mollkg % DP 27.0 79.5 0.102 41.0 87.7 0.132 44.0 90.1 0.137 43.1 89.1 0.135
two phases present in tank no. 1, 2 1, 2,3 1, 2,3 1, 2,3
Table V. Two-Phase Zone Results
/
60-
PDI 1.97 2.12 2.17 2.14
I
d
0
d.0.5
A
a+;l.O
v Variabla L
50-
-----
ed1=1,62zO~5 Variablou kl*l. u p 0
I
I
1
I
I
I
I
I
Figure 4. Average degree of polymerization as a function of residence time.
(250-260-270 "C)
lia stream leaving 3rd tank conv HzO, lactam, mol / ke % DP 24.6 93.8 0.071 33.0 100.6 0.08 36.9 103.3 0.085
two phases present in tank a no. 1, 2 0.0,1, 2 0.5,1, 2 1.0,-
70
2.2 Tamporaturo: 230
PDI 1.92 1.98 2.0
a.0
A
a.O.5 6.1.0 Yoriobk d C41~1~0,62*0~5 Varioblr d LY,lil.O, d2.0-0
0
2.1 -
- 260-253.5 C
0
v 0
-----
g 2.070
1.9
-
60
V
30
*o/
I
I
lo
;
I
0
d1.10 62iOO Varioble d Vorlobe cz+lO 6 2 ' 0 5
cc
0
0
1
2
3
L
5
y1
;,2;
I
6
7
8
1.8
0
I
I
I
1
2
3
I L
I
I
I
I
I
5
6
7
8
9
I 10
I
-----
9
10
Time, hrs
Figure
3. Monomer conversion as a function of residence time.
lected. A similar procedure also eliminated the possibility of only the first tank containing two phases for all the cases considered. The procedure was then extended to the case where two phases exist in the first and second tanks and finally to the case where two phases exist in all three tanks to eliminate unrealistic solutions. Another check on the validity of the solutions was that all the vapor flow rates should be positive. For instance, the case of generation of vapor phase in all three tanks for the 250-260-270 "C set was eliminated as some vapor flow rates came out to be negative. Thus for this set, two phases exist only in the top two tanks. It must be noted that when vapor phase is not generated in the last tank, there would be no vapor stream to or from the tank and correspondingly the recycle liquid stream flow rate would be zero. With all the cross checks, the solutions to these equations were established. The results obtained for the two phase zone are shown in Tables 111-V. The results for the entire reactor, Le.,
TemperolurP. 2b0-253.5-265 TemperolurP 2b0-2535-265 0 Oc.0 0 n a.05 a.0.5 A OI.l.0 &=lo
c
v Vorloble a 61.10
~2'05
70t Time, hrs
Figure 6. Average degree of polymerization as a function of residence time.
up to the end of PFR, are shown in Figures 3-7. In these figures the results of the two-phase zone are indicated by points located at 3 h residence time and those for the PFR are indicated by lines. Effect of Backmixing. Tables 111-V show the effect of backmixing on the two-phase zone performance. The
332 Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985 I
2501
! I
ZOOC
0
uro 6.05
b
6.10
0
1
I
I
I
0
1
2
3
I L
I
I
I
1
I
I
5
6
7
8
9
io
Tlmo. hrs
Figure 7. Average degree of polymerization as a function of residence time.
second columns of the tables indicate the tanks which contain two phases. The parameter a, which is the fraction of the liquid stream being recycled, is proportional to the extent of backmixing. The first column gives the two values of a which were used to make the calculations. If two phases exist in all three tanks there should be two recycle streams, one each between the first and second and second and third tanks. The two numbers given for a are the fractions of the recycle stream between the first and second and second and third, respectively. If two phases do not exist in the last tank, e.g., see the set 250-260-270 "C, there would be no vapor stream from the third tank, and so only one value of a , corresponding to the recycle stream between the first and the second tank, was used. The flow rate of the vapor stream coming out of the last tank would be less compared to the other tanks. Because of this, it is not necessary that the extent of backmixing be the same in all tanks. To reflect this effect, calculations were also performed with two different values of a for the two recycle streams. These are shown in Tables I11 and IV. Now looking at the results presented in Table I11 for the set 230-240-253.5 "C, it is apparent that as a increases or as backmixing increases, the conversion increases. This result is contrary to usual expectations, but it can be understood once the crucial role of water is kept in mind. Water enhances the reaction rates tremendously and it is known that its presence is essential in the initial portions of the reactor. Backmixing brings in a stream containing less monomer from subsequent stages into the earlier stages, thus lowering the activity of monomer there. This effectively increases the water-holding capacity of the mixture since a t the boiling point the decrease in the contribution by the monomer to the total pressure has to be compensated by water only. This effect is augmented by the fact that the streams from subsequent stages are at a higher temperature and hence when they enter an earlier stage at a lower temperature their water-holding capacity is increased. The counter-balancing effects of backmix streams from earlier stages are not that important as they carry more water anyway. For instance, for a = 0.5,0.5, it was found that the liquid streams leaving tanks 1,2, and 3 contained 0.24,0.214, and 0.177 mol of water/kg of mixture, respectively. It might also be observed that the results for a = 1.0, 0.5 lie between those for 1.0, 1.0, and 1.0,O.O. These also confirm the above findings that increased backmixing causes increased conversion. We can
therefore conclude that backmixing does tend to promote conversion due to the increased presence of water. Increased conversion implies an increased value of DP as the amount of water consumed by reactions was found to be approximately the same in all cases. Thus as backmixing increased, the DP was also found to increase as shown in Table 111. Further, backmixing implies more heterogeneity in the molecular weight distribution. Thus the polydispersity index should increase with increased backmixing. This is confirmed by the figures presented in Table 111. Figure 3 shows the results of conversion as a function of residence time for the entire reactor. The PFR results are different for different cases only because the inputs to it are different. The output from the last tank has more water and less monomer content for larger values of a. This continues to be so as the mixture passes through the PFR as well. However, as water content has only a minor effect on the equilibrium conversion, essentially 90% conversion is reached in all the cases. Figure 4 shows the results for DP as a function of residence time for the whole reactor. Streams with lower water content will polymerize to a higher value of DP a t equilibrium since polycondensation occurs to a greater extent. Thus more backmixing will lead to a lower DP. This is illustrated in Figure 4, where DP continuously decreases as a is increased and in the sequence 0.0, 0.0; 1.0, 0.0; 0.5,0.5; 1.0,0.5; 1.0, 1.0. In other words, the trend at the end of the two-phase zone is exactly reversed at the end of the PFR. It can be seen from Figure 4 that the effect of backmixing in the top portion of the reactor on the final product properties is quite significant. Figure 5 shows the values of PDI for the entire reactor as a function of the residence time. As the conversion reaches equilibrium values and as polycondensation occurs to greater and greater extents, the polydispersity index should attain a value of 2 according to theory. This is confirmed by the calculations. Thus PDI of the product is relatively unaffected by the backmixing in the top portion of the column. Tables IV and V and Figures 5-7 show the results obtained for the sets of temperatures 240-253.5-265 "C and 250-260-270 "C. It should be noted that for a = 0, two phases exist only in the first two tanks for the former set while two phases exist only in the first two tanks for all the values of a considered for the latter set. For this reason, consideration of two different values of a is not meaningful for these cases. Except for this difference, the effects of backmixing are very similar to those discussed earlier. Effect of Temperature. Temperature has a very interesting but predictable effect on the two-phase zone results. It can be seen from Tables I11 and IV that the conversions for 240-253.5-265 "C are higher than those for 230-240-253.5 "C. However, when compared with the results for 240-253.5-265 "C, the conversions are lower for the set 250-260-270 "C. These results can be rationalized once the rate accelerating effects of water and temperature are understood. At higher temperatures, the water-holding capacity of the streams is low and this can be seen from Tables III-V. Thus the decreased water contents tend to lower the rates as temperature is increased. However, as temperature is increased, the rate of the reactions should increase. Thus the two opposing effects produce a maximum in the conversion at an intermediate range of temperatures. The effect of temperatures in the two-phase zone on the final product properties was also assessed. It was found that PDI was 2 in all the three cases considered after about
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985 333
8 h of residence time. The conversions reached an equilibrium value of about 90% for the three sets of temperatures. However, there were significant differences in the DP of the product and are shown in Figures 4 , 6 , and 7. Thus for LY = 1, the DPs achieved at 230-240-253.5 OC, 240-253.5-265 OC and 250-260-270 “C, respectively, are 155, 172, and 214. Thus for approximately equal conversions, there is considerable disparity in the molecular weight of the final product produced. The difference in the product properties is also partly due to the fact that the PFR was assumed to be at the temperature of the last tank. To eliminate this effect, PFR calculations were also performed assuming that the PFR was maintained at 253.5 “C. From this calculation the DPs achieved at roughly the same conversions for a = 1 were 155, 178, and 225 for 230-240-253.5 OC, 240-253.5-260 “C, and 2EdF26&270 “C, respectively. Clearly, temperatures employed in the tanks determine the end product’s properties very significantly. It might be noted that lowering temperature in the PFR would naturally result in increased residence times to reach the same conversions. Conclusions A stagewise model with backmixing has been proposed to simulate polymerization in the top portion of the VK column reactor. Results indicate that backmixing in the top portion of the reactor affects the properties of the final product significantly. Increased backmixing leads to decreased degree of polymerization of the final product but a slightly higher conversion. Calculations indicate that the temperatures employed in the top portion also have a major effect on the end products properties. Increased temperatures in the top portion lead to higher degrees of polymerization. Nomenclature A = frequency factor B, = Bernoulli numbers as defined by Jolley (1961) c = concentration of acid end groups, mol/kg E = activation energy F = liquid feed rate, kg/h g = acceleration due to gravity G = vapor flow rate in two-phase zone, kg/h h = height of CSTRs, m, taken to be 1.2 m H = enthalpy k = reaction rate constants K = equilibrium constants M, Mv = concentration of caprolactam, in liquid and gas phases, respectively, mollkg P, = pressure above melt, 1 atm PwSat, Pcyt= vapor pressures of water and caprolactam, respectively, atm r = rate of formation of the respective quantities indicated by the subscript
R = gas constant S = entropy s, = concentration of linear n-mer, H(HN(CH,),-CO),OH, mol/kg; s1 in particular refers to aminocaproic acid t = residence time, h T = temperature, “C V = volume of CSTR, m3 w,wv= concentration of water in liquid and gas phases respectively, mol/kg z = concentration of amide groups, mollkg p = density of reaction mixture, kg/m3 T = residence time in CSTR, h LY = parameter characterizing the degree of backmixing = Zth moment of the s, distribution, C&lnls, Subscripts and Superscripts 1,2,3 = subscripts on A, E, H, S, k , K refer to the reaction number; superscript u and c on A, E, k refer to the catalyzed and uncatalyzed parts 0, 1 , 2 , 3 = subscripts on all quantities other than above refer to conditions in the space above the top of the reactor, in the first CSTR, in the second CSTR, and in the third CSTR, respectively Registry No. t-Caprolactam, 105-60-2. Literature Cited Brown, K. M. ”Numerical Solution of Systems of Nonllnear Aigebralc Equations”; Byrne, G. D.; Hall, C. A,, Ed.; Academic Press: New York, 1973. Chlanese. A,; Annesini, M. C.; Marelli, L. Chem. Eng. J. 1981, 22, 151. Glori, C.; Hayes, B. T. J. Polym. Sci. A-7 1970, 8 , 351. Gupta, A.; Gandhi, K. S. J. Appl. Polym. Sci. 1982, 27, 1099. Gupta, A.; Gandhi, K. S. I n “Frontiers In Chemical Reactlon Engineering”; Doraiswamy, L. K.; Mashelkar, R. A,, Ed.; Wlley-Eastern: New Delhi, 1984; Vol. I,p 667. Hermans, P. H.; Heikens, D.; Van Velden, P. F. J. Pokm. Sci. 1958, 30, 81. Hoftyzer, P. J.; Hoogschagen, J.; Van Krevelen, D. W. Chem. Eng. Sci. 1985, 20, 247. Jacobs, H.; Schweigman, C. Proceedings of V European/II International Symposium on Chemical Reaction Engineering, Amsterdam, May 2-4, 1972. Jolley, L. 8. W. “Summation of Series”; Dover: New York, 1961. Kwatra, G. Synth. Fibers OctlDec 1980, 27. Mecklenburgh, J. C.; Hartland, S. “The Theory of Backmixing”; Wiley: New York, 1975. Naudin ten Cate, W. F. H. “Proceedings International Congress on Use of Electronic Computers in Chemical Engineering”; Paris, April 1973. Ralston, A.; Wilf, H. S. “Mathematical Methods for Dlgltal Computers”; Wlley: New York, 1962 Vol. I. Relmschuessei, H. K. J. Polym. Sci. Mscromoi. Rev. 1977, 72,65. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. “The Properties of Gases and Liquids”; McGraw-Hill: New York, 1977. Sittig, M. “Polyamide Fiber Manufacture”; Noyes Data Corp.: New Jersey 1972. Sneil, F. D.; Ettre, L. S. “Encyclopedia of Industrial Chemical Analysis”; Interscience: New York, 1969; Vol. 8. Tai. K.; Arai. Y.; Tagawa, T. J. Appi. Pdym. Sci. 1982, 27, 731. Tal, K.; Tagawe, T. Ind. Eng. Chem. Prod. Res. D e v . 1983, 22, 192. Tirrell, M. V.; Pearson, G. H.; Welss, R. A,; Laurence, R. L. Polym. Eng. Sci. 1975, 75, 386.
Received for review June 4, 1984 Accepted January 10, 1985
