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Ind. Eng. Chem. Res. 1996, 35, 1550-1555
Transient and Steady-State Behavior of Wiped-Film Reactors for Reversible Condensation Polymerization B. R. Choi and H. H. Lee* Department of Chemical Engineering, Seoul National University, Seoul 151-742, Korea
An improved model is used to investigate the transient and steady-state behavior of the wipedfilm reactors. An efficient numerical scheme is developed that decouples heavily coupled conservation equations. A simple forward marching is involved. The transients provide a valuable piece of information regarding the optimal reactor size for the maximum possible degree of polymerization that cannot be obtained from steady-state consideration. The reactor can be designed in such a way that the deleterious effects of degradation reactions are minimized. The model together with the efficient numerical scheme should provide a sound basis for designing and analyzing the wiped-film reactors. Introduction A unique aspect of condensation polymerization is that a volatile byproduct has to be removed continuously from the reaction mixture to make the reaction proceed to the product side. This need is particularly acute for reversible polymerization reaction. In addition, there is a need to remove the volatile byproduct to enhance the quality of the final product. Wiped-film reactors have been used to facilitate the removal. In these reactors, a polymer film is generated continuously from the bulk on the wall of a rotating cylinder, and after an exposure time, it is wiped and mixed with the bulk. The film is exposed to high vacuum to facilitate the removal of the volatile byproduct by diffusion through the film while polymerization takes place there. In the bulk, polymerization also takes place while being transported axially. There have been two models presented for the wipedfilm reactors (Ault and Mellichamp, 1972a,b; Amon and Denson, 1980). Ault and Mellichamp (1972) considered the amount of polymer mixture of the film to be much larger than the amount in the bulk and took the bulk to be well-mixed, i.e., steady-state CSTR. On the other hand, Amon and Denson (1980) took the amount of the film to be small compared with that of bulk and treated the bulk as plug flow. The two models are identical for the film except for a boundary condition. Both neglected the convective term. For the bulk and the mixing between the wiped film and the bulk, Ault and Mellichamp (1972a,b) simply introduced a contact time for a correlation based on an approximate solution. In the case of Amon and Denson (1980), the steady-state plug flow mass balance included the disappearance of the volatile product from the film. While these two models contributed significantly to our understanding of the wiped-film reactors, they do not contain all the pertinent design parameters. As a result, a rather complete picture does not emerge for the design and analysis, in particular for the design. In the model to be presented, the convective term for the film is included and the unsteady-state nature of the bulk as well as that of the film is taken into consideration. The resulting balance equations are made dimensionless in the same vein as in Steppan et * To whom correspondence should be addressed.
Table 1. Kinetic Model (Steppan et al., 1991)a polymerization
A+CaL+W
degradation reactions
C f SE + W L f SE + A SE f CO2 + SB SB + 2A f X + 2NH3
rp ) cTkp(xAxB - xLxW/Kp r1 ) cTk1xC r2 ) cTxL(k2 + k2CxA) r3 ) cTk3xAxSE0.1 r4 ) cTk4xSB0.3xA
a A: amine end group. C: carboxyl end group. L: amide linkage. W: water. SE: stabilized end group. SB: Schiff base. X: cross-link. xi ) 1/cT ) 1/(cA + cC + cL + cW + cSE + cSB + cX) (i ) A, C, L, W, SE, SB).
Figure 1. Wiped-film reactor.
al. (1990) so that all the pertinent design parameters could be included in a concise manner. Model Formulation In general, polycondensation reactions involve strongly nonideal species that cause the apparent rate and reaction equilibrium constants to vary with composition. Another aspect to consider is that degradation reactions should be included in the kinetic model. A specific example of these aspects is given by Steppan et al. (1987, 1990, 1991) for Nylon 6,6 polymerization. Although the reactor model to follow is applicable to other polycondensation reactions, the kinetic model of Steppan et al. is used here for specific simulation results. The model is summarized in Tables 1 and 2. In the reactor model, the bulk flow whose direction is perpendicular to the rotating direction of the cylinder is assumed to be plug flow. It is also assumed that the thickness of the film is so small compared with the cylinder perimeter (or diameter) that the curvature effect is negligible, i.e., a planar film. An additional assumption is that polymer diffusion in the film is negligible. Taking the direction of film movement to be x (refer to Figure 1), the thickness direction to be y, and the direction of the bulk flow to be z, the dimensionless conservation equations for the film and the bulk
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1551 Table 2. Kinetic Parameters (Steppan et al., 1987, 1991)
{
ki ) k0 exp -
(
Eapp 1 1 R T T0
)}
(i ) 1, 2, 2c, 3, 4, p)
k0 (1/h) k1 k2 k2c k3 k4 kp
Eapp (kcal/mol)
T0 (°C)
30 30 30 10 50 21.44
293 305 305 305 305 200
0.06 0.005 0.32 0.35 10.0 exp[2.55 - 0.45 tanh{25(xw - 0.55)}] + 8.58[tanh{50(xw - 0.101)} - 1](1 30.05xc)
{
Kp ) K0 exp -
K0 ) exp
[{
(
∆Hp 1 1 R T T0
( )}
1 - 0.47 exp -
x1/2 w 0.2
Cj )
X ) x/W
Y ) y/δ
Z ) z/L
(concentration in the film)j cW0 (j ) A, C, W, L, SE, SB, X) (concentration in the bulk)i cW0 (j ) A, C, W, L, SE, SB, X)
]
φ)
(8.45 - 4.2xw)
(
θ ) t/(δ2/D)
Yi )
)}
∆Hp ) 7650 tanh{6.5(xw - 0.52)} + 6500 exp -
The first terms on the right-hand sides of eqs 3 and 4 represent the difference between the amount wiped into the bulk from the cylinder and that picked up onto the cylinder from the bulk. The dimensionless quantities are defined as follows:
)
xw - 800 0.065
DaZ )
T0 ) 200 °C
Rj )
can be written as follows:
( ) δ2/D 1/kpcW0
1/2
DaX )
L/vZ 1/kpcW0
W/vX 1/kpcW0
f ) Vf/Vb
rj (j ) A, C, W, L, SE, SB, X) kpcW0
(8)
Film 2 ∂CW φ2 ∂CW ∂ CW + ) + φ2RW 2 ∂θ DaX ∂X ∂Y
(1)
∂Ci φ2 ∂Ci + ) φ2Ri (i ) A, C, L, SE, SB, X) (2) ∂θ DaX ∂X Bulk ∂YW φ2 ∂YW fφ2 + ) (C h - YW) + φ2RW ∂θ DaZ ∂Z DaX W
(3)
∂Yi φ2 ∂Yi fφ2 + ) (C h - Yi) ∂θ DaZ ∂Z DaX i φ2Ri (i ) A, C, L, SE, SB, X) (4) where
C h W(θ,Z) ≡
∫0 CW(θ,Y,Z)|X)1 dY
(5)
C h i(θ,Z) ≡
∫01Ci(θ,Y,Z)|X)1 dY
(6)
1
and
RA ) R2 - 2R4 - Rp
CW ) CWs at Y ) 0;
RL ) -R2 + Rp RW ) R1 + Rp RSE ) R1 + R2 - R3 RSB ) R3 - R4 (7)
Refer to Table 1 for the notation of various species i and the rate expressions R1(r1) through R4(r4) and Rp(rp).
∂CW/∂Y ) 0 at Y ) 1
(9)
where CWs is the dimensionless equilibrium concentration of the volatile byproduct above the film and is determined by Ogata’s (1960) correlation. The X boundary condition is that the concentrations in the film at the points the film is picked up onto the cylinder from the bulk are those in the bulk:
CW(θ,Z) ) YW(θ,Z) Ci(θ,Z) ) Yi(θ,Z)
RC ) -R1 - Rp
RX ) R4
As can be seen from the definitions, the time is scaled with respect to the film diffusion time (thickness2/ diffusivity), x with respect to the perimeter of the cylinder above the bulk, y with respect to the film thickness, and z with respect to the reactor length L. The Thiele modulus φ is of the usual form, and the Damkoehler number for the film, DaX, contains the linear rotating velocity of the cylinder vX, while the number for the bulk, DaZ, does the linear fluid velocity vZ of the bulk flow. The factor, f, which is the ratio of the total volume of the film to that of the bulk, is an important design parameter. The Y boundary conditions for the film are the usual ones (Amon and Denson, 1980):
}
for all Y, at X ) 0
(10)
The boundary condition for the reactor (bulk) is that, at Z ) 0, the concentrations are those at the inlet which are assumed to be at their equilibrium values, i.e., an equilibrium feed. Time zero is taken as the time the cylinder makes one full rotation after the reaction mixture in the bulk reaches the end of the reactor. The concentrations in the bulk remain at their equilibrium values until the cylinder makes the first full rotation and thus the initial values are the equilibrium ones. The prime quantity of interest here is the numberaveraged degree of polymerization Xn, which is given by
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1552 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
number of repeated unit number of polymer chain YL + YP ) YP
Xn )
(11)
where YL is the concentration of amide linkage in the bulk and YP is the concentration of polymer in the bulk that is approximated as follows:
YP )
YA + YC + YSE + YSB - YX 2
(12)
An assumption here is that there is at most one crosslink per polymer molecule. Numerical Method The equations for the film and those for the bulk are coupled through YW(θ,Z) and Yi(θ,Z) in eq 10 and through C h W(θ,Z) and C h i(θ,Z) in eqs 3 and 4. In order to solve these equations simultaneously, the characteristic method has been used. Along the characteristic lines given by
dθ DaX ) 2 dX φ
∂CW DaX ∂2CW ) 2 + DaXRW ∂X φ ∂Y2
(14)
dCi ) DaXRi dX
(15)
Also, along the characteristic lines given by
(16)
the equations for the bulk become
dYW DaZf ) (C h - YW) + DaZRW dZ DaX W
(17)
dYi DaZf ) (C h - Yi) + DaZRi dZ DaX i
(18)
The characteristic lines for both the film and the bulk are shown in Figure 2. Consider simultaneous numerical solutions of eqs 14, 15, 17, and 18 along the characteristic lines in Figure 2. Let θ1 be the time that the front of the first plug of the reaction mixture arrives at the reactor outlet. Since the cylinder does not rotate until time θ1 has elapsed and the feed is at its equilibrium, no further reaction takes place and the values of Y at θ1 throughout the reactor are those at the inlet, i.e.,
YW(θ1,Z) ) YWi
Yi(θ1,Z) ) Yii
trations in the bulk do not change. As a result, one has
YW(θ1,Z) ) YWi
Yi(θ,Z) ) Yii
θ1 e θ e θ2 (20)
(13)
the equations for the film become
dθ DaZ ) 2 dZ φ
Figure 2. Characteristic line: (a) bulk; (b) film.
(19)
where the subscript i is for the inlet. As soon as the reactor is filled, the cylinder starts rotating. Let θ2 be the time that the front of the film picked up onto the cylinder at θ1 reaches the other side of the cylinder and gets wiped into the bulk. During this time, the concen-
Now that the initial conditions are set for the numerical solution along the characteristic lines, one can proceed as follows: 1. The values of YW(θ1,Z) and Yi(θ1,Z) on the base line a (Figure 2a) are obtained by solving eqs 17 and 18 along the characteristic lines. Using these values as the boundary conditions, eqs 14 and 15 are solved along the characteristic lines in Figure 2b for each Z to obtain the value of C h W(θ2,Z) and C h i(θ2,Z) at X ) 1. 2. Now that the values of C h W(θ2,Z) and C h i(θ2,Z) at X ) 1 on the base line b in Figure 2a are known, eqs 17 and 18 can again be solved along the characteristic lines with initial conditions to obtain the values of YW(θ2+∆θ,Z+∆Z), Yi(θ2+∆θ,Z+∆Z). This way, the time is incremented by ∆θ. Note that ∆Z here is the increment corresponding to ∆θ that satisfies eq 16. By doing so, the rectangular grid is generated, which is more efficient on a digital computer (see the point marked b in Figure 2a). 3. The number-averaged degree of polymerization Xn at the reactor outlet is calculated. 4. The starting point has now been incremented by ∆θ. Thus starting with the point A (Figure 2b), the procedures 1, 2, and 3 can be repeated, each time advancing by ∆θ in time. 5. Repeat all procedures until steady states are reached. The numerical problem as posed by eqs 1-10 is a formidable one because of the coupling between the film and the bulk all along the reactor length. The numerical scheme presented here on the basis of the characteristic method, in effect, decouples the film from the bulk and allows one to proceed with the numerical solution in a straightforward manner, i.e., a simple foward-marching numerical problem. This numerical scheme is a powerful tool not only for the analysis and design of wiped-film reactors including startup but also for other reactors of similar configurations. Behavior of the Wiped-Film Reactor All the design parameters for the reactor are contained in Damkoehler numbers (DaX, DaZ), Thiele
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1553
Figure 3. Effect of P on the degree of polymerization.
Figure 5. Transient behavior of the degree of polymerization: (a) T ) 250 °C; (b) T ) 280 °C.
Figure 4. Effect of T on the degree of polymerization.
modulus, the volume fraction of the film relative to the bulk volume, and the equilibrium concentration of the volatile byproduct that pertains to the operating level of vacuum. In addition, temperature and pressure are the important operating parameters. The prime quantity of interest for the reactor is the number-averaged degree of polymerization Xn at the reactor outlet. The behavior of this degree of polymerization as affected by the dimensionless design and operating parameters is shown in Figures 3-8. The feed is considered to be an equilibrium mixture of which the number-averaged molecular weight is 6200 and the degree of polymerization is 54.87 (Du Pont, 1975). It is assumed for the feed that the side products by the degradation reactions are negligible. The reference conditions around which temperature and pressure are varied are those of Steppan et al. (1990):
DaZ ) 2
DaX ) 0.02
φ)1
Figure 6. Effect of DaX on the degree of polymerization.
f ) 5 (21)
Shown in Figure 3 is the degree of polymerization at the reactor outlet against dimensionless time when the pressure is varied at 250 °C. As one might surmise from physical reasoning, the steady-state value of the degree of polymerization increases with decreasing pressure. A lower pressure means a lower concentration of the volatile byproduct at the polymer film-gas phase interface. More efficient removal of the volatile byproduct should result in a higher degree of polymerization.
Figure 7. Effect of f on the degree of polymerization.
More interesting features are shown in Figure 4 where the outlet degree of polymerization as affected by temperature is given against dimensionless time. At low temperature (in this case 250 °C), the degradation
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1554 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
sponding to the line a-a′. In this way, the maximum possible degree of polymerization can be obtained. The design parameter that is pertinent to the reactor size is the Damkoehler number for the bulk (reactor), DaZ, which contains the reactor length L. A smaller Damkoehler number corresponds to a smaller residence time relative to the reaction time. Therefore, an optimum DaZ can be selected for the maximum possible outlet degree of polymerization. The value of DaZ that yields the maximum is in Figure 5b at the reactor outlet is found to be 0.4. In this light, the effects of the design parameters are investigated with DaZ fixed at 0.4. The other reference conditions used in Figures 6-8 are as follows:
DaX ) 0.002
Figure 8. Effect of φ on the degree of polymerization: (a) DaZ ) 0.4, DaX ) 0.002; (b) DaZ ) 0.4, DaX ) 0.01.
reactions do not come into play. As the temperature is raised, however, the degradation reactions start affecting the degree of polymerization, and the effect is most prominent at the highest temperature (300 °C in the figure). It is seen that, at high temperature, the outlet degree of polymerization increases rapidly with time, reaches a maximum, and then decreases rather slowly with time due to the degradation reactions. This transient behavior is shown to reveal a valuable piece of information regarding the maximum in the outlet degree of polymerization that would not be available from steady-state behavior. This aspect is to be more fully exploited later. The transient reactor profiles of the degree of polymerization are shown in Figure 5. It is seen in Figure 5a that the time progression at low temperature is continuous, whereas it is reversed, i.e., a higher value of the degree of polymerization initially and then gradually decreasing with time, at high temperature (Figure 5b), due to the degradation reactions. Both figures show that, in the entrance region, the polymerization is dominant; in the rest of the reactor, the degradation reactions come into play. The transient behavior in Figure 4 together with that in Figure 5 suggests that the reactor be designed in such a way that the reactor mixture leaves the reactor before the degradation reactions start taking place. This point would correspond to the line a-a′ shown in Figure 5b. The reactor is then in effect reduced to the size corre-
φ)1
f)5
T ) 280 °C P ) 100 Torr (22)
The effect of the film Damkoehler number, DaX, on the outlet degree of polymerization is shown in Figure 6. A smaller film Damkoehler number means more exposure of the bulk via the film formation (faster cylinder rotation speed) relative to the reaction time. Under the condition that the driving force for the removal of the volatile byproduct, CWs (or the sink concentration), is maintained constant regardless of the cylinder rotation speed, the degree of polymerization would increase with more removal of the volatile byproduct, which a smaller Damkoehler number implies. The most important design parameter next to DaZ is the volume fraction of the film relative to the bulk volume, f. This effect is shown in Figure 7. A larger f corresponds to a case where more of the total reaction mixture is in the film on the cylinder than in the bulk. Since the Thiele modulus φ is fixed, the film thickness is constant. Therefore, a larger f means a larger exposed area and thus more removal of the volatile product and a higher degree of polymerization. One might be surprised to find in Figure 8a that the effect of the Thiele modulus φ is negligible. A larger Thiele modulus means a more diffusion-controlled reaction in the film. However, this effect has to be examined in conjunction with the DaZ/DaX ratio, which is large for the reference condition, i.e., 200. The ratio corresponds to the number of times the reactor mixture on the cylinder is turned over. When the mixture is exposed to the vacuum on the cylinder so many times, in this case 200 times, the diffusion effect is not significant. This result is consistent with that of Steppan et al. (1990). The effect is more significant, however, when the ratio is smaller, as shown in Figure 8b (DaZ/DaX ) 40). As expected, a smaller Thiele modulus results in a larger degree of polymerization. Concluding Remarks The unsteady-state wiped-film reactor has been modeled, including the convective term for the film, that can yield a rather complete picture of the reactor. Two Damkoehler numbers, Thiele modulus, and the volume fraction of the film relative to the bulk amount completely describe the reactor along with two physical parameters, the pressure and the temperature. An efficient numerical scheme involving only forward marching has been developed that decouples heavily coupled film and bulk conservation equations all along the reactor length. The scheme would enable one to design and analyze the reactor in a routine manner.
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1555
The transient behavior is shown to be essential in the design of the reactor in that the reactor size yielding the maximum possible degree of polymerization can be found from the transient but not from the steady-state behavior. The key parameter for the design is the Damkoehler number for the bulk (reactor). When the number is properly chosen, the deleterious effect of degradation reactions can be almost completely eliminated. The model together with the efficient numerical scheme developed here would provide a sound basis for designing and analyzing the wiped-film reactors. Nomenclature CW ) concentration of volatile byproduct in the film, dimensionless Ci ) concentration of species i in the film (i ) A, C, L, SE, SB, X), dimensionless C h W ) average concentration at X ) 1, dimensionless C h i ) average concentration at X ) 1, dimensionless CWs ) equilibrium concentration of volatile byproduct above the film, dimensionless cW ) concentration, mol/cm3 ci ) concentration, mol/cm3 cW0 ) concentration in the initial state in which all the species are present as monomer, mol/cm3 ci0 ) concentration in the initial state in which all the species are present as monomer, mol/cm3 D ) diffusivity of the byproduct, cm2/s DaX ) Damkoehler number in the direction of X, dimensionless DaZ ) Damkoehler number in the direction of Z, dimensionless f ) ratio of the total volume of the film to that of the bulk, dimensionless Kp ) equilibrium constant, dimensionless kj ) reaction rate constant (j ) 1, 2, 3, 4, p), cm3/mol s L ) reactor length, cm P ) pressure, Torr W ) perimeter of the cylinder above the bulk, cm Rj ) polymerization reaction rate (j ) 1, 2, 3, 4, p or A, C, L, W, SE, SB, X), dimensionless rj ) polymerization reaction rate (j ) 1, 2, 3, 4, p), mol/cm3 s t ) time, s Vb ) volume of the bulk, cm3 Vf ) volume of the film, cm3 vX ) velocity in the direction of X, cm/s VZ ) velocity in the direction of Z, cm/s
X ) coordinate direction of the film movement dimensionless x ) coordinate direction of the film movement, cm Xn ) number-averaged degree of polymerization Y ) coordinate direction of the film thickness, dimensionless y ) coordinate direction of the film thickness, cm YW ) concentration in the bulk, dimensionless Yi ) concentration in the bulk, dimensionless YWi ) concentration in the bulk at the reactor inlet, dimensionless Yii ) concentration in the bulk at the reactor inlet, dimensionless Z ) coordinate direction of the bulk flow, dimensionless z ) coordinate direction of the bulk flow, cm Greeks δ ) film thickness, cm φ ) Thiele modulus, dimensionless θ ) time, dimensionless
Literature Cited Amon, M.; Denson, C. D. Simplified Analysis of the Performance of Wiped-Film Polycondensation Reactors. Ind. Eng. Chem. Fundam. 1980, 19, 415. Ault, J. W.; Mellichamp, D. A. A Diffusion and Reaction Model for Polycondensation. Chem. Eng. Sci. 1972a, 27, 2229. Ault, J. W.; Mellichamp, D. A. Complex Linear Polycondensation. II. Polymerization Rate Enhancement in Thick Film Reactor. Chem. Eng. Sci. 1972b, 27, 2233. Du Pont. Preparation of Polyamides by Continuous Polymerization. U.S. Patent 3,900,450, 1975. Ogata, N. Studies on Polycondensation Reactions of Nylon Salt. I. The Equilibrium in the System of Polyhexamethylene Adipamide and Water. Makromol. Chem. 1960, 42, 52. Steppan, D. D.; Doherty, M. F.; Malone, M. F. A Kinetic and Equilibrium Model for Nylon 6,6 Polymerization. J. Appl. Polym. Sci. 1987, 33, 2333. Steppan, D. D.; Doherty, M. F.; Malone, M. F. Wiped Film Reactor Model for Nylon 6,6 Polymerization. Ind. Eng. Chem. Res. 1990, 29, 2012. Steppan, D. D.; Doherty, M. F.; Malone, M. F. A Simplified Degradation Model for Nylon 6,6 Polymerization. J. Appl. Polym. Sci. 1991, 42, 1009.
Received for review July 7, 1995 Revised manuscript received February 8, 1996 Accepted February 16, 1996X IE940351K X Abstract published in Advance ACS Abstracts, April 15, 1996.
