Polymerizations in the Presence of Vaporization: Experimental Results

Jul 3, 2008 - This study deals with the hydrolytic step-growth polymerization of ε-caprolactam to produce nylon-6 in a semibatch reactor at near indu...
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Ind. Eng. Chem. Res. 2008, 47, 9061–9071

9061

Polymerizations in the Presence of Vaporization: Experimental Results on Nylon-6 Manojkumar Ramteke and Santosh K. Gupta* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

This study deals with the hydrolytic step-growth polymerization of ε-caprolactam to produce nylon-6 in a semibatch reactor at near industrial conditions. ε-caprolactam is polymerized in a 1.6 L stainless steel reactor at three different initial water concentrations, 4.43% (by mass), 2.52%, and 3.45%, respectively. During the polymerization, the values of the temperature and the pressure are controlled and recorded. Samples of the liquid reaction mass are taken from the reactor at different times and analyzed. The monomer conversions are obtained gravimetrically (in terms of water extractibles) as well as by using gas chromatography. The samples are also analyzed for the degree of polymerization using amide and acid end-group concentrations. The parameters are tuned using one set of data with genetic algorithm. The tuned parameters are then used to predict the second set of data. In the simulation, the poly-NRTL model is used to describe the vapor-liquid equilibria. The simulated values match well with the experimental values. The tuned model gives reasonably good results. Introduction Several industrial polymerizations of commodity and specialty plastics, e.g., nylons, polyethylene terephthalate, polycarbonates, etc., involve the vaporization of low molecular weight compounds. Reaction engineering principles and information are reasonably well-established for the liquid-phase kinetics of these systems. However, information on the vapor-liquid equilibrium (VLE) and mass transfer aspects for such systems are not as well-developed. Some experimental data are available in the open literature, e.g., Giori and Hayes1,2 for nylon-6, on the VLE of polymer-monomer-solvent systems in the absence of polymerization, but the experimental scatter is considerable. The Flory-Huggin’s theory,3 with a curve-fitted value of the χ1 parameter, has been used by Laubriet et al.4 for modeling vaporization in wiped-film PET reactors. More recently, Seavey et al.5,6 have applied the poly-NRTL model7 to the nylon-6-εcaprolactam-water system and have analyzed the data of Giori and Hayes1,2 to tune the model parameters. They then carry out the simulation of an industrial nylon-6 reactor train but do not provide any industrial data for proprietary reasons. A better method for tuning model parameters characterizing VLE is to use experimental/industrial data on polymerizing systems. This is the focus of the present study. Four sets of experimental data have been generated for nylon-6 polymerization under nearindustrial conditions and in the presence of vaporization, and the data have been used to better our understanding of such reactors. The methodology is quite general, and the knowledge gained can easily be used for studying other systems. Most of the nylon-6 plants use the hydrolytic polymerization process.8,9 This process consists of the polymerization of ε-caprolactam (C1 or CL) in the presence of water (W) at temperatures ranging from 230 to 280 °C. Water is used to open the ε-caprolactam ring (reversibly) to give a linear bifunctional molecule, aminocaproic acid (ACA, S1; see Table 1). Polymerization then proceeds by the step-growth mechanism to give linear polymers (Sn; n ) 1, 2,...), with water as the condensation byproduct. In addition to these reactions, the amino end-group of any linear polymer molecule (Sn) can also open the ε-caprolactam ring reversibly. This is the polyaddition reaction. The

linear dimer, S2, however, can cyclize reversibly to form cyclic dimer (C2). The amino end-group of any linear polymer (Sn) can attack a molecule of the cyclic dimer and add it on, giving a longer linear molecule (Sn+2). Higher cyclic oligomers (C3, Table 1. Kinetic Scheme of Nylon-6 Polymerization

* To whom correspondence should be addressed. E-mail: skgupta@ iitk.ac.in. Tel.: 91-512-259 7031; 7127. Fax: 91-512-259 0104. 10.1021/ie800287d CCC: $40.75  2008 American Chemical Society Published on Web 07/03/2008

9062 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 2. Rate and Equilibrium Constants11,12

i

Ai0 (kg/(mol h))

Ei0 (J/mol)

1 2 3a 3b 3c 4 5

5.9874 × 10 1.8942 × 1010 2.8558 × 109 2.8558 × 109 2.8558 × 109 8.5778 × 1011 2.5701 × 108

8.3198 × 10 9.7389 × 104 9.5606 × 104 9.5606 × 104 9.5606 × 104 1.7577 × 105 8.9141 × 104

a

5

Aic (kg2/(mol2 h) 4

4.3075 × 10 1.2114 × 1010 1.6377 × 1010 1.6377 × 1010 1.6377 × 1010 2.3307 × 1012 3.0110 × 109 7

Eic (J/mol)

∆Hi(J/mol)

7.8703 × 10 8.6504 × 104 8.4148 × 104 7.5733 × 104 8.4182 × 104 1.5652 × 105 8.5374 × 104 4

∆Si(J/(mol K))

8.0268 × 10 -2.4883 × 104 -1.6923 × 104 -1.5231 × 104 -1.5150 × 104 -4.0176 × 104 -1.3263 × 104 3

-3.2997 × 101 3.9496 × 10° -2.9068 × 101 -2.9068 × 101 -2.9068 × 101 -6.0766 × 101 -2.4384 × 10°

Values from Tai et al.11 b Values from Wajge et al.12 c This work.

Figure 1. Schematic of the reactor assembly. Table 3. Details of the Four Experimental Runs values

variable [C1]0 (mol kg-1) [W]0 (mol kg-1) F (kg) [Nv]0 (mol m-3) Dr (m) ds (m) 103 Vg (m3) a

Run 1a Run 2 Run 1 (near-replicate) [W]0 ) 4.43%a [W]0 ) 4.43% [W]0 ) 3.45%

Run 3 [W]0 ) 2.52%

8.458

8.458

8.544

8.627

2.461

2.461

1.917

1.40

0.522 59.56 (N2)

0.622 84.36 (Ar)

0.600 81.56 (Ar)

0. 586 81.56 (N2)

0.109 0.105 1.358

0.109 0.105 1.2

0.109 0.105 1.2056

0.109 0.105 1.2456

Mass fraction.

C4,...) are also formed, but these are not included in Table 1 since their concentrations are much smaller, and data on their rates of formation are not available. This table also does not include side-reactions like decarboxylation, desamination, and peroxidation of the ε-caprolactam.

Hydrolytic polymerization is usually carried out until nearequilibrium conditions. The quantity of monomer present in the equilibrium polymer product depends upon the reaction temperature and the amount of water present in the reaction mass. Under industrial conditions, the amount of residual monomer present in the polymer is about 8-9%, while that of the cyclic oligomers is about 3-6%. It is necessary to remove the monomer, water, and cyclic oligomers from the product since the subsequent processing of the polymer is adversely affected by their presence. This is achieved by a hot water extraction process or vacuum evaporation. Several studies have been reported on nylon-6 polymerization. Reimschuessel8 and his group were among the early workers in the field. This group described the mechanism and the kinetics of polymerization and reported values of several rate constants. Their constants, however, were estimated using only a small amount of experimental data. They did not measure the concentration of aminocaproic acid (they estimated the equilibrium concentration of S1 from the equilibrium end-group concentration and the number-average degree of polymerization by assuming a Flory-Schulz distribution3,9). In fact, very little

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9063

Figure 2. Experimental data and model predictions for [W]0 ) 4.43% (Run 1).

Figure 3. Experimental data and model predictions for [W]0 ) 3.45% (Run 2).

data on the concentration of S1 exists under a variety of experimental conditions except those of Hermans et al.,10 and that, too, under comparatively limited conditions. Tai and Tagawa11 carried out a very extensive study of nylon-6 polymerization in sealed tubes under isothermal conditions (230-280 °C) and with several initial concentrations (0.421-1.18

mol/kg) of water. They measured the concentrations of aminocaproic acid by high-pressure liquid chromatography, of -NH2 and -COOH end-groups by titration, and of the cyclic dimer by gas chromatography. They found that the rate constants depend upon the initial water concentrations to some extent. Average values of the rate constants, useful over a limited

9064 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

Figure 4. Experimental data and model predictions for [W]0 ) 2.52% (Run 3).

Figure 5. Experimental data and model predictions for the near-replicate Run 1a ([W]0 ) 4.43%).

though relevant range of initial water concentrations (0.42-1.18 mol/kg), have been reported. These11 are listed in Table 2 along with recently updated values.12 It is observed from Tables 1 and 2 that all the reactions are reversible and catalyzed by the carboxyl end-groups of the polymer chains. The Arrhenius form is used for both k0i and kci , the rate constants for the uncatalyzed

and catalyzed components of ki, respectively. The equilibrium constants are expressed in terms of the enthalpies and entropies of reaction. Recently, Mallon and Ray13 applied a more fundamental approach to fit the data11,14 in the literature. They assumed that the water in the nylon melt is present in two forms: the bridged

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9065 Table 4. Error Analysis of Run 1a t (h)

grav Xm

0.933 1.550 2.089 3.017 4.033 5.033 6.033 7.033 8.017 9.017 10.000 11.083 12.050 13.060 14.000 14.585

0 0.7322 ( 1.08 × 10-4 0.8638 ( 1.17 × 10-4 0.8553 ( 1.17 × 10-4 0.8427 ( 1.16 × 10-4 0.8558 ( 1.15 × 10-4 0.8426 ( 1.17 × 10-4 0.8480 ( 1.15 × 10-4 0.8426 ( 1.16 × 10-4 0.8372 ( 1.16 × 10-4 0.8367 ( 1.14 × 10-4 0.8371 ( 1.15 × 10-4 0.8365 ( 1.16 × 10-4 0.8335 ( 1.14 × 10-4 0.8455 ( 1.16 × 10-4 0.8457 ( 1.14 × 10-4

µn (amide) 34.96 ( 0.48 45.85 ( 0.65 52.30 ( 0.77 53.50 ( 0.82 58.12 ( 0.95 61.63 ( 1.10 63.97 ( 1.45 57.79 ( 1.45 62.79 ( 1.86 71.50 ( 1.47 76.58 ( 2.79 97.22 ( 2.74 104.42 ( 4.74 143.00 ( 4.99

µn (acid) 30.43 ( 0.49 46.21 ( 0.84 53.42 ( 1.22 52.66 ( 1.20 53.59 ( 1.25 62.12 ( 1.95 59.95 ( 1.82 61.20 ( 2.27 64.71 ( 2.59 68.41 ( 3.43 72.51 ( 3.16 81.98 ( 2.42 105.27 ( 7.65 126.63 ( 6.04

and the free forms. Only the free water takes part in the polymerization. Schaffer et al.15 carried out similar experiments on nylon-6-12. Gupta and co-workers9,12,16,17 have reported several studies on the modeling of nylon-6 reactors. One of these is on an industrial semibatch reactor (see Figure 1), in which C1 and W vaporize and build up the pressure in the (closed) vapor space above the liquid reaction mass. The vapors are released later so as to lower the pressure in a programmed manner. This controls the liquid-phase concentration of W and, thus, controls the rate of polymerization (the initial concentration of W in the liquid is kept high so as to speed up the ring-opening step and is lowered subsequently to drive the reversible polymerization in the forward direction). Unfortunately, trustworthy industrial data was available to us only for three initial water concentrations. The temperature histories, T(t) (for a constant jacket fluid temperature, TJ) were available for these three runs, as were the pressure histories, P(t) [the latter was controlled, so only the part before the release of vapors was useful for parameter estimation]. In addition, the final values of the monomer conversion and the number-average chain length were available (instead of the values at several intermediate times) for the three cases. This was insufficient to “tune”, properly, the parameters used in the simulation model,12,17 particularly those associated with the rates of vaporization of C1 and W. This was a severe handicap. In fact, very little experimental/ industrial data are available in the open literature to estimate the parameters describing the rates of vaporization in any polymerization reactor. The present study generates experimental data on a 1.6 L stainless steel nylon-6 reactor in which vaporization and vapor release take place continuously so as to achieve a desired pressure history. Samples of the liquid are taken at different times during the polymerization. The liquid samples are analyzed using gas chromatography, gravimetric analysis, and end-group analysis to give the number-average molecular weight, Mn, of the polymer and the monomer conversion, Xm. Our detailed experimental study provides more data to carry out a meaningful fitting of the model12 parameters. Experimental Details Design of the Reactor. The reactor used in this study is specially designed to carry out the polymerization of nylon-6 at temperatures of about 250 °C. A 1.6 L semibatch 304-stainless steel (SS) reactor is used (see Figure 1). The reactor is mounted on a movable aluminum trolley. A 2000 W electric heater is wound around the reactor. This is insulated on the outside with asbestos rope. There are several SS pipes connected to the

reactor head. One (0.64 cm φ) is fitted with a needle valve (N2 in Figure 1), which controls the rate of release of vapor and, thus, the pressure inside the reactor. Two more openings in the reactor head include one for feeding inert nitrogen or argon to the reactor through the needle valve (N1 in Figure 1) [the nitrogen/Ar exits through the needle valve (N2) during purging] and the other for charging the monomer/water. The latter is closed by a SS stopper. The reactor is fitted with a pivoted anchor stirrer, a thermowell, and a pressure gauge (0-14 kg/ cm2 on the inlet nitrogen/argon line). A calibrated J-type (iron-constantan) thermocouple is used to measure the temperature of the liquid inside the reactor continuously. The temperature of the reaction mass is controlled by a timeproportioned proportional-integral-derivative (PID) controller (Fusi Electricals, Japan). The stirrer is operated by a 0.187 kW motor and its speed is measured by a digital tachometer. A ball valve (V1) is fitted on another SS tube on the reactor head. This tube is open at the bottom and dips into the reaction mass. A 250 W heating tape is wound around this tube outside the reactor. This tube is used for taking out samples at the beginning when the reaction mass is not too viscous. There is an opening at the bottom of the reactor connected to a 1.4 cm o.d. SS pipe with a ball valve (V2). This is used to take out polymer samples during later stages of polymerization when the reaction mass is more viscous. This is also wrapped with a 250 W heating tape. The first two (initial) samples are taken out through the ball valve V1 while the remaining samples are taken out through the ball valve V2. This is because the viscosity of the polymer in the early stages is low and the pressure inside the reactor is quite high. If these early samples are taken out through the valve V2, it would lead to a very rapid and complete draining of the liquid from the reactor. The liquid samples are taken out periodically from the reactor over short periods of time in empty beakers that have been chilled in ice-water. Raw Materials. ε-Caprolactam and water are the main raw materials for the production of nylon-6. GC-grade (purity >98%) ε-caprolactam (Fluka, Belgium) is used without further purification, along with double-distilled water. meta-Cresol (Loba Chemicals, Mumbai, India) having a purity greater than 98% is used. Extrapure-grade benzyl alcohol (Qualigens, Mumbai, India) having a purity greater than 98.5% is used in the analysis. Ethylene glycol (Qualigens, Mumbai, India) having a purity greater than 98% is used. Polymerization. A measured amount of solid ε-caprolactam (>500 g) is put inside the reactor when it is in the tilted position so that the solid does not fill up the seat of the pivot at the bottom of the stirrer. The reactor is then made vertical, with the head placed on top and sealed. The remaining C1 and the prescribed amount of W are introduced through the opening in the head, which is then closed using the SS stopper. The contents of the sealed reactor are flushed four times using either nitrogen (Grade-I, Pawan Gases, Kanpur, India) or argon (Grade-I, Pawan Gases, Kanpur, India). Initially, a (gauge) pressure around 101 kPa of nitrogen/argon is maintained in the reactor. The reactor is then heated (over about 38 min) to 245 °C. Thereafter, the temperature is maintained constant using the controller. The temperature of the molten reaction mass is measured by a calibrated J-type thermocouple. The ε-caprolactam starts melting as the temperature increases. Since the melting point of ε-caprolactam is 69 °C, the stirrer is started as soon as the temperature of the reaction mass goes above this value. The stirrer speed is about 100 rpm at the start of a run and decreases to about 40 rpm near the end. As the reaction proceeds, the pressure builds up in the vapor space as ε-caprolactam and water

9066 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 5. Mass Balance and Moment Equations12

vaporize while there is no release of vapors. High water concentrations are required at the beginning of the process, while low water contents are needed toward the end so as to drive the polycondensation reaction forward and obtain nylon-6 having a high value of Mn. To achieve this, the reactor is operated in five different stages. In the first stage, the pressure is allowed to build up to a set value with the needle valve (N2) closed. In the second stage, the needle valve is opened in such a way that the pressure remains almost constant for a desired period. In the third, fourth, and fifth stages, the needle valve is operated in such a way that the pressure drops (with three different slopes of the pressure with respect to time). The pressure is recorded periodically over time. The pressure history is similar to that used industrially.12 Temperature Control. A time-proportioned PID controller is used to control the temperature of the reaction mass. The temperature is measured by a calibrated J-type thermocouple inserted in the thermowell. The controller is operated in the autotuned mode. Autotuning is done using silicone oil (370-390 mPa s at 20 °C) in the reactor at 200 °C. The tuned parameters are given below: (1) Proportional band (P) ) 11.6 (2) Integral action time (I) ) 906 s (3) Derivative action time (D) ) 174.3 s (4) Proportional time cycle (TC) ) 30 s (5) Insensitive zone (HYS) ) 2%

Analytical Procedure. Gravimetry11 is used to analyse the polymer samples. The quenched (solid) polymer samples collected in the chilled beakers are crushed to a powder by a hammer or shaved by a drill. The weighed samples are then extracted with 20 times their weight of freshly distilled water at 80 °C for 4 h in an oven, after which they are filtered onto weighed filter papers (No. 42, Whatman, Maidstone, U.K.) and dried to a constant weight by heating to 105 °C at 1 atm pressure in an oven. Under these conditions, extraction equilibrium can be assumed to have been achieved, and the hydrolysis of C1 to S1 is negligibly small. The total hot water-soluble content is equal to the loss in weight of the sample caused by extraction. grav Thus, the gravimetric conversion, Xm , is given by Xgrav m ) )

weight of dried polymer sample after extraction weight of polymer sample before extraction kg of (only) polymer produced kg of C1,W used (corrected for vaporization)

(1)

The liquid hot-water extract is analyzed to get the concentration of the unreacted ε-caprolactam and the monomer conversion, GC 11 Xm . A Nucon-5765 (Nucon, New Delhi, India) gas chromatograph (GC) equipped with a dual-flame ionization detector is used for the analysis of C1. The GC is fitted with a 1 m long stainless steel column packed with Tenax GC of 80/60-mesh size11 (Josco, New Delhi, India). Nitrogen (Grade-I, Sigma Gas

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9067

Service, New Delhi, India) is used as the carrier gas. In the analysis, the oven temperature is set at 200 °C and the carrier gas flow rate is set at 30 mL/min. The inlet temperature is maintained at 230 °C. A Hamilton (Hamilton, Reno, NV) microliter syringe is used. Generally, 0.5 µL samples are analyzed each time. The peak for C1 is observed to occur at 38 s. Initially, a sample of known concentration (made by dissolving a known amount of C1 in 100 mL of water) is analyzed. The area for this sample is used as a calibration for the calculation of the concentration of C1 in the extract, and then the total mass of C1 GC in the extract is obtained. The monomer conversion, Xm , is calculated using Table 6. Correlations and Equations Used5,6,12,21–25a

a

Note: Additional details can be found in refs 12, 17, and 25.

(

XGC m ) 1)

mass of caprolactam in extract mass of polymeric sample before extraction

kg of polymer and C2 produced kg of C1,W used (corrected for vaporization)

) (2)

Molecular Weight Determination. The end-group analysis technique18–20 is used commonly for the determination of the number-average molecular weights of condensation polymers. The molecular weights of linear condensation polymers are normally below about 20 000, and end-group methods are quite accurate for such ranges. It is possible to estimate the concentration of the end-groups by chemical analysis and use these to

9068 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 7. Poly-NRTL5,6 and Poly-NRF21 Parameters for the Water-ε-Caprolactam-Nylon-6 System (Randomness Factor (rij) ) 0.3) component i/j interaction parameters

water/ ε-caprolactam

ε-caprolactam/ nylon-6 segment

water/ nylon-6 segment

Poly-NRTL Model5,6 -0.313 0.628 -15.4 -13.7 0.0495 -0.0898

aij aji bij bji cij cji

0 0 297 -601 0 0

0 0 265 207 0 0

dissolved in 60 mL of benzyl alcohol at 150 °C. The solution is then titrated against a 0.02 N potassium hydroxide solution with ethylene glycol, using phenolphthalein as the indicator, at this temperature. The correct normality of a glycol-KOH solution used for analysis is found by titrating with a standard base solution. The following equation19 is used to compute Mn using either of the two methods: Mn )

(sample weight × 1000) (titervolume [mL] × normality)

µn )

Mn (3) M0

where M0 is the molecular weight of the repeat unit () 113).

Poly-NRF Model (tuned; this study) -0.336 0.596 -16.399 -13.993 0.033 -0.083

aij aji bij bji cij cji

0 0 287.773 -604.632 0 0

0 0 269.528 197.242 0 0

Table 8. GA Parameters Used for Tuning parameter

value

Np Pcross Pmut Ngen,max lchr Nseed w1 w2 w3

20 0.9 0.05 20 60 0.889 100 1 1

Table 9. Bounds and the Optimal Values of the Parameters no.

parameter

lower limit

upper limit

tuned

Kinetic Parameters (Poly-NRTL) (10-4 E3c) (J/mol) (-10-4 ∆H3) (J/mol) δ

1 2 3

7.5312 1.4770 0.001

8.7864 1.5983 0.005

8.4182 1.5150 0.0011

Poly-NRF Parametersa 1 2 3 4 5 6 7 8 9 10 a

-0.25 0.5 -12.0 -10.0 0.03 -0.08 280.0 -590.0 240.0 190.0

a32 a23 b32 b23 c32 c23 b31 b13 b21 b12

-0.35 0.6 -18.0 -15.0 0.06 -0.10 310.0 -610.0 270.0 210.0

-0.336 0.596 -16.399 -13.993 0.033 -0.083 287.773 -604.632 269.528 197.242

Table 10. Statistical Analysis of the Two Models R2 value poly-NRTL run no. 1 1a 2 3

µn 0.9003 0.8657 0.9551 0.7713

Xm

0.9852 0.8177 0.9853 0.9906

poly-NRF Xm

GC

0.3662 0.9924 0.0207

Experimental Results. In this study, three polymerization runs (and a near-replicate of one run) are carried out with initial water concentrations of 4.43%, 2.52%, and 3.45% (w/w), respectively. The details of these experiments are given in Table 3. The temperature and the pressure are recorded during polymerization. The near-isothermal set-point temperature (set point ) 245 °C) and the pressure histories are maintained similar to those used in the industrial reactor.12 These experimental histories are shown in Figures 2–5 for the three values of [W]0. The temperature of the reaction mass increases from about 21 to 245 °C in about 38 min and then remains almost constant. Figures 2–5 also show the experimental results for the two GC grav conversions, Xm and Xm , as well as the experimental values of µn obtained using the amide and/or acid end-group concentraGC tions. It is observed that the values of Xm are slightly higher grav than (or almost equal to) those of Xm , as expected from eqs 1 and 2. Also, the values of µn obtained using the concentrations of the two end-groups are in good agreement (in cases for which both are carried out). The information on the errors is given for Run No. 1a in Table 4. The errors in the values of the monomer conversion are below about 1%, while those for µn are below about 3.5% for most cases. Modeling. The kinetic scheme and the corresponding rate constants for nylon-6 polymerization have already been discussed earlier and are given in Tables 1 and 2. The modeling of such reactors12 poses severe problems primarily because of lack of good correlations to model the vaporization of C1 and W. Indeed, the following empirical equations have been used in our earlier studies12,17 to model the activity coefficients of the monomer, C1 (m), and water (w) for this reactor βmf - βmo (monomer conversion)γw 0.95 βwf - βwo (monomer conversion) (4) ) βwo + 0.95 In eq 4, βmo, βmf, βwo, and βwf are curve-fit parameters, which were tuned earlier12 to explain the (small amount of) available industrial data. In the present study, two models that are more detailed, namely, the poly-NRTL7 and the poly-NRF21 (nonrandom factor) models, are used instead of eq 4 to describe the vapor-liquid equilibrium at the interface of the vapor and the liquid reaction mixture. Tables 5 and 65,6,12,21–25 summarize the complete set of model equations with these two models incorporated. The parameters of the poly-NRTL model are taken from Seavy et al.,5,6 while the parameters of the poly-NRF model have been tuned in this study. These are given in Table 7. In both the models, the entropic contribution is calculated using the Flory-Huggins’ equation (eq T6-g in Table 6). The γm ) βmo +

Polymer (1)-caprolactam (2)-water (3).

grav

Results and Discussion

µn

Xmgrav

XmGC

0.9020 0.8662 0.9542 0.7729

0.9850 0.8180 0.9848 0.9907

0.3693 0.9925 0.0161

estimate the value of Mn. In the amide end-group method, the polymer is dissolved using phenolic solvents. m-Cresol (60 mL) is used to dissolve 0.5 g of polymer. Methanol (20 mL) and water (10 mL) are added to ensure that the solution is homogeneous. The titrant is 0.02 N aqueous hydrochloric acid, and the indicator is a neutral 0.1% solution of thymol blue in water. In the acid end-group method, 0.5 g of the polymer is

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9069

enthalpic contributions are calculated using the NRTL (eq T6e) or the NRF (eq T6-f) models for the two cases. These two contributions are added to give the final expressions for γm and γw (eq T6-h). The temperature dependence is built-in in terms of the interaction parameter, τij (eq T6-i), in the two models. The interfacial concentrations of C1 and water are calculated using eqs T6-a-T6-c. These equations involve the activity coefficients. It is assumed that the gas-phase resistances are negligible. The same method cannot be used for calculating the interfacial concentration of the polymer (an information piece needed in the poly-NRTL/NRF models). We assume, empirically, that the interfacial concentration of the polymer is δ times its bulk (liquid-phase) concentration, i.e., µ0* ) δ × µ0, where δ is an empirical parameter. Since the polymer molecules are large compared to the sizes of the monomer and water molecules, the concentration of the polymer at the interface will be very low so that there is no vaporization (there is some kind of an inconsistency in this argument since then there should be mass transfer from the bulk of the liquid to the interface, but there is no vaporization of the polymer). The parameter, δ, is expected to be small and will be a function of the rpm of the stirrer, as well as the size and geometry of the reactor. This parameter is estimated (tuned) by minimizing the mismatch between experimental data and model-predicted values (as explained in the next section). An increase in the value of δ leads to lower amounts of vaporization of the monomer and water, but the sensitivity is small. An iterative procedure (successive substitutions/Picard iteration) is required to compute the activity coefficients since the interface concentrations themselves are functions of the activity coefficients. The activity coefficients, γm and γw, are first assumed. These are used to calculate the interfacial concentrations (eqs T6-a-T6-c). The latter are then used to recalculate γm and γw using eqs T6-e-T6j. If the mismatch between either of the calculated and assumed values is >0.0005, the iterations continue. The activity coefficients are evaluated at time t, using the experimental values of the temperature and pressure at that time (this enables the successful convergence of the iterative scheme). The detailed model is given in Tables 5 and 6. Most of the equations (except those describing the vapor-liquid equilibrium using the poly-NRTL/NRF model) are the same as given in our earlier papers.12,17 Table 5 consists of 14 ordinary differential equations (ODEs) of the initial-value kind (mass balances and moment equations) that describe the system completely. In addition to these, several correlations describing the vaporization of low-molecular-weight species and various physical properties like viscosity, vapor pressure, density of the reaction mass, etc. are required for integrating these stiff ODEs. These are given in Table 6.12,17 The driving force for the vaporization is the difference between the concentration in the “bulk” of the liquid reaction mass and that at the interface. The interfacial concentrations are calculated using eqs T6-a-T6-d. The reactions are driven in the forward direction by vaporization that takes place by two phenomena, namely, quiescent and bubbly desorption. In quiescent desorption, water and ε-caprolactam vaporize from the surface of the reaction mass at the top. In bubbly desorption, bubbles of low-molecular-weight species (water and ε-caprolactam) form in the entire bulk of the reaction mass by supersaturation of the liquid phase associated with the increase in temperature, and then these bubbles rise to the surface. Bubbly desorption is further divided into low- and high-bubbly regimes. Above the critical relative supersaturation, σc, the high-bubbly regime starts. A more detailed description of these mass transfer correlations is given in ref 25. The effect of temperature on the

various physical properties of the mixture is also given in Table 6. The stiff ODEs given in Table 5 are integrated using the D02EJF subroutine (based on Gear’s algorithm) of the NAG library. A tolerance, TOL, of 10-7 is used. Parameter Estimation. The experimental temperature and pressure histories are used as inputs in the model to obtain the model-predicted (simulated) results. The equations in Tables 5 and 6 are used to obtain the model-predicted values. Data on GC grav Xm , Xm , and µn for only one run, namely, for [W]0 ) 4.43% (Run No. 1), are used for tuning the parameters. Single-objective genetic algorithm (GA) is used to minimize the weighted normalized sum-of-square deviations of (a) the experimental (subscript, expt) values of µn, GC (b) the monomer conversion, Xm , obtained using GC, and grav (c) the gravimetric conversion, Xm from the model-predicted (subscript, sim) values: ND

min I ) w1

∑ i)1

(

µn,expt - µn,sim µn,expt

)

2

NC

+ w2

∑ i)1

(

GC XGC m,expt - Xm,sim

NC

w3

∑ i)1

(

XGC m,expt

)

2

grav Xgrav m,expt - Xm,sim

Xgrav m,expt

+

)

2

(5)

In eq 5, ND and NC are the number of experimental data points available for µn and the conversions, respectively. w1, w2, and w3 are the weightage factors. The tuning of the parameters is carried out in two stages. The poly-NRTL model is used in the first stage. Different sets of parameters from among the several in Table 6 are tried for tuning, and finally three (the ones to which the results are most sensitive) are identified that give the best agreement: two parameters, Ec3 and ∆H3, characterizing the polyaddition reaction, and δ (the results were found to be much less sensitive to the NRTL parameters). Since the kinetic parameters reported by Tai and Tagawa11 in Table 2 are averaged over several initial water concentrations, and since the values in the present study lie outside this range, the two most sensitive kinetic parameters are used for tuning. The best set of computational parameters used in the GA code and obtained by trial is given in Table 8, along with the weightage factors, while the upper and lower bounds used for the three parameters are given in Table 9. Table 9 also gives the tuned values of these parameters. These three tuned parameters are then used unchanged (so as to ensure that the same kinetic constants are used in both the VLE models) to obtain the 10 interaction parameters of the poly-NRF model. The same objective function as given in eq 5 is used. The bounds of these as well as the tuned values are given in Table 9 (the latter are also given in Table 7). It may be emphasized that the parameters of the poly-NRF model for the system nylon6-caprolactam-water are being reported for the first time. Predictions of the Tuned Model. The model-predicted results using the tuned parameters and the poly-NRTL model are shown in Figure 2 for Run No. 1 ([W]0 ) 4.43%). These tuned parameters are then used unchanged to predict the results for the other three runs. The model-predicted results are shown in Figures 3–5. It is observed that the experimental results are in reasonable accord with the model predictions, except for µn near the final stages of polymerization. Possibly, diffusional resistance26 assumes importance during this stage (and the apparent rate constants change with time). The results of the tuned poly-NRF model are almost indistinguishable from those obtained with the poly-NRTL model. Table 10 compares the two models using R2 values. It is observed that the predictions of the two models are nearly the same. The grav predictions of µn and Xm are quite good. However, the low

9070 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

values of R2 for XGC m for all cases except Run No. 2 are because of the significant deviations of a few initial experimental points. Conclusions Four sets of polymerizations of nylon-6 have been carried out at three different initial water concentrations. These polymerizations are associated with the simultaneous vaporization of two low-molecular-weight compounds. Either the polyNRTL or the poly-NRF models can be used to describe the vapor-liquid equilibrium. The polymerizations are carried out using pressure and temperature histories that are similar to those used in industry. Experimental data on only one run, namely, for [W]0 ) 4.43%, are used to obtain the best-fit (tuned) values of the parameters. The tuned model is then used unchanged to predict the other results. Reasonable agreement is observed for the monomer conversion, the conversion of the hot waterextractible material, and the number-average chain length. Acknowledgment We dedicate this paper to Professor Arvind Varma. Financial support from the Department of Science and Technology, Government of India, New Delhi (through Grant SR/S3/CE/ 46/2005-SERC-Engg, dated November 29, 2005), is gratefully acknowledged. Nomenclature a ) specific interfacial area (m2 m-3) A ) jacket area (m2) Ai ) frequency factor for the ith rate constants (kg mol-1 h-1) c, Csi ) concentration of polymer in solution (kg/100 kg, g cm-3) [Ci] ) concentration of ε-caprolactam (1) and cyclic dimer (2) in liquid phase (mol (kg of mixture)-1) v Cp,i ) specific heat of pure i in vapor phase (kJ kg-1 K-1) 1 Cp,mix ) specific heat of liquid reaction mixture (kJ kg-1 K-1) ds ) diameter of stirrer (m) Di ) diffusivity of component i; i ) m (monomer), w (water) (m2 h-1) Dr ) diameter of reactor (m) DP ) degree of polymerization of polymer product Ei ) activation energy of ith reaction (J mol-1) F ) mass of liquid in reactor at time t (kg) hi ) heat transfer coefficient of liquid (kJ m-2 h-1 K-1) ∆Hi ) enthalpy change for ith reaction (J mol-1) k ) thermal conductivity of reaction mass (kJ m-1 h-1 K-1) ki ) rate constants for ith reaction (kg mol-1 h-1) (kl,ia)j ) liquid-phase mass transfer coefficient of component i; i ) m (monomer), w (water); for desorption from j; j ) f (free surface); b (bubble surface) (m h-1) Ki ) equilibrium constants for the ith reaction mI ) ratio of polymer free volume to segment free volume [Mv] ) concentration of ε-caprolactam in vapor phase (mol m-3) Mn ) number-average molecular weight Mw ) weight-average molecular weight n ) rate of rotation of stirrer (rev min-1) Nj-i ) normality of the ith solution with the jth solute [Nv] ) concentration of nitrogen or argon in the vapor phase (mol m-3) NRe ) Reynolds number NSc ) Schmidt number NSh ) Sherwood number P ) total pressure (kPa) Pisat ) vapor pressure of component i (kPa) Q ) polydispersity index

ri ) net forward rate for ith reaction (mol kg-1 h-1) ri, I ) number of segments of i per polymeric species I R ) universal gas constant (kPa m3 mol-1 K-1 or J mol-1 K-1) Rvm ) rate of evaporation of ε-caprolactam (mol h-1) Rvw ) rate of evaporation of water (mol h-1) [Si] ) concentration of linear oligomers in liquid (mol kg-1) ∆Si ) entropy change for ith reaction (J mol-1 K-1) t ) time (h) tf ) total (final) reaction time (h) T ) temperature (K) Tr ) reference temperature () 473.15 K) Vg ) volume of the vapor space (m3) VT ) rate of the vapor escape from the reactor (mol h-1) Xi ) segment fraction of species i at time t {} Xm ) monomer conversion grav Xm ) monomer conversion obtained using gravimetry GC Xm ) monomer conversion obtained using gas chromatography xi ) mole fraction of species i at time t [W] ) water concentration in liquid (mol/(kg of mixture)) [Wv] ) water concentration in vapor (mol m-3) w/w ) mass fraction Greek Symbols R ) nonrandomness factor (poly-NRTL/NRF model) γi ) activity coefficient of i η ) viscosity of liquid mixture (Pa s or poise; 1 poise ) 10-1 Pa s) [η] ) intrinsic viscosity of ε-caprolactam-nylon-6 mixture (100 (kg of mixture)/(kg of polymer)) τij ) poly-NRTL/NRF binary interaction parameter for components i and j ζi ) total moles of monomer (i ) 1), water (i ) 2), or both (i ) 3) vaporized till time t (mol) µi ) moments of the Sn distribution (i ) 0, 1, 2) µn ) number-average chain length (≡ µ1/µ0) F ) density of the liquid mixture (kg m-3) φi ) volume fraction of species i σ ) relative supersaturation σc ) critical value of the relative supersaturation at which the highbubbling regime starts φ ) enhancement factor for the volumetric liquid-phase mass transfer coefficient for the free liquid surface due to bubbling Ωb ) supersaturation in the liquid phase that determines the start of bubbly desorption (when Ωb > P) (kPa) Subscripts/Superscripts b ) bubble/bubbly expt ) experimental value f ) free surface J ) jacket l ) liquid m ) monomer o ) feed conditions q ) quiescent sim ) simulation values v ) vapor w ) water *) interfacial Note: Additional details can be found in refs 12, 17, and 25.

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ReceiVed for reView February 19, 2008 ReVised manuscript receiVed April 29, 2008 Accepted April 30, 2008 IE800287D