# Effects of diffusional water removal and heat transfer in Nylon 6 reactors.

INTRODUCTION

Nylon-6, a polymer of considerable commercial significance, is produced by water initiated polymerization of [Epsilon]-caprolactam. There has been significant research in the area of design and optimization of industrial nylon-6 reactors. Reimschuessel et al. (1, 2) considered the following three major reversible reactions: (a) ring opening reaction, in which the ring of the monomer, [Epsilon]-caprolactam, is opened by water to form aminocaprolc acid; (b) polycondensation reaction in which the amino and carboxylic acid end groups react to form large polymer chains via amide linkages, with water formed as a condensation product; (c) and polyaddition reaction in which monomer adds on to the growing polymer chains. Later, Tirrell et al. (3) incorporated the reaction with monofunctional acids (modifiers) in their kinetic scheme. Arai and co-workers (4) have coupled the ring-opening and polyaddltion reaction of cyclic dimer into Reimschuessel et al.'s basic reactions. They have obtained precise data and have curve-fitted their results to obtain a better set of rate constants for nylon-6 polymerization. They used their data to simulate the thermal effects for plug flow reactor (5). Gupta and Tjahjadi (6) simulated an industrial nylon 6 tubular reactor that has an introductory section for preheating the reactants with the heat produced by the reaction. They have shown the effects of various operating conditions and parameters on the temperature and molecular weight profiles.

The reaction that matters for the control of the degree of polymerization in polycondensation reaction is where water is formed as a condensate. This water greatly affects the chemical reaction equilibrium so that we cannot obtain polymer with high molecular weight, as desirable for industrial application. Therefore, this condensation water must be removed from the reaction mass. One approach to accomplish this goal is the design of reactors that effectively remove the condensated water by contacting inert dry gas to the reaction mass. Gupta et al. (7) have studied the design of three-stage nylon-6 reactors with intermediate mass transfer by solving the mass transfer equations numerically. Their study considered only the diffusional mass transfer of water through the reaction mass. Gupta and Gupta (8) also studied a semi-batch nylon 6 reactor in which gas bubbles move upward while removing water vapor from the reaction mass.

[TABULAR DATA FOR TABLE 1 OMITTED]

There is heat transfer involved in the polymerization process in the special case of a wetted-wall column type tubular reactor. In this study, idealized two-stage models of contacting inert dry gas with the reaction mass are considered to examine the effect of both mass and heat transfer on the polymerization reaction of [Epsilon]-caprolactam.

PROCESS MODELING

The three basic reactions, as briefly mentioned in the Introduction, are coupled with the ring-opening and polyaddition reaction of cyclic dimer to form a complete reaction system. The kinetic scheme used is summarized in Table 1, The rate constants are those corresponding to the reaction between functional groups. The polymerization is well described by an autocatalytic reaction, and the forward reaction rate constant [k.sub.i](i = 1, 2, 3, 4, 5) is expressed in terms of the acid end group concentration (in g-mole/kg mixture) as

[Mathematical Expression Omitted] (1)

where [[S.sub.n]] is the concentration of linear functional compounds and [[Mu].sub.0] is the zeroth moment of the linear functional compounds. The v-th moment of the linear functional compounds is generally expressed by

[[Mu].sub.v] = [summation of] [n.sup.v] [[S.sub.n]] where n = 1 to [infinity] (2)

The values of v are 0, 1, 2, and 3 for the four different kinds of moments used in the formulation. [k.sub.i] and [Mathematical Expression Omitted] [TABULAR DATA FOR TABLE 2 OMITTED] have units of kg/mol-hr, while [Mathematical Expression Omitted] has units of [kg.sup.2]/[mol.sup.2]-hr. Both [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are functions of temperature expressed by an Arrhenius-type relation as follows:

[Mathematical Expression Omitted] (3)

where the superscript * is to be replaced by o or c for each corresponding rate constant. A and E are the frequency factor and the activation energy, respectively. R is the gas constant and T is the absolute temperature expressed in units of K. The equilibrium constant, [K.sub.i], is similarly given as a function of temperature:

ln [K.sub.i] = [Delta][S.sub.i] / R - [Delta][H.sub.t] / RT (4)

Here, [Delta]S and [Delta]H denote the change of entropy and the change of enthalpy, respectively. Values of [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Delta][S.sub.t] and [Delta][H.sub.t] for the various reactions have been given by Arai et al. (4) and are summarized in Table 2. The ring opening of [Epsilon]-caprolactam is endothermic since the enthalpy of linear aminocaproic acid produced is higher than that of [Epsilon]-caprolactam. On the other hand, the ring opening of cyclic dimer is exothermic since the equilibrium concentration of cyclic dimer increases with increasing temperature as observed by Arai et al. (4).

Two kinds of gas-reaction mass contacting systems are considered. One in which inert gas bubbles are injected into an intermediate position of the tubular reactor and removed from the outlet position. The other is a wetted-wall type tubular reactor in which reaction mass contacts with inert gas after an intermediate position. For analysis, the former can be idealized as a hollow sphere model and the latter as a hollow cylinder model. They are schematically described in Fig. 1. In the hollow sphere model, we consider the bubble size as average value for calculation. Mass balance equations are now written for the idealized systems of hollow sphere and hollow cylinder. For nylon 6 polymerization, the mass and energy balance equations are given in Table 3, where [D.sub.w] is diffusivity of water, W, through the polymeric reaction mass, and k/[Rho][C.sub.t] (= [Alpha]) is thermal diffusivity and [[Mu].sub.v] (v = 0, 1, 2, 3) is the v-th moment of linear functional compounds defined by Eq 2. In writing these equations, the following assumptions have been made:

(1) Only the condensation product, water W, can diffuse. The monomeric and polymeric species do not diffuse at all.

(2) The Interfacial water concentration [[W].sub.1] and temperature [T.sub.1] at r = [r.sub.1] are taken to be constant and treated as parameters.

(3) The flow pattern of the reaction mass is plug flow.

The equations for the moments [[Mu].sub.0], [[Mu].sub.1], and [[Mu].sub.2] can be obtained by using the z-transform defined by

D(z, t) = [summation of] [z.sup.-n][[S.sub.n]] where n = 1 to [infinity] (5a)

through differentiation of the following form

[[Mu].sub.v] = [[Delta].sup.v]D(z, t) / [Delta][(ln [z.sup.-1]).sup.v] [where] z = 1 (5b)

The differential equation governing the time change of D is first obtained according to Eq 5a using exact expression of [Delta][[S.sub.n]]/[Delta]t. The Equation is subsequently differentiated using Eq 5b to obtain the balance equations of the various moments. In Table 3, the mass balance equation for the component [[S.sub.1]] contains the components [[S.sub.2]] and [[S.sub.3]], and the equation for [[Mu].sub.2] contains the component [S.sub.2]] and [[Mu].sub.3]. This requires additional balance equations for the components [[S.sub.2]], [[S.sub.3]] and [[Mu].sub.3]. This, in turn, constitutes the hierarchy of equations, which makes the solution of the system of equations impossible. To solve these difficulties, the first hierarchy has been broken by assuming that [[S.sub.2]] and [[S.sub.3]] are equal to [[S.sub.1]] (2). The second has been broken by the fact that the Schultz-Zimm distribution [TABULAR DATA FOR TABLE 4 OMITTED] function is suitable for representing the distribution of polymerization degree of poly-[Epsilon]-caprolactam (9). According to the properties of the Schultz-Zimm function, the following relations can be derived:

[Mathematical Expression Omitted] (6)

The following expressions for the calculation of density, [Rho], and specific heat, [c.sub.p], of reaction mass have been reported by Jacobs and Schweigman (10) and adopted in this study.

[Rho] = [10.sup.3][{1.0065 + 0.0123[[C.sub.1]] + (T - 495)(0.00035 + 0.00007[[C.sub.1]])}.sup.-1] (7)

[c.sub.p] = 238.09(2.76 [[C.sub.1] / [[[C.sub.1]].sub.0] + [[[C.sub.1]].sub.0] - [[C.sub.1]] / [[[C.sub.1]].sub.0] (2.035 + 0.00141T)) (8)

where [[C.sub.1]] denotes the concentration of [Epsilon]-caprolactam and subscript 0 denotes the feed. The units of [Rho] and [c.sub.p] are in kg/[m.sup.3] and cal/kg-K. T is in units of K. The set of equations in Table 3 are solved along with the above relations. The initial and boundary conditions required for the two systems are

t = 0, 0 [less than] r [less than or equal to] [r.sub.2]: y = [y.sub.0]

0 [less than] t [less than] [t.sub.0], r = 0: [Delta]y / [Delta]r = 0

0 [less than] t [less than] [t.sub.0], r = [r.sub.2]: [Delta]y / [Delta]r = 0

t = [t.sub.0], [r.sub.1] [less than] r [less than] [r.sub.2]: [Delta]y / [Delta]r = 0 (9)

t [greater than] [t.sub.0], r = [r.sub.1]: [W] = [[W].sub.1],

[Mathematical Expression Omitted]

t [greater than] [t.sub.0], r = [r.sub.2]: [Delta][W] / [Delta]r = 0, [Delta]T / [Delta]r = 0

where y denotes all concentrations and temperature. The concentrations and temperature are displayed as spatially averaged values in the Figures. The spatial average concentrations and temperature are obtained using the following equations.

For hollow sphere:

[Mathematical Expression Omitted] (10a)

For hollow cylinder:

[Mathematical Expression Omitted] (10b)

The number- and weight-average chain lengths are computed as follows:

[[Lambda].sub.n] = [[Mu].sub.1] / [[Mu].sub.0] (11)

[[Lambda].sub.w] = [[Mu].sub.2] / [[Mu].sub.1]

The polydispersity index, [Delta], is given by

[Mathematical Expression Omitted] (12)

The equations in Table 3 are transformed into finite difference form by using forward difference for the time- and first order space-derivative terms, and central difference for the second order space derivative terms. These difference equations can be solved explicitly with a time increment [Delta]t of 0.005 h. The spatial increment [Delta]r should satisfy the following convergence criterion (11).

[([Delta]r).sup.2] / ([Delta]t)[D.sub.w] [greater than or equal to] 4 (13)

[([Delta]r).sup.2] / ([Delta]t)[Alpha] [greater than or equal to] 4

Thus, the radial distance from [r.sub.1] to [r.sub.2] is divided into 10 equal meshes and each mesh point is numbered from 1 to 11 in an increasing order.

RESULTS AND DISCUSSION

As mentioned in the Introduction, the focus of this study has concentrated on investigating how greatly the mass and heat transfer affect the reactor performance and how much the properties of the polymer produced are to be influenced by them with respect to the monomer conversion, the molecular weight and its distribution, and the content of cyclic dimer in the product.

To investigate these effects, different sets of geometries and dimensions have been selected. The typical dimensions of hollow sphere systems have been chosen in such ways as (1) inner radius [r.sub.1] of 0.005 m and outer radius [r.sub.2] of 0.030 m, and (2) inner radius [r.sub.1] of 0.010 m and outer radius [r.sub.2] of 0.060 m, and those of hollow cylinder systems in such ways as (3) [r.sub.1] of 0.025 m and [r.sub.2] of 0.050 m, (4) [r.sub.1] of 0.050 m and [r.sub.2] of 0.100 m. These are summarized in Table 4 along with the calculated results for several process conditions. The contact between the inert gas and the reaction mass has been designed to occur 5 h after the inlet in those systems. The inlet temperature of the feed was fixed at 240 [degrees] C and the inlet water concentration at 0.333 mol/kg for all of the cases, which will permit the reaction mass to reach reaction equilibrium within 5 h, before contacting with the inert gas takes place. It was assumed that the interfacial water concentration be kept at 0.088 mol/kg (= 0.001 mol/mol-caprolactam). The values of the diffusivity of water in the reaction mass, [D.sub.w], and the thermal conductivity of the reaction mass, k, used for computation were assumed to be 0.9 x [10.sup.-4] [m.sup.2]/hr and 165 cal/[m.sup.2]-hr-K, respectively.

The calculated results for the temperature and the water concentration distribution of the hollow cylinder type reactor(Case-4) are graphically shown in Figs. 2 and 3. From these three-dimensional Figures we can basically Figure out the process characteristics of these models. In the Figures that follow, the radially averaged values of each variable will be plotted along the axial direction scaled on a time basis.

The monomer conversion profiles for the systems are shown in Fig. 4. From this Figure, it can be said that all of the systems are well conditioned in terms of the monomer reaction equilibrium since the conversions show almost the same profile for all of the cases and reach the equilibrium conversions of [approximately]91%. Therefore, the mass and heat transfer will only affect the temperature, the water concentration, and the polymer properties including the content of cyclic dimer. The temperature profiles are shown in Fig. 5. For the first three cases in which no water removal and hollow sphere systems are concerned the temperature profiles are almost the same since heat transfer is not considered. On the other hand, the hollow cylinder systems, in which both heat transfer and diffusional water removal are considered, show sharp decreases in temperature. The higher the interfacial area per unit volume of the reaction mass is increased, the sharper the drop in the temperature profile becomes.

The effects of diffusional water removal are shown in Fig. 6. For the case of no water removal, the water content slightly increases 4 h after the reaction starts, since polycondensation reaction dominates in the later portion of the reactor. This condensated water is in equilibrium with the growing polymers, which will prohibit the growth of polymer chain. Therefore, it is desirable to remove this condensated water from the reaction mass. For hollow sphere model, case-3 has a bigger bubble size than case-2, but has a smaller interfacial area per volume than case-2. This results in less diffusional water removal in case-3 than in case-2 as can be seen in Fig. 6. The same discussion is possible for the hollow cylinder model when the cases 4, 7, and 11 are compared. However, when we compare cases 2 and 7, we see that even if case-7 has an interfacial area per unit volume as large as 13.33 [m.sup.2]/[m.sup.3], much larger than 2.79 [m.sup.2]/[m.sup.3] of case-2, the former has less diffusional effect than the latter, since the former has a diffusional depth of 0.050 m which is longer than 0.025 m of the latter.

The number average chain length for the selected cases are shown in Fig. 7. By comparing with Fig. 6, one can say that higher water removal leads to a polymer with higher molecular weight as can be expected by the inherent nature of polycondensation reaction.

To investigate the heat transfer effect, two hollow cylinder systems were chosen in such a way that one has the interfacial area per unit volume as large as 26.67 [m.sup.2]/[m.sup.3] and the other has that as large as 13.33 [m.sup.2]/[m.sup.3]. The interfacial temperature for these systems were increased from 240 to 280 [degrees] C and the changes in number average chain length and polydispersity were observed. The results are presented in Figs. 8 and 9. As the interfacial temperature increases, the molecular weight increases for the former while it decreases for the latter. When the interfacial area per unit volume is large, the temperature of reaction mixture is relatively low. As a result, the equilibrium conversion of [Epsilon]-caprolactam becomes low since the ring opening of [Epsilon]-caprolactam is endothermic. As the temperature increases, the conversion increases and dominates the exothermic feature of the polycondensation and the polyaddition reactions such that equilibrium of these reactions moves in reverse as the temperature increases. Hence, the number average molecular weight will increase. However, when the interfacial area per unit volume is small, the reaction mixture has sufficiently high temperature and the equilibrium conversion reaches the highest value at [approximately]240 [degrees] C of the interfacial temperature. However, because of the exothermic nature of the polycondensation and the poly-addition reactions, the equilibrium of these reactions moves in the reverse direction at high temperature and this will dominate over the conversion of [Epsilon]-caprolactam, and hence, the number average molecular weight decreases as the temperature increases. Poly-dispersity increases for both cases as the interfacial temperature increases.

As far as molecular weight is concerned, it can be mentioned from Table 4 by comparing the cases 4, 10 and 11 with each other that to obtain a polymer of higher molecular weight higher interfacial area per unit volume (i.e., a reactor with larger diameter) is preferred if the diffusional depths are fixed the same. On the other hand, if the interfacial areas per unit volume are fixed the same, a smaller diffusional depth (i.e., a reactor with smaller diameter) is preferred for obtaining a polymer of higher molecular weight.

The cyclic dimer has a higher melting point than the polymer and low solubility in water. Since the cyclic dimer remains in the polymer even after extraction, it acts like an impurity in the polymer and is a defect in processing of nylon 6. As a result, it is desirable to keep it from being produced in the polymerization stage. Table 4 shows that the heat transfer has a profound effect in reducing the content of the cyclic dimer in polymer product. When there is no water removal and no heat transfer, the cyclic dimer content reaches 0.93 wt%. This figure does not seem to be prominently reduced by introducing dry inert gas bubbles into the system. This means that the diffusional water removal alone does not change the content of cyclic dimer. On the other hand, the cyclic dimer content decreases prominently in the hollow cylinder type reactor, where the heat transfer is easily applied along with the diffusional water removal. The low temperature in the later portion of the reactor is preferred for the reduction of the amount of cyclic dimer in the product. By setting the interfacial temperature as low as 240 [degrees] C, the content of the cyclic dimer was reduced to 0.44 wt%, which is less than half for reference case-1. Larger interfacial area per volume is also shown to be beneficiary for reducing the amount of the cyclic dimer.

CONCLUDING REMARKS

The combined effects of diffusional water removal and heat transfer were investigated for the hollow sphere and hollow cylinder system by solving the system differential equations using finite difference technique. By designing the process conditions so that the monomer conversions were not influenced by heat transfer and diffusional water removal, we could investigate only the effects of these transport phenomena on polymer properties. Increasing the interfacial area per unit volume brings about a sharp decrease in temperature and water content of the reaction mass, and consequently an increase in molecular weight. Smaller diffusional depth is beneficiary for the removal of heat and water from the reaction mass, and will Increase the molecular weight. For the hollow cylinder system with a larger Interfacial area per unit volume, the higher Interfacial temperature brings about higher molecular weight. However, for the system with a smaller interfacial area the molecular weight decreases as the interfacial temperature increases. Both the Increased interfacial temperature and the decreased Interfacial area per volume will broaden the molecular weight distribution. For the Interfacial area per unit volume of 26.67 [m.sup.2]/[m.sup.3], the number average chain length reaches as high as 277.04 and the polydispersity as high as 2.06 when the interfacial temperature is 270 [degrees] C. The heat transfer has a profound effect on the content of cyclic dimer so that the hollow cylinder type reactor is preferred to the hollow sphere type one since the former is a lot easier to incorporate heat transport phenomena.

ACKNOWLEDGMENTS

This study was partially supported by the research grant from the Kyungnam University, Korea. The author sincerely expresses his gratitude for the support.

REFERENCES

1. H. K. Reimschuessel and K. Nagasubramanian, Chem. Eng. Sci., 27, 1119 (1972).

2. H. K. Reimschuessel, J. Polym. Sci. Macromol. Rev., 12, 65 (1977).

3. M. V. Tirrell, G. H. Pearson, R. A. Weiss, and R. L. Laurence, Polym. Eng. Sci., 15, 386 (1975).

4. Y. Arai, K. Tai, H. Teranishi, and T. Tagawa, Polymer, 22, 273 (1981).

5. K. Tai, Y. Arai, and T. Tagawa, J. Appl. Polym. Sci., 28, 2527 (1983).

6. S. Gupta and M. Tjahjadi, J. Appl. Polym. Sci., 33, 933 (1987).

7. S. Gupta, A. Kumar, and K. K. Agrawal, J. Appl. Polym. Sci., 37, 3089 (1982).

8. A. Gupta and S. K. Gupta, Chem. Eng. Comm., 113, 63 (1992).

9. K. Tai, Y. Arai, H. Teranishi, and T. Tagawa, J. Appl. Polym. Sci., 35, 1789 (1980).

10. H. Jacobs and C. Schweigman, Proc. 5th European/2nd International Symposium on Chemical Reaction Engineering, p. B7.1, Amsterdam. (1972).

11. W. H. Ray and J. Szekeley, Process Optimization, 1st Ed., Wiley-Interscience, New York (1973).

Nylon-6, a polymer of considerable commercial significance, is produced by water initiated polymerization of [Epsilon]-caprolactam. There has been significant research in the area of design and optimization of industrial nylon-6 reactors. Reimschuessel et al. (1, 2) considered the following three major reversible reactions: (a) ring opening reaction, in which the ring of the monomer, [Epsilon]-caprolactam, is opened by water to form aminocaprolc acid; (b) polycondensation reaction in which the amino and carboxylic acid end groups react to form large polymer chains via amide linkages, with water formed as a condensation product; (c) and polyaddition reaction in which monomer adds on to the growing polymer chains. Later, Tirrell et al. (3) incorporated the reaction with monofunctional acids (modifiers) in their kinetic scheme. Arai and co-workers (4) have coupled the ring-opening and polyaddltion reaction of cyclic dimer into Reimschuessel et al.'s basic reactions. They have obtained precise data and have curve-fitted their results to obtain a better set of rate constants for nylon-6 polymerization. They used their data to simulate the thermal effects for plug flow reactor (5). Gupta and Tjahjadi (6) simulated an industrial nylon 6 tubular reactor that has an introductory section for preheating the reactants with the heat produced by the reaction. They have shown the effects of various operating conditions and parameters on the temperature and molecular weight profiles.

The reaction that matters for the control of the degree of polymerization in polycondensation reaction is where water is formed as a condensate. This water greatly affects the chemical reaction equilibrium so that we cannot obtain polymer with high molecular weight, as desirable for industrial application. Therefore, this condensation water must be removed from the reaction mass. One approach to accomplish this goal is the design of reactors that effectively remove the condensated water by contacting inert dry gas to the reaction mass. Gupta et al. (7) have studied the design of three-stage nylon-6 reactors with intermediate mass transfer by solving the mass transfer equations numerically. Their study considered only the diffusional mass transfer of water through the reaction mass. Gupta and Gupta (8) also studied a semi-batch nylon 6 reactor in which gas bubbles move upward while removing water vapor from the reaction mass.

[TABULAR DATA FOR TABLE 1 OMITTED]

There is heat transfer involved in the polymerization process in the special case of a wetted-wall column type tubular reactor. In this study, idealized two-stage models of contacting inert dry gas with the reaction mass are considered to examine the effect of both mass and heat transfer on the polymerization reaction of [Epsilon]-caprolactam.

PROCESS MODELING

The three basic reactions, as briefly mentioned in the Introduction, are coupled with the ring-opening and polyaddition reaction of cyclic dimer to form a complete reaction system. The kinetic scheme used is summarized in Table 1, The rate constants are those corresponding to the reaction between functional groups. The polymerization is well described by an autocatalytic reaction, and the forward reaction rate constant [k.sub.i](i = 1, 2, 3, 4, 5) is expressed in terms of the acid end group concentration (in g-mole/kg mixture) as

[Mathematical Expression Omitted] (1)

where [[S.sub.n]] is the concentration of linear functional compounds and [[Mu].sub.0] is the zeroth moment of the linear functional compounds. The v-th moment of the linear functional compounds is generally expressed by

[[Mu].sub.v] = [summation of] [n.sup.v] [[S.sub.n]] where n = 1 to [infinity] (2)

The values of v are 0, 1, 2, and 3 for the four different kinds of moments used in the formulation. [k.sub.i] and [Mathematical Expression Omitted] [TABULAR DATA FOR TABLE 2 OMITTED] have units of kg/mol-hr, while [Mathematical Expression Omitted] has units of [kg.sup.2]/[mol.sup.2]-hr. Both [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are functions of temperature expressed by an Arrhenius-type relation as follows:

[Mathematical Expression Omitted] (3)

where the superscript * is to be replaced by o or c for each corresponding rate constant. A and E are the frequency factor and the activation energy, respectively. R is the gas constant and T is the absolute temperature expressed in units of K. The equilibrium constant, [K.sub.i], is similarly given as a function of temperature:

ln [K.sub.i] = [Delta][S.sub.i] / R - [Delta][H.sub.t] / RT (4)

Here, [Delta]S and [Delta]H denote the change of entropy and the change of enthalpy, respectively. Values of [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Delta][S.sub.t] and [Delta][H.sub.t] for the various reactions have been given by Arai et al. (4) and are summarized in Table 2. The ring opening of [Epsilon]-caprolactam is endothermic since the enthalpy of linear aminocaproic acid produced is higher than that of [Epsilon]-caprolactam. On the other hand, the ring opening of cyclic dimer is exothermic since the equilibrium concentration of cyclic dimer increases with increasing temperature as observed by Arai et al. (4).

Two kinds of gas-reaction mass contacting systems are considered. One in which inert gas bubbles are injected into an intermediate position of the tubular reactor and removed from the outlet position. The other is a wetted-wall type tubular reactor in which reaction mass contacts with inert gas after an intermediate position. For analysis, the former can be idealized as a hollow sphere model and the latter as a hollow cylinder model. They are schematically described in Fig. 1. In the hollow sphere model, we consider the bubble size as average value for calculation. Mass balance equations are now written for the idealized systems of hollow sphere and hollow cylinder. For nylon 6 polymerization, the mass and energy balance equations are given in Table 3, where [D.sub.w] is diffusivity of water, W, through the polymeric reaction mass, and k/[Rho][C.sub.t] (= [Alpha]) is thermal diffusivity and [[Mu].sub.v] (v = 0, 1, 2, 3) is the v-th moment of linear functional compounds defined by Eq 2. In writing these equations, the following assumptions have been made:

(1) Only the condensation product, water W, can diffuse. The monomeric and polymeric species do not diffuse at all.

(2) The Interfacial water concentration [[W].sub.1] and temperature [T.sub.1] at r = [r.sub.1] are taken to be constant and treated as parameters.

(3) The flow pattern of the reaction mass is plug flow.

The equations for the moments [[Mu].sub.0], [[Mu].sub.1], and [[Mu].sub.2] can be obtained by using the z-transform defined by

D(z, t) = [summation of] [z.sup.-n][[S.sub.n]] where n = 1 to [infinity] (5a)

through differentiation of the following form

[[Mu].sub.v] = [[Delta].sup.v]D(z, t) / [Delta][(ln [z.sup.-1]).sup.v] [where] z = 1 (5b)

The differential equation governing the time change of D is first obtained according to Eq 5a using exact expression of [Delta][[S.sub.n]]/[Delta]t. The Equation is subsequently differentiated using Eq 5b to obtain the balance equations of the various moments. In Table 3, the mass balance equation for the component [[S.sub.1]] contains the components [[S.sub.2]] and [[S.sub.3]], and the equation for [[Mu].sub.2] contains the component [S.sub.2]] and [[Mu].sub.3]. This requires additional balance equations for the components [[S.sub.2]], [[S.sub.3]] and [[Mu].sub.3]. This, in turn, constitutes the hierarchy of equations, which makes the solution of the system of equations impossible. To solve these difficulties, the first hierarchy has been broken by assuming that [[S.sub.2]] and [[S.sub.3]] are equal to [[S.sub.1]] (2). The second has been broken by the fact that the Schultz-Zimm distribution [TABULAR DATA FOR TABLE 4 OMITTED] function is suitable for representing the distribution of polymerization degree of poly-[Epsilon]-caprolactam (9). According to the properties of the Schultz-Zimm function, the following relations can be derived:

[Mathematical Expression Omitted] (6)

The following expressions for the calculation of density, [Rho], and specific heat, [c.sub.p], of reaction mass have been reported by Jacobs and Schweigman (10) and adopted in this study.

[Rho] = [10.sup.3][{1.0065 + 0.0123[[C.sub.1]] + (T - 495)(0.00035 + 0.00007[[C.sub.1]])}.sup.-1] (7)

[c.sub.p] = 238.09(2.76 [[C.sub.1] / [[[C.sub.1]].sub.0] + [[[C.sub.1]].sub.0] - [[C.sub.1]] / [[[C.sub.1]].sub.0] (2.035 + 0.00141T)) (8)

where [[C.sub.1]] denotes the concentration of [Epsilon]-caprolactam and subscript 0 denotes the feed. The units of [Rho] and [c.sub.p] are in kg/[m.sup.3] and cal/kg-K. T is in units of K. The set of equations in Table 3 are solved along with the above relations. The initial and boundary conditions required for the two systems are

t = 0, 0 [less than] r [less than or equal to] [r.sub.2]: y = [y.sub.0]

0 [less than] t [less than] [t.sub.0], r = 0: [Delta]y / [Delta]r = 0

0 [less than] t [less than] [t.sub.0], r = [r.sub.2]: [Delta]y / [Delta]r = 0

t = [t.sub.0], [r.sub.1] [less than] r [less than] [r.sub.2]: [Delta]y / [Delta]r = 0 (9)

t [greater than] [t.sub.0], r = [r.sub.1]: [W] = [[W].sub.1],

[Mathematical Expression Omitted]

t [greater than] [t.sub.0], r = [r.sub.2]: [Delta][W] / [Delta]r = 0, [Delta]T / [Delta]r = 0

where y denotes all concentrations and temperature. The concentrations and temperature are displayed as spatially averaged values in the Figures. The spatial average concentrations and temperature are obtained using the following equations.

For hollow sphere:

[Mathematical Expression Omitted] (10a)

For hollow cylinder:

[Mathematical Expression Omitted] (10b)

The number- and weight-average chain lengths are computed as follows:

[[Lambda].sub.n] = [[Mu].sub.1] / [[Mu].sub.0] (11)

[[Lambda].sub.w] = [[Mu].sub.2] / [[Mu].sub.1]

The polydispersity index, [Delta], is given by

[Mathematical Expression Omitted] (12)

The equations in Table 3 are transformed into finite difference form by using forward difference for the time- and first order space-derivative terms, and central difference for the second order space derivative terms. These difference equations can be solved explicitly with a time increment [Delta]t of 0.005 h. The spatial increment [Delta]r should satisfy the following convergence criterion (11).

[([Delta]r).sup.2] / ([Delta]t)[D.sub.w] [greater than or equal to] 4 (13)

[([Delta]r).sup.2] / ([Delta]t)[Alpha] [greater than or equal to] 4

Thus, the radial distance from [r.sub.1] to [r.sub.2] is divided into 10 equal meshes and each mesh point is numbered from 1 to 11 in an increasing order.

RESULTS AND DISCUSSION

As mentioned in the Introduction, the focus of this study has concentrated on investigating how greatly the mass and heat transfer affect the reactor performance and how much the properties of the polymer produced are to be influenced by them with respect to the monomer conversion, the molecular weight and its distribution, and the content of cyclic dimer in the product.

To investigate these effects, different sets of geometries and dimensions have been selected. The typical dimensions of hollow sphere systems have been chosen in such ways as (1) inner radius [r.sub.1] of 0.005 m and outer radius [r.sub.2] of 0.030 m, and (2) inner radius [r.sub.1] of 0.010 m and outer radius [r.sub.2] of 0.060 m, and those of hollow cylinder systems in such ways as (3) [r.sub.1] of 0.025 m and [r.sub.2] of 0.050 m, (4) [r.sub.1] of 0.050 m and [r.sub.2] of 0.100 m. These are summarized in Table 4 along with the calculated results for several process conditions. The contact between the inert gas and the reaction mass has been designed to occur 5 h after the inlet in those systems. The inlet temperature of the feed was fixed at 240 [degrees] C and the inlet water concentration at 0.333 mol/kg for all of the cases, which will permit the reaction mass to reach reaction equilibrium within 5 h, before contacting with the inert gas takes place. It was assumed that the interfacial water concentration be kept at 0.088 mol/kg (= 0.001 mol/mol-caprolactam). The values of the diffusivity of water in the reaction mass, [D.sub.w], and the thermal conductivity of the reaction mass, k, used for computation were assumed to be 0.9 x [10.sup.-4] [m.sup.2]/hr and 165 cal/[m.sup.2]-hr-K, respectively.

The calculated results for the temperature and the water concentration distribution of the hollow cylinder type reactor(Case-4) are graphically shown in Figs. 2 and 3. From these three-dimensional Figures we can basically Figure out the process characteristics of these models. In the Figures that follow, the radially averaged values of each variable will be plotted along the axial direction scaled on a time basis.

The monomer conversion profiles for the systems are shown in Fig. 4. From this Figure, it can be said that all of the systems are well conditioned in terms of the monomer reaction equilibrium since the conversions show almost the same profile for all of the cases and reach the equilibrium conversions of [approximately]91%. Therefore, the mass and heat transfer will only affect the temperature, the water concentration, and the polymer properties including the content of cyclic dimer. The temperature profiles are shown in Fig. 5. For the first three cases in which no water removal and hollow sphere systems are concerned the temperature profiles are almost the same since heat transfer is not considered. On the other hand, the hollow cylinder systems, in which both heat transfer and diffusional water removal are considered, show sharp decreases in temperature. The higher the interfacial area per unit volume of the reaction mass is increased, the sharper the drop in the temperature profile becomes.

The effects of diffusional water removal are shown in Fig. 6. For the case of no water removal, the water content slightly increases 4 h after the reaction starts, since polycondensation reaction dominates in the later portion of the reactor. This condensated water is in equilibrium with the growing polymers, which will prohibit the growth of polymer chain. Therefore, it is desirable to remove this condensated water from the reaction mass. For hollow sphere model, case-3 has a bigger bubble size than case-2, but has a smaller interfacial area per volume than case-2. This results in less diffusional water removal in case-3 than in case-2 as can be seen in Fig. 6. The same discussion is possible for the hollow cylinder model when the cases 4, 7, and 11 are compared. However, when we compare cases 2 and 7, we see that even if case-7 has an interfacial area per unit volume as large as 13.33 [m.sup.2]/[m.sup.3], much larger than 2.79 [m.sup.2]/[m.sup.3] of case-2, the former has less diffusional effect than the latter, since the former has a diffusional depth of 0.050 m which is longer than 0.025 m of the latter.

The number average chain length for the selected cases are shown in Fig. 7. By comparing with Fig. 6, one can say that higher water removal leads to a polymer with higher molecular weight as can be expected by the inherent nature of polycondensation reaction.

To investigate the heat transfer effect, two hollow cylinder systems were chosen in such a way that one has the interfacial area per unit volume as large as 26.67 [m.sup.2]/[m.sup.3] and the other has that as large as 13.33 [m.sup.2]/[m.sup.3]. The interfacial temperature for these systems were increased from 240 to 280 [degrees] C and the changes in number average chain length and polydispersity were observed. The results are presented in Figs. 8 and 9. As the interfacial temperature increases, the molecular weight increases for the former while it decreases for the latter. When the interfacial area per unit volume is large, the temperature of reaction mixture is relatively low. As a result, the equilibrium conversion of [Epsilon]-caprolactam becomes low since the ring opening of [Epsilon]-caprolactam is endothermic. As the temperature increases, the conversion increases and dominates the exothermic feature of the polycondensation and the polyaddition reactions such that equilibrium of these reactions moves in reverse as the temperature increases. Hence, the number average molecular weight will increase. However, when the interfacial area per unit volume is small, the reaction mixture has sufficiently high temperature and the equilibrium conversion reaches the highest value at [approximately]240 [degrees] C of the interfacial temperature. However, because of the exothermic nature of the polycondensation and the poly-addition reactions, the equilibrium of these reactions moves in the reverse direction at high temperature and this will dominate over the conversion of [Epsilon]-caprolactam, and hence, the number average molecular weight decreases as the temperature increases. Poly-dispersity increases for both cases as the interfacial temperature increases.

As far as molecular weight is concerned, it can be mentioned from Table 4 by comparing the cases 4, 10 and 11 with each other that to obtain a polymer of higher molecular weight higher interfacial area per unit volume (i.e., a reactor with larger diameter) is preferred if the diffusional depths are fixed the same. On the other hand, if the interfacial areas per unit volume are fixed the same, a smaller diffusional depth (i.e., a reactor with smaller diameter) is preferred for obtaining a polymer of higher molecular weight.

The cyclic dimer has a higher melting point than the polymer and low solubility in water. Since the cyclic dimer remains in the polymer even after extraction, it acts like an impurity in the polymer and is a defect in processing of nylon 6. As a result, it is desirable to keep it from being produced in the polymerization stage. Table 4 shows that the heat transfer has a profound effect in reducing the content of the cyclic dimer in polymer product. When there is no water removal and no heat transfer, the cyclic dimer content reaches 0.93 wt%. This figure does not seem to be prominently reduced by introducing dry inert gas bubbles into the system. This means that the diffusional water removal alone does not change the content of cyclic dimer. On the other hand, the cyclic dimer content decreases prominently in the hollow cylinder type reactor, where the heat transfer is easily applied along with the diffusional water removal. The low temperature in the later portion of the reactor is preferred for the reduction of the amount of cyclic dimer in the product. By setting the interfacial temperature as low as 240 [degrees] C, the content of the cyclic dimer was reduced to 0.44 wt%, which is less than half for reference case-1. Larger interfacial area per volume is also shown to be beneficiary for reducing the amount of the cyclic dimer.

CONCLUDING REMARKS

The combined effects of diffusional water removal and heat transfer were investigated for the hollow sphere and hollow cylinder system by solving the system differential equations using finite difference technique. By designing the process conditions so that the monomer conversions were not influenced by heat transfer and diffusional water removal, we could investigate only the effects of these transport phenomena on polymer properties. Increasing the interfacial area per unit volume brings about a sharp decrease in temperature and water content of the reaction mass, and consequently an increase in molecular weight. Smaller diffusional depth is beneficiary for the removal of heat and water from the reaction mass, and will Increase the molecular weight. For the hollow cylinder system with a larger Interfacial area per unit volume, the higher Interfacial temperature brings about higher molecular weight. However, for the system with a smaller interfacial area the molecular weight decreases as the interfacial temperature increases. Both the Increased interfacial temperature and the decreased Interfacial area per volume will broaden the molecular weight distribution. For the Interfacial area per unit volume of 26.67 [m.sup.2]/[m.sup.3], the number average chain length reaches as high as 277.04 and the polydispersity as high as 2.06 when the interfacial temperature is 270 [degrees] C. The heat transfer has a profound effect on the content of cyclic dimer so that the hollow cylinder type reactor is preferred to the hollow sphere type one since the former is a lot easier to incorporate heat transport phenomena.

ACKNOWLEDGMENTS

This study was partially supported by the research grant from the Kyungnam University, Korea. The author sincerely expresses his gratitude for the support.

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Author: | Ahn, Young-Cheol |
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Publication: | Polymer Engineering and Science |

Date: | Feb 1, 1997 |

Words: | 3660 |

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