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노형준, 이현구

서울대학교 응용화학부·화학공정신기술연구소

Generalized Two-Step Method for Molecular Weight Control

in a Batch Polymerization Reactor

Hyung-Jun Rho and Hyun-Ku Rhee

School of Chemical Engineering & Institute of Chemical Processes

Seoul National University

Introduction

A generalized two-step method was developed for the molecular weight control in a batch MMA polymerization reactor. The two-step method is one of the optimization methods that provide the reaction temperature trajectory which would lead to the polymer product having desired average- and weight-average molecular weights. The generalized two-step method was evolved so as to deal with the constraint or minimize the reaction time. The simulation result showed that the polymer product with desired average molecular weights was indeed obtained at a specified monomer conversion after the minimum reaction time.

Polymerization Reaction Model

The system considered in this study is the batch solution polymerization process of methylmethacrylate (MMA) using benzoylperoxide (BPO) as the initiator and ethylacetate (EA) as the solvent; i. e.,

Initiation       [eqn: &I`buildrel rarrow {k_d```}{2 phi CDOT }# &phi CDOT ``+`M~~ buildrel rarrow {k_i``````}`R_1 CDOT]       Propagation      [eqn: R_j CDOT +`M~~ buildrel rarrow {k_pj```}`R_j+1 CDOT]

Termination by [eqn: R_i CDOT +`R_j CDOT buildrel rarrow {k_tc```}`P_i+j]       Termination by   [eqn: R_i CDOT +`R_j CDOT buildrel rarrow {k_td`}`P_i `+`P_j]

combination                                  disproportionation

Chain transfer  [eqn: R_i CDOT +`M~~buildrel rarrow {k_trm}`P_i `+`R_1 CDOT]  Chain transfer    [eqn: R_i CDOT +`S~~buildrel rarrow {k_trs`}~P_i `+`S CDOT]

to solvent                                    to monomer

From the kinetic mechanism of free radical solution polymerization, it is straightforward to formulate the following differential equations which describe the dynamics of the polymerization reactor.

[eqn: 1 OVER V_r {d(IV_r`)} OVER {dt} `=`-k_d I`]	[eqn: 1 OVER V_r {d(H_0 V_r`)} OVER {dt}`=`2f`k_d`I`-`0.5k_tc G_0^2`+`+`k_trm MG_0`+`k_trs SG_0`]

[eqn: 1 OVER V_r {d(MV_r`)} OVER {dt}`=`-2fk_d I`-`k_p MG_0 ` -`k_trm MG_0]   [eqn: 1 OVER V_r {d(H_1 V_r`)} OVER {dt}`=`2f`k_d`I`+`k_p MG_1`+`k_trm MG_0`-`k_trs SG_0`]

[eqn: 1 OVER V_r {d(SV_r`)} OVER {dt}`=`-`k_trs SG_0`]	[eqn: =-`1overV left[ d(MV) over dt`-`d(SV) over dt right]]

[eqn: 1 OVER V_r {d(G_0 V_r`)} OVER {dt}`=`2fk_d I`-`k_t G_0 sup 2`]	∀ [eqn: H_1 V_r`=`(MV_r`)_0`-`MV_r`+`(SV_r`)_0`-SV_r`]

[eqn: 1 OVER V_r {d(G_1 V_r`)} OVER {dt} &=`2fk_d I`+`k_p MG_0 `-`k_t G_0 G_1# &+`k_trm M(G_0 `-`G_1 )`+`k_trs S(G_0 `-`G_1 )][eqn: 1 OVER V_r {d(H_2 V_r`)} OVER {dt} &=`2f`k_d`I`+`k_p M(G_0`+`2G_1`)`+`k_tc G_1^2# &+`k_trm MG_0`+`k_trs SG_0`]

where I, M and S represents the concentrations of initiator, monomer and solvent, respectively. Here, [eqn: G_k] and [eqn: H_k] are the [eqn: k]-th moments of living and total (living and dead) polymer concentrations, respectively. The monomer conversion, the number- and weight-average molecular weights, and the polydispersity index can be determined as follows:

[eqn: X_m`=`1`-`MV_r over {(MV_r`)_0`}],  [eqn: Mn``=``MW_m {H_1`V_r}over{H_0`V_r}`],  [eqn: Mw``=``MW_m {H_2`V_r}over{H_1`V_r}`]  and  [eqn: P`=`Mw over Mn]

In this study, the gel effect was considered by using the free volume correlation suggested by Schmidt and Ray [1]. The reaction mixture volume can be calculated by the equation:

[eqn: V_r`=`V_m`+`V_s`+`V_p`=`{MV_r}over rho_m MW_m`+`{SV_r}over rho_s MW_s`+`{H_1 V_r}over rho_p MW_m]

The kinetic rate constants were determined by applying the parameter estimation technique to the experimental data [2] while the physical properties were taken from the literature [3].

Generalized Two-Step Method

In the first step, the profile of the instantaneous average chain length to give the desired molecular weights is examined. In the second step, the time profile of the reactor temperature is obtained by matching the instantaneous average chain length determined in the first step to the value predicted by the model equations. This procedure is repeated until the converged profile was obtained[4].

We divide the first and the second moment equations by the zero-th moment equation of the total polymer concentration (coordinate transformation):

[eqn: {d(H_1 V_r )}over{d(H_0 V_r )}`=`{2f`k_d (IV_r )V_r +[k_p (MV_r )+k_trm (MV_r )+k_ktrs (SV_r )](G_0 V_r )}over{2f`k_d (IV_r )V_r +[k_trm (MV_r )+k_ktrs (SV_r )-0.5k_tc (G_0 V_r )](G_0 V_r )}`]

[eqn: {d(H_2 V_r )}over{d(H_0 V_r )}`=`{d(H_1 V_r )}over{d(H_0 V_r )}+{[2k_p (MV_r )+k_tc (G_1 V_r )](G_1 V_r )}over{2f`k_d (IV_r )V_r +[k_trm (MV_r )+k_ktrs (SV_r )-0.5k_tc (G_0 V_r )](G_0 V_r )}`]

For easy reference, we introduce several notations:

[eqn: u`==`{d(H_1 V_r )}over{d(H_0 V_r )}`],  [eqn: u`>`0`]    and    [eqn: p`==`{(dH_0 V_r`)(dH_2 V_r )}over{(dH_1 V_r )^2}]

[eqn: x`==`H_0 V_r] ,    [eqn: y_1`==`H_1 V_r]    and    [eqn: y_2`==`H_2 V_r]

where [eqn: u] and [eqn: p] are the instantaneous average chain length and the instantaneous polydispersity index, respectively. The two equations are then expressed as:

[eqn: dy_1 over dx `=`u,~~~``~~~y_1`=`left{{0~~at~x=0}atop{y_1^* ~~at~x=x_f}}`]    and    [eqn: dy_2 over dx `=`p``u^2`,~~~~~y_2`=`left{{0~~at~x=0}atop{y_2^* ~~at~x=x_f}}`]

where * represents the desired value of output. We assume that the instantaneous polydispersity index remains constant and denote it by [eqn: p_c`].

[eqn: y_1^* `=`int_0^x_f u`dx`]      and      [eqn: y_2^* over p_c `=`int_0^x_f u^2``dx`]

The [eqn: u] profiles that satisfies these equations should have two dimension. Therefore, we suggest a shape function and suppose that the [eqn: u] profile has two-adjustable parameters having the roles of re-scaling and shifting of the shape function, respectively; i. e.,

[eqn: u(x)`=`{v(x)`-`c_2}over c_1 ,~~~c_1`>`0`]

Inserting this into the governing equations gives the parameters, [eqn: c_1] and [eqn: c_2]:

[eqn: c_1`=`1over y_1^* sqrt{{p_c (x_f A_2 -A_1^2`)}over{P^* -p_c}}]    and    [eqn: c_2`=`{(A_1 -c_1 y_1^*`)}over x_f]

where [eqn: A_1] and [eqn: A_2] are the integrals of [eqn: v(x)] and [eqn: v(x)^2], respectively.

[eqn: A_1`=`int_0^x_f v(x)`dx]     and     [eqn: A_2`=`int_0^x_f v(x)^2`dx]

The parameters exist in real space if [eqn: P^*`>=`p_c] and [eqn: x_f`A_2`>=`A_1^2]. The first condition represents the necessary physical condition that the desired polydispersity index should be larger than the instantaneous polydispersity index for free radical polymerization reaction. The second condition represents the shape limitation but it is satisfied automatically because of the Cauchy-Schwarz inequality for integral. For the second step, the reaction temperature is calculated by applying the root searching algorithm to the following equation:

[eqn: u`=`{2f`k_d (IV_r )V_r +[k_p (MV_r )+k_trm (MV_r )+k_ktrs (SV_r )](G_0 V_r )}over{2f`k_d (IV_r )V_r +[k_trm (MV_r )+k_ktrs (SV_r )-0.5k_tc (G_0 V_r )](G_0 V_r )}`]

The [eqn: p_c] values is fixed at 1.70 in this study.

By appropriate designing the shape function, we can imitate various optimizing actions that the optimal control method may perform. First, the generalized two-step method can deal with the system constraint that is often resulted from the mechanical or physical limitation in the process. Second, it can give the minimal time solution. With the optimal control theory, exact solutions are very difficult to obtain since the nonlinear programming is essential with a large number of iterations.

To treat the constraint problem, we propose the following iteration rule:

[eqn: v(x)`=`u(T_r`)~~~~~~~if~T_r`>=`T_r^min ~~ rm or ~ it T_r`<=`T_r^max]

[eqn: v(x)`=`u(T_r^min`)~~~~~if~T_r`<`T_r^min]

[eqn: v(x)`=`u(T_r^max`)~~~~~if~T_r`>`T_r^max]

The correlation between the reaction temperature and [eqn: u] is obtained in the second step. The minimum and the maximum temperature constraint are 64 ℃ and 75 ℃, respectively. To solve the minimal reaction time problem, the zero-th moment is transformed into the following:

[eqn: {d`t_R}over {dx} `=`V_r over{2f`k_d (IV_r )V_r +[k_trm (MV_r )+k_ktrs (SV_r )-0.5k_tc (G_0 V_r )](G_0 V_r )}`=`f_t`]

we propose the following update rule:

[eqn: v`=`u`-`alpha`f_t over u {partial u}over{partial f_t}]    and    [eqn: {partial u} over {partial f_t}`=`{partial u} over {partial T_r}{partial T_r} over {partial f_t}]

The term [eqn: f_t`/`u] represents the size effect of the partial derivative. The partial derivative term indicates the amount of decrease in [eqn: u] per unit increase of the time slope [eqn: f_t]. The term [eqn: partial u`/{partial f_t`}] is determined by the chain rule since the reaction temperature is the only manipulated variable.

Fig.  1.  Simulation result by the generalized two-step method when the constraint treatment rule is applied.

Fig.  2.   Simulation result by the generalized two-step method when the minimal time rule is applied.

Conclusions

We developed the generalized two-step method to obtain the minimal time trajectories and also to treat the problems with constraints on the reaction condition. The polymer product having desired average molecular weights could be obtained effectively by tracking the temperature trajectory computed by the generalized two-step method.

94

Acknowledgment

This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Automation Research Center (ARC) at Pohang University of Science and Technology.

References

1. Schmidt, A. D. and Ray, W. H., Chem. Eng. Sci., 36, 1401(1981).

2. Chang, S.-S., Ph.D. thesis, Seoul National Univ. (1995).

3. Brandrup, J. and Immergut, E. H., "Polymer Handbook", 3rd ed., Wiley, NY (1989).

4. Takamatsu, T., Shioya, S. and Okada, Y., Ind. Eng. Chem. Res., 27, 93(1988).