Multiscale modeling of syndiospecific styrene polymerization

Abstract

A detailed mathematical model for syndiospecific styrene polymerization based on combining features of the multigrain model (MGM) and the polymeric multigrain model (PMGM). This model has been established to predict the radial monomer concentration within the growing macro particles and the rate of polymerization. The latter, the parameters, have an effect on the molecular weight distribution (MWD). In this model, the effect of intraparticle diffusion resistance and the radius of catalyst particles on the rate of polymerization and MWD were studied. The model simulation showed the presence of a large distribution of monomer concentration across the radius of particles. It was further noticed that the diffusion resistance was most intense at the beginning of the polymerization process. For MWD, the model simulation showed that the existence of diffusion resistance led to have an increase in the molecular weight within a period of time similar to the one needed in the catalyst decay. Moreover, the validation of the model with experimental data given a good agreement results and show that the model is able to predict a correct monomer profile, polymerization rate, particle growth factor and MWD, an algorithm, which embeds physicochemical effects, has been developed to model the industrial reactors.

Appendix 1

The changes in the shells volume, ∆V_i and the location of the grid points R_i with time are given in this section. As show in Fig. 2, the hypothetical shell can be defined as R_h,_i−1 ≤ r ≤ R_h,i such that the entire polymer produced by the catalyst particles of radius R_c are accommodated in it . In the interval t to t+∆t, the total volume of polymer and the volume of micro particle produced at ith shell are given by:

$$ \frac{{{\text{d}}{{\text{V}}_{\text{i}}}}}{\text{dt}} = \frac{{0.001\,{{\text{k}}_{\text{p}}}{\text{C*}}\,{{\text{M}}_{{\mu, {\text{i}} + 1}}}\left( {{{\text{N}}_{\text{i}}}\frac{4}{3}\pi {\text{R}}_{\text{c}}^3} \right){\text{m}}{{\text{w}}_{\text{sty}}}}}{{{\rho_p}}} $$

(1.1)

$$ \frac{{{\text{d}}{{\text{V}}_{{{\text{s}},{\text{i}}}}}}}{\text{dt}} = \frac{{0.001\,{{\text{k}}_{\text{p}}}{\text{C*}}\,{{\text{M}}_{{\mu, {\text{i}}}}}\left( {\frac{4}{3}\pi {\text{R}}_{\text{c}}^3} \right){\text{m}}{{\text{w}}_{\text{sty}}}}}{{{\rho_p}}} $$

(1.2)

With V_i(t = 0) and V_S,i(t = 0) being the initial total volume and volume of every polymer micro particle of ith volume, respectively.

$$ {{\text{V}}_{\text{i}}}({\text{t}} = 0) = \frac{{{{\text{N}}_{\text{i}}}\left( {\frac{4}{3}\pi {\text{R}}_{\text{c}}^3} \right)}}{{1 - \varepsilon }} $$

(1.3)

$$ {{\text{V}}_{{{\text{s}},{\text{i}}}}}\left( {{\text{t}} = 0} \right) = \frac{4}{3}\pi {\text{R}}_{\text{c}}^3 $$

(1.4)

We can now define the hypothetical shells at any time by:

$$ {{\text{R}}_{{{\text{h}},{\text{i}}}}} = {\left( {\frac{3}{{4\pi }}\mathop{\sum }\limits_{{{\text{j}} = 1}}^{\text{i}} {{\text{V}}_{\text{j}}}} \right)^{{1/3}}} $$

(1.5)

Where R_h,o = 0 and the radius of micro particle at ith shell being:

$$ {{\text{R}}_{{s,{\text{i}}}}} = {\left( {\frac{3}{{4\pi }}{{\text{V}}_{{{\text{s}},{\text{i}}}}}} \right)^{{\frac{1}{3}}}} $$

(1.6)

The catalyst particles are assumed to be placed at the mid points of each hypothetical shell. Thus:

$$ {{\text{R}}_{{1,{\text{i}}}}} = {{\text{R}}_{{{\text{h}},{\text{i}} - 1}}} + \frac{1}{2}\left( {{{\text{R}}_{{{\text{h}},{\text{i}}}}} - {{\text{R}}_{{{\text{h}},{\text{i}} - 1}}}} \right) $$

(1.7)

Then the computational grid points are related to R_1,i by:

$$ {{\text{R}}_1} = 0 $$

(1.8)

$$ {{\text{R}}_2} = {{\text{R}}_{\text{c }}} $$

(1.9)

$$ {{\text{R}}_{{{\text{i}} + 1}}} = {{\text{R}}_{{1,{\text{i}}}}} + {{\text{R}}_{{{\text{s}},{\text{i}}}}} $$

(1.10)

$$ {{\text{R}}_{{{\text{N}} + 2}}} = {{\text{R}}_{{{\text{h}},{\text{N}}}}} $$

(1.11)

The values of ∆r_i to be used in the equation (6a, b & c) are given by:

$$ \Delta {{\text{r}}_{\text{i}}} = {{\text{R}}_{{{\text{i}} + 1}}} - {{\text{R}}_{\text{i}}} $$

(1.12)