Effect of Internal Diffusional Restrictions on the Hydrolysis of Penicillin G: Reactor Performance and Specific Productivity of 6-APA with Immobilized Penicillin Acylase

Abstract

A mathematical model that describes the heterogeneous reaction–diffusion process involved in penicillin G hydrolysis in a batch reactor with immobilized penicillin G acylase is presented. The reaction system includes the bulk liquid phase containing the dissolved substrate (and products) and the solid biocatalyst phase represented by glyoxyl-agarose spherical porous particles carrying the enzyme. The equations consider reaction and diffusion components that are presented in dimensionless form. This is a complex reaction system in which both products of reaction and the substrate itself are inhibitors. The simulation of a batch reactor performance with immobilized penicillin G acylase is presented and discussed for the internal diffusional restrictions impact on effectiveness and productivity. Increasing internal diffusional restrictions, through increasing catalyst particle size and enzyme loading, causes impaired catalyst efficiency expressed in a reduction of effectiveness factor and specific productivity. High penicillin G initial concentrations decrease the impact of internal diffusional restrictions by increasing the mass transfer towards porous catalyst until product inhibition becomes significant over approximately 50 mM of initial penicillin G, where a drop in conversion rate and a maximum in specific productivity are then obtained. Results highlight the relevance of considering internal diffusional restrictions, reactor performance, and productivity analysis for proper catalyst and reactor design.

Appendix

Simulation was done by solving Eqs. 22–24 and 32–34. Dimensionless equations were discretized using Crank–Nicolson method to achieve stability during solution processing in computer programs. The number of equations depends on N obtaining N + 1 equations for each substance. In this study, N = 20 was used obtaining a system of 63 equations. Algebraic form of discretized equations are as follows:For j = 0:

$$ - \frac{3}{2}{s_0} + 2{s_1} - \frac{1}{2}{s_2} = 0 $$

$$ - \frac{3}{2}{p_0} + 2{p_1} - \frac{1}{2}{p_2} = 0 $$

$$ - \frac{3}{2}{q_0} + 2{q_1} - \frac{1}{2}{q_2} = 0 $$

For j = 1,…, N −1:

$$ s_j^{{n + 1}} - \frac{{\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]s_{{j - 1}}^{{n + 1}} - 2s_j^{{n + 1}} + \left[ {1 + \frac{1}{j}} \right]s_{{j + 1}}^{{n + 1}}} \right) = s_j^n + \frac{{\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]s_{{j - 1}}^n - 2s_j^n + \left[ {1 + \frac{1}{j}} \right]s_{{j + 1}}^n} \right) - \Delta \tau {\Phi^2}\sigma_j^n $$

$$ p_j^{{n + 1}} - \frac{{{D_{{e1}}}\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]p_{{j - 1}}^{{n + 1}} - 2p_j^{{n + 1}} + \left[ {1 + \frac{1}{j}} \right]p_{{j + 1}}^{{n + 1}}} \right) = p_j^n + \frac{{{D_{{e1}}}\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]p_{{j - 1}}^n - 2p_j^n + \left[ {1 + \frac{1}{j}} \right]p_{{j + 1}}^n} \right) - \Delta \tau \Phi_1^2\sigma_j^n $$

$$ q_j^{{n + 1}} - \frac{{{D_{{e2}}}\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]q_{{j - 1}}^{{n + 1}} - 2q_j^{{n + 1}} + \left[ {1 + \frac{1}{j}} \right]q_{{j + 1}}^{{n + 1}}} \right) = q_j^n + \frac{{{D_{{e2}}}\Delta \tau }}{{2{{(\Delta \rho )}^2}}}\left( {\left[ {1 - \frac{1}{j}} \right]q_{{j - 1}}^n - 2q_j^n + \left[ {1 + \frac{1}{j}} \right]q_{{j + 1}}^n} \right) - \Delta \tau \Phi_2^2\sigma_j^n $$

For j = N:

$$ s_N^{{n + 1}} + \frac{{\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}s_{{N - 2}}^{{n + 1}} + 2s_{{N - 1}}^{{n + 1}} - \frac{3}{2}s_N^{{n + 1}}} \right) = s_N^n - \frac{{\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}s_{{N - 2}}^n + 2s_{{N - 1}}^n - \frac{3}{2}s_N^n} \right) $$

$$ p_N^{{n + 1}} + \frac{{{D_{{e1}}}\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}p_{{N - 2}}^{{n + 1}} + 2p_{{N - 1}}^{{n + 1}} - \frac{3}{2}p_N^{{n + 1}}} \right) = p_N^n - \frac{{{D_{{e1}}}\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}p_{{N - 2}}^n + 2p_{{N - 1}}^n - \frac{3}{2}p_N^n} \right) $$

$$ q_N^{{n + 1}} + \frac{{{D_{{e2}}}\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}q_{{N - 2}}^{{n + 1}} + 2q_{{N - 1}}^{{n + 1}} - \frac{3}{2}q_N^{{n + 1}}} \right) = q_N^n - \frac{{{D_{{e2}}}\Delta \tau }}{{2\gamma \Delta \rho }}\left( { - \frac{1}{2}q_{{N - 2}}^n + 2q_{{N - 1}}^n - \frac{3}{2}q_N^n} \right) $$