Analytical expression for the model that describes the heterogeneous reaction-diffusion process with immobilized enzyme (penicillin G acylase)

Abstract

This research article examines the reaction-diffusion process in an immobilized enzyme batch reactor. The model incorporates strongly non-linear factors that are associated with standard Michaelis-Menten kinetics. The non-linear reaction-diffusion equations for substrate and product concentrations have been approximated analytically. Employing two different semi-analytical methods, Akbari-Ganji's method (AGM) and the modified Adomian decomposition method (MADM), to compute the dimensionless steady-state solutions to the system of non-linear differential equations for all values of reaction parameters. In addition, the dynamics of the mean integrated effectiveness factor of penicillin acylase in porous spherical particles have been presented for the determination of the local effectiveness factor. In order to gauge the potency of our proposed solution, we compare two semi-analytical results with a numerical result that are in good agreement across the whole concentration range. The proposed formulation aims to simulate the dynamic performance of the system utilizing the parameters and would enhance the determination of the optimum particle size for enzyme catalysts.

1. Introduction

Several factors, including pH, temperature, enzyme concentration, and substrate concentration, can influence enzyme activity. An enzyme's activity is measured by the amount of enzyme required to convert a fixed amount of substrate into a product. It can be measured by the initial phase of its progress curve [1]. Fermentation is a potential method for mass-producing enzymes for use in industry. Enzymes are synthesized via fermentation, which employs microorganisms like yeast and bacteria. Enzymes can be produced using either of two distinct fermentation processes. Submerged fermentation and fermentation on solid surfaces are two examples. Enzymes produced by microorganisms are widely used in many different industries. Enzymes from microorganisms can be genetically modified, making them a potential low-cost alternative to enzymes from plants and animals. Growing bacteria, mould, and yeast is an integral part of the fermentation process used to create microbial enzymes [2].

The significance of biological catalysis in the framework of sustainable chemical manufacturing is raised, as is the demand for enzyme immobilization as a crucial facilitating technology and for large-scale processing. Immobilization is necessary for boosting the stability and recyclability of the biocatalyst in comparison to free enzyme. The immobilization used is further impacted by the reactor configuration, such as stirred tank, fixed bed, and fluidized bed [3]. Immobilization is usually the key to boosting an enzyme's operational performance in industrial processes, particularly for use in non-aqueous media [4]. Enzyme immobilization techniques are often categorized into two distinct forms based on the technique by which the chemical process is bound through support binding or entrapment. To achieve stable attachment, covalent bonds must be formed, typically through reactions involving functional groups on the protein surface. Like non-covalent adsorption, these methods can be applied to unmodified proteins [5]. The process of immobilizing Penaeus merguiensis alkaline phosphatase (PM ALP) onto gold nanorods (GNRs) was achieved through the use of ionic exchange and hydrophobic contacts. The optimal pH and temperature for achieving maximal enzyme activity in the immobilized PM ALP have been determined to be 11.0 and 60 °C, respectively, for the hydrolysis of para-Nitrophenylphosphate (p-NPP), were discussed by Homaei. A et al. [[6], [7], [8]]. The use of immobilized enzyme-based catalytic constructs has the potential to substantially improve a wide range of industrial processes, according to their distinct catalytic activity and reaction specificity. The enhancement of key functions in the preparation of nano-enzymes and their catalytic features will result in advantageous micro-environments for biocatalysts that are significant in industrial applications [9].

Both the bulk liquid phase, which contains the substrate (and products), and the solid enzyme catalyst phase, which is where the reaction and transformation of substrate into products take place, are modeled separately during heterogeneous enzymatic catalysis in a batch reactor. Substrate mass is moved from the bulk liquid phase into the interior of the carrier, where the enzyme is immobilized during the reaction. Fick's law and the general rate equation that characterizes enzyme catalysis are used to model the mass transfer process taking place within the catalyst particles [10].

A rate-determining step is only one of several that occur in succession in immobilized (bio) catalyst systems, particularly in porous particles. There are several steps involved in film diffusion, such as pore diffusion to the active site of the substrate molecule, bio catalytic steps (s), pore diffusion to the surface of the product molecule, and film diffusion to the bulk of the product molecule. Based on the circumstances of the reaction, each transport phase could replace the bio catalytic step as the rate-determining step. Under the current conditions, determining which step is rate-limiting is crucial [11]. In a porous catalyst, the reaction rate is regulated not only by intrinsic kinetics but also by intraparticle mass and heat transport. A reaction's efficiency is measured as the ratio of the observed rate to the rate if intraparticle mass and heat transfer resistances do not exist [12]. Penicillin acylase is a rare case of an enzyme being used effectively in the industrial manufacture of a high-value drug [13,14]. Recently, penicillin acylase has gained a lot of attention because it has expanded its role from that of a simple hydrolase in the synthesis of 6-amino penicillanic acid (6APA) from penicillin G or V to that of a valuable biocatalyst in a number of different organic synthesis reactions [15].

Immobilized enzymes undertake operations in consecutive batches, with the enzyme being retrieved after a certain point of conversion has been obtained and reused until the catalyst replacement requirement has been fulfilled [16]. Besides substrate concentration, the rate-determining variables in the rate equation are the enzyme's catalytic potential, which is in turn proportional to the enzyme's concentration and the reaction rate constant. The amount of immobilized enzyme and the amount of catalyst in the reactor both influence the enzyme concentration in the reactor. The aim of the equations is to correlate the diffusion and reaction processes within the catalyst particles. For spherical enzyme catalyst particles, a flexible system of equations is derived that permits the determination of the concentration profile of substrate within the catalyst in terms of particle geometry (radius) and concentrations of substrate in the bulk liquid phase [17]. Recently, there has been a significant focus on finding the analytical solution for the case of strongly nonlinear differential equations using various analytical approaches. Exponential approaches were used by J.I. Ramos to solve the reaction diffusion equations [18]. J. He's energy balance approach [19] was used to generate approximate solutions by D. D. Ganji et al. Work on finding exact solutions to nonlinear diffusion-reaction equations, including quadratic and cubic nonlinearities, has been done by R.S. Kaushal et al. Furthermore, it is stated that analytical solutions are used to describe many physical phenomena and are very helpful in enhancing the system [20]. Analytical solutions for Cauchy reaction-diffusion problems have been obtained by M.S.H. Chowdhury and I. Hashim by HPM; they compared HPM's efficiency to that of ADM and HAM in a study published in Ref. [21]. We present in this article a semi-analytic method for solving nonlinear differential equations, both individually and in sets, which enables the quick and simple determination of the final solution of each differential equation in the form of an algebraic function. By choosing an answer function for a differential equation with constant coefficients that can be calculated by using particular initial or boundary conditions, Akbari-Ganji's method (AGM) aims to solve all nonlinear differential equations algebraically [22]. In the steady-state regime, no rigorous analytical solution to the reaction-diffusion equations has yet been presented for the concentration of substances. Using a modified Adomian decomposition method (MADM) and Akbari-Ganji's method (AGM), we obtained an analytical expression for dimensionless substrate and product concentration. Several diffusional constraints are shown, and their corresponding dynamics are described. In addition, a range of parameter values has been analyzed to determine the evolution of the mean integrated effectiveness factor response. Error analysis of expressions for substrate concentrations and effectiveness factor response by T. Praveen et al. Performance calculations have also been done for batch reactors [23].

The key objective of this work is to validate a reaction-diffusion model by comparing the numerical solution to two solutions (the AGM and the MADM). We employed both the AGM and the MADM to compute the mean integrated effectiveness factor, with the AGM approach favouring ease of calculation. When comparing the AGM result to the numerical result, the present model's graphical depiction is more consistent with the AGM results than the MADM's approach.

2. Formulation of the problem

The modeling of reaction-diffusion phenomena within immobilized enzyme catalysts considers a uniform catalyst particle size and a uniform enzyme distribution. The equations involve substrate and product concentrations, the catalytic potential of the enzyme, which depends on the enzyme concentration, and the reaction rate constant. The equations are modeled to understand the process of diffusion and reaction inside the catalyst particles, the diffusion and reaction phenomena inside the catalyst particles are modeled to understand the dynamics of reaction-diffusion. The heterogeneous nature of catalysis causes substrate and product concentration profiles inside the catalyst particle. Reaction time (t) and radial distance within the enzyme catalyst particle (r) are the variables to be considered for modeling. Now, considering a steady-state regime, time is no longer a variable, and the concentration profiles are plotted against the radial distance in Fig. 1.

Fig. 1.

Graphical representation of immobilized enzyme on a spherical catalyst particles.

The model equations have been developed with the diffusion and reaction components. The diffusion component was modeled with Fick's law, and the kinetic component was modeled considering uncompetitive inhibition by penicillin G (Pen-G), competitive inhibition by phenyl acetic acid (PAA), and non-competitive inhibition by 6-aminopenicillanic acid (6-APA) [17], [23–24]. This hypothesis is formulated on the premise that greater the volume of immobilization, the more will be overall efficacy of the result obtained using Pen-G. The chemical structure of the reactant Pen-G hydrolysis gives the product 1 as PAA and product 2 as 6-APA, as depicted in the following mechanism.

The governing reaction diffusion equations are as follows [17,24]:

$$\frac{\partial S}{\partial t}=D_e\left(\frac{{\partial }^{2}S}{\partial r^2}+\frac{2}{r}\frac{\partial S}{\partial r}\right)-v$$ (1)

$$\frac{\partial P_1}{\partial t}=D_{e1}\left(\frac{{\partial }^2{P}_1}{\partial r^2}+\frac{2}{r}\frac{\partial P_1}{\partial r}\right)+v$$ (2)

$$\frac{\partial P_2}{\partial t}=D_{e2}\left(\frac{{\partial }^2{P}_2}{\partial r^2}+\frac{2}{r}\frac{\partial P_2}{\partial r}\right)+v$$ (3)

$$v=\frac{kES}{K\left[1+\left(S/K\right)+\left(S^2/K{K}_S\right)+\left(P_1/K_1\right)+\left(P_2/K_2\right)+\left(S{P}_2/K{K}_2\right)+\left(P_1/K_1\right)\left(P_2/K_2\right)\right]}$$ (4)

Where $S$ is the Pen-G concentration inside biocatalyst, $P_1$ is the PAA concentration, $P_2$ is the 6-APA concentration, $k$ the reaction rate constant, $E$ the enzyme concentration inside the catalyst, $K$ substrate Michaelis constant, $K_S$ is the Pen-G inhibition constant, $K_1$ and $K_2$ are PAA, and 6-APA inhibition constants respectively, $r$ is the radius inside the biocatalyst particle, $D_e,D_{e1},D_{e2}$ are the effective diffusion coefficients for Pen-G, PAA, and 6-APA respectively. Substrate diffusion occurs inside the catalyst particles in the liquid-filled cavities of the pores. This complex interaction is modeled by considering an overall diffusion process, which is controlled by a coefficient $D_e$, known as the effective diffusion coefficient, which considers the porous shape of the support were discussed by Valencia P et al. [17,25]. A property of both the solute and the solvent, $D_e$ can be experimentally determined [26]. The values for the parameters involved in the equations (1)–(4) are listed in Table 9 [17,24].

Table 9.

Kinetic and diffusional constants for penicillin G hydrolysis model [17,24].

Substance	Parameter	Value	Units
Enzyme	k	43	1/s
Penicillin G	K	0.13	mM
Penicillin G	Ks	821	mM
PAA	K1	1.82	mM
APA	K2	48	mM
Penicillin G	De	5.30 × 10−10	m2/s
PAA	De1	7.33 × 10−10	m2/s
APA	De2	5.89 × 10−10	m2/s

In order to study a reaction-diffusion model, the following assumptions are made:

• i)There is no concentration dependent interaction between the support and reactants (constant diffusivity); ii) isothermal conditions are present during reaction; iii) reactants are transported through the porous support by diffusion according to Fick's law; iv) enzyme is uniformly distributed inside catalyst particle; and v) there is no external diffusional restrictions.

The initial and boundary conditions for the equations (1)–(3) are as follows:

The initial concentrations of substrate and both products inside the catalyst particle are:

$$S\left(r,0\right)=S_{b0}$$ (5)

$$P_i\left(r,0\right)=0,i=\mathrm{1,2}$$ (6)

The boundary condition for substrate and both products concentration on a spherical particle at $r=0$ are:

$$\frac{\partial S}{\partial r}\left(0,t\right)=0$$ (7)

$$\frac{\partial P_i}{\partial r}\left(0,t\right)=0,i=\mathrm{1,2}$$ (8)

This condition reflects that there is no substrate nor products flux through the center of the sphere. For all t > 0, concentration at the catalyst surface (r = R) is:

$$S\left(R,t\right)=S_b$$ (9)

$$P_i\left(R,t\right)=P_{bi},i=\mathrm{1,2}$$ (10)

The above system of equations can be converted into the dimensionless form by defining the following variables:

$$\begin{array}{l}{\mathrm{\Phi }}_1=R\sqrt{\frac{k\cdot E}{K\cdot D_e}},{\mathrm{\Phi }}_2=R\sqrt{\frac{k\cdot E}{K_1\cdot D_{e1}}},{\mathrm{\Phi }}_3=R\sqrt{\frac{k\cdot E}{K_2\cdot D_{e2}}},\alpha =\frac{K}{K_S},{\beta }_b=\frac{S_b}{K},\\ {\gamma }_{1b}=\frac{P_{b1}}{K_1},{\gamma }_{2b}=\frac{P_{b2}}{K_2},\beta =\frac{S}{K},{\gamma }_1=\frac{P_1}{K_1},{\gamma }_2=\frac{P_2}{K_2},x=\frac{r}{R}\end{array}$$ (11)

Here the typical dimensionless concentration are denoted as $\beta ,{\gamma }_1$ and ${\gamma }_2$ for $S,P_1$ and $P_2$ respectively. $x$ is the dimensionless radial distance, ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2$ and ${\mathrm{\Phi }}_3$ are the Thiele modulie. The dimensionless form of the equations (1)–(3) for the steady-state condition is as follows:

$$\frac{d^2\beta }{d{x}^2}+\frac{2}{x}\frac{d\beta }{dx}={\mathrm{\Phi }}_1^{2}N$$ (12)

$$\frac{d^2{\gamma }_1}{d{x}^2}+\frac{2}{x}\frac{d{\gamma }_1}{dx}=-{\mathrm{\Phi }}_2^{2}N$$ (13)

$$\frac{d^2{\gamma }_2}{d{x}^2}+\frac{2}{x}\frac{d{\gamma }_2}{dx}=-{\mathrm{\Phi }}_3^{2}N$$ (14)

where

$$N=\frac{\beta }{1+\beta +\alpha {\beta }^2+{\gamma }_1+{\gamma }_2+\beta {\gamma }_2+{\gamma }_1{\gamma }_2}\text{.}$$ (15)

The boundary conditions for the system of non-dimensional equations (12)–(14) are as follows:

$$\frac{d\beta }{dx}=\frac{d{\gamma }_1}{dx}=\frac{d{\gamma }_2}{dx}=0\text{; when }x=0$$ (16)

$$\beta ={\beta }_b,{\gamma }_1={\gamma }_{1b},{\gamma }_2={\gamma }_{2b};\text{when }x=1$$ (17)

The relation between $\beta$, ${\gamma }_1$ and $\beta$, ${\gamma }_2$ are as follows

$$\beta =\frac{{\mathrm{\Phi }}_1^2}{{\mathrm{\Phi }}_j^2}\left({\gamma }_{ib}-{\gamma }_i\right)+{\beta }_b,\text{where }i=1,2\text{and }j=i+1$$ (18)

Using Eq. (18), the relation among $\beta$, ${\gamma }_1$ and ${\gamma }_2$ is obtained in Eq. (19)

$$\frac{2\beta }{{\mathrm{\Phi }}_1^2}+\frac{{\gamma }_1}{{\mathrm{\Phi }}_2^2}+\frac{{\gamma }_2}{{\mathrm{\Phi }}_3^2}=\frac{2{\beta }_b}{{\mathrm{\Phi }}_1^2}+\frac{{\gamma }_{1b}}{{\mathrm{\Phi }}_2^2}+\frac{{\gamma }_{2b}}{{\mathrm{\Phi }}_3^2}$$ (19)

3. Materials and methods

In recent years, it has become a major research focus to develop approximate analytical methods for the reaction diffusion equations to solve nonlinear partial differential equations that are both simple and accurate. These methods include He's variation iteration method [27], Akbari-Ganji's method [22], homotopy perturbation method [28], Adomian decomposition method [[29], [30], [31], [32]], modified Adomian decomposition method [33], homotopy analysis method [34], new improved generalized decomposition method [35], Duan-Rach modified Adomian decomposition method [[36], [37], [38]], as well as numerical techniques like the boundary element method [39], finite difference method [40], and finite element method [41]. The most significant advantage of choosing Akbari-Ganji's method and the modified Adomain decomposition method compared to other analytical methods is that they diminish the level of difficulty in solving nonlinear differential equations.

3.1. Akbari Ganji's method and its advantages

Catalysts are commonly employed in mass transfer chemical reactions to control the reaction rate. The catalyst's form and physical characteristics are crucial parameters in determining the differential equation that governs the chemical system. A powerful algebraic approach, Akbari-Ganji's method [22] provides semi-analytic approximation solutions to nonlinear differential equations, as mentioned in the introduction. Solutions are given in the form of convergent series, and linearization is not necessary in this method.

3.1.1. Analytical expression of dimensionless concentration of substrate and products under steady state condition using Akbari Ganji's method

In this paper, the Akbari-Ganji's method (see Appendix A) is used to solve the system of non-linear differential equations. The closed form of analytical expression of dimensionless concentrations (see Appendix B) of substrate $\beta \left(x\right)$ and products ${\gamma }_1\left(x\right)$ and ${\gamma }_2\left(x\right)$ are

$$\beta \left(x\right)=\frac{{\beta }_b\sinh \left(\frac{{\mathrm{\Phi }}_{1}x}{\sqrt{1+{\beta }_b+{\gamma }_{1b}+{\gamma }_{2b}+\alpha {\beta }_b^2+{\beta }_b{\gamma }_{2b}+{\gamma }_{1b}{\gamma }_{2b}}}\right)}{x\sinh \left(\frac{{\mathrm{\Phi }}_1}{\sqrt{1+{\beta }_b+{\gamma }_{1b}+{\gamma }_{2b}+\alpha {\beta }_b^2+{\beta }_b{\gamma }_{2b}+{\gamma }_{1b}{\gamma }_{2b}}}\right)}$$ (20)

Using the relation Eq. (18), the product concentrations ${\gamma }_1\left(x\right)$ and ${\gamma }_2\left(x\right)$ are obtained as follows

$${\gamma }_i\left(x\right)={\gamma }_{ib}-\left[{\beta }_b\left(\frac{\sinh \left(\frac{{\mathrm{\Phi }}_{1}x}{\sqrt{1+{\beta }_b+{\gamma }_{1b}+{\gamma }_{2b}+\alpha {\beta }_b^2+{\beta }_b{\gamma }_{2b}+{\gamma }_{1b}{\gamma }_{2b}}}\right)}{x\sinh \left(\frac{{\mathrm{\Phi }}_1}{\sqrt{1+{\beta }_b+{\gamma }_{1b}+{\gamma }_{2b}+\alpha {\beta }_b^2+{\beta }_b{\gamma }_{2b}+{\gamma }_{1b}{\gamma }_{2b}}}\right)}-1\right)\right]\cdot \left[\frac{{\mathrm{\Phi }}_j^2}{{\mathrm{\Phi }}_1^2}\right]\text{,}$$

$$\text{where }i=\mathrm{1,2}\text{and }j=i+1$$ (21)

The Eqs. (20) and (21) satisfy the boundary conditions (16)–(17). These equations represent the new analytical expression of the concentration of substrate and products for all possible values of the parameters ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3,\alpha ,{\beta }_b,{\gamma }_{1b}$ and ${\gamma }_{2b}$. The solution of ${\gamma }_1$ and ${\gamma }_2$ are same except the bulk concentration and Thiele modulus. Thus, product concentrations behave identically depending on diffusional limitations.

3.2. Modified Adomian decomposition method and its advantages

In this work, the modified Adomian decomposition method is adopted to obtain the analytical solution for the above Eqs. (12)–(14) for the conditions (16)–(17). In recent years, many authors handled MADM to solve strongly non-linear problems. MADM provides more realistic series solutions that generally converge very rapidly in real physical problems. The rapid convergence shows that this method is reliable [33,37]. The constructive procedure and choice of handling the operator are provided by M. M. Hosseini and M. Jafari [[42], [43]]. The decomposition method is straightforward, without restrictive assumptions, and the components of the series solution can be easily computed using any mathematical symbolic package [44].

3.2.1. Solution of dimensionless concentration of substrate and products under steady state condition using modified Adomian decomposition method

We have also employed the modified Adomian decomposition method (see Appendix C) to solve the system of non-linear differential equations. The simple and closed form of analytical expression of dimensionless concentrations (see Appendix D) of substrate $\beta \left(x\right)$ and products ${\gamma }_1\left(x\right)$ and ${\gamma }_2\left(x\right)$ are

$$\beta \left(x\right)={\beta }_b+A\cdot \left[{\mathrm{\Phi }}_1^2\left(x^2-1\right)\right]$$ (22)

$$A=\frac{1}{6}\left[\frac{{\beta }_b}{1+{\beta }_b+\alpha {\beta }_b^2+{\gamma }_{1b}+{\gamma }_{2b}+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{2b}}\right]$$ (23)

Using the relation Eq. (18), the product concentrations ${\gamma }_1\left(x\right)$ and ${\gamma }_2\left(x\right)$ are obtained as follows

$$\begin{gathered} {\gamma }_i\left(x\right)={\gamma }_{ib}-\left[\beta \left(x\right)-{\beta }_b\right]\cdot \left[\frac{{\mathrm{\Phi }}_j^2}{{\mathrm{\Phi }}_1^2}\right], \\ \text{where }i=\mathrm{1,2}\text{and }j=i+1 \end{gathered}$$ (24)

The Eqs. (22)–(24) satisfy the boundary conditions (16)–(17). These equations represent the new analytical expression of the concentration of substrate and products for all possible values of the parameters ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3,\alpha ,{\beta }_b,{\gamma }_{1b}$ and ${\gamma }_{2b}$. The solution of ${\gamma }_1$ and ${\gamma }_2$ are same except the bulk concentration and Thiele modulus. Therefore behaviors of the both product concentrations are almost same, varies with the diffusional restrictions.

3.2.2. Solution with degree of conversion

Consider a degree of conversion for the concentration of substrate and products respectively as,

$$X=\frac{{\beta }_{b0}-{\beta }_b}{{\beta }_{b0}};X=\frac{P_{1b}}{S_{b0}}=\frac{P_{2b}}{S_{b0}}=\frac{{\gamma }_{1b}{K}_1}{{\beta }_{b0}K}=\frac{{\gamma }_{2b}{K}_2}{{\beta }_{b0}K}$$ (25)

Bulk concentrations in function of degree of conversion:

$${\beta }_b={\beta }_{b0}\left(1-X\right);{\gamma }_{1b}=\frac{{\beta }_{b0}K}{K_1}X;\text{and }{\gamma }_{2b}=\frac{{\beta }_{b0}K}{K_2}X$$ (26)

Replacing in Eqs. (20), (22), we get the concentrations inside catalyst as

$$\beta \left(x\right)={\beta }_{b0}\left(1-X\right)+A{\mathrm{\Phi }}_1^2\left(x^2-1\right)$$ (27)

$${\gamma }_1\left(x\right)=\frac{{\beta }_{b0}K}{K_1}X+A{\mathrm{\Phi }}_2^2\left(1-x^2\right)$$ (28)

$${\gamma }_2\left(x\right)=\frac{{\beta }_{b0}K}{K_2}X+A{\mathrm{\Phi }}_3^2\left(1-x^2\right)$$ (29)

$$A=\frac{1}{6}\left[\frac{{\beta }_{b0}\left(1-X\right)}{1+{\beta }_{b0}\left(1-X\right)+\alpha {\left[{\beta }_{b0}\left(1-X\right)\right]}^2+\frac{{\beta }_{b0}K}{K_1}X+\frac{{\beta }_{b0}K}{K_2}X+\frac{{\beta }_{b0}K}{K_2}X{\beta }_{b0}\left(1-X\right)+\frac{{\beta }_{b0}K}{K_1}X\frac{{\beta }_{b0}K}{K_2}X}\right]$$ (30)

The limitation of the proposed solution for substrate concentration $\beta \left(x\right)$ is given as follows by the following constrain:

$$6\left(\begin{array}{l}{K}_1{K}_2+{\beta }_{b0}\left(1-X\right)K_1{K}_2+\alpha {\beta }_{b0}^2{\left(1-X\right)}^2{K}_1{K}_2+\\ {\beta }_{b0}XK{K}_2+{\beta }_{b0}XK{K}_1+{\beta }_{b0}X\left(1-X\right)K{K}_1+{\beta }_{b0}^2{X}^2{K}^2\end{array}\right)-{\mathrm{\Phi }}_1^2{K}_1{K}_2>0$$ (31)

Based on the combination of ${\beta }_{b0}$ and ${\mathrm{\Phi }}_1$ as determined by Eq. (31), Fig. 9(a and b) illustrates the acceptable and unacceptable areas of the proposed solution. The white zone denotes the valid solution region, while the red zone denotes the invalid solution region.

Fig. 9.

Limitation of the proposed solution using (MADM, Eq. (31)) at different degree of conversions $X$(a) 0 and (b) 0.9 where valid region (depicts the white zone) and invalid region (depicts the red zone).

4. The global integral effectiveness factor is as follows

From the present study, we can investigate the effectiveness of a catalyst or reactor in promoting a chemical reaction, considering the influence of reaction kinetics and mass transport (diffusion) within the system. Additionally, it assesses the efficiency of substrate transportation to the catalyst and the removal of products from it. The usage of this technique facilitates the optimization of reaction speeds while simultaneously minimizing the occurrence of inefficient side reactions. Its value can be used to determine the potential of a particular reactor or catalyst for performing the intended chemical conversion. For a spherical catalyst particle, the global integral effectiveness factor is [45].

$${\eta }^{\prime }=3\underset{0}{\overset{1}{\int }}\eta \cdot x^2\cdot dx$$ (32)

The local effectiveness factor depends on the reaction rate equation. The general definition is:

$$\begin{gathered} \eta =\frac{\text{reaction}\text{rate}\text{inside}\text{catalyst}}{\text{reaction}\text{rate}\text{at}\text{catalyst}\text{surface} \\ \text{conditions}\left(\text{bulk} \\ \text{conditions}\right)}=\frac{v\left(S\left(r\right),P\left(r\right)\right)}{v\left(S_b,P_b\right)} \end{gathered}$$ (33)

For the penicillin acylase reaction-diffusion equation, the local effectiveness factor is obtained by replacing N in the above definition.

$$\eta =\frac{\beta \left(1+{\beta }_b+\alpha {{\beta }_b}^2+{\gamma }_{1b}+{\gamma }_{2b}+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{2b}\right)}{{\beta }_b\left(1+\beta +\alpha {\beta }^2+{\gamma }_1+{\gamma }_2+{\gamma }_2\beta +{\gamma }_1{\gamma }_2\right)}$$ (34)

4.1. Mean integrated effectiveness factor for the concentration of substrate using AGM solution

The mean integrated effectiveness factor obtained using Eqs. (32), (34) is as follows:

$${\eta }^{\prime }=\left(\frac{3}{{\mathrm{\Lambda }}^2}\right)\left[\mathrm{\Lambda }\coth \left(\mathrm{\Lambda }\right)-1\right],\text{where}\mathrm{\Lambda }=\frac{{\mathrm{\Phi }}_1}{\sqrt{1+{\beta }_b+{\gamma }_{1b}+{\gamma }_{2b}+\alpha {\beta }_b^2+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{1b}}}$$ (35)

The dynamics of the mean integrated effectiveness factor is provided in the Table 7(a - d).

Table 7.

Mean integrated effectiveness factor for uncompetitive substrate inhibition by Pen-G ${\eta }^{\prime }$ against the initial bulk concentration ${\beta }_{b0}$ (a) 0.1 (b) 1 (c) 10 and (d) 100 using (AGM, Eq. (35)) for various values of Thiele modulus ${\mathrm{\Phi }}_1$ and $\left({\mathrm{\Phi }}_2={\mathrm{\Phi }}_3=0.1,\alpha =0.000158\right)$.

${\beta }_{b0}=0.1$		${\beta }_{b0}=1$		${\beta }_{b0}=10$		${\beta }_{b0}=100$
${\mathrm{\Phi }}_1$	${\eta }^{\prime }$	${\mathrm{\Phi }}_1$	${\eta }^{\prime }$	${\mathrm{\Phi }}_1$	${\eta }^{\prime }$	${\mathrm{\Phi }}_1$	${\eta }^{\prime }$
0.1	0.9995	0.1	0.9999	1	0.9997	1	1
0.5	0.9876	1	0.9891	3	0.9974	20	0.9987
0.8	0.9691	2	0.9582	6	0.9898	40	0.9948
1	0.9529	2.4	0.9413	8.5	0.9798	55	0.9902

4.2. Mean integrated effectiveness factor for the concentration of substrate using MADM solution

The mean integrated effectiveness factor obtained using Eqs. (32), (34) is as follows:

$${\eta }^{\prime }=\frac{1}{12}\frac{\left\{\begin{array}{l}\sqrt{DE-2E^2-Q_1}\left[\begin{array}{l}\sqrt{2}{Q}_3{A}_{1}DE-\sqrt{2}{Q}_{3}B{D}^2+2\sqrt{2}{Q}_{3}BDE-\sqrt{2}{Q}_{3}BD\sqrt{D^2-4EC}-2\sqrt{2}{Q}_3{E}^2{A}_1\\ +2\sqrt{2}{Q}_{3}BE\sqrt{D^2-4EC}+2\sqrt{2}{Q}_{3}BCE+\sqrt{2}{Q}_3{A}_{1}E\sqrt{D^2-4EC}-2\sqrt{2}{Q}_3{E}^{2}B\end{array}\right]+\\ \sqrt{DE-2E^2+Q_1}\left[\begin{array}{l}\sqrt{2}{Q}_{2}B{D}^2-2\sqrt{2}{Q}_{2}BDE-\sqrt{2}{Q}_{2}BD\sqrt{D^2-4EC}-\sqrt{2}{Q}_2{A}_{1}DE+2\sqrt{2}{Q}_2{E}^{2}B\\ +\sqrt{2}{Q}_2{A}_{1}E\sqrt{D^2-4EC}-2\sqrt{2}{Q}_{2}BCE+2\sqrt{2}{Q}_{2}BE\sqrt{D^2-4EC}+2\sqrt{2}{Q}_2{E}^2{A}_1\end{array}\right]+\\ 2B\sqrt{D^2-4EC}\sqrt{DE-2E^2-Q_1}\sqrt{DE-2E^2+Q_1}\end{array}\right\}}{E\sqrt{D^2-4EC}\sqrt{DE-2E^2-Q_1}\sqrt{DE-2E^2+Q_1}}$$ (36)

where $Q_1=E\sqrt{D^2-4EC}$; $Q_2={\tan }^{-1}\left[\frac{\sqrt{2}E}{\sqrt{D-2E-E\sqrt{D^2-4EC}}}\right]$; $Q_3={\tan }^{-1}\left[\frac{\sqrt{2}E}{\sqrt{D-2E+E\sqrt{D^2-4EC}}}\right]$; $A_1=\left[18\left(1+{\beta }_b+\alpha {{\beta }_b}^2+{\gamma }_{1b}+{\gamma }_{2b}+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{2b}\right)-3{\mathrm{\Phi }}_1^2\right]$; $B=3{\mathrm{\Phi }}_1^2$; $C=\left(1+{\beta }_b+\alpha {{\beta }_b}^2+{\gamma }_{1b}+{\gamma }_{2b}+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{2b}\right)$;

$$\begin{gathered} D=\left(M_1+2\alpha {\beta }_b{M}_1+{\gamma }_{2b}{M}_1-M_2-M_3-{\beta }_b{M}_3-M_3{\gamma }_{1b}-M_1{\gamma }_{2b}\right);E=\left(\alpha M_1^2+M_2{M}_3-M_1{M}_3\right); \\ {M}_i=\frac{{\mathrm{\Phi }}_i^2{\beta }_b}{6\left(1+{\beta }_b+\alpha {\beta }_b^2+{\gamma }_{1b}+{\gamma }_{2b}+{\gamma }_{2b}{\beta }_b+{\gamma }_{1b}{\gamma }_{2b}\right)};i=\mathrm{1,2,3}; \\ {\beta }_b={\beta }_{b0}\left(1-X\right);{\gamma }_{1b}=\frac{{\beta }_{b0}K}{K_1}X;\text{and }{\gamma }_{2b}=\frac{{\beta }_{b0}K}{K_2}X \end{gathered}$$ (37)

when comparing the MADM with the AGM for calculating the mean integrated effectiveness factor, the AGM provides a simpler computation, whereas the MADM gives a more comprehensive mathematical expression. The dynamics of the mean integrated effectiveness factor is provided in the Table 8(a - c).

Table 8.

Mean integrated effectiveness factor for uncompetitive substrate inhibition by Pen-G for ${\eta }^{\prime }$ against the initial bulk concentration ${\beta }_{b0}$ (a) 0.1 (b) 1 and (c) 10 using (MADM, Eq. (36)), for various values of Thiele modulus ${\mathrm{\Phi }}_1$ and $\left({\mathrm{\Phi }}_2={\mathrm{\Phi }}_3=0.1,\alpha =0.000158,X=0,K=0.13,K_1=1.82,K_2=48\right)$.

${\beta }_{b0}=0.1$		${\beta }_{b0}=1$		${\beta }_{b0}=10$
${\mathrm{\Phi }}_1$	${\eta }^{\prime }$	${\mathrm{\Phi }}_1$	${\eta }^{\prime }$	${\mathrm{\Phi }}_1$	${\eta }^{\prime }$
0.2	0.9977	0.2	0.9988	0.2	0.9993
1	0.9475	1	0.9824	3	0.9941
2	0.8071	3.5	0.7043	6	0.9714
3.4	0.3208	4.6	0.1119	8.5	0.7545

5. Results and discussion

The concentration of substrate (Pen-G) and products (PAA, 6-APA, respectively) depends upon the Thiele modulus ${\mathrm{\Phi }}_i;i=\mathrm{1,2,3}$. In addition, we analyze the behavior of graphs depicting the relationship between dimensionless substrate concentration $\beta$ and product concentrations ${\gamma }_i;i=\mathrm{1,2}$ over dimensionless radial distance $x$, as a function of the Thiele modulus $\mathrm{\Phi }$ for various values of ${\beta }_{b0}$. The graphs based on Fig. 2(a–d), 3 (a - d), and 4 (a - d) represent the mechanisms of uncompetitive substrate inhibition by Pen-G, competitive product inhibition by PAA, and non-competitive product inhibition by 6-APA (see Table 1, Table 2).

Fig. 2.

Comparison of dimensionless substrate concentration $\beta$versus dimensionless radial distance$x$with numerical simulation results (FDM) for various diffusional restrictions${\mathrm{\Phi }}_1$and different initial bulk substrate concentration${\beta }_{b0}$(a) 0.1 (b) 1 (c) 10 and (d) 100 using(12), (20), (22).

Table 1.

(a) Comparison of dimensionless uncompetitive inhibition by Pen-G concentration of substrate $\beta$ (Eq. (20), (22)) and numerical simulation against the dimensionless radial distance $x$ for different values of Thiele modulus ${\mathrm{\Phi }}_1$ and (${\beta }_{b0}={\gamma }_{1b}={\gamma }_{2b}=100\text{,}\alpha =0.000158$).

${\mathrm{\Phi }}_1=0.1$						${\mathrm{\Phi }}_1=1$
X	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)
0.0	100	100	100	0	0	99.9992	99.9991	99.9991	0.0001	0
0.2	100	100	100	0	0	99.9992	99.9992	99.9992	0	0
0.4	100	100	100	0	0	99.9993	99.9993	99.9993	0	0
0.6	100	100	100	0	0	99.9995	99.9994	99.9994	0.0001	0
0.8	100	100	100	0	0	99.9997	99.9997	99.9997	0	0
1.0	100	100	100	0	0	100	100	100	0	0
Average % of Error				0	0	Average % of Error			0.00003	0
(b)
${\mathrm{\Phi }}_1=5$						${\mathrm{\Phi }}_1=10$
X	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)
0.0	99.979	99.979	99.975	0.004	0.004	99.92	99.92	99.92	0	0
0.2	99.982	99.981	99.981	0.0001	0	99.92	99.92	99.92	0	0
0.4	99.983	99.983	99.983	0	0	99.93	99.93	99.93	0	0
0.6	99.987	99.987	99.987	0	0	99.95	99.95	99.95	0	0
0.8	99.994	99.994	99.994	0	0	99.97	99.97	99.97	0	0
1.0	100	100	100	0	0	100	100	100	0	0
Average % of Error				0.0007	0.0006	Average % of Error			0	0

Table 2.

(a) Comparison of dimensionless uncompetitive inhibition by Pen-G concentration of substrate ${\gamma }_1$ (Eq. (21), (24)) and numerical simulation against the dimensionless radial distance $x$ for different values of Thiele modulus ${\mathrm{\Phi }}_2$ and (${\beta }_{b0}={\gamma }_{1b}={\gamma }_{2b}=100\text{,}{\mathrm{\Phi }}_1=0.1\text{,}\alpha =0.000158$).

${\mathrm{\Phi }}_2=0.1$						${\mathrm{\Phi }}_2=1$
X	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)
0.0	100	100	100	0	0	100	100.0008	100.0008	0.0008	0
0.2	100	100	100	0	0	100	100.0008	100.0008	0.0008	0
0.4	100	100	100	0	0	100	100.0007	100.0007	0.0007	0
0.6	100	100	100	0	0	100	100.0005	100.0005	0.0005	0
0.8	100	100	100	0	0	100	100.0003	100.0003	0.0003	0
1.0	100	100	100	0	0	100	100	100	0	0
Average % of Error				0	0	Average % of Error			0.00052	0
(b)
${\mathrm{\Phi }}_2=5$						${\mathrm{\Phi }}_2=10$
X	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)	AGM (20)	MADM (22)	Numerical (FDM)	% of Error (20)	% of Error (22)
0.0	100	100.020	100.020	0.0199	0	100	100.08	100.08	0.0799	0
0.2	100	100.019	100.019	0.0189	0	100	100.08	100.08	0.0799	0
0.4	100	100.017	100.017	0.0169	0	100	100.07	100.07	0.0699	0
0.6	100	100.014	100.013	0.0129	0.0009	100	100.05	100.05	0.0499	0
0.8	100	100.007	100.007	0.0007	0	100	100.03	100.03	0.0299	0
1.0	100	100	100	0	0	100	100	100	0	0
Average % of Error				0.0116	0.0002	Average % of Error			0.0516	0

5.1. Effect of thiele modulus ${\mathrm{\Phi }}_1=R\sqrt{\frac{kE}{K{D}_e}}$ for the concentration of uncompetitive substrate inhibition

The concentration of substrate within the catalyst decreases in all cases (a) – (d) as the Thiele modulus increases, as depicted in Fig. 2(a–d), and shown in Table 3, Table 5. Moreover, the effectiveness factor ${\eta }^{\prime }$ will decrease in all cases, referring to Fig. 5(a–d). The observation signifies that an increase in substrate concentration in the bulk ${\beta }_{b0}$ leads to a decrease in diffusional restriction $\left(\mathrm{i}.\mathrm{e}.{\mathrm{\Phi }}_1\ll 0.1\Rightarrow \frac{\beta }{{\beta }_{b0}}\to 1\right)$. For all ${\beta }_{b0}$ values with a lower Thiele modulus have the highest effectiveness factor, which is tabulated in Table 7(a - d) and 8 (a - c), i.e., $\left(\forall {\beta }_{b0}\left({\mathrm{\Phi }}_1\ll 0.1\Rightarrow {\eta }^{\prime }\to 1\right)\right)$ which indicates that the catalyst is highly efficient. As diffusional restriction increases, the effectiveness factor begins to reduce. Diffusional control governs the response when the Thiele modulus is high $\left({\mathrm{\Phi }}_1\gg 0.1\right)$. This phenomenon is observed under conditions of intense catalytic activity with minimal reaction rate constants.

Table 3.

Concentration of uncompetitive substrate inhibition by Pen-G $\beta$ (AGM, Eq. (20)) against the dimensionless radial distance $x$ for various values of Thiele modulus ${\mathrm{\Phi }}_1$ and ($\text{at}x=0\text{,}\alpha =0.000158$).

${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=0.1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=10$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=100$
${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$
0.1	0.09987	0.1	0.9997	0.1	10.000	1	100.00
0.3	0.09887	1	0.9728	1	9.993	10	99.92
0.5	0.09691	1.5	0.9401	3	9.935	20	99.67
0.8	0.09235	2	0.8970	6	9.745	40	98.70
1	0.08841	2.4	0.8563	8.5	9.497	55	97.56

Table 5.

Concentration of uncompetitive substrate inhibition by Pen-G $\beta$ (MADM, Eq. (22)) against the dimensionless radial distance $x$ for various values of Thiele modulus ${\mathrm{\Phi }}_1$ and ($\text{at}x=0\text{,}\alpha =0.000158$).

${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=0.1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=10$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=100$
${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$	${\mathrm{\Phi }}_1$	$\beta$
0.1	0.09987	0.1	0.9997	0.1	10.000	1	100.00
0.3	0.09886	1	0.9722	1	9.993	10	99.92
0.5	0.09684	1.5	0.9375	3	9.935	20	99.67
0.8	0.09192	2	0.8889	6	9.740	40	98.69
1	0.08737	2.4	0.8400	8.5	9.479	55	97.52

Fig. 5.

Dynamics of effectiveness factor during batch reactor operation ${\eta }^{\prime }$versus dimensionless substrate concentration${\beta }_{b0}$with different diffusional restrictions${\mathrm{\Phi }}_1$for different initial bulk substrate concentration${\beta }_{b0}$(a) 0.1 (b) 1 (c) 10 and (d) 100 using (AGM,Eq. (35)).

5.2. Effect of thiele modulus ${\mathrm{\Phi }}_2=R\sqrt{\frac{kE}{K_1{D}_{e_1}}}$ for the concentration of competitive product inhibition

As the concentration of competitive product inhibition drops, the Thiele modulus ${\mathrm{\Phi }}_2$ is also minimally driven in all cases (a)–(d), as depicted in Fig. 3(a–d), and in the Table 4, Table 6. The majorities of the reactant concentration and the Thiele modulus are predominantly influenced by the reaction kinetics and the diffusivity characteristics within the catalyst particle. As the concentration of the competitive product rises, it begins to retard the reaction by occupying the active sites on the surface of the catalyst. As a result, there is a reduction in the value of the reaction rate constant ‘k', leading to an appropriate decrease in the Thiele modulus ${\mathrm{\Phi }}_2$. The inhibitory effect can be intensified by a higher Thiele modulus in the presence of competitive product inhibition. The reaction rate may be limited due to the reduced availability of the reactant at the catalyst surface, which is caused by the concentration gradient of the reactant resulting from internal diffusion resistance.

Fig. 3.

Comparison of dimensionless product concentration${\gamma }_1$versus dimensionless radial distance$\mathit{x}$with numerical simulation results (FDM) for various diffusional restrictions${\mathrm{\Phi }}_2$and different initial bulk product concentration${\gamma }_{1b}$(a) 0.1 (b) 1 (c) 10 and (d) 100 usingEq. (13),(21)and(24).

Table 4.

Concentration of competitive product inhibition by PAA and non-competitive product inhibition by 6-APA (${\gamma }_1,{\gamma }_2$) (AGM, Eq. (21)) against the dimensionless radial distance $x$ for various values of Thiele moduli ${\mathrm{\Phi }}_2$, ${\mathrm{\Phi }}_3$ and ($\text{at}x=0\text{,}\alpha =0.000158\text{,}{\mathrm{\Phi }}_1=0.1$).

${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=0.1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=10$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=100$
${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$
0.1	0.1001	0.1	1	1	10.01	1	100
1	0.1126	1	1.028	3	10.06	25	100.5
2	0.1505	2	1.111	7	10.35	45	101.7
2.5	0.1788	3.4	1.321	10	10.72	60	103

Table 6.

Concentration of competitive product inhibition by PAA and non-competitive product inhibition by 6-APA (${\gamma }_1,{\gamma }_2$) (MADM, Eq. (24)) against the dimensionless radial distance $x$ for various values of Thiele moduli ${\mathrm{\Phi }}_2$, ${\mathrm{\Phi }}_3$ and $\left(\text{at}x=0,\alpha =0.000158,{\mathrm{\Phi }}_1=0.1\right)$.

${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=0.1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=1$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=10$		${\beta }_{bo}={\gamma }_{1b}={\gamma }_{2b}=100$
${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$	${\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$	${\gamma }_1,{\gamma }_2$
0.1	0.1001	0.1	1	1	10.01	1	100
1	0.1126	1	1.028	3	10.06	25	100.5
2	0.1505	2	1.111	7	10.35	45	101.7
2.5	0.1789	3.4	1.321	10	10.72	60	103

5.3. Effect of thiele modulus ${\mathrm{\Phi }}_3=R\sqrt{\frac{kE}{K_2{D}_{e_2}}}$ for the concentration of non-competitive product inhibition

A decrease in the Thiele modulus ${\mathrm{\Phi }}_3$ entails a maximum reaction rate relative to the diffusion rate. Given the Thiele modulus ${\mathrm{\Phi }}_3\approx 0.1\left(\text{or}\right)1$, it can be inferred that the rates of reaction and diffusion are of similar magnitude. The Thiele modulus ${\mathrm{\Phi }}_3$ may have an impact on the concentration of the non-competitive product inhibitor. The concentration of the inhibitor may become monotone depending on the interplay between reaction and diffusion rate in a specific system as the Thiele modulus increases $\left({\mathrm{\Phi }}_3\gg 0.1\left(\text{or}\right)1\right)$. As represented in Fig. 4(a–d), and in the Table 4, Table 6, it is apparent that an increase in Thiele modulus triggers a corresponding increase in product concentration across all cases, which leads to the diffusion rate exceeding the reaction rate.

Fig. 4.

Comparison of dimensionless product concentration${\gamma }_2$versus dimensionless radial distance$\mathit{x}$with numerical simulation results (FDM) for various diffusional restrictions and different initial bulk product concentration${\gamma }_{2b}$(a) 0.1 (b) 1 (c) 10 and (d) 100 usingEq. (14),(21)and(24).

The flexibility of the reaction-diffusion model's solution is determined by alterations in reaction and mass transfer rates. The two phenomena are impacted by two common factors, which are the bulk substrate concentration ${\beta }_{b0}$ and the Thiele modulus ${\mathrm{\Phi }}_i;i=\mathrm{1,2,3}$. The reaction rate of each analyzed mechanism is influenced by the substrate concentration. The impact is prominent until the concentration of ${\beta }_{b0}=10$, at which point the enzyme becomes saturated with substrate and the reaction rate no longer exhibits significant revisions. However, substrate concentration has a consistent and substantial effect on mass transfer. Increasing substrate concentration diminishes mass transfer limitations driven by diffusional restrictions depicted in Fig. 6(a–d), for different degree of conversions $X=\mathrm{0,0.9}$ respectively. The Thiele modulus describes the effects, with higher values indicating greater mass transfer limitations. Inhibition decreases product accumulation within the catalyst, thereby reducing the reaction rate and substrate consumption in Fig. 7(a–d) and Fig. 8(a–d), for various degree of conversions $X=\mathrm{0,0.9}$ respectively.

Fig. 6.

Dynamics of dimensionless substrate concentration $\beta$versus dimensionless radial distance$x$with different diffusional restrictions${\mathrm{\Phi }}_1$for initial bulk substrate concentration (a)${\beta }_{b0}=10\text{,}$when degree of conversions$X=0$, (b)${\beta }_{b0}=100\text{,}$when$X=0$, (c)${\beta }_{b0}=10\text{,}$when$X=0.9$, and (d)${\beta }_{b0}=100\text{,}$when$X=0.9$, using theEq. (27)and$\alpha =0.000158\text{, }K=0.13\text{, }{K}_1=1.82\text{, }{K}_2=48$.

Fig. 7.

Dynamics of dimensionless product concentrations ${\gamma }_1$versus dimensionless radial distance$x$with different diffusional restrictions${\mathrm{\Phi }}_2$for initial bulk substrate concentration (a)${\beta }_{b0}=10\text{,}$when degree of conversions$X=0$, (b)${\beta }_{b0}=100\text{,}$when$X=0$, (c)${\beta }_{b0}=10\text{,}$when$X=0.9$, and (d)${\beta }_{b0}=100\text{,}$when$X=0.9$, using theEq. (28)and$\alpha =0.000158\text{, }K=0.13\text{, }{K}_1=1.82\text{, }{K}_2=48$.

Fig. 8.

Dynamics of dimensionless product concentration ${\gamma }_2$versus dimensionless radial distance$x$with different diffusional restrictions${\mathrm{\Phi }}_3$for initial bulk substrate concentration (a)${\beta }_{b0}=10\text{,}$when degree of conversions$X=0$, (b)${\beta }_{b0}=100\text{,}$when$X=0$, (c)${\beta }_{b0}=10\text{,}$when$X=0.9$, and (d)${\beta }_{b0}=100\text{,}$when$X=0.9$, using theEq. (29)and$\alpha =0.000158\text{, }K=0.13\text{, }{K}_1=1.82\text{, }{K}_2=48$.

The proposed MADM solution, which involves truncating the Taylor expansion, exhibits divergence at certain ${\beta }_{b0}$ and ${\mathrm{\Phi }}_1$ combinations, leading to negative substrate concentrations within the catalyst. The error affects the effectiveness factor calculations. Negative values of the effectiveness factor are obtained under conditions of significant diffusional limitations. This is due to the negative substrate concentration values obtained in the invalid solution approximation zones. The validity range of the solution is reliant upon the kinetic mechanism, and is denoted by Eq. (31). Fig. 9(a and b), depicts the restricted area (white), wherein the solutions for the mechanisms yield positive values for all possible combinations of ${\beta }_{b0}$ and ${\mathrm{\Phi }}_1$. Invalid solutions are typically observed when diffusional restrictions are significant. Fig. 9(a and b), displays the acceptable and unacceptable regions with ${\beta }_{b0}$ and ${\mathrm{\Phi }}_1$ combinations at different degree of conversion $X=\mathrm{0,0.9}$ respectively. In cases of competitive inhibition, an increase in conversion and product concentration results in a decrease in the validity zone of the solution. The observed effect is a result of elevated diffusional limitations caused by increased product inhibition.

To analyze the precision of the analytical approach utilizing a finite amount of terms, the system of differential equations ((12)–(14)) was also solved numerically by the PDE-solving function, pdepe (finite difference method), FDM [40]. Matlab/Scilab coding [46] is also mentioned in Appendix E. To show the efficiency of our proposed solution, we compared our two analytical results, the AGM equations (20) and (21) and the MADM equations (22)–(24) with the numerical simulation of equations (12)–(14). In addition, we have provided the tables for error percentage on plots in Table 1(a - b) and 2 (a - b), which demonstrate a satisfactory level of agreement. In all the cases, the concentration of substrate and products reflects average relative errors of (0.0007 % in AGM, 0.00067 % in MADM) and (0.0516 % in AGM, 0.0002 % in MADM) respectively, as determined by comparing our analytical results with numerical results. It can be inferred from our analysis that there is no significant difference between numerical and analytical solutions across a range of Thiele modulus values ${\mathrm{\Phi }}_i;i=\mathrm{1,2,3}$. The AGM solution provides a closed-form analytical solution, despite the MADM method is more comprehensive. The AGM method exhibits greater efficiency than the MADM method, as evidenced by the analysis of the graphs and tables. The MADM method is quite diverse compared to the AGM method for establishing the mean integrated effectiveness factor.

6. Conclusion

The steady state condition of immobilized enzymes (penicillin G acylase) under internal diffusional restrictions is analyzed over a wide range of parameters using a mathematical model. This paper outlines the derivation for substrate and product concentrations using two approximate semi-analytical approaches, using AGM and the MADM. The effect of the parameters on the concentration profiles and Thiele moduli's impact pertaining to the substrate and product concentrations were discussed. Our analytical solutions were compared to numerical simulations, and a satisfactory level of agreement was observed. This study analyzed the limitations of a proposed strategy at varying degrees of conversion, considering both valid and invalid regions, and identified suitable combinations of initial bulk concentration and Thiele modulus that result in dependable concentration profiles within the catalyst and effectiveness factor. The analytical expressions derived in this study enable quick computation of the impact of diffusional limits on concentration profiles within the catalyst particle and effectiveness factor values. Based on the aforementioned, it can be concluded that the solution obtained by the AGM method is better for computation. Thus, this tool plays an important role in the design and effectiveness of immobilized enzyme reactors, particularly in determining the optimal radius of catalyst particles.

Data availability statement

We have added the “Supplementary Material” in the “Attach file” section.

CRediT authorship contribution statement

L. Niyaz Ahmed: Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Formal analysis, Conceptualization. Praveen T: Writing – review & editing, Writing – original draft, Supervision, Software, Resources, Methodology, Investigation, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

List of Symbols. Nomenclature

D_e

effective diffusion coefficient of substrate (cm² s⁻¹)

D_e1

effective diffusion coefficient of PAA (cm² s⁻¹)

D_e2

effective diffusion coefficient of 6-APA (cm² s⁻¹)

E

enzyme concentration inside catalyst (mol cm⁻³)

$k$

reaction rate constant (mol cm⁻³ s⁻¹)

K₁

product inhibition constant for PAA (mol cm⁻³)

K₂

product inhibition constant for 6-APA (mol cm⁻³)

K_S

Pen G substrate inhibition constant (mol cm⁻³)

K

substrate Michaelis constant (Pen G model is present) (mol cm⁻³)

S

substrate concentration inside biocatalyst (mol cm⁻³)

P₁

concentration of product PAA (mol cm⁻³)

P₂

concentration of product 6-APA (mol cm⁻³)

S_b

bulk concentration of a substrate (mol cm⁻³ s⁻¹)

S_b0

initial bulk concentration of a substrate

P_i

concentration of products i (mol cm⁻³)

P_bi

bulk concentration of a products i (mol cm⁻³ s⁻¹)

P_b0

initial bulk concentration of a product

$r$

variable radius inside the biocatalyst particle (cm)

R

biocatalyst particle radius (cm)

$t$

reaction time (s)

X

degree of conversion

Dimensionless parameters (Greek symbols)

$\beta =\frac{S}{K}$

dimensionless concentration of a substrate

${\beta }_b=\frac{S_b}{K}$

dimensionless concentration of a substrate outside the support

${\beta }_{b0}$

dimensionless initial bulk concentration of a substrate

${\gamma }_i=\frac{P_i}{K_i},i=\mathrm{1,2}$

dimensionless concentration of the products i

${\gamma }_{ib}=\frac{Pi}{K_i},i=\mathrm{1,2}$

dimensionless concentration of the products outside the support i

$x=\frac{r}{R}$

dimensionless radial distance

$\alpha =\frac{K}{K_S}$

dimensionless inhibition degree

${\mathrm{\Phi }}_1=R\sqrt{\frac{k\cdot E}{K\cdot D_e}}$

Thiele modulus for substrate

${\mathrm{\Phi }}_{i+1}=R\sqrt{\frac{k\cdot E}{K_i\cdot D_{ei}}},i=\mathrm{1,2}$

Thiele modulus for products i

$\eta$

local effectiveness factor

${\eta }^{\prime }$

mean integrated effectiveness factor

Abbreviations

AGM

Akbari-Ganji's method

MADM

Modified Adomian decomposition method

FDM

Finite difference method

Pen G

Penicillin G

PAA

Phenyl acetic acid

6-APA

6-aminopenicillanic acid

Appendix A. Supplementary data

The following is the Supplementary data to this article: