A Kinetic Non-Steady-State Analysis of Immobilized Enzyme Systems Without External Mass Transfer Resistance

. In this paper, a non-steady-state non-linear reaction di ﬀ usion in immobilized enzyme on the nonporous medium is considered for its mathematical analysis. The non-linear terms in this model are related to the Michaelis-Menten kinetics. For the considered model, the approximate analytical expressions of the substrate concentration and the e ﬀ ectiveness factor for the various geometric proﬁles of immobilized enzyme pellets are obtained using homotopy perturbation method (HPM). The obtained approximate analytical expressions proved to be ﬁt for all values of parameters. Numerical solutions are also provided using the MATLAB software. When comparing the analytical and the numerical solutions, satisfactory results are noted. The e ﬀ ects of Thiele modulus and Michaelis-Menten kinetic constants on the e ﬀ ectiveness factor are also analyzed.


Introduction
The process of immobilizing the enzymes on the support materials has its extended use in the continues bioreactors and batch reactors.Immobilized enzymes exhibit distinct kinetic behaviours due to factors such as diffusion limitations within and between particles, substrate distribution between the support and the surrounding solution, structural changes during immobilization, and microenvironmental alterations stemming from support interactions, leading to deviations from the intrinsic kinetics observed in free enzymes.The extent of these effects is contingent on the characteristics of the support material, the nature of the substrate and its concentration, and the specifics of the immobilization technique employed.[1][2][3].Mass transfer limitations impacting the observed reaction rates stem from two key factors: external mass transfer resistance, which hinders the transport of substrate from the bulk fluid phase to the external surface of support pellets, and internal mass transfer resistances associated with pore diffusion [4,5].For the design, modelling, simulation, and process of development of batch reactors, a crucial step involves studying the intrinsic kinetics of enzymatic reactions.This involves a better understanding of the reaction kinetics without the interference of mass transfer constraints [6].
In recent times, researchers prefer the analytical solution for non-linear differential equations since it offers a distinct advantage over numerical solutions by providing precise, closed-form expressions, enabling deeper insights into the underlying mathematics and yielding results that can be rigorously proven and interpreted, which is essential for advancing our understanding of the system.Karthika et al. [7] utilized the homotopy perturbation method for obtaining the analytical expression for the mathematical model of packed bed tubular reactor for lactose hydrolysis.
Praveen et al. [8] provided analytical expressions for the immobilized enzymes with the competitive and uncompetitive substrate and product inhibitions with the reversible Michaelis-Menten reactions by utilizing the modified Adomian decomposition method for all the possible values of the presented parameters.The analytical expressions for effectiveness factor of batch reactor performance are also presented and analysed.Jeyabarathi et al. [9] utilized the semi-analytical methods namely Adomian decomposition method and Taylors series method for solving the non-linear differential equation arising in the reaction diffusion kinetic model of Langmuir-Hinshelwood-Hougen-Watson (LHHW) type for various geometries and the effectiveness factor.Sivakumar and Senthamarai [10,11] derived the substrate concentration of steady-state immobilized enzyme system with and without the mass transfer resistance for the various geometries of the catalytic pellets immobilized on to the nonporous medium.These approaches are limited to the steady-state conditions of the immobilized enzymes.To the authors best of the knowledge, there is no existing analytical expression for non-steady-state condition of reaction-diffusion kinetics of immobilized enzyme on the nonporous medium without the mass transfer resistance which can be used for the better understanding of the reaction kinetics of immobilized enzyme.
In this paper, the non-steady-state condition model of the substrate concentration of immobilized enzymes on the nonporous medium which is not affected by the mass transfer is analysed mathematically.The non-steady-state reaction-diffusion equation of the substrate concentration is a non-linear differential equation with the non-linear terms related to the Michaelis-Menten kinetics.The approximate analytical expressions for the substrate concentration with the geometries of planar, cylindrical and spherical pellets are obtained by utilizing the homotopy perturbation method and Laplace transform method for various values of parameters.The closed-form analytical expression of the effectiveness factors of planar, cylindrical, and spherical geometry of the enzyme pellet is also provided.Numerical solution of the system is obtained using the MATLAB software and are compared with the obtained analytical results.

Mathematical formulation of the model
A non-steady-state nonlinear differential equation with the initial and boundary conditions of substrate concentration of immobilized enzyme on the nonporous medium is considered and it is given as [12]: where C(x, t) = 0 when t = 0, (2.2) where C(x, t) is the concentration of the substrate and α depict the Thiele modulus and β is the dimensionless Michaelis-Menten constant.
Further, to describe the mass transfer limitation effect on the overall reaction rate, the overall effectiveness factor η is derived by using η = reaction rate reaction rate in the absence of internal and external resistances (2.5) 3. An approximate analytical expression of the non-steady state concentration using HPM with Laplace transform technique Nonlinear equations serve as essential tools for representing challenges across various domains, including applied mathematics, physics, chemical engineering, and biological sciences.The persistent challenge faced by researchers in these fields pertains to obtain the exact solutions.In recent years semi-analytical methods like homotopy perturbation method (HPM) [13] - [16], Akbari-Ganji method (AGM) [17] - [19], Taylor's series method [10,11,20,21], Adomian decomposition method (ADM) [19,22], variational iteration method (VIM) [23] - [25] are utilized for obtaining the approximate analytical solution.
In this article, we have utilized HPM and Laplace transform technique for obtaining the approximate analytical expression for the substrate concentration of immobilized enzyme on the nonporous material.The approximate analytical expression is obtained as follows (see Appendix A): + a sin where a = α 2 1 + β applying the solution (3.1) in (2.6), we will get the required effectiveness factor as 4. Influence of geometry 4.1.Planar geometry.When g = 1, the pellet shape in Eq. (2.1) becomes planar and the resulting concentration flow is as follows: + a sin and the corresponding effectiveness factor is obtained by taking g = 1 and applying (4.1) in (2.6), we get 4.2.Cylindrical geometry.When g = 2, the pellet shape is a cylindrical geometry and the resulting concentration flow accordingly becomes and the corresponding effectiveness factor is obtained by taking g =2 and applying (4.3) in (3.2), we get 4.3.Spherical geometry.When g = 3, the pellet shape is a spherical geometry and the resulting concentration flow accordingly becomes and the corresponding effectiveness factor is obtained by taking g = 3 and applying (4.5) in (3.2) and found as     The dimensionless concentration profiles according to the spherical geometrical results so obtained have clearly been exhibited by the graphs given below and the approximate analytical results obtained by the HPM have been compared with their numerical simulation graphs plotted with the help of MATLAB software for various values of the parameters α, β and the time factor t have a sense of ensuring how far our method of solution yields results much coherent to the numerical simulations.In view of the comparison manner, we could have a clear vision that the approximate analytical results even in non-steady state are almost coinciding with the results of numerical simulations.In Table .3, the mean errors have been calculated between the analytical solutions and the numerical solutions for the three sets of parameter values which have been collected from the previous articles and it describes that the approximate analytical solutions and the corresponding numerical results almost coincide under this geometry even in non-steady state.The dimensionless concentration levels and the overall effectiveness factors in different concentration levels have been represented by the graphs, Figure.

Sensitivity parameters
Derivatives of the system respect to the parameters quantify the effect of the parameters on the solution of the system.The obtained results of parametric sensitivity analysis reveal the amount of parametric impact on the effectiveness of batch reactor performance.The proposed non-steady-state result is utilized to investigate the effect of various diffusion and kinetic parameters on the substrate and the effectiveness factor of the system.The above proposed method can be employed to solve the time-dependent reaction-diffusion problems arising in the various fields of chemistry.
8. Appendix-A: Obtaining the approximate analytical solution of equation (2.1) using HPM By constructing the homotopy for the Eq.(2.1), we have Applying the initial condition Eq. (2.2) in Eq. (8.1), we get Then we shall obtain the approximate solution in the form Substituting Eq. ( 8.3) in Eq. (8.2) and equating the coefficients of the zeroth power of p, we have where a = α 2 1+β .Taking Laplace transform on Eq. (8.4), we have (8.12) The residues of Eq. (8.12) can be obtained at s = 0, we obtain a simple pole and the solution of cosh s+a g = 0 generates infinitely many poles given by s n = −g (2n+1)

Figure 3 .
Figure 3.Comparison graph of Dimensionless substrate concentration C(x, t) versus the dimensionless distance x (3a) for the fixed parametric values β = 10, t = 1 and for various values of α, (3b) for the fixed parametric values α = 0.1, t = 1 and for various values of β, (3c) for the fixed parametric values α = 1, t = 5 and for various values of β, where '• • • ' represents the numerical results and ' * * * ' represents the analytical results by HPM Eq. (4.3) for cylindrical pellets.

Figure 5 .
Figure 5.Comparison graph of Dimensionless substrate concentration C(x, t) versus the dimensionless distance x (5a) for the fixed parametric values β = 10, t = 1 and for various values of α, (5b) for the fixed parametric values α = 0.1, t = 1 and for various values of β, (5c) for the fixed parametric values α = 1, t = 5 and for various values of β, where '• • • ' represents the numerical results and ' * * * ' represents the analytical results by HPM Eq. (4.5) for spherical pellets.

Figure. 7
symbolizes the influence of the effect on the effectiveness of the reaction diffusion process of the enzymatic reaction inside the batch reactor.The parametric values utilized for the analysis are α = 0.6, β = 5.From Figure.7, it is seen that the Thiele modulus has the most effect on the effective working of the batch reactor with 95%, and the dimensionless Michaelis-Menten constant has the least impact on the effectiveness factor of the batch reactor with 5%.

Figure 7 .
Figure 7. Effect of parameters on effectiveness factor

Table 1 .
Comparison table of the approximate analytical results by HPM with the corresponding numerical results for fixed values of parameters α, t and for various values of β for planar geometry.Here also we have a clear vision that the approximate analytical results even in non-steady state are almost coinciding with the results of numerical simulations.In Table.2, the mean errors have been calculated between the analytical solutions and the numerical solutions for the various parameter values which have been collected from the previous works and it describes that the approximate analytical solutions and the corresponding numerical results almost coincide under this geometry even in non-steady state.The dimensionless concentration levels and the overall effectiveness factor variations corresponding to the variations in concentration levels have been represented by the graphs, Figure.
results so obtained have clearly been demonstrated by the graphs given below and the approximate analytical results obtained by the HPM have been compared with their numerical simulation graphs plotted with the help of MATLAB software coding for various values of the parameters α, β and the time factor t to have a sense of ensuring how far our method of solution yields results coherent to the numerical simulations.

Table 2
. Comparison table of the approximate analytical results by HPM with the corresponding numerical results for fixed values of parameters α, t and for various values of β for cylindrical geometry.

Table 3 .
Comparison table of the approximate analytical results by HPM with the corresponding numerical results for fixed values of parameters α, t and for various values of β for spherical geometry.