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In this paper, mathematical model of Martens and Hall (Analytical chemistry 66, 2763-2770 (1994)) for an immobilized oxidase enzyme electrode is discussed. The model involves the system of non-linear reaction diffusion equations under the steady state conditions. A simple and closed-form of approximate analytical expressions for the concentrations of the immobilization of three enzyme substrates has been derived by solving the system of non-linear reaction diffusion equations using new approach of homotopy perturbation method. Approximate polynomial expression of concentration of substrate, oxygen and oxidized mediator and current was obtained in terms of the Thiele moduli and the small values of parameters
B
_{s},
B
_{o} and
B
_{m} (normalized surface concentration of substrate, oxygen and oxidized mediator). Furthermore, in this work the numerical simulation of the problem is also reported using Matlab program. An agreement between analytical expressions and numerical results is noted.

There have been many publications on models for enzyme electrodes. Schulmeister et al. [_{x}) electrode Leypoldt et al. [

The details of the model adopted have been fully described in Mertens and Hall [

We assume that the concentrations of all reactants and enzyme intermediates remain constant for all time. Also the concentration of total active enzyme [E_{t}] and the reactants in the bulk electrode remain constant. We can consider that the diffusion of the reactants can be described by Fick’s second law and the enzymes are assumed to be uniformly dispersed throughout the matrix. The enzyme activity is not a function of position. The coupled three non linear reaction/diffusion equations in normalized form are

The boundary conditions becomes

The normalized parameters are

where F_{s}, F_{o}, and F_{m} represent the normalized concentrations of substrate, oxygen and oxidized mediator and B_{s}, B_{o}, and B_{m} are the corresponding normalized surface concentrations. The surface concentration is the ratio of the bulk concentration and the reaction constants. Φ_{s}, Φ_{o}, and Φ_{m} denote the Thiele moduli of substrate, oxygen and oxidized mediator, respectively. Thiele modulus Φ^{2} represents the ratio of the characteristic time of the enzymatic reaction to that of substrate diffusion. d is the thickness of the enzyme layer. The normalized current J_{OX} is given by,

Recently, many authors have applied the Homotopy perturbation method to solve the various non linear problems in physical and chemical engineering sciences [

Recently Anitha and Rajendran [

Equation (11) to Equation (13) represents the new simple and closed-form of approximate analytical expression of concentrations of substrate, oxygen and oxidized mediator.

Equation (11) to Equation (13) represent the new closed approximate analytical expression of the non-steady state concentration of substrate, oxygen and oxidized mediator for all values of kinetic and diffusion parameters. The concentration depends on parameters such as B_{s}, B_{o} and B_{m} and Φ_{s}, Φ_{o} and Φ_{m} (Thiele moduli).

_{s}. From this figure, it is inferred that the concentration of substrate and oxygen decreases when B_{s} (surface concentration of substrate) increases. Also concentration mediator decreases due to consumption by the enzyme reaction and reaching the minimum at the centre of the membrane (x = 0.5). Then the concentration of the mediator increases from x = 0.5 to x = 1 due reoxidation of the electrode.

_{i} less than 0.1).

_{o}. From this figure, it is inferred that the concentrate of substrate and oxygen increases when B_{s} decreases and become uniform for very small values of B_{o}. Here also concentration mediator decreases slowly from x = 0 to x = 0.5. Then from x = 0.5 to x = 1 the concentration increases due to reoxidation at the electrode.

_{ox} verses Thiele modules/B_{o} for various values of dimensionless parameter B_{s}, B_{o}, B_{m} and Φ_{m}_{. }From this figure it is observed that, the current density increases when B_{s}, B_{o}, B_{m} (surface concentration of substrate, oxygen, mediator) decreases. The most accessible parameters in the design of a sensor are the thickness of the membrane and the actual loading of active enzyme in the matrix. Also the maximum current decreases with decreases of membrane thickness or actual loading of active enzymes due to decrease in the total amount of enzyme presence in the system.

Tables 1-3 represent the comparison of analytical expression of concentration of the substrate, oxygen and mediator (F_{s}, F_{o}, F_{m}) for various of Thiele modules. The maximum average relative error between the analytical results and numerical results is 1.62%. This error is less than pervious published analytical result [

In this paper, steady state nonlinear differential equations in biofiltration model have been solved analytically. Approximate analytical expressions pertaining to the concentrations of substrate, oxygen and oxidized mediator are derived using homotopy perturbation method. These analytical solutions are compared with the numerical simulation results. These analytical results provide a good understanding of the system and