Mathematical Investigation of Non-Linear Reaction-Diffusion Equations on Multiphase Flow Transport in the Entrapped-Cell Photobioreactor Using Asymptotic Methods

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Introduction
Sustainable energy is produced from resources that can support present activities without endangering the climate or the energy demands of future generations.Wind, solar, and hydropower are some of the most widely used renewable energy sources.The need for sustainable energy is rising due to the depletion of fossil fuels.Humans and the environment have suffered dramatically due to an over-reliance on fossil fuels.Hydrogen is frequently seen as an appealing replacement for fossil fuels because of its efficient conversion and clean burning, which produces no CO2.Biological hydrogen generation is a secure, affordable, and sustainable method of producing hydrogen owing to its environmental dependability, capability to prevent ecological damage, and inexpensive equipment investment costs [1,2].Photosynthetic bacteria (PSB), one of the microorganisms used to produce biological hydrogen, is a feasible option due to its high theoretical considerations, changing yield, and ability to prevent the oxygen suppression of biological systems brought on by oxygen-evolving activity [3].In addition, PSB can treat waste by biodegrading a wide range of organic substrates and a broad spectrum of solar radiation [4].As a result, this topic has attracted the attention of numerous researchers worldwide.
Many experiments on PSB photohydrogen generation are currently being conducted in the suspended-cell photobioreactor because of the excellent mass transfer.The potential to repair and reuse cell mass and advances in cell concentration and operational stability make techniques for immobilizing cells, including biofilm, cell entrapment, and self-flocculation, promising [5].
Particularly, cell entrapment is a more rational approach because it can offer a steady and beneficial environment for the growth of PSB trapped by gel granules.In the entrapped-cell photobioreactor, it is essential to note that multiphase flow, mass transfer, and biochemical reactions all coincide.This has an impact on the features of the substrate concentration distribution and the efficiency of photo-hydrogen production.Therefore, it is advantageous to encourage the scale-up use of photo-hydrogen generation systems by providing complete analyses of the intricate biochemical responses and transport processes.However, it is exceedingly challenging to experimentally assess the multiphase flow and mass transfer concerning the biochemical activities in the entrapped-cell photobioreactor.
The mathematical model is a potent tool that can explain how biochemical reactions and transfer mechanisms interact.Numerous mathematics models have been developed recently to forecast the creation of photohydrogen.The characteristics of Rhodopseudomonas palustris CQK 01's substrate consumption and hydrogen production in an immobilized-cell photobioreactor were examined by Wang et al. [6,7].An innovative annular fiber-illuminating biological hydrogen reactor was investigated as part of a two-dimensional steady-state model developed by Zhang et al. to simulate substrate degradation and diffusion in the biofilm zone and substrate convection and distribution in the bulk fluid zone [8,9].Yang et al. [10] used the lattice Boltzmann method to model a biological reaction in a substrate solution passing around a circular cylinder.
Liao et al. [11] developed a two-phase flow and mass transfer model to forecast the production of photohydrogen and substrate biodegradation.Guo et al. [12] conducted a simulation study to examine the features of photo-hydrogen generation and substrate deterioration across different stacking configurations.Even if we have numerical solutions for every type of model, a thorough analysis of the system requires identifying the analytical solution.Recently, there has been a lot of focus on determining the reaction-diffusion equations' straightforward approximation analytical expression.Ganji et al. [13] used He's Energy Balance Method to derive approximations of solutions.Two methods are employed in the mathematical model of two-phase flow transfer in an immobilized-cell photobioreactor: the homotopy perturbation approach by Shirejini et al. [14] and the Adomian decomposition method by Praveen et al. [15].Using Bernoulli Wavelets and a spectral method, Venkata Subrahmanyam Sajja et al. [16] solved reaction-diffusion equations in gel-granules.The problem based on electroactive polymer film placed on an inlaid micro disc electrode was solved analytically by Meena and Rajendran using HPM [17].Jeyabharathi et al.
used Akbari-Ganji and Taylor's series methodology to develop a packed bed immobilized-cell electrochemical photobioreactor [18].Guo et al. outlined a multiphase mixture model for the entrapped-cell photobioreactor's photo-hydrogen generation performance and substrate concentration distribution features [19].
An analytical solution for the multiphase mixture model has yet to be published.In the current work, the homotopy perturbation approach and the Akbari-Ganji method are used to derive the analytical solution for a multiphase mixture model for the concentrations of substrate, hydrogen production, solid and liquid phase flows in the entrapped-cell photobioreactor.Furthermore, the effects of reaction-diffusion and saturation parameters on substrate (glucose) and product (hydrogen) concentrations are also investigated.

Model construction and analysis
Guo et al. [19] designed a cell-photobioreactor that is entrapped and filled with gel granules carrying PSB (Rhodopseudomonas palustris CQK 01), as shown in Fig. 1.The h-direction refers to the substrate flow direction in the primary channel, and the r-direction relates to the product transport direction inside the gel granule.The entrapped-cell photobioreactor's peristaltic pump provides the substrate medium, and the only carbon source used is glucose.Once the glucose has diffused into gel granules from the primary channel, the entrapped cells biodegrade it.The produced gases and metabolites eventually diffuse out of the gel granules under the flow of the substrate medium in the entrapped-cell photobioreactor.Evidently, in the entrapped-cell photobioreactor, multiphase flow, biochemical reactions and mass transfer all coincide.transfer processes are viewed as a one-dimensional flow along the h-direction in the significant channel.Darcy's law can be used to describe substrate and product transfer processes because it assumes that fluids' physical and thermal properties stay constant.The biochemical processes are limited to gel granules.The PSB's consistent distribution and activity within the gel granules are considered.The mole ratio of carbon dioxide to hydrogen, the sole gaseous product produced by substrate biodegradation, is 2:1 [20].
In the entrapped-cell photobioreactor, it is anticipated that the multiphase mixture model can adequately represent the processes of mass transfer and multiphase flow in the gel granules.
Diffusion is crucial in the reactant and the product's mass movement.Applying Fick's law, the mass transfer equation for the substrate and hydrogen inside the gel granules is as follows [19]: Here, (2. 3) The The associated boundary conditions would change to: = 0 when  = 0 (2.9) Thus, in the entrapped-cell photobioreactor, the substrate biodegradation rate ∅  ̃ can be determined by: Where α is the elemental volume's specific gel granule area.
Furthermore, the gas and liquid phase interfacial mass transfer rate can be written as follows: Where the rates of generation of hydrogen and carbon dioxide, respectively, are denoted by ∅  2 ̃ and ∅  2 ̃, which can be defined as: d̃│ =  ̃ (2.14) Here,  3 denotes the molecular weight of carbon dioxide.
The following gives the expressions for the liquid and gas phases under steady-state conditions: )

An approximative analytical expression for the concentration of the substrate and product using HPM
Recent techniques for solving non-linear differential equations include using hyperbolic functions [21][22], Taylor's series approach [23][24], Adomian decomposition method [25], the homotopy analysis method [26], and variational iteration method [27].Among these techniques, Ji-Huan's homotopy perturbation method (HPM) uses the linearization process to solve nonlinear equations [28][29][30].This method would find and add a small parameter to the equation, utilizing the traditional perturbation method while eliminating its limitations.Numerous studies have used HPM to formulate an analytical solution for non-linear equations in various engineering and physical problems [31][32].This approach combines traditional perturbation techniques with the topological concept of homotopy.The current study uses the HPM and Akbari-Ganji approach to analytically approximate equations (2.7) and (2.8).
By utilizing the HPM on Eqs.(2.7) and (2.8), analytical computations were generated for the concentrations, liquid, and gas phases for every parameter value (Appendix-A).

Analytical solutions for the substrate and product concentrations utilizing the Akbari-Ganji method
M. Akbari and D. Ganji proposed the Akbari-Ganji method (AGM) which has been effectively employed for determining analytical solutions for non-linear equations.The Akbari-Ganji Method is the new name for this method, formerly known as the Algebraic Method.Recently, this technique was applied to solve a non-linear problem in the chemical sciences [33][34][35].
Using the AGM to solve equations (2.7) and (2.8), analytical computations were generated for the concentrations and source terms of the liquid and gas phases (Appendix-B). ) ) ℎ () ( Eqs. (4.3) -(4.6) represents the analytical expressions obtained using the homotopy perturbation method.
The analytical expressions are derived as follows after using the AGM to solve the coupled equations of (4.1) and (4.Where the constant coefficients n and k are obtained by solving these equations: [( 2 )() +  2 ] = 0 (4.12)

Analytical expressions with saturated kinetics
The limiting condition, where  ≫ 1, occurs when the substrate utilization rate remains constant concerning the substrate concentration.The simplified equations of (2.7) and (2.8) can be defined below:  After resolving these equations, the constant coefficients p and q can be determined: +  2 ] = 0 (4.24)

Analytical results validation
Numerical techniques were utilized to solve the non-linear reaction-diffusion equations (2.7) and (2.8), along with the appropriate boundary conditions (2.9) and (2.10).The MATLAB software's pdex4 function, a tool for resolving boundary value issues for partial differential equations, was utilized to solve these equations numerically.There was a satisfactory level of agreement when comparing the numerical solutions with the HPM and AGM.Two iterations of the HPM and simple algebraic calculations of AGM were used to calculate the approximate concentrations.The reader might initially conclude that the AGM yields more accurate results than the HPM.As we only used the zeroth and first-order iterations in the latter technique, remember that the accuracy can be significantly boosted if additional iterations are encountered.
The overall relative error for the HPM is 0.42 %, and AGM is 0.25 % for the various reactiondiffusion parameter values ( 1 ,  2 ) displayed in Tables 1 and 2.

Conclusion
This work presents a thorough theoretical analysis of multiphase flow transport in the entrapped-cell photobioreactor within the gel granules.The concentration expressions of the substrate, product, and liquid and gas phases were determined analytically using asymptotic methods of homotopy perturbation and the Akbari-Ganji method to solve the non-linear reaction-diffusion equations.The effects of several parameters, including the saturation and reaction-diffusion parameters, were discussed.Furthermore, two limiting situations with saturated and unsaturated enzyme kinetics were given closed-form analytical calculations.The resulting approximation expressions of the concentrations were proved to be highly accurate by direct comparison with dependable numerical data obtained by the MATLAB programme.The proposed solutions can accurately replicate the system's dynamic performance with the parameters, as demonstrated by the close correspondence between the simulated and analytical data.It is helpful to comprehend and forecast the system's behaviour with the expressions provided in this work.It is also a valuable tool for catalyst particle optimization.

Appendix A (Homotopy perturbation method)
The homotopy was constructed for the Eqs.(2.7) -(2.8) as follows: The following are the analytical solutions to equations (8.1) and (8.2): Using the same powers of p-terms to rearrange the results of substituting Eqs.

FIGURE 1 .
FIGURE 1. Schematic representation of the entrapped-cell photobioreactor.The model's development takes into account the following presumptions.Operating conditions for the entrapped-cell photobioreactor are steady-state.The substrate and product

Table 1 .
Comparison of the numerical simulation for the substrate concentration () with the analytical results of Eqs.(3.1) and (3.5) for various values of parameter  1 .

FIGURE 4 .
FIGURE 4. The liquid phase's source term   is determined by using Eq.(3.3) and considering reaction-diffusion parameters  1 and the saturation value .

FIGURE 5 .
FIGURE 5. Change of the source term gas phase   for various values of parameter a)  2 , b)  obtained by Eq. (3.5).

Figure. 5
Figure.5(a-b) supports this inference.Figures. 4 and 5 show that increasing  1 and  2 values and decreasing  are necessary to achieve a high mass transfer rate between the gas and liquid phases.
solution can be represented as follows by HPM:

( 9 . 1 )Where 7 ) 1 [ 𝜑 1 ( 2 [(
-(9.2), the values of constants are obtained as:  and  are the constant coefficients.The equations (2.7) -(2.8) can be rearranged in the following manner to get the values of l and m: can be expressed as follows when the equations (9.4) and (9.5) have been substituted in equations (9.6) -(9.7) and at  = 1.(1) =  2 + 2  tanh () − 1  1+) +  1 ] = 0 (9.8) (1) =  2 + 2  tanh () − 1   2 ) (1+) +  2 ] = 0 (9.9) concentrations of substrate and hydrogen in the gel granules are represented by ̃.The substrate biodegradation rate for PSB trapped in the gel granules is indicated by    ̃, whereas the hydrogen production rate is represented by    2 ̃.The specific growth rate of PSB and the maintenance coefficient are denoted by the symbols  ̃ and  ̃ , respectively; ᴪ ̃ is the cell density increasing coefficient, and   ̃ is the cell density. ̃  represents the cell yield, The monod constant is   ̃, the operation temperature is  ̃, and  ̃ stands for the medium pH values. ̃ is the highest possible specific growth rate.For hydrogen production,  ̃ is the non-growth-associated kinetic constant, and  is the growth-associated kinetic constant. 1 and  2 represents the molecular weight of glucose and hydrogen respectively.The value of  ̃ and  ̃ are considered to be 7 and 30℃ according to the experimental data.The corresponding boundary conditions for the dimensional Eqs.(2.1) and (2.2) are listed below:

Table 2 .
Examine the numerical simulation for various reaction-diffusion parameter values  2 for