Mathematical Analysis of Nonlinear Reaction Diffusion Process at Carbon Dioxide Absorption in Concentrated Mixtures of 2-Amino-2-Methyl-1-Proponal and 1,8-Diamino-p-Methane

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Introduction
According to the fact that carbon dioxide in exhaust gases produced by burning fossil fuels is the primary source of air pollution, acid rain, global warming and other environmental issues, the chemical immobilization of CO2 has recently acquired close consideration as a research area [1].Chemical reaction engineering practice often deals with multiphase reaction systems.
These systems are commonly encountered in industrially significant processes such as oxidation, hydroformylation, gas purification, hydrogenation and oxidation.A lot of attention was aid to one of them, the removal of carbon dioxide employing amines in aqueous solutions.The chemical absorption technique is a standard approach to accomplish the CO2 removal and recovery on an industrial scale.Alkanolamines are chemical compounds that are significant to industry [2].
Refineries, natural gas and synthetic gas sectors commonly employ aqueous alkanolamine solutions to remove CO2 and other acidic gases from gas streams.
Numerous investigations involving basic mass balance analysis have been conducted on the principles and kinetics of carbon dioxide's interaction with different alkanolamines, leading to the discovery of zwitterion mechanism [3][4].A class of sterically hindered amines has recently been discovered that exhibits a high amine capacity [5][6][7][8] and a reasonably substantial uptake rate, even under enhanced carbon dioxide loading.A prime illustration is 2-amino-2-methyl-1proponal.Blender amines have been proposed as potentially helpful in facilitating the uptake of acid gases through leveraging the positive aspects of amine [9] K.J. Oh et al. presented a theoretical and experimental analysis of the absorption of carbon dioxide into aqueous solutions [10].Paul et al. established the CO2 extraction in the solution of 2-(1-piperzinyl) ethylamine [11].Subramaniam et al. presented the study of mass transfer along CO2 absorption through phenyl Glycidyl ether solution by Adomian decomposition method [12].Anitha et al. examines the carbon dioxide concentration in the solutions that used Homotopy analysis method [13].
The primary objective of this study is to obtain the analytical approximations regarding the absorption of carbon dioxide into the aqueous solutions along with 2-amino-2-methyl-1proponal (AMP) and 1,8-diamino-p-methane (DAM), making use of Akbari Ganji Method (AGM) and Differential Transform Method (DTM).Quantitative and graphical statistics are presented to demonstrate these approaches.

Mathematical formulation
The mathematical model of CO2 absorption in the stirred semi batch tank with a planar gas liquid was built on the Zwitterion reaction mechanism presented out by K.J. Oh [10].The following pre-assumptions form the basis of the mathematical equations that govern for the assimilation of CO2 over the aqueous solution.(i) The circumstance essentially isothermal (ii) Henry's law constitutes applicable (iii) AMP and DAM species are non-removal (iv)First order reactions exist concerning P and Q as well as between P and ′ with regard to  and ′ respectively.
The following set of nonlinear differential equations describes the fundamental mass balance computations of CO2 absorption in the solution of AMP with DAM.The relevant dimensionless boundary conditions are often expressed as 12) The normalized flux becomes,

Akbari Ganji Method
AGM signifies a striking progress in the field of nonlinear sciences.Analytically simulating nonlinear differential problems can be considerably more difficult than addressing linear differential equations.This method is an extremely innovative approach to overcoming such issues in this sense.Within the AGM, a solution function containing a newly discovered constant coefficient satisfies the initial and boundary limitations.Employing this approach, nonlinear equations (2.8) -(2.10) may be solved to get the straightforward analytical expressions for the concentration of species.
The AGM proceeds by presumed that the hyperbolic function governs the solution to the equations (2.8) -(2.10).
When boundary conditions (2.11) -(2.12) have been substituted in (3.1) -(3.3), the value of the constants are The analytical expression for the concentration of carbon dioxide, 2-amino-2-methyl-1-proponal and 1,8-diamino-p-methane for all dimensionless parameters is derived by substituting equations The nondimensional current is determine as

Differential Transform Method
The differential transform method had been first proposed by Zhou From (3.17In DTM Approach, the exact solution set is considered as, 2 )  2 (3.26) At  = 1, we obtain the values, After substituting the constants in (3.24) -(3.26), we are able to compute the analytical formulation that follows.
We determine the analytical equation for the current expressed as,

Evaluation of analytical findings with numerical simulation
The numerical technique provides an approximation to solve a mathematical issue.
Additionally, the validation of analytical findings is also advantageous.Figure (2)(3)(4)(5)(6) summaries the findings of concentration predictions for the nonlinear differential equation that was numerically solved using MATLAB software to examine the appropriateness of these analytical approaches.Figure 2 represents that the nondimensional concentration of carbon dioxide  ̃ 2 versus non dimensional diffusion coordinate of gas  for the various values of the dimensionless parameter  1 using (3.13).Figure 2 demonstrates that with a sense of decreasing amount of  1 , the concentration of carbon dioxide reduces into an aqueous solution.That means, the reaction rate constant  1 is inversely proportional to the concentration profile of CO2 for all minimal amount of  2 .

Equations
The concentration  ̃ 2 approaches the steady state regarding the range of  1 ≥ 100 whereas the gas's diffusion coordinate exceeds in the bound 0.4 ≤  ≤ 1.At the initial state of the diffusion coordinate of gas,  ̃ 2 attains its optimum value.A linear tandem occurs among the concentration and reaction rate constant  2 meanwhile  ̃ 2 is minimal and closes in steady state.

Conclusion
Mathematical formulation of Carbon dioxide absorption into an aqueous solution were discussed.A steady state behavior of system of non-linear equations were solved analytically by using Akbari Ganji Method and Differential Transform Method in order to obtain the analytical solution for the concentrations of CO2, AMP and DAM for all parameter values.The normalized current was also analytically expressed in simple closed form.A remarkable outcome might be observed graphically while comparing the new analytical results of AGM and DTM along with numerical simulation results for differing parameter reliability.The absorption rate of CO2 in an aqueous solution was examined by the new approximate result of the diffusion model.These excellent outcomes are used to estimate a removal of gases released from power plant flues for the purpose of using Carbon dioxide.There was an extensive comprehension of the system while these outcomes were in good accordance our two techniques are simple to use in addition to the possibility that solve other nonlinear equations.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.

Figure 1
Figure 1 established the schematic representation of the carbon dioxide absorption into an aqueous solution containing two reactants of 2-amino-2-methyl-1-proponal and 1,8-diamino-pmethane in a stirred semi batch tank with a liquid of planar gas.The range of concentrations of

Figure 1 .
Figure 1.Diagrammatic depiction of the carbon dioxide absorber

Figure 2 :
Figure 2: Graph of Analytical and Numerical solutions for different values of parameter  1 and the fixed value of  2 .

Figure 3 :
Figure 3: Plot of the solutions both Analytically and Numerically various amounts of values of  2 for all  1 .

Figure 3
Figure 3 shows that the normalized concentration of carbon dioxide for the different values of  2 and for fixed value of  1 .Figure 3 leads to the conclusion that  ̃ 2 decreases when  2 falls depends on the diffusion coordinate of gas  .Even with increasing diffusion film thickness or the reaction rate constant of CO2 and the maximum value of  2 ≥ 100 , the concentration of CO2 remains constant.Additionally considering the diffusion gas approaches  ≤ 0.1, the inclined curve of the concentration reflects that the elevated level of concentration of CO2.For any significant amount of the reaction rate constant, the curve succeeds the value of steady state within 0.5 ≤  ≤ 1.

Figure 4 (
Figure 4(a)-(b).Plot of dimensionless concentration of AMP versus dimensionless diffusion coordinate of gas.(a)for fixed e and various  3 (b)fixed value of  3 and numerous value of e.

Figure 4 (
Figure 4(a)-(b) depicts that the compact influence on the concentration of 2-amino-2methyl-1-proponal due to the nondimensional parameter  3 and the stoichiometric coefficient e.From Figure4(a), it evident that the concentration of AMP decreases for maximal value of reaction rate constant  3 and the stoichiometric coefficient e = 1.The analytical and numerical outcomes are coincide at the equilibrium value of the diffusion coordinate of gas  = 0.5.The Concentration profile  ̃ attains its peak value whenever the range of  is both of  ≤ 0.01 and  ≥ 1.

Figure 4 (
Figure 4(b) illustrates that the dimensionless concentration of 2-amino-2-methyl-1proponal according to the various amount of stoichiometric coefficient e with the fixed value of  3 = 3.It is observable that  ̃ decreases for large values of  3 and the stoichiometric coefficient rises.For the very minimal amount of e≤0.01, it clear that the AMP concentration is uniform.The effect of the stoichiometric coefficient which is inversely proportional to the amount of 2-amino-2-methyl-1-proponal.

Figure 5 (
Figure 5(a)-(b).Graph of nondimensional DAM concentration with the dimensionless distance for numerous amount of constant  4 .

Figure 5 (
Figure 5(a)-(b) exhibits a graph of the concentratoion of 1,8-diamino-p-methane contrasted with the gas's diffusion coordinate considering various reaction rate constant  4 and fixed values of the stoichiometric coefficient e'.It is clear that DAM concentration drops at the highest  4 values.A comparison of figure 5(a) and 5(b) shows that for all maximal e', the concentration profile  ̃ ≤0.5.According to figure 5(a)-(b), the stoichiometric coefficient e' which is inverted with respect to  ̃ but immediately correlated to the reaction rate constant 4 .

Figure 6 (
Figure 6(a)-(b) suggests the normalized current response fluctuations under many different kinds of parameter values  1 and  2 .It represent how the non dimensional parameters

Non-dimensional version of the problem
=  ′  4  ̃ 2 ()  ̃ ()(2.10) in 1986 and since then, it has been explained in several literatures to solve different types of integral and differential problems.With the help of differential transform approach, one may find the coefficient of the Taylor series expansion explored for determining differential equations.

Table 1 .
Deviation Table (1-3) compares the simulation results with AGM and DTM results.The maximum average error of CO2, AMP and DAM is reported as 0.01%, 0.01%, 0.02% by applying AGM and 0.01%, 0.02% and 0.2% using DTM respectively.It provides a good agreement for all parameter values upon comparison.between Numerical result (2.8) and Analytical results (3.13) and (3.27) of the concentration of Carbon dioxide for different parameters.  = .,   = .   = .,   = .  Num

Table 3 .
Comparative analysis of numerical solution (2.10) and analytical solutions (3.15) and (3.29) of the concentration of 1,8-diamino-p-methane for different parameter values.  = ,  = .,  =    = .,  = .,  = . It represent how the non dimensional parameters  1 and  2 impact current circumstances profiles.The following information indicates that as the reaction rate constant  1 is increased, the molar flux climbs at a modest diffusion coefficient   2 .The variation of normalized current Ω  2 ≤ 1 for numerous values of reaction rate constant  2 and   2 ≥0.1.For any maximum value of the diffusion coefficient   2 ≤0.01, the normalized current becomes lower in the range of Ω  2 ≤ 0.1. 2 is exactly correlated to the diffusion coefficient   2 and the reaction rate constant  2 .