Akbari-Ganji‟s method “AGM” to chemical reactor design for non-isothermal and non-adiabatic of mixed flow reactors

In this paper, the chemical reactor with the energy changing (non-isotherm) and non-monotonic reactions in a mixed flow is investigated and also the sets of complex nonlinear differential equations obtained from the reaction behavior in these systems are solved by a new method. The set of differential equations are consisted of the mass equilibrium and energy (temperature) changing at concentration governing the reactors. Our purpose is to enhance the ability of solving the mentioned nonlinear differential equation with a simple and innovative approach which entitled „‟Akbari-Ganji's Method‟‟ or „‟AGM‟‟. Comparisons are made between AGM and numerical method (RungeKutta45). The results show that this method is very effective and simple and can be applied for other nonlinear problems.


INTRODUCTION
In this study, a mixed flow reactor is investigated through resolving a set of nonlinear differential equations governing its chemical reactions by a new analytical method. The interaction of chemical reactions in chemical reactors attracted the attention of many researchers according to the complexity of the reaction kinetics since the knowledge of real performance of reactions in reactors is helpful in the analysis and design process of reactors and are of great importance for researchers in the fields of mechanical and chemical engineering. In the context of designing a reactor, the reaction properties of chemical reactants especially in the presence of catalyst are very important since most of these reactions are complex and completely nonlinear which makes the designing process to be critical. In fact, analytical resolution of nonlinear differential equations governing these processes is a very difficult task which requires a precise and convenient analytical method. Therefore, in this paper, a new analytical *Corresponding author. E-mail: akbari_hamid46@yahoo.com. Tel: +22997016126, +22921360199.
Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License method capable of solving nonlinear differential equations governing the design of chemical reactors has been presented entitled AGM (Akbari-Ganji's method) which can be applied in all of the fields of engineering and basic sciences because of its high precision and convergence in dealing with problems with high nonlinearity. Consequently, many new techniques have been presented in the literature namely Homotopy Perturbation Method (HPM) (Ganji, et al. 2009a;Ganji, et al. 2009b), the Elliptic Lindstedt-Poincare method (LP) (Masdemont* 2005;Navarro 2008), Adomian Decomposition Method (Abbasbandy 2007;Adomian 1994), EXP-function Method (Chang, 2011;He and Wu, 2006) and variational Iteration Method (Ganji, et al. 2007;He 1999). For instance, the classical Blasius' equation was solved using VIM and also this method was utilized to give approximate solutions for some well-known non-linear problems. Another analytical method called the He's Amplitude Frequency Formulation method (Akbari, et al. 2014c) which was first presented by Ji-Huan He. This method rapidly gives convergent successive approximations of the exact solution. Other methods do not have this ability to precisely and accurately solve the problems, so nonlinear differential equations such as the problem presented in this case study should be solved by utilizing new approaches like AGM (Akbari, et al. 2014a;Akbari, et al. 2014b;Akbari, et al. 2014d).

The analytical method [Akbari-Ganji's Method (AGM )]
Boundary conditions and initial conditions are required for the calculation procedure used in analytical methods for each linear and nonlinear differential equation. In order to comprehend the given method, two differential equations governing engineering processes will be solved in this new manner. In accordance with the given boundary conditions, the methodology of AGM can be presented in the following form: To solve the first differential equation in respect to the boundary conditions in x=L in Equation 2, the series of letters in the nth order with constant coefficients which is the answer of the first differential equation is considered as follows: With more precise answer from Equation 1, the more we have choice of series sentences from Equation 3. Approximately five or six sentences from the series are enough to solve nonlinear differential equations. In the answer of Equation 3 regarding the series from degree (n), there are (n+1) unknown coefficients that need (n+1) equations to be specified. The boundary conditions of Equation 2 are used to solve a set of equations which consisting of (n+1) ones.
The boundary conditions are applied on the functions as follows: a) The application of the boundary conditions for the answer of differential Equation 2 is presented in the following form: And when x = L: , the application of the boundary conditions on differential Equation 1 is done according to the following procedure: With regard to the choice of n sentences from Equation 3 and in order to make a set of equations consisting of (n+1) equations and (n+1) the unknown, we confronted a number of additional unknowns which are indeed the coefficients of Equation 3. Therefore, to resolve this problem, we should derive m times from Equation 1 according to the additional unknowns in the aforementioned set of differential equations and then this is the time to apply the boundary conditions of Equation 2 as follows.
c) Application of the boundary conditions on the derivatives of the differential equation in Equation 7 is done in the form of: (n+1) Equations can be made from Equation 4 to Equations 8 and 9, so that (n+1) unknown coefficients of Eq.(3), for example will be computed. The answer of the nonlinear differential Equation 1 will be gained by determining coefficients of Equation 3. Akbari et al. 3 THEORICAL FORMULATION

Mixed flow to chemical reaction
The mixed chemical reactor is considered for chemical reaction is shown in Figure 1. The chemical reaction is considered as follows: Supposing the kinetics of the reaction are non-monotonic for component (A) which is considered as: Using the energy and mass balance equations for the reactor leads to the following equations: where and are the heat of chemical reaction, the total coefficient of heat transmission, the heat exchange rate, the external wall temperature of the reactor, the volume of the reaction mixture, the volume of the metal parts of the reactor (stirrer and cooling coil), the density of the reaction and metal parts, volumetric flow rate of feed, the feed concentration of reactant A, feed temperature and area of transfer between the reactor and the cooling coil, respectively.
The dimensionless form of Equations 13 and 14 can be written as follow: The dimensional parameters are defined as follows: , , It is worthy to note that the subscript ref in Equation 17 denotes the reference parameter.
The initial conditions in plug chemical reactor are are described as follows:

Solving the nonlinear differential equation by AGM
In the AGM calculation procedure, there is no need to use the Taylor series expansion for the exponential term ; whereas in the other methods such as Homotopy Perturbation Method (HPM), (Adomian's Decompostion Method (ADM), VIM and DTM (Differntial Transform Method), we should convert such terms into polynomials using Taylor series expansions. As a result, the obtained solution by AGM is more accurate and precise in comparison with the afore-mentioned methods. On the basis of the given explanations in the previous section, the answer of Equation 16 in this method is considered by the AGM as polynomials series with constant coefficients as follows:

t a a t a t a t a t a t a t
The constant coefficients of Equation 20 which are

Applying initial or boundary conditions in AGM
Based on the given explanations in the previous section, the constant coefficients b) The application of initial or boundary conditions on the main differential equations which in this case are Equations 15 and 16 and also on it their derivatives is done in the following general form: where () IC is the abbreviation of initial conditions. Therefore, after substituting Equation 20 which has been considered as the answer of the main differential equation into f(t) and g(t), the initial conditions are applied on the obtained equation and also on its derivatives as follows: And for the higher derivative, we have: By solving a set of algebraic equations which is consisted of fourteen equations with fourteen unknowns, the constant coefficients  into Equation 20, the solution of the presented problem will be obtained as follows:

 
, the numerical solution of the mentioned problem is presented in Table 1.

Comparing the analytical and numerical solutions
The obtained analytical solution from Equations (30 and 31) and the numerical solution which its results have been presented in Table 1 can be compared as shown in Figures 9 and 10.

Conclusion
Due to the complexity of solving nonlinear differential equations for chemical reactions under the presented conditions, AGM (Akbari-Ganji's Method) is used and successfully applied for the governing equations. In this study, a complicated set of nonlinear differential equations have been introduced and analyzed completely by (AGM) and also the obtained results have been compared with the numerical method (Runge-Kutta) to show the ability of AGM for solving such problems. It is concluded that AGM is a reliable and precise approach for solving miscellaneous differential equations. Moreover, a summary of the AGM excellence and its benefits is explained as: By solving a set of algebraic equations with constant coefficients, we are able to obtain the solution of nonlinear differential equation very easily which applying this procedure is possible even for researchers with intermediate mathematical knowledge. On the other hand, it should be mentioned that AGM is able to solve linear and nonlinear differential equations directly in most of the situations that means the final solution can be obtained without any dimensionless procedure. Therefore, AGM can be considered as a significant progress in nonlinear sciences. Since the shortage of boundary condition(s) in this method for solving differential equation(s) is completely terminated, AGM is operational for miscellaneous nonlinear differential equations especially in the field of chemical engineering and we are hopeful         that this method will be applied by enthusiastic young researchers in the near future.