An Approximate Analytical Solution Of Nonlinear Equations In N-Aminopiperidine Synthesis: New Approach Of Homotopy Perturbation Method)

the synthesis of N-aminopiperidine (NAPP) using hydroxylamine-O-sulfonic acid (HOSA) is based on system of nonlinear rate equations. The new approach to homotopy perturbation method is applied to solve the nonlinear equations. A simple analytical expression for concentrations of hydroxylamine-O-sulfonique acid (HOSA), piperidine (PP), N-aminopiperidine (NAPP), sodium hydroxide (NaOH) and diazene (N2H2) along with NAPP yield is obtained and is compared with numerical result. Satisfactory agreement is obtained in the comparison of approximate analytical solution and numerical simulation. The obtained analytical result of NAPP yield is compared with the experimental results. The influence of reagents ratio p and rate constants ratio r on yield has been discussed. Keywords—Kinetic modelling; Nonlinear rate equations; Mathematical modelling; Synthesis of NAPP; Homotopy perturbation method

An Approximate Analytical Solution Of Nonlinear Equations In N-Aminopiperidine Synthesis: New Approach Of Homotopy Perturbation Method) 596 R. Swaminathan, K. Lakshmi Narayanan, V. Mohan, K. Saranya and L. Rajendran [19] reported the usage of new approaches to solving nonlinear partial differential equations in microdisk biosensors using the homotopy perturbation method (HPM) for the first time. K. Saranya, V. Mohan, R. Kizek, C. Fernandez and L. Rajendran [20] discussed the theoretical model of glucose oxidation and hydrogen peroxide production in the chitosanalginate microsphere and also obtained approximate analytical expression for the concentration of glucose, oxygen, gluconic acid and hydrogen peroxide using HPM. K. Saranya, V. Mohan, R. Kizek, C. Fernandez and L. Rajendran [21] solved the mathematical problem corresponding to a one-electron reversible electron transfer at a rotating disk electrode by using HPM under transient and steady-state conditions. E. Labarthe, A.J. Bougrine, V. Pasquet and H. Delalu [22] developed the mathematical model of N-Aminopiperidine (NAPP) synthesis and solved the nonlinear equations using Runge-Kutta methods. To the author's knowledge no analytical expression for concentration in NAPP synthesis and yield has been reported. In this paper, we find a consistent approximate analytical expression of concentrations and yield in the synthesis of NAPP using a new approach to homotopy perturbation method for all values of rate constant and time. The obtained analytical results is used to predict the evolution of the system (yield) as a function of the reagent concentrations ratio and rate constants ratio. The accuracy of the derived expressions for concentration is compared with MATLAB generated numerical simulations and satisfactory agreement is noted. In the next section, the basic concept of the new approach to homotopy perturbation method is described.
II. BASIC CONCEPT OF NEW APPROACH TO HOMOTOPY PERTURBATION METHOD The HPM was proposed first by He [8] for solving differential and integral equations, both linear and nonlinear. In this section, we present a new approach to homotopy perturbation method (NAHPM) as a tool for solving nonlinear ordinary differential equations with initial conditions. HPM provides an approximate solution to a wide range of nonlinear problems, but NAHPM provides the approximate analytical solution with a minimum number of iterations. Consider a general system of nonlinear equation in the form: The boundary conditions are where A is any differential operator and B is a boundary operator and Γ is the boundary of the domain Ω. f(u,v) and g(u,v) are nonlinear terms. In this method, first we can write u(t) in Taylor series expression.
Using this Taylor series expression, we can bring the nonlinear equation (1) where p ∈ [0,1] is an impeding parameter, t ∈ Ω and u0(t) is the initial approximation to the solution of (1) that satisfies the boundary conditions. The NAHPM uses the parameter p as a small parameter, and assumes that the solutions of (5) can be written as a power series in p: where u0, u1, u2, … and v0, v1, v2,… are unknown functions to be determined. Substituting (7) and (8) into (5) and (6) and matching identical powers of p terms, provides the values of the sequence u0, u1, u2, … and v0, v1, v2,… iteratively. Then, as p → 1, the solution of (1) and (2) is given by: Here we can apply this method to solve the nonlinear problem in NAPP synthesis.
III. MATHEMATICAL FORMULATION OF NAPP SYNTHESIS PROBLEM N-aminopiperidine (NAPP) is used for plant protection, in cosmetology and photographic chemistry. Other uses include the manufacturing of paper and transparent recording films, inhibitor composition, and amine-597 based solvent-resistant compounds. The synthesis process of hydroxylamine-O-sulfonique acid (HOSA) has many advantages: it is as pure as the Raschig (water chemistry with no pollutant reagents) process. HOSA is more durable than monochloramine, easier to transport, more condensed solutions can be stored, and the synthesis can be operated batch-wise or continuously in a thinner fashion. E. Labarthe, A.J. Bougrine, V. Pasquet and H. Delalu [22] developed the kinetic modelling of synthesis of NAPP from HOSA and piperidine (PP) using differential equations. A reaction sequence for the NAPP synthesis with HOSA is described as follows: NAPP synthesis using hydroxylamine-O-sulfonic acid and piperidine is described by the system of nonlinear differential equations as follows [22]: The initial conditions are where x,a,u,b and f are the concentrations of hydroxylamine-O-sulfonique acid (HOSA), piperidine (PP), Naminopiperidine (NAPP), sodium hydroxide (NaOH) and diazene(N2H2), at instant t, respectively and k1 and k2 are rate constants. From (11) and (14), it is evident that Integrating on both sides and using the boundary From (11), (12) and (15), it is noticeable that From the above equation we get, Using this relation, the system of five ordinary differential equations (11)- (15) can be reduced to just three: An Approximate Analytical Solution Of Nonlinear Equations In N-Aminopiperidine Synthesis: New Approach Of Homotopy Perturbation Method) This is the nonlinear differential equation system that we are going to solve analytically using the new approach to homotopy perturbation method in a simple and closed form. The NAPP yield is defined as [22]: where uss denotes the steady state concentration of NAPP. IV. CONCENTRATIONS IN NAPP SYNTHESIS USING NEW APPROACH TO HOMOTOPY PERTURBATION METHOD Consider the functions x(p,t),a(p,t) and u(p,t) of the form: so that three homotopies can be constructed as: By substituting (26)-(28) into (29)-(31), and matching terms of the same power of p, we can compute the functions x0,x1,x2,… , a0,a1,a2,… and u0,u1,u2,… .The explicitly solvable sequence of ODEs can be obtained for (29)-(31) as follows: The corresponding initial conditions are Solving (32)-(35) using the boundary conditions (36), the following results can be obtained.
1 R. Joy Salomi, 2 S. Vinolyn Sylvia, 3 L. Rajendran where E1 represents exponential integral, ( 1 . The approximate solution of (21) -(23), when p→1, will be as follows: The approximate analytical expressions of b(t) and f(t) can be obtained easily by using x(t) and a(t) in (18) and (20) V. ANALYTICAL EXPRESSION OF STEADY STATE CONCENTRATIONS When t=∞, the concentration of HOSA, PP, NAPP, NaOH and diazene N2H2 are The analytical expression of NAPP yield using (25) is only on reagents ratio p and rate constants ratio r which is a function of pH and temperature.
VI. NUMERICAL SIMULATION The system of nonlinear rate equations was solved numerically to examine the accuracy of the solution obtained using a new approach to homotopy perturbation method. The numerical solution can be obtained using the MATLAB software program. In Figs. (1)-(5), analytical results of concentration of HOSA, PP, NAPP, NaOH and diazene are compared with the corresponding numerical results graphically for different values of the parameter. The comparison verified that the numerical results match very well with our analytical results obtained. The average deviation between the analytical and numerical results using the NAHPM is less than 1% (Refer to Table 1).

VII. RESULT AND DISCUSSION
Equations (42)-(44) represent the approximate analytical expressions of concentration of HOSA, PP, NAPP. The obtained analytical results are compared with numerical solution, which is provided in Figs. (1)- (6). From the figures it is observed that the concentration of NAPP and N2H2 is an increasing function, while that of HOSA, PP and NaOH are decreasing functions with respect to time.  Fig. 1 it is observed that the concentration of HOSA increases as the rate constants k1 and k2 decreases. Also it is noted that at t=100min, the concentration of HOSA reaches its steady-state value.  Fig. 2, it is inferred that the concentration of piperidine increases when the rate constant k1 and k2 decreases for some fixed values of x0 * and a0 * . As time increases it is observed that a(t) approaches a0*exp(-x0*/a0*) and after that it becomes steady.  Fig. 3(a), it is investigated that the 601 concentration of NAPP increases when k1 increases for some fixed values of x0 * and a0 * . The concentration of NAPP reaches the steady-state value when t=20min for different values of k1. From Fig. 3(b), it is noticed that u(t) and k2 are inversely related. When t=150min for varying values of k2, the NAPP concentration approaches steady-state value. This concentration stabilizes for larger value of t and it is observed to be b0*-2x0*. . From these figures, it is noted that the concentration of diazene increases when k1, k2 increases. As time approaches larger value, f(t) approaches a constant value which is given in (49). Fig. 6. Plot of concentration of HOSA (1), PP (2), NAPP (3), NaOH (4) and N2H2 (5) against time t for k1=3.39lmol -1 min -1 , k2=1.17lmol -1 min -1 , x0 * =1.23x10 -3 mol/l, a0 * =1.23x10 -2 mol/l, b0 * =5.33x10 -2 mol/l. Solid line represents numerical result and dotted line represents analytical result Fig. 6 illustrates the behaviour of all concentration profiles. From this figure, it is observed that the concentration of HOSA (x(t)) is a strictly decreasing function, whereas the concentration of NAPP (u(t)) is a strictly increasing function. A. Influence of reagent ratio p & ratio of rate constants r on yield Fig. 7 (a). Comparison of analytical result (50) of yield with experimental result [22] at k1=3.39lmol -1 min -1 and k1=1.17lmol -1 min -1 (b). Effect of ratio of rate constants r and reagents ratio p on yield (50) (c). Profile of yield φu over reagent ratio p and ratio of rate constants r. The comparison of analytical results of NAPP yield with experimental results for fixed values of k1 and k2 is shown in Fig. 7(a) The satisfactory agreement is noted. From Fig. 7(b) it is observed that yield decreases for a higher ratio of rate constants r. The NAPP yield reaches 90% at p=8. For p>8, there is a negligible increase in the yield. It is noted that the NAPP yield increases for higher reagents ratio p. It can be found from Fig. 7(c) that when the rate constant ratio r is very small and the reagent ratio p exceeds 15, the peak value of yield is obtained.
An Approximate Analytical Solution Of Nonlinear Equations In N-Aminopiperidine Synthesis: New Approach Of Homotopy Perturbation Method) 602 B. Differential sensitive analysis of parameters: ratio of rate constants r and reagents ratio p Fig. 8. Sensitivity analysis of parameters, ratio of rate constants r and reagents ratio p : Percentage change in yield φu. Equation (50) represents the new approximate analytical expression for NAPP yield in terms of the parameters such as the ratio of rate constants r and reagents ratio p. By differentiating the NAPP yield partially for these parameters, the impact of the parameters over NAPP yield can be determined [23,24]. The percentage of change in NAPP yield for r and p are 86% and 14 %. From this, it is evident that the ratio of rate constants r has more impact on NAPP yield when it is varied. In contrast, the remaining parameter, reagents ratio p accounts for only small changes in yield. The analysis of these results is described in Fig. 8 and confirmed in the Figs. 7(a) -7(c). VIII. CONCLUSION The new analytic approach of the homotopy perturbation method is being used because of the difficulties in solving the nonlinear differential equations in NAPP synthesis. In this paper, the time-dependent nonlinear rate equation was solved analytically. The obtained results have a good agreement with those numerical and experimental results. This work's primary result is a simple approximate calculation of concentration and yield for all possible parameters such as the ratio of rate constants r and reagents ratio p. It is beneficial and simple, and only a few iterations are needed to find an approximate solution. The analytical results are useful to characterize the reaction mechanism and predict the behaviour of NAPP synthesis.