Analytical expressions of concentration of nitrate pertaining to the electrocatalytic reduction of nitrate ion
Highlights
► Theoretical analysis of electrochemical reduction of nitrate ion is presented. ► Demonstrated the applicability of the HAM for nonlinear reaction equations. ► Our results are in good agreement with the numerical results.
Introduction
Interest in the electrochemical reduction of nitrate and nitrite ions has increased in the past several years [1], [2], [3], [4], [5]. It is to decrease the concentration of the contaminants in drinking water. At present, the major health concern for nitrate in water is that it can lead to methemoglobinemia and gastric cancer in humans and animals. The uptake of nitrate in the human body causes oxygen-deficiency such as cyanosis and breathing problems. In particular, infants are seriously affected, and this disease in infants is known as blue-baby syndrome. Such toxicity is not due to nitrate itself but to nitrite formed from it. The nitrite oxidizes hemoglobin to methemoglobin, resulting in loss of the capability of carrying oxygen. The most common problem of the reduction of nitrate is the formation of side-products such as nitrite, ammonia and hydroxylamine, which are more toxic than the removed nitrate [6], [7]. It is also believed that N-nitrosamines formed from nitrite and amines cause an increased possibility of cancer and diabetes. Despite this concern, investigations of nitrate electroreduction have been limited to acidic or alkaline systems without consideration of neutral pH solutions that may well reflect typical drinking water. Several of the reported investigations were carried out in either acidic or alkaline concentrated nitrate and nitrite ion solutions for environmental reasons [3], [5], [8].
Lu et al. [2] used different hydrogen storage cathodes to reduce nitrate to ammonia in alkaline media and found that the efficiency of ammonia production depended on the nitrate ion concentration, electrode type and applied current density. Both galvanostatic and potentiodynamic electroreduction were investigated. The reduction of nitrate ion has received considerable attention over the last four decades, mainly because of the possibility of utilizing abundant and inexpensive sources for the production of useful chemicals [9], [10], [11], [12], [13], [14]: nitrous oxide (N2O), a compound used as an anesthetic in medical applications; ammonia (NH3), a nitrogen source in fertilizers; hydroxylamine (NH2OH), a material very important in the manufacture of caprolactam. Numerous efforts have been made so far, concerning the electrochemical reduction of nitrate on various metallic electrodes [15], [16], [17], [18] and graphite. This method is applicable when nitrite, nitrogen and ammonia are the only products of the reduction of nitrate. The reduction of nitrate also gives other products such as hydroxylamine and nitrous oxide.
An area in which the conversion of nitrate into NH3 may also find application is the recycling of strongly alkaline solutions used in the neutralization of nitric acid waste clean-up solutions in the nuclear industry [19]. Moreover, the increasing interest in the reduction of nitrate is also connected with environmental problems and the increasing cost of the purification of drinking water [20]. Several water treatment methods for nitrate removal are available [21], but R&D work is still needed to improve both process performances and economics [22], [23], [24]. In fact the challenge between different technologies depends on desired water quality, plant capacity, process automation or access to manpower of suitable technical skill.
Electro-catalytic reduction of nitrate has been experimentally studied in the last 30 years [1], [3], [4], [14], [25], [26], [27], [28]. Moreover, the concentration of nitrate can influence the rate of the reduction and the distribution of the products, as it has been previously reported [29], [30], [31], [32], [33], [34]. To our knowledge, no general analytical expressions for the concentration of nitrate, nitrite, ammonia and nitrogen against the electrolysis time t and current have been reported for all values of the rate constants k1, k2 and k3 [10]. The purpose of this paper is to derive an analytical expression for the concentration of nitrate, nitrite, ammonia, nitrogen and current for all values of the rate constants k1, k2 and k3 using Homotopy Analysis Method (HAM).
Section snippets
Mathematical formulation of the problem
The first order kinetics for both nitrate and nitrite was performed assuming the following reduction scheme [10]:The nonlinear differential equations for the concentrations of the various products in (1) may be written as follows [10]:Here and [N2] denote the concentration of nitrate, nitrite, ammonia and nitrogen against the electrolysis time t. k1, k2 and k3
Homotopy Analysis Method (HAM)
The system of Eqs. (3), (4), (5) cannot be solved using rigorous standard analytical method because of nonlinearity. Recently a general approach to obtain a series solution of nonlinear differential equation is discussed in [35]. Liao [36], [37] proposed a powerful analytical method for solving nonlinear problems, namely the homotopy analysis method (see Appendix A). Different from all reported perturbation and nonperturbative techniques, the homotopy analysis method [38], [39], [40], [41], [42]
Discussion
Concentration of nitrate and by-products depend upon the rate constants k1, k2, k3 and C0 (initial concentration of nitrate). The initial concentration of and [N2] = 0. The nonlinear differential Eqs. (3), (4), (5), (6) are solved analytically and numerically. The function ode45 in Scilab software is used to solve two-point boundary value problems (BVPs) for ordinary differential equations. Fig. 1, Fig. 2, Fig. 3 illustrate the comparison of analytical result
Conclusions
In this work, we obtained analytical expressions of concentration of nitrate, nitrite, ammonia and nitrogen for all values of rate constant. Time-dependent nonlinear reaction equations have been formulated and solved analytically. The closed analytical expressions of concentrations and current are obtained using homotopy analysis method. An excellent agreement with the numerical result is noted. The information gained from this theoretical model can be useful for the kinetic analysis of the
Acknowledgements
It is our pleasure to thank the referees for their valuable comments. This work was supported by the Council of Scientific and Industrial Research (CSIR No.: 01(2442)/10/EMR-II), Government of India. The authors also thank Mr. M.S. Meenakshisundaram, Secretary, The Madura College Board, The Principal, Madura College, Madurai, India for their constant encouragement.
Dr. L. Rajendran received his M.Sc. in Mathematics in 1981 from Presidency College, Chennai, TN, India. He obtained his Ph.D. in Applied Mathematics from Alagappa University, Karaikudi, TN, India during 2000. At present, he is an Assistant Professor in Mathematics at Madura College, Madurai, TN, India. Before this position (1986–2007), he was working as a Post Graduate Assistant in Mathematics at SMSV Higher Secondary School, Karaikudi, TN, India. He has 20 years teaching experience and 15 years
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