Self-Similar Solutions and Global Existence for Nonlinear Reaction-Diffusion Systems in Industrial Ammonia Synthesis

TL;DR

This paper analyzes nonlinear reaction-diffusion models for ammonia synthesis, constructing self-similar solutions, establishing conditions for global existence, and illustrating results with numerical simulations relevant to industrial processes.

Contribution

It introduces a novel application of Lie group analysis to derive self-similar solutions and provides rigorous conditions for global existence in different diffusion regimes.

Findings

Existence of self-similar solutions for the reaction-diffusion system.

Conditions for global-in-time solutions in slow and fast diffusion regimes.

Numerical simulations confirming theoretical asymptotic behaviors.

Abstract

This paper investigates a system of nonlinear reaction-diffusion equations modeling the industrial synthesis of ammonia. By applying Lie group analysis, we construct self-similar solutions and derive a reduced system of ordinary differential equations. Using comparison principles and barrier techniques, we establish sufficient conditions for the existence of global-in-time solutions in both slow-diffusion ($\gamma_i > 0$) and fast-diffusion ($\gamma_i < 0$) regimes. Detailed asymptotic analysis near the diffusion front reveals power-law behavior of concentration profiles, with explicit expressions for the decay exponents. The theoretical results are illustrated by numerical simulations, demonstrating the spatio-temporal evolution of reactant concentrations under realistic parameter values. The study provides rigorous mathematical foundations for predicting and optimizing ammonia production in catalytic reactors, with potential extensions to other chemically reacting systems.