ANALYSIS OF ENZYMATIC HYDROLYSIS OF UREA IN A SINGLE PARTICLE: EFFECTS OF pH-DEPENDENT KINETICS, IONIC EQUILIBRIA, PRODUCT INHIBITION, AND NERNST-PLANCK DIFFUSION

Abstract

A mathematical model is developed to analyze urea hydrolysis in immobilized urease particles. The reaction rate is described by a modified Michaelis-Menten form which takes into account inhibition by ammonium ions and pH-dependent kinetics. The products of urea hydrolysis may exist in several ionic states, which are assumed to be at local equilibrium. A Nernst-Planck diffusion flux expression is used to describe the transport of the charged species, whose diffusivities differ by an order of magnitude. The coupled nonlinear differential equations are solved numerically using orthogonal collocation. Our simulation results indicate that at a bulk urea concentration less than K_M (Michaelis-Menten constant), the effectiveness factor approaches that for product-inhibited Michaelis-Menten kinetics. At higher urea concentrations, the reaction rate of urea hydrolysis within a single particle is dominated by the pH dependence of the kinetics. Ionic equilibria of the product species cause the solution pH to increase and to buffer at about pH = 9.2. The effectiveness factor can be reduced to about 20% of that for the product-inhibited Michaelis-Menten kinetics (for Thiele modulus near unity). The effect of noncompetitive product inhibition is negligible compared to the pH effects. Nernst-Planck diffusion effects are important only at ionic strengths less than 1 mM. Nernst-Planck diffusion results in a smaller flux of NH₄ ⁺, and thus a higher NH₄ ⁺concentration within a particle; the net effect is n small decrease in pH and subsequent increase in the reaction rate within the particle (the pH change is due to a shift in the NH₃/NH₄ ⁺equilibrium). At a urea concentration higher than K_M external mass transfer resistance is significant even at Biot number > 20, because the reaction is severely limited by the high pH within the particle.