Competitive effects between stationary chemical reaction centres: A theory based on off-center monopoles

Abstract

The subject of this paper is competitive effects between multiple reaction sinks. A theory based on off-center monopoles is developed for the steady-state diffusion equation and for the convection-diffusion equation with a constant flow field. The dipolar approximation for the diffusion equation with two equal reaction centres is compared with the exact solution. The former turns out to be remarkably accurate, even for two touching spheres. Numerical evidence is presented to show that the same holds for larger clusters (with more than two spheres). The theory is extended to the convection-diffusion equation with a constant flow field. As one increases the convective velocity, the competitive effects between the reactive centres gradually become less significant. This is demonstrated for a number of cluster configurations. At high flow velocities, the current methodology breaks down. Fixing this problem will be the subject of future research. The current method is useful as an easy-to-use tool for the calibration of other more complicated models in mass and/or heat transfer.

The Steady-State Convection-Diffusion Equation at High Péclet Numbers for a Cluster of Spheres: An Extension of Levich's Theory

“…The question to be addressed now is how to use the single-sphere's concentration field to model competitive effects between two (or more) spheres. A brief review is given below of the methodology developed in our previous paper [6]. A crucial ingredient in this method is the knowledge of a Green's function for the corresponding equation.…”

“…In a recent paper [6] we developed an approximate analytical theory for such a system in the case of pure diffusion and in the case of diffusion combined with a relatively weak convective component. It was found that the range of applicability of the former theory in the convection-diffusion (CD) case was very limited; beyond Péclet numbers of about 0.6, our theory was found to be unreliable.…”

“…The reader is cautioned that with this definition, Pe is 2 times larger than the variable η used in our previous publication [6] and half as large as the Péclet number as used in many engineering texts (which usually take the sphere's diameter rather than its radius as the unit of length). Following Levich, we are primarily interested in the case that Pe 1, in other words, that 1.…”

“…The motion of the microscopic species consists of a combination of stochastic movements (diffusion) and externally imposed motion by a carrier fluid (convection). The macroscopic absorption centers are assumed to be spheres of which the locations are known and fixed; they are assumed to be all of equal size.In a recent paper [6] we developed an approximate analytical theory for such a system in the case of pure diffusion and in the case of diffusion combined with a relatively weak convective component. It was found that the range of applicability of the former theory in the convection-diffusion (CD) case was very limited; beyond Péclet numbers of about 0.6, our theory was found to be unreliable.…”

“…The velocity field − → u ( − → r ) is assumed to be known; c( − → r ) is the concentration field that is to be computed. In [6] it was assumed that the velocity field is constant and unidirectional everywhere in space outside the sphere (for another series of papers that is also based on a constant velocity field, we refer the reader to [9,12,13]). This choice, however, leads to unrealistic mass distributions.…”