Characteristic Time for Gas-Phase Diffusion to a Particle

Let us assume that we introduce a particle of radius Rp consisting of a species A in an atmosphere with a uniform gas-phase concentration of A equal to c. Initially, the concentration profile of A around the particle will be flat and eventually after time xdg it will relax to its steady state. This timescale, xdg, corresponds to the time required by gas-phase diffusion to establish a steady-state profile around a particle. It should not be confused with the timescale of equilibration of the particle with the surrounding atmosphere. We assume that cx remains constant and that the concentration of A at the particle surface (equilibrium) concentration is cs and also remains constant. 

This problem was studied in Section 12.1.1, and the change of the concentration profile with time is given by (12.8). The characteristic timescale of the problem can be derived by nondimensionalizing the differential equation describing the problem, namely, (12.4). The characteristic lengthscale of the problem is the particle radius Rp, the characteristic concentration cx, and the characteristic timescale, our unknown xdg. We define dimensionless variables by dividing the problem variables by their characteristic values, 

FIGURE 12.5 Schematic of gas- and aqueous-phase steady-state concentration profiles for the case where there are gas-phase, interfacial, and aqueous-phase mass transport limitations. Also shown is the ideal case where there are no mass transport limitations. 

Note that all the scaling in the problem is included in the dimensionless term (RjJzjgDg) and, as the rest of the terms in the differential equation are of order one, this term should also be of order unity. Therefore 

This timescale can also be obtained from the complete solution of the problem in (12.8). Note that the time-dependent term approaches zero as the upper limit of the integration approaches zero or when l /Dgi r - Rp or equivalently when (r - RP)2/4 Dg. For a point close to the particle surface (r - Rp f Rp and 

---