DIFFUSION-CONTROLLED QUASI-STATIONARY MASS TRANSFER FOR AN ISOLATED SPHERICAL PARTICLE IN AN UNBOUNDED MEDIUM

Abstract

A consolidated mathematical formulation of the spherically symmetric mass transfer problem is presented, with the quasi-stationary approximating equations derived from a perturbation point of view for the leading-order effect. For the diffusion-controlled quasi-stationary process, a mathematically complete set of the exact analytical solutions is obtained in implicit forms to cover the entire parameter range. Furthermore, accurate explicit formulas for the particle radius as a function of time are also constructed semi-empirically for convenience in engineering practice. Both dissolution of a particle in a solvent and growth of it by precipitation in a supersaturated environment are considered in the present work.

Keywords:

• Diffusion
• Dissolution
• Mass transfer
• Mathematical analysis
• Particle
• Precipitation

Acknowledgment

The author is indebted to Professor Richard Laugesen of the University of Illinois for his helpful discussions and skillful illustration of mathematical manipulations. The author also wants to thank Yen-Lane Chen, Scott Fisher, Ismail Guler, Cory Hitzman, Steve Kangas, Travis Schauer, and Maggie Zeng of BSC for their consistent support.

Notes

¹Here the solubility [Cbreve] _S is treated as a constant, implying that the particle size effect on solubility, as may be observed for submicron particles (often due to significant surface energy influence), is ignored for theoretical simplicity.

²It seems though the analytical solution to (16) obtained by Krieger et al. (1967) was not noticed by Chen and Wang (1989) and Rice and Do (2006). However, Krieger et al. (1967) used their “highly nonlinear” implicit formula merely to iteratively determine the value of diffusion coefficient, while Chen and Wang (1989) solved the same equation for an analytical solution with the application in drug particle dissolution in mind. The recent work of Rice and Do (2006) again obtained “an exact analytical solution” by solving the same mathematical problem, although with a slight change in the form of a parameter to account for the “bulk flow effect” as in (3).

³Actually, if the identity is used, we can have (31) in a similarly looking form to (18) as

where coth⁻¹ denotes the inverse hyperbolic cotangent function.