On a Model for Generating Theoretical Crystal Size Distributions (CSDs) in Igneous Systems: A Moment Transformation Approach

摘要

A moment transformation is applied to the batch population balance equation (BPBE), a number continuity equation describing the evolution of crystal populations in closed magmatic systems such as sills. The BPBE is a key component of crystal size distribution (CSD) theory that describes a quantitative petrographic framework for investigating crystalline rocks. The moment transformation yields a system of nonlinear ordinary differential equations (ODEs) that is often more easily solved (usually numerically) when terms that are functions of the evolving population are included, and when they are augmented with expressions for mass and energy balances. Solutions to the system of nonlinear ODEs yield moments of the crystal population as a function of time. Moments of an evolving crystal population are generated here using the moment-transformed BPBE. Crystal growth rate, G, is inversely proportional to the second CSD moment (a measure of total surface area in a crystal population) and directly proportional to the amount of mass available for solidification provided by an energy balance expression incorporating latent heat. Nucleation rate, I, is a function of cooling rate. This combination of G and I generates CSDs and properties of suites of CSDs observed in natural rocks (Resmini 2007). The set of equations is solved numerically with a fourth-order Runge–Kutta scheme. The moments of the crystal population are virtually identical to those generated by the numerical