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Methods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established method to generate these random chords, and we describe a geometric randomness model associated with it. We verify our results by computing measures for hyper-ellipsoids and certain non-convex sets.

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Multistaged crystallization systems are used in the production of many chemicals. In this article, employing the population balance framework, we develop a model for a column crystallizer where particle agglomeration is a significant growth mechanism. The main part of the model can be reduced to a system of integrodifferential equations (IDEs) of the Volterra type. To solve this system simultaneously, we examine two numerical schemes that yield a direct method of solution and an implicit Runge–Kutta type method. Our numerical experiments show that the extension of a Hermite predictor–corrector method originally advanced in Khanh (1994) for a single IDE is effective in solving our model. The numerical method is presented for a generalization of the model which can be used to study and simulate a number of possible operating profiles of the column.

The framework of population balance equations has been employed widely in the modeling of particulate processes. When considering particle growth dispersion and agglomeration, one obtains a second-order nonlinear partial integrodifferential equation. In this article, we develop a rather simple numerical scheme using a Rothe method coupled with a modified simple shooting method (MSSM) to solve initial and boundary value problems for such equations on rectangular regions in the plane. We examine the implementation of this scheme and its performance on some examples. The use of the MSSM is effective in removing the nonphysical oscillations that arise using the Rothe method. Moreover, a diffusion-convection equation arises as a special case of our model. We show that the Rothe-MSSM method effectively removes the oscillations that are typically reported with the Rothe method or the method of lines for such equations.