Optimization of the Numerical Solution of the Population Balance Equation
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The population balance equation (PBE) is a frequently applied equation in the modeling of multiphase reactors. Accurate solution of the PBE is important in order to have a precise estimate of the interfacial area, which governs the rate of mass, heat and momentum transfer between the phases. Solving the PBE has a relatively large computational demand. A reactor model often also takes into account chemical reactions and fluid flows, creating a multitude of equations that need to be solved simultaneously. A solution of the PBE that is numerically efficient is desirable. This thesis uses the orthogonal collocation numerical method in order to solve the PBE. The solution of an implemented spectral element method is compared to the standard spectral solver. The spectral element numerical mesh is constructed using a hp adaptive procedure. The procedure is based on calculating the error and smoothness of the various elements. The error is calculated based on a difference in solution between a high resolution spectral method and the elements of a the spectral element solver. The smoothness is calculated based on the Sobolev regularity index. In addition, the properties of the hp adaptive method is compared to a h adaptive method and a p adaptive method. Results show that for most cases, the total error of the PBE decreases at a similar rate for h, p and hp adaptive methods. The error is more uniformly distributed in the entire computational domain in the cases for h and hp refinement. This leads to a larger amount of numerical solution points for p refinement in order to achieve the same local error tolerance. Inaccuracies in the calculated error and smoothness are identified as key points for improvement a hp adaptive method. Results also show that the ability to combine elements or decreasing polynomial order is an important feature to achieve the most computationally efficient grid.