Mathematical Modeling
Authors: Bockhorn Henning 
Published: December 2006

The article contains sections titled:

1.Introduction1.1.Terminology1.2.Application Areas of Mathematical Modeling in Industrial Chemistry and Chemical Engineering1.3.Limitations of Mathematical Models2.Construction and Classification of Mathematical Models2.1.Construction of Mathematical Models2.2.Classification of Mathematical Models3.Empirical Models3.1.Linear Empirical Models3.1.1.Linear Empirical Models with One Variable3.1.1.1.Parameter Estimation (Linear Regression)3.1.1.2.Assessment of Estimated Parameter Values3.1.1.3.Sensitivity Analysis3.1.1.4.Example of a Linear Model3.1.1.5.Concluding Remarks3.1.2.Linear Empirical Models with Several Variables3.1.2.1.Parameter Estimation3.1.2.2.Assessment of Estimated Parameter Values3.1.2.3.Concluding Remarks3.2.Nonlinear Empirical Models3.2.1.Parameter Estimation (Nonlinear Regression)3.2.1.1.Transformation of the Model into Linear Form3.2.1.2.Direct Search Methods3.2.1.3.Gradient Methods3.2.2.Assessment of Estimated Parameter Values3.2.3.Example of a Nonlinear Model3.2.4.Concluding Remarks3.3.Further Calculation Methods and Model Types3.3.1.Further Methods of Parameter Estimation3.3.2.Other Types of Models3.3.2.1.Models with Constraints on the Parameters3.3.2.2.Models Based on Differential Equations4.Models Based on Transport Equations for Probability Density Functions4.1.Terminology4.1.1.One‐Dimensional Distribution and Probability Density Functions4.1.2.Multidimensional Distribution and Probability Density Functions4.1.3.Conditional Probability Density Functions4.2.Transport Equations for Single‐Point Probability Density Functions4.2.1.General form of the Transport Equations for Probability Density Functions4.2.2.Limitations of Single‐Point Probability Density Functions4.3.Examples of Calculating Probability Density Functions4.3.1.Solutions for Deterministic Systems4.3.1.1.Age (Residence Time) Distributions in Chemical Reactors4.3.1.2.Size Distribution in Continuously Operating Crystallizers4.3.2.Solutions for Statistical Systems4.3.2.1.Closure of the Transport Equation for Probability Density Functions4.3.2.2.Solution Methods of the Transport Equation for Probability Density Functions4.3.2.3.Example: Combustion of Propane in a Turbulent Diffusion Flame5.Models Based on Physicochemical Principles (Transport Phenomena)5.1.Application of the Principle of Conservation of Momentum5.1.1.Laminar Tube Flow5.1.2.Turbulent, Nonreactive Free Jets5.1.2.1.Models for Reynolds Stresses5.1.2.2.Solution Method of the Resulting System of Partial Differential Equations5.1.2.3.Example: Turbulent Flow of Nitrogen in Air5.2.Applications of the Principle of Conservation of Enthalpy5.2.1.Heat Conduction5.2.2.Heat Transfer5.2.2.1.Exact Solution for a Boundary Layer Problem5.2.2.2.General Principles of Modeling Heat Transfer5.3.Applications of the Law of Conservation of Mass5.3.1.Mass Transfer without Chemical Reaction5.3.1.1.Exact Solution for a Boundary Layer Problem5.3.1.2.General Principles for Modeling Mass Transfer5.3.1.3.Examples of Simultaneous Mass and Heat Transfer: Dynamic Models5.3.1.4.Examples of Simultaneous Mass and Heat Transfer: Static Models5.3.2.Mass Transfer with Chemical Reactions5.3.3.Chemical Reactions in the Homogeneous Phase5.3.3.1.Isothermal Reactors with Frictionless Flow, Constant Density, and Reactions Without Volume Changes5.3.3.1.1.Stability of Isothermal Reactors5.3.3.1.2.Sensitivity Analysis5.3.3.2.Nonisothermal Reactors5.3.3.2.1.Heterogeneous Catalytic Reactions5.3.3.2.2.Stability Analysis of Nonisothermal Reactors5.3.3.2.3.Use on Statistical Processes