Mathematical Modeling

Abstract The article contains sections titled: 1. Introduction 1.1. Terminology 1.2. Application Areas of Mathematical Modeling in Industrial Chemistry and Chemical Engineering 1.3. Limitations of Mathematical Models 2. Construction and Classification of Mathematical Models 2.1. Construction of Mathematical Models 2.2. Classification of Mathematical Models 3. Empirical Models 3.1. Linear Empirical Models 3.1.1. Linear Empirical Models with One Variable 3.1.1.1. Parameter Estimation (Linear Regression) 3.1.1.2. Assessment of Estimated Parameter Values 3.1.1.3. Sensitivity Analysis 3.1.1.4. Example of a Linear Model 3.1.1.5. Concluding Remarks 3.1.2. Linear Empirical Models with Several Variables 3.1.2.1. Parameter Estimation 3.1.2.2. Assessment of Estimated Parameter Values 3.1.2.3. Concluding Remarks 3.2. Nonlinear Empirical Models 3.2.1. Parameter Estimation (Nonlinear Regression) 3.2.1.1. Transformation of the Model into Linear Form 3.2.1.2. Direct Search Methods 3.2.1.3. Gradient Methods 3.2.2. Assessment of Estimated Parameter Values 3.2.3. Example of a Nonlinear Model 3.2.4. Concluding Remarks 3.3. Further Calculation Methods and Model Types 3.3.1. Further Methods of Parameter Estimation 3.3.2. Other Types of Models 3.3.2.1. Models with Constraints on the Parameters 3.3.2.2. Models Based on Differential Equations 4. Models Based on Transport Equations for Probability Density Functions 4.1. Terminology 4.1.1. One‐Dimensional Distribution and Probability Density Functions 4.1.2. Multidimensional Distribution and Probability Density Functions 4.1.3. Conditional Probability Density Functions 4.2. Transport Equations for Single‐Point Probability Density Functions 4.2.1. General form of the Transport Equations for Probability Density Functions 4.2.2. Limitations of Single‐Point Probability Density Functions 4.3. Examples of Calculating Probability Density Functions 4.3.1. Solutions for Deterministic Systems 4.3.1.1. Age (Residence Time) Distributions in Chemical Reactors 4.3.1.2. Size Distribution in Continuously Operating Crystallizers 4.3.2. Solutions for Statistical Systems 4.3.2.1. Closure of the Transport Equation for Probability Density Functions 4.3.2.2. Solution Methods of the Transport Equation for Probability Density Functions 4.3.2.3. Example: Combustion of Propane in a Turbulent Diffusion Flame 5. Models Based on Physicochemical Principles (Transport Phenomena) 5.1. Application of the Principle of Conservation of Momentum 5.1.1. Laminar Tube Flow 5.1.2. Turbulent, Nonreactive Fre