Process Control and Optimization

2.

Mathematical Modeling of Chemical Processes

2.1.

Rationale for Process Modeling

2.1.1.

Predicting Process Behavior

2.1.2.

Design and Optimization

2.1.3.

Control System Design

2.1.4.

Training and Education

2.2.

Types of Models

2.2.1.

Theoretical Models (First-Principles)

2.2.1.1.

Physical Laws and Principles

2.2.1.2.

Model Assumptions and Simplifications

2.2.2.

Empirical Models (Black-Box)

2.2.2.1.

Data-Driven Approaches

2.2.2.2.

Regression and System Identification

2.2.3.

Semi-Empirical Models (Grey-Box)

2.2.3.1.

Combining Theory and Data

2.3.

Development of Dynamic Models

2.3.1.

Conservation Laws

2.3.1.1.

Conservation of Mass

2.3.1.1.1.

Total Mass Balances

2.3.1.1.2.

Component Material Balances

2.3.1.2.

Conservation of Energy

2.3.1.2.1.

Energy Balances

2.3.1.2.2.

Enthalpy and Heat Effects

2.3.1.3.

Conservation of Momentum

2.3.2.

Constitutive Relationships

2.3.2.1.

Equations of State

2.3.2.2.

Reaction Kinetics

2.3.2.3.

Transport Equations

2.3.2.3.1.

2.3.2.3.2.

2.3.2.3.3.

Momentum Transfer

2.3.3.

Degrees of Freedom Analysis

2.3.3.1.

Definition and Purpose

2.3.3.2.

Application to Process Systems

2.4.

The Laplace Transform

2.4.1.

Definition and Properties

2.4.1.1.

2.4.1.2.

2.4.1.3.

Frequency Shifting

2.4.1.4.

Initial and Final Value Theorems

2.4.2.

Inverse Laplace Transform

2.4.3.

Solving Linear Ordinary Differential Equations

2.4.3.1.

Application to Process Models

2.5.

Transfer Functions

2.5.1.

Definition and Derivation

2.5.1.1.

From Differential Equations to Transfer Functions

2.5.2.

Poles and Zeros

2.5.2.1.

Physical Interpretation

2.5.2.2.

Effect on System Response

2.5.3.

Steady-State Gain

2.5.3.1.

Calculation and Significance

2.5.4.

Transfer Functions for Common Elements

2.5.4.1.

First-Order Systems

2.5.4.2.

Second-Order Systems

2.5.4.3.

Integrators and Differentiators

2.5.4.4.

Pure Time Delay

2.6.

Dynamic Behavior of First-Order Systems

2.6.1.

Mathematical Representation

2.6.2.

2.6.2.1.

Analytical Solution

2.6.2.2.

Graphical Interpretation

2.6.3.

Impulse Response

2.6.4.

2.6.5.

2.6.5.1.

Physical Meaning

2.6.5.2.

Determination from Data

2.6.6.

Dead Time in First-Order Systems

2.7.

Dynamic Behavior of Second-Order Systems

2.7.1.

Mathematical Representation

2.7.2.

Overdamped Response

2.7.3.

Critically Damped Response

2.7.4.

Underdamped Response

2.7.5.

Damping Coefficient

2.7.6.

Natural Frequency and Period

2.7.7.

Step and Impulse Responses

2.8.

Systems with Time Delays (Dead Time)

2.8.1.

Causes of Dead Time

2.8.2.

Mathematical Representation

2.8.3.

Effects on System Behavior

2.8.4.

Approximation Methods

2.9.

Linearization of Nonlinear Models

2.9.1.

Taylor Series Expansion

2.9.2.

Validity and Limitations

2.9.3.

Application to Process Models

2.9.4.

Deviation Variables

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