# Chapter 3 Dynamic Modeling.

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Chapter 3 Dynamic Modeling

Overall Course Objectives
Develop the skills necessary to function as an industrial process control engineer. Skills Tuning loops Control loop design Control loop troubleshooting Command of the terminology Fundamental understanding Process dynamics Feedback control

Process Dynamics Chemical Engineering courses are generally taught from a steady-state point-of-view. Dynamics is the time varying behavior of processes. Chemical processes are dynamically changing continuously. Steady-state change indicates where the process is going and the dynamic characteristics of a system indicates what dynamic path it will take.

Uses of Dynamic Process Models
Evaluation of process control configurations For analysis of difficult control systems for both existing facilities and new projects Process design of batch processes Operator Training Start-up/shut-down strategy development

Classification of Models
Lumped parameter models- assume that the dependent variable does not change with spatial location within the process, e.g., a perfectly well mixed vessel. Distributed parameter models- consider that the dependent variable changes with spatial location within the process.

Example of a Lumped Parameter Process

Example of a Distributed Parameter Process

Modeling Approaches Lumped parameter processes- Macroscopic balances are typically applied for conservation of mass, moles, or energy and result in ODE’s. Distributed parameter processes- Microscopic balances are typically applied yielding differential equations for conservation of mass, moles, or energy for a single point in the process which result in PDE’s.

Conservation Equations: Mass, Moles, or Energy Balances

Mass Balance Equation

Accumulation Term

Other Terms in Mass Balance Eq.

Mole Balance Equation

Accumulation Term

Other Terms in Mass Balance Eq.

Thermal Energy Balance Equation

Accumulation Term

Other Terms in Energy Balance Eq.

Constitutive Relationships
Usually in the form of algebraic equations. Used with the balance equations to model chemical engineering processes. Examples include: Reaction kinetic expressions Equations of state Heat transfer correlation functions Vapor/liquid equilibrium relationships

Degree of Freedom Analysis
The number of degrees of freedom (DOF) is equal to the number of unknowns minus the number of equations. When DOF is zero, the equations are exactly specified. When DOF is negative, the system is overspecified. When DOF is positive, it is underspecified.

Different Types of Modeling Terms
Dependent variables are calculated from the solution of the model equations. Independent variables require specification by the user or by an optimization algorithm and represent extra degrees of freedom. Parameters, such as densities or rate constants, are constants used in the model equations.

Dynamic Models of Control Systems
Control systems affect the process through the actuator system which has its own dynamics. The process responds dynamically to the change in the manipulated variable. The response of the process is measured by sensor system which has its own dynamics. There are many control systems for which the dynamics of the actuator and sensor systems are important.

Dynamic Modeling Approach for Process Control Systems

Dynamic Model for Actuators
These equations assume that the actuator behaves as a first order process. The dynamic behavior of the actuator is described by the time constant since the gain is unity

Heat addition as a Manipulated Variable
Consider a steam heated reboiler as an example. A flow control loop makes an increase in the flow rate of steam to the reboiler. The temperature of the metal tubes in the reboiler increases in a lagged manner. The flow rate of vapor leaving the reboiler begins to increase. The entire process is lumped together into one first order dynamic model.

Dynamic Response of an Actuator (First Order System)

Dynamic Model for Sensors
These equations assume that the sensors behave as a first order system. The dynamic behavior of the sensor is described by the time constant since the gain is unity T and L are the actual temperature and level.

Dynamic Model for an Analyzer
This equation assumes that the analyzer behaves as a pure deadtime element. The dynamic behavior of the sensor is described by the analyzer deadtime since the gain is unity

Dynamic Comparison of the Actual and Measured Composition

Model for Product Composition for CSTR with a Series Reaction

Model for Cell Growth in a Fed-Batch Reactor

Class Exercise: Dynamic Model of a Level in a Tank
Model equation is based on dynamic conservation of mass, i.e., accumulation of mass in the tank is equal to the mass flow rate into the tank minus the mass flow rate out.

Class Exercise Solution: Dynamic Model for Tank Level
Actuator on flow out of the tank. Process model Level sensor since the level sensor is much faster than the process and the actuator

Sensor Noise Noise is the variation in a measurement of a process variable that does not reflect real changes in the process variable. Noise is caused by electrical interference, mechanical vibrations, or fluctuations within the process. Noise affects the measured value of the controlled variable; therefore, it should be included when modeling process dynamics.

Modeling Sensor Noise Select standard deviation (s) of noise. s is equal to 50% of repeatability. Generate random number. Use random number in a correlation for the Gaussian distribution which uses s. This result is the noise on the measurement. Add the noise to the noise free measurement of the controlled variable.

Numerical Integration of ODE’s
Accuracy and stability are key issues. Reducing integration step size improves accuracy and stability of explicit integrators The ODE’s that represent the dynamic behavior of control systems in the CPI are not usually very stiff. As a result, a Euler integrator is usually the easiest and most effective integrator to use.

Development of Dynamic Process Models for Process Control Analysis
It is expensive, time consuming, and requires a specific expertise. It is typically only used in special cases for particularly difficult and important processes.

Overview Dynamic modeling for process control analysis should consider the dynamics of the actuator, the process, and the sensor as well as sensor noise.

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