Chapter 3: Study Notes on Linear Transfer Functions for Chemical Processes

Chapter 3: Development of Linear Transfer Functions for Chemical Processes

1. Necessity of Modeling

• Modeling is essential for:

  • Production Scheduling

  • Supervisory Control

  • Plant-wide Optimization

  • Optimal Control of a Process Unit

  • Control through DCS System (including PID-tuning, Model-based Feedback Control, Feedforward Control)

2. Hierarchical Structure of Process Control Systems

• Levels of Process Control:

  • Level 0: Field Level (involves sensors and actuators directly)

  • Level 1: Direct Control (immediate control actions)

  • Level 2: Supervisory Control (monitoring and coordinating controls)

  • Level 3: Production Control (resource allocation and scheduling)

  • Level 4: Production Scheduling (overall planning)

3. Examples of Transfer Functions in Chemical Processes

3.1. Transfer Function Development

• Example: Transfer function for temperature measurement

  • Output: Measured temperature by a thermometer.

  • Input: Outside temperature affecting the thermometer.

  • Energy balance equation:

  • $m imes c<em>p rac{dT</em>t}{dt} = h imes A (T_t - T)$

    • Where:

    • $m$ = mass of liquid in thermometer (kg)

    • $c_p$ = specific heat (kJ/(kg K))

    • $A$ = area of thermometer bulb

    • $h$ = heat transfer coefficient (kJ/(s m² K))

  • Transfer function: Steps to derive

  1. Energy balance applied (no need for mass balance here).

  2. Simplifying energy equations leads to:

  • $G(s) = rac{T(s)}{T<em>{in}(s)} = rac{K</em>p}{ au s + 1}$,

  • where $K<em>p = rac{1}{m imes c</em>p}$ and $au = rac{1}{h imes A}$.

4. Non-Linear Models and Linearization

• Challenge with Non-Linear Models:

  • Example: Liquid storage tank with a control valve.

  • Mass Balance Equation:

  • $rac{dm(t)}{dt} = <br>ho rac{dV(t)}{dt} = A<br>ho rac{dl(t)}{dt} = <br>ho F<em>{in}(t) - ho F</em>{out}(t)$ where the effective outflow is influenced by the open valve.

  • Key Terms:

  • $<br>ho$ = density of the liquid

  • $A$ = cross-sectional area of the tank

  • Dynamic Model Relationships:

  • Desired relationships:

    • Process Dynamic Model between the controlled variable (liquid level) and manipulated variable (valve opening).

    • Disturbance Dynamic Model involves understanding how disturbances impact controlled variables.

4.1. Linearization Techniques

• Tools for Linearization:

  • Taylor Series Expansion

  • For single independent variable:
    $f(x) = f(x<em>0) + rac{df}{dx}(x</em>0)(x - x<em>0) + rac{1}{2!} rac{d^2f}{dx^2}(x</em>0)(x - x_0)^2 + ext{higher order terms}$

  • For multiple independent variables, the expansion of $f(x,y)$ becomes cumbersome with mixed derivatives.

5. Steps to Derive Transfer Functions

• Procedure:

  1. Write mass and energy balances on the system.

  2. Linearize non-linear terms using Taylor Series.

  3. Express all variables in terms of deviation from steady state.

  4. Rearrange the equation to isolate the output variable.

  5. Apply Laplace Transform to both sides to derive the transfer function.

6. Case Studies: Cylindrical Stirred Tank Heater

• Mass and Energy Balance for Simplified STH:

  • Mass balance: $rac{dV(t)}{dt} = F<em>{in}(t) - F</em>{out}(t)$

  • Energy balance incorporates temperature impacts, potentially using heat exchanger dynamics if applicable (assumes insulated systems).

7. Control Systems and Stability Analysis

• Poles and Zeros: Understanding poles in the s-domain:

  • Stability Criteria:

  • If all system poles are real and negative → stable

  • Complex poles with negative real parts → stable and oscillatory

  • At least one positive pole → unstable

• MATLAB Commands for Stability:

  • Use commands like pzmap to analyze the locations of poles and zeros in the s-plane.

8. Process Identification Techniques

• Numerical Models and System Identification:

  • Use methods like least squares for parameter estimation.

  • Implement Excel or MATLAB tools for real-time data analysis and model fitting.

• Graphical Methods:

  • Steps to approximate processes dynamically through empirical data analysis and graphical interpretations.

9. Representation of Common Transfer Functions and Dynamics

• **Common Forms of Transfer Functions:
  ** - FOPTD, SOPTD, and mechanisms for system response characterization.

• Control system modeling and behavior must be understood through various transfer function forms.

Conclusion

• Modeling and analysis lead to practical applications in chemical processes emphasizing linear transfer function dynamics. Understanding the principles of transfer functions, stability, linearization techniques, and empirical modeling ensures robust control strategies in industrial applications.