Chapter 3 : Simple Process Dynamics and Transfer Function Professor Shi-Shang Jang Department of Chemical Engineering National Tsing-Hua University Hsinchu, Taiwan March, 2013

Motive of Developing First Principle Models Improve understanding of the process Train plant operating personnel Develop a control strategy for a new process Optimize process operating conditions

3-1 Introduction Theoretical (First Principle) models are developed using the principles of chemistry , physics, and biology. Theoretical models offer process insight into process behavior, and they are applicable over wide ranges of conditions They trend to be expensive and time-consuming to develop

Example - Industrial Furnace TT 23 TC Stack gases Fuel Air FY QY FT 24 Set point CV: temperature of the furnace MV: fuel flow rate to the furnace Figure 1-1

Temperature Profile of TT23 Time (min)

Plant Dynamics Plant DV CV MV Flow rate Flow rate time time temperature temperature time time

The Concept of Deviation Variables Flow rate Flow rate Plant MV CV DV time time temperature temperature yd=y - ys time time

The Essence of Process Dynamics - Continued The feedback process control needs to understand the relationships between CV and MV, on the other hand, feedforward process control needs to understand the relationships between DV and CV. The relationships are called process models. For the ease of mathematical analyses, the process modeling only implements a linear model and Laplace transform instead of direct use of time domain process model. Implementation of deviation variables is needed as indicated below.

3-1 Introduction- Continued Empirical models are obtained by fitting experimental data. Empirical models typically do not extrapolate well, and their range is typically small. Empirical models are frequently used in the industrial environment since a theoretical model is basically not precisely available.

3-1 Introduction- Continued Semi-empirical models are a combination of the models of theoretical and empirical models; the numerical values of the parameters in a theoretical model are calculated from experimental data. Semi-empirical models can (i) incorporate theoretical knowledge, (ii) extrapolate wider range than empirical range, (iii) require less effort than theoretical models.

3-2 General Modeling Principles

3-2 General Modeling Principles: Constitution Equations Heat Transfer： Reaction Rate： Flow Rate： Equation of State： Phase Equilibrium:

3-3 Transfer Functions - Conventions － on the top of a variable= steady state of a variable, example: Capital = deviation variable, example: Capital with (s)= Laplace transform of a variable to the deviation variable, example:

3-3. Transfer Function Transfer function is a mathematical representation of the relation between the input and output of a system. It is the Laplace transform of the output variable, y(t), divided by Laplace Transform of the input variable, x(t), with all initial conditions equal to zero. The term is often used exclusively to refer to linear, time-invariant systems (LTI), and non-linear, real-system are linearize to obtain their Transfer Function. So, Transfer Function G(s) for a system with input x(t) and output y(t) would be-

More over Transfer Function As for previous equation, it could be said that if transfer function for the system and input to the system is known, we can obtain the output characteristics of the system. Transfer Function for the system could be easily obtained by dynamic study of the system and making balances for quantities like energy, mass etc. We take inverse Laplace Transform to obtain time-varying output characteristics of Y(s). In block diagram: G (s) X(s) Y(s)

3-3 Transfer Functions – Example: Thermal Process Ts Inputs: f(t), Ti(t),Ts(t) Output: T(t)

3-3 Transfer Functions – Cont. Let f be a constant V= constant, Cv=Cp      

3-3 Transfer Functions – Cont. Let f be a constant V= constant, Cpi=Cp =time constant

3-3 Transfer Functions – Cont. where, Gp(s) is call the transfer function of the process, in block diagram: Gp(s) i(s) (s)

Step Response of a First Order System

Process Identification

Example: Mercury thermometer A mercury thermometer is registering a temperature of 75F. Suddenly it is placed in a 400F oil bath. The following data are obtained. Time (sec) 1 2.5 5 8 10 15 30 Temp. (F) 75 107 140 205 244 282 328 385 Estimate the time constant of the temperature using Initial slope method 63% response method From a plot of log(400-T) versus time

Solution ((1) =9sec)    

Comparisons Fit 1 Fit 2 Fit 3

3-3 Transfer Functions – Cont. By including the effect of surrounding temperature:

3-3 Transfer Functions – Cont. Gp1(s) Gp2(s) + i(s)  s(s) (s) Σ

Numerical Data

Examples (1) Thermal Process- Continued Deviation Variables

Simple Systems  

Simple Systems  

Block Diagram K1 1s+1 M(s) Y1(s) Y(s) + K2 2s+1 D(s) Y2(s)

3-3 Transfer Functions-An Example Cross-sectional=A2 h2 V2 f1 f2 h1 V1 Cross-sectional=A1

3-3 Example Non-Interactive Tanks

3-3 Example Non-Interactive Tanks – Cont. F0(s) H1(s) H2(s)

3-4 Transfer Functions and Block Diagrams – Cont.

3-4 Transfer Functions and Block Diagrams – Cont.

3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.2)

3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.3)

3-4 Transfer Functions and Block Diagrams – Cont. (Example 3-5.3)

3-5 Gas Process Example

3-5 Gas Process Example – Cont.

3-5 Gas Process Example – Cont. Σ s+1 K1  s+1 K2 K3 + - Mi(s) Mo(s) P1(s) P(s)

3-5 Gas Process Example – Cont.

3-6 Dead Time

3-4 Dead Time – Cont. Time delay:

3-4 Dead Time – Cont.

3-4 Causes of Dead Time - Cont. Transportation lag (long pipelines) Sampling downstream of the process Slow measuring device: GC Large number of first-order time constants in series (e.g. distillation column) Sampling delays introduced by computer control

3-4 Effects of Dead-Time - Cont. Process with large dead time (relative to the time constant of the process) are difficult to control by pure feedback alone: Effect of disturbances is not seen by controller for a while Effect of control action is not seen at the output for a while. This causes controller to take additional compensation unnecessary This results in a loop that has inherently built in limitations to control

3-5 Transfer Functions and Block Diagrams Consider a general transfer function for an input X(s) and an output Y(s): Note that the above case is always true , although many mathematical manipulating is needed as shown below:

3-7 Chemical Reactor

3-7 Chemical Reactor – Cont.

3-7 Chemical Reactor – Cont.

3-8 Effects of Process Nonlinearity Real processes are mostly nonlinear The approximate linear models are only valid in local about the nearby of the operating point In some cases, process nonliearity may be detrimental to the control quality (e.g. high purity column) Process nonliearity plays important role to control quality in control systems

3-8 Effects of Process Nonlinearity – Cont.

3-9 Additional Comments Reading assignment P96-98

Homework Page 99 3-1, 3-2, 3-3, 3-4, 3-9, 3-10 (April 17th), 3-13, 3- 14(SIMULINK) 3-20, 3-21 (April 24th)

Process-Supplemental Material Inputs: f(t), Ti(t),Ts(t) Output: T(t) Ts

3-3 Transfer Functions – Cont. Gp1(s) Gp2(s) + i(s)  s(s) (s) Σ Gp3(s) F(s)

Thermal Process-Supplemental Material