Dynamic models - 2014

Class date(s):	13 to 07 February 2014
Download video: Link (plays in Google Chrome) [361 M]
Download video: Link (plays in Google Chrome) [364 M]
Download video: Link (plays in Google Chrome) [227 M]
Download video: Link (plays in Google Chrome) [367 M]
Download video: Link (plays in Google Chrome) [305 M]
Download video: Link (plays in Google Chrome) [317 M]
Download video: Link (plays in Google Chrome) [294 M]
Download video: Link (plays in Google Chrome) [271 M]
Download video: Link (plays in Google Chrome) [322 M]
Download video: Link (plays in Google Chrome) [287 M]
Download video: Link (plays in Google Chrome) [267 M]
Download video: Link (plays in Google Chrome) [215 M]

Readings and preparation for class; video and audio files

• Main reference: Marlin textbook, Chapter 3, Chapter 4, Chapter 5, and Appendix B
• Alternative reference: Seborg textbook, Chapters 2, 3, 4 and 5

Date	Class number	Video and audio files		Main concepts	Reading (other)	Handout
13 January	02A	Video (361M)	Audio (42M)	Why we need process models What is "steady state" General balance equations Derived the tank height model	Your 3E4 notes on dynamic models and ODEs. Any math textbook on the Integrating Factor (or Appendix B in Marlin) MATLAB software for numerical integration	None
15 January	02B	Video (364M)	Audio (43M)	Use of the integrating factor Analytical solution of the tank height model MATLAB solution of ODEs		None
17 January	02C	Video (227M)	Audio (42M)	Coffee cooling example What is the Laplace transform Using Laplace to solve the coffee cooling problem	An amazing description of where the Laplace transform comes from: Video 1 and continues with video 2	Table of Laplace transforms (from Seborg)
20 January	03A	Video (367M)	Audio (43M)	Review the coffee example Initial value and final value theorems Using deviation variables Solve coffee example in deviation form
22 January	03B	Video (305M)	Audio (42M)	Reviewed deviation variables, IVT and FVT The tank height problem in deviation form Deriving a transfer function for the tank height for varying inlet flow	None	None
24 January	03C	Video (317M)	Audio (42M)	Confirming understanding of a transfer function (TF) How to find system's output, for a given input Chaining up TFs for systems in series Learning about "pure gain" systems	None	None
27 January	04A	Video (294M)	Audio (41M)	combining transfer functions in series linearization of nonlinear functions basic block diagrams	None	None
29 January	04B	Video (271M)	Audio (43M)	linearization block diagrams for series and parallel systems time delays	None	None
31 January	04C	Video (322 M)	Audio (44 M)	time delay derivation and interpretation first order system theory composite input example	See code below for class 04C	None
03 February	05A	Video (287 M)	Audio (42 M)	review of all work covered in weeks 1 to 4 assignment 3, question 3	None	Developing process models
05 February	05B	Video (267 M)	Audio (45 M)	composite inputs formal introduction to second order systems question period	Types of damping	None
07 February	05C	Video (267 M)	Audio (45 M)	wrapped up second order systems re-discussed the objective of control systems		None

* The book by Seborg et al. is easily available new or second hand, as it was the prescribed textbook in 2013 (Marlin's book was prescribed in 2012). I will make reference to the chapters from Seborg on the website as well.

Test your understanding before and after class with these resources from Dr. Thomas Marlin. This website also contains full Powerpoint slides for each chapter from his textbook. Use this as a resource to get a different teaching perspective on the same topic. It's quite OK if someone else's approach is a "better fit" for you than my approach.

Computer code: class 02B, 15 January

In a file called cstr_height.m:

function d_depnt__d_indep = cstr_height(indep, depnt)
 
% Dynamic balance for the CSTR height

%    indep: the independent ODE variable, such as time or length or the reactor
%    depnt: a VECTOR of dependent variables
% 
%    Returns:
%
%    d(depnt)
%   ---------- = a vector of ODEs
%    d(indep)
 
% Assign some variables for convenience of notation: one row per DEPENDENT variable
h = depnt(1);
 
% Constant and other equations
A = 0.5;      % m^2
F_i = 0.8;    % m^3/min
R   = 15;     % min/m^2
F_o = h/R;    % m^3/min

% Output from this ODE function must be a COLUMN vector, with n rows
% n = how many ODEs in this system?
n = numel(depnt); d_depnt__d_indep = zeros(n,1);

% Specify every element in the vector below: 1, 2, ... n
d_depnt__d_indep(1) = 1/A * (F_i - F_o);

The call the above model file from the "driver"; you can call this file anything, e.g. ODE_driver.m:

% The independent variable always requires an initial and final value:
indep_start = 0.0;  % s
indep_final = 50.0; % s
 
% Set initial condition(s): for integrating variables (dependent variables)
h_depnt_zero = 4.0;    % i.e. h(t=0) = 3.0
 
IC = [h_depnt_zero];
  
% Integrate the ODE(s):
[indep, depnt] = ode45(@c02B_linear, [indep_start, indep_final], IC);
 % Plot the results:
clf;
plot(indep, depnt(:,1))
grid('on')
hold('on')

xlabel('Time [min]')
ylabel('Tank height')
legend('h')
title('Tank height with time')

% Does it match the analytical equation?
height = 12 - 8.*exp(-indep/7.5);
hold on
plot(indep, height, 'r.')

Computer code: class 04C, 31 January

t = 0:0.01:18;
y_regular = 3.*(1-exp(-t./4));

step_function = ones(size(t));
step_function(t<2) = 0;
y_delayed = 3.*(1-exp(-(t-2)./4)) .* step_function;

y_combined = y_regular + y_delayed;

% Show the plots
f = figure;
set(f,'Position',[440 79 841 638])
plot(t, y_regular,'b.-')
hold on
%plot(t, step_function,'k.-')  % if you want to visualize the step function
plot(t, y_delayed,'r.-')
plot(t, y_combined,'m.-')

legend('Response from 1 tank', 'Response from delayed tank', ...
    'Sum of the two responses', 'location','NorthWest')
grid on