IT1005 Lab session on week 11 (7 th meeting) 1 or 2 more weeks to go…

Lab 7 – Quick Check Have you all received my reply for lab 7? – Of course not. I have not finished grading your submissions yet >.< – I have important research paper deadline >.< Abstract due: 4 April 2008, Paper due: 8 April – My future reply should contains: Remarks on your M files (look for SH7 tags again) – For CSTR_input.m, CSTR_matrix.m, and CSTR_soln.m, etc Remarks on your Microsoft Word file – For the other stuffs Your marks is stored in the file “Marks.txt” inside the returned zip file! – I do not think I will be strict mode, the answers are standard for L7! Note that my marking scheme is slightly different from the standard one, as I put emphasize on coding style (indentation, white spaces), efficiency, things like proper plotting, etc…

L7.Q1 – 4 tanks CSTRs Q1. A. Transform the non standard set of Linear Equations into standard format: -(Q+k*V)*CA1 + 0*CA2 + 0*CA3 + 0*CA4 = -Q*CA0 Q*CA1 -(Q+k*V)*CA2 + 0*CA3 + 0*CA4 = 0 0*CA1 + Q*CA2 -(Q+k*V)*CA3 + 0*CA4 = 0 0*CA1 + 0*CA2 + Q*CA3 -(Q+k*V)*CA4 = 0 Q1. B. Convert the standard format into matrix format. Another straightforward task: [-(Q+k*V)000] *[CA1] = [-Q*CA0] [Q-(Q+k*V)00][CA2] [0] [0Q-(Q+k*V)0][CA3] [0] [00Q-(Q+k*V)][CA4] [0] Q1. C. You just need to do: V = 1; Q = 0.01; CA0 = 10; k = 0.01; A = [-(Q+k*V) 0 0 0; Q -(Q+k*V) 0 0; 0 Q -(Q+k*V) 0; 0 0 Q -(Q+k*V)]; b = [-Q*CA0; 0; 0; 0]; x = A\b % you should get x = [5; 2.5; 1.25; 0.625], x = inv(A) * b is NOT encouraged! % Using fsolve for this is also not encouraged!

L7.Q2 – n tanks CSTRs Q2. A. CSTR_input.m Answer is very generic, similar to read_input.m in L6.Q2 (Car Simulation) Q2. B. CSTR_matrix.m % This is my geek version, do NOT USE THIS VERSION (too confusing for novice)! function [A b] = CSTR_matrix(Q, CA0, V, k, N) A = diag(repmat(-(Q+k*V),1,N)) + diag(repmat(Q,1,N-1),-1); % This is a bit crazy :$ b = zeros(N,1); b(1) = -Q*CA0; Q2. C. CSTR_soln.m clear; clc; clf; % New trick, but important ! Clear everything before starting our program! [Q CA0 V k N] = CSTR_input(); [A b] = CSTR_matrix(Q, CA0, V, k, N); plot(A\b,'o'); % I prefer not to connect the plot with line, but it is ok if you do so. title('CA of each tank'); xlabel('Tank no k'); ylabel('CA_k'); % Good for CSTR_plot.m axis([0.5 N CA0]); % Fix y axis so that it is consistent across 4 plots (same CA0!)

L7.Q2 – Good Plot Should just stop here (n = 4) for case 1 Remember: Plot A against B means that A is the Y axis, B is the X axis!

L7.Q3 and Q4 Q3. Type these at command window: syms x y; % no need to say syms f1 f2, the next two lines will create f1 and f2 anyway f1 = x^2 * y^2 - 2*x - 1; f2 = x^2 - y^2 - 1; [a b] = solve(f1,f2); % or >> s = solve(f1,f2); a = s.x; b = s.y; eval(a), eval(b) % convert the symbolic values to numeric values, these are the roots Q4. Create this function function F = lab07d(x) F(1) = sin(x(1)) + x(2)^2 + log(x(3)) - 7; F(2) = 3*x(1) + 2^(x(2)) x(3)^3; F(3) = x(1) + x(2) + x(3) - 5; At command window (wild guesses will likely give you many imaginary numbers): [1 1 1])  x=0.5991, y=2.3959, z= [5 -1 1])  x = , y = , z= Tips for guessing logically: 1.Plot the functions with some range, see the region of zero intercepts… 2.Guess from the easiest equation! e.g. x + y + z = 5 3. log(z)… hm… z should be > 0

Application 5: IVP (revisited) Equation – A statement showing the equality of two expressions usually separated by left and right signs and joined by an equals sign. Differential Equation – A description of how something continuously changes over time. Ordinary Differential Equation – A relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. Initial Value Problem – An ODE together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.

Application 5: IVP (revisited) We have seen several examples of IVP throughout IT1005 – Spidey Fall example (Lecture 2, Lecture ODE 1) How velocity v change over time dt? (dv/dt) – Depends on gravity minus drag! How displacement s change over time dt? (ds/dt) – Depends on the velocity at that time. The Initial Values for v and s at time 0  v(0) = 0; s(0) = 0; – Car accelerates up an incline example (Lab 6, Q2) How velocity v change over time dt? (dv/dt) – Depends on engine force on some kg car minus friction and gravity factor! How displacement s change over time dt? (ds/dt) – Depends on the velocity at that time. The Initial Values for v and s at time 0  v(0) = 0; s(0) = 0; – And two more for Term Assignment (Q2 and Q3)

Application 5: IVP (revisited) Solving IVP (either 1 ODE or set of coupled ODEs): – Hard way/Traditional way/Euler method: Time is chopped into delta_time, then starting from the initial values for each variable, simulate its changes over time using the specified differential equation! What Colin has shown in Lecture 2 for Spidey Fall is a kind of Euler method. What you have written for Lab 6 Question 2 (Car Simulation) is also Euler method. – Matlab IVP solvers: mostly numerical solutions for ODE. Create a derivative function to tell Matlab about how a variable change over time! function dydt = bla(t,y) % always have at least these two arguments % explain to Matlab how to derive dy/dt! Can be for coupled ODEs! dydt = dydt'; % always return a column vector! Call one of the ODE solver with certain time span and initial values [t, y] = [tStart tEnd], IVs); % IVs is a column vector for coupled ODEs! plot(t,y(:,1)); % we can immediately plot the results (also in column vector!)

Term Assignment – Admin This is 30% of your final IT1005 grade... Be very serious with it. – No plagiarism please! Even though you can ‘cross check’ your answers with your friends (we cannot prevent that), you must give a very strong ‘individual flavor’ in your answers! The grader will likely grade number per number, so he will be very curious if he see similar answers across many students. Do not compromise your 30%! – Who will grade our term assignment? I may not be the one doing the grading! Perhaps all the full time staff… dunno yet. – Submit your zip file (containing all files that you use to answer the questions) to IVLE “Term Assignment” folder! NOT to my Gmail! – Your zip file name should be: yyy-uxxxxxx.zip, NOT according to my style! – Strict deadline, Saturday, 5 April 08, 5pm That IVLE folder will auto close by Saturday, 5 April 08, 5.01pm Be careful with NETWORK CONGESTION around these final minutes… To avoid that problem, submit early, e.g. Friday, 4 April 08, night.

Term Assignment – Q1 Question 1: Trapezium rule for finding integration – A. Naïve one. Explain your results! – B. More accurate one. Explain your results! – References: help quad – Revision(s) to the question: Symbol ‘a’ changed to ‘c’ inside function f(t)! In Q1.B, the rows in column ‘c’ are [ ] not [ ]! In Q1.B, the range of k is changed from k = 2:n-1 to k = 1:n-1, but it is ok if you stick with the old one!

Term Assignment - Q2 Question 2: Zebra Population versus Lion Population – A. IVP, coupled, non-linear ODEs – B. Explain what you see in the graph of part A above. – C. Steady state issue. – D. IVP again, but change the IVs according to part C above. Comment! – E. IVP with different IVs, and different plotting method. Comment! – References: Google the term ‘predator prey’ as mentioned in the question. help odeXX (depends on the chosen solver) – Revision(s) to the question: No change so far…

Term Assignment – Q3 Question 3: Similar to Q2, Predator-Prey: n = 4 species – A. IVP again, 4 coupled, non-linear ODEs. Dr Saif said that we must use ode15s! (See ODE 3 & 4 lecture note) – B. IVP, same IVs, years, 3D plot x1-x2-x3 (x4 is not compulsory), and explain. – C. Explain plot in B as best as you can. – References: (mentioned in the question). Google ‘Matlab 3D plot’ help odeXX (depends on the chosen solver) – Revision(s) to the question: The ODE equations are updated! Read the newest one! The coefficient r(3) is changed from 1.53 to 1.43!

Free and Easy Time Now, you are free to explore Matlab to: – Do your Term Assignment (all q1, q2, and q3) – You should NOT use me as an oracle, e.g. I cannot find the bug in my program, can you help me? Are my 2-D/3-D plots correct? Are my …. bla bla … correct?