Chapter I: Hands-on experience in SCILAB operations

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1 Chapter I: Hands-on experience in SCILAB operations Introduction to SCILAB SCILAB is a numerical mathematical equation solver which involves a powerful open computing environment for engineering and scientific applications. SCILAB includes hundreds of mathematical functions with the possibility to add interactive functions from various languages. SCILAB is available free of cost and can download from When you start up SCILAB, you can see a console window. The user enters SCILAB commands at the prompt (--->). Tutorial #1: Simple operations using SCILAB Objective: To learn the SCILAB syntax like arithmetic operation, built-in functions, creating matrices, simple matrix operations, polynomials, graph generation and programming commands and solve simple problem in chemical engineering using SCILAB. (a) Arithmetic operation and built-in functions: Type the following commands in the SCILAB platform and fill the answer in the appropriate space in the table below ->2+3 ans = 5. ->a = 2 -->sqrt(4) exp(2.3025) -->cos(a) ->a = 5; b = 10; c = 15; -->disp([a b c]) -->2/3 -->b = 3 -->log(10) -->a = 4 -->%pi -->a = 10, b = 20, c = 30 -->2^3 -->c = a+b -->log10(10) -->sin(a) sin(%pi/4)

2 (b) Creating matrices and simple matrix operations ->a = [1 2 3] -->x = [1 2 3]; y = [ ]; -->a = [x y] --> c = a + b -->e = a*b (matrix multiplication) -->a = [1;2;3] -->a = [1 2 3;4 5 6;7 8 9] -->d = a - b -->f = [3 1 2; 1 5 3; 2 3 6] -->a = [1 2 3]' -->b = a' -->g = inv(f)

3 -->f*g -->a.*b -->a.+b -->c = eye(3,3) -->a.-b -->det(f) -->a^2 (matrix multiplication) -->a = zeros(3,2) -->c = eye(3,3)*10 -->log(a) -->a.^2 (elements by elments) -->b = ones(4,3) -->y = rand(2,2) -->y = [1:5;6:10;11:15] -->y(6) ->a = [] -->size(y) -->length(y) -->y(1) -->y(2) -->y(3,2) ->y(1,5) -->1:5 ->a = [1:5] ->c = [0:2:10] -->size(a) ->a = [1 2 3;4 5 6] -->b = a(2,2) -->a(2,2) = 100

4 (c) Submatrices -->a = int(rand(5,8)*100) -->c = a(1:2:$,1:2:$) -->b = a(3:4,2:5) ->d = a(:,$:-1:1) (reverse the order of column) -->a(3:4,2:5) = zeros(2,4) -->a = rand(5,3) -->c = a(:,2:3) -->s = sum(a,'r') -->a(3:4,2:5) = b(1:2,1:4) -->p = sum(a,'c') -->a(:,3:$) -->m = mean(a,1) -->sd = stdev(a,1)

5 Instructions: Submit the answers to these questions along with chapter correction The answer should write or print on A4 sheets Write or print the answer after check your answer using SCILAB platform Answer all the questions 1. What is the command to clear the screen? 2. When do you think it is useful to use the semicolon (;)? 3. What is the size of an empty matrix a = []? 4. What is the command to extract the diagonal elements of a square matrix into a vector? 5. Extract the off-diagonal terms of a square matrix into a vector 6. Extract the last column of a matrix and store it in matrix b 7. Replace the even numbered columns of matrix a having size 3x5 with ones 8. What is the sub-matrix of a extracted by the following command a (1:3,$-2:$)? 9. Assuming a to be a 5x8 matrix, are the following valid commands? If so, what do they do? If not, what is the correct command? a. A(1:, 5) b. A(:, 5) c. a(1:3,$-1:$) d. a(:$,3:6) 10. What are the commands used for the following functions? a. Identity matrix b. Diagonal matrix c. Determinant of square matrix d. Square root of each element of a matrix Inferences: The following SYNTAX of SCILAB practiced and understood the operations a. Arithmetic operation and build in functions b. Creating matrices and simple matrix operations c. Sub-matrices

6 Tutorial #2: Polynomial operations, Graph generation, and Programming language SCILAB has support for operations on polynomials. You can create polynomials, find their roots and perform operations on them such as addition, subtraction, multiplication, and division. The graph generated can do in the SCILAB platform which can be enhanced and annotated. You can add grid lines, labels, a legend for the different lines, etc. Three-dimensional plots of surfaces can plot with the plot3d() functions. The programming language offers many features of a high-level language, such as looping (for, while), conditional execution (if-then-else, select) and functions. a. Polynomial operations -->p1 = poly([3,2],'x') ->p4 = p1 * p2 -->roots(p6) (or) -->p1 = poly([3,2],'x','r') -->p2 = poly([6-5 1],'x','c') ->p5 = p1/p2 -->coeff(p1) -->x = poly(0,'x') -->p = (1+2*x+3*x^2)/(4+5*x+6*x^2) -->roots(p2) -->derivat(p1) -->x = poly(0,'x') -->numer(p) -->denom(p) -->p3 = p1+p2 -->p6 = 6-5*x + x^2

7 b. Plotting x-y and 3D graphs ->x = [0:%pi/16:2*%pi]'; -->y = [cos(x) sin(x)]; -->plot2d(x,y) ->u = linspace(-%pi/2,%pi/2,40); -->v = linspace(0,2*%pi,20); -->x = cos(u)'*cos(v); -->y = cos(u)'*sin(v); -->z = sin(u)'*ones(v); -->plot3d(x,y,z); -->x = [0:%pi/32:2*%pi]'; -->y(:,1) = cos(x); -->y(:,2) = sin(x); -->y(:,3) = cos(x)+sin(x); -->plot(x,y); -- >xtitle('triginometricfunctions',' x','f(x)'); >legend('cos(x)','sin(x)','cos(x) +sin(x)',1,%f); -->clf(); -->subplot(121); -->plot3d3(x,y,z); subplot(122); -->plot3d2(x,y,z); ->x= [0:%pi/16:2*%pi]'; size(x) ans = >z = sin(x)*ones(x)'; -->plot3d(x,x,z); -->x(:,1) = [ ]'; -->x(:,2) = [ ]'; -->p = [ ]'; -->t = [ ]'; -->y(:,1) = [ ]'; -->y(:,2) = [ ]'; -->clf(); -->subplot(221); -->plot(x,p); -->subplot(222); -->plot(y,t); -->subplot(223); -->plot(y(:,1),y(:,2));

8 c. SCILAB Programming Language ->for i=1:10 -->disp(i) -->end ->x = 10; -->if x<0 then disp('negative') -->elseif x==0 then disp('zero') -->else disp('positive') -->end -->a = int(rand(3,4)*100); -->a ==0 -->disp(a) -->a <20 -->find(a < 20) I planted a Christmas tree in 2005 measuring 1.2 m. It grows to 30 cm per year. I decided to cut it when it exceeds 7 m. In what year will I cut the tree? ->h = 1.2; -->y = 2005; -->while h<7 -->h = h+0.3; -->y = y+1; -->end -->disp(y) -->function y = f(x); -->y = 36/(8+exp(-x)); -->endfunction -->f(10) -->f(12) -->f(1) -->function y = g(x); -->y = 4*x/9+4; -->endfunction -->g(12.5) Solving linear equations -->A = [1 2 3;4 5 6]; -->B = [1;1]; -->X = A\B Numerical Integration Find a numerical approximation to t dt and compare it to the analytical solution. Construct a grid over the range of t, dense enough to make the integrand approximately constant over each interval, and use the function sum. -->dt = 0.001; -->f = 0; -->for t = 0:dt:1 -->y = 1/(sqrt(t^2+2)); -->f = f+y; -->end -->val = f*dt

9 Instructions: Submit the answers to these questions along with chapter correction. The answer should write or print on A4 sheets. Write or print the answer after check your answer using SCILAB platform. Answer all the questions 1. A mixture of phenol and water forms two separate liquid phases one rich in phenol, and other rich in water, the composition of the layer is 70% and 9% (by weight) phenol respectively. If 500 kg of phenol and 700 kg of water are mixed and layer allowed to separate, what will be the weight of two layers. (Hint: make two simultaneous equations and solve these equations using SCILAB ) moles of benzene (A) and toluene (B) mixture containing 50 mole% of benzene is subjected to a differential distillation at 1 atm. Pressure till the composition of the benzene in their residue is 33%. Calculate the total moles of the distillate. The equilibrium relationship between liquid and vapor phase is: x y = 1 + x ( 1) α = 2.1, x is the mole fraction of benzene in liquid and y is the mole fraction of benzene in the vapor. 3. Calculate the volume occupied by one mole of n-octane vapor at K where the saturated pressure is Mpa. Assume the n-octane follows the Vander Waal s equation of state. The Vander Waal s constant a and b are pa(m 3 /mol) 2 and 2.37 x 10-4 m 3 /mol respectively. 4. Solve the system of equations x 1 + x 2 = 2 x 2 + x 3 = 3 x 1 + x 3 = 4 5. Find the roots of the equations: (a) p(x) = x 3 + 2x 2 +3x+4 = 0 and (b) p(x) = x 4 +3x 3 +5x 2 +7x The enthalpy of the binary liquid system of species 1 and 2 at fixed temperature T and pressure P is represented by the equation. H = 400x x 2 +x 1 x 2 (40x 1 +20x 2 ), where H is in J/mol and x 1 is mole fraction of species 1 and x 2 is the mole fraction of species 2. Determine the enthalpy of the binary mixture

10 for 0.1 x and also plot the enthalpy of binary mixture (H) as function of mole fraction of species 1 (x 1 ). Inferences: The following SYNTAX of SCILAB practiced and understood the operations a. Polynomial operations. b. Plotting x-y and 3D graphs. c. SCILAB Programming Language.

11 Tutorial #3: Polynomial Curve Fitting The function polyfit() can be written in SCIPAD and then loaded into SCILAB workspace: function[a, yf]=polyfit(x, y, n) xx=zeros(length(x),n+1) fori=1:n+1 xx(:,i)=x.^(i-1); end X=xx'*xx; b=xx'*y; a=inv(x)*b; yf=xx*a; endfunction To test the above polyfit() function, the steady state temperature distribution within a plane wall 1 m thick with a thermal conductivity of 8 W/m K is measured as a function of position as shown in table below. Z (m) T ( C) >exec('c:\users\sky\documents\polyfit.sci', -1) -->z = [ ]'; -->T = [ ]'; -->[a,yf] = polyfit(z,t,4)

12 Instructions: Submit the answers to these questions along with chapter correction. The answer should write or print on A4 sheets. Write or print the answer after check your answer using SCILAB platform. Answer all the questions 1. Consider the sample data given below: x y Get the coefficients of fourth order polynomial equation using SCILAB platform. 2. The irreversible isomerization reaction A B was carried out in a batch reactor and the following concentration time data were obtained Time (min) C A (mol/dm 3 ) Fit a second order polynomial equation to the above experimental data and also estimate the rate of reaction (-dc A /dt) in each concentration - time data. 3. The reaction rate constant for the decomposition of a substituted dibasic acid has been determined at various temperatures as given in the table below. Use the method of least squares to determine the activation energy E in the equation, k = e -E/RT, where T is measured in Kelvin T ( C) k x 10 4 (h -1 ) Inferences: The Polynomial curve fitting practiced and understood the operations in SCILAB.

13 Chapter 2: Hands-on experience in commercial process simulators (ASPENPLUS )

Solving Equations and Optimizing Functions

Solving Equations and Optimizing Functions At the end of this lecture, you will be able to: solve for the roots of a polynomial using polyroots. obtain approximate solutions to single equations from tracing