COMPUTER APPLICATIONS IN CHEMICAL ENGINEERING

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES363 P. Casal St., Quiapo, Manila

CHEP 530D1

COMPUTER APPLICATIONS IN

CHEMICAL ENGINEERING

Crispulo G. MarananInstructor

Laboratory Exercise No. 1Familiarization with Matlab Environment, Built-in Functions, Matrices and Plotting

Bautista, Keziah Lynn S. Laboratory Exercise No. 1Familiarization with Matlab Environment, November 9, 2013Built-in Functions, Matrices and Plotting

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1. Objective(s):The activity aims to familiarize the students with matlab environment, built-in functions, matrices and plotting.2. Intended Learning Outcomes (ILOs):

The students shall be able to:2.1 get acquainted with matlab environment and its various features.2.2 understand the built-in functions of matlab.2.3 Operate on the matrices.2.4 Plot different graphs using matlab.

3. Discussion: Matlab is a powerful language for technical computing. Its basic data element is matrix (array).It can be used for math computations, modeling and simulations, data analysis and processing, visualization and graphics, and algorithm development.

The standard Matlab program has tools (functions) that can be used to solve common problems.The array is a fundamental form that Matlab uses to store and manipulate data. An array is a list

of numbers arrange in rows or in columns. The simplest array (one-dimensional) is a row, or a column of numbers. A more complex array (two-dimensional) is a collection of numbers arranged in rows and columns. One use of array is to store information and data, as in a table. In science and engineering, one-dimensional arrays frequently represent vectors and two-dimensional arrays represent matrices.

Once variables are created in Matlab they can be used in a wide variety of mathematical operations. Matlab is designed to carry out advanced array operations that have many applications in science and engineering. Addition and subtraction are simple operations. The other basic operations, multiplication, division and exponentiation can be done in Matlab in two different ways. One way, which uses the standard symbols (*,/ and ^), follows the rules of linear algebra. The second way, which is called element-by-element operations, uses the symbols .*,./ and .^ ( a period is typed in front of the standard operation symbol).In both types of calculations, Matlab has left division operator (.\ or \).

4. Resources:Matlab

5. Procedure:1.Identify the different matlab windows and write its corresponding purpose.2.Note the different symbols used in the command window and write its corresponding use.3.Use matlab as a calculator and show the results in the accompanying table.4.Note the different built-in functions and show the results in the accompanying table.5.Evaluate the results after pressing the enter key for the assignment operator (=).6.Evaluate the results after pressing the enter key for the creation of vectors (row vector and column vector) from a known list of numbers, with constant spacing by specifying the first term, the spacing, and the last term, with constant spacing by specifying the first and last terms,and the number of terms7.Evaluate the results after pressing the enter key for the creation of two-dimensional array (matrix).8.Evaluate the results after pressing the enter key using colon (:) in addressing arrays.9. Identify the different built-in functions for handling array and indicate its description and give an example.10.Evaluate the results after pressing the enter key that involves strings and strings as variables.11. Evaluate the results after pressing the enter key that involves the operations of matrices.12.Evaluate the values of x, y and z of the three equations three unknowns :


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4x – 2y + 6z = 82x + 8y + 2z = 46x + 10y + 3z = 0

13.Evaluate the results after pressing the enter key that involves element-element operations.14.Identify the different built-in functions for analyzing arrays and indicate its description and give an example.

Course: Laboratory Exercise No.:1Group No.: Section:CH51FA1Group Members: Date Performed: November 9, 2013

Date Submitted: November 11, 2013Instructor:Engr. Crispulo Maranan

6. Data and Results:1.

Window Purpose1.Command Window Inputs the commands to be use. It is used to

execute commands or aliases directly in the


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Visual Studio integrated development environment (IDE). You can execute both menu commands and commands that do not appear on any menu. To display the Command window, choose Other Windows from the View menu, and select Command Window.

2.Figure Window It is directed to a window that is separate from the Command Window. This window is referred to as a figure. The characteristics of this window are controlled by your computer's windowing system and MATLAB figure properties.

3.Editor Window It is to create your own custom editor window that can float free or be docked as a tab, just like the native windows in the Unity interface. Editor windows are typically opened using a menu item.

4.Help Window Helps you about the commands and other terms to be used.

5.Launch Pad Window Access help, tools, demos and documentation6.Command History Compiles the history of the commands7.Workspace Window It consist of common mathematical figures,

graphs, etc.8.Current Directory The directory (folder) that MATLAB is currently

working in. This is where anything you save will go by default, and it will also influence what files MATLAB can see. You won't be able to run a script that you saved that you saved in a dierent directory.

2.Symbol Purpose>> Relational operators perform element-by-

element comparisons between two arrays. They return a logical array of the same size, with elements set to logical 1 (true) where the relation is true, and elements set to logical 0(false) where it is not.

; Used inside brackets to end rows. sssUsed after an expression or statement to suppress printing or to separate statements.

% Percentage. Used to indicate that the number preceding it should be understood as a


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proportion multiplied by 100clc It clears all input and output from the Command

Window display, giving you a "clean screen."After using clc, you cannot use the scroll bar to see the history of functions, but you still can use the up arrow key, ↑, to recall statements from the command history.

3.Mathematical Expression Result>> 8 + 5/9 8.5556>> (8 + 5)/9 1.4444>> 8^5/9 3.6409e+03>>29^1/5 + 35^0.7 17.8461

4.Built-in Function Result>>sqrt(144) 12>>exp(7) 1.0966e+03>>abs(-99) 99>>log(100000) 11.5129>>log10(100000) 5>>factorial(10) 3658800>>sin(pi/4) 0.7071>>round(19/6) 3>>rem(16,5) 1>>sign(-19) -1

5.>>x= 10 10>>x=4*x -15 25>>a = 10 10>>B= 9 9>>C= (a –B) +50 – a/B *16 33.2222>>a=10,B=9; C= (a –B) +50 – a/B *16 33.2222>>x = 0.99; 0.9900>>E = sin(x)^3 + cos(x)^4 0.6750

6.>>yr = [ 2001 2002 2003 2004 2005] yr =

Columns 1 through 2

2001 2002

Columns 3 through 4

2003 2004

Column 5


5

2005>>yr = [ 256; 299; 350; 402; 503] yr =

256 299 350 402 503

>>y = [1:2:15] y =

Columns 1 through 5

1 3 5 7 9

Columns 6 through 8

11 13 15>>y = [1.5:0.1;2.0] 2>>y=[-5:15] y =

Columns 1 through 5

-5 -4 -3 -2 -1

Columns 6 through 10

0 1 2 3 4


5 6 7 8 9


10 11 12 13 14

Column 21

15>>b = [21:-3:6] b =

Columns 1 through 5

21 18 15 12 9

Column 6


6

6>>a = linspace(0,8,6) a =

Columns 1 through 3

0 1.6000 3.2000

Columns 4 through 6

4.8000 6.4000 8.0000>>b=linspace(30,10,11) b =

Columns 1 through 5

30 28 26 24 22


20 18 16 14 12

Column 11

10>>c=linspace(49.5,0.5) c =

Columns 1 through 3

49.5000 49.0051 48.5101

Columns 4 through 6

48.0152 47.5202 47.0253

Columns 7 through 9

46.5303 46.0354 45.5404


45.0455 44.5505 44.0556


43.5606 43.0657 42.5707


42.0758 41.5808 41.0859


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40.5909 40.0960 39.6010


39.1061 38.6111 38.1162


37.6212 37.1263 36.6313


36.1364 35.6414 35.1465


34.6515 34.1566 33.6616


33.1667 32.6717 32.1768


31.6818 31.1869 30.6919


30.1970 29.7020 29.2071


28.7121 28.2172 27.7222


27.2273 26.7323 26.2374


25.7424 25.2475 24.7525


24.2576 23.7626 23.2677


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22.7727 22.2778 21.7828


21.2879 20.7929 20.2980


19.8030 19.3081 18.8131


18.3182 17.8232 17.3283


16.8333 16.3384 15.8434


15.3485 14.8535 14.3586


13.8636 13.3687 12.8737


12.3788 11.8838 11.3889


10.8939 10.3990 9.9040


9.4091 8.9141 8.4192


7.9242 7.4293 6.9343


6.4394 5.9444 5.4495


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4.9545 4.4596 3.9646


3.4697 2.9747 2.4798


1.9848 1.4899 0.9949

Column 100

0.5000

7.>>a = [2 35 6;5 67 88;22 56 89] a =

2 35 6 5 67 88 22 56 89

>>b = [23 56 78 73 6835 98 54 32 1599 34 23 12 2]

b =

23 56 78 73 68 35 98 54 32 15 99 34 23 12 2

>>cd = 9 ;e 6;h=8;>>Ram=[e, cd*h,cos(pi/3);h^2,sqrt(h*h/cd),15]

Ram = 6.0000 72.0000 0.5000 64.0000 2.6667 15.0000

>>Z= [1:2:11;0.0:5:25;linspace(10,60,6)] Z =

Columns 1 through 5

1 3 5 7 9 0 5 10 15 20 10 20 30 40 50

Column 6


10

11 25 60

>>zr=zeros(4,6) zr =

Columns 1 through 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Column 6

0 0 0 0

>>on=ones(3,4) on =

1 1 1 1 1 1 1 1 1 1 1 1

>>we=eye(5) we =

1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

>>aa=[4 8 9]>>bb= aa’

aa =

4 8 9bb =

4 8 9>>B=[3 6 7 8; 8 7 6 4;2 7 9 3]>>C=B’

B =

3 6 7 8 8 7 6 4 2 7 9 3C =

3 6 7 8 8 7 6 4 2 7 9 3

>>D=[ 3 5 6 8 23 67] D =


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>>E=D(3) Columns 1 through 5

3 5 6 8 23

Column 6

67E =

6>>D(2)=69 D =

Columns 1 through 5

3 69 6 8 23

Column 6

67>>D(2) + D(5) 92>>D(3)^3 + D(4)^4 4312>>M=[3 11 6 5;4 7 10 2;13 9 0 8] M =

3 11 6 5 4 7 10 2 13 9 0 8

>>M(2,3)=18 M =

3 11 6 5 4 7 18 2 13 9 0 8

>>M(3,2)-M(4,1) Index exceeds matrix dimensions.

8.>> v=[23 56 34 45 67 54 23 12 21]>>w=v(2:6)

v =

Columns 1 through 5

23 56 34 45 67

Columns 6 through 9

54 23 12 21w =

56 34 45 67 54>>Q=[1 3 4 5 6 8 ;4 6 7 8 2 1;1 1 4 6 8 9; Q =


12

23 56 7 8 34 2; 21 45 67 83 2 3] Columns 1 through 5

1 3 4 5 6 4 6 7 8 2 1 1 4 6 8 23 56 7 8 34 21 45 67 83 2

Column 6

8 1 9 2 3

>>R=Q(:,3) R =

4 7 4 7 67

>>S=Q(2,:) S =

Columns 1 through 5

4 6 7 8 2

Column 6

1>>T=Q(2:4,:) T =

Columns 1 through 5

4 6 7 8 2 1 1 4 6 8 23 56 7 8 34

Column 6

1 9 2

>>U=Q(1:3,2:4) U =

3 4 5


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6 7 8 1 4 6

>>V=4:3:34 V =

Columns 1 through 5

4 7 10 13 16


19 22 25 28 31

Column 11

34>>A=[10:-1:4;ones(1,7);2:2:14;zeros(1,7)] A =

Columns 1 through 5

10 9 8 7 6 1 1 1 1 1 2 4 6 8 10 0 0 0 0 0

Columns 6 through 7

5 4 1 1 12 14 0 0

>>B=A([1,3],[1,3,5:7]) B =

10 8 6 5 4 2 6 10 12 14

9.Function Description Example

length(A) Designate the number of column

10 9 8 7 6 5 41 1 1 1 1 1 12 4 6 8 10 12 140 0 0 0 0 0 0Ans =7

size(A) Number of row and number of column.

10 9 8 7 6 5 41 1 1 1 1 1 12 4 6 8 10 12 140 0 0 0 0 0 0Ans = 4 x 7


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reshape(A,m,n) B = reshape(A,m,n) returns the m-by-n matrix B whose elements are taken columnwise from A. An error results if A does not have m*n elements.

Reshape a 3-by-4 matrix into a 2-by-6 matrix:

A = 1 4 7 10 2 5 8 11 3 6 9 12 B = reshape(A,2,6) B = 1 3 5 7 9 11 2 4 6 8 10 12

diag(v) Diagonal matrices and diagonals of a matrix.

23 0 0 0 0 0 0 0 00 56 0 0 0 0 0 0 00 0 34 0 0 0 0 0 00 0 0 45 0 0 0 0 00 0 0 0 67 0 0 0 00 0 0 0 0 54 0 0 00 0 0 0 0 0 23 0 00 0 0 0 0 0 0 12 00 0 0 0 0 0 0 0 21

Diag(A) Diagonal of matrices and diagonal of matrix.

10 9 8 7 6 5 41 1 1 1 1 1 12 4 6 8 10 12 140 0 0 0 0 0 0Ans = 10 1 6 0

where: A is a matrix and v is a vector

10.>> b = ‘Matlab Programming’ No result>>c= ‘ My name is Richard Schooling’ No result>>c(5) 47.5202>>c(12:18) Columns 1 through 5

44.0556 43.5606 43.0657 42.5707 42.0758


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Columns 6 through 741.5808 41.0859

>>Info=char(‘Student Name:’,’Richard Schooling’,’Grade:’,’A+’)

No result

11.>>VecA=[ 8 6 7];VecB=[2 3 6];>>VecC= VecA + VecB

VecC =

10 9 13>>A=[3 -5 7;7 8 3];B=[2 4 5; 1 2 2];>>C= A - B

C =

1 -9 2 6 6 1

>>D= A + B D =

5 -1 12 8 10 5

>>A=[2 3 4; 5 4 7; 3 6 9; 5 3 1];>>B=[3 4 ; 3 2 ; 7 8];>>C=A*B

C =

43 46 76 84 90 96 31 34

>>D=B*A Inner matrix dimensions must agree.>>F=[6 7; 4 3]; G=[1 2; 4 5];>>H=F*G

H =

34 47 16 23

>>I=G*F I =

14 13 44 43

>>AV=[ 2 5 7];BV=[3;4;1];>>AV*BV

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>>BV*AV ans =

6 15 21 8 20 28 2 5 7

>>A=[2 6 7 9; 3 2 1 4; 4 6 3 1]; b=2;>>b*A

ans =

4 12 14 18 6 4 2 8 8 12 6 2

>>A*b ans =

4 12 14 18 6 4 2 8 8 12 6 2


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>>D=5*A D =

10 30 35 45 15 10 5 20 20 30 15 5

>>A=[3 -2 5; 3 2 6;7 4 2];>>B=inv(A)

B =

0.1408 -0.1690 0.1549 -0.2535 0.2042 0.0211 0.0141 0.1831 -0.0845

>>A*B ans =

1.0000 0 0.0000 -0.0000 1.0000 0 -0.0000 0.0000 1.0000

>>A*A^-1 ans =

1.0000 0 0.0000 -0.0000 1.0000 0 -0.0000 0.0000 1.0000

12.>>A = [4 -2 6;2 8 2;6 10 3];>>B= [8;4;0];>>X = A\B

X =

-1.8049 0.2927 2.6341

>>Xb=inv(A)*B Xb =

-1.8049 0.2927 2.6341

>>C=[4 2 6;-2 8 10;6 2 3] C =

4 2 6 -2 8 10 6 2 3

>>D=[8 4 0] D =

8 4 0>>Xc=D/C Xc =

-1.8049 0.2927 2.634113.

>>A=[3 6 8; 3 5 6] A =

3 6 8 3 5 6

>>B=[2 4 3; 6 3 4] B =


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2 4 3 6 3 4

>>C=A.*B C =

6 24 24 18 15 24

>>D=A./B D =

1.5000 1.5000 2.6667 0.5000 1.6667 1.5000

>>E=B.^B E =

Columns 1 through 2

4 256 46656 27

Column 3

27 256

>>F=A*B Inner matrix dimensions must agree.>>x=[1:8] x =

Columns 1 through 5

1 2 3 4 5

Columns 6 through 8

6 7 8>>y=x.^2 + 5*x y =

Columns 1 through 5

6 14 24 36 50

Columns 6 through 8

66 84 104>>x=[1:2:15] x =

Columns 1 through 5

1 3 5 7 9

Columns 6 through 8


18

11 13 15>>y=(x.^3 + 5*x)./(4*x.^2 – 10) y =

Columns 1 through 3

-1.0000 1.6154 1.6667

Columns 4 through 6

2.0323 2.4650 2.9241

Columns 7 through 8

3.3964 3.8764>> x =

Columns 1 through 3

0 0.5236 1.0472

Columns 4 through 6

1.5708 2.0944 2.6180

Column 7

3.1416>>y=cos(x) y =

Columns 1 through 3

1.0000 0.8660 0.5000

Columns 4 through 6

0.0000 -0.5000 -0.8660

Column 7

-1.000014.

Function Description Examplemean(A) It returns the mean values of

the elements along different dimensions of an array.

A = [1 2 3; 3 3 6; 4 6 8; 4 7 7];

mean(A)


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ans =

3.0000 4.500 6.0000

mean(A,2)

ans =

2.0000

4.0000

6.0000

6.0000

C=max(A) It returns the largest elements along different dimensions of an array.

Return the maximum of a 2-by-3 matrix from each column:X = [2 8 4; 7 3 9];max(X,[],1)ans =

7 8 9Return the maximum from each row:max(X,[],2)ans =

8 9Compare each element of X to a scalar:max(X,5)ans =

5 8 57 5 9

(d,n)=max(A)min(A) It returns the smallest elements

along different dimensions of an array.

Return the minimum of a 2-by-3 matrix from each column:X = [2 8 4; 7 3 9];min(X,[],1)ans =

2 3 4Return the minimum from each


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row:min(X,[],2)ans =

2 3Compare each element of X to a scalar:min(X,5)ans =

2 5 4 5 3 5

(d,n)=min(A)sum(A) It returns the sum of the

elements of A along the first array dimension whose size does not equal 1

Create a 3-by-3 matrix.A = [1 3 2; 4 2 5; 6 1 4]A =

1 3 2 4 2 5 6 1 4Compute the sum of the elements in each row.S = sum(A,2)S =

6 11 11

sort(A) It sorts the elements along different dimensions of an array, and arranges those elements in ascending order.

Sort horizontal vector A:A = [78 23 10 100 45 5 6];

sort(A)ans = 5 6 10 23 45 78 100

median(A) It returns the median value of A. Define a 4-by-3 matrix.A = [0 1 1; 2 3 2; 1 3 2; 4 2 2]A =

0 1 1 2 3 2 1 3 2 4 2 2Find the median value of each column.M = median(A)M =1.5000 2.5000 2.0000


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For each column, the median value is the mean of the middle two numbers in sorted order.

std(A) It is a function of X, where X is a vector, returns the standard deviation using (1) above. The result s is the square root of an unbiased estimator of the variance of the population from which X is drawn, as long as X consists of independent, identically distributed samples.

For matrix XX = 1 5 9 7 15 22s = std(X,0,1) s = 4.2426 7.0711 9.1924s = std(X,0,2)s = 4.000 7.5056

det(A) It returns the determinant of the square matrix X.

The statement A = [1 2 3; 4 5 6;

7 8 9]

Produces

A =

1 2 3

4 5 6

7 8 9

This happens to be a singular

matrix, so det(A) produces a very

small number.

Changing A(3,3) with A(3,3) =

0 turns A into a nonsingular

matrix. Now d = det(A) produces d = 27 The

statement A = [1 2 3; 4 5 6; 7 8

9]produces

A =

1 2 3


22

4 5 6

7 8 9

This happens to be a singular

matrix, so det(A) produces a very

small number. Changing A(3,3)

with A(3,3) = 0 turns A into a

nonsingular matrix. Now d =

det(A) produces d = 27.

dot(a,b) It returns the scalar product of the vectors A and B. A and B must be vectors of the same length. When A and B are both column vectors, dot(A,B) is the same as A'*B.

The dot product of two vectors is calculated as shown:a = [1 2 3]; b = [4 5 6];c = dot(a,b)c = 32

cross(a,b) It returns the cross product of the vectors A and B. That is, C = A x B. A and B must be 3-element vectors. If A and B are multidimensional arrays, cross returns the cross product of A and B along the first dimension of length 3.

The cross and dot products of two vectors are calculated as shown:a = [1 2 3]; b = [4 5 6];c = cross(a,b) c = -3 6 -3d = dot(a,b)d = 32

inv(A) It returns the inverse of the square matrix X. A warning message is printed if X is badly scaled or nearly singular.

Here is an example demonstrating the difference between solving a linear system by inverting the matrix with inv(A)*b and solving it directly with A\b. A random matrix A of order 500 is constructed so that its condition number, cond(A), is 1.e10, and its norm, norm(A), is1. The exact solution x is a random vector of length 500 and the right-hand side is b = A*x. Thus the system of linear equations is badly conditioned, but consistent.On a 300 MHz, laptop computer


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the statementsn = 500; Q = orth(randn(n,n));d = logspace(0,-10,n);A = Q*diag(d)*Q';x = randn(n,1);b = A*x;tic, y = inv(A)*b; tocerr = norm(y-x)res = norm(A*y-b)produceelapsed_time = 1.4320 err = 7.3260e-006 res = 4.7511e-007while the statementstic, z = A\b, tocerr = norm(z-x)res = norm(A*z-b)produceelapsed_time = 0.6410err = 7.1209e-006res = 4.4509e-015It takes almost two and one half times as long to compute the solution with y = inv(A)*b as with z = A\b. Both produce computed solutions with about the same error, 1.e-6, reflecting the condition number of the matrix. But the size of the residuals, obtained by plugging the computed solution back into the original equations, differs by several orders of magnitude. The direct solution produces residuals on the order of the machine accuracy, even though the system is badly conditioned.The behavior of this example is typical. Using A\b instead of inv(A)*b is two to three times as fast and produces residuals on


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the order of machine accuracy, relative to the magnitude of the data.

7. Conclusion:I therefore conclude that Matlab can be used as an engineering devise in solving common mathematical problems such as mathematical computations, modelings, graphs, etc.

8. Further Readings: Ferraris, G. and Manenti, F. (2010). Interpolation and regression models for the chemical engineer: solving numerical problems. Germany: Wiley-VCH Verlag Filo, O. (2010). Information processing by biochemical systems: neural network type configurations. New Jersey: Wiley. Gopal, S. (2009). Bioinformatics: a computing perspective. India: McGraw-Hill Science/Engineering Math. Jaluria, Y. (2012). Computer methods for engineering with MATLAB applications (2nd ed.). Boca, Raton,Florida: CRC Press. Knopf, F. C. (2012). Modeling, analysis and optimization of process and energy systems.Hoboken, New Jersey: John Wiley and Sons. Velten, K. (2009). Mathematical modeling and simulation: introduction for scientists and engineers. Singapore: Wiley-VCH.

9. Assessment (Rubric for Laboratory Performance):TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES

RUBRIC FOR MODERN TOOL USAGE(Engineering Programs)

Student Outcome (e): Use the techniques, skills, and modern engineering tools necessary for engineering practice in complex engineering activities.Program: Chemical Engineering Course: CHE 530D1 Section: _______ ____Sem SY ________


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Performance Indicators

Unsatisfactory1

Developing2

Satisfactory3

Very Satisfactory4

Score

1. Apply appropriate techniques, skills, and modern tools to perform a discipline-specific engineering task.

Fails to identify any modern techniques to perform discipline-specific engineering task.

Identifies modern techniques but fails to apply these in performing discipline-specific engineering task.

Identifies modern techniques and is able to apply these in performing discipline-specific engineering task.

Recognizes the benefits and constraints of modern engineering tools and shows intention to apply them for engineering practice.

2. Demonstrate skills in applying different techniques and modern tools to solve engineering problems.

Fails to apply any modern tools to solve engineering problems.

Attempts to apply modern tools but has difficulties to solve engineering problems.

Shows ability to apply fundamental procedures in using modern tools when solving engineering problems.

Shows ability to apply the most appropriate and effective modern tools to solve engineering problems.

3. Recognize the benefits and constraints of modern engineering tools.

Does not recognize the benefits and constraints of modern engineering tools.

Recognizes some benefits and constraints of modern engineering tools.

Recognizes the benefits and constraints of modern engineering tools and shows intention to apply them for engineering practice.

Recognizes the need for benefits and constraints of modern engineering tools and makes good use of them for engineering practice.

Total ScoreMean Score = (Total Score / 3)

Percentage Rating = (Total Score / 12) x 100%Evaluated by: ______________________________________ _______________ Printed Name and Signature of Faculty Member Date


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COMPUTER APPLICATIONS IN CHEMICAL ENGINEERING