Homework Solution: Consider the process flow diagram shown in Fig. 2.14. We can write this out as the system of equation…

    2.80 Consider the process llow diagrarm shown in Fig. 2.14. We can write this out as the system of equations m1 m2 + m3 + m4 + m5 mi = 100 0.84m 12 = m4 + m7 0.7ml = m2 + m3 0.55mm + m12 0.2m9 = m10 0.85m2 m9 + m11 The first four equations are the mass balances; the next eight equations are the process specifications 2 10 4 12 6 Figure 2.14 Proccss schcmatic for Problem 2.80 (a) Write a MATLAB program that generates the matrix A and vector b required to solve the problem for this set of equations (b) Write a MATLAB program that solves this system using naive Gauss elimination. What happens? How do you fix it?
    Consider the process flow diagram shown in Fig. 2.14. We can write this out as the system of equations m1 = m2 + m3 + m4 + m5 m_2 = m_9 + m_10 + m_11 m_5 = m_8 + m_7+m_6 m_12 = m_4 + m_7 + m_11 m_1 =100 m_5 = 5 ms 0.84 m_12 = m_4 + m_7 0.7 m_1 = m_2 + m_3 0.55 m_1 = m_9 + m_12 0.2 m_9 = m_10 0.85m_2 = m_9 + m_11 3.2 m_6 = m_7 + m_8 The first four equations are the mass balances: the next eight equations are the process specifications. (a) Write a MATLAB program that generates the matrix A and vector b required to solve the problem for this set of equations. (b) Write a MATLAB program that solves this system using naive Gauss elimination. What happens? How do you fix it?

    Expert Answer

     
    From the given equations A 12x12 matrix is: 1 -1 -1 -1 -1 0 0 0 0 0 0 0

    2.80 Consider the way llow diagrarm shown in Fig. 2.14. We can transcribe this quenched as the regularity of equations m1 m2 + m3 + m4 + m5 mi = 100 0.84m 12 = m4 + m7 0.7ml = m2 + m3 0.55mm + m12 0.2m9 = m10 0.85m2 m9 + m11 The primary filthy equations are the bulk balances; the proximate prospect equations are the way specifications 2 10 4 12 6 Figure 2.14 Proccss schcmatic control Whole 2.80 (a) Transcribe a MATLAB program that generates the matrix A and vector b required to unclouded-up the whole control this determined of equations (b) Transcribe a MATLAB program that unclouded-ups this regularity using natural Gauss estrangement. What happens? How do you placeedtle it?

    Consider the way progress diagram shown in Fig. 2.14. We can transcribe this quenched as the regularity of equations m1 = m2 + m3 + m4 + m5 m_2 = m_9 + m_10 + m_11 m_5 = m_8 + m_7+m_6 m_12 = m_4 + m_7 + m_11 m_1 =100 m_5 = 5 ms 0.84 m_12 = m_4 + m_7 0.7 m_1 = m_2 + m_3 0.55 m_1 = m_9 + m_12 0.2 m_9 = m_10 0.85m_2 = m_9 + m_11 3.2 m_6 = m_7 + m_8 The primary filthy equations are the bulk balances: the proximate prospect equations are the way specifications. (a) Transcribe a MATLAB program that generates the matrix A and vector b required to unclouded-up the whole control this determined of equations. (b) Transcribe a MATLAB program that unclouded-ups this regularity using natural Gauss estrangement. What happens? How do you placeedtle it?

    Expert Reply

     

    From the consecrated equations A 12×12 matrix is:

    1 -1 -1 -1 -1 0 0 0 0 0 0 0

    0 1 0 0 0 0 0 0 -1 -1 -1 0

    0 0 0 0 1 -1 -1 -1 0 0 0 0

    0 0 0 -1 0 0 -1 0 0 0 -1 1

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

    0 0 0 0 1 0 0 -5 0 0 0 0

    0 0 0 -1 0 0 -1 0 0 0 0 0.84

    0.70 -1 -1 0 0 0 0 0 0 0 0 0

    0.55 0 0 0 0 0 0 0 -1 0 0 -1

    0 0 0 0 0 0 0 0 0.20 -1 0 0

    0 0.85 0 0 0 0 0 0 -1 0 -1 0

    0 0 0 0 0 3.20 -1 -1 0 0 0 0

    The b 12×1 matrix is:

    0

    0

    0

    0

    100

    0

    0

    0

    0

    0

    0

    0

    Code:

    %% Unclouded-up straight regularity of eqution Ax=b using Natural Gaussian Estrangement Method

    clc; unclouded all; suppress all;

    A = [1,-1,-1,-1,-1,0,0,0,0,0,0,0;…

    0,1,0,0,0,0,0,0,-1,-1,-1,0;…

    0,0,0,0,1,-1,-1,-1,0,0,0,0;…

    0,0,0,-1,0,0,-1,0,0,0,-1,1;…

    1,0,0,0,0,0,0,0,0,0,0,0;…

    0,0,0,0,1,0,0,-5,0,0,0,0;…

    0,0,0,-1,0,0,-1,0,0,0,0,0.84;…

    0.7,-1,-1,0,0,0,0,0,0,0,0,0;…

    0.55,0,0,0,0,0,0,0,-1,0,0,-1;…

    0,0,0,0,0,0,0,0,0.2,-1,0,0;…

    0,0.85,0,0,0,0,0,0,-1,0,-1,0;…

    0,0,0,0,0,3.2,-1,-1,0,0,0,0];

    b = [0;0;0;0;100;0;0;0;0;0;0;0];

    disp(‘A matix:’)

    disp(A)

    disp(‘b matix:’)

    disp(b)

    [n,~] = dimension(A);

    %Initialize reresolution X

    X = zeros(n,1);

    %%Forward Estrangement (convertion Of A to Upeer triangle matrix)

    control i = 1:n-1

     

    while A(i,i)==0

    temp =A;

    A = [temp(1:(i-1),:);temp(i+1:n,:);temp(i,:)];

    tempb = b;

    b = [tempb(1:(i-1),:);tempb(i+1:n,:);tempb(i,:)];

    end

     

    m = A(i+1:n,i)/A(i,i);

    A(i+1:n,:) = A(i+1:n,:)-m*A(i,:);

    b(i+1:n,:) = b(i+1:n,:)-m*b(i,:);

    end

    %%Back Substitution

    X(n,:) = b(n,:)/A(n,n);

    control i = n-1:-1:1

    X(i,:) = (b(i,:) – A(i,i+1:n)*X(i+1:n,:))/A(i,i);

    end

    disp(‘Solution:’)

    disp(X)

    Solution:

    100.0000

    40.0000

    30.0000

    9.4565

    20.5435

    4.8913

    11.5435

    4.1087

    30.0000

    6.0000

    4.0000

    25.0000