Question

use the linearize model to estimate k and C0 based on the following data using matlab

Required information Linear regression provides a powerful technique for fitting a best line to data. The below figure shows the transformations can be used to express the data in a form that is compatible with linear regression. Apart from the given figure, there are other models that can be linearized using transformations. For example, the following model applies to third-order chemical reactions in batch reactors expression error) = (expression error) Standard deviation = sy = V (28-1) where c-concentration (mg/L), co = initial concentration (mg/L), k = reaction rate (L2/(mg? d), and t= time (d). (d) 0 0.5 11. 52345 3.26 (mg/L) 2.09 1.62 1.48 1.171.06 0.9 0.85 Linearization Linearization < Prey 9 of 9 Next Sok Air

Answer 1

Screenshot of the program:  linearizeModel.m

24 moliz. anti koji Data : Initialize time vector t = [0 0.5 1 1.5 2 3 4 5); Initialize concentrations vector at each time c = [3.26 2.09 1.62 1.48 1.17 1.06 0.9 0.85); X =t; Y = 1./c.^2; a = polyfit(X, Y, 1); a = a (2); al = a (1); k = a1/2; CO = 1/sqrt (a0); fprintf('The value of kis 8.4f L^2/ (mg2 d)', k); fprintf('and the value of co is %.4 mg/L. \n', c0); &find error e = (Y-a0-a1.*x).^2; find sum of squares of error sr = sum (e); find standard deviation Sy = sqrt (sr/(28-1)); C=C0./(sqrt (1+2*k*c0^2*t)); c_e = polyval (a,x); $plot the data plot(t,c, 'o',t,C); xlabel('Time (d)'); ylabel('Concentration'); legend ("Data', 'Untransformed Model'); grid on figure (2) plot (X, Y, 'o',X,C_e) xlabel ('Time (d)'); ylabel('Concentration'); legend ('Transformed Data', 'Transformed Model'); grid on

Sample output:

Command Window >> linearizeModel The value of k is 0.1332 L^2/ (mg 2 d) and the value of co is 3.0324 mg/L. fx >>

- O X Figure 1 File Edit View Insert Tools Desktop Window Help Data -Untransformed Model Concentration 0 0.5 1 1.5 2 3 3.5 4 4.5 5 2.5 Time (d)

- O X Figure 2 File Edit View Insert Tools Desktop Window Help o Transformed Data - Transformed Model Concentration 0 0.5 1 1.5 2 3 3.5 4 4.5 5 2.5 Time (d)

Code to copy:

%Data : Initialize time vector
t = [0 0.5 1 1.5 2 3 4 5];
%Initialize concentrations vector at each time
c = [3.26 2.09 1.62 1.48 1.17 1.06 0.9 0.85];
X =t;
Y = 1./c.^2;
a = polyfit(X,Y,1);
a0 = a(2);
a1 = a(1);
k = a1/2;
c0 = 1/sqrt(a0);
fprintf('The value of k is %.4f L^2/(mg^2 d)', k);
fprintf('and the value of c0 is %.4f mg/L.\n', c0);
%find error
e = (Y-a0-a1.*X).^2;
%find sum of squares of error
sr = sum(e);
%find standard deviation
Sy = sqrt(sr/(28-1));
C =c0./(sqrt(1+2*k*c0^2*t));
C_e = polyval(a,X);
%plot the data
plot(t,c, 'o',t,C);
xlabel('Time (d)');
ylabel('Concentration');
legend('Data','Untransformed Model');
grid on
figure(2)
plot(X,Y,'o',X,C_e)
xlabel('Time (d)');
ylabel('Concentration');
legend('Transformed Data','Transformed Model');
grid on