Doehlert.m

% Doehlert Design for Wet Pelletization in High Shear Mixer
% Author's Data: Housam Binous
% Department of Chemical Engineering
% National Institute of Applied Sciences and Technology, Tunis, TUNISIA
% Email: [e-mail]



% Acknowledgement :
% This problem was presented in the excellent book by 
% Lewis G. A., D. Mathieu and R. Phan-Than-Luu, 
% Pharmaceutical Experimental Design, 
% Drugs and the Pharmaceutical Sciences, Vol 92, 1999, 
% Marcel Dekker, New York.


% Three Factors:
% 1/ Impeller Speed (rpm)= 
% Associated variable: X1, Lower level: 250, Upper level: 400
% 2/ Amount of binding solution (%)= 
% Associated variable: X2, Lower level: 60 Upper level: 70
% 3/ Pelletization time (min)= 
% Associated variable: X3, Lower level: 5 Upper level: 20

% One Response: 
% Mean size of pellets (mm)

% Plan and Experimental Results

A=[   1, 0, 0, 0, 325, 65.0, 12.5, 1594; ...
    2, 1, 0, 0, 400, 65.0, 12.5, 1398; ...
    3, 0.5, 0.866, 0,362, 69.3, 12.5, 1352; ...
    4, -0.5, 0.866, 0, 288, 69.3, 12.5, 1466; ...
    5, -1, 0, 0, 250,65.0, 12.5, 1750; ...
    6, -0.5, -0.866, 0, 288, 60.7,12.5, 1064; ...
    7, 0.5, -0.866, 0, 362, 60.7, 12.5,1337; ...
    8, 0.5, 0.289, 0.816, 362, 66.4, 18.6, 1533; ...
    9, -0.5, 0.289, 0.816, 288, 66.4, 18.6, 1617; ...
    10, 0, -0.577, 0.816,325, 62.1, 18.6, 1332; ...
    11, 0.5, -0.289, -0.816, 326,63.6, 6.4, 1409; ...
    12, -0.5, -0.289, -0.816, 288, 63.6, 6.4, 1560; ...
    13, 0, 0.577, -0.816, 325, 68.9, 6.4, 1431]

% Responses' vector is labelled y

y=A(:,end)

% Forming M the Computation Matrix

X1=A(:,2)'

X2=A(:,3)'

X3=A(:,4)'

X1SQR=(X1.*X1)

X2SQR=(X2.*X2)

X3SQR=(X3.*X3)

X12=(X1.*X2)

X13=(X1.*X3)

X23=(X2.*X3)

M=[1,1,1,1,1,1,1,1,1,1,1,1,1;X1;X2;X3;X1SQR;X2SQR;X3SQR;X12;X13;X23]'

% Determination of the quadratic model by multilinear regression 
% of the mean particle size:

% yp = 1594 - 97.5*x1 + 117.6*x2+ 16.7*x3- 20.0*x1^2- 379.0*x2^2 
% - 70.8*x3^2 - 223.4*x1*x2+ 120.1*x1*x3 + 83.0*x2*x3

% Results are in agreement with the model obtained 
% by Lewis G. A., D. Mathieu and R. Phan-Than-Luu 
% in Pharmaceutical Experimental Design, 
% Drugs and the Pharmaceutical Sciences, Vol 92, 1999, 
% Marcel Dekker, New York. 

a=inv(M'*M)*M'*y

% Contour plots of the obtained mean size versus impeller speed (x1) 
% and pelletization time (x3) for x2 (binding solution) =0.5 and 0. 
% We get the same figures as plotted in Figure 5.14 page 229 
% of the book by Lewis G. A., D. Mathieu and R. Phan-Than-Luu 
% in Pharmaceutical Experimental Design, 
% Drugs and the Pharmaceutical Sciences, Vol 92, 1999, 
% Marcel Dekker, New York.  

x2=zeros(201,201);

[x1,x3] = meshgrid([-1:.01:1]);

yp = 1594 - 97.5*x1 + 117.6*x2+ 16.7*x3- 20.0*x1.^2- 379.0*x2.^2 ...
- 70.8*x3.^2 - 223.4*x1.*x2+ 120.1*x1.*x3 + 83.0*x2.*x3;

figure(1)
contour(x1,x3,yp,30)

x2=0.5*ones(201,201);

[x1,x3] = meshgrid([-1:.01:1]);

yp = 1594 - 97.5*x1 + 117.6*x2+ 16.7*x3- 20.0*x1.^2- 379.0*x2.^2 ...
- 70.8*x3.^2 - 223.4*x1.*x2+ 120.1*x1.*x3 + 83.0*x2.*x3;

figure(2)
contour(x1,x3,yp,30)

x2=-0.5*ones(201,201);

[x1,x3] = meshgrid([-1:.01:1]);

yp = 1594 - 97.5*x1 + 117.6*x2+ 16.7*x3- 20.0*x1.^2- 379.0*x2.^2 ...
- 70.8*x3.^2 - 223.4*x1.*x2+ 120.1*x1.*x3 + 83.0*x2.*x3;

figure(3)
contour(x1,x3,yp,30)