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Matlab Homework #1


ChE 231


Spring 2016



Consider the production of an ethylene-propylene-diene (EPDM) polymer in an isothermal CSTR.


The reactor has 7 adjustable inputs x1 . . . x7 denoting the residence time, ethylene monomer


feed concentration, propylene monomer feed concentration, diene monomer feed concentration,


catalyst feed concentration, hydrogen feed concentration, and co-catalyst feed concentration, respectively. All inputs have been normalized to unity with their nominal value and are therefore


non-dimensional. The 4 measured outputs y1 . . . y4 represent the number average molecular weight


(kg/mol), the polymer ethylene fraction (%), the polymer diene fraction (%), and the polymer


production rate (kg/L/h), respectively. A Matlab function cstrdesign is provided along with


supporting ?les cstrsys and cstrsf to implement experimental designs on the polymerization


system. The cstrdesign function has the following usage:








where X is a n × 7 matrix of inputs with each row corresponding to a single run, Y is a n × 4


matrix of outputs with each row corresponding to a single run, D is a m × 7 experimental design


matrix, and ncp is the number of center points to augment the design matrix (n = m + ncp).


Since a center point (all zeros) corresponds to a run with all inputs at their nominal value (one),


speci?cation of ncp alone allows a number of repeated experiments to be run with the nominal


inputs. The cstrdesign function expects a design matrix D with values between ?1 and 1 and


scales the CSTR inputs X from 0.75 to 1.25 (±25%) such that a zero in the design matrix D


corresponds to a nominal input value of 1.


1. Perform 40 polymerization experiments with the nominal inputs and collect the steadystate molecular weight for each run. Use Matlab and the collected molecular weight data


(mw=Y(:,1);) to perform each of the following tasks:


(a) Generate a plot of the data versus the sample number (plot).


(b) Generate a normal (Gaussian) probability plot (normplot). Is the plot linear? Assess


whether the data could have come from a normal distribution.


(c) Calculate the sample mean µ (mean).




(d) Calculate the sample standard deviation ? (std).




(e) Calculate the sample variance ? 2 (var).




(f) Perform a mean test (t-test) to test if the data comes from a normal distribution with


mean µ (ttest). Use ? = 0.03. Can the hypothesis that the data comes from a




normal distribution with mean µ be rejected at the 3% signi?cance level? Report a




97% con?dence interval on the mean.





(g) Perform a variance test to test if the data comes from a normal distribution with




variance ? 2 (vartest). Use ? = 0.03. Can the hypothesis that the data comes from a




normal distribution with variance ? 2 be rejected at the 3% signi?cance level? Report


a 97% con?dence interval on the variance.


(h) Perform maximum likelihood estimation to estimate the mean, µ, and standard de?


viation, ? , of a normal distribution data (normfit). Report the estimated mean and




variance along with 97% con?dence intervals for both.


2. Generate the following experimental designs for the polymerization system. In each case


comment on the design, D, and report the number of runs required to implement the design.


It is not necessary to report the design matrix D.


(a) 2-level full factorial design: D=fullfact([2 2 2 2 2 2 2])*2-3;


(b) 4-level full factorial design: D=fullfact([4 4 4 4 4 4 4])*(2/3)-(5/3);


Note: The fullfact function returns designs with levels assigned the numbers


1, 2 . . . n where n is the number of levels. The additional math operations


to generate D scale the full factorial design to the interval ?1 . . . 1 to be


consistent with designs generated with other Matlab functions and to meet


the requirements of the cstrdesign function.


(c) 2k fractional factorial designs for k=3,4,5,6:


gens=fracfactgen(?a b c d e f g?,k);




3. Implement a 2-level full factorial design with 15 additional center points on the polymerization system. Use the cstrdesign function to ?nd the predictor X and response Y matrices


from the design D ([X,Y]=cstrdesign(D,15)). Analyze the ?t and response of a linear


model with the interactive Matlab tool rstool (rstool(X,Y)).


(a) Produce a printout with all of the predictors X1. . .X7 at their nominal values (1).


(b) Export the ?tted parameters, ?, and report their values. Each column of the matrix


beta corresponds to one of the four outputs. The ?rst row of beta contains the bias


terms, ?0 , and the remaining rows contain the linear regression coe?cients, ?1 . . . ?7 ,


for each of the 7 outputs. Write the linear model equation that relates the molecular


weight output to the 7 inputs (x1 . . . x7 ) using the ?tted parameter values.


(c) Comment on the e?ect of each predictor X1. . .X7 (inputs) on each response Y1. . .Y4


(outputs). Recall what each of the inputs and outputs physically represents as you


discuss the results.


(d) Adjust the nominal predictor values by either clicking on the graphs or entering new


numbers in the boxes. Suggest a new set of inputs based on your model that increase


both the molecular weight and the polymer production rate while maintaining the


polymer monomer fractions approximately constant.






4. Pick any input (Xj , j ? [1, 2 . . . 7]) and perform correlation analysis on the outputs using the


data from your design implementation. Calculate p-values for the correlation coe?cients and


use the test p < 0.03 to determine if the correlations are signi?cant. Report the correlation


matrix R and the p-value matrix P , and list all of the outputs that have a signi?cant


correlation with your chosen input.


[R, P] = corrcoef([Y X(:,j)])


[r, c] = find(P < 0.03); rc = [r c]; k = find(rc(:,1) == 5);




5. Implement one of the following reduced design:


? 25 fractional factorial design with 3 additional center points (ncp=8)


? 26 fractional factorial design with 4 additional center points (ncp=16)


? D-optimal design with 20 runs and 6 additional center points for ?tting a linear model


(D=rowexch(7,20,?linear?); ncp=6)


? D-optimal design with 60 runs and 10 additional center points for ?tting a model with


interactions (D=rowexch(7,60,?interaction?); ncp=10)


? D-optimal design with 60 runs and 10 additional center points for ?tting a purely


quadratic model (D=rowexch(7,60,?purequadratic?); ncp=10)


(a) Comment on the reduced design matrix D. What is the advantage of a reduced design?


(b) Use rstool to produce a printout with all of the predictors X1. . .X7 at their nominal


values (1).


(c) How do the results for the reduced design compare to the results from the full factorial


design? Does the linear model from the reduced design capture the same e?ects as the


full design?


(d) Compare the 95% con?dence intervals (two red curves) obtained from the two designs.







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Oct 15, 2019





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