12 Regression and Correlation of Data.pdf

8/11/2019 12 Regression and Correlation of Data.pdf

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Cutlip and Shacham: Problem Solving in Chemical and Biochemical EngineeringChapter 3

Regression and Correlation of Data

Cheng-Liang Chen

P SELABORATORYDepartment of Chemical Engineering

National TAIWAN University


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Estimation of Antoine Equation ParametersUsing Nonlinear Regression

Concepts Utilized: Direct use of the Antoine equation to correlate vaporpressure versus temperature data.

Numerical Methods: Nonlinear regression of a general algebraic expression withdetermination of the overall variance and condence intervals of individualparameters.

Problem Statement:The Antoine equation is a widely used vapor pressure correlation that utilizes threeparameters, A, B, and C . It is often expressed by

P v = 10A + B

T +

C (3-1)

where P v is the vapor pressure in mm Hg and T is the temperature in oC.

Vapor pressure data for propane ( P a ) versus temperature ( K ) is found in TableB-5 in Appendix B. Convert this data set to vapor pressure in mm Hg andtemperature in oC. Then


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(a) Determine the parameters of the Antoine equation and the corresponding 95%condence intervals for the parameters from the given data set by using nonlinearregression on Equation (3-1).

(b) Calculate the overall variance for the Antoine equation.

(c) Prepare a residual plot for the Antoine equation.

(d) Assess the precision of the data and the appropriateness of the Antoine equation

for correlation of the data set.

Solution:The form of the Antoine equation to be considered in this problem is to beregressed in its nonlinear form. An alternative treatment is to use multiple linear

regression on the logarithmic form of this equation, but this transformation is notfully suitable, as discussed in Problem 2.8.

General nonlinear regression can be used to determine the parameters of anexplicit algebraic equation, or model equation, as dened by Equation (3-2):

y = f (x1 , . . . , x n ; a1 , . . . , a m ) (3-2)


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where the single dependent variable is y, the n independent variables arex1 , . . . , x n , and the m parameters are al , . . . a m . It is usually assumed that theexperimental errors in the preceding equation are normally distributed withconstant variance. Nonlinear regression algorithms determine the parameters for a

particular model by minimizing the least-squares objective function (L8) given by

LS =N

i =1yi (obs) yi (calc)

where N is the number of data points and (obs) and (calc) refer to observed andcalculated values of the dependent variable.

The overall estimate of the model variance, 2 , is calculated from

2 =

N

i =1yi (obs) yi (calc)

= LS

where is the degrees of freedom, which is equal to the number of data pointsless the number of model parameters, (N m). Thus an algorithm that minimizes the least-squares objective function also minimizes the variance .Often the variance is used to compare the goodness of t for various models.


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A widely used graphical presentation that indicates any systematic difficulties witha particular model is called the residuals plot. This is simply the error, i , in themodel plotted versus the observed value of the independent variable, yi (obs) withthe error given by

i = yi (obs) yi (calc)

Nonlinear Regression of the Antoine Equation The rst step is to enter thedata by assigning a name to each variable (column). The column for the vaporpressure in P a will be designated as P and the column for the temperature in Kwill be denoted as TK. Since the Antoine equation is to correlate the vaporpressure in mm Hg and the temperature in oC, then a new column denoted as Pvand another new column denoted as TC can be created that calculate the pressureand temperature in units of mm Hg and oC by

Pv = .0075002 P

TC = TK 273.15


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function P3_01_CCLclear, clc, format short g, format compactxyData=[-187.68 1.268E-06

-183.15 7.268E-06

-173.15 0.0001883-163.15 0.0025884-153.15 0.0221106-143.15 0.1315085-133.15 0.5900482-123.15 2.117681-113.15 6.352669-103.15 16.50044-93.15 37.87601-83.15 78.7521-73.15 150.754-63.15 269.2572-53.15 453.7621-42.08 757.5202-33.15 1110.03-23.15 1635.044-13.15 2332.562

-3.15 3225.086


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6.85 4365.11616.85 5767.65426.85 7485.236.85 9525.254

46.85 1.2E+0456.85 1.485E+0466.85 1.823E+0476.85 2.213E+0486.85 2.663E+04];

X=xyData(:,1); m=size(X,1); %Determine the number of data pointsY=xyData(:,2);prob_title = ([\bf Antoine Eq. Par.s, Nonlinear Reg.]);dep_var_name= [\bf Vapor P (mmHg) ];ind_var_name= [\bf Temperature (C)];parm=[6 -1000 200]; % Initial guess for A, B, Cnpar=size(parm,2); % Determine the number of the parametersoptions = optimset(MaxFunEvals,1000); % Change the default value fBeta = fminsearch(@AntFun,parm,options,X,Y); % Find optimal param[f,Ycalc]= AntFun(Beta,X,Y); % Compute Y (calculated) at the optimumdisp( Results, Antoine Eq. Par.s, Nonlinear Regression );

Res=[];


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for i=1:nparRes=[Res; i Beta(i)];enddisp( Parameter No. Value );

disp(Res);s2=sum((Y-Ycalc)*(Y-Ycalc))/(m-npar); %variancedisp([ Variance , num2str(s2)]);ymean=mean(Y);R2=(Ycalc-ymean)*(Ycalc-ymean)/((Y-ymean)*(Y-ymean));%linear correldisp([ Correlation Coefficient , num2str(R2)])subplot(2,1,1)

plot(X,Ycalc,r-,X,Y,b+,LineWidth,2) %Plot of experimental andset(gca,FontSize,14,Linewidth,2)

title([\bf Cal./Exp. Data prob_title],fontSize,12)xlabel([ind_var_name],fontSize,14)ylabel([dep_var_name],fontSize,14)

subplot(2,1,2)plot(Y,Y-Ycalc,*,LineWidth,2) % Residual plot

set(gca,FontSize,14,Linewidth,2)title([\bf Residual Plot, prob_title ],fontSize,12)xlabel([dep_var_name \bf (Measured)],fontSize,14)

ylabel(\bf Residual,fontSize,14)

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function [f,Ycalc]=AntFun(parm,X,Y)a=parm(1);b=parm(2);c=parm(3);

for i=1:size(X,1);Ycalc(i,1)=10^(a+b/(X(i)+c));endresid(:,1)=Y-Ycalc;f=resid*resid;

Results, Antoine Eq. Par.s, Nonlinear RParameter No. Value

1 7.26442 -1046.33 281.61

Variance 507.9109Correlation Coefficient 0.99922

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Correlation of Thermodynamic andPhysical Properties of n-Propane

Concepts Utilized: Correlations for heat capacity, thermal conductivity,viscosity, and latent heat of vaporization.

Numerical Methods: Linear and nonlinear regression of data with linearizationand transformation functions.

Problem Statement:Tables B-7 through B-10 present values for different properties of propane(heat capacity, thermal conductivity for gas, viscosity, and latent heat of vaporization for liquid) as a function of temperature.

Determine appropriate correlations for the properties of propane listed in TablesB-7 through B-10 using suggested expressions; given in the chemicalengineering literature.

Solution:Heat Capacity for a Gas Heat capacity for propane gas is given in TableB-7 for temperatures between 50 K and 1500 K. According to Perry et al., the

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heat capacities of gases are most commonly represented as a simple polynomial:

C p = a0 + a1 T + a2 T 2 + a3 T 3 + . . .

Thermal Conductivity Thermal conductivity for gaseous propane is given inTable B-8 for the temperature range 231 K to 600 K. Perry et al. note that oversmall temperature ranges the thermal conductivity of low-pressure gases can befairly well correlated by a linear equation, which is also a rst degree polynomial.

However, we can try polynomials here. For a wide range of temperature, Perry etal. recommend the correlation of thermal conductivity, k, with T n , where T is theabsolute temperature and n 1.8. This correlation can be directly evaluated withnonlinear regression using the form below, where both c and n are parameters.

k = cT n

Liquid Viscosity The recommended correlation for viscosity of liquids by Perryet al. is similar to the Antoine equation for vapor pressure:

log() = A + B/ (T + C )

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where is the viscosity and A, B and C are parameters. If T is expressed inKelvin, parameter C can be approximated by C = 17 .71 0.19T b, where T b is thenormal boiling point in Kelvin. For n-propane the normal boiling point is 231 K;thus the approximate value of C is 26.18.

A four-parameter equation used in Reid et a1. provides another possiblecorrelation equation for viscosity of liquids:

log() = A + B/T C log(T ) + DT 2

Heat of Vaporization Heat of vaporization data for propane are shown inTable B-10 for the temperature range of 85.47 K to 360 K. Heat of vaporizationcan be correlated by an equation based on the Watson relation (Perry et al.):

H = A(T C T )n

Watsons recommended value for n is 0.38, but n can be found by regression of the experimental data. The critical temperature of propane is 369.83 K. Theequation can be directly used in nonlinear regression, or it can be linearized bytaking the log of each side yielding

log( H ) = log( A) + n log(T C T )

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function P3_03A_CCLclear, clc,format short g, format compactprob_title = ([ Heat Cap. of Gaseous Propane]);ind_var_name= [\bf Temperature (K)];

dep_var_name= [\bf Cp (kJ/kg-mol-K) ];xyData=[ 50 34.06100 41.3150 48.79200 56.07273.16 68.74298.15 73.6300 73.93400 94.01500 112.59600 128.7700 142.67800 154.77900 163.351000 174.61100 182.671200 189.74

1300 195.85


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xlabel([\bf Temperature (K)],FontSize,14)ylabel([dep_var_name],FontSize,14)subplot(2,1,2)

plot(y,y-ycal,*,LineWidth,2) % residual plot

set(gca,FontSize,14,Linewidth,2)title([\bf Residual plot, prob_title],FontSize,12)xlabel([dep_var_name \bf (measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ycal,ConfInt, Var, R2, n]=PolyReg(x,y,degree,freeparmtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabiif freeparm==1

n=degree+1;else

n=degree;end

m=size(x,1);

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for i=1:mfor j=1:n

if freeparm==1p=j-1;

else p=j;end

X(i,j)=x(i)^p; %Calculate powers of the independent variable and putend

end

Beta=X\y; % Solve XBeta = Y using QR decompositionycal=X*Beta; % Calculated dependent variable valuesVar=sum((y-ycal)*(y-ycal))/(m-n); % variance

if (m-n)>45t=2.07824-0.0017893*(m-n)+0.000008089*(m-n)^2;

elset=tdistr95(m-n);

endA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixfor i=1:n

ConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervals

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endymean=mean(y);R2=(ycal-ymean)*(ycal-ymean)/((y-ymean)*(y-ymean));%linear corr

Input 1 if there is a free parameter, 0 otherwise 1Enter the degree of the polynomial 3

Warning: Matrix is close to singular or badly scaled.Results may be inaccurate. RCOND = 4.135355e-020.> In P3_03A_CCL>PolyReg at 92

In P3_03A_CCL at 30Results, Heat Capacity of Gaseous Propane

Parameter No. Beta Conf_int

0 20.524 4.63861 0.19647 0.0278032 -2.6848e-005 4.2358e-0053 -1.5129e-008 1.8159e-008


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function P3_03B1_CCLclear, clc,format short g, format compactprob_title = ([ Thermal Conductivity of Gaseous Propane ]);ind_var_name=[\bf Temp. (K)];

dep_var_name=[\bf Thermal Conductivity (W/m*K) ];xyData=[ 231.07 1.14E-02240 1.21E-02260 1.39E-02280 1.59E-02300 1.80E-02

320 2.02E-02340 2.26E-02360 2.52E-02380 2.78E-02400 3.06E-02420 3.34E-02440 3.63E-02460 3.93E-02480 4.24E-02500 4.55E-02520 4.87E-02

540 5.20E-02

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560 5.53E-02580 5.86E-02600 6.19E-02];x=xyData(:,1);

y=xyData(:,2);[m,n]=size(x);freeparm=input( Input 1 if there is a free parameter, 0 otherwisedegree=input( Enter the degree of the polynomial );[Beta, ycal,ConfInt, Var, R2, n]=PolyReg(x,y,degree,freeparm);disp([ Results, prob_title]);

Res=[];for i=0:n-1

if freeparm==0; ii=i+1; else ii=i; endRes=[Res; ii Beta(i+1) ConfInt(i+1)];

enddisp( Parameter No. Beta Conf_int);disp(Res);disp([ Variance , num2str(Var)]);disp([ Correlation Coefficient , num2str(R2)]);subplot(2,1,1)

plot(x,ycal,r-,x,y,bo,Linewidth,2) %Plot of experimental and

set(gca,FontSize,14,Linewidth,2)

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title([\bf Cal/Exp Data prob_title],FontSize,12)xlabel([ind_var_name],FontSize,14)ylabel([dep_var_name],FontSize,14)

subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2) % Residual plotset(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title ],FontSize,12)xlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ycal,ConfInt, Var, R2, n]=PolyReg(x,y,degree,freeparmtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...


n=degree+1;else

n=degree;

end

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m=size(x,1);for i=1:m

for j=1:nif freeparm==1

p=j-1;elsep=j;

endX(i,j)=x(i)^p; %Calculate powers of the independent variable and putend

endBeta=X\y; % Solve XBeta = Y using QR decomposition

ycal=X*Beta; % Calculated dependent variable valuesVar=sum((y-ycal)*(y-ycal))/(m-n); % variance

if (m-n)>45t=2.07824-0.0017893*(m-n)+0.000008089*(m-n)^2;


endA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

for i=1:n

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ConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervalsendymean=mean(y);R2=(ycal-ymean)*(ycal-ymean)/((y-ymean)*(y-ymean));%linear corr


Warning: Matrix is close to singular or badly scaled.Results may be inaccurate. RCOND = 2.731195e-020.> In P3_03B1_CCL>PolyReg at 86

In P3_03B1_CCL at 31Results, Thermal Conductivity of Gaseous Propane

Parameter No. Beta Conf_int0 0.0062337 0.000919151 -4.7525e-005 7.2333e-0062 3.4436e-007 1.8141e-0083 -1.8414e-010 1.4585e-011

Variance 1.1612e-009Correlation Coefficient 1


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function P3_03B2_CCLclear, clc, format short g, format compactxyData=[231.07 1.14E-02240 1.21E-02

260 1.39E-02280 1.59E-02300 1.80E-02320 2.02E-02340 2.26E-02360 2.52E-02

380 2.78E-02400 3.06E-02420 3.34E-02440 3.63E-02460 3.93E-02480 4.24E-02500 4.55E-02520 4.87E-02540 5.20E-02560 5.53E-02580 5.86E-02

600 6.19E-02];

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X=xyData(:,1); m=size(X,1); %Determine the number of data pointsY=xyData(:,2);prob_title = ([ Thermal Conductivity of Gaseous Propane]);

dep_var_name=[\bf Thermal Conductivity (W/m*K) ];ind_var_name=[\bf Temperature (K)];parm=[0.001 1.8];npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@NonlinFun,parm,options,X,Y); % Find optimal paramete

[f,Ycalc] = NonlinFun(Beta,X,Y); % Compute Y (calculated) at the optidisp([ Results, prob_title ]);Res=[];for i=1:nparRes=[Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);s2=sum((Y-Ycalc)*(Y-Ycalc))/(m-npar); %variancedisp([ Variance , num2str(s2)]);ymean=mean(Y);

R2=(Ycalc-ymean)*(Ycalc-ymean)/((Y-ymean)*(Y-ymean));%linear correl

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disp([ Correlation Coefficient , num2str(R2)])subplot(2,1,1)

plot(X,Ycalc,r-,X,Y,bo,Linewidth,2) %Plot of experimental anset(gca,FontSize,14,Linewidth,2)


subplot(2,1,2)plot(Y,Y-Ycalc,*,Linewidth,2) % Residual plot


title([\bf Residual, prob_title ],FontSize,12)xlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [f,Ycalc]=NonlinFun(parm,X,Y)c=parm(1);n=parm(2);for i=1:size(X,1);

Ycalc(i,1)=c*X(i)^n;endresid(:,1)=Y-Ycalc;

f=resid*resid;

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Results, Thermal Conductivity of Gaseous PropaneParameter No. Value

1 7.2698e-0072 1.7762

Variance 7.1848e-008Correlation Coefficient 0.9943

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-4.018181 0.003652];X=xyData(:,2:end);y=xyData(:,1);[m,n]=size(X);

freeparm=input( Input 1 if there is a free parameter, 0 otherwise >[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm

if freeparm, ii=i+1; else ii=i; endRes=[Res; ii Beta(i+1) ConfInt(i+1)];


plot(X(:,1),ycal, r-,X(:,1),y,bo,Linewidth,2)set(gca,FontSize,14,Linewidth,2)

title([\bf Cal/Exp Data prob_title],FontSize,12)

xlabel([ind_var_name],FontSize,14)

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ylabel([dep_var_name],FontSize,14)subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2)set(gca,FontSize,14,Linewidth,2)

title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columns

if freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl

% Calculate the confidence intervals

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A=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabiif (m-npar)>45

t=2.07824-0.0017893*(m-npar)+0.000008089*(m-npar)^2; % t for degre

elset=tdistr95(m-npar);

endfor i=1:npar

ConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervalsend

Input 1 if there is a free parameter, 0 otherwise > 1Results, Propane viscosity with Antoine Eq

Parameter No. Beta Conf_int1 -4.5104 0.067488

2 159.94 8.9729

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function P3_03C2_CCLclear, clc, format short g, format compactxyData=[100 3.82E-03110 2.29E-03120 1.53E-03130 1.08E-03140 8.32E-04150 6.59E-04160 5.46E-04170 4.53E-04

180 3.82E-04190 3.36E-04200 2.87E-04220 2.24E-04240 1.80E-04260 1.44E-04270 1.30E-04280 1.17E-04290 1.05E-04300 9.59E-05];X=xyData(:,1);

m=size(X,1); %Determine the number of data points

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Y=log10(xyData(:,2));prob_title = ([ Visc. Propane, Antoine Eq. Params, Nln. Regr]);dep_var_name=[\bf log (viscosity) ];ind_var_name=[\bf Temperature (K)];

parm=[-4.51 159.9 -26.18];npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@AntFun,parm,options,X,Y); % Find optimal parameters[f,Ycalc]=AntFun(Beta,X,Y); % Compute Y (calculated) at the optimumdisp([ Results, prob_title ]);

Res=[];for i=1:nparRes=[Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);s2=sum((Y-Ycalc)*(Y-Ycalc))/(m-npar); %variancedisp([ Variance , num2str(s2)]);ymean=mean(Y);R2=(Ycalc-ymean)*(Ycalc-ymean)/((Y-ymean)*(Y-ymean));%linear correldisp([ Correlation Coefficient , num2str(R2)])

subplot(2,1,1)


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Results, Visc. Propane, Antoine Eq. Params, Nln. RegrParameter No. Value

1 -4.93792 308.263 23.967


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f


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function P3_03D1_CCLclear, clc, format short g, format compactprob_title = ([ Propane Heat of Vap., Linear Regr.]);ind_var_name=[\bf log(TC-T)];dep_var_name=[\bf log(deltaH) ];xyData=[7.394452 2.4538697.390935 2.4468947.383815 2.431097.374748 2.4146897.367356 2.397645

7.359835 2.3799047.352183 2.3614077.344392 2.3420877.33646 2.3218687.32838 2.3006617.320146 2.2783657.311754 2.2548627.303196 2.2300147.294466 2.2036587.285557 2.1755997.274158 2.142264

7.262451 2.113375

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2 042 2 0 8 66


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7.25042 2.0785667.238046 2.0407217.222716 1.9992617.206826 1.9534217.187521 1.9021667.164353 1.8440427.139879 1.7769197.10721 1.6974917.071882 1.600217.021189 1.474653

6.951823 1.2973236.833147 0.9925535];X=xyData(:,2:end);y=xyData(:,1);[m,n]=size(X);freeparm=input( Input 1 if there is a free parameter, 0 otherwise >[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm

if freeparm, ii=i+1; else ii=i; end

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f


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Res=[Res; ii Beta(i+1) ConfInt(i+1)];enddisp( Parameter No. Beta Conf_int);disp(Res);

disp([ Variance , num2str(Var)]);disp([ Correlation Coefficient , num2str(R2)]);subplot(2,1,1)


title([\bf Cal/Exp Data prob_title],FontSize,12)

xlabel([ind_var_name],FontSize,14)ylabel([dep_var_name],FontSize,14)

subplot(2,1,2)plot(y,y-ycal,*,Linewidth,2)

set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,14) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columns

if freeparm

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[ ( 1) ] % Add l f if h i f


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X=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl

% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...


t=2.07824-0.0017893*(m-npar)+0.000008089*(m-npar)^2; % t for degre

else

Chen CL 43

di 95( )


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t=tdistr95(m-npar);endfor i=1:npar


Input 1 if there is a free parameter, 0 otherwise > 1Results, Propane Heat of Vap., Linear Regr.

Parameter No. Beta Conf_int1 6.4664 0.00761872 0.3765 0.0036327


Chen CL 44


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Chen CL 46

280 1 61E+07


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280 1.61E+07290 1.54E+07300 1.46E+07310 1.38E+07320 1.28E+07330 1.18E+07340 1.05E+07350 8.95E+06360 6.81E+06];X=xyData(:,1);

m=size(X,1); %Determine the number of data pointsY=xyData(:,2);prob_title = ([ Heat of Vaporization of Propane, NlN Reg.]);dep_var_name= [\bf Heat of Vaporization (J/kmol) ];ind_var_name= [\bf Temperature (K)];parm=[2.926e6 0.3765];npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@NonlinFun,parm,options,X,Y); % Find optimal paramete[f,Ycalc]=NonlinFun(Beta,X,Y); % Compute Y (calculated) at the optimudisp([ Results, prob_title ]);

Res=[];

Chen CL 47

for i=1:npar


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for i=1:nparRes=[Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);s2=sum((Y-Ycalc)*(Y-Ycalc))/(m-npar); %variancedisp([ Variance , num2str(s2)]);ymean=mean(Y);R2=(Ycalc-ymean)*(Ycalc-ymean)/((Y-ymean)*(Y-ymean));%linear correldisp([ Correlation Coefficient , num2str(R2)])

subplot(2,1,1)plot(X,Ycalc,r-,X,Y,bo,Linewidth,2) %Plot of experimental an

set(gca,FontSize,14,Linewidth,2)title([\bf Cal/Exp Data prob_title],FontSize,12)xlabel([ind_var_name],FontSize,14)ylabel([dep_var_name],FontSize,14)


set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title ],FontSize,12)xlabel([dep_var_name \bf (Measured)],FontSize,14)

ylabel(\bf Residual,FontSize,14)

Chen CL 48

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [f,Ycalc]=NonlinFun(parm,X,Y)A=parm(1);n=parm(2);for i=1:size(X,1);Ycalc(i,1)=A*(369.83-X(i))^n;endresid(:,1)=Y-Ycalc;f=resid*resid;

Results, Heat of Vaporization of Propane, NlN Reg.Parameter No. Value

1 2.9686e+0062 0.37362



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Chen CL 50

function P3 03D3 CCL


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function P3_03D3_CCLclear, clc,format short g, format compactprob_title = ([ Heat of Vaporization of Propane, Polynomial ]);ind_var_name=[\bf Temperature (K)];dep_var_name=[\bf Heat of Vaporization (J/Kmol) ];xyData=[ 85.47 2.48E+0790 2.46E+07100 2.42E+07110 2.37E+07120 2.33E+07

130 2.29E+07140 2.25E+07150 2.21E+07160 2.17E+07170 2.13E+07180 2.09E+07190 2.05E+07200 2.01E+07210 1.97E+07220 1.93E+07231.07 1.88E+07

240 1.83E+07

Chen CL 51

250 1 78E+07


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250 1.78E+07260 1.73E+07270 1.67E+07280 1.61E+07290 1.54E+07300 1.46E+07310 1.38E+07320 1.28E+07330 1.18E+07340 1.05E+07

350 8.95E+06360 6.81E+06];x=xyData(:,1);y=xyData(:,2);[m,n]=size(x);freeparm=input( Input 1 if there is a free parameter, 0 otherwisedegree= input( Enter the degree of the polynomial );[Beta, ycal,ConfInt, Var, R2, n]=PolyReg(x,y,degree,freeparm);disp([ Results, prob_title]);Res=[];for i=0:n-1

if freeparm==0; ii=i+1; else ii=i; end

Chen CL 52

Res=[Res; ii Beta(i+1) ConfInt(i+1)];


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Res=[Res; ii Beta(i+1) ConfInt(i+1)];enddisp( Parameter No. Beta Conf_int);disp(Res);disp([ Variance , num2str(Var)]);disp([ Correlation Coefficient , num2str(R2)]);%Plot of experimental and calculated data%% for i=1:m% index(i)=i;

% endsubplot(2,1,1)plot(x,ycal, r-,x,y,bo,Linewidth,2)

set(gca,FontSize,14,Linewidth,2)title([\bf Cal/exp data prob_title],FontSize,12)xlabel([\bf Temperature (K)],FontSize,14)ylabel([dep_var_name],FontSize,14)subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2) % residual plotset(gca,FontSize,14,Linewidth,2)

title([\bf Residual, prob_title],FontSize,12)

xlabel([dep_var_name \bf (measured)],FontSize,14)

Chen CL 53

ylabel(\bf Residual FontSize 14)


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ylabel( \bf Residual , FontSize ,14)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ycal,ConfInt, Var, R2, n]=PolyReg(x,y,degree,freeparmtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...


n=degree+1;else

n=degree;end m=size(x,1);for i=1:m

for j=1:nif freeparm==1

p=j-1;else

p=j;

end

Chen CL 54

X(i j)=x(i)^p; %Calculate powers of the independent variable and put


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X(i,j) x(i) p; %Calculate powers of the independent variable and putend

endBeta=X\y; % Solve XBeta = Y using QR decomposition

ycal=X*Beta; % Calculated dependent variable valuesVar=sum((y-ycal)*(y-ycal))/(m-n); % variance

if (m-n)>45t=2.07824-0.0017893*(m-n)+0.000008089*(m-n)^2;


endA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixfor i=1:n

ConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervalsendymean=mean(y);R2=(ycal-ymean)*(ycal-ymean)/((y-ymean)*(y-ymean));%linear corr


Warning: Matrix is close to singular or badly scaled.

Chen CL 55

Results may be inaccurate. RCOND = 5.894599e-018.


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Results may be inaccurate. RCOND 5.894599e 018.> In P3_03D3_CCL>PolyReg at 100

In P3_03D3_CCL at 40Results, Heat of Vaporization of Propane, Polynomial

Parameter No. Beta Conf_int0 3.3335e+007 1.5284e+0061 -1.4048e+005 242492 602.57 117.083 -1.1448 0.17534

Variance 40745270933.3095

Correlation Coefficient 0.99847

Chen CL 56


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Chen CL 57

Heat Transfer Correlations


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Heat Transfer Correlationsfrom Dimensional Analysis

Concepts Utilized: Correlation of heat transfer data using dimensionless groups.

Numerical Methods: Linear and nonlinear regression of data with linearizationand use of transformation functions.

Problem Statement:

An important tool in the correlation of engineering data is the use of dimensionalanalysis. This treatment leads to the determination of the independentdimensionless numbers, which may be important for a particular problem. Linearand nonlinear regression can be very useful in determining the correlations of dimensionless numbers with experimental data.

A treatment of heat transfer within a pipe has been considered by Geankoplisusing the Buckingham method, and the result is that the Nusselt number isexpected to be a function of the Reynolds and Prandtl numbers.

Nu = f (Re,Pr ) or hD

k = f

Dv

,

C p

k(3 19)

Chen CL 58

A typical correlation function suggested by Equation (3-19) is


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A typical correlation function suggested by Equation (3 19) is

Nu = aRe bP r c(3 20)

where a, b, and c are parameters that can be determined from experimental data.A widely used correlation for heat transfer during turbulent ow in pipes is theSieder-Tate equation:

Nu = 0 .023Re 0 .8 P r 1 / 3 (/ w )0 .14 (3 21)

in which a dimensionless viscosity ratio has been added. This ratio (/ w ) is theviscosity at the mean uid temperature to that at the wall temperature. TableB-15 gives some of the data reported by Williams and Katz for heat transferexternal to 314-inch outside diameter tubes where the Re, P r , (/ w ) and Nu

dimensionless numbers have been measured.

(a) Use multiple linear regression to determine the parameter values the functionalforms of Equations (3-20) and (3-21) that represent data of Table B-15.

(b) Repeat part (a) using nonlinear regression.

Chen CL 59

(c) Which functional form and parameter values should be recommended as a


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(c) Which functional form and parameter values should be recommended as acorrelation for this data set? Justify your selection.

Solution:

For convenience, let the functional forms of Equations (3-20) and (3-21) bewritten asNu = ai Re bi P r ci (3 22)Nu = di Re e i P r f i Mu gi (3 23)

where i = 1 indicates parameter values from linear regression and i = 2 indicates

parameter values from nonlinear regression. Mu represents the viscosity ratio.(a) A linear regression of either Equation (3-22) or (3-23) requires atransformation into a linear form.

ln(Nu ) = ln(a i ) + bi ln(Re) + ci ln(P r ) (3 24)

ln(Nu ) = ln(di ) + ei ln(Re) + f i ln(P r ) + gi ln(Mu ) (3 25)

(b) Direct nonlinear regressions of both Equations (3-22) and (3-23) can becompleted where the converged values from the linear regressions of part (a) canbe used as initial parameter estimates.

Chen CL 60function P3_05A1_CCL


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clear, clc, format short g, format compactprob_title = ([ Heat Transfer Correlation - Linear Regr. 1]);ind_var_name=[\bf Re ];dep_var_name=[\bf Nu ];xyData=[5.624018 10.79958 0.83290915.852202 11.13605 0.82417546.042633 11.34805 0.81977985.407172 10.43998 0.84156725.17615 10.03889 0.8586616

4.743191 7.186144 5.5053324.563306 6.836259 5.5093884.22391 6.249975 5.5254533.893859 5.846439 5.6094724.025352 4.811371 7.3251493.686376 3.988984 7.3714893.850148 4.437934 7.3271234.54542 7.130099 4.676564.60417 6.928538 5.2257474.420045 6.142037 6.0258663.580737 4.00369 7.171657];X=xyData(:,2:end);

Chen CL 61

y=xyData(:,1);


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y y ( , );[m,n]=size(X);freeparm=input( Input 1 if there is a free par., 0 otherwise );[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm


enddisp( Parameter No. Beta Conf_int);disp(Res);sigmar=0;Var=0;for i=1:m

Var=Var+ ((exp(ycal(i))-exp(y(i))))^2;sigmar=sigmar+((exp(ycal(i))-exp(y(i)))/exp(y(i)))^2;

enddisp([ Variance , num2str(Var/(m-(nparm+1)))]);disp([ Relative variance , num2str(sigmar/(m-(nparm+1)))]);

plot(y,y-ycal,*,Linewidth,2)

Chen CL 62



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(g , , , , )title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columnsif freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;

elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

tdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

Chen CL 632.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..


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2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...


t=2.07824-0.0017893*(m-npar)+0.000008089*(m-npar)^2; % t for degreelse

t=tdistr95(m-npar);end

for i=1:nparConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervalsend

Input 1 if there is a free parameter, 0 otherwise > 1Results, Heat Transfer Correlation - Linear Regr. 1

Parameter No. Beta Conf_int1 -0.41204 1.35572 0.53954 0.116063 0.24535 0.11394

Variance 265.5792

Relative variance 0.010278

Chen CL 64


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Chen CL 65function P3_05A2_CCL


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clear, clc, format short g, format compactprob_title = ([ Heat Transfer Correlation - Linear Regr 2]);ind_var_name=[\bf Re ];dep_var_name=[\bf Nu ];xyData=[5.624018 10.79958 0.8329091 -0.05445625.852202 11.13605 0.8241754 -0.04709166.042633 11.34805 0.8197798 -0.04186425.407172 10.43998 0.8415672 -0.0586895.17615 10.03889 0.8586616 -0.0661398

4.743191 7.186144 5.505332 -0.52424864.563306 6.836259 5.509388 -0.53956814.22391 6.249975 5.525453 -0.54645283.893859 5.846439 5.609472 -1.2378744.025352 4.811371 7.325149 -1.2241763.686376 3.988984 7.371489 -1.2765433.850148 4.437934 7.327123 -1.3205074.54542 7.130099 4.67656 -0.32296394.60417 6.928538 5.225747 -0.4910234.420045 6.142037 6.025866 -0.66943073.580737 4.00369 7.171657 -1.298283];X=xyData(:,2:end);

Chen CL 66y=xyData(:,1);


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[m,n]=size(X);freeparm=input( Input 1 if there is a free parameter, 0 otherwise >[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm


enddisp( Parameter No. Beta Conf_int);disp(Res);sigmar=0;Var=0;for i=1:m

Var=Var+ ((exp(ycal(i))-exp(y(i))))^2;sigmar=sigmar+((exp(ycal(i))-exp(y(i)))/exp(y(i)))^2;

enddisp([ Variance , num2str(Var/(m-(nparm+1)))]);disp([ Relative variance , num2str(sigmar/(m-(nparm+1)))]);plot(y,y-ycal,*,Linewidth,2)

Chen CL 67set(gca,FontSize,14,Linewidth,2)


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title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columnsif freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;

elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

Chen CL 682.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..


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2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...





Input 1 if there is a free parameter, 0 otherwise > 1Results, Heat Transfer Correlation - Linear Regr 2

Parameter No. Beta Conf_int1 -0.62602 1.72862 0.55883 0.150673 0.25237 0.122994 -0.067722 0.31632

Variance 234.7091

Chen CL 69Relative variance 0.011346


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Chen CL 70function P3_05B1_CCL


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clear, clc, format short g, format compactxyData=[277 49000 2.3348 68600 2.28421 84800 2.27223 34200 2.32177 22900 2.36114.8 1321 24695.9 931 24768.3 518 251

49.1 346 27356 122.9 151839.9 54 159047 84.6 152194.2 1249 107.499.9 1021 186

83.1 465 41435.9 54.8 1302];X=xyData(:,2:end); m=size(X,1); %Determine the number of data pointsY=xyData(:,1);prob_title = ([ Heat Transfer Correlation - Noninear Regr 1]);

Chen CL 71dep_var_name=[\bf Nu ];


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ind_var_name=[\bf RE];parm=[0.6623 0.5395 0.2454];npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@NonlinFun,parm,options,X,Y); % Find optimal paramete[f,Ycalc]=NonlinFun(Beta,X,Y); % Compute Y (calculated) at the optimudisp([ Results, prob_title ]);Res=[];for i=1:npar

Res=[Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);sigmar=0;Var=0;for i=1:m

Var=Var+ ((Ycalc(i)-Y(i)))^2;sigmar=sigmar+((Ycalc(i)-Y(i))/Y(i))^2;

enddisp([ Variance , num2str(Var/(m-npar))]);disp([ Relative variance , num2str(sigmar/(m-npar))]);

Chen CL 72plot(Y,Y-Ycalc,*,Linewidth,2)


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set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [f,Ycalc]=NonlinFun(parm,X,Y)a=parm(1);b=parm(2);c=parm(3);

for i=1:size(X,1);Ycalc(i,1)=a*X(i,1)^b*X(i,2)^c;endresid(:,1)=Y-Ycalc;f=resid*resid;

Results, Heat Transfer Correlation - Noninear Regr 1Parameter No. Value

1 0.165672 0.663533 0.34136

Variance 68.2755

Chen CL 73Relative variance 0.016832


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Chen CL 74function P3_05B2_CCLl l f h f


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clear, clc, format short g, format compactxyData=[277 49000 2.3 0.947348 68600 2.28 0.954421 84800 2.27 0.959223 34200 2.32 0.943177 22900 2.36 0.936114.8 1321 246 0.59295.9 931 247 0.58368.3 518 251 0.579

49.1 346 273 0.2956 122.9 1518 0.29439.9 54 1590 0.27947 84.6 1521 0.26794.2 1249 107.4 0.72499.9 1021 186 0.612

83.1 465 414 0.51235.9 54.8 1302 0.273];X=xyData(:,2:end); m=size(X,1); %Determine the number of data pointsY=xyData(:,1);prob_title = ([ Heat Transfer Correlation - Noninear Regr 2]);

Chen CL 75dep_var_name=[\bf Nu ];i d [\bf RE]


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ind_var_name=[\bf RE];parm=[0.5347 0.5588 0.2524 -0.06772];npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@NonlinFun,parm,options,X,Y); % Find optimal paramete[f,Ycalc]=NonlinFun(Beta,X,Y); % Compute Y (calculated) at the optimudisp([ Results, prob_title ]);Res=[];for i=1:npar

Res=[Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);sigmar=0;Var=0;for i=1:m

Var=Var+ ((Ycalc(i)-Y(i)))^2;sigmar=sigmar+((Ycalc(i)-Y(i))/Y(i))^2;

enddisp([ Variance , num2str(Var/(m-npar))]);disp([ Relative variance , num2str(sigmar/(m-npar))]);

Chen CL 76plot(Y,Y-Ycalc,*,Linewidth,2)

( F Si 14 Li id h 2)


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set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)%%%%%%%%%%%%%%%%%%%%%%%%%function [f,Ycalc]=NonlinFun(parm,X,Y)a=parm(1);b=parm(2);c=parm(3);

d=parm(4);for i=1:size(X,1);Ycalc(i,1)=a*X(i,1)^b*X(i,2)^c*X(i,3)^d;endresid(:,1)=Y-Ycalc;f=resid*resid;

Results, Heat Transfer Correlation - Noninear Regr 2Parameter No. Value

1 0.149152 0.67329

3 0.32857

Chen CL 774 -0.17765

Variance 66 6846


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Variance 66.6846Relative variance 0.014764

Chen CL 78

Correlation of Binary Activity Coefficients


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Using Margules Equations

Concepts Utilized: Estimation of parameters in the Margules equations for thecorrelation of binary activity coefficients.

Numerical Methods: Linear and nonlinear regression, transformation of data forregression; calculation and comparison of condence intervals, residual plots, andsum of squares.

Problem Statement:The Margules equations for correlation of binary activity coefficients are

1 = exp x22 (2B A) + 2 x32 (A B ) (3-30) 2 = exp x21 (2A B ) + 2 x31 (B A) (3-31)

where xl and x2 are mole fractions of components 1 and 2, respectively, and 1and 2 are activity coefficients. Parameters A and B are constant for a particularbinary mixture.

Equations (3-30) and (3-31) can be combined to give the excess Gibbs energy

Chen CL 79expression:


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g = GE RT

= x1 ln( 1 ) + x2 ln( 2 ) = x1 x2 (Ax 2 + Bx 1 ) (3 32)

Activity coefficients at various molefractions are available for the benzeneand n-heptane binary system in thefollowing Table, from which g inEquation (3-32) can be calculated.A multiple linear regression withoutthe free parameter can be used toestimate the parameter values of A

and B. Another method is to sumEquations (3-30) and (3-31) and usenonlinear regression on this sum todetermine A and B.

Activity Coefficients forBenzene(1) and n-Heptane(2)

No. x1 1 21 0.0464 1.2968 0.9985

2 0.0861 1.2798 0.99983 0.2004 1.2358 1.00684 0.2792 1.1988 1.01595 0.3842 1.1598 1.0359

6 0.4857 1.1196 1.06767 0.5824 1.0838 1.10968 0.6904 1.0538 1.16649 0.7842 1.0311 1.2401

10 0.8972 1.0078 1.4038

Chen CL 80(a) Use multiple linear regression on Equation (3-32) with the data of Table to

determine A and B in the Margules equations for the benzene and n heptane


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determine A and B in the Margules equations for the benzene and n-heptanebinary system.

(b) Estimate A and B by employing nonlinear regression on a single equation thatis the sum of Equations (3-30) and (3-31).

(c) Compare the results of the regressions in (a) and (b) using parameter condenceintervals, residual plots, and sums of squares of errors (least-squares summationscalculated with both activity coefficients).

Solution:Equation (3-32) can be rearranged to a linear form as

g = Ax 21 xx 2 + Bx 1 x22 = a1 X 1 + a2 X 2

Lets create a column for the summation equation, ( 1 + 2 ), and call it gsum,as it will provide the function values during the nonlinear regression.

gsum = exp x22 (2B A) + 2 x32 (A B ) + exp x

21 (2A B ) + 2 x

31 (B A)

Chen CL 81function P3_08AC_CCLclear clc format short g format compact


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clear, clc, format short g, format compactprob_title = ([ Multiple Linear Regression for Par.s of Margules Eq.ind_var_name=[\bf X1 and X2 ]; dep_var_name=[\bf g ];xyData=[0.0106279 0.042194 0.00205310.0210584 0.0719119 0.00677490.0478473 0.1281278 0.03211210.0619954 0.1450591 0.05618830.0786764 0.1456923 0.0908980.0885122 0.1284698 0.1213257

0.0902979 0.1015646 0.14164560.0838331 0.0661763 0.14757150.0704555 0.0365199 0.13271040.041839 0.0094815 0.0827507];x1=[0.0464 0.0861 0.2004 0.2792 0.3842 0.4857 0.5824 0.6904 0.78420.8972];

gamma1=[1.2968 1.2798 1.2358 1.1988 1.1598 1.1196 1.0838 1.0538 1.0311.0078];gamma2=[0.9985 0.9998 1.0068 1.0159 1.0359 1.0676 1.1096 1.1664 1.2401.4038];X=xyData(:,2:end);y=xyData(:,1);

Chen CL 82[m,n]=size(X);freeparm=input( Input 1 if there is a free parameter 0 otherwise >


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freeparm=input( Input 1 if there is a free parameter, 0 otherwise >[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm


end

disp( Parameter No. Beta Conf_int);disp(Res);disp([ Variance , num2str(Var)]);disp([ Correlation Coefficient , num2str(R2)]);errsum=0;for i=1:m

gamma1c(i)= exp((1-x1(i)) ^ 2 * (2 * Beta(2) - Beta(1)) + 2 * (1-gamma2c(i)= exp(x1(i) ^ 2 * (2 * Beta(1) - Beta(2)) + 2 * x1(i)^errsum=errsum +(gamma1(i)-gamma1c(i)) 2+(gamma2(i)-gamma2c(i))^2

enddisp([ Sum of squares of errors , num2str(errsum)]);plot(y,y-ycal,*,Linewidth,2)

Chen CL 83set(gca,FontSize,14,Linewidth,2)

title([\bf Residual prob title] FontSize 12) % residual plot


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title([ \bf Residual, prob_title], FontSize ,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columnsif freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;

elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % variance

ymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

Chen CL 842.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2 1098 2 1009 2 093 2 086 2 0796 2 0739 2 0687 2 0639


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2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...





Input 1 if there is a free parameter, 0 otherwise > 1Results, Multiple Linear Regression for Par.s of Margules Eq.s

Parameter No. Beta Conf_int1 0.00011101 0.00157392 0.25048 0.0128583 0.46044 0.011462

Variance 6.4045e-007

Correlation Coefficient 0.99938

Chen CL 85Sum of squares of errors 0.2963


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Chen CL 86function P3_08BC_CCLclear, clc, format short g, format compact


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clear, clc, format short g, format compactx1=[0.0464 0.0861 0.2004 0.2792 0.3842 0.4857 0.5824 0.6904 0.78420.8972];gamma1=[1.2968 1.2798 1.2358 1.1988 1.1598 1.1196 1.0838 1.0538 1.0311.0078];gamma2=[0.9985 0.9998 1.0068 1.0159 1.0359 1.0676 1.1096 1.1664 1.2401.4038];%xyData=[];X=x1;

m=size(X,1); %Determine the number of data pointsY=gamma1+gamma2;prob_title = ([ Nonlinear Regression for Par.s of Margules Eq.s]);dep_var_name=[\bf gsum ];ind_var_name=[\bf x1];parm=[0.25 0.46];

npar=size(parm,2); % Determine the number of the parametersoptions=optimset(MaxFunEvals,1000); % Change the default value forBeta=fminsearch(@NonlinFun,parm,options,X,Y); % Find optimal paramete[f,Ycalc]=NonlinFun(Beta,X,Y); % Compute Y (calculated) at the optimudisp([ Results, prob_title ]);Res=[];

Chen CL 87for i=1:nparRes=[Res; i Beta(i)];


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Res [Res; i Beta(i)];enddisp( Parameter No. Value );disp(Res);s2=sum((Y-Ycalc)*(Y-Ycalc))/(m-npar); %variancedisp([ Variance , num2str(s2)]);ymean=mean(Y);R2=(Ycalc-ymean)*(Ycalc-ymean)/((Y-ymean)*(Y-ymean));%linear correldisp([ Correlation Coefficient , num2str(R2)])errsum=0;for i=1:m

gamma1c(i)= exp((1-x1(i)) ^ 2 * (2 * Beta(2) - Beta(1)) + 2 * (1-gamma2c(i)= exp(x1(i) ^ 2 * (2 * Beta(1) - Beta(2)) + 2 * x1(i)^errsum=errsum +(gamma1(i)-gamma1c(i))^2+(gamma2(i)-gamma2c(i))^2;

end

disp([ Sum of squares of errors , num2str(errsum)]);subplot(2,1,1)

plot(X,Ycalc,r-,X,Y,bo,Linewidth,2) %Plot of experimental anset(gca,FontSize,14,Linewidth,2)

title([\bf Cal/Exp Data prob_title],FontSize,12)xlabel([ind_var_name],FontSize,14)

Chen CL 88ylabel([dep_var_name],FontSize,14)

subplot(2 1 2)


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set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title ],FontSize,12)xlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%%function [f,Ycalc]=NonlinFun(parm,X,Y)A=parm(1);

B=parm(2);for i=1:size(X,1);Ycalc(i,1)=exp((1-X(i))^2*(2*B-A)+2*(1-X(i))^3*(A-B))+exp(X(i)^2*

endresid(:,1)=Y-Ycalc;f=resid*resid;

Results, Nonlinear Regression for Par.s of Margules Eq.sParameter No. Value

1 0.260772 0.45159

Variance 5.2323e-005

Chen CL 89Correlation Coefficient 0.88483Sum of squares of errors 0.00079956


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q

Chen CL 90

Regression of Rate Data


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checking Dependency Among Variables

Concepts Utilized: Correlation of reaction rate data with various reaction ratemodels.

Numerical Methods: Multiple linear regression with determination of parametercondence intervals, residual plots, and identication of linear dependency amongregression variables.

Problem Statement:The following Table presents rate data for the reaction A R, as reported byBacon and Downie. They suggested tting the rate data with two reaction ratemodels. An irreversible model has the form of a rst-order reaction

r R = k0 C A (3 38)

and a reversible model has the form of reversible rst-order reactions

r R = k1 C A

k2 C R (3

39)

Chen CL 91

where rR is the rate of generation of component R (gm-mol/dm 3 s); C A and C Rare the respective concentrations of components A and R (gm-mol/dm 3 ); and k0


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are the respective concentrations of components A and R (gm mol/dm ); and k0 ,kl and k2 are reaction rate coefficients (s

l).

Reaction Rate DataNo. rR C A C R

1 1.25 2.00 7.982 2.50 4.00 5.953 4.05 6.00 4.004 0.75 1.50 8.495 2.80 4.00 5.996 3.57 5.50 4.507 2.86 4.50 5.47

8 3.44 5.00 4.989 2.44 4.00 5.99

r R : gm-mol/dm 3 s 108 ;C A,R : gm-mol/dm 3 104

(a) Calculate the parameters of bothreaction rate expressions using the datain Table.

(b) Compare the two models and

determine which one better correlatesthe rate data.

(c) Determine if the two variables, C A andC R , are correlated.

(d) Discuss the practical signicance of any correlation among the regressionvariables.


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Chen CL 93

% rR = k0 CAfunction P3_11A1_CCL


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_ _clear, clc, format short g, format compactprob_title = ([ Regression of Rate Data]);ind_var_name=[\bf Concentration (g-mol/dm^3 ];dep_var_name=[\bf Reactoion Rate, g-mol/(dm^3-s) ];xyData=[1.25E-08 0.00022.50E-08 0.00044.05E-08 0.00067.50E-09 0.00015

2.80E-08 0.00043.57E-08 0.000552.86E-08 0.000453.44E-08 0.00052.44E-08 0.0004];X=xyData(:,2:end);

y=xyData(:,1);[m,n]=size(X);freeparm=input( Input 1 if there is a free parameter, 0 otherwise >[Beta, ConfInt,ycal, Var, R2]=MlinReg(X,y,freeparm);disp([ Results, prob_title]);Res=[];

Chen CL 94

if freeparm==0, nparm = n-1; else nparm = n; endfor i=0:nparm


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subplot(2,1,2)



%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Chen CL 95

function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columns


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( )if freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable values

Var=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

tdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabi

Chen CL 96

if (m-npar)>45t=2.07824-0.0017893*(m-npar)+0.000008089*(m-npar)^2; % t for degre


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gelse

t=tdistr95(m-npar);endfor i=1:npar


Input 1 if there is a free parameter, 0 otherwise > 0

Results, Regression of Rate DataParameter No. Beta Conf_int

0 6.5514e-005 2.7399e-006Variance 2.3399e-018Correlation Coefficient 0.83395

Chen CL 97


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Chen CL 98

% rR = k1CA - k2 CRfunction P3_11A2_CCL


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clear, clc, format short g, format compactprob_title = ([ Regression of Rate Data]);ind_var_name=[\bf Concentration (g-mol/dm^3 ];dep_var_name=[\bf Reaction Rate, g-mol/(dm^3-s) ];xyData=[1.25E-08 0.0002 0.0007982.50E-08 0.0004 0.0005954.05E-08 0.0006 0.00047.50E-09 0.00015 0.0008492.80E-08 0.0004 0.0005993.57E-08 0.00055 0.000452.86E-08 0.00045 0.0005473.44E-08 0.0005 0.0004982.44E-08 0.0004 0.000599];X=xyData(:,2:end);


Chen CL 99



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end

disp( Parameter No. Beta Conf_int);disp(Res);disp([ Variance , num2str(Var)]);disp([ Correlation Coefficient , num2str(R2)]);subplot(2,1,1)



subplot(2,1,2)



%%%%%%%%%%%%%%%%%%%%%%%

Chen CL 100



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if freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

tdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabi

Chen CL 101



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endfor i=1:npar


Input 1 if there is a free parameter, 0 otherwise > 0Results, Regression of Rate Data

Parameter No. Beta Conf_int0 6.8667e-005 4.5085e-0061 -2.6304e-006 3.1766e-006



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Chen CL 103

% CR = a0 + a1 CAfunction P3_11B_CCL


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clear, clc, format short g, format compactprob_title = ([ Regression of Rate Data]);ind_var_name=[\bf CA (g-mol/dm^3) ];dep_var_name=[\bf CR (g-mol/dm^3) ];xyData=[0.000798 0.00020.000595 0.00040.0004 0.00060.000849 0.000150.000599 0.00040.00045 0.000550.000547 0.000450.000498 0.00050.000599 0.0004];X=xyData(:,2:end);


Chen CL 104


if f ii i 1 l ii i d


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end




subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2)set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%

Chen CL 105



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if freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

tdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabi

Chen CL 106


l


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endfor i=1:npar



Parameter No. Beta Conf_int1 0.00099746 3.9891e-0062 -0.99784 0.0092954


Chen CL 107


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Chen CL 108

% rR = a0 + a1 CAfunction P3_11D_CCLl l f h f


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clear, clc, format short g, format compactprob_title = ([ Regression of Rate Data]);ind_var_name=[\bf Concentration (g-mol/dm^3 ];dep_var_name=[\bf Reaction Rate, g-mol/(dm^3-s) ];xyData=[1.25E-08 0.00022.50E-08 0.00044.05E-08 0.00067.50E-09 0.000152.80E-08 0.00043.57E-08 0.000552.86E-08 0.000453.44E-08 0.00052.44E-08 0.0004];X=xyData(:,2:end);


Chen CL 109


if freeparm ii=i+1; else ii=i; end


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end




subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2)set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%

Chen CL 110

function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columnsif f


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if freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;

elsenpar=n;endBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable valuesVar=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl% Calculate the confidence intervalsA=X*X;Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrix

tdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...

2.0195 2.0181 2.0167 2.0154 2.0141]; % 95 percent probabi

Chen CL 111


else


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end

for i=1:nparConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervals

end


Parameter No. Beta Conf_int1 -2.6263e-009 3.1668e-0092 7.1298e-005 7.3792e-006


Chen CL 112


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Chen CL 113

Calculation of Antoine Equation ParametersUsing Linear Regression


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Using Linear Regression

Concepts Utilized: Direct use of the Antoine equation to correlate vaporpressure versus temperature data.

Numerical Methods: Multiple linear regression with determination of theoverall variance and condence intervals of individual parameters.

Problem Statement:Calculate the Antoine equation parameters of Equation (3-1) and the variousstatistical indicators for the propane vapor pressure data of Table B-5. Reportthe parameters for the vapor pressure in psia and the temperature in oF. (Thisproblem is similar to Problem 3.1, but the parameters have different units.) Thefundamental calculations for linear regression are to be carried out during the

solution. The following sequence is to be used:

(a) Transform the data so that the Antoine equation parameters can be calculatedusing multiple linear regression.

(b) Find the matrices X T X and X T y.

Chen CL 114

(c) Solve system of equations to obtain the vector A.


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(d) Calculate the variance, the diagonal elements of (X T y) l and the 95% condenceintervals of the parameters (use the t distribution values provided in Table A-

4).

(e) Prepare a residual plot (plot of i versus yi (obs) ).

(f) Assess the precision of the data and the appropriateness of the Antoine equationfor correlation of the data.

Solution:An alternative linear form of the Antoine equation is

T log(P v) = ( AC + B) + AT C log(P v)

The original data should be entered and transformed into the variables (columns)shown in Table as specied by y = T log(P v), x1 = T and x2 = log( P v).

Chen CL 115

Transformed Variables for theAntoine Equation Regression


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For the case of multiple linear regression withtwo independent variables x1 and x2 and one

dependent variable y, the matrix X T X and thevector X T y can be written

X T X =N x1 ,i x2 ,ix1 ,i x21 ,i x1 ,i x2 ,ix2 ,i x2 ,i x1 ,i x22 ,i

=20 500 34.5131500 79000 1383.28

34.5135 1383.28 63.6854

X T y =yi (obs)

x1 ,i yi (obs)x2 ,i yi (obs)

=1383.281592813339.67

y x1 x2-60.7227 -70 0.867467-59.26 -60 0.987666-55.0185 -50 1.10037-48.3806 -40 1.20952-39.2249 -30 1.3075-28.0967 -20 1.40483-14.9693 -10 1.49693

0. 0 1.5820616.6276 10 1.6627634.8859 20 1.7442954.6454 30 1.8215175.6838 40 1.8920998.1421 50 1.96284

121.787 60 2.02979146.54 70 2.09342172.378 80 2.15473199.336 90 2.21484227.184 100 2.27184256.122 110 2.32838

285.625 120 2.38021

Chen CL 116

function P3_14_CCLclear, clc, format short g, format compactprob title = ([ Antoine Parameters by Linear Regression]);


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prob_title ([ Antoine Parameters by Linear Regression ]);ind_var_name=[\bf T (C) ];dep_var_name=[\bf Tlog(Pv) ];

xyData=[-60.72272 -70. 0.8674675-59.25998 -60. 0.9876663-55.01853 -50. 1.100371-48.3806 -40. 1.209515-39.22488 -30. 1.307496-28.06241 -20. 1.403121-14.9693 -10. 1.49693

0 0 1.58206316.62758 10. 1.66275834.88586 20. 1.74429354.64541 30. 1.821514

75.68378 40. 1.89209598.14213 50. 1.962843121.7874 60. 2.029789146.5395 70. 2.093422172.3783 80. 2.154728199.3359 90. 2.214844


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Chen CL 118

title([\bf Cal/Exp Data prob_title],FontSize,12)xlabel([ind_var_name],FontSize,14)ylabel([dep var name] FontSize 14)


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ylabel([dep_var_name], FontSize ,14)subplot(2,1,2)

plot(y,y-ycal,*,Linewidth,2)

set(gca,FontSize,14,Linewidth,2)title([\bf Residual, prob_title],FontSize,12) % residual plotxlabel([dep_var_name \bf (Measured)],FontSize,14)ylabel(\bf Residual,FontSize,14)

%%%%%%%%%%%%%%%%%%%%%%%function [Beta, ConfInt, ycalc, Var, R2]=MlinReg(X,y,freeparm)[m,n]=size(X); % m-number of rows, n-number of columnsif freeparmX=[ones(m,1) X]; % Add column of ones if there is a free parameternpar=n+1;else

npar=n;endA=X*XXty=X*yBeta=X\y; % Solve XBeta = Y using QR decompositionycalc=X*Beta; % Calculated dependent variable values

Chen CL 119

Var=((y-ycalc)*(y-ycalc))/(m-npar); % varianceymean=mean(y);R2=(ycalc-ymean)*(ycalc-ymean)/((y-ymean)*(y-ymean));%linear correl


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R2 (ycalc ymean) (ycalc ymean)/((y ymean) (y ymean));%linear correl% Calculate the confidence intervalsA=X*X;

Ainv=A\eye(size(A)); %Calculate the inverse of the XX matrixtdistr95=[12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.306...

2.2622 2.2281 2.2010 2.1788 2.1604 2.1448 2.1315 2.1199..2.1098 2.1009 2.093 2.086 2.0796 2.0739 2.0687 2.0639...2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 2.0395 2.0369...2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211...



t=tdistr95(m-npar);

endfor i=1:nparConfInt(i,1)=t*sqrt(Var*Ainv(i,i)); %confidence intervals

end

Chen CL 120

Input 1 if there is a free parameter, 0 otherwise > 1A =

20 500 34.511


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20 500 34.511500 79000 1383.3

34.511 1383.3 63.681

Xty =1383.3

1.5928e+0053339.8

Results, Antoine Parameters by Linear RegressionParameter No. Beta Conf_int

1 677.87 7.95442 5.2294 0.0409413 -428.52 5.1958


Chen CL 121


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