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In case you missed previous parts, the links to them are listed below.

In this part, I tried to recreate the examples in section A.2.3 of the computational appendix in the reaction engineering book (by Rawlings and Ekerdt).

## Function Minimization

In part 2, the reaction equilibrium conditions was determined by solving a set of nonlinear equations. Alternately, it could be determined by minimizing the Gibbs free energy of the system. The function to be minimized is (refer to part2 for information on what the variables refer to):

The following constraints need to be satisfied for and

First I used constrOptim function for minimization. We need to specify the function to be minimized

# function to be minimized eval_f0=function(x){ dg1=-3.72e3; dg2=-4.49e3; T=400; R=1.987; P=2.5 K1=exp(-dg1/(R*T)); K2=exp(-dg2/(R*T)) yI0=0.5; yB0=0.5; yP10=0; yP20=0; d=1-x[1]-x[2] yI=(yI0-x[1]-x[2])/d yB=(yB0-x[1]-x[2])/d yP1=(yP10+x[1])/d yP2=(yP20+x[2])/d f=-(x[1]*log(K1)+x[2]*log(K2))+(1-x[1]-x[2])*log(P)+yI*d*log(yI)+ yB*d*log(yB)+yP1*d*log(yP1)+yP2*d*log(yP2) return(f) }

The constraints need to be specified in the form

# constraint A=matrix(c(-1,-1,1,0,-1,0,0,1,0,-1),ncol=2,byrow=TRUE) > A [,1] [,2] [1,] -1 -1 [2,] 1 0 [3,] -1 0 [4,] 0 1 [5,] 0 -1 b=c(-0.5,0,-0.5,0,-0.5) > b [1] -0.5 0.0 -0.5 0.0 -0.5

Next, the function is minimized using constrOptim (starting from an initial guess of (0.2,0.2)). Here Nelder-Mead method is used since BFGS method requires specifying the gradient of the function by the user. R taskview optimization lists other options.

# initial guess xinit=c(0.2,0.2) # minimize function subject to bounds and constraints xans2=constrOptim(theta=xinit, f=eval_f0, grad=NULL, ui=A, ci=b, method = "Nelder-Mead") > xans2 $par [1] 0.1331157 0.3509254 $value [1] -2.559282 $counts function gradient 100 NA $convergence [1] 0 $message NULL $outer.iterations [1] 3 $barrier.value [1] 8.695209e-05

The solution can be accessed from xans2$par and is (0.1331,0.3509)

Next, I also tried function minimization with ConstrOptim.nl from the package alabama. Here the constraints are specified in terms of . Even if gradient is not supplied, this function will estimate it using finite-differences.

Definition of constraints in the format for constrOptim.nl

# load library alabama library(alabama) library(numDeriv) h_ineq=function(x){ h=rep(NA,1) h[1]=-x[1]-x[2]+0.5 h[2]=x[1] h[3]=-x[1]+0.5 h[4]=x[2] h[5]=-x[2]+0.5 return(h) } > xans3=constrOptim.nl(par=xinit,fn=eval_f0,hin=h_ineq) Min(hin): 0.1 par: 0.2 0.2 fval: -2.313951 Min(hin): 0.01602445 par: 0.1332729 0.3507026 fval: -2.559282 Min(hin): 0.01597985 par: 0.1331951 0.350825 fval: -2.559282 Min(hin): 0.01597985 par: 0.1331951 0.350825 fval: -2.559282 > xans3 $par [1] 0.1331951 0.3508250 $value [1] -2.559282 $counts function gradient 97 14 $convergence [1] 0 $message NULL $outer.iterations [1] 4 $barrier.value [1] 0.0008697741

The solution can be accessed from xans3$par and is (0.1332,0.3508).

Since this is a 2 dimensional problem, the solution can also be visualized using a contour plot of the function.

# Region of interest: 0.01<=x1<=0.49, 0.01<=x2<=0.49 x1=seq(0.01,0.49,by=0.01) x2=seq(0.01,0.49,by=0.01) # vectorizing function eval_f0 so that it can be evaluated in the outer function fcont=function(x,y) eval_f0(c(x,y)) fcontv=Vectorize(fcont,SIMPLIFY=FALSE) > z=outer(x1,x2,fcontv) There were 50 or more warnings (use warnings() to see the first 50)

The warnings are produced in regions where

# filled.coutour and contour are overlaid with the minimum point (0.133,0.351) filled.contour(x1,x2,z,xlab="x1",ylab="x2", plot.axes={axis(1); axis(2); contour(x1,x2,z,levels=c(-3,-2.5,-2,-1.5,-1,0),vfont=c("sans serif","bold"),labcex=1,lwd=2,add=TRUE); points(0.133,0.351,pch=15,col="blue")})

MATLAB/Octave functions for function minimization (fmincon) have been used in Chemical Engineering computations for a long time and are robust. R has traditionally not been used in this domain. So it is hard to say how the functions I have used in this blog will perform across the range of problems encountered in Reaction Engineering.

> sessionInfo() R version 2.15.1 (2012-06-22) Platform: i386-apple-darwin9.8.0/i386 (32-bit) locale: [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8 attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] alabama_2011.9-1 numDeriv_2012.3-1 rootSolve_1.6.2

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