\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1) kill(all)$

part 2 : intro staged-process models ( problem model )
[email protected],31/01/2017

problem :(n+1)= log(phi)/log(theta)

theta = L/(k*V)

1e)if log(theta)= 0 and log(phi)= value : (n+1) → infinity

2e)if log(theta)= 0 and log(phi)= 0     : (n+1) → indeterminate

  theta = 1 = L/(k*V)

(%i5) eq1:phi1 = -(y[i+1]-x[0]*k)*theta-y[i+1]+y[1];
eq2:phi2 = x[0]*k-y[1];
eq3:phi = rhs(eq1)/rhs(eq2);
eq4:theta = L/(k*V);
eq: m = n+1;
\[\tag{eq1}\label{eq1}\mathit{phi1}=\left( {{x}_{0}}k-{{y}_{i+1}}\right) \theta-{{y}_{i+1}}+{{y}_{1}}\] \[\tag{eq2}\label{eq2}\mathit{phi2}={{x}_{0}}k-{{y}_{1}}\] \[\tag{eq3}\label{eq3}\phi=\frac{\left( {{x}_{0}}k-{{y}_{i+1}}\right) \theta-{{y}_{i+1}}+{{y}_{1}}}{{{x}_{0}}k-{{y}_{1}}}\] \[\tag{eq4}\label{eq4}\theta=\frac{L}{Vk}\] \[\tag{eq}\label{eq}m=n+1\]

indeterminate ⇒ L'Hopital rule = 0/0 = phi'/theta'

(%i8) eq5:diff(rhs(eq3),theta,1);
eq6:diff(lhs(eq4),theta,1);
eq7:rhs(eq) = eq5/eq6;
\[\tag{eq5}\label{eq5}\frac{{{x}_{0}}k-{{y}_{i+1}}}{{{x}_{0}}k-{{y}_{1}}}\] \[\tag{eq6}\label{eq6}1\] \[\tag{eq7}\label{eq7}n+1=\frac{{{x}_{0}}k-{{y}_{i+1}}}{{{x}_{0}}k-{{y}_{1}}}\]

cascade: plate :  distillation operation

1e) operating line       : y[i+1]=a*x[i]+b
2e) relative volatility  : alpha = y[i]/x[i]/((1-y[i])/(1-x[i]))

1e) operating line :

(%i10) eq8:a = L/V;
eq9:b = y[1]-x[0]*L/V;
\[\tag{eq8}\label{eq8}a=\frac{L}{V}\] \[\tag{eq9}\label{eq9}b={{y}_{1}}-\frac{{{x}_{0}}L}{V}\]
(%i11) eq10:y[i+1]=rhs(eq8)*x[i]+rhs(eq9);
\[\tag{eq10}\label{eq10}{{y}_{i+1}}=\frac{L\,{{x}_{i}}}{V}-\frac{{{x}_{0}}L}{V}+{{y}_{1}}\]

2e) relative volatility = alpha :

(%i12) eq11:alpha = y[i]/x[i]/((1-y[i])/(1-x[i]));
\[\tag{eq11}\label{eq11}\alpha=\frac{\left( 1-{{x}_{i}}\right) \,{{y}_{i}}}{{{x}_{i}}\,\left( 1-{{y}_{i}}\right) }\]
(%i13) eq12:solve(eq11,y[i])[1];
\[\tag{eq12}\label{eq12}{{y}_{i}}=\frac{\alpha{{x}_{i}}}{\left( \alpha-1\right) \,{{x}_{i}}+1}\]
(%i14) eq13:subst(i+1,i,eq12);
\[\tag{eq13}\label{eq13}{{y}_{i+1}}=\frac{\alpha{{x}_{i+1}}}{\left( \alpha-1\right) \,{{x}_{i+1}}+1}\]
(%i15) eq14:rhs(eq10)=rhs(eq13);
\[\tag{eq14}\label{eq14}\frac{L\,{{x}_{i}}}{V}-\frac{{{x}_{0}}L}{V}+{{y}_{1}}=\frac{\alpha{{x}_{i+1}}}{\left( \alpha-1\right) \,{{x}_{i+1}}+1}\]
(%i16) eq15:lhs(eq14)-rhs(eq14)=0;
\[\tag{eq15}\label{eq15}-\frac{\alpha{{x}_{i+1}}}{\left( \alpha-1\right) \,{{x}_{i+1}}+1}+\frac{L\,{{x}_{i}}}{V}-\frac{{{x}_{0}}L}{V}+{{y}_{1}}=0\]

total condenser : y[1]=x[0]

(%i17) eq16:subst(x[0],y[1],eq15);
\[\tag{eq16}\label{eq16}-\frac{\alpha{{x}_{i+1}}}{\left( \alpha-1\right) \,{{x}_{i+1}}+1}+\frac{L\,{{x}_{i}}}{V}-\frac{{{x}_{0}}L}{V}+{{x}_{0}}=0\]
(%i18) eq17:radcan(-(alpha*x[i+1])/((alpha-1)*x[i+1]+1)+(L*x[i])/V-(x[0]*L)/V+x[0]);
\[\tag{eq17}\label{eq17}\frac{\left( \left( L\alpha-L\right) \,{{x}_{i}}+\left( \left( {{x}_{0}}-1\right) V-{{x}_{0}}L\right) \alpha-{{x}_{0}}V+{{x}_{0}}L\right) \,{{x}_{i+1}}+L\,{{x}_{i}}+{{x}_{0}}V-{{x}_{0}}L}{\left( V\alpha-V\right) \,{{x}_{i+1}}+V}\]
(%i20) eq18:factor(part(eq17,1));
eq19:factor(part(eq17,2));
\[\tag{eq18}\label{eq18}L\alpha{{x}_{i}}\,{{x}_{i+1}}-L\,{{x}_{i}}\,{{x}_{i+1}}+{{x}_{0}}V\alpha{{x}_{i+1}}-V\alpha{{x}_{i+1}}-{{x}_{0}}L\alpha{{x}_{i+1}}-{{x}_{0}}V\,{{x}_{i+1}}+{{x}_{0}}L\,{{x}_{i+1}}+L\,{{x}_{i}}+{{x}_{0}}V-{{x}_{0}}L\] \[\tag{eq19}\label{eq19}V\,\left( \alpha{{x}_{i+1}}-{{x}_{i+1}}+1\right) \]
(%i23) eq20:coeff(eq18,x[i]);
eq21:coeff(eq18,x[i+1]);
eq22:ev(eq18,x[i]=0,x[i+1]=0);
\[\tag{eq20}\label{eq20}L\alpha{{x}_{i+1}}-L\,{{x}_{i+1}}+L\] \[\tag{eq21}\label{eq21}L\alpha{{x}_{i}}-L\,{{x}_{i}}+{{x}_{0}}V\alpha-V\alpha-{{x}_{0}}L\alpha-{{x}_{0}}V+{{x}_{0}}L\] \[\tag{eq22}\label{eq22}{{x}_{0}}V-{{x}_{0}}L\]
(%i26) eq23:factor(ratsubst(t,x[i]*x[i+1],eq18));
eq24:coeff(eq23,t);
eq25:t=x[i+1]*x[i];
\[\tag{eq23}\label{eq23}L\alphat-Lt+{{x}_{0}}V\alpha{{x}_{i+1}}-V\alpha{{x}_{i+1}}-{{x}_{0}}L\alpha{{x}_{i+1}}-{{x}_{0}}V\,{{x}_{i+1}}+{{x}_{0}}L\,{{x}_{i+1}}+L\,{{x}_{i}}+{{x}_{0}}V-{{x}_{0}}L\] \[\tag{eq24}\label{eq24}L\alpha-L\] \[\tag{eq25}\label{eq25}t={{x}_{i}}\,{{x}_{i+1}}\]

Ricatti : equation
part 3: How to solve Ricatti equation

x[i+1]*x[i]+A*x[i+1]+B*x[i]+C = 0

(%i30) eq26: A = eq21/eq24;
eq27: B = eq20/eq24;
eq28: C = eq22/eq24;
\[\tag{eq26}\label{eq26}A=\frac{L\alpha{{x}_{i}}-L\,{{x}_{i}}+{{x}_{0}}V\alpha-V\alpha-{{x}_{0}}L\alpha-{{x}_{0}}V+{{x}_{0}}L}{L\alpha-L}\] \[\tag{eq27}\label{eq27}B=\frac{L\alpha{{x}_{i+1}}-L\,{{x}_{i+1}}+L}{L\alpha-L}\] \[\tag{eq28}\label{eq28}C=\frac{{{x}_{0}}V-{{x}_{0}}L}{L\alpha-L}\]
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