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

analytical method : binary distillation
[email protected],04/04/2017
part1: enriching-section 'operating line'

alpha :constant relative volatility

(%i1) assume(alpha >= 1);
\[\tag{\%{}o1}\label{o1} [\alpha\mbox{\ensuremath{\ensuremath{>}}=}1]\]

p1:hyperbole       : equilbrium equation.
p2:linear equation : operating line.

(%i3) p1:y=(alpha*x)/(1+(alpha-1)*x);
p2:y=m*x+b;
\[\tag{p1}\label{p1}y=\frac{\alphax}{\left( \alpha-1\right) x+1}\] \[\tag{p2}\label{p2}y=mx+b\]

removes (p1,p2),solve for x

(%i5) p3:ratsimp(rhs(p1)-rhs(p2))=0;
p4:(-1)*num(part(p3,1))=0;
\[\tag{p3}\label{p3}-\frac{\left( \alpha-1\right) m\,{{x}^{2}}+\left( m+\left( \alpha-1\right) b-\alpha\right) x+b}{\left( \alpha-1\right) x+1}=0\] \[\tag{p4}\label{p4}\left( \alpha-1\right) m\,{{x}^{2}}+\left( m+\left( \alpha-1\right) b-\alpha\right) x+b=0\]
(%i8) p5:coeff(part(p4,1),x^2);
p6:coeff(part(p4,1),x);
p7:ev(part(p4,1),x=0);
\[\tag{p5}\label{p5}\left( \alpha-1\right) m\] \[\tag{p6}\label{p6}m+\left( \alpha-1\right) b-\alpha\] \[\tag{p7}\label{p7}b\]

solve : quadratic equation ,two roots.

(%i10) p8:float(solve([p4], [x])[1]);
p9:float(solve([p4], [x])[2]);
\[\tag{p8}\label{p8}x=-\frac{1.0\left( \sqrt{{{m}^{2}}+\left( \left( 2.0-2.0\alpha\right) b-2.0\alpha\right) m+\left( {{\alpha}^{2}}-2.0\alpha+1.0\right) \,{{b}^{2}}+\left( 2.0\alpha-2.0{{\alpha}^{2}}\right) b+{{\alpha}^{2}}}+m+\left( \alpha-1.0\right) b-1.0\alpha\right) }{\left( 2.0\alpha-2.0\right) m}\] \[\tag{p9}\label{p9}x=\frac{\sqrt{{{m}^{2}}+\left( \left( 2.0-2.0\alpha\right) b-2.0\alpha\right) m+\left( {{\alpha}^{2}}-2.0\alpha+1.0\right) \,{{b}^{2}}+\left( 2.0\alpha-2.0{{\alpha}^{2}}\right) b+{{\alpha}^{2}}}-1.0m+\left( 1.0-1.0\alpha\right) b+\alpha}{\left( 2.0\alpha-2.0\right) m}\]

first root : assume (alpha ≥1)
          : x=(0 = < first root =< 1)

first  value :interval 'x=0' in p4

(%i11) p10:ev(p4,x=0);
\[\tag{p10}\label{p10}b=0\]

second value :interval 'x=1' in p4

(%i12) p11:factor(ratsimp(ev(p4,x=1)));
\[\tag{p11}\label{p11}\alpha\left( m+b-1\right) =0\]

reason : alpha > 0

(%i13) p12:part(p11,1,2)=0;
\[\tag{p12}\label{p12}m+b-1=0\]

rem : A=(alpha-1)*m          → p5
     B=m+(alpha-1)*b-alpha  → p6
     C=b                    → p7
A*x^2+B*x+C = 0

same: p11 : A+B+C=0

(%i14) p13:factor(ratsimp(p5+p6+p7))=0;
\[\tag{p13}\label{p13}\alpha\left( m+b-1\right) =0\]

which root  p8 or p9 stay
interval :        0 < x = 'root' < 1 (negative root,p8)
may be other root,x > 1 (positive root,p9)'enriching-section'

x=1,y(1)=m*x+b → y=m+b, because  y < 1 → m+b < 1
y= mol fraction of vapor  , 0 = < x = < 1 (p1)
x= mol fraction of liquid , 0 = < y = < 1 (p1)
y = x       :  azeotrope
reason :
y <> x = 1  :  distillation ⇒ y < 1 ⇒ m+b < 1

rem: y=m*x+b , general equation
-------------------------------
x=0 ⇒  b>0
x=1 ⇒  m+b < 1

(%i15) p14:subst(1, x, p2);
\[\tag{p14}\label{p14}y=m+b\]
(%i16) assume(b>0,part(p14,2)<1);
\[\tag{\%{}o16}\label{o16} [b\mbox{\ensuremath{\ensuremath{>}}}0,-m-b+1\mbox{\ensuremath{\ensuremath{>}}}0]\]
(%i19) m:0.501;
b:0.43;
alpha:5.5;
\[\tag{m}\label{m}0.501\] \[\tag{b}\label{b}0.43\] \[\tag{alpha}\label{alpha}5.5\]

negativ root,quadratic equation (p8)

(%i20) assume(b>0,part(p14,2)<1);
\[\tag{\%{}o20}\label{o20} [\mathit{redundant},\mathit{redundant}]\]

p15: first  intersection 'green line' and 'blue line'
p16: second intersection 'green line' and 'blue line'

(%i22) p15:ev(part(p8,2));
p16:ev(part(p9,2));
\[\tag{p15}\label{p15}0.1589233204289739\] \[\tag{p16}\label{p16}1.200136338031882\]

y=x ,45 degree line.'ref line'

(%i23) p15:y=x;
\[\tag{p15}\label{p15}y=x\]

find : intersection between 'green line ' and 'red line'
green line : y=m*x+b  ' enriching-section operating line.
red   line : y=x      ' above in the tower y=x=xd'

(%i26) p16:solve([p2, p15], [x,y])$
p17:xd=float(p16[1][1]);
p18:yd=float(p16[1][2]);
\[\mbox{}\\\mbox{rat: replaced -0.43 by -43/100 = -0.43}\mbox{}\\\mbox{rat: replaced -0.501 by -501/1000 = -0.501}\] \[\tag{p17}\label{p17}\mathit{xd}=\left( x=0.8617234468937875\right) \] \[\tag{p18}\label{p18}\mathit{yd}=\left( y=0.8617234468937875\right) \]
(%i27) plot2d([part(p1,2),part(p15,2),part(p2,2)],
    [x,-.5,1.3],[y,-.5,1.3],
[plot_format, gnuplot])$
\[\mbox{}\\\mbox{plot2d: some values were clipped.}\]
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