\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

BATCH DISTILLATION :part 2(equilbrium y=(alpha*x)/((alpha-1)*x+1))
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part2 : Integral batch distillation.
[email protected] , 18/02/2017

principle : integral

(%i1) kill(all)$
(%i1) eq1:'integrate(x^2,x)=integrate(x^2,x)+constant;
\[\tag{eq1}\label{eq1}\int {\left. {{x}^{2}}dx\right.}=\frac{{{x}^{3}}}{3}+\mathit{constant}\]
(%i2) eq2:'integrate(x^2,x,a,b)=integrate(x^2,x,a,b);
\[\tag{eq2}\label{eq2}\int_{a}^{b}{\left. {{x}^{2}}dx\right.}=\frac{{{b}^{3}}}{3}-\frac{{{a}^{3}}}{3}\]

equilibrium equation :' y '
alpha = relative volatility ( = constant )
info: https://en.wikipedia.org/wiki/Relative_volatility

(%i3) eq11:y=(alpha*x)/(1+(alpha-1)*x);
\[\tag{eq11}\label{eq11}y=\frac{\alphax}{\left( \alpha-1\right) x+1}\]

A : alpha >  1

(%i7) assume(x >= 0 and x <= 1);
assume(y >= 0 and y <= 1);
assume( alpha >1,b>a);
assume( a >0 and a < 1 and b < 1);
\[\tag{\%{}o4}\label{o4} [x\mbox{\ensuremath{\ensuremath{>}}=}0,x\mbox{\ensuremath{\ensuremath{<}}=}1]\] \[\tag{\%{}o5}\label{o5} [y\mbox{\ensuremath{\ensuremath{>}}=}0,y\mbox{\ensuremath{\ensuremath{<}}=}1]\] \[\tag{\%{}o6}\label{o6} [\alpha\mbox{\ensuremath{\ensuremath{>}}}1,b\mbox{\ensuremath{\ensuremath{>}}}a]\] \[\tag{\%{}o7}\label{o7} [a\mbox{\ensuremath{\ensuremath{>}}}0,a\mbox{\ensuremath{\ensuremath{<}}}1,b\mbox{\ensuremath{\ensuremath{<}}}1]\]

1e) integral for batch distillation.

(%i8) eq22:op1='integrate(1/(rhs(eq11)-x),x,a,b);
\[\tag{eq22}\label{eq22}\mathit{op1}=\int_{a}^{b}{\left. \frac{1}{\frac{\alphax}{\left( \alpha-1\right) x+1}-x}dx\right.}\]

alpha = value given.

(%i9) eq23:ev(eq22,alpha=2.48);
\[\tag{eq23}\label{eq23}\mathit{op1}=\int_{a}^{b}{\left. \frac{1}{\frac{2.48x}{1.48x+1}-x}dx\right.}\]

integration values  a<b,a<1,b<1

(%i10) eq24:ev(eq23,a=0.4,b=0.7);
\[\tag{eq24}\label{eq24}\mathit{op1}=\int_{0.4}^{0.7}{\left. \frac{1}{\frac{2.48x}{1.48x+1}-x}dx\right.}\]
(%i11) eq25:ev(eq24,nouns);
\[\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\mbox{}\\\mbox{rat: replaced 0.4 by 2/5 = 0.4}\mbox{}\\\mbox{rat: replaced 0.7 by 7/10 = 0.7}\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\mbox{}\\\mbox{rat: replaced 2.48 by 62/25 = 2.48}\mbox{}\\\mbox{rat: replaced 1.48 by 37/25 = 1.48}\mbox{}\\\mbox{rat: replaced -2.48 by -62/25 = -2.48}\mbox{}\\\mbox{rat: replaced -1.48 by -37/25 = -1.48}\mbox{}\\\mbox{rat: replaced 2.48 by 62/25 = 2.48}\mbox{}\\\mbox{rat: replaced 1.48 by 37/25 = 1.48}\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\mbox{}\\\mbox{rat: replaced 0.4 by 2/5 = 0.4}\mbox{}\\\mbox{rat: replaced 0.7 by 7/10 = 0.7}\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\mbox{}\\\mbox{rat: replaced 2.48 by 62/25 = 2.48}\mbox{}\\\mbox{rat: replaced 1.48 by 37/25 = 1.48}\mbox{}\\\mbox{rat: replaced 2.48 by 62/25 = 2.48}\mbox{}\\\mbox{rat: replaced 1.48 by 37/25 = 1.48}\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\mbox{}\\\mbox{rat: replaced 0.4 by 2/5 = 0.4}\mbox{}\\\mbox{rat: replaced 0.7 by 7/10 = 0.7}\mbox{}\\\mbox{rat: replaced 0.2999999999999999 by 3/10 = 0.3}\] \[\tag{eq25}\label{eq25}\mathit{op1}=1.539608645759518\]

⇒ ' op1 (= eq22) = op2 '
input : alpha >1
       (begin = a,end = b of the interval for integration).

(%i12) eq25:op2=integrate(1/(rhs(eq11)-x),x,a,b);
\[\tag{eq25}\label{eq25}\mathit{op2}=\frac{\log{(b)}}{\alpha-1}-\frac{\alpha\log{\left( 1-b\right) }}{\alpha-1}+\frac{\log{\left( 1-a\right) }\alpha}{\alpha-1}-\frac{\log{(a)}}{\alpha-1}\]
(%i13) eq26:logcontract(eq25);
\[\tag{eq26}\label{eq26}\mathit{op2}=\frac{\log{\left( \frac{b}{a}\right) }+\alpha\log{\left( \frac{a-1}{b-1}\right) }}{\alpha-1}\]

example :

B : alpha = 2.1
   a     = .4
   b     = .7

(%i14) eq26:ev(eq25,alpha=2.1,a=0.4,b=0.7);
\[\tag{eq26}\label{eq26}\mathit{op2}=1.832022606464825\]

alpha = 1 ? division by zero.

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