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

BATCH DISTILLATION :part 1( equilibrium eq: y=k*x)
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part1 : Integral batch distillation.
[email protected] , 18/02/2017

(%i1) kill(all)$
(%i1) eq1:integrate(x^2, x);
\[\tag{eq1}\label{eq1}\frac{{{x}^{3}}}{3}\]
(%i2) eq1:integrate(x^2,x,a,b);
\[\tag{eq1}\label{eq1}\frac{{{b}^{3}}}{3}-\frac{{{a}^{3}}}{3}\]
(%i6) assume(x >= 0 and x <= 1);
assume(y >= 0 and y <= 1);
assume( b>a);
assume( a >0 and a < 1 and b < 1);
\[\tag{\%{}o3}\label{o3} [x\mbox{\ensuremath{\ensuremath{>}}=}0,x\mbox{\ensuremath{\ensuremath{<}}=}1]\] \[\tag{\%{}o4}\label{o4} [y\mbox{\ensuremath{\ensuremath{>}}=}0,y\mbox{\ensuremath{\ensuremath{<}}=}1]\] \[\tag{\%{}o5}\label{o5} [b\mbox{\ensuremath{\ensuremath{>}}}a]\] \[\tag{\%{}o6}\label{o6} [a\mbox{\ensuremath{\ensuremath{>}}}0,a\mbox{\ensuremath{\ensuremath{<}}}1,b\mbox{\ensuremath{\ensuremath{<}}}1]\]
(%i7) eq2:y=k*x;
\[\tag{eq2}\label{eq2}y=kx\]
(%i8) eq3:op1='integrate(1/(rhs(eq2)-x),x,a,b);
\[\tag{eq3}\label{eq3}\mathit{op1}=\int_{a}^{b}{\left. \frac{1}{kx-x}dx\right.}\]
(%i9) eq4:combine(ev(eq3,nouns));
\[\tag{eq4}\label{eq4}\mathit{op1}=\frac{\log{\left( b\,\left| k-1\right| \right) }-\log{\left( a\,\left| k-1\right| \right) }}{k-1}\]
(%i10) eq5:radcan((log(b*abs(k-1))-log(a*abs(k-1)))/(k-1));
\[\tag{eq5}\label{eq5}\frac{\log{(b)}-\log{(a)}}{k-1}\]
(%i11) eq6:op1=logcontract(eq5);
\[\tag{eq6}\label{eq6}\mathit{op1}=\frac{\log{\left( \frac{b}{a}\right) }}{k-1}\]

example :

k= 2.3
a= 0.4
b= 0.7

(%i12) eq7:ev(eq6,k=2.3,a=0.4,b=0.7);
\[\tag{eq7}\label{eq7}\mathit{op1}=0.4304736830272483\]
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