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Differential Distillation

Rayleigh Equation

( )

i

i i

dxdL

L y x

(1)

Integrating:11

( )

i

o io

xL

i

i iL x

dxdL

L y x

1

1ln( )

i

io

x

i

o i ix

dxL

L y x

(2)

Where:oL Initial amount of liquid in pot, ,moles 1

L Remainingamount of liquid in pot, ,moles dL Amount of liquid vaporized in time d

,i idx dy Change in concentration for time interval d Solution of the relation depends on the form of the equilibrium relationship: ( )i iy f x

a) When i i iy m x 1 1 1

11

ln( ) ( ) ( 1)

i i i

io io io

x x x

i i i

o i i i i i i ix x x

dx dx dxL

L y x m x x m x

111

ln ln( 1)

i

o i io

xL

L m x

(3)

Rewritten in another form:1

111

imi

o io

xL

L x

(4)

b) For i i i iy m x c

1 1 1

1ln( ) ( ) ( 1)

i i i

io io io

x x x

i i i

o i i i i i i i i ix x x

dx dx dxL

L y x m x c x m x c

11( 1)1

ln ln( 1) ( 1)

i i i

o i i io i

m x cL

L m m x c

(5)

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c) When the relative volatilityij is constant and the equilibrium expressed is as:

1 ( 1)

ij i

i

ij i

xy

x

1 1

1 1

1ln( )

( )1 ( 1)

(1 ( 1) ) (1 ( 1) )

( 1 ) ( 1)(1 )

i i

io io

i i

io io

x x

i i

ij io i ix xi

ij i i

x x

ij i i ij i i

i ij ij i i i ij ix x

L dx dx

xL y xx

x

x dx x dx

x x x x x

1 1 ( 1)

( 1)(1 ) ( 1)(1 )

i i

io io

x x

ij i ii

i ij i i ij ix x

x dxdx

x x x x

1 11

( 1) (1 ) (1 )

i i

io io

x x

i i

ij i i ix x

dx dx

x x x

The integration becomes:

11

1 1

1 11ln ln ln( 1) 1 1

io i io

o ij i io i

x x xLL x x x

1

1 1

1 11ln ( 1) ln

( 1) 1 1

io i ioij

ij i io i

x x x

x x x

1

1

11ln ln

( 1) 1

i ioij

ij io i

x x

x x

And finally:

1 1111

ln ln ln( 1) 1

i iij

o ij io io

x xL

L x x

(6)

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d) Graphical integration is applied when the equilibrium data ( , )i ix y is given in tabular

form:

For each equilibrium point ( , )i ix y the value1

i iy x

is calculated:

1

i iy xi iy xiyix

----

----

----

A plot of 1i iy x

versus[ ]ix is then made and the area under the curve between 1[ ]ix and

[ ]iox gives1lno

L

L.