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1
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 1L Remaining amount 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
1 1ln
( ) ( ) ( 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
11 1ln 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)
2
c) When the relative volatility ij 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 xL
L 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 11 11ln ln ln
( 1) 1
i iij
o ij io io
x xL
L x x
……… (6)
3
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 value 1
i iy x
is calculated:
1
i iy x
i iy x iy ix
- - - -
- - - -
- - - -
A plot of1
i iy x
versus[ ]ix is then made and the area under the curve between 1[ ]ix and
[ ]iox gives 1lno
L
L.