### Multi-component Distillation Prepared by Dr.Nagwa El

```Multi-component Distillation
Prepared by Dr.Nagwa El – Mansy
Chemical Engineering Department
Cairo University
Faculty of Engineering
Fourth year
Multi-component Distillation
Introduction:As we do with binary columns, we’ll work with ideal
stages which can be converted to real stages using an
efficiency factor.
The limiting cases of total and infinite reflux apply to
multi-component columns just as they do to binary
systems.
The overall approach to solving multi-component
problems is the same as we use for all equilibrium stage
system. Use the equilibrium relationships and the
operating relationships. Review multi-component
bubble point and dew point calculations.
1-Single Stage Multi-component Distillation:-
Multi-component flash distillation Calculations:Overall Material Balance:F = L + V ------(1)
Component Material Balance:F. x Fi = L . x i + V . yi --------(2)
 ( L + V ) x Fi = L . x i + V . yi
 L ( x Fi - x i ) = V ( yi - x Fi )
(yi - x Fi )
L
 =
------ (3)
V
( x Fi - x i )
yi = k i x i
(operating line equation)
(equilibrium relation)
L
( +1) x Fi
(yi - x Fi )
(k i x i - x Fi )
L
 =
=
 xi  V
---(4)
L
V
( x Fi - x i )
( x Fi - x i )
( + ki )
V


xi 

L
( + 1) x Fi
V
= 1 ---(5)
L
( + ki )
V
Also by substitution with x i = y i /k i
L
( + 1) x Fi
V
L
(
+1)
Vk i
 yi 

y
i

------(6)
L
( + 1) x Fi
V
 1 ------(7)
L
ki (
+1)
Vk i
Steps for calculations:1- Calculate Ki values for each component ( ki = P◦i/PT)
2- Assume L / V
3- Calculate ∑yi = ∑xFi (1+L/V)/(1+L/Vki)
If ∑yi ≠1 repeat your assumption of L/V
If ∑yi=1 → your assumption is correct then calculate
∑ xi =1 .
(solution is by trial and error)
Calculations of dew point and bubble point:A-Bubble point:yi = k i . x i
y
i
=1=
(for the first bubble x i = x F )
k
i
. x Fi
given, x Fi , P assume Tbubble and calculate k i
PCo
Pio
PAo
PBo
 
x Fi =
x FA +
x FB 
x FC      1
PT
PT
PT
PT
Pio
if 
x Fi = 1  your assumption is correct
PT
if not repeat your assumption
B- Dew point :yi
xi =
ki
 xi = 1 =

yi
ki
given, x Fi , P assume Tdew and calculate k i
yC
yA
yB
  xi =
+
+
     1
o
o
o
PA
PB
PC
( )
( ) ( )
PT
PT
PT
if

yi
ki
= 1  your assumption is correct
if not repeat your assumption
2-Multi-Stage Multi-component Distillation:Key Components:Suppose a four component mixture A-B-C-D in which A is
the most volatile and D is the least volatility is to be
separated as shown in the following table.
Feed
A
B
C
D
Top
A
B
C = HK
_
Bottom
_
B = LK
C
D
Then B is the light key appearing in the bottoms and is
termed light key (LK) component ꞊ xwLK
And C is the heaviest component appearing in the distillate
and is called the heavy key (HK) component ꞊ xDHK
All other components are called the non-keys components.
Plate to plate calculations for multi-component distillation:Calculation from plate to plate are based upon the
bubble-point and dew-point calculations coupled with mass
balances at each plate.
There are many methods for calculation number of plates
necessary for given separation and composition on each
plate. From these methods:
1-Lewis-Matheson Method (equimolar flow rates).
The method proposed by Lewis Matheson is essentially the
application of Lewis-Sorel method to the solution of multi
component problems (general method).
Constant molar overflow is assumed and the material balance
and equilibrium relationship equations are solved stage by
stage starting at the top or bottom of the column. In this
method we must specify the following variables:1)Feed composition, flow rate, reflux ratio and condition (q ).
2)Distribution of key-components.
3)Products flow rates.
4)Column pressure.
5)Assumed values for the distribution of non-key components.
The usual procedure is to start the calculations at the top and
bottom of the column and proceed toward the feed point. The
initial estimates of the component distributions in the
products are then revised and the calculations repeated until
the compositions calculated from the top and bottom match
at the feed point.
Notes:Lewis Matheson Method:1-Similar to Lewis method.
2-Tray to tray calculations are done
with the assumption of constant
molar flow rates of liquid and
vapour in each section.
3-Top section tray to tray calculations
are done till xi ≤ xFi
4-Bottom section tray to tray
calculations are done till yi ≥ xFi
V
y1
L
Xoi
V
D
xDi
L
F
xFi
V’ L’
V’
yri
L’
x’1i
W
xWi
Top section:L1 = L2 = L3 = --------- = L
V1 = V2 = V3 = -------- = V
Bottom section:L’1 = L’2 = L’3 = --------- = L’
V’1 = V’2 = V’3 = -------- = V’
(where molal latent heats are mainly the same)
• Reflux ratio = R = L/ D
• V = ( L + D ) = D ( L/D + 1) = D ( R + 1 )
• From overall (M.B) on the column , calculate D ,W.
• Then calculate L & D.
• From feed conditions ( q )calculate L’ & V’.
• q = L’ – L / F → calculate L’ .
• q – 1 = V’ – V / F → calculate V ‘ .
Calculation steps:Top section:1- Assume total condenser
conditions
i.e y1i = x Di = x oi
2-Knowing key components
compositions assume xDi′ s
3-Calculate x1i ′s from y1i = K1i x1i
( assume T1 , calculate ki1 = P◦1i / PT then check T1 at
∑ x1i = 1 if not repeat )
4- Substitute in the overall material balance equation of
the top section:L
D
y n+1 i = x n i + x D i
V
V
 for n = 1  first stage or plate
L
D
y 2 i = x1 i + x D i  cal. y 2 i 's
V
V
 for n = 2
 second stage or plate
L
D
y3 i = x 2 i + x D i  cal. y3 i 's
V
V
y2 i
(knowing x 2 i =
from the previous step )
k2 i
Repeat your calculations till reaching x n i  x F i
Bottom section:First start with the reboiler(partial
vaporizer is considered as one
theoretical stage)
L'
W
ym i =
x m+1 i xw i
V'
V'
 for m = 0  Reboiler
L'
W
yr i =
x1i 
x w i  cal. x1i's
V'
V'
( after calculating  y r i = k r i x w i )
 for n = 1
 first stage or plate
L'
W
y1i =
x2 i 
x D i  cal. x 2 i 's
V'
V'
(knowing y1i = k1i x1i from the previous step )
Repeat your calculations till reaching y m i  x F i
At feed entrance we make matching between top and
bottom and feed streams to check whether the
assumption of xDi’s is correct or not.
If not repeat your assumption, but if it match calculate
the number of stages.
To perform these calculations we must know the
equilibrium relations (calculate K=f(T,P)) and the
operating pressure .
BUT operating temperature varies from tray to another,
so each tray calculations will be done by assuming T
and checking it from Sx or Sy (as if it’s a normal
flashing problem).
2- Constant relative volatility method:By calculating the equilibrium composition of vapor
and liquid at a single plate, K-values must be known,
but these cannot be determined until the stage
temperature is determined which is a function of
composition.
Trial and error procedure is required.
Much of trial and error can be eliminated if the relative
volatility is used in place of K.
The relative volatilities are referred to one key
components(heavy key) .
If the system is ideal or nearly ideal
pio
α i = relative volatility for component (i) = o
p ref
Where , pio = vapor pressure of component (i)
p oref = vapor pressure of a reference component ( heavy key = HK )
pio
αr i =
p
o
ref
pT
pT
Ki
=
, where p T = total pressure
K ref
The equation on which the calculations is made:yi = K i x i
 yi =
Ki
K ref x i
K ref
-------(8)
Ki
 yi =
K ref x i = α r i K ref x i
K ref

y
i

1

=
K ref

yi 
α
ri
α
ri
K ref x i = K ref  α r i x i = 1.0
xi
or K ref =
α r i xi
 αr i xi
-------(9)
Also
yi
xi =

Ki
yi
Ki
K ref
K ref
yi
=
α r i K ref
α
1
ri
xi
xi
yi
yi
yi
=
=
Ki
Ki
α r i K ref
K ref
K ref
yi
1
 xi =  α K = K
ri
ref
ref

yi
= 1.0
αr i
yi

αr i
xi 
yi
α
ri
--------(10)
The number of stages is calculated by using the operating line
equations for top and bottom
Top Section:- (for any component (i)) : 
1-Assume total condenser conditions: y1i = x D i = x 0 i
Knowing key components compositions assume x Di ' s
2- For n = 1 (for first plate)
calculate x1i ' s from y1i ' s
y1i
αr i
x1i =
y1i
α
ri
Ki
where α r i =
Kr
( K r =K HK )
3- Substitute in the top operating line equation :y n+1 i
L
D
=
xn i +
xD i
V
V
For n = 1
 y2 i
L
D
=
x1 i +
x D i (calculate y 2 i ' s after calculating x1i ' s
V
V
from equilibrium relations)
4- For n = 2
L
D
y3 i =
x2 i +
xD i
V
V
( calculate y3 i ' s after calculating x 2 i ' s
from equilibrium relations)
y2 i
where x 2 i =
αr i
y 2i
α
ri
(Repeat your calculations till reaching feed entrance)
Bottom Section:1- Reboiler where ( m = 0)
yr i =
x w i αr i
 x w i αr i
2- Substitute in the bottom operating line equation:L'
W
x m+1 i xw i
V'
V'
L'
W
 yr i =
x1 i x w i ( calculate x1 i ' s )
V'
V'
3- For m = 1 (first plate from the bottom)
ym i =
L'
W
x2 i xw i
V'
V'
x α
where y1 i  1 i r i
 x1 i α r i
y1 i =
( calculate x 2 i ' s)
Repeat your calculations till reaching feed entrance
then make matching between top and bottom and feed
streams to check whether the assumption of xDi’s is
correct or not.
If not repeat your assumption, but if it match calculate
the number of stages.
3-Short-cut methods for stage and reflux
requirement:Most of the short-cut methods were developed
for the design of separation columns for hydrocarbon
Systems in the petroleum and petrochemical system
industries. They usually depend on the assumption for
severely non-ideal systems.
From these methods:1- Pseudo-Binary system method = Hengstebeck’s
method
2-Gilliland, Fenske , Underwood Method.
1-Pseudo-Binary system method = Hengstebeck’s
Method:Changes the multi-component system to binary
system.
Using Mc-cabe Thiele Equations:Upper Section:- Vn+1 = L n + D
V = L+D
For any component (i)  vi = li + d i
For equilibrium relation  yi = k i x i
v i / V = k i li / L
vi = k i li ( V/ L )
Bottom Section:L'm+1 = V'm + W
L' = V' + W
For any component (i)  l'i = v'i + w i
For equilibrium relation  y'i = k'i x'i
v'i / V' = k'i l'i / L'
v'i = k'i l'i ( V'/ L' )
To reduce the multicomponent system to an equivalent binary
system we must estimate the flow rates of the key components:Upper Section:L e  L -  li
Ve = V -  vi
where L e &Ve are the flow rates of key components of upper section
li & vi are the flow rates of components lighter
than key components in the upper section.
Bottom Section:L'e  L' -  l'i
V'e = V' -  v'i
where L'e &V'e are the flow rates of key components
of upper section l'i & v'i are the flow rates of
components heavier than key components
in the bottom section.
Upper Section:-
v i = li + d i
v i = k i li ( V/ L )
For any component
For equilibrium relation
Substitute in the MB equation:k i li ( V/ L ) = li + d i
 ( k i ([ V/ L] -1 ) li  d i
For heavy key:di
(k HK ([ V/ L] -1 ) =
 zero
li
L
 k HK 

V
di
li 
=
V
ki ( ) - 1
L
V
v i = li + d i  k i ( ) l i = d i  l i
L
di
di
ki
=
where αi =
ki
αi - 1
k HK
-1
k HK
 For Top Section:di
li =
αi -1
& v i = d i + li
L e  L -  li
& Ve = V -  vi
For Bottom Section:l'i = v'i + w i
L'
(
) v'i  v'i + w i
V'K'i
L'
(
-1 ) v'i  w i
V'K'i
For light key:wi
L'
L'
(
-1)=
= zero & K'LK =
V'K'LK
v'i
V'
For any component(i) :wi
L'
(
-1)=
V'K'i
v'i
 v'i =

K'LK
wi
(
- 1) =
Ki
v'i
wi
K'LK
(
- 1)
Ki
Ki
wi (
)
K HK
w i αi
v'i =
=
K'LK
Ki
α LK - α i
(
)
K HK K HK
w i αi
 v'i =
&
α LK - αi
l'i = v'i + w i
L'e  L' -  l'i & V'e = V' -  v'i
Equilibrium Relations:α LK x
y=
(α LK - 1) x +1
Also a new compositions
f LK
xF =
f LK + f HK
d LK
xD =
d LK + d HK
xw =
w LK
w LK + w HK
Gilliland, Fenske , Underwood Method:1- Gilliland Equation for calculation the number
of stages at operating reflux:A simple empirical method is used for
preliminary Estimates. The correlation requires
knowledge only of the minimum reflux ratio.
This is shown in the following figure, where the
group :(N - N min)/(N + 1) is plotted against
( R – R min ) / ( R + 1 ) .
Where N = no. of plates.
R = reflux ratio.
N min=minimum no. of plates. R min= minimum reflux ratio.
2-Fenske equation for calculation minimum
number of plates :-
3- Minimum reflux ratio(R min):- Underwood equation:-
Location of feed tray
• Is critical to column efficiency:• Basis for estimate:-
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