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Multi-component DistillationPrepared by Dr.Nagwa ElMansy

Chemical Engineering DepartmentCairo University

Faculty of Engineering

Fourth year


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Multi-component Distillation

Introduction:-As we do with binary columns, well work with idealstages which can be converted to real stages using anefficiency factor.

The limiting cases of total and infinite reflux apply tomulti-component columns just as they do to binarysystems.

The overall approach to solving multi-component

problems is the same as we use for all equilibrium stagesystem. Use the equilibrium relationships and theoperating relationships. Review multi-componentbubble point and dew point calculations.


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1-Single Stage Multi-component Distillation:-


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Multi-component flash distillation Calculations:-

Fi i i

Fi i i

Fi i i Fi

i Fi

Fi

Overall Material Balance:-

F = L + V ------(1)

Component Material Balance:-

F. x = L . x + V . y --------(2)

( L + V ) x = L . x + V . y

L ( x - x ) = V ( y - x )

(y - x )L=

V ( x -

i

i i i

Fii Fi i i Fi

Fi i Fi i

i

------ (3) (operating line equation)x )

y = k x (equilibrium relation)L

( +1) x(y - x ) (k x - x )L V= = ---(4)

LV ( x - x ) ( x - x )( + k )

V

ix


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Fi

i

i

i i i

Fi

i

i

Fi

i

ii

L ( + 1) x

V x = 1 ---(5)

L( + k )V

Also by substitution with x = y /k

L ( + 1) x

Vy ------(6)L

( + 1 )

VkL

( + 1) xV y 1 ------(7)

L

k ( + 1 )Vk


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Steps for calculations:-

1- Calculate Kivalues for each component ( ki = Pi/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)


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Calculations of dew point and bubble point:-

A-Bubble point:-

i i i i F

i i Fi

Fi bubble i

oo o o

i A BFi FA FB FC

T T T T

o

iFi

T

y = k . x (for the first bubble x = x )

y = 1 = k . x

given, x , P assume T and calculate kPP P P

x = x + x x 1P P P P

Pif x = 1 your a

P

C

ssumption is correct

if not repeat your assumption


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B- Dew point :-

ii

i

ii

i

Fi dew i

A Bi o o o

A B

T T T

i

i

yx = k

y= 1 =

k

given, x , P assume T and calculate kyy y

= + + 1P P P

( ) ( ) ( )P P P

yif = 1 your assumption is correct

k

if not repeat your a

C

C

x

x

ssumption


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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 beseparated as shown in the following table.

Then B is the light key appearing in the bottoms and istermed light key (LK) component xwLKAnd C is the heaviest component appearing in the distillateand is called the heavy key (HK) component xDHKAll other components are called the non-keys components.

Feed Top Bottom

A

B

C

D

A

B

C = HK

_

_

B = LK

C

D


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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 massbalances 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).


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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. Theinitial 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.


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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 traycalculations are done till yi xFi

L

x1i

FxFi

V

y1

V

yriW

xWi

L

Xoi

D

xDi

V L

V L


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Top section:-

L1= L2= L3= --------- = L

V1= V2= V3= -------- = VBottom section:-

L1= L2= L3= --------- = L

V1= V2= V3= -------- = 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.


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From feed conditions ( q )calculate L & V.

q = L L / F calculate L .

q1 = V V / F calculate V .

Calculation steps:-

Top section:-

1- Assume total condenser

conditions

i.e y1i = x Di= x oi2-Knowing key components

compositions assume xDi s

3-Calculate x1is from y1i = K1ix1i( assume T1 , calculate ki1= P1i/ PT then check T1at

x1i= 1 if not repeat )


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4- Substitute in the overall material balance equation of

the top section:-

i i i

i

n+1 n D

2 i 1 i D i 2 i

3 i 2 i D 3 i

2 i2 i

2 i

L Dy = x + x

V V

for n = 1 first stage or plate

L Dy = x + x cal. y 'sV V

for n = 2 second stage or plate

L D

y = x + x cal. y 'sV V

y(knowing x = fr

k

n i F i

om the previous step )

Repeat your calculations till reaching x x


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Bottom section:-

First start with the reboiler(partial

vaporizer is considered as onetheoretical stage)

i

i

m i m+1 w i

r i 1i w i 1i

r i r i w i

1i 2 i D

L' Wy = x - x

V' V'

for m = 0 ReboilerL' W

y = x x cal. x 'sV' V'

( after calculating y = k x )

for n = 1 first stage or plateL' W

y = x xV' V'

2 i

1i 1i 1i

m i F i

cal. x 's

(knowing y = k x from the previous step )

Repeat your calculations till reaching y x


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At feed entrance we make matching between top and

bottom and feed streams to check whether the

assumption of xDisis correct or not.If not repeat your assumption, but if it match calculate

the number of stages.

To perform these calculations we must know theequilibrium 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 Tand checking it from Sx or Sy (as if its a normalflashing problem).


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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) .


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If the system is ideal or nearly ideal

o

i

i oref

o

i

o

ref

o

i

T ir i To

ref ref

T

p = relative volatility for component (i)=

p

Where , p = vapor pressure of component (i)

p = vapor pressure of a reference component ( heavy key = HK )

pp K

= = , where pp K

p

i i i

ii ref i

ref

= total pressure -------(8)

The equation on which the calculations is made:-y = K x

K y = K x

K


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ii ref i r i ref i

ref

i r i ref i ref r i i

r i i ref

ref r i i

r i ii

r i i

K y = K x = K x

K

y K x = K x = 1.0

1 1 = x or K =

K x

x y

x

i i ii

ii r i ref ref

ref

-------(9)

Also

y y y x = =

KK KK

K


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i i ii

ii r i ref ref

ref

i ii

r i ref ref r i

i

r ii

i

r i

y y y x = =

KK KK

Ky y1

x = = = 1.0 K K

y

x --------(10)y

The number of stages is calculated by using the operating line

equat

ions for top and bottom


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1i D i 0 i

Di

1i

Top Section:- (for any component (i)) :

1-Assume total condenser conditions:-

y = x = xKnowing key components compositions assume x '

2- For n = 1 (for first plate)

calculate x ' fro

s

s

i

1i

1i

r i i1i r i r HK

1i r

r i

n+1 n i D i

m y 'y

Kx = where = ( K =K )

y K

3- Substitute in the top operating line equation :-

L D

y = x + xV V

s


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2 1 i D i 2 1i

3 i 2 i D i

For n = 1

L D y = x + x (calculate y ' after calculating x '

V V from equilibrium relations)

4- For n = 2

L Dy = x + xV V

i i s s

i

3 i 2 i

2 i

r i2 i

2

r i

( calculate y ' after calculating x '

from equilibrium relations)

y

where x = (Repeat your calculations till reaching feed enty

s s

rance)


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w i r ir i

w i r i

m i m+1 i w i

r i 1 i w i

Bottom Section:-

1- Reboiler where ( m = 0)

x y =

x

2- Substitute in the bottom operating line equation:-

L' Wy = x - x

V' V'L' W

y = x - x ( calculate xV' V'

1 i

1 i 2 i w i 2 i

1 i r i1 i

1 i r i

' )

3- For m = 1 (first plate from the bottom)

L' Wy = x - x ( calculate x ' )

V' V'

x where y

x

s

s


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Repeat your calculations till reaching feed entrance

then make matching between top and bottom and feed

streams to check whether the assumption of xDisiscorrect or not.

If not repeat your assumption, but if it match calculate

the number of stages.


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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 forseverely non-ideal systems.

From these methods:-

1- Pseudo-Binary system method = Hengstebecksmethod

2-Gilliland, Fenske , Underwood Method.


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1-Pseudo-Binary system method = Hengstebecks

Method:-

Changes the multi-component system to binary

system.

Using Mc-cabe Thiele Equations:-n+1 n

i i i

i i i

i i i

Upper Section:- V = L + D

V = L + D

For any component (i) v = l + dFor equilibrium relation y = k x

v / V = k l /

i i i

L

v = k l ( V/ L )


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m+1 m

i i i

i i i

i i i

Bottom Section:-

L' = V' + W

L' = V' + W

For any component (i) l' = v' + w

For equilibrium relation y' = k' x'

v' / V' = k' l' / L'

i i i v' = k' l' ( V'/ L' )To reduce the multicomponent system to an equivalent binary

system we must estimate the flow rates of the key components:-

Upper Section:-

Le i

e i

e e

L - l

V = V - v

where L &V are the flow rates of key components of upper section


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i i

e i

e i

e e

l & v are the flow rates of components lighter

than key components in the upper section.

Bottom Section:-

L' L' - l'

V' = V' - v'where L' &V' are the flow rates of key components

of upper sectio

i in l' & v' are the flow rates of

components heavier than key components

in the bottom section.


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i i i

i i i

i i i

Upper Section:- v = l + d For any component

v = k l ( V/ L ) For equilibrium relation

Substitute in the MB equation:-k l ( V/ L ) = l + di

i i i

iHK

i

HK i i i i i i

i i i ii i

i i HKi

HK

( k ([ V/ L] -1 ) l d

For heavy key:-d

(k ([ V/ L] -1 ) = zerol

L V

k v = l + d k ( ) l = d lV L

d d d k l = = where =

V k - 1 k k ( ) - 1 - 1

L k

i


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ii i i i

i

e i e i

i i i

i i i

i

i ii

LK

For Top Section:-

dl = & v = d + l

-1L L - l & V = V - v

For Bottom Section:-

l' = v' + w

L'( ) v' v' + w

V'K'

L' ( -1 ) v' w

V'K'

For light key:-

L'( - 1

V'K'

iLK

i

w L') = = zero & K' =

v' V'


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i LK i

i i

ii

LK

ii

HK i ii

LK i LK i

HK HK

i ii i i i

LK i

e

For any component(i) :-

w K' wL'( - 1 ) = ( - 1) =

V'K' v' v'

wv' =

K'( - 1)

K

Kw ( )

K w v' = =

K' K - ( - )

K Kw

v' = & l' = v' + w -

L' L

i i

i

K

i e i

' - l' & V' = V' - v'


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LK

LK

LKF

LK HK

LKD

LK HK

LKw

LK HK

Equilibrium Relations:-

xy =

( - 1) x +1

Also a new compositionsf

x =f + f

dx =

d + dw

x =w + w


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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

( RRmin ) / ( R + 1 ) .


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Where N = no. of plates. R = reflux ratio.

N min=minimum no. of plates. R min= minimum reflux ratio.


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2-Fenske equation for calculation minimum

number of plates :-


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3- Minimum reflux ratio(R min):- Underwood equation:-


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Location of feed tray

Is critical to column efficiency:-

Basis for estimate:-

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