Multi-component Distillation Prepared by Dr.Nagwa El – MansyChemical Engineering Department

Cairo UniversityFaculty 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:-

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 ii

------ (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

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

V y ------(6)L

( + 1 )Vk

L ( + 1) x

V y 1 ------(7)L

k ( + 1 )Vk

Steps for calculations:-1- Calculate Ki values for each component ( ki = P◦

i/PT)2- Assume L / V3- Calculate ∑yi = ∑xFi (1+L/V)/(1+L/Vki)If ∑yi ≠1 repeat your assumption of L/VIf ∑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:-

i i i i F

i i Fi

Fi bubble i

oo o oi A B

Fi FA FB FCT T T T

oi

FiT

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

y = 1 = k . x

given, x , P assume T and calculate k

PP P P x = x + x x 1

P P P P

Pif x = 1 your a

P

C

ssumption is correct

if not repeat your assumption

B- Dew point :-

ii

i

ii

i

Fi dew i

A Bi o o o

A B

T T T

i

i

y x =

k

y = 1 =

k

given, x , P assume T and calculate k

yy y = + + 1

P P P( ) ( ) ( )

P P P

y if = 1 your assumption is correct

k

if not repeat your a

C

C

x

x

ssumption

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.

Then B is the light key appearing in the bottoms and is termed light key (LK) component x꞊ wLK

And C is the heaviest component appearing in the distillate and is called the heavy key (HK) component x꞊ DHK

All other components are called the non-keys components.

Feed Top BottomABCD

AB

C = HK_

_B = LK

CD

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 theapplication of Lewis-Sorel method to the solution of multicomponent 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

L’x’1i

FxFi

Vy1

V’yri

WxWi

LXoi

DxDi

V L

V’ L’

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 conditionsi.e y1i = x Di = x oi

2-Knowing key components compositions assume xDi s′

3-Calculate x1i s from y′ 1i = 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 ofthe 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 's

V V for n = 2 second stage or plate

L Dy = x + x cal. y 's

V V y

(knowing x = frk

n i F i

om the previous step )

Repeat your calculations till reaching x x

Bottom section:-First start with the reboiler(partial vaporizer is considered as one theoretical 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 Reboiler

L' Wy = x x cal. x 's

V' V'( after calculating y = k x )

for n = 1 first stage or plate

L' Wy = x x

V' 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

At feed entrance we make matching between top andbottom and feed streams to check whether theassumption of xDi’s is correct or not.

If not repeat your assumption, but if it match calculatethe number of stages. To perform these calculations we must know the equilibrium relations (calculate K=f(T,P)) and theoperating 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 vaporand liquid at a single plate, K-values must be known,but these cannot be determined until the stagetemperature is determined which is a function ofcomposition. Trial and error procedure is required.Much of trial and error can be eliminated if the relativevolatility is used in place of K.The relative volatilities are referred to one keycomponents(heavy key) .

If the system is ideal or nearly idealoi

i oref

oi

oref

oi

T ir i To

ref ref

T

pα = 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

α = = , 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

ii ref i r i ref i

ref

i r i ref i ref r i i

r i i refref r i i

r i ii

r i i

K y = K x = α K x

K

y α K x = K α x = 1.0

1 1 = α x or K =

K α x

α x y

α x

i i i i

ii r i refref

ref

-------(9)

Also

y y y x = =

KK α K KK

i i i i

ii r i refref

ref

i i i

r i ref ref r i

i

r ii

i

r i

y y y x = =

KK α K KK

y y1 x = = = 1.0

α K K α

yα

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

α

The number of stages is calculated by using the operating line

equat

ions for top and bottom

1i D i 0 i

Di

1i

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

1-Assume total condenser conditions:-

y = x = x

Knowing 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 '

yα K

x = where α = ( K =K )y Kα

3- Substitute in the top operating line equation :-

L Dy = x + x

V V

s

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 + x

V 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)

yα

where x = (Repeat your calculations till reaching feed ent y

α

s s

rance)

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 αy =

x α

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 α

where yx α

s

s

Repeat your calculations till reaching feed entrancethen make matching between top and bottom and feedstreams to check whether the assumption of xDi’s is

correct or not.If not repeat your assumption, but if it match calculatethe number of stages.

3-Short-cut methods for stage and reflux requirement:-Most of the short-cut methods were developedfor the design of separation columns for hydrocarbonSystems in the petroleum and petrochemical systemindustries. They usually depend on the assumption forseverely non-ideal systems.From these methods:-1- Pseudo-Binary system method = Hengstebeck’s method2-Gilliland, Fenske , Underwood Method.

1-Pseudo-Binary system method = Hengstebeck’sMethod:-Changes the multi-component system to binarysystem.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 + d

For equilibrium relation y = k x

v / V = k l /

i i i

L

v = k l ( V/ L )

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

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.

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 ) = zero

l

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

V Ld d d k

l = = where α = V k α - 1 kk ( ) - 1 - 1L k

i

ii i i i

i

e i e i

i i i

i i ii

i ii

LK

For Top Section:-

dl = & v = d + l

α -1

L 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'

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' w L'( - 1 ) = ( - 1) =

V'K' v' v'

wv' =

K'( - 1)

K

Kw ( )

K w αv' = =

K' K α - α( - )K K

w α v' = & l' = v' + w

α - α

L' L

i i

i

K

i e i' - l' & V' = V' - v'

LK

LK

LKF

LK HK

LKD

LK HK

LKw

LK HK

Equilibrium Relations:-

α xy =

(α - 1) x +1

Also a new compositions

fx =

f + f

dx =

d + d

wx =

w + w

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:-