McCabe-Thiele Method 1

5/19/2018 McCabe-Thiele Method 1

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Jawaharlal Nehru TechnologicalUniversity Kakinada

III Year B. Tech. Petrochemical EngineeringII em.

!ass Trans"er #$eration % I

&ITI''(TI#NE)UI'IB*IU! T(+E ,(',U'(TI#N

!c,(BE-TIE'E !ET#&

Presentation /y

Pro". K. 0. *ao(cademic (dvisor 1 0isiting Pro"essor

chool o" Petroleum ,ourses


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!c,a/e-Thiele !ethod

It is a gra$hical method andinvolves calculation o" total num/ero" e2uili/rium stages re2uired "or a

given se$aration using material/alance and e2uili/rium relations.

The following notations shall be used :

y : mole fraction of more volatile component in vapour

phase

x : mole fraction of more volatile component in liquid

phase

V : molar flowrate of vapour, mole/time

L : molar flowrate of liquid, mole/time

F : molar flowrate of feed vapor or liquid or mixed!,

mole/time


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*e"er to the 3igure that shows several $latesinside a distillation column. (ssume that the$lates are num/ered serially "rom the to$ downand that the $late under consideration is the n-th $late "rom the to$. Then the $late

immediately a/ove this $late is $late 4n-56 andthe $late immediately /elow this $late is $late4n756.


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"treamTotal #olar

Flowrate

$omposition

#ole Fraction

#V$!

Vapor leaving

plateVn yn

Liquid leaving

plate

Ln xn

Vapor entering

plateVn%& yn%&

Liquid entering

plateLn'& xn'&

There are four streams ( vapor and (

liquid! associated with this plate, eachwith its own flow rate and concentration:


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Ideal Plate1Tray1tage(n ideal $late is one where the va$orleaving the $late is in e2uili/rium withthe li2uid leaving the same $late 4seethe e2uili/rium diagram 8 9

nand y

nare

in e2uili/rium6.


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)n each plate, the vapor rising to it and the liquid

flowing down to it are not in equilibrium* Thus there is

a concentration difference driving force! for masstransfer between the two phases

The system tends to reach equilibrium on each tray

because some of the less volatile component

condenses from the rising vapor into the liquid, thusincreasing the concentration of the more volatile

component in the vapor as it moves upwards, e*g* yn+

yn%&

*

some of the more volatile component is vaporiedfrom the liquid on the tray, thus decreasing the

concentration of the more volatile component in the

liquid as it moves downwards, e*g* xn'&

+ xn*


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The changes in va$or and li2uid $hase mole"ractions are shown in the 3igure. The com/inede:ect is a gradual increase in concentration o"

the more volatile com$onent in the va$or as itmoves u$ the column; and a gradual increase inconcentration o" the less volatile com$onent inthe li2uid as it moves down the column. ee the3igure


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-ubble .oint Temperature

The temperature at which the first bubble of vapor is

formed*

ew .oint Temperature

The temperature at which the first dew of liquid is formed*

The vapor and liquid streams inside the column areassumed to be saturated at their respective dew points

and bubble points corresponding to the position in the

column*

The heat released by one mole of vapor uponcondensation is approximately equal to the heat required

to vaporie one mole of the liquid: the number of

molecules passing from the vapor phase to the liquid

phase and vice versa will be approximately the same*


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0f the molar heats of vaporiation are approximately

constant, we can assume that the flows of liquid and

vapour are nearly constant in each section of the

column,i.e.

L&1 L

(1 L

21 ********* 1 L

n1 constant

V& 1 V

( 1 V

2 1 ********* 1 V

n 1 constant

This is the important concept of constant molaloverflow*

"eparation is achieved with the vapor rich in the

more volatile component leaving the top of the

column, and the liquid rich in the less volatilecomponent leaving the bottom of the column*

The temperature decreases as one moves up the

column, i*e* Tn%&

+ Tn+ T

n'&*


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$alculation of Total 3umber of 4quilibrium "tages

Procedure:

The VLE data must be available at the operating pressure of the

column.

Separation must be specified.

feed condition (temperature, composition), distillate and bottom

compositions and the reflu! ratio, "hich is defined as the ratio of

reflu! li#uid over the distillate product. This is sho"n in the $igure

belo".


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e num er o eore ca s ages requ re or a g ven


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e num er o eore ca s ages requ re or a g venseparation is then the number of triangles that can bedrawn between these operating lines and theequilibrium curve. The last triangle on the diagramrepresents the reboiler.

To obtain the number of theoretical trays using theMcCabe-Thiele Method, we shall use the section analysisthat is rst carried out by partitioning the column into 3sections rectifying, feed and stripping sections asshown the gure below


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The McCabe-Thiele Method involves the steps givenbelow to determine the number of theoretical stages

% !nalysis of the "ectifying section, and determine the"#$ using %

&and "

% !nalysis of the 'eed section, and determine the feedcondition (q)

% &etermination of the feed line (q-line) using %'and q

% $ocate the intersection point between "#$ and q-line% !nalysis of the *tripping *ection, and determine the

*#$using (+) and %

% 5.*ecti"ying ection #$erating 'ine 4*#'6% Consider the rectifying section as shown in the 'igure

below. (*ystem shows a total condenser and thereu% is at bubble point)
http://www.separationprocesses.com/Distillation/DT_Chp04l.htmhttp://www.separationprocesses.com/Distillation/Fig044.htmhttp://www.separationprocesses.com/Distillation/Fig044.htmhttp://www.separationprocesses.com/Distillation/DT_Chp04l.htm


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Material balance around the envelope / 0 #1T

Thus, we have($

n2 &) y

n20 $

n%

n2 & %

&

1nder constant molal overow assumption$0 $

40 .......... $

n-0 $

n0 $

n20 $ 0 constant

OMB: Vn+1 = Ln+ D

CMB: Vn+1 yn+1 = Lnxn+ D xD


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50 5

40 .......... 5

n-0 5

n0 5

n20 5 0 constant

The subscripts can be dropped. Thus, the equationsimplies to($ 2 &) y

n

2 0 $ %n

2 & %&

"e-arranging in the form y 0 f(%), we have

ntroducing "eu% "atio " 0 $ 6 &,

This is the )perating Line 4quation for the rectifying

section, or 5)L in short*


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,haracteristics8 traight 'ine E2uation

slo$e *14*756; constant "or given value o" *

Interce$t 4514*756 9&; constant "or given * and$urity o" distillate 9

&

The o$erating line $asses through the $oint49

&; 9

&6 on the


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&, diagonal (note y

0 %&), draw a

hori8ontal line tothe left until it

touches theequilibrium curvethis gives the point(%

, y

).

'rom this point (%,

y) draw a vertical

line down to the"#$ this gives thepoint (%

, y

4). n

this manner we

had obtained onetriangle (no.)where thehori8ontal distanceis (%

& - %

) and the

vertical distance is

(y - y4). #netrian le is


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The di7erence (%&

- %) represents

the decrease in

the concentrationof the morevolatilecomponent in theliquid phase as itsmoves down one

tray, i.e. from tray to tray 4. Thedi7erence (y- y4)

represents theincrease in the

concentration ofthe more volatilecomponent in thevapor phase as itsmoves up one

tray, i.e. from tray4 to tra . *ee


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4 t d ti f ' d th li


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4. ntroduction of 'eed the q-line

Consider the section of the distillation column ( see the'igure below) at the tray where the feed is introduced(9nown as the feed tray location), say tray f

3ig. ,ase a. ,old

3eed

!s an e%ample, considerthe 'igure above wherebythe feed is a cold liquid. nthis case, all the liquidfeed will go to thestripping section. naddition, because thefeed is cold, it will also

condense some of therising vapor. !s a result,the amount of liquid owin the stripping section $:is much larger than the

liquid ow in the rectifyingsection $. The vapour owin the rectifying section 5,is lower than the vapourow in the stripping

section 5: because of thecondensation into the


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*imilar evaluation can be carried out for the other feed conditions

operating line (or


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operating line (orsimply the q-line) canbe obtained byperforming a materialbalance around the

feed tray. =lotting ofthe q-line requiresthe q-value and thefeed M5C molefraction, %

'. !s shown

above, q 0 .> forsaturated liquid andq 0 >.> for saturatedvapour. 'or vapour-liquid mi%ture, q 0

fraction of feed thatis liquid. 'or otherconditions, we needto calculate the q-values. The feed traylocation can beidentied once the


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3eed ection #$erating 'ine 42-line6

liquid ow 0 q ' moles6hr? vapour ow 0 (-q) 'moles6hr

#verall material balance$: 0 $ 2 q '5 0 5: 2 (-q) 'Component balance for the more volatile component (*ee the 'igure below)


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& tif i ti V L *


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&ectif'ing section : V ' L ! * !*

Stripping section : V+ ' L+ ! - !-

t the feed point "here the t"o lines operating lines intersect:

( V V+ ) ' ( L L+ ) ! * !* - !

-

"e have:

V V+ ( / # ) $

L L+ # $

0n addition, from component balance around the entire column:

$ !$ * !

* - !

-

Thus, ( / # ) $ ' # $ ! $ !$

&earranging in the form ' f(!), "e have:

$or a given feed condition ! and # are fi!ed therefore the #line is


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$or a given feed condition, !$and # are fi!ed, therefore the #line is

a straight line "ith slope # 1 (/#) and intercept !$1 (/#).

0f ! !$ , then ' !$.

i.e. the #line passes through the point (!$, !$) on the 23o diagonal.

*ifferent values of # "ill result in different slope of the #line. See the

$igurebelo".

4ote that the #line passes through the point (!$, !$) on the 23o

diagonal for all values of #.


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The 2-values

f the condition of the feed is 9nown to be either saturatedliquid or saturated vapour, then the value of q is either

or >. ;owever, if we are not certain of the feed condition,then we must calculate the value of q. @e can do so byderiving a formula for q using enthalpy balance aroundthe feed plate f. This is shown in the 'igure below

0


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,

' ;'2 $

';f-2 5:

;5,f20 $:

;$,f2 5

;5,f

where

;'0 enthalpy of feed, evaluated at T';

50 enthalpy of vapour, and

;$0 enthalpy of liquid

!ssume that ;$,f- 0 ;$,f0 ;$, and, ;5, f2 0 ;5,f 0 ;5

Then,

' ;'2 $ ;

$2 5: ;

50 $: ;

$2 5 ;

5

' ;'0 (5 - 5:) ;

52 ($: - $) ;

$

' ;'0 ( - q ) ' ;

52 q ' ;

$

;'0 ;

5- q ;

52 q ;

$

; - ; 0 ; - ;


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@e now have the equation for calculating q

5alues of ;', ;

5and ;

$can be obtained from enthalpy-

concentration diagram for the mi%ture concerned.

! typical e%ample is shown in the 'igure below. /otethe regions for vapour only, liquid only, and vapour-liquid mi%ture.


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/ot all mi%tures havethe enthalpy-concentrationdiagram convenientlyavailable. /or is suchinformation easilyobtained. Thus, valueof q cannot be

calculated using theprevious formula. Theequation for q can bere-written as

!lt ti l f th f l b i t t d


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!lternatively, from the formula, q can be interpreted asthe heat required to convert mole of feed from itsentering condition to a saturated vapour?divided by themolal latent heat of vapori8ation. The above relationshipcan be illustrated using the temperature-enthalphydiagram shown in the 'igure below ased on thisdenition, we can

derive the formula forthe case whereby q < (cold liquid feed)and q B >(superheated vapourfeed).Thus, we have forcold liquid feed,

for superheatedvapour feed


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A. (nalysis o" tri$$ing ection


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A a ys s o $$ g ec o

!nalysis of *#$ is presented below using the 'igure belowwhich shows the stripping section of a distillation column.The re-boiled vapor in equilibrium with bottoms liquid

leaving the column. Material balance / 0#1T!ssuming constant molaloverow$:m 0 $:m2 0 .... 0 $: 0constant5:m 0 5:m2 0 ..... 0 5:0 constantMaterial balance#verall $: 0 5: 2 :M5C $: %

m0 5: y

m22

%

*ubstituting, and re-arrange in the form y 0f(%), we obtain

!gain dropping the subscripts m2 and m we


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!gain, dropping the subscripts m2 and m wehave

*ubstituting 5: 0 $: - we have the stripping operating line (

This is straight line with slope ( $: 6 $: - ) and intercept (

%6 $: - )

n addition, when % 0 % , y 0 %

, i.e. the operating line

passed through ( %, %

) on the +Dodiagonal line.

1sing the equilibrium diagram, the stripping section

operating line can be drawn and the number of theoretical

stages in the stripping section can be done in the same

manner.

"eminder The last sta e on the ra hical construction


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5suall' the S6L is the last line to dra", after both &6L

and #line are dra"n. $i!ing the &6L and the #line

automaticall' fi!es the S6L.

6n the completed design (e#uilibrium diagram): Thenumber of triangles dra"n 4umber of theoretical

tra's / &eboiler (last triangle).

feed plate locationcan

also be determined. 0n

the e!ample above, it is

Tra' 78.


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Thank You

Documents

McCabe-Thiele Method 1