EXTRACTIVE DISTILLATION CALCULATIONS BY MODIFIEDRELAXATION METHOD'

TAKESHI ISHIKAWA AND MITSUHO HIRATADepartment of Industrial Chemistry, Faculty of Engineering,Tokyo Metropolitan University, Tokyo, Japan

A modified relaxation procedure is developed for calculations of the multicomponentdistillation column. The method is applied to extractive distillation to calculate the product

distribution and the composition and temperature profile in the column when feed con-ditions, flow rates, number of plates, locations of feed plates, and reflux ratio arespecified.

The method is simple and numerically stable. Some problems are discussed to demon-strate the feasibility of the procedure.

Introduction

Extractive distillation is an important process for theseparation and purification of chemicals produced syn-thetically and in the separations of other closely boiling

components, and especially of azeotropic mixtures,which cannot be separated by ordinary distillation pro-

cedure.Broadly, extractive distillation is distillation in which

the volatilities of the key components are altered byaddition of a third component, the so-called solvent.Therefore, the system is composed of at least more thanthree components. Such a system deviates markedly

f

rom an ideal mixture.

Several methods have been proposed for solving non-ideal multicomponent distillation problems and ex-tractive distillation problems. These methods maybe classified into following three categories :

(1) Graphical solutions, namely, the Benedictmethod3^, Bonilla method4), Atkins method2), Lee

m

ethod8^ and Tanaka method16).

(2) Plate-to-plate calculation procedures, namely,The Lewis-Matheson method9) and the Thiele-Geddesmethod17), with a d method developed by Lyster10).

Hirose6) presented a calculational procedure forsolving non-ideal multicomponent distillation problemsby combining the Lewis-Matheson method with the

m

odified 6 method.

In recent years, Sadamoto13) developed a method forthe calculation of number of plates in an extractive

d

istillation column by using the modified 6 method.

Finally, (3) there is the analytical method proposedby Sugie et al.15\ This procedure applies the analyt-ical method developed by Acrivos1) to extractive dis-

tillation calculations.Received on August 25, 1971Presented at the 4th Autumn Meeting of the Soc.of Chem. Engrs., Japan, Oct. 1970

Graphical methods of type (1) are intuitive and lendthemselves to an easy understanding of the idea of

distillation calculations, but the procedures are rather

complicated and time-consuming. Moreover, in these

methods the calculations are carried out under thespecial assumption that solvent concentration in the en-riching section of the column is constant and determi-nation of the locations of the feed plates is approximate,hence the results obtained by these methods are not

always rigorous.On the other hand, the plate-to-plate calculation

methods of type (2) are excellent. However, in thesemethods the values of the liquid compositions in theenriching section of the extractive distillation column

are occasionally negative, because the initial values ofthe terminal compositions are not adequate. There-

fore these methods often lead to no solution, unless agreat deal of experience and judgement are applied in

choosing initial values. Also, the results in each ite-ration obtained from plate-to-plate calculationmethods have no physical meanings, and only theconvergence values are useful.

Finally, the analytical method of type (3) is also aninteresting procedure. The method, however, is com-

plicated in comparison with others and involves the

special assumption that solvent concentration in theenriching section of the column is constant. Further-more, it is questionable whether this analyticalmethod can be applied to systems deviating largely

f

roman ideal mixture.

In this paper, a new method is developed for cal-culation of the multicomponent distillation column.

The method was applied to extractive distillation forcalculating the product distribution and the composi-tion and temperature profile in the column when feedconditions, flow rates, number of plates, locations of

f

eed plates, and reflux ratio are specified.This method employs the modified relaxation algo-

VOL.5 NO, 2

972

'27' 125

f condenser

r -D. x D i

s, q s

2

s

�"s i

F , q P

s * 1

j

t - 1

f

Z f if +1

N

Fig. 1 A model for the extractive distillation column

rithm for the solution of non-linear simultaneousequations. Therefore, the method cannot be clas-sified into any one of the three categories described

a

bove. Weconsider that it belongs to a fourth type.This procedure, as described later, makes use of un-

steady state equations for the determination of thesteady state solution. Therefore, this method is equi-valent to having a pilot plant column and actuallyoperating it to obtain the answers. Also, problemsin the sellection of initial values are simple, and nospecial consideration for the least amount of compo-

nents involved in distillate and/or bottoms is necessary.

Mathematical Model

The extractive distillation column shown in Fig. 1isconsidered as an ideal model. This column has N

equilibrium plates, including a solvent feed plate and

a feed plate, and has a condenser at the top and areboiler at the bottom. For the convenience of cal-

culations the stages are numbered from top to bottom,with the condenser as zeroth stage and reboiler as theAM- lth stage.

Fig. 2 shows such an ideal equilibrium stage, ex-cluding condenser, solvent feed plate, feed plate andreboiler.

For the time period or interval from t to t-\-Jt, thecomponent-material balance on the yth stage asshown in Fig. 2 is given by the integral-difference

equation with the law of conservation of mass.

^[HPtjl^-lHjXtjl (1)

where j refers to the stage number and i refers to thecomponent number.

Hfj refers to the total moles of liquid holdup on the

////II

¥

¥¥

¥¥

¥I

J /

///

H ^ x.

VVH yi.M L j X 'J

Fig. 2 An ideal equilibrium stage in the column

yth stage and V and L refer to the vapor and liquidflow rates, respectively. The vapor holdup, H^, in Fig. 2 is neglected in thematerial balance because it is usually small relative tothe liquid holdup. By use of the mean value theorem of integral andthe mean value theorem of differential calculus, theleft-hand and right-hand sides, respectively, of Eq.(l)may be stated in the following form.

t+Jt [{Vj+1yitj+1JrLj-1xitj-1)-(Vjyij+LjXij)'}dt

=[(Vj+1yi,j+1+Lj-lxi)j-l)-{Vjyij+LjXij)lavdt (2)

Substituting Eqs.(2) and (3) into Eq.(l) yields

L dt-1 =[(VJ+1yitj+t+Lj-iXi,y_!)-{Vjytj+Ljxij)'\av

(4)

It is assumed that the liquid holdup on each stageremains constant with respect to time, that is,

dHl=,dt

(5)

Furthermore, by taking the limit of each term ofEq.(4) as At approaches zero, following differentialequation is obtained.

Hi~rr-) =t<yj+iVt.j+i+Lj-i*i.j-i)

dt /t. -(VjVij+LjXi^l

(6)

If all the compositions and the flow rates at time tareknown, it is then possible to calculate all the con-centration gradients, dXij/dt.

However, an implicit form of the finite difference

approximation for the differential equation isdx

~~dtij

_(%ij)t+Jt-{$i

t At

jit.(7)

F

or the convenience of calculation, consider time tto beiteration n and t-\-At to be iteration n-\-l. Then

a combination ofEqs.(6) and (7) leads toc»+i=xfj+ftjKVj+iyt,j^+Lj^Xt,y_!

-{Vjytj+LjX^T

where pj is equal to (AtjH1-) and the relaxationfactor. Eq.(8) is also the working equation of the

126 (28) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

procedure proposed by Rose13). For the condenser,

the solvent feed plate, the feed plate and the reboiler,respectively, similar but slightly different equations

canbe written as follows,xTi^xlt+PJL Vtfto-LoVu-DxDtY (9)x1+,1 =xrl,+ftsUSzSi+Vt+iyt,l+1+L,-1xitS.i)

~(Vsyis+Lsxis)T (10)xni}1 =xntf+(2/[(FzFi+Vf+1yi,f+1+Lf.1xiif-1)

-{Vfyif+Lfxif)T (ll)

-WxmT (12)

If the compositions and flow rates at any iteration(n) are known, Eqs.(8)~(12) may be used to calculatethe compositions at iteration (n+1), and the process

repeated to obtain values for iteration (w+2), and soon.In this calculation, the values of vapor composi-tions, yij, are obtained from the corresponding valuesof liquid compositions, Xtj, by use of the vapor-liquidequilibrium relation.

Method of Computations

Assuming constant molal overflow and that theliquid holdup is equal for all stages of the column, thevapor and liquid flow rates in each section of the col-umnare the following,

for the solvent recovery section,L=RD (13)V=(R+1)D (14)

for the enriching section,L" =RD+Sqs (15)V"=(R+l)D+S(l-qs) (16)

for the recovery section,L' =RD+Sqs+FqF (17)

V'=(R+l)D-S(l-q8)-F(l-qI) (18)

where R is reflux ratio.When the thermal conditions, the compositions and

flow rates offeeds, the amounts of the distillate stream,number of plates, locations of feed plates, and refluxratio are specified, calculations may be started usingfeed composition as the initial values of the liquid com-

positions for all stages of the column. The first stepis the use of the basic equation for the condenser tocalculate the compositions of the liquid on this stageat the first iteration. The second step is to use thebasic equation to calculate the compositions of the

f

irst stage at the first iteration.In the same manner the compositions at first itera-

tion are calculated for each of the other stages fromtop to bottom. Alternatively the calculations can

b

egin at the bottom and proceed upward.In this method, as the calculation for each stage is

independent of that for all other stages, the abovecalculations can start from any arbitrary stage and to

p

roceed upward or downward.

Now consider calculation from top to bottom. InRose's method, when the compositions of the 7th

stage at nth iterations, a??,y_i, had been used as thecompositions of the above stage. Therefore, the

method may be called mathematically a simultaneouspoint relaxation method.

This method is stable, but the convergence rate isgenerally too slow for practical use, and if feed compo-sitions are used as the initial values of the liquid compo-sitions for all stages of the column, further calculationcannot proceed because the correction term of theright-hand side in Eqs.(8)~(12) becomes zero.

Accordingly, to increase the rate of convergenceand to be able to apply for distillation calculations ofthe non-ideal multicomponents, wepropose the follow-

i

ng procedures.Instead of x7l>j^1 in Eq.(8), the values of the liquid

compositions of the above stage calculated at (^+l)th

iteration, that is, #?*/-i> were used as the composi-

t

ions of the j-lth stage. Consequently, Eq.(8) maybe written as follows.

'>iV =x'"<j+PjU Vj+1ylj+1 +LJ-1x'itj1-l) -( VjV*tJ+LjX*tt)] (19)

This is the basic equation for our method. Relativeto Rose's method this may be called a successive pointrelaxationmethod. It is as stable as Rose's method,

but the rate of convergence is more rapid. For thecondenser, the feed plates and the reboiler, an equationof this same kind can be written.

Alternatively, when the calculations are begun atthe bottom of the column and proceed upward, bymeans ofa manner similar to that described above, butslightly different, the basic equations were easily deriv-ed. Briefly, instead of y\j+1 in Eq.(8), y7Cj'+i areused as the vapor compositions of the y+lth stage.Therefore, Eq.(8) may be written as

;?}l =*?y+Ao[(V',+iy":y+i+£/-i*" y-i)

-<yjvi,+Ljx«ti)]

(20)

In this case, it is emphasized that the vapor compo-sitions, 2/i^+i, in equilibrium with the liquid, a??*/+i,

on the i+lth stage at (n+l)th iteration must beevaluated before Eq.(20) is applied to the 7th stage.Thus, for simplicity, calculation from top to bottom of

t

he column may be recommended.

For the calculation of the vapor compositions to bein equilibrium with the liquid, relation for the non-

ideal multicomponent vapor-liquid equilibrium hasto be used, because in extractive distillation a solventis added to increase the nonideality of the system. Thecalculations of such a system may be carried out by

t

he following procedure.In this system, the vapor-liquid equilibrium ratio

depends on the temperature, the pressure and thecompositions. Hence when the liquid compositions

are given, by use of the applicable activity coefficient-liquid composition relation, for example, the Wilsonequation18^ or the Margules equationll), the activitycoefficients were computed.

Then assuming a temperature and proceeding until(22/tj-']_) is less than a specified tolerance, the tem-

perature was corrected by the Newton-Raphson itera-tion method. For the purpose of decreasing the num-ber ofiterations in this calculation, the following device

VOL.5 NO. 2

972

[29] 127

S T A R T

R E A D

V ^ ^ l SｻSPECI F IC A -

by Eq sS ~14 | T IO N S

K = 1

X jj b y E q .19

IS! ｻrfi

n1i= W xfp b y E q .1 5

0 J> N *1

ye s

N O R M A L IZ Eo F x K+1i K = K *1

｣ i: by E q .2 0

& 2 b y E q .2 1

｣.i-tr< S. y esW R IT ER E S U

S T O P

Fig. 3 Flow diagram for the computer calculations

is used. The initial values of the temperature for thefirst stage calculated, for example at top or bottom ofthe column, is taken equal to the temperature definedby the relation

NCi1hi

(21)

where Tbi is a boiling point for the pure components consisting of the system at the pressure and x\ is theinitial values of the liquid compositions. If the con-verged value of the temperature was determined by means of the procedure interpreted before, that value is used as the initial value of the stage below and sameprocedure is continued.

Computational Procedure

Computational procedure may be written as follows.Step 1. Relaxation factor, fij, is given as the recipro- cal number of two to five times the summation ofamounts of solvent feed and feed per unit time, froma large number of trials to avoid divergence and in-stability in calculations, that is,

Pj~ (2~5)x(S+F) (22)

Step 2. Calculate the vapor and liquid flow rates ofeach section by use ofEqs.(13) to (18).Step 3. Assume an initial composition on each stage.It is convenient to use overall feed composition present-ed by the following equation as the initial values,

iJ S+F

(23)

Table 1 Specifications of Example 1

F 100.0 NT 10 V-L relationD40.0 NF 4 "CompV ~au TFi

W60.0 Total 1 3.0 0.3

q 1.0 condenser 2 2.0 0.4R 3.0 3 1.0 0.3

NT: number of total platesNF : feed plate numbered from top

Comp.: componenta: relative volatility

Step4. Evaluate vapor composition yá"j, that isin equilibrium with the liquid, x^j.Step 5. Solve the basic equations, Eq.(19), for x%y\using the successive relaxation method.Step 6. If the values of the liquid compositions deter-mined in Step (5) are negative because of over-correc-tion, to decrease the values of the relaxation factor theapplicable constant is multiplied by it and then thecalculation proceeds, using the new value in Step (5).On the other hand, if the value is not negative, goon to the succeeding step.Step 7. Normalize the liquid compositions deter-mined in Step (5).

Step 8. Repeat Step (4) through Step (7) unitil thefollowing equations are satisfied.

\

FzPi+Szsi-Dxlt- Wxlj 1

FzFt +SzSi

£Bu (24)

ATC N+l(2 2 [(^;i-iy/ii;i]!^! (25)

where £H and £2 are tolerances. Fig.3 shows the flowdiagram for computer calculations of such a procedure.In Fig.3, the repetitions for the component and the

stage were omitted, to avoid complexity on the dia-gram. Also, the flow diagram for calculation of thevapor compositions was omitted because of space limi-tations.

Illustrative Examples

This method has been programed for a FACOM270-30 digital computer at Computer Center ofTokyoMetropolitan University, and many test problemshave been successfully solved without relying on spe-cial experience.

The computer running time for the method is pro-portional to the number of components and the num-ber of stages in the column. For the problems tested,the computing time was about 0.04 seconds per com-ponent per stage per iteration.

Example 1In confirmation of availability of the new method,

firstly, the same problem as was described in our pre-vious paper14) was attempted.

The specifications of the problem are shown inTable1. Briefly, consider the distillation of an idealsystem consisting of three components. The columnhas 12 stages, including a total condenser and a re-boiler.

128 [30) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

Table 2 Results of Example 1

S t a e re N o . j Composition p r o fi le x S um x Composition p r o fi l e x S u m x

1 2 3 1 2 3

5 t h I t e r a ti o n 3 0 th I t e r a t io n

D is t i ll a t e 0 0.63572 0 .3 1 8 1 4 0 .0 4 6 1 4 1. 0 0 0 0 0 0.661 19 0 . 3 2 4 8 8 0 . 0 1 3 9 3 1 . 0 0 0 0 0

1 0.50803 0 .3 8 1 3 6 0 . 1 1 0 6 2 1. 0 0 0 0 1 0.55548 0.40941 0 . 0 3 5 1 1 1 . 0 0 0 0 0

2 0.43034 0 . 4 0 6 9 8 0 . 1 6 2 6 8 1. 0 0 0 0 0 0.46414 0.46490 0 . 0 7 0 9 6 1 . 0 0 0 0 0

3 0.37729 0 . 4 1 4 3 7 0 . 2 0 8 3 4 1. 0 0 0 0 0 0.38661 0.48600 0 . 1 2 7 3 9 1 . 0 0 0 0 0

F e e d 0.33940 0 . 4 0 8 4 0 0 . 2 5 2 2 1 .0 0 0 0 1 0 . 3 2 0 5 6 0 .4 7 1 1 7 0 .2 0 8 2 6 0 .9 9 9 9 9

5 0.33678 0 . 4 0 8 8 3 0 . 2 5 4 3 9 1. 0 0 0 0 0 0.29002 0 .4 9 8 2 5 0 .2 1 1 7 3 1 .0 0 0 0 0

6 0.33430 0 . 4 0 9 1 6 0 . 2 5 6 5 3 0 . 9 9 9 9 9 0.25597 0 .5 2 6 7 1 0 .2 1 7 3 2 1 .0 0 0 0 0

7 0.30002 0 . 4 3 6 0 0 0 . 2 6 3 9 8 1. 0 0 0 0 0 0.21901 0 .5 5 3 5 2 0 .2 2 7 4 7 1 .0 0 0 0 0

8 0.24841 0 .4 7 0 5 2 0 .2 8 1 0 7 1. 0 0 0 0 0 0.17978 0 . 5 7 2 7 4 0 .2 4 7 4 8 1 .0 0 0 0 0

9 0. 18694 0 . 4 9 3 8 4 0 . 3 2 0 2 2 1. 0 0 0 0 0 0 . 1 3 8 9 7 0 .5 7 3 2 7 0 .2 8 7 7 6 1 . 0 0 0 0 0

1 0 0 . 1 2 2 8 3 0 .4 7 6 9 1 0 .4 0 0 2 6 1. 0 0 0 0 0 0.09768 0 . 5 3 7 4 0 0 . 3 6 4 9 2 1 .0 0 0 0 0

B o t t o m s 1 1 0.06877 0 . 3 9 6 7 6 0 . 5 3 4 4 6 0 . 9 9 9 9 9 0.05857 0 . 4 4 6 6 2 0 .4 9 4 8 1 1 . 0 0 0 0 0

1 0 t h I te r a ti o n 4 2 n d I t e r a t io n ( F in a l r e s u l ts )

D is t ill a t e 0 0.68720 0 . 2 8 8 6 8 0 .0 2 4 1 2 1. 0 0 0 0 0 0.66105 0 . 3 2 5 2 5 0 .0 1 3 7 0 1 . 0 0 0 0 0

1 0.57623 0 . 3 6 3 1 0 0 . 0 6 0 6 7 1. 0 0 0 0 0 0.55551 0.40997 0 . 0 3 4 5 2 1 . 0 0 0 0 0

2 0.48362 0 .4 1 2 3 6 0 . 1 0 4 0 2 1. 0 0 0 0 0 0.46443 0 . 4 6 5 4 2 0 .0 7 0 1 4 0 .9 9 9 9 9

3 0.40077 0 .4 4 0 9 8 0 . 1 5 8 2 5 1. 0 0 0 0 0 0.38701 0 .4 8 6 4 1 0 . 1 2 6 5 8 1 .0 0 0 0 0

F e e d 0 . 3 2 7 8 6 0 .4 4 5 8 6 0 . 2 2 6 2 9 1. 0 0 0 0 1 0.32091 0 . 4 7 1 3 5 0 .2 0 7 7 5 1 .0 0 0 0 1

5 0.29284 0 .4 7 5 7 1 0 .2 3 1 4 5 1 . 0 0 0 0 0 0.29047 0 .4 9 8 4 2 0 .2 1 1 1 1 1 .0 0 0 0 0

6 0.252 17 0 . 5 0 8 4 6 0 .2 3 9 3 7 1. 0 0 0 0 0 0.25652 0 .5 2 6 9 6 0 .2 1 6 5 3 1 .0 0 0 0 1

7 0.20873 0 . 5 3 8 3 7 0 .2 5 2 8 9 0 . 9 9 9 9 9 0.2 1 9 6 2 0 . 5 5 3 9 8 0 .2 2 6 4 0 1 .0 0 0 0 0

8 0. 16499 0 . 5 5 6 9 9 0 . 2 7 8 0 1 0 . 9 9 9 9 9 0. 1 8 0 4 4 0 . 5 7 3 6 1 0 .2 4 5 9 5 1 .0 0 0 0 0

9 0 . 1 2 2 7 0 0 .5 5 1 7 1 0 .3 2 5 6 0 1. 0 0 0 0 1 0. 1 3 9 6 4 0 . 5 7 4 8 0 0 .2 8 5 5 6 1 .0 0 0 0 0

1 0 0.08344 0 . 5 0 5 8 6 0 . 4 1 0 7 0 1. 0 0 0 0 0 0.09832 0 .5 3 9 7 5 0 .3 6 1 9 3 1 .0 0 0 0 0

B o t t o m s 1 1 0.04852 0 .4 0 7 6 6 0 .5 4 3 8 2 1. 0 0 0 0 0 0.05907 0.44957 0 . 4 9 1 3 6 1 . 0 0 0 0 0

2 0 t h I t e r a t i o n Results o f S h o n o

D is t i ll a t e 0 0.66445 0 . 3 2 0 3 1 0 .0 1 5 2 4 1. 0 0 0 0 0 0.6613 0 . 3 2 5 0 0 .0 1 3 7 1 .0 0 0 0

1 0.55806 0 . 4 0 3 5 3 0 . 0 3 8 4 1 1. 0 0 0 0 0 0.5558 0 . 4 0 9 8 0 .0 3 4 4 1 .0 0 0 0

2 0.46468 0 . 4 5 9 8 2 0 .0 7 5 5 0 1. 0 0 0 0 0 0.4647 0 . 4 6 5 3 0 .0 7 0 0 1 .0 0 0 0

3 0.38604 0 . 4 8 2 1 0 0 . 1 3 1 8 6 1. 0 0 0 0 0 0.3872 0 .4 8 6 3 0 . 1 2 6 5 1 .0 0 0 0

F e e d 0.31997 0 .4 6 9 0 2 0 .2 1 1 0 0 0 . 9 9 9 9 9 0.3210 0 . 4 7 1 3 0 .2 0 7 7 1 .0 0 0 0

5 0.28918 0 . 4 9 5 9 2 0 .2 1 4 9 0 1. 0 0 0 0 0 0.2906 0 .4 9 8 3 0 .2 1 1 0 1 . 0 0 0 0

6 0.25487 0 .5 2 3 9 8 0 .2 2 1 1 5 1. 0 0 0 0 0 0.2567 0 .5 2 6 9 0 .2 1 6 4 1 . 0 0 0 0

7 0.21761 0 . 5 5 0 0 2 0 . 2 3 2 3 7 1. 0 0 0 0 0 0.2198 0 . 5 5 3 9 0 .2 2 6 3 1 .0 0 0 0

8 0 . 1 7 8 0 5 0 .5 6 7 8 5 0 .2 5 4 1 0 1. 0 0 0 0 0 0. 1806 0 . 5 7 3 6 0 .2 4 5 7 1 .0 0 0 0

9 0.13697 0 . 5 6 6 1 2 0 .2 9 6 9 1 1. 0 0 0 0 0 0. 1 3 9 8 0 .5 7 4 9 0 .2 8 5 2 1 . 0 0 0 0

1 0 0.09563 0 .5 2 7 3 6 0 .3 7 7 0 1 1. 0 0 0 0 0 0.0985 0 . 5 4 0 0 0 . 3 6 1 5 1 . 0 0 0 0

B o t t o m s 1 1 0 . 0 5 6 8 6 0 . 4 3 4 5 8 0 .5 0 8 6 7 1. 0 0 0 0 1 0 . 0 5 9 2 0 . 4 5 0 0 0 . 4 9 0 7 1 . 0 0 0 0

Fig. 4 Liquid compositionprofile for Example 1

D

r r r i

0

a c a a a . c ^jm a

1 o c a A > V ^ c m r xi

2

3

t 4

j K h C o m p o n e n t Ite ra t io n

N u m b e r

蝣jjt t i a a 5 t h

1 0 t h( F e e d ) ^ > o o ｫ f c a a A u a �" 3 0 t h

nE=> 5z :

F i n a lA > R e s u l t

5 6 a m

n 7a l l o o

8 a r

9 a j T

1 0 n ｫ ^ r

w agio a a * - 一 �" o o

I

1 0 2 0 3 0 A O 5 0 6 0

L i q u id C o m p o s i t i o n ( m o le p e r c e n t )

VOL. 5 NO. 2 1972 (3i; 129

Starting with feed composition used as initial valuesof the liquid compositions for all stages of the column,

a solution was obtained in 42 iterations. Table 2and Fig.4 give the computational results of this prob-

lem. In Table 2 liquid composition profiles xij9computed at the end of the 5th, 10th, 20th, 30th, and

last iterations. It is observed that xis changed onlyslightly after the 30th iteration.

Table 3 Variation for errors of the material balance ofeach component for Example 1

I te ra tio n E r ro r s

n u m b e r ｣ 2

5 3.89xlO"2 7. 2 3 x l O " 2 1 .4 2 x l O " 1 1 .5 7 x lO " 2

10 3 .9 1 x lCT 3 9 .5 5 x l O - 2 1 .2 3 x l O " 1 2 .2 0 x lO ~ 3

2 0 1. 3 4 x lO - 3 2 .9 4 x lO " 2 3 .7 8 x l O " 2 8 .9 3 x lO " 5

3 0 1 .14 x lO - 3 5. 3 4 x lO " 3 8 .2 6 x l O "3 3 .6 8 x l O "

4 0 6 . 1 3 x lO " 4 7 .5 6 x lO " 4 1 .6 2 x l O - 3 1 .3 6 x l O " 7

4 2 4.65xl(T4 3 .9 1 x lO " 4 9 .8 7 x l O - 4 5 .18 x l O " 8

e -\ft-(dt)ea-{Wi)ea\U ft

NC NT+1,X^J1-^jVe2-2j2j I t^)

Table 4 Specifications and physical data of Example 2

F= 1 . 0 q F = ¥ ,0 N T = 1 5 F e e d s c o m p o s it i o n s

S = 2 .0 G s = 1 . 0 N S = 3 2 ^ = 0 .9 ^ ^ 1 = 0 .0

D=0.5 R = 3 .0 N F = 1 0 z F 2 = O A z s 2 = 0 .0

W = 2 . 5 > = 0 . 1 z F 3 = 0 .0 2 5 3 = 1 .0

Marseilles p a r a m e t e r s a n d A n t o in e 's c o n s t a n t s u s e d

C o m p . A n t o i n e 's M a r seil le s

No. c o n s t a n t 1 7* p a r a m e t e r s 1 7 ^

B i

7 . 2 3 9 6 7 1 2 7 9 .8 7 0 1 2 3 7 .5 A 1 2 = A 2 1 =

0.2885 0 .2 5 0 1

7 .2 4 2 9 9 1 3 9 7 .9 2 9 0 2 3 8 .9

0.3197 0 . 0 8 2 8

7.96681 1688 . 2 0 9 9 2 2 8 .0 A 2 3 = A 3 2 =

0.0019 0 . 0 2 4 2

^ �"1 2 3 =

0 . 2 4 5 4

T

he results obtained by Shono14) are also shown in thelast columns of Table 2.

It can be seen from this table that the results of thiswork agree very well with those of Shono. Conse-

q

uently, it is confirmed that this new method is useful.

Table 3 shows the variation for the errors of eachcomponent in this problem. There are convergencedifferences depending on the component. As shown

i

n Fig.4, however, the results obtained at the end ofthe 30th iteration could be considered sufficiently ac-curate in practical use.

In this problem, whenSxj = 10~3and £2 = 1CT6, thecomputing running time, including the printout ofinput data and detailed results of the final iteration,

was about 30 seconds. When the same problem was

s

olved by Rose's method, 349 iterations were requiredfor a converged solution and the computing time wasabout 300 seconds.

E

xample 2

Nowconsider separation of acetone from acetone(l)-acetonitrile(2) mixture by extractive distillation using

water(3) as solvent. The problem specifications and

p

hysical data are given in Table 4.

Briefly, the column has 15 equilibrium plates witha total condenser and a reboiler. The solvent entersat the 3rd plate, and the feed enters at the 10th plate.

The column is operated at atmospheric pressure.Starting with feed composition used as the initial

values of the liquid compositions for all stages of thecolumn, and assuming relaxation factor is 0.1, a solu-tion was obtained in 42 iterations. Table 5 gives thecomputational results of this problem.

Conclusions

A newmethod for calculation of the multicomponentdistillation column has been developed. The calcula-

tion solves all unsteady-state equations simultaneouslyby a modified relaxation method. The method was

Table 5 Final results of Example 2

Sta ge N o . 3 T e m p .

[ - C ]C o m p o s i t i o n p r o f i l e x tj 2

C o m p o s i t i o n p r o f i l e 2V t j 3

(D ) 0 0.96723 0 .0 0 0 1 4 0 . 0 3 2 7 3

1 6 0 . 1 0 0.84949 0 . 0 0 0 1 9 0 . 1 5 0 3 2 0.96723 0 .0 0 0 1 4 0 . 0 3 2 8 3

2 6 7 . 6 5 0.59301 0 . 0 0 0 2 1 0 . 4 0 6 7 8 0.87902 0 .0 0 0 1 7 0 . 1 2 0 8 1

(S ) 7 9 . 4 5 0.25436 0 . 0 0 0 2 0 0 . 7 4 5 4 4 0.6 8681 0 . 0 0 0 1 8 0 . 3 1 3 0 2

4 7 9 . 4 2 0.25489 0 .0 0 0 4 1 0 . 7 4 4 7 0 0.68720 0 .0 0 0 3 7 0 . 3 1 2 4 3

5 7 9 . 3 6 0.25613 0 .0 0 0 8 5 0 . 7 4 3 0 2 0.68815 0 . 0 0 0 7 5 0 . 3 1 1 1 1

6 7 9 . 2 2 0.25898 0 .0 0 1 7 0 0 . 7 3 9 3 2 0.69035 0 . 0 0 1 5 0 0 . 3 0 8 1 6

7 7 8 . 9 0 0.26552 0 .0 0 3 4 3 0 . 7 3 1 0 5 0.69537 0 . 0 0 2 9 9 0 . 3 0 1 6 8

8 7 8 . 1 9 0.28091 0 .0 0 7 0 0 0 . 7 1 2 0 9 0.70687 0 . 0 0 6 0 1 0 . 2 8 7 3 4

9 7 6 . 4 4 0.32086 0 .0 1 4 8 4 0 . 6 6 4 3 0 0.73375 0 . 0 1 2 2 7 0 . 2 5 3 9 7

(F ) 1 0 7 1. 5 8 0 .44945 0 . 0 3 3 8 0 0 . 5 1 6 7 5 0.80369 0 . 0 2 5 9 8 0 . 1 7 0 3 4

l l 7 1 . 5 8 0.44918 0 . 0 3 3 8 9 0 . 5 1 6 9 3 0.80334 0 . 0 2 6 0 5 0 . 1 7 0 3 7

1 2 7 1 .6 2 0.44798 0 . 0 3 4 1 6 0 . 5 1 7 8 6 0.80283 0 .0 2 6 2 7 0 . 1 7 0 9 0

1 3 7 1. 7 7 0.44308 0 . 0 3 4 8 8 0 . 5 2 2 0 4 0.80013 0 . 0 2 6 8 7 0 . 1 7 3 0 2

1 4 7 2 .4 3 0.42296 0 . 0 3 6 6 9 0 . 5 4 0 3 5 0.78918 0 . 0 2 8 4 8 0 . 1 8 2 5 2

1 5 7 5 . 0 7 0.34736 0 . 0 4 0 2 5 0 . 6 1 2 3 9 0.74383 0 . 0 3 2 5 6 0 . 2 2 3 6 2

(W ) 1 6 8 3 . 8 4 0 . 1 6 6 2 4 0 . 0 3 9 9 9 0 . 7 9 3 7 7 0 . 5 7 3 7 3 0 . 0 4 0 5 7 0 . 3 8 5 7 1

130 :m JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

applied to the calculation of extractive distillationproblems.

T his method has the following advantage;(1) Computational procedure is simple and does notrequire a very large and fast computer.

(2) Each trial gives the exact answer for distillationunder the chosen conditions. Accordingly, using this

method is equivalent to obtaining the answers.(3) For multicomponent problems, the choice of

successive trial value is very simple, and no specialconsideration for very small quantities of a compo-

nent in distillate or bottoms stream is necessary.(4) If the real starting composition, holdup and cor-responding flow rate and time lags are used, the calcu-lations describe the approach to steady state.

(5) This method can be used to solve multiple-column problems where the columns are interrelated.

(6) The method converges relatively rapidly andstably.

Nomenclature

Aij - Margulesparameters [-]Ait Bit Ct~ Antoine constants [-]D å =distillate [mole/hr]F =feed rate [mole/hr]HJ = vaporholdup [mole]H*f = liquid holdup [mole]L = liquid flow rate [mole/hr]N = numbersof plates [-]R - refluxratio [-]S - solvent feed rate [mole/hr]T = temperature [°C]V = vaporflowrate [mole/hr]W = bottomproduct [mole/hr]q = thermal conditions offeeds [-]x =liquid composition, mole fraction [-]y =vaporcompostion, mole fraction [-]z =feed composition, mole fraction [-]

<Greek letters)y = activity coefficientAij = Wilsonparametersp. = relaxation factorA - increment£ = tolerance errors

( Superscripts )n -iteration numberL =referstoliquidV =referstovapor

0 = initial value/ = refers to recovery section

// =refers to enriching section(Subscripts)

1 =componentnumberj =stagenumber

t =timeF -refers to feed/ =referstofeedplateS =refers to solvent feed5 =refers to solvent feed plate

Literature Cited

1) Acrivos, A. and N. R. Amundson: Chem. Eng. Sci., 4,29 (1955)

2) Atkins, G. T. and C. M. Boyer: Chem. Eng. Progr., 45,553 (1949)

3) Benedict, 3VI. and L. Rubin: Trans. Am. Inst. Chem.Engrs., 41, 353 (1945)

4) Bonilla, C. F.: Chem. Eng. Progr., 37, 669 (1941)5) Hala, E.: "Vapor-Liquid Equilibrium Data", Per-

gamon Press (1968)6) Hirose, Y. and H. Hiraiwa: Kagaku Kogaku, 32, 998

(1968)7) Ishikawa, T. and M. Hirata: Kagaku Kogaku, submitted8) Lee, K. T. and K. Kammermeyer: Chem. Eng. Progr.

Symp. Ser., 49, 99 (1953)9) Lewis, W. K. and G. L Matheson: Ind. Eng. Chem., 24,

496 (1932)10) Lyster, W.N., S.L. Sullivan, D. S. Billingsley and

C. D. Holland: Petrol. Refiner, 38, 221 (1959)ll) Margules, M.,: Akad. Wiss. Wien, Math. Naturw. Kl.

II, 104, 1243 (1895)12) Rose, A., R. F. Sweeny and V.N. Schrodt: Ind. Eng.

Chem., 50, 737 (1958)13) Sadamoto, S., M. Kodama and M. Yorizane: Kagaku

Kogaku, 33, 557 (1969)14) Shono, H. and M. Hirata: Kagaku Kogaku, 33, 1236

(1969)15) Sugie, H. and I. Yamada: Kagaku Kbgakur 32, 273

(1968)16) Tanaka, S. and I. Yamada: Kagaku Kogaku {Chem.

Eng.. Japan), 28, 661 (1964)17) Thiele, E. W. and R. L. Geddes: Ind. Eng. Chem., 25,

290 (1933)18) Wilson, G.M.: J. Am. Chem. Soc, 86, 127 (1964)

VOL. 5 NO. 2 1972 (33) 131