Coker A.K. Fortran Programs for Chemical Process Design, Analysis, and Simulation

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Mass Transfer 517

bottoms products is specified by setting limits on two key components

between which it is desired to effect separation. Also, fixing one com-

ponent composition does not uniquely determine the other component

compositions and the stage temperature.

Key Components

Before the start of the column design, the process designer must

select the two key components between which it is desired to effect the

separation. In cases where pure components are being produced, the

key components are the compounds boiling adjacent to one another on

the temperature scale. The material having the lower boiling point is

known as the light key component and the next heavier as the heavy

key component. The light key component is the component that is

desired to keep out the bottoms product, and the heavy key is the com-

ponent that is desired to keep out the top product. Specifications are

often set on the maximum concentrations of the keys in the top and

bottoms products.

The keys are known as "adjacent keys" if they are adjacent in a list-

ing of the components in order of volatility. Sometimes, the keys

selected are not adjacent but have an intermediate boiling component

between them. They are known as "light key," "heavy," key and "inter-

mediate" (boiling) or "distributed key." The non-key components that

appear in both top and bottoms products are known as "distributed"

components. Those that are not present to any significant extent, in one

or other product, are known as the "non-distributed" components.

Equations Surveyed

The increasing availability of personal computers with acquired simu-

lation software packages has helped design engineers to optimize a num-

ber of equilibrium stages in multipurpose fractionating towers and

absorbers. Designers adept in mathematical modeling are still engaged

in the use of vigorous, iterative plate-to-plate computation to study a

wide range of process conditions. However, preliminary design with

graphical correlations has often helped to arrive at an approximate opti-

mum number of stages before reverting to established design methods.

Figure 7-15 shows a schematic of a fractionating column with two or

more multicomponent and associated equipment items. Alternatives to

the preliminary design are the short-cut methods to achieve a realistic

optimum number of theoretical stages. Generally, short-cut methods were

proposed to establish the minimum number of stages at total reflux, and

518 Fortran Programs for Chemical Process Design

ENRICHING

SECTION

FEED

STRIPPING

SECTION

V~ L o

i___r.-~, L

~ (--r- 2

(--f- 3

~]i][[]"

-]--1

-I---r-'

- t---g- 2

(--rl

---- - 5

v k g

7 s ---II ,

c[i

CONDENSER [

I T

W=ATER

DISTILLATE

- I

1REFLUX

:J'~ ACCU MULAT OR

9 , V

0 ~ T 1

REFLUX PUMP

STEAM

.L -~

Q ldr ....

I r- ....

T"E O'LE t T

.,.

BOTTOMS

Figure 7-15. Factionation column with a reboiler and refluxing auxiliaries.

also the minimum reflux at an infinite number of theoretical stages.

Many authors have correlated the minimum reflux and minimum stages

with operating reflux and corresponding theoretical equilibrium stages

required. These methods have enabled designers to vary the reflux ratio

and plates in order to achieve an optimum relationship based on invest-

ment and operating costs.

Mathematical Modeling

The distribution of components between the distillate and bottoms is

given by the Hengstebeck-Geddes equation [35,36].

[di]

log bi - Ai + B~ log

(7-94)

where d~ = moles of component i in the distillate

b~ = moles of component i in the bottoms

(zi = relative volatility of component i

A~, B~ = correlation constants

Mass Transfer 519

A material balance for the

i th

component in the feed is:

fi = di + bi (7-95)

The quantity of component i in the distillate can be expressed as mole

fraction recovered, or d~/fi. Alternatively, in the bottoms, the mole frac-

tion of component i recovered is bi/f i.

If Equation 7-94 is expressed with respect to the heavy key compo-

nent, then

IdHK

log ~HK -- A~

+ B 1

log

0~HK

(7-96)

The relative volatility of the heavy key component (that is, 0~HK) is 1.0

Therefore,

A~ - 1og[~UKl

dHK

(7-97)

but

fHK = dHK + bHK

(7-98)

and

fLK-" dLK + bLK

(7-99)

Therefore

If IlK

bHK ]

A,- log -bH K

(7-100)

Equation 7-100 can be expressed in terms of mole fraction recovered as:

A, -log[ (1" 0 - bHK/fHK) ]~-b-~K)~HK;

(7-101)

Substituting Equation 7-97 into Equation 7-94 and expressing in terms

of the light key component.

1og[~LLKKI--1Oglb~KKI+B , 1og~LK

(7-102)

520 Fortran Programs for Chemical Process Design

Therefore

B 1

,ogl/ dLK

log O~LK

Expressing Equation 7-103 in terms of fractional recoveries.

dLK/fLK bHLK/fnK

log fix - di~K fnK - dnK

fLK f.~:

B~ =

log ~'L~

(7-103)

(7-104)

B 1 ~--

log dLK/fLK bHLK/fnK

1 dtx 1 bnK

fix fnK

log CZLK

(7-105)

The recoveries of the i 'h component in the distillate and bottoms"

Equation 7-94 and Equation 7-95 are

E di ]

log ~ - A 1 +B~log~

(7-94)

fi - di + bi

(7-95)

Expressing Equations 7-94 and 7-95 in terms of the recoveries of the

i th component.

That is

di/f i

log 1 - d i/fi

- A~ + B 1 logc~

(7-106)

Hence,

Mass Transfer 521

I d~/f~ ]

_ 10A,. 10(log,,,a,~,)

1-di/f i

l0 A,

-- .(Xi

(7-107)

di = 10A'fi

" i~B'(

1- ~d~)

The recovery of the i ~h component in the distillate is given by

(7-108)

di = (10A,. @B,) (7-109)

-~- lOAl . 0~, B' )

And the recovery of the

i th

component in the bottoms is

bi =

d i

(7-110)

fi fi

B 1

= 1- (10A'" ~a ) (7-111)

(1+10 A'.c~ )

b i _ 1

(7-112)

fi 1 + 10 A' . ~i

The Fenske's Method for Total Reflux

Fenske's [37] equation for determining the minimum equilibrium

stages at total reflux was based on an ideal mixture. This suggests that

the ratio of vapor pressures or the ratio of equilibrium vaporization of

the key components is constant over the range of temperatures (that is,

the relative volatilities are constant). Fenske expressed the minimum

number of equilibrium stages as:

,og[/XL 1

Nmin =

XHK D XLK B (7-113)

522

Fortran Programs for Chemical Process Design

The Gilliland Method for Number of Equilibrium Stages

The number of theoretical equilibrium stages required for a given

separation at a given reflux ratio is often determined by empirical corre-

lations [38,39]. The abscissa, X, represents a reflux function as:

(7-114)

Correspondingly, the ordinate, Y, represents a stage function as given by

y _ (N - N~ ) (7-115)

(N+I)

The ratios used for the axes of abscissa and ordinate were chosen

because they provide fixed end points for the curve (X = 0.0, Y = 1.0

and at X = 1.0, Y = 0.0). The two functions were found to give good

correlations. These correlations are shown in Figures 7-16 and 7-17.

Gilliland's correlation has produced relevant results that offer the

following advantages:

9 They represent an optimum solution concerning the location of the

feed plate.

9 The splitting for the two key components is verified.

9 The maximum deviation using Gilliland's correlation in terms of

tray number is within the7% range [39].

The Gilliland correlation tends to be conservative for feeds with

low values of the thermal condition of the feed (q). It can give inaccu-

rate results when there is a large difference in tray requirements above

and below the feed. The correlation is adequate for preliminary designs

before a detailed analysis is carried out. Great caution should be

exercised when used for final designs. Hines and Maddox [5] have

described an improved correlation for determining the number of theo-

retical stages.

Underwood's Method

If an infinite or nearly infinite number of equilibrium stages is

involved, a zone of constant composition must exist in the fractionating

column. In this instance, there is no measurable change in the composi-

tion of liquid or vapor from stage to stage. Under these conditions, the

reflux ratio can be defined as the minimum reflux ratio,

Rmin,

with

Mass Transfer 523

1.0

r I

ili~i

8 I

,-- O.

.+.

z 0.6

I-,-.I

I--,

=,.,

0.4

,,.,...

l--

iV3 9 ,.--

O.2

,,...

0.0 J

0.0

..-,,,

9 (i,

",t'

L L

t l l

0.2 0.4

0.6 0.8 1.0

REFLUX FUNCTION (R-Rm)I(R+I)

Figure 7-16. Linear coordinates based on Gilliland's correlation.

respect to a given separation of two key components (that is, the light

key and heavy key) [40].

Therefore, for component i, in the distillate

Rmi n -+- 1

- ~ (O('i -- Xi'D)

(7-116)

where 0 is Underwood's constant (or root of the equation), and must lie

between the relative volatilities of the light and heavy keys ((ZLK and

0~HK ). The number of components is n.

Correspondingly, for the feed,

(~i ~ Xi, F )

1-q - ~)

i=l (O('i --

(7-117)

524

Fortran Programs for Chemical Process Design

1.0

0.8

0.6

0.5

,- 0.4

" 0.3

E 0.2

o 0.1

0.06

u_ 0.051

0.04

1--.

I J 4

: ! 1 tt I I t

! r

!11!1!ti It

t t tt tt)

, s II t Ill

~ -;-' ,'* ' - - ' ' . .1 ~ . . ' !

t ; f ': 0 l i ~

: I

, , , [

. . t , . , . 1 , . ,

[

0.03 - ! ~ ~~ !~'!!~ J ~ !! ! 1, t !., ~ J1~!

i i.

~ ~ ] i~! i il~! ! ! i i l!J! l!

0.02t

' , s i i II! i]!i ..! ! !..i i tiii]..j

]

0.01 0.02 0.03 0.05 0.08 0.1 0.2 0.3 0.5 0.8 1.0

REFLUX FUNCTI ON (R-Rm)

/ (R+I)

Figure 7-17. Logarithmic coordinates based on Gilliland's correlation.

Equation 7-117 shows the relationship for the feed, where q is the frac-

tion of feed that is liquid at the feed tray temperature and pressure. For

a bubble point feed, q = 1, and for a dew point feed, q = 0. The minimum

reflux ratio is determined from Equation 7-117 by substituting into Equa-

tion 7-116. Coker [41] developed a numerical method for computing

0 and

Rmi n

respectively. However, other methods should be tried, if

Rmi n

gives a negative value. Also, it may be that the separation between

the feed and the overhead can be accomplished in less than one equilib-

rium stage.

Equations for Describing Gilliland's Graph

Many equations have been proposed to describe Gilliland's curve for

multicomponent distillation. However, the difficulty with some of these

equations has been in meeting the end conditions of X = 0, Y = 1 and

X = 1, Y = 0. A review of the many equations proposed by these authors

is as follows:

Mass Transfer 525

Proposed Equations:

1. Hengstebeck [42] (1961)

log Y- A + B(log X) + C(log X) 2 + D(log X) 3 + E(log X) 4 (7-118)

where A =-1.3640187

B =

-3.0920489

C = -3.407344729

D = -1.74673876

E =-0.33268897

2. Liddle [43] (1968)

For 0.0 < X < 0.01, Y - 1.0 - 18.5715X

(7-119)

For 0.01 < X < 0.90, Y - 0.545827 - 0.591422X

+ 0.002743/X

(7-120)

For 0.90 < X < 1.0, Y - 0.16595 - 0.16595X

(7-121)

3. Van-Winkle and Todd [44] (197 l)

For 0.0078 < X < 0.125, Y - 0.5039 - 0.5968X

- 0.0908(log X)

(7-122)

For 1.25 < X to X - 1.0, Y- 0.6257 -0.9868X + 0.516X

- 0.1738X 3 (7-123)

4. Molokavov [45] (1972)

0 _< (x,Y) < 1.0

y_ l_exp[ / 1+54.4X )/X-1)I

ll+ii i k x

5. Hohman and Lockhart [46] (1972)

(7-124)

X - 0, Y- 0.65

X - 1.0, Y- 0.067

0.65 - 0.50X

1.0+ 1.25X

(7-125)

526

Fortran Programs for Chemical Process Design

6. Eduljee [47] (1975)

X =0, Y= 1.0

X= 1.0, Y = 0

Y = 0.75 - 0.75X 0.5668

(7-126)

7. Huan-Yang Chang [48] (198 l)

For X = 0, Y = 1.0

X = 1.0, Y-0

Y - 1 - exp(1.49 + 0.315 X

1.805 )

k-~ (7-127)

8. Harg [49] (1985)

For X = 0, Y = 1.0

X= 1.0, Y = 0

Y = 1 - X 1/3 (7-128)

9. McCormick [50] (1988)

X=0, Y= 1.0

X= 1.0, Y = 0

y= 1-X B

where

B = 0.105(log X) + 0.44

(7-129)

From the equations listed, McCormick's gives a good agreement in the

normal operating range of real towers. The reflux ratio, R, is calculated

as a multiple of the minimum reflux ratio,

Rmi n-

That is,

R = FACTOR.Rmi n

(7-130)

The multiplier "FACTOR" generally varies from 1.2 to about 1.5 for

conventional columns, but because of economic situations, the range is

now between 1.05 and 1.20.