Liquid Extraction, First Edition, McGraw-Hill Book Company, Inc

>

ft

?OU_1 68043 g

UJ

*

,

This bp* should be rm Z.O/VJL

=

,

and

B

(3.42)

effects of the various molec-

Thus,

xA x B k AB

+ .

x B xA k BA 7

+

~T X B X A X A K BAA

+

xA xA x B kAAB x A x B x A k ABA 7 X B X A X B K BAB -f- XA X B X B K ABB x B x B x A k BBA ,

+

j

+

(3.43)

where the subscripts of the constants A; indicate merely the type of concenSince order of multiplication is unimtration term associated with each. = x k k x x x x kAAB = xA x B xA kABA) etc., and therex x B A B B AB A A A BA} portant, fore

23RT

(3.44)

LIQUID EXTRACTION

50 or since XA

+

=

XB

1.0,

+

B

Letting

and

3k ABBxA x%

(3.45)

AAB = 2kAB

+

3k ABB

(3.46)

ABA = 2k AB

+

3k AAB

(3.47)

rr "'"

=

2ZRT

XAXB ^XA ^ BA

+ ^AAB)

(3.48)

-+-

Since

UB

,

and

nA

nB

;

= XB

therefore

n B )FE 2.3#!T

nA n B + n*

=

Dijfferentiation in accordance

-

]

(3.50)

yB

=

xiU^ +

2(^ A5

- ^^)xj

(3.51)

three-suffix

TA log TB

= =

=

1),

log

A^B = Similarly,

Margules equations, which, since XA

ABA =

~ A AB )xl + - A BA )x\ +

(2A B A (2A AB

+

XB

=

1

,

--

2(A AB 2(A BA

- A BA )xl

(3.52)

AAB)S?A

(3.53)

-

for the constants

A = ABA

0(x B

AAB)XA

(5)

A 5 = A^

=

with Eq. (3.40) then leads to

2(A BA

Or they may be solved

^

'

+

log

If

.

^

xl[A AB

also be written

,

\

=

log

may

n B A AB nA + w

n A A BA + n*

log V A

and These are the

I

riA

(XB

(XA

-

-

-

X A ) log 7A -^ XB

+

x B ) log 7 B

log y A + ~~~~

,

21og-y/j -

(3.54)

XA

2

(3 55) '

y A = A^ 5 or ,

y A as X A > 0, or terminal value of log y Al or log y'A limit of log

limit of log

y B as z*

-

0, or

terminal value of log y B) or log 7^

1

( \

)

,

,

j

As examples

of the use of these equations to fit activity-coefficient data, refer to Figs. 3.5 and 3.6, where the activity coefficient-mole fraction

curves drawn are given by Eqs. (3.52) and (3.53), with the indicated values of the constants A AB and A BA .

may be worked out on the basis of different assumptions the size of the molecular groups which affect the excess free respecting Other equations

PREDICTION OF DISTRIBUTION

51

energy and the relative magnitudes of the molar volumes of the constituThe most important of these for present purposes

ents of the solution. are:

The

two-suffix van

Laar equations

(5,

23)

:

AAB -

log

7*

[1

+

[I

+

=

(3.58)

j

(3.59)

;

or (3.60)

(3.61)

Figures

and 3.4 show examples

3.2, 3.3,

of the use of these equations to

activity-coefficient data.

fit

The

three-suffix

Scatchard-Hamer equations

log

yA

=

log

yB

=

4

AAB

+

2 (ABA

ZA\

ABA

+

2

f

AAB

(5,

30, 31, 32, 35):

VA ~ A AIi

A

(3.62)

^ - ABA)ZB

(3.63)

y^

)z

or

=

(3.64)

-^log7*-

(3.65)

where

z

= volume

V =

fraction based on the volumes of the pure components molar volume of the pure constituent t*

+ V Bx B

VA xA + V7 B x B

,

w\

(,0.00)'

In each of these cases, differentiation of the equations shows that they are true integrations of the Gibbs-Duhem equation, and the values of AAB and ABA are the terminal values of log y A and log JB [Eqs. (3.56) and More complex equations with constants other than the terminal (3.57)].

7 values are also possible (35). Redlich and Kister (29) have shown that

log

/^-l (3.67)

/

JAX-Q

and that 108

fe) 5(1

-

(3.68)

LIQUID EXTRACTION

52

where

M

data.

A

,

N, and S are constants may be used to represent experimental plot of log (TA/TB) against X A

is

consequently a convenient

smoothing and correlating vapor-liquid equilibrium data. The constants may be determined from such a plot very readily, and the net area between the curve representing the data and the line log (JA/JB) = must, by Eq. (3.67), equal zero. If S = 0, the equation becomes identical

method

of

with the Margules equations previously given. 1. The Margules equations are quantitatively most useful Limitations. for relatively symmetrical systems, i.e.j where AAB nearly equals ABAThey are very flexible in that they can show maximum or minimum values

7f with concentration. Differentiation of the equation shows that Wohl has (or minimum) can be expected ABA/AAB > 2.0. for such the that of shown, however, dissymmetry Margules equasystems tions fail quantitatively to follow the observed data and that a better criterion for the probable appearance of a maximum is when ABA/ AAB > 2 + of log

a

maximum

It is then clear that for systems of considerable dissymmetry (2.3 A BA/4). without maxima the Margules equations will be inapplicable. If AAB = ABA) the equations reduce to the simple forms

log 7A

= A AB x2B

,

log

7*

= A BA xA2

(3.69)

The van Laar equations can satisfactorily follow data showing high values of AAB and ABA and greater dissymmetry than the Margules equations, but very large ratios of ABA to AAB cannot be handled. They fit cases where ABA/ AAB is in the neighborhood of 2 better than the Margules equations. These equations are more limited in the shapes of curves that can be obtained, however, in that they cannot show maxima or minima If ABA = A A/?, or change in sign of log 7 with changing concentration. 2.

and consequently for nearly symmetrical systems there made between the van Laar or Margules equations. For unsymmetrical systems beyond the capabilities of the van Laar

Eq. (3.69) is little

3.

results,

choice to be

equations, the Scatchard-Hamer equations, although less convenient, are If VA = VB, they reduce to the Margules equations, while if better. AAB/ABA = VA/VB, they reduce to those of van Laar. Carlson and Col-

burn (5) consequently suggest that the ratio of VA/VB may be taken as a guide as to which of the equations are applicable. For systems that cannot be handled by any of these, the more complex equations suggested by Wohl (35) may be tried. 4. As has been pointed out, the equations are all integrations of the

Gibbs-Duhem relationship. They consequently cannot be applied to systems which when treated in the ordinary fashion apparently do not follow this basic relation, as in the case of dissociation of electrolytes in solution. t If log

yA shows a maximum, then log 75 shows a minimum at the same concentration.

PREDICTION OF DISTRIBUTION Effect of Temperature.

temperature ships

(5, 18,

is

27)

The dependency

53

of activity coefficient

upon

readily derived from the basic thermodynamic relation:

6T

(370) (6 7()) '

RT*

Jp

HA = partial molal enthalpy of component A in soln. HQA = molal enthalpy of pure component A at the same may be simplified to

where This

Affs

d(l/T)

2.30312

^"'7n; ,

AH 3 =

A similar equation partial molal heat of solution of A. be written for component B. At infinite dilution, Eq. (3.71) becomes

where

may

d logy A

temp.

dA AB ^ d(l/T)

where A//s

=

heat of solution of

A

Affg 2.303fl

at infinite dilution.

,~

v

^''72 } Plotting log 74

against l/T at any value of XA should therefore produce a curve whose = 0, to the heat slope is related to the differential heat of solution, or, at XA is solution infinite dilution. of at Conversely, it possible to estimate the

temperature on the activity coefficients, and for this purpose Eq. (3.72) is most useful. Unfortunately, not too many heat-of-solution data are available. Ordinarily, solutions which exhibit positive deviations from Raoult's law are formed from their constituents with an absorption of heat. AH& is

effect of

For positive, therefore, and 7^ will be smaller at higher temperatures. mixtures with negative deviations, the AHs is ordinarily negative. In both cases, therefore, the solutions ordinarily more nearly approach Raoult's law as the temperature is increased. Obvious exceptions to this rule are systems with lower critical solution temperatures, where, at least

neighborhood of the lower C.S.T., the Raoult's law deviations become greater with increasing temperature. Benedict, et al. (2), and many others have included the effect of tem-

in the

perature as T log 7 when expressing 7 as functions of x. At values of X A = 0, this quantity becomes TA^s (27), which leads to the simple approximation

This simplification cannot be general but could serve as a useful approximation if more reliable data are lacking. Berg and McKinnis (3) have proposed another empirical method for so-called "regular" solutions. Data for a typical system are shown in Fig. 3.7, where the activity coefficients for ethanol in

aqueous solution at 20, 40, 55, and 75C.

(21),

LIQUID EXTRACTION

54

as well as values calculated from constant-pressure vapor-liquid equilibria 1 atm. over the temperature range 78.3 to 100C. (22), are plotted.

at

Clearly in this system the variation of activity coefficient with temperature 7.0 r

0.4

0.2

0.6

0.8

1.0

Mole fraction ethanol FIG. 3.7.

Effect of temperature

on activity

coefficients of ethanol in

aqueous solutions.

not great. This will be the case for most systems, and for present purposes the temperature dependency can be frequently ignored. Figure 3.8 shows the values of log yA at XA = 0, or AAB, for this system as a function is

of temperature in accordance with Eq. (3.72). Illustration 4.

From

the value of

AAB

at

40C.

for ethanol

(A)-water (B) and heat-

of-solution data, estimate the value of AAB at 55C., and compare with observed data. Solution. AAB at 40C. (Fig. 3.7) = log 5.90 = 0.7709. "International Critical

Tables" (Vol. V) lists integral heat-of-solution data for ethanol in water for 0, 17.33, and 42.05C. as a function of ethanol concentration. Extrapolation of these data to zero concentration of ethanol results in the following heats of solution at infinite dilution:

PREDICTION OF DISTRIBUTION From Eq.

55

(3.72),

/^r

=

AABI

d

Jl

T] \T

2.303ft

00305 .'

A AB at 55C. =

0.7709

+

1

2.303(1.987)

F

003195

0.90

0.85

0.80

0.15

0.70

0.65

0.60 2.6

2.8

3.0

3.2

3.4

3.6

3 Reciprocal temperature,0/o K )x 10

FIG. 3.8.

Temperature dependency of AAB

for ethanol (^l)-water (J5).

integration is performed graphically by plotting &H'S against 1/T, Fig. 3.9, and determining the area under the curve between 1/T = 0.00305 and 0.003195. The area is 0.270 cal./(gm. mole)(K).

The

.'.

=

A AB at 55C. =0.7709+-

0.829,

(XA

0)

which

=

6.75.

compared with 0.8195 and 6.60,

0.270

2.303(1.987) corresponds to JA at

These the

results

may be

observed

values,

resp.

Moderate temperature changes in such minor changes in

result

activity coefficient that constantpressure data are ordinarily satisfactory for application of the

various integrated

forms of the

Gibbs-Duhem equation.

Heat of solution of ethanol in water at infinite dilution.

FIG. 3.9.

LIQUID EXTRACTION

56

Applications of the Integrated Equations. The usefulness of the Gibbsequation for establishing the thermodynamic consistency of, and The various integrated forms for smoothing, data has been pointed out.

Duhem

most useful for extending limited data, sometimes from even measurements, and it is these applications that are most important

are probably

single for present purposes.

1. Calculation of Activity-coefficient Curves from a Single Vapor-Liquid Datum. If a single vapor-liquid equilibrium measurement has been made (pi, x, and t/), the activity coefficients may be readily calculated by Eqs. (3.14) or (3.15), concentrations and activity coefficients may be substituted in the appropriate Gibbs-Duhem equation, and the constants AAB and ABA

The

computed.

integrated equations

may

then be used to calculate activ-

ity coefficients of each component of the solution over the entire concentraGreatest precision will result if the single datum is known at a tion range. x between approximately 0.25 and 0.75. value of special case, which frequently arises, is knowledge of the composition, temperature, and pres-

A

sure of an azeotrope.

For an azeotrope, x

=

The

?/,

recent indexed compilation of Horsley (17) data of this sort.

and consequently

is

most useful

for obtaining

An azeotrope is reported (17) for the system ethyl acetate (A)-ethanol Illustration 6. (B) as follows: 760 mm. Hg, 71.8C., 30.8 wt. per cent ethanol. From this, calculate the van Laar constants and activity coefficients for the system. Compare these with activity from complete vapor-liquid equilibria at 1 atm. 30.8 wt. per cent ethanol = 0.46 mole fraction ethanol.

coefficients calculated

Solution.

xB XA

0.460

=

1

0.46

=

0.540 mole fraction ethyl acetate

At 71,8C., the vapor pressure of ethyl acetate = PA = 631 mm. Hg, and that of = PB ~ 581 mm. Hg. p = 760 mm. Hg. By Eq. (3.17), at the azeotrope

ethanol

t

---"

By

Eqs. (3.60) and (3.61):

The van Laar

equations [Eqs. (3.58), (3.59)] are therefore

0.4029 1

log

TA

__

0.3848

,

i'

g 7B

-

PREDICTION OF DISTRIBUTION

57

a comparison of activity cofrom these equations with those calculated from the vapor-liquid data of Furnas and Leighton [Ind. Eng. Chem. 29, 709 (1937)] at 760 mm. Hg.

Figure 3.10

is

efficients calculated

2.

Calculation of Activity-coefficient

Curves from Boiling Points of Solutions or Isothermal Total-pressure Data.

Frequently, boiling points or total pressures of solutions as a func-

tion

of

concentration

are known,

without information concerning the concentration of the equilibrium vapor.

If

the appropriate integrated

equation can be chosen, it is not difficult to determine the activity coefficients of

each component over the

entire concentration range

by followa ing procedure suggested by Carlson and Colburn (5). From Eq. (3.15),

FIG. 3.10. Activity acetate-ethanol.

for ethyl = O, azeotrope. data of Furnas and Leighton. [Ind. Eng. Chem. 29, 709 (1937).] Curves calculated

X

coefficients

=

D

.

from azeotrope.

(3.74)

and (3.75)

Since, as XB approaches 1.0,

75 also approaches

1.0,

then as a

first

approxi-

mation, (3.76)

Similarly, (3.77)

Knowing the boiling points of solutions dilute in component A permits calculation of approximate values of 74. By plotting these on semilogarithmic will give an approximate value of the curve paper against XA, extrapolation

JA at XA = 0, or AAB. Similarly, a value of ABA can be obtained. Substitution in the appropriate integrated Gibbs-Duhem equation permits better estimation of the activity coefficients in the dilute range, and in this of

way, by successive approximations, obtained.

may

The

reliable values of

AAB and ABA

are

activity coefficients over the entire concentration range

then be calculated.

LIQUID EXTRACTION

58 Illustration 6.

"

International Critical Tables

solutions of ethanol (A)-toluene (B) at

"

32.3C. as

(Vol. Ill) reports total pressures of follows:

Calculate the van Laar constants from these data. Solution.

At 32.3C., p A

=

219.5

mm. Hg, p B = Pt

n, as a

first

approximation.

-

93.0

mm. Hg.

Eq. (3.76):

93.0SB

Therefore, approximate values of 74

may

be computed as

follows:

The approximate values of 7 A are plotted on semilogarithmic paper against ZA, and by ex7A at (ZA=O) is 6.65. A trial value of AAB = log 6.65=0.8228.

trapolation, the value of Similarly,

approximate values of 75 are computed from Eq.

(3.77):

In a manner similar to that for 7A, a trial value of ABA is found to be log 5.88 = 0.7694. trial values of AAB and ABA are tested by computing 7*8, using the van Laar equations, and with these calculating p from the relationship

The

t

Pt

The

= yApAXA

results of these calculations are

+

:

For most purposes, the computed values of p agree satisfactorily with the observed data, and the trial values of AAB and ABA may be considered final. Better values may be obtained by repeating the calculations, using Eqs. (3.76) and (3.77) together with the values of 7^ and 75 computed above. t

PREDICTION OF DISTRIBUTION

59

Calculation of Activity Coefficients from Compositions of Liquid and method Vapor at Known Pressures, without Knowledge of Temperature. 3.

A

suggested by Carlson and Colburn (5). As a first approximation, a temperature is found by a>-=urnmg that Raoult's law applies to the component present in the larger amount. This temperature permits a first approximation of the activity coefficients of the component present

of calculation

is

amount, and plotting these on semilogarithmic coordinates permits approximation of AAB and ABA- Substitution in the appropriate integrated form of the Gibbs-Duhem equation will give more reliable activity coefficients, and by successive approximations final values of AAB and

in lesser

a

first

ABA

are obtained.

Calculation of Activity Coefficients from Mutual Solubility Data (5, 7, As the components of a solution become more and more dissimilar 31). chemically, their solutions show greater and greater deviations from 4.

Large positive deviations ultimately lead to partial immisThus, in the systems alcohol-water, we observe that methyl, cibility. and ethyl, propyl alcohols are completely miscible but with increasing deviations from Raoult's law. Butyl alcohol, with very large deviations when mixed with water, is only partially soluble. Similar observations can be made in the ketone-water systems, with acetone-water showing moderate deviations and complete miscibility, while methyl ethyl ketone and water show a solubility gap and large positive deviations. In any two phases at equilibrium, including two liquid phases, the fugaciProvided that the ties of each component are the same in the two phases. same standard state for a substance is chosen for its condition in each phase, the activities of the substance in each phase are also equal. Thus, Raoult's law.

aAA

=

aAB,

asA

_ JAB

XBA XBB

=

ass

(3.78)

Consequently,

XAA XAB

.

yAA

of concentrations between the solubility limits the apparent activity coefficients will vary inversely as the concentrations based on the mixture as a whole. Elimination of y's between Eq. (3.79) and any of the

Over the range

Gibbs-Duhem equations therefore permits the estimation of AAB and ABA from the mutual solubility. Thus, the Margules equations integrated lead to

logg? AA

A ** = T~TT.

^15 ABA

or/^.2

T7

__2

\^/^.3

_^3

\i

w

/A~.

\

i

zz* XAB ^

(3 -

\

*!

2 T Ik^A r

/

\_ ^JS/ _2

*

8

)

(3.81)

LIQUID EXTRACTION

60

The van Laar equations

give

logf^ (3.82)

1

I

+

A A BX A A\

AAB ABA XBB

log

i\l#BA

+

(3.83)

The Scatchard-Hamer equations

__

L

2x AA x A B log --XAA

XBA

/2A BA VA

2

_

4^]

XBAXBB log

~

give

~

_

2

tt

(

ABAVA [\LI A -

(3.84) z

l\

"

:!+K^-^)-2(^-^)]iog^

A AB

(3.85)

In Eqs. (3.80) to (3.85), the concentrations are those of the equilibrium saturated solutions. In each case, the second of the equation pairs per-

mits calculation of AAB/ A B A from the data, and the first then gives AAB. Colburn and Schoenborn (7) have given a graphical solution of the van Laar

A

similar graphical solution could be worked out for the Margules equations, but the additional parameter of VA/VB in the Scatchard-Hamer set.

equations would

For systems

make such a method awkward.

moderately symmetrical solubility, i.e., where XAB is AAB will nearly equal ABA, and either the Margules or van Laar equations would be expected to be applicable. For moderate dissymmetry, the van Laar equations ought to be better, whereas for strong dissymmetry the Scatchard-Hamer equations should be used. As the mutual solubility decreases, these equations show larger and larger values of

nearly equal to XAA

of

}

AAB and ABA and

correspondingly increasing activity coefficients.

XAB and XBA = XBB, Eqs. Hildebrand (15) has shown, (3.80) to (3.85) become indeterminate. 2 2 however, that at the critical point (d In a)/dx and (d In a)/dx both equal zero. Applying these criteria to the van Laar equations, as an example,

At the

critical-solution point,

there result

since XAA

=

PREDICTION OF DISTRIBUTION

~

=

4^ ABA

61

o* "^A

(3-86)

*!*' %A

and 5.862(1

-AAB

which then system the

is

\ 2

/2 __

- sA )t Q _ #2)

be used with critical-solution data.

may

(,o.o/;

For example,

symmetrical, so that the critical-solution composition

AAB = ABA =

is

x

if

=

the 0.5,

0.868.

by these methods agree

Activity coefficients calculated

fairly well for

systems where the original equations, i.e., Eqs. (3.50) to (3.66), apply. Carlson and Colburn (5) and Colburn, Schoenborn, and Shilling (8) have shown that the van Laar constants cannot be calculated from solubility data for n-butanol-water and isobutanol-water, but the van Laar equations satisfactorily describe the activity coefficients obtained from vaporliquid data in these systems either.

do not

The mutual solubility of methyl ethyl ketone (A) and water (B) at and 90.1 wt. per cent ketone (21). Calculate the constants AAB and ABA for the system, and compare with those calculated from vapor-liquid data. At 18.6% methyl ethyl ketone and a. Calculation from solubility data. Solution, 81.4% water, the mole fractions of ketone and water are 0.054 and 0.946, resp. At 90.1% ketone and 9.9% water, the mole fractions of ketone and water are 0.695 and Illustration 7.

40C.

are 18.6

0.305, resp.

Therefore,

Water layer: Ketone layer: Since the solubility

Eq.

_ XAB XAA

is

(3.83):

=

XBB XBA

0.054,

0.695,

0.946 0.305

moderately unsymmetrical, the van Laar equations

A

0.054\/

70.695

V0.305

0.946/1

2(

0.695^0.054 +

0.305

-"

will

be used.

0.054V

10g

"*"

-_^ . ABA

= =

0.695 \ 0.305

.

1

.

.

!

860

0.695)(0.054)logg;g|

0.946

(0 305)(0 .

.

946)log

O_||

Eq. (3.82): 0.054

.

10

= , AAB [1

+

~ 1.86(0.695/0.305)J

2

[1

+

1.86(0.054/0.946)]*

ABA " i:si = 781 y A at (XA =0) = 28 51 ys at (XA = 1) = 6.04 '

-

These correspond to

Othmer and Benenati Und. Eng. Chem. 37, 6. Calculation from vapor-liquid data. 299 (1945)] report vapor-liquid data for the system at several total pressures. At 200 mm. Hg, the temperatures most closely approach 40C., and these will be used. The

LIQUID EXTRACTION

62

the reported temperatures and compositions, and activity coefficients calcu-

table

lists

lated

by Eq.

(3.15).

A semilogarithmic plot of yA and ya against XA shows that at XA \ogy A = AAB = 1.462; and at X A = 1, ys = 8.2, and log -ys = ABA compare very favorably with the results of the solubility calculations. Illustration 8. is

The

yA

=

and These

29,

0.914.

methyl ethyl ketone (A) and water (B) Calculate the van Laar constants at this

critical-solution point for

45 wt. per cent ketone, at 150C.

(21).

temperature.

=

At 45 wt. per cent methyl ethyl ketone, XA Solution. Substitution in Eq. (3.86):

AAB _ ABA

(1

-

0.1696)'

2(0.1696)

-

5.862(1

A

Eq. (3.87):

(2 .'.

0,

=

ABA

-

1.258

0.1696)

-

3.125

=

(0.1696)

2

[1

2

3.125

0.1696)*

-

0.1696 mole fraction ketone.

(0.1696)

= 2

1.258

]

0.402

Comparison of these constants with those of the previous example

will illustrate

again

the effect of temperature on actiyity coefficients.

NONIDEAL TERNARY MIXTURES The law

general principles established for ideal solutions, such as Raoult's

any number Gibbs-Duhem equation is applicable to any number of components, and as in the case of

in its various forms, are of course applicable to solutions of

of components.

Similarly, the

nonideal solutions of

binary mixtures various relationships can be worked out relating the This problem has now been activity coefficients for ternary mixtures. attacked from several points of view, a most excellent summary of which is presented by Wohl (35). His most important results pertinent to the

problem at hand are summarized here. Two-suffix Ternaries Composed of Three Symmetrical Binaries. equations for this case, worked out by Benedict, et al. (2) are:

VA log y B log y c

log

= A AB XB = AABXA = AACXA

+ A ACx*c + x Bx c (A A B + A A c - ABC) + A B cXc + xA xc (A A B + ABC A AC) + ABC^?B + XAx B (A A c + ABC AAB)

The (3.88) (3.89) (3.90)

PREDICTION OF DISTRIBUTION The constants

in these equations are those obtained

63

from the end values which is

of the activity coefficients of the three binary systems, each of

symmetrical:

AAB =

= AAC = = 4i?c

= =

and XB and XA

JA as XA log 75 as X B

limit of log

limit of

74 as XA limit of log 7c as XG

limit of log

>

1

(3.91)

and Xc*l and XA 1 and z? 1 1 and X B

> >

limit of log y B as X B limit of log 70 as XG

> 1

* >

(3.92)

(3.93)

Since there are no additional constants which are characteristic only of the ternary systems, then knowledge of the three binary systems alone permits calculation of the ternary-mixture activity coefficients. It is most

important to note that, for even this most simple of situations, ordinary interpolation of the binary-solution data is not possible. Simple interpolation of the log 7*8 would be valid only if the AAB Asc, AAC, and terms in the equations were omitted. If, as is ordinarily the case, the constants are positive, values of log 7 will be lower than the arithmetic average of values from the binaries. Linear interpolation of log 74 will be valid only if the B-C mixtures are ideal. Two-suffix van Laar Equations. These are perhaps the next most simple equations

:

The

expressions for log ys can be obtained by a "rotation" principle, by substituting subscripts B for A, C for B, and A for C throughout the equation. Thus in the equation for log 75, XB in Eq. (3.94) is changed

i.e.,

to x c

,

(3.94)

in

to ABC, A AC to ABA, etc. Similarly, in that for yc, x c in Eq. to to etc. The constants are again defined ABA XB, changed AAC,

AAB is

terms of binary systems only:

AAB =

i-

=

J ! limit of log

/ICB

.

f

i

limit of log

74 as XA yB as x^

>

>

"^

=

-* g 74 7c as x c > limit of log 7fl as X B -

=

limit of log

c CA

limit of log

=

Wohl has shown

7c as Xc

^

0,

A 0,

J 0,

XB

1

au

Binary if 1

c

A-B

j

^

I

} 1

XA

0,

^-

1

0,

XB

1 } J

>

Binary

4-C

fi

fl _

(3.95)

J

that these equations are limited to those cases where

ABC

\AAC\AB.

(3.96)

LIQUID EXTRACTION

64

a limitation which may not be too serious. As in the previous case, the ternary data can be predicted from information on the binary systems alone. Equations Containing Ternary Constants. In the more complex cases, constants derivable from ternary data are included in the equations, which are in turn more generally useful. Thus, log y A

=

zl\A AB

+

+

2z A (A B A

~

A - A AB )}

+

2

z c [A AC

qn

z B z c [A BA

9A qs

q + + A AC - A CB -

+

2z A (A CA

2z c (A CB

-

^

- A AC )}

qc qA -

-

AAC)

qc

qc

+

2z A (A CA

- ABC ~ -

C(l

-

2z A )]

(3.97)

qn

qc

If, in this equation, various values are assigned to the "q fractions," the equations resulting are as follows:

-~ qB

~ qB

24 qB

=

-==r

VB

- -

-^r

Vc

qc

= -j^; A BA 24 qc

=

;

-

= qc i; z

-

:

AC ~:

AA CA

three-suffix

Scatchard-Hamer equations

three-suffix

van Laar equations

stricted

by

replaced

by Eq

.

[use re-

(3. 96)]

x: three-suffix

Margules equations

In each of these cases, equations for log y B and log yc can be obtained from in accordance with the rotation prinThe A constants are defined by Eq. (3.95), ciple previously described.

Eq. (3.97) by changing subscripts

and a ternary constant C, requiring at

least one ternary measurement, is In the absence of ternary data, Colburn (6) has suggested as an approximation that C can be estimated in the following manner:

included.

C = %[(A B A - A AB + )

(A AC

- A CA + )

(AcB

-

ABC)}

(3.98)

The

general applicability of these equations may be inferred from the discussion of the corresponding binary equations.

More complex equations containing additional constants are also preby Wohl. A convenient graphical approximation of the ternary data from binaries is offered by Scheibel and Friedland (33). sented

PREDICTION OF DISTRIBUTION be recalled, as pointed out at the beginning of the chapter, that desirable to be able to predict the distribution of a solute between The discussion partially miscible solvents from a minimum of data. It will

it is

now

offers a means of attacking this problem in fairly sysIn what follows, B is considered the extracting solvent for removing a distributed substance C from A-C solutions.

presented above tematic fashion.

PREDICTION OF DISTRIBUTION As previously

indicated, provided that the

65

same standard

state

is

chosen

for a substance for its condition in each phase, the activities of the substance in each phase at equilibrium are equal: O,AA

=

GAB,

O.BA

=

QBB

(3,99)

Also, O>CA

The

=

=

O>CB

ycBXcs, etc.

(3.100)

general procedure which presumably could be followed would be to

establish values of activity coefficients in the three binary systems (A-B A-Cj and B-C) from data of the sort previously described, using the binary integrated forms of the Gibbs-Duhem equation to assist in extending meager y

data as necessary.

From

these, predic-

tions of activity coefficients

and

activi-

the ternary systems can be made use of the ternary integrated Gibbs-

ties in

by

Duhem equations. Equilibrium ternary liquid layers then exist where activities of all three components are Reference to

equal.

Fig.

3.11

will

make

the last step clearer. Here there is shown a typical ternary liquid-phase diagram for a Type 1

system, with an heterogeneous area

MKPLN

and a typical

tie

line

KL.

FlQ 3

n

Act i vitie8

in ternary systems.

A, B, and C in the solution at K must equal those of A, B, and C, respectively, in the solution at L. Lines VK and LR represent solutions of constant activity of C. Similarly, TK and LS are solutions of constant activity of B, and UK and L those of constant activity of A By means of the ternary activitylocate points such as and L to it should be possible coefficient equations

The

activities of

W

.

K

where the three constant activity curves intersect. In this fashion it should be possible to locate both the tie lines and the solubility curve. Unfortunately, the activity-coefficient equations cannot conveniently be made explicit in terms of x, and the location of the constant activity curves on the triangular diagram is possible only by a lengthy series of interLocation of the triple intersection points becomes an even more polations. While this can be done, for practical difficult trial-and-error procedure. purposes use of the ternary activity-coefficient equations is ordinarily limited to cases where the solubility curve of the ternary liquid system is known. For such a situation, the calculations become relatively simple, since it is then merely necessary to compute activities of C along the solubility

curve and to join equal values on opposite sides of the curve by the

tie lines.

LIQUID EXTRACTION

66 Illustration 9.

Predict the distribution in the

Type

1

system water (A)-ethyl acetate

20C.

(C), at

(B)-thanol

Solution. Ethyl acetate (B)-ethanol (C). Refer to Illustration 5 and Fig. 3.10. The vapor-liquid data at 1 atm. of Furnas and Leigh ton (loc. cit.) are plotted as activity coefficients. Smooth curves drawn through the 5

(

1

1

1

points and extended to the 7 axes give

1

1

ACB = ABC =

4.0

log 2.20 log 2.30

= =

0.3424 0.3617

(NOTE: These are not the constants calculated from azeo tropic data.)

fethanol 3.0

Water (A)-ethanol (C). Vapor-liquid data at 20C. are available in " International Critical Tables," Vol. III. These have been calculated in the form of activity coefficients and are plotted in Fig. 3.12.

2.0

Extension of the curves to the 7 axes gives

S'-

ACA =

5

A AC =

= =

0.6334 0.3802

Water (A)-eihyl acetate (B). Solubility data for this binary are available in "InAt ternational Critical Tables," Vol. III.

1.0

02

0.4

06

08

20C., the saturated solutions contain 7.04 and 96.99 wt. per cent ethyl acetate. These

1.0

Mole fraction ethanol, x c

Fia. 3.12. water,

log 4.30 log 2.40

Activity coefficients for ethanol-

20C.

concentrations correspond to the following

mole fractions: XBA XBB

= =

0.01738, 0.8679,

= =

XAA XAB

0.9826 0.1321

Using the van Laar equations, Eq. (3.83):

AAB ABA 0.9826

0.01738

Eq.

0.565

0.1321 0.1321

2(0.9826) (0.1321) log

+ 0.8679

09826

(3.82):

0.1321

g

AAB

0.9826

1.030

0.565

Summary: ACA =* 0.6334 AAC - 0.3802

=

=

1.788

ACB - 0.3424 ABC = 0.3617

AAB =

1.030

-

1.788

A^

1.030

PREDICTION OF DISTRIBUTION

67

Eq. (3.96):

ACB ^ IAcA\IAAB\ ABC \AAC/\ABA) 0.3424

=

0.3617 0.948

/0.6334\/1.030\

\0.3802/\L780/

=

0.960

This is sufficiently close to an equality that the two-suffix ternary van Laar equations can be used [Eq. (3.94)]. XA '

A CA

log TC

Substitution of the constants leads to

0.3815x|

6

'

(xc

+ 0.601XA +

1.055s*)

1

(/. Chem. Soc. 1938, 67) provide ternary-solubility data for this system at 20C. In the table below, their weight fractions have been converted to mole Activities ac are calculated from fractions and yc calculated from the above equation.

Beech and Glasstone

the relation ac

A plot of (ICA vs. XCA and of OCB vs. XCB and XCB are read at constant values of

is

shown

activity:

in Fig. 3.13.

Equilibrium values of XCA

LIQUID EXTRACTION

68

These are plotted as the curve on Fig. 3.14 and compared with the measured equilibrium data of Beech and Glasstone (loc. tit.) plotted as points. The agreement between predicted and observed data is seen to be very satisfactory.

Wafer layer Predict the distribution

Illustration 10. 0.20

I

in the

2 system w-heptane (A)-aniline (B)-cyclohexane (C), at 25C.

Type

n- Heptane Solution, (A)-aniline (B). Mutual-solubility data are available from

0.15

/*Esfer layer

Hunter and Brown



93.20 wt. per cent heptane, XAA 6.80 wt. per cent aniline, XBA

I'0.05

= =

0.9272

= =

0.0607

0.0728

Aniline-rich solution: 010 0.15 0.05 Mole fraction ethanol, x c

020

G.50 wt. per cent heptane, XAB 93.50 wt. per cent aniline, XBB

0.9393

FIG. 3.13. Calculated activities of ethanol in the system water (A) -ethyl acetate

Substitution of these data in the van Laar

(B)-ethanol (C).

equations [Eqs. (3.82) to (3.83)] give

AAB =

,,.,,.

ABA =

1.359,

,

*

,,

1.290

Mutual-solubility data of Hunter and

Aniline (B)-cyclohexane (C). are used: at 25C., Cyclohexane-rich solution:

T

,

83.50 wt. per cent cyclohexane, Xcc 16.50 wt. per cent aniline, XBC

Brown

(loc. tit.)

0.8488

=

0.1512

= =

0.2820

'

Aniline-rich solution:

26.20 wt. per cent cyclohexane, XCB 73.80 wt. per cent aniline, XBB

The van Laar equations

0.7180

[Eqs. (3.82) to (3.83)] give

ABC = n-Heptane (A)-cyclohexane (C). this system nearly ideal.

ACB = 0.867

1.120,

No

data are available, but

it is

not unreasonable to

assume

ACA =

A AC =

The same ternary van Laar equation simplification,

it

is

^

0,

=

1.0

used as in Illustration

9.

After substitution and

becomes g yc

_ "

1-4473%

~

(XC+XA

0.1691x^xg

+

1.292**)'

Ternary-solubility data at 25C. of Hunter and Brown (loc. tit.) are used in the of Illustration 9 to calculate activity coefficients and activities.

manner

PREDICTION OF DISTRIBUTION

0.04 0.06 0.10 0.02 0.05 Mole fraction ethanol in woifcr layer, X CA FIG. 3.14.

69

0.12

Distribution of ethanol between water and ethyl acetate,

20C.

LIQUID EXTRACTION

70

Figure 3.15 shows the plot of acA vs. XCA and acs vs. XCB, from which equilibrium values of XCA and XCB were read at equal values of activity, as follows:

XCA

XCB Figure 3.16 shows these data as a curve, compared with the measured tie-line concentraHunter and Brown. While agreement is not so good as in the previous illustration, it should be noted that the entire prediction was based only on solubilities in two of tions of

the three binaries. Furthermore, there is some question concerning the accuracy of the aniline-heptane solubility since it is not in exact agreement with the measurements of Varteressian and Fenske [Ind. Eng. Chem. 29, 270 (1937)]. For use in these types of calculations, very accurate data are desirable.

/ /

H

I

->

H

JA.

- H

I

hydrogen can coordinate between nitrogen or fluorine, as well as " " and these are termed donor atoms. Hydrogen can also coordioxygen, nate between any of the donor atoms and carbon, provided that there is a Similarly,

sufficiently effective negative

grouping attached to the carbon.

Some

of

these bondings are strong and others comparatively weak. Solution of a substance into one which associates through hydrogen bonding may involve either breaking the hydrogen bonds or forming new

Thus, if an alcohol dissolves in water, the hydrogen bonds between water molecules and between alcohol molecules may be broken and new ones.

LIQUID EXTRACTION

80

bonds formed between alcohol and water molecules; we

may

obtain a con-

figuration of the following sort:

H

R

H

H

I

I

I

I

H O -> H

H

->

->

H

R - H

I

If the size of the hydrocarbon chain of the alcohol is great, the water-towater hydrogen bonding is so strong that the barely negative portion of the The alcohol then does not dissolve alcohol is relatively unimportant. An excellent description of the role of the hydrogen bond in appreciably. such cases is provided by McElvain (26).

Ewell, Harrison, and Berg (12) have classified liquids into five groupings,

based on their possibilities of forming hydrogen bonds. Their listing is reproduced in Table 3.2. They have also pointed out that when liquids are mixed, positive deviations from Raoult's law result if hydrogen bonds are broken, negative if they are formed. The extent to which the hydrogen bonds are involved will, in general, indicate the extent of the Raoult's law deviations. Table 3.3 reproduces their summary of these effects.

TABLE

3.2.

CLASSIFICATION OP LIQUIDS ACCORDING TO Class

HYDROGEN BONDING*

I

Liquids capable of forming three-dimensional networks of strong hydrogen bonds, e.g., water, glycol, glycerol, ammo alcohols, hydroxylamine, hydroxy acids, polyphenols, amides, etc. Compounds such as nitromethane and acetonitrile also form three-dimensional networks of hydrogen bonds, but the bonds are much weaker than those involving

OH and NH

Therefore these types of compounds are placed in Class

groups.

II.

Class II

Other liquids composed of molecules containing both active hydrogen atoms and donor atoms (oxygen, nitrogen, and fluorine), e.g., alcohols, acids, phenols, primary and secondary amines, oximes, nitro compounds with a-hydrogen atoms, nitriles with a-hydrogen atoms, ammonia, hydrazine, hydrogen fluoride, hydrogen cyanide, etc. Class III

Liquids composed e.g.,

of molecules containing

donor atoms but no active hydrogen atoms, amines (including pyridine type), nitro

ethers, ketones, aldehydes, esters, tertiary

compounds and

nitriles

without a-hydrogen atoms, Class

etc.

IV

Liquids composed of molecules containing active hydrogen atoms but no donor atoms. These are molecules having two or three chlorine atoms on the same carbon as a hydrogen atom, or one chlorine on the same carbon atom and one or more chlorine atoms on adjacent carbon atoms, e.g., CHC1, CH 2 C1 2 CH 8 CHC1 2 CH 2 C1 CH 2 C1, ,

CH C1 CHC1 CH C1, CH C1 CHC1 2

2

2

2,

Class All other liquids,

i.e.,

liquids

,

etc.

V

having no hydrogen-bond-forming

capabilities, e.g.,

hydrocarbons, carbon disulfide, sulfides, mercaptans, halohydrocarbons not in Class IV, nonmetallic elements such as iodine, phosphorus, and sulfur. * Ewell, Harrison, and Berg, Ind. Eng. Chem. 86, 871 (1944). Society.

With permission of the American Chemical

PREDICTION OF DISTRIBUTION TABLE

3.3.

Classes

I+V +V

DEVIATIONS PROM IDEALITY BASED ON HYDROGEN BONDING* Deviations

Hydrogen bonding

Hydrogen bonds broken only

Always

+ IV

Hydrogen bonds formed only

Always

I

II

+IV + IV

Hydrogen bonds both broken and

Always

+

+

deviations; I V, quently limited solubility

II

III

81

fre-

deviations

+

+

deviations; I IV, quently limited solubility

formed, but dissociation of Class I or II liquid is the more impor-

fre-

tant effect

I

+1 + 11

I

+III

II

+ 11 + III

I

II

Hydrogen bonds both broken and formed

give

+ III No hydrogen bonds involved +V IV+IV

III III

IV

+

deviations, very compliUsually deviations cated groups, some

some maximum azeotropes

Quasi-ideal systems, always ations or ideal; azeotropes, will

+ deviif

any,

be minima

+V

V +V *Ewell, Harrison, and Berg, Ind. Eng. Chem. 36, 871 (1944).

With permission of the American Chemical

Society.

It will be noted that for all mixtures of Class III, IV, and V liquids, with the exception of Class III with Class IV, no hydrogen bonds are involved. Hildebrand (15) and Scatchard (30) show that in such cases deviations

from Raoult's law are a function

of the square of the difference in square

roots of the internal pressures of the constituents, the importance of this effect being minimized when hydrogen bonding is extensive. Large deviations for such solutions, in other words, result from large differences in interThe internal pressure, in turn, can be estimated from the heat of vaporization and molar volumes:

nal pressures.

P.

where p*

&EV

AHV R T

V

= = = = = =

=

41 .3 AS,

V

41.3(Ag,

- RT) (3.118)

V

internal pressure, atm. internal energy of vaporization, cal./gm. enthalpy of vaporization, cal./gm. mole

mole

the gas const., 1.987 cal./(gm. mole)(K) abs. temp.,

K

molar vol. of the liquid, cu. cm./gm. mole Table 3.4 is a short list of internal pressures for substances included in Class III, IV, and V liquids, arranged in order of magnitude of internal pres-

LIQUID EXTRACTION

82 TABLE

The enthalpy

3.4.

INTERNAL PRESSURES AT 25C.

used in calculating these were taken if necessary by usual methods or estimated 28) (21, two farther are on the list, the greater should The substances (19). apart be the deviations from ideality of their solutions.

sure.

of vaporization

from various tabulations

Recommend a solvent potentially useful for extracting acetone from aqueous solutions. Acetone is a Class III liquid and water a Class I liquid (Table 3.2). AqueSolution. ous acetone solutions can be expected to give positive deviations from Raoult's law (Table 3.3). For a most favorable distribution coefficient, therefore, solutions of acetone in the extracting solvent should show negative deviations. Table 3.3 indicates that liquids of Class IV, such as certain of the chlorinated hydrocarbons, should provide favorable distribution coefficients. Halohydrocarbons of Class V would also be satisfactory, since their acetone solutions would show at most weakly positive deviations. Illustration 15.

its

Typical distribution coefficients are indicated in the following tabulation:

Consideration other than distribution coefficient alone must be kept in mind before a solvent is chosen, however, as indicated in Chap. 4.

PREDICTION OF DISTRIBUTION Illustration 16.

83

Predict the direction of distribution of acetic acid between benzene

and water. Solution. Acetic acid (Class II)-benzene (Class V) solutions should show strong Acetic acid positive Raoult's law deviations, or large escaping tendency for the acid. (Class II)-water (Class I) solutions should show less strong positive deviations, or a lesser

escaping tendency for the acid. On distribution, the acetic acid should favor the waterrich phase. This is confirmed by the data of Hand (14), at 25C., which show a distribution coefficient of 30.4 in weight per cent units at low acid concentrations, favoring the

water phase. Illustration

Predict the direction of the distribution of 1,6-diphenylhexane

17.

(Class V) between furfural (Class III) and docosane (Class V), at 45C. Since in both furfural-diphenylhexane and docosane-diphenylhexane soluSolution. tions,

hydrogen bonds are not involved, recourse can be had to internal-pressure data to

indicate the distribution.

Mol. wt.

Furfural:

=

Density at '

^=

A#v = ,

96.03

45C. =

96 03 t TOO l.loo

84.8 cu. cm./gm. mole

12,140 cal./gm. mole at

[Eq. (3.118)]

Mol. wt.

Docosane:

=

1.133

=

45C.

(estd.)

V40-

=

41.3

=

[1 1,218 atm.

273)

1^87(45+

J

310.36

Density at 44C. = 0.7782 V = 391 cu. cm./gm. mole AHV = 15,440 cal./gm. mole at 45C. /. p t [Eq. (3.118)] = 1,565 atm. .*.

1,6-Diphenylhexane:

= 238 Density at 45C. = 0.95 (estd.) /. V - 251 cu. cm./gm. mole A#v = 16,250 cal./gm. mole at 45C. /. pt [Eq. (3.118)] = 2,575 atm.

(estd.)

Mol. wt.

Ap for diphenylhexane-furfural = 2,575 Ap for diphenylhexane-docosane = 2,575

-

(estd.)

1,218 1,565

= =

1,357 atm. 1,010 atm.

+

.*. The furfural solutions should show the larger deviations, and diphenylhexane should favor the docosane-rich phase on distribution. This is confirmed by Briggs and Comings [Ind. Eng. Chem. 35, 411 (1943)] whose data

at

45C. show

resp., at

distribution coefficients of 1.021

and 2.72

in

weight and mole per cents,

low concentrations of diphenylhexane. Notation for Chapter 3

A -

constant in Margules, van Laar, and Scatchard equations

= limit of log 7 as x 0. = component of a solution. a = activity. B = component of a solution. = second virial coefficient of the equation = (RT/pt) + B, cu. cm./gm. mole. t;

of state

LIQUID EXTRACTION

84

C

component of a

solution.

= constant [Eq. (3.97)]. D = component of a solution. d = differential operator. &EV = internal energy of vaporization, cal./gm. mole. e = base of natural logarithms = 2.7183. F free energy, cal./gm, mole. F = partial free energy of a substance in solution, cal./gm. mole. F = total free energy of a solution, cal. FB = excess free energy of a nonideal solution, cal./gm. mole. F# = total excess free energy of a nonideal solution, cal. / = fugacity of a pure substance, atm. / = partial fugacity of a substance in solution, atm. 7/ = enthalpy of a pure substance, cal./gm. mole. H = partial enthalpy of a substance in solution, cal./gm. mole. &Hs ~ partial heat of solution, cal./gm. mole. A//v = enthalpy of vaporization, cal./gm. mole. = constant. In = natural logarithm, = common logarithm. log M = constant. m = distribution coefficient = ratio of concentrations of solute A;

MW

liquid phases.

=

N

molecular weight. constant.

n = number of moles p p Pi

q

R r

S

T V v

x

X

= = = = = = = = = = = = =

y

Z = = = ft 7 = d = z

of a

component.

constant [Eq. (3.111), (3.112)]. pressure; vapor pressure of a pure substance, atm. partial pressure of a component of a solution, atm. internal pressure, atm. arbitrary factor for Eq. (3.97). universal gas constant, cal./gm. 4

mole K.

constant. constant.

absolute temperature, specific specific

volume volume

K.

of a liquid, cu. cm./gm. mole. of a gas, cu. cm./gm. mole.

mole fraction

in a liquid. weight fraction in a liquid. mole fraction in a gas or vapor. - ZRT. compressibility factor in the equation of state pv volume fraction, based on volumes of pure components. selectivity.

activity coefficient. partial differential operator.

Subscripts:

A = B =

A.

C

C.

D AB

component component = component component component

B.

D.

A

in

a B-rich solution,

etc.

hi equilibrium

PREDICTION OF DISTRIBUTION

86

Superscripts: '

= =

standard state. zero concentration.

LITERATURE CITED 1.

2.

Beech, D. G., and S. Glasstone: J. Chem. Soc. 1938, 67. Benedict, M., C. A. Johnson, E. Solomon, and L. C. Rubin: Trans.

Am.

Inst.

Chem.

4.

Engrs. 41, 371 (1945). Berg, C., and A. C. McKinnis: Ind. Eng. Chem. 40, 1309 (1948). Briggs, S. W., and E. W. Comings: Ind. Eng. Chem. 35, 411 (1943).

5.

Carlson, H. C.,

6.

Colburn, A. P.: "Azeotropic and Extractive Distillation," paper presented at the Am. Soc. Eng. Educ. Summer School, Madison, Wis., 1948. and E. M. Schoenborn: Trans. Am. Inst. Chem. Engrs. 41, 421 (1945).

3.

7. 8. 9.

,

,

Dodge, B.

Company, 10.

11. 12.

13. 14. 15.

and A. P. Colburn: Ind. Eng* Chem.

F.: Inc.,

34, 581 (1942).

and D. Shilling: Ind. Eng. Chem. 35, 1250 (1943). "Chemical Engineering Thermodynamics," McGraw-Hill Book

New

York, 1944.

and A. N. Hixson: Trans. Am. Inst. Chem. Engrs. 40, 675 (1944). Elgin, J. C.: Ind. Eng. Chem. 39, 23 (1947). Ewell, R. H., J. M. Harrison, and L. Berg: Ind. Eng. Chem. 36, 871 (1944). Francis, A. W.: Ind. Eng. Chem. 36, 764, 1096 (1944). Hand, D. B.: /. Phys. Chem. 34, 1961 (1930). Hildebrand, J. H.: "Solubility of Non-electrolytes," 2d ed., Reinhold Publishing Drew, D.

A.,

Corporation, New York, 1936. Hixson, A. W., and J. B. Bockelmann: Trans.

Am. Inst. Chem. Engrs. 38, 891 (1942). Horsley, L. II.: Ind. Eng. Chem., Anal. Ed. 19, 508 (1947); 21, 831 (1949). 18. Hougen, 0. A., and K. M. Watson: "Chemical Process Principles," Part 2, John 16.

17.

Wiley 19.

&

New

Sons, Inc.,

and

:

York, 1947. Part 1.

Ibid.,

20. Hunter, T. G., and T. Brown: Ind. Eng. Chem. 39, 1343 (1947). 21. "International Critical Tables," McGraw-Hill Book Company, Inc., 1926. 22. Jones, C. A., A. P. Colburn, and E. M. Schoenborn: Ind. Eng. Chem. 35, 23. Laar, J. J. van: Z. physik. Chem. 72, 723 (1910); 185, 35 (1929).

24. Lewis, G. N.: Proc. Am. Acad. Arts Sci. 37, 49 (1901). and M. Randall: "Thermodynamics and the Free 25.

Substances," McGraw-Hill 26.

Book Company,

Inc.,

New

New

York,

666 (1943),

Energy of Chemical

York, 1923.

McElvain, S. M.: "The Characterization of Organic Compounds," Chap. The Macmillan Company, New York, 1946.

27. Mertes, T. S.,

Ill,

and A. P. Colburn: Ind. Eng. Chem. 39, 787 (1947). "Chemical Engineers' Handbook," 3d ed. (1950), McGraw-Hill

28. Perry, J. H., Ed.:

Book Company, 30. 31.

32. 33.

34. 35. 36.

Inc.,

New

York, 1941.

and A. T. Kister: Ind. Eng. Chem. 40, 341 (1948). Scatchard, G.: Chem. Rev. 8, 321 (1931). and W. J. Hamer: /. Am. Chem. Soc. 57, 1805 (1935). and S. S. Prentiss: /. Am. Chem. Soc. 56, 1486 (1934). Scheibel, E. G., and D. Friedland: Ind. Eng. Chem. 39, 1329 (1947). Treybal, R. E.: Ind. Eng. Chem. 36, 875 (1944). Wohl, K.: Trans. Am. Inst. Chem. Engrs. 42, 215 (1946). Woodburn, H. M., K. Smith, and H. Tetervsky: Ind. Eng. Chem. 36, 588 (1944).

29. Redlich, O.,

CHAPTER

4

CHOICE OF SOLVENT In choosing a solvent for a liquid-extraction process, there are several These are frequently conflicting, principles which can be used as a guide. and certainly no single substance would ordinarily possess every desirable

Compromises must be made, and in what follows an be made to indicate the relative importance of the various

characteristic.

attempt

will

factors to be considered. Selectivity.

This

is

the

applicability of a solvent,

first

and

property ordinarily studied in deciding the refers to the ability of a solvent to extract

it

one component of a solution in preference to another. The most desirable solvent from this point of view would dissolve a maximum of one component and a minimum of the other.

M

4.1, where point represents a planned to separate by use of solvent B. a simple extraction process on this diagram.

Consider the ternary system of Fig. solution of

A

and

C which

Let us follow the course of

it is

B

to the solution Af, point S would represent the comof the resulting two-phase mixture as a whole. After vigorous position agitation to ensure the attainment of equilibrium and settling of the liquid

Upon

addition of

layers, the line

RT is,

customary

two If

B

two insoluble equilibrium layers at

R and T would

B

and recover the extracted substances in solvent-free form. removed completely from R, the binary solution at D results, while

solutions is

E

represents the solvent-free solution corresponding to T. has been separated into cedure, the original solution at

M

By

this pro-

two solutions

D

and E, the first of these richer in A and the second richer in C. be a maximum distance apart obviously desirable that E and

at

The

result.

of course, a tie lute joining the conjugate concentrations. It is in extraction operations to remove the solvent from the

D

It is if

Ke

most effective. If, for the same system, A is used as a solvent to separate a solution of C and 5, as shown in Fig. 4.2, it is clear that while a separation is indeed possible since the ultimate solutions E and D still have different composioperation

is

to be

has not been nearly so effective as that first described. of tie-line slope has been responsible for the differthe direction Obviously it is ence. Indeed, entirely conceivable that in Fig. 4.2 the tie line utilized tions, the separation

by the operation would

coincide precisely with the line

AM representing

the original addition of solvent, in which case the solutions

when

stripped

CHOICE OF SOLVENT

87

A

would both have compositions identical with the original mixture. We may say, in the example shown, that B is more selective in separating solutions of A and C than A for solutions of B and C. Similarly, several solvents for separating the same binary pair may be compared. As a rough guide, an extended tie line will intersect that side of the triangle representing binary mixtures of distributed substance and the more selective of the two nonconsolutes. of solvent

FIG. 4.1.

A

Selective extraction of of solvent B.

C

from

FIG. 4.2.

B

by means

Selective extraction of of solvent A.

C

from

by means

Quantitatively, the property of selectivity may be demonstrated more by plotting the concentrations of the distributed substance C

satisfactorily

D

in corresponding solutions at and E against each other, or, in other words, by plotting a distribution curve for C on a solvent-free basis. Two selectivity diagrams of this sort may be prepared for each ternary, depending

upon which nonconsolute is considered the solvent. The curves so obtained are somewhat similar to simple distribution curves in that they start at the origin of the diagram, ordinarily but not necessarily pass through a maximum, and finish on the 45 diagonal at the plait point.

The more the 45

highly selective the solvent, the greater will be the space between diagonal and the curve. If the curve coincides with the 45 di-

In many respects^ these curves are agonal, no separation is possible. to the analogous constant-pressure McCabe-Thiele diagrams used for vapor-liquid equilibria in distillation studies (8). Figure 4.3 shows typical selectivity curves for

shown

in Fig. 4.1: benzene-water-ethanol at

two systems of the type (9) and ethyl acetate-

25C.

water-ethanol at 2GC. (1), ethanol being the distributed substance in each case. In these systems, a simple distribution curve indicates that the distribution of the ethanol favors the water layers in both cases. Figure 4.3 shows that at low concentrations water is much more selective for separating ethanol from benzene than is benzene for separating ethanol and water. Similarly in the ethyl acetate systems, water is the more selective solvent.

LIQUID EXTRACTION

88

if a choice were to be made between benzene and ethyl acetate as solvents for separating ethanol and water, clearly benzene is the more selective of the two. Other considerations, however, might show neither

However,

to be very desirable. c

(Ben2ene(A)-Wafer(B)-Ethanol(C).2S C(9)

\

Ethyl acetate as >

0.1

0.2

solvent-

0.4

0.3

Weight fraction ethanol

0.5

in non-solvent layer,

solvent -free basis

FIG. 4.3.

Selectivity diagrams,

Type

1

systems.

In Type 2 ternary systems, similar comparisons of the selectivity of In the case of Fig. 4.4, B is the solvent that is used solvents are possible. to separate solutions of A and C, and the diagram is lettered in the same

manner as

Consequently the selectivity diagram is plotted as on a B-free basis. Selectivities are ordinarily somewhat lower in this type of system than in those of Type 1, and two examples are shown in Fig. 4.5. As in the case of vapor-liquid equilibria, numerical values of the selectivity, designated as 0, are desirable, and these may be calculated in the Fig. 4.1.

concentrations of

C

in the equilibrium phases

same fashion as the processes (8).

aiuilouou* property, relative volatility, for distillation for C is defined as follows: Thus, the selectivity ft of

B

CHOICE OF SOLVENT

XCB XCB 1

XCB or

XCA + XAA

+ XAB

-

89

XCA 1

+

(4.1)

-

+

more simply, (4.2)

Similarly, the selectivity of

A

for

C

is

(4.3)

A FIG. 4.4.

B Selective extraction in

Type 2 systems.

o n-Hepfane(AhAniline(B) tetiylcyclohexane(C),2

a Refined cottonseed oil(A)-

0.4

0.2.

Wright fraction Fio. 4.5.

C

in A-rich

0.8

0.6

phase, B-free basis

Selectivity diagrams for

Type 2

systems.

1.0

LIQUID EXTRACTION

90

for the A- and B-rich phases are equilibrium concenvalue for /3 will be the same whether weight or numerical the trations, and Like relative volatility, for concentration. used units are fraction mole

The concentrations

ft

has been shown to be substantially constant for a few systems C 040

Weight fraction of

020

in

Weight fraction of C

Fio. 4.6.

Selectivities in

(2, 7),

the A-rich phcise=XcA 0.60

in

0.80

the B-nch phase =

Type

1

1.0

X CB

and Type 2 systems.

can be used as a correlating device for tie-line data [Eqs. In most cases, varies widely with concentrations, as (2.14), (2.15)]. in shown Fig. 4.6, where selectivities for the systems of Figs. 4.3 and 4.5 are shown. in

which case

it

CHOICE OF SOLVENT

91

The importance of good selectivity for extraction processes parallels that of relative volatility for distillation. Practical processes require that ft exceed unity, the more so the better. Selectivities close to unity will result in large plant

and

stages,

ration

is

equipment, large numbers of extraction contacts or investment and operation. If ft = 1, sepa-

in general, costly

impossible.

(2.7) and (4.2), we see that selectivity of coefficient in the following manner: distribution related to the

Combining Eqs.

ft

=

B

for

m^

C

is

(4.4)

Since XAA/XAB is always greater than unity, then for systems with a favorable distribution coefficient (m > 1), ft will always exceed unity.

Systems with an unfavorable distribution coefficient (m < 1) or systems for which m varies from less than 1 to greater than 1 (characterized by a reversal of tie-line slopes) will not necessarily give values of ft less than unity except if the mutual solubility of the nonconsolutes A and B is considerable.

In quaternary systems, where B and C are distributed between immiscible A and D, the selectivity and distribution coefficients are related

solvents

in the following

manner:

mB = f^, A BD

m c = $2d A CD

(4-5)

and P

where tems,

ft ft

is

= ma~ = XsAXcp

y

the selectivity of the solvent pair for B.

must exceed unity

Recoverability.

In

all

,

^

A

>.

(4-6)

As with ternary

sys-

for a successful process.

liquid-extraction processes,

it

is

necessary to

remove the extracting solvent from the two products resulting from the This is important not only to avoid contamination of the with the solvent but also to permit reuse of the solvent in order products In practically every instance, the recovery to reduce the cost of operation. fractional is of one distillation, and the relative volaultimately process and to be solvent substance of the separated must be high in order tility separation.

that this

may

their presence tion means.

The existence in the system must be checked particularly, since

be carried out inexpensively.

of azeotropes involving the solvent

may

prevent separation of the solvent by ordinary

distilla-

A very complete, indexed list of azeotropes which has recently

most convenient for at least initial studies of recoverquestion as to whether the solvent or the components ability (4). which are separated by the extraction process should be the more volatile is an important one. In most extraction processes, the quantity of solvent been compiled

is

The

LIQUID EXTRACTION

92

If, in the recovery by greater than that of the desired products. is the more solvent the volatile, large quantities will be vapordistillation,

used

is

and the process will be costly. Therefore in such cases it is preferable that the solvent be the less volatile, and distillation will involve vaporiza-

ized

tion of the desired products which are present in smaller

amounts-

If the

possible that very small amounts will be used, very selective, in which case it may be advantageous if the solvent is the more volatile.

solvent

it is

is

If the solute in the solvent-containing solution is nonvolatile, it

may be

necessary to recover the solvent by evaporation. In either case, if the solvent must be vaporized, its latent heat of vaporization should be low to reduce the cost of recovery. 1.5

Trichloro ethane - rich I solutions

1.4

1.3

TJ

!''

o

1.1

1.0

P/a/t

po/nt

Wafer-rich, solutions 0.9 10

20

40 30 50 Weight per cent acetone

60

70

Fio. 4.7. Densities of equilibrium solutions in the system water (A)-l,l,2-trichloroethane (B)-acetone (C), at 25C. (6).

The

possibilities of using

methods

of solvent-product separation other

than vaporization should not be overlooked. Crystallization of the product from the solvent solution and removal of the solute by adsorption are occasionally worth serious investigation. Density.

A

difference in densities of the contacted phases is essential Not only is the rate of disengaging possible.

and should be as great as

of the immiscible layers thereby enhanced, but also the capacity of the contacting equipment is increased. It is insufficient to examine merely the relative densities of the solution to be extracted and the pure extracting solvent, since on admixture mutual solubility of the two will alter the densities; for continuous contacting equipment, it is important to be certain

that a satisfactory density difference for the contacted phases exists

CHOICE OF SOLVENT Figures 4.7

throughout the entire range of the contemplated process.

and

4.8 indicate desirable

and potentially undesirable

situations respec-

saturated layers in the Type 1 system acetone-l,l,2-trichloroethane-water (acetone as distributed substance) tively.

In Fig.

4.7, the densities of

are plotted, with equilibrium layers joined by tie lines (6). The densities of the water-rich layers are always less than those of trichloroethane-rich is layers, but the change in density difference with acetone concentration necessarily great, since at the plait point the densities of the conjugate solutions are identical. Figure 4.8 is the same sort of plot for the Type 2 Note that here a methyl ethyl ketone-water-trichloroethylene (5)

system

.

1.5

1.4 ,

j

Tnchloroethylene-nch solutions

1.3

VI.2

i.o

Water-rich solutions 0.9

0.8

I

10

20

30

40

50

60

70

80

90

Weight percent methyl ethyl ketone Fia. 4.8.

Densities of equilibrium solutions in the system water (A)-trichloroethylene (C), at 25C. (5).

(B)-methyl ethyl ketone

reversal of sign in the density difference between equilibrium layers occurs, and while a stagewise contacting operation could work across the conjugate

solutions of equal density, a continuous contacting operation could not. The interfacial tension between immiscible phases Interfacial Tension. which must be settled or disengaged should preferably be high for rapid interfacial tension on the other hand ~inay lead to the adequate dispersion of one liquid in the otheiv white too low a value may lead to the formation of stable emulsions. Unfortunately,

action.

Too high an

difficulties in

few liquid interfacial-tension measurements for complete ternary been made. As an extremely rough guide, the differences have systems in the surface tensions with air of the contacted liquids may be used to estimate the order of magnitude of interfacial tension, but this will be at relatively

LIQUID EXTRACTION

94

In Type 1 systems, the interfacial tension between equilibrium layers will fall to zero at the plait point. Chemical Reactivity. Chemical reactions between solvent and com-

best a very crude indication.

ponents of the solution yielding products extraneous to the process are undesirable, since only in their absence will the yield of products be high and complete solvent recovery be possible. On occasion, the formation of a chemical compound as part of the extraction process may be considered desirable, since then the rate and even the extent of extraction may be enhanced. Ordinarily any such reaction product should be capable of easy decomposition so that solvent recovery is possible, unless the

reaction product itself is the desired substance. Polymerization, conat or of the solvent any temperature attained densation, decomposition in the process, including the recovery

equipment,

is

not desirable.

Mutual Solubility with Solution to Be Extracted. The extracting solvent and solution to bo extracted should be highly immiscible. In a Type 1

(b)

(cO

FIG. 4.9.

Effect of solvent solubility on extraction.

system, for example, this will ordinarily mean that high concentrations of distributed solute can be attained before complete solution of the immis-

This in turn increases the ultimate extent of separa-

cible liquid occurs.

In both of the systems shown, it is tion possible, as indicated in Fig. 4.9. for use to separating solutions of A and C within possible component to since only in this range of concentration will the limits of pure

B

A

M

,

immiscibility occur on addition of B. Clearly the possibilities with the much with the system (6) are than those Furthermore, system (a) greater .

solvent recovery in highly insoluble systems is simpler, and, for a given distribution coefficient, the selectivity will be better [Eq. (4.4)]. Corrosiveness. In order to reduce the cost of equipment, the solvent

should cause no severe corrosion construction.

be required.

Expensive alloys

difficulties

with

common

materials of

and other unusual materials should not

CHOICE OF SOLVENT Viscosity.

Low power

95

requirements for pumping, high heat-transfer

rates, high rates of extraction, of low viscosity, and hence this

and general ease is

of handling are corollaries in extrac-

a desirable property of solvents

tion processes.

Vapor Pressure. The vapor pressure of a proposed solvent should be sufficiently low so that storage and extraction operations are possible at atmospheric or at most only moderately high pressure. This requirement may of course conflict with the requirement of high relative volatility with the solution being extracted, and a compromise may be necessary. Freezing Point. The solvent should have a sufficiently low freezing point so that it may be conveniently stored and otherwise handled at outdoor temperatures in cold weather. Inflammability. Low inflammability is of course desirable for reasons of safety, and the flash point is frequently used as a numerical indication If the solvent can be burned, it should have a high flash of the property.

point and close concentration limits for explosive mixtures with air. Toxicity. Highly poisonous materials are difficult to handle industrially.

Unless elaborate plant safety devices are planned, with frequent medical inspection of personnel, the more toxic substances must be avoided.

Low

and ready availability in adequate quantities usually and are of course desirable solvent attributes. While it is true that solvents are recovered from product solutions, nevertheless make-up solvent to replace inevitable process losses must be expected. Furthermore, large quantities of expensive solvent which are retained in the plant represent sizable sums invested. Interest on such money is Cost.

cost

parallel each other

chargeable directly to the process. Of all the desirable properties described, favorable selectivity, recoverability, interfacial tension, density, and chemical reactivity are essential

The remaining properties, while for the process even to be carried out. not necessary from the technical point of view, must be given consideration in good engineering work and in cost estimation. Notation for Chapter 4

m

= -

X

=

/3

selectivity, defined by Eqs. (4.1) and (4.2). distribution coefficient = ratio of concentration of a solute in equilibrium

liquid phases.

concentration, weight fraction.

Subscripts:

A B t

t

D = components A, B, C, and AB = A in a J5-rich solution.

C,

D, resp.

LITERATURE CITED 1.

2.

Beech, D. G., and S. Glasstone: J. Chem. Soc. 1938, Brown, T. F.: Ind. Eng. Chem. 40, 103 (1948).

67.

LIQUID EXTRACTION

96 3. 4.

5. 6. 7. 8.

Hixson, A. W., and J. B. Bockelman: Trans. Am. Inst. Chem. Engrs. 38, 891 (1942). Horeley, L. H.: Ind. Eng. Chem., Anal. Ed. 19, 608 (1947); 21, 831 (1949). Newman, M., C. B. Hayworth, and R. E. Treybal: Ind. Eng. Chem. 41, 2039 (1949). Treybal, R. E., L. D. Weber, and J. F. Daley: Ind. Eng. Chem. 38, 817 (1946). Varteressian, K. A., and M. R. Fenske: Ind. Eng. Chem. 29, 270 (1937).

Walker, W. H., W. K. Lewis, W. H. McAdams, and E. R. Gilliland: "Principles Chemical Engineering," 3d ed., Chap. XVI, McGraw-Hill Book Company, Inc., New York, 1937. Washburn, E. R., V. Huizda, and R. Void: J. Am. Chem. Soc. 63, 3237 (1931). of

9.

CHAPTER

5

DIFFUSION AND MASS TRANSFER Consider a liquid flowing through a pipe. Extensive study of the velocity distribution in such a fluid has revealed that at the pipe wall the fluid is motionless and that at increasing distances from the pipe wall the fluid velocity gradually increases, reaching a maximum value at the center. Near the pipe wall the flow is viscous or laminar, characterized by a velocity which is a linear function of the distance from the pipe wall, with no general mixing in the direction of the pipe radius. If the average velocity is sufficiently large, laminar flow is confined to a relatively thin layer adjacent In the central core of the fluid, flow is turbulent, characto the pipe wall. terized by eddy currents with large velocity components perpendicular to the axis of the pipe and considerable mixing of the fluid in the direction of the pipe radius. Similar phenomena are observed whenever fluids move

rapidly past solid boundaries of any sort and also when the boundary is an interface between two immiscible fluids, although in the latter case the relative interfacial velocity in the is

two phases and not the absolute velocity

zero.

In extraction processes, where moving immiscible liquids are brought into contact for the purpose of causing the diffusion of a substance from

one liquid to the other across the phase boundary,

it is

clear that the diffus-

ing substance must pass through various portions of the fluid that are in viscous or turbulent flow. The rates at which the diffusion occurs through these zones are of major importance in determining the size of equipment for carrying out the extraction.

MOLECULAR DIFFUSION Molecular diffusion

is

the

mechanism

which

of transfer of

a substance either

the ffuTcTis in laminar flow, in a is motionless~or, through a fluid of the fluid. The phenomenon direction perpendicular to the velocity has been studied from many points of view7frequently conflicting, the most if

important of which are those of Fick and of Maxwell-Stefan. Fick (7) applied the well-known Fourier equation for rate of heat flow to the problem of diffusion. Unfortunately the mechanism of the two processes is not in the penetration of a liquid by a diffusing solute there will since identical, be displacement of the liquid and consequent volume changes necessarily which the Fourier equation does not account. As an approxifor arising 97

LIQUID EXTRACTION

98

mation, however, the Fick concept is very useful, particularly since the Fourier equation has been integrated for many situations which parallel problems in diffusion. Maxwell (17) and later Stefan (27) considered the simultaneous movement of both components of the solution through which the solute is diffusing and arrived at equations which are much more general. These have been reviewed and extended by Lewis and Chang (15), Sherwood (24), and Arnold (2). Maxwell-Stefan Concept. In substance, the resistance to diffusion of component A through a solution of A and B is assumed to be proportional to the relative velocity of A with respect to B, UA UB', to the distance dl through which the diffusion occurs; and to the number of molecules of A

and

B

path of the diffusion, in turn proportional to the concentrations CA and CBThe resistance must be overcome by a concentration in the

gradient in the direction of diffusion dcA-

dCA

Thus,

~"bCACB(UA

Us)dl

(5.1)

This basic equation may then be integrated for various situations. 1. Equimolal Counterdiffusion. This leads to Pick's law. If NA and NB are the number of moles of A and B, respectively, diffusing per unit time through a cross section S,

NA = cA uA S, N A =-NB

and

NB

=

c B u BS

(5.2) (5.3)

Substitution in Eq. (5.1) then leads to

NA =The

,

b

^

diffusion coefficient, or diffusivity

D* -

(5.4)

.

~j

D is defined as (5 5) -

^TT^j'

whence

NtDJ** which

is

Pick's law.

2. Steady-state

or

(5.6)

Conditions

Not Diffusing (UB

.

=

dc A

0).

=-

(N A =

constant),

Equation

bNACsdl

(5.1)

=-

One Component Stationary

then becomes

N A c B dl

.

(5 g)

Integration of Eq. (5.8) then depends upon the effect of concentration on the volume of the liquid solution and upon D.

DIFFUSION AND MASS TRANSFER a.

Solution very dilute

(CA

~

CB),

=

D=

99

const. (24).

const.

(5.9)

and A B

=

dcA

or bU A (CA .

/

r /

,

~r CB)

NA Cs(li Letting

(I,

-

li)

=

A

6.

The volume

(5.10)

k)

,-

,

1N

(5.12)

I,

JA,

the constituents,

dl

of the solution

D=

is

-

(5.13)

CA,)

an additive function

of the

volumes of

const. (15, 24).

CA

+ CB dcA

= c = const. = ~ dcB

(5.14) (5.15)

Therefore, ,

dc A

=-

=-

,

dc B

-^-

AB dl

,K

N

(5.16)

(5.17)

cDAln^fto c^

(5.18)

2

Define

.

-cD A (c~Bl -

c ft )

(5.19)

_ -

N

A ~TT n Vi

j. y

O

CBAf

Since

+

Cj5i

and

CAI

.

--

c^ 2

= - CA,) N A =

(Z*

cDA (cAi ~

=

CBM

Zi)

+

CA,

I

(5.14)

(5.12) l

~o" >->

or (5 22) '

Converting to mole fraction units x for concentration, CA

=

XA C,

CBM

=

^BMC

(5.23) (5.24)

LIQUID EXTRACTION

100 Since

(5 25) '

Equations

(5.22), (5.24),

and moderate variations c

=

(ci

and

most

(5.26) are

useful for ordinary purposes, taken care of by use of an average,

in c are usually

+ c )/2. 2

Illustration

I/ Calculate the rate

of diffusion of ethanol across a film of water solution

20C., when the concentrations on either side of the film are 14 and 9.6 Under these conditions, the diffusivity of the ethanol may be taken wt. per cent ethanol.

0.2 cm. thick at

as 0.74

X

10~ 5 sq. cm. /sec.

Use Eq. (5.22). Mol. wt. ethanol = 46.05, mol. wt. water 20C., the density of the 14% solution = 0.9764 gm./cu. cm. Solution.

- -

Consequently,

/( ~_ 46.05

Cl

The

-

0.00297

9.6%

+ 0.0466 =

solution

"

=

=

-

ct

+ca ^l

2 I

=

0.0496

-

gm cu.

'

m

le

cm. soln.

0.9824 gm./cu. cm.

=

;

0.0466

c Bt

-

mole . n R gm. cu. cm. n Arn , "gm molo U.UoUO cu. cm.

0.0492

=

In (0.0466/0.0492)

0.0496 -h 0.0513 ^ 2

"

0.2 cm. A?)

=

"

;

'

ln(c ft /c&) C ==

---- ---

gin. -

gin, mole ethanol n ftnOA _ " U.UUZUO cu. cm. soln. 0.9824(0.904) = _ . , Art gni. mole water 0492 cu. cm. soln. 18.02 gm. mole = 0.00205 4- 0.0492 0.0513 cu. cm. soln. CBI

CBM

-

At

;

0.9824(0.096) TTTT^ 46.05

CAI

nnoo7

cu.

0.9764(0.86) o A^ 18.02

density of the

Cfl>

.

.

CB\

18.02.

mole ethanol cm. soln. mole water ^^..^gm. a U.U4bo cu. cm. soln.

0.9764(0.10 CA\

=

3.54

X

_

0.74(10-)(0.0505)(0.00297

-

0.00205)

0.0485(0.2) 10~ 8 gm. mole ethanol/sq. cm. cross section

3. Unsteady-state Conditions. Arnold (2) has integrated the Maxwell" Stefan equation for gaseous diffusion in the case of the semi-infinite or diffusion from a which the concentrations are kept at column/' plane constant into a space filled with gas extending to infinity, both for vapori-

zation of a liquid into a gas and absorption of a gas by a liquid. It is the that to be could possible resulting equations applied successfully liquid diffusion for similar circumstances, provided that an assumption analogous

D

to Dalton's law for gases can be made and that is assumed to remain The direct application to extraction operations of such equa-

constant.

DIFFUSION AND MASS TRANSFER tions is unlikely, but of the diffusivity.

it

101

could be useful in the experimental determination

There is perhaps more direct application for an integration of the diffusion equation for unsteady-state diffusion from spheres, since frequently extraction operations involve the dispersion of one liquid in the form of more or less spherical drops into another immiscible liquid. Isolated drops rising or falling through the continuous liquid will undergo unsteadystate extraction of their solute.

Newman

(21) presents the integration

Fick equation for this case, for a sphere with an initial uniform concentration of solute c, a constant surface concentration CAI, and a of the

radius

r.

The

concentration

WA -

fraction unextracted at

is

WA,

-CAI

is

any time

0,

when

the final average

given by

_ - 6/\c

(5.27)

Numerical solutions of the equation are given by Newman. Experimental Determination of Diffusivity. Reviews of the various methods of determining D are given by Williams and Cady (33) and Cohen and Bruins (3). Two general techniques are used: (a) those in which average concentrations are determined in different zones of a liquid by sampling and analysis after diffusion of a solute into the various zones has occurred, and (6) those in which the course of the diffusion is followed without disturbing the liquid, using optical or similar means of analysis. Aside from the obvious requirements of constant temperature and absence of vibration and eddy currents, the various experimental techniques require

means

of obtaining a sharp demarcation between the dilute and concentrated solutions at the beginning of the experiment. Cohen and Bruins

(3) and others accomplished this by placing several plates in contact with each other, each with a hole bored through it. These were so arranged that by rotating the plates the holes would line up to form a diffusion column. A concentrated solution in the hole of one plate could then be

rotated into contact with solvent in the other holes with a

minimum

of

At the end

of the experiment, rotating the plates would slice from the diffusion column without disturbance effectively samples of the liquid. McBain and Liu (18) and others have used a porous, sin-

disturbance.

membrane to separate the concentrated and dilute solutions, with the concentration gradient and diffusion confined to the pores of the membrane. Gels have frequently been used to keep the solutions free of eddy currents and to facilitate the clean slicing of samples, but it is tered, glass

doubtful whether the rate of diffusion through a gel is the same as through a solution. The data resulting from these experiments have almost invari-

ably been converted to diffusivities by application of several integrated f onri& of Pick's law. Since the diffusivities in liquid solutions are functions

LIQUID EXTRACTION

102

of concentration, unless the concentration gradient hasJ^een kept small the resulting diffusivities are average values over the range of concentration

" " encountered, rather than instantaneous values. Many data are tabu" International Critical Tables" (Vol. V, pp. 63-75). lated in the Empirical Estimation of Diffusivities for Nonelectrolytes. Dilute Solu-

Two reasonably successful approaches have been made to the problem of estimation of diffusivities in the absence of measured data, based on an extension of the kinetic theory to liquids and on the theory of absolute

tions.

reaction rates.

Kinetic theory approach. By application of the kinetic theory of paralleling gases to the liquid phase, Arnold (1) obtained an expression of in form that obtained previously for gases: 1.

D

MW B at 20nopat

ABA where MW = = /i V = A =

D

,,

s

(5 9R 28)

,

'

B

mol. wt. viscosity at 20C., centipoises molar vol. at the boiling point, cu.

cm./gm. mole

abnormality factor diffusivity at

Subscripts

sq. cm. /sec. refer to the diffusing solute

20C.,

A and B

and the

solvent,

resp.

The equation

is limited to very dilute solutions, to a temperature of diffusion through relatively low-boiling solvents.

and to

TABLE

ATOMIC AND MOLECULAR VOLUMES

5.1.

Atomic

Mol.

vol.

Carbon ..... Hydrogen ....

..... 14.8

........

........

Chlorine

.................

The

31.2

37.0

.........

15.6

SO 2 ..............

44.8

NO NO

36.4

.

methyl

.

.

.

.

10.5

........... 12.0

................

esters

..........

methyl ethers

in higher ethers

..........

ring: subtract

Naphthalene t

2

Air ............... 29.9

9.1

in higher esters ................... 11.0 in acids ..................... 12.0

Benzene

N ............

27.0

............... 7.4

in

14.3

25.6

.....

2

CO ................ 30.7 CO ............... 34.0

.

in

O

vol.

2

25.6

Nitrogen Nitrogen in primary amines Nitrogen in secondary amines

Oxygen Oxygen Oxygen Oxygen Oxygen Oxygen

H

.................

Iodine Sulfur

3.7

......... 24.6

.....................

Bromine

20C.,

ring: subtract

listed units

9.9

.11.0

......... 15 ..... 30

must be used

in

Eq.

(5.28).

2

......... 23.6 ........

2

NH ........ 25.8 H ............ 18.9 .32.9 H S ......... 8

2

2

.

COS .........

51.5

C1 2

48.4

........

Br2 ................. 53.2 la ....................

71.5

DIFFUSION AND MASS TRANSFER

V V is

103

can be obtained conveniently from Kopp's law, which of the atomic volumes of the constituents of a molecule. The contributions of each atom given by Le Bas are used by Arnold, and these are listed in Table 5.1. For complex molecules, these are then added together. Thus, for toluene CyHs, V = 7(14.8) For the smaller molecules such as those of the gases, 15 = 118.2. 8(3.7) the table lists directly the values to be used. The abnormality factors A A and AB are corrections included in the equation to account for " associValues of AB are strictly ation," or unduly large intermolecular attraction. functions of the solvent, paralleling in numerical magnitude the association factors obtained from other anomalous properties of liquids, and are listed Values of

states that

an additive function

+

TABLE

AB, ABNORMALITY FACTORS FOR SOLVENT*

5.2.

AB

Solvent

Ethyl ether Benzene Toluene Acetone

0.90 .

1.0

.....

1.0

1.15

Water Methanol

4.70

...

Ethanol

2.0 2.0

.

1.36

Propanol

Amyl

alcohol

1.36

.

m-Xylenc

0.97

Chloroform Carbon tetrachloride

0.94

Carbon

1 .0

disulfide

.1.0

.

Ethyl acetate

Heptane

1.06

0.66

.

.

Acetic acid

.

Ethyl benzoato Nitrobenzene * J. H. Arnold, J.

in

Table

5.2.

Am. Chem.

1.0

1.35

Soc. 52, 3937 (1930).

Values of

AA

1.86

With permission

of the

American Chemical Society.

represent abnormality of the solute and are of A B for the solvent, as indicated

somewhat dependent upon the value

Table 5.3. Additional substances are given by Arnold. For short ranges in temperature, Arnold recommends that the temperature effect on D be estimated by the expression

in

D=

A>o C .(l

+

W)t

(5.29)

and ft

t

The

listed units

must be used

=

0.020

in Eqs. (5.29)

(5.30)

and

(5.30).

LIQUID EXTRACTION

104

TABLE

* J. H. Arnold, /.

where

D

D = *c. t

MB

= = = =

5.3.

Am. Chem.

A A ABNORMALITY ,

Soc.

52?,

3937 (1930).

diffusivity at temp, diffusivity at temp., C.

FACTORS FOR SOLUTES*

With permission

of the

American Chemical Society.

t

0C.

solvent viscosity at 20C., centipoises

solvent density at 20C., gm./cu. cm. PB It is an observed fact that the larger the value of Z), the less is its temperature dependence, although Eqs. (5.29) and (5.30) do not include this effect.

Figure 5.1 compares some observed values of by the Arnold correlation.

D

with those calculated

2. Absolute-rate-theory Approach. Eyring and others (6, 8, 14, 23, 29) have extended the theory of absolute rates to the problem of liquid diffusion and viscosity with considerable success. Viscosity is a measure of the force per unit area required to overcome the frictional resistance between two

layers of molecules of a liquid in maintaining unit relative velocity of the two layers. In diffusion, molecules of the diffusing solute move past those

DIFFUSION AND MASS TRANSFER of the solvent,

The phenomena

105

and a force must be applied to maintain

their velocity.

are thus closely related.

10.0r 8.0 6.0

4.0

J2.0

y.8 -10.6

V

0.2

O.I

0.2

O.I

0.4

D

0.6

observed

FIG. 5.1.

Arnold

Companson

0/8 1.0

xI

5

S(l-

2.0

cm -/ sec

4.0

6.0

8.0 10

-

of observed difTusivities with those calculated

by the method

of

(1).

In the Eyring concept, a liquid is regarded as being made up of a conof matter interspersed with holes. The same energy is required to make a hole in the liquid the size of a molecule as the energy of vapori-

tinuum

zation per molecule.

where &Ev = AHy = RT =

Thus, per mole,

energy to

A#F = A# F - RT make a hole

(5.31)

enthalpy of vaporization

work of vaporization for an ideal gas one layer of a liquid relative to another requires the movement of molecules from one equilibrium position to another, for which it is necessary that a hole be available. This in turn requires expenditure of energy to make the hole. This concept leads to the relationship

Movement

external

of

XtkT X 2A 3A 2 A;'

(5.32)

LIQUID EXTRACTION

106

where

X2

= =

X3

=

average distance between adjacent molecules in a direction

X

=

perpendicular to the movement distance between equilibrium positions in the direction of

Xi

distance between two layers of molecules distance between adjacent molecules in the direction of move-

ment

movement k = Boltzmann r

k

const.

specific rate const.

Application of the theory of absolute rates then leads to E "* /RT M = k"c

where k"

=

(5.33)

a constant including the molar volume, energy of vaporization,

and temperature

Ev =

energy of activation for viscous flow k" is fairly independent of temperature, and hence a plot of log n vs. l/T will result in a reasonably good straight line for most liquids, the slope It has been found that of which will permit determination of E vls ia

.

tf vb

= AEv

-

(5.34)

fli

where n

=

a number, between 3 and 4 for most unassociated liquids, but it is somewhat lower and varies with temperature. This indicates that the hole-size required for the

for highly associated liquids such as water

movement

of

a molecule

is

only J^ to

Y that of the molecule

itself.

Diffusion, as indicated above, is a related problem, differing however in the fact that molecules of different sizes are involved. Many years ago, (5), in considering' the Brownian movement of colloids where the particles are very large in comparison to the molecules of the solvent and

Einstein

assuming that Stokes' law described the motion of the

what

is

known

particles, arrived at

as the Stokes-Einstein equation:

This relation has been found inadequate to describe ordinary diffusion data perfectly, since the Stokes' law assumption that the liquid through which the diffusing molecules move is continuous is inapplicable when the molecules of the diffusing solute actually approach those of the solvent in size. Application of the Eyring "hole" theory for ideal liquids leads to

DA = and on the assumption that X and

fc'

X 2 fc'

for diffusion

(5.36)

and viscosity are the same, (5.37)

DIFFUSION AND MASS TRANSFER which Xi/X2X

107

form the Stokes-Einstein equation. Clearly the quantity and hence the product DM should be the molar volume of the system. Furthermore, it is known

parallels in 8

related to the molar volume,

is

a function of

that the temperature coefficient for the diffusion of large molecules is the same as the temperature coefficient of viscosity of the solvent, indicating that the Stokes-Einstein equation is a limiting case of Eq. (5.37). This interpreted as meaning that in the case of large diffusing molecules the

is

determined by a jump of the solvent molecule from one equilibrium position to another, followed by movement of the diffusing molecule into

rate

is

the space thus

The theory

made

available.

of absolute rates,

when

applied to the rate constant

leads to

DA = where k'"

a constant including

^dw = energy Since k"'

is

k'"eX, k,

E

W

k',

then

RT (5.38)

temp., and the free vol. of the liquid

of activation for diffusion

substantially independent of temperature, a plot of log Dvs.l/T

results in reasonably

good straight lines. Because of the similarity of the diffusion and viscosity phenomena, the energies of activation of the two processes are apparently the same, so that

EMtt = E

via

=

(5.39)

and -Bdifl can thus be calculated either from latent heats of vaporization or from the temperature dependency of viscosity. Since two unlike molecules are involved, it has been found that &E V should be calculated from the values for the pure constituents by the relationship

&E V =

(X A

*E$A

+

XB

AE^) 2

(5.40)

Thus, by a combination of Eqs. (5.38), (5.39), and (5.40), a single determination of D permits estimates of the constant k" and hence the temperature dependency of D. f

Direct application of Eq. (5.38) by determination of fc'" from its conwhich are too large, differing from stituent constants leads to values of the observed values by a factor frequently larger than 2. Wilke (32) has

D

employed an empirical modification, however, which permits estimates of D. Rearranging Eq. (5.37) to read

where

T=

DA = MB = t

The

temp.,

K

diffusivity of

A, sq. cm./sec., viscosity of solvent, centipoises

listed units

must be used

in

Eq.

(5.41).

much

closer

LIQUID EXTRACTION

108

Wilke found that F for a given solvent correlated well as a function of the molal volume of the solute. The effect of different solvents is expressed which at values of solute molal volumes less than through a parameter 150 cu. cm./gm. mole equals the ratio of F for the solvent to F for water at constant molal volume of solute. Furthermore, F for a given solution is

7"

s

Absolute temperature. *K DiffUsivity ofsolufe.sq cm/sec Viscosity of solvent, centipofses Function ofso/verrf

0.1

20

30

40506080100

300400 6008001000 200 V, solute moloi volume, CG/CJ mole

2000

4000

6000

I

FIG. 5.2.

C. R. Wilke [Chem. Diffusivity correlation for dilute solutions of nonelectrolytes. Reproduced with the permission of the American Institute of

Eng. Progress 45, 218 (1949). Chemical Engineers.]

relatively independent of temperature. The solute molal volume correlation.

Figure 5.2 summarizes the Wilke VA should be calculated from the

data of Table 5.1, in the manner described for the Arnold correlation. Values of for water, methanol, and benzene are 1.0, 0.82, and 0.70, respecValues of for other solvents may be obtained by plotting all tively. available data for diffusion in the solvent on the figure and drawing the best line through the plotted points. In the absence of such data, it is



recommended that be taken as 0.9. Figure 5.3 compares observed data with those calculated by the Wilke method.
AM)concd

d In a A d In XA

(5.43)

values of Xi and X 2Xs are used, and if these vary linearly with mole fraction concentration, it follows that the quantity If

mean

(DAI*) coned

d In aA /d In XA should be a linear function of mole fraction concentration at constant temperature, which has been shown to be the case for several systems.

LIQUID EXTRACTION

110

This relationship

may

be expressed in the following convenient man-

ner (32):

"liit

(2*) \ 1 /coned =

where (DA ^/T) COQC(i

fc +11 _IYJi) I

PhjaseE

c#t,

laminar-flow portions of the R phase to the interface, the equivalent molecuis

s

the effective film thick-

I

lar diffusional resistance for

represented by ness ly. Equilibrium

which

concentrations Irtferface

pertain at the interface, so that CE% is in equilibrium with CR IJ the relationship between them given by the equilibrium-distribution curve for the sys-

tem.

There

is

consequently no diffu-

sional resistance at the interface,

and

the diffusing substance passes into the E phase. The concentration gradient CE then provides the driving force for diffusion into the main body of the CE\

Distance through phases

FIG. 5.8.

Concentration gradients.

E phase, where the average concentration

is CE> The equivalent molecular represented by the effective film thickness 1# be thought that the concentration difference CEI CR

diffusional resistance

is

.

might at first would represent a barrier or a concentration gradient operating in a direction such as to oppose the forward movement of the diffusing substance. It should be recalled, however, that if all portions of the two phases were in equilibrium, so that the concentrations of the distributed substance were represented by horizontal lines at CRi in the R phase and CE* in the E phase, the concentration difference CEI CR{ would still exist, and yet there would be no net diffusion from E to R. The apparent concentration barrier would disappear if activities were used rather than concentrations, and the driving forces of Fig. 5.8 would be represented by a continuous drop as we moved from R to E. It is instructive to plot the concentrations on a distribution diagram, as It

in Fig. 5.9.

The

equilibrium-distribution curve will contain the point

(cEi,CRi), and the point represented by main-body concentrations (CE,CR) will be below the equilibrium curve. The driving-force concentration for are as each indicated ACE and Ac#, respectively. In phase gradients

applying Eq. (5.526), let us consider the differential rate of transfer dN which will occur through differential interfacial surface dS. Further,

LIQUID EXTRACTION

120

since the thickness of the fictitious film through which most of the concentration gradient occurs is very small, so that the solute content of the films is negligible with respect to the total solute present, conditions the rate of transfer of solute through the

E

R

through the

phase.

k R dS(cR - Cft.) = k E dS(cE i - CR) k R dS Ac/z = k E dS &C E k R dSc(x R - x Rt ) = k E dSc(xE > - X E )

_

CEJ

CE

CR

cR i

kE

(5.53)

Acg Aca

(5.54)

In most practical situations, the ratio k R /k B

Over-all Coefficients.

known.

phase equals that

Consequently,

= =

dN =

kit

under steady-state

Ordinary sampling of the liquids

dinates c E and C R on Fig. 5.9, but

it is

and analysis

is

not

will give the coor-

ordinarily impossible to approach the

UJ

Equilibrium

V3

1

Equilibrium distribution

"'distribution

curve 'OE

i

I

C*R C R| CR Concentration of distributed substance in phase

FIG. 5.9.

Over-all

Concentration of distributed

substance

R

and individual

film

driving forces.

in

phase R

Fia. 5.10. Principal diffusional resistance in phase R.

and to sample the liquids accurately enough can be determined. The true film driving forces Ac# and Ac/z cannot then be obtained. If the equilibrium-distribution curve is a straight line, so that at all concentrations encountered c E i is proportional

interface sufficiently closely

so that csi

tO

and

c Rl

CjZi,

(5.55)

we may

define a concentration CE

which would be

mca

in equilibrium with C R (5.56)

DIFFUSION AND MASS TRANSFER

We may then represent

121

the complete transfer process in both phases in the

following manner:

dN = KB dS(c*E where

KB

is

an

=

CR)

KB dS &COE

E

concentration gradient AC O E in the phase. concentration in equilibrium with c^,

where

KR dS(cR

dN = KR

is

an

Elimination of

CE*, c Rl

R

,

(5.58)

dN =

+

KR

l/fc#,

ances of the individual films.

(m/k R )

=

JL =

and the terms 1/ks, m/kn,

(5.53), (5.55),

dS(cl

and

(5.56) (16)

-

(5.60)

shows therefore that _1_

Similarly,

(5.59)

on an over-all con-

phase.

1

(5.57)

defined as a

- 4) = KR dS ACOR

and CR from Eqs.

results in

Comparison with Eq.

is

Similarly CR

over-all mass-transfer coefficient based

centration gradient in the

on an over-all

= mcl

CE

and

(5.57)

over-all mass-transfer coefficient based

m

_j_

(5.61)

JL kR

* .

(5.62)

mk E

and \/mk E

may

be considered the

resist-

the equilibrium distribution of the diffussubstance favors the will be very large; provided ing strongly phase, k E and k R are of comparable order of magnitude, R will nearly equal k Rj and we may conclude that the principal resistance to diffusion lies in the If

E

m

K

R phase.

Graphical interpretation of such a situation

is

shown

in Fig. 5.10,

Equilibrium distribution curve\

C RJ C R Concentration of distributed substance in phase

FIG. 5.11.

R

Principal diffusional resistance in phase E.

In such instances Eq. (5.59) is is very nearly equal to Ac#. is very small Similarly if ordinarily used to describe the operations. 12 will substance favors the the distributed and phase, KB nearly equal A^, the resistance in the J5-phase controls, and Eq. (5.57) is used. Figure

where Aco

m

5.11 represents this situation.

LIQUID EXTRACTION

122

In continuously operating extraction processes, each phase undergoes wide concentration changes as it flows through the equipment. The point on the concentration diagrams whose coordinates are (CR,CE) therefore represents only one position in such equipment, and the rate equations (Eqs. (5.53), (5.57), and (5.59)) are "instantaneous," or point conditions.

In applying them to actual operations, therefore, they will have to be integrated over the concentration range which the liquids experience. Clearly if the simple distribution law, Eq. (5.55), is not applicable, wide variation with concentration of the over-all coefficients KE and R can be expected even if k E and /c# remain constant. Since the factors upon which ks and

K

depend are also

fc/z

likely to vary, the use of the over-all coefficients is

definitely limited.

Notation for Chapter 5 Since this chapter deals largely with diffusivities and in most cases these are reported and used in C.G.S. units, the following table lists only C.G.S. units for the various quantities.

well, all

Most of the equations of the chapter can be used with consistent English units as except where there have been included certain dimensional constants. These are

marked.

A = a 5 c

D d

E #ciifr

Ev

\a

&Ey e

F /

&Hy

K k

abnormality factor, Eq.

over-all

= mass

mass transfer

-

k" =

coefficient,

transfer film coefficient,

k = Boltzmann k'

(5.28).

= activity. = temperature coefficient of diffusivity, Eqs. (5.29), (5.30). = concentration, gm. moles/cu. cm. = molecular diffusivity, or diffusion coefficient, sq. cm. /sec. = differential operator. = eddy diffusivity, sq. cm. /sec. = energy of activation for diffusion, cal./gm. mole. = energy of activation for viscous flow, cal./gm. mole. = internal energy of vapprization, cal./gm. mole. = base of natural logarithms, 2.7183. = defined by Eq. (5.41), K sec./sq, cm. centipoises. = function. = enthalpy of vaporization, cal./gm. mole.

--

^-

^ sec. sq.

1

-

cm. (gm. moles/cu. cm.)

sec. sq.

cm.

!

(gin.

moles/cu. cm.)

16 constant, 1.38 X 10~ crgs/K. specific rate constant for viscous flow and diffusion. constant.

= constant. L = relative partial molal enthalpy, cal./gm. mole. I = distance in direction of diffusion, cm. = effective or fictitious film thickness, cm. In = natural logarithm, = common logarithm. log MW = molecular weight. m = distribution coefficient, for concentrations expressed k"^_

1

as gm. moles/cu. cm.

DIFFUSION AND MASS TRANSFER

m

=

concentration, molality. rate of diffusion or mass transfer, constant.

N= n

R = r

5

cross-sectional area perpendicular to the direction of diffusion, sq. cm. absolute temperature, K. temperature, C.

T =

t

gm. moles/sec.

the gas constant, 1.987 cal./(gm. mole)(K). the gas constant, 8.314 X 10 7 ergs/(gm. mole)(K). radius of a sphere, cm.

=

R'

123

U =

absolute velocity of an ion at infinite dilution, cm./sec. dyne. velocity of diffusion, cm./sec.

u

F = V =

molal volume, cu. cm./gm. mole. partial molal volume, cu. cm./gm. mole. average velocity in a fluid in the general direction of flow, cm./sec.

v = w = final average liquid concentration, gm. moles/cu. cm. x = concentration, mole fraction. Z = valence of an ion. a = temperature coefficient of ionic velocity (Table 5.4). 7 = activity coefficient. 7 = mean ionic activity coefficient for an electrolyte, referred e = eddy viscosity, gm/cm. sec. 6 = time, sec.

to molality.

X

=

1

intermolecular distances, Eq. (5.32), cm.

A2

X3

=

/n

viscosity,

gm./cm.

sec.

In Eqs. (5.28), (5.30), (5.41), and (5.44), and in

Fig. 5.2, the unit must be centipoises. number of ions per molecule.

= = 3.14159. = density, gm./cu. cm. - shearing stress parallel to direction of flow, gm./cm. = F for a solvent/F for water (Fig. 5.2).

v

w p T

=

d

partial differential operator.

Subscripts:

A = B = E = = = R dil = coned = = 1, 2

phase E.

mean. over-all.

phase R. dilute, infinite dilution

concentrated. positions

Superscripts:

-f

= = =

diffusing substance. solvent.

interface.

i

M

component A, the component B, the

initial,

cation.

anion.

1

and

2, resp.

sec.

LIQUID EXTRACTION

124

LITERATURE CITED 1.

3. 4. 5. 6. 7.

8.

9.

10.

:

13.

14. 15. 16.

Am. Chem. Soc. 52, 3937 (1930). Am. Inst. Chem. Engrs. 40, 361 (1944).

H.: J.

Trans.

Cohen, E., and H. R. Bruins: Z. physik. Chem. 103, 337 (1923). Dryden, H. L.: Ind. Eng. Chem. 31, 416 (1939). Einstein, A.: Ann. Physik. 17, 549 (1905); 19, 371 (1906). Eyring, H.: /. Chem. Phys. 4, 283 (1936). Pick, A.: Ann. Physik. 94, 59 (1855). Glasstone, S., K. J. Laidler, arid H. Eyring: "The Theory of Rate Processes," McGraw-Hill Book Company, Inc., New York, 1941. Gordon, A. R.: /. Chem. Phys. 5, 522 (1937). Earned, H.

S.: Chem. Rev. 40, 461 (1947). and B. O. Owen: "The Physical Chemistry

11.

12.

J.

Arnold,

2.

of Electrolytic Solutions/'

Reinhold Publishing Corporation, New York, 1943. Higbie, L.: Trans. Am. Inst. Chem. Engrs. 31, 365 (1935). Kalinske, A. A., and C. L. Pien: Ind. Eng. Chem. 36, 220 (1944). Kincaid, J. F., H. Eyring, and A. E. Stearn: Chem. Rev. 28, 301 (1941). Lewis, W. K., and K. C. Chang: Trans. Am. Inst. Cfiem. Engrs. 21, 135 (1928). Lewis, W. K., and W. G. Whitman: Ind. Eng. Chem. 16, 1215 (1924). J.

C.: Phil. Trans. Royal Soc. 167, 49 (1866).

17.

Maxwell,

18.

McBain, J. W., and T. H. Liu: /. Am. Chem. Soc. 63, 59 (1931). Murphree, E. V.: Ind. Eng. Chem. 24, 727 (1932). Nernst, W.: Z. physik. Chem. 2, 613 (1888). Newman, A. B.: Trans. Am. Inst. Chem. Engrs. 27, 310 (1931). Partington, J. R.: In "Treatise on Physical Chemistry/' H. S. Taylor, Ed., 2d ed., p. 673, D. Van Nostrand Company, Inc., New York, 1930. Powell, R. E., W. E. Roseveare, and H. Eyring: Ind. Eng. Chem. 33, 430 (1941). " Sherwood, T. K.: Absorption and Extraction," McGraw-Hill Book Company,

19.

20. 21. 22.

23.

24.

Inc.,

New

25.

:

York, 1937.

Am. Inst. Chem. Engrs. 36, 817 (1940). and B. B. Woertz: Trans. Am. Inst. Chem. Engrs. Trans.

35, 517 (1939). Wien. Sitzungsber. 63, 63 (1871). 28. Taylor, G. L: Proc. Roy. Soc. (London) A 161 (873), 421 (1935); A 166 (888), 307 26.

27. Stefan, J.:

(1936). 30. 31.

32. 33.

H.

Chem. Phys. 6, 331 (1938). and T. K. Sherwood: Ind. Eng. Chem. 31, 457 Whitman, W. G.: Chem. Met. Eng. 29, 147 (1923). Wilke, C. R.: Chem. Eng. Progress 46, 218 (1949). Williams, J. W., and L. C. Cady: Chem. Rev. 14, 171 (1934).

29. Taylor,

Towle,

W.

S.: J.

L.,

(1939).

CHAPTER

METHODS OF CALCULATION

I.

6

STAGEWISE CONTACT WITH

A SINGLE SOLVENT The

separation of the components of a solution

by extraction may be number of in a about ways, depending upon the nature of the solbrought vent system and the physical arrangement of the apparatus employed. A convenient classification is the following: 1. Single solvent systems, including all those which consist of, or which may be reduced to the equivalent of, three components: the two to be separated, a.

and the

solvent.

Stagewise contact.

In this category are included those arrange-

equipment where solvent and mixture to be separated are intimately contacted, allowed to approach equilibrium, and sepThe operation may then be repeated and with a variety arated.

ments

of

of flowsheets. b.

Continuous contact. This includes arrangements where solvent and mixture to be separated are continuously in contact within the equipment for the entire operation. Equilibrium is not ordinarily approached.

2.

Mixed

This includes those arrangements employing a solsolvents. vent solution consisting of at least two components, where the solubility relationships are such that simplification to the equivalent of a ternary system is not feasible. a.

Stagewise contact.

Continuous contact. Double solvents (fractional extraction). This category includes those arrangements where the mixture to be separated is distributed between two immiscible solvents. The systems contain at least four b.

3.

components. Stagewise contact. b. Continuous contact.

a.

In this chapter only Stagewise contact with single solvents is considered. In studying the various processes of extraction, Distillation Analogy. it is frequently helpful to keep in mind the parallel processes of distillation,

which are generally more familiar. The analogy between the two methods In distillaof separation has been pointed out by several writers (13, 14). tion processes, a mixture of two substances is separated by creation of two 125

LIQUID EXTRACTION

126

phases, one liquid and one vapor, by the addition of heat, and the separation is brought about by virtue of the fact that the relative concentration

two phases. Subsequent condensation about brought by removal of heat. In extraction, two liquid phases are formed by the addition of an immiscible solvent, which then becomes the analog of heat. Since the relative concentration of the substances to be separated is different in the two phases, physical Resettling of the liquid layers produces the desired degree of separation. moval of solvent from the solvent-rich phase is then analogous to condensation of the vapor in the case of distillation. Table 6.1 indicates the analoof the substances of the

is

vapor phase

different in the is

gous situations.

TABLE

EXTRACTION-DISTILLATION ANALOGY

6.1.

Operation or Condition in Extraction

Distillation

Addition of solvent Solvent mixer

Removal

Analogy

Addition of heat Reboiler

Removal

of solvent

of heat

Condenser

Solvent separator Solvent-rich solution saturated with solvent

Vapor at the boiling point

Solvent-rich solution containing more solvent than that required to saturate it

Superheated vapor

Solvent-lean solution, containing less solvent than that required to saturate it

Liquid below the boiling point

Solvent-lean solution saturated with sol-

Liquid at the boiling point

vent

Two-phase

Mixture of liquid and vapor

liquid mixture

Relative volatility

Selectivity Change of temperature

Definitions.

A

Change

stage is

of pressure

a ihechanical device or

series of devices

wherein

the solution to be separated and an immiscible solvent are intimately mixed, allowed to approach equilibrium, and then settled or separated into two

immiscible liquid phases which are then withdrawn. The solvent-rich phase leaving the stage is termed the extract, the solvent-lean phase the

SoM/on fo be separated Extracting so/ve/if

^*+.

^fr-t ^ >/ IdealAY^

raffinate. is

A

^^

s\sfacte)^ ^+*^*r FIG. 6.1.

The

^

I

phases

ideal stage.

theoretical or ideal stage is

sufficiently intimate

c // \Eqi////br/i/m

+-Paff/nate\

one where contact between phases

and maintained

for a sufficient period of time that established, so that raffinate and extract are

distribution equilibrium is equilibrium solutions. Although there are

many varieties and combinaused for this purpose, in this chapter the ideal stage will be designated diagrammatically by a circular figure, as in Fig. 6.1. tions of mixers

and

settlers

STAGEWISE CONTACT WITH A SINGLE SOLVENT

127

Design Diagrams. The interrelationship among the phase equilibria, of stages, and concentrations and weights of various streams in an extraction process can best be worked out graphically on a diagram showing the phase equilibria. Many types of diagrams can be used, and certain coordinate systems are more convenient than others, depending upon the nature of the phase equilibria and the process. Three types will be

number

considered here

:

Customarily, ternary-phase-equilibrium Triangular Coordinates. data are described in the literature on triangular diagrams, and since they 1.

concentration characteristics of such systems, general properties of these coordinates are In connecto which reference should now be made.

are capable of depicting they are most useful.

all

The

described in Chap. 2, tion with the use of such diagrams in this chapter, the following scheme of notation will be applied :

A, B, and

A

system.

C

will represent the three

and

pure components of the ternary

C

are the principal components of the solution to be the pure extracting solvent. In Type 1 ternary systems, A

separated, B and B are partially miscible; in

Type 2 systems, the pairs A-B and B-C are The symbols will also represent the weight of these Thus, A will mean A pounds of component A B will mean

partially miscible.

components. pounds of component #,

B

and

}

etc.

and R will designate feed, extract, both with respect to weight and com-

F, E,

raffinate solutions, respectively, Thus, the symbol will represent

F pounds of a feed solution of F composition indicated by point F on the triangular diagram. Other solutions and mixtures will be designated by appropriate symbols, in each case the symbol representing not only the weight but also the composition as indicated on the phase diagram. Weight fraction compositions will be with appropriate subscripts as described in Chap. 2. Thus, designated by position.

X

weight fraction of A in a J5-rich solution, XCE = weight fraction of C in the extract, XBF = weight fraction of B in the feed, etc. It should be noted that moles and mole fractions may be substituted throughout for

XAB =

weights and weight fractions, if desired. Stages are numbered, and when these numbers are used as subscripts they ordinarily indicate streams issuing from the indicated stage. Thus,

XcR = weight fraction t

raffinate

from the nth

of

C

in the raffinate

from stage

2.

Rn =

weight of

stage, etc.

The length of a line on a phase diagram between two points and P will be represented by OP, as in Fig. 6.2a. A mixture at 0, when added to one at P, results in a new mixture at on the straight line OP, such that

K

will

K

on the line PK extended. removing P from yields be shown lighter in weight than other construction lines.

Similarly,

Tie lines

LIQUID EXTRACTION

128

2. Janecke Diagram. This system of coordinates, described in Chap. 2, involves plotting X, the weight fraction of C on a 5-free basis as abscissa against N, the weight of B per unit weight of B-f ree solution as ordinate, as

in Fig. 6.26.

Thus,

X =

Xc

XA

N

Xc

=

(6.2)

CB

V (d) FIG. (c)

6.2.

x

c

(b)

Cooidmate systems

for

extraction

calculations,

(c) (a)

triangular,

(b)

Janecke,

distribution.

As

in the case of triangular diagrams, the symbols E, A, R, etc. will represent both weight and composition as indicated on the diagram of liquids E, A, and R. Boldface symbols will represent the 5-free weights of the desig-

nated streams.

Thus, the raffinate from stage

related in the following

R

2

represents the combined weight of

The

2.

total

of

B

jB-free

A and C

weights of a stream

in

P are

manner:

P = The weight

and

in the solution

+ NP

P(l

P

will

(6.3)

)

be

B P = PN

(6.4)

P

and will be indicated by OP and, length of the line between points as in the case of imnjsul.'U coordinates, moles and mole fractions may be

The

1

used in place of weights and weight fractions. On the Janecke diagram, as on triangular coordinates, if a mixture at will be on the straight line is added to one at P, the resulting mixture

K

OP, and the weights are

in the following relationship:

O P

P

K

KP OK

(6.5)

PK

from results in on the line extended. Tie Likewise, removing and construction lines will be shown light and heavy, respectively.

lines

STAGEWISE CONTACT WITH A SINGLE SOLVENT

129

3. Simple Distribution Diagram. On these coordinates XCA is plotted as abscissa against XCB as ordinate, as in Fig. 6.2c. Modification of these coordinates, in cases where the liquids A and B may be considered sub-

stantially insoluble, will be described as they are needed.

Calculations

may

ordinarily be

worked out on any

of these diagrams,

but

for certain systems, because of overcrowding of the phase relationships in a given region of the plot, one or the other of the diagrams may not be con-

venient for graphical computation. It is therefore desirable to be familiar with the use of all. Properties of the diagrams other than those used here, the complete interrelationship between them, and other useful coordinate systems are described at length in a series of articles by Randall and Longtin (13). Flowsheet Arrangements. Stagewise single-solvent extraction can be brought about in several fashions, depending upon the arrangement of the Thus, we have stages, each yielding different results. 1. Single contact. This involves the use of a single stage, where solution to be separated and extracting solvent are contacted once and extract and raffinate phases separated. Operation may be batchwise or continuous. The distillation analog is flash vaporization or equilibrium distillation. 2.

Differential extraction.

amount

of solution to

This

be extracted

is

is

a batch operation wherein a definite contacted with differential portions

of extracting solvent, the differential portions of extract being

fast as formed.

The operation

removed as

exactly analogous to differential distillation and has been termed "cocurrent infinite stage" by Varteressian and is

Fenske (22). 3. Cocurrent multiple contact. This may be batch or continuous and is an extension of the single contact wherein the raffinate is contacted repeatedly with fresh extracting solvent. In the limit represented by an infinite number of stages, this becomes the same as differential extraction. 4. Countercurrent multiple contact. This method involves the use of a cascade of stages, extracting solvent and solution to be extracted entering at opposite ends of the cascade. Extracts and raffinates flow counterThe operation is more analogous to gas absorption than to currently. distillation It is necessarily continuous but may be simulated any practice. in batch fashion, in the laboratory for example, as "pseudo countercurrent multiple contact. 5.

"

Countercurrent multiple contact with

reflux.

This

is

a continuous

A

cascade of stages is emoperation annlogoii^ to fractional distillation. with feed solution to be ployed, separated customarily entering somewhere in the middle of the cascade

Extract and both ends of be used only at one end of the cas-

and extracting solvent at one end.

raffinate phases flow countercurrently, with reflux provided at

the cascade.

Alternatively, reflux may cade, corresponding to the enriching or stripping practices of distillation.

LIQUID EXTRACTION

130

SINGLE CONTACT Triangular Coordinates. The flowsheet, with indicated weights of the various streams, is shown in Fig. 6.3a (22). While ordinarily the solution to be extracted consists solely of a mixture of A and C, and the extracting solvent

is

in the general case all three

pure B,

components

will

be present in

C

Solution

tobe

\

V

/

extracted

T~~"Y

^ffaff/nate *

R

Extracting

solvent S (a)

(b) Fio. G.3.

Single-contact extraction.

these streams, as indicated by the positions of the points F and S on Fig. Here the curve RDEG is a typical binodal-solubility curve of a

6.36.

Type

1

system, enclosing an area of two liquid phases.

for the operation

is

F

M

+

S = R

A material balance

M

+E=

(6.6)

and solvent, as indicated on the diagram. Its location can be determined graphically on the line FS through the relationship ,

representing the mixture of feed

is

in the two-liquid-phase

region,

*">

& f= PM or analytically C balance is

by

calculating its composition

by material balances.

Thus,

a

+

SXcs =

MXcM

r77f\r ACF -r

CV ^^cs =

rJ?V ACF ~r

FXcF from which

v = XCM Similarly,

by a

B

1

(6.8) I

dV OA^S

j^

jjj

//^

n\

(6.9)

balance,

V BM X

-

j

^v#j* ~r *J^*

as

=

x

^v HI* ~r

&X BS

^r^s

j

M

/ a in \ (6 10 ) '

Equations (6.9) and (6.10) will locate point Alternatively, to determine the quantity of solvent needed for a given feed in order to produce a predetermined Af, Eq. (6.10) may be solved for S:

S = *-*:"* **BS

.

" B" **BM

(6.11)

STAGEWISE CONTACT WITH A SINGLE SOLVENT

131

In the mixer of the ideal stage, equilibrium is established, so that the twophase mixture produces extract and raffinate solutions E and /J, the

M

products of the operation, located

M

.

The compositions

on the diagram (which

cally

Their weights

tion).

on opposite ends of the tie line through E and R must be determined graphi-

of the streams

may

may

involve trial-and-error tie-line interpola-

be computed either graphically,

EM RM

R

E or

Thus, a

by material balance.

EXcs

~\~

(6.12)

C balance: RXcB =

(6.13)

Solving simultaneously with Eq. (6.6) gives

E - M(XCM

3 (6.14)

^CR

(6.6) then permits calculation of R. In the operation of such a plant, it is clear that if the mixture is not within the two-liquid-phase area, no separation by extraction will occur.

Equation

M

Consequently there at point

Z),

is

a minimum amount of solvent which

DS which There

XBS

maximum of raffinate maximum amount which

results in a is

will locate

M

given by

also a

of composition will locate

" v( GF\ -

c

X BD

(6.15)

D and

M at G,

no

extract.

(6.16)

\GS)~

maximum of extract of composition G and no raffinate. must use an amount of solvent between these limits. Stage Efficiency. The theoretical stage produces as effluent two streams, extract and raffinate, which are in equilibrium. However, if agitation is resulting in a real plant

inadequate or if insufficient time of contact is maintained between solvent

and

solution

to

be

extracted,

the

streams issuing from the stage will not be in equilibrium, or, as it is expressed, the stage efficiency is less than 100 per Refer to Fig.

cent.

S

6.4.

When

sol-

added to feed F, at first two saturated phases at D and G are very vent

is

FIG. 6.4.

Diffusion in a theoretical stage.

LIQUID EXTRACTION

132

which

be dispersed one within the other if agiof the various components of Diffusion vigorous. the mixture, a relatively slow process, then gradually changes the compositions of the phases along the solubility curve through L and K, quickly produced tation

(6),

will

is sufficiently

M

E

at opposite ends of the tie line through repreeventually to R and the mixture as a whole. Only if the compositions R and E are senting realized is the stage considered a theoretical stage. ,

Solvent Recovery. Although not strictly part of the extraction operasuch separations are ordinarily followed by removal of solvent from

tion, all

the extract and raffinate solutions to give the finished .products. Figure shows a typical extraction solvent recovery scheme. Distillation is

6.5

Fin/shed raffinate Raffinate

R'

Extracting solvent

Finished extract E'

Make-up solvent B FIG. 6.5.

Single-contact extraction with solvent recovery.

ordinarily used to remove the solvent from the solutions, and if it is assumed that relative volatilities are comparatively low so that incomplete solvent separation results, two recovered solvents, SB from the extract and SR from

E

f

and R' also carry the raffinate, are produced. The finished products some of the unremoved solvent. The recovered solvents are mixed to give the recycle solvent S', make-up solvent (usually pure B) is added to balance the loss of solvent in the finished streams E' and R', and the resulting mixture

S used

again in extraction.

The

entire series of operations can be followed conveniently on the triangular diagram, as in Fig. 6.6. When solvent SR is removed from raffinate

Material balances on the raffinate solvent

R, finished raffinate R' results. recovery: Over-all:

Component B:

R'

+

+ SR = R

(6.17)

(6.18)

STAGEWISE CONTACT WITH A SINGLE SOLVENT

133

Solving simultaneously, (6.19)

E

results in finished extract Similarly, removal of solvent S E from extract E', the quantity of which may be calculated in a similar manner:

B>

=

(6-20)

A

B

Fia. 6.6.

If desired, the

Single-contact extraction with solvent recovery.

products .R'and E'

may

be computed graphically:

K 'Further,

and

SR =

R- R'

(6.22)

SB =

E-

(6.23)

Recycle solvent: S'

Extraction solvent:

= SK

S =

S'

E'

+

SE

(6.24)

+B

(6.25)

For the plant as a whole, including extraction and solvent recovery,

B

balance:

F + B = E' + R' = K FXBF + B = E'X BS + R'X BR .

Consequently the make-up solvent

B = E'XBB>

(6.26) -

(6.27)

is

+

R'XBR- -

FX Bf

(6.28)

LIQUID EXTRACTION

134

result of the complete operation, therefore, the feed F has been separated into two solutions, one at E' rich in C, the other at R', rich in A.

As a

In the frequently arising situation where the feed consists solely of A and C, where pure B is the solvent and solvent recovery is substantially

complete so that

S =

= B

the diagram

much

is

(6.29)

simplified, as

For such a situation, E' and R are on the A-C axis of the diagram, and all of the equations developed above apply with in Fig. 6.7. r

the additional simplifications that FIG. 6.7. Simplified diagram for single-contact extraction with solvent recovery.

XBS = XBB XBE>

==

1>

=

XBR>

and Xcs

==

XBF

0.

Consider the situation described in the diagram of Fig. 6.8a. The point where E'B is tangent to the binodal curve, is clearly the maximum

E',

possible concentration of C in a finished extract, and can be realized for all It is also clear that feeds with C concentrations between XCF> and XCE'.

XCE* in this diagram represents the maximum concentration of C in any feed that can be processed. Similarly, referring to Fig. 6.86, the maximum

Tieh'ne

(M

Maximum possible concentration of

the

maximum amount

A

concentrations: (a) extract,

in the finished raffinate

of solvent

with zero yield of

(&) raffinate.

R'

is

obtained by using

raffinate.

There

is

no

absolute upper limit to the A concentration in the raffinate, but the actual maximum depends on the feed composition.

Type 2 ternary systems 1,

are handled in the

and the construction diagrams

same fashion as those of Type and simplified cases are

for the general

STAGEWISE CONTACT WITH A SINGLE SOLVENT shown

All the equations previously developed apply. For no absolute upper limit to the purity of C in the

in Fig. 6.9.

such systems there

is

finished extract or of

A in the finished

raffinate,

with the actual maxima for

a given feed depending upon the tie lines through points G and D. Calculations Janecke Diagram. for the flowsheet of Fig. 6.4 can be followed on the Janecke diagram if it is

135

(a)

recalled that the S-free weights

of the various streams are F, S, E, E',

R, R',

feed

NF (Ib.

S/j,

stream,

(lb.

B)/(lb.

A +

Ib.

For the 5*, and S'. the coordinates are

A+

C),

Ib.

C),

etc.

X F (lb. C)/ 6.10

Figure

shows the construction for a Type 1 system (the construction is identical

Type 2 systems, but the solubility curve has the appearance shown in Points F and S correFig. 2.206.

for

sponding to feed and oximriinn solvent are first located, and the point located on the line FS either

M

graphically,

F = | S

MS ^ FM

(6.30)

B

or analytically by a series of material balances. Thus,

(A

B

+

+ SN S

balance:

balance:

case, (b) simplified case.

F+S=E+R=M

C) balance:

/.

C

Single-contact extraction with solvent recovery, Type 2 systems: (a) general FIG. 6.9.

NM

(6.31)

=

(6.32)

+ SN S

=

M

F

+

S

SX S = Er

+ SX a

M

(6.33)

(6.34) X.r

F

+ SX a +S

(6.35)

Equations (6.33) and (6.35) give the coordinates of M, with which the point may be located. In case the amount of solvent must be determined in

LIQUID EXTRACTION

136

order to provide a predetermined location for

M

,

Eq. (6.33)

may be

TM--NF)

solved .

f

(6.36)

N

E'

FIG. 6.10.

Single-contact extraction and solvent recovery on Janecke coordinates.

Extract and raffinate concentrations are found graphically by locating tie line through M, as shown in the figure. The weights of these

the

streams

may

be calculated graphically:

or

by material balances:

B

balance:

ENE + RN R =

Solving simultaneously with Eq.

E = Equation

(6.38)

(6.31),

M (^~Jr)

(6 *

(6.31) then permits calculation of R.

The minimum amount

D

MNM

of solvent that

can be used

will place

3Q )

M at point

(Fig. 6.9):

(6.40)

The maximum amount

of solvent places mai

As

before, real plants

~

M at G:

N 8 - Na

must use solvent quantities between these

(6.41) limits.

STAGEWISE CONTACT WITH A SINGLE SOLVENT As

Solvent Recovery. finished raffinate

(A

B

+

R

'

in the case of the triangular coordinates, the from removal of solvent SR from raffinate R.

results

+

SR

C) balance:

balance:

R

+

R'

= R

(6.42)

R'##

(6.43)

N# - N SR Similarly,

E'

and

+

SE

The

recycle solvent:

E' S'

S'N S

>

S'Y AS' S'

is fortified

137

by addition

of

= E(N E

-

(6.44)

N 8E

(6.45)

= E = SE + SR = S ENs E + S RN SR Q Y & E Y ~T~ Q &R**s

(6.46) (6.47)

(6.48) A(\\ v.O.*y/ /"A

I

ASfl

R

pure B, to give the extraction solvent S.

NB

Fig. 6.10, the addition of pure B, for which vertical line S'S. Consequently Xs' = X$ S = S', ,

B = S(N a -

Ns

On

shown by the and make-up solvent B: ,

is

>)

(6.50)

K

(6.51)

For the plant as a whole,

F

plant

B

=

R'

+

E'

=

K

is on the line R'E', vertically above (or below) F, so Xf. Further, the make-up solvent B can be calculated by a balance:

Consequently point that X/c

+B

=

B = E'N E

>

Having weights of

(6.52)

calculated all

B-free

the

streams,

the total

weights and B content may be determined by Eqs. (6.3) and (6.4). In the simplified case, feed and finished extract

and

raffinate solu-

B

is the tions are B-free, and pure solvent. Refer to extracting Fig. Addition of solvent to the 6.11.

R'

F

E'

shown by the vertical line """ """*" Single-contact extraction, simpliFM and removal of solvent from raffinate and extract streams by vertical lines RR' and EE' Equations feed

is

'.

(6.31) to (6.35), (6.37) to (6.39)

all

apply, with the simplification that

LIQUID EXTRACTION

138

S =

0,

Ns =

oo,

SN S =

5,

and

of Eqs. (6.36), (6.40),

Xs =

(6.41),

F = M,

0,

B may

E'

B = V(N M - N F

Figure 6.11 also shows that the finished extract will result

when

maximum

at Q, the point of

=

E, R'

-

R.

In place

be calculated by

maximum

)

(6.53)

#,) #F)

(6.55)

(6.54)

possible concentration of C in a used to give an extract

sufficient solvent is

abscissa on the solubility curve. C=Acetone

E'

A=Water FIG. 6.12. [Ind. Eng.

B'Mik.

Solution of Illustration

Chem. 33, 1240

XF

,

E,

narily fixed

The

principal quantities in the extraction process Of these, F, Fl and s are ordis R, S, and the process. Only one of the remaining quantities can be

XB

by

Acetone-water-methyl isobutyl ketono at 25 C.

(1941)].

Types of Problems. are F,

1.

,

XR

,

X

X

.

X

whereupon the remainder are no longer under control but by the characteristics of the equilibrium diagram. Distribution Diagram. Although computations for single-contact extraction can be carried out on distribution-diagram coordinates, it is incon-

arbitrarily fixed,

are determined

venient because a trial-and-error solution this

method

is

(A)

required; hence description of

One hundred pounds of a solution containing 25% acetone (C), by weight are to be extracted with methyl isobutyl ketone (MIK) (B) at

Illustration 1.

75% water

is

omitted.

STAGEWISE CONTACT WITH A SINGLE SOLVENT

139

25C.

Calculate (a) minimum quantity of solvent, (b) maximum quantity of solvent, the weights of solvent-free extract and raffinate for 100 Ib. of solvent and the per cent acetone extracted, (d) the maximum possible purity of C in the finished extract, (c)

and

A

(e)

maximum

the

possible purity of

in the raffinate.

Solution. Equilibrium data of Othmer, White, arid Treuger will be used [Ind. Eng. Chem. 33, 1240 (1941)]. Calculations will be made on triangular and Janecke

coordinates. a.

Point is

Triangular coordinates (Fig. 6.12). F corresponding to the feed solution

Pure

located.

B

The

the solvent.

is

minimum quantity

of solvent will give a mixture with the feed at D. F = 100;

XBD =

0.036.

Eq. (6.15):

XBF)

>

XBS

-

_

100(0.036 0) 1 - 0.036

=

3.74

Ans.

Ib.

Janecke diagram (Fig. 6.13):

-

X, -

F = =

ND

100

0.25,

Ib.,

0.04

02

Eq. (6.54): Tz>

=

-

100(0.04

NF -

FIG.

0.6

1.0

X of

Illustration

1

on

Janecke coordinates.

)

0)

Solution

6.13.

0.6

0.4

=

4

Ans.

Ib.

(NOTE: Discrepancies by the two methods are due entirely to errors incurred

in graphical

construction.) b.

Triangular coordinates.

X BG =

Janecke diagram: Eq. (6.55): c.

Eq.

Maximum

No =

B mHX

32.

= *(Na - NF) =

Triangular coordinates.

F

(6.6):

+

(6.10):

XBM

100

S =

M

+

100

FX BF + SX BS

M

-

100(32

S = B = 100

Eq.

solvent will give a mixture with the feed at G.

0.97.

-

3,200 Ib.

Ans.

Ib.

E '

0)

M

+R -

100(0)

200

+ 200

Ib.

100(1)

0.50

LIQUID EXTRACTION

140

XCB -

0.15,

XBB -

0.818;

3?

0.09,

~

_

fl=M-

(6.6):

Lines

XCR =

~ E - M (XcAf Xcfi)

Fn (6.14). Eq. /fiUV Eq.

M

with the help of a tieXCM = 0.125. Locate the tie line through curve and the available data, to give points E and R. From the graph,

M.

Locate point

line correlation

XBR =

0.023.

-

( Q 125

20

-

0>15

= 200-

'

09 )

- H6.8 1ft ft

=

116.8

-

-

_ 83.2

1

IK Ib.

Ib.

BE and BR are extended to E' and 72'. From the graph, XCR' = =

=

0.829, EBB'

E'

Eq.C6.20): M

= =

/T

-

.

AW

-XtfSjj

R'

0.093,

XBR'

=

0;

0.

E'

=

100

_

a

]b

1.0

-

=

21.3

Ans.

78.7 Ib.

Per cent of the acetone of the feed which was extracted

~ (100)

=

c irvr

=

21. 3(0.829) (100)

=

70.5%.

100(0.25)

Ans.

Janecke diagram:

B = p Eq.

100

S =

Ib.,

w-

SN S =

0,

+ SAr s

vM N

f* (6.33):

XM =

Eq. (6.35):

M and the

,

Eq. (6.31): (6.3):

_ E -

R= E =

_

tf*)

100

_ =

=F =

100

Ib.

, n 1.0

0.25

the coordinates of

E(l -f iV ) R(l AT)

-

0.852, or

The

_

Janecke diagram.

78.5(1

Draw

_ -

0.0235)

Ib.

+ 4.55) = + 0.0235)

E

and

R

are obtained:

=

21.5

Ans.

R'

119.3

=

Ib.

80.3

the tangent from acetone. Ans.

Ib.

Ib.

B

to the binodal curve.

The

85.2%

greatest value of

corresponds to 84.0% acetone. e. Triangular coordinates. R and R'. XAR' for this point or

100(1

21.5(1

=

Triangular coordinates.

corresponding XCE' Janecke diagram.

0.5% acetone

_ -

M-E = 100- 21.5 = 78.5 +

R d.

tie line,

-

+

100(0)

M

0,

N

En (6.39). 63QV Eq.

Eq.

_ -

= X^ =

-

After locating the point = 4.55, R = 0.0235.

NE

X5 =

B,

Xc on the binodal curve

at

Q

is

0.840.

This

Ans.

Draw the tie line through (?, locating the corresponding = 0.995, or 99.5% water. Ans. The tie line through G gives XCR = XCR' = 0.005, corresponding to

99.5%

water.

Ans.

Part (c) of Illustration 1 was recalculated for quantities of solvent other than the 100 Ib. used in the illustration, with the interesting results shown

The percentage of acetone extracted from the feed rises rapidly with increasing solvent up to roughly 90 per cent with 300 Ib. Additional solvent results in little additional extracsolvent/100 Ib. feed tion and correspondingly greater dilutions of the extract. The maximum in Fig. 6.14.

STAGEWISE CONTACT WITH A SINGLE SOLVENT

141

concentration of acetone in the solvent-free Extract occurs at point P.

These curves are

fairly typical of calculations with this type of system, alin mind that with other systems different equilibria

must be kept

though may have great it

100

80

influence.

r

s

lOOXce' weight per cent' acefane in so/vent- free -extract

\

60

40

Percentage of

*

or/gmcnI acetone/ xv' extracted

20

extract//OO Ib. feed\ hvx/'f, win solvent

4 Fiu 6.14 at

6

810

20 40 6080100 200 400600 Lbs solvent per 100 Ibs feed solution

1000

2000

4000

Extraction of a 25 per cent acetone-water solution with methyl isobutyl ketone

25C.

DIFFERENTIAL EXTRACTION Differential extraction, as previously pointed out, is analogous in many It is not used industrially and is of respects to differential distillation. interest principally as a laboratory procedure and because it represents the

number of stages of a cocurrent extraction. counterpart, actual operations can probably only approach the more or less ideal situation described below.

limiting result of increasing the

Like

its distillation

Imagine the container of Fig. 6.15 filled initially with a solution F to be extracted. Solvent S is admitted slowly to the bottom of the container (if the solvent has a lower specific gravity than the If the initial solution is not saturated, solution). not located on the binodal curve of a phase diagram, the first portions of solvent added will Further dissolve until the solution is saturated.

i.e.,

addition of solvent results in the formation of an extract layer which collects at the top where

it

Solvent in FIG. 6.15.

Differential ex-

immediately withdrawn. Thorough agitation traction. should be maintained within the vessel so that the extract leaving is at all times in equilibrium with the raffinate solution is

maining behind.

re-

LIQUID EXTRACTION

142

coordiCalculations will be described in connection with the tr::i!i:l nates of Fig. 6.16. The solvent required to saturate the feed F, bringing it to the condition of Ro, will be by a B balance, :

F

(6.56)

XBS

At a

later stage in the process, after ex-

tract has

begun to form, nate in the container be R. of

a differential amount

let

the

raffi-

By addition

of solvent

dS

is reduced by an amount dR, producing an extract layer at E of an amount dE, at the opposite end of a tie line. The new raffinate,

the raffinate layer

X

R Fio. 6.16. Differential extraction on angular coordinates.

dR, has a composition AR dXA R, XBR dXsR, XCR dXcR. A total material balance for the process:

tri-

R + dS = (R - dR) + dE dS = dE - dR C

balance:

XcsdS = (R -'dR)(Xca - dXc*) + XCE dE R dXc R Xcs dS = XCE dE XCR dR

RXCR A

(6.59) (6.60)

balance:

XAS dS = (R- dK)(X AR - dXAR + XAE dE XA s dS = XAE dE R dXAR XAR dR

RXAR

)

_

Eliminating dS and 7

(6.57)

(6.58)

aiv

(Xca

V

XCR)/(XCS

XCE)

(X A s

^J

"\

V

V

CL^AR/\^-AS

//

(6.62)

Eqs. (6.58), (6.60), and (6.62), -A CE)

CL

R

dE from

A CR/ \

(6.61)

//

"V

^

^AEJ XAR )/(XA s

(6.63)

Integrating between appropriate limits, t

In

RQ

p*'

rRo

=

I

/

A/

dR

-pfi

XAR)~\

v \ XAE)J

dXAR

-I

(X A s

XAE) [(Xcs

XCE)

XCR)/ (Xcs

(X A s

XAR) I (X A s

(6.64)

analogous to Rayleigh's law for distillation and may be evaluated The first integral is the area under a curve of XCR as abscissa, graphically.

This

is

1

(XA s

XAR )/(XA s

XA E)]

STAGEWISE CONTACT WITH A SINGLE SOLVENT

143

as ordinate, the concentrations (XCE,XAE) and opposite ends of

(XC R,XA R) being taken on The second integral can be evaluated graphically

tie lines.

in similar fashion.

which

E/, the final extract

is

(not in equilibrium with #/),

a composite of all the extracts withdrawn can be obtained by eliminating dS from

Eqs. (6.58) and (6.60):

dE = Since

XCs

is

a constant,

=

f

/

yo

This can be evaluated by determining the area under a curve of XCR)R as abscissa, obtaining the data l/(Xcs XCE) as ordinate, (XC s

To

during the evaluation of Eq. (6.64). Sf

To

this should

Eq.

(6.56).

The

= Ef

be added the final

obtain the solvent used,

+R -R ff

initial

(6.67)

Q

solvent required to saturate feed,

composite extract composition can be obtained by a

series of material balances:

+

RoXcRo -

~ RfXcRf (a ark

^.

(6.69) 1

ABE/

-&CEf

In

y

r

7-^

I

^

\Q.7\j)

***AE/

For the special case where the solvent Eq. (6.64) reduces to ^

pure B

is

(6.66)

becomes

V

=

^

-

Y^

=

^

-

0,

(6.71)

A

(6.72)

CE

ft

= XAS =

^

[R*X CRQ j fv [K&CR

For certain Type 2 systems,

7^

/

JXARf

and Eq.

(3, 22), X C s

const

-

(

6 73 ) -

where ft = selectivity, as described in Chaps. 2 and 4. For this special case and for pure B as solvent (23), substitution in Eq. (6.71) leads to In

2 = f tt/

-5^-r 1 (in p \

fe& AC/

-

ft

In

(6.74)

LIQUID EXTRACTION

144

which

is analogous to the integration of Rayleigh's law for distillation at constant relative volatility.

One hundred pounds of a 50% acetone (C)-50% water (A) solution 10% acetone by differential extraction with 1,1,2-trichloroe thane at

Illustration 2. is

to be reduced to

25C.

Calculate the quantity of solvent required and the concentrations and weights of

extract and raffinate. Solution. Equilibrium data are available [Treybal, Weber, and Daley, Ind. Eng. Chem. 38, 817 (1946)]. See Fig. 6.17. Point F corresponding to the initial solution is

G'Acetene

BTrichloroethane FIG. 6.17. Differential extraction in the system acetone-water-trichloroethane. from Ind. Eng. Chem. 38, 817 (1946).]

located

and

=

XBRO

at

1.0;

10%

line

FB drawn, thus locating R Q 0.045,

XC R

O

X CR, =

acetone.

=

0.478,

0.100,

(X Bs /.

,

Ro Ro

-

100 .

XA

X A Rf

F =

.

=

R

=

104.7

=

S = B\ XBF =

0;

XBS = XBB

Rf is located on the solubility curve

0.895.

- 0) = - TTT^T 0.045)

100(0.045

- X BRo )

+ 4.71

100 lb.;

0.477.

[Equilibria

.

-r

(1.0

104.7

71 n 4.71 lb. .

lb.

r^CR -0478 o

?

-

477

XAE[(XAR/XAS)

Tie lines are drawn on the figure corresponding to the computations to be made, a portion of which are tabulated on page 145. Data of columns 1 to 4 are taken from the ends of the

STAGEWISE CONTACT WITH A SINGLE SOLVENT

O

00 ^H Ci

-

t

CP *H

SCO CO

C^l

00 Se*

oo "t" co 10 i-H CO CO CO *O iO

f-H

00

I

*

s

0000000

:ss

32 O OO O "^