RECYCLE IN LIQUID EXTRACTION R0 BE RT E
.
T R EY BA L
, New York University, New York 53, N . Y .
In mixer-settler liquid extractors, the effect on stage efficiency of recycling the settled liquids back to the mixer was studied mathematically. Recycling that liquid favored by solute distribution increases the stage efficiency, whereas recycling the other, or recycling both, reduces it. When solute distribution favors the dispersed phase and stage efficiency i s low without recycle, a minimum of cascade volume may result. These effects are in addition to any benefit which may result from better settling or inversion of the dispersion.
IN
MIXER-SETTLER type liquid extractors, it has been known for some time that the settling characteristics of the dispersion issuing from the mixing vessel can be favorably influenced by recycling one of the settled liquids back to the mixer of the same stage-see, for example, (2, 4, 6). In this manner it is possible, for example. to maintain the liquid whose net flow through the stage is in the minority as the continuous phase, whereas without recycle the minority liquid tends to be dispersed. In other cases, merely diluting the dispersion by recycling the continuous liquid is beneficial to settling. Similarly, in certain leaching operations where the ratio of finely divided solids to leaching solvent is too high for ease in handling, recycling of liquid has been resorted to. Whereas in all of these cases mechanical handling problems prompted the use of recycle, there seems to have been no assessment of the influence of recycling on the stage efficiencies, with the thought that recycling may (or may not) be beneficial irrespective of its influence on mechanical handling. This discussion is limited to estimating this effect for liquid extraction, although the ,argument can be readily extended to other operations. T h e influence of recycle on dispersion settling characteristics cannot yet be predicted without recourse to experiment.
dispersed or continuous liquid are more meaningful for present purposes than those expressed in terms of extract or raffinate. Thus the Murphree dispersed-phase stage efficiency is
E,,
=
(1)
-
where the distribution coefficient, which will be considered constant, is m = y * / x . If the same stage, fed with the same liquids, is now operated with recycling of the liquids, the flowsheet is that in Figure 2. Generally only one liquid is recycled, but for purposes of establishing a general expression, recycle of both is shown. The solute concentrations in the effluents are now altered to xnR and y n R as a result of the recycling. It will be assumed that settling is complete in the settler (no backmixing of liquids to other stages in the cascade). The efficiency of the mixersettler alone, within the confines of Envelope I, is I
EMD
-
YnR YnR*
-Y
YnR
=
-y
mX?LR
-Y -y
(2)
For the stage as a whole, within Envelope 11, however, the efficiency is
-
EMDR-
Influence of Recycling on Stage Efficiency
Consider in Figure 1 a single stage of a cascade of stages operated without recycle, Neglecting any possible effect of direction of extraction, stage efficiencies expressed in terms of
Yn - Yn+l - Y n - Yn+l mxn Yn+l Yn* - y n + l
Y ~ R Yn+l YnR* - Y n + 1
-
-
YnR Yn+l mxnR - Yn+ 1
(3)
Solute material balances about the point of mixing VDRwith incoming V Dand for the stage as a whole are, respectively,
V D y n + l f VDRynR = (VD f VDR)Y
x
-
aMIXER-
-
p
(4)
and SETTLER
VDyn+ 1
+ Vcxn-
I
=
VDYnR
+
(5)
vCxTLR
r, Yn+i
Piaure 1 .
Single stage of a cascade, no recycle
' ~
SETTLER Figure 2.
xnR I
I
I
'
I
Stage with recycle
yo
VOL. 3
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AUGUST 1964
185
CHANGE OFSCALE
-..
MCHANGE OF SCALE
0
4
Iy)
RD = AMOUNT DISPERSED PHASE RECYCLED/AMOUNT
FORWARD
0
0.4
0.2
0.1
1.0
2
Figure 3. Effect of recycle of dispersed phase (Rc = 0) on stage efficiency
0.1
0.2
These assume inappreciable volume change of the liquids with passage through the stage. Combining Equations 2 to 4, and letting R, = V,,/V,,
Since Z is the same for both,
FORWARD
E M , is stage efficiency a t RD = 0. V C / V D = 10
VD 9
~
vD
and with recycling for the same vessel,
I
4
impeller power is above a certain minimum, as it usually is (7,9 ) , 6D =
from which it is evident that, since E,' < 1, E,, > E,' for all values of R, exceeding zero. The problem is then to determine how EM,' (and hence In previous work (8) reasonable E,,,) is related to E,,. success was had by assuming that with vigorous agitation the continuous phase in the mixing vessel is thoroughly mixed, especially if the vessel is baffled. This is in accord also with observations of dye dispersions in mixing vessels (5). For a mixing vessel of height Z, and assuming that all mass transfer occurs within the mixer,
E
Figure 4. Effect of recycle of dispersed phase (Rc = 0) on cascade volume rn large, rnVn/Vc = 1.4.
Equilibrium distribution strongly favoring dispersed phase
1.0
0.4
R~ = AMOUNT OISPEUSEO PHASE RECYCLED/AMWNT
f
+DR =
vC'
VD
VD + VDR + VDR + Vc + VCR
(13)
A simplified form of Vermeulen's (77) expression for interfacial area in liquid-liquid mixing is a =
25.4 c $ ~ ~ . ' N ~ e ~ . ' / d ~
(14)
where the Weber number, d,3hnp/u, is computed with a mean density. This has been derived from data taken from two liquid phases in equilibrium, in the absence of mass transfer, and for a condition of no flow-Le., without continuous flow of the contacted liquids through the agitated vessel. Furthermore, it may not correctly represent the variation of interfacial area with position throughout the vessel resulting from coalescence and redispersion of the dispersed phase droplets. Relationships which include such effects are not yet available, and what follows must therefore be subject to such restrictions as these matters may impose. If the small variation in p is ignored as +D is altered, then for a given vessel, impeller speed, and chemical system,
a=
(9)
(E) 0.6
aR Further,
where
S=
VD~RKDR = ~RKDR ( V D V D J ~ K D (1 R D M D
+
+
-1 -- -1+ - m (10)
For spherical dispersed particles (70), kDa = 4ir2 D D / d p a
(11)
KD
kD
kc
and a similar expression applies to the case of recycle. If m is small (distribution strongly favors the continuous phase), K D A k,,and
and since dp = 6 +,/a,
k,
= 4
DDa/36 6,'
ir2
A
D,a/+na
A similar expression applies in the case of recycle. 186
I&EC FUNDAMENTALS
(12)
If the
Alternatively, if m is large (distribution strongly favors the continuous phase), only the k,'s need be considered. As a
Figure 5.
Effect of recycle of continuous phase (Ro = 0) on stage efficiency Equilibrium distribution strongly favoring continuous phase
good approximation (7), k , is essentially independent of dp and @ ,, so that for this case,
settler is kept constant. without recycle,
Combining the foregoing with the definition of S provides two cases : m small.
or nR with recycle,
Thus the number of real stages is n
The cross-sectional areas of the stages are proportional to volumetric liquid handling requirements, or
m large.
‘-4,_ - VC(I
+
A Equations 19 and 20 may then be combined with Equation 9. This assumes, of course, that recycling has caused no inversion of the dispersion. Distribution Favors Dispersed Phase ( m Large). It is is negative, and consequently readily shown that bE,,,/bRc the stage efficiency will always be reduced by recycling the continuous phase. Therefore only the case for R, = 0 need be considered. At R, = 0, there are two situations which may usefully be studied. The first is that where a n existing cascade is overdesigned in so far as flow capacity of the settlers is concerned, so that the additional liquid handling resulting from recycling the dispersed phase is possible (Figure 3). Here, for example, a stage operated a t V,/V, = 10, and providing E,, = 0.4 with no recycle (point A ) , should show E.MDR= 0.52 with dispersed phase recycle R , = 4 (point B ) . Significant improvement in stage efficiency clearly occurs. The second is that in .which a cascade, as yet not built, is to be designed. It is clea,r that the number of stages for a given separation decreases as E,,,, increases by recycling dispersed phase, but the cross-sectional area and therefore the volume per stage increase, assuming that the velocity of flow in mixer and
RC)
VC
+ +
VD(l
+ R,)
vD
and the cascade volumes, the product of stages and crosssectional areas, are in the ratio
T.= T [Vc(1
+ Rc) + VD(1 + RD)1 log [1 + EM,(mVdVc(Vc
+ VD)log [I + EMDR (m v ~ / V c- 111
111
(21) which reflects the ratio of costs of the compared cascades. Taking a typical extraction factor m V,/Vc = 1.4, a likely economical value, with rn very large, V c / V , will be large. For V,/V, = 10, and R, = 0, the volume (or approximate cost) ratio of cascades with and without dispersed-phase recycle is shown in Figure 4. I t is clear that a pronounced optimum R, exists for cases where stage efficiency is low without recycle. Thus, recycling may be planned to good advantage in design for such cases. Distribution Favors Continuous Phase ( m Small). In this case, bE.bf,R/bR, is negative and stage efficiency will be reduced by recycle of the dispersed phase. At R , = 0. therefore, Figure 5 shows (in the manner of Figure 3) that where existing cascades are capable of handling larger liquid VOL. 3
NO. 3
AUGUST 1964
187
flows, significant improvement in stage efficiency may be expected from recycling the continuous phase. At small values of V,/VD, which are likely for small m’s, the liquid flowing in the minority can be kept as continuous only by extensive recycling, and these curves are not extended to R, = 0. The economic advantage of recycling in the case of new designs which was found for large m does not develop here, and the volumes (and cost) of new cascades are always increased by the recycling. Recycle of Both Liquids. If R , = R,, so that both liquids are recycled to keep the same proportion of continuous to R,) for either dispersed liquids as in the net flow, S = 1/(1 type of solute distribution. In this case, bE,,,/bR, is strongly negative, at least a t R, = 0, and recycling is harmful to stage efficiency. The entire analysis was repeated on the assumption that Calderbank’s relation (3) for interfacial area applies. For six-bladed turbine impellers, for example,
+
diffusion coefficient, sq. ft./hr. Murphree dispersed-phase stage efficiency without . recycle, fractional Murphree dispersed-phase stage efficiency of mixer with recycle, fractional Murphree dispersed-phase stage efficiency of entire stage with recycle, fractional over-all stage efficiency of a cascade = n,/n over-all mass-transfer coefficient, lb. moles/hr. (sq. ft.) (lb. moles/cu. ft.) individual phase mass-transfer coefficient, Ib. moles/ hr. (sq. ft.) (lb. moles/cu. ft.) equilibrium distribution coefficient = y * / x impeller speed, l / h r , impeller Weber number = dt3;Y2p/u,dimensionless number of real stages number of ideal stages recycle ratio = V,/V defined by Equation 10 volume, cu. ft. superficial velocity of flow, cu. ft./hr. (sq. ft.) solute concentration in continuous phase, Ib. moles/ cu. ft. solute concentration in dispersed phase, Ib. moles/ cu. ft. height of mixer, ft. density, Ib./cu. ft. interfacial tension, lb. mass/(sq. ft.) volume fraction dispersed phase in the mixer
Eo K k m
N NW,
n ni
R S
T V X
(22)
Y
This leads to somewhat different values of 5‘ and corresponding Qualitatively the conclusions are numerical values of E,,. the same as before.
P
Z
Conclusions
If an existing mixer-settler cascade is operating a t less than its liquid-handling capacity, recycling that liquid which the equilibrium distribution favors should increase the stage efficiency, whereas recycling the other liquid will reduce it. When distribution favors the dispersed phase, recycling that liquid may result in important economies in the design of new cascades. Recycling of both liquids to keep the same ratio of flows as in the net flow reduces stage efficiency. However, superimposed upon this may be the greatly improved settling characteristics, and corresponding increased liquid-handling capacities or reduction in backmixing owing to improved settling, which recycling may bring about. This is particularly true if inversion of the dispersion from one liquid dispersed to the other is sought. These effects, which may be very large, cannot now be estimated except through experiment. Nomenclature
d,
specific interfacial area, sq. ft./cu. ft. mixer cross-sectional area, ft. = impeller diameter, ft. = drop diameter, ft.
188
l&EC FUNDAMENTALS
a
=
A d,
=
U
4D
SUBSCRIPTS = continuous phase D = dispersed phase n = stage number = recycle, or with recycle R C
SUPERSCRIPT
*
=
a t equilibrium
literature Cited
I
1) Barker, J. J., Treybal, R. E., A.Z.Ch.E. J . 6 , 289 (1960). 2) Black, K., Koslov, J., Chem. Eng. Progr. Symp. Ser. 5 5 , No. 22, 105 (1959). (3) Calderbank, P. H., Trans. Znst. Chem. Engrs. (London) 36, 443 (1958). (4) Lash, L. D., Mining Eng. 10, 1161 (1958). (5) Matten, R. V., Bilous, D., Piret, E. L., A.Z.Ch.E. J . 3, 497 (1957). (6) Ryon, A. D., Lowrie, R. S., U.S. At. Energy Comm. ORNL-3381 M., M.Ch.E. thesis, New York University, 1960. R. E., A.Z.CI2.E. J . 4, 202 (1958) ; 6 , 5M (1960). R. E., “Liquid-Extraction,” 2nd ed., McGraw-Hill, , , New York, 1963. 10) Vermeulen, T., Znd. Eng. Chem. 45, 1664 (1953). G. E., Chem. 11) Vermeulen. T.. Williams, G. M., Langlois, ‘ Eng. Progr. 51,’85F’ (1955).
I
RECEIVED for review November 6, 1963 ACCEPTEDFebruary 24, 1964
