Nomenclature
YI
= mole fraction of product = mole fraction of feed
Y
= average mole fraction of the diffusing component in
= capital investment required for cascade = operating cost of cascade
z
= coordinate length of channel, measured in the direction
D
= product withdrawal rate, moles/hr.
F I J K L LIP .Y P
= = = = =
0
A B
c, c,
= ratio of comoonent b to comoonent d
= total area of cascade, sq. ft.
= =
=
Q =
R
s
yr,
=
=
phase V
feed rate, molesjhr. value of integral in Equation 30 diffusion rate, moles,;hr.-sq. ft. equilibrium constant, dimensionless average flow rate of one phase. moles/hr. average flow rate of one phase, cu. ft./hr. total number of stages in cascade product rate of cascade moles/hr. feed rate of component a , molesjhr. reflux ratio L!D cross section area of a stage average flow rate of one phase? moles/hr. total height of one stage of the cascade. ft. mass transfer area per unit volume. sq. ft./cu. ft. variable defined in terms of v , Equation 29 height of a theoretical stage based on flow rate L: ft. mass transfer area based on the flow rate of phase L . moles/hr.-sq. ft. integration variable stage number average mole fraction of the diffusing component in phase L equilibrium average mole fraction in phase L mole fraction of product
v = z =
a
=
P
=
hr, kL
= =
r
=
s x
= =
x*
= =
XD
v ?r
p p
w
Q
of flow. ft. fraction of solution removed in product stream fraction of component Q removed in product stream area to flow rate ratio sq. ft. ’(cu. ft.,/hr.) molar density. moles ’cu. ft. cascade separation factor. Equation 37 = separation factor for a single component = separation factor for a stage = = = = =
Literature Cited
(1) Ahelson. P. H.. Rosen. N.. Hoover. J. I.. “Liquid Thermal Diffusion,“ TID 5229, Technical Information Service Extension, CSAEC. Oak Ridge. Tenn.. 1958. (2) Benedict. M..Pigford, T. H., “Nuclear Chemical Engineering.“ McGraw-Hill, New York. 1957. (3) Cohen. K.. “The Theory of Isotope Separation.” National Nuclcar Enerqv Series, Division 111, Vol. l B , McGraw-Hill, New York. 1951. (4) Fisher. G. T.. Prados, J . \V., Bosanquet, L. P.. “Thermal Diffusion of Salt Solutions in Single Stage Cells and in Continuous Horizontal Columns: T h e Sgstem CLISO,-H,O.” .1.I.Ch.E. J . 48, No. 6, in press. (5) Grass,elli. R.. Brown? G. R.: Plymale, C. E., Chem. En?. Proqr. 57, 59 (1901)
RECEIVED for review January 21, 1963 ACCEPTED Julv 15, 1963
DIFFUSION A N D BACK-FLOW M O D E L S FOR TWO-PHASE AXIAL D I S P E R S I O N T E R U K A T S U M I Y A U C H I , L’nzversity of Tokyo, Tokyo. Japan T H E 0 D 0 R E V E R M E U L E N , Unia’ersity of Californzn, BPrkeiPy, Car?. Two-phase flow operations are described by a generalized model which assumes back flow, superimposed on the net flows through a column, with perfectly mixed stages in cascade. The diffusion model, which is used extensively to describe longitudinal dispersion, is derived as an extreme case of the back-flow model. The perfectly mixed stage (or cell) model is derived as another extreme. It is shown that the dispersed phase for these models may be treated as a second continuous phase. The nature of the longitudinal dispersion coefficient is also examined.
XIAL MIXIVG EFFECTS in agitated countercurrent equipAment may be described by a detailed analysis of back flo~vs between discrete segments of the ’.cascade.” frequently with greater rigor than bv assuming a differentially continuous diffusion model with a constant axial dispersion coefficient for each phase. For single-phase operations. the relations bet\veen a multicompartment (or mixing-cell) nonequilibrium model and the diffusion model have been explored widely. although usually tvithout considering back flow For tnophase operations. relative to the diffusion model. the cell model is underdefined if back flowc are neglected, and it is overdefined if they are specified for both phases. Thus. it is worthwhile to explore the mathematical relation between these models in some detail. .4n added justification for this Lvork is the one of providing adequate background for design calculations that can take into account the axial dispersion effects in countercurrent operations. For over-all calculations under conditions of constant mass transfer coefficients and linear equilibrium. integrated solutions based on the diffusion model are now available to describe the system. If. however, the parameters are not held
304
I&EC FUNDAMENTALS
constant. a stepivise numerical calculation must be undertaken lvhich inherently resembles the cell-model treatment. Figure 1 shorvs the “back-flow” model schematically. It consists of n,, perfectly mixed stages with stage height LO,each having the same volume. Exchange of material bettveen two adjacent stages is due to net floivs. F , and F,, of main streams and an additional back floiv, F. of the mixed phases. which occurs in each direction and is the sum of individual-phase back flotvs of F, and F,. Thus. the total flows bet\veen adjacent stages are ( F , f F, f F,) and ( F , iFu f F,). For the limiting case of F ( = F, f F , ) + 0, this system reduces to a “stage model” (of perfectly mixed cells in cascade) ty-pified by the usual mixer-settler extractor. For another limiting case. ivith n p >> 1, it will be shown later that the system reduces to the “diffusional model” rvhich assumes mean diffusivities and mean velocities for both continuous and dispersed phases (77. 2 2 ) . X particular case of this model has been utilized by Hill (7) for calculations o n salt-metal extraction processes. Sherivood and Jenny (27) and Colburn (2) have utilized a similar concept to treat the effect of entrainment on tray efficiency. For
single-phase flow Latinen and Stockton (70) have discussed the relation benveen the model and the diffusion model. Sleicher (23) has developed a similar treatment for a mixer-settler extractor with interstage entrainment. Dispersed Phase Behavior
For two-phase floir operations in a perfectly mixed stage, one phase is usually dispersed into the other in the form of bubbles or droplets. If enough coalescence and redispersion take place, the concentration of each droplet is the same, and the dispersed phase may be considered as a second continuous phase. If not. the over-all rate process in the stage should be treated on the basis of the residence-time distribution of droplets and of [heir concentration distribution as they enter. To formulate the rate process, phase X i s taken as continuous and phase Y as dispersed. The direction of mass transfer is from phase X to phase Y. (The final conclusion is independent of these arbitrary choices.) iVe consider first the limiting case of no concentration varialion from droplet to droplet in a given stage and later, in less detail: the case of no coalescence (and hence no redispersion) between droplets. Calculation Using a Mean Concentration. This case is consistent with the assumption that the dispersed phase behaves as a second continuous phase. When the equilibrium mp), the material balance relation is linear (that is, x * = b and the rate equation taken for the j t h stage give the following dimensionless relations :
+
!+here CY+ = F , F,. cyii = FY,’Fy SOrO = k,,aLa,’Pr. .Vou~ = k,,aLo F,; 27, is the mean Concentration of dispersed phase Y in (or leaving) thejth stage (the mean concentration is taken on a volume basis) ; and ko,a is assumed constant throughout the column
-
Behavior without Coalescence and Redispersion.
+
+
Two-Phase Flow Systems
Countercurrent Back-Flow Model. From the assumption that the dispersed phase can be characterized by mean concentration values and from a material balance around the j t h stage as diagrammed in Figure 1, the basic rate equation has the form of Equation 1. Solving this equation is tedious, as the solution contains five variable parameters: a,. cyy. n p , iV,O and b. A machine computation and an approximate calculation method have been presented by Sleicher ( 2 3 ) . Here, hoivever, instead of the equations being solved, they will be used to develop the diffusion model and to sho\v that the model is applicable to the behavior of the dispersed phase (\vith some restrictions) even vithout coalescence and redispersion of liquid droplets. Diffusion Model. The diffusion model equations (77: 22) utilize the assumption that the dispersed phase can be treated as a second continuous phase. Since this assumption has been found reasonably satisfactory for the back-flow model: its use in the diffusion model will be particularly justifiable if the latter model can be derived from the general back-Ao\v case. Such a derivation is shown in this section. ‘The diffusion-model equations in dimensionless form are as follo\vs: 0 ( 1 I P z B ) d2X/dZ2- dx/dZ - -VOz[x- ( b nly)] (2) ( l / P , B ) d2y/dZ2 dy/dZ Nov [ x - ( b my)] = 0 where -To, = k,,aL/F, and .Toy = k,,aL/F,. If lowest order central differences are used, Equation 2 becomes. for phase X :
+
F y tF
“P
X
nP
+ (1/2)1
=
.Vor[X - ( b
+ mgjl
(31
- xl) - X j - 1 ) = ~ ~ 7 0 z[XI o - (6 Wj)l (4) This equation is essentially the same as Equation 1. Ishen the following equality is satisfied for phase X (and also for phase Y; i = io r p ) : [(npiPrB)
\YF
a b Back-flow model for countercurrent operation
a. Multicompartment contoctor b. lndentification of flows, with internal flow F = Fz F,
+
-
Lvith another similar equation for phase Y . For a total number of segments t i p , the size of each segment (AZ) is equal to l,’n,. iVith this equality, Equation 3 is transformed to : i
Figure 1,
- X~-l)/(2IZ)
(Xj+l
+ +
+
+ xj-i),’(AZ)’
( I I P z B )( ~ j - 1- 2xj
n -I
P
For
this case, in any given stage the concentration of each droplet is different, depending on the time it has been in the stage, its size, and its entering concentration. T h e equations developed here (see Appendix) are a generalization of previous work (75, 76). Uniform Drop Size, Consider the I t h stage, under a steady continuous operation with flow rates of F , F: and Fy F , where the contents are mixed perfectly, with a uniform drop diameter d, and a uniform volume fraction cu for the dispersed phase. Assume further that the partition coefficient m is constant and that the over-all coefficient of mass transfer is a constant. As shown in the .4ppendix. any assumed concentration distribution in a stage will determine the mean concentration. and integration of the changes that occur in the input concentration distribution leads to an output distribution which conforms to Equation 1, If some coalescence and redispersion d o occur. as has been observed for agitated liquid-liquid systems (74: 24, 25): it may be possible to relax some of the restrictions just stated. and still apply Equation 1. These conclusions apply even for the stage model (ay = a , = 0). Thus. if the foregoing conditions are satisfied. it is entirely permissible to treat the dispersed phase as if it were a second continuous phase.
(xj-1
[ ( % J / P z B )- ( I D 1 1
+
(XI
-1- - L + 3 P;B or
272,
np
lim ( P i B ) = n P / a i
nr +
(5)
m
VOL. 2 NO. 4 N O V E M B E R 1 9 6 3
305
The boundary conditions (77, 22) a t the two ends of the column for solving Equation 2 are derived from the end conditions for the back-flow model, by putting AZ-, 0. For single-phase flow with n p >> 1, Latinen and Stockton (70) have derived Equation 5 from Einstein's "random walk" diffusion equation (5?79), and they thus relate a longitudinal dispersion coefficient to the rate of change of a series of discrete fluid displacements. For a finite number of stages: they adopt the following form for both physical transients and homogeneous first-order reaction :
Use of the term 2 ( n p - 1) is based upon Kramers and Alberda's treatment for the cell model ( 8 ) . Equation 5 or 6 also carries out the reverse reduction of Equation 1 into Equation 3. which is then converted into Equation 2 with a sufficiently large number of segments (n, >> 1) ; this procedure justifies applying the diffusion model to the dispersed phase and also renders the diffusion model applicable to a stagewise system. Kevertheless, these two models are basically different from each other; hence, the conversion relation will vary somewhat, depending upon what basis is taken for comparison. Note that Equation 5 applies only for n p >> 1. Obtaining more accurate conversion relations is discussed below in three special cases to examine the conditions under which the two models behave identically.
a t infinite With this limiting condition the diffusionmodel solution. as given by Miyauchi and Vermeulen (77), is:
with 1
A
1
P,,B=P,B+p,B and the back-flow model solution by Sleicher (22), originally obtained for a multistage mixer-settler extractor with entrainment and with each stage at equilibrium, rearranges to the form :
with
p
=1
- (1 - A ) / ( A a z
+ + 1) CYy
at = F t / F t
and
Equating the above two relations. the exact conversion relation for phase 1 ( = 1 o r ) ) is:
where
Conversion Relation for Transient Behavior
One workable and representative link between the two models is provided by comparing the variance for residencetime distribution of fluid elements. The procedure used by van der Laan ( 9 ) gives the variance u D > of residence times for phase i, based on the diffusion model: (1/2)
UDL2
=
1 !P,B - (1/P,B)2 (1 -
e-?
(7)
withi = x o r y . The basic transient equations for the back-flow model are written without difficulty for phase i from the material balance taken for each stage. The variance uB: of residence times for phase i is then given by solving the transient equations in a similar manner:
where
As a matter of definition. the mean residence time B T z for phase z is erL = L / ( F ,'ei) in the foregoing treatment. The conversion relation for phase 2 , based on the variances, is obtained by setting uD? = ug2?. The following simple empirical equations express the equality almost exactly for the entire range of n p , a,. and ay:
For n p >> 9, Equation 9 reduces to fiquation 5 as expected. Conversion Relation for Two-Phase Mass Transfer
Another workable link between the two models is provided by equating the extents of mass transfer for countercurrent flow 306
I&EC FUNDAMENTALS
The correction factor f T has essentially the same form as J and is shown in Figure 2 as a function of .i and Aa, ay With (h, a,) > 0 5. fr is nearly equal to 1. irrespective 01 A. Equation 12 may be used to compare the countercurrentdiffusion model solution with the back-flow model result at For this comparison, calculations \\ere finite values of .Yo, made at the relatively severe conditions of n p = 2 over the range of variables of 1 -VOz m , 0.56 < (.la, a,) 32. 16. Under these conditions. the fraction and 0.0625 < 1 uneatracted usually agreed to well within i 5 % in its absolute the approximation imvalue. At increasing n p , a ? . and -Yo=, proves rapidly. Agreement to \tithin *lo% was obtained under the same conditions by taking fr = 1 throughout. Application to Liquid Extraction. Take as an example the operation of pulsed perforated-plate columns; a similar treatment should be applicable to rotating-shaft equipment such as the RDC or Mixco extractors.
+
+
