SUBSCRIPTS
D
= overhead (collapsed foam)
L = liquid draining into liquid pool at bottom of column W = liquid pool a t bottom of column
literature Cited (1) Brunner. C. A.. “Foam Fractionation.” Ph.D. dissertation. \
t
6) Lemlich, R., Lavi, E., Science 134, No. 3473, 191 (1961). 7 ) McGauhey, P. H., Klein, S. A., Palmer, P. B., “Study of Operating Variables as They Affect ABS Removal by Sewage Treatment Plants,” Sanitary Eng. Res. Lab. Rept., Univ. Calif., October 1959. (8) Schnepf, R. W., Gaden, E. L., Jr., J . Biochem. Microbiol. Tech. Eng. 1, 1 (1959). (9) Schnepf, R. W., Gaden, E. L., Jr., Mirocznik, E. Y., Schonfeld, E., Chem. Eng. Progr. 55, No. 5, 42 (1959).
,
Universit; of Cincinnati. June 1963. (2) Chem. Eng. 68, No. 7 , 100 (1961). (3) Zbid., 69, No. 9, 62 (1962). (4) Husman, W., Textil-Rundschau 17, 88 (1962). (5) Lauwers, A., Joos, P., Ruyssen, R:, Vortraege Originalfassung Intern. Kongr. Grenzflaechenaktive Stoffe, 3rd, Cologne, Vol. 3, p. 195, 1960.
RECEIVED for review March 15, 1963 ACCEPTEDJune 7, 1963 Division of Water and Waste Chemistry, 143rd Meeting, IACS, Cincinnati, Ohio, January 1963. Work supported directly by U. S. Public Health Service Research Grant WP-161 and indirectly by a National Science Foundation Cooperative Fellowship.
A DIFFERENTIAL STAGE CASCADE DESIGN
FOR SEPARATION OF T W O COMPONENTS
IN A DILUTE SOLVENT G ERA LD T
.
FISH ER
’
AND J 0
HN W
.
P R A D 0 S, The University of Tennessee, Knoxville, Tenn.
The equations of a differential stage cascade are formulated for the separation of two components in a dilute solvent. The solution is presented as a definite integral of unknown general solution; a program is presented for the solution of the cascade differential equation on an analog computer. A specific solution is presented for a thermal diffusion cascade separating CuSOl from Cos04 in HzO. HE SEPARATION of mixtures in mass transfer cascades is Tquite common. The distillation tower is a n example of a cascade of equilibrium stages; it operates usually with all stages of equal size and constant flow rate from stage to stage. A single tower operates with no internal reflux on each stageLe., all of the products from one stage go initially to another stage. A series of distillation towers can be operated as a cascade with unequal size stages and unequal flow rates from stage to stage (7); the purification of heavy water has been performed by such a system ( 2 ) . The advantage of such a system is that it minimizes the total size of the equipment required to perform the separation; this is obtained a t the expense of increased heat loads for a distillation cascade (2,3). In this article, the equations of a cascade for separation of two components which are in a dilute solution with a third component, a solvent, will be considered. The equations were developed for the thermal diffusion separation by horizontal columns (Jury-Von Halle type) of two salts in water solution (4), but the equations will apply to any system which can be characterized by a constant mass transfer coefficient and a linear equilibrium relation.
Equations for a Single Stages of the Cascade I t is assumed that the rate of mass transfer for each component is J = ~ L ( x *- X ) (1) and that the equilibrium relation is y = Kx* If a single unit of the cascade is operated as shown in Figure 1, a material balance on a differential section for constant L is L dx J = - - (3) aS dz 1
300
Present address, Vanderbilt University, Nashville, Tenn. l&EC FUNDAMENTALS
The material balance over the top section of the unit is V y = LX
+D
x ~
(4)
By combining the first four equations, one obtains
where
Equation 5 may be integrated from x ( 0 ) = x , to x ( Z ) = xD, to obtain, for constant L, V , K , and h,,
(6) where R = L / D and Z is the total length of the column. If one defines the overhead fraction 0 by
e = -D
v
1 R+1
(7)
the equation becomes
For component a, this quantity is defined as w : thus w = - aXL oXD
(9)
If the ratio of the component concentrations, A , is defined as
A L ( s - 1) =
n(s- l]AD(a-1)
If Equations 14, 15>and 16 are combined, one has
If one adds and subtracts 1 4 L ( 8 ) / Q ( s - then one has Single stage of the cascade
Figure 1.
If the number of stages is large and the chznge in concentration ratio per stage is small, then the system may be described with small loss of accuracy by considering the concentration ratio, A L ( E ) ,to be a continuous function of the variable s, and the difference in concentration ratios becomes approximately equal to the derivative ofA,,(,j ltith respect to s. Thus,
I
I I
I I I
I
I I I I
I
I
I
!I I I
I I
I
I
I
I I
II I
I
I I
I I
I
I
I
I
I I
?=$
STAGE S t I
Figure 2.
The cascade
and then
A material balance on the feed line of stage s + 1 is I1 -
~,s+?!lQ(sA?)
+
~ts)Q(s)= Q ( ~ + I )
(21 1
An examination of Equations 5: 8: 10, 2nd 12 shows that each stage of the cascade can be opereted with a constant cut 0 and have a constant stage sepzration factor Q for eech stage if the mass transfer area-to-flow-rate rz-tio P ( P = ZSa/L) is constant for each stage. The restriction of constant 0 for each stage of the cascade will be made in the development to follow. TVith this restriction, Equation 21 is a second order difference equation Ivith constant coefficients, 2nd the "initial conditions" are Q(o) = 0
then the separation of the column may be expressed in terms of the stage separation factor, Q, as AL(8)
=
n AD(.)
(11)
T h e quantities therefore define the separation in the L stream. Let the fraction of component a that appears in the D stream leaving a stage be denoted by V ; thus
The complete solution of the difference Equation 21 is then
Q(8) P
=
zs
Y
= 1/2
(24)
The differential Equation 20 then becomes
If Q(\) is the quantity of component a fed to stage
s. then Y ( \ ) is the quantity removed in the D stream and [I - Y ( ? ) ] Q is (~) the quantity removed in the L stream. The quantity Y ( is related to 0 by Q(s)
foru # 112,andforv = 11'2,
2 dA = - (n ds
The usual arrangement for cascades is foi the products of stage s to be fed to stage s f 1 and to stage s - 1. ds sho\\n in Figure 2. A material balance of component n over the first s - 1 stagesis vis)lQ(r)
=
v(s-IjQ(s-1)
\\here P is the o\ erhead product rate. component b is A ~ r a 11 i
-
Y t
] Q(aj
=
1)AL
+n
[AL(l)
-
ALI
s - 1
IVith total cascade reflux. P = 0, and the differential equation is
Equations for the Cascade
-
-
A D ( * -i j Q ( a -
+P
(14)
A material balance for
1 1 ~ ( a 1) -
+ PAL,])
dAL ds
-
(a- l ) A L
(27)
This total cascade reflux can be obtained for arbitrary v by returning the product of the first stage to its feed. By use of the integrating factor method. it may be shown that the solution of Equation 25 is. for Y # 1 2
(15)
The s e p r a t i o n ofsrage (s - 1) is expressed by the relation VOL. 2 NO. 4 N O V E M B E R 1 9 6 3
301
Figure 3. Analog computer circuit for the solution of the cascade differential equation
I -
A
where
g
1-lJ = y
(29)
-100
for v # 1/2, and for v = 1/2 is
and
= 2 ( N f 1)
The quantity A L ( N + l )is the composition of the feed to the cascade, and A L ( l )is the product composition. For the case where Y = 1/2, thesolution of Equation 26 is
(35)
Equipment costs-and in the case of thermal diffusion, processing costs-are proportional to 2 Q ( 8 J / Pwhile , feed costs are proportional to F/P. Hence, one wishes to minimize both of these quantities. Consideration of an Ideal Cascade
As no analytical expression for either of the above integrals is known, the integrals must be evaluated numerically for specific cases. An alternate attack on the solution of the differential equation is to program the problem for a computer; the problem was programmed for a n analog computer and the effect of variable Y on solutions of the equation was investigated. The basic computer circuits for Equation 2 5 are shown in Figure 3. The total flow in the cascade per unit of product is given
The operation of an ideal cascade means a method of operation such that no streams of unequal concentration ratios are mixed. This implies that AL(a+l)
A quantity
(p
= Arc,) =
AD(s-1)
(36)
may be defined by AI7($) = 'PAD(,)
(37)
It can be shown that p is constant for all stages if 0 is constant. If the definition of p is combined with the relations for a n ideal cascade, then one obtains
by
AD(a+n)
= 'P-"AD(a)
(38)
AV(,+,,)
= 'P-nAV(*)
(39)
and AL(a) = 'P2AD(s)
for Y # 1/2, and for N
Y =
Qy
=
5
=1
1/2 is
-& =
N ( N + 1)
(33)
s= 1
The quantity of component a that must be fed to the cascade per unit mass ofa in the product stream is
I&EC FUNDAMENTALS
nAD(8)
(40)
If Equation 4 is written for component a and for component 6, the ratio of the two equations may be rearranged to give w
e=fJJ
302
=
+ llco
If Equation 41 is substituted into Equation 13, then one has
A Cascade Example The differential equation for the cascade W F . S solved with the following operating conditions, which were determined experimentally for a horizontal thermal diffusion device (JuryVon Halle column) for the separation of CuSO4 and Cos04 in H20 by Fisher, Prados, and Bosanquet ( 4 ) . KCuSO,
= 0.944
KCoSO4
=
0.962
Z/hL-cuS04
=
6.54
Z/hL-CaSOr
= 7.08
T
With a n area-to-flow-rate ratio of s = 0.053 sq. ft./(mL/hr.), one has that the total area, B j is
B = 12'6 IO6liters/day X 0.053 sq. ft./(ml./hr.) X 24 hr./day l o 3 ml./liter =
28.8 X 106sq. ft.
= 0.288 X 106 sq. ft./(lb. of product/day)
At a construction cost of $50 per sq. ft. the capital investment C, would be Cz = 28.8 X 106 sq. ft. X $SO/sq. ft. = $1.38 X 109
or per unit of product
= A / ( L / p ) = 0.053 sq. ft./(ml./hr.)
CZ = $13.8 X 10B/(lb. of product/day)
After the separation parameters are determined from the operating conditions, the number of stages required for the desired separation may be calculated from the solution of differential Equation 25. The values of the cascade variables for a fixed separation are given in Table I.
Values of
c%
and F / P are calculated from Equations 32 through 35 for the corresponding values of v and R; the number of stages, A', indicated \vi11 give a change in salt ratio between feed and For exprodurt b>- a factor of lO--i.e.? A L ( l , = 10A,(,+;). ample, such a cascade could produce a product containing 0.05% of CoSO, from a feed containing 0.5% of Cos04 in CuS04. Table I shows that
cy
or =
12.6 X 106 liters/day
Table 1.
Values of the Size-Determining Factors of a Cascade
V
0.534 0.528 0.521 0.518 0.512 0.508 0.504 0,500 0,480
1.036 1.037 1.038 1.038 1.039 1.039 1.040 1.040 1.042
'0
75 7':
80 87 100 125 250 m
C, = 28.8 X 106sq. ft. X 53.00 X 10-3/hr.-sq. f t . X 24 hr./day = $2.00 X lOe/day
or C,
207
75.6 18.9 12.8 5.6 5.2 5.1 6.2 m
264,000 80,500 15,500 8,900 4,160 1 ,440 915 502 25
=
520.0 X 103/lb. of product
The feed required would be F = 50.2 X 103/lb. of CuS04/day
goes through a rela-
tively flat minimum in the region 0.500 < v < 0.518. F / P decreases with decreasing v. Hence, unless feed costs were inordinately low, it would appear desirable to operate the cascade in the neighborhood of v = 0.500 in order to minimize the total cost of a unit of product. If v is increased greatly, the number of stages would increase, as the sta.ge separation factor R decreases Ivith increasing V. C n the other hand, a situation of minimum reflux is reached if v is decreased; a value of u = 0.48 in this problem prevented the product to feed composition ratio from becoming larger than 1.5. In order to calculate the area of a cascade, the quantity of product desired must be known. For the example, let P = 100 pounds per day = 181 gram-moles of CuSO4 per day. If the maximum permissible concentration in the feed is restricted to 0.90 molar, then the total interstage flow, for v = 0.5, is
1 - X 6.25 X IO4 X 181 gram-moles/day 0.90
With the capital investment amortized over a 5-year period, the sum of fixed and operating costs on a daily besis. C,, is computed from a value of fixed plus operating costs of 3.00 mills per hr.sq. ft.
or F
=
502 Ib./lb. of product
For the cascade considered, interstzge dilution would be required since total salt concentration tends to increase toward the product end. It was assumed that process water would be available a t a sufficiently low cost to have no effect o n the large separation cost above. Comments on Economics
The costs involving a thermal diffusion process for this separation in a horizontal column ere extraordinarily high. This separation is a three-component sepzration in \+hich the two solutes (salts) diffuse in the same direction with respect to the solvent (water) ; the separation is obtained onl) because of the slight differences in thermal diffusion rates. The separation of the solutes from the solvent is more feasible. but this was not the separation of interest in this case. The economics of the separation of one component from a mixture is treated by Grasselli, Brown, and Plymale ( 5 ); they considered the purification of hydrocarbon oil in a Clusius-Dickel unit. Their costs are in line with other commercial processes. The separation parameters used in the example are somewhat limited in magnitude because of the temperature limits of the solution; the cold temperature is governed by the freezing point and the hot temperature by boiling temperature. A hydrocarbon separation would probably not suffer such limitations. Also, the parameters are limited by diffusion through a membrane in the Jury-Von Halle column; considerable room remains for improvement of membrane characteristics. Acknowledgment
This work was carried out under a resezrch contract with the Chemical Technology Division of Oak Ridge National Laboratories (operated by the Union Carbide Nucleer Company). The authors wish to express their gratitude for this support and for permission to publish results of the study. VOL. 2 NO. 4 N O V E M B E R 1 9 6 3
303
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 Fu f F,). adjacent stages are ( F , f F, f F,) and ( F , iFor 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
