LONGITUDINAL DISPERSION IN TWO-PHASE CONTINUOUS-FLOW OPERATIONS T E: R U K A T S
U M
I Y A UC H I
,
University of Tokjo, Tokyo, Japan
T I4 E 0 D 0 R E V E R M E U L E N , Department of Chemical Engineering, C‘niaersity
oj’ Caltfornia, Berkeley, Calif.
In two-phase flow operations, longitudinal dispersion (or axial mixing) has an undesirable effect on equipment performance, (especially if a high degree of completion i s needed. Mathematical solutions are developed here for general and specific cases of longitudinal dispersion accompanying mass transfer or heat transfer, calculated from a simplified diffusion model that assumes a mean longitudinal-dispersion Coefficient and a mean velocity for each phase. Concentrations within the equipment and at the outlet are shown to depend upon four dimensionless parameters. These are functions of the dispersion rates and velocities, the equilibrium partition (coefficient, and the “true” over-all mass-transfer coefficient. Based upon computed tables now available, graphs are included to show how the parameters are related. An empirical correlation i s given for “exterior apparent NTU,” which serves as a useful design factor.
s CONTISUOUS-flo~ operations such as absorption, extraction,
I and chemical reactors, it is generally recognized that the
effective rate coefficieni: is lowered by the phenomenon of longitudinal dispersion. Although much attention has been paid to dispersion in single-phase flow operations, very limited study- has been made so far in continuous-flow two-phase operations, where a reduction of the effective rate coefficient again results. This effect is specifically mentioned by various authors in connection with countercurrent solvent-extraction columns of the spray type (78! 79: 22, 25; 47), stirred-compartment type ( 3 4 , rotating-disk type (38, 39), pulsed packed and sieve-plate types (3>7, 75, 47, 44: 4 6 ) , and the Scheibel type (401 ; much additional \vork has been reported recently. A similar effect also should be observed for other continuousflow two-phase column operations such as absorption, distillation, heterogeneous reaction, and even heat transfer. I n relation to the behavior of spray extraction columns, Geankoplis and co\vorb:ers (78: 79, 25, 47) and Gier and Hougen (20) have undertaken to obtain the coefficient of mass -ransfer by measuring concentration distribution within the columns. ‘This approach should be more adequate than the alternative of using a logarithmic-mean driving force, computed only from the end concentrations of the incoming and outsoing streams. A theoretical approach to permit evaluating local behavior of pulsed columns has been reported by \.ermeulen, Lane, Lehrnan, and Rubin (46); experimental measurements of dispersion in packed extraction columns have been performed by Jacques and l-ermeulen ( 2 3 ) .and in packed absorption columns by Dunn, Vermeulen, \Vilke, and Word ( 72). T o correlate the behavior of t\vo-phase-flo\v column opsralions, it is necessary to analyze their dependence upon such variables as the true coefficient of mass transfer, rate of longitudinal dispersion of fluid in each phase, flo\v rate of each stream, partition coefficient of the transferring component, and extent of approach to equilibrium. This paper gives the results of such analysis: applied mainly to countercurrent systems and also briefly ti> cocurrent rases.
Flow Model
A simplified model of flolv behavior has been used: similar to that assumed previously (7-5, 70, 7 7 , 73, 77. 17-24, 26. 27, 33, 37, 49, 50) for a one-dimensional homogeneous-phase flow system. This model, like the simplest mathematical approaches to packed columns, can be described as ”differentially continuous.” I t has been applied previously to multicompartment columns by Yagi and Miyauchi (50) and by Nagata and coworkers (33). The model is also applied to nvo-phase counterflow operations by- Eguchi and Nagata ( l J >75)and by Otake and coworkers ( 3 5 , 3 6 ) ,although the analysis is confined to including the influence of longitudinal dispersion of one phase only. Recently, Sleicher (43) applied essentially the same concept as in the present work, using a computer for numerical computation but without deriving analytical solutions. .4n alternate approach to longitudinal-dispersion problems is provided in the “equivalent completely mixed stage” concept advanced by Young (57), Epstein (76). and others. These two methods should supplement each other, each being perhaps more applicable to certain types of equipment. Mathematical relations between them are discussed else\vhere (37). The analysis developed here from the ”differentially continuous” model should be useful to correlate experimental data, to apply them in scale-up, and to interpret the behavior of various kinds of columns. In this model. the t\vo phases flow in generally opposite directions Lvith uniform superficial velocities F, and F,, with each phase undergoing longitudinal dispersion. In many actual two-phase-flow operations, one phase remains dispersed in the other in the form of liquid droplets, gas bubbles, or a spreading liquid film, depending on the combination of fluids employed. The basic equation expressing the behavior of the continuous phase has been shown to explain the experimental results for the L h c o and other similar columns (29, 57). There is more question as to the conditions under \vhich the model fits the dispersed phase, because the basic equation VOL. 2
NO. 2
MAY
1963
113
requires that all of the dispersed phase (in the form of droplets, bubbles, or liquid film) present a t a given level in a column have the same concentration. If the droplets or gas bubbles coalesce and break up so rapidly that there is no fluctuation in their concentration at any particular point, the model fits the actual situation. If not, the concentration variation between droplets or bubbles at a section will reflect the gradient of average concentration under conditions of constant E, and E,. Even in the latter case the simplified model is believed to provide a workable approach. Basic Equations
For single-phase continuous-flow systems Damkohler (9) has given the equation of continuity, d c j h =
- div( - D j grad
-div
cj)
-
(u cj)
- $(e7)
=
0
Two-phase countercurrent systems can be treated by a n extension of this equation, using the following dimensional variables : FOR EACHPHASE : superficial velocity, Fi; superficial axial dispersion coefficient, Ei, analogous to a diffusivity ; mixing length, li = Ei/Fi; concentration, ci, or temperature, t i (subscript i designating the ith phase). FOR THE T w o PHASESTOGETHER:over-all transfer coefficient, k,i> or height of over-all transfer unit, H,i = Fi/k,ia, expressed relative to either phase; partition coefficient rn = dc,*/dc, (which is unity in heat transfer); length within the column, z , measured from the X-phase inlet; characteristic length, d-for instance. particle diameter d, in packed beds; interfacial area? a , per unit volume; total column length, L. When the reaction rate, $(cj), is replaced by a mass-transfer rate linking the two phases X and Y , we obtain: Ezd2c,/dr2
- F,dc,/dz - k,,a
E,d2c,ldzZ
+ F,dc,/dz
[e,
- ( q 4-
f kora [e, - ( q
me,)] = 0
+ me,)]
(1)
= 0
with equilibrium distribution represented by the linear relation c,* = q mc,. From the foregoing dimensional variables, nine dimensionless variables are defined :
+
1,2. Column P6clet numbers, P,B and P,B> with B = L/d, and Pi = d, 'It. 3. Capacity ratio, or extraction factor, A = mF,/F,. 4. Number of true over-all transfer units (relative to the X phase), No, = L 'Hoz= k,,aL/F,. 5. Fractional length, Z = z / L . 6,7. Dimensionless "concentrations," Ci = ci/c,o or tL/t.(J, where superscript 0 denotes the feed value; also, Q = q/c.O. 8,9. Generalized concentrations X and Y ?with
X = Cz - ( Q 1 - (Q
P,B
=
F,L/EH,, P,B
=
F,L/EH,, h
-?To= = uaL/cHzp.Fz, k,, =
=
C H ~ ~ ~ F ~ / C H ~ P ~ F ~
~ / C H ~ m P ~=,
1.0, Q = @
where EHr and E H , are the effective longitudinal thermal diffusivities for phases X and Y , respectively, and U is the over-all coefficient of heat transfer across the phase boundary. Boundary Conditions. The rate of longitudinal dispersion in a column is assumed to be much higher than that in the incoming and outgoing streams away from the column. Suitable boundary conditions are obtained as follows. Integration of Equation l for phase X,over an arbitrary length of column, h, gives
- il.(ci)
where u is the linear velocity of the fluid, c 1 is the concentration of the j t h component at the point of interest, D, is its diffusion coefficient, T is time, and $(cj) is a reaction-rate term. For one-dimensional steady-state flow systems in which constant velocity and constant diffusivity of the j t h component can be assumed. Damkohler's equation becomes Djd2Cj/dz2 - udc,/dz
be applicable under certain circumstances. The dimensionless parameters in Equation 2 should be modified as follows. as can be seen from the equation of continuity:
+ mC:) + mCi)
where c z 0 = ( c ~ ) ~ - ~ cih ; = ( ~ * ) ~ , h ; and n is the total amount of component (in moles) transferred from phase X to phase Y between the point of interest and the X inlet, per unit cross section of flow. At the X-inlet end of the column. the net flow is given in dimensionless terms by the sum
-
Outside the column, the net flow is -CZo. or -1. because m and (dC,/dZ)o 0 for the model assumed. Equating the net flows, a t Z = 0, gives
PJ3
-f
-
(g) =
P,B (1
-
Czo)
At the X-outlet end, a comparison of the net flows inside and outside yields the relation
&.(!g)
- C,'
= C,1
1
where C,1 = 1, and C ', is the concentration of outgoing phase X. The coefficient l,'P,B is always positive or zero. At the boundary. the concentration gradient calculated from the left-hand term is opposite in sign to the gradient given by the right-hand term. Thus the only condition allowed by this equation is C,i = Cs'. (dC,/dZ)j
= 0
(3b)
Because the boundary region is small. mass transfer in this region is neglected. and the boundary condition becomes identical with the result given by Danckwerts (70) and others
(49,57). The boundary conditions for phase Y are derived from considerations similar to those described above. Thus, the boundary conditions for phase X and phase Y are Z
=
0 : --(dC,/dZ) = P,B (1 - C,o),
Z
=
1 : -(dC,/dZ)
=
- (dC,/dZ)
=
0
0 , -(dC,,/dZ) = P,B(C,i - Cul)
(4)
(2)
Characteristics of Longitudinal Dispersion. Before we develop the solution for Equation 2, it will be helpful to examine some of its characteristic properties. Figure 1 shows the concentration distributions calculated for two particular sets of parameters that illustrate the effect of changing A',,, and indicate the following general behavior :
When heat transfer takes place from phase X to phase Y , even through a n intervening pipe wall, Equation 2 will again
The concentration driving force between two phases is obviously lowered by longitudinal dispersion, but not so much as was expected originally. Actually, calculation shows that
Rearranging Equation 1 into dimensionless form, we have : d2C,/dZ2 - P,B dC,/dZ - -V,zPzB r[Cz d2C,/dZ*
114
+ PUBdC,/dZ + -!r,,uP,B
l&EC
FUNDAMENTALS
- (Q
f mC,)]
=
0
[C, - ( Q 4- mC,)] = 0
the concentration driving forces a t both ends of the column taken here are higher than those for piston flow case under the same Thus, the poor completion of mass transfer due to longitudinal dispersion is partly attributable to the decrease of concentration driving force; and partly also to longitudinal transport of the transferring component. induced by longitudinal dispersion. T h e concentration of each incoming stream increases or decreases abruptly (‘.juinps’’) at the time it enters the column. I n contrast, the concentration pattern for each outgoing stream becomes flat as it approaches its outlet, and no discontinuity is observed a t this end. At increasing Cz0and mC,1 (resulting from the jumps) and finally are controlled rapidly become insensitive to only by P,B or P,B. At increasing .TOT, the extent of completion of mass transfer (indicated by CZ1and Cw0)approaches a n upper-limit value that also is controlled by P,B, PUB, and A. If an over-all column PCclet group P, ,B is defined by the relation
0
0
0.2
0.4
0.0
0.6
1.0
Z Figure 1 .
Concentration distribution P,B = P U B = 4
A = l No, = 5
- - - No, =
the independent variables (besides LVoz)are reduced from three to two (PouBand A ) , These two serve to fix the values of = m. the extent of completion. X:that apply a t Z = 1 and Such values of X ( = .Yl) can be calculated from equations given below (Table I. Case 9) and are plotted in Figure 2. If a n available column with known P,B and PUBbehavior is proposed for use on a system with specified A and XI,Figure 2 will indicate in some cases that the desired separation is impossible when considered solely from the standpoint of longitudinal dispersion.
Analytic Solution. The solution of Equation 2: satisfying the boundary conditions (Equation 4), is given as follows. Details of computation, including those for various special cases, have been published (30). GENERALCASE. Capacity Ratio or Extraction Factor A( = mF,/F,) # 1.
100
0 99
709b -
z 2 096 I-
o 094 Q
K 092 090 X W LL 0 080
+
E$
070 060
w 040 020
(5)
-
1 1
Figure 2.
No,
1 1 1 1
I
’
I
1
Limiting extent of approach to completion for
03
where
( 3 = 1, 2, 3, and 4 ) A,
=
0
+2 +2 (213 + 2
X l = a/3 AT = A4
=
a/3
P 4
and
4 cos ( u / 3 )
with
+ 2*/3) 4 cos ( u / 3 + 4 s / 3 )
4
COS
6
cos u = q/$?“
=
*/ =
T.
such that
+ P/3 + aP/6 + r/2
(a/3)?
- P,B ATozPzB P,BP,B (1 - A ) iVo,PzBP,B
a = P,B
(u/3
where u is determined as a n angle between 0 and
= (a/3)2
+
+ .\-,rP,B
The solution is obtained as Equation 5 only for q2
-
p3
=
zs1 (a3? - cu2P/4 + 9 a c y l 2 VOL. 2
NO. 2
- B2
+ 27 r2/4) < 0
MAY 1963
115
This relation is satisfied for ordinary mass-transfer operations, except when both phases are mixed perfectly. I n this case, the relation becomes zero. USIT CAPACITY RATIO(OR EXTRACTION FACTOR). A X ( = mF,/F,) = 1. I n this case the solution takes a different form :
Types of possible application for each case in Table I may be summarized as follo\vs. Further extensive investigations are needed to complete the description.
CaJt
Types of Possible Application
1
Liquid-liquid contactors such as Mixco, pulsed, rotatingdisk, Scheibel, and packed columns. Gas-liquid contactors such as gas bubbles through long column with mechanical agitation or gas-liquid contact by packed columns with liquid phase dispersed Dispersed liquid or gas phase in noncoalescing free flow through long column, without mechanical agitation. Gas-liquid contact with minute longitudinal dispersion of liquid gas phase by packed column Perfect countercurrent piston-flo\v operation ( Colburn equation) Larye gas bubbles of similar size through continuous mixing tank Small gas bubbles through continuous mixin? tank Perfect mixing \Vhoie rate-process controlled only- by either of P,B and PUB; one-dimensional homogeneous-flow reactors Maximum attainable completion of mass-transfer for given pattern of longitudinal dispersion
2
3 4 3
6 -7
8
"Apparent" and "True" HTU and NTU
SPECIALCASES. Solutions for various special cases, each involving a t least one less degree of freedom than in the general case, are summarized in Table I. Like the foregoing solutions, they are obtained in the general form
T h e analysis of longitudinal dispersion in a continuous column operation leads to three different definitions of HTU (height of a transfer unit), depending upon how one defines the effective concentration driving force. "True" Values. Based upon the original definition of HTU by Chilton and Colburn (6). "rrue" HTU must be defined as the ratio of the volumetric flow rate per unit cross section of column to the true over-all coefficient of mass transfer : Hoz = Fz/koza
(8)
A' = X(A',,, A , P,B, P U B ,2 )
Y
=
Likewise, the true number of over-all transfer units ( S T U ) is
Y(,Vox,.i, P,B, PUR,Z )
Since the definition of the X and Y phases is arbitrary, the solution for P,B = m and P,B = 0, for example, should be derived from the one for P,B = 0 and PUB = 00 (Case 4, Table I) by interchanging the X and Y phases. As is easily seen. this interchange satisfies the follou-ing inversion relations:
X
=
1 - Yt. (P,B) = ( P U B ) ' ; A = l/At; Z = 1 - Zt
Y
=
1 - X t ; ( P U B )= ( P z B ) t ; So,= .itAV,='
(7)
lvhere superscript denotes the value for X and Y phases interchanged. Therefore, the solutions for the above example can be derived by introducing the relations into the solution for Case 4-i.e., Xt
.\'[I =
for (PzB)t =
I
00
(-.v,>=~)I + exp ( -.v,,~z~) + &it- >itexp ( -.yo*+)
- exp
and PUB)'
=
0
I&EC F U N D A M E N T A L S
+
(?a)
Some other special cases omitted from Table I may be derived by a similar procedure. 116
"Measured" or "Interior Apparent" Values. If a column behaves in the same manner as the model assumed here, the concentration distribution for phase S in the column will follow curve ABDE of Figure 3. and curie FGHK will correspond to phase Y . These t\vo curves may be known by measuring the concentration distriburions in the column, as was done by Geankoplis and coworkers (78. 79, 23, 47) and Gier and Hougen (20). Thus one can define a different kind of NTU and HTU values from the measured curves for phase X and Y. These are called "measured" values and may be expressed in terms of the distance between the curves--Le., cz - (Q mC,):
Geankoplis and co\vorkers used this expression to discuss the behavior of spray columns, although their measurement was confined to the continuous phase. T h e reason it does not give the "true" N T U is that mass transfer is not the only
Table 1. Case
I
Finite
Finite
-_
Solutions
A
px p:, B --
-
Solutions for Various Cases (Counterflow Operation) X
Y
and
5 6
Equation Equation
# I = I
for
a = PxB + AN,
b = ( l-A)NoxPx5
----_ _ _ _ -- - -_ - - D = DI t I - A 2 / PxB ) exp A - A2( I - A /Px 5 1 exp A , 2
Finite
co
A
-
D,= [Nox/Px B
3
co
CD
4
0
CO
5
Finite
0
-
_All
6
0
0
7
Finite
F i nl t e
8
Finite
Finite
9
All
all
Note
:
A1.0
X=
I
Cx-
X=
Inversion R e l a t i o n s
I
( Q+ mC:)
I - (Q+mC$)
:
I
+
+ a ( l + l / N o x ) ] exp a , D3 = - I
ANOx
Y =
;
I + ( I + A ) No,
A Nox I +(l+A)Nox
A’s a s above , w i t h a
Y .
,
3=
(a/2) -
a = (P,B+
v ‘ m
No,)
( Single - s t a g e m i x e r )
Px B a n d b = No,
PxB
G
m ( CyI-(Q+mC:)
(PxBjt= (PYB) (P,Bjt=
factor producing concentration changes-that is, it is not a complete integral for Elquation 2. “Piston-Flow” or ‘LExterior Apparent” Values. Still other apparent values of HTU and NTU can be defined in terms of the logarithmic-mean driving force computed from the exterior incoming and outgoing concentrations a t both ends of a column:
At= I / A
( px 6 )
Nix
AN,,
may be understood by referring to Figure 3. C,P and C,P are related simply by material balance; integration of Equation 10 then gives the well-known Colburn equation (8, 4 2 ) , at A # 1 :
or, a t A = 1:
where C,p and C,p are the hypothetical concentrations for phases X and Y , that would be found in the column if both phases flowed through it in piston-type flow. This situation
These results correspond to Case 3 in Table I. I n general, one will find Hoip 2 Hod&2 H,, (z’ = x and y ) , because both Ha,, and H,i.u include the effect of longitudinal dispersion in each phase. Numerical Example. To illustrate the relation among VOL 2
NO. 2 M A Y 1 9 6 3
117
c x or mcy
t
Reverting to dimensional length we can write
A
c f ,a
= E
CXL
mcyL 1
m c YL
2.0
2.1
Figure 3. Concentration distribution column (schematic)
+
in extraction
ABDE. Actual distributon of c g FGHK. Actual distribution of c y AD‘€. Hypothetical distributian of c, assuming piston flow FH’K. Hypothetical distribution of c y assuming piston flow
Solid curves correspond to A = 0.8, No, = 3, N o r p = 1.3, PzB = 1.5, PUB = 3
these three HTL’s, the following numerical values are taken : A = 1.0, P,B = P,B = 4, N , , = 5, Q = 0, and C j = 0. Equation 6 yields the following concentration distributions.
C,
=
mC,
=
0.8110 - 0.51762
+ 0.3862 X
+
exp(7.49Z) 0.0209 exp( -7.492)
0.7080 - 0.51762 - 1.168 X l0-5exp(7.49Z) 0.0691 exp( -7.492)
(13)
Hence C,O = 0.832, and C,] = 0.362. The concentration pattern thus obtained is shown in Figure 1. may be obtained by graphical integration of Equation 9, using the numerical values for C, and mC, calculated from Equation 13: P0.832 J n
iVorpis calculated from Equation 12 for A
rnc; = 0 :
C,I Therefore,
=
0.362 = (1
= 1, Q = 0, and
+ NoZp)-’or N o z p = 1.76
2.
and setting .V,,dZ = d.Yoz,
Hence, based upon an experimental profile of C, us z , graphical differentiation to get the second derivative will enable the righthand term of Equation 15 to be integrated, so as to convert the “interior apparent” value, N,,vI, to the true value, No,, and subsequently to the coefficient. k,, Figure 4 indicates that the second derivative is generallv positive. consistent with .v,, > *V”*lI. Based upon the extent of separation calculated from parameter values with and without longitudinal dispersion. Figure 5 sho\\s the influence of P,B and on the ratio H r ~ , / H oatz A = 1.0 and P,B = P,B. The ratio increases \$ith decreasing values of the PCclet group, and with increasing -To=. This figure can also be considered a plot of the ratio .YOz/N,,~ against similar information for another value of A (A = 0) is given in Figure 12. Numerical Concentration Values
General Case. The foregoing solutions for longitudinal dispersion yield the relative outlet concentration or temperature values and also the profile of values for each phase throughout the length. Various sets of values of the parameters correspond to the same “exterior“ performance. or combination of inlet and outlet concentrations; but each set of parameter values corresponds uniquely to a particular pair of “interior” profiles Experimental knowledge of the complete profiles is therefore essential in proving that the theory does apply adequately to given types of flow equipment, and in confirming the numerical rates of dispersion and of interphase transport that are assigned. Numerical evaluation of the analytic solution is likewise essential in order to apply the theory either to the interpretation of experimental profiles or to the design of new columns. The solutions to the general and linear cases. Equations 5 and 6, were therefore calculated on an IBM 701 computer. Figure 6, in block-diagram form, shows the computation program that was used. Values of X and Y were developed and are available (ZB),to the fourth decimal place, for the following range of variables:
So, = 1, 2, 4, 8 ; in some cases also 16, 32, 64
Thus H o z and ~ Hozp express apparent H T U values which include the effect of longitudinal dispersion; H,, corresponds to the true coefficient of mass transfer, rather than actual column performance. Figure 4, based on the same numerical values of the parameters, shows how the different terms in Equation 2 (as written for the X phase) sum to zero a t each plane Z in a column. The positive terms represent the rate of net supply of solute a t each plane, and the negative term the rate of its removal by mass transfer. Without longitudinal dispersion the secondderivative term would be negligible compared with the others; the concentration term and the first-derivative term would then have equal and opposite slopes, and would each be more uniform than in the case illustrated. We note that Equation 2 can be written in the form No‘,.[C,- ( Q 118
+ mC,)]dZ
=
- dC,
l&EC FUNDAMENTALS
1 d2C, +P,B dZ2 dZ -
(14)
1 PzB = --No,, 4
No,, 4 N o ,
1 1 P U B = - P z B , - P,B, P z B , 2 P z B , 8 P z B 8 2
Z
=
0, 0.05, 0.15, 0.50, 0.85, 0.95, 1.00
The fractional-length, Z , values were selected to permit construction of the entire profile by graphical interpolation. The curvature of any profile increases toward the outlet end, since a t this point a zero slope is reached. Plots of typical profiles (with P,B = PUBin each) are given for the following cases : A = 1 : iVoz= 1 AND 4 (FIGURE 7). The uniformly varying piston-flow case is shown by the P,B = m lines. Perfect mixing in both phases (P,B = PUB = 0 ) , not shown, corresponds to a horizontal line a t X = (A X O z l)/(A Xoz .IToz 1)-i.e., 0.667 and 0.555, respectively. Because of the inversion relations discussed above, Figure 2 is also a plot of Y against Z but with the coordinates reversed.
+
+
+
-5
Ii,J 0
0.2
0.4
0.6
0.8
1.0
01
0
2
=
PUB
=
4
P,B =
-1 = 'i4' . N o , = 1 AND 4 (FIGURE 8). Again the range from no mixing to almost complete mixing is given; now the horizontal-line limits are X = 0.555 and X = 0.333. With the capacity ratio inverted, this becomes a plot of Y us. Z a t a new pair of AT,, values. A = 4: = 1 AXD 4 (FIGERE9 ) . The horizontal-line limits for X are 0.833 and 0.810. The curves for :YC,= = 1, here, are the complements of those for AToz= 4 in Figure 3that is: if one figure describes the X-phase behavior in a column, the other describes the accompanying Y-phase behavior.
T h e separate effects of P,B and P,B can be shown graphically if the other variables are reduced in number-for instance, by setting Z = 1, and plotting only the outlet concentration, X I . Figure 10 shows the variation in X I that can occur a t a specific pair of parameter values: -1 = 1, 'VOz = 4. The abscissa is P,B, on a nonlinear scale, while the contours correspond to different P,B values. Perfect mixing in one phase, ivith entire
INITIAL L'ALUES
30
40
J
50
Figure 5. Ratio of apparent piston-flow HTU to true HTU as a function of P,6
No, = 5 h = 1
S E T COUNTERS i , j , k, 1 , m T O
I
I
20
Nox
Figure 4. Local accumulation and depletion effects for transferring material in X phase
P,B
I
IO
>
COMPUTE a, P, y. FROM i th No,, -
P,B
absence of mixing in the other, is represented by the uppermost right-hand point and the lowermost left-hand point; if A were not equal to unity, the corresponding X I values would not be equal, Special Case of Negligible Concentration Change in One Phase. If one of two phases (say, the Y phase) is supplied a t large flow rates-i.e., F , +. w -or provides a n irreversible outlet for the solute ( m +- O), the capacity ratio becomes very small (A + 0 ) . Under these conditions. the Y phase has, either actually or effectively, an entirely uniform concentration from entrance to exit. Within the apparatus. regardless of the flow pattern, the Y phase acts as if it underwent perfect mixing (PUB= 0). Therefore it is convenient to use the equations for perfect mixing in one phase (Table I, Case 5) as a starting point for introducing the condition that h = 0. The X-phase concentration will continue to show the inlet discontinuity and subsequent continual decrease characteristic of longitudinal dispersion. I t is possible to describe its behavior by a still simpler equation than for PUB = 0. If the condition that A = 0 is added, then in the Case 5 equation
./
-
VOL. 2
NO. 2
MAY
1963
119
E = 0 and X = F / D . The equation for outlet concentration. with some rearrangement. thus becomes
--Z’
+
Y’ at
Z‘
with v = (1 42V,,z,’f‘zB)1 z. A similar result has been obtained for first-order chemical reaction in single-phase flow (70, 50). Exterior-apparent N T U , i V o i ~ ,is related to outlet concentration by the Colburn relation, Equation 11. In the case discussed here this becomes
( X ~ )+ Ao =
-
N o z ~= - h(X1)A o (18) with (XI).,0 given by Equation 16. (For A = 0, the singlephase values .Vz and N,P are equal, respectively, to the over-all values _Vozand .Vorp, and may replace them in the above three equations.) Thus. for .I = 0, either X I or i V o r ~(or .V,P) can be calculated from P,B and AVoz(or S,) by use of Equations 16 and 17. The resulting relations are plotted in Figure 11 (upper). (Concentration values for 0 < Z < 1 can be calculated as shown in Case 5, but are not given here.) The reverse calculation. of from h T o r p and P,B, is needed in data interpretation but cannot be made explicitly from Equation 16 ; for computer use, the following empirical relation (due to J. S. Moon, Lawrence Radiation Laboratory) is a convenient replacement for Figure 11 (upper) :
-
0 00
Figure
02
04
zL
0 6
, 0 8
10 :
7. Typical concentration profiles in countercurrent
flow for A = 1
-
2‘
(17)
e-Nozp
or
The tables described above for the general solution do not to 16. Beyond this cover A values outside the range of range, it is desirable to examine the behavior at -1 = 0 (if for high A, by inverting the phase designations), and to use the latter result if it appears less extreme. The more general case of a constant concentration in phase Y , but a t a higher value than Y = 0:is represented by PUB= 0. For this system Case 5 of Table I again applies. In the mathe. matical solution, A enters separately from the term in .VoZand P,B. The calculation can therefore be reduced to graphical form, as indicated by the sequence of steps -4,B, C, D in Figure 11 (lower). With .VoZpgiven by this figure, the incompletion fraction, XI,can be determined by Equation 11. 0-2’
PX B 0
0
5
1
2
,
4
8
16
32
0.5
0.4
XI 0.3
02
I
1
0 Pya= ‘/4?B
Z-
Figure 9. Typical concentration profiles in countercurrent flow for A = 4 120
l&EC FUNDAMENTALS
A PyB=PxS
0 PyB=4PxB
I
Inlet Discontinuity as a Measure of Performance
If the complete concentration profiles in a coluinn are not specified, a good indication of its behavior can still be had by knowing the interior conceiitrations at the inlet ends (Xu, Y J , together with the outle-t values (XI = Xi>Yo = YO). In any experiment. the niatlie~naticaldiscontinuity a t the inlet is not achieved complrtely. brcause the dispersion coefficient is likely not to remain constant a t the ends of the equipment. Hence it is best to estimate Xo and Y1 by extrapolation from the X and Y values at other interior points. Each discontinuity depends primarily upon the three variables -1, -V,,*.and PB for the phase involved, and secondarily upon PB for the other phase. As a result, any one set of the four independent variables corresponds uniquely to some one set of four terminal interior concentrations, .YO- XI: Y1,and YO. \C’ith previous kno~vledgeonly of I,and not of either dispersivities or transfer coefficients, it is necessary to apply trialand-error procedures to find the values of the controlling variables that match the experimental concentration data. Plots of X o and Y],Direct plotting of X o or Y, is one evident method. Figure 12 shoxvs Xo as a function of lYOz and P,B, with PUB and -1 specified (both zero); here X o is determined from the equation X = F:’D: which corresponds as above to Case 5 of Table I. Similar diagrams for other cases can be calculated from the respective equations, or from the numerical tables available (28).
“Jump Ratio” Plots. Better groundii.ork could be laid for trial-and-error procedures if the multiplicity of graphs like Figure 1 2 could be reduced. This would require finding a variable to which concentration is relatively insensitive, such as P,B for the opposite phase, as already cited. Further simplification can be brought about by using a secondary function of experimental concentration values. the jump ratio, defined as follo\vs for the two phases : ri = (1 - X o ) / ( l ry
=
-
SI)
(19)
YdYO
For either phase, then. r is the difference between the feed concentration and the inlet-end interior concentration, divided by the difference between feed and outlet concentrations. This ratio is much less sensitive to .\-oz than the X Qor Y I values themselves. Figure 13 shows a particularly favorable case for using the r , and r y values, a t A = 1. ?‘he plotted values have been obtained from the available concentration-profile tables (28). ,4 complete grid of contours is given here for = 2 ; and a partial grid, ivith every second value of P,B, for N o , = 8. If ‘VOz is known only to within joy,, the P,B values can be determined to Lvithin about I OYc. The resulting estimates can be used with the Xivalue, by methods described below, to provide a better estimate of This in turn gives better values of the P,B’s, and the process can be repeated as many times as necessary. A similar plot for .i = 4 is given in Figure 14. Here an accurately known pair of r values, taken with a 50% error in No,>can give a 257, error in P,B or PUB. Figure 15 shows the relationships for A = 16: and gives about the same error in PiB’s as is introduced in .Voz. Even this large discrepancy does not prevent convergence after iteration through several cycles. The inversion relations are utilized in Figures 14 and 15, to make them applicable also to the case of -1 = 1/4 and 1!16.
I
I
I
I
I
10 0.9
0.5
1
5
2
20
Nox
0.8
07 0.6
05
04 03 02
p 0
NoxP
I
= 0.
Lower.
2
4
8
6
i
IO
12
14
16
px B
Figure 1 1. Effect of longitudinal dispersion in the X phase with constant concentrcition in the Y phase Upper. Outlet concentration XI, for NoZp, for P,B = 0
A=O
I
0.1
Calculation of
Figure 12. Effect of longitudinal dispersion in the X phase with constant concentration in the Y phase Inlet-end interior concentration XI,
VOL. 2
NO. 2
MAY 1963
121
1.01
I
I
0 8
I IO
For such use, it should be remembered that PZtB = P,B, and PViB = P,B. Plots at intermediate values of .i-i.e., 2 and and 8 and 1/8-could also be prepared from the tables available. For a A applicable to any particular experiment, interpolation between the graphs already available would probably be preferable to resolving the exact equation by machine computation to develop a new table. I n the event column performance is determined entirely by dispersion, Figure 16 shows the jump-ratio behavior (for N , , .+ co, at all A’s). For this situation, the mixing effects in the two phases are combined in the term P,,B, defined above between Equations 4 and 5. I n any given case, a threeway confirmation is obtained that the infinite-,Y,, approximation is satisfactory, if the P,,B values deduced from ra. from 7y, and from 21 are all equal. Rapid Calculation Method for Extent of Completion
I
1
02
0
06
04
‘Y Figure 13. Jump ratios at X-phase and Y-phase inlets for A = 1 10
aa 06 04
02
The calculation of some one point on the concentration profile, such as X , or X I , is inherently simpler than a computation of the entire profile. I n this paper, it has been possible to describe X O by a series of graphs relating it to X I . Elsewhere, Sleicher has given graphs illustrating the behavior of X1 (43). An algebraic calculation method of X1 will now be given which is less exact, but much faster. then the complete analytical solution represented by Equation 5. The relation selected for this purpose takes its form from the equation for N o , +- m (Table I, Case 9 ; cf. Figure 2). Combination of that relation a t A # 1, with Equation 11 for iVov0+p, yields, only for the case where Aroz = 03 :
rx(=ry V O l [This relation is continuous through the value of .i = 1 ; there In A/(A - 1) is unity, and also X = 1/(1 ArOJp).] Since .lroZp is a function of No,, A, P,B, and PUB,it can be broken up arbitrarily into parts that retain this functional dependence. The behavior of .VOzpin the t\vo limiting cases of mass transfer controlling and of longitudinal dispersion controlling is consistent with the addition of reciprocals,
+
005
002
Pp’8 0 01
I
where nos^ is the “number of over-all dispersion units,” referred to the X phase-evidently a function of A> P,B, P,B, and probably also Nor. Equation 21 retains its validity if every term is multiplied by the column height, L. The quotient H,, = L/lY,, is recognized as the customary true H T U . H,,p = L / N , , p becomes the exterior apparent H T U ; and H,,D = L / , $ ’ o zis~ “height of over-all dispersion unit.” Thus, H,,p = H,, H,,D. This concept has been utilized in fixed-bed separation operations by Van Deemter et al. (45). Combination of Equations 20 and 21 indicates that the m . This former gives the limiting value of N,,D. as N o , relation is therefore assumed to provide a general form for the evaluation of N,,D:
+
Naz= ~
Figure 15. Jump ratios at X-phase and Y-phase inlets for A = 16 122
I&EC FUNDAMENTALS
In A
@
+ (PB)Y
(22)
As a first step toward possible correlation, Equation 22 was used with 9 = 1 and with (PB), = P,,B as calculated by a relation given earlier. Figure 17 shows the resulting correlation, in dashed curves, compared with the computer values as solid curves. Since the agreement seems adequate a t A = 1 but not elsewhere, it was concluded that the correlation had
I
I
m - 1 2 -
---
O a t
4
1i
A=4
32
8
I --
I
XI
04
tc ---_
PxB.05
A= I
02-
= 0.25
,\
w-
No, = 4 01
,
,
lo'
2
5
5
,o', 2
5
2
10
32 ---__ 5
2
I
102
poy
r, Figure 16. Jump ratios at X-phase and Y-phase inlets for No, = a
Figure 17. Trial use of mixing in both phases
been put in suitable algebraic form but that a different definition of the over-all Ptclet group would be necessary. For this reason, (PB),.has been defined by the equation
P,,B to correlate effect of
5
2 fx
Here the weighting factmors, fz and f U , are functions of and -1;and 4 may be a function of (PB),, No,, and A. In practice, 0 is assumed to deviate from unity only in the region of low Ptclet numbers where, because perfect mixing is approached, the mass-transfer framework of calculation is put to its severest test. The factors fz and fZIare not true constants, but only average values which provide a reasonable approximation over the entire range of possible behavior. Thus the values of N O zto~ be developed from Equations 22 and 23 must always be viewed as approxima.te. For each available XI value, known as a function of the independent variables, Equation 11 can be solved explicitly to obtain the corresponding nor^; Equation 21 then leads to the exact N , , D . At given A and A',,, i:wo different combinations of P,B and P,B can be solved simu1i:aneously (with 6 = 1) to obtain values of fz and fu. I t was noted that, since the definitions of the X and Y phases are arbitrary, fz and fv should be closely related. The inversion properties given in Equation 7 require that
or
I
fY 05
02
01
5
IO
Figure 1 8. Weighting factors for calculating as functions of A and No,
(PB),
01
02
0 5
2
I
h
Table il.
Comparison of Approximate and Exact Raffinate Compositions (A = 0.25)
x1 PzB 1. o
A pair of empirical functions having this property, and fitting the limiting conditions a t N o , = m and also at A = 1, is
4.0 8
1 .o
4.0 A very reasonable fit c'f these functions to the computed f. and f y values is given by the constants cy = 0, K = 6.8. The behavior of fi and f,, calculated by use of Equation 25 as a smoothing function, is shown in Figure 18. Other exponents on .2 were investigated, which did not improve the correlation;
16.0
p,B 0.25 1 .o 4.0 1 .0 4.0 16.0 0.25 1. o 4.0 1. o 4.0 16.0 4.0 16.0 64.0
VOL. 2
Estimated 0.3860 0.3628 0.3520 0.3179 0.2819 0.2689 0.2213 0.1807 0.1569 0.1354 0,0760 0.0540 0.0396 0.0156 0.0105
NO. 2
Computed 0,3844 0.3728 0.3518 0.3190 0.2883 0.2690 0.2178 0.1906 0.1468 0.1465 8.0796 0,0474 0.0451 0,0149 0.0106
MAY 1963
123
variations up to 15% in K did not cause much change in the general fit. Determination of 0, as the correction for the difference between correlation values of (PB), (from Equation 23) and "exact" . ~ ~ o i ~leads ' s . 10 the relation
where a,, A,, and u take entirely the same form as in Equation 5 ; but ,b, g, a. 0,y , and A , in these factors take the following forms :
P
=
(43)s
q = (a/3)3 a = P,B
The entire correlation becomes unsatisfactory in the range where 4 becomes negative. The correlation has been checked against every value of XI in the computed tables (28). Typical values of X I estimated by means of this correlation, using in turn Equations 25, 23, 26, 22. 21, and 11. are shown in Table I1 along with comparative values from the exact computer calculations. The estimated values generally agree within 1% of the feed-concentration level, and therefore appear satisfactory for use in many design calculations. In this correlation, (Pi?),is insensitive to small changes in iV,,. Hence it can be evaluated with an inaccurate iVor, and used in Equations 21 and 22 to develop a more accurate value. \Vith the recognition that longitudinal dispersion is often a significant factor in equipment performance, and with the development of calculation methods to account for it, a pressing need has developed for measuring complete concentration profiles in operating equipment--not merely outlet values-which will supply more accurate values of transport rates than were heretofore available, along with new quantitative information about dispersion rates.
+ P/3
+ .Plh
- PUB
6 = N,,P,B y =
A,
=
(1
- -{If2
- P,B
PUB
+ 1 .Y,*PYB
+ A ) ,Vo,PzB PUB
D,,/D,
( 3 = 1. 2, 3, and 4 )
and
Xsa?eXr
Dpl
=
+
(1 - $ ) a 2
XaaseXJ
Xlareh
X3eX1
XIeh
(1 - p$)@
(1 -
&)a4
General Solution for Parallel-Flow Operation
A general solution for parallel-flow (cocurrent) operation has been developed from considerations similar to those in the counterflow case. Let the direction of mass transfer be from phase X to phase I-, Then.
A = I /4, A' = 4
Tables similar to those described above under "Numerical Concentration Values. General Case" (28) have been calculated on the IBM 701 computer for parallel flow of the phases in contact (32) corresponding to the solution just given. In this case the inversion relation for fractional length is Zt = Z,while the other variables still conform to Equation 7. Because Yt is now related to X at the same Z.conversion of X to Yt is facilitated; the tabulated range of variables has therefore been reduced (compared with the range covered for countercurrent flow) by eliminating A values greater than 2. Typical profiles are given in Figure 19, for .i = 114 and A = 4. It appears that cocurrent flow will always give plots that are concave upward on X - Z coordinates, unlike the countercurrent profiles with A > 1 typified by Figure 9. Conclusions
A'
L
Figure 19. for A =
'14
124
Typical concentration profiles in cocurrent flow
I&EC FUNDAMENTALS
A general theoretical treatment based on the proposed model permits evaluation of the over-all behavior of two-phase continuous-flow operations by taking into consideration the effect of longitudinal dispersion of both fluids. The behavior is expressed as a function of four dimensionless parameters. Various special cases of the pattern of longitudinal dispersion have been solved, and actual applications are indicated. Three kinds of over-all height of transfer unit have been distinguished, and the interrelation between them is shown. Two of them, both used in previous work, reflect the influence of longitudinal dispersion. Illustrative numerical examples indicate that longitudinal dispersion produces a n extremely undesirable effect when a high completion of mass transfer is desired. and that the lowering of the completion is attributable partly to the lowering
of concentration driving force and partly to accumulation and depletion effects. There is a maximum attainable completion of mass transfer under a given pattern of longitudinal dispersion. Solutions for this maximum comp1.etion have been derived by use of a n infinite value for the true over-all mass-transfer coefficient. A discontinuity in concentration between the inside and outside of a column is predicted at the inlet end for each feed. Its magnitude is plotted as an aid to evaluation of individualphase Ptclet number from experimental data on absorption or extraction operations. An empirical fit is developed for the computed values of extent of extraction, by 'converting the exterior apparent STL(AVoz~) into a function of separate terms, each involving a smaller number of independent dimensionless variables. Although countercurrent flow is emphasized, a few results are given for parallel flo\v.
Acknowledgment
. 4 c e E;. hfc?r/lullen programmed the computation. and S o r m a n Nian-tze Li prepared the jump-ratio plots. The authors express their thanks for this assistance; and also for helpful discussions, in the course of this \vork. with Thomas Baron. Charles A. Sleicher. and Octave Levenspiel.
solute concentration in I- phase, m C f ) ] ,dimensionless length within column, measured from X-phase inlet in direction of flow, cm. fractional length in column, z, L , dimensionless factors in general solution, dimensionless adjustable exponent in Equation 25. dimensionless fractional void-volume in packing occupied by fluid phase, dimensionless numerical coefficient in Equation 25. dimensionless factor in general or special solution, dimensionless capacity ratio, mF,/F,, dimensionless factor in solution for A = 1, dimensionless (1 4 -Vor/PzB)1/2, dimensionless density, gramsjcc. time, sec. correlation factor, in Equation 22. dimensionless reaction rate, moles/cc. sec.
I.'
= generalized
2
=
m(C, - C,O)/[l - (Q
z CY, CY
6, y
= =
=
E
=
h'
x
= =
A
=
ir
= = = = = =
v P T
P
$
+
+
SUBSCRIPT':
D
=
H
= thermal
i
=
i
= = = = = = = =
M 0
P x,
Y
0 1 1. . . 5
dispersion designates phase concerned, either -1-or I' component concerned; also, index in series measured (interior apparent) value over-all piston-flow model, or exterior apparent value X or Y phase feed-inlet end, inside column feed-outlet end. inside column terms in algebraic sequence
SUPERSCRIPTS
*
Nomenclature
factors in general solution: dimensionless interfacial area per unit column volume? sq. cm.,/cc. factors in solution for A = 1, dimensionless L,/d: dimensionless heat capacity. cal./gram deg. K. concentration of solute in ith phase, gram-molesi'cc. c i 'c! (dimensionless)? Lvith cl the feed-stream concentration length characteristic of equipment, cm. molecular diffusion coefficient, sq. cm.,/sec. calculation parameters in evaluation of X a t PUB
=o
factors in general or special solution, dimensionless effective longitudinal dispersion coefficient (superficial) in the ith phase, sq. cm./sec. weighting factor, in Equation 23, dimensionless superficial volumetric flow rate of ith phase through unit cross section of apparatus, cm.,'sec. specified length or height in column, cm. height of over-all transfer unit based on ith phase, crn. over-all coeflicient of mass (or heat) transfer based on ith phase cm.,'sec. mixing length, E , / F i , cm. effective column length in direction of mean f l o ~ , cm. solute partition coefficient, dcz*/dcy, dimensionless amount of solute transferred, moles/sec./sq. cm. number of over-all transfer units in column, based on ith phase, L/H,i, dimensionless factors in general solution, dimensionless local Ptclet number for ith phase, d / l i , dimensionless intercept on linear equilibrium plot, cz = q mcy, grams-moles/cc. q ~ ~ dimensionless 0 , jump ratio in ith phase; see Equation 19, dimensionless temperature in ith phase, O C . above base value linear velocity, F I E ,cm./sec. over-all heat-transfer coefficient, cal./sq. cm. deg. K. sec. generalized solute concentration in X phase, [Cz- (Q mC,O)I/[l - (Q mcp)], dimensionless
+
+
+
t
0 1
= equilibrium =
inversion of phase definitions
= feed-inlet end, outside column = feed-outlet end, outside column
literature Cited
(1) Aris, R., Amundson, N. R., A.I.Ch.E. J . 3, 230 (195-). (2) Askins, J. W., Hinds, G. P., Kunreuther. F.. Chem. Eng. Progr. 47, 401 (1951). (3) Burger, L. L., Swift, W. H., U. S. At. Energy Cornm.. Hanford Works, Rept. HW-29010 (1953). (4) Carberry, J. J., A.I.Ch.E. J . 4, 13M (1958). (5) Carberry, J. J., Bretton, R. H., Ibid., 4, 367 (1958). (6) Chilton, T. H., Colburn, A. P., Ind. Erzg. Chem. 27, 255 (1935). (7) Cohen, R. M., Beyer, G. H., Chem. En?. Prog. 49, 279 (1953). (8) Colburn, A. P , Trans. Am. Inst. Chem. Engrs. 29, 174 (1939); lnd. Eng. Chem. 33, 459 (1941). (9) Damkohler, G., "Der Chernie-Ingenieur," .I. Eucken and M. Jakob, eds., Vol. 3 , Part 1. p. 366, .;\kadernische Verlagsgesellschaft, Leipzig, 1937. (10) Danckwerts, P. V.. Chem. Ene. Sci. 2. 1 11953): Trans. Fuaruduy Sor. 46; 300 (1950). (11) Deisler, P. F., Jr., IVilhrlm, R. H.. Ind. En:. Chem. 45, 150 (1953). (12) Dunn, W. E., Vermeulen. T., \l.'ilke, C . R.. \Yard: T. T., Univ. California Lawrence Radiation Lab.. Rept. UCRL10394 (1962). (13) Ebach, E. A , , White, R. R., A.I.Ch.E. J . 4, 161 (1958). Nagata, S., Ci-em. Eng. ( J a p a n ) 22, 218 (1958). (14) Eguchi, W., fl5) Ibid.,23, 146 (1959). (16) Epstein, N.: Can. J . Chem. Eng. 36, 211) (19581. (17) Forster, T., Geib, K. H., Ann. Phys. 20, 250 (1934). (18) Geankoplis, C. J., Hixson, A. N., l n d . Eng. Chem. 42, 1141 (1950). (19) Geankoplis, C. J., h'ells, P. L., Hawk. E. L.. Ibi'd.,43, 1848 (1951). (20) Gier, T. E., Hougen, J. 0..Ibid.,45, 1362 (1953). (21) Gilliland, E. R., Mason, E. A , . Ibid.. 41, 1191 (1949); 44, 218 (1952). (22) Hulburt, H. M., Ibid.,36, 1012 (1944) : 37, 1063 (1945). (23) Jacques, G. L.: Vermeulen, T., Univ. California Radiation Lab., Kept. UCRL-8029 11957) ; Revision by Hennico, A , , Jacques, G. L., Vermeulen: T., Ibid.,UCRL-10696 (1963). (24) Kramers, H., .4lberda, G.: Chem. Eng. Sri. 2, 173 (1953). (25) Krieger, R. M., Geankoplis, C. J., Ind. En8) used modified spheres made by pelletizing powder. T h e pellets consisted of a short cylindrical section with hemispherical caps. In this rvork? true spheres were made by casting. Figure 2 shows the two halves of the copper mold for casting the spheres. T h e sphere mold was made by cutting approximate hemispheres about 0.115 inch deep in each of the two matching plates. Steel ball bearings 0.250 inch in diameter \yere inserted in the holes of one plate, and the two plates were clamped together. T h e plates were pressed slowly together several times using a pressure of 200,000 p.s.i. to form 0.250-inch holes. While still clamped together. holes were drilled for steel taper pins
