Ind. Eng. Chem. Fundam. 1980, 19, 358-363
358
Calculations for Separations with Three Phases. 1. Staged
Phllllp C. Wankat School of Chemical Engineering, Purdue University, West Lafayefte, Indiana 47907
Graphical and analytical solution methods for staged, three-phase separations with two cocurrent streams flowing countercurrent to the third stream are developed. With constant flow rates and linear equilibria a modified Kremser equation of the form
+
where (Y = V / ( W / K 2 L / K , ) results. For nonlinear equilibria two graphical techniques are developed. One method uses variable operating lines while the other method develops a pseudo-equilibriumcurve. The analysis is extended to nonequilibrium stages by defining appropriate forms of the Murphree stage efficiency. The analysis is also extended to a countercurrent cascade with cross-flow of the third phase.
Since diffusional type separations involving three phases are not nearly as common as separations with two phases, they have not been very extensively studied. The threephase systems do occur in distillation when both organic and aqueous liquids are present, in slurry adsorption (Maslan, 1972), in slurry scrubbers (Rochelle and King, 1977), in extraction with three liquid phases (Meltzer, 1958) and in liquid membrane systems (Li, 1971; Mauch, 1976). Three-phase systems are relatively common in mechanical separation systems (Jacques et al., 1979). With the increased interest in three-phase systems, particularly liquid membranes (where the membrane can be considered the third phase) the time seems to be ripe to develop generalized calculation techniques for the three-phase systems. In this paper we will consider simplified analyses for staged systems while a second paper will consider continuous contact systems. First, the three-phase analogue of the Kremser equation will be developed. Then a variable operating line graphical solution technique for countercurrent flow and for countercurrent cascades with cross flow will be developed and simplified for special cases. Finally, an alternate graphical solution for countercurrent cascades using pseudo-equilibrium lines will be developed. Kremser Equation Development The staged system for a three-phase, countercurrent process is shown in Figure 1. Two phases, labeled L and W, flow cocurrently and flow countercurrent to the third phase, V. We will assume that each stage is an equilibrium stage, and that temperature and pressure are constant. Thus xi, yi, and zi are solute mole fractions leaving stage i in equilibrium. In addition, to develop the Kremser equation we will assume that total flow rates L, V, and W are constant, and that the equilibrium is linear y = Klx + bl y = K ~+ z bz (1) The linear forms in eq 1 imply that the equilibrium between x and z must also be linear. Equations l are written for a single solute. The development is also valid for multisolute systems if all solutes are independent. For a system with four components (phases L, V , and W, and one solute) there is one remaining degree of freedom.
The constant flow rate assumption requires either constant molal overflow or immiscible phases with dilute solutions. Equations l will be valid for dilute solutions, or for limited ranges of concentration the equilibrium curve can be linearized. A mass balance around stage i is vyi
+ wzi + LXi = WZi+l + LXi+l + vyi-1
(2)
Values for x and z leaving a stage can be eliminated using eq 1.
After substituting this equilibrium result into eq 2 and after algebraic manipulation, the result is y;+1 - ( a + 1)Yi
+ ffyi-1 = 0
(4)
where a=
V
W K2
L - + - K1
The W and L streams entering the column are not in equilibrium, but for convenience a fictitious value of y may be defined. If eq 3 is written for Z N + ~and xN+l, the equilibrium value is y*N+1. This value is
-+K2
K1 The interesting thing about eq 4 is that it is exactly the same equation as we obtain for a countercurrent two-phase system with the same assumptions (Mickley et al., 1957, p 321). The term CY is defined differently; however, if W = 0, a from eq 5 reduces to the appropriate result for two-phase systems. The general solution to eq 4 is (Mickley et al., 1957, p 325) yi = co + C l d (7)
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Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980 359 W
.
)iJ . e
W
Figure 1. Three-phase countercurrent cascade.
where Co and C1 are constants of integration. The constants are determined from the boundary conditions y = yin a t i = 0 ( 8) y = y*N+l at i = N + 1 (9) The value y*N+lis the value of y in equilibrium with the entering streams xin and zin and is given by eq 6. Utilizing the boundary conditions, eq 8 and 9, to determine the constants of integration, the solution is Y*N+1 - YinffN+l Yo - Y*N+1 (10) Yi = 1- p + 1 1 - aN+l a'
+
This can be rearranged to Yi
- Yin
Y*N+I
-.
Yin
=--- ak- 1 aNfl- 1
If we let i = N then yN = youtand the resulting equation is aN- 1 Yout - Yin =(12) aN+l- 1 Y * N + ~- Yin Further manipulations of eq 1 2 are simplified if we define a' = ( l / a ) . Then eq 1 2 can be written as
or be rearranged to
Upon solving for N we have
Equations 11 to 15 are all forms of the Kremser equation for three-phase systems. If W = 0 these equations all reduce to the appropriate forms of the Kremser equation (Mickley et al., 1957; King, 1971). The existing graphical plots of the Kremser equation (e.g., King, 1971, p 395) can be used for three-phase systems if Y * N + ~is defined by eq 6 and a' is used instead of L / m V . The predicted behavior of the cascade is essentially the same as for a two-phase system if we use y*N+ldefined by eq 6 and a' instead of L/mV. For the case where we wish to remove solute from V (analogous to a three-phase absorption) with yb, YN, x N + 1 , and Z N + ~fixed, increases in a' (or decreases in a ) will increase the separation a t a given N o r require fewer stages for a specified separation. In the system with equilibrium stages more separation can be obtained by varying either downward moving phase independently. If the stages are not equilibrium stages this may not be true. Although developed for a single solute, the forms of the Kremser equation can be used for multisolute problems if we can assume that the solute equilibrium equations are all independent. Then each solute can be solved for independently. Equation 15 is used to find N for the key component, and one of the equations 11 to 14 is used to find the distribution of the other components using the calculated value of N. Variable Operating Line Graphical Solution Method Although it is extremely useful when there are large numbers of stages, the Kremser equation is somewhat limited because of the restrictive assumptions which are required. Some of these assumptions can be relaxed when a graphical solution method is used. The graphical technique also provides a convenient visual impression of the separation. We will fiist develop the general solution using variable operating lines for nonlinear equilibria and then consider two special cases. Then a pseudo-equilibrium curve method will be developed. If one of the concentrations is constant or if the equilibrium is linear, the graphical method is simplified. In all cases temperature and pressure are assumed to be constant and overall flow rates will be assumed to be constant. The procedure will be developed for a single solute, but it is easily extended to multiple solutes if the solute equilibria are independent. For the countercurrent system shown in Figure 1 a mass balance around stage i + 1and the top of the column can be rearranged to give an operating equation of the form
where we have assumed that overall flow rates are constant. For any stage i + 1 where xi+l and zi+l are known, eq 16 can be used as an operating line on a y vs. x diagram. + ~Y N The slope is L / V and the intercept [ ( W / V ) Z ~ + ( L / v ) x ~-+ (~w / V ) Z N + l ] will change for each stage. With yNknown, the following calculation procedure is easily used to step off stages down the column: (1)Use equilibrium (eq 1or other expressions) to calculate xN and zN from y N . (2) Calculate Y N - ~ = ( L / V ) X N+ [ ( W / V ) z , + y N - ( L / V ) x N + , - ( W / V ) Z N + ~ ] . (3) Calculate xN-l and Z N - ~from equilibrium. (4)Calculate yN-2 = ( L / v ) x ~+- [~( W / v ) z ~ - l + YN - ( L / V ) X N -+ ~(W/V)ZN+~I etc. This procedure can be done analytically or graphically. The graphical construction with McCabeThiele diagrams is illustrated in Figure 2. Step 1 determines X N and zN. Step 2 plots an operating line from which YN-1 is determined. Step 3 determines xN-l and zN-l from equilibrium.
360
Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
Slope = L / V
2’
E 2
Figure 2. “Countercurrent” variable operating line graphical solution for three-phase countercurrent cascade with nonlinear equilibria: A = ( W / V ) X N+ Y N - (L/V)XN+I - ( ~ / V ) Z N + IB ; = (~/V)ZN-I + Y N - (L/V)XN+~ - ( ~ / V ) Z N +Numbered ~. arrows refer to steps in procedure.
Step 4 plots a new operating line and determines YN-2. The procedure is continued to the bottom of the column. If desired, the operating lines can be plotted on the y-z diagram. Alternatively, a y-x and z-x diagram can be used with the z equilibrium value determined from the latter plot. This stage-by-stage calculation procedure in the direcworks well since tion of the cocurrent phases, L and two variables, z, and x,, are calculated from the two equilibrium relationships. If we try to reverse the direction of the calculation and work our way up the column (against the two cocurrent phases), a trial-and-error calculation is required. Thus stepping off stages in the cocurrent flow direction is preferred. If the problem is specified with one outlet concentration given at the bottom of the column (say xl), use equilibrium to calculate y1 and zl,and use an overall mass balance to calculate Y N . Now proceed in the cocurrent flow direction. Equation 16 has the form of a countercurrent equation. An equation relating z to x which has cocurrent form can also be derived. Starting with a mass balance around the bottom of Figure 1 and solving for z
XI
x2 1 3
X
Figure 3. “Cocurrent” variable operating line graphical solution for three-phase countercurrent cascade with nonlinear equilibria: A = (V/W)Y,- (V/W)Y, + ( L / W ) x , + 21; B = (V/W)Y, - (V/W)Y, + ( L /W)x1 + zl. Numbered arrows refer to steps in procedure.
v,
v
zi+l
v
L -y. - -y. + w x ~+ 2 1 1 (17) W ‘ W’”
This equation relates the two unknowns xi+l and zi+l, These streams are also in equilibrium so the second relationship between them is the equilibrium relationship. On a z vs. x diagram eq 17 plots as a straight line with a slope -L/ W and a z intercept given by the bracketed term. Stepping off stages from the bottom up yi will be known when zi+l and xi+l are being calculated. This procedure is illustrated in Figure 3 where we assume x1 is specified. The y-x graph is used (1)to calculate y1 at equilibrium and this value is used (2) to calculate new intercepts (point A) on the z-x graph. Then (3) the operating line can be drawn and (4)( z ~ +xi+J ~ , is at the intercept of the equilibrium and operating lines. If we try to use this cocurrent calculation procedure going down the column, the calculation will be trial-and-error. Simplified Graphical Procedure for Special Cases. The variable operating line graphical procedure shown in Figure 2 is not trial-and-error, but is somewhat inconvenient since a new operating line must be drawn for each stage. Two special cases can be solved easily. Special Case 1. If one of the phases, say W, saturates so that zi = constant, then the analysis can be simplified. Now the term in brackets in eq 16 is constant since z ; + ~ (i # N)is constant. (The W feed may not be saturated.)
r
X
Figure 4. Graphical solution for three-phase countercurrent cascade with linear equilibria.
Thus eq 16 represents a single operating line and the usual McCabe-Thiele procedures can be used. In Figure 2 the y-z equilibrium curve would be a vertical line and there would be only one operating line on the y-x plot. Special Case 2. Linear Equilibrium. With one degree of freedom and with L and W in equilibrium then zi+l = f(x,+J, This expression can be used in eq 16 to remove the variable If the equilibrium relationships are linear, then ~i = K x ~ b (18)
+
and the operating equation becomes
yi =
(++
[ (++ y)xl] yin-
(19)
Equation 19 represents a straight line with a slope of [ ( L / V) + ( W K / V)] and a y intercept of [yin- ((L/ V) + ( W K / V ) ) x l ] .This is illustrated in Figure 4. The y-x and y-z equilibria are also linear and of the form shown in eq 1. The equilibrium constants are related by K = K1/K2 (20) 1 b = -(b1 - b2) Kz This case then has exactly the same assumptions as the
Ind. Eng. Chern. Fundam., Vol. 19, No. 4,
1980 361
YH
YN.,
YN-2
Y
A B
Figure 6. Variable operating line Murphree efficiency calculation for general case where none of phases are in equilibrium. Use ELy and Ew,: A = ( W / ~ ) Z N+ YN - (L/V)xN+1 - ( W / V ) Z N t l ; A' = ( L / v ) x N + YN - ( L / v ) x N t 1 - (w/v)zh't+l;B = ( w / v ) z N - l + YN ( L / v ) x N t L - ( W / v ) z N + l ; B ' = (L/V)XN-l + Y N - ( L / v ) x N t l - (w/ V)ZN+1.
is assumed to be well mixed. In the usual definition of a Murphree efficiency y* and x* terms are defined for each stage. With three phases present several different definitions are possible. Thus y*x = y value in equilibrium with actual xout (23A) Y * ~= y value in equilibrium with actual zOut
(23B)
= y value in equilibrium with weighted mixed average of xout and zOutallowed to equilibrate (23C)
Y*mir
Figure 5. Three-phase countercurrent cascade with cross-flow.
Kremser equations. Utilizing eq 20 the slope of the operating line is
The variable a' is thus the slope of the operating line divided by the slope of the y-x equilibrium curve ( K J . In two-phase systems a' = L / m V has the same meaning. Countercurrent with Cross-Flow. With three phases present other cascade arrangements may be advantageous. One example is the countercurrent system with cross-flow shown in Figure 5. A similar arrangement with two phases was studied by Gunn 1977). T o analyze this as a countercurrent system utilizing the variable operating line method we can do a mass balance around stage i 1 and the top of the cascade. Upon rearrangement the mass balance becomes L
+
Yi
=
-Xi+l
V
(Equation 23C is the procedure used in eq 3 and 6.) Similar definitions are used for x* , x * ~ x*,k, , and z*,, z*?, z * , ~ Note that if the L and W pgases are in equilibrium the three definitions in eq 23A, B, and C will give the same y* value. With different definitions of the equilibrium concentrations several different values of the Murphree efficiency can be defined. For the V phase - Yin
Yout
= y * x - yin
E, =
-
- Yin
Yout
~ * z Yin Yout
- Yin
Evm, - y*m u - Yin ,
and for the L phase
+ YN +
with similar definitions for EL,and ELd. The definition for the W phase is similar where L and V have been assumed to be constant, but the cross-flow rate, Wj,on each stage can vary. Starting at the top of the cascade with Y N known, we can calculate Z N and x N from equilibrium. Then eq 22 can be used to calculate YN-1. Graphically, the calculations will look like those in Figure 2 except the y intercept will be the term in brackets in eq 22. The cascade can also be analyzed as a cross-flow cascade but the procedure is somewhat inconvenient. If the equilibria are linear, Gunn's (1977) solution can easily be extended to three phases. Stage Efficiencies. Up until now the stages have been considered to be equilibrium stages. If equilibrium is not attained on each stage we can use stage efficiencies to design the system. We will use a modified Murphree stage efficiency to study the system. Each stream, V ,L , and W ,
Using these Murphree efficiencies to calculate the number of stages requires a slight modification of the usual method for two-phase systems. For the general case none of the phases is in equilibrium and the equilibria are nonlinear. The easiest way to do this calculation is to use EL, and EW? (note Ew # EL,) and work down the column shown in Figure 1. dtilizing other Murphree efficiencies or working up the column will require a trial-and-error calculation. The procedure is illustrated in Figure 6. We start with YN, Z N + ~ and , x N + 1 known. Then determine X * N ~ and Z * N ~ from equilibrium. Now calculate X N from ELy and Z N from Ew . This allows us to plot the operating line from eq 16 and cdcdate YN-1. Since the operating lines both represent
362 Ind. Eng. Chem. Fundam., Vol. 19, No. 4, 1980
the same mass balance, only one operating line need be drawn (or calculated analytically). Then proceed to the next stage. The general case is somewhat cumbersome and can be simplified for certain special cases. Special Case 1. Phase W saturates and zi = constant. Use Evz from eq 24A on a y vs. x plot and use the usual technique for a Murphree vapor efficiency for a two-phase system. Special Case 2. Phases L and W i n equilibrium and the equilibrium is linear (eq 18). The solution for y in equilibrium with x and z was shown in Figure 4. Equation 19 can again be used for the operating line. Since L and Ware in equilibrium, Evx = Evy = EV . = Ev. The vapor efficiency now represents the fraction 3 the distance from the operating line to the equilibrium line which we should step up in Figure 4. This case also corresponds to the Kremser equation case. With constant E v the corresponding Kremser equation is \
Special Case 3. Phases L and Ware in equilibrium, but the equilibrium is nonlinear. As in the general case the easiest calculation procedure is to start at the top of the column and work down. Either EL, or Ewycan be used. Since z and x are in equilibrium, only one of these need be used. The calculation procedure still requires construction of an operating line for each stage but is somewhat simpler than the general case shown in Figure 6 since only one Murphree efficiency calculation is required for each stage. Special Case 4. Phases W and V flowing countercurrently are in equilibrium, and the equilibrium is nonlinear. The easiest way to solve this case is to work down the column using ELy. In Figure 6 the y-x diagram would be the same while only the equilibrium curve would be used in the y-z graph. This special case is of interest for liquid membranes where W is the membrane phase, L is the encapsulated phase, and V is the countercurrently flowing continuous phase. Here it makes sense to use E, since the slow mass transfer is between the L and W phases. Unfortunately, this choice of efficiency requires an awkward calculation procedure. However, since W and V are in equilibrium x*, = x*y and EL, = EL,. Thus we will get the same result using EL, and the calculation procedure outlined in the previous paragraph. Pseudo-Equilibrium Line Graphical Solution Method An alternate graphical analysis is to develop a pseudoequilibrium curve. Define the average mole fraction in the cocurrent phase as
Now for the countercurrent system shown in Figure 1the operating eq 16 becomes
Equation 29 is a straight line on a McCabe-Thiele y-x‘ diagram with a slope of ( L + W)/ V and y intercept of CyN
Figure 7. Pseudo-equilibrium curve analysis for three-phase countercurrent cascade: A = y N - ( ( L+ W ) / V ) Z ~ + ~ .
+
W / V ) x h + l ) .The pseudo-equilibrium curve is represented by eq 28 with yi, xi,and zi in equilibrium. The pseudo-equilibrium values can be plotted for a given L / W by finding the x and z values in equilibrium with a given y and calculating the corresponding x ’ value. The McCabe-Thiele diagram is shown in Figure 7. If the equilibrium is linear as in eq 1, then Lbi Wb, - (L
-+K1
K2
-+-
K1
K2
or
y = K’x‘+ b’
(31)
Using eq 31, the Kremser equation can be developed directly. This procedure is obviously more convenient to use than the variable operating line procedure for the system shown in Figure 1for equilibrium stages. If phases L and Ware in equilibrium but phase V is not, then the Murphree efficiency E v = Vvz = Evy = E , can be used directly in Figure 7. This procedure would be equivalent to the use of Murphree efficiency for binary systems. The pseudo-equilibrium procedure would be extremely awkward to use for a countercurrent system with crossflow. A new pseudo-equilibrium curve would be required for each stage plus a new operating line would be required for each stage. The pseudo-equilibrium procedure would also be difficult to use with nonequilibrium stages of phases L and W are not in equilibrium. This would make plotting a pseudo-equilibrium curve very difficult. Discussion The Kremser equations developed for the three-phase systems all simplify to the usual forms if either W or L is zero. Both graphical solution techniques for the countercurrent systems also simplify to the usual two-phase techniques if either W or L is zero. If the equilibria are linear, the graphical methods and the Kremser equation will give the same results. The relationship between these two approaches is shown in eq 21. For the graphical solution techniques the operating lines are shown below the equilibrium curves. If solute is being removed from phase V then the operating lines will be above the equilibrium curves in Figure 2, 4, 6, and 7. If the number of stages is specified, the stage-by-stage calculations will be trial-and-error. The graphical procedure was derived for constant overall flow rates. For systems with a single solute this assumption can be relaxed if the flow rates of the inerts in each
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Ind. Eng. Chem. Fundam. 1980, 19, 363-367
phase are constant. Then the usual procedure (King, 1971, p p 264-276) of defining flow rates as flow of inerts and defining compositions as weight or mole ratios can be used. Figures 2,3, and 7 will be unchanged except the variables having different meanings. Figure 4 will be valid if the equilibria are linear in the ratio units. The stage efficiency analysis can still be used if the efficiencies are defined in the new composition units. The analysis of continuous contact systems will be considered in part 2. Acknowledgment
Discussions with Drs. Daniel Tondeur and Alden Emery, Jr., were most helpful. The pseudo-equilibrium method was suggested by Professor C. Judson King in a review of the original manuscript. The hospitality of ENSIC in Nancy, France, is gratefully acknowledged. This research was partially supported by NSF Grant Eng. 77-21069. Nomenclature
b, bl, b2 = linear equilibrium parameters in eq 1 and 18 b' = average linear equilibrium parameter, eq 31 and 30 Co, C1 = constants of integration ELy, E L , EL . = Murphree efficiencies in L phase, eq 25 EvmiX, d v X ,Fv2 = Murphree efficiencies in V phase, eq 24 Ewmk,E,, E w = Murphree efficiencies in W phase, eq 26 i = stage number K , K1, K2 = linear equilibrium parameters in eq 1 and 18 K' = average linear equilibrium parameter, eq 31 and 30 L = phase flow rate, mol/h N = total number of stages V = phase flow rate, mol/h W = phase flow rate, mol/h x = composition of solute in phase L , mole fraction x' = average mole fraction in cocurrent phases, eq 28 x * ~= x value in equilibrium with actual yout,mole fraction x*, = x value in equilibrium with actual zout, mole fraction
x * , ~ = x value in equilibrium with weighted mixed average of xout and youtallowed to equilibrate, mole fraction y = composition of solute in phase V , mole fraction y*N+l= y in equilibrium with xh and zh, see eq 6, mole fraction y*, = y value in equilibrium with actual xOut,mole fraction = y value in equilibrium with actual zoUt,mole fraction y*,& = y value in equilibrium with weighted mixed average of .routand Fout,allowed to equilibrate, mole fraction z = composition of solute in phase V , mole fraction z , , ~= composition of solute in entering cross-flow stream in Figure 5 , mole fraction z*, = z value in equilibrium with actual xoUt,mole fraction z * ~= z value in equilibrium with actual yout,mole fraction zmi = z value in equilibrium with weighted mixed average of xout and youtallowed to equilibrate, mole fraction Greek Letters a = V / ( W / K ,+ L/Kd (eq 5) a' = 1/a = (W/VK,) + ( L / V K , ) (see also eq 21) Subscripts and Superscripts i , j , N = stage number in = inlet streams (to column or to stage) out = outlet streams (from column or from stage) * = equilibrium value L i t e r a t u r e Cited Gunn, D. J., Chem. Eng. Sci., 32, 19 (1977). Jacques, M. T., Hovarongkura, A. D., Henry, J. D., AIChE J., 25, I 6 0 (1979). King, C. J., "Separation Processes", McGraw-Hill, New York, 1971. Li, N. N., Ind. Eng. Chem. Process Des. Dev., IO, 125 (1971). Maugh, T. H., Science, 193, 134 (July 9, 1976). Maslan, F., Ind. Eng. Chem. Fundam., 11, 238 (1972). Meltzer, H. L., J. Bo/. Chem., 233, 1327 (1958). Mickley, H. S., Sherwood, T. K., Reed, C. E., "Applied Mathematics in Chemical Engineering", 2nd ed, McGraw-Hili, New York, 1957. Rochelle, G. T., King, C. J., Ind. Eng. Chem. Fundarn., 16, 67 (1977).
Received for review November 26, 1979 Accepted June 18, 1980
Adsorption Chromatography Measurements. Parameter Determination N. Wakao, S. Kaguei, and J. M. Smith" School of Engineering, Yokohama National University, Yokohama, Japan 240
Pulse-response data were obtained at 293 K and atmospheric pressure for the physical adsorption of nitrogen in beds of activated carbon particles. Error analysis of the response curves in the real-time domain confirmed that an accurate value of the adsorption equilibrium constant could be obtained from a single measurement. Data as a function of gas velocity were required in order to establish reliable values of the axial dispersion coefficient and intraparticle effective diffusivity (De). Steady-state measurements of De were also made. For particles of uniform porosity the effective diffusivity obtained by the dynamic and steady-state measurements were in good agreement. For particles with a central core of lower porosity, the dynamic method gave higher De values. There is a particular advantage of error maps obtained by analysis of response data in the real-time domain. Such maps establish the range of operating conditions in which to make measurements for most accurate evaluation of parameters. I f data are analyzed only by the moment method, such optimum operating conditions are not known.
Chromatography has been applied by many investigators (Kubin, 1965; Kucera, 1965; Schneider and Smith, 1968; Clements, 1969; 0stergaard and Michelsen, 1969; Anderssen and White, 1970; Gangwal et al., 1971; Boxkes and Hofmann, 1972; Suzuki and Smith, 1972; Wakao, 1976a)
* Department of Chemical Engineering, University of California-Davis, Davis, Calif., 95616.
to the determination of kinetic and transport rate parameters. In the conventional model for reversible adsorption of a tracer in a packed bed of adsorbent particles, five parameters are involved: the axial fluid dispersion coefficient D,, particle-to-fluid mass transfer coefficient kf, intraparticle effective diffusivity De, adsorption rate constant k,, and adsorption equilibrium constant KA. In a recent article (Wakao e t al., 1979) the authors showed that K A and a relation between D, and De were
0196-4313/80/1019-0363$01.00/00 1980 American Chemical Society
