Table II.
Steps in Column Design
Specifications Breakthrough, C/CO, 1% Particle diameter 0.011 om. Diffusivity, Ds, 1 'X 1O-ssq. cm./sec. Flow rate R 10 l./hr. Feed con&n&ation c o 0.10 A' Resin capacity, q m p b , 1.0 meq./ml. Eduilibrium constant, K = m ( P = 0 ) I/R? Uqmpb, Co(v UfE), - U,fE? 9/Z L: Hrs. I , L. G. Equivs. G. Equivs. L.
-
Step in calculation Calculated value
0.80
Step in calculation Calculated value
4 9 o1, 9OOa 9 71)
Step in calculation Calculated value
4 0.974
a
Vol. 45, No. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY
1664
1
v
Case 1. Utilize 80% of Resin Capacity 2 3 4 J 6 3 . 1 0.026 0.26 0.26 0.21 Case 2 . Operate on 16-13011r Schedule 8 7 6 5 3 192 1.6 16 16 16 3 24
The result of Kunin and Barry also suggests strongly t'hat, in the nonaqueous systems involving sulfonic acid resins, the solid diffusion rates for exchange of ions other than hydrogen will be higher than t'hose found in the present work-Le., nearer to those found when water is the solvent. Resin Utilization, Cycle Time, or Column Volume Are Possible Design Specifications
h wide latitude exists in the specification of operating conditions for a column to carry out a desired exchange. Usually the feed concentration and flow rate are determined by the supply and the limiting concentration a t breakthrough is fixed by purity requirements. The choice of resin is a p t t o depend upon cost, availability, and general suitability for the exchange. When it is once chosen, its particle size, capacity, diffusivity, and equilibrium for the exchange become fixed. Table I1 shows a series of sample calculations based upon a n arbitrary combination of solution and resin, where the latter has a diffusivity representative of anhydrous glycol or ethanol. Three possible design cases are considered. First, economic factors (including the chemicals consumption in regeneration) may
- (EfE/R),
7 2.1 2 160
Case 3 . Given the Column Yolume, 2 L. 2 1 J 6 0.2 2 2 1.95
Trial estimate, step 4.
7
7 19.5
HI%
Y 0.21 1
16 8
1.95
fix a lower limit for the resin utilization and the shoitest process period which will give such a ut,ilization must be calculated, as in case 1. Here e/ Z is specified. Then Z is determined from Figure l , v/R is calculated by Equation 1, and v q , p b follows by way oi u from the known values of R and of q-pb. Combination of Equat,ions 1 and 2 w-it,h the relation, DG = qmpb/co,fz, gives Co(J'
- Vjc)
0 ~-V q m p b
z
(17)
Division by co a,nd then by 12 yields 7 - ( ~ f f i / R )the > useful time per cycle of operation, during which effluent of the requisite purity is being produred by the column. I n case 2 the useful t'iine of the operating cycle is specified, and the order of calculations (as shown by the numbers in parenthesis) is completely reversed. I n order to apply Equat,ion 17 a trial value of e/Z must be assumed. A better value is obtained from Figure 1 a t the end of the sequence and, if it is appreciably clifferent,, it must then be used in a repetition of steps 5 through 9 to arrive a t a still more reliable value. It often happens that a column of knorvn volume is available or is specified as a matter of particular convenience. A still different order of calculation is indicated here (Case 3), Trith the useful time as the end result. Acknowledgment
The work reported was performed under the auspices of the
U. S. Atomic Energy Commission. RECEIVED for review Novenlber 12, 1952. I ~ C C E P T E D hIay 6, 1958. Preseiit,ed as part of t h e Symposiuin on Nonaqueous Applications of I o n Exchange Resins before the Division of Colloid Chemistry a t t h e 122nd Meeting of the AliERIcas C H E I I I C A L S O C I E T l r , Atlantic C i t y , S . J.
Theory for Irreversible and
Constant-Pattern Solid Diffusion THEODORE VERMEULEN Department o f Chamisfry and Chemical Engineering, Universify o f California, Berkeley 4, Calif.
C
OXCENTRATIOS history calculations for fixed-bed adsorption and ion exchange columns must make use of one or another of a group of specialized results which take the place of a general solution t o the problem. The specialized results can be identified on the basis of controlling mechanism (external diffusion, internal pore diffusion, internal solid-phase diffusion, or longitudinal diffusion) and on equilibrium (unfavorable, linear, favorable. or completely irreversible). The mechanism of internal solid-phase diffusion applies especially t o synthetic organic gel-type exchange resins. This mechanism has been treated in the case of favorable or irreversible equilibrium with a constant pattern by Glueckauf and Coates (11), who assumed the driving force for mass transfer t o be proportional t o the difference between the outer surface concentration and the mean concentration of the entire particle. The
same assumption was utilized by Hiester and T'ermeulen (14) in solving the unfavorable and linear equilibrium cases of solid diffusion by an extension of Thomas' kinetic method ( 2 5 ) . Recently, the linear equilibrium case has been treated exactly by Thomas ( 2 6 ) , Edeskuty and hmundson (P), and, with numerical results, by Rosen (25). h exact infinite series solution for extent of particle saturation (but not explicitly for effluent concentration) has been given by Boyd, Adamson, and Myers ( b ) , Barrer ( I ) , Geddes (Q),and Eagle and Scott (6) for the case of constant surface concentration, as an extension of the related case in heat transfer. This paper uses an empirical approximation to the infinite series result that is easily handled, is more accurate than the Glueckauf-Coates assumption, and can be extrapolated to caws where the infinite series result does not apply.
August 1953
1665
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
Constant Surface Concentration I s Key to Irreversible Exchange
I n column operation under irreversible conditions, the surface of the solid particles retains a zero concentration as long as the solution concentration is zero. When internal diffusion is taken as rate controlling, the external diffusion rate is considered to be infinitely fast. The resin surface will then remove solute from the liquid phase as Ion gas any portion of it remains unsaturated, and solute will not be able t o accumulate in the liquid adjacent t o a particle unless the surface of the particle is entirely saturated, After that the concentrations in the particle and in the solution are independent of one another, except for material balance considerations; the rate of saturation of the particle will thus be independent of the actual solution concentrations. Thus the case of constant surface concentration provides the key t o the problem of irreversible equilibrium. For this case, the authors cited above (1, I ,6, 9) have given the following relation for the fraction of particle saturation attained:
In the case of irreversible exchange, breakthrough of the feed solute occurs discontinuously with an initially infinite slope a t a I
l
l
l
02
04
I
0.8
I
l
I
l I
l
06
08
I O
P cc 0.6 Y
0
E
0.4
c
a
gfn 0.2 '0
12
Figure
1.
I6
14
DIMENSIONLESS TIME PARAMETER,
I8
20
Ae
Comparison of Exact and Approximate Solutions for Solid Diffusion
Tb, measured from the first flow of influent a t concentration ( = co). For any later time, 7,a solution capacity parameter 0, can be defined:
time, (cA):
I n this equation, qa is the concentration of solute A on the resin, in moles per unit weight of dry adsorbent, a t a time, ( A T ) ; ( qA)" represents the value reached by the equilibrated particle prior t o the time when ( A T ) = 0, a t which the feed concentration was changed from (cA*)', in equilibrium with (yAA)o,to a new value, (e&'; and (yA*)' is the concentration of A on the solid t h a t would 0 is a dimensionless parameter be in equilibrium with (ea,,'). given by
(7) where vjs/R is the time required t o fill a column of void volume, a volumetric flow rate, R. The solution capacity parameter a t breakthrough, 8b,can be similarly defined with respect to 7. Thus, for column operation, U ~ E a, t
A0 = 8
Equation 1 indicates t h a t the fractional saturation y' is a single valued function of A@, As a close approximation, it can be replaced by
- 8a
(8)
Material balance relations can be used t o evaluate 0 6 a t moderate to large column volumes. Conditions are favorable for a constant pattern t o develop (14),so t h a t (9)
which is equivalent t o (4)
This can be differentiated t o obtain
(5)
where CA is the effluent concentration a t time T from a bed of volume v, and hence XA can be calculated by using Equation 1, 4, or 4a with the help of Equations 2, 8, and 9. A column capacity parameter, 8,equivalent t o a number of transfer units, is defined : VfE x = - 4D,,rr2 dp2 Da
(10)
The assumption of a linear driving potential leads instead to where the distribution ratio or partition coefficient, DG,is given by which integrates t o 1
In ___ = A@ 1 - y' 03'
Figure 1 compares the values of y1 calculated by Equations 1, 4, and 4a, and indicates that Equation 4 is much the better approximation. Column Operation. When the exchange reaction is irreversible and the equilibrium constant, K , is very large ( K + m), Equation 1 may be simplified somewhat. Here (ca*)" = (yA)" = 0 and (yA*)' = ym, the ultimate capacity or total concentration of all solutes in the resin phase. Then,
A column of a given volume, v, for an exchange whose diffusivity is D,, will correspond t o a particular value of Z that will measure the sharpness of breakthrough, The solution capacity parameter 8,will have a value, eatoio., equal to the value of B when the equivalents of solute introduced in the feed, co7R, are stoichiometrically equal to the exchange capacity of the column, P r n P b f f , plus the quantity that would be held in the voids, (ca)ovfa; t h a t is, when Co(TR - VjE) = From Equations 7 and 12,
q m pbv
(12)
1666
INDUSTRIAL AND ENGINEERING CHEMISTRY
By matching this with Equations 10 and 11, t h i o . = 8. Under - e,) will be constant, constant-pattern conditions, ( and thus €)b can be related to 2.
AREA
a E Q U I V A L E N T S OF
At small values of 2,y a t any e will have nearly the same value as a t 2 = 0 and may therefore be treated as a constant. Integration gives
I n the constant-pattern region,
Jl-
1-
eb
eSTOIC
COLUMN-CAPACITY P A R A M E T E R ,
Figure 2.
Vol. 45, No. 8
e
Positioning of Theoretical Breakthrough Curve
by Integration to Find
Ostoio. (=
Z)
At estolo. = 2 , also, the amount of solute that has escaped from the column must be just equal to the remaining unused exchange capacity. For a concentration history plot of X A against 0, as shown in Figure 2, the area between 5 and unity must represent the total exchange capacity of the column-Le., Z: or ~ B t o l c . in dimensionless units. Algebraically, m
12: =
1
-
i/
Thus there is need of an expression for (1 - 5) that will approach Equation 20 as 2: -+ 0 and Equation 21 as Z becomes large. Exact solutions can be made for the corresponding problem in external diffusion ( 5 ) and for the thin-bed irreversible case of the Glueckauf-Coates approximation. An exact solution is still needed for the present problem. However, the author of this paper has obtained an approximate solution by empirical means. The function, (1 - y2))/[(2y/x) 1 y], was found to satisfy the limits of Equations 20 and 21 and was first considered as a possible solution. However, it did not converge rapidly enough upon the constant-pattern form, as the curves calculated from it showed too high values of breakthrough 8 for any given slope in the high concentration region, compared to the experimental curves of the accompanying paper. The necessary convergence was approached by including a factor of e-I: in the first term of the denominator and was exceeded if a factor of e-lz was used. Accordingly, the following equation is proposed:
+ +
Sumerical or algebraic integration yields the following equations, applicable only to constant-pattern conditions ( 2 > 4): This relation can be used u-ith Equations 14 and 18 to solvc~for eb and the result is:
The respective values of Ba are seen to be Z - 0.64 in the exact case, Z - 0.614 in the present quadratic approximation, and Z - 1 in the linear driving force approximation. These values were used for locating the abscissas of the curves given in Figure 1, and emphasize the relative utility of Equation 16 as an approximation t o Equation 15. Thin Beds. The plots for resin concentration, y, against the increment in solution capacity parameter, A 8 = 8 - Ob, dl have the same pattern a t or near the entrance to a column as at points downstream where a constant-pattern solution concentration will develop-that is, Equation 3 will apply throughout. However, a t the influent end of the column, & = 0 = 2, whereas, in 0.614. Thus the rate of the constant-pattern region, 8 6 = Z advance of the exchange zone through the bed increases gradually until constant-pattern conditions are attained and then remains uniform. The solution concentration does not display a constant shape in this region, but starts out infinitely steep a t the bed entrance when x increases discontinuously from zero t o 1. The limiting behavior here is given by the conservation equation which can be written ( 1 5 ) :
This converges to the constant-pattern value of 8 b = Z - 0.614, as would be expected. Values of z were calculated for a series of values of @/E and 2 , from Equation 22, with y determined by Equation 3 and A 0 ( = 8 - ea) obtained through the use of Equation 23. The results are shoTvn as solid curves in Figure 3 and were found to agree closely with Equation 18 a t several check points. Figure 3 also shows the marked deviation of the linear gradient treatment in the low Z region. Rate Equation I s Extended lo Partially Reversible (Favorable) Equilibrium
Equation 5 will again be used as the starting point.
-
I n the event that q m > q A I * , the latter will still represent the concentration a t the particle surface, in equilibrium with the solution concentration, C A , a t t h a t point. After division by q m , there results
If this is combined with Equation 5, there results
-(”)
ax e
=--1 - y2
2u
where y* = q A ’ * / q m .
August 1953
INDUSTRIAL AND ENGINEERING CHEMlSTRY
It will be assumed that Equation 26 applies also to cases where the surface concentration, PAT*, is not constant; and the definition of y * is extended t o include any instantaneous ratio, qa*/qm. Actually, the rate will depend not only upon the instantaneous g ~ * but , also on the previous history of this variable. As in the irreversible case, however, the equation is offered as a good general approximation, somewhat more accurate than Glueckauf and Coates's relation for a linear driving potential,
*
The suitability of Equation 26 can be judged by inspecting the limiting case of linear equilibrium, where y * = x and, a t large 2, y differs from x by only second-order terms. I n column operation, Equation 26 then becomes
1
This is now the same result as would be given by Equation 26. Introducing Equation 2, the rate itself is given by
and substitution of this into a typical solution of the linear equilibrium case (8) yields
The resulting solutionsare identical in shape, but Z or e calculated by the approximation is 41r2/15 ( = 2.64) times Rosen's presumably correct values. The breakthrough curves change in shape rather slowly with variations in 2, so that this discrepancy is not much greater than the inaccuracies in interpreting experimental data. The deviation from an exact result is probably under 10% in the region where a constant pattern is most likely to prevail-that is, where the equilibrium constant, K , is greater than 4 or where the equilibrium parameter, r ( = 1/K, for ion exchange), lies between 0.25 and zero. Integration of Equation 26 has only been carried out under this constant-pattern condition. The following steps are involved. At equilibrium between the solution phase and the surface of the resin particles, the surface concentration, y*, is given by the equilibrium relation
provided the exchanging ions have the same valence. If the valences are not equal, an effective second-order equilibrium function, K", must be calculated by Equation 30, using suitable average values of x and g that correspond to the true equilibrium. By transformation of this equation, y*
B y comparison, the result of Rosen ( 2 5 ) for large L: can be put into the form:
5:
Kx
(K - 1 ) +~ 1
Also, the material balance for constant-pattern operation ( 1 4 ) requires that $1
(32)
X
Equation 26 then becomes
0.99
0.95
The variables are separable, and integration yields a relation implicit in x:
X
z
pa
1667
KQ 0.80
-In=)
(K
+ 1) - 2x1 - (K - 1)~: + - 2x2 - (K - 1)xg
K
I-
,
z
w
0.50 0 0
t-
z
w
=)
or, rearranging, 0.20
-1 LL tL
w
K2+K 1 K 2 - 1 I n r x
0.05
0.0 I 0.2
01
0.5
I
2
THROUGHPUT R A T I O , 9/x
Figure 3.
Breakthrough Behavior in Irreversible Exchange
---
Predicted from quadratic gradient From linear gradient
+2 In -1 = e Ka-1
(35)
x
This equation has been solved a t values of r = 0.2, 0.5, 0.8, and a t r = 0, where i t reduces t o Equation 16. The resulting curves are shown in Figure 4,which applies only t o the constantpattern conditions of large Z. The error in C A / C , will be less than 1% if Z exceeds the values given in Table I. The relation of 8, to Z,as determined graphically by use of Equation 14,is also shown. The generalized breakthrough values from Equation 35 have also been cross plotted in Figure 5, which is valid in the same
1668
Vol. 45, No. 8
INDUSTRIAL AND ENGINEERING CHEMISTRY Table 1. Equilibrium Constant, K m
6 2 1.25
centration difference, a is the transfer area per unit volume of the bed as packed, subscript i refers to t,he liquid-solid interface, and subscripts F and P denote external (or film) and internal (or particle) values. I n the initial stages of breakthrough, at low 2, zi* will tend to be zero and yi" will be less than unity. Here Equation 36a, in first-order irreversible form, will apply. Later, z i * becomes finite and yi* becomes unity; henceforth, Equation 36b (or 5 ) applies. At the abrupt transition from one mechanism to the other, denoted by subscript FP,
Constant-Pattern Solutions Equilibrium Parameter, T 0 0.2 0.6 0.8
Minimum Applicable 2 4
10 25
76
Z - e, 0.614 0.5s 0.64 1.0
and, as a result,
From z = 0 to z =
z ~ pthe ,
=
breakthrough curve will be given by
eeF-(ei;)c
(39)
and, from z = X F P to z = I,by
where (ep),and ( e p ) , are constants which fix the location of the will be proportional to, but not equal to, curve. Increments of increments of O p ; the proportionality a t the transition point, which must be consistent with values of ( Q F ) ~and ( e p ) , , is
-15
-10
-5
0
+5
+IO
+I5
+20
Equation 14 applies and can be evaluated with the aid of Equation 36:
R E L A T I V E V A L U E S OF S O L U T I O N CAPACITY PARAMETER, e S
-
Figure 4.
Effect of Equilibrium Parameter on Breakthrough
region of large Z. This cross plot supersedes the similar region in more extensive cross plots previously prepared (14), for which the mathematical solutions were based upon the assumption of a linear driving force ( 11). As the new equation approaches the linear driving force result when K decreases toward 1, it would appear t h a t use of the new equation could not be justified a t values of K below 1. Therefore, the values previously calculated for the latter region must still be considered the best t h a t are available. Slow External Diffusion Modifies Calculations
The results just presented are oversimplified because internal diffusion is not always slow compared t o external diffusion. Where the two mechanisms have similar rates, external diffusion will tend to predominate a t low extents of breakthrough and internal diffusion will have more of a retarding effect as full saturation is approached (14). Irreversible Equilibrium, Constant Pattern. The rate of external diffusion, given by Drew e l al. (6) and Michaels (bd), may be equated to the rate of internal diffusion adapted from Equation 26:
where k is the mass transfer coefficient per unit area per unit con-
Xo
XFP
From Equation 39, (33) Hence
(44) If the ratio k F & / k p a p D G is specified, X F P can be found by Equation 38. Then, if Z p is specified, ( e . P ) P P and ( e p ) F p can be calculated by Equations 44 and 41. Then Equations 39 and 40 can be used to determine ( O F ) , and (ep),, and, from these, the entire breakthrough curve for the combined mechanism can be calculated. Figure 6 gives a concentration history curve calculated in this way for Z p = 4 and k p a F / k p a p l ) G = 1; the resulting Curve may be compared with those for iZp = 4 when k p a p = and for Z p = 4 when k s a p = m . The results of this calculation method cannot all be put into a single plot. A separate family of curves, like Figure 3, would have t o be constructed for each constant value of k F a F / k p U p D G , with varying Z F , or for each constant Z with varying ratios of kFaF/kPaPDG.
August 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
- 20
1669
and the column capacity parameter will be
- 10 W
J
-5
in keeping with the discussion of an earlier paper by Hiester and Vermeulen (14). Here D is the solute diffusivity in the gas (or liquid) phase and L is a constant for any given adsorbent, which depends upon its porosity. Although a rigorous derivation has not been made, empirical evidence is accumulating that Equations 16 and 35 will apply (6).
VI W
Ea
-2
0 W
z
w
-I
-0.5
I
CD
Conclusions
+ 20 w _I
a + 10 u m W
->
s
+ 5
E
m 0
+ 2
P
+ I
0
0.1
0.2
0.3
EQUILIBRIUM
Figure 5.
0.4
0.5
0.6
0.7
0.8
PARAMETER, r ( = I / K )
Crossplots of Constant-Pattern Breakthrough Curves with Solid Diffusion Controlling
Partially Reversible Equilibrium, Constant Pattern. At any one value of the equilibrium constant the results for this case can, in principle, be put into the same form as those for irreversible equilibrium. However, the algebra is not as simple a s before because both rates must be taken into account throughout the calculations. Equation 36 again applies, together with Equation 31 in the form
vi* =
Kzi * ( K - l)zi*
+1
(45)
I n any specific instance, vi* and zi* can be eliminated algebraically or numerically between these equations, in order to give ( d z / d ~ )as , a complicated function of zalone. I n many cases of combined mechanisms, a sufficiently accurate result can be obtained by solving for an effective mean reaction kinetic constant ( I d ) , which must then be used with the solutions of Sillen and Ekedahl ( 2 4 ) or Thomas (25). For example, a SillBn curve with a value of the kinetic column capacity parameter, s, of 4, a t K = 03, falls almost midway between the pure external and pure internal diffusional curves of Figure 6 . Mathematical Treatment M a y Be Applied to Physical Adsorption
This matheniatical treatment holds for adsorption as well as ion exchange by substitution of r = I / K . It can be used directly with the T that applies to adsorption calculations which is defined in terms of the Langmuir equilibrium constant as Tad.
=
1
1
As in other methods of interpreting and predicting effluent concentration histories from fixed-bed columns, generality of mathematical treatment is obtained by defining a dimensionless solution capacity parameter, 8 (Equation 7 ) ,and a dimensionless column capacity parameter, Z (Equation 10). With the aid of these parameters, the new rate relation for solid diffusion has been used to obtain a n approximate solution for thin beds a t X < 4 in the irreversible case (Equations 22 and 23 and Figure 3 ) . This result is utilized in connection with the data of the accompanying paper. The new relation, rewritten in the form of Equation 26, has been found reasonably correct even a t K = 1. It is therefore used to improve upon previous treatments of the constantpattern effluent behavior a t values of K between 1 and 0 3 , resulting in Equation 35 and Figures 4 and 5. The results, in a related form, may apply also t o similar problems of physical adsorption in fixed beds.
+ Kad.
co
I n using Equation 35 for constant-pattern adsorption (or Equations 22 and 23 for thin-bed irreversible adsorption) definitions of 8 and z1 similar to those of Equations 7 and 10 apply if the internal mass transfer occurs mainly as two-dimensional diffusion along the inner pore surfaces. If gas-phase diffusion inside the pores is the controlling step, on the other hand, the solution capacity parameter will be
SOLUTION-
CAPACITY
PARAMETER,
e
Figure 6. Breakthrough Curves under Irreversible Conditions (r = 0,K = ).. for Film, Particle, and Combined Resistances Controlling Nomenclature
Dimensions of the variables are indicated in terms of a selfconsistent set of units. up = effective area for mass transfer through fluid film surrounding adsorbent particles, sq. cm./cc. up = effective area for mass transfer inside adsorbent particles, sq. cm./cc. ad. = adsorption, used as a superscript A , B = solute components C A , CB = concentration of solute A or B in fluid phase at a specified point in the column, gram equivalents/cc. eo = total concentration of solutes in fluid phase entering column; in ion exchange, co = C A C B dp = mean diameter of adsorbent particles, cm. D = effective ionic diffusivity for A diffusing against B (or for A diffusing in adsorption), sq. cm./sec. D, = effective ionic diffusivity in the solid phase Da = ratio of concentrations in solid and in fluid phase at saturation, q m p b / C o f E A E y / V , = internal pressure or ratio of molar internal energy of vaporization to molar volume of liquid, cal./cc.
+
erf. = the integral error function,
INDUSTRIAL AND ENGINEERING CHEMISTRY
1670
f~
=
IO
=
ratio of void space outside particles of adsorbent to total volume of packed column, dimensionless Bessel function of zero order and first kind, with imaginary argument (16,25,27) J = solut’ion function giving c/c, a t r = 1, dimensionless k~ = mass transfer coefficient for case of fluid-phase transfer controlling, cni./sec. l i p = mass transfer coefficient for case of solid-phase transfer controlling, cm./sec. : k p a p = 4 D , . ? r 2 / d p 2 K A E = chemical equilibrium constant for exchange of A with B , usually identical with K& K.& = apparent second-order equilibrium constant for exchanging ions of unequal valence, dimensionless
Kii
=
~AX:B/XA!/B
+
[ K ( q m / c o ) a - l ] ~ / ( a1)
L = proportionality constant in the pore-diffusion case, Equations 47 and 48 of the second paper n = an integer; used in infinite series Q A , q B , etc. = concentrat,ion of solute ion in the solid phase a t a specifiedpointin the column, gram equivalent/gram of air-dried adsorbent (q,#, etc. = uniform concentration level in solid phase as reached prior to start of current run ( Q A * ) ’ , etc. = concentration of solute in the solid phase, in equilibrium with the coexisting liquid phase a t concentration C A , etc., during the time of constant feed conditions designated by I q m = total concentration of solute in the solid phase when complete saturation is reached R = volumet,ric floir- rat,e of fluid through fixed solid, cc./sec. T = equilibrium parameter, dimensionless; for ion exchange, T = 1 / K if ions have equal valence, or T = l j k ’ ” ; for adsorpt,ion, T = 1/(1 Kad.co). s = column capacity parameter for the kinetic case, dimensionless t = solution capacit,y paramet’er for the kinetic case, dimensionless = hulk-packed volume of column., cc.: v”~f w is the void 2’ volume of t’hecolumn 2)pb = m i g h t of dry adsorbent charged to column, grains = volume of saturating fluid fed to column, cc.; V - v f s is V the volume of saturating fluid that has reached adsorbent a t a volume, v, downstream from the column inlet, x = c / c , for the sit,urating component zi* = local value of z a t the fluid-particle interface, in equilibrium \Tit,hthe surface of the solid, Z F P = value of z a t the transition from external to internal diffusion in the irreversible case xo = value of x in the first effluent leaving the columnLe., a t V - v f 8 = 0 = q / q m for the saturating component y1 = y during the time of constant feed conditions designated by I = equilibrium value of ?/ in equilibrium with a feed value T/* -of x less than x = 1 * = local value of y at the outer surface of the solid, in equilibrium TTith the adjacent fluid 8 = solution capacity parameter for diffusion controlled cases (see Equation 7 ) ; the value for internal or particle diffusion, ep, is indicated unless O p (film diffusion) or is specified; 8, is a constant of
+
I
Vol. 45, No. 8
integration; ( 0 p ) F p or (B&p are values a t the transition from film to particle diffusion e/Z = throughput parameter pb = density of dry adsorbent as packed in the column and saturated with carrier fluid, grams/cc. bulk volume Z = column capacity parameter for diffusion controlled cases (see Equation 10); the value for internal or particle diffusion, Zp, is indicated unless Z p (film diffusion) or is specified 7 = elapsed time from start of run. min. literature Cited (1) Barrer, R. >I., “Diffusion in and through Solids,” p. 29, University Press, Cambridge, England, 1941, (2) Boyd, G. E., Adamson, A. W., and Myers, L. S., Jr., J . Am. Chem. Soc., 69,2836 (1947). (3) Brinkley, S . R., Jr., Edwards, H . E., and Smith, R. W., ,JI., Math. Tables Aids Comv.. 6.40 (1952). (4) Chance, F. S., Boyd, G. E., and Garber, H. S., IND. ENG.CHEV, 45,1671 (1953). (5) Drew, T. B., Spoonei, F. hI.,and Douglas, J. (1944), repoited by Klotz, I. M., Chem. Revs., 39,241 (1946). (6) Eagle, S.,and Scott, J. W., IND.ENG.CHEM..42, 1287 (1950). (7) Edeskuty, F. J., and Amundson, N. R., J . Phis. Chem., 56, 148 (1952). (8) Furnas, C. C., Trans. Am. Inst. Ckem. Engrs., 24, 142 (1930). (9) Geddes, R. L., Ibid., 42,79 (1940). (10) Glasstone, S.,Laidler, K. J., and Eyring, H., “Theory of Kate Procesees,” pp. 516-22, New York, NcGraw-Hill Book Go., 1941. (11) Glueckauf, E., and Coates, J. I., J . Chem. SOC., 1937, p. 1315. (12) Grossman, J. J., and Adamson, A. W., J . Phys. Chem., 56, 97 (1952). (13) Hammett, L. P., “Physical Organic Chemistry,” pp. 251-72, New York, XcGraw-Hill Book Co., 1940. (14) Hiester, N. K., and Vermeulen, T., Chem. Eng. Proyr., 48, 505 (1952); Amer. Doc. Inst., Doc. 3665 (1952). (15) Hiester, N. K . , and Vermeulen, T., J . Chem. Phys., 16, 1087 (1948). (16) Hildebrand, J. H., and Scott, R. L., “Solubility of Non-Electrolytes,” 3rd. ed., New York, Reinhold Publishing Gorp., 1950. (17) Hougen, 0. A,, and Marshall, W. R., Chem. Eng. Progr., 43, 197 (1947). (18) Klinkenberg, A., IND. ENG.CHEM.,40, 1992 (1948). (19) Kunin, R., and Barry, R. E., Ibid., 41,1269 (1949). (20) Matheson, L. A, private communication, reported by Tompkins, E. R., J . Chem. Educ., 26, 92 (1949). (21) Mayer, S. W., and Tompkins, E. R., J . Am. Chem. Soc., 69,2866 (1947). (22) Michaels, A. S., IND. EFG.CHEM.,44,1922 (1952). (23) Rosen, J. B., J . Chem. Phys., 20,387 (1952). (24) Sill&, L. G., and Ekedahl, E., Arkiv Kemi, Mineral. Geol., 22A, Nos. 15 and 16 (1946). (25) Thomas, H. C., J . Am. Chem. Soc., 66,1664 (1944). (26) Thomas, H. C., J . Chem. Phys., 19,1213 (1951). (27) Vermeulen, T., and Hiester, N. K., IND.ENG.CHEM.,44, 636 (1952). (28) Walter, J. E., J.Chem. Phys., 13,229 (1945). RECEIVED for review November 12, 1952.
ACCEPTED May 6 , 1963.
