An Elementary Theory of Adsorption in Fluidized Beds J
J
MATHEMATICS OF ADSORPTION IN BEDS PAUL R. KASTEN AND NEAL R. AMUNDSON University of Minnesota, Minneapolis 14, M i n n .
way without recourse to numerical methods some restrictive assumptions must be made regarding this mechanism. I n general, where either an equilibrium or kinetic relation is required, it is assumed linear. Certainly in the case of dilute solutions this assumption is not offensive. The use of other than linear isotherms for the equilibrium case will be made the subject of a further communication. I n the analysis which follows it is necessary that the concentration of adsorbate in the fluid phase in the bed be uniform in time and space; the zoning procedure of Kalbach (6) could be used here also with subsequent modification of the problem. The present authors assume that the whole operation is isot]hermal, in the steady state, and that the introduction of adsorbent is continuous. The latter implies that the spheres are so small that in so far as the adsorbent stream is concerned it behaves essentially as a fluid.
T h e process of adsorption of a solute from a fluid stream by a bed of fluidized adsorption is considered, assuming that the solid particles are uniform, porous spheres. Adsorbent moves in the bed in a random fashion so that elementary probability can be applied to determine the residence time of the spheres. Adsorbate diffuses into the spheres, and is adsorbed either according to an equilibrium theory, or, if the rate of adsorption is slow, according to a kinetic relation. Mass transfer at the fluid-solid interface is considered. No experimental confirmation or rejection is given and the theory, since it assumes linear relationships for equilibrium and kinetic cases, holds only for special substances or dilute solutions. Qualitative conclusions can be drawn for other cases from the results obtained here.
S'
XCE the advent of fluidized beds for the catalytic cracking
of petroleum fractions many catalytic and heat transfer processes have been re-examined to determine the applicability of the fluidization technique, and new chemical processes using the fluidization principle are in the offing. Campbell et al. ( 8 ) and Arnold and Baxter ( 1) have considered the processes of selective adsorption in a fluidized bed of adsorbent. Kalbach (6) analyzed fluidized beds from an over-all point of view emphasizing solid-gas reactions. Numerous papers (8, 11, 13, 16) have appeared in the literature on the mechanical and fluid mechanical aspects of fluidization. Mass transfer from particles in fluidized beds was treated by Resnick and White ( 1 2 ) and the reaction of methane on a fluidized copper oxide catalyst was considered by Lewis, Gilliland, and Reed (9) in an experimental way. However, theoretical development of design equations for fluidized solid-gas reactors has not appeared in the literature. In this paper the authors are dealing with adsorption, which is rlosely related to heterogeneous catalysis, in idealized fluidized beds of adsorbent, They assume that a stream of solid adsorbent and a fluid stream containing adsorbate are simultaneously admitted to a vessel. Two streams are withdrawn, a solid stream of adsorbent containing adsorbed material removed from the fluid stream and a fluid stream which is poorer in the adsorbate than when admitted. The analysis in the paper, however, is also applicable to the case of stripping the adsorbate from the adsorbent in a fluidized bed, if it is assumed that the adsorbate is uniformly adsorbed. The adsorbent is assumed to be in the form of porous spheres of constant diameter, and these are kept in a state of suspension and random motion by the fluid stream which enters the vessel. It is conjectured that the adsorbate passes from the fluid phase through a resisting film a t the surface of the spheres and thence diffuses into the spheres where it is adsorbed on the internal surface. The case of equilibrium a t the interface is not excluded and can be obtained as a special case. The adsorption mechanism which takes place inside the sphere has recently been examined by Laidler ( 7 ) t o determine the conditions under which equilibrium or kinetic relations apply. In order to solve the problem in a mathematical
PROBABILITY CONSIDERATIONS IN THE BED
If one assunies that the motion of spheres in the bed is a completely random one there is no a priori reason why one sphere should leave the bed before any other one. Consequently a sphere which has just been admitted has the same chance of leaving as one which has resided in the bed for some time. Because it is assumed that the bed is operating in a steady state one must assume for the sake of mathematical convenience that the bed has been operating for infinite time. Hence there is a probability that some particles may have a residence time 15-hich is very large, although it can be shown that this is extremely improbable. I n this section the original idea of MacMullin and Weber (10) will be used, although it will be extended somewhat. Consider a box of N spheres. If one more sphere is added, the robability of withdrawing that sphere in one draw is 1/(N 1). Hence the probability that that sphere remains in the box after 1). If the process is continued by adding one draw is N / ( N other spheres one a t a time and removing an equivalent number one a t a time, after n simultaneous additions and withdrawals, the probability that the original sphere is still in the box is (N/N 1)n. I n an analogous fashion, if the bed has a mass, W , and a mass, Am, is added and then a mass, Am, withdrawn, the probability that the added mass is still in the bed is W / (W Am). If the process is repeated as above, after n withdrawals the probability that the original mass, Am, is still present
+
+
+
+
is
(+m)n
If the n withdrawals each occur at intervals of time At, the total time is t = n At, and the above can be written (1
+
-t -
$)At
If one assumes that s = W / Am, then
1341
Vol. 42. No. 7
INDUSTRIAL AND ENGINEERING CHEMISTRY
1342
Now if Am +0 in such a way that A m / At has a limit, then s + m and the above expression reduces to
lim
since s -+- (1
Am dm + 1,’~)~ e and --+ = w ,where e is the At dt =
base of natural-logarithms. Hence the probability that a mass, dm, introduced a t time t = 0 is still in the bed a t time t = t is given by Equation 1. This shows clearly that it is improbable that a given mass of adsorbent will stay in the bed for long periods of time. It is evident then that if an element of mass, wdt, is admitted a t time t = 0, the part of this element which resides in the bed a t t = tis given by - tw
-
we
dt
If Equation 2 is integrated from zero to infinity, one should obtain the mass in the bed. Direct integration reveals that this is the case. In order t o calculate the average time a sphere (considered as an element) would remain in the bed, form the weighed mean of the time as follows:
W
Integration by parts yields - = 1 lvhich is usually called the W nominal holding time for continuous stirred tank reactors. In considering a single sphere one sees tiiut the a,dsorbate diffuses through the surface film to the sphere surface and then must diffuse through the fluid phase in the void .volume inside the sphere and be adsorbed on the internal, surface area. By Ficks l a ~ vone knows that, the rate of diffusion 1s glven by the product of diffusivity, effective area for diffusion, and concentration gradient. Consequently, the rate at which adsmbate leaves the sphere is
and the rate a t which it leaves per unit mass of spheresis -3D,a
np (%, This is a function of the time only since the derivative is to be evaluated a t the surface. If an element of mass ~ d ist admitted a t time t = 0, the part of this element still present a t time t = 1 is given by Equation 2 and the rate at which adsorbate 1s leaving this element is
It follows that the rate a t which adsorbate is leaving all the spheres in the bed is given by
Equation 3 is the fundamental formula on which this paper is based. One must now calculate what the concentration of adsorbate will be as a function of time and position inside the sphere. Once this is done the concent,ration gradient can be evaluated a t the surface. DERIVATIOR- OF DIFFERENTIAL EQUATION AND BOUNDARY COiVDITIONS
Consider a single sphere Tvhich is suspended in a fluid of uniform concentration cl. Adsorbate diffuses into the sphere and is adsorbed on the interior surface area. If one imagines a spherical shell of inside radius T and thickness ar, by equating rate of inflow minus rate of outflow to the rate accumulation in the shell, there results
where T
< I. < r + A?.
Letting
37
-+
0, the above reduces to cy
at
(4)
when the diffusion coefficient of the adsorbate in the fluid phase in the sphere void volume is a constant. This equation contains two dependent variables, n, the amount adsorbed, and e, the concentration in the fluid in the sphere. Both are functions of t and r. If there is a resistance to mass transfer a t the fluid-solid interface this resistance can be characterized by a mass transfer coefficient, k,-, such that the mathematical condition at the boundary can be expressed as
Physically, the left-hand side is the rate of outflow of adsorbate per unit of area from the point of view of an observer inside the sphere vhile the right hand side is the same thing and defines mathematically the mass transfer coefficient, k l . If one allows kf in Equation 5 to become infinite then this boundary condition reduces t o c = c1 when r
=
R
(6)
A boundary condition of this kind can in ,some cases be used as an approximaticn, if the mass transfer coefficient is very large. Often this is a valuable maneuver since the solutions of mathematical problems with this type of condition are somewhat simpler than when the former type is used. In addition to the above, the init’ial condition of the spheres must be known. As the spheres ent,er the vessel, it will be assumed that there is some solute adsorbed uniformly on the internal surface, and also a uniform solute concentration in the fluid in the void volume of the spheres. Then, c = eo\ when I = 0 12 =
(7)
In the case of adsorption, co and no rvould be zero or very close thereto. If the equations are to be used for stripping, then cs and no are not zero. Before a solution can be found for the problem some relationship between n and c inside the sphere must be assumed. In what follows two possibilities will be considered-equilibrium and kinetic relations. Case A, Equilibrium. Assuming that once adsorbate diffuses into the sphere equilibrium is established between adsorbed material and adsorbate in void volume, then the relation between n and c is that of an adsorption isotherm. In general
in which case Equation 4 reduces to
For the majority of cases f(c) is a complicated function so that this equation is nonlinear and standard analytical methods of mathematics will not lead to a solution and numerical methods must be used. For the present, this difficulty is circumvented by assuming the isotherm is of the form
+ K2
n = KIC
where K , and K2 are constants which characterize the adsorbate, solvent, and adsorbent as well as the temperature which has
1343
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1950
been assumed t o be uniform throughout. these conditions Equation 8 can be written
Under
1
I
I
I
I
I
I
I
where
In order to find the adsorbate concentration in the sphere, one must solve the mathematical system consisting of Equations 5, 7, and 9. This can be done easily if one makes the change of variable v = c - CI. The system then reduces to
D,
2+ v
F i g u r e 1, Plot of B 2)s. 6 for Various
k f v = 0, when r = R
- clJ when t
= co
0
=
This is a standard problem and can be found in the book Of Carslaw and Jaeger (s, page 202)* After making a few minor changes in notation, the solution can be written as
c=c1+
2(ce
-
C1)EE
r
-pf + + (E
w.
€(E
1'7
- 1)
and m, and the corresponding values for wi are given in Table I. I n general, the first four terms of the infinite series are sufficient to give an accuracy of a t least three significant figures. The values of B when E = corresponds to an infinite value of ,bf, and under this condition the absolute concentration in the fluid a t the very surface of the sphere is the same as that in the main body of fluid in the bed. As mentioned earlier, this would serve as an approximation for very large mass transfer coefficients.
TABLE I. ROOTSOF where w i is a root of
1/z
w
cot
w =
1-
(11)
E
and where the summation in Equation 10 is over the infinite number of roots of Equation 11. Making use of Equations 10 and 11,
(2);
=R =
-
R
1
+ 4 E - 1)
-rw?t e Rz
from Equations 14 and 15
E
1
2 5 10
WI
w2
1.1656 1.5708 2.0288 2.5704 2.8363 3.1416
4.6042 4.7124 4.9132 5.3540 5.7172 6.2832
w
cot
wn
7.7899 7.8540 7.9787 8.3029 8.6587 ~ . 4 2 ~
w
+ -1=0 6
w4
w1
we
10.9499 10.9956 11.0856 11.3349 11.6532 812.5664
14.1017 14.1372 14.2075 14.4080 14.6870 15,7080
17.2498 17.2788 17.3364 17.5034 17.7481 18.8496
From Equation 11 it is evident that wn -L nrr as an infinite k / Equation 14 reduces to
E
-
m , SO
for
m
If this is substituted into Equation 3 and the indicated integration is performed with respect to t, one obtains
X = 6wWG
+
c
= A X B
m
01)
(co
- cl)
P
1
1
[w:
+ - l ) l ( w ? + 8) E(€
(12)
This is the over-all rate a t which adsorbate is leaving the spheres and hence must be to the rate at which fluid passing through the adsorber must be picking it up. Therefore
in the
x = V(c1 "
(13)
c2)
assuming V i s sensibly constant from inlet to outlet. Equations 12 and 13 gives ci co
- ca
- c1
=
w(Ki
+ a)
The convergence of the infinite series in this case is not as rapid as it was in Equation 14; however, it is still rapidly convergent mathematically. From the point of view of computation the convergence is poor. The values on the graph were obtained by using fifty terms of the infinite series and estimating the remaining terms by the integral approximation
862
Combining
(14)
= A X B where A and B are two dimensionless functions. Examination of Equation 14 shows that the infinite series converges very rapidly since the nth term behaves essentially as The values of w i for various values of E are tabulated in Carslaw and Jaeger (8,page 378) and as shown there (3, page 203) the roots are real and simple. For large values, con is approximately equal to nr. In Figure 1 are the calculated values of B in graphical form. These curves were obtained by assuming values of 6 of 0,l/2, 1, 3, 5, 10, and 20. The values of E are l / 2 , 1, 2, 5, 10,
The above integral is approximately 6/50rrz since 6 is small comin this range of 2. pared to ?r2z2 As an application of the above theory, the following problem is considered.
One hundred thousand (100,000) cubic feet per hour of air at 77" F. and 1-atmosphere pressure, containing 0.90% by volume of solute A , is to be treated in a solvent recovery plant. The solute is t o be adsorbed by a fluidized bed of lbmesh, spherical particles, originally containing no solute, with 99% recovery desired. Determine the feed rete w, the bed mass W , and the size of vessel required, given the following: n = 5 X lO4c a t 77" F. and atmospheric pressure, for low concentration of adsorbate in fluid; the particle apparent density is 55 pounds per
cubic foot; the particle porosity is 0.57; the diffusion coefficient of adsorbate A in air under above conditions is 0.31 square foot per hour; the superficial velocity of the fluid in the vessel is to be 2 feet per second, corresponding to a bed density of 20 pounds
INDUSTRIAL AND ENGINEERING CHEMISTRY
1344
per cubic foot, as determined by an economic balance between solid recovery costs and solid recirculation costs. SOLUTIOK. From above, co = 0; c1 = 0.01 c2; D , = 3.2.5 X IOw3feet; K1 = 5 X 10'; p = 55 pounds per cubic foot; CY = 0.57; D, = 0.31 square foot per hour; ug = 2 feet per second; p~ = 20 pounds per cubic foot; and T' = 106 cubic feet per hour. From Equation 14 m
(0.01-1 )e: 0-0.Olcp
Vol. 42, No. ?
Case B, Nonequilibriurn. If the rate of adsorption is slow compared with that of diffusion, one cannot assume that the adsorption isotherm is valid inside the sphere and a relation describing this rate must be used. In general, the rate of adsorption will be a function of temperature, pressure, the system itself, the adsorbate concentration in the void volume, and also of the amount already adsorbed. For a given system at constant temperature and pressure, the relationship would be of the form:
- 99
Let B represent the infinite series summation, so
Generally, g ( c , n ) would he given i r k graphical form arid no analytic expression would be available. If, for example, one were to zssuine kinetics of Imigmuii type, then
If S is the cross-sectional area of the vessei u,S(3600) = lo6 cubic feet per hour:
s = loo -Oo0 L= 13 9 squ:ire feet Z(3600)
The diameter of vessel required i s d $ 1 3 . 9
G
=
=
where h, and hz are constants and -To is the saturation capacity of the adsorbent. This would be desirable, hut the analytical difficulties are insuperable a t present and the solution can only be obtained in some approximate form. Hence the discussion will be limited to the linear relation
4.2 feet.
29 ~,p,(3600) = 2(3600) X 3 ~ X 9
"_5; -
- 533pounds per hour per square foot
arid modified Reynolds number
D G A
=
P
3.25 x 10-3(533) = 39.1 0.0183(2.42)
Assuming the mass transfer data of Resnick and White ( l a ) for fluidized beds is applicable here for any depth bed, then k j = 580 pound moles per hour per square foot. Thus
Also
D"CY
y=--Ki
+
CY
-5
0.31(0.37) ._ 0.57
X lo4
+
- 3.54 x
10-6
so 3.54 x 10-6 (1.625 X
Y
W
-1
=
1.333*
If the rate of adsorption inside the sphere is proportional to the difference betveen the adsorbate eoncentration in the void volume and the equilibrium concentration corresponding to the amount adsorbed, then
;(2)
= ka(C
- c,)
If the equilibrium relationship is of the form n = Lac,, then this formula is a special case of Equation 16. To obtain the adsorbate concentration inside the sphere for the linear kinetic relation assumed, Equations 4,5, 7 , and 16 must be solved simultaneously. In so far as the authors are aware, this solution does not eyist in the literature; however, it can be obtained by the use of the Laplace transform. The definition of the t i ansform as given by Churchill (4)is used here
From Figure 1, for a given E and 6 one can obtain the relation between w and w/W. In this problem, the following table can be constructed: w/
w
1 5 10 20
d
0.75
3.75 7.5
16.0
B
0.88 0.60
0.44
0.29
W
12,380
Applying the operator to c and n, one defines
18,140
24,750 37,550
The proper w/W to use is determined by an economic balance between recirculat,ing solid cost for high w ,and cost of niaintaining preesure drop across fluidized bed for high TV. If w / W = 5 , then w = 18,140 pounds per hour and W- = 18,140/5 = 3630 pounds. hfinimum height of vessel = height of fluidized bed = 3620 = 13.1 feet. Study of Figure 1 reveals that w decreases as IC/ increases, for any given 6. Since there is no film resistance a t the solid-fluid interface for infinite k f , the residence time required to saturate the spheres with adsorbate is less than for finite kf; or, less TV is required. For low values of the mass transfer coefficient, high values of W are advantageous. The relation bet.lveen 6 and z u / W is very dependent upon the values of K , and R. An adsorbent, having a very high adsorbate capacity (high K 1 ) can advantageously have a high W , even for small particle sizes. If K1 is comparatively small, the size of particles have to be larger if there is t o be any advantage in long residence time. Thus for any K1 the larger the particle size, the more expedient the use of large TV. Conversely, dynamical considerations in the bed limit the particle size.
The notation L-1 is used to denote the inverse operator. Applying the operator to Equations -1 and 16, making use of Equation 7 at the same time, results in
and
Upon eliminating W betxeen these two equations, there results d2C dr2 -
dC' a +;2 (&) - ojc =
(-CYCD
+ ha) f pn0. + kza)
-no) ( p
d " ( P
where
a = p ( p +kl p
+ k2a)
+ kaa
(17)
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1950
The left-hand side of Equation 17 is of the form of Bessel's equation (14, page 64) of order one half, and the right-hand side is a constant. Making use of relations between Bessel's functions of order one half and trigonometric functions (14, page 61), the general solution of Equation 17 can be written as
where A and B are functions of p whose form must be determined. On a physical basis the solution must remain finite a t the value r = 0. This is impossible if B # 0, and therefore B = 0. To determine A , take the transform of Equation 5, obtaining dC
-D~dr
=
kf
(c - );
Consider first the pole corresponding to p = 0. I n the first term of Equation 19, substitute for the trigonometric terms their respective iniinite series representations, and cancel all factors common to numerator and denominator. Applying Equation 20 to the resulting terms of Equation 19, the resulting residue a t p = 0 is cl. Repeating the same procedure, the residue is zero for the pole corresponding to p = -kl - kza, for the residue of the first term is the negative of the residue of the second term. Equation 21 is used to find the residues corresponding t o the remaining poles since the roots of Equation 22 are real and simple as mentioned previously. Denote these roots by w,. From the definition of w,, define b, as
This reduces to
when r = R
P2
4
Substituting Equation 18 into this equation, an equation in A is obtained. After solving for A and substituting its value in Equation 18, there results:
where
w
+
+ + +
rw (no ape?(p k z a ) - pno sin R pa(p kl kea) = lid?. This is the transform of the solution of
+
1345
the problem, and to find the solution itself the inverse transform must be obtained. This can be accomplished by an extension of the Heaviside expansion theorem ( 4 , page 167) as follows:
+ + bn) + bnkza
fP ( ~ I
Solving for p ,
1
-2
PI,
[kt
f kza
~ZCY
0
+ 6 , - d ( k l + k z e + bn)'
- 4bnakzI
It is easy to show that p l n and p z n are negative. Having already treated the poles corresponding to p = 0 and p = -kl - kZa,let p ( p ) = (1 -
E)
sin
w
- w cos w
Then
Consider a function of f ( p ) which is analytic in the finite, complex plane of variable p , except for a set of simple poles P I , P% PSI . . . I Pn, . . . The inverse transform off@) is
. Letting
m
Nin = (pzn
where p , ( l ) is the residue of e p t f ( p ) at p = pn. When a pole is simple the residue can be written as lim ( p - pn)f(p)ePt P-Pn In particular, if f ( p ) has the fractional form pn(t) =
+ no? - pinno - aCi(Pcn + kl +i k 21,2 ~~)1
+
( ~ C O
=
apply Equation 21 obtaining the inverse of Equation 19 in the form
(20)
c(r,t) = c1
+
w,,
20,~ &r 2 b n ( u n sin
sin
an
TWn R
- e cos w,)
Pn
n= 1
where
+ hN+ukza + bn
where g ( p ) and g ( p ) are analytic at p = p , and g(p,) # 0, the residue a t the simple pole p n is
P -
where p'(pn) is the derivative of p(p) evaluated at p = p,. Hence,
Although an essential singularity occurs a t p = kze, it can be shown that the resulting residue is zero. Now calculate the concentration gradlent and substitute in Equation 3. The integration can be performed easily to obtain a rather cumbersome expression which can be reduced by making use of some relations between the roots of a quadratic equation. Combining the resultant expression with Equation 13 gives
- 2pin
p a t
2pzn
+ klNzn+ kza + bn -
m
m
Therefore, under the above conditions, when all the poles of f ( p ) are simple, the inverse transform is given by this formula. Examination of the denominators in Equation 19 reveals poles a t p = 0, p = -kl - k2a, and the zeros of
- e) sin w - w cos w
0 (22) It can easily be shown t h a t the poles at p = 0 and p = -kl kza! are simple poles. The zeros of Equation 22 have been examined by Carslaw (5) who has shown t h a t these zeros are all real and simple for the values of E considered in this paper. Note that the pole p = -kl - kza corresponds to the pole w = 0 so that only the nonzero roots of Equation 22 need be considered. (1
where
=
and
epznt
1346
INDUSTRIAL AND ENGINEERING CHEMISTRY
Equation 23 gives the relation between inlet and outlet fluid concentrations but its use is not quite so simple as in the equilibrium case. The application of this formula t o a particular case is made difficult by the paucity of literature on adsorption kinetics. I n passing, i t might be noted that the w. appearing in Equation 23 are roots of the same equation occurring in the equilibrium case. The solution for the case of infinite mass transfer coefficient is obtained by letting E ~ 3 . As before wn -+ nr, so in the limit Equation 3 becomes +
m
SUMMARY AND CRITIQUE
I
Adsorption in a fluidized bed has been considered assuming uniform, porous spheres as the adsorbent medium. The spheres have been assumed to move in a completely random fashion in the bed with the supporting fluid of uniform concentration. Two cases were considered. If equilibrium is attained within the spheres, the relation between the adsorbate concentration of the fluid and of the solid must be according t o the isotherm for that system. I n this paper the isotherm had to be linear. In the second case the nonequilibrium condition was discussed and a linear, reversible kinetic relation was used. This equation is the most general relation for the problem involved which can be solved by elementary analytical methods. It was supposed there existed a resistance to mass transfer a t the fluid-sphere interface. It might be that the adsorbate concentration is not uniform throughout the fluid since there might be a continuous concentration gradient in the fluid bed from inlet to outlet. Gilliland and Mason (6)found back-mixing of gas in fluidized beds relatively low, although more data is needed for vessels of low length-diameter ratios. Perhaps a better approximation would be to assume the spheres are in contact with a fluid whose adsorbate concentration is the average of the inlet and outlet concentrations. This would have the effect of replacing the lefthand side of Equation 14 by 2(Cl 2co -
- c2) c1 - c2
D,
diffusion coefficient of adsorbate in fluid in the sphere void volume, sq. ft./hr. G = mass velocity of fluid in vessel, Ib./hr./sq. ft. kj = mass transfer film coefficient, Ib. molcs/sq. ft./hr./unit concentration difference kl, kz = constants in the kinetic equation in reciprocal hours K1, K L= constants in the adsorption isotherm n = adsorbate on adsorbent, Ib. moles/unit apparent volume of adsorbent no = initial adsorbate on adsorbent, Ib. moles/unit apparent volume of adsorbent N = Laplace transform of n p = Laplace transform parameter corresponding to t T = radius variable in sphere, ft. R = external radius of spheres, ft. S = cross-sectional area, sq. ft. t = time, hr. uQ = superficial velocity of fluid, ft./sec. v =c-c1 V = fluid feed rate to bed, cu. ft./hr. w = adsorbent feed rate, lb./hr. R’ = mass of adsorbent in fluidized bed, Ib. 01 = porosity of spheres, cubic feet of void volume per cubic foot of apparent volume of solid or square feet of open area per square foot of sphere surface Y = D,,oi/(K, N) = wR;/bF@, dimensionless e = kfR/D,, dimensionless (modified Nusselt number) p = apparent density of spheres, Ib./cu. ft. P B = bulk density of fluid bed, Ib./cu. ft. p = viscosity of fluid, Ib./ft.-hr. w = a root of the transcendental equation: w cot w = 1 - E =
+
LITERATURE CITED
Arnold, h l . H. hI., and Baxter, D., C . 6. Patent 2,476,472 (July 19, 1949). Campbell, D. L., Tyson, C. W., Martin, H. Z., and hIurphree, E. V., U. S. Patent 2,428,690 (Oct. 7, 1947); Zbid., 2,446,076 (July 27, 1948).
Carslaw, H. S., and Jaeger, J. C., “Conduction of Heat in Solids,” London, Oxford University Press, 1947 Churchill, R. V., “Modern Operational Mathematics in Engineering,” New York, McGraw-Hill Book Co., 1944. Gilliland, E. R . , and Mason, E. A , , IND.ENG.CHICM.,41, 1191 (1949).
Kalbach, J. C., Chsm. Eng., 54, No. 1, 105 (1917); 54, No. 2, 136 (1947).
Laidler, K. J., Bull. soc. chim. France, 1949, D 171-6. Leva, M., Grummer, hl.. Weintraub, M., and Pollchik, AM., Chem. Eng. Progress, 44, 511, 619 (1948).
Lewis, W. K., Gilliland, E, R., and Reed, W. A., IND.ENQ. CHEM.,
with similar changes in the other formulas. The assumption of completely random motion of the particles with the resultant use of probability to determine the residence time of spheres is probably valid since the amount of adsorbent used would be large and the particle size small. For the case of a linear equilibrium isotherm, Equation 14 reveals that for industrial adsorbents, whose K1 values run from lo3 to 104, there is some doubt about the utility of fluidized beds for adsorption purposes. This is readily apparent since the assumption of a uniform fluid concentration means that the exit fluid has the concentration of the bed fluid and hence the driving force for diffusion into the particles is small with the result that the quantity adsorbed per unit mass of adsorbent is not very great, The results of this paper, however, should be applicable to the much more important process of catalysis. NOMENCLATURE
c = adsorbate concentration of fluid phase, Ib. moles/cu. f t . co = initial adsorbate concentration of spheres in fluid phase
in the void volume, Ib. moles/cu. f t .
c1
= adsorbate concentration of fluid phase in reactor vessel, Ib.
moles/cu. ft. = inlet adsorbate concentration of fluid stream, Ib. moles/cu. ft. C = Laplace transform of c
c2
Vol. 42, No. 7
41, 1227 (1949).
MacMullin, R. B., and Weber, M., Jr., Trans. Am. Znst. C h m . Engrs., 31, 409 (1935). Parent, J. D., Jagol, No,and Steiner, C. S., Chem. E ~ QProgrcea, . 43, 429 (1947).
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RECEIVED January 16,1950.
Dust and Fume Standards-Correction I n the article on “Dust and Fume Standards” [McCabe, Rose, Hamming, and Viets, IND.ENG.CHEM.,41, 2388 (1949)l the formula on page 2390 should be corrected by changing the minua signs to plus signs. LOUISC. MCCABE
