INDUSTRIAL AND ENGINEERING CHEMISTRY
1698
of Cubberley and Mueller ( 5 ) to determine the alcohol-ketone ratio allowed calculation of a value of 2.0 for
K = -P(pheno1) X P2(H2) P( cyclohexanone) Equilibrium constants at other temperatures were calculated on the assumption of 40,000 calories as the heat of reaction. The curves in Figure 1 are based on these constants, together with those of Cubberley and Mueller. literature Cited ( 1 ) Bagnall, mi. H., Goodings, E. P., and M’ilson, C. L., J . Am. Chem.Soc.,73,4794 (1951). (2) Bailey, W.A., and Peterson \V. H., Brit. Patent 559,682 (March 1, 1944). (3) Bartiett, E. P , and Field, E., U. S. Patent 2,291,585 (July 28, 1942).
EngFnyring
Vol. 44, No. 7
(4) Berkman, S., Morrell, J. C., and Egloff, G., “Catalysis,” New York, Reinhold Publishing Corp., 1940. (5) Cubberley, A. H., and Mueller, M. B., J . Am. Chem. Soc.. 69, 1535 (1947). (6) Dowden, D. A., J . Chem. SOC.,1950,242. (7) Field, E., U. S. Patent 2,265,939 (Dec. 9, 1941). (8) Griffith, R. H., “Contact Catalysis,” 2nd ed., Oxford, Oxford University Press, 1946. (9) Knoevenagel, E., and Klages, A., Ann., 281,94 (1894). (10) Loder, D. J., U. S. Patent 2,321,551 (June 8, 1943). (11) Lukes. R. M., and Wilson, C. L., J. Am. Chem. Soc., 73, 4790 (1951). (12) -Veerwein, H., Schoeller, W., Schwenk, E., and Borgwardt, E., Ger. Patent 574,838 (April 22, 1933). (13) Sabatier, P., Ber., 44, 1984 (1911). (14) Sabatier, P., and Gaudion, C., Compt. rend., 168, 670 (1919). (15) Skita, A., and Ritter, H., Be?., 44, 668 (1911). RECEIVED for review November 13 1951.
ACCEPTEDMarch 10, 1962
ffect of Intra article Diffusion
&cess development
Agitated Nonflow Adsorption Systems I
FRED
J.
EDESKUTY
AND
NEAL
R. AMUNDSON
University o f Minnesota, Minneapolis, Minn.
T
HE problem of adsorption of a solute from a solution onto a solid adsorbent has received such widespread and exhaustive treatment a6 almost t o defy efforts t o catalog the work t h a t h a been done. Any investigator is soon struck by the fact t h a t of all this mass of research little has been done on the rate of adsorption. Amis ( 1 ) has given a short review of some rate studies and Harris (9) in three review articles covers t h e recent literature. T h e many factors which may influence the rate have been discussed by the present writers (6). Of all the factors the one which is most amenable to a simple treatment is that of the diffusion of solute inside the particles. It is this aspect of the kinetic process which will be dealt with here although the resistance to mass transfer a t the particle surface may be included with little effort. Recently, Foster and Daniels (7)have concluded that for adsorption of nitrogen dioxide from air on silica gel the rate-determining step is the diffusion of the solute inside the particles. Eagleton (6) has found also that diffusion inside the particles seems t o control the adsorption of water vapor on alumina. The general problem will be analyzed first, following which a special case-adsorption on cylindrical-shaped adsorbent particles-iyill be considered Let it be assumed that a vessel containing a volume, V ,of solution contains AT particles of adsorbent, all of which are the same size and shape. Let CY be the fractional void volume of the solid particles and co be the concentration of the solution in the vessel as well as t h a t of the solution in the void volume of the porous adsorbent, At time t = 0 additional solute is admitted t o the vessel so that the concentration of the solution exterior t o the particles is Co. It is then desired t o obtain a formula which mill predict the variation of solution concentration with time The agitation level in the vessel is assumed t o be high enough t o enBure a uniform concentration throughout the external solutionLe., there are no concentration gradients in the bulk of the s o h tion. On the other hand, the agitation is not so violent as t o reduce to zero the resistance to mass transfer a t the particle surface. If D is the diffusivitv. c the concentration of solution in the
void volume of a particle, and n the amount adsorbed in the particle per unit of volume of adsorbent, a simple rate balance over an element of volume in the particle gives CY
div. ( D grad. c )
=i
a
dc an +at at
Note that if there is no adsorption, n is zero, and this equation reduces t o the ordinary equation for diffusion. The quantities n and c are not independent and are related either by an equilibrium relationship, Le., isotherm, or more generally by a kinetic relation which describes the adsorption process alone,
where f(c,n)is generally an empirical expression. The condition a t t h e particle surface may he stated in the form
where S 1s the surface which bounds the particle, il. is the mas8 transfer coefficient characterizing the maas transfer resistance, and &/as is the normal derivative taken on S The initial conditions can be atat,ed as c
C
= co = C,\
n
no^
when
t =
0
(4)
In addition, the relation between the internal and external concentrations must be obtained. The rate a t which solute crosses the bounding surface of N particles is
where the integrations are t o be taken over the surface of a particle and S is the unit outward normal t o 8.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
July 1952
Equations 1 through 5 make up a complete mathematical description of the system under the hypothesis that the fractional void volume in the particle is not affected by the adsorption. Stated with this degree of generality no direct solution to the problem has been obtained. Numerical solutions are possible but the tedium attendant on such calculations precludes their use on general problems although modern high speed computing machines may rectify this situation. In theory, then, one must solve Equations 1 and 2 simultmeously with the boundary condition Equation 3, Equation 5, and the initial condition os given in Equation 4. This problem is of great mathematical interest since it poses a problem in partial differential equations which has been investigated but little. Special cases have been considered by Lowan ( l a ) ,Jaeger (IO), Wilson (l7),Crank (S),and Crank and Godson ( 4 ) .
1699
The transform of Equation 6 is
The solution of this equation which remains finite when r = 0 is
where A is a constant whose value must be determined, and J o ( z ) is the Bessel function of zero order and the first kind. To determine A , the transforms of Equations 7 and 9 must be taken,
-D
= k(h
($?)v-R
- H ) , whenr
=
R
Theoretical It is the purpose of this paper t o solve another special case and t o supply the experimental confirmation of the assumed mechanism. The problem t o be considered here is that of cylinders in which diffusion in the axial direction may be neglected. This would be the case for an infinite cylinder or for the finite cylinder in which the ends of the cylinder are made impervious to the diffusing material. I n these cases the general problem reduces to the following
-D
(g)
IDR
= k(c
- C), when r
=
R
Between these two equations H can be eliminated t o give
(D
+
ps)($) + 7-R
kCo
kh = P
If Equation 10 i R substituted into this equation, an equation in
A is obtained which can be solved t o obtain
where
w =
where
e =
R
d y y ,
and hence
(7)
c = col
n = n;} when t = 0 C = CoJ
It is necessary t o make an assumption about the,adRorption mechanism, and it will be assumed that the rate of adsorption is much faster than the rate of diffusion so that equilibrium obtains at each point inside the particle. In order to obtain analytical formulas the isotherm is taken to he a linear funrtion of the concentration. This is a serious limitation but suffices to illustrate the diffusional effect, which is the primary purpose of this paper. In the previous reference t o Foster and Daniels the adsorption isotherm of nitrogen dioxide on silica gel is practically a linear one in the range of concentrations considered by these authors. Therefore
n
=
+ kz
klc
In this case Equation 5 reduces to
D kR'
To find the inverse transform of Equation 11 the standard procedure ( I )can be used. The residues of eP%(r,p) must be obtained a t the poles of h ( r , p ) . It is clear that these poles occur a t p = 0 and the nonzero zeros of -wJo(w)
+
(€W*
- 2P)Jl(W)
=0
(12)
The residue corresponding to the pole p = 0 can be obtained by evaluating the limit, lim. [pe%(r,p)]. This is accomplished by p-0
expanding the numerator and denominator in Equation 13 into their respective infinite series, canceling like terms, and letting p approach zero. This gives
co - co l+p+ca
where W is the total mass of particles and pis the particle density. This can be written as
where
P=-
To obtain the sum of the residues a t the remaining poles the extended Heaviside expansion theorem is used ($). The manipulative details are omitted since they are straightforward. The final aolution is
WaDy PY
y = - kl
+
Or
OrD
"hie Bystem is considerably simpler than the general one and may be solved by the Laplace transform with ease. Details of this procedure wvill be omitted and the solution will merely be sketched. If L denotes the Laplace operator, then
L[c(r,t)l = h(?,P) = h L[C(t)] = H ( p ) = H
where w,, is a root of Equation 12 and the summation is over all of the root,$ of thip equation J ! ( z )and J ? ( z )are the Bessel function8 of the first, and second order, resppctively, and of the first kind. Sinre. emerimentally, i t is desirable to have C(t),the concentration in the main body of the solution, rather than d r , t ) , the concentration inside the particle, Equation 9 is intepated
1700
INDUSTRIAL AND ENGINEERING CHEMISTRY
C = C O - 'r R PJ~($)
TFR
dt
The operations indicated in this equation performed on Equation 13 produce
Vol. 44, No. 7
It may have only nonpositive zeros under broader conditions. This condition will be satisfied for the calculations made later in this paper. It can be shown, in general, t h a t all roots are simple. If E = 0 it is clear that the above condition is always valid.
.. -
rived in the previous section a series of runs was made using activated carbon particles as the adsorbent and phenol as adsorbate in a water solution. when the steady state obtains, is Carbon used was obtained from the Carbide and Carbon Chamicals Corp. and was identified as activated carbon Grade- SXA, c o COP cm = c, = mesh 6 / 8 , drum No. 8-567 and Grade CXA, mesh 4/13,drum No. 1+P 9-185. The material as obtained consisted of cylindrical particles with jagged ends. The 6- to 8-mesh particles varied in diameter Also from Equation 14, C, may be obtained by letting t m. from 0.25 to 0.31 em. while the 4- to 6-mesh particles had diameters from 0.40 to 0.47 em. The lengths varied from 1 to m 6 mm. Jl(wn) Particles were first sized into narrow diameter ranges = " 4P(c0 wn[wn(l 2€)Jl(Wb) - (Ew7~' - 2P)Jz(Wn)] for example, one size range was from 0.278 to 0.282 cm. n=l I n order to bring all particles of a given size range t o the same length a jig was made by drilling holes in a brass plate, By equating these values of C, the infinite series in ~~~~~i~~ 14 the diameter of the holes being essentially the same as the dican be split into two, thereby giving after some manipulation ameter of the mrticles. The Darticles were then Dlaced in the holes and ihe ends sanded i o a plane surface. *The jig was then turned over and the same operation performed on the Jo(wn)e*,Lt c- = - co other ends. I n this way practically perfect finite cylinders 2e)Jo(wn) - (Eon' - 2P)2J2(wn) C 0 - - & 1 + P n = l wn2(1 of uniform length and diameter could be prepared. I n a given size range all particles were within 1% of the mcan in both diameter and length. The particles were then washed in Making use of the relation defining the roots wn and also the redistilled water to remove all dust, dried in an oven a t 125" C. currence relation for Bessel functions, for 4 hours, cooled in a desiccator, and weighed. 2 Because the theoretical part dealt with finite cylindcrs some Jz(wn) Jo(wn) = JdWn) method had t o be obtained t o minimize the end effects. After a good deal of experimentation it was found that coating the ends reduces the above equation t o with a spray-on plastic sold under the name of m exp. Krylon was completely satisfactory. The plastic 1 (I5) was sprayed on with the particles in the jig so co-& I + P n=l that, it was hoped, no coating would be applied t o the cylindrical surface. The ends were given two coats, This equation is the general solution for the problem of infinite the first being a very light covering t o seal the pores and a second cylinders or for finite cylinders with ends impervious to diffusing a heavier one t o ensure this. Microscopic examination of broken solute. It takes into consideration the effect of mass transfer reparticles revealed practically zero penetration of the plastic. Exsistance at the particle surface as occasioned by imperfect haustive tests were run t o determine .ir.hether the coating was agitation and intraparticle diffusion but neglects the effect of the attacked by phenol and also whether it prevented the passage of finite rate of adsorption. If the level of agitation is sufficiently phenol. When the total surface of a cylinder was coatcd in this high the surface resistance t o mass transfer may become negligimanner no adsorption took place. ble, in which case E + 0 since lc -P m . Equation 15 in this inThe fractional void volume, a, of the particles was determined stance reduces t o in the standard manner by ascertaining the amount of water taken up by the thoroughly evacuated cylinders. A simplc calcum exp. I c ---co lation, knowing the volume of cylinders, then gives the void (16) 46 4P2 Co co 1 P + 4 P n = l wn' volume. Runs were carried out in a 1-liter round-bottomed flask with where the wn are roots of standard taper joints. The sides of the flask were crimped to provide baffling t o give efficient agitation. Early work indicated that (17) WJdW) 2PJd-w) = 0 the carbon particles would not stand up under agitation sufficient to give a uniform concentration throughout the flask. Although Equation 16 agrees with that of Wilson ( 1 7 ) . Equation 16 is the the particles did not break up, attrition was enough to cause the solution which will be used in the experimental part of this paper. formation of a black powder after about 4 hours. This difficulty was circumvented by surrounding the glass impeller with a fine Some consideration must be given t o Equations 12 and 17 in order (14-mesh) wire screen. The screen was coated with the spray-on t o ensure t h a t all the roots are accounted for. If one lets -wz = plastic, The agitator was run a t about 1300 r.p.m. - z = -R2py, then z is a constant positive multiple of p. By I n order t o follow the course of the adsorption 1-ml, samples expanding the Bessel's functions in Equation 12 into their inwere withdrawn a t intervals and phenol content was determined finite series representations, there results bv the method of Komeschaar (11). m This method is b a s e i b n the knovm zk r ' 28) -wJo(w) ( € 0 2 - 2P)JdW) = --w 2 2 k q k f 1 ) q k + 2) [2EIc2 + IC(' fact that phenol in solutions can be k=O brominated quantitatively accordBy a direct application of Laguerre's theorem ( 1 4 ) it can be ing to the reactions shown t h a t the right-hand side will have real nonpositive zeros in C G H ~ O H 3Br2 +CdhBr30H 3HBr (1) z and hence in p if CsHEBrSOH Brz -zCsHzBrrOBr HBr (11) (I 2E)2 If, then, a known excess of bromine is generated in a reaction P
