Engrnyring
(Effect of In traparticle Diffusion)
ANALYTICAL SOLUTION FOR SIMPLE SYSTEMS IN MOVING BED ADSORBERS
Rocess development
PAUL R. KASTEN
AND
N E A L R. A M U N D S O N
University o f Minnesota, Minneapolis, Minn.
URING the past decade considerable interest has been shown in the application of selective adsorbents to the separation of mixtures not readily amenable t o the more common techniques of distillation andextraction. Successful use of adsorption columns for the separation of hydrocarbons from liquid or vapor phase mixtures has been reported ( 4 ,8,10,11,16,17,22). Theoretical analyses for stationary beds of adsorbent are abundant in the literature (IS,18, 20, 24). The adsorption of a component material flowing countercurrent to a moving granular bed has become of great commercial interest through the medium of the hypersorption process ( 5 , 6), which is essentially a fractionating technique employing activated carbon particles to adsorb selectively the desired hydrocarbons from lean gas streams. The
1
I-+
moves downi\urd in rodlike flow countercurrent to the fluid flow, eimultaneously adsorbing solute from the fluid. The fluid moves through the bed with a constant velocity and a horizontal velocity profile with its adsorbate concentration decreasing as it moves upward. However, the analysis can be applied to stripping operations as long as the adsorbent originally has a uniform adsorbate concentration. Although countercurrent flow is considered, the same equations are applicable t o parallel flow of adsorbent and fluid if proper adjustment is made to the algebraic signs of the flow rates. The adsorbate is assumed to move from the fluid phase through a film resistance a t the surface of the spheres and then to diffuee through the interconnected pores in the solid where it is adsorbed on the internal surface. The case of no resisting film a t the surfare is not excluded and can be obtained as a special instance of the general equations. In order to solve the problem mathematically without recourse t o numerical methods some restrictive asiumptions must be made regarding the adsorption mechanibm which takes place inside the spheres. Specifically, where either a n equilibrium or kinetic relation is required it is assumed to be linear. Also, the authors assume that the whole operation is isothermal, isobaric, and in the steady state with respect t o time and space, that rodlike particle and fluid flow occur, and that the introduction and removal of adsorbent are continuous. The latter imply that the spheres are so small that the adsorbent stream behaves essentially as an ideal fluid in its flow properties.
Derivation of Differential Equation and Boundary Conditions
d
Figure 1
I
I
Diagrammatic Sketch of
Adsorber
mathematical development of adsorption in moving beds, not neglecting diffusion in the particles, has not been considered in the literature, although for the case of a linear equilibrium isotherm, the analogous equations for heat transfer in moving beds have been solved approximately by Love11 and Karnofsky (21) and exactly by Amundson and Rlunro (3). Heterogeneous catalysis, which is closely associated with adsorption, is similarly carried out in moving beds as employed in some catalytic cracking processes (18,,%3). Here adsorption is considered in a moving bed of adsorbent composed of uniform, porous spheres. It is assumed that a stream of solid adsorbent enters the top of the vessel (Figure l ) ,
If a single sphere is considered, the differential equation expressing the relation between the adsorbate concentration of the fluid, g ( ~ , t )and , of the adsorbent, w ( ~ , t )in, the sphere, assuming spherical symmetry, is
as shown in a previous paper (2). Isolating a column thickness Ax, let At be the time interval required for a sphere to traverse this distance, Since rodlike particle flow is assumed, the spheres will have the constant velocity, u = diz/dt. Equation 1 can now be written
The rate at ahich adsorbate is acquired by the particles is dependent on the rate the adsorbate diffuses through the porous area at the very sphere surface. If a n elemental volume of unit cross-sectional area of tower and thickness dx is considered, under steady state conditions a rate balance over the adsorbate yields that the adsorbent moving through the dx volume acquires all the adsorbate lost by the fluid passing through the same volume. Thus, if G,is the rate flux of adsorbent, by applying Fick’s first
1704
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1952
1000
I
I
I
I
,‘d=15.0
/
1705
In the case of adsorption yo and 20% would / I
I
/13=150.’,
I
usually be zero, or nearly
so, whereas in stripping this would not be true.
I
Before the problem can be solved a relationship between y and w inside the sphere must be known. I n what follows, linear equilibrium and kinetic relations will be considered.
Equilibrium If pointwise equilibrium is attained between the adsorbed material and that present in the sphere void volume as adsorbate diffuses through the sphere, the relation between y and w is that of an adsorption isotherm. For mathematibal simplicity one can assume an isotherm of the form
IO0
io
.
w =KyfKo
(6)
where K and KOare constants. Under this condition Equation 2 can be written as
IO (7)
6 I n order to find the adsorbate concentration in the sphere, the mathematical system of Equations 3, 4,5, and 7 must be solved. This probIem is similar to one presented by Amundson and Munro (3) on the corresponding problem of heat transfer and heat release in moving beds. In the present paper a detailed analysis of the problem will not be presented since by a mere change of symbols one problem may be transformed into the other. The method used there ( 3 ) was the Laplace transformation. The results of that paper which are needed here are embodied in cases E and F. If the following notation is used,
4 3 2 I
0.04
0.08
0.12
0.16
0.20
*
L
law of dilfusion the rate of adsorbate entering the particles passing through dx volume is
and the rate of adsorbate leaving the fluid flowing through the same volume is G/p(dS),where G is the rate flux of gas containing S moles of adsorbate per cubic foot of gas. Equating yields (3)
*
(the individual terms are defined in the table of nomenclature) the problem is resolved into two cases: (A) p 3 and (B) P = 3. In ( 3 ) the differences encountered in these two cases are fully exploited. A. p 3. This case is further divided into the two cases p > 3 and p < 3. For p > 3,
+
which is a function of x only since the partial derivative is evaluated a t the sphere surface. It is known that if there is a resistance t o mass transfer a t the fluid-solid interface such resistance can be characterized by a mass transfer coefficient, IC,, defined by the equation
D.($)
r-R
= k/(S
- y),
when r = R
(4)
where the left-hand side is the rate flux of material adsorbed by the adsorbent per unit of area. If k f is very large one can assume as an approximation that it is infinite, and in this case Equation 4 reduces to
where the summation is over the positive roots of the equation w
cot
w
=
p -I-( € -
P
+
and m is the positive root of
y(R,z) = S(X) which corresponds to negligible resistance to maas transfer a t the surface. In addition t o the above, the initial condition of the spheres and the final adsorbate concentration in the outgoing fluid must be specified. It is assumed that the entering spheres and the exit fluid have constant adsorbate concentrations, or
-
s = so y
yo, when
w = wo
x =0
(5)
For 6
< 3,
where the io{ are as defined previously.
1)w2
€W2
1706
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 7
From the experimental data of Ahlberg (1)on the adsorption of water v a v r from air by silica gel, Hougen and Marshall (18) established an equation for the over-all heights of mms transfw units from which
5 00
G' k/ = pjM,(1.42)
(t)(3500)(492) D,G
-0.51
530
where G' is thr mags velocity of the air with respect to the bed, p f is the partial pressure (atmospheres) of the inert gas, and M# is the molecular weight of the inert gas. The superficial air velocity hare is (7.5)(60)/0.075 = 6000 feet per hour with respect to the stationary vessel and 6010 feet per hour with reapect to the particles. Thus G' = (7.5) 6010/6000 = 7.52 pounds per foot per minute, and
IO0
Q6O
D,G' - 0.0128(7.52)(106) = 129,2 ~/1 74.5
IO
so
6
k
4 3 2 1
(7'52'(60) (129.2)(359)(%) = 354 pound (1)(29)(1.42) nioles per hour per square foot per (pound mole per cubic foot)
'
-
I
Because of the tortuous path a vapor follows inside the particle rather than along radii, an effective diffusion coPfficient must be used. For the present, assume this is one-half that as calculated. More will be said of this later. From the data given,
0.04
0.08
0.12
0.16
G
0.20
-
450 pounds per square foot per hour
= 10 feet per hour G, = 350 pounds per square = 0.1 S(S)
ZJ
so
R = 0.0064 foot
= 69 pounds per = 0.54 ?In = 0 il" = 0.0750 pound p K = 4560
pe
foot per hour
cubic foot
a
B. p = 3. This is a rather special case but since it is the dividing region for the two subcases under equilibrium it is included for completeness.
per cubic foot
D, = 0.160 square foot per hour k f = 354 pound moles per square foot per hour per (pound mole per cubic foot)
Thus
(9C) For high mass transfer coefficients-Le., negliglble surface resistance-ne can let c + 0 with resulting simplifiration of Equations 9A, 9B, and 9C. Application of Equations. As an apphcatlon of these equations consider a modification of lllustratlon 2 of Hougrn and 1Iarshall ( 1 8 ) applied to a moving bed of adsorbent. Problem. Benzene vapor is t o be adsoibed from air in a solvent recovery plant by passing the air upward through a downward moving bed of silica gel at 70' F. and atmospheric pressure. The bed moves in rodlike flow a t 10 feet per hour while the air mass velocity (solvent-free basis) is to be 7.5 pounds per square foot per minute. Any tendency toward fluidization of the solid is eliminated by a downward moving head of solid, and the upward flowing gas is removed by a disengaging tray construction. The entering air contains 0.90% benzene by volume and a minimum benzene recovery from the air of 90% is desired. The silica gel is 4- t o 6-mesh in size, has a particle density of 69 pounds per cubic foot, a bulk density of 35 pounds per cubic foot, a particle porosity of 0.54, and originally contains no adsorbate. For adsorption up to 20% by weight, adsorbate-free basis, the weight of benzene adsorbed is directly proportional to the partial pressure for the benzene in the air; at 70" F. and atmospheric pressure for the dilute mixture of this problem the isotherm can be expressed by w = 4560 y. Calculate the bed depth required. Solution. The effective particle diameter of 4- to 6-mesh silica gel is 0.0128 foot. At 70 OF.and atmospheric pressure, the density of air is 0.0750 pound per cubic foot, the viscosity is 74.5 X pound per foot per minute, and as calculated from the Gilliland equation (16, 8 5 ) the diffusion coefficient of benzene vapor in air is 0.320 square foot per hour.
R 2 = 10.0064)2 = 4.10 X 10-6 squarr foot Here p = 11.58, eo Equation 9-4 must be used. To find the value of m one must find the root of rn ctnh m =
+
11.58 0.9296 mp 11.58 - 0.0704 mz
~
By Newton's method one finds nz = 6.892. In order to find the values of w1 in the summation in Equation 9 h the positive roots of w
cot
w =
11 58 - 0.9296~' 11.58 0 . 0 7 0 4 ~ ~
+
must be found. Again by Newton's method, the following roots were obtained: Wt
WI Wg
= =
4.874 8.218
= 11.420 = 14.585
% a 6
W?
= 17.718
= 20.840 = 23.959
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1952
IO00
For this problem Equation 9A becomes
-
0.01396 ,3-9.796
888.2
e--14.46
+
[m+294.9+518.3+ e--1.094
e--3.110
2
1391 + 2146
+
e-B.Ol6 Z
z
e-2K.43
e--40.Q4
t
1702
5 00
z
-mTI + e2.2462
-
1
13.66 Solving by trial and error, the bed depth, x, required is 0.810 foot. On neglecting the entire infinite series portion in Equation 9A the bed depth required is 0.820 foot, or an error of 1.47%. Equation 9A, neglecting the infinite series, may be written in the form: R2 x = m2 y log, * :[oQ + 11
IO0
50
0
where
IO
6
and for design purposes this equation should be useful. It is clear that the positive exponential in Equation 9A will practically always be the dominating term. I n order t o show the effect of the mass transfer at the particle mrface let it be supposed temporwily that E = 0, that is, the mass transfer resisthtnce is negligible. I n this case it is necessary t o obtain the roots of the equations
m ctnh ?n =
4 3 2 I
0.04
B 4-m2 B Figure 4.
cot w =
u, =
W)
UP =
We
4.933 8.399 = 11.744 = 15.026
As
-
(;)z
+
(%)z
-
sso( =x YbO oversus L
0.20
= zXfor
R2
= 0.2
(%)z+
Az
Integrating and inserting the boundary conditions S = So and w = w owhen x = 0, one obtains S = bw
= 18.269 = 21.486 = 24.687
+ So - bwo
(11)
The increase in adsorbate per cubic foot of bed per hour can be written as
where C U I is the bed “solidity” and a is the interfacial area for mass transfer between solid and fluid per unit volume of bed. For a linear isotherm y = w/K. Since z = ut Equation 12 can be written a$
Subfitituting the values of w and dw/dx as given in Equations 10 and 11, there resuIts
Solving this equation for S and inserting the condition S = SO when x = 0 results in
=
Since G, p , G,, and p e are msumed constant and S and w we functions of x alone, dividing through by Ax and letting Ax + 0, one obtains in the limit
where
A =
G - d_S = Gsdw p ds pa & or
0.16
and E = 0.5
Solving Equation 9A for x in the same manner as above gives x = 0.312 foot, while neglecting the infinite series portion and using the approximation form gives x = 0.320 foot. Hence it is seen that the effect of maxs transfer at the particle surface is important and cannot be neglected. The effect of the diffusivity is likewise important and may be illmtrated 88 follows: If a diffusivity of 0.320 square foot per hour is used in the above cal(aulations, the bed depth is 0.02 foot, while if one fifth of the Gilliiand diffusivity is used the bed depth increases t o 1.3 feet. For further comparison consider now the previous rase where diffusion of adsorbate through the porous particle is neglected, Le., as soon m the solute crossea the surface boundary film it distributes itself uniformly inside the particle. Under steady state conditions by taking a rate material balance on the adsorbate around an elemental section of unit cross-sectional area and Az thicknm, (%>z+
Plot of @ =
p - wt R
* The root of the first is 10.475 while the first seven roots of the second are
W4
0.12
L
and w
0.08
~
AIBO
5
SO
=
so - bWo
Bo(Kb - 1)’
=
k r d ( K b - 1) va’h’b
Kse+ (So - bwo)x
1; Kb = 1
INDUSTRIAL AND ENGINEERING CHEMISTRY
1708
II
100
I
I
p = 15.0:
6 4 3
w/-/-/0.04
-
Plot of @
Figure 5.
0.12
0.08 =
~
so
0.20
L versus
- YO
and e
0.16
L
= YX - for
R2
E
= 0.5
= 1.0
Using Equation 14A to solve this problem, 0
35/69 = 0.507 354 pound moles per square foot per hour per (pound mole per cubic foot) 4
2
e
3otcr‘
$ R3
R
0.0064
(0.54)(0.507) = 128.3 square
1
feet per cubic foot 10 feet per hour 4560
So
lo-*)
gives a mass transfer coefficient which is too low, since the mass transfer coefficient of Hougen and Marshall was based on a larger effective area and also included the resistance of the solid to mass transfer. If the mass transfer coefficient used here were higher, the effect of diffusion on the length of the adsorber would be even greater. Since rates of diffusion in porou.; materials are generally much slower than in free solutions, the effect of diffusion on the length of a moving bed adsorber is even more pronounced. Since the transfer coefficient used probably contains resistance of the solid to mass transfer, it would be desirable to calculate the effective diffusion coefficient required to give an adsorber length of 0.363 foot using a true value of the transfer coefficient. Hougen and Watson ( 1 9 ) state that the correlation obtained from Ahlberg’s data give transfer unit heights 3.57 greater than the values calculated from the gas film alone, obtained by evaporating water from particles (14, 98). Thus consideiing li/ = 354(3.57) = 1264 pound moles per (hour)(square foot)(pound mole per cubic foot), by trial and error the value of D, necessary t o give a bed depth of 0.363 foot in Equation 9A is 0.24 square foot/hour. This value is in the direction to be expected and seems to substantiate the equations developed here. Thiele (26) states that intraparticle diffusion is usually much slower than film diffusion and the present work also seems to bear this out. Of course it has been assumed in the development of Equations SA, SB, and 9C that the mass transfer coefficient constitutes the resistance offered by the fluid film alone and involves no other factors. I t is not always clear t h a t this is the case in literature values. For convenience, graphs of Equations SA, 9B, and 9C have been prepared, obtaining the vhlues in the same manner as in the problem illustration. These are presented in Figures 2 through 5 for various values of the parameters. These charts have been prepared with overlapping values of E so that interpolation is facilitated to some extent. Because of the nonlinearity of the curves, two other sample charts presenting essentially .the same calculations have been prepared.. I n Figure 6 the relationship for a fixed value of L is given using d, and E as coordinates for various B’s, while in Figure 7 + versus p is plotted for various E’S at a fixed L. Sinre L is determined by the length of the adsorber and some of the physical parameters, Figures 6 and 7 show the effect of the p and E on the concentration change ratio for a fixed physical system. Kinetics
-
2
4560(8.454 X
Vol. 44, No. 7
=
3.855
= 0.18
If equilibrium is not attained within the particles between the adsorbed solute and that present in the sphere void volume, the ratio relation between the two needs to be known to solve the problem. Generally, this kinetic expression is a function of temperature, pressure, adeorbate concentrations, and the system itself. At constant temperature and pressure a given system would have in general a kinetic relation not expressible by R simple equation, which a t present ~ o u l dresult in insuperable mathematical complexities. Therefore the following discussion is limited to the special case where a linear kinetic relation exists of the form
Substituting in Equation 14A one ultimately obtains e5.61
7.67
or 5 = 0.363 foot, which is the bed depth assuming all resistance to adsorption is in the surface film. It is clear from this example t h a t both the rate processes of diffusion inside the particles and mass transfer a t the particle surface are of importance. Hougen and Marshall ( 1 8 ) considered the experimentally determined surface area as t h a t available for mass transfer, implying that the adsorbate must pass through this area, after which i t is adsorbeduniformly throughout theparticle. It has been assumed here that each sphere has a surface available for mass transfer of 4xR2a and it is on this basis t h a t k j has been used. This, in all probability,
Since
1:
=
vi, this can be written as
To find the adsorbate concentration inside the sphere for the above condition, Equations 2, 3, 4, 5, and 15 must be solved simultaneously. The solution to this problem has not been found in the literature; however, it can be obtained by means of the Laplace transform. This operation has been discussed briefly in the literature (2,5) and a complete treatment is given by Churchill (9). Here Churchill’s definition of the transform is used, Le.,
July 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
1709
1000
500
too 50
0 and
la.’ ( p W - W O ) = klY - kzW Eliminating W between these two equations results in
IO 6
4
3 2
where
I The left-hand side of Equation 16 is in the form of Bessel’s equation of order one half ($7) and the right-hand side is a constant. Making use of the relations between Bessel’s functions of order one half and trigonometric functions ($7) the general solution of Equation 16 can be written as ~ ( r , p= )
A 7 sin
(r
vZ)+
0.2
0.0
0.4
0.6
E: Figure 6.
Sample Plot of 0 versus E for
[ p ( ~- 1) - 61 sin w
where A and B are functions of p which must be determined. On a physical basis, the solution must remain finite a t T = 0, which cannot be if B =k 0; therefore, B = 0. To determine A , take the transforms of Equations 3 and 4,and eliminate T ( p )between them, obtaining
I .o
0.8
L
=
!X!
R2
+ w cos w ( b - p . )
= 0.08
= 0
To find the residue corresponding to p = 0, substitute for the trigonometric terms their respective infinite series representations, multiply by p , and cancel all factors common to numerator and denominator. Since w = 0 when p = 0, and ]im-wa = P
- R“
Du
w+o
(kl
+
kza) kza
P-+O
the resulting residue a t p = 0. can be written as
Evaluating Equation 17 and its derivative with respect t o r a t r = R and substituting their values into Equation 18, an equation in A results. Solving for A and substituting its value in Equation 17, one obtains
where w =
R d z a and
b =
R2p&v
3Dv0rGa~
This is the transform of the solution to the problem, and in order to obtain the solution the inverse transform must be found. This can be accomplished by an extension of the Heaviside expansion theorem (9) which is discussed brieffy in ( 9 ) . There i t is pointed out that the inverse transform is the summation of the residues remaining a t the poles of the denominators of the terms of Equation 19. Examination of the denominator of Equation 19 reveals poles -kzff atp = 0,p = and the zeros of
Repeating the same procedure, the residue is zero for the pole corresponding t o p = kza) , for the residue of the first
-(a + 2)
term is the negative of the residue of the second term. For the special case when
a double pole occurs a t the origin. The remaining poles occur when [p(e
- 1) - b ] sin w
+ w cos w ( b - p e ) = 0
excepting the pole a t o = 0. By definition, R 1 / y a = w, so let ai and
wi
be defined by
(21)
INDUSTRIAL AND ENGINEERING CHEMISTRY
1710
1000
Vol. 44, No. 7
p = --k2a. However, it can be shown t h a t the residue a t this essential singularitjr IS * _ zero. I n order t o find a relation between the inlet and outlet adsorbate concentration, Equation 3 must be integrated. Aftei some manipulation there results
I
500
IO0
The use of this equation is somewhat more difficult than in the equilibrium case because of the complex nature of the poles. It, would be desirable t o illustrate the use of this equation but the paucity of data on adsorption kinetics precludes this. The sohtion for the case of an infinite mass transfer coefficient is obtained by letting e approach zero.
50
(P
Summary
IO 6
4
3 2 I
4
2
0
8
6
IO
14
12
P Figure 7.
Sample Plot of
@
versus fi for L =
16
': R
= 0.12
where w i and pi are the corresponding roots defined by Equation 21. The residue at each of the above poles can be written a ~ ;
where g and y are the numerator and denominator, respectively, of the transform term, evaluated a t the root p,. Thus, ~ ' ( p , is~ ) the derivative with respect to p of Equation 21 evaluated at p = p,. Now
Adsorption has been considered in a moving bed, assuming uniform, porous spheres as the adsorbent medium, and rodlike flow of both fluid and particles. If equilibrium is attained within the particles, the relation between the adsorbate concentration of the fluid and solid muRt be arcording t o the isotherm for that system. A linear isotherm is considered here. If nonequilibrium relations apply, a kinetic expression must be used. A linear, reveraible kinetic relation h assumsd, which is the most general relation for the problem involved, which can be solved by elementary analytical methods. A resistance to mms transfer is assumed to exist a t the fluid-sphere interface. I6 is pointed out that the value of the mass trander coefficients must not include resistance to mass transfer bemuse of adsorbate diffusion inside the particle. It is felt that many coefficients given in the literature are not true values, but contarin such resistanca. Graphs have been made for various values of parameters considering adsorbate diffusion into the solid and a linear isotherm. These are applicable t o the analogous case of heat transfer by making the appropriate symbol substitutions. I n any actual process, nonspheriral particles of varying sizes are used. However, for a practical mathematical solution, an effective particle diameter must be chosen. Also, the fluid seldom has a truly horizontal velocity profile, but channeling of the fluid is undesirable and aome effort is made to prevent it. Rodlike particle flow has been virtually attained in a commercial process ( 5 ) and in the laboratory (7). The results of this paper should be applicable t o the processes of catalysis where adsorption is controlling for the limited c a s e which follow assumptions analogous to those made here.
where
Acknowledgment The authors are indebted to Thomas E. Guentor who made the calculations and prepared the graphs. and where
Nomenclature = -p v ( p v
On performing all the operations indicated in Equation 22 and summing over all the poles there finally results after a good d e d of manipulation
+ CYMO)GSP v ( r , x ) = ( k , + k2a)Gsp - k,p,G -
(WB
where
Ic2
2
+
(pne
- bjsin run R
d~nepnz
,-
-
T-QYo) p t ~ k2a
Irz(lc:O
+
++ k:
!//opnt
G ~ (for P the c a m d case of kinetics); b = P~G equilibrium) D, = effective diffusion coefficient of adsorbate in fluid in Rphere void volume, square feet per hour G = upward mms velocity of inert carrier fluid in vessel, pounds inert fluid per hour per square
L32p.G~
(23)
C-
-.
IUUL
G,
. n -
k.1)
=--3D'aGsp (for the
b
F , sin
72-1
+ k2a + +ha)
(PV
SO
It can be shown that all roots p n are real. One point which needs explanation is the exibtence of an essential singularity a t
downward mass velocity of adsorbate-free adsorbent in vessel, pounds &orbate per hour per square foot D, n2 JlL = RZ IC,, ICs = constants in the kinetic relation in reciprocal hours k, = transfer film cmfficient in pound moles per s q a r e foot per unit adsorbate concentration difference =
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1952
K , K O = constants in the adsorption isotherm L = E! Ra L, = Laplace transform symbol, with respect t o z Q = La lace transform parameter corresponding t o 2 r = r&us variable in spheres, feet R = external radius of spheres, feet s = adsorbate concentration of fluid phase in moving bed, pound moles per cubic foot of fluid so = adsorbate concentration of fluid !caving top of vessel, pound moles per cubic foot of flmd X t = time, hours; t = ; T = Laplace transform of S downward velocity of particles in moving bed, feet per 2 = hour z L ‘ = adsorbate on adsorbent, pound moles per apparent cubic foot, of adsorbate-free adsorbent UO = initial adsorbate or adsorbent the Laplace transform of UI 2 = ut = distance measured in direction of particle flow from top of vessel, feet s i = adsorbate concentration of fluid phase in sphere void volume, pound moles per cubic foot of fluidinitial adsorbate concentration of fluid phase in sphere Ye void volume, pound moles per cubic foot of inert fluid y = the Laplace tranvform of y a = porosity of spheres, cubic feet of void volume per cubic foot of apparent solid volume, or square feet of open area per square foot of apparent sphere surface = solidity of bed, cubic feet of appctrent solid per cubic foot of bed 3Gap(K CY) B = GP~ D*CY
w =
+
Y
=
E
=
P
=
+ 00
v(K
Do -
,
kfR density of adsorbate-free carrier fluid, pounds per cubic foot = apparent density of adsorbate-free solid, pounds per cubic foot = a root of a transcendental equation
a
= S(z)
P4
so
- Ye
- yo
1711
Literature Cited E.,TND. ENQ.CHSM.,31, 988 (1939). (2) Amundson, N. R.,arid Kaaten. P. R., Zbad., 42, 1341 (1950). (3) Amundson, N. R..and Munro, W. D., Ibid., 42, 1481 (1950). (4) Barrer. R. M.. W. S. Patent 2,306,610(1942). (1) Ahlberg, J.
(5) Berg, C., Trans. Am. I n s t . Chcm. Enprs., 42, 665 (1946). (6) Berg, C., Fairfield, R. G., Imhoff, D. H.. and Multer, H. S., Petrokmm Refiner, 28, No. 11, 113 (1949). (7) Brinn, M.S.. Friedman, 6 . J., Gluckert, F. A., and Pigford, R. L.. IND.ENO.CHEM.,40,1050 (1948). (8)Burrell, G. H., and Guild, L. V., U. S. Patent 2,399,095(1946). (9) Churcahill, R. V., “Modern Oporlttional Mathematics in Engineering,” p. 167,New York, McGraw-Hill Book Co.. 1944. (10) Claussen, W. H., and Shiffler, W. H., U. S. Patent 2,470,339 (1949). (11) Dinneen, G. V., Bailey, C . W., Smith, J. R., and Ball, J. S., Anal. Chem., 19, 992 (1947). (12) Eastwood, S. C., Hornberg, C. V., and Potae, A. E., IND.ENCI. CHEM.,39, 1685 (1947). (13) Furnas, C. C.,Trans. Am. Inst. Chem. Engrs., 26, 142 (1930). (14) Gamson, B. W.,Thodos, G., and Hougen, 0. A., Ibid., 39, 1 (1943). (15) Gilliland, E.R.,TND. ENG.CHEM.,26, 681 (1934). (16)Hibshmann, H. J., U. S. Patent 2,434,535(1948). (17) Hirschler, A. E., and Amon, S., IND.ENG.CHEM.,39, 1585 (1947). (18) Hougen, 0.A., and Marshall, W. R., Jr., C h m . Eng. Progress, 43, 197 (1947). (19) Houge:,, 0 . A., and Watson, K. M., “Chemical Process Principles, Vol. 111, p. 1085, New York, John Wiley and Sons, 1947. (20) Klinkonberg, A., TND. ENG.CHEM.,40, 1992 (1948). (21) Lovell. C . L..and Karnofskv. G..Ihid.. 35,391 (1943). (22j Mair, OB, J.,’Gaboriault,A: ‘L., and Rossini, F. D., Ibid., 39, 1072 - - .- (1947). ,- - - . (23) Newton, R. HI,Dunham, G . S., and Simpson, T. P., Trans. Ant. Inst. Chem. Engrs., 41, 215 (1945). (24) Schumann, T. E.S., J . Franklin I d . , 208,405 (1929). (25) Sherwood, T. K.,“Absorption and Extraction,” New York, McGraw-Hill Book Co., 1937. ENQ.CHEM.,31,,?16 (1939). (26) Thiele, E. W.,IND. (27) Von Karmm, T.,and Biot, M. A., Mathematical Methods in Engineering,” pp. 61,64,New York, McGraw-Hill Book Co., 1940. (28) Wilke, C.R.,and Hougen, 0. A., Trans. Am. Inst. Chem. Engrs.. 41,441 (1945). R ~ C ~ I Vfor ED review September 27, 1951. ACCEPTED April 9, 1952.
* * * * * I&EC readers will be interested to know of the forthcoming publication of a series of six papers on high vacuum evaporation that are the result of several years of study and research by IC. C. D. Hiclrman and D, J. Trevoy a t the ICodalr Research Laboratories. The importance of these papers, whioh contain much new and valuable information on the industrial practice of distillation, is evident in the following comments quoted from their two reviewers: I believe that Hickman’s beautiful observations on the appearance of evaporating liquid surfaces are a real high point.. .he introduces the concept of torpidity in the most logical and convincing fashion. The observations are essentially qualitative, but I do not believe this can detract from their great scientific value. The papers contain a great deal of data and description of new techniques. I n addition, the novel ideas presented regarding the character and behavior of the surfaces of the liquids may well be classic. These ideas probably mark a turning point in general scientific thinking as far as mass transfer between liquid and vapor is concerned. I am positive they will be very stimulating to other workers in the field. Four of the papers will be published i n the August issue of INDUSTRIAL AND ENGINEERING CHEMISTRY: The Falling-Stream Tensimeter Projective and Equilibrium Vapor-Liquid Relationships for Two Binary Systems Over a n Extended Range Surface Behavior in the Pot Still Systematic_Comparison of High Vacuum Stills and Tensimeters and two in the August issue of Analytical Chemistry:
A Shaft Seal for Vacuum Apparatus A Directional Cold Finger Condenser
