j-factor for mass transfer based on interstitial velocity, = -l’sr1Ss,2/3 (dimensionless) j-factor for heat transfer based 011 interstitial velocity, = N s ~ ~(dimensionless) \ ~ ~ ~ ~ ’ ~ mass transfer coefficient, cm/second thermal conductivity of gas, cal/(second) (em) ( O K ) thermal conductivity of screen material, cal/ (second) (em) ( O K ) constant in Equation 13 mesh size, wires per lineal em in Equations 9 to 12 Prandtl number, = c,p/kr (dimensionless) Reynolds iiumber bared on hydraulic radius and i n t e r 4 t i a l relocity, = 4 r . & i p (dimensionless) Reynolds iiuniber baqed on wire diameter and interstitial velocity, = dG, l p (dimensionless) Reynold.; iiuniber based on wire diameter and superficial velocity, = dG,!p (dimensionless) Schmidt number, = v / D (dimensionless) Sherwood number, = k , Stanton number for ma stitial velocity, = k,: Stanton number for heat transfer based oii superficial velocity, = /i/c,G, (dimensionless) Stanton nuniber for heat transfer based on interstitial velocity, = h/c,Gt (dimensionless) number of heat transfer units, = hd,,/Tl*c, (dimensionless) hydraulic radius for screen, = €/a,cni interstitial gas velocity, cni/second
GREEKLETTERS
V
= =
p
=
p
gas viscosity, grams/(cm) (second) kinematic gas viscosity, cm2/second gas density, grami;/cni3
literature Cited
Altschuller, A . P., Cohen, I. R., Anal. Chem. 32, 802-10 (1960). Armour, J. C., Cannon, J. N., ii.I.Ch.E. J . 14,415-20 (1968). Coppage, J. E., London, A. L., Chem. Eng. Progr. 52(2), 57F-63F IlQ.56) j - ” _ _ j .
Cortez, 11. H., Sc.D. thesis, Massachusetts Institute of Technology, Cambridge, >lass., 1968. Dixon, J. K., Longfield, J. E., “Catalysis,” Vol. 7 , P. H. Emmett, ed., pp. 281-304, Reinhold, New York, 1960. Gay, B., IIaiighan, R., Int. J . Heat Jlass Transfer 6 , 277-87 IlS63i.
JakiL:-lI,, “Heat Transfer,’’ Vol. I I > pp. 530-47, Wiley, Yew York, 1957. Levempiel, O., “Chemical Reaction Engineering,,’ pp. 275-78, Wile\., New York, 1962. London. ii. L., hIitchel1, J. R., Sutherland, W. A., J . Heat Transjer82,199-213 (1460). I..-. c A...~.. R \R. ~$; ~ .H.. ed.. uu. McAdams, H., “Heat Transmisiion.” Transmission,” 3rd ed., pp. 258-60. 258-60, -? AIcGraw-Hill, A IcGra~~-H New i l l ~York, ~ e w1954. lIondt, J. R . , “International Ilevelopments in Heat Transfer,” A.S,lI,E,,Seu-York, 1961. Part 111,pp. 614-21, A.S.lI.E., Nowak. J., Cheni. Ena. Eng. Sei. 21, 19-27 11966). Nowak, E. J.. Perry’s Chemical Engineers’ Handbook, 4th ed., pp. 14-19, Perry’s’ >IcGraw-HiIl. lIcGraw-HiIl, New York. York, 1963. IPetrovic, L. J., Thodos, G., IKD.ESG.CHIX. FVSDAM. 7 , 274-80 (1968). Pucci, P. F., Howard, C. P., Piersall, C. H., Jr., A.S.M.E. Trans., ~
~
~
I
A
I
J . Eng. PowerA89,29-40 (1967). Reid, 11, C., Sherwood, T. K., “Propertie. of Gases and Liquids,” 2nd ed., p. 523, >IcGiaw-H~II,Ken- York, 1966. J . FranklznZnst. 208,405 (1929). Schumann, T. K. W., Tong, L. S., London, A . L., J . Heat Transfer79, 1558-70 (1957). Vogtlander, P. H., Bakker, C. A. P., Chem. Eng. Sei. 18, 583-89 (1963).
E
em
A
screen porosity = effective screen porosity for longitudinal heat ductioii = loiigitudinal heat conduction parameter, k,d, TT.c,L (dimen.ionless) =
(’011-
-
Wendlandt, R . , Z. Elektrochern. 53,307 (1949). RCCI:IVCDfor review October 16, 1969 ACCEPTEDJuly 30, 1970 Financial iupport provided by the Xational Center for Air Pollntion Control under Grant AP 00513.
Fixed-Bed Adsorption Kinetics with Pore Diffusion Control Ralph S. Cooper and David A. Liberman University of California, Los Alamos Scientific Laboratory, Los Alamos, N. M.87544
A general analytic solution has been obtained for the time-dependent behavior of a fixed-bed absorption or ion exchange column under the conditions of pore diffusion control and irreversible equilibrium, without the constant pattern assumption. This i s a single dimensionless solution which does not depend explicitly on the column or rate parameters, but only on two dimensionless combinations of them. The constant pattern as calculated b y Hall is fully developed in a finite time and finite column length.
PERFORXASCE
of fixed-bed absorbers and ion eschangers under “const,ant,pattern” conditions has been well developed by many authors (Glueckauf, 1955; Hall et al., 1966) under a variety of controlling mechanisms and equilibrium 11arameters. The analysis of the time-dependent, early stages (or equivalently, thin beds) has received much less attent’ion. Glueckauf and Coates (1947) and Vernieuleii (1953) obtained approximate solutions and Cooper (1965) found analytic 620 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
solutions for solid diffusion control and irre\-er*ible equilibrium. Thi3 paper presents an exact solution for pore diffusion control and irreversible equilibrium. A mass balance for a point in the column, neglecting axial diffusion, gives the fised-bed equation:
ax
z +cax , z = -Ae bY-at -
(1)
where X and Y are the normalized solution and solid concentrations, 1' is t h e superficial fluid velocity, E is the void fraction, and A i* the Hall et al. (1966) distribution parameter
For pore diffusion arid irreversible equilibrium one can repeat t h e analysis of Acrivos and Vernieulen described by Hall et a2. (1966) (Equations 7 to 14),without the constant pattern assumption ( X = Y), to obtain d Y / d t applicable for the lionsteady state. This gives dy -
at
(I - E)k ____ 5h
X [(l - Y)-1'3 - I]
e
(3)
where k = l5DP,,,,r p 2 , a pore diffusion parameter. T h e boundary conditions for flow into a n initially empty column are
X(t,O) = 1 t X ( O > t )= 0 Z Y(0,z)
=
0 z
1 1 y(e,o)
INTEGRATION
,
3 705 N
20 >0 20
Figure 1 .
Regions in 8, N plane
Region A. Saturated solid and solution Region B. Variable X and Y Region C. Empty solid and completely depleted solution Constant pattern occurs in region B above broken line 6' = 5 / 2
Equations 1 and 3 can be put into dimensionless form by choosing variables
(4)
c
(5)
0.8
[These differ b y consrants from 0 and N , defined b y Cooper (1965) for solid diffusion control.] These give Y
-X b-X - 5[(1 - Y)-1'3 - 11 bN
(6)
bY - -- de 5[(i
(7)
f X
- y)-l'a - 11
0.61
/
/
with boundary conditions
x(e,o) =
1
X(0,Z)
=
0
e20 z >0
Y(0,Z)
=
0
Z
20
8
These equations are valid for all time (if it is understood t h a t the right' sides of Equations 6 and 7 are zero in regions where Y = 1 or X = Y = 0), including the transient and constant pattern regimes, and the explicit absence of the parameters k , A, E, and v shows t h a t t'here exists a single general solution for X and Y in terms of 0 and N . This should coincide wit,h Hall's constant pattern solut'ion for large 8 or N. T h e 0 , N plane (Figure 1) has several possible regions: A (complete saturat'ion), B (variable X and Y), and C (completely depleted solution). On this plane a breakthrough curve is a vertical line (N = constant) and the column concentration a t a fixed time is given b y a line with 8 intercept (1 e)k/A and slope -€/A. First we can calculate Y a t the inlet of the columii, Y,(eo), from Equation 7 , since there X z 1. For brevity define the denominator
g(Y) = 5[(1 - Y)-l'a - 11
Figure 2.
Equation loa is a cubic in 0 (Figure 2 ) and can be solved explicitly b y selecting the appropriate one of its three real roots. K e can obtain the general solution in regions A and B as follows: Define
(8) We can rewrite Equation 7 as
Int'egrating Equation 7 y(e,o)
e
=
Y(8, 0 )
!?(y)dyJ
5
(9)
g(Y)
bY
=
aG(Y)
7-- x
Ind. Eng. Chem. Fundom., Vol. 9, No.
4, 1970 621
0 5 5/2, we can integrate from N e (Figure 1).
Substitute for X in Equation 6
=
0 along a line of constant
Interchange the order of differentiation
”(”)=-dY a~
de
Integrate with respect to
e5
e
dG - = -Y bN
-
H(N)
(15)
where H is a n arbitrary function of N only. Consider G along the line where X = 0-Le., along N = p(0). On this line Y must be zero, since the solid has not been exposed t o solute. Thus from Equation 11 G = 0, and from Equation 15
dG dN = H ( N ) , N = p(0)
(16)
Thus given a value of 8, we can calculate Y(0,O) from Equation 10a and Y(N,B) implicitly from Equation 20. The equation of the front, N = p(e), can be computed from Equation 20 by letting Y(0,N) = 0. The constant terms are then
15 - 415 tan-’ 4;+ -In 3 2 ~
0
dG dG dG=O=-ddB+-dN de dN
dG bN
S
Y(B,N)
Y(B,O)
9 7
Y
3
=
=
de
- dN
(21)
Integrating and using the boundary condition that 8 = N = O =
0 along this line
N Thus in the region 8
=
.(e)
2
=
e-
j,2
Equation 18 becomes
-y
dy =
-
(19)
Integrating and rearranging,
A general integration would include a n arbitrary function of 0 to be evaluated, b u t since we know Y(0,O) along N = 0 for
i
I
I
,1.0
I
I
N
Figure 3.
-0.82940
For 8 > 5 / 2 we should integrate from t8hecurve N = .(e) where Y = 1, X = 1. Along this line G = 6 1 2 and dG = 0. Using Equat’ion 17, and evaluating t’he partial derivatives by Equations 12 and 18 gives
Along this line dG = 0
Using Equation 12 and remembering X shows that H ( N ) = 0. T h u s
25 (20)
2
de
N
X(e, N)
622 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
Figure 4.
Y(e, N)
5/2
at
(22)
This is exactly the constant pattern solution for Y, including the constant of integration, derived by Hall et al. (Equation 25). Also the form of Equation 24 show? t h a t lines of constant Y will be parallel straight lines of slope equal to 1.0 for 8 2 5 / / z . Now let us solve for X by showing that a constant ratio exists between X aiid Y. Differentiate Equations 19 aiid 23 with respect to 8 (N constant).
to the value a t the inlet, given b y Equation 10. Figures 3 and 4 present the detailed forms of Y and X in the transient region. Ac knowledgrnent
The authors thank Thoma5 Weber, State University of Kew York, for suggesting this study. Nomenclature
wheref'(0) is some function only of 8, not of N . Using Equation 12 with Equation 25 shows that the ratio X , Yitself is a function of 8 only, and evaluating this rat,io a t N = 0 gives
thus
where Y(0,O) is known from Equation 10 and X ( N , 8 ) can be evaluated numerically by substituting Equation 27 into Equation 20. This then shows t h a t the constant pattern rewlt holds for X as well as Y for 8 2 j ?, and holds also for all N > 3.705, the value of N at which the front (X = 0) reaches the coiist'aiit pattern region (poiiit D in Figure 1). Therefore all breakt,hrough curves with N > 3.705 will have the comtant pattern shape. Equation 26 also leads to a useful approsinintioil for most of the transient period in typical columns. The concentrations over the column a t a given time are given by a line of slope -€,'A, which is usually miall compared to 1.0. Then the entire column has approximately the same value of 8 , and the ratio of Y/X over the column is approximately const,ant and equal
equilibrium concentration in solid inlet solution concentration pore diffusivity function of e function of Y (Equation 8) function of Y (Equation 11) a function of N pore diffusion parameter diniensionless variable (Equation 5 ) particle radius time superficial velocity normalized solutiop concentration normalized solid phase concentration dummy variable for Y distance along column fuiictioii of e, at saturation function of 8,a t t h e front void fraction dimensionless variable (Equation 4) distribution parameter (Equation 2) literature Cited
Cooper, R . s.,I i i D . ENG.CHEM. FUZD.Ih1. 4, 308 (1963). Glueckauf, E., T r a n s . Faraday SOC.51, 34 (1955). Glueckauf, E., Coates, J., J . Chern. SOC.1947, 1313. Hall, K. I{., -4crivos, A,, Eagleton, L. C., Vermeulen, T., ISD. EXG.FLSDIM. 5 , 212 (1966). Vermeulen, T., I n d . Eng. Chem. 45, 1664 (1953).
RECEIVED for review October 27, 1969
ACCEPTED June 23, 1970
Work performed under the auspices of the U. S. Atomic Energy Commission.
Ind. Eng. Chern. Fundam., Vol. 9, No. 4, 1970
623
