F I X E D BED A D S O R P T I O N W I T H V A R I A B L E G A S V E L O C I T Y D U E TO P R E S S U R E D R O P IMRE ZWIEBEL Worcester Polytechnic Institute, Worcester, Mass. 01609 An explanation has been derived for the oft-observed premature breakthroughs and the long extended “tails” of the experimental concentration-time curves obtained in isothermal adsorption. This modification of the Schurnan-Furnas model considers velocity deviations due to pressure drop. The resulting correlations are given in terms of dimensionless variables which include configurational parameters.
H E solution of mathematical models describing adsorption Tin fixed beds dates back to the works of Anzelius (1926), Schuman (1929), and Furnas (1932), who described the analogous process of heat transfer in a packed column. The analogy is applicable within the framework of the assumptions made, the most significant of which are: Dilute feed systems are used, so that heat generation effects are insignificant and linear isotherms describe the equilibrium. Plug-like flow regime prevails, so that no radial profiles exist, the pressure drop is negligible, and the velocity remains constant throughout the column. The mass transfer mechanism is described by a first-order model based upon a constant coefficient and a concentration difference between the bulk phase and the solid phase. Recently a number of papers have discussed the effects of some of these assumptions. Edeskuty and Amundson (19521, Masamune and Smith (1964), Needham et al. (1966), and others studied the mass transfer mechanism by using more complicated rate models, particularly those which include diffusion within the solid phase. Vermeulen (1958) and Fosberg et al. (196i) considered the case of nonlinear equilibrium behavior, specifically the Langmuir isotherm. Carter (1966) and RIeyer and Weber (1967) attempted to solve the nonisothermal problem numerically. But heretofore little attempt has been made to determine the effects of the variable gas velocity. Fukunaga and coworkers (1968) included an empirical expression for pressure drop in the model they solved numerically, but failed to elaborate on its effects upon the breakthrough curves. Cooney (1966) considered a comprehensive isothermal problem in which he accounted for pressure variation due to hydrodynamic and composition effects. His primary concern was to establish criteria which determine the stability of the moving profiles, and with the aid of perturbation analysis he developed an expression for the stability coefficient. This method also provides the means for deriving equations which would predict breakthrough curves; however, these expressions can be so complex as to be impractical. Furthermore, the validity of the relationships thus developed has not yet been established by comparing the resulting profiles with experimental data. From the design point of view the effects of the variable velocity would seem important, especially in solid-gas adsorption, since velocity variations are caused by either the pressure drop or the change in mass flow due to the removal of appreciable amounts of adsorbate from a nondilute feed. I n this paper the velocity effects due to pressure drop using dilute feeds are discussed.
Since low concentration of the adsorbate is assumed, there is effectively no change in the mass flow rate and the physical properties of the gas stream throughout the column-Le., the flow regime is independent of the adsorption process. Hence, the over-all material balance and the momentum balance can be written as steady-state expressions, using the distance along the axial direction as the only independent variable. Under these special conditions, as shown below, these equations can be integrated to give relationships for the gas velocity as a function of the axial distance in the column
V’
= f(x)
This can be used in conjunction with the adsorbate material balance and mass transfer rate equations to develop a modified and more realistic description of the pertinent concentration profiles. Integration of Adsorbate Material Balance Equation
In the present analysis the adsorption process is described by a simplified adsorbate component balance and film-type controlling rate expression, as was done by Acrivos (1956) and others.
.
with the following boundary conditions:
C(v = 0, u) = 1.0
(4%)
0) = 0
(4b 1
TY(v, u = 0) = 0
(4c )
V ( v = 0 , u) = 1.0
(4d )
C(V,U =
In Equations 2 through 4,the dependent variables have been normalized with respect to the inlet conditions to the column, the independent distance variable v = kq’Vo, and the second independent variable, the dimensionless time u = kKt/p,. Thus Equations 1 through 4, which assume a dilute feed, isothermal operations, linear isotherm, no diffusive dispersion, constant mass transfer coefficient, and velocity distribution dependent upon axial distance alone, represent the modified mathematical model for the adsorption process. By Laplace transformation these equations can be solved to give the following integral expressions for the gas concentration and VOL.
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the solids loading, respectively,
C-V = 1- e4L
heretofore considered :
Y
e-81~(2fi)ds
(5 )
(2d/yr)dr
(6 )
0
W .V
= e - y L e-'lo
where lois the zeroth-order modified Bessel function of the first kind, Y is the modified dimensionless distance parameter defined as follows
Y = UA/ , ' $
dP
- = qiv dv
+ $PV
where qi and $ are constants dependent upon the physical configuration of the packed column and other nonvarying parameters. Equation 9 is applicable over a wide range of velocities, since it is the result of the combination of the BlakeKozeny equation for laminar flow and the Burke-Plummer equation for turbuleiit flow, as presented by Bird el al. (1960). In an essentially isothermal constant mass flow system of an ideal gas the over-all material balance is a steady-state equation and can be written as follows:
d(P'V') --0 dx
and 8 is the modified dimensionless time parameter given by
where the primed variables are the actual pressure and velocity a t any point in the column. After integration and normalizing with respect to inlet conditions, Equation 10 becomes (11 1
PV = 1 Both parameters reduce to the form in which they appear in the literature for the constant velocity problem when V = 1 is imposed upon Equations 7 and 8. Hence, Y is a modified expression for the number of transfer units, and 8 is a modified performance parameter, as defined by Vermeulen (1958). The distance integral of Equation 7a represents an equivalent distance that is traversed by a parcel of inert gas in a given time. Since the velocity increases with distance as a result of the pressure drop, the actual distance traveled in a given time is larger, and the equivalent distance for adsorption, and thus the distance parameter, Y , is less than it would be in the constant velocity problem because of the reduced residence. Also, the distance integral as introduced in Equation 8 represents the time required to fill a given length of column-Le., the residence time. Thus, a t a given operating time u,the performance parameter, 8, is larger for the variable velocity case than for the constant velocity case because of the smaller value of Y . By observing the expressions for concentration and loading in Equations 5 and 6 it can be noted that trends which increase 8 and reduce Y give a larger value of the product CV and a lower value of TVV a t a given point in the column a t a given operating time. Hence, an inferior performance is indicated when the variable velocity problem is compared to the constant velocity case. The products of CV and TYV can be obtained by evaluating the integrals of Equations 5 and 6 a t predetermined values of 8 and Y . Since the integrals do not lend themselves to simple integration, they have been evaluated numerically and their values presented in graphical form in the literaturefor example, Eteson and Zwiebel (1969), Hougen and Watson (1947), or Vermeulen (1968). The values of C or W can readily be obtained for any adsorption process, provided the normalized gas velocity, V , is known a t the point of interest in the column.
Equations 9 and 11 can be solved readily with the boundary condition of constant inlet pressure, P(v = 0) = 1, to give the following expression for the velocity variation along the axial dimension, or as a function of the distance parameter,
This is of the form of Equation 1, where N' = 2(6 4- $) and = (A/B)N', both dimensionless. The quantity N , termed the packing coefficient, is a function of particle shape, particle diameter, column dimensions, gas viscosity, inert molecular weight, and temperature, all of which are set or assumed constant in the current analysis. (By considering other flow models such as the Darcy equation, or assuming nonspherical
N
Evaluation of Gas Velocity
As stipulated above, in the adsorption from dilute mixtures the material traversing within the column is essentially all inert, and the flow regime is effectively independent of the adsorption process. Hence, the pressure-velocity relationship can be described by the Ergun equation written here in the dimensionless form consistent with the other relationships 804
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FUNDAMENTALS
3
NORMALIZED
DISTANCE,
Z
Figure 1. Velocity variation within column as a function of packing coefficient
I
oo Table 1.
/
*
Summary of Experimental Data Used for Figure 3
Alonso Experimenter Adsorbent Adsorbate Inert gas carrier Inlet adsorbate mole fraction Inlet velocity, V,, s.cc./sec. Bed length, L,cm. Particle diameter, d,, cm. Void fraction in bed, f
aW
IW
z
a a
a
a
10 0
Collentro
Fukunaga
(1967)
(1968)
(1968)
Charcoal
Charcoal
5A sieve
0.011 170 180 0.305 0.55
0.0034 600 80 0.224 0.50
0.027 50 18.3 0.159 0.5
W V
z a IE
Y and 8, in terms of known constants or available design conditions
0
2
Y = - [ (1 3N‘
D
w
LL
n 0
+
N’u)3’2
- 11
(13)
n
I 1.0
IO0
10.0
1.0
D I S T A N C E , Cr
DIMENSIONLESS
Figure 2. Deviations in distance parameter as a result of velocity variation
particles, nonideal gases, or nonhomogeneous packing, etc., different forms of the velocity equations could be derived in place of Equation 12.) Figure 1 is a plot of the variable velocity, V , as a function of the normalized distance, z = z / L a t various values of N . With z as the abscissa, the plots of Figure 1 are independent of operating conditions but directly related to column configuration within the framework of the different values of the packing coefficient as the parameter. Hence, in conjunction with Equation 12 the gas concentrations, C, and solids loadings, W , can be evaluated by Equations 5 and 6, respectively. Evaluation of Performance Parameters
As stated above, the values of the integrals in Equations 5 and 6 are readily available once modified parameters 0 and Y are available. These can be evaluated once the velocity equation has been established. Substituting Equation 12 in 7b and 8, and integrating, gives the desired expressions for
E
1.0
2
0.9
Figure 2 gives a plot of Y us. u, in which N’ is used as the parameter. This choice was also made with application versatility in mind, since a single set of curves suffices to describe Equation 13. Experimental Verification
To establish the validity of the newly derived expressions, the values predicted by Equation 5 were compared in Figure 3 with experimental breakthrough curve data obtained a t this laboratory by Alonso (1967) and Collentro (1968), those available in the literature (Fukunaga et al., 1968), and the predictions from the original, constant velocity model. Table I lists the operating conditions for each run used in this illustration. Discussion
The modified parameters Y and 0 obtained above provide the means for improved designs of adsorption systems treating feeds containing trace quantities of impurities. Typical examples may be preferential SO2 removal from stack gases, where large quantities of effluent containing usually no more than 1% pollutant are to be treated under a severe pressure drop limitation, the removal of nitrogen oxides from auto-
I-
z E!
0.8
I- 0 . 7 4
a
z
$
0.6 0.5
z
2
0.4
I-
0.3
W 2
0.2
z -1 U
-
u 0.I
W
0
0
e Figure 3.
(MODIFIED
PERFORMANCE P A R A M E T E R )
Comparison of experimental data with predicted breakthrough curves VOL.
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mobile exhausts, and the removal of trace quantities of water to obtain bone-dry gas streams. As evident from Figure 2, the most significant improvements are realized as the absolute value of coefficient N’ increases. This increase may be due to larger numerical value of the packing coefficient, N , or to smaller value of ratio A / B . An increased N represents smaller particles and/or tighter packing in the column, both of which cause greater resistance to flow. The seeming contradiction, that the numerical value of N’ also increases with reduced value of ratio A / B , which corresponds to a shorter bed and hence to reduced pressure drop, is overcome by the fact that as A / B decreases the origin is approached along the abscissa of Figure 2. In the region of low values of the distance parameter v all the curves approach the straight line Y = v . In other words, regardless of the packing coefficient, in very short beds or a t very low feed rates, the constant velocity problem is approached. When Equation 13 is differentiated with respect to v and the derivative is set equal to zero, maxima are indicated in the Y us. v plots of Figure 2. These would occur a t zi = - l/N’, which corresponds to z = - l/N. Also, the curves terminate a t this maximum, since a t greater values of v the modified distance parameter, Y, becomes imaginary. These maxima have no physical significance, since z > 1 corresponds to points beyond the outlet end of the adsorption column, and since I -Ar I is always less than unity, the maxima occur outside the bed. Similarly, the velocity curves reach their largest significant point at z = I, or V,, = Tiexit = (1 N)-’I2. Equations 5 and 6 relate products CV and WV to the integrals whose values range between 0 and 1. The effect of the parameters, whose values the current analysis modifies for the variable velocity problem, represents a shift along the
+
1
0 998
1 I
I
,
I
I
I
I
axes of the familiar profile curves which appear in the literature. Since the velocity increases because of the pressure drop, resulting in values of V are greater than unity, the values of C and W decrease throughout the range under study. At first glance this seems like a contradiction of the stipulated hypothesis that inferior performance results from the variable velocity. However, as seen in Figure 4, the change in the values of the performance parameters more than offsets the reduction in the effluent concentration values, and a t low values of O-i.e., a t the early part of the breakthroughhigher concentrations are obtained than predicted by the constant velocity case. This effect was substantiated by the experimental data, and indicated in Figure 3, where the approximately 6% premature breakthrough (at C = 0.01) is evident on some of the curves. At the other extreme of the breakthrough curves C = 1 is never reached by concentratidn profiles for the variable velocity problem; in fact, the maximum effluent concentration attainable is equal to 1/V. While this may indicate violation of the conservation of matter, it is the result of the dilution effect caused by the reduced density, since the concentrations are expressed in moles per volume units. Hence, as shown in Figure 4, the often observed “tail” is developed in the breakthrough curves, and their existence is also substantiated by the experimental results. Heretofore this phenomenon was often written off as experimental imprecision in measuring small concentration changes, or the result of thermal effects or axial diffusion, all of which may contribute to its development. Figure 4 presents a series of calculated breakthrough curves, plotted on probability paper to accentuate the deviations caused by the velocity variations both a t the start and a t the tail of the profiles. At low values of Y-Le., in short bedslittle or no premature breakthrough is noticeable. However, in longer beds, or under operating conditions which increase the effective bed length, the deviations in the concentration due to velocity variation become significant a t both ends of the breakthrough curves. Figure 4 is a modified version of the design curves previously presented in the literature. In addition to the profiles a t constant velocity ( N = O.O), those obtained a t various values of the packing coefficients are also plotted. Conclusions
The solution of the modified mathematical model for the isothermal adsorption of a dilute solute in a fixed bed under variable velocity conditions provides a means for improved design of adsorption columns. While improvements of relatively nominal magnitude (up to 5 to 10%) will be realized in most applications, the negligible amount of added effort in terms of calculations-Le., the evaluation of the modified parameters Y and &justifies their utilization. Of considerable significance, however, is the demonstration of the “tail” a t the end of the breakthrough curves, which were not predicted by the simple model. The existence of this tail may assume major proportions in the consideration of the desorption process. e
(MODIFIED PERFORMANCE PARAMETER)
Figure 4. Calculated breakthrough curves at various values of packing coefficient Plotted on probability coordinates
806
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FUNDAMENTALS
Acknowledgment
The author expresses his gratitude to C. A. Keisling and D. C. French for assistance in the preparation of the manuscript. Special thanks are due to the staff of the Worcester Area College Computation Center.
Nomenclature
t$
A = dimensionless coefficient, distribution parameter, p ~ / K f
B
= dimensionless coefficient, AVo/kL
C = normalized gas phase concentration, c/Co Co = constant inlet gas phase concentration, moles/volume d p = particle diameter f = void fraction, gas volume/column volume k = mass transfer coefficient, moles/time/bed volume/concentration K = equilibrium coefficient, concentration/solids loading L = total length of adsorption column M = adsorbate molecular weight N = packing coefficient, (2d 2$)* ( A / B ) = ( A / B ) N ’ P = normalized total column pressure, P’/Po Po = inlet total pressure to column, atm. q = solids loading, moles/weight of solids T = dummy variable in Equation 6 R = gas constant s = dummy variable in Equations 5, 7 , and 8 t = time T = absolute gas temperature u = dimensionless time, k K t / p ~ u = dimensionless distance, old distance parameter, kx/Vo V = normalized gas velocity, V’/Vo Vo = constant inlet superficial gas velocity, distance/time W = normalized solid phase loading, p/p, z = distance along bed axis Y = modified distance parameter (see Equations 7 and 13) z = normalized distance, x / L a = dummy variable in Equation 7a 0 = modified performance parameter (see Equations 8 and 14) p = gas viscosity p~ = bulk density of bed, weight of solids/column volume
+
= portion of packing coefficient significant during laminar
flow C0.145 p VoL(l - f)2/Podp2gcf3when atm. c.g.s. units are employed] , J, = portion of packing coefficient significant during turbulent flow [0.00169LV>M(l - f)RTdpg,f3 when atm. c.g.s. units are employed] literature Cited
Acrivos, Andreas, Ind. Eng. Chem. 48, 703 (1956). Alonso, J. R., M.S. thesis, Worcester Polytechnic Institute, 1967. Anzelius, A., 2. Angew. Math. Mech. 6, 291 (19:f). Bird, R. B;: Stewart, W. E., Lightfoot, E. N., Transport Phenomena, Wiley, New York, 1960. Carter, J. W., Trans. Inst. Chem. Engrs. (London) 44, T253 ( 1966). Collentro, W. V., M.S. thesis, Worcester Polytechnic Institute, 1968. 6, 426 (1966). Cooney, D. O., IND.ENG.CHEM.FUNDAMENTALS Edeskuty, F. J., Amundson, N. R., J . Phys. Chem. 66, 148 (1952). Eteson, D. S., Zwiebel, Imre, A.I.Ch.E.J. 16, 124 (1969). Fosberg, T. M., Phillips, C. E., “Adsorption Dynamics in Granular Solid Beds,” Paper 14d, 62nd National hfeeting, A.I.Ch.E., Salt Lake City, Utah, 1967. Fukunaga, Paul, Huang, K. C., Davis, S. H., Jr., Winnick, Jack, Ind. Eng. Chem. Process Design Develop. 7, 269 (1968). Furnas, C. C., U . 8.Bur. Mines,liBull. 361 (1932). Hougen, 0.A,, Watson, K. M., Chemical Process Principles,” Part 111, W-iley, New York, 1947. Masamune, Shinobu, Smith, J. M., A.I.Ch.E.J. 10, 246 (1964). Meyer, 0. A., Weber, T. W., A.I.Ch.E.J. 13, 457 (1967). Needham, R. B., Campbell, J. M., McLeod, H. O., Znd. Eng. Chem. Process Design Develop. 6 , 122 (1966). Schuman, T. E. W., J . Franklin Inst. 208, 405 (1929). Vermeulen, Theodore, in (‘Advances in Chemical Engineering,” T. B. Drew, ed., 5’01. 2, pp. 147-208, Academic Press, New York, 1958. RECEIVED for review October 21, 1968 ACCEPTED April 24, 1969
EXPERIMENTAL TECHNIQUES
V A P O R I Z A T I O N OF L I Q U I D DROPLETS IN HIGH T E M P E R A T U R E A I R STREAMS GEORGE C. F R A Z I E R , JR.’,
A N D WILLIAM W. HELLIER, JRH2
Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Md. 21 218 An experimental method for the study of rapid vaporization of liquid droplets in high temperature gas streams is described. Data from a prototype model are presented and discussed in terms of data obtained by other methods.
HE phenomenon of vaporization of small droplets, as well Tas their condensation, is fundamental to the formation, stability, and dispersal of aerosols. In many situations of interest, the characteristics and behavior of the droplets may influence or determine the gross characteristics of larger systems of which they are a part. In particular, in the area of combustion of liquid fuels the vaporization rate of small fuel droplets may be the rate-limiting step in the total process. Present address, Chemical and Metallurgical Engineering, University of Tennessee, Knoxville, Tenn. 37916 Present address, Department of Chemical Engineering, University of hlaryland, College Park, Md. 20740.
Interest in this, as well as other areas, has prompted a considerable number of investigations into various aspects of small drop phenomena (Bitron, 1955; Essenhigh and Fells, 1960; Ranz and Marshall, 1952). However, basic questions are still unanswered, especially with respect to the rapid vaporization of droplets in high temperature air streams, aside from the purely hydrodynamic processes. Among these are the extent of the deviation from vapor pressure-temperature equilibrium at the droplet surface during vaporization, the sensitivity of the vaporization rate to the form of the mixture “rule” for the transport properties of both polar and nonpolar vapor mixtures, and VOL.
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