COMMUNICATIONS
Effect of Adsorption Characteristics on Pulse Retention Times
The retention time (pl)for a pulse of adsorbable gas passed through a bed of adsorbent particles is shown to depend upon the reversibility and rapidity of the adsorption process. For reversible, first-order adsorption that is rapid with respect to the retention time, p1 is directly proportional to the adsorption equilibrium constant and independent of any rate constants. It is shown that when irreversible adsorption also occurs the retention time is reduced and may become less than p1 for a pulse of inert gas. In this case 1.11 is a function of the irreversible rate constant and mass transfer coefficients. For reversible but slow adsorption the retention time is increased and becomes a function of the ratio of adsorption and desorption rate constants. These distinctions suggest that pulse experiments on fresh and used catalyst particles may yield quantitative information on the characteristics of the adsorption process.
When a gaseous pulse containing an adsorbable component is passed through a bed of adsorbent particles, the retention time 111 depends upon the characteristics of the adsorption as well as on the gas velocity, bed void fraction, and bed length. If the adsorption and desorption processes are linear and rapid with respect to the retention time, 1 1 is directly proportional to the equilibrium constant K, and involves no individual rate constants. Experimental data for the adsorption of NO on activated carbon particles a t 18OOC exhibited such rapid, reversible adsorption after the carbon surface had been treated with NO-helium mixtures. With a fresh surface the retention time was greater than that for the treated surface (Chiu et al., 1974). This was first thought to be due to some purely irreversible adsorption, on especially active sites of the surface, superimposed on the rapid, reversible process. However, theoretical studies (Miller and Bailey, 1973) indicate that the retention time should be less than that for rapid reversible adsorption, with the decrease in 111 increasing as the irreversible rate constant increases. In order to resolve this difference between experimental and theoretical results, equations have been derived for adsorption processes of different characteristics. The pertinent differences in these processes are whether the processes are rapid or slow with respect to the retention time and whether they are reversible or irreversible. It is shown that retention times can be either greater or less than that for rapid, reversible adsorption, depending on the characteristics of the processes involved. All processes are assumed to be linear since only then can explicit expressions for 111 be obtained. Equations for the retention time are derived by solving, in the Laplace domain, the differential, mass conservation equations within the adsorbent particles and in the bed of length z. Then 111 is related to the first derivative of the concentration c ( z , p ) in the Laplace domain. The procedure is described in detail by Schneider and Smith (1968). The conservation equations include the effects of axial dispersion, gas-to-particle mass transfer, intraparticle diffusion, and rate of adsorption a t a site within the particle. We are interested here only in how the characteristics of the adsorption rate equation affect the first moment.
In order to satisfy the first-order nature of this expression, experimental measurements must be conducted with low enough concentrations of adsorbable component in the pulse for the isotherm to be linear. For this rate equation Schneider and Smith (1968) and others have shown that 111 is a function of K, = k,dkd;
pi-;=-
" [1 + - l - " P ( l + p a ) ] 2,
(Y
(2)
Rapid, Reversible plus Slow Reversible Adsorption For this case it is postulated that rapid, reversible adsorption occurs on some of the sites and that a slower, reversible, adsorption can take place on the remaining sites. For each type of site the adsorption rate is supposed to be linear, that is, low concentration of adsorbable component in the input pulse. Then the net rate of adsorption is given by R
- pp [ ( k q C i - kd,nf) A - P
-t
(k+Ci -
(3)
where nfand n, are the adsorbed concentrations on the two kinds of sites and k, and kds are the adsorption and desorption rate constants for the slower process. The solution of the conservation equations, using eq 3, gives
Note that eq 3 and 4 in themselves do not require that k, is greater than k,, but only that different sites are involved.
Rapid Reversible plus Irreversible Adsorption If the rapid, reversible adsorption is accompanied by completely irreversible adsorption, the rate equation becomes
Rapid, Reversible Adsorption In this case all the adsorption is rapid and reversible with respect to 111. The rate equation is written Id.Eng. Chem., Fundam., Vol. 14, No. 3, 1975
273
Table I. Adsorption of Nitric Oxide in a Bed of Activated Carbon Particles Temperature Bed void fraction, a Particle radius, R , cm Bed length, cm Diameter of bed, cm Particle void fraction, p Particle density, p,, g/cm3 Effective intraparticle diffusivity, D,, cm2/sec Axial dispersion coefficient, E A , cm2/sec Fluid-to -particle mass transfer coefficient, k , R, cmz/sec
-
N
,
0
1
2
3 z v
.
4
5
6
ret
Figure 1. Retention time for various types of adsorption.
where ki is the first-order, irreversible rate constant. Now the retention time (first absolute moment) can be shown to be a function of kj and also mass transport rate coefficients E A , k,, and De, according to the equation
+ -32- 1f -f @ (0 +
pDKJ
where
Equation 6 is similar to the equation derived for the first moment for the case of a first-order surface reaction (Suzuki and Smith, 1971). In principle, eq 6 can be used to evaluate the irreversible rate constant, but, practically, this may be very difficult for the reasons given in the Discussion section. If only irreversible adsorption takes place, eq 6 is applicable with K , = 0. This form of the expression for p1 is similar to that derived by Miller and Bailey (1973) for a firstorder irreversible reaction. These authors did not consider fluid-to-particle mass transfer resistance ( k , a),which and K , is usually small for gaseous systems. When k , = 0, eq 6 becomes identical with that derived by Miller and Bailey.
--
Discussion The derived equations for retention time are examined here for the conditions corresponding to the adsorption of nitric oxide (in 3% NO pulses in helium) a t 18OoC in a bed of activated carbon (Filtrasorb, Calgon Division of Merck and Co.). The properties of the system, as shown in Table I, are those obtained by Chiu et al. (1974) except for E A , k,, 274
Ind. Eng. Chem., Fundam., Vol. 14, No. 3,1975
22.0
0.55 0.535 0.956 0.01 0.47
1.00
and De. Values for these latter three quantities were estimated to be appropriate for the system and conditions of the Chiu experiments. The heavy solid line corresponds to rapid, reversible adsorption (eq l),with K , = 4.35 cm3/g. The data points were obtained by Chiu et al. from p1 measured after one kind of sites were eliminated by pretreatment with NO. The experimental points for untreated carbon particles show greater retention times (Figure 1)and follow the linear form of eq 4.Thus the data suggest that another reversible adsorption, for example, a slow process, is superimposed upon the rapid reversible process. Using the known K,, the ratio of rate constants ka,lkd, which gives agreement between eq 4 and the data points is 3.12 cm3/g. For irreversible adsorption, eq 6 shows that the relationship between p1 and z/v is no longer linear. This is shown _ coth _h _ , -
1
180 "C 0.438 0.0135
(y)'
h,
( h , cot; h,
-
cosech2 h ,
1
+
De
in Figure 1 by the dashed curves for ki values of 10 and lo4 cm3/(g)(sec).Also, the first moments are less than those for rapid, reversible adsorption. Such reduced pl are physically understandable since irreversible adsorption would reduce the concentration in the tail of the response peak. Several systems and temperatures were studied (NO, SOz, and HzS with silica gel and activated carbon) in an attempt to find a situation where eq 6 was applicable and, hence, where ki could be evaluated. In each instance the observed moments were either too short to be measured accurately or the response peaks were poorly defined. From a practical viewpoint the retention time in the volumes between pulse injection and entrance to bed, and between bed exit and detector, must be subtracted from the observed values in order to evaluate p1 for the bed itself, that is, to evaluate the p1 values corresponding to eq 2,4, or 6. This is most accurately accomplished by subtracting, from the observed p1, the retention time when a pulse of nonadsorbable gas (for example helium in nitrogen carrier gas) is passed through the bed. The moment vs. z/v relationship for such an inert system is also shown in Figure 1by the dotted line. It is seen that for some values of ki the first moment for irreversible plus rapid reversible adsorption is even less than that for a nonadsorbable gas, again because of reduction in the tail of the response peak when irreversible adsorption occurs.
Since retention times would be so low, and since the observed p1 values would be similar in magnitude to those for a nonadsorbable gas, it is doubtful that irreversible rate constants could be measured by the pulse technique. Reversible adsorption tends to increase the retention time making evaluation of rate constant ratios feasible. Within limits it appears possible to determine such ratios for cases where the adsorption and desorption processes are slow with respect to the retention time.
Acknowledgment A Fulbright Award to one of the authors is gratefully acknowledged.
Nomenclature
ci
= concentration of adsorbable component A in the intraparticle pores, mol/cm3 De = effective intraparticle diffusivity, cm2/sec E A = axial dispersion coefficient of A in the bed, cmz/sec ho = Thiele modulus, defined by eq 7 k,, = adsorption rate constant for rapid, reversible adsorption, cm"/(g)(sec) kdr = desorption rate constant for rapid, reversible desorption, sec-' k,,kd, = adsorption and desorption rate constants for slow adsorption and desorption, cm3/(g)(sec) and sec-l, respectively ki = rate constant for irreversible adsorption, cm3/(g)(sec)
k , = fluid-to-particle mass transport coefficient, cm/sec
K A = adsorption equilibrium constant for rapid, reversible adsorption, cm3/g
nf,ns = adsorbed concentration on sites for rapid, reversible adsorption and for slow, reversible adsorption, respectively, moVg R A = total adsorption rate of A, mol/(cm3)(sec) R = particle radius, cm t o = pulse injection time, sec u = gas velocity in interparticle space, cm/sec z = bedlength,cm Greek Letters a = bed void fraction p = intraparticle void fraction p1 = retention time of adsorbable component in bed, sec pp = particle density, g/cm3
Literature Cited Chiu, H.-M., Hashimoto. N., !Smith, J. M., Ind. Eng. Chern., Fundarn., 13, 282 (1974). Miller, G. A,. Bailey, J. E., AIChEJ., 19, 876 (1973). Schneider. P., Smith, J. M.. AIChEJ., 14, 762 (1968). Suzuki, M., Smith, J. M.. Chern. Eng. Sci., 28, 221 (1971).
University of California Davis, California 95616
M. A. Galan M. Suzuki J. M. Smith*
Receiued for review November 8,1974 Accepted April 4,1975
An Improved Model for Analyzing Platelet Deposition on Glass Surfaces Experimental values of blood platelet deposition on glass as a function of time and flow rate were correlated by use of mass transport theory by Grabowski et al. (1972). Their theory was modified in the present investigation; the new theoretical results are more consistent with the experimental data over a wider range of surface coverage by platelets than previously.
In a recent publication, Grabowski et al. (1972) have developed a theory for analyzing the deposition of platelets from flowing blood onto glass surfaces. They proposed pseudo-second-order kinetics between the platelet concentration and surface site availability and obtained a numerical solution to the transport equations. The authors recognized that the solution, based upon the applicability of continuum theory, might not be valid a t high degrees of surface coverage with platelets. In fact, they resorted to analyzing the deposition data only for low surface coverages (less than 50%). They used a modified version of a previously available theory for first-order reaction kinetics which did not consider the influence of surface availability (Solbrig and Gidaspow, 1967). The analysis of Grabowski and coworkers then indicated that the deposition rate for platelets on glass may be diffusion controlled. Under diffusion controlled conditions of transport, it is obvious that the numerical solution of Grabowski et al. fails to describe adequately the experimental data a t high surface coverage. The theory predicts a linear increase of platelet deposition with time until surface saturation is attained (Figure 2 of Grabowski et al., 1972), whereas the experimental data show an increase in platelet deposition which approaches the surface saturation level asymptotically. The failure of the theory appears to consist in the lumping of the surface availability factor with the dimen-
sionless reaction rate parameter (defined as the ratio of the platelet reaction rate and the platelet diffusivity). When the reaction rate factor is infinite, that is for diffusion-controlled kinetics, the product of the available surface area factor and the reaction rate parameter is always infinite for less than complete surface coverage. Such a condition is physically implausible since the surface coverage is not solely dependent on the reaction rate parameter, but actually exhibits an inhibiting effect on further platelet deposiition, regardless of the transport parameter. An alternate method which separates the surface and transport factors and predicts a gradual approach to platelet saturation level can be developed directly from the theory of Solbrig and Gidaspow (1967). Initial platelet fluxes are predicted from this theory; however, the surface influence is incorporated by consideration of the probability of a platelet arriving at an occupied surface site. Thus there are two probabilities: the probability of reaction at a surface site calculated from the theory of Solbrig and Gidaspow for various reaction rates, and the probability of reaching an unoccupied site. The former probability can be considered constant in time; however, the latter depends on surface availability or free surface sites, which change with time. The time dependence of the surface coverage factor can Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
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