Frequency response of liquid-phase adsorption on polymeric adsorbents

Pergamon

Chemical Engineerin 0 Science, Vol. 52, No. 10, pp. 1589-1608, 1997 © 1997 Elsevier Science Ltd. All fights reserved Printed in Great Britain P I I : S001D-2509(96)00513-1 0009-2509/97 $17.00 +0.00

Frequency response of liquid-phase adsorption on polymeric adsorbents D. S. Grzegorczyk and G. Carta* Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442, U.S.A. (Received 4 September 1996; in revised form 25 November 1996; accepted 27 November 1996) Abstraet--A frequency response technique is applied to the study of mass transfer in liquidphase adsorption on polymeric adsorbents. The technique uses an imposed sinusoidal temperature cycle to elicit an adsorbate concentration response in a stirred vessel. Mass transfer parameters (intraparticle diffusivities and film coefficient) are obtained from the steady periodic concentration response by matching experimental phase angle lag and amplitude ratio to model predictions. The technique is applied to the adsorption of the amino acids phenylalanine and tryptophan from aqueous solution onto two commercial polymeric adsorbents with varying pore size. In all cases, the technique yields results that are consistent with the behavior observed in isothermal batch uptake experiments with a finite concentration step. However, the frequency response technique is found to provide the advantage of yielding mass transfer parameters for near equilibrium conditions and of detecting deviations from Fickian diffusion behavior, such as the presence of surface barriers. © 1997 Elsevier Science Ltd. All rights reserved Keywords: Frequency response; liquid-phase adsorption; polymeric adsorbents; amino acids; diffusion; mass transfer.

INTRODUCTION The frequency response (FR) method is a relaxation technique in which the equilibrium state of a system is perturbed periodically in order to measure the response of the state variables under periodic steadystate conditions, allowing one to determine characteristic time constants of the system (Stephanopoulos, 1984). When applied to the study of transport phenomena, measurement of the response over a perturbational frequency range allows one to identify the magnitude and nature of the controlling transport mechanisms (Jordi and Do, 1993). The technique was first used by Angstrom in the last century to measure the thermal conductivity of a solid bar (Carslaw and Jaeger, 1959) and has been more recently extended to the measurement of mass transfer parameters in adsorption systems by measuring either the pressure response (Naphtali and Polinski, 1963; van-den-Begin and Rees, 1989; Yasuda, 1976, 1982; Yasuda and Matsumoto, 1989; Yasuda et al., 1991) or the thermal response (Sun et aL, 1994; Bourdin et al., 1996) of a closed adsorption system subjected to a sinusoidal volume change. A frequency response method has also been applied to open gas adsorption systems by Boniface and Ruthven (1985). These authors deter*Corresponding author.

mined axial dispersion coefficients, diffusivities, and adsorption equilibrium constants in zeolites by sinusoidally varying the input concentration of a gas chromatograph and determining the outlet concentration response. Compared to classical single-step methods, the FR technique has several advantages. Since measurements are carried out in a periodic steady state, the results are not dependent upon the initial phase of contact of the adsorbent with the fluid phase, which is often significantly affected by experimental error. Moreover, the technique allows differential measurements near equilibrium conditions, giving results that are insensitive to small changes in experimental conditions (Bourdin et al., 1996) and which are reflective of the repeatability of adsorption and desorption (Bulow and Micke, 1995). In addition, frequency response techniques can discriminate between the different rate-limiting mechanisms of mass transfer, such as surface barrier and Fickian diffusion, as a result of the high sensitivity of the method to the nature of the governing transport equations. In fact, when multiple resistances occur in series, it is often possible to determine each individually by studying the system response over different frequency ranges. While applications to gas-phase adsorption systems have been extensive, liquid-phase adsorption applications have been more limited, probably

1589

1590

D. S. Grzegorczyk and G. Carta

because of experimental difficulties in imposing sinusoidal perturbations. For example, while cyclic adsorption~lesorption can be obtained in a gas system by cyclic pressure of volume changes, this is not possible in liquid-phase adsorption. In some systems, however, the same effect can be obtained by a cyclic perturbation of the system temperature. In a closed, constant-volume system, an imposed temperature perturbation causes a concentration response whose frequency response is directly related to transport limitations. When heat transfer is fast compared to the rate of mass transfer and the heat of adsorption can be neglected because of the large heat capacity of the liquid, the adsorbent particles can be considered to be uniform in temperature. In this case, the concentration response is dependent only on mass transfer effects that can be determined by comparison with appropriate models. Such an approach has been used by Hsu and Pigford (1991) for the determination of mass transfer rates in thermally regenerated ion-exchange resins. In this paper, we extend this technique to adsorption from dilute aqueous solutions onto porous polymeric adsorbents using amino acids as model solutes. A general model that considers external mass transfer resistances or a surface barrier along with diffusion in the pores is considered and compared with experimental results for the adsorption of the amino acids phenylalanine and tryptophan in two distinctly different types of polymeric adsorbents. A comparison of dynamic parameters, effective diffusivities and external film coefficients obtained from frequency response method with those obtained independently from single-step batch uptake and empirical correlations is made in order to assess the validity of the approach. THEORETICAL DEVELOPMENT When a linear system is subjected to a sinusoidal temperature perturbation of frequency 09 and amplitude AT, the response in the periodic steady state is also a sustained sinusoidal wave of the same frequency, but with amplitude Ac. This steady-state response can be characterized through the amplitude ratio, Ac/AT, and the phase angle or phase lag, 4~, obtained from the difference in phase shifts of the input and response waves. These two parameters are directly related to the rates and mechanisms of the adsorption process.

Mathematical formulation The mathematical analysis is based on the following system: a fixed volume of spherical adsorbent particles is placed within a constant-volume wellmixed vessel whose temperature is varied sinusoidally. A relatively small periodic perturbation of temperature causes a change in the equilibrium distribution coefficient, which causes the solute to diffuse in and out of the adsorbent particles. The system contains a non-reacting solute that physically adsorbs onto the adsorbent. The following assumptions are made: (1) the temperature changes sinusoidally around a mean

value T, as T = T + ATexp(i~ot); (2) the slope of the adsorption isotherms can be approximated as constant over the induced concentration range; and (3) the temperature throughout the particle is uniform and equal to the temperature of the fluid. Assumption (2) is dependent on the choice of a temperature amplitude which is sufficiently small that the isotherm can be approximated as linear. Thus, even if the adsorption isotherms are non-linear, experimental conditions can be chosen so that the frequency spectrum is dependent only on the local gradient of the isotherm, and not on its global nature (Jordi and Do, 1993). Assumption (3) is valid for a system where the characteristic time for heat transfer is much smaller than the characteristic time for mass transfer. It can be easily verified that this assumption is usually satisfied for liquid-phase adsorption. Under conditions of intraparticle transport control, the characteristic time for heat tranfer is zh r~/O~eand that of mass transfer 2 is to = rp/epDp, where ~e and epDp are the effective thermal and molecular diffusivities. The effective thermal conductivity of a liquid-impregnated porous material can be estimated from the Russell equation (Liley and Gamble, 1973). However, as a conservative approximation, c~e ~ ep~, where ~ is the thermal diffusivity of the liquid. Thus, we obtain rh/zo <. Dv/a. For typical liquids the thermal diffusivity is of the order of 1-5 x 10-3 cmZ/s (Cussler, 1984), while the pore diffusivity for a small solute is of the order of 1 x 10 -5 cm2/s. As a result, it is apparent that the ratio of heat and mass transfer time constants is typically much less than 1 and that the particle temperature may be assumed to be uniform in most cases. Furthermore, since only small perturbations are used and the heat capacity of the liquid is large, heat of adsorption effects can also be neglected (see Ruthven, 1984). Finally, it is also possible to show that the characteristic time constant for extraparticle heat transfer is also much lower than the corresponding time constant for mass transfer. Using typical correlations for heat and mass transfer to small particles suspended in agitated contactors, the ratio of time constants is approximated by rh/~O = (ppCp/piC I) (Pr/Sc) 2/3. For our experimental system ppCp/pfCf 1, Pr ~ 1 - 10, and Sc ,,~ 1000. Thus, it is apparent that also in the case of extraparticle transport control zh/vo <<1. In previous work, both surface and pore diffusion mechanisms have been found to be important in liquid-phase adsorption from dilute aqueous solutions on polymeric adsorbents (Komiyama and Smith, 1974; van Vliet et al., 1980; Cornel et al., 1986; Costa and Rodrigues, 1985; Casillas et al., 1993; Grzegorczyk and Carta, 1996b) with the relative importance of the two mechanisms dependent on the adsorption affinity of the solute (Komiyama and Smith, 1974). Thus, for the general case, we consider a model with external film mass transfer resistance and parallel pore and surface diffusion within the particle. The system conservation equations are given by the following. =

Frequency response of liquid-phase adsorption

159 1

easily obtained analytically, yielding

F o r the particle:

1,1(

dcp dn ~P ~ - + PP dt

r 2dr

r2 e p D p + p ~ D , ~ c

/ dr

1

c' = where

(2)

f(09") = 3Ag(09*)/{3Ag(09*) + v/-~-;-/2 x (sinh 2~//2~ - s i n 2~/2-~)(cosh 2 ~

%Dp + poD~-~c ]-~r j . . . . = k f ( c -- cpl,=,).

- cos 2 x / ~ ) } (3)

F o r the fluid: 3k I V v rv -~ (c -- cpl,=~).

(4)

(13)

h(09") = {09*g(09*)/Bi + (cosh ~ x [~(sinh

Solution method

The perturbation solutions are described by the following equations: (5)

c o = 6 o + ATc'p(r)e i°'

c = 6 + A T c ' e i'°t

- (cosh 2 w / ~ - cos ~/209")}/{3A9(09")

(dn3

+ \dTJ

x (cosh ~

- sin - cos 2 ~ / ~ ) }

(14)

9(09") = [ ~ l ~ ( s i n h 2 x / / ~ + sin 2 x / / ~ )

L42

1

(cosh 2x//~-~ _

+i1 (sinh 2x//~-~ _

cos 2x/~-~-~)] 2

sin 2x//~)2.

(15)

In these equations 09* = m09zo is a dimensionless frequency, Bi = ksrffDe the Biot number, and A = Vvm/V. The imaginary part of A T c ' e i°'~ is Im { A T c ' e i~'} = Ac sin (tot - ¢)

(16)

where Ac and 4>are the amplitude and phase angle lag of the concentration response. These are obtained by algebraic manipulation of eq. (12) yielding

(6)

where 6o = 6 is the mean value of the pore- and fluid-phase concentrations. Since

dn (dne dc

- cos 2x/2-~)

2V/2-~ + sin 2x//2~)

+ ?(sinh In these equations, D o and D~ are the pore- and solidphase diffusivities, dne/dc is the local slope of the adsorption isotherm, ky the external film mass transfer coefficient, and Vp and V the adsorbent and fluid volumes. The pore diffusivity is related to the particle tortuosity, z o, by D v = Dolt v. These equations assume that the pore fluid and the adsorbed phase are always at equilibrium at each point with the particle. Moreover, it is assumed that when equilibrium is finally reached the external and pore-fluid solute concentrations are identical. In this case, the apparent diffusivity D~ = cod p + ppD~(dne/dc)T accounts for parallel transport in the two phases.

dt = \OC ]T dt

i09ATei°*

Ac

~f(09")

AT

x/1 + h(09.) z

~b = t a n - 1 [h(09")].

(dne dc

ff[ = \ d c J r dt + \ d T J

(12)

(1)

dr d,=o

dc dt -

-- ?f(09")

1 + ih(09*)

(7)

eqs (1)-(5) can be written as

(17) (18)

Limiting solutions are found in the extreme cases of particle or film mass transfer control. When Bi ~ intraparticle mass transfer is dominant. When Bi --* O, the film resistance is dominant and the following simplified expressions are obtained for f(09") and h(09"):

Oe/m d ( 2 dc; r 2 drr kr

dr ] = i09c~, + i097

(8)

dr/,=o = 0

(9)

D. -d~r ],=,. = k f [ c ' -- c~(ro) ]

(10)

dc~-]

3kf V o

i09c' = - - - - - [c' - c~(ro) ] ro V

(11)

m = ep + po(dn'lac)f and ~ = po(dne/dT)d [81, + pn(dne/dc)f]. A solution of these equations is

where

f(09,) = h(09") =

A I+A

= constant

09*

3Bi(1 + A)

=

mogro/3k I

(1 + A)

(19) (20)

The general behavior predicted by these equations is shown in Fig. 1 in terms of amplitude ratio and phase angle lag as a function of dimensionless frequency for values of Bi = l, 10, and 100. The limiting solution for Bi = oo (particle diffusion control) is also shown over the entire frequency range. At low frequencies, the solution for combined resistances is seen

1592

D. S. Grzegorczyk and G. Carta

"F Particle diffusion 0.1 "O

Q.

E

0.01 0

,

.

-

.

-

,

10

20 "G-

30 GI

_¢

40

O~

== m cfl.

50

\

Particle diffusion

60 70

B'=I ~

~

80 9O 10 o

10 +

102

10 3

104

10 s

Dimensionless frequency, ~"

Fig. 1. Dynamic response to a sinusoidal temperature cycle in a well-mixed vessel calculated from eqs (17) and (18) with A = t. to approach the solution for particle diffusion control. The latter is limited to a maximum phase lag of 45 °. At a dimensionless frequency of about 100, however, significant deviations from the particle mass transfer control case become apparent, even if the Biot number is as large as 100. This occurs because at high frequencies the concentration oscillations are limited to a short depth of penetration from the particle surface, making the external film resistance comparatively more important. Thus, at high frequencies the complete solution approaches the limiting case for film resistance, gradually attaining a maximum phase lag of 90 ° . It is interesting to compare the phase-lag behavior in FR experiments with the transient response of batch uptake experiments. Film resistance is, of course, always important in the initial phase of the latter experiments, when clean particles first come into contact with the solution. However, in practice, when intraparticle rates are limited by pore diffusion, it is generally difficult to determine the magnitude of the film resistance from these experiments, since the times when this resistance is important are typically extremely short. In contrast, the behavior shown in Fig. 1 demonstrates that the FR method is sensitive to the nature of the controlling resistance. Experimental

data can be fitted in the frequency domain with the appropriate model to determine the controlling rate parameter. In addition, in certain cases it is possible to determine both resistances independently by carrying out experiments first at cycling frequencies sufficiently low that particle diffusion becomes controlling, and then at cycling frequencies sufficiently high that the film resistance becomes dominant. Of course, on the basis of the maximum phase angle lag attained, it is also possible to discern the presence of surface barriers that result in non-Fickian behavior. EXPERIMENTAL

Materials Two distinctly different polymeric adsorbents, XUS-40323 and XUS-43444, (Dow Chemical Company, Midland, MI) are considered in this study. The properties of these materials are described in detail by Grzegorczyk and Carta (1996a). XUS-40323 is a copolymer of ethylvinylbenzene and divinylbenzene that is characterized by a highly hydrophobic, unfunctionalized backbone and a matrix of permanent pores. The average pore size is 92/~ and the surface area is 680 m2/g. XUS-43444 is also styrenic, but is synthesized in a different way yielding a much finer pore

Frequency response of liquid-phase adsorption structure. This material has a significantly smaller pore size (28/~) and a larger surface area (1200 m2/g) than XUS-40323. The polymeric backbone of XUS43444 is chloromethylated and dimethylamine functionalized to enhance its water-wetting properties. XUS-43444 also has a translucent physical appearance that is consistent with the existence of only very small pores. Two lots of XUS-43444 were used in this study, designated lot A and lot B. These two lots appeared similar with regard to equilibrium properties but had a somewhat different particle size and a different dynamic behavior. Samples of these adsorbents with a fairly uniform particle size were used in the experiments. The volume-average particle size of each sample was determined from microphotographs and is included in Table 1. The particle size of the XUS-40323 sample used here is different from that used in our previous work, since a sieved fraction was used, eliminating most of the fines present in the original sample. The amino acids L-phenylalanineand L-tryptophan were used as model solutes. They were obtained from Sigma Chemical Co. (St. Louis, MO). Both possess a hydrophobic side chain and are adsorbed from aqueous solutions onto hydrophobic adsorbents. At neutral pH, both amino acids exist primarily in their zwitterionic form. They both absorb UV light, which allows their spectrophotometric detection.

Adsorption isotherms Adsorption isotherms were obtained by a batch method. Isotherms for phenylalanine are given by Grzegorczyk and Carta (1996a). Those for tryptophan are given in Fig. 2 for XUS-40323 and XUS-43444 (lot B). The isotherms for lot A were essentially coincident with those for lot B. In each case, the isotherms were found to conform well to the ideal Langmuir model:

nsbc n - l + bc.

(21)

The parameters were determined by fitting the entire data set with a single value of ns by nonlinear leastsquares fitting. A plot of In b vs I/T yielded an essentially straight line for each adsorbent-solute pair.

1593

Thus, b is related to an apparent energy of adsorption by

b = boe-alt°/Rr.

(22)

A summary of these parameters and other adsorbent properties is given in Table 1 and a comparison of calculated and experimental isotherms is given in Fig. 2. For both solutes the uptake is greater on XUS43444, partly as a result of the larger surface area. The effect of temperature on the isotherm is stronger for tryptophan on both adsorbents and in the case of phenylalanine, somewhat weaker for XUS-43444 than for XUS-40323.

Apparatus for frequency response study The experimental apparatus used to carry out the frequency response experiments is shown schematically in Fig. 3. It consists of a ca. 125 cm 3 jacketed glass vessel with a slightly upward-pointing conical bottom. The vessel is equipped with a magnetically driven, four-blade Teflon impeller ( ~ 3.5cm diameter) rotated at about 275 rpm. The jacket temperature is maintained constant at a temperature in the range 0-45 °C, dependent on the experimental conditions chosen, by circulating a water-ethylene glycol mixture with an M G W Lauda Model RC 20 circulator. The coolant acts as a heat sink for the vessel. A sinusoidal temperature profile within the vessel is elicited by a 150W immersion heater (Watlow, St. Louis, MO, Firerod Mod. 9418G) whose power output is controlled by a solid-state relay and a VARIAC voltage transformer. Opening and closing of the solidstate relay is in turn controlled by a microcomputer through a Data Translation analog-digital interface board (Model DT-2805). The computer uses a control algorithm based on a model that describes heat transfer in the vessel. The parameters of this model (heat transfer coefficients to the jacket and from the immersion heater) were determined experimentally from dynamic experiments and used to set up the control program. In this way, it was possible to preset the operating parameters that yield the desired amplitude of the temperature oscillation and the cycling frequency. Concentration changes in the vessel are continuously monitored by circulating a stream, withdrawn through a stainless-steel filter, through a UV detector

Table 1. Summary of adsorbent properties Physical properties

Equilibrium parameters*

Adsorbent

ep

pp (g/cm3)

dp (mm)

Phenylalanine

Tryptophan

XUS-40323

0.62 __+0.03

0.392

0.54 + 0.05

n, = 0.79 AH0 = - 3.00 bo = 1.72x 10-4

n~= 1.04 AHo = - 4.41 bo = 3.4x 10-5

XUS-43444

0.66 + 0.02

0.389

0.60 + 0.05

n, = 1.14 AHo = - 2.01 bo = 2.30 × 10-3

ns = 2.33 AHo = -- 4.44 bo = 6.77 × 10- 5

*n~ (mmol/g), AHo (kcal/mol), bo (l/mmol).

1594

D.S. Grzegorczyk and G. Carta 0.8

_

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.

.

.

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,

.

.

.

.

,

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-

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(a)

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0.6 A

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O

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p.

0.3 0.2 0.1 & 0.0

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C (mM) 1.4

_

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-

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-

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(b) 1.2 1.0 A

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0

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0.2

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•

0.0 0

|

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2

4

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.

6

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° C

45 °C

!

8

10

12

c (mM)

Fig. 2. Equilibrium isotherms for the adsorption of tryptophan on (a) XUS-40323and (b) XUS-43444from aqueous solutions.

(Pharmacia, Mod. UV-2) at a wavelength of 254 nm. Millivolt signals from the detector are converted to absorbance units via the data acquisition system. Flow rates of 12 cm 3 per minute were obtained with a recirculation pump (FMI, Mod. QD1). The response time of the detection system, including the residence time of the recirculation loop, was experimentally determined to be 1.0 s. For each experimental run, the adsorbent (1.5- 4 g, dry basis) and the amino acid solution (100cm 3) were placed in the vessel and allowed to reach equilibrium at the desired mean operating temperature. The temperature cycle was then initiated and allowed to continue while monitoring both temperature and concentration. When a periodic state was reached, the data were collected and stored for later analysis. The collected data (both temperature and concentration) were fit to sinusoidal functions with MathematicaT M using a period equal to the one imposed experimentally. Generally, two to four consecutive

periods were used in the data fit. The phase lag was determined from the difference of the fitted temperature and concentration phase shifts. The amplitude ratio was obtained by dividing the fitted concentration amplitude by the fitted temperature amplitude. The same apparatus was also used for isothermal batch uptake experiments with the procedure previously used by Grzegorczyk and Carta (1996b). In this case, average effective diffusivities were determined by matching the transient uptake curves with the numerical solution of the conservation equations (1)-(4), with the experimentally determined adsorption isotherms. RESULTSAND DISCUSSION

Frequency response Two representative samples of raw data are shown in Figs 4 and 5 for the adsorption of tryptophan on XUS-40323 at mean temperatures of 11 and 58°C for

Frequency response of liquid-phase adsorption

1595

THERMOCOUPLE

1 CIRC. PUMP

I..°.~,°~....I 1

MICROCOMPUTER FOR DATAACQUISITION AND CONTROL

LN DETECTOR

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VARIAC

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Fig. 3. A p p a r a t u s for f r e q u e n c y r e s p o n s e studies.

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400

800 T i m e (s)

1200

1600

t e m p e r a t u r e a n d c o n c e n t r a t i o n t i m e series f o r t h e a d s o r p t i o n X U S - 4 0 3 2 3 . P e r i o d = 6 0 0 s , T = l l ° C , A T = 5°C, ~ = 4.9 m M .

of tryptophan

on

1596

D. S. Grzegorczyk and G. Carta

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64 62

.......

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800 1200 Time (s)

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Fig. 5. Experimental temperature and concentration time series for the adsorption of tryptophan on XUS-40323. Period = 600s, T = 58°C, AT = 5°C, ? = 7.22 mM.

a cycling period of 600 s. The imposed temperature amplitude was + 5°C, which is in good agreement with the experimental result. Except for noise associated with the detection system, both the imposed temperature cycle and the concentration response are essentially purely sinusoidal for these conditions. This result is expected if the amplitude of both temperature and concentration oscillations are sufficiently small that the system behaves linearly. The concentration response is, of course, shifted in time relative to the temperature cycle as a result of the finite rate of interphase mass transfer. The possible effect of non-linearities was considered by carrying out a full numerical simulation of the adsorption process via the numerical solution of the system conservation equations (1)-(4), with an imposed sinusoidal temperature oscillation. The equations were solved by orthogonal collocation, using a method similar to that of Saunders et al. (1989). A typical result is shown in Fig. 6 for conditions similar to the adsorption of tryptophan on XUS40323. Starting with a clean adsorbent sample, the concentration drops offfrom the initial value and then begins to oscillate, quickly approaching a pure sine wave. The dashed line shows the sinusoidal response

predicted from eq. (6), with amplitude ratio and phase angle lag predicted from eqs (17) and (18) for the parameter values used in the simulations. This result shows that a steady periodic state is reached quickly and the response of the system can be treated as linear. Experimental amplitude ratios and phase angle lags are shown in Figs 7-9 for XUS-40323 and in Figs 10-12 for XUS-43444 (lots A and B). In each case, the best-fit line obtained for optimum values of the parameters De and kf is also shown. A summary of these parameter values is given in Table 2. The fit was obtained from the phase angle lag, since this parameter is more sensitive to the mass transfer kinetics. The absolute value of the amplitude ratio, on the other hand, depends directly on 7 which, in turn, is dependent on the isotherm derivatives (One/c~T)~ and (One/Oe)r. Of these two derivatives, the second is known accurately from the experimental isotherms at the temperature of interest. F o r a Langmuir isotherm, neglecting the accumulation of solute in the pore fluid and assuming that the heat of adsorption is independent of loading, 7 is approximated by

7

(dne/~T)~ (One/Oc)¢

_ AHo

c -R--~.

(23)

Frequency response of liquid-phase adsorption

1597

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5.4 A

5.2 0

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6000

T i m e (s) Fig. 6. Simulated a p p r o a c h to periodic steady state for the adsorption of t r y p t o p h a n on XUS-40323 with Vp = 1 0 c m 3, V = 100cm 3, period = 1000s, T = 24°C, A T = 5°C, Co = 1 5 m M , De = 2.7x 10-6cm2/s, ks = 0.0037 cm/s. The dashed line is the sinusoidal response predicted with amplitude and phase-angle lag calculated from eqs (17) and (18) (Ac/AT = 0.0573 m M / ° C , q~ = 23.9°).

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i

i

!

!

i

i i i!

!

i

!

....................T............i.... i..........;TTi .......................T............!........Hit

~

~

i

iT

e,.

®

30

n

40

10 .4

10 "~

1 0 "~

Frequency, Hz Fig. 7. Experimental and fitted amplitude ratio and phase angle lag for the a d s o r p t i o n of phenylalanine on XUS-40323, T = 24°C, g = 6.0raM.

1598

D.S. Grzegorczyk and G. Carta 1.0

...........................................

,,'.......:..... '.,....-.....:.. ,.- ..! ..................................

~ ........ :......,'....,

...........................................

!'""'~'"'i""~'"'.~'"

f"! ..................................

~ ........ !"'"'i""'i'""

.'t "'~ ......................

~ ............

~ ........ ~.'"'"!""'"

......................

~ ............

~ ........ ? ""'" ~.""'1""'~.'"'.~'"

...........................................

I

F-

d

~....:.....- .. ~.. f ""~'" f" ~.'"" !""-~"i "'"

?'"'"

? "'"i"""

":'"'.."" ? "'~ ...........................................

":'"'"'.."""

:"'"'..'"-~'"

? ""

i

i

~

~

i

!

i

i

i!

!

!

i

0.1

(9 "10

.................................................... ..........................................

" : ' " " ' i ~...... " ' " " !? " ...~....:,...? " ' ! " " ' : " " ? "'~ ..! ............................................ ........................................... ........................................................................ . . . . . .


•

&T=3 °C

[]

AT=5°C AT=7°C

•

":'""" ! " ' " ' ! " " ...?..-!--~. " ":'"'Y • : ?.....?....? ~"-"T'"'T'"?'"~"T'"

......i " i i i ~ !

............................................! i i i ' i ~

...... ~:'"'!"'"?"?"?"!

......................

~i ~i ! ~ i~i i ii ii ii ii

...... T'"'~""~'"'F"T"; :

:

:

:

:

!

.:

~

!

~

!!

:

:

:

:

:

:

~ ............

~ ........ ~ : - ~ - - ! ? ? ' t

i i i ~i i !i ii i ii ii i

............................................

T""~-"'T"T'"~"T'"

:

:

:

:

,.

,

;

i

i

!

!

i

i

0.0

0

i

:

10 --I

_¢ t
i" ,,;,oc

20 30 40

....................!i............ ......!-----'----i-i ...................................

50 60 10 . 4

1 0 "~

1 0 "~

Frequency, Hz Fig. 8. Effect of temperature amplitude on experimental amplitude ratio and phase angle lag for the adsorption of tryptophan on XUS-40323, 6 = 5.0 mM.

Using this equation the experimental and predicted amplitude ratios were found to be in good agreement in most cases. In a few cases, however, a discrepancy of as much as 40% was observed. This is probably the result of inaccuracies in the estimation of the derivative of the isotherm with respect to temperature. Figure 8 shows that the frequency response is independent of the temperature amplitude over the range 3-7°C, confirming that these values are sufficiently small to yield a linear response. Figure 9 shows the effect of the average temperature for the adsorption of tryptophan on XUS40323. As temperature is reduced, the experimental phase angle lag increases. At higher frequencies, phase lags exceeding 45°C are approached, since for these conditions the external resistance becomes important. Thus, it is possible to determine both intraparticle diffusivity and external film coefficent over this frequency range. Values of the mass transfer coefficient, kl, determined from the frequency response and given in Table

2 are compared with predicted values from literature correlations. External mass transfer coefficients for small particles suspended in agitated contactors are generally correlated in terms of the power input per unit mass of fluid, g. In this work we considered two correlations by Levins and Glastonbury (1972) and by Armenante and Kirwan (1989) which express the Sherwood number ( S h = k l d p / D o ) as

Scl/3

= x

(24)

where S c = v/Do is the Schmidt number, dp the particle diameter, and v the kinematic viscosity. Levins and Glastonbury give x = 0.50 and ~/= 0.62 while Armenante and Kirwan give x = 0.52 and r / = 0.52. Film mass transfer coefficients were independently measured with ion-exchange particles of size similar to the adsorbent used in this work by Borst et al. (in press) in our experimental contactor. The experimental krvalues were within experimental error in

Frequency response of liquid-phase adsorption

1.0

1599

:::::::::::::::::::::i:::::::::::~:~::::::~::::::i:::::~::::~:::i!::i::i::::::::::::~:~::::i • looo :~:::::::::~::~ :t ......................

~ ............

~. . . .

. Ti

i i;~

i

!

o

...................... i............ ~ i i i i i - i i i _ : . . . . . . . ~ . ~ ..................... i............ i

,

2,octt ,~oo

I--

~5 ¢= 'ID

E

iiiiii i iiilii

0.1 0 lO 20

.~.....~

o~ t-

30

e-"

40

•

10°o i.........i

50

•

4 s ° c ti......... i

I1.

A 6o°c . . . . . .

60 10 4

....

~ -i..-~

................

i ............

, .........

~------i-----~----i---,.'---i--

.....

-

-

-

-

-

~

~

"

~

.

~

-

i

-

~ i iiii

i

10 .3

.....

10 "2

Frequency, Hz Fig. 9. Effect of temperature on experimental and fitted amplitude ratio and phase angle lag for the adsorption of tryptophan on XUS-40323, # = 5-7.1 raM. The mean concentration was constant within each temperature set.

1600

D.S. Grzegorczyk and G. Carta

1.0

:

:

:

:

:

:

v

:

:u

..................... ~............ ~......... i i i f i i i

..................... I............ i ......... ~......~.....i...-~-i.--i..

.....................

.....................

•.'............

~.........

~------~-----~----i---'.'---i---]-

..................... i ............ i ........ i...i..i....i..".i...i

I ............

..................... i ..........

.....................

o

•

..

i i i iii

i

i

i

~ ............

4 .........

~..-..-i.....i-...i.--~,..-~--4

•

:

.....................

24

I'q'"

c

.i.

t

,,'=oo°c!.--,-t

.........................

~~..._i_~....~...~_i. d

I-

..................... i ............ i.

.

.

.

.

.

.

.

.

1

.

i

i

i i if!

' ........

~, .-....~---..~..-.i...J,--.~-.

Q) "0

.= E

0.1 0

:

:

:

~-

-;---

'

:

:

r

v

:

r

:

;

;

;

10

m m ¢1 ¢1 O.

20

. . . . . . . . . . . . . . . . . . . . .

~..................

-~---~.--~--" .....................

~............

~- ........

L--.--~.-...i....i-..~

.....

30 40 50 60

10-4

10 .3

10 .2

Frequency (Hz) Fig. 10. Effect of temperature on experimental and fitted amplitude ratio and phase angle lag for the adsorption of phenylalanine on XUS-43444 lot A, ? = 3.3-4.9 raM. The mean concentration was constant within each temperature set.

Frequency

response

of liquid-phase

adsorption

1601

1.0 .................... -.............. ........... !------~-.--i..--~---i--~--i ..................... --.:............ i ....

~

~

~0~

.................... " ............ i........ i"'"'"':.'""i'""+'"~ "'2.'"i ..................... " ............ i'"" t .................... + ............ . ........ . " ' " ! ' " " i " ' } " } " H ...................... ~............ } - t

• 0

10 C ' 24 °C

..................... t .................... ! t i t i i i ...................... i ............ i t i ! ~ i i i ! i~ i

• A

43oc

.....................

7 . . . . . . . . . . . . {. . . . . . . . i " ' " '

............

TI"'T"'~"T"~

.....................

[........ k.i_..i....i...i.l.i

.oS

~

6000

]

.................... i......~....i....-:."...i ....

. -:.: ..................... ]............ [........ ]T--ii::

.................... - ............ ~........ i i 0 "0 D.

. . . . . . ............. . . . ........i iiiii! !. . . . . . . .

E

<:

ii

0.1 0

10 -o-

20 O~

c

¢1 u) ¢1

'-

I:L

.................. : ......... :. . . . i-.- :..-.i.-..~---L~--i ...................... ; ............ i........ ~------~----~:----~---~--.~-.

30 ,,

40

50

q

.; . . . . . . . . i - - - . . . $ ---i-...~-...~---~-.~ . . . . . . . . . . . . . . . . . . . . . .

0 •

240C i i i i :: ! ! :: 430c i ........ ]TTTii--I

a

e0°ci

i

i

i iili

~-

-i-....-~----'----~----';---~-,

i i i

i i ; . i i i i i i i iii

60 10 "4

10 "3 Frequency, Hz

10 2

F i g . 11. E f f e c t o f t e m p e r a t u r e on experimental and fitted amplitude ratio and phase angle lag for the a d s o r p t i o n o f p h e n y l a l a n i n e o n X U S - 4 3 4 4 4 l o t B, 6 = 4 . 8 - 8 . 6 m M . T h e m e a n c o n c e n t r a t i o n was constant w i t h i n e a c h t e m p e r a t u r e set.

1602

D.S. Grzegorczyk and G. Carta 1.0

..................................

i ........ i..-..;.....i-;i.iJ

..................... " ............ i ........ i+J+i

...................... . ............ i-i i

...................... i ............ i

...................... !~............ i~........ i-----.:,----i....4....;..-~..i:: ! i i ! ::i ...................... ::~............ ii'" ..................... ~ .......................... ':'*'"'?'"?"!'"i ........................ i ............ i'"

i

i iii,.I

•

,oooliJ

O ='

24C° 43°C

"'::1" ~'1

#-

"o

._=

...................... .~............ ~........ ~..-...4....-~----4....I

¢z.

'. ...................

~........ ~..-..-i----J-.--J,...i...i..

E <

0.1 0 10

....................

~

.

, ............

::

~i

,

i

........

i

i

': i

i i i i :: :: =; i i i ] =:

i

i i i i

~------4------:----4----~--4-.4

. . . . . . .....................

• .~ ............

i

;..

::

i i ! i iiii i i .......J......i....J....i...i...i..i ...................... L............ [..

20

0

•

10°0 24 C 43°c 0

o°o

¢p

30

i

i

~i

'~iii

i

i

e3

=g n

40 50 60 10-4

10 -3

10-2

Frequency, Hz

Fig. 12. Effect of temperature on experimental and fitted amplitude ratio and phase angle lag for the adsorption of tryptophan on XUS-43444 lot B, ~ = 3.3-5.9mM. The mean concentration was constant within each temperature set.

agreement with values predicted from eq. (24) with ~=450-1-190cm2/s 3, which was estimated from power number correlations, indicating that eq. (24) should be valid for prediciting mass transfer coefficients in our contactor. Predicted ks-values for the conditions of the frequency response experiments are given in Table 2. In most cases, the kfvalues obtained from the frequency response experiments are in good agreement with the predicted ones, essentially within the scatter of the data that form the basis for the correlations used in this estimation. An exception is the results for the adsorption of phenylalanine on XUS-43444 lot A, which gave kfvalues a factor of two to three times lower than the estimated ones. Mass transfer rates observed in isothermal batch experiments were also generally lower for this lot of XUS43444 than for the other. To ascertain the origin of this discrepancy, we examined in detail the surface features of this adsorbent by taking scanning electron micrographs of fractured particles. An example is given in Fig. 13. The XUS-43444 lot A particles appear to have a skin layer with a thickness of the order

of 10 #m enveloping a core material. No such skin was found in XUS-40323 particles. These appeared to have a much coarser structure, similar to that of typical macroreticular resins (see Grzegorczyk and Carta, 1996a). With lot B of XUS-43444 only very few particles exhibited a skin layer, while most of them appeared to be rather homogeneous. The exact causes for the presence of a skin layer in lot A of XUS-43444 are not known, but they are likely to be related to the manufacturing process. The presence of this layer appears to reduce mass transfer rates giving apparent kl-values that are much lower than the true boundary layer mass transfer coefficient values. Interestingly, however, the pore diffusivities for the two lots, as determined from the frequency response analysis, appear to be quite similar. Similar effects have been noted by Kumar and Sircar (1986) for adsorption onto extruded zeolite pellets. While the origins of a skin barrier are obviously quite different for this case, the general conclusion that a skin resistance may be important in some adsorption systems is the same. As noted by Kumar and Sircar, in general, it

1603

Frequency response of liquid-phase adsorption Table 2. Mass transfer parameters obtained from frequency response experiments System

T (°C)

~ (mM)

ti (mmol/g)

De (10-6cm2/s)

ks (10-3cm/s)

k~r't(10-3cm/s)*

XUS-40323/Phe

24

6.0

0.10

2.9

4.1

3.5-4.3

XUS-40323/Trp

10 24 24 45 60

5.0 3.0 5.0 5.5 7.1

0.32 0.16 0.24 0.18 0.17

1.5 2.7 2.7 4.2 5.5

2.3 3.5 3.7 5.2 6.3

2.3 2.8 3.4-4.2 3.4-4.2 4.9-6.4 6.7-8.9

XUS-43444A/Phe

24 40 60

3.3 3.5 4.9

0.21 0.19 0.21

1.9 2.8 3.7

1.6 1.9 2.2

3.3-4.2 4.7-6.3 6.6-8.9

XUS-43444B/Phe

10 24 24 24 24 45 60

5.6 1.8 3.5 4.8 7.3 7.5 8.6

0.35 0.12 0.22 0.28 0.38 0.34 0.33

1.4 2.1 1.7 1.9 1.8 2.4 3.1

3.7 3.4 3.4 3.5 3.6 4.0 4.4

2.6-3.2 3.3-4.2 3.3-4.2 3.3-4.2 3.3-4.2 4.8-6.3 6.(~8.8

XUS-43444B/Trp

10 24 24 24 24 45 60

3.3 0.75 0.90 3.8 6.4 4.8 5.9

0.85 0.20 0.23 0.74 1.01 0.63 0.56

0.60 1.6 1.5 1.2 0.60 1.9 3.0

1.4 3.0 3.0 2.9 1.9 3.6 5.0

2.1-2.7 3.24.1 3.2~1.1 3.24.1 3.24.1 4.7-6.2 6.4-8.6

*Lower value from Armenante and Kirwan (1989), upper value from Levins and Glastonbury (1972).

v

Fig. 13. Scanning electron micrographs of a fractured XUS-43444 lot A particle.

would be difficult to establish the existence of such a resistance a priori. T h e frequency response method, with its sensitivity to the n a t u r e of the mass transfer m e c h a n i s m (film or surface barrier vs intraparticle diffusion) appears to provide a n i n d e p e n d e n t way of assessing the i m p o r t a n c e of film barriers.

Comparison with batch uptake A c o m p a r i s o n of effective intraparticle diffusivities,

De, d e t e r m i n e d from the frequency response m e t h o d with diffusivities d e t e r m i n e d from isothermal b a t c h u p t a k e experiments is s h o w n in Fig. 14 for XUS40323. This g r a p h includes results for phenylalanine

1604

D.S. Grzegorczyk and G. Carta 1.0

A

Sp/Xp=0.35

E 0

%

Phe - batch (ref. a) Phe - batch in 20% methanol (ref. a) Phe - batch (this work Phe - FR Trp - batch (this work)

a®

Trp - FR

0.1 0.1

1.0

10.0

• Do ( 1 0 "s cm2/s)

Fig. 14. Effective diffusivities for phenylalanine and tryptophan in XUS-40323 plotted as a function of the free solution diffusivity. Ref. a data are from Grzegorczyk and Carta (1996b) adjusted for the correct particle size.

from Grzegorczyk and Carta (1996b), which were adjusted for the correct particle size (see Grzegorczyk and Carta, 1997), as well as results for phenylalanine and tryptophan from this work. The solute concentration steps in this case were between 5 and 30 raM. The data are plotted as a function of the solution diffusivity, Do, estimated from the Wilke and Chang equation (Reid et al., 1977) for each experimental condition. The data for the different solutes appear to be linearly related to the free diffusivity and independent of adsorbate loading and concentration step, indicating that pore diffusion is probably the controlling mechanism in this adsorbent. The ratio De~Do = ep/rp is equal to 0.35 for this adsorbent, which corresponds to a tortuosity factor of about 1.8. This value is consistent with our previous determination (Grzegorczyk and Carta, 1996b) when the correct particle diameter is used in the analysis, and with values reported by CasiUas et al. (1992) for the adsorption of phenylalanine in Amberlite XAD-4, an adsorbent with properties similar to XUS-40323. In the case of XUS-43444, the effective diffusivity was found to vary with the average adsorbate loading. The results from frequency response experiments are shown in Fig. 15 as DdDo vs the adsorbate loading ri x M, (g/g). The graph includes data at different concentrations and temperatures. It appears that in the case of XUS-43444, the ratio De/Do decreases with adsorbate loading. This behavior is consistent with our previous observations (Grzegorczyk and Carta, 1996b), where we determined that the apparent diffusivity determined from isotherm batch uptake experiments was not linearly related to the free diffusivity and varied with the initial concentration. The origin of this adsorbate loading dependence of the effective diffusivity is not exactly known. One possibility is that surface diffusion acts in parallel to pore diffusion as indicated by other authors (Komiyama and Smith, 1974; Cornel et al., 1986) for the adsorption of volatile organics on polymeric adsorbents. Recalling that the

effective diffusivity De is related to pore and surface diffusivities by De = epDp + pp(dne/dc)~Ds, assuming that the surface diffusivity remains constant, a plot of D e vs the isotherm slope should be linear (Costa et al., 1985; Yoshida et al., 1991). Such a plot is shown in Fig. 16 for the phenylalanine and tryptophan data at 24°C. Numerals in parenthesis represent the adsorbate loading in g/g. A straight line is not obtained with either adsorbate and it appears that similar diffusivities are obtained at similar adsorbate loadings for the two adsorbates even if the isotherm slopes are very different. This behavior is not consistent with surface diffusion unless the surface diffusivities of phenylalanine and tryptophan are assumed to be very different. In either case, however, the experimental D,values are not higher than what would be expected for pore diffusion alone, with the typical values of the tortuosity factor in the range 2-6 (Ruthven, 1984, p. 134). Thus, another possibility is that as the adsorbate loading is increased, diffusion in the small pores of this adsorbent becomes increasingly hindered. Hindered diffusion in liquid-filled pores is generally dependent on the ratio of molecule and pore size and is generally significant when this ratio is larger than 0.05-0.1 (Deen, 1987). The mean pore size of this adsorbent is about 28 A, while the size of phenylalanine and tryptophan molecules is of the order of 6-7 A, as estimated from the free solution diffusivity using the Stokes-Einstein equation. Therefore, it appears possible that diffusion would be significantly hindered. As the adsorbate loading is increased, diffusional hindrance should increase as the space available for diffusion within the pores is reduced by the adsorbed molecules. Such a mechanism would be analogous to those discussed by Hossain et al. (1986) and Hossain and Do (1987). In the case of XUS-40323, an effect of adsorbate loading on the effective diffusivity was not observed. This is probably because the pore size is much larger ( ~ 92.~). In addition, the adsorbate loading is generally smaller with this adsorbent.

Frequency response of liquid-phase adsorption 0.5

.....

O.4

£

- - - - , -

- - , . . . . . . . . o Tr0, 10 *C & Trp, 24 °C 0 Trp, 45 °C

XUS-40323 z~ v

0

• .---..,,.~...,~. e• • - A~-..~...,,..~ • ~ 9

0.3

a(~:

0.10"2

•

"

V

Trp, 60 °C

•

Phe, 10 °C

•

Phe, 24 °C

• Phe,45 °C • Phe,60 °C rn Uracil,24 °C 40%CH3OH X

0.0 . . . . . . . . . . . 0.00 0.05

1605

U

~

i

' 0.15

0.10

Adsorbete loading, n x

. . . . . . 0.20

' 0.25

M r (g/g)

Fig. 15. Effective diffusivities for phenylalanine and tryptophan obtained from frequency response experiments plotted as a function of the mean adsorbate loading for XUS-40323 and XUS-43444. Data for uracil in 40% methanol were obtained chromatographically.

3.0

2,5

+

Z~

Trp

•

Phe

• (0.02)

2,0

(o.o~), (o.o~)*' & (o.o3s)

1.5

A

A (0.041)

(o.o47)

v

Z~

a" 1.0 A

0.5 0 , 0

.

0

20

.

.

.

(o.ls)

(0,21)

.

.

.

40

'

60

"

"

+

.

80

.

.

.

.

.

100

.

120

Isotherm slope, pp(dn*/dc)r

Fig. 16. Effective diffusivities for phenylalanine and tryptophan obtained from frequency response experiments plotted as a function of the isotherm slope for XUS-43444.

To explore further the nature of the controlling intraparticle transport mechanism, we also determined the intraparticle diffusivity chromatographically. F o r this purpose, a sample of XUS-43A.A.A, lot B was slurry packed in a 0.9 cm ID, 2 0 c m long glass chromatography column (Spectrum Scientific). The column was operated with a 40/60 v/v methanol/water eluent at flow rates between 0.25 and 1.25 cm3/min with a ProSys C h r o m a t o g r a p h y Workstation (BioSepra Inc., Marlborough, MA). Uracil (Eastman Kodak, Rochester, NY) was used as a test

solute. This c o m p o u n d has a molecular weight (M, = 112) only slightly lower than phenylalanine, but is less hydrophobic. It was found to be only very slightly retained (chromatographic retention factor ~2.7) in the 40/60 v/v methonol/water eluent. Feed pulses with a volume of 200 m m 3 were applied to the column with uracil in a concentration of 0.5 mg/cm 3. F o r these conditions, the adsorbate loading and the isotherm slope are very close to zero. The van Deemter curves obtained by this procedure were analysed with the classical moment method to

1606

D. S. Grzegorczyk and G. Carta 1.0

0.8

0.6 LL

0.4

0.2

0.0

0

1000

2000

3000

4000

5000

6000

Time (s)

Fig. 17. Comparison of numerically predicted and experimental batch uptake curves for the adsorption of tryptophan on XUS-43444 lot B for different concentration steps. The solid lines represent the model prediction with the diffusivity data in Fig. 14 and kI = 0.003cm/s. The dashed lines correspond to numerical results with a fitted effective diffusivity assumed independent of adsorbate loading. T = 24°C.

determine the intraparticle diffusivity (Ruthven, 1984, pp. 245-247). The resulting diffusivity is shown in Fig. 15 as De~Do at ri = 0. In this calculation, we used the value Do = 5.7 x 10-6cm2/s that was estimated for uracil in the 40/60 v/v methonol/water mixture from the Wilke-Chang equation (Reid et al., 1977) using the viscosity data for methanol/water mixtures obtained by Colin et al. (1978). From Fig. 15, it is clear that the De/Do-value obtained chromatographically for uracil is in good agreement with the values for phenylalanine and tryptophan extrapolated to zero adsorbate loading. This seems to confirm that pore diffusion is the dominant intraparticle transport mechanism for these solutes. The values of De/Do obtained for these solutes at finite adsorbate loadings are in fact always lower than the limiting value at n=0.

In any event, whatever the mechanism, the effective diffusivity of the two adsorbates considered in this study in XUS-43444 appears to be a decreasing function of adsorbate loading expressed in Fig. 15. Thus, we consider the feasibility of predicting isothermal batch uptake curves with finite concentration steps using the isotherm data and the adsorbate loadingdependent De-values. For this prediction, the adsorbate loading dependence of De for tryptophan was expressed empirically by De/Do = 0 . 2 7 - 0 . 1 6 n T M which fits the data in Fig. 15. Equations (1)-(4) were then solved numerically as described earlier with the experimentally determined value of ks = 0.003 cm/s. Numerical results are shown in Fig. 17 for two different cases in comparison with experimental results for the adsorption of tryptophan on XUS-43444 lot B. The two cases correspond to low (run A) and a high (run B) final adsorbate loadings. In each case, the

model prediction is in good agreement with the experimental, confirming the validity of the diffusivities determined from the frequency response experiments. Lines corresponding to the numerical solution of the equations with a constant fitted diffusivity value are also shown in this graph. An approximate fit of the experimental results is possible in each case with a constant De. However, De-values that vary with the initial concentration and decrease with the average adsorbate loading are required in order to obtain this fit, indicating that the model assuming a constant efffective diffusivity would not be mechanistically correct for these conditions. In constrast, in the case of XUS-40323, an excellent agreement was found between numerically predicted and experimental batch uptake curves when a constant, adsorbate concentration-independent diffusivity was used (data not shown), reflecting the good agreement between Devalues obtained from frequency response and batch uptake experiments shown in Fig. 14.

CONCLUSIONS The frequency response method appears to provide some advantages over traditional batch-uptake methods to determine mass transfer rates in adsorbents. In particular, it is possible to determine individual resistance parameters (film and intrapartide diffusivity) and detect the presence and effect of surface barriers. In addition, the technique allows measurements for near-equilibrium conditions, so that concentration-dependent diffusivity values can be obtained. When applied to two polymeric adsorbents, the technique yields pore diffusivities that are consistent with those obtained from batch-uptake

1607

Frequency response of liquid-phase adsorption measurements for an adsorbent with large pores. However, diffusivity values that are dependent on the mean adsorbate concentration are found when the technique was applied to a small-pore adsorbent. Measurement of diffusivities as a function of adsorbate loading allowed a prediction of batch-uptake curves with a finite concentration step.

Zh zp ~o ~o*

heat conduction time (= r2/ote) tortuosity factor cycling frequency, rad/s dimensionless frequency (= com~i))

Superscripts e equilibrium value mean value

Acknowledgments

This research was supported in part by the Dow Chemical Company, Ajinomoto Co. Inc., and NIH grant No. T32GM08401. NOTATION

b bo Bi C CO Cp

CI Cp dp Do De Dp Ds kl m Mr n ns n~ er r

rp

R Sc Sh t T V

v~

adsorption parameter, cma/mmol pre-exponential factor, cm3/mmol Biot number i (= kyrp/De) fluid-phase concentration, mmol/cm 3 initial fluid-phase concentration, mmol/cm 3 concentration within pores, mmol/cm 3 fluid-phase heat capacity, cal/g K particle heat capacity, cal/g K particle diameter, cm fluid-phase diffusivity, cmZ/s effective pore diffusivity, cm2/s pore diffusivity, cmE/s adsorbed-phase diffusivity, cm2/s film mass transfer coefficient, cm/s slope of adsorption isotherm including pore fluid molecular weight adsorbed-phase concentration, mmol/g saturation capacity, mmol/g equilibrium adsorbed-phase concentration in batch, mmol/g Prandtl number (= v/~) radial coordinate, cm particle radius, cm ideal gas constant, cal/mol K Schmidt number (= v/Do) Sherwood number (= kydp/Do) time, s temperature, K solution volume, cm 3 adsorbent volume, cm 3

Greek letters thermal diffusivity, cm2/s effective thermal diffusivity, cm2/s ~e concentration amplitude, mmol/cm 3 Ac heat of adsorption, cal/mol AHo temperature amplitude, K AT adsorbent void fraction power input per unit mass, cm2/s 3 phase angle lag, deg ratio of isotherm derivatives partition ratio (= Vpm/V) A kinematic viscosity, cm2/s fluid density, g/cm 3 Pf particle density, g/cm 3 Pp diffusion time (= rZp/De) "CD

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