Dynamics of a semi-batch adsorber with constant molar supply rate: a method for studying adsorption rate of pure gases

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Chemical Engineerin(¢ Science, Vol. 50, No, 3, pp. 549 553, 1995 Elsevier Science Ltd Printed in Great Britain (X)09 2509/95 $950 ~ 0.00

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Dynamics of a semi-batch adsorber with constant molar supply rate: a method for studying adsorption rate of pure gases (Received 20 April 1994; accepted for publication 13 July 1994)

I. I N T R O D U C T I O N

Adsorption uptake into a porous particle is usually controlled by the rate of diffusion of adsorbate molecules from the bulk into the interior of the particle, where the local adsorption rate is much faster than the diffusion rate. Understanding this overall uptake rate is one of the key concerns to engineers in designing large-scale adsorber. To this end, a number of techniques are available in the literature, for example, the microbalance method (Gray and Do, 1991 l, the fixed bed breakthrough, the chromatography (Schneider and Smith, 1968), the zero-length column (Eic and Ruthven, 1988), the differential adsorption bed (Do et al., 1991), the N M R pulsed field gradient (Karger and Ruthyen, 1989), the frequency response (Yasuda, 1982), etc. These methods are very useful in their own way, but they all have their limitations and disadvantages. In this paper, we will present a method as an alternative to the other available methods to understand the adsorption dynamics. This method basically deals with a semi-batch adsorber, in which solid adsorbents are loaded and the adsorber is cleaned and put under vacuum initially. At time t = 0 +. a constant molar flow ofadsorbate is introduced into the adsorber. This molar supply rate should not be too slow or not too fast because if it is too slow the system is always at equilibrium and hence dynamics can not be learnt from such a small introduction of adsorbate. However, if the molar supply rate is too fast, the process is simply a filling process of the gaseous space of the reservoir, i.e. the pressure in the adsorber would increase linearly with the a m o u n t added according to the ideal gas law. When the supply rate is comparable to the adsorption rate into the particle, the pressure response will not follow a linear relationship with the a m o u n t added but rather will be dictated by the interplay between the rate of the a m o u n t added to the reservoir and the rate of adsorption into the particle. It is this interplay that will help us to study dynamics in a simple way. In what is to follow, we will present a complete analysis for the case of Langmuir kinetics, and then briefly present the results for the case of micropore diffusion control and the case of combined pore and surface diffusion control. The Langmuir kinetics case is relevant for very small particle and slow adsorption kinetics. The second case is relevant for zeolite powders or crystals, while the last case is for solids such as activated carbon.

Since the time constant of adsorption is not known a priori to the experiment, one will not know what supply rate should be used. But one should know the two limits of the pressure response curve versus amount loaded into the adsorber. In one limit, where the molar supply rate is too fast. the pressure response curve vs a m o u n t loaded will be controlled by the ideal gas law, i.e.

NRT

P = -

(1)

V

where N is the molar amount loaded and V is the volume of the reservoir. In this limit, the a m o u n t adsorbed will be practically zero, i.e, C¢ = O, where C, is the adsorbed concentration. Thus, a plot of the pressure vs N R T / V i n this limit of very high molar supply rate will give a straight line passing through the origin with a slope of unity. O n the other hand, in the same limit the plot of the a m o u n t adsorbed vs N R T/V will be the horizontal axis, meaning no adsorption is taking place within the particle. In the other limit, where the molar supply is very slow, the pressure response curve vs the a m o u n t loaded will follow the mass balance equation: N = N~ + Ns

(2)

where N is the molar amount added, Ng is the molar a m o u n t left in the gas phase and N~ is the molar a m o u n t adsorbed in the solid. The a m o u n t in the gas phase and the a m o u n t in the solid, in the case of very slow molar supply rate, are not independent but rather they depend on each other via an equilibrium relationship. This equilibrium relationship is C. = f ( P )

N~ with C~, = - - ; V~,

P -

N~ R T V

(3)

where C. is the adsorbed phase concentration, and P is the gas pressure. Thus, if the equilibrium relationship takes a linear form (Henry law), C. = KpP, we combine this with eqs (2) and (3) to obtain the following equations for the gas phase pressure and the adsorbed concentration vs the a m o u n t loaded:

P where

C~ K.

1 NRT [1 + K v ~ / v ] v

K = KpRT.

(4)

This means that when the molar rate is very slow, the plot of pressure vs NR 7"/'V will give a straight line passing through the origin with a slope of

2. THEORY In this theory section, we will present a theory to a semibatch adsorber operating under isothermal condition. The batch adsorber contains an a m o u n t of adsorbent, initially cleaned and is left under vacuum before the process of adsorbate introduction is initiated. At time t = 0 ÷, a constant molar supply of adsorbate molecule is introduced into the adsorber, possible with the use of a mass flow controller.

1

[1 + K v . / v ]

< 1.

(5)

The plot of Cu/K ~ vs N R T / V also gives the same straight line [see eq. (4)], and the slope of these two plots contains only the equilibrium parameter, namely K = KpR T. 549

Shorter Communications

550

Thus, we have established the limits of the pressure response curve and the limits of the adsorbed concentration response curve. Figure 1 shows these two limits graphically. The hatched area is the possible region where a pressure response curve at moderate molar supply rates will lie, and the cross-hatched area is the possible region where an adsorbed concentration response curve at moderate molar supply rate will be found. The preliminary result obtained so far suggests that if the pressure measured, when plotted vs NR T! V, gives a straight line passing through the origin with a slope of unity, then one can conclude that the supply rate is too fast for the study of adsorption dynamics. On the other hand, if the same plot gives a straight line passing through the origin with a slope given in eq. (5), one can conclude that the supply rate is too slow for adsorption dynamics studies. If the molar supply rate is comparable to the rate of adsorption dynamics, the pressure response as well as the adsorbed concentration response will be a function of N R T, and their patterns will depend on the interplay between the supply rate and the adsorption rate. These patterns are dictated by the mechanism of adsorption at the particle level. To show this, we will treat the case of Langmuir kinetics first, and then present the results for the case of micropore diffusion control, and the combined pore and surface diffusion control. 2.1. Langmuir kinetics The mass exchange between the two phases is controlled by a Langmuir kinetics at the adsorption sites, taking the following equation:

dCu_kaC_kaCu=ka(KC_Cu); dt

K=k~

ka

V dC~ /Q "dt =

t* = (1 + fl)/ka. Thus, to ensure that the pressure in the batch adsorber is such that the assumption of Henry law is valid, the choice of N~ V is

p* V

t*RT

For example, if a system has a Henry law valid at pressure below 20 Torr and the dynamic time scale of 20 s, the constraint on N/Vis 5.38 × 10- s mol/cc/s. If the batch adsorber has a volume of 500 cc, the suggested molar supply rate is then 36.2 cc STP/min. The long-term solution for the adsorbed concentration is lim Cu , ~ Kp

(I~t)RT/V (1 + fl)

1

NRT

(10)

(l + fl)2 Vka

It is interesting here that the long-time solution for the adsorbed phase concentration vs N R T / V has exactly the same slope as the plot of pressure [see eq. (9)], but the intercept is negative for the case of adsorbed concentration. This negative is due to the fact that it takes finite time (according to the mass exchange kinetics) for the adsorbed phase to pick up mass from the gaseous phase. The short-time behaviour of the gas phase pressure is lira P = (ffIt)RT.

(6)

where Cu is the sorbate concentration inside the solid, defined as moles per unit volume of solid (mol/cc of solid). Here, we have assumed that the partition between the two phases follows a linear relationship, valid usually at very low pressure. Carrying out the mass balance of the whole reservoir yields: dC V~-+

and from the intercept, the dynamic parameter, kd, can be obtained. The long-time solution given in eq. (9) will intersect the curve corresponding to very fast molar supply [eq. (1)] at the time t*, given by

t~

V

This simply states that the pressure response when plotted vs NR T/V will have a unity slope, indicating that in the initial period the process is simply the filling process of the gaseous space of the reservoir. We finally determine the time at which the pressure response curve approaches the linear asymptote. This is to find the range of validity of the asymptote. The last term of eq. (9) can be ignored when

(7)

(1 +fl) kat>4. Rearranging this, we get

where N is the molar supply rate of adsorbate (mol/s), and C is the sorbate concentration in the vessel (mol/cc), related to pressure according to the ideal gas law, C = P/R T. The I.C. of eqs (6) and (7) are: t = 0;

C = C~ = O.

(8)

The model equations are linear, and are solved by the Laplace transform method. The solution for the gas phase pressure is then given by

P _ (IVt)RT/V - - + (1 +fl) xexp[-(1

fl (NRT) (1 + fl)2 V k a +fl)kat],

fl (IVRT) (1 + fl)2 V k a

wherefl=

GK V

(9)

This pressure response at long-time approaches to a straight line asymptote given by the first two terms in the RHS ofeq. (9). We note that the first term of the asymptote is simply the limit when the molar supply rate is extremely slow [see eq. (4)], i.e. under the condition of local equilibrium between the two phases. The asymptote carries two important information: equilibria (the first term) and dynamics (the second term). Thus, by plotting of the gas phase pressure versus NRT/V, we get

slope

1 1 + fl;

fl intercept

(NR T)

(1 + fl)a Vka

From the slope, we can obtain the slope of the isotherm, K,

- - T v

_ _ 4 IVR__T_ 4(1 + fl) Intercept. (l + ~) Vk~

Thus, the linear asymptote is valid when the amount loaded (NR T/V) is greater than 4 times the intercept multiplied by (i + fl)/fl. This means that when extrapolating the linear part of the experimental data to the pressure axis to obtain the intercept, we need to make sure that amount loaded at which the linear part of the curve starts to happens should satisfy the above relation. A typical plgt of the pressure response vs time is shown in Fig. 2 for N = 0.005 tool/rain, V = 1000cc, f l = l, k a = I m i n - l , and T = 2 9 8 K . W e n o t e that the pressure response curve vs time at very short-time asymptotes to the curve corresponding to the fast filling, and as time progresses the pressure approaches a linear line parallel to the equilibrium line. This linear asymptote line (long time solution) has an intercept of

NRT -- 0.03 atm. (I + fl) Vk~ To confirm experimentally whether the straight line of the long-time behaviour is properly drawn in the range of validity, we plot the pressure response versus NR T/V as shown in Fig. 3. The linear part of the curve starts at N R T / V = 0.2atm, which is about 8 times the intercept value of 0.03 atm. Thus, this is a useful check to make sure that the linear part of the pressure response is properly drawn.

q

f-

Possible r

concentra~

NRT/V Fig. 1. Typical diagram of the pressure response and the adsorbed concentration vs NR T/V, where N is the a m o u n t loaded. 0.2

0.16

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Longtimesolution\

~

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E 0.12 Actualpressureresponse

~=

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0.2

0.4

0.6

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12

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Time (rain) Fig. 2. Plot of the pressure response vs time (rain) for the case of Langmuir kinetics.

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NRT/V (atm) Fig. 3. Plot of the pressure response vs hiRT/V (atm) for the case of Langmuir kinetics.

0.2

Shorter Communications

552

We have completed the analysis of the case where the Langmuir kinetics controls the mass exchange between two phases. We now turn to the case where the mass exchange kinetics is controlled by the micropore diffusion. 2.2. Micropore diffusion The mass balance equation describing the distribution of mass within the zeolite crystal is

OCu 1 ~ f sOCu~ O~f-= D u - ~ r ~ r --~-r)

r=O;

--=0C" 0 and r = R ; C.=KC. Or The mass balance over the whole vessel is

(12)

(s + 1) I R = ~ Jo r~C' dr.

(1 + fl)

fl

(I~RT) R 2

(1 + fl)2

- 2fl [r"s~ ~-)

V

~ (R~I ~

3D,

exp(-22Dut/R2) ~ ;t2,(f12+fl+22.)

\D,/].:

(l + fl) - 6fl

fl

+

2 + fl) Intercept

fl

(19)

where the intercept is given in eq. (17). 2.3. Combined pore and surface diffusion in activated carbon For solids such as activated carbon where the mass transfer mechanism inside the particle is controlled by the combination of the pore and surface diffusion as activated carbon has surface area necessary for surface diffusion flux to be comparable to the pore diffusion flux. The mass balance equation for describing the adsorbate distribution in the particle is

OCu

1 8 /

OCu\ 1 O / sOC,\ ~-r)

+(1-e,,)D,p~r~r

(20)

where s is the particle shape factor, C is the concentration in the gas space inside the particle, C, is the sorbate concentration in the adsorbed phase, defined as moles per unit volume of solid (particle volume excluding macropore volume). The two diffusion coefficients in eq. (20) are assumed constant. We assume that within the activated carbon particle, there is a linear partition between the gas phase and the adsorbed phase inside the particle, i.e.

C~, = K C

(IVR. T) R 2 V

(21)

(r0C 3 z=m'""rnOG\ Or)'

15Ou

n=~l 2"2(9fl2 +

~

9fl + 22)

for spherical zeolite crystals. The eigenvalues 2. are obtained from the following transcendental equations: SLAB:

9 (1

2,~

Combining eqs (20) and (21), we get the following diffusion equation:

(1 + tip

~

>

(14)

for slab zeolite crystals, and

p=(NtRoT)/V

v

(13)

These mass balance equations subject to the initial conditions [eq. (8)] can be again solved by the method of Laplace transforms, and the solution for the gas phase pressure in the reservoir is

P = (blt)RT/V +

- -

?~C

V d (dCt~) = N ;

dt

NRg T

(11)

where C. is the sorbate concentration inside the crystal, defined as moles per unit volume of crystal, and s = 0 and 2 for slab and sphere. Here, we assume isothermal system as well as constant micropore diffusivity. Next, we assume that there is no mass transfer resistance at the exterior surface of the crystal and the local isotherm between the gas and crystal phases at the surface is linear, we have the boundary conditions:

v -d- C+

where 21 is the first root of the transcendental equation (16). This means that once the diffusivity is obtained from the intercept of the linear part of the plot of pressure vs N R T~ V, the criterion (18) is then checked to make sure that the region of linearity of the plot is satisfied by the experimental data. Since the diffusivity is not known a priori to the experiment, we rearrange eq. (18) to obtain the following useful criterion, parallel to the one we obtained earlier for the case of Langmuir kinetics:

titan 2 + 2 = 0;

SPHERE: 2 cot 2 - 1 = 22/3fl.

(16)

Similar to the Langmuir kinetic case, the plot of long time solution [the first two terms in eqs (14) and (15)] of the pressure versus the amount loaded will give a straight line with a slope and an intercept: 1

Slope = ~ ; 1 +

fl (]VR T) 1 Intercept = (1 + fl)~2 V(DjR 2~)a

(17)

where a is 3 for slab and 15 for sphere. The slope in this case is the same as that in the case of Langmuir kinetics. This is not surprising as this group of parameter relates to equilibrium, hence, it should be the same irrespective to the kinetic mechanism. The intercept will give information about the dynamic parameter D,. The third term in eqs (14) and (15) contributes only in the initial transient period, and is smaller than the other two terms when

.2 D~t

,t 1 ~ -

> 3

(18)

where

Dapp:

G.D v + (1 -- em)KD~

(22)

gin+(1 --cm)K

If we further assume that there is no mass transfer resistance at the exterior surface of the particle, the boundary conditions to the mass balance eq. (22) are r=0;

0C --=0 Or

and r = R ;

C = Cb

(23)

where Cb is the adsorbate concentration in the vessel. The mass balance over the whole reservoir gives the following equation:

vdCb (m,~[ed dt

d7 +\~/L~+ll-~]=N

(241

where , are the volumetric mean concentration in the particle, given as (s + 1) ~R = R(~+I)Jo r~Cudr and

(s + 1) ~R (C> = R~+1~ Jo r~Cdr"

(25)

Solving the mass balance equation using the method of Laplace

Shorter Communications transform, we obtain the following solution:

p=(Nt)RT/V+ (1 + f l ' ) - 2(1 + ~, ×

fl'

(I~IRT)

R2

(l+fl')2

V

(s+l)(s+3)D,p v

s)/7'L\~Y\~-~:-.)J.r

exp [ - 2 ~z (D, pptlR2)] ~2 [(1 + s)2[J2 + (1 + s)2/7 + ,C. , z. ~]

(26)

where s is the particle shape factor with s = 0 for slab, s = 1 for cylinder and s = 2 for sphere. The eigenvalues 2 are obtained from the following transcendental equation: SLAB:/7' tan (2) + ). = 0; CYLINDER: 2]7' J1 (2) + .~Jo(2) = 0; SPHERE: 2 cot (2) - 1 = 22/3/7 ' where

/7' = V.pv/ V = (rn.lp.)[e + (1 -- e)K]lV. The long-time behaviour of the pressure response is simply the first two terms in the RHS of eq. (26). 3. DISCUSSION We have presented in the paper a new procedure of determining the dynamic time scale of a system. This method entails a simple batch adsorber and a means of introducing constant mass flow rate, which is possible with a mass flow controller. The system pressure is monitored by using a pressure transducer, such as the Baratron capacitance pressure transducer. The monitored pressure exhibits a concave shape in the early stage of operation. The initial slope of such curve follows the pressure curve if there is no adsorbent inside the adsorber (gas filling only). At the time which is comparable to the time scale for mass transfer into the particle, the pressure response will asymptote to a straight fine, of which the slope is the same as the slope of the equilibrium line. The equilibrium line is the line whereby the equilibrium is always obeyed between the gas and solid phases. The equilibrium line is the line whereby the equilibrium is always obeyed between the gas and solid phases. If the linear part of the pressure response curve is extrapolated to the pressure axis, the non-zero intercept can be obtained, Since this intercept contains the information of the dynamic time scale, and the dynamics constants can be readily determined. This procedure presented in this paper is useful when the following assumptions are observed: (1) The system is isothermal, (2) the adsorption isotherm is linear, and (3) the dynamics constants are invariant. The first assumption can be satisfied experimentally with steady flow into the adsorber and the adsorption process proceeds gradually rather than abruptly like in the case of single particle analysis where the bulk concentration is raised instantaneously from 0 to some finite value. Moreover, with the system arrangement, the particles can be mounted in a stirrer and the stirrer can be stirred with

553

a means such as the magnetic stirrer, hence facilitating the heat transfer from the particle. Except for systems where the Henry law is not possible (this is rare), the second assumption can be easily satisfied in the experiment because the experiment is started from vacuum and the size of the adsorber can be chosen such that the pressure at the end of the experiment will be lower than the threshold pressure below which the Henry law applies. The third assumption is closely linked to the second. This means that if the Henry law is obeyed, the dynamics constant such as the diffusivity is usually invariant. Thus, in summary, one can design the experiment such that the three assumptions can be readily satisfied, and the method proposed provides a very quick means of determining the dynamic parameters. 4. CONCLUSION We have presented in this paper a simple and quick procedure to determine parameter for dynamic processes. Four cases are considered: Langmuir kinetics, slab zeolite crystal, spherical zeolite crystal and activated carbon of general shape. Simulations were carried out using parameters normally encountered in practice to check for the validity of the procedure, and it is concluded that the procedure is very versatile and we propose it as an addition to many methods currently available for adsorption studies. Moreover, because the procedure is very quick, it is useful in circumstances such as adsorbent screening where there are a lot of solids to be tested.

Acknowledgement-This work is supported by a grant from Australian Research Council. D.D. DO

Department of Chemical Engineering University of Queensland Queensland 4072 Australia REFERENCES

Do, D. D., Hu X. and Mayfield, P., 1991, Multicomponent adsorption of ethane, n-butane and n-pentane in activated carbon. Gas Sep. Pur!f~ 5, 35-48. Eic, M. and Ruthven, D., 1988, A new experimental technique for measurement of intracrystalline diffusivity. Zeolites 8, 40-45. Gray, P. and Do, D. D., 1991, Dynamics of carbon dioxide sorption on activated carbon particles. A.I.Ch.E.J. 37, 1027-1034. Karger, J. and Ruthven, D., 1989, On the comparison between macroscopic and NMR measurements of intracrystalline diffusion in zeolites. Zeolites 9, 267-281. Schneider, P. and Smith, J., 1968, Adsorption rate constants from chromatography. A.I.Ch.E.J. 14, 762-771. Yasuda, Y., 1982, Determination of vapor diffusion coefficients in zeolite by the frequency response method. J. Phys. Chem. 86, 1913 1917.