Retention of toxic gases by modified carbon in fixed beds

@X3-6223/90 13X0+ .Ml copyright 0 1990Pergamon Pressplc

CarbonVol. 28. No. 6. pp. 839-848. 1990 Printed in GreatBritain.

RETENTION OF TOXIC GASES BY MODIFIED CARBON IN FIXED BEDS Department

SIDDHARTH G. CHATTERJEE and CHI TIEN of Chemical Engineering and Materials Science, Syracuse University, Syracuse, NY 13244, U.S.A.

(Received 6 November 1989; accepted in revised form 16 February 1990) Abstract-A new model for the retention of toxic gas by modifed carbon in fixed-bed operations is presented. The retention of the toxic gas is assumed to be effected through two independent pathways:

physical adsorption and chemical reaction between the toxic gas and carbon impregnants with the reaction taking place instantaneously. Reasonable agreement was obtained between experiments and predictions on breakthrough time and effluent concentration histories based on the model. Key Words-Activated

carbon, toxic gases, adsorption, gas protection.

1. INTRODUCTION

It is commonly known that regular activated carbon is not effective in removing toxic chemical agents such as phosgene, hydrogen cyanide, and arsine from ambient air. Carbon’s performance in retaining these agents, however, can be significantly improved by impregnating carbon with certain metal compounds (for examle, copper-silver-chromium as an ASC whetlerite). With the presence of impregnants, retention of toxic gas by carbon pellet is enhanced because of the reaction between toxic gas and the impregnants. The purpose of the present work is the formulation of a model for the retention of toxic gas by modified carbon in fixed-bed operation. The model is intended as an idealized approximation for the operation of gas mask canisters. Although there exists a large body of information on the modelling of adsorptive processes in the literature, consideration of adsorption processes in which physical adsorption and chemical reaction occur simultaneously has been noticeably lacking except for the recent work of Graceffo et al.[l]. These investigators assumed the reaction between the toxic gas and carbon impregnants to be first order with respect to both toxic-agent concentration in the pore space of the pellets and impregnant concentration of the pellet. Uptake of the toxic gas was by a combination of the reaction and physical adsorption. Their model was found to fit well with data on the retention of cyanogen chloride by ASC whetlerite. The model of Graceffo et al. is based on the assumption that the reaction between the toxic gas and impregnants is first order with respect to both the impregnant and toxic gas (or second order overall). As the model is based on a particular kinetic expression, its use must be considered restrictive. Equally important, there does not exist any method or procedure by which the rate constant can be independently predicted or determined. 839

In the present study, we present a new model for toxic gas retention by impregnanted carbon that does not have the above-mentioned deficiencies. This is accomplished by assuming that the reaction between the toxic gas and impregnants proceeds instantaneously. Using this assumption, the contribution to toxic gas retention by impregnants can be determined from stoichiometric considerations alone and does not require any kinetic data. While the use of this assumption represents an additional restriction, its validity in the case of modified carbon is reasonably assured since any practically useful modified carbon must have the feature of fast reaction between its impregnants and the toxic gases it removes. The scope of the work carried out under the present study may be summarized as follows: First, a model was formulated for the removal of toxic gases by modified carbon in fixed-bed operation under the condition that the reaction between the toxic gas and impregnants is instantaneous. This aspect of the work represents a generalization of the single-pellet analysis of toxic gas retention reported recently by Chatterjee and Tien[2]. Second, a numerical method for the solution of the governing equations was developed. Finally, predictions based on the present model were compared with the earlier model of Graceffo et al .[ l] as well as experimental data for model validation.

2. GOVERNING EQUATIONS

The problem considered in this work is the removal of toxic gas from gas streams by modified carbon pellets in fixed-bed operation. Consider the flow of a gas stream with toxic species through a bed packed with pellets (spherical in shape) of modified carbon. Under the usual assumptions that the flow is isothermal and noncompressible, the superficial gas velocity is constant, and there is negligible axial dispersion effect, the macroscopic conservation

S. G. CHATI’ERIEEand C. TIEN

840

equation of the toxic species is given as u,!$+(l-r)[p,$+ii]

=0

(1.a)

with

c=o,

OIZ’L,

~=o,

c = c,S”9

e=o e>o

(1-b) (14

where the independent variables are the axial distance, z, and the corrected time 8 is defined as t (ez/uJ. The dependent variables are c, the toxic gas concentration of the gas stream, Zj, the average amount of toxic gas retained within a pellet per unit mass of pellet, and x, the average rate of reaction between the toxic gas and carbon impregnant per unit volume of the pellet. u,, pP, and l are the superficial gas velocity, pellet density, and bed porosity, respectively. Finally, L is the bed length while c,, is the influent concentration of the toxic species. The toxic gas may be retained within a @let in two forms: that present within the pore spaces of the pellet and that physically adsorbed by the pellet. By definition, Ij can be written as

q=3

a$,

I” 0

r2(qpP + e,c’) dr

(2)

where q is the local adsorbed phase concentration of the toxic gas and c’ the local gas phase concentration of the toxic gas in the pore space. lP is the porosity of the pellet. Generally speaking, qpP 9 E&‘, and hence eqn (2) can be simplified to

haustion of the impregnants. For a pellet exposed to a gas stream containing toxic gas, the pellet may be considered to be composed of two parts: an interior core, 0 s r 5 r, throughout which the impregnant concentration remains the same as its initial value (i.e., n = n,) and an exterior layer, r, I r I up throughout which the impregnants are exhausted (i.e. n = 0). The radius of the interior core, r,, is a function of time. The decrease in r, reflects directly the reaction between toxic gas and impregnant. A schematic diagram demonstrating the dist~bution of the toxic gas and impregnant within a pellet under the assumption of an instantaneous reaction is shown in Fig, 1. Consider a carbon pellet with a uniform impregnant concentration initially. As it begins to be exposed to toxic gas, the uptake of the toxic gas is due to the combined action of physical adsorption and chemical reaction. The contribution from chemical reaction, 75, can be written as

where v is the stoichiometric ratio between the impregnants and toxic gas in their reaction. This contribution of chemical reaction continues until the impregnants are totally exhausted. To obtain expressions for q and r,, the moving boundary intraparticie diffusion equation needs to be solved; it is given as

a4 D, a ‘dc’+p-=riar pa0 pae in

r,(u, z) < r < a,.

(6)

The initial boundary conditions are In an analogous way, the average reaction rate, g, can be written as

c’ = 0,

q = 0

for

0 Fc r 22 up, 8 = 0

(7)

For a pellet located at an arbitrary position in the bed, both q and R are local, time-dependent variables. In other words, q and R are functions oft, 8 as well as the radial distance from the center of the pellet. In order to obtain suitable expressions for q and R, the relationship between q and c’ and the mechanism of toxic gas uptake by carbon pellets must be known. The assumptions used by Graceffo et al. are also adopted here, namely, the toxic gas diffuses into the peIlet and becomes adsorbed as well as reacts with the impregnant present within the carbon. 2.1 Instantaneous reaction Since the reaction is considered instantaneous, physical adsorption takes place only after the ex-

Fig. 1. Schematic diagram representing the uptake of toxic gas by a modified carbon pellet for the case of an instantaneous reaction.

in fixed beds

Retention of toxic gases by modified carbon

c’ = 0,

.

841

..(- 8) =Dp(?$ at

r=Q,

8>0

(8)

at

r=a,,

8 > 0

(9)

(14.d)

e=o

rr=O,

(10) where c,, is the influent toxic gas concentration.

where D, is the pore diffusivity of the toxic gas and k, the external mass transfer coefficient. The intraparticle diffusion applies only to that part of the pellet where the impregnant is exhausted. Thus, the toxic gas uptake process may be divided into two stages: The first begins with 8 = 0 until the time when r, = 0. The second stage starts when there is no longer any impregnant present within the pellet. As stated before, during the first stage, both physical adsorption and chemical reaction contribute to the uptake process. On the other hand, only physical adsorption is operative during the second stage. A single pellet analysis of toxic gas uptake has recently been considered by Chatterjee and Tien[2]. These investigators solved the moving-boundary intraparticle diffusion equation using the approximate method developed by Goodman(3). Within the pellet, local equilibrium between the gas and adsorbed phases was assumed. Furthermore, the equilibrium relationship was assumed to be linear because in practical applications c’ is usually of the order of ppm), or q = Kc’

2 +x =?

[c - cfI,J

The 6 represents the thickness of the exterior a pellet in which the impregnants are ex6 is a function of time (or T) and is found solution of the following equation:

as z=-1-S -+-RI

2

N

where y = c/c,, N = 2

0

(p,K

(12)

P

The value of c’],., can be found directly from the ’ solution of eqns (6j through (10). The results of Chatterjee and Tien may be summarized as (1) First stage, 0 5 T 5 T, For cases where N, = k, a,lD,, f 1

lp)

(17)

+ C?C(- C&h + c,$

(18.a) (18.b)

(19) $ = +dci 4Ns, Y ” = N,,

- 4C?C>

(20)

1

2 -++ NB, - 1

(21.a)

? C? =-+6 IV”, -1 1

I

2

(21.b)

(2l.c) I

1

where

6 = 1 - (rJa,)

+

A = ; [2c,c, - 2c,c, - c,cJ

+(S. X, T) =

x = 1 - (r/a,)

(16)

(11)

The results of Chatterjee and Tien provide a direct relationship between the total uptake rate, pP (@/ de) + R and the gas phase concentration, c, to which the pellet is exposed. Notice that the total uptake rate is directly related to the external mass transfer process, or pp

quantity layer of hausted. from the

(14.a)

Ch = 2

(14.b)

2 -+6 NB, - 1

I[

(21 .d)

1 4

-++ NB, - 1

(21.e)

(21.f) I

S. G. CHATTERTEEand

C.

TIEN

For NBi # 1, (21 *g)

I 2N, c’=N,_l%

i?y 1 - 6 4NBiY +N, I- N I

(,$ +wa, I1 1 I/

(21.h)

+

-

In Equation (20), one should choose the positive or negative sign depending upon whether NBi > 1 or Nr,>< 1. For cases where NBi = 1 Q0

tNBi

l)pj -

-

-1)

4llY(S,

=

(22)

(27.a)

7,) p, - t t II

y

Najy(s, 7,)

+ I

&(s. x, 7) = (6 - x)y + b,

$

I

0%~- 1)

(27.b)

where 6, = 6,(s) and Q,,,‘sare the roots of the following equation.

(23.a)

tanB,

--h+

-

-

Ns,

_

1v

m = 1, 2, 3, . . .

(27.~)

(23.b) (23.~)

For Na, = 1, f = L

cow-

w%lI - cosPm Pm

(24)

Brn = (;?m - 1) f,

m = 1,2,3,

.. .

(29)

(25.a) K, = 26 - $ [261-t, - 4SZy].

(25.b)

In principle, the first stage ends when 6 = 1, or T, should be the time when 6 = 1. As pointed out by Chatterjee and Tien, Goodman’s method becomes less accurate as the value of 6 is close to unity. T, was, therefore, chosen to be the time when 6 = S, = 0.9. This is tantamount to ignoring the contribution in toxic gas uptake due to 0.1% of the impregnants initially present in the pellet. The error introduced can be expected to be insignificant, (2) Second stage T > T, At a specific location s( = z/L) in the bed, the toxic gas profile within a pellet during the second stage is Y’(S,

x. 7) = (d/c,,) = y(s, 7)

-I-

2 i

B1 + (hi - 1)’

’ y(s.

1): + (NB, - 1)

p.)e-bilT #) dcL - y(s, T)

- $

41 =

fW

[sin pm - sin((1 - 6,)f$J] I)(

(31.a)

c=[(l - wm.1 + 2 I% Pm

x sin((1 - S&3,] + $

m

x fcos Bnr- cos{(l - M PiJl.

(31.b)

2.2 Second-order reaction For convenience of subsequent discussions, the governing equations of the previous model of Graceffo et d. are presented below. The intrapehet diffusion equation with simultaneous chemical reaction (second order) is given as

m-l

xPb + (Net -

where T, is the time when the first stage ends when 6 = 6,. II, p, and 4, are given as

II 0 s r 5 a,

(32)

843

Retention of toxic gases by modified carbon in fixed beds Table 1. Basic parameters used to obtain the results shown in Figs. Za-5 9.29 x lo-’ cmz/s 6.5 cm/s 2.923 cm/s 3.36 X 10-Rgmol/cm’ 2.79 x lo-’ gmol/cm’ 0.5 0.6 0.075 cm 0.6 g/cm’ 10.0 cm 1.0 1.857 x lO’cm’/g

1;(

ICI

,n B

%

an

-!

ae=

v

kc’n.

(33)

i5 u I( E f

The initial

C1

=

0,

and boundary

q = 0,

ad ar

c w

are

n = n,,

for -=0

conditions

at

0 5 r 5 up, r=O,

8 = 0

O>O

(34) (35)

c Time, t (mid

(9) at the pellet surface. The reaction rate by k while n represents the local impregnant concentration in the pellet. The relation between q and c’ is given by eqn (11). As a matter of convenience, the macroscopic conservation equation (i.e., eqn (la)) may be written in terms of dimensionless variables to give

and eqn

constant is denoted

Fig. 2(a). Effluent concentration ratio versus time-low

concentration range.

where s = ZIL,

7 = D,8/(a;(p,K

+ E,)]

(37.a) (37.b)

z

+ 3(1 - 4$%[Y

-

y:.,1

=

0

(36)

(37.c)

Time,? (min) Fig. 2(b). Effluent concentration ratio versus time.

S. G. CHATIERJEEand C. TIEN

844

The corresponding are

initial and boundary conditions

y=o,

04SI,

y = 1,

s= 0,

r=o, 7 > 0.

(3&a) (38.b)

Equations (36) together with its initial and boundary condition of eqn (38.a) and (3&b) and the results of the single pellet analysis (instantaneous or the second-order reaction cases) are the governing equations of the problem under consideration, the solution of which yields the effluent concentration history (or y at s = l.O), which is indicative of the performance of the carbon column in retaining toxic gas.

3. NUMERICAL

RESULTS

To obtain the effluent history and other related results, eqn (36) was solved numerically using an implicit finite difference method similar to that used by Graceffo et al. previously. Briefly speaking, the spatial derivative aylas was approximated by a backward difference approximation. The total toxic gas uptake rate 0, - y’]l,_,] was determined by the appropriate solutions to the intrapellet diffusion equation. In other words, for pellets in the first stage, y’l,=, was found from eqn (13) (or eqn (22) if Ns, = 1.0). On the other hand, eqn (26) was used for pellets in the second stage. (A procedure for evaluating the integral of eqn (26) was developed and is given in the Appendix.) For pellets in the first stage, it was necessary to consider the shrinkage of the core (namely, the increase in 6) as part of the calculation. A large number of sample calculations were made and some of these results are presented below. In addition, similar calculations based on the model of Graceffo et al. were made since the results based on the instantaneous reaction rate assumption in this work can be considered as the limiting case of the Graceffo et al. model with k + m. The conditions used for the sample calculations shown in Figs. 2a5 are given in Table 1 and are typical of those used in testing the breakthrough of toxic agents of gas mask canisters. The results shown in Figs. 2a and 2b are effluent concentration histories (or c,Jc,, versus time) at short times and long times. In both Figs. 2a and 2b, effluent concentration histories based on the present model (k = 30) and the previous model of Graceffo et al. and corresponding to different values of k are included. It is obvious that the results obtained from the present work indeed correspond to a limiting situation of the work of Graceffo et al., namely, the results as k -+ 33. Consequently, the effluent concentration breakthrough predicted by the present model occurs at a later time than that based on the earlier work of Graceffo et al. On the other hand, corresponding to a fixed isotherm expression (i.e., K), the same initial impregnant concentration n,, and

the stoichiometric ratio, u, the ultimate retention capacity of carbon pellets is the same and independent of the rate of the reaction. A later breakthrough also implies an earlier saturation; consequently the various effluent concentration history curves intersect with each other (except that with k = 0) as shown in Fig. 2b. The fixed-bed retention performance can also be seen from the gas phase concentration profiles throughout the fixed-bed. In Fig. 3, the toxic gas concentration profiles, C/q, versus z corresponding to the conditions of Table 1 at t = 80 min predicted according to the results of the study as well as those of the earlier work are shown. As the values of the rate constant increases, the steepness of the profile increases. On the other hand, the concentration profile does not display the step-function like behavior that is expected under the condition of local equilibrium because of the mass transfer effect considered in this work. The extent of the exhaustion of the impregnants and that of the saturation of modified carbon pellet are shown in Fig. 4. The results shown in Fig. 4 correspond to a carbon pellet situated at 4 cm into the bed after 80 minutes of operation. The results are those of q/q0 versus rla, and n/n,, versus rla, where q. is the value of q in equilibrium with c,,. Both the present study and the earlier model of Graceffo et al. were used to perform these calculations. The importance of the impregnants and that of the reaction rate in retarding the saturation of carbon pellets can be seen by comparing the q/q0 versus rla, curves predicted according to the results of this study with those based on the work of Graceffo et al.

Previously of a modified composed of (t,) when all

it was pointed out that the useful life carbon pellet may be considered to be two stages. In Fig. 5 the transition time, pellets in a fixed-bed are exhausted of

a

Axial Distance. z(m) Fig. 3. Concentration

profiles of toxic gas in bed.

845

Retention of toxic gases by modified carbon in fixed beds

impregnants, is shown as a function of bed height. Under the conditions listed in Table 1, this relationship is almost linear. 4. COMPARISON OF MODEL PREDICTIONS WITH EXPERIMENTS

Milius et af.[4] have examined experimentally modified carbon’s effectiveness in retaining toxic gases. In particular, they have measured the breakthrough time (defined as the time when the effluent concentration reaches a certain value) of cyanogen chloride (CNCI) on the modified carbon, ASC whetlerite, as a function of bed depth. The experimental conditions of the above investigators along with other estimated parameters are shown in Table 2. The bulk diffusivity D, of cyanogen chloride in air was estimated from the correlation of Chen and Othmer[S] while the mass transfer coefficient (k,) was calculated from the empirical expression of Sherwood et al. [6]. The pore diffusivity D, was estimated from the expression D, = Dq,lT, (using T, = 2) where T, is the tortuosity factor varying from 2 to 6. At low concentrations (i.e., c’ = 0 to 6.51 x 10eR gmol/cm3), the physical adsorption isotherm expression of CNCl on ASC whetlerite was approximated by q (in gmol/gram

carbon)

= 1.54 X lOQ’(in gmol/cm3).

This expression was based on the experimental data of Reucroft and Chiou[7]. Table 3 shows comparisons between the experimental observations of Milius et al. and predictions assuming finite reaction rate (model of Graceffo et al.) and instantaneous reaction rate (present model) between cyanogen chloride and the impregnants in ASC whetlerite. The breakthrough time was defined

1.0

IO

00

4 8 6 Axial Distance. z km)

2

Fig. 5. Time required for the exhaustion of pellets in the case of an instantaneous reaction as a function of bed depth.

by Milius et al. as the time when the effluent concentration reaches 0.5% of the influent value, or c,J C,” = 0.005. For a fixed value of v, an increase in either the reaction rate constant, k, or bed length, L, delays the breakthrough of the toxic gas. For large values of k(rlW), both models give a higher estimate of the breakthrough time for smaller bed heights but a lower estimate for larger bed heights. For a given reaction rate constant, breakthrough occurs at a later time as u increases. This behavior, of course, is expected because an increase in v implies

Table 2. Conditions of the experiments of Milius et a/.[41 and the estimated mass transfer and chemical kinetics parameters* 2.79 x lo-* cm’/s 6.5 cm/s 2.923 cm/s 6.51 x 10eXgmol/cm? 0.5 0.6 0.075 cm 0.6 g/cm’ 1.0 25°C

B

P 0.6

ae” .o

c,. E lp aIt PP Wt Temperature %

t

IO

Adsorption isotherm at 25°C 4 = 1.54 x 1O’c’

f

0.2% Q

0.2

0.0 0.0

0.2 Dimensicdess

0.4 Radii

0.0 I.0

06 Dirtonce.

r/r+

Fig. 4. Concentration profiles of impregnant and toxic gas (in adsorbed phase) in pellet.

4 is in gmol/gram carbon c’ is in gmollcm’ *The relative humidity of the influent air and CNCI mixture was 80%. The ASC whetlerite was equilibrated with air at 80% relative humidity and 25°C before being loaded into the adsorption chamber. $Value used by Graceffo ef a[.[ l]

846

S. G. CHKII-ERJEEand C. TIEN Table 3. Comparison of model predictions and experimental data of Milius et a/.[41 Breakthrough time’ (min) Bed Depth (cm) 1.2 1.7 2.3 2.9 3.5 4.7

k = lo-’

Calculated k = 1P

v=l 0.6 4.0 26.8 49.2 71.4 115.8

v=l 9.1 27.8 49.9 72.1 94.3 138.7

k== v=l 15.4 33.5 55.6 77.8 100.0 144.4

Experimental v=2 28.37 59.44 %.76 134.8 171.57 245.98

5 24 49 88 107 175

Note: k is in cm’l(gmo1.s) *Defined as the time when c,,,/c,. in 0.005

0

0

Experimental Ootoof Miliu otd(l9741

-Predictions

L= 2.3cm

Time (min) Fig.

6. Effluent concentration history for chloride on ASC whetlerite carbon.

cyanogen

Retention of toxic gases by modified carbon in fixed beds an increase

in the retentive capacity of the adsorbents. With Y = 2, the predicted breakthrough time was greater than experiment. The difference, however, decreases as bed height increases. A more stringent and complete test of the models is the comparison of the breakthrough curve in its entirety. Figure 6 gives the comparisons between the predicted effluent concentration histories and experiments. The predictions include results based on the present model (with u = 1 and 2) and those based on the model of Graceffo et al. (with v = 1, k = lo’, 104, and 106). All the predictions with u = 1 are found to the left of experimental data. The agreement between experiments and predictions based on the present model was found to be much better with the use of v = 2. This degree of accuracy, in the authors’ opinion, represents what can be expected with the present model in view of the lack of knowledge about the nature of the reaction between the toxic gas and impregnants, as well as the rather approximate method in estimating the various model parameters. In formulating models for the retention of toxic gases with impregnated carbon, the major difficulty resides in the lack of any quantitative information between the toxic gas and substances that may be used in preparing impregnated carbon. This lack of information necessitates the use of relatively simple expressions in describing the rate of interaction due to chemical reaction. It also means that the model so formulated must contain a number of parameters that are difficult to determine independently and therefore must be regarded as adjustable parameters. In the case of the model of Graceffo et al., these parameters are the reaction rate constant, k, and the stoichiometric ratio, u. For the present model, only v remains to be the adjustable parameter. The presence of one less parameter represents the major improvement of the present model over the previous one. This fact may become more significant in the future if some independent method is devised for the determination of Y. One can then apply the present model as a truly predictive tool for screening the effectiveness of different candidate substances as possible impregnants in carbon preparations. Acknowledgement-This study was conducted as part of the research activities of the Air Purification Technology Center, a consortium composed of the State University of New York at Buffalo, University of Minnesota, and Syracuse University under contract with U.S. Army Chemical Research Development and Engineering Center, Aberdeen Proving Ground, Maryland. REFERENCES 1. L. Graceffo, H. Moon, S. G. Chatterjee, and C. Tien, Carbon, 27,441 (1989). 2. S. G. Chatterjee and C. Tien, Chem. Eng. Sci., 44,2283 (1989).

847

3. T. R. Goodman, Application of integral methods to transient nonlinear heat transfer. Advances in Heut Transfer, Vol. 1, Academic Press, New York (1964). 4. J. W. Milius, J. S. Greer, and A. J. Johola, Edgewood Arsenal Contract Report. Advanced Collective Protection System Design Studies, Edgewood Arsenal, Aberdeen Proving Ground, MD (1974). 5. R. H. Chen and D. F. Othmer, J. Chem. Eng. Data, 7, 37 (1962). 6. T. K. Sherwood, R. L. Pigford, and C. R. Wilke, Mass Transfer, p. 242, McGraw Hill, New York (1975). 7. P. J. Reucroft and C. T. Chiou, Carbon, 15,285(1977).

NOMENCLATURE

a, = quantity defined by eqn (27.b) a, = quantity

defined by eqn (19) radius of pellet defined by eqns (18.a) and (18. b) , respectively b, = defined by eqn (23.a) defined by eqns (21.a) through c4, = (21.h), respectively CR gas phase concentration of toxic c= gas in bed (per unit volume) c’ = gas phase concentration of toxic gas in the pore space of pellet (per unit volume) Ci. = inlet concentration of toxic gas effluent concentration of toxic Ceff =

A,;= cl, 6.

c2, Cb,

c3, Cl,

gas D= bulk diffusivity of toxic gas D, = pore diffusivity of pellet I,, 4 = parameters in eqn (26) k= second-order reaction rate constant between impregnant and toxic gas k, = fluid-phase mass transfer coefficient K= linear adsorption isotherm constant K,, K2 = defined by eqns (25.a) and (25.b). respectively L= length of bed n= impregnant concentration in pellet n, = initial impregnant concentration in pellet N= defined as ci, (p,,K + q)l(q,v) NB, = defined as k,a,lD, PI = quantity defined by eqn (31.a) adsorbed phase concentration of 9= reactant (per unit mass) 4” = value of q in equilibrium with ci” (1= average concentration of toxic gas retained by a pellet (per unit mass) quantity defined by eqn (31.b) 41 = radial coordinate measured from r= center of pellet r, = radius of unreacted core

S. G. CHATEIUEEand C. TIEN

848

R=

local rate of reaction (per unit

R=

volume) in pellet average rate of reaction (per unit volume) in pellet

APPENDIX Method used for evaluating the integral in equation (26)

Dropping the letters for convenience, the integral present in eqn (26) can be written as

ZlL

S= t=

time 1, = time when the first stage ends T. = tortuosity factor u, = superficial fluid velocity in bed 1 - rla, x= Y= y'

=

.Z=

(AlI Letting T = 7: = 7, + AT, we get (here AT is a small time step)

C/C,. c’ lc,,

axial distance in bed

Greek Letters Pm = Y= 6= 6, = e= 4 = T)ll 7l2 =

eigen value in eqn (17) WJ(4aj) defined as 1 - rClap value of 6 when T = T, porosity of bed porosity of pellet defined by eqns (23.b) and (23.~). respectively t - zdu, 0= integration variable in eqn (26) k= v= number of moles of toxic gas reacting with one mole of impregnant Pp = density of pellet 7= WMV + 41 7, = value of T when the first stage ends defined

as (c’/c,,)(r./a,) 4= JI= quantity defined by eqn (20)

=

y(T’)2;2y(T2) [ 1

_

e

B!. AT],

W)

m

Letting T = 7) = 72 t AT. we have

x

[l

-

,-Ph]

=

+ Y(71)2;*Y(T?)

[I

J,e-Ob _

,-B.?+]

(A3)

m

In general, it can be shown that J,

zz J_,e-Pi+

+

‘(“+ :‘p2+ y(T’) 1, m

where (I = 2, 3. 4.

.

_

e-PhA.]

(Ad)