Column adsorption in multi-solute water

291

Chemical Engineering and Processing, 32 (1993) 291-299

Column adsorption Jarmo

Reunanen*

in multi-solute

water

and Seppo Palosaari**

Department of Chemical Technology, Lappeenranta University of Technology, Lappeenranta (Finland)

Minoru

Miyahara

and Morio Okazaki

Department of Chemical Engineering, Kyoto University, Kyolo (Japan) (Received March 1, 1993; in revised form April 20, 1993)

Abstract Adsorption equilibrium in multi-solute water is described via a multi-component Langmuir equation. Of the two possible variables, one is assumed to be constant whereas the other is specific to each component. This coefficient is assumed to be a distributed variable. The relationship between the amount of adsorbent and the equilibrium concentration in the solution in experiments where different quantities of adsorbent are in contact with the same amount of liquid is called the integral adsorption equilibrium curve, and is described here by a skew Gaussian distribution. The coefficients have been determined by parameter fitting to the experimental measurements. The obtained distribution was then used to calculate the breakthrough curve in a packed bed adsorber. It has been found that the new distribution function, when used in the simulation of the breakthrough curve, gives better results than the parabolic distribution reported earlier.

Introduction The use of adsorption for municipal and industrial wastewater treatment has become more and more prevalent in recent years and a number of studies have been made on the packed bed adsorbers. Rosen [l] has published a basic solution to the calculation of the breakthrough curve when it is used for one solute only and when a linear adsorption isotherm is assumed. Thomas [2] has used the Langmuir isotherm. Liapis and Rippin [3] have developed a numerical solution for calculating the breakthrough curve in the case of two adsorbates. Tien [4] has used IAS theory for the case of several adsorbates. Normally it has been assumed that all the components and their single component isotherms are known. Since wastewater generally contains many kinds of chemicals, it is difficult in most cases to be able to identify the individual contaminants and their composition. Hence, previous studies can scarcely be applied to the design of an adsorption system for the treatment of unknown multi-solute wastewater. It is therefore important to develop a new design procedure for the adsorption treatment of unknown multi-solute wastewater. Kage and Tien [5] and

*To whom correspondence should be addressed. **Temporary address: Department of Chemical Engineering, Kyoto University, Kyoto, Japan.

0255-2701/93/$6.00

Ceresi and Tien [6] used a lumped component model to calculate the multi-component breakthrough curves. Nevertheless, knowledge regarding at least one component was still essential. However, Okazaki et al. [ 71 used a method to calculate the multi-component breakthrough curve where information regarding the individual components was not essential. The purpose of the present study is to develop this latter method further. The present paper consists of two parts. In the first part, a new approximate method representing the characteristics of adsorption equilibrium for unknown multi-solute wastewater is proposed, while by applying this approximation in the second part, a procedure for predicting the breakthrough curve of such wastewaters is presented assuming instantaneous adsorption equilibrium between the water and the adsorbent.

Approximate representation equilibrium of wastewater Characteristic

distribution

of multi-solute

adsorption

curve

The concentration of organic chemicals or pollutants in water is usually expressed as TOC, BOD or COD. TOC is used here, but the method discussed is not dependent on the measure used. Correlation formulae for multi-solute adsorption equilibrium [S] usually include several kinds of coefficients for the description of the adsorption characteristic of the individual solutes.

0 1993 -

Elsevier Sequoia. All rights reserved

292

These coefficients sets may be assumed to have a oneto-one correspondence to each solute in the wastewater. If one of these coefficients can be assumed to be far more predominant than the others, we can accordingly describe the solutes in terms of this coefficient and then regard the coordinate of this coefficient on the axis as a kind of substance [9]. Furthermore, if there are numerous solutes, it is reasonable to assume that the wastewater is a distributed system on this ‘substance’ axis and a distribution curve of the concentration probability density relative to the axis of this, the most predominant, coefficient may be defined. This distribution curve is then called the characteristic distribution curve (CDC) of the wastewater [9]. The Langmuir equation As a simple correlating equation of multi-solute adsorption equilibrium, we have adopted the Langmuir equation for multi-component systems, i.e.

1 +Ck,c, which includes two kinds of coefficient, q, and k. The adsorption characteristic of the ith component can then be specified by the values of these two coefficients. To simplify the following mathematical calwe have assumed that one of these culations, coefficients exerts a much greater controlling influence than the other. Taking account of the fact that q,, rather than k, is monotonic between the various solutes over the whole concentration range, we have coefficient. Furtherchosen qbj as the controlling more, we have assumed that k is constant for all the solutes. When q,,i is rewritten as x, the distributed expression of eqn. (1) is given by: kxc(x) &n;

q(x) = I+k

Adsorption equilibrium curve To describe the adsorption equilibrium, we can use either the adsorption isotherm or the curve relating the amount of adsorbent used to the concentration in the liquid. These two relations are interchangeable. As far as the latter relationship, which is used in the present work for the identification of the characteristic distribution curve (CDC) is concerned, two different types exist. The first is the integral adsorption equilibrium curve (IAEC), which is the relationship between the amount of adsorbent and the equilibrium concentration in solution experiments where different quantities of adsorbent, yi, are in contact with the same amount of wastewater (Fig. 1). The second type is the differential adsorption equilibrium curve (DAEC). In this case, a small amount of adsorbent, dyD, is added to the wastewater. After equilibrium has been achieved, the wastewater is separated from the adsorbent by filtration and the TOC concentration of the solution phase measured. Successive repetition of this operation gives the DAEC as the relationship between the accumulated amount of adsorbent, y,, and the concentration of the solution phase (Fig. 1). Determination of the characteristic distribution curve Although the DAEC describes the multi-solute adsorption behaviour of the wastewater more explicitly than the IAEC, we have adopted the IAEC as a means of identifying the CDC, taking into account the simplicity of the experimental procedure and, in comparison with the DAEC, the small accumulation of experimental errors. Okazaki et al. [7] have demonstrated how the CDC may be determined from the IAEC. They have derived the following expression: -%ax

G(Y)

=

Xmax

J C(X,Y) = J Xnll”

11 + kG(yMx, 0) & 1 + kc,(y) + ykx

(4)

xmin

c(x) dx J Xrni”

where c(x) is a probability density function of the concentration of the x component. The TOC concentration of the wastewater, C,, is given by: c, =

c(x) dx s Xmin

(3)

when c(x) is known, the adsorption equilibrium under any conditions can be calculated readily from the c(x) value of the initial wastewater, and the problem essentially becomes the determination of c(x).

(b) Fig. 1. (a) Integral and (b) differential adsorption equilibrium curves.

293

The term c(x, 0) is the initial concentration distribution of the solute, and is the CDC of the wastewater. The CDC can then be determined from the observed IAEC, C,(y), by using eqn. (4). Although there are various methods of solving eqn. (4) for a given C,(y), we have applied a parameter fitting method. Okazaki et al. [7] used a parabolic distribution curve; however, in this work, a skew Gaussian distribution, studied by Heal [lo], has been used as the CDC curve. This is given by the expression: &, 0) = e-‘“(W%)

(5)

Fig, 2. Distribution column.

of the components

in the fluid

along

the

where f(x)

(6)

= ln( v) /A

4j-I,,-

V=l+2A(x-B)/D

(7)

Using this distribution it is possible to reduce the number of parameters to be determined which could make the calculation easier and faster.

Calculation

of the breakthrough

curve

Generally speaking, to calculate the breakthrough curve of a packed bed adsorber, it is necessary to solve the mass balance equation and the transfer rate equations simultaneously. However, for the unknown multisolute wastewater considered, a rigorous solution of these equations would seem somewhat impossible. To overcome this, Okazaki et al. [7] introduced an approximate instantaneous equilibrium between the liquid phase and the adsorbent phase, so that it is possible to calculate an approximate breakthrough curve using only information from the adsorption equilibrium of the wastewater, i.e. the CDC. If the flow velocity of liquid in the column is low, then equilibrium may be assumed and the experimental results may be compared with those predicted by a method where the mass transfer rate is neglected, Dividing the range of ‘components’ into N intervals, they approximated the wastewater to a water containing N kinds of solute. When the initial concentrations of these solutes in the water change in the are c 1,i.J~C2,N, *. ’ 3 CN,N, the longitudinal concentration distribution is stepwise as shown in Fig. 2. The final solution for the unknown solutes is then as follows:

cj-

,,j_,

=

(8)

i=l

/

The derivation of the above equation is given in Appendix A. The Langmuir-type equilibrium equation may be expressed as:

I=

kXj_lcj_,,i_,

(9)

j-l

In the subscripts in the above equations, the left-hand term refers to the components and the right-hand term to the adsorption steps. The concentrations in the Nstep are given by the CDC. The breakthrough curve can then be calculated successively from the (N - l)step to the l-step by solving the terms cJ_ l,j_1 and qj_ ,,i_ 1 simultaneously by an iterative method and by calculating the moving velocities from each step.

Results Experimental

procedure

The experimental work in this study has been undertaken in the same manner as by Okazaki et al. [7]. The adsorption experiments in this work have been effected using two kinds of synthetic wastewater, viz. wastewaters A and C of Okazaki et al. The wastewaters and their content as used by Okazaki et al. are listed in Table 1.

TABLE 1. Composition based on mol carbon)

of wastewaters

Wastewater A

0.179

Wastewater B p-Cresol Nitrobenzene p-Nitrophenol Phenol Polyethylene

fraction

Aniline Benzene p-Cresol Cyclohexanol p-Nitrophenol Phenol

0.167 0.166 0.166 0.167 0.167 0.167

Wastewater D 0.095 0.092

0.093 0.093

glycol

(mole

Wastewater C 0.160 0.158 0.178 0.172 0.153

Benzamide Benzoic acid p-Cresol Hydroquinone p-Nitrophenol Phenol

studied

0.627

p-Chlorophenol p-Cresol /?-Dinitophenol Nitrobenzene m-Nitrophenol

0.141 0.282 0.113 0.211 0.253

294 TABLE

2. Experimental

conditions

Adsorbent Activated carbon Shape Mesh Density (kg m-‘)

Takeda x-7000 granule 14-16 724.1

IAEC Initial cont. Temperature

(mol C m-‘) (“C)

Column test Wastewater Column diameter (cm) Initial cont. (mol C m-‘) Superficial velocity (m h ‘) Bed voidage Temperature ( “C)

used by Okazaki CAL

ef al. [7]

granule 14-16 797.0

Takeda x-7000 granule IO-12 724.1

Takeda x-7000 granule 10-12 724.1

12.58 35

26.89 35

10.00 35

17.75 35

A 1.2 12.66 2.2 I 0.4220 35

B 1.2 26.56 5.44 0.4580 35

C 1.2 16.33 5.15 0.4602 35

D 1.2 17.33 5.10 0.4602 35

temperature of 20 “C. Column experiments were undertaken on a column of 4 cm diameter, the temperature being kept at 20 “C. The fluid velocity was kept constant in each experiment and the void volume in the column was also determined. The wastewaters were analyzed by liquid chromatography (Varian model 5000). For this purpose, an ODS column was used, the detector being a UV filtrator at 280 nm. The total concentration for each species present was obtained by integrating each peak and adding the integral values This method was employed both for the determination of the IAEC and the breakthrough curve. The experimental conditions used by Okazaki et al. [7] and in this work are presented in Tables 2 and 3, respectively. Equilibrium

TABLE

3. Experimental

conditions

used in this work

Adsorbent Activated carbon Shape Pellet diameter (mm) Apparent density (kg m-‘)

Norit RBl granule I.0 490

Norit RBI granule 1.0 490

Norit RBI granule 1 .o 490

Norit RBl granule 1.0 490

IAEC Initial cont. Temperature

12.7 20

12.7 20

16.3 20

16.3 20

A 4.0 12.7 2.2 0.469 0.50 20

A 4.0 12.7 2.2 0.418 0.79 20

C 4.0 16.3 5.2 0.354 0.22 20

C 4.0 16.3 5.2 0.394 1.05 20

(mol Cm-‘) (“C)

Column test Wastewater Column diameter (cm) Initial cont. (mol C m 3, Superficial velocity (m h-‘) Bed voidage Bed length (m) Temperature (“C)

Batch experiments were carried out to determine the integral adsorption equilibrium curve. The wastewater was placed in several flasks ( 1 dm3) and into these flasks was introduced a measured amount of pretreated activated carbon. The flasks were stirred for 7d by means of a magnetic stirrer and maintained throughout at a

TABLE

4. Parameters

Parameter

of the IAEC curve calculated

(%)

results

work

in which a skew Gaussian

distribution

was employed

Wastewater B

A

A B D k Error

in the present

and column experiment

The measured values of the batch experiments were fitted to eqn. (4) to obtain the values of various parameters. The results presented by Okazaki et al. [7] (parabolic) are also included in the following figures to enable an effective comparison to be made. It was found that the method employed in the present work took only c. 30% of the CPU time required by the method used by Okazaki et al. The values of the fitted parameters are listed in Table 4. Figures 3-8 illustrate the characteristic distribution curves for wastewaters A, B, C and D, respectively. These figures also show the measured and fitted batch data for each component. Figures 9-14 illustrate the breakthrough curves for wastewaters A, B, C and D, respectively. Figures 3-8 show that a successful fit of the CDC curve has been achieved. The measured data and fitted values are almost identical except for wastewater C and the experiment conducted in the present work. It can also be seen that fitting by the Gaussian-type CDC curve is better than in the results presented by Okazaki et al. [7]. It should also be noted that the calculated IAEC curve depicted in Fig. 7 exhibits discontinuities. The reason for this is that the data set used in the

Okazaki

This work

I .4464282 12.721377 3.6141063 1.1030570 0.558

0.0013784 10.003341 12.320743 0.0520743 3.331

0.2573093 13.357526 24.984612 9.9985507 0.761

D

C Okazaki

This work

0.5230537 19.668601 6.3489530 0.1415882 0.048

8.272lE-9 10.000000 20.472918 0.3166836 10.914

0.0312588 18.229578 16.545848 1.2620612 2.458

295

-d

Pmabolk

--Th1a

work

-. -.__ ,

o

0

10

r--;--,--,_.,___

,

20

30

40

50

00

60 (b)

X

(a) Fig. 3. IAEC (integral parameters are listed

adsorbed in

0.5

Table

equilibrium

curve)

and CDC

Y,

(characteristic

distribution

2

1.5

1

Wm31

curve)

A. The values

for wastewater

of the

4.

15-

-

Pamholii

--This

0

10

20

30

40

work

50

0.” 0

60

X

(a)

”

”

” 2

”

3

Y, Mh31

(b)

Fig. 4. IAEC (integral adsorbed equilibrium curve) and CDC (characteristic present work. Values of the parameters are listed in Table 4.

”

1

distribution

curve) for wastewater

A. The data are from the

‘\ ‘. ..__ I

0 (a)

10

I

20

30

40

.-‘i---__

.

50

60

0

1

equilibrium

curve)

and

CDC

calculations is not very good and in such cases eqn. (4) produced discontinuities since the measured total TOC concentration is used on both sides of that equation. From Figs. 9-14, it can be seen that the calculated breakthrough curves follow the measured breakthrough

(characteristic

1.5

2

2.5

3

Y* Mh31

(b)

X

Fig. 5. IAEC (integral adsorbed parameters are listed in Table 4.

0.5

distribution

curve)

for wastewater

B. Values

of the

curves relatively well. The agreement is quite good, especially for wastewaters A and D. However, poorer agreement was achieved for wastewater B. For wastewater C and the experiment conducted by Okazaki et al. [7], the agreement is poor; however, for the same

296

l--

?, :: :' 1,

O-8-

:

s &0.6-

:

-

/

20

0

Pmabolk

- - lhk

30

(a)

40

work

50

80

0

X

0.5

1 Y,

(b)

Fig. 6. IAEC (integral adsorbed parameters are listed in Table 4.

equilibrium

curve)

I.81

and

CDC

(characteristic

distribution

1.5

2

kdm31 curve)

for wastewater

C. Values

of the

I

-

Panbdk

--TM*

WOh

\I ‘> -.

o0

10

I

20

----___

/

30

40

50

0

60

X

(4

Fig. 7. IAEC (integral adsorbed equilibrium curve) and CDC (characteristic present work. Values of the parameters are listed in Table 4.

1 -..

....I-. .' .. ,' '\

.....

-

(a)

10

20

30

40

6

8

1012

Y. kdm31 distribution

curve) for wastewater

C. The data are from the

..

work

50

60

X

Fig. 8. IAEC (integral adsorbed parameters are listed in Table 4.

4

Panbalk

-_Thk

0

2

(b)

Y, [kg/m31

(b) equilibrium

curve)

and

CDC

wastewater the agreement achieved in present work is better. In all cases, it may be stated that the agreement between the calculated and observed breakthrough curves is better using the method presented in the present work than using that presented by Okazaki et al. [7].

(characteristic

distribution

curve)

for wastewater

D. Values

of the

Discussion

This work presents a method for describing the characteristic distribution curve of the pseudo-components in multi-component adsorption via a skew Gaussian distribution. Previous work has employed a parabolic

297

_-; 0 0204080

OL 0

,6 80

Time, [h] Fig. 9. Breakthrough

OP 0

curves

for wastewater

A.

Fig. 13. Breakthrough from the present work.

120

curves

for wastewater

C. The data

are

_-‘0

50

108

200

150

Time, [h] Fig. 10. Breakthrough from the present work.

curves

0 for wastewater

A. The data

0t 0

50

100

150

200

for wastewater

B.

Time, [h]

I 1. Breakthrough

50

are

curves

‘.*~

108

150

208

for wastewater

D.

Time, [h] Fig. 14. Breakthrough

I.*~

Fig.

100

Time, [h]

curves

type of distribution to describe the characteristic distribution curve. The distribution curve presented here is continuous with no cut-off point as for the parabolic type of distribution (see Fig. 3). Parabolic distribution requires extra coefficients to determine the limits over which integration occurs. Hence, the suggested distribution is more reliable. Because of the suggested type of distribution, the calculated breakthrough curve approaches saturation more gradually (see Fig. 9). The suggested distribution also has a lower number of variables and hence the calculations are simpler and faster. The new distribution function also produces a better agreement between the predicted and measured breakthrough curves.

Nomenclature A

B c “0

50

100

150

ci,j

:

Time, [h] Fig. 12. Breakthrough

curves

for wastewater

c(x) C.

fitted variable, dimensionless fitted variable, mol C kg-’ AC TOC concentration of component, mol C mm3 W TOC concentration of the ith component in the jth step, mol C mm3 W probability density of TOC concentration, mol C rnw3 W

298

total TOC concentration, mol C mV3 W fitted variable, mol C kg-’ AC fitted variable, dimensionless Langmuir coefficient, m3 W (mol C)-’ amount of adsorbed TOC of component, mol C (kg AC) -’ amount of adsorbed TOC of the ith component in the jth step, mol C (kg AC)-’ probability density of amount of adsorbed TOC, dimensionless amount of saturated adsorbed TOC of component, mol C (kg AC) -’ superficial velocity of wastewater in packed bed, m h-’ moving velocity of step, m h-l fitted variable, dimensionless value of amount of saturated adsorbed TOC, mol C (kgAC)-’ amount of adsorbent, kg AC ( m3 W) - ’

CT AX)

k 4

#

u

V X

Y

Abbreviations AC C W

adsorbent (activated organic carbon wastewater

carbon)

Appendix A All symbols employed are shown in Fig. 2. The mass balance for the jth component is: UCj,j= Uj[ECj,j +

4

5

6

7

8

9

10

J. B. Rosen, Kinetics of a fixed bed system for solid diffusion into spherical particles, J. Chem. Phys., 20 (1952) 387-394. H. C. Thomas, Heterogeneous ion exchange in a flowing system, J. Am. Chem. Sot., 66(1944) 1664-1666. A. I. Liapis and D. W. T. Rippin, The simulation of binary adsorption in activated carbon columns using estimates of diffusional resistance within the carbon particles derived from batch experiments, Chem. Eng. Sci., 33 (1978) 593-600. C. Tien, Recent advances in the calculation of multicomponent adsorption in fixed beds, in M. J. McGuire and F. H. Suffet (eds.), Treatment of Water by Granular Activated Carbon, Am. Chem. Sot., Washington, DC, 1983, pp. 167-199. H. Kage and C. Tien, Further development of the adsorption affinity characterization procedure for aqueous solutions with unknown compositions, Ind. Eng. Chem., Res., 26 (1987) 284-292. J. E. Ceresi and C. Tien, Carbon adsorption of phenol from aqueous solutions in the presence of other sorbates, Sep. Technol., I(l991) 273-281. M. Okazaki, H. Kage, M. Kusuoka, J. Tsubota and R. Toei, Prediction of the breakthrough curve of a packed bed adsorber used for treatment of unknown multi-solute wastewater, Chem. Eng. Process,, 26 (1989) 247-255. M. Okazaki, H. Kage and R. Toei, Prediction of liquid phase adsorption equilibria of multi-solutes in water, J. Chem. Eng. Jpn., 13 (1980) 286-291. M. Okazaki, H. Kage, F. Iijima and R. Toei, Approximate descrption of multi-solute adsorption rate in organic aqueous solution, J. Chem. Eng. Jpn., 14 (1981) 26. G. R. Heal, Modifications of the Dubinin-Radushkevitch equation for microporous adsorption, 4th In?. Conf. Fundament. Adsorp., Kyoto, Japan, May 17-22, 1992, pp. 51-62.

(Al)

By solving the velocity of the jth component (Al), we get:

vj=

from eqn.

ucj j

ECj,J

+

(Aa

( 1 - E)qj, i

The mass balance be written as

for the (j - 1)th component

may

U(Cj_~,j-Cj_],j-~)=Uj[E(Cj_I,j-Cj_~,j_l)

+(1-~)(qj-~,j_qj-~,j-~>l (A3) and on dividing

eqn. (A3) by eqn. (Al),

we get:

u(cj-I,j-cj-I,j-l) ucj,I = uj[E(Cj-l,j-Cj-I.j-l) +(l -E)(qj-l,* --4j-l,j-I)1 vjiECj, j + ( ’ - E)qj, jl (A41 Rearrangement

References

(1 - E)qj,,i]

of eqn. (A4) leads to:

cj- 1.j -“j-l,j-I

Ci. j =E(Cj-t,j-Cj-*,~-l) $-Cl+E)(Yj-I,j-Y,-l,j-l)l ECj,j + ( 1 + E)4j,j (A51 On multiplying we get: (cj-l,j

eqn. (A5) by c,,,(Ecj,, + (1 + E)gj,,j),

-cj-I,j-I)(Ecj,j + t1 -c)qj,j)

= Cj,j[E(Cj-l,j-Cj_l,j_1) +(l -E)(qj-I,j-qj-I.j-l)l (A6) and by rearranging

we obtain:

cj-l,jEcj,,+ ci-l,j(l -E)Qj,j - Cj- L.j-IECj,j- Cj- I,j-1(l = Cj,,PCj-I,j-I,j+cj,j(l

Cj,JECj_ I,j-1

-c)qj-I,j-cj,j(l

and be rewriting

E)qj, j

LE)qj-I,j-*

(A7)

as follows, we get:

cj- I,pcj,j- cj- *,j- lccj,j - Cj,,PCj-I,j-1 + Cj.pCj_I,j_* =

-Cj_I,j(l-E)qj,j+Cj-I,j-l(l-E)q~,,i +Cj,j(l

-E)qj-I,jL’cj,~jtl

-')qj-I.j-*

C-W

299

O=(l

by Cj-l,j-

and then multiplying

which is equal to:

-~)[-cj-I,j~j,j+cj-,,j-,~j.j +cj,,~j-l,j-cj,,~j-l,j-,l cj.i (A9)

By dividing eqn. (A9) by (1 - E) and rearranging, get: C~,/&l,,-r

qj_1,j-

q,,j(-Cj1 =

Y;-

I,j

I,i-I)

*,j +‘J-

=

equilibrium

equation

(A=)

j-l 1 + k c ci,j_ 1 i=,

eqn. (All)

4j-,,j+ cj. j

and (Al%

and by rearranging qJ,jCj-l,j I

we get:

- j-1

4j,j(-C/-,,j+C/-,,j-l)

_kxj-lCj-l,j-l l+k

'j-1,/-1

ci,i

ci, i

- qj,j

Cj,j

kxj-lcj-19J-’

=

cd

and on rewriting

1 + k C Ci,j-, i=l j-1 C Ci,j_l i=l >

1 +k

1 + kji Ci,j- 1 i= 1

as follows we get:

- qj,jCj,,j) i= I >(qj-,,jCj,j

(A14)

j-l

1 + k C Ci,j-, cj-l,J-I

kx,_, (A13 I %.j _ ~_ ,,jI ci,j1 cjl,j1 cj,jcjby cj _ 1, j _ , , this gives:

(A201

=

>

4j,jcj-1.j

e-LJ

+kJs

which, on rearranging, 1

gives:

we get: 4j,jcj-l,j-I

‘*’

qj,,jcj,j - qj,jcjI,j (A19 I,j-, = j-l > ( C,,jkxj_,

C Cf,j_,

(A181

_4j,i

C c~,~_] i=l

which on rearranging

(A13)

‘j.J

j-l

l+k

Cj_

cj,,j-I 1

_%A-I,/

kXj_ 1

is,

%-I,/

on dividing

=

may be ex-

kxj-, Cj_ ,,j_ 1

By combining

.J

I

(Al 1)

‘j’j

the term ci_ r,,._ I) we get: q,~,

‘J.i

_4i,i

(A17)

leads to:

+

The Langmuir-type pressed as: 4j-l,j-1

we

Separating

the term qj- I,j-l

which on separating

kXj_, j-l 1 +k c ci,j_I i=,

qj_ , j _ p4i.ici- l,.i _

(AlO) =q,,,Cwcj-,,, +cj-,,j-,) +‘j,jqj-,,j

I gives:

i=l

Finally,

ci,,_,

1 +k

kx,_,

1 + k C Cl,j- , i-l

“j

(A16)

Cj _ ,,j-,

by the term cj,j we get:

j-1

leads to:

,,j= j-1 _4i,i ,,j- 4j,jcjj-1qj‘j. j1

cj- 1,

by dividing

=

C Ci,j_, 4,-,,jFEcj-l,j i=, >( j-1

kx,_,

-E

.(

1+ k c

is I

Ci,j_,

> >

(A211