Nonlinear driving force approximations of intraparticle mass transfer in adsorption processes

Pergamon

Int. Comm. HeatMass Transfer, Vol. 23, No. 3, pp. 367-376, 1996 Copyright © 1996 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/96 $12.00 + .00

PII SO735-1933(96)OOO22-X

NONLINEAR DRIVING FORCE APPROXIMATIONS OF INTRAPARTICLE MASS TRANSFER IN ADSORPTION PROCESSES

Andreas Georgiou and KrzysztofKupiec Technical University of Cracow, Institute of Chemical Engineering and Physical Chemistry, 31-155 Krakow, ul.Warszawska 24, Poland

(Communicated by P.J. Heggs) ABSTRACT A new methodology for the derivation of driving force approximations and a new approximation of adsorbate uptake rate are presented. The performed tests show that the accuracy of this approximation is excellent considerably improving on other available approximations.

Introduction

The substantial simplification of computations achieved by the use of driving force approximations in transient diffusion and adsorption modelling has motivated numerous studies whose objective was to develop such expressions. In most cases the methodology of intraparticle concentration profile approximaUon has been applied. An analysis of this methodology is given elsewhere [1]. In this communication a different approach, based on an analysis of existing analytical solutions, is presented. It is shown that the obtained nonlinear driving force (NLDF) model substantially improves on available approximations.

Approximate expressions for the case with negligible external mass-transfer resistances

We consider the case of a spherical particle with a uniform initial concentration Q0 subject at t=0 to a unit step change of the concentration at r=R. This system is described by 367

368

A. Georgiou and K. Kupiec

a0Q:D 1

Vol. 23, No. 3

8 (r2 8Q ~

O)

with the conditions t = 0;

Q = Qo,

(2a)

r = 0;

~ = 0, Or

(2b)

r = R;

Q = QR.

(2c)

The meaning of the variables and parameters depends on the diffusion model and is given in Table 1.

TABLE 1 Definitions of Variables and Parameters Variables and parameters

Pore diffusion model

Solid diffusion model

Q

C

q

D

Dp

D,

a

8p+K

l

b

1

K

The change of variables A-

Q-Qo

(3a)

QR -- Q0 r x

(3b)

z -

R

Ot RZa

(3c)

leads to the equation c3A_ 1 8 (

& with the conditions

x~~

x~ ~ )

(4)

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INTRAPARTICLE MASS TRANSFER IN ADSORPTION

x = 0;

A = 0,

x = o;

--:

x = 1;

A = 1,

0A 0x

369

(5a) o,

(Sb)

(5c)

The analytical solution is [2]: = 1 - re-65-~~ ~n1- exp

(_n27~2z)

(6)

At short times, upon a step change, the particle can be treated as a semiinfinite medium and this assumption leads to the following approximation [2]: X : 6~f~

(7)

Let us now define the following function F: dA F=

d~ 1-A

(8)

Taking into account Eq(6) we have : ~ exp(-n2~Zz ) F=7~ 2

n=l

(9)

~i ~ f exp( -nz~2x ) It can be shown that

limF =

(10)

a2

,r---> oo

Furthermore let us define the function G = G ( A )

_~2 G=

1

A I-A

(11)

Taking into account that 1

the function G can be written in the form:

7~z

(12)

370

A. Georgiou and K. Kupiec

\n=l

G=71:2

a.-

-

aob n

o-

n=l

Vol. 23, No. 3

anb, (13)

where an = exp(-n27z~z),

(14a)

1

b,

(14b)

n 2 .

In Fig. 1 the dependence G = Cr(A) is shown.

G

4

~ [rll}ll]lJll

0.0

0.2

The dependence G : Ca(A)

0.4

0.6

FIG. 1 (1)Exact, (2) QDF,

0.8

~

(3) DM,

1.0

(4) NLDF

Taking into account Eq.(7) it can be shown that (3(0) -- 18 ~ 5729

(15)

7~

Some earlier published approximations can be considered as resulting from suitable approximations of the functions F and G:

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INTRAPARTICLE MASS TRANSFER IN ADSORPTION

371

I. Linear Driving Force approximation [3]. The LDF model is equivalent to the approxtmation FLDF = 15.

(16)

II. Vermeulen's Quadratic Driving Force approximation (QDF) model [4]. In this case FQD F = 7t 2 1

+_A

2A

(17)

IimFQDF = n 2

OQDF:I 2I+X )X 2A

2

1---A -

(18)

2

~ 4.934

(19)

Therefore Vermeulen's QDF model can be obtained by approximating the function G with a constant such that at the large time region Eq(18) is satisfied. III. Do and Mayfield [5] approximation (DM). In this case: FDM = 9 + 4.797

1-A __ A

(20)

Neglecting the difference between 9 and n 2 we can see that GDM ~ 4.797

(21)

A comparison of Eq.(19) and Eq.(21) shows that the last two approximations are almost identical and comparable accuracies can be expected. A study of Fig. 1 shows that improvements are possible with a better approximation of the function G. In this paper the function G is approximated with a linear function: Gr~DF = 1 8 ( 1 - A) 7~-

(22)

and therefore the proposed nonlinear driving force aproximation is:

W--/n' For small fractional uptakes n2 <<--18 (1 - _ A ) 2 and A << 1. Therefore Eq(23) can be written in n A the form dA 18 1 = -- = dz n A

--

The integration of this equation leads to Eq(7). For large fractional uptakes n 2 >>

(24)

18 (1- n)2 n

A

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A. Georgiou and K. Kupiec

Vol. 23, No. 3

and Eq.(23) reduces to a linear driving force approximation, In dimensional terms Eq.(23) can be rewritten as (25)

R--~a 7t + ~ (Q, _QoXQ_Qo)),..~,

Applications to batch adsorption Negligible external mass-transfer resistances

Equation (23) was integrated and the obtained A was divided by Aox (Eq.(6))~ The ratio A -- A/Aox is shown in Fig.2 For comparison purposes analogous results for the LDF approximation ALDF = 1-- exp(-15X)

(26)

.10

A -1.05

1.5-

1.0-

•

3

0.5-

0.0 I . . . . . . . . t 1 E-4 1 E-3

'"l

1E-2

........

I

1E-1

FIG.2 Comparison of various approximations, (1) LDF,

......

1~

;~-0.90

1E+O

(2) DM,

(3) NLDF

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INTRAPARTICLE MASS TRANSFER IN ADSORPTION

373

and the Do and Mayfield [5] approximation ADM = 1 - e x p ( - ( 9 x + 1.153x°'3z))

(27)

are shown. As we can see both the LDF and the Do and Mayfield approximations lead to over 50% errors at the short time region while the maximum error resulting from the application of the proposed NLDF model is 2.7 %.

Batch adsorption in a bath with finite volume Intraparticle diffusion controlling

We consider the case of a spherical particle of volume Vp and an initial concentration Q0=0 plunged at t=0 into a well stirred medium of volume V and initial concentration Cbo. In this case the governing dimensionless equations become 0x

(28a)

x 2 0x x2 dX b dx

1 dA et dx

(28b)

with the conditions x=0;

A= ~.=0,

Xb=I

x=0,

c3A --=0 0x

(29b)

x=l;

A=Xb,

(29c)

A-

(30a)

(29a)

where

Q

bCbo

X b = Cb Cbo

(30b)

The analytical solution is [2]:

X A~

6or(or + 1) .~-~9 +-~e~+--13~-etz exp(-13:x)

(31)

where 13. is the n-th nonzero root of tan [3°-

313. 3 +~13~.

(32)

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A. Georgiou and K. Kupiec

Vol. 23, No. 3

A simple mass balance gives

~_

(33)

l+c~

The approximate solution is obtained by integrating the system of ordinary differential equations

dX

x

_

dx = rt2 ~

dX b d~

(34a)

Xb

1 dA c~ d~

(34b)

with the initial conditions defined in Eq(29a). In Fig.3 the results of these computations are shown for a few values of the parameter ec The errors resulting from the application of the approximation (25) do not exceed 8%.

1.10

A 1.05-

1.00

0.950.90 1 1E-5

1E-4

1E-3

1E-2

1E-1 't 1E+0

FIG 3 Results of numerical computation for batch adsorption in a finite bath. (1)ot=O.l, (2) ot=0.3, (3)ot-~l.O, (4)ot=2.0, (5)ot=9.0

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INTRAPARTICLE MASS TRANSFER IN ADSORPTION

375

Conclusions

In this communicaUon a nonlinear driving force approximation of intraparticle mass transfer in adsorption processes with a linear isotherm and negligible external mass transfer resistances has been presented. The derivation of this approximation was based on an analysis of the available analytical solution of the problem, resigning therefore from the much exploited methodology based on intraparticle concentration profile approximations. The obtained approximate expression was applied to batch adsorption systems under various operational conditions. The obtained results were compared with the analytical solution of each problem and it was shown that in the studied cases the maximum errors do not exceed 8%. The accuracy was particularly good in the small time region.

Nomenclature

a

defined in Table 1

A

fractional uptake

b

defined in Table 1

C

gas-phase concentration

D

defined in Table 1

Dp

effective pore diffusion coefficient

D~

effective surface diffusion coefficient

K

linear equilibrium adsorption constant

k~

film mass transfer coefficient

q

solid-phase concentration

Q

defined in Table 1

r

spatial dimensional variable

R

particle radius

t

time

V

bulk volume

Vp

particle volume

X

dimensionless spatial variable

X

dimensionless gas phase concentration

376

Y

A. Georgiou and K. Kupiec

Vol. 23, No. 3

dimensionless solid-phase concentration

Greek letters (X

= V/(Vp ab)

A

ratio of approximate value to the exact = relative error + 1

£p

porosity of particle

T

dimensionless time

Subscripts 0

value at t = 0

1

value at x = 1 value at t ~

b

refers to bulk

ex

exact value

R

value at r = R

Superscripts -

-

average value

References

1. A.Georgiou and KKupiec, Inz. Chem. Proc. 16, 75 (1995)

(in polish).

2. J.Crank, The Mathematics of Ditthsion Oxford University Press, Oxford (1956). 3. C H L i a w , J.S.PWang, RAG-reencorn and K C C h a o , MChE Journal 25,376 (1979). 4. T.Vermeulen, IndEngng Chem. 45, 1664 (1953). 5. D D D o and P.LJ. Mayfield, MChE Journal 33, 1397 (1987).

Received October 17, 1995