An investigation of the linear driving force approximation to diffusion in spherical particles

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AN INVESTIGATION APPROXIMATION

OF THE LINEAR TO DIFFUSION PARTICLES

DRIVING FORCE IN SPHERICAL

J. H. HILLS Department of Chemical Engineering, University Park, Nottingham

NG7

ZRD, U.K.

(Receiued 14 Auyust 1985) Abstract-The linear driving force approximation is studied for three cases of batch adsorption and one of leaching in a CSTR, for which exact analytical solutions are available. Common pitfalls in its application are underlined and it is shown that, for adsorption from a finite volume of fluid, very large errors can arise when the fractional uptake approaches unity.

Glueckauf was interested in chromatographic columns, where the medium composition (and hence qs) is not constant with time. For this case,eq. (3) can be modified to give

INTRODUCTION

Unsteady-state diffusion in spherical particles plays an important role in the design of adsorption and ion exchange equipment, and in the theory of chromatography. The appropriate partial difTerentia1 equation is:

Analytical solutions are available (see Crank, 1956) for a number of simple boundary conditions and constant D, but they are often in the form of infinite series and are not easy to use. For this reason, the linear driving force (LDF) approximation, first suggested by Glueckauf and Coates (1947) is often used. In this method, the rate of adsorption by the sphere is written as the product of surface area. an effective mass-transfer coefficient k, and a driving force consisting of the difference between the bulk average concentration in the sphere 4 and the surface concentration (1.. Rate = i 5tu3 g

x

Z$exp

[-n2n2D(t--)/a’]

Differentiating eq. (4) and solving the integral in both eq. (4) and its derivative by parts leads to eq. (5) after some manipulation. (It is assumed that higher derivatives are negligible

dq, with respect to dt and that the

series can be truncated at its first term at the lower limit where t = 0)

(5)

In a chromatographic column, the solid and fluid are always close to equilibrium, so that

= 47~~ k, (qs - ij)

or

dq,_% dq

-3k”(&--81. dt- a

B-4.3 -= qm - 40

1-s $

PI=1

dc -

(2) da dtThis LDF

$exp(-n2a2Dt/a2) (3)

where F is the fractional approach

dt

(6)

Substituting into eq. (5) we obtain

The ordinary differential eq. (2) is much easier to handle than the partial differential eq. (1) and gives easier analytical solutions. Glueckauf (1955) presented a justification of eq. (2) and an indication of the conditions in which it is valid. He started with the known analytical solution to eq. (1) for the case of a sphere of initial concentration q,, plunged at t = 0 into a well-stirred fluid of constant composition. If the final steady-state concentration in the solid (in equilibrium with the fluid) is 4,. the solution is: F=

dz.

to equilibrium. 2779

-

15D

__ a”

CL-- 4).

is the most commonly quoted form of the solution. Comparison with eq. (2) shows that k,

= 5Dja.

Glucckauf (1955) compared the predictions of eqs (5) and (7) with the exact solution (4) for two analytical forms of qE(t) corresponding to a diffuse front and a sharp front in a chromatographic column. For realistic values of the parameters, agreement with eq. (5) was almost perfect, while agreement with eq. (7) was adequate in the case of the diffuse front. Nakao and Suzuki (1983) compared the performance of eq. (2) with a numerical solution of eq. (1) for a cyclic adsorption process corresponding to pressure

2780

I. H. HiLLS

swing adsorption. Spherical pellets were subjected alternately to a medium of concentration c, and a medium ofconcentration zero until the oscillations in q became steady. To match the two steady-state predictions, k, for the LDF eq. (2) had to vary with the cycle time, and even when the steady-state predictions agreed, the LDF still gave a more rapid approach to equilibrium. Rice et ul. (1983) compared the LDF solution with the exact solution for the case of leaching of solute from solid to fluid within a CSTR. They concluded that. for the range of parameters studied, eq. (7) gives an adequate representation of the behaviour of the exact solution over the whole range of times, and a near-perfect fit for reduced times greater than 0.4. Peel and Benedek (1981) compared the LDF with experimental data for a batch experiment. They found a very poor fit at short times (see below) and suggested use of the quadratic driving force model of Vermeulen (1953) as being more accurate in this region. The majority of workers have used the LDF model eq. (7) uncritically. While this may give results acceptable for chromatographic columns and long packed bed adsorbers (see Hills and Pirzada, 1984) the evidence above suggests it should be checked for other configurations, and this paper aims to make such a check for three cases of batch adsorption, for which analytical solutions of eq. (1) are available. Rice’s work (1983) will also be extended and discussed.

CASE

I

This involves a solid with concentration q0 plunged at t = 0 into a well-stirred infinite medium. The analytical solution is given by eq. (3). For values of Or/a’ > 0.1, only the first term of the sum is significant, and the equation can then be rewritten: In (1 - F) = In (6/n*) - n’Dr/a*. fntegrating the commonly at t = 0 and remembering qs is constant, 1x1(1-F)=

(8)

quoted eq. (7) using q = q. that for an infinite medium -lSDt/a’.

(9)

A plot of In (1 - F) against t, as suggested by eqs (8) and (9) is commonly used to estimate D from batch adsorption data. The asymptotically exact eq. (8) and the LDF approximation (9) differ in two important respects. First, the gradient of the plot is x2 D/a’ according to eq. (8) and 15D/u* according to eq. (9). Use of the approximation (9) to estimate D would lead to a value smaller by a factor of 0.66 than that given by the more correct solution (8). The origin of the discrepancy can be found in the approximations leading to the LDF eq. (7) in particular the assumption of near equilibrium between particle and fluid represented by eq. (6). While true for the chromatographic column, this is quite untrue for the present batch experiment, and (6) should be

replaced by

-dqs = dt

Substitution

0.

@a)

in eq. (5) then gives dq dl-

z2D a2

-

(qs-4).

Or, on integration, In (1 -F)

= - a2Dt/a2

(10)

which now has the same gradient as eq. @I. Secondly, the intercept according to the LDF model [eq. (9) or (lo)] is zero, where the asymptotically correct value is In (6/7c*) according to eq. (8). This discrepancy arises from the fact that the boundary condition used in the integration of eq. (7) was imposed at t = 0, which is outside the range of validity of the approximation (Dt/az > 0.1). No significance can thus be attached to the intercept for the LDF models. Smith and Tsao (1971) failed toappreciate this point. Finding a non-zero intercept in their experimental plots of In (1 - F) against t, they attributed it to an error in the initial concentration 90 used in the calculation of F. They even suggested use of the intercept to estimate the “true” value of q. in cases where experimental measurement is difficult. In fact, examination of their graphs shows that the intercepts scatter round the theoretical value of - 0.50 obtained from eq. (8). Use of Vermeulen’s (1953) quadratic driving force approximation avoids these problems, since it gives a reasonable fit over all values oft, as shown by Peel and Benedek (1981). However, its arbitrary parameter cannot be related to D in any simple manner, and it has not found much favour in practice. The “standard” LDF eq. (7) was derived by Liaw et al. (1979) and Rice (1982) by assuming a parabolic concentration profile within the particle. The disadvantage of this approach compared to Glueckauf’s is that it hides the nature of the approximations involved: concentration profiles do look roughly parabolic, but it is by no means clear that the result (7) should only be used in near equilibrium conditions when eq. (6) may be expected to hold. Rice (1982) in fact uses the result k, = 5D/a even for batch systems where k, = n2D/3a would be more appropriate.

CASE

II

In this case, a solid with concentration q. plunged at t = 0 into an infinite medium and a fluid side masstransfer coefficient k, and linear adsorption isotherm qs = Kc are considered. The exact solution [which can be derived from Crank (i956)] is: F=

4-90 p= 40x--4,

1_

f II=1

6L2 exp (- /3iDt/a2)

B.‘lx+~(~--1)1

(l’)

LDF

and /3” are the roots of fi, cot &

L = ak,/DK

where

+L-1

approximation

=o.

For large values of Dt/a’, we can again ignore all but the first term of the sum, and write:

In (1 -F)

Table range

= In

6L2 II/3: [j?: + L(L - 1) ]

1 gives values of values of

1

- B:Dt’a2.

to diffusion

2781

than 5’ for small L, but in most practical cases the external mass-transfer resistance is small (L + cc) so 5’ is usually the better choice. A more accurate solution can be obtained from Glueckauf’s original eq. (5). If c, is the bulk fluid concentration and ci is the interfacial concentration,

(12)

of the intercept and of 0: for a L from 0 (fluid mass-transfer

control) to co (solid mass-transfer control). For the LDF approximation we have only to combine internal and external mass-transfer coefficients in the usual way:

4,

= Kc,

q, = Kc; 4na2kT K (4,-99s)

Rate=~rra3~=4na2k,(c,-cJ=--

. -dq dt

= 3k, aK

(9,

- 9,).

Differentiating eq. (14) and eliminating qs and 2 Using k, = 5D/a

eq. (5),

from eq. (6), 1 -= K,,

$+E =-

a

D5[

from

T

-+-. 1 1

1

L

The

solution

of eq. (15) is

Liaw et al. (1979) and Rice (1982) define a parameter

4 3, where ml,

5 = ,=$[&+&I 3;

_ 4 = AemIt + ~~~~~

(16)

m, are given by

so that eq. (2) becomes + J(3L-or In (I -F)

=

-

t/c.

(13)

corresponding to /3: in the asymptotically exact eq. (12), is also listed in Table 1. Using the value of k,appropriate to an infinite batch process [k, = n2D/3a, eq. (lo)] we could define a modified parameter: The

parameter

7rz)I +0.8n4L

Taking m1 as the larger root, the term Be’“z’is negligible for large t and all values of L, and eq. (16) may thus be written as

a2/D<.

A

In (1 --F)

z In-----+ m,t. 4 m - 40

(17)

Table 1 also shows the value of - m1 a2/D as a function of L: it can be seen to be in very close agreement with the asymptotically exact /?:. CASE 111

of a2/D<’ are also listed in Table 1. Comparing the two LDF solutions with the asymptotically exact one, we see that they both agree at L = 0 (external mass transfer controlling), while only a2/Dc’agrees at L = cc (internal mass transfer controlling). 5’ has a maximum error of x 10% when L = 3.5, which may be acceptable for engineering purposes. 5 is more accurate

Values

This involves a solid with concentration go plunged at t = 0 into a well-stirred medium of volume V. Adsorption follows a linear isotherm q, = Kc. The exact solution (Crank, 1956) is F=

4-s ------=

l-

41a-xJ

g

6cc(cr+ 1)

n=l 9+9a+y.2a2

exp (-

y,‘Dt/a’) (18)

Table 1. Effect of external mass-transfer resistance

7 aJ/D5 a’/DC’ --m,a2/D

L:

0

0.01

[eq. (12)]

0 0 0 0

0.02994 -0 0.02994 0.0299 1 0.02994

0.1

- o.ooo2 0.2941 0.2941 0.2912 0.2941

1.0

-0.0146 2.4674

2.5000 2.3007 2.4617

3.5 -0.1059 5.6675 6.1765 5.0875 5.6174

10.0 -0.2735 8.04% 10.0000 7.4264 7.9715

100 - 0.4685 9.6733 14.2857 9.5552 9.6582

00 - 0.4977 9.8696 15.OoOo 9.8696 9.8696

.I.

2782

H. HILLS

where volume of solution

results of a batch experiment in which a was 2.92. (Note that his paper has inadvertently misprinted a as V, K/V rather than the inverse: the arithmetic, how-

V

c( = K x volume of solid = e

3Y?l tan Yn= ~ 3+ay,Z’ The fractional uptake of adsorbate (fraction of that originally in the solution which is adsorbed onto the solid) is given by +1.

l+o!

As t becomes large, eq. (18) can be written as ln(l-F)

6a(a + 1) 9+9a+y:o?

= In

1- y:m,M. (19)

Table 2 lists values of the intercept and Y: for various values of the fractional uptake 4. Note that as 4 -+ 1, the intercept --+ - co. A simple mass balance gives < 4+

V&/K

1 + A as a> ( 20.13, so using eq. (22) rather than eq. (19) would underestimate D by a factor of 1.73. Foo and Rice (1979) using the same experimental data but analysing by eq. (19) found a value of D 1.88 times larger than that found by Rice (1982). Rice suggests there was a factor of two mathematical error in the earlier work; the calculations here show that the discrepancy is caused by using the LDF eq. (22) rather than the asymptotically exact (19). As a check, the intercept in Rice’s Fig. 1 is In (0.59), while calculation from eq. (19) yields In (0.51). Considering the scatter in the data, this is satisfactory agreement, so the experiment supports the use of eq. (19). As with case II, we can solve the more accurate eq. (5) for this case. Differentiating eq. (20), ever, is correct.) This gives y: as 11.64 and 15

and Y,, are the non-zero roots of

= V,qm + Vq,lK

d4s- _ dt

4,--@=((4z

(20)

ln(l--F)

= -

Substituting this into the LDF eq. (2), dq

3k

(21)

a

Thus, the kc values are multiplied by Depending on whether k, is taken as SD/a rc2D/3a, eq. (21) will integrate to ln(l--F)=

-15

ln(1 --F)

= -x2

or

(22)

or

Table x2

2 compares

the values of

Dt/a’.

(231

15

and

rt’(l+a)

l-

Dt

1 +a-7r2/15

a”

(24)

The factor 7~’(1-t a)/( 1 + a - 7r2/15) is also listed in Table 2. It can be seen to follow Y: much more closely, with a maximum deviation of 43 0/0 when a = 0. Because none of the approximate methods can predict the correct intercept, and particularly since the intercept + - co as a + 0, these methods are not suitable for predicting the time needed to attain a given approach to equilibrium (e.g. 1 - F = 0.01) in a batch adsorption. When such calculations are needed in the design of batch adsorbers, the asymptotically correct solution (19) should be used.

CASE

IV: LEACHING

IN A CSTR

A solid initially charged uniformly with adsorbate to a concentration q, is plunged at time t = 0 into a wellstirred tank through which flows an adsorbate-free fluid. The isotherm is linear, qS = Kc, and there is no externa1 mass-transfer resistance. This case is a development of ease III, and is solved by similar methods. The solution was first given by

from these two LDF solutions with the

asymptotically correct y:. As expected, eq. (23) agrees exactly as a - co (infinite medium) but diverges seriously as a -+ 0. Rice (1982) used eq. (22) to analyse the Table 2.

Effect of fractional

uptake

6:

0

tL:

m

0.2 4.00

0.4 1.5ooo

0.6 0.6667

0.8 0.2500

-0.4977 9.8696 15.ooocI 9.8696 9.8696

-0.6259 11.2126 18.7500 12.3370 11.3950

-0.8283 12.8944 25.cKKm 16.4493 13.3950

- 1.1784 14.9853 37.5000 24.6740 16.3076

- 1.8844 17.4819 75.m 49.3480 20.8386

Intercept [eq. (19)] Y: 15(1+ l/a) d(1 + l/a) Factor in eq. (24)

1 d@ a dt

substituting into (5) and integrating,

whence

-_=e dt

---

1.0 0.0000

20.1s7 cc 28.g63

LDF

approximation to diffusion

Rice et al. (1983). The mass balance equation is:

2783

with a different definition of T and c:

dc dq V~+V++QC=O

r = J(1

or

i= (25)

where Q is the dimensionless elapsed time (= Dt/a’), dimensionless residence time of fluid and c( = V/K V,, as in case III. [Note that Rice et al. (1983) used K = 1 and N, = u/3, N, = or/30, as their parameters.] The exact solution, given by Rice, is, with our terminology, OR is the ( = D Y/Qa’)

4 -z= 40

where

6 (A,’- l/Q,) exp ( - Ai 0)

IL .Ci A,

(26)

3/0, - 9A,” (1 + l/or) - a(A,’- l/0,)’

are A.

the

non-zero

roots

of

tan

A.

= l+a/3(*l/0,)’ Substitution of eq. (26) into eq. (25) and integration gives the fractional approach to equilibrium of the solid phase:

1-c

6c& d,’[a (2 -

- l&# l/Q2

exp (-

AZ@

+ 9A,’(1 + l/a) - 3/6a 3’

(27)

Rice et al. based their LDF solution on eq. (7), which they derived from the parabolic velocity profile assumption. They used Laplace transforms to eliminate 4 between eqs (7) and (25), and obtained, on inversion, 4s -= %

(-sO)sinh(T

+c?-7r~/15)aa/7c20r (30)

rC7c2(1+c()+oL/esl 2(1+ d!- 7c2/15) .

Substitution of eq. (28) into eq. (25) and integration gives the approximate solution for the fractional approach to equilibrium:

p.=%-4 __

=

40

1 -exp(

-kV)cosh(v@)

tanh(yH)).

I+>$!!$

(31) Rice et al. (1983) compared their approximate solution [eqs (28) and (29)] with the exact eq. (26) in two ways: (a) direct plots of dimensionless fluid concentration, proportional to 4,/q,,, against 19for selected values of the parameters 01and 0,; and (b) comparison of the values of tl where qs/qo passes through a maximum, plotted as a function of 0x for fixed a. Our approximate solution [eqs (28) and (30) ] gives emax values consistently larger than Rice’s, giving a better fit to the exact solution for large 0,, and a worse fit for small 0,. However, since the approximate solutions are valid only for large 8, i.e. for 0 9 e,,,, any agreement in the prediction of emaXmust be to a certain extent fortuitous. More relevant is the comparison of approximate and exact solutions as 8 -+ co. For this case, eq. (26) becomes:

f3)

4 2% 40

(28)

6(A: - l/0,) exp ( - A%,) 3/C?, - 9n: (1 + l/a) - a (2: - l/8,)2

(32)

and eq. (28) becomes:

where r=Je,lls

(29)

5 = 1/2r [15(1+

l/a)+

4 Axq”

l/O,].

We can apply the same method to Glueckauf’s eq. (5) in place of eq. (7). Equation (28) again results, but

0, 2at J<_

exp

(

- ‘-me

z

>

.

(33)

In both cases, a plot of In (qs/qo) against 6 is linear, and Table 3 compares the gradients and intercepts for the

Table 3. Leaching with DL = 4

Gradient exact Rice This work Fluid intercept Exact Rice This work Exact Rice This work

2

cz

0.040 0.040

0

0.040

0

0.002

0.02

0.2

9.870 15.000 9.870

9.839 14.885 9.837

9.508 13.599 9.482

3.713 3.750 3.707

1.179 1.180 1.179

--co -* --oo

- 5.788 -4.871 - 5.293

- 3.289 - 2.405 - 2.829

-1.464 - 1.466 - 1.326

- 1.562 - 1.563 - 1.520

- 1.594 - 1.594 - 1.580

- 1.608 - 1.608 - I.606

-1.609 - 1.609 - 1.609

-0.498 0.000 0.000

-0.493 -2 X 10-d 2 X 10-J

-0.454 -0.034 0.010

-1.137 - 1.179 -0.992

- 1.477 - 1.481 - 1.477

- 1.566 - 1.567 - 1.552

- 1.605 - 1.605 -1.604

- 1.609 -1.609 - 1.609

Solid intercept

0.667

20

ep: 0

0.398 0.398 0.398

0

J. I-l. HILLS

2784

Table 4. Leaching with OL= 0.04 0.002

c&:0 Gradient Exact Rice This work

7.244 8.508 7.244

9.870 15.000 9.870

Fluid intercept Exact Rice This work

-* -cc -cc

Solid intercept Exact Rice This work

- 0.498 O.OW 0.000

-7

0.0067 4.043 4.199 4.021

z In

6a(l: A: [a(if

-

l/Q*

0.190 0.190 0.190

0.019 0.019 0.019

co

0 0 0

-0.431 -0.349 -0.290

-0.164 -0.152 -0.105

-0.051 -0.051 -0.045

- 0.040 -0.040

-0.039 - 0.039 -0.039

-0.208 X 10-a 9 X 10-Z

-0.431 -0.021 0.083

-0.037 -0.031 0.023

- 0.038 -0.038 -0.032

- 0.039 - 0.039 -0.038

- 0.039 - 0.039 - 0.039

-

l/e,)’

+ 9if (1 + l/a) - 3/Q,] j

and ln(l --F)

1.701 1.711 1.699

2

- 1.209 -0.845 -0.921

exact and approximate solutions. We have used a = 4, following Rice, but have covered the complete range 0 -C 0, -=zco where Rice used only 0.2, 2/3 and 2.0. It can be seen that for 0, > 1 both the gradient and intercept are predicted well by both approximations. As OR decreases below 1, our approximation for the gradient remains within 1 y0 while Rice’s gradually diverges; as OR+ 0, the exact solution and our approximation predict a gradient of IL’where Rice predicts 15. Both approximations over-predict the intercept at small f3,: the Rice version by ln(15/16) = 0.916 and ours by In (7c2/6) = 0.498. Instead of following efAuent fluid concentration, y,/q,, we can also follow the fractional approach of the sollid to equilibrium, as given by eqs (27) and (3 1). As 0 + co, both equations give a linear plot of In (1 -F) against 8, as do the other three cases studied earlier in this paper. The results are:

ln(l-F)

0.2

0.02

-0.040

between the approximations and the exact solution continues down to 8, = 0.01. As a rule of thumb, Rice’s a parameter N, = 38, can be taken, and the approximate solutions used whenever N, > 1. CONCLUSION

The LDF approximation should be used with caution in the design of batch adsorbers. In the case of non-zero fractional uptake, it can lead to serious errors. NOTATION a

A, B C

-n:e

radius of spherical particle constants in eq. (16) fluid-phase concentration

(34)

value of c in bulk fluid value of c at interface diffusion coefficient in solid phase

z

(35) The gradients are the same as those in eqs (32) and (33) for the fluid concentration; the intercepts are also listed in Table 3. Again, both approximations agree with the exact solution for 19~> 1, but diverge for smaller values; as 8, -+ 0, both approximations have an intercept of In (1) = 0, while the exact solution has In (6/7c2) = - 0.498. By considering only the case where K = 1, Rice et al. restricted their study to cases where solute is held in the pores rather than adsorbed onto the pore surface: adsorption would normally give K s 1 and hence a -$ 4. We repeated the above analysis using a = 0.004 to see how it affected the conclusions (Table 4). As can be seen, the pattern is similar, but agreement.

4-q fractional approach to equilibrium = ----E 4CX-4, effective mass-transfer coefficient in solid effective mass-transfer coefficient in fluid overall mass-transfer coefficient distribution coefficient Biot number for mass transfer = akJDK parameters in eq. (16) solid-phase concentration initial value of q equilibrium value of CJ value of q at surface of particle bulk average value of q over whole particle volumetric flow-rate of fluid radial distance time

2785

LDF approximation to diffusion V Y

Greek

volume of fluid volume of solid letters

volume of fluid/K x volume of solid roots of Bncot&+L-1= 0 3% roots of tarry, = ~ 3+ay,2. -fractional uptake of adsorbate mass-transfer parameters dummy variable in eq. (4) [also the function defined in eqs (29) and (30)] roots of tan i, = AzJ(I+ l/30( (A,’- l/0,)) dimensionless elapsed time Dt/a’ dimensionless residence time D V/Qa’ function defined in eqs (29) and (30)

REFERENCES

Crank,J., 1956, The Mathematics

of Diffusion,1st edn. Oxford University Press, Oxford. Foo, S. C. and Rice, R. G., 1979, Ind. Engng Chem. Fundnm. 18, 68. Glueckauf, E., 1955, Trans. Faraday Sot. 51, 1540. Glueckauf, E. and Coates, 1947, .I. Chem. Sot. 1315. Hills, J. H. and Pirzada, I. M., 1984, Chem. Engng Sci. 39,919. Liaw, C. H., Wang, J. S. P., Greenkorn, R. H. and Chao, K. C., 1979, A.Z.Ch.E. J. 25, 376. Nakao, S.-I. and Suzuki, M., 1983, J. them. Engng Jap. 16, 114. Peel, R. G. and Benedek, A., 1981, Can. J. them. Engng 59,688. Rice, R. G., 1982, Chem. Engng Sci. 37, 83. Rice, R. G., Nadler, K. C. and Knopf, F. C., 1983, Chem. Engng Commun. 21,55. Smith, F. B. and Tsao, G. T., 1971, Chem. Engng Prog. Symp. Ser. 67 (No. 108). 24. Vermeulen, T., 1953, Znd. Engny Chem. 45, 1664.