On gradient dependent diffusivity

ChemicalEngineering

Science. 1973, Vol. 28, pp. 1565- 1576.

Pergamon Press.

Printed inGreat

Britain

On gradient dependent difhhity W. W. KRUCKELS Institut fiir Chemische Verfahrenstechnik Universitat Stuttgart, Stuttgart, West Germany (Received

24August

1972; in revisedform

16 November

1972)

Abstract-It is a known fact that surface diffusion is dependent on temperature and concentration. Data on adsorption of water in microporous silica gel indicate that an additional quantity is influencing ditfusivity in such a system. These observations can be explained by a surface diffusivity depending on concentration gradient. Such a behaviour is analogous to gradient dependency of viscosity, both being deducible from Eyrings theory of rate processes. 1. INTRODUCTION

IT HAS been known for a long time that some in-

adequacy with respect to mass transport in adsorbents exists. Twenty years ago Ednie and Groves [l, 21 stated that it is impossible to describe the process as simple Fick’s-law diffusion with a constant or concentration dependent diffusivity. Most studies on transport in porous systems however were made by measuring steady state diffusion through porous plugs, using the Wicke-Kallenbach method[3]. This may be the reason why the observed marked effect did not attract attention earlier. The computer allows us to handle extensive numerical calculations, so transient diffusion measurements at present have become more popular. Ruthven et a1.[4,5] made adsorption studies by such a method and showed that solid diffusion into micro-crystalites is rate determining for adsorption of normal hydrocarbons in zeolites. Our own measurements on adsorption of water vapour in silica and alumina gel particles demonstrated that it is impossible to explain the observed rate behaviour by a conventional gas or surface diffusivity[6]. Non-linearity of surface transport with respect to concentration gradient was already postulated by Babitt[7]. He and later Gilliland et al. [8] succeeded in matching their data with a surface flow concept where the flux is proportional to dC?/dx; but in respect to our own data this relation does not hold. Recent data published by Kast and Jokisch on adsorption of water vapor in silica gel [ 181 seem to contradict our own results.

The authors show that it is possible to calculate the adsorption rate in their system by a constant gas-side diffusivity. This is possible if their gel has a mean pore radius which is considerably larger than 10 A. In such a case Knudsen diffusion may be rate determining. Unfortunately the authors did not give information on pore size distribution in their gel. There are sophisiticated, detailed new approaches to describe surface diffusivity by aid of statistical thermodynamics or simpler mechanistic models[9-111. None of these corresponds to our own observations. In all of these cases diffusivity is increasing with concentration, whereas we observed a marked decrease over a wide concentration range; none of these gives a nonlinear relation between flux and concentration gradient, which apparently caused the inconsistency of adsorption rate in regard to Fickian diffusion.

2. SYSTEM

The adsorbents, in which the discussed effect was observed, are different silica and alumina gels with hydrated surfaces all having a unimodal pore system with mean pore radii of about 10 A and 95% of pore volume below a pore radius of 100 A;, which is regarded as upper limit for micropores. One of these, a silica gel, whose pore size distribution is shown in Fig. 1, is regarded here in detail, although the others in principle show the same behaviour. The adsorbate is water, whose specific properties, especially the high energy of

1565

W. W. KRiiCKELS

p 80 n f ; 60 6 2 t 40 $ ; 20 B L

0 loo

lo2

10'

lo3

lo4

rP A

Fig. 1. Pore size distribution of microporous silica gel (G 127) calculated from the isotherm - measured by aid of porosimeter-. X

evaporation are characteristic for the system and may be the cause of the special observed effect.

1 P ,_*~p

0.6

3. EXPERIMENTAL

/

Adsorption rates were measured by means of a vacuum electro-balance (Cahn RG) according to the following principle: A probe of about 10 equal sized particles (ca. 100 mg) as they are used in technical adsorbers is placed on the pan of the balance, heated (15O’C) and evacuated (10e5 torr) until equilibrium is reached. The temperature of the probe is then adjusted. By opening a valve to a thermostated water vessel a step-change of water vapour pressure can be applied to the probe. During the process of adsorption, weight vs time is recorded and the temperature of the probe is held as constant as possible by means of a thermostated mantle around the weighing chamber. The results of such an experiment are shown in Fig. 2 for one pressure step in terms of a plot of the saturated fraction x vs time t. After reaching equilibrium further pressure steps can be applied until the particles are saturated. To simplify matters only the submonolayer concentration range is regarded here. In order to hold the temperature inside the particles constant, pressure steps were made as small as possible. Generally the concentration range up to mono-

,

1;; 0

10

20

30

40

50 t

min

Fig. 2. Mass increase of adsorbent probe after step change of sorbate pressure vs. time, x adsorption of pure water vapor; o adsorption of water vapor from air; pa = 10m4torr, pm = 8.5 torr, T = 21°C. R = 1.46 mm.

layer coverage was subdivided into 5 equal intervals, corresponding to an increase of loading during one step of 3.50%. Monolayer concentration corresponds to about half of the saturation loading of the particles. Experiments were made at different temperatures between 0 and lOO”C, but temperature influence is not considered here. 4. CALCULATION OF THE DIFFUSIVITY

EFFECTIVE

It can be shown that in the case where adsorbents are microporous and the activation energy of surface diffusion, E,,is markedly lower than the isosteric energy of adsorption, Eiso, the rate

1566

On gradient dependent difbivity

determining process is surface diffusion[l2]. In such a case the analysis is simplified by the fact that adsorption only takes place at the outer surface of the adsorbent particles. Penetrating matter is in the adsorbed phase. The analysis is based on the following simplifying assumptions: (1) Particles have a spherical shape. (2) The pore system inside the particles is homogeneous and isotropic. (3) The effective diffusivity is constant. These assumptions mean that adsorption is regarded as a diffusion process into the spherical space. The continuity equation in polar coordinates for concentric diffusion with initial and boundary conditions according to the adsorbate pressure step is

(1) I.C. c = c, for0
x

erfc (2n+ l)R--x 2ah *t)

_erfc(2n+ l)R+x 2V(D,-t)

1 -

&=%[I-+;x)].

(5)

With Eq. (5) the effective diffusivity D, and correspondingly the apparent diffusivity D, was calculated. 5. RESULT OF EXPERIMENTS

The results of a series of successive pressure steps at constant temperature are plotted as log D, versus mean concentration in Fig. 3. It is obvious that for one rate experiment the effective diffusivity is by far not constant but shows a significant increase after having passed a minimum D, at the very beginning of the process until near the end a maximum value of D, is reached. The ratio D,/Di, is increasing with concentration and may reach values of about 10 near monolayer coverage (see Fig. 4). This fact is in striking contradiction to the third assumption namely that the effective diffusivity be constant. This means that the calculated diffusivity, D,, has lost its direct physical significance. Therefore we call it apparent dilfusivity, D,, but being a differential quantity in contrast to the integral quantity x it is very sensitive to any change in the rate of adsorption, so it is retained as a type of diagnostic quantity.

(2)

The concentration c has to be integrated over the particle volume to get the mean concentration. This yields X=

Schilling[ 141 showed that as long as x is less than O-928 this solution does not deviate more than 0.5% from the exact solution Eq. (3), so that in regard to the inevitable experimental error under the restriction x G 0.928 it is appropriate to use Eq. (4) to calculate effective diffusivities:

6. CONCLUSION

c-cc, _ 6 --~@e4(&+2~ c,-cc,

the results for one pressure step with the simple Fick’s law solution, one is urged to the following conclusion: At the beginning of the process, when the concentration gradients inside the particles are high, the apparent diffusivity is low, in contrast near the end of the process, when the gradients in the particles are low, the apparent difTusivity is high. This leads one to suspect that there exists some inverse relation between diffusivity and concentration gradient. Comparing

(3) The infinite series inside brackets can be neglected for small dimensionless times, which results in (4)

1567

W. W. KRiiCKELS

PI

10.' 0.0

0.2

0.4

0.6

0

0.6

I.1

-c Fig. 3. Adsorption kinetics for different pressure step changes in terms of apparent diffusivity vs relative monolayer concentration. --- calculated with intrinsic ditfusivity act. Eq. (23) -calculated taking account of Knudsen diffusion. 1. Measured points for different pressure steps. 2. -a- calculated with same parameters for all steps. 0 10-“-l a74torr Temperature 40°C 0 1.74-4.80 torr Particle radius I .05 mm A 4.80~8.00 torr Number of particles I9 n 8~00-11~1torr Silicagel Grace 127 cl 11.1-15.3 torr

%

I 2.10-s cm2 s 10-s

7

5.10-7

2.10-7

10-r 0.0

L 0.2

F 0.4

0.6 -

66 c/tco-

1.0 C,)

Fig. 4. Comparison between adsorption and desorption rate experiments. 0 Adsorption c, = 0.592 c, = 0.759 0 Desorption c, = 0.757 c, = 0.592 T = 60°C R = 1.38 mm, 8 Particles -calculated with ditfusivity according F!q (23)

1568

On gradient dependent diisivity

To test this speculation adsorption and desorption processes were compared. If it is true that the concentration gradient influences diffusion, the apparent diffusivity in desorption experiments should also be low initially and increase when concentration approaches equilibrium. The result of such a comparison is shown in Fig. 4. The expected inverted behaviour of desorption affirms that the concentration gradient is an additional variable influencing the process of surface diffusion.

one dimensional model of surface diffusion, which can be regarded as an activated hopping of molecules. A molecule is fixed at its adsorption site by the electrostatic adsorption forces. Vibrating with a frequency z+,it is able to leave the site only if it has sufficient energy to surmount the energy barrier having the mean height Es. The probability W for such a jump is given by the Arrhenius expression

7. EXPLANATION OF GRADIENT DEPENDENCY

so that the jumping frequency (number of jumps per unit time) in one of the two possible directions is

Gradient dependency of transport coefficient is a well known fact in momentum transport, being explicable by means of Eyring’s theorie of rate processes [ 151. It is possible to explain gradient dependency of surface ditfusivity in an analogous way. Figure 5 a illustrates the simple %O-------

_-

FLUID

P-.

----

-----

J--

PHASE

fi+yjp

“-

“I

A

----

---

0

----

lmEKTA\AEL OF -----_

W = exp (- E,/WT)

v=y.exp

(6)

(-E,/%‘T).

(7)

This is a simplified form of the Eyring equation for rate processes. The mean jumping distance is assumed to be constant as long as concentration does not change, so that the rate of transport of molecules passing through a balanced boundary (see Fig. 6) is given by: J,= NL(C-;*~)u+.a.

(

L-0-J

(8)

-

________ ----------I _ DISTANCE

X

L t

%a

“-

/

u+_

\t/ POTENTIAL

OF ADSORBED

DISTANCE

X

-

!i

MOLECULE

Fig. 5. Surface diffusion, schematic (model of hopping molecules). (a) Without interaction,(b) With interaction of.adsorbed molecules.

-

X

Fig. 6. Surface element with diierent jumping frequencies in forward and backward direction resulting from concentration and activation energy gradient.

1569

W. W. KRtiCKELS

The backward

flow

through the boundary is

rate

J1=NL(C+;$+~

(9)

and the resulting flux under the assumption V+= V_ = vis J res= JI - Jz =-NL.CiZ.y.dC/dx

pendency of activation energy on concentration it is possible to calculate y+=~.exp(_E.(C)+(aE~~~~(aC/ax).n13 (16) and Y_ correspondingly diffusivity

(10) Ds=DsO

which compared with the implicit definition of the surface ditIusivity ( Jr,, = - NL . D,, *dCldx gives the well known expression diffusivity

which yield a surface

(11)

for the surface

&, = 4 . v.

[

l-i*&tanh

(aE&C)@C/~x) W-T

. @/2) >I

.

(17)

The initial assumption that the jumping distance ii as well as the gradient dE,/dC are constant with respect to concentration allows one to introduce the following simplifying constant parameters

(12) k,=z

and

k,=-e.f

(18)

a

If there is a superposition of the surface potential attracting adsorbed molecules with the lateral interaction potential of neighbouring molecules this may result in a decrease of activation energy for surface diffusion as shown in Fig. 5b. If in addition there is a concentration gradient superimposed, i.e. if a molecule has more neighbours on one side than on the other this will cause a difference in its jumping frequencies in forward and backward direction. Balancing both fluxes in the same way as before results in a total flux C ir2 J,,=-N,~(v++v_)~~-C~7(v+

-v-)

(13)

which can be simplified by putting Dso =;

(v++v_)

which yields an expression sivity D

* =D

al

1-g.

a

(14)

for the surface difIu-

D,= Dso.

[

l+ks.C

aclax 1

tanh (k4 *X/W

.

( 19)

This expression allows one to make the following important conclusions: (1) The expression for the surface ditfusivity consists of two parts: the constant or concentration dependent diffusivity Dso; to which is added the linear combination of Da with the gradient dependent function f (dC/dx). (2) High concentration gradients cause a diminuition of the surface ditfusivity and vice versa as it is shown in Fig. 7 where the gradient functionf( dC/dx) is plotted vs dC/d.x. (3) In the submonolayer range the influence of gradient dependency must increase with surface concentration because the gradient dependent term is proportional to C. 8. VERIFICATION OF GRADIENT DEPENDENT DIFFUSIVITY

(v+-v;;;~x++v-) .

Under the assumption

so that finally D, becomes

I

To prove that a surface diffusivity as describ-

(15) ed by Eq. (19) is able to explain the observed

that there is a linear de-

adsorption kinetics, D, has to be inserted into the continuity equation Eq. (1). Introduction of a

1570

On gradient dependent diisivity

stant ditR.tsivity, corresponding to the Fick’s law diffusion and also with a simple expression for a I concentration dependent diffusivity according =t I Eq. (22). In Fig. 8 the resulting apparent diRusivities are plotted vs the saturated fraction x. In any case there are high values of D, at the beginning when x is low which are caused by the imperfection of the numerical calculation, but then the expected trends of D, are to observe which are in close agreement with the intrinsic diffusivity. Trying to introduce an intrinsic surface diffusivity, D,, according Eq. (19) a difficulty arises because D, is related to the coordinate x, on the internal surface, whereas the continuity equation refers to the radial coordinate xr. It was assumed Fig. 7. Gradient function for different parameters k, depending on surface concentration gradient. (a) f(ac/ar) = 1/(~3c/~W), that it is possible to combine both by a surface (b)f(ac/ar) = tanh (k&/ar)/(ac/ar). tortuosity factor X, so that X, = A, - x,.. There is no way to determine As,therefore it is regarded to variable concentration and a concentration gra- be included in the parameters k3 and k4 where the real jumping distance d has to be replaced by n/A,. dient dependent diffusivity in Eq. (1) results in The adsorption rate data in Fig. 3 show that there is not only a dependency of the apparent diffusivity on concentration gradient but also on concentration. Connecting the minimum values where of D, gives approximately a straight line representing the relation aD,e_ aD,, . E+ a% a% f cac/ar)

ax-

ac ax a(ac/ax) 'Xi?* (21)

This nonlinear partial differential equation with the same initial and boundary conditions as Eq. (1) was solved by aid of an explicit finite difference method. The radius of the sphere was subdivided into 12 increments, time intervalls of 10T4 time units had to be taken to ensure stable calculation procedure. Influences of discretisation and round-off errors were tested in the usual manner. The calculated concentration profiles were integrated by aid of Simpsons rule. Mean concentrations together with the appropriate times were used to calculate apparent diffusivities according Eq. (5) as it was done with the experimental data. Now arbitrary functions for concentration and gradient dependent intrinsic diffusivities could be inserted into this mathematical model. This was done at first with a con-

Da= D,.exp(-k,*c).

(22)

A corresponding equation for the surface diffusivity D,, was used as empirical expression for representation of concentration dependency, so that the final formula for calculation of transient diffusion of water vapor into microporous hydrated adsorbents is D,=k,.exp(k,*c)*

aclar

1*

(23)

This function contains several free parameters. These were fixed by a nonlinear least square fit between mathematical model and experimental data. The parameters for the set of experiments shown in Fig. 3 are given in Table 1. The para-

1571

W. W. KRUCKELS Table

No.

1. Parameters for surface ditfusivity according Eq. (23) calculated with data given on Fig. 3.

clz--c6l

k, (1V cm*/sec)

b -

ks (l/cm)

kd (cm)

Ax,, %

NOBS -

-9685 -8.677 - 6.201 -6.576 -3.827 -3863 -3.356 -3.571 -3.073 -3.352

45.95 120.8 25.12 75.54 2380 41.75 27.77 5294 3289 49.97

0.3945 0.2683 0.4187 0.3109 0.3999 0.3355 0.3213 0.2668 0.2612 0.2287

340 4.20 2.04 2.65 160

13

31.12

0.2647

3.00

54

Standard error (X 1oJ)

Final sum of squares (X 10s)

Least squarefitfor single steps 1 lat 2

0800

3 3at 4 4at 5 5at

0.3830-0.5575

2at

-0.2012

2.621 1.365 3.178 1.742 1.630 1.111 1-358 0.999 1.343 1.260

0.2102-0.3861

O-5573-0.7531 0*7526-09684

14 15 16 16

5.81 1.25 6.85 1.59 5.63 4.34 5.43 3.56 5.05 2.35

0.304 0.013 0.516 0.025 0.349 0.193 0.384 0.156 0.332 0.066

Least squarejitfor 5 steps simultaneously l-5

c0 = 0.2814

1.287

Temperature: 40°C Adsorbens: ta with Knudsen diffusion

-2.592 Grace 127

No. of particles: 19

meters are listed as well for single adaption as for common adaption of all five experiments. The results for calculated adsorption with best parameters for each step are shown in Fig. 3 as full lines. 9. DlSCUSSlON

OF

RESULTS

The comparison of measured and calculated adsorption kinetics shown in Fig. 3 allows one to assume that gradient dependency of surface diffusivity accounts for the observed effect. However, there is an inconsistency at the beginning of the process. The intrinsic diffusivity, as can be seen from Eq. 19, is starting with a minimum value and ending with a maximum of D,, whereas calculation due to the numerical method begins with a high value of D, which falls down to a minimum and finally shows the expected increase. The reason for this initial effect as mentioned before is the unsteady increase of concentration in the outmost particle shell at time zero. By subdividing the outmost shell and calculation of starting values with diminished time increments, the influence of this effect could be reduced. It is somewhat startling that the adsorption rate experiments show a similar behaviour. One might suppose that an effect comparable to that in calculation causes the coincidence. There must

20.43

29.21

Radius: 1a05 mm

be a rapid uptake of adsorbate at the very beginning of the experiments which also causes high values of D,. This is possible if there is a mechanism of transport which is orders of magnitude faster than surface diffusion. This could be Knudsen diffusion in the few macropores. It is possible to verify this suspicion by augmenting the instantaneous adsorbate increase at time zero which corresponds to the introduction of a AX,, into the calculation. The xe to be used to calculate the apparent diffusivity by Eq. (5) then is xe = Ax&l -xs)

+ XS.

(24)

With respect to the mathematical model of adsorption rate experiments this means one more free parameter. Table 1 allows a comparison of parameters for one pressure step with and without unsteady initial mass increase. It can be seen that the values of some parameters are markedly influenced by introduction of Ax0 although its amount is never more than 4.2% of the total mass increase during one experiment. Figure 3 allows one to compare the apparent dith.rsivity calculated with and without initial mass increase with experimental data. The coincidence of measurements and calculation for single steps is nearly perfect. Deviations

1572

On gradient dependent ditkivity

are in the range of experimental errors. The parameters for different steps however are not constant as to be expected. Especially the first two pressure steps give much higher values for parameter k3. This can be explained as follows. It is a well known fact that water is not completely desorbed from silica gel by degassing at 150”, so experiments are actually starting with a surface concentration cO. According the data given by Egorow et al. [ 193 this concentration is c,, = O-275 referred to monolayer concentration. Inserting c* = (c,+ c) in Eq. (23) in front of the gradient term leads to the last set of parameters in Table 1. After this correction it is possible to describe adsorption rate from zero up to monolayer concentration with a single set of parameters whose values also are given in Table 1. In Fig. 3 the apparent diffusivities for this case are presented too and show satisfactory agreement with measured data. The parameters given in Table 1 have a physical meaning. k3 allows one to calculate the mean jumping distance d. With the average value k3 = 50 cm-’ d of 4 - lo6 A is obtained. This cannot be correct, because the value of D, then would be several orders of magnitude larger than the measured one. So the question arises if the effective particle diameter may not be much smaller than the real particle diameter. Assuming the effective jumping distance is equal to the mean pore diameter of 10.5 A one obtains a value for the particle diameter of about 55 A. This speculation is strongly sustained by Acker[ 161 who summarizes his studies on the physical structure of dried silica gels as follows: “Dessicant gel is composed of spherical primary particles of about 70 A in diameter. The finer pores are probably located within these primary particles, the larger ones are located between the tightly bonded primary particles”. This model of pore structure is confirmed by the own adsorption rate experiments. If the radius of primary particles is 35 A the effective diffusivity however is several orders of magnitude lower than initially calculated. Instead of values of about 10m5cm%ec one obtains those of lo-l6 cm2/sec which correspond to values for solid diffusion at low temperatures.

The product k,k, is equal to (dEJdC)/(ST). It can be assumed that dE,/dC is constant. Then it is possible to calculate the decrease of activation energy between zero and monolayer surface concentration. Using the values of line 4 a in Table 1 this value is AE, = 8780 cal/mol. This corresponds to the value of the internal energy of evaporation of water (9730 cal/mol) and may confirm the idea that the mutual attraction of molecules which must be overcome in evaporation also accounts for the observed gradient dependency of surface ditfusivity. 10. COMPARISON BETWEEN GRADIENT DEPENDENT VISCOSITY AND DIFFUSIVITY For a gradient approaching zero the diffusivity reaches a limit which can be calculated using de I’Hopitals rule lim o,=D,(l+k,.k,.C)=D,,. dC,d.U+0

(25)

Extracting D., from Eq. (19) results in I _ tanh (k&/ax) k,aClax

)I’ (26)

The product k,k, generally is large, i.e. about 15. This means that for sufficiently large values of C it is possible to neglect l/(k,k,C) compared with one, so D, becomes

D = D tanh (k&X3x) I

11

(27)

.

k&33x

This equation is very similar to the corresponding expression for gradient dependent viscosity which is shown as follows: For non-Newtonian flow the so-called Prandtl-Eyring equation in conventional form is $=-C,.sinh

5 0

where C, and A are properties independent of shear stress and only dependent on temperature. Introducing the viscosity defined according to Newton’s law dw r=-ndy where 7 is assumed to be a gradient dependent variable, after some arrangements one obtains

ldw -.C dy

.

(30)

There is a close relation between Eq. (27) for gradient dependency of surface diffusion and Eq. (30) for gradient

1573

W. W. KRijCKELS

LdU6

Y3dcP

J 2dcP

-0.2

-0.L L 0.

,’ 0.L

0.2

0.6

0.6

X

1.0

10-6

/

Fig. 8. Adsorption kinetics calculated by means of dierent functions for D, compared with measureddata. (a) D, = const (b) 0, according Eq. (22) (c) D, according Eq. (23) (d) a measured apparent diffusivity.

0.

0.

dependent viscosity. This is shown in Fig. 9 where both functions are plotted vs the driving force gradient. The correspondence is not surprising because the physical principle for deduction of both formula is the same. Diffusivity is not a basic physical quantity. The results of these adsorption rate studies may also be expressed in terms of the surface flux. Using the parameters of Table 1 and the pertinent mean concentrations the flux was calculated and is plotted in Fig. 10 vs the concentration gradient with the concentrations as parameters. Observations made by Gomer[ 171 on surface diffusion of Hz and CO on tungsten by aid of field emission and field ion microscopy seem to confirm in some way the results in Fig. 10. For deposits of adsorbate in between 0.3 to 1.0 of monolayer, diffusion with a sharp boundary was observed, this f,(fJ 1.0 I 0.8

0.6

0.4

0.2

0.0 0.0

1.0

2.0

3.0

4.0

5.0 -

6.0

7.0

8.0

aC/ax(dwldy)

Fig. 9. Comparison of gradient functions for surface diffusivity fi = D,/D,, and viscosity f&= q/q,,according Eq. (27) and Eq. (30).

1

2.

3.

4.

5

- e._ g

Fig. 10. Surface flux depending on concentration and concentration gradient calculated with a diffusivity according Eq. (23) and parameters given in Table 1. c, = 040 c3= 0.470 Cl = 0.101 c* = 0.665 c, = 0.298 c, = O-861 corresponds to a flux independent of concentration gradient. For deposits insufficient for this type of diffusion, migration with a diffuse boundary i.e. Fickian diffusion occurred with a higher energy of activation. This is in global agreement with our results. 11. SUMMARY

While studying adsorption kinetics of water vapor on microporous hydrated gels an unexpected effect was observed. The apparent diffusivity calculated according Eq. (5) showed a marked increase with increasing degree of saturation. Such a behaviour can be explained by assumption of a gradient dependency of diffusivity which may be caused by the lateral interaction of adsorbed molecules. A qualitative equation for the intrinsic diffusivity which explains the gradient dependency is deduced. This expression for a concentration and gradient dependent diffusivity is inserted into the continuity equation, which was solved numerically. The solution is representing a mathematical model for the adsorption process. The four free parameters of

1574

On gradient dependent dilkivity

the model were fixed by a least square fit between model and observations. Corrections referring to the influence of Knudsen diffusion at the beginning of the process and to the fact that experiments were starting with a non-zero surface concentration had to be introduced. After these improvements it was possible to describe the rate behaviour for different experiments between zero and monolayer concentration with a single set of parameters which affirms the proposed model. The. measured adsorption kinetics also allow one to make conclusions on the pore structure of the gels. There must be a few macropores causing a rapid mass increase of about 3 % at the beginning of the transient process, followed by a slow concentration gradient dependent diffusion into primary particles of about 70 A dia, of which the gel is composed. The gradient dependency of surface diffusion will probably also be observed in other systems if the surface is and adsorbate energetically heterogeneous interaction energy is high. Acknowledgement-I am very grateful to Dip].-lng. B. Staehle for his help in calculation and performing accurate measurements.

Dll apparent ditfusivity, cmz/sec

De effective diffusivity, cm2/sec D8 intrinsic surface dxusivity, cm2/sec &? effective surface diffusivity, cm2/sec 6 activation energy of surface ditTusion, cal/mol surface mass flux, moleculeslsec resp. cmlsec k,, k,, ks,k4 parameters in Eq. ( 19) NL Avogadro number, molecules/mole relative radius, dimensionless r=x/R R particle radius, cm 92 gas constant, cal/mol grd t time, set T temperature, “K velocity, cmlsec W X9Y space coordinates, cm J

Greek symbols

shear stress, g/(cm sec2) 77 viscosity, g/(cm set) X saturated fraction, dimensionless, Eq. (3) 7

Superscripts . ”

maximum value minimum value mean value

NOTATION H

c C

mean jumping distance, cm sorbate concentration in particles, g/cm3, or dimensionless surface concentration, moles/cm2

Subscripts a w

value at beginning value at the end of the transient process

REFERENCES

VI EDNIE N. A., Ph.D. Thesis, University of Wisconsin, 195 1.

GROVES F. R., Ph.D. Thesis, University of Wisconsin, 1955. R., Kolloid-Z. 194197 135. RUTHVEN D. M. and LOUGHLIN K. F., Chem. Engng Sci. 197126 577. RUTHVEN D. M. and LOUGHLIN K. F., C/rem. Engng Sci. 197126 1145. DENGLER W. and KRUCKELS, W. Chem. Zng. Tech. 1970 42 1258. BABITT J. D., Canad. J. Phys. 195 129437. GILLILAND E. R., BADDOUR R. F. and RUSSEL, J. L., A.1.CL.E. Jf 1958,4,90. HIGASHI K., IT0 H. and OISHI J., J. Nuclear Sci. Tech. 1964,1,298. REYES R. R.; Ph.D. Thesis, University of Michigan, 1966. WEAVER J. A., Ph.D. Thesis, University of Delaware, 1965. KRUCKELS W. W., Habilitationsschrif, Uniuersitiit Stuttgart, 1972. CARSLAW H. S. and JAEGER J. C., Conduction of Heat in Solids, Oxford University Press 1959. SCHILLING J. D., Dr. Diss., Technische Hochschule Aachen, 1966.

t:; WICKE E. and KALLENBACH, [41 [A [61 171 PI r91 1101 1111 t::; 1141

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