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Multicomponent diffusion in porous media
Chemical
Engineering Science,
Press. 1973, Vol. 28, pp. 807-8 18. PCIWUIIO~
Printed ia
Great Britain
Multicomponent diffusion in porous media D. HESSE and J. KijDER Institut filr Physikalische Chemie der Universitiit Milnster, Germany (Received7 July 1972) Abstract-A transport equation for multicomponent dithrsion through porous media, which has been proposed in the literature, is tested experimentahy by studying the pressure dependence of ternary diffusion (pH,/oH,/He resp. pH,loH,/Arl through a fritted glass disk with a mean pore radius of P = 1~. The diffusion resistance has been measured in a pressure range between 2 and nearly 600 mm Hg using the experimental method of the “diffusion-reaction cell”. The linear dependence on pressure predicted theoretically has been observed for higher pressures only (p > 200 mm Hg). At lower pressures considerable deviations from the expected linear behavior can be found. In particular, at low pressures a minimum is observed in the resistance vs. pressure relationship. Taking into account the additional resistances due to the finitelength of the transportpores one can partiaIIyexplain these deviations. Finally the results are used to calculate the pressure dependence of the effectiveness factor for diffusion controlled o-p-H, conversion. At low pressures lower values
for this factor are obtained than predicted by the equation mentioned above. A surprising result is that the pressure dependence of the effectiveness factor shows a minimum for the o-p-I-&-He mixtures. 1. INTRODUCTION IN and
TECHNOLOGICAL solids
the
determined
by
solids
are used
and therefore
prOCeSSeS
involving
rate of conversion diffusion with small
effects.
large internal mean
RaSeS
is frequently Often surface
pore radii (P =
porous areas
100 A, [ 1,2]). In this case the rate of conversion is limited by diffusional transport in the small pores. If the conversion conditions (pressure, temperature) are chosen such that the mean free paths of the gases are comparable to the mean pore diameters, besides gas-gas, gas-wall collisions become important in determining the transport (diffusion in the transition region). Moreover, surface diffusion may contribute to the mass transport; this transport mechanism will be ignored in the present paper, however. In addition, heterogeneous conversions often occur (in the case of chemical reaction nearly always) in the presence of multicomponent mixtures. These mixtures-especially in the case of those in which the components have different molecular transport properties - show a number of characteristics which cannot be described by a simple Fick type equation[3,4]. For example, in the ethylene hydrogenation, which is controlled
by boundary layer diffusion, the different diffusion velocities of the reactants lead to a gas composition at the solid surface which is different from that expected from the stoichiometry of the reaction [S ,6]. If the reaction takes place at the internal surface of the catalyst, besides these often considerable concentration shifts other effects caused by transport process are possible. One expects, for example, that the different diffusion velocities of the components for “gas-wall diffusion” produce a total pressure gradient [7] which increases with decreasing mean pore radius. These effects may be important in the case of heterogeneous hydrogenation reactions, especially if parallel and simultaneous reactions can take place. These problems have hitherto not been discussed in the literature. In connection with reactor design for heterogeneous catalysis, questions of effectiveness factor, selectivity and poisoning of the catalyst used are important in practice. These questions are coupled-besides others -with the pore structure of the catalyst and the prevailing transport mechanism. Two catalysts having the
807
D. HESSE and J. KODER
same chemical nature, but different pore structure, can be completely different in those properties[8]. Thus, recent results show that the Thiele modulus is a function of F/i, the ratio of the mean pore radius to the mean pore length [9,101 (see also[ll, 121). To clarify the questions posed above, a reliable transport equation is needed which takes into account both the characteristics of multicomponent diffusion and the peculiarity of the catalyst’s pore structure. At present a transport equation for multicomponent diffusion in the transition region is discussed in the literature, which is derived by using the “pseudo-capillary” model [ 13, 141 or the “dusty gas” model [ 15,161. This equation, however, has not yet been tested experimentally. The corresponding law for binary diffusion has been found to hold relatively well in some cases [17], but one has also observed substantial discrepancies between experimental results and theoretical calculations. Some measurements can only be interpreted by assuming a pressure dependence of the structural factor[ 18,1]. Moreover, the mean pore radius calculated from the diffusion equation does not always agree with the value obtained from the measured pore-size distribution [ 191. Often the experimental data are not sufficient to verify the theory satisfactorily [20]. The present results, however, show thatwith certain restrictions-the diffusion equation for binary mixtures is useful in practice. It can only be decided by experiments, however, whether the equation proposed for multicomponent diffusion is also reliable. The general case of multicomponent diffusion in the transition region is very complex and therefore this case is not suitable to test the validity of the transport equation. Therefore the simple example of diffusion controlled o-p-H, conversion in the presence of the inert gases He or Ar was studied. The pressure dependence of the diffusion resistance for ternary di&.rsion through a porous fritted glass disk, which was mounted in a “diffusion-reaction cell” [2 l] was measured in the pressure range between 2 and nearly 600 mm Hg. To test the transport
equation the experimental results obtained are compared to values calculated using the structural factor of the glass disk. 2. TRANSPORT EQUATION MULTICOMPONENT DIFFUSION POROUS MEDIA
FOR THROUGH
If it is assumed that the porous solid can be approximated by a bundle of very long capillaries with a mean radius P (“pseudo-capillary” model), the transport equation for multicomponent diffusion through porous media can be derived by the momentum balance method[22] with the following assumptions: (a) Reflection of molecules colliding with the capillary wall is diffuse, (b) molecular diffusion and Knudsen diffusion are independent of one another, (c) the diffusional transport is not influenced by external forces; pressure diffusion is neglected This derivation results in the following equation [131: -gradci=&+$1’e,i=1,2,..N
(1) 11
IK
Here @if denotes the effective coefficient for Knudsen diffusion of component i and Dil is the binary diffusion coefficient for the components i and 1, assumed to be independent of concentration. One further assumes that the structural factor I/J(permeability) is independent of the diffusing species and the gas pressure. Usually this factor can only be determined experimentally. One often sets: Dr;(”= ~v~F$ Equation (1) therefore contains two parameters P and JI. The derivation of the transport equation for multicomponent diffusion using the “dusty gas” model leads to the same Eq. (1). This equation is only valid for diffusion through porous media with a uniform pore-size distribution [20,2 1] and pores with inflnitely long length (iB F)[9, lo]. A summation of Eq. (1) over all i gives:
808
-grad c = 2“1 i=l
Deff.
iK
Multicomponent diffusion in porous media
in a Fickian form:
According to this the fluxes cannot be chosen freely under a given pressure gradient and vice versa. If one sets, for example in a WickeKallenbach experiment [23], the pressure gradient equal to zero, the fluxes are coupled in the following way:
; = - Dieff grad
with 1 Qeff -=
i=l
With this constraint
the diffusion process thus
produces
mass
a
total
flux:
-$&+
1 rlr&z
The essential feature of this equation is that the generalized effective diffusion coefficients Dieffare functions of the mole fractions of all components in the mixture, the molecular transport properties of the gases, the stoichiometric
m = i$1 Miji.
Cunningham et al.[24] and Remick et al. [25] solved the transport Eq. (1) with this constraint for ternary diffusion through a capillary (J, = 1) and showed that besides the effects typical for multicomponent diffusion (osmotic diffusion, reverse diffusion) other peculiarities are possible in the transition region. However, the theoretical treatment of this transport problem is very complex even for this relatively simple case. Moreover, generally valid statements for multicomponent diffusion in the transition region cannot be derived from these considerations. In the case of the practically important questions mentioned above, the fluxes are coupled by the stoichiometry of a chemical reaction:
if1ViAi
coefficients
of the reaction
*$I v& = 0 and the
parameters P and $I. Since the pressure varies with position the treatment of this transport problem is in general very complex. The case of diffusion controlled ethylene hydrogenation which was mentioned above [6] is therefore too complex to be used in an experimental verification of the transport Eqs. (1) or (6) respectively. A reaction that is better suited for this purpose is the o-p-H, conversion in the presence of an inert gas. We have the reaction: pHe ---*OH,.
(8)
= 0:
From Eq. (4) follows (vl = ---
.ii _
jk
Vi
Vk'
;=-;;ff=o
(4)
Equation (4) says that in the steady state the reactants are transported to the place of conversion in relative amounts corresponding to the stoichiometry of the reaction. Therefore the fluxes are uniquely determined. From Eq. (2) there results a stationary total pressure gradient along the transport path [7]: -gradc=gzs.
(6)
C(
(5)
v2, v3 =
0): (9)
where the indices denote: 1 + pH2, 2 + OH,, 3 + inert gas. Since OH, and pHZ have the same molecular transport properties, for diffusion in the t direction there is always: de z=
0.
(10)
For this diffusion problem there exists only an effective diffusion coefficient Deffwhich is given by Eq. (7) to be:
In this case there is no total mass flux: i$1 Mdi = 0. Using Eq. (4), Eq. (1) CEUI be written 809
D. HESSE
and J. KiiDER
where DT1is the self-d8usion coefficient for H, at 297 K and p. = 76Oinm Hg, D& the binary diffusion coefficient Hz/inert gas at the same conditions and D& = 3 Lvl the coefficient for the Knudsen diffusion of hydrogen. According to Eq. (11), a plot of diffusion resistances, l/Pff vs. pressure should yield straight lines with slope: 1 -=i[l-,. +QDc” $D&
(l-%)I-$
(12)
The common intersection of these straight lines with the ordinate is l/t,!&. The structural factor, $, which is still unknown, and the axis intersection, 1/I,@&,can be determined experimentally from the pressure dependence of the pure o-p-H, diffusion (_x~= 0) ifOf is given. In these experiments it was found that the pressure dependence of the self diffusion of hydrogen through the fritted glass can not be described by a simple Bosanquet formula[26]:
&
(x3 = 0) =
l+Le. Wi
W&PO
(13)
The measurements yielded higher transport resistances than predicted from this equation for small gas pressures (p < 200 mm Hg) [ 101. The deviations from the Bosanquet straight line were attributed to the effect that in the fritted glass, which has a fairly uniform pore-radius distributation[27], short pores determine the transport. In this case, the finite pore length has to be considered in the transport equation by “end correctionsj’ [9]. The pore models mentioned in the introduction can thus not be used without modification to describe the pressure dependence of the diffusion resistance. Instead, the fritted glass was approximated by a system of pores of various lengths and various radii, both of the same order of magnitude connected in series, where the pores were separated by open volumes having the shape of cylindrical disks. This pore model represents quite well the pressure dependence of the diffusion resistance, but its disadvantage is that it is very unhandy for
practical calculations with its seven model parameters which are difficult to assess. In the following the model will therefore be brought into a form that makes it suitable for practical applications. For this we assume the pore structure of the fritted glass to be a system of pores connected in series and having the same mean length, I; and mean pore radius, F. Due to the finite pore length the transport equation has to consider the influence of “end corrections” on the diffusion process as explained in earlier papers [9, lo]. It was found that the additional resistances due to “end corrections” are different for the two limiting cases of Knudsen diffusion and molecular diffusion, since they are caused by different effects. In the case of Knudsen diffusion the theory of Clausing[28] and de Marcus [29] yields smaller diffusion coefficients for the shorter pores than for the longer ones (both having the same radius), because for short pores the back scattering of molecules through the entrance opening due to gas-wall collisions is more important than for long ones. This effect yields a greater transport resistance for diffusion than calculated from the relationship: 0, = 3 FU. In the regime of molecular diffusion the additional resistance is due to the fact that in the entrance and end openings of the pores the diffusion flux lines are no longer parallel to each other but converge or diverge on entering the neighbouring medium. These deformations of the concentration field result in an additional transport resistance which is not considered in Eq. (1) and respectively Eq. (11) (see also [30]). Let (14) be the resistances caused by these effects for Knudsen diffusion (RKZi) or molecular diffusion (RGZi), resp., of component i, with
810
Multicomponent
diffusion in porous media
through a porous replaced by
and 6li = jr/ji. These additional resistances are only valid for the given limiting cases. Weighted with the expected probabilities for Knudsen diffusion (gK) and molecular diffusion (gc) in the pore, they give the additional resistance in the region between the limiting cases:
medium,
Eq. (1 1) is then
P
P,
(16)
With the assumptions that (Yand j3 do not depend on pressure and diffusing species and that gK and gG are the same for all components i of the mixture, the transport equation for multicomponent diffusion in z direction through a porous medium with finite pore length is given by:
The unknown quantities $, (Yand p in Eq. (19) can be determined from the pressure dependence of the pure o-p-H, diffusion (x, = 0) if ? is known. For those gas pressures for which 1 s DK/DT2 . p/p0 holds, a linear behavior follows from Eq. (19): 1 1’ D*e” =$z++&
i=1,2
,....
N.
(17)
The assumptions made here for gx and gG appear reasonable since the occurrence of one of the limiting mechanism, Knudsen or molecular diffusion, is controlled by the entire gas mixture because of the coupling between the diffusion of the components i. It has yet to be studied how g, and gG are to be defined in general. For the present example of inert gas impeded o-p-Hz-diffusion, g, and gG can be given relatively easily in analogy to the pure o-p-Hz diffusion since the ternary mixture is characterized by only one coefficient for Knudsen diffusion and molecular diffusion. The functions gK and gG further have to yield the case for the pure o-p-Hz ‘diffusion for the limiting case xg + 0. Therefore, it was set: (18)
For the pressure dependence of the dill&ion resistance of inert gas impeded o-p-H, diffusion
(19)
(20)
(1 +@pt’
Extrapolation of this straight line to p + 0 gives, for known r, the value of the structural parameter +. With the diffusion coefficient, D&, known the slope of the straight line yields p. If the experimental results for small pressures are plotted as l/D*eff vs. p and extrapolated to p 4 0, the intersection with the ordinate is: ~(P-fo)=-&(l+a)
(21)
K
from which (Y can be calculated. With the parameters thus determined and the values of Dy3 given, the diffusion resistance l/D*eff can be calculated from Eq. (19) for all values of x3 and p. 3. EXPERIMENTAL EVALUATION
APPARATUS OF DATA
AND
Since the measuring apparatus has been described in detail elsewhere [3 1,321, the principle of the measurements will only briefly be discussed. To study the pressure dependence of the inert gas impeded o-p-H, diffusion, the “difisionreaction” cell stretched in Fig. 1 was used, in which a f&ted glass disk (characteristics: see
811
D. HESSE and .I. KODER Reaction with
cp
chamber catalyst
Table 1. Characteristics of the fritted glass disk
&-Par
-
ay
glass disk
I
I Mixing
0.5 < L< 0.85 /.L
chamber
Fig. 1.Diffusion-reaction cell, schematically.
Table 1) separates a mixing chamber from a reaction chamber. Through the mixing chamber there is a constant flow of the ternary mixture pHJoHJinert gas. The input flow (concentration co) contains for all measurements pH,-oHe mixtures of the composition of the low temperature equilibrium which is established on charcoal at liquid nitrogen temperature [33]: clO: czo = 1: 1. Pure helium or argon were used as inert gas. Mixtures containing 10, 50 and 90 per cent of helium and 10, 29,48,7 1 and 9 1 per cent of argon were studied. The hydrogen diffusing through the fiitted glass into the reaction chamber is converted spontaneously to the high temperature equilibrium (cl1 : czl = 1: 3) at the supported platinum catalyst? (5% Pt on SiOJ even at room temperature. The added. inert gas does not partake in the reaction and is therefore not transported in the steady state. Thus the hydrogen diffuses through inert gas standing in the fritted glass. Due to the o-p-H, conversion taking place in the reaction chamber a time independent partial pressure gradient for pH2 and OH, is created. From the mass balance in the mixing chamber one obtains: s(ciO-ciA)
L=0.6cm F = 7.07 cm* rz= 0.350
Thickness Area Total porosity Interval of pore radii [27]
=D*effF
.v
A--.I (22)
where B is the volume velocity of the gas flowing through the mixing chamber and cio, Ci”, c: are the concentrations of component i in the feed gas, exhaust gas and in the reaction chamber (see Fig. 1). With the gas pressure given and with
constant volume velocity d, the diffusion resistance l/D*eff can easily be calculated from Eq. (22) if the concentrations ci have been determined. This was done utilizing a heat conductivity cell which makes use of the different heat conductivities of OH, and pH2 at liquid nitrogen or oxygen temperature. Since the area F of the fritted glass disk and its thickness L could not be determined as accurate as the concentrations, they were, together with the permeability $, combined to a factor f= L/F+. From Eq. (22) follows then: (23) The diffusion resistances calculated according to Eq. (23) from the concentration data were reproducible to 1~5%. Measurements were performed at 297 K in the pressure range of 2 to about 600 mm Hg. 4. EXPERIMENTAL
RESULTS
1. Determination of the parameters In order to determine the parameters CY,~ and f (see Eq. 19) the pressure dependence of the pure o-p-H, diffusion was measured first. The results are plotted as dots in Fig. 2. The slope of the straight line expected for high pressures is ( p = p. = 760 mm Hg) : &- (l+p)
=0*484
(24)
12
and the intersection gives:
tThe authors thank Dr. Krahl of Heraeus, Hanau, for donating the catalyst used.
812
#
with the ordinate
K
= O-068.
(p + 0)
(25)
Multicomponent
diffusion in porous media Table 3. Binary difhtsion coefficients forp,=76OmmHgand T=297K
045
040 t
1445 cm2 set-1 [34] O-785cm2 set-r [34] 1.51 cmzsec1[34]
Fig. 2. o-p-H,
_Eq.(lS) EF.pelimnt
diffusion through the glass disk, T = 297°K.
The extrapolation of the experimental points at low pressures is not unambiguous since after passing through a minimum a steep rise of the diffusion resistance is observed with decreasing gas pressure. This behavior cannot be reconciled by Eq. (19). IfflD* (p + 0) is chosen to be 0.1, the parameters given in table 2 are calculated with P = 7*10-5 cm (see Table 1). With these data, thef/D* vs. p curve plotted in Fig. 2 can be calculated from Eq. (19), which apart from the pressure region p < 100 mm Hg fits the data within experimental error. 2. Pressure dependence of the ternary difusion through the fritted glass disk With the parameters thus obtained and the binary diffusion coefficients listed in Table 3, the curves shown in Fig. 3 for the mixture pH,/oH,/Ar with x3 = 0.1, 0.48 and 0.91 can be calculated from Eq. (19). In particular at gas pressures of p > 150 mm Hg the experimental points deviate from the calculated curve more and more with increasing x3. Even at low pressures marked deviations from the plotted
o-01
’
d0
1
’
200
Fig. 3. o-p-H,-Ar
1
1
1
1
MO
P hmHg1
diision
I
400
I
500
I
1
,600
’
through the glass disk, T = 297”K.
curve can be seen. Qualitatively, however, the theory predicts correctly the pressure dependence. The pressure dependence of the diffusion resistances for the pH,/oH,/He mixtures (x3 = 0.1, O-5, 0.9) are drawn in Fig. 4 through 6. Whereas Eq. (19) predicts the f/D* values for p > 100 mm Hg within experimental error, a considerable increase of the f/D* values is observed with decreasing gas pressure particularly for the mixtures with x3 = 0.5 and 0.9.
Table 2. Parameters for diffusion through the glass disk Mean pore radius Structural factor Permeability End corrections
f= f= $= a! = /‘3=
O-7/L(see Table 1) 0.5617 0.151 0.4706 0.2708
0.05
1
,&
’
1
’
1
’
1
1
1
1
I
I
P hmHg1
Fig. 4. o-p-Hz-He
diffusion through the glass disk,x, = 0.100; T = 297°K.
D. HESSE
0.03
’
,t,
2,
’
’
&
’
’
’
P[mmHgl Fig. 5. o-p-H,-He diffusion
0.05
’
,k
’
’ 300
1
’
and J. KGDER
’
through theglass disk,x, = 0.500; T = 297”K.
,b,
’
&
’
4&j
’
&
’
&
’
p [mmHgl
Fig. 6. o-p-Hz-He
diffusion through the glass disk,x, = 0900; T = 297°K.
5. DISCUSSION
OF RESULTS
“end corrections” in the By considering transport equation for diffusion through porous media, it is possible to interpret qualitatively the deviations from linearity of the function f/D* vs. p. (Eq. 11). Equation (19) does not quantitatively describe, however, the results for low gas pressures, p < 100 mm Hg. A comparison of the pH,/oH,/He diffusion data with those for pH,/oH,/Ar suggests that the observed deviations can not be explained by the iniluence of the pore structure alone. The assumption made for Eq. (1) that molecular diffusion and Knudsen diffusion are independent might not be true. Probably the directional distribution of the molecules after gas-gas collisions has at low pressures a different influence on the mass transport than assumed in Rothfeld’s deviation of Eq. (1). This conjecture is suggested by the
finding that the light helium atoms, being anisotrop scaterers [ 3 5,3 61, give larger deviations from the theoretical curves than the heavy argon atoms which scatter isotropically. Also, it can not yet be established to date to what extent the pore structure can modify the differences in scattering behavior of the inert gas molecules. Apart from these deviations, higher JTD* values than calculated from Eq. (19) are found for pH,/oH,/Ar mixtures even at gas pressures of p > 150 mm Hg. Assuming that these differences are due to the value of the binary diffusion coefficient D,0,(H,/Ar) used in the calculations, the deviations can be removed by using for DTS (H,/Ar) the data indicated by dots in Fig. 7. These data are obtained from the slope of the straight lines found at higher pressures. According to Fig. 7, Ofa qualitatively show the same concentration dependence as measured already by Ker1[37]. The absolute values for D&, however, are smaller by about 11 per cent than Kerl’s data and smaller by about 12 per cent than Waldmann’s data [3 81. The concentration dependence obtained in the present study for DtS (H,/Ar) agrees with the theoretical expectation [39], thus the given interpretation of the deviation is reasonable. In Fig. 8, the flD* vs. p curves have been recalculated from Eq. (19), taking into account the DL (H,/Ar) values indicated in Fig. 7. Now theory and experiment agree within experimental error for pressures of p>80mmHg. If the f&ted glass is assumed to be a system of pores connected in series, with mean pore 0.86 OM-
omo-a-
0
0.78 076 -
-
06-
x0
x
x
0.74-
EID on.oy)1 *g o-7068o-660.64 O-62-
0
0
0
x
x
.
XKei [371 OWoldnmd381 -This wxk
. .
l
.
O!I
I 0.2
I 0.3
I 04
I DS
I 06
I 01
I o*
I 09
I to
13
Fig. 7. Concentration dependence of the binary diffusion coefficient for the H,/Ar mixture; p = 670mmHg; T = 297°K.
814
Multicomponent diffusion in porous media
obtained from the micrographs is probably not accidental. Moreover, from geometrical considerations for the packing of spheres the ratio of the mean radius, P, of the channels between the spheres to the mean length of the channels, 1, is expected to be 9 -&[40,41]. To a tist approximation fritted glass can be considered to be a packing of spheres[42], thus making the obtained value reasonable also from this point of view. The micrographs give another maximum of the probability distribution around y = O-5 (see Fig. 9). Transport pores with this J value exhibit a minimum in the f/D* vs. p curve (see[lOl). However, the theory does not give the same steep rise as measured after passing through the minimum.
Fig. 8. o-p-Hz-Ar diffusion through the glass disk, T = 297°K; D& values from Fig. 7.
radius P and mean pore length I, and separated by volumes with the shapes of cylindrical disks, then one obtains approximately (for r/r= ji s 2.5)[9,10]:
6. PRACTICAL
SIGNIFICANCE
OF RESULTS
In the following we shall briefly discuss the significance of the obtained results for the 2.9 y = (Y. (26) practical problems mentioned in the introduction. With (Y= O-471 one therefore obtains: J = As an example, the determination of the effectiveO-16. The detailed pore model developed in an ness factor for the pores shall be considered. earlier paper for the f&ted glass yielded 9 = We assume that the o-p-H, conversion takes O-178.From a series of scanning electron micro- place in a porous catalyst plate of thickness L graphs of the f&ted glass the relative probability and area F, and that the reaction is of fnst order H(y) = N(y)/N,,, to find pores with a para- with pressure-independent rate constant kO.The meter value of y was determined, with the result reaction rate shall be large enough so that the that nearly one third of the measured pores N,, conversion is controlled by diffusion in the had a value of 7 = O-2 (see Fig. 9). Although interior of the catalyst. At the location, z = L, the determination of 7 values is problematic the gas composition always corresponds to the (e.g. because of parallax errors), the rather good high temperature equilibrium: cl : c, = 1: 3, agreement between the calculated value and that and let (dcl/dz)s=L= 0. Under isotherm steady state conditions the effectiveness factor (see also 0.50 [43]) is given by r) c 6: 0.40
s I
-
0.30f
gj
0.20-
l
OGQiI ,,v,
]
t
,,,,,,,,, 0.2
0.4
06
0.8
I.0
I.2
I.4
Y
Fig. 9. Relative probability for finding pores with the value y in the fritted glass disk.
According to Eq. (27), the effectiveness factor depends apart from the constant k0 and the geometry of the catalyst (L, F), in particular on the diffusion resistanceflD* and thus on the pore structure of the catalyst. Generally it is assumed that the influence of pore structure on the mass transport is identical for the case of a chemical reaction taking place as for that with
815
D. HESSE
and J. KODER
no conversion[44,45], as long as the pore structure is not modified during the reaction. With this assumption, the results obtained in this study can be used to discuss the influence of the pore structure on the value of thehffectiveness factor. Therefore we first assume that the diffusional transport is done by the Knudsen mechanism. If one compares the effectiveness factors for different jj = r/l, one obtains from Eq. (19): 7,=_-12e Vi-G
application of the “pseudo-capillary” model can be demonstrated particularly clearly when the results of the present study are used to calculate the pressure dependence of the reduced effectiveness factor
(28)
where q. is the effectiveness factor for catalysts in which the length, t of the transport pores is large compared to their radius, J. If (Y= 2.99 is introduced (Eq. 26), the relationship d/q0 vs. jj is obtained as plotted in Fig. 10. Porous catalysts with J values different from zero can thus have considerable lower effectiveness factors than those with pores of length t% P (see also [l 11). This result should be of interest for the production of effective catalysts. In practice such catalysts are frequently used in which at about 700°C and not too high pressures (p < 5 atm) mass transport is mainly done by Knudsen diffusion. Under these conditions the influence of “end corrections” explained above could be important for the calculation of the effectiveness factor. If, for instance, the effectiveness factor is calculated with a structural factor determined from diffusion studies in the regime of molecular diffusion, then the q. value given by the “pseudo-capillary” model would indicate a higher efficiency of the catalyst than could be derived from the actual conversion data. This could give rise to erroneous interpretation of the results. The errors introduced by uncritical
If r~,.vs. l/Gis calculated without considering “end corrections”, the curve plotted as a solid line in Fig. 11 is obtained. For p + 0 this yields a value of vT( p + 0) = 1.86. Considering the finite pore length one finds, on the other hand, r),= 1.54 (for flD* (p+ 0) =O.lOO). This value is smaller by about 20 per cent than that given above.
lX3=0oocl xX,=0900
Fig. 11. Pressure dependence of the reduced effectiveness factor; (o-P-Hz-He mixtures); T = 297°K.
If helium is added to the o-p-H, mixture one would expect according to the model that the function qr (p) is practically the same as for the pure o-p-H, mixture even for the case x3 = 0.9 considered here, since 0:: and D& differs only slightly. Figure 11 shows that this is only true for higher gas pressures. At pressures of p < 100 mm Hg, 7,. goes through an unexpected maximum and drops then sharply with decreasing gas pressure. Although presently no physical interpretation can be given for this behavior, it can be concluded that a prediction of the effectiveness factor can lead to largely erroneous of ditfusional Fig. IO. Reduced effectiveness factor as a function off = i//1. values unless the characteristics 816
Multicomponent diffusion in porous media
F
transport in porous media are properly considered. In particular, effective diffusion coefficient measured at atmospheric pressure can generally not be extrapolated to higher or lower pressures using the “pseudo-capillary” model if accurate values are to be obtained (see also[ 17,461).
area of the glass disk, cm2 structure factor, cm-‘, f = L/F+ f molar flux of component i, mole cm-2 set-1 i thickness of the glass disk, cm i mean pore length, cm molar mass of component i, g mole-’ Mi P total pressure, mmHg mean pore radius, cm absolute temperature, K Vi mean molecular velocity of component i, cm set-’ volume velocity of the gas, cm3 see-’ mole fraction of, component i quotient: F/i spatial coordinate, cm
T’
Acknowledgement-We are grateful to Prof. Dr. E. Wicke for many helpful discussions. One of us (D. H.) thanks the Deutsche Forschungsgemeinschaft for financial support. NOTATION
total concentration, mole cme3 concentration of component i, mole cme3 effective coefficient for Knudsen diffusion of component i, cm2 see-’ binary diffusion coefficient for component i and 1,cm2 set-’ 0: Dilfor p0 = 760 mm Hg, T = 297 K 0;" generalized effective diffusion coefficient of component i, cm2 set-’ DK coefficient for Knudsen diffusion of component i through a pore with mean pore radius F,cm2 set-l DFff 1G effective coefficient for molecular diffusion of component i (Eq. 15), cm2 see-’
Greek (Y
j3 E JI Q vf
symbols
parameter in Eq. ( 14) parameter in Eq. ( 16) total porosity of the porous glass disk permeability of the glass disk reduced effectiveness factor, Eq. (29), sec-1’2 stoichiometric coefficient of component i, vt > 0 for products, yf C 0 for reactants vi = 0 for inert components.
REFERENCES [l] SAlTERFIELD C. N., Mass Transfer in Heterogeneous Catalysis, M.I.T. Press, London, 1970. [2] RI’ITER H. L. and DRAKE L. C., Znd. Engng Chem. Anal. Ed. 1947 17 787. [31 TOOR H. L..A.Z.Ch.E. Jl1957 3 198. [4] DUNCAN J: B. andTOORH. L.,A.Z.Ch.E.J/1962838. [5] HUGO P., Chem. Engng Sci. 1965 20 187; 385; 975. [6] HESSE D. and HUGO P., to be published in Chem. Znge. Techn. [7] RESHETOV V. A., 2. Fiz. Khim. 1970 44 3057. [S] WHEELER A., Advances in Catalysis, Vol. 3, p. 250. Academic Press, New York 195 1. [9] HESSE D., to be published in Z. phys. Chem. [lo] HESSE D., to be published in Chem. Znger. Techn. [ 1 l] VERHOFF F. H. and STRIEDER W., Chem. Engng Sci. 197 126 245. [12] PENTENERO A. and LE GOFF P., J. chim. Phys. 197168 1027. [13] ROTHFELD L. B.,A.Z.Ch.E. Jll963 9 19. [14] SCOmD.S.andDULL1ENF.A. L.,A.Z.Ch.E.Jl 19628 113. [15] EVANS R. B., WATSON C. M. and MASON E. A. J. them. Phys. 196135 2076. [16] MASON E. A., MALINAUSKAS A. P. and EVANS R. B., J. them. Phys. 1967 46 3 199. [17] YOUNGQUIST G. R., Znd. Engng Chem. 197162 52. [I81 JUST?. J. and GELBIN D., Chem. Technik 197123466; 581; 648. [ 191 HORAK Z. and SCHNEIDER P., Chem. Engng Jll97 12 26. [20] CUNNINGHAM R. S. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1968 7 535. [21] HUGO P. and WICKE E., Chem. Znge Techn. 1968 40 1133. [22] PRESENT R. D., Kinetic Theory of Gases. McGraw-Hill, New York, 1958. [23] WICKE E. and KALLENBACH R., KolloidZ. 194197 135. [24] CUNNINGHAM R. S. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1968 7 429. [25] REMICK R. R. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1970 9 206. [26] BOSANQUET C. H., British TA Rept., BR-507,1944.
817
D. HESSE
and J. KGDER
[271 FRANK W., GZT-Zeitschr. 1967 11683. [281 CLAUSING P., Ann. Phys. 1932 12 961. 1291 DE MARCUS C., ORGDP Rep. K-1302, Part 3,1957. i3oj 1 HUGO P., unnublished. t3 11 BEYER H. D:, Diploma&it, Miinster 1966. [=I KODER J., DIPLOMARBEIT, Miinster, 1972. [331 CREMER E. Handbuch der Katalyse, Vol. 6. Springer, Wien 1943. MULLER R., Chem. Znge Techn. 1968 48 344. tiz; HIBY W. and PAHL M., Z. Phys. 1951 129 517. [361 HIBY W. and PAHL M., Z. Phys. 1951138 348. KERL K., Dissertation, Braunschweig, 1968. ::i; WALDMANN L., Z. Phys. 1948 1242. MASON E. A.. .I. chem. Phvs. 1957 27 75: 782. ::z; MANEGOLD’E. and SOLF K., Kolloidi. 1939 89 36. 1411 HIBY W. and PAHL M., Z. Narurf. 1956 lla 80. t421 PRAUSNITZ P. H..?KolloidZ. 1943 104 246. [431 WICKE E. and BROTZ W., Chem. Znger. Tech. 1949 21219. [441 HUGO P. and PARATELLA A., Simposio di Dinamica delle Reazioni Chimiche, Padova 1966, Vol. V. Roma 1966. [451 HAWTIN P., Chem. Engng Sci. 197126 1783. [461 BROWN L. F., HAYNES H. W. and MONOQUE W. H.,J. Cntal. 1969 14 220.
818
Engineering Science,
Press. 1973, Vol. 28, pp. 807-8 18. PCIWUIIO~
Printed ia
Great Britain
Multicomponent diffusion in porous media D. HESSE and J. KijDER Institut filr Physikalische Chemie der Universitiit Milnster, Germany (Received7 July 1972) Abstract-A transport equation for multicomponent dithrsion through porous media, which has been proposed in the literature, is tested experimentahy by studying the pressure dependence of ternary diffusion (pH,/oH,/He resp. pH,loH,/Arl through a fritted glass disk with a mean pore radius of P = 1~. The diffusion resistance has been measured in a pressure range between 2 and nearly 600 mm Hg using the experimental method of the “diffusion-reaction cell”. The linear dependence on pressure predicted theoretically has been observed for higher pressures only (p > 200 mm Hg). At lower pressures considerable deviations from the expected linear behavior can be found. In particular, at low pressures a minimum is observed in the resistance vs. pressure relationship. Taking into account the additional resistances due to the finitelength of the transportpores one can partiaIIyexplain these deviations. Finally the results are used to calculate the pressure dependence of the effectiveness factor for diffusion controlled o-p-H, conversion. At low pressures lower values
for this factor are obtained than predicted by the equation mentioned above. A surprising result is that the pressure dependence of the effectiveness factor shows a minimum for the o-p-I-&-He mixtures. 1. INTRODUCTION IN and
TECHNOLOGICAL solids
the
determined
by
solids
are used
and therefore
prOCeSSeS
involving
rate of conversion diffusion with small
effects.
large internal mean
RaSeS
is frequently Often surface
pore radii (P =
porous areas
100 A, [ 1,2]). In this case the rate of conversion is limited by diffusional transport in the small pores. If the conversion conditions (pressure, temperature) are chosen such that the mean free paths of the gases are comparable to the mean pore diameters, besides gas-gas, gas-wall collisions become important in determining the transport (diffusion in the transition region). Moreover, surface diffusion may contribute to the mass transport; this transport mechanism will be ignored in the present paper, however. In addition, heterogeneous conversions often occur (in the case of chemical reaction nearly always) in the presence of multicomponent mixtures. These mixtures-especially in the case of those in which the components have different molecular transport properties - show a number of characteristics which cannot be described by a simple Fick type equation[3,4]. For example, in the ethylene hydrogenation, which is controlled
by boundary layer diffusion, the different diffusion velocities of the reactants lead to a gas composition at the solid surface which is different from that expected from the stoichiometry of the reaction [S ,6]. If the reaction takes place at the internal surface of the catalyst, besides these often considerable concentration shifts other effects caused by transport process are possible. One expects, for example, that the different diffusion velocities of the components for “gas-wall diffusion” produce a total pressure gradient [7] which increases with decreasing mean pore radius. These effects may be important in the case of heterogeneous hydrogenation reactions, especially if parallel and simultaneous reactions can take place. These problems have hitherto not been discussed in the literature. In connection with reactor design for heterogeneous catalysis, questions of effectiveness factor, selectivity and poisoning of the catalyst used are important in practice. These questions are coupled-besides others -with the pore structure of the catalyst and the prevailing transport mechanism. Two catalysts having the
807
D. HESSE and J. KODER
same chemical nature, but different pore structure, can be completely different in those properties[8]. Thus, recent results show that the Thiele modulus is a function of F/i, the ratio of the mean pore radius to the mean pore length [9,101 (see also[ll, 121). To clarify the questions posed above, a reliable transport equation is needed which takes into account both the characteristics of multicomponent diffusion and the peculiarity of the catalyst’s pore structure. At present a transport equation for multicomponent diffusion in the transition region is discussed in the literature, which is derived by using the “pseudo-capillary” model [ 13, 141 or the “dusty gas” model [ 15,161. This equation, however, has not yet been tested experimentally. The corresponding law for binary diffusion has been found to hold relatively well in some cases [17], but one has also observed substantial discrepancies between experimental results and theoretical calculations. Some measurements can only be interpreted by assuming a pressure dependence of the structural factor[ 18,1]. Moreover, the mean pore radius calculated from the diffusion equation does not always agree with the value obtained from the measured pore-size distribution [ 191. Often the experimental data are not sufficient to verify the theory satisfactorily [20]. The present results, however, show thatwith certain restrictions-the diffusion equation for binary mixtures is useful in practice. It can only be decided by experiments, however, whether the equation proposed for multicomponent diffusion is also reliable. The general case of multicomponent diffusion in the transition region is very complex and therefore this case is not suitable to test the validity of the transport equation. Therefore the simple example of diffusion controlled o-p-H, conversion in the presence of the inert gases He or Ar was studied. The pressure dependence of the diffusion resistance for ternary di&.rsion through a porous fritted glass disk, which was mounted in a “diffusion-reaction cell” [2 l] was measured in the pressure range between 2 and nearly 600 mm Hg. To test the transport
equation the experimental results obtained are compared to values calculated using the structural factor of the glass disk. 2. TRANSPORT EQUATION MULTICOMPONENT DIFFUSION POROUS MEDIA
FOR THROUGH
If it is assumed that the porous solid can be approximated by a bundle of very long capillaries with a mean radius P (“pseudo-capillary” model), the transport equation for multicomponent diffusion through porous media can be derived by the momentum balance method[22] with the following assumptions: (a) Reflection of molecules colliding with the capillary wall is diffuse, (b) molecular diffusion and Knudsen diffusion are independent of one another, (c) the diffusional transport is not influenced by external forces; pressure diffusion is neglected This derivation results in the following equation [131: -gradci=&+$1’e,i=1,2,..N
(1) 11
IK
Here @if denotes the effective coefficient for Knudsen diffusion of component i and Dil is the binary diffusion coefficient for the components i and 1, assumed to be independent of concentration. One further assumes that the structural factor I/J(permeability) is independent of the diffusing species and the gas pressure. Usually this factor can only be determined experimentally. One often sets: Dr;(”= ~v~F$ Equation (1) therefore contains two parameters P and JI. The derivation of the transport equation for multicomponent diffusion using the “dusty gas” model leads to the same Eq. (1). This equation is only valid for diffusion through porous media with a uniform pore-size distribution [20,2 1] and pores with inflnitely long length (iB F)[9, lo]. A summation of Eq. (1) over all i gives:
808
-grad c = 2“1 i=l
Deff.
iK
Multicomponent diffusion in porous media
in a Fickian form:
According to this the fluxes cannot be chosen freely under a given pressure gradient and vice versa. If one sets, for example in a WickeKallenbach experiment [23], the pressure gradient equal to zero, the fluxes are coupled in the following way:
; = - Dieff grad
with 1 Qeff -=
i=l
With this constraint
the diffusion process thus
produces
mass
a
total
flux:
-$&+
1 rlr&z
The essential feature of this equation is that the generalized effective diffusion coefficients Dieffare functions of the mole fractions of all components in the mixture, the molecular transport properties of the gases, the stoichiometric
m = i$1 Miji.
Cunningham et al.[24] and Remick et al. [25] solved the transport Eq. (1) with this constraint for ternary diffusion through a capillary (J, = 1) and showed that besides the effects typical for multicomponent diffusion (osmotic diffusion, reverse diffusion) other peculiarities are possible in the transition region. However, the theoretical treatment of this transport problem is very complex even for this relatively simple case. Moreover, generally valid statements for multicomponent diffusion in the transition region cannot be derived from these considerations. In the case of the practically important questions mentioned above, the fluxes are coupled by the stoichiometry of a chemical reaction:
if1ViAi
coefficients
of the reaction
*$I v& = 0 and the
parameters P and $I. Since the pressure varies with position the treatment of this transport problem is in general very complex. The case of diffusion controlled ethylene hydrogenation which was mentioned above [6] is therefore too complex to be used in an experimental verification of the transport Eqs. (1) or (6) respectively. A reaction that is better suited for this purpose is the o-p-H, conversion in the presence of an inert gas. We have the reaction: pHe ---*OH,.
(8)
= 0:
From Eq. (4) follows (vl = ---
.ii _
jk
Vi
Vk'
;=-;;ff=o
(4)
Equation (4) says that in the steady state the reactants are transported to the place of conversion in relative amounts corresponding to the stoichiometry of the reaction. Therefore the fluxes are uniquely determined. From Eq. (2) there results a stationary total pressure gradient along the transport path [7]: -gradc=gzs.
(6)
C(
(5)
v2, v3 =
0): (9)
where the indices denote: 1 + pH2, 2 + OH,, 3 + inert gas. Since OH, and pHZ have the same molecular transport properties, for diffusion in the t direction there is always: de z=
0.
(10)
For this diffusion problem there exists only an effective diffusion coefficient Deffwhich is given by Eq. (7) to be:
In this case there is no total mass flux: i$1 Mdi = 0. Using Eq. (4), Eq. (1) CEUI be written 809
D. HESSE
and J. KiiDER
where DT1is the self-d8usion coefficient for H, at 297 K and p. = 76Oinm Hg, D& the binary diffusion coefficient Hz/inert gas at the same conditions and D& = 3 Lvl the coefficient for the Knudsen diffusion of hydrogen. According to Eq. (11), a plot of diffusion resistances, l/Pff vs. pressure should yield straight lines with slope: 1 -=i[l-,. +QDc” $D&
(l-%)I-$
(12)
The common intersection of these straight lines with the ordinate is l/t,!&. The structural factor, $, which is still unknown, and the axis intersection, 1/I,@&,can be determined experimentally from the pressure dependence of the pure o-p-H, diffusion (_x~= 0) ifOf is given. In these experiments it was found that the pressure dependence of the self diffusion of hydrogen through the fritted glass can not be described by a simple Bosanquet formula[26]:
&
(x3 = 0) =
l+Le. Wi
W&PO
(13)
The measurements yielded higher transport resistances than predicted from this equation for small gas pressures (p < 200 mm Hg) [ 101. The deviations from the Bosanquet straight line were attributed to the effect that in the fritted glass, which has a fairly uniform pore-radius distributation[27], short pores determine the transport. In this case, the finite pore length has to be considered in the transport equation by “end correctionsj’ [9]. The pore models mentioned in the introduction can thus not be used without modification to describe the pressure dependence of the diffusion resistance. Instead, the fritted glass was approximated by a system of pores of various lengths and various radii, both of the same order of magnitude connected in series, where the pores were separated by open volumes having the shape of cylindrical disks. This pore model represents quite well the pressure dependence of the diffusion resistance, but its disadvantage is that it is very unhandy for
practical calculations with its seven model parameters which are difficult to assess. In the following the model will therefore be brought into a form that makes it suitable for practical applications. For this we assume the pore structure of the fritted glass to be a system of pores connected in series and having the same mean length, I; and mean pore radius, F. Due to the finite pore length the transport equation has to consider the influence of “end corrections” on the diffusion process as explained in earlier papers [9, lo]. It was found that the additional resistances due to “end corrections” are different for the two limiting cases of Knudsen diffusion and molecular diffusion, since they are caused by different effects. In the case of Knudsen diffusion the theory of Clausing[28] and de Marcus [29] yields smaller diffusion coefficients for the shorter pores than for the longer ones (both having the same radius), because for short pores the back scattering of molecules through the entrance opening due to gas-wall collisions is more important than for long ones. This effect yields a greater transport resistance for diffusion than calculated from the relationship: 0, = 3 FU. In the regime of molecular diffusion the additional resistance is due to the fact that in the entrance and end openings of the pores the diffusion flux lines are no longer parallel to each other but converge or diverge on entering the neighbouring medium. These deformations of the concentration field result in an additional transport resistance which is not considered in Eq. (1) and respectively Eq. (11) (see also [30]). Let (14) be the resistances caused by these effects for Knudsen diffusion (RKZi) or molecular diffusion (RGZi), resp., of component i, with
810
Multicomponent
diffusion in porous media
through a porous replaced by
and 6li = jr/ji. These additional resistances are only valid for the given limiting cases. Weighted with the expected probabilities for Knudsen diffusion (gK) and molecular diffusion (gc) in the pore, they give the additional resistance in the region between the limiting cases:
medium,
Eq. (1 1) is then
P
P,
(16)
With the assumptions that (Yand j3 do not depend on pressure and diffusing species and that gK and gG are the same for all components i of the mixture, the transport equation for multicomponent diffusion in z direction through a porous medium with finite pore length is given by:
The unknown quantities $, (Yand p in Eq. (19) can be determined from the pressure dependence of the pure o-p-H, diffusion (x, = 0) if ? is known. For those gas pressures for which 1 s DK/DT2 . p/p0 holds, a linear behavior follows from Eq. (19): 1 1’ D*e” =$z++&
i=1,2
,....
N.
(17)
The assumptions made here for gx and gG appear reasonable since the occurrence of one of the limiting mechanism, Knudsen or molecular diffusion, is controlled by the entire gas mixture because of the coupling between the diffusion of the components i. It has yet to be studied how g, and gG are to be defined in general. For the present example of inert gas impeded o-p-Hz-diffusion, g, and gG can be given relatively easily in analogy to the pure o-p-Hz diffusion since the ternary mixture is characterized by only one coefficient for Knudsen diffusion and molecular diffusion. The functions gK and gG further have to yield the case for the pure o-p-Hz ‘diffusion for the limiting case xg + 0. Therefore, it was set: (18)
For the pressure dependence of the dill&ion resistance of inert gas impeded o-p-H, diffusion
(19)
(20)
(1 +@pt’
Extrapolation of this straight line to p + 0 gives, for known r, the value of the structural parameter +. With the diffusion coefficient, D&, known the slope of the straight line yields p. If the experimental results for small pressures are plotted as l/D*eff vs. p and extrapolated to p 4 0, the intersection with the ordinate is: ~(P-fo)=-&(l+a)
(21)
K
from which (Y can be calculated. With the parameters thus determined and the values of Dy3 given, the diffusion resistance l/D*eff can be calculated from Eq. (19) for all values of x3 and p. 3. EXPERIMENTAL EVALUATION
APPARATUS OF DATA
AND
Since the measuring apparatus has been described in detail elsewhere [3 1,321, the principle of the measurements will only briefly be discussed. To study the pressure dependence of the inert gas impeded o-p-H, diffusion, the “difisionreaction” cell stretched in Fig. 1 was used, in which a f&ted glass disk (characteristics: see
811
D. HESSE and .I. KODER Reaction with
cp
chamber catalyst
Table 1. Characteristics of the fritted glass disk
&-Par
-
ay
glass disk
I
I Mixing
0.5 < L< 0.85 /.L
chamber
Fig. 1.Diffusion-reaction cell, schematically.
Table 1) separates a mixing chamber from a reaction chamber. Through the mixing chamber there is a constant flow of the ternary mixture pHJoHJinert gas. The input flow (concentration co) contains for all measurements pH,-oHe mixtures of the composition of the low temperature equilibrium which is established on charcoal at liquid nitrogen temperature [33]: clO: czo = 1: 1. Pure helium or argon were used as inert gas. Mixtures containing 10, 50 and 90 per cent of helium and 10, 29,48,7 1 and 9 1 per cent of argon were studied. The hydrogen diffusing through the fiitted glass into the reaction chamber is converted spontaneously to the high temperature equilibrium (cl1 : czl = 1: 3) at the supported platinum catalyst? (5% Pt on SiOJ even at room temperature. The added. inert gas does not partake in the reaction and is therefore not transported in the steady state. Thus the hydrogen diffuses through inert gas standing in the fritted glass. Due to the o-p-H, conversion taking place in the reaction chamber a time independent partial pressure gradient for pH2 and OH, is created. From the mass balance in the mixing chamber one obtains: s(ciO-ciA)
L=0.6cm F = 7.07 cm* rz= 0.350
Thickness Area Total porosity Interval of pore radii [27]
=D*effF
.v
A--.I (22)
where B is the volume velocity of the gas flowing through the mixing chamber and cio, Ci”, c: are the concentrations of component i in the feed gas, exhaust gas and in the reaction chamber (see Fig. 1). With the gas pressure given and with
constant volume velocity d, the diffusion resistance l/D*eff can easily be calculated from Eq. (22) if the concentrations ci have been determined. This was done utilizing a heat conductivity cell which makes use of the different heat conductivities of OH, and pH2 at liquid nitrogen or oxygen temperature. Since the area F of the fritted glass disk and its thickness L could not be determined as accurate as the concentrations, they were, together with the permeability $, combined to a factor f= L/F+. From Eq. (22) follows then: (23) The diffusion resistances calculated according to Eq. (23) from the concentration data were reproducible to 1~5%. Measurements were performed at 297 K in the pressure range of 2 to about 600 mm Hg. 4. EXPERIMENTAL
RESULTS
1. Determination of the parameters In order to determine the parameters CY,~ and f (see Eq. 19) the pressure dependence of the pure o-p-H, diffusion was measured first. The results are plotted as dots in Fig. 2. The slope of the straight line expected for high pressures is ( p = p. = 760 mm Hg) : &- (l+p)
=0*484
(24)
12
and the intersection gives:
tThe authors thank Dr. Krahl of Heraeus, Hanau, for donating the catalyst used.
812
#
with the ordinate
K
= O-068.
(p + 0)
(25)
Multicomponent
diffusion in porous media Table 3. Binary difhtsion coefficients forp,=76OmmHgand T=297K
045
040 t
1445 cm2 set-1 [34] O-785cm2 set-r [34] 1.51 cmzsec1[34]
Fig. 2. o-p-H,
_Eq.(lS) EF.pelimnt
diffusion through the glass disk, T = 297°K.
The extrapolation of the experimental points at low pressures is not unambiguous since after passing through a minimum a steep rise of the diffusion resistance is observed with decreasing gas pressure. This behavior cannot be reconciled by Eq. (19). IfflD* (p + 0) is chosen to be 0.1, the parameters given in table 2 are calculated with P = 7*10-5 cm (see Table 1). With these data, thef/D* vs. p curve plotted in Fig. 2 can be calculated from Eq. (19), which apart from the pressure region p < 100 mm Hg fits the data within experimental error. 2. Pressure dependence of the ternary difusion through the fritted glass disk With the parameters thus obtained and the binary diffusion coefficients listed in Table 3, the curves shown in Fig. 3 for the mixture pH,/oH,/Ar with x3 = 0.1, 0.48 and 0.91 can be calculated from Eq. (19). In particular at gas pressures of p > 150 mm Hg the experimental points deviate from the calculated curve more and more with increasing x3. Even at low pressures marked deviations from the plotted
o-01
’
d0
1
’
200
Fig. 3. o-p-H,-Ar
1
1
1
1
MO
P hmHg1
diision
I
400
I
500
I
1
,600
’
through the glass disk, T = 297”K.
curve can be seen. Qualitatively, however, the theory predicts correctly the pressure dependence. The pressure dependence of the diffusion resistances for the pH,/oH,/He mixtures (x3 = 0.1, O-5, 0.9) are drawn in Fig. 4 through 6. Whereas Eq. (19) predicts the f/D* values for p > 100 mm Hg within experimental error, a considerable increase of the f/D* values is observed with decreasing gas pressure particularly for the mixtures with x3 = 0.5 and 0.9.
Table 2. Parameters for diffusion through the glass disk Mean pore radius Structural factor Permeability End corrections
f= f= $= a! = /‘3=
O-7/L(see Table 1) 0.5617 0.151 0.4706 0.2708
0.05
1
,&
’
1
’
1
’
1
1
1
1
I
I
P hmHg1
Fig. 4. o-p-Hz-He
diffusion through the glass disk,x, = 0.100; T = 297°K.
D. HESSE
0.03
’
,t,
2,
’
’
&
’
’
’
P[mmHgl Fig. 5. o-p-H,-He diffusion
0.05
’
,k
’
’ 300
1
’
and J. KGDER
’
through theglass disk,x, = 0.500; T = 297”K.
,b,
’
&
’
4&j
’
&
’
&
’
p [mmHgl
Fig. 6. o-p-Hz-He
diffusion through the glass disk,x, = 0900; T = 297°K.
5. DISCUSSION
OF RESULTS
“end corrections” in the By considering transport equation for diffusion through porous media, it is possible to interpret qualitatively the deviations from linearity of the function f/D* vs. p. (Eq. 11). Equation (19) does not quantitatively describe, however, the results for low gas pressures, p < 100 mm Hg. A comparison of the pH,/oH,/He diffusion data with those for pH,/oH,/Ar suggests that the observed deviations can not be explained by the iniluence of the pore structure alone. The assumption made for Eq. (1) that molecular diffusion and Knudsen diffusion are independent might not be true. Probably the directional distribution of the molecules after gas-gas collisions has at low pressures a different influence on the mass transport than assumed in Rothfeld’s deviation of Eq. (1). This conjecture is suggested by the
finding that the light helium atoms, being anisotrop scaterers [ 3 5,3 61, give larger deviations from the theoretical curves than the heavy argon atoms which scatter isotropically. Also, it can not yet be established to date to what extent the pore structure can modify the differences in scattering behavior of the inert gas molecules. Apart from these deviations, higher JTD* values than calculated from Eq. (19) are found for pH,/oH,/Ar mixtures even at gas pressures of p > 150 mm Hg. Assuming that these differences are due to the value of the binary diffusion coefficient D,0,(H,/Ar) used in the calculations, the deviations can be removed by using for DTS (H,/Ar) the data indicated by dots in Fig. 7. These data are obtained from the slope of the straight lines found at higher pressures. According to Fig. 7, Ofa qualitatively show the same concentration dependence as measured already by Ker1[37]. The absolute values for D&, however, are smaller by about 11 per cent than Kerl’s data and smaller by about 12 per cent than Waldmann’s data [3 81. The concentration dependence obtained in the present study for DtS (H,/Ar) agrees with the theoretical expectation [39], thus the given interpretation of the deviation is reasonable. In Fig. 8, the flD* vs. p curves have been recalculated from Eq. (19), taking into account the DL (H,/Ar) values indicated in Fig. 7. Now theory and experiment agree within experimental error for pressures of p>80mmHg. If the f&ted glass is assumed to be a system of pores connected in series, with mean pore 0.86 OM-
omo-a-
0
0.78 076 -
-
06-
x0
x
x
0.74-
EID on.oy)1 *g o-7068o-660.64 O-62-
0
0
0
x
x
.
XKei [371 OWoldnmd381 -This wxk
. .
l
.
O!I
I 0.2
I 0.3
I 04
I DS
I 06
I 01
I o*
I 09
I to
13
Fig. 7. Concentration dependence of the binary diffusion coefficient for the H,/Ar mixture; p = 670mmHg; T = 297°K.
814
Multicomponent diffusion in porous media
obtained from the micrographs is probably not accidental. Moreover, from geometrical considerations for the packing of spheres the ratio of the mean radius, P, of the channels between the spheres to the mean length of the channels, 1, is expected to be 9 -&[40,41]. To a tist approximation fritted glass can be considered to be a packing of spheres[42], thus making the obtained value reasonable also from this point of view. The micrographs give another maximum of the probability distribution around y = O-5 (see Fig. 9). Transport pores with this J value exhibit a minimum in the f/D* vs. p curve (see[lOl). However, the theory does not give the same steep rise as measured after passing through the minimum.
Fig. 8. o-p-Hz-Ar diffusion through the glass disk, T = 297°K; D& values from Fig. 7.
radius P and mean pore length I, and separated by volumes with the shapes of cylindrical disks, then one obtains approximately (for r/r= ji s 2.5)[9,10]:
6. PRACTICAL
SIGNIFICANCE
OF RESULTS
In the following we shall briefly discuss the significance of the obtained results for the 2.9 y = (Y. (26) practical problems mentioned in the introduction. With (Y= O-471 one therefore obtains: J = As an example, the determination of the effectiveO-16. The detailed pore model developed in an ness factor for the pores shall be considered. earlier paper for the f&ted glass yielded 9 = We assume that the o-p-H, conversion takes O-178.From a series of scanning electron micro- place in a porous catalyst plate of thickness L graphs of the f&ted glass the relative probability and area F, and that the reaction is of fnst order H(y) = N(y)/N,,, to find pores with a para- with pressure-independent rate constant kO.The meter value of y was determined, with the result reaction rate shall be large enough so that the that nearly one third of the measured pores N,, conversion is controlled by diffusion in the had a value of 7 = O-2 (see Fig. 9). Although interior of the catalyst. At the location, z = L, the determination of 7 values is problematic the gas composition always corresponds to the (e.g. because of parallax errors), the rather good high temperature equilibrium: cl : c, = 1: 3, agreement between the calculated value and that and let (dcl/dz)s=L= 0. Under isotherm steady state conditions the effectiveness factor (see also 0.50 [43]) is given by r) c 6: 0.40
s I
-
0.30f
gj
0.20-
l
OGQiI ,,v,
]
t
,,,,,,,,, 0.2
0.4
06
0.8
I.0
I.2
I.4
Y
Fig. 9. Relative probability for finding pores with the value y in the fritted glass disk.
According to Eq. (27), the effectiveness factor depends apart from the constant k0 and the geometry of the catalyst (L, F), in particular on the diffusion resistanceflD* and thus on the pore structure of the catalyst. Generally it is assumed that the influence of pore structure on the mass transport is identical for the case of a chemical reaction taking place as for that with
815
D. HESSE
and J. KODER
no conversion[44,45], as long as the pore structure is not modified during the reaction. With this assumption, the results obtained in this study can be used to discuss the influence of the pore structure on the value of thehffectiveness factor. Therefore we first assume that the diffusional transport is done by the Knudsen mechanism. If one compares the effectiveness factors for different jj = r/l, one obtains from Eq. (19): 7,=_-12e Vi-G
application of the “pseudo-capillary” model can be demonstrated particularly clearly when the results of the present study are used to calculate the pressure dependence of the reduced effectiveness factor
(28)
where q. is the effectiveness factor for catalysts in which the length, t of the transport pores is large compared to their radius, J. If (Y= 2.99 is introduced (Eq. 26), the relationship d/q0 vs. jj is obtained as plotted in Fig. 10. Porous catalysts with J values different from zero can thus have considerable lower effectiveness factors than those with pores of length t% P (see also [l 11). This result should be of interest for the production of effective catalysts. In practice such catalysts are frequently used in which at about 700°C and not too high pressures (p < 5 atm) mass transport is mainly done by Knudsen diffusion. Under these conditions the influence of “end corrections” explained above could be important for the calculation of the effectiveness factor. If, for instance, the effectiveness factor is calculated with a structural factor determined from diffusion studies in the regime of molecular diffusion, then the q. value given by the “pseudo-capillary” model would indicate a higher efficiency of the catalyst than could be derived from the actual conversion data. This could give rise to erroneous interpretation of the results. The errors introduced by uncritical
If r~,.vs. l/Gis calculated without considering “end corrections”, the curve plotted as a solid line in Fig. 11 is obtained. For p + 0 this yields a value of vT( p + 0) = 1.86. Considering the finite pore length one finds, on the other hand, r),= 1.54 (for flD* (p+ 0) =O.lOO). This value is smaller by about 20 per cent than that given above.
lX3=0oocl xX,=0900
Fig. 11. Pressure dependence of the reduced effectiveness factor; (o-P-Hz-He mixtures); T = 297°K.
If helium is added to the o-p-H, mixture one would expect according to the model that the function qr (p) is practically the same as for the pure o-p-H, mixture even for the case x3 = 0.9 considered here, since 0:: and D& differs only slightly. Figure 11 shows that this is only true for higher gas pressures. At pressures of p < 100 mm Hg, 7,. goes through an unexpected maximum and drops then sharply with decreasing gas pressure. Although presently no physical interpretation can be given for this behavior, it can be concluded that a prediction of the effectiveness factor can lead to largely erroneous of ditfusional Fig. IO. Reduced effectiveness factor as a function off = i//1. values unless the characteristics 816
Multicomponent diffusion in porous media
F
transport in porous media are properly considered. In particular, effective diffusion coefficient measured at atmospheric pressure can generally not be extrapolated to higher or lower pressures using the “pseudo-capillary” model if accurate values are to be obtained (see also[ 17,461).
area of the glass disk, cm2 structure factor, cm-‘, f = L/F+ f molar flux of component i, mole cm-2 set-1 i thickness of the glass disk, cm i mean pore length, cm molar mass of component i, g mole-’ Mi P total pressure, mmHg mean pore radius, cm absolute temperature, K Vi mean molecular velocity of component i, cm set-’ volume velocity of the gas, cm3 see-’ mole fraction of, component i quotient: F/i spatial coordinate, cm
T’
Acknowledgement-We are grateful to Prof. Dr. E. Wicke for many helpful discussions. One of us (D. H.) thanks the Deutsche Forschungsgemeinschaft for financial support. NOTATION
total concentration, mole cme3 concentration of component i, mole cme3 effective coefficient for Knudsen diffusion of component i, cm2 see-’ binary diffusion coefficient for component i and 1,cm2 set-’ 0: Dilfor p0 = 760 mm Hg, T = 297 K 0;" generalized effective diffusion coefficient of component i, cm2 set-’ DK coefficient for Knudsen diffusion of component i through a pore with mean pore radius F,cm2 set-l DFff 1G effective coefficient for molecular diffusion of component i (Eq. 15), cm2 see-’
Greek (Y
j3 E JI Q vf
symbols
parameter in Eq. ( 14) parameter in Eq. ( 16) total porosity of the porous glass disk permeability of the glass disk reduced effectiveness factor, Eq. (29), sec-1’2 stoichiometric coefficient of component i, vt > 0 for products, yf C 0 for reactants vi = 0 for inert components.
REFERENCES [l] SAlTERFIELD C. N., Mass Transfer in Heterogeneous Catalysis, M.I.T. Press, London, 1970. [2] RI’ITER H. L. and DRAKE L. C., Znd. Engng Chem. Anal. Ed. 1947 17 787. [31 TOOR H. L..A.Z.Ch.E. Jl1957 3 198. [4] DUNCAN J: B. andTOORH. L.,A.Z.Ch.E.J/1962838. [5] HUGO P., Chem. Engng Sci. 1965 20 187; 385; 975. [6] HESSE D. and HUGO P., to be published in Chem. Znge. Techn. [7] RESHETOV V. A., 2. Fiz. Khim. 1970 44 3057. [S] WHEELER A., Advances in Catalysis, Vol. 3, p. 250. Academic Press, New York 195 1. [9] HESSE D., to be published in Z. phys. Chem. [lo] HESSE D., to be published in Chem. Znger. Techn. [ 1 l] VERHOFF F. H. and STRIEDER W., Chem. Engng Sci. 197 126 245. [12] PENTENERO A. and LE GOFF P., J. chim. Phys. 197168 1027. [13] ROTHFELD L. B.,A.Z.Ch.E. Jll963 9 19. [14] SCOmD.S.andDULL1ENF.A. L.,A.Z.Ch.E.Jl 19628 113. [15] EVANS R. B., WATSON C. M. and MASON E. A. J. them. Phys. 196135 2076. [16] MASON E. A., MALINAUSKAS A. P. and EVANS R. B., J. them. Phys. 1967 46 3 199. [17] YOUNGQUIST G. R., Znd. Engng Chem. 197162 52. [I81 JUST?. J. and GELBIN D., Chem. Technik 197123466; 581; 648. [ 191 HORAK Z. and SCHNEIDER P., Chem. Engng Jll97 12 26. [20] CUNNINGHAM R. S. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1968 7 535. [21] HUGO P. and WICKE E., Chem. Znge Techn. 1968 40 1133. [22] PRESENT R. D., Kinetic Theory of Gases. McGraw-Hill, New York, 1958. [23] WICKE E. and KALLENBACH R., KolloidZ. 194197 135. [24] CUNNINGHAM R. S. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1968 7 429. [25] REMICK R. R. and GEANKOPLIS C. J., Znd. Engng Chem. Fundls 1970 9 206. [26] BOSANQUET C. H., British TA Rept., BR-507,1944.
817
D. HESSE
and J. KGDER
[271 FRANK W., GZT-Zeitschr. 1967 11683. [281 CLAUSING P., Ann. Phys. 1932 12 961. 1291 DE MARCUS C., ORGDP Rep. K-1302, Part 3,1957. i3oj 1 HUGO P., unnublished. t3 11 BEYER H. D:, Diploma&it, Miinster 1966. [=I KODER J., DIPLOMARBEIT, Miinster, 1972. [331 CREMER E. Handbuch der Katalyse, Vol. 6. Springer, Wien 1943. MULLER R., Chem. Znge Techn. 1968 48 344. tiz; HIBY W. and PAHL M., Z. Phys. 1951 129 517. [361 HIBY W. and PAHL M., Z. Phys. 1951138 348. KERL K., Dissertation, Braunschweig, 1968. ::i; WALDMANN L., Z. Phys. 1948 1242. MASON E. A.. .I. chem. Phvs. 1957 27 75: 782. ::z; MANEGOLD’E. and SOLF K., Kolloidi. 1939 89 36. 1411 HIBY W. and PAHL M., Z. Narurf. 1956 lla 80. t421 PRAUSNITZ P. H..?KolloidZ. 1943 104 246. [431 WICKE E. and BROTZ W., Chem. Znger. Tech. 1949 21219. [441 HUGO P. and PARATELLA A., Simposio di Dinamica delle Reazioni Chimiche, Padova 1966, Vol. V. Roma 1966. [451 HAWTIN P., Chem. Engng Sci. 197126 1783. [461 BROWN L. F., HAYNES H. W. and MONOQUE W. H.,J. Cntal. 1969 14 220.
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