Argönül, Technology
L. Argönül ,and
F. J.Metallurgy,
Keil
Journal of the University ofA.Chemical
42, 2, 2007, 181-186
THE INCONSISTENCY IN MODELING SURFACE DIFFUSION
WITH ADSORPTION EQUILIBRIUM
A. Argönül1, L. Argönül2, F. J. Keil1
1
Hamburg University of Technology (TUHH),
Institute of Chemical Reaction Engineering,
Eissendorferstr. 38, 21073, Hamburg, Germany
Tel: 0049-40-42878-2545, Fax: 0049-40-42878-2145
E-mail: [email protected],
[email protected]
2
Ethem Efendi Cad., No:52/18, 34738, Istanbul, Turkey
Received 05 March 2007
Accepted 10 May 2007
ABSTRACT
The commonly used adsorption equilibrium assumption in modeling simultaneous gas phase flow and surface
diffusion is shown to contradict the existence of surface flow. Through mass balances, it has been revealed that at steadystate it is not possible to achieve throughout adsorption equilibrium for such systems. A brief discussion about consequences concerning modeling is given at the end. Beside others, may be the most important consequence is that the
calculated values based on this assumption, such as surface diffusion coefficient, may be unrealistic.
Keywords: adsorption, surface diffusion, equilibrium assumption, mass balance, Knudsen flow.
INTRODUCTION
It is known for a long time that surface diffusion
can contribute significantly to the total transport in a
porous medium [1, 2]. This contribution is more pronounced for small pore diameters, at lower pressures,
that is, in the region where Knudsen diffusion prevails
[3]. Macroscopically, the rate of surface diffusion may be
measured using the steady-state permeability method or
by using a diffusion (Wicke-Kallenbach) cell [2, 4, 5].
For example, in a diffusion cell, the surface
diffusion flux can be calculated as follows [6]:
First, the flux of a gas, which is expected not to
be adsorbed, is measured. The gas often employed is
helium. Then, expecting that non-surface diffusion will
occur by the Knudsen mechanism, the expected flux for
the test gas, A, is calculated from Graham’s law:
J A (nonsurface ) = − J He
MwHe
Mw A
(1)
where J is the flux and Mw stands for the molecular
mass.
The difference between the experimental measurement and the nonsurface estimate is the surface diffusion flux:
J A (surface ) = J A (experimental ) − J A (nonsurface ) (2)
Typically, the flux inferred for the surface diffusion is less than half of that total flux measured experimentally [6].
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Journal of the University of Chemical Technology and Metallurgy, 42, 2, 2007
The determination of the surface diffusion coefficient from the surface flux requires the estimation
of the surface driving force, that is, the corresponding
concentration gradient. Commonly for this purpose, an
adsorption isotherm (Henry’s Law, Langmuir isotherm,
etc.) is chosen, and to facilitate the corresponding calculations, it is assumed that the adsorption-desorption
equilibrium has been achieved. For example, Jackson
[7] states his assumption as follows: “If we assume that
the rates of adsorption and desorption are both large
compared with the surface migration rate, the surface
and bulk concentrations of each species will be almost
in equilibrium, and hence will be related by the equilibrium adsorption isotherm, ...”, Do [8] mentions that
“...By local equilibrium here, we mean that at any given
point within the particle the gas and solid phases are in
equilibrium with each other, despite the gradients of
concentration in both phases are present. This is acceptable if the rates of adsorption and desorption at any
point are much faster than the rates of diffusion in both
phases”. In some cases explicitly, often also implicitly,
equilibrium assumption is commonly used in modeling
of transport phenomena in porous media [1, 3, 4, 9-14],
where surface diffusion is assumed to take place. As a
consequence, the surface coverage and the surface concentration gradient are straightforwardly determined
based on the adsorption equilibrium equation.
Although at first sight it seems reasonable, the
equilibrium assumption is inappropriate and creates a
misconception in modeling. An equilibrium between
the gas phase and the surface allows no mass interchange
between the two phases and sets the surface flow value
to zero. This contradicts the existence of the surface
diffusion. The following sections clarify the above arguments.
INCONSISTENCY IN THE EQUILIBRIUM MODEL
Writing a (steady-state) mass balance for each
flow separately will reveal that the equilibrium assumption contradicts the existence of surface flow. The mass
transport streams and the control volumes are illustrated
in Fig.1.
Mass balance for the gas phase flow has to be
made around the core control volume:
gas
FAgas
, z − FA, z + ∆z = (rads , z − rdes , z )⋅ 2 π r pore ⋅ ∆z (3)
182
Fig. 1. Differential control volumes in the pore.
where F is the flow rate [mol s-1], rpore is the pore radius
[m], and rads and rdes are, respectively, the adsorption
and desorption rates [mol m-2 s-1].
Whereas, the mass balance for the surface flow
shall be expressed around the annular control volume:
Surf
FASurf
, z − F A, z + ∆z = (rdes , z − rads , z )⋅ 2 π r pore ⋅ ∆z
(4)
By rearranging eq.3 & eq.4 and knowing that for
an infinitesimally small ∆z, the derivative dFA, z dz
will replace the difference FA, z + ∆z − FA, z ∆z , it follows that:
(
(r
ads , z
− rdes , z )⋅ 2 π rpore =
)
dFASurf
,z
dz
=−
dFAgas
,z
dz
(5)
Furthermore, the equilibrium assumption implies
that the rates of adsorption and desorption are equal,
thus the term on the left-hand side of eq.5 becomes
zero:
rads, z − rdes, z = 0
(6)
(
)
Consequently, through eq. 5 & eq.6, the first derivatives of the gas phase and the surface flows become
zero as well:
dFASurf
,z
dz
=
dFAgas
,z
dz
=0
(7)
This leads to constant gas phase and surface flows
for the entire length of the pore:
Surf
FASurf
= FASurf
, 0 = F A, z
, L = constant
(8)
gas
gas
FAgas
, 0 = F A, z = F A, L = constant
(9)
A more detailed drawing of the porous structure
model and flow through a single pore is given in Fig. 2.
As can be seen from the above figure, there is an incoming surface flow into the pore originating from the
A. Argönül, L. Argönül , F. J. Keil
Fig. 2. Schematic drawing of surface flow and gas phase flow.
left outer-face of the porous structure. This is due to
the adsorption and desorption on this facial surface.
Owing to mass balance on that surface, this surface flow,
FASurf
, 0 , has to be equal to the difference between the
corresponding adsorption and desorption rates on the
face
facial surface area per porei, Asolid / pore :
(r
ads , 0
face
− rdes , 0 )⋅ Asolid/pore
= FASurf
,0
(10)
For the above derivation, it was assumed that the
surface concentration on the corresponding face was
constantii.
Combination of eq.10 and the equilibrium assumption sets the surface flow at the entrance of the
pore to zero.
(r
ads , 0
− rdes ,0 ) = 0 = FASurf
,0
(11)
From eq. 9 and eq. 11, it follows that the surface flow has to be zero throughout the pore. Consequently, the only way of transport will be the gas phase
flow.
FAgas = FAtotal
and
FASurf = 0
(12)
This result is inconsistent with the statement
claiming the existence of the surface flow.
DISCUSSION
For a flow system, there has to exist a source and
a sink, between which the material is transported by
means of different transport mechanisms. For the system in question, although the source and the sink are
not given explicitly, it is commonly taken for granted
that there are two reservoirs on the left and on the right
of the porous medium (or the pore). Based on the flow
of species A, the left and right reservoirs become the
source and the sink, respectively. The two transport
mechanisms, gas-phase and surface flows, serve as connections between these reservoirs. The connection
through the gas-phase flow is direct; the molecules directly flow from one reservoir to the other via this
mechanism. On the other hand the surface flow connects the two reservoirs indirectly. For this connection,
first the molecules have to get onto the surface (adsorption), then they have to be transported (surface diffusion) and finally they have to be released (desorption).
The adsorption equilibrium assumption hinders this
The facial surface area per pore can be calculated by making use of the porosity, ε. The ratio of solid area (facial
surface area) to the free cross-sectional area is (1− ε ) / ε . The facial surface area per pore can then be calculated from
(1 − ε )
face
2
the multiplication of this ratio with the cross-sectional area of a pore: Asolid
.
/ pore = π r pore ⋅
ε
i
ii
It may be worth noting that if a definite geometrical shape, e.g., circle, would be assigned to the facial surface area per
pore, a concentration gradient on the face could be calculated and the surface flow at the entrance of the pore could be
found by integration. Consequently the constant facial surface concentration assumption could be omitted.
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Journal of the University of Chemical Technology and Metallurgy, 42, 2, 2007
transport mechanism by preventing the first and the last
steps to occur. Thus physically under equilibrium conditions, it is not possible that a surface flow appears
(see eq.12). On the other hand, a surface concentration
gradient and consequently a surface flow can be calculated through the same assumption by the utilization of
the gas-phase concentrations. These two contradictory
findings demonstrate the inconsistency and the resulting misconception in modeling surface diffusion with
adsorption equilibrium. Throughout adsorption equilibrium and surface flow cannot exist simultaneously,
although a finite value for the surface flow can be calculated. Thus equilibrium assumption becomes inappropriate for such systems.
The rates of adsorption and desorption can still
be expected to be close to each other. If they are really
much higher than the diffusion rates, they should not
be very different in magnitude. This is due to the fact
that their difference is equal to the rate of change of
diffusion rates (see eq.5). This rate of change is expected
to be comparable with the diffusion rates themselves.
On the other hand, very high adsorption and desorption
rates means also that any small difference relative to
them might be not that small compared to the diffusion
rates. Thus small differences between the adsorption and
desorption rates may cause considerable changes in the
diffusion rates, and affect accordingly the gas phase and
surface concentration profiles. Consequently, precise
estimations of these rates are required for realistic modeling.
To the best of our knowledge, there is no physical reason for the special case where there exists adsorption equilibrium only in the pore and non-equilibrium on the facial surface. The gas phase concentrations over the left facial surface and at the entrance of
the pore are actually the same (see Fig.2). A similar
argument is valid as well for the exit side. Since the
facial surface and the pore surface are a continuation of
the same solid phase, they will most probably behave
similar under similar conditions. So it can be expected
that they have both equilibrium or both non-equilibrium conditions, and not equilibrium conditions
throughout the pore and net adsorption/desorption only
at the facial surfaces.
If we indicate the rate difference between adsorption and desorption rates as ADRD (Adsorption
184
and Desorption Rate Difference), it can be noted that
the ADRD will be positive on the left facial surface
since (net-)adsorption is expected to be dominant on
this side. The molecules are expected to adsorb on the
facial surface and in the pore, and serve to maintain the
surface flow. Along the pore, ADRD will cease slowly
and at an arbitrary point it will become zero indicating
a local equilibrium point. The exact place of this local
equilibrium point is expected to be dependent on many
parameters, which affect the rates of diffusion and
ADRD as well, such as; surface properties, gas concentrations, temperature, etc. Beyond this point, desorption will be expected to be dominant (a negative ADRD)
and the adsorbed species will be released back to the
gas phase from the remaining part of the pore and finally from the outlet (right) facial surface. All of these
imply that up to the local equilibrium point the surface
will have below-equilibrium and beyond it above-equilibrium concentrations. Owing to the definition of the
steady-state there can be no accumulation of mass, therefore the total amount adsorbing before the equilibrium
point should exactly be equal to the total amount desorbing after it. In other words, the integral of ADRD
over the whole surface has to be zero.
Since the gas phase flow and surface flow are
coupled through the adsorption and desorption rates,
the gas phase profile cannot be the same for the cases
without and with surface diffusion. Thus determination
of the gas phase profile as if there were no surface diffusion (decoupled flows), and then calculation of the surface concentration profile based on this determined profile will most probably give unrealistic values. Although
the total flow stays constant, the individual rates of surface flow and gas phase flow will change along the pore.
As a consequence, the individual flows in the system
cannot be represented by single values, therefore the
utilization of eq. 2 is not as straightforward as it is usually interpreted. The surface concentration profile and
consequently the surface flux calculated by this way can
thus lead to unrealistic results. Consequently, any parameter (e.g., surface diffusion coefficient) based on that
calculation will be questionable as well, as long as the
error associated with the utilization of the assumption
is not estimated.
As a summary, all the above points indicate that
the behavior of the concentrations and the fluxes have
A. Argönül, L. Argönül , F. J. Keil
to be much more complex than the commonly estimated
behavior. Investigation (by experiments and/or simulations) of these expected complex behaviors is necessary
for accurate understanding and modeling of such systems and various phenomenon occurring therein.
CONCLUSIONS
For a gas phase diffusion systemiii with surface
diffusion, under steady-state conditions, it has been
shown through mass balances that;
Physically, it is not possible for the entire system
to have adsorption-desorption equilibrium and surface
flow simultaneously. Therefore, the equilibrium assumption (i.e., relating the gas-phase and surface concentrations through adsorption isotherms) should not be used
for such systems unless an estimation of the error involved is known.
An accurate modeling of such a system has to
include the calculation of the rates of adsorption and
desorption or at least their difference.
As a consequence of the above points, it is expected as well that;
There should exist an equilibrium point in the
pore where adsorption and desorption rates are equal.
Up to that point the adsorption rate and beyond the
desorption rate would be dominant. In other words, up
to the equilibrium point the surface concentrations
would be below-equilibrium and beyond above-equilibrium values. Additionally, the total amount adsorbing
up to the equilibrium point should exactly be equal to
the total amount desorbing beyond it.
One needs to consider the adsorption and desorption on the inlet-side and outlet-side faces of the
porous structure in order to determine the boundary
values of the surface flow.
The gas phase diffusion rate and surface diffusion rate cannot be constant along the pore length.
They both will be functions of the axial pore coordinate, while their sum being constant. Thus, they can-
not be calculated independently, such as straightforward utilization of eq. 2.
Investigation of the above points by experiments
and/or simulations would serve for better understanding of the phenomenon occurring in systems with surface diffusion.
Furthermore it should be noted that being able
to estimate the rates of adsorption and desorption, or at
least their difference, accurately turns out to be vital in
realistic modeling of such systems.
Acknowledgements
One of the authors would like to thank to Mr.
Kerem Ali Günbulut for his valuable comments.
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