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www.rsc.org/pccp | Physical Chemistry Chemical Physics
Nonequilibrium thermodynamics—A tool to describe heterogeneous
catalysis
Dick Bedeaux,bc Signe Kjelstrup,bc Lianjie Zhua and Ger J. M. Kopera
Received 13th July 2006, Accepted 10th October 2006
First published as an Advance Article on the web 23rd October 2006
DOI: 10.1039/b610041d
In the study of multi-component mass transfer it is common to use the film model, in which all
the resistance to mass transfer towards a catalytic surface is assumed to be localized in a diffusion
layer in front of the surface. At the surface one furthermore assumes that the temperature and
chemical potentials are continuous, while the coupling of a possible heat flux to the mass fluxes is
assumed to be negligible. Both these assumptions are questionable. Using nonequilibrium
thermodynamics we discuss how to integrate the coupling between heat and mass fluxes in the
description of the film. Furthermore, following Gibbs, we introduce the surface as a separate
thermodynamic system where the coupling between the vectorial heat flux and the scalar reaction
rate is allowed and can be significant in heterogeneous catalysis. Non-equilibrium thermodynamic
theory for surfaces allows one to find the proper rate equations. It allows for a consistent and
complete description of mass and heat transfer through the film and subsequently from the film to
the surface where the reaction takes place. Fast endo- or exothermic surface reactions in
heterogeneous catalysis may give significant temperature gradients between a catalyst surface and
the media, which will, when not accounted for, lead to an incorrect evaluation of the activity,
stability and selectivity of a catalyst. Non-equilibrium thermodynamics is a useful tool for
predicting the surface temperature as well as for analyzing the system. In this contribution we
sketch how to systematically set up the complete description, in which the film and the surface
‘‘sum up’’ to one effective surface.
1. Introduction
In the study of multi-component mass transfer to and from a
catalytic surface it is convenient to use the film model, see for
instance ref. 1 and references therein. In this model all the
resistance to mass transfer is assumed to be localized in a film
between the surface and the well-mixed flow region. The
reactants diffuse through this film to the surface, where they
react, and the products diffuse back. Though there may be
heat exchange with the catalyst carrier, we will always consider
running conditions such that the temperature of the catalyst
pellets is uniform and equal to the temperature of the surface.
In that case there is no heat exchange with the catalyst carrier.
Two assumptions are commonly made when applying the
film model. In the first place, one usually assumes that the
temperature and chemical potentials of the surface are equal to
those in the adjacent bulk region. Secondly, when describing
the heat and mass fluxes through the film one neglects the
coupling of these fluxes and as a consequence, the film then has
neither a Soret nor a Dufour effect. Both assumptions are
questionable. The reaction enthalpy can be substantial, which
a
Delft ChemTech, Delft University of Technology, Julianalaan 136,
2628 BL Delft, The Netherlands
Department of Chemistry, Norwegian University of Science and
Technology, 7491 Trondheim, Norway
c
Process and Energy Department, Delft University of Technology,
Leeghwaterstr.44, 2628 CV Delft, The Netherlands
b
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results in large heat flows. Coupling effects may therefore be
significant. In a recent paper2 some of us documented the
importance of the Soret effect for catalytic hydrogen oxidation, H2 + (1/2)O2 - H2O. A systematic method of combining heat and mass transfer is provided by nonequilibrium
thermodynamics.3,4 Following nonequilibrium thermodynamics, one may easily incorporate the coupling between heat
and mass fluxes in the description. How to do this is in fact
explained by Taylor and Krishna.1 Similarly one may, following Gibbs,5 introduce the surface as a separate thermodynamic
system.6–9 The nonequilibrium thermodynamic theory for
surfaces allows one to find proper rate equations in these
heterogeneous systems. It allows for a consistent and complete
description of catalytic reactions at surfaces far from equilibrium. As we shall show, the surface features as an ‘‘additional
film’’. As it is often sufficient to use a thin film approximation
in the bulk phase, which is most easy to implement, we shall
follow this procedure. It makes the similarity between the
description of the film and of the surface all the more
apparent.
One of the aims of this contribution is to sketch how to
integrate the description of the film with that of the surface. In
the second section we discuss the thermodynamic description
of the surface. In the third section we use nonequilibrium
thermodynamics for the surface and derive expressions for the
temperature and chemical potential differences between the
surface and the film and for the Gibbs energy of the catalytic
Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5421
The surface excess internal energy density is
reaction at the surface in terms of the fluxes. In the fourth
section we discuss the description of the film. In the fifth
section we combine the film and the surface to one effective
film, for which explicit expressions are given for the temperature and chemical potential differences across this whole
region. Expressions are also given for the resistances involved,
in terms of the resistances of the films and the surface.
Concluding remarks regarding the practical use of the description are made in the last section.
and Gibbs-Duhem’s equation becomes
2. Thermodynamic variables for a surface
2.2.
We choose the x-axis perpendicular to the planar surface. The
thermodynamic properties of the surface are given by the
values of the excess mass and energy densities, which were
defined by Gibbs.5 Following Gibbs we define the location of
the dividing surface, which separates the homogeneous phases,
such that the excess concentration of a reference component is
zero. In the present case the logical choice of the reference
component is the carrier material. The surface as described by
the excess mass and energy densities can be regarded as a 2-D
thermodynamic system with properties that are given per unit
of surface area. The dependence on the coordinates y and z
remains.
We have so far considered systems that are in global equilibrium, i.e. the temperature and the chemical potentials are
constant throughout the whole system. When the system is not
in global equilibrium we need to introduce the assumption of
local equilibrium. For a surface element, we say that there is
local equilibrium when the local thermodynamic relations
(4)–(6) are valid in each point along the surface and at each
moment in time. Similarly, local equilibrium implies that all
the usual thermodynamic relations are valid locally. The
intensive thermodynamic variables for the surface, indicated
by superscript s, are therefore given by the derivatives:
s
s
du
du
s
Ts ¼
and
m
¼
ð7Þ
j
dss Gj
dGj ss ;Gk
2.1.
Local thermodynamic identities for the surface
When excess surface densities are defined in this manner, the
normal thermodynamic relations, like the first and the second
law and derived relations apply for the densities.5 The Gibbs
equation for the total excess internal energy, Us, of the
equilibrium surface becomes:
dU s ¼ TdSs þ gdO þ
n
X
mi dNis
ð1Þ
i¼1
where Ss, O and Nsi are the total excess entropy, the surface
area and the total excess density of component i. Furthermore
T, g and mi are the temperature, the surface tension and the
chemical potential of component i, respectively. In view of the
extensive nature of Us, Ss and Nsi we can use Euler’s theorem
and obtain
U s ¼ TS s þ gO þ
n
X
mi Nis
ð2Þ
i¼1
Gibbs–Duhem’s equation for the surface follows by differentiation of this equation and subtracting eqn (1):
0 ¼ Ss dT þ Odg þ
n
X
Nis dmi
ð3Þ
i¼1
For the surface, we need the local variables given per unit of
surface area. These are the excess internal energy density us =
Us/O the adsorptions Gi = Nsi /O and the excess entropy
density, ss = Ss/O. When we introduce these variables into
eqn (1), and use eqn (2), we obtain the Gibbs equation for the
surface:
dus ¼ Tdss þ
n
X
mi dGi
i¼1
5422 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427
ð4Þ
us ¼ Tss þ g þ
n
X
mi Gi
ð5Þ
i¼1
0 ¼ ss dT þ dg þ
n
X
Gi dmi
ð6Þ
i¼1
Definition of local equilibrium for the surface
The temperature and chemical potentials, defined in this
manner, depend only on the surface excess variables, not on
the value of bulk variables close to the surface. By introducing
these definitions we therefore allow for the possibility that the
surface has a different temperature and/or chemical potentials
from the adjacent homogeneous systems. Molecular dynamics
simulations support the validity of the assumption of local
equilibrium for surfaces.10,11 The assumption of local equilibrium, as formulated above, does not imply that there is local
chemical equilibrium.12 In that case the Gibbs energy of the
reaction is also zero. The thermodynamic variables for the
surface depend on the position along the surface and the time.
As we shall not consider transport along the surface, we shall
further restrict ourselves to cases where the variables are
independent of y and z. Also we shall restrict ourselves to
stationary states. This implies that all the excess densities have
time independent values. This simplifies the description rather
considerably.
An essential and surprising aspect of the local equilibrium
assumption for the surface and the adjacent homogeneous
phases is the fact that the temperature and chemical potentials
on both sides of the surface may differ not only from each
other, but also from the values found for the surface, see
Fig. 1.
3. The excess entropy production rate for the
surface
Consider a surface s between two phases i and o. We take the
origin of the x-axis to coincide with the surface s. The phase i is
located on the left of the surface, x o 0, and the phase o is
located on the right of the surface, x 4 0. In our case i is the
phase in which the diffusion takes place while o is the carrier
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minus the total heat flux out of the surface to the right, Jo,i
q .
Both the molar and the heat fluxes in the above equations
should be taken in a frame of reference in which the surface is
at rest. These heat fluxes are related to the measurable heat
fluxes by
X
0
0
Jqi ¼ Jqi þ
hij Jji and Jqo ¼ Jqo
ð11Þ
j
where hij is the partial enthalpy density of component j in the
i-phase. The measurable heat fluxes are independent of the
frame of ref. 12.
3.2.
Fig. 1 Schematic diagram of the temperature variation in heterogeneous exothermic catalytic reactions, which can be modeled by nonequilibrium thermodynamics.
phase. The change of the entropy in a surface area element is a
result of the flow of entropy in and out of the surface element,
and of the entropy production rate inside:
d s
s ¼ Jsi;o Jso;i þ ss
dt
ð8Þ
where Ji,o
s is the asymptotic value of the entropy flux in the
adjacent phase i left of the surface and into the surface, and Ji,o
s
is similarly the entropy flux in the phase o to the right of the
surface and out of the surface. All fluxes are taken in a frame
of reference in which the surface is at rest (the surface frame of
reference). The first (or the only) roman superscript gives the
phase, i, s or o in this case. The second superscript, o or i,
indicates a value close to phase o or i. The combination i, o
means therefore the value in phase i as close as possible to the
o-phase at the interface. The excess entropy production rate is
ss Z 0. We shall find explicit expressions for ss by combining:
mass balances
the first law of thermodynamics
the local form of the Gibbs equation.
In the derivation we follow ref. 6–8 We shall see that ss can
be written as the product sum of thermodynamic fluxes and
forces in the system. These are the conjugate fluxes and forces
for the surface.
3.1.
Balance equations
The balance equation for the excess molar density of component j is:
d
Gj ¼ Jji;o þ n j rs
dt
ð9Þ
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n
dss
1 dus
1 X
dGj
ms
¼ s
s
dt
T dt T j¼1 j dt
ð12Þ
By introducing eqns (9) and (10) into eqn (12), and comparing
the result to the entropy balance eqn (8), we find the excess
entropy production rate in the surface frame of reference:
1
1
1
1
o;i
ss ¼Jqi;o
þ
J
q
T s T i;o
T o;i T s
"
!#
ð13Þ
n
X
msj
mi;o
D r Gs
j
i;o
s
þ
Ji þ
r
T s T i;o
Ts
j¼1
where DrGs =
surface.
3.3.
P s
njmj is the reaction Gibbs energy for the
Stationary states
We will restrict the further analysis to stationary states. In that
case the pellets carrying the catalyst have a uniform temperature equal to the temperature of the surface, To,i = Ts. This
implies that
0
Jqo;i ¼ Jqo;i ¼ 0
ð14Þ
The excess entropy production simplifies to
"
!#
X
n
msj
mi;o
1
1
Dr Gs
j
i;o
s
i;o
s
þ
r
þ
J
s ¼Jq
j
T s T i;o
T s T i;o
Ts
j¼1
X
n
h
m i
1
Dr Gs
j
þ
Jji;o Di;s
þ rs s
T
T
T
j¼1
ð15Þ
ð10Þ
The change of the excess internal energy density of the surface
is given by the total heat flux into the surface from the left, Ji,o
q ,
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The time derivative of the entropy density is given by the
Gibbs equation in its local form:
¼Jqi;o Di;s
is the molar flux into the surface and nj the
where Ji,o
j
stochiometric coefficient of component j, while rs is the reaction rate per unit of surface area. The stochiometric coefficients are taken negative for the reactants and positive for the
products. For simplicity we only consider one reaction. The
first law of thermodynamics for the surface is:
dus
¼ Jqi;o Jqo;i
dt
The excess entropy production rate
In the last equality we introduced a short hand notation for
the difference of a variable. For stationary states one furthermore finds that the molar fluxes, Jj, are constant throughout
the diffusion layer while the total heat flux, Jq, is constant
everywhere. Using the measurable heat flux J 0 iq, defined by eqn
(11) and the thermodynamic identity @(mj/T)/@(1/T) = hj, one
obtains after some algebra
X
n
Di;s mj;T ðT s Þ
1
Dr Gs
0
s
ss ¼ Jqi Di;s
Jj þ
r
ð16Þ
þ
Ts
Ts
T
j¼1
Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5423
The subscript T implies that the difference is calculated at a
constant temperature, which in this case is given by Ts.
3.4.
Linear force-flux relations for the surface
The excess entropy production given in eqn (15) results in the
linear relations
n
X
1
¼rsee Jq þ
rsek Jk þ rser rs
Di;s
T
k¼1
Di;s
m j
T
¼rsje Jq þ
n
X
s;e s
rs;e
jk Jk þ rjr r
ð17Þ
k¼1
n
Di;s mj;T ðT s Þ s 0 i X
s;q s
¼rjq Jq þ
rs;q
jk Jk þ rjr r
s
T
k¼1
ð18Þ
Eji;o Jj with T i;o ¼ T s and rs ¼ 0
j¼1
Jq0i;o
¼
n
X
ð20Þ
Qi;o
j Jj
with T
s;q s
s
s
s
rsee ¼rsqq ; rs;e
rr ¼ rrr ; rre ¼ rer ¼ rrq ¼ rqr
i;o
s
s
¼ T and r ¼ 0
j¼1
E*i,o
and Q*i,o
are the total and measurable heats of transfer.
j
j
Comparing this equation to eqns (11), (17), (18) and (19), it
follows that
rsej
¼ Qi;o
þ hi;o
j
j
rsee
Qi;o
¼
j
rsqj
rsqq
ð21Þ
Using that the measurable heat flux is invariant when one
changes the velocity of the frame of reference,12 it follows that
n
X
i;o
xi;o
¼ hi;o and
j Ej
j¼1
n
Dr Gs s 0 i X
s;q s
¼r
J
þ
rs;q
rq
q
rk Jk þ rrr r
Ts
k¼1
s;q
s;e
s;e
o;i s
s
rsek ¼rske ¼ rsqk ho;i
k rqq ; rrk ¼ rkr ¼ rrk hk rrq
n
X
Eji;o ¼ where the resistance matrices satisfy the Onsager symmetry
relations. The resistances in these matrices are related by
n
X
i;o
ci;o
¼0
j Qj
ð22Þ
j¼1
i,o i,o
and ci,o
where xi,o
j cj /c
j are the average mole fractions and
molar concentrations and hi,o the enthalpy density in the
i-phase. All these quantities of transfer are the same in the
bulk when one uses a frame of reference in which the surface is
at rest. One may therefore use the same values as those found
in the next section, see e.g. eqn (38).
ð19Þ
s;q
s;e
o;i s
o;i s
o;i o;i s
rs;e
jk ¼rkj ¼ rjk hk rjq hj rqk þ hk hj rqq
This follows using eqn (11) and the thermodynamic identity
@(mj/T)/@(1/T) = hj. Expressions for rsqq, rsqk = rskq, rsjk have
been found for the liquid–vapor interface using kinetic
theory.14–17 For the diagonal coefficient due to the reaction
one can use a form typical for reaction rate theory. Molecular
dynamics simulations for a one-component liquid–vapor interface seem to agree with kinetic theory for sufficiently short
range interaction potentials.18 For longer range potentials19
and for binary mixtures20 molecular dynamics simulations
indicate that the kinetic theory values are too small. For the
off-diagonal coefficients due to the reaction nothing is known.
Much work remains to be done to find reliable coefficients. On
the basis of eqn (18) one of us attempted to calculate the
resistance coefficients for a CO oxidation catalytic system and
found that the coupling between the heat flow and the reaction
rate in the 2-D reaction surface is significant and responsible
for the surface temperature excess.25 By evaluating the reaction resistance coefficients rrr for the same reaction but using
different catalysts, we are able to judge which catalyst is more
efficient and therefore it is helpful for catalyst design.
3.5.
Jq ¼
n
X
Dr Gs s
s;e s
¼rre Jq þ
rs;e
rk Jk þ rrr r
s
T
k¼1
while the excess entropy production given in eqn (16) results in
the linear relations
n
X
1
0
¼rsqq Jqi þ
Di;s
rsqk Jk þ rsqr rs
T
k¼1
for the surface. For the total and measurable heat fluxes, when
the temperature difference to the surface is zero and the
reaction is in equilibrium, we can write
4. The film model
In the usual studies of mass and heat transport to a surface,
one uses the film model. The coupling of mass and heat flux is
usually neglected, however. The derivation of the entropy
production in the films has been discussed in many places,
see for instance, ref. 1 and 21 and we just give the result in the
form that is most convenient for our present purpose, see
also:13
X
n
@ 1
@ mj
si ¼ Jq
Jj ð23Þ
þ
@x T
@x T
j¼1
The expression in the o-phase only contains the first term. It
will not be needed to consider this phase in our further
discussion of stationary states. For stationary states the total
heat flux, Jq, and the molar fluxes, Jj, are constant. We already
used this property in our analysis of the surface above. This
makes it possible to integrate the entropy production across
the film, which results in
X
n
m
1
j
sf ¼ Jq Df
ð24Þ
Jj Df þ
T
T
j¼1
Quantities of transfer
It is common to introduce quantities of transfer in the
description of transport in mixtures.12 This can be done also
5424 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427
where Df(. . .) gives the difference of the quantity between
brackets across the film (the value on the right hand side
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minus the value on the left hand side). The resulting linear laws
are
n
X
1
¼rfee Jq þ
rfek Jk
Df
T
k¼1
ð25Þ
n
m X
j
f;e
f
¼rje Jq þ
Df
rjk Jk
T
k¼1
where the resistance matrix satisfies the Onsager symmetry
relations.
Using eqn (23) one may similarly give a continuous description, in which the gradients of 1/T and m/T are expressed in the
energy and mass fluxes. It follows that the resistances in eqn
(25) are the integrals of the resistances in the continuous
description across the film. It is important to realize that the
validity of eqn (25) is not restricted to the thin film case, which
we will use below.
In order to express the resistances in terms of coefficients
familiar from the film theory, we replace the total energy flux
by the measurable heat flux on the right-hand side of the film,
J0fr
q , by introducing eqn (11) into eqn (24) the result is the
alternative expression:
X
n
mj 1
1
0
þ
ð26Þ
þ hfj Df
Jj Df sf ¼ Jqfr Df
T
T
T
j¼1
Here hfj is the average specific enthalpies in the film. In writing
eqn (26) we assumed that the film thickness is small so that the
variation of the specific enthalpies across the box is negligible.
The film thicknesses are between 0.1 and 1.0 mm in a vapor
film.1 In practice a choice is made that fits the data. The thin
film approximation made above implies the assumption that
this film thickness is small compared to the length scales which
follow from the diffusion of mass, which are given by the
diffusion coefficients divided by the center of mass velocity,
and the length scale which follows from the diffusion of heat,
which is given by the thermal conductivity divided by the heat
capacity at constant pressure times the center of mass velocity.
We will not go into the issue which film thickness to use any
further. We only wanted to clarify what the assumption that a
film is thin exactly means.
Using the thin film approximation, one may write eqn (26)
in the form
!
X
n
Df mj;T T fl
1
0
sf ¼ Jqfr Df
þ
ð27Þ
Jj T
T fl
j¼1
is the heat flux at the right end of the box
where J0fr
q
and Dfmj,T(Tfl) is the difference of the chemical potential across
the box at the constant temperature Tfl, the temperature at the
left end of the box, which is equal to the temperature of the
mixed fluid region. For cases when the film is too thick for this
approximation we refer to Taylor and Krishna.1 The linear
laws which follow from eqn (27) are
n
X
1
0
Df
rfqk Jk
¼rfqq Jqfr þ
T
k¼1
ð28Þ
n
Df mj;T ðT fl Þ f 0 fr X
f
¼rjq Jq þ
rjk Jk
T fl
k¼1
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The relation between the resistances in eqns (25) and (28) is
rfee ¼rfqq
rfek ¼rfke ¼ rfqk hfk rfqq
ð29Þ
f;e
f
f f
f f
f f f
rf;e
jk ¼rkj ¼ rjk hk rjq hj rqk þ hk hj rqq
where the enthalpies are the average values in the film.
4.1.
Extended Maxwell–Stefan equations
In recent years the Maxwell–Stefan equations have become the
favourite way to describe diffusion in a multi-component
system. Traditionally the coupling of the mass fluxes to the
heat flux, leading to the Soret and the Dufour effect, is
neglected. Extensions of the Maxwell–Stefan equations to
include these effects have been given. We will now give the
relevant expressions, to clarify the relation between the resistances above and the Maxwell–Stefan diffusion coefficients.
First we rewrite eqn (28) in the alternative form
X
n rf
1
1
0 fr
qk
Jq ¼ f Df
Jk
f
rqq
T
r
k¼1 qq
ð30Þ
"
#
X
n
rfjq rfqk
Df mj;T ðT fl Þ rfjq
1
f
Jk
¼ f Df
rjk f
þ
T fl
rqq
rqq
T
k¼1
There is a heat flux due to the mass fluxes across the film, the
Dufour effect. The total and measurable heats of transfer of
the film are defined in the usual way, similar to the definitions
given above. They are given (see eqn (21)) by
rfej
f
¼ Qf
j þ hj
rfee
Ejf ¼ Qf
j ¼
and
rfqj
rfqq
ð31Þ
and satisfy (see eqn (22))
n
X
xfj Ejf ¼ hf and
j¼1
n
X
cfj Qf
j ¼0
ð32Þ
j¼1
where xfj cfj /cf and cfj are the average mole fractions and
molar concentrations in the film, see ref. 12, pages 281–284.
Furthermore we define
lf df
and Rfjk rfjk rfqq T fl T fr
rfjq rfqk
rfqq
ð33Þ
Here lf is (the harmonic average of) the heat conductivity of
the film for given mass fluxes and df is the thickness of the film.
Using these parameters eqn (30) becomes
0
Jqfr ¼ lf
Df T þ
df
n
X
Qf
k Jk
k¼1
X
n
Df mj;T ðT Þ
1
f
¼
Q
D
Rfjk Jk
þ
f
j
T fl
T
k¼1
fl
ð34Þ
Now we use the thermodynamic property
n
X
cfj Df mj;T ðT fl Þ ¼ 0
ð35Þ
j¼1
Not all the thermodynamic forces are as a consequence
of eqn (35) independent. This results, using also eqn (32b),
Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5425
in the following property of the coefficients in eqn (34)
n
X
j¼1
cfj Qf
j ¼ 0 and
n
X
cfj Rfjk ¼
j¼1
n
X
cfj Rfkj ¼ 0
ð36Þ
j¼1
two sequenced surfaces. Below we shall discuss how to combine these two surfaces into ‘‘one effective surface’’.
In order to calculate the total differences of 1/T and mj/T
across both films and the surface we write
Here we also used the Onsager symmetry relations.
The Maxwell–Stefan diffusion coefficients, Djk, are now
defined by
1
1
1
D
Df
þ Di;s
T
T
T
m m m j
j
j
Df
þ Di;s
D
T
T
T
f
Rfjk dR
for j 6¼ k
cf Djk
ð37Þ
The diagonal coefficients are found using eqn (36). The
Maxwell–Stefan diffusion coefficients are symmetric, Djk =
Dkj. The heats of transfer can be written in terms of the
thermal diffusion coefficients, DTj, such that they satisfy eqn
(36), as
!
n
X
RT fl xfk DTj
DTk
f
Qj ð38Þ
f
Djk
rfj
rk
k¼1
Here rfj cfjMj is the average mass density in kg m3 and Mj
the molar weight in kg mol1 of component j. Using eqns
(36)–(38) we can write the linear laws (34) in the following
form
!
n
X
cf RT fl xfk xf‘ DTj
DTk f cfj Df mj;T ðT fl Þ
lf
0 fr
Jq ¼ f Df T þ
f v‘ Djk
rfj
rk
df T fl
d
k;‘¼1
! T
n cf RT fl xf xf
n cf Rxf xf
X
X
DTk
1
j k Dj
j k f
D
¼
ðvk vfj Þ
f
Djk
T
rfj
rfk
df Djk
k¼1
k¼1
ð39Þ
where vfj Jj/cfj is the average velocity of the j-th component in
the film. For the Maxwell–Stefan and the thermal diffusion
coefficient one should also use average values.
The Maxwell–Stefan diffusion coefficients are reasonably
well known.1,21 The heats of transfer are not so well known.
Approximate expressions have been given by Kempers22 and
by Haase23 for binary mixtures. Given these coefficients and
the heat conductivity one can first calculate the resistances rfqq,
rfqk = rfjq, rfjk and then the resistances rfee, rfek = rfje, rf,e
jk for the
film. For an analysis of the influence of these coefficients we
refer to Jenkinson and Pollard.24 In a recent paper2 we
documented the importance of the Soret effect for catalytic
hydrogen oxydation, H2 + (1/2)O2 - H2O.
Equations similar to eqn (39), in which Df/df is replaced by a
gradient and without the Latin superscripts, are given by
Taylor and Krishna,1 page 268, in the context of the continuous description. In that case one may, analogously to the
analysis given above, obtain expressions for the resistances ree,
rek = rje, rejk. When the film is not thin, this is the appropriate
procedure. As pointed out in the first remark in this section,
the resistances rfee, rfek = rfje, rf,e
jk should then be obtained by
integrating ree, rek = rje, rejk across the cell.
5. Combining the films with the surface
A very important feature of the description of the film in the
fourth section is its great analogy with the description of the
surface in the third section. The film and the surface are like
5426 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427
ð40Þ
Using eqn (17) for the surface and (25) for the film we find
n
X
1
1
1
ðrfek þ rsek Þ Jk þ rser rs
¼ s fl ¼ ðrfee þ rsee Þ Jq þ
D
T
T
T
k¼1
ree Jq þ
n
X
rek Jk þ rser rs D
k¼1
¼
m j
T
n
X
msj
mflj
s;e
s;e s
þ fl ¼ ðrfje þ rsje Þ Jq þ
ðrf;e
jk þ rjk Þ Jk þ rjr r
s
T
T
k¼1
rje Jq þ
n
X
s
rejk Jk þ rs;e
jr r k¼1
¼rsre Jq þ
n
X
Dr Gs
Ts
s;e s
rs;e
rk Jk þ rrr r
k¼1
ð41Þ
These relations make it possible to replace the films and the
surface by an ‘‘effective surface’’. One may, similar to the
procedure above, eliminate the total heat flux Jq in favor of the
measurable heat flux at either one end. We will not do this as it
serves no real purpose.
6. Conclusions
As shown, it is straightforward to integrate the contributions
due to the surface into the usual film theory. It is therefore
advantageous to use the full theory, which follows from and is
compatible with nonequilibrium thermodynamics. This makes
a systematic description of catalytic transitions on surfaces far
from equilibrium possible, and relatively easy to integrate in
already existing routines. The linear laws given in eqn (41)
make it possible to obtain combinations of the various resistances from measurements in which fluxes, the reaction rates
as well as the chemical potentials and temperatures of the
surface are known. It is not straightforward to obtain these
values for the surface. In the thesis of one of us25 a method is
described to obtain the surface temperatures from the reaction
rate using the Arrhenius equation.26
For both transport limited and kinetically controlled catalytic
reactions, coupling effects are found to be significant25 and
influence considerably the catalyst surface temperatures by
using non-equilibrium thermodynamics. Therefore, non-equilibrium thermodynamics is a convenient tool to model a heterogeneous catalytic system including the coupling effects both in
the diffusion boundary layer and the 2-D reaction surface.
This journal is
c
the Owner Societies 2006
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