Chemistry Reference
In-Depth Information
where
q
i
,sat,
X
denotes the saturation capacity of species
i
on site
X
,
b
i
,
X
is the affinity
constant and
f
i
is the gas phase fugacity of species
i
. The continuum level of an
entire pellet is calculated according to the partial differential equation
@
q
i
@
1
r
1
z
2
@
1
r
n
i
r
2
N
i
Þþ
t
¼
@z
ðz
;
i
¼
1
;
2
; ...;
n
(37)
where
q
i
is the loading of species
i
,
r
is the zeolite framework density,
z
the
diffusion path,
N
i
the molar flux of species
i
,
n
i
the stoichiometric coefficient and
r
is the rate of reaction. As the composition of reactants and products along the
pores changes continuously due to reaction, the adsorption equilibria for arbitrary
compositions inside the pores need to be calculated. This was achieved by means of
ideal adsorbed solution theory (IAST) [
194
] which requires only the pure compo-
nent isotherm data as input. These isotherms are taken from CBMC simulations.
The suitability of the IAST has to be checked by a limited number of MC simula-
tions of the multi-component adsorption equilibria. Therefore, IAST is a linking
model for MC adsorption results.
MD simulations were carried out in a rigid zeolite framework for a variety
of loadings and temperatures, employing the same force field as was used in the
CBMC simulations. From these data, self-diffusivities and Maxwell-Stefan diffu-
sivities were extracted. These diffusivities were used in the
N
i
terms of (37). Again
one has to check the validity of the Maxwell-Stefan approach by means of some
multi-component MD simulations. As the composition of the molecular mixture
inside the pores changes along the pore and with time, one has also to refer to a
linking model for diffusivities, as a complete calculation by MD would be by far too
time consuming. The Maxwell-Stefan approach serves as a linker. For details
see [
182
].
The rate of reaction of the one-step mechanism (see Fig.
6
) is given by
r
r
¼ ~
r
¼
k
f
q
EþB;H
þ
k
r
q
EB;H
þ
;
(38)
where
k
f
and
k
r
are the rate coefficients for the forward and reverse reaction of
ethene and benzene to form ethylbenzene, respectively,
q
E
þ
B
;
H
þ
is the amount of co-
adsorbed “ethene + benzene” at the active sites and
q
EB
;
H
þ
is the amount of adsor-
bed ethylbenzene at the active sites. As benzene and ethylbenzene are mostly sited
in the zeolite cages whilst the ethene molecules can move into the channels and the
cages, the assumptions made for Langmuir-Hinshelwood kinetics are not fulfilled.
For
q
E
þ
B
;
H
þ
and
q
EB,H
analytical expressions are needed in order to calculate them
from species loadings
q
E
,
q
B
, and
q
EB
. As was outlined in [
182
], such terms may be
obtained from MC simulations of the multi-component adsorption isotherms. As
benzene and ethylbenzene are mostly sited inside the cages, fitting to Langmuir-
Hinshelwood kinetics is not possible.
To conclude, by employing suitable linking models based solely on mole-
cular data, one can obtain results on a macro level, i.e. composition profiles inside
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