inggg

1. Analysis and optimal design of an ethylene oxide reactor
Andreas Peschel a
, Florian Karst a
, Hannsj¨org Freund a,n
, Kai Sundmacher a,b
a
Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
b
Process Systems Engineering, Otto-von-Guericke University Magdeburg, Universit¨atsplatz 2, 39106 Magdeburg, Germany
a r t i c l e i n f o
Article history:
Received 16 May 2011
Received in revised form
30 August 2011
Accepted 31 August 2011
Available online 12 September 2011
Keywords:
Chemical reactors
Optimization
Design
Process intensification
Elementary process functions
Ethylene oxide
a b s t r a c t
In this work, a recently proposed multi-level reactor design methodology (Peschel et al., 2010) is
extended and applied for the optimal design of an ethylene oxide reactor. In a first step, the optimal
reaction route is calculated taking various process intensification concepts into account. The potential
of each reaction concept can be efficiently quantified, which is the economic basis for the design of
advanced reactors. Based on these results, a promising concept is further investigated and a technical
reactor is designed. As an extension to the design method, reactor design criteria for external and
internal heat and mass transfer limitations are directly included in the optimization approach in order
to design the catalyst packing. The derived reactor concept is investigated with a detailed 2D reactor
model accounting for radial concentration and temperature gradients in addition to a radial velocity
profile.
The example considered in this work is the production of ethylene oxide which is one of the most
important bulk chemicals. Due to the high ethylene costs, the selectivity is the main factor for the
economics of the process. A membrane reactor with an advanced cooling strategy is proposed as best
technical reactor. With this reactor design it is possible to increase the selectivity of the ethylene
epoxidation by approximately 3% compared to an optimized reference case.
& 2011 Elsevier Ltd. All rights reserved.
1. Ethylene oxide: industrial process and intensification
aspects
1.1. Industrial production
Since ethylene oxide (EO) is a bulk scale product with an
annual production of about 19 million tons in 2010, cost effi-
ciency in the production is very important. Even a selectivity
increase in the range of 1% has a large effect on the economy of
the process since ethylene makes up for approximately 68% of the
total cost of the EO process (Baerns et al., 2006).
The reaction network of the silver catalyzed ethylene oxide
production is shown in Fig. 1. It consists of the desired partial
oxidation of ethylene (E) to ethylene oxide (reaction 1), the total
oxidation of ethylene to carbon dioxide (reaction 2), and the
consecutive oxidation of EO to carbon dioxide (reaction 3).
In general, ethylene is oxidized over a promoted Ag-catalyst
either using air (air based process) or pure oxygen (oxygen based
process). In addition to the different process technologies, different
types of Ag-catalysts can be used. In both processes, tube bundle
reactors with a constant cooling temperature are commonly used.
Due to explosion hazards, the E, O2 and EO concentrations as well as
the conversion are limited. In addition, the used reactor types
cannot provide optimal component concentration and temperature
profiles along the reactor length. This may give rise to problems
with an axial hot spot as well as ethylene and oxygen depletion
along the channel, which results in a lower reactor productivity. For
a general process description and more details, refer to the literature
(e.g. Rebsdat and Mayer, 2007).
1.2. Process intensification aspects for ethylene oxide reactors
In order to increase the EO production selectivity, many
reactor types such as membrane reactors or micro-reactors have
been investigated.
The influence of different dosing options (ethylene or oxygen)
for the air based process was experimentally studied by Lafarga
and Varma (2000) using a fixed bed membrane reactor. They
compared the performance to a classical fixed bed reactor and
their results indicate that concentration manipulation along the
reaction channel can increase the selectivity.
In order to investigate the potential of micro-reactors for the
EO process, Kestenbaum et al. (2002) constructed several micro-
reactors and compared their performance to industrial EO reac-
tors. While the space time yield (STY) and the conversion were in
the same range as the conventional reactors, the selectivity was
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Chemical Engineering Science
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2011.08.054
n
Corresponding author. Tel.: þ49 391 6110275; fax: þ49 391 6110634.
E-mail address: freund@mpi-magdeburg.mpg.de (H. Freund).
Chemical Engineering Science 66 (2011) 6453–6469

2. much lower compared to the industrial process. An important
aspect when using micro-reactors is that the reaction mixture can
lie within the explosive range, which may allow for higher
ethylene and oxygen inlet and higher EO outlet concentrations.
Increasing the EO outlet concentration can improve the produc-
tivity of the plant. However, in the connecting pipes the composi-
tion will also be in the explosive range and hence such a reactor
will require additional safety measures in the plant. In addition,
even in a micro-reactor the maximum safe diameter might be
exceeded as shown by Fischer et al. (2009). Therefore, the gas
mixture must never be within the explosive regime for the EO
process.
Besides these approaches to design new micro- or membrane
reactors, also other reaction routes to EO were investigated.
Berndt and Br¨asel (2009) worked on a completely new reaction
route, where the epoxidation of olefins is performed using ozone
in the gas phase. Lee et al. (2010) presented a concept for the EO
production based on gas-expanded liquids similar to the HPPO
process for propylene. For both reaction concepts, the process
efficiency must account for the production of ozone or hydrogen
peroxide, respectively, and hence these concepts are not directly
comparable to the established process.
To sum up, selectivity optimization for the EO reactor is of
major industrial importance. Different trends for temperature,
pressure, and component concentrations on the selectivity can be
observed depending on the used catalyst. As shown by various
authors, process intensification concepts such as membrane
reactors or micro-reactors seem to be an interesting option for
the production of EO. However, no study was performed which
considers all PI measures in a systematic manner. Hence, it can be
concluded that the question on how an optimal EO reactor has to
be designed and what would be the benefit compared to the
standard design has not yet been answered satisfactorily.
Therefore, this contribution is intended to design an improved
EO reactor, which performs optimal from both the reaction
concept and the technical point of view. The selectivity of the
reactor is maximized taking component dosing and removal
concepts, advanced temperature control, the dimensioning of
specific exchange areas for heat and mass transport, the choice
of catalyst, and the choice of the catalyst packing into account.
Since most of the reaction kinetics available from the literature
are only valid for the air based process, this study focuses on the
air based EO process. This work extends successfully our recently
published reactor design methodology (Peschel et al., 2010) to a
selectivity problem taking more advanced process intensification
methods into account and providing more insight into the
modeling based on elementary process function concept. In
addition, design criteria for external and internal mass and heat
transport are considered directly in the optimization based design
framework. These criteria—to the best of our knowledge—are not
yet rigorously considered in any published reactor design
method. Furthermore, the derived reactor design is validated by
optimizing a 2D reactor model. Such an optimization can also not
be found in the literature and can be considered as proof of
principle. The followed design approach considers non-idealities
of the technical approximation in order to derive best possible
technical reactors and not only design reactors based on simpli-
fied models.
2. Design of an optimal ethylene oxide reactor
The state-of-the-art reactor design methods can be classified
into heuristics (e.g. Hanratty and Joseph, 1992; Schembecker
et al., 1995; Jacobs and Jansweijer, 2000), attainable region
methods (e.g. Glasser et al., 1987; Hildebrandt and Glasser,
1990; Feinberg and Hildebrandt, 1997), and rigorous optimization
approaches such as superstructure optimization (e.g. Balakrishna
and Biegler, 1992; Kokossis and Floudas, 1994; Lakshmanan and
Ethylene
C2H4
Ethylene Oxide
(CH2)2O
2·CO2 + 2·H2O
+ 2.5·O2
Consecutive
Oxidation (r3)
ΔHr3 = -1216.3 kJ/mol
Total Oxidation (r2)
ΔHr2 = -1323 kJ/mol
Partial Oxidation (r1)
ΔHr1 = -106.7 kJ/mol
Fig. 1. Simplified macroscopic reaction scheme.
Level 1
Level 2
Level 3
• Integration &
enhancement concept
• Schematic reactor
concept, catalyst support
• Transport mechanisms
• Control variables
• Approximation concept
• Type of model for
detailed investigation and
design
Decisions Model Results
• Balance equations
• Reaction kinetics
• Thermodynamics
• Intrinsic bounds
• Mass & energy transport
• Exchange areas
• Reaction eng. bounds
• Balance equations for all
supporting phases
• 2D or 3D model
• pseudo-homogeneous,
heterogeneous
• Optimal route
• Potential of intensification
concept
• Best reactor concept
• Best control variables
• Losses due to limited mass &
energy transport
• Best technical approximation
• Losses due to non-ideal
control variable profiles
• Losses due to non-ideal flow
field, radial gradients,
diffusion effects
Detailed
Design
Fig. 2. Generalized decision structure for the development of an optimal reactor.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696454

3. Biegler, 1996), the dynamic optimization approach (e.g. Horn and
K ¨uchler, 1959; Bilous and Amundson, 1956; Aris, 1960), or the
systematic staging of reactors (Hillestad, 2010). However, all of
these methods do not directly allow the design of advanced and
innovative reactors.
In addition to the classical reactor design approaches, process
intensification options (Freund and Sundmacher, 2008) become
more and more important for the design of tailor-made reactors,
which are superior to standard reactors for the specific task. Here,
a three step approach (Peschel et al., 2010) is used to develop a
reactor design which ensures the optimal process conditions
along the reaction coordinate. The general decision structure is
shown in Fig. 2.
On the first level, the optimal route is calculated considering
the equations of change, reaction kinetics, thermodynamic limita-
tions, and system inherent boundaries. This optimal route is
obtained by balancing a fluid element and manipulate the fluid
element along the reaction coordinate such that the reaction
conditions are optimal all along the reaction coordinate. Here,
different integration and enhancement options are compared
with each other and the potential of every option is quantified
by the comparison with an optimized technical reference case.
On the second level, a schematic reactor set-up is chosen based
on the result of the first level. The kinetic expressions for mass
and energy transport are added and the influence of limited mass
and energy transport is quantified. By choosing the schematic
reactor set-up, bounds for the specific exchange areas and the
catalyst density are defined and correlations for heat and mass
transfer from the literature can be used.
On the third level, the best possible technical reactor is derived
by approximation of the control variable profiles and the design is
validated taking non-idealities into account.
Summarizing the applied reactor design method, all possible
intensification options are screened with relatively simple models
on the first level and only the most promising reactor set-ups are
investigated in detail. This approach enables the model based
development of an optimal technical reactor on the basis of the
optimal route in state space taking a wide range of physical and
chemical phenomena into account. The optimal reactor is
designed following a structured, knowledge- and optimization
based approach, which yields the best reactor considering the
objective function, reaction engineering criteria, and decisions
taken by the engineer.
2.1. Catalyst selection
Before the best reaction concept for the air based EO process
can be derived, a suitable catalyst must be chosen. The choice of
the catalyst has a major influence on the performance of the
reaction system and determines how the selectivity changes with
temperature, pressure, and concentration of reactants and reac-
tion products.
From the many investigated catalysts for the EO process, the
kinetics published by Petrov et al. (1986), Stoukides and Pavlou
(1986), Al-Saleh et al. (1988), Borman and Westerterp (1995),
Schouten et al. (1996), and Lafarga et al. (2000) are compared in
order to identify the best catalyst from this selection.
Referring to Fig. 3, the selectivities of the investigated systems
are quite different and also the reaction rates (only r1 shown) vary
strongly. The comparison is performed at typical inlet conditions
for the component partial pressures (pE ¼ pO2
¼ pCO2
¼ 1:5 bar,
pEO ¼ pH2O ¼ 0 bar) and a wide temperature range. The qualitative
trends and the differences between the various reaction rates are
the same for other operating conditions.
As can be observed in Fig. 3 the catalyst investigated by
Al-Saleh et al. (1988) exhibits by far the highest differential
selectivity and hence this catalyst system with the according
reaction kinetics is chosen for the further investigations.
2.2. Level 1: optimal route in state space
On the first level, the potential of different integration and
enhancement concepts is investigated. For this purpose, a refer-
ence case must be specified, which is a tube bundle reactor with
constant cooling temperature as used in the industrial process.
For the investigated catalyst, the temperature profile in the
reactor for a constant cooling temperature is nearly uniform since
the selectivity is high and the reaction rates are relatively low.
Hence, the temperature and pressure for a fluid element with
fixed inlet composition (xE ¼ xO2
¼ xCO2
¼ 0:075, xEO ¼ xH2O ¼ 0)
are directly optimized. The inlet mole fractions are the mean
values of the industrial process range (Rebsdat and Mayer, 2007).
For this study, we chose a STY of 0.27 mol/(m3
s) within a
residence time of 30 s for all investigated reaction concepts.
Specifying the STY and the residence time is better suited than
comparing different integration concepts for a fixed conversion
since the amount of reactants can vary depending on the initial
composition and on the dosing of reactants. The optimal
operation parameters for the reference reactor are T¼625.7 K
and p¼20 bar. With this reactor a selectivity of up to 79.10% can
be achieved.
In Section 2.2.1 each investigated integration and enhancement
concept is explained. Each case is a dynamic optimization problem,
where the optimal route in state space with respect to the
integrated and enhanced fluxes is determined in order to maximize
the selectivity. The required model equations are presented in
Section 2.2.2. Bringing the model equations and the investigated
concepts together, the arising optimization problems are stated in
550 560 570 580 590 600
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Selectivity[−]
Temperature [K]
550 560 570 580 590 600
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Reactionrate[mol/kg/s]
Temperature [K]
Fig. 3. Catalyst comparison (Al-Saleh et al., 1988, ; Stoukides and Pavlou, 1986, ; Borman and Westerterp, 1995 (model 2 including long time deactivation
factors), —; Lafarga et al., 2000, Á Á Á Á; Schouten et al., 1996 (model 3, tubular reactor), – –; Petrov et al., 1986, - Á -). (a) Differential selectivity and (b) reaction rate r1.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6455

4. Section 2.2.3 before the results are presented and discussed in
Section 2.2.4. As discussed in Section 3, the numerical solution
approach can only yield locally optimal solutions and hence every
solution must be taken as locally optimal.
2.2.1. Investigated integration and enhancement concepts
For the EO production process, controlling the heat flux and
applying dosing and removal strategies are of great interest to
increase the EO selectivity. The different integration and enhance-
ment concepts which are investigated are shown in Fig. 4. Here,
all investigated cases from level 1 to level 3 are shown. In all cases
the inlet composition and the system pressure are degrees of
freedom for the optimization. We refer to all investigated cases as
intensified air based processes even if ethylene or oxygen is dosed
individually to the reaction mixture since the concentration
range, the inert gas, the conversion, and the space time yield
are chosen in the range of the industrial air based process.
Considering assist oxygen (enrichment in oxygen) is an important
aspect in the industrial application of process intensification.
For safety reasons, the ethylene and oxygen concentration
must not exceed the explosion limit. Since a mathematical
formulation of the explosive range at elevated pressure and
temperature for the multi-component mixture in the EO reactor
is not available in the open literature, it is assumed that the mole
fractions must always stay in the industrial range of the inlet
conditions given in Table 2 in order to stay out of the explosive
range. The concentration limit of each component is considered as
an intrinsic bound and hence considered directly on level 1 (and
all subsequent levels) in our approach.
Case 1: Optimized inlet composition, isothermal. This case is
similar to the reference case except that the inlet composition is
optimized. This does not affect the reactor itself, but the technical
process must be adjusted to meet the optimal inlet conditions and
hence this case is regarded as an intensified concept.
Case 2: Inlet composition and heat flux optimized (q). In addition
to the optimization parameters of case 1, the heat flux is adjusted
along the reaction coordinate in order to obtain an optimal
temperature profile. This case yields the potential of an advanced
heat flux profile.
Case 3: Inlet composition, heat and ethylene flux optimized ðq, jEÞ.
Besides the optimal heat flux profile, an optimal ethylene dosing
profile is calculated. This case quantifies the potential of an
advanced heat flux profile combined with an ethylene dosing
strategy.
Case 4: Inlet composition, heat and oxygen flux optimized ðq, jO2
Þ.
This case is similar to case 3, but here an optimal oxygen dosing
strategy instead of an ethylene dosing strategy is applied.
Case 5: Inlet composition, heat and CO2 flux optimized (q, jCO2
Þ.
Here, an optimal removal strategy for CO2 is calculated combined
with an advanced heat flux profile. Although CO2 in principle
reduces both reaction rates owing to the adsorption term, the CO2
partial pressure has a large influence on the selectivity in case the
STY is fixed. Due to the STY constraint the reaction temperature
must be increased in case of higher CO2 partial pressure. This
reduces the selectivity due to the higher activation energy of r2
compared to r1.
Other removal strategies will not be investigated in this
contribution, even if the in situ removal of EO is of great interest,
since the consecutive oxidation of EO and the downstream
absorption can be avoided. The influence of the EO removal on
the selectivity is not investigated in the chosen reaction rate laws
(Al-Saleh et al., 1988) since EO does not affect r1 and r2, and r3 was
neither observed nor determined for this catalyst. The removal of
CO2 can be considered as a general show case to exemplify the
proposed methodology for product removal.
Case 6: Inlet composition, heat, ethylene and oxygen flux opti-
mized ðq, jE, jO2
Þ. This case investigates the potential of an optimal
ethylene and oxygen dosing strategy combined with an optimal
heat flux profile.
Case 7: Inlet composition, heat, ethylene, oxygen and CO2 flux
optimized ðq, jE, jO2
, jCO2
Þ. Heat, ethylene, oxygen, and CO2 flux are
optimized in addition to the inlet composition for this case. This
concept yields an upper bound for all intensified reaction con-
cepts since all factors influencing the reaction rates, namely the
mole fractions of E, O2, CO2, the pressure, and the temperature,
are optimally manipulated. All degrees of freedom of the cases
1-6 are included in this case. Nevertheless, the cases 1-6 are of
great interest since it may be sufficient to influence only a certain
set of the mole fractions in addition to the temperature. Hence,
the potential of each set must be known in order to decide which
concept should be further investigated. In addition, the results of
this case answer the question whether a reactor with CSTR
characteristics can approximate the optimal profiles.
In principle, optimal pressure profiles and apparent catalyst
density profiles—which can be obtained by catalyst dilution
or by realizing different void fractions using different catalyst
packings—are of interest for the optimal reactor design. In case of
EO, the optimal pressure is always at the upper boundary since a
high pressure helps to fulfill the required STY and increases the
Level 2
Objective: Selectivity
Heat and reaction
flux integrated
Heat, reaction and component fluxes integrated
Case1:
isothermal
Level 1
Level 3
Techn.
Appr.
Detailed
Model
Tc, Per
Case 2:
q
Case 3:
q, jE
Case 4:
q, jO2
Case 5:
q, jCO2
Case 6:
q, jE, jO2
Case 7:
q, jE, jO2, jCO2
Fig. 4. Decision structure for the development of an optimal EO reactor.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696456

5. selectivity. In addition, the apparent catalyst density is also
always at its upper limit specified by the lower limit for the void
fraction since the heat flux can be ideally controlled, and hence a
high catalyst density helps to fulfill the required STY.
2.2.2. Model equations of level 1
The component mole balance for every component is given by
the following equation:
dni
dt
¼ si Á ji þrp Á
1Àe
e
Á Vgas Á
XNR
j ¼ 1
ni,j Á rj ð2:1Þ
The case selection variables si are used in order to obtain a
comprehensive and clear representation in the balance equation.
The value of si depends on the investigated case explained in
Section 2.2.1, and it can either be zero if the flux is not considered
or one if the flux is considered as optimization function.
The void fraction depends on the catalyst packing and may
vary in a wide range depending on the type and shape of the
catalyst support. On the first level a constant void fraction is
assumed ðe ¼ 0:4Þ. The chosen value represents a typical technical
void fraction in case of a randomly packed bed with uniform
spheres.
The reaction rates along with the reaction rate constants and
adsorption constants are given by Eqs. (2.2)–(2.4) and the para-
meters are summarized in Table 1.
rj ¼
kjp
nE,j
E p
nO2,j
O2
1þKjpCO2
, j ¼ 1; 2 ð2:2Þ
kj ¼ k0,j exp À
EA,j
RT
, j ¼ 1; 2 ð2:3Þ
Kj ¼ K0,j exp
Tads,j
T
, j ¼ 1; 2 ð2:4Þ
The energy balance is written in terms of temperature and is
simplified by assuming no technical work, negligible influence of
the pressure change and of the dosed components on the
temperature change (refer to Eq. (2.5)). The heat flux into and
out of the fluid element qtot is the total heat flux. While the second
term on the right hand side is bounded, qtot is assumed to be an
unrestricted control function. Hence, every temperature profile
can be obtained on level 1 and the temperature profile can
directly be taken as an optimization function on this level.
n Á cp
Vgas
Á
dT
dt
¼ À qtot þrp Á
1Àe
e
Á
XNCOM
i ¼ 1
hi
XNR
j ¼ 1
ni,j Á rj
0
@
1
A
0
@
1
A ð2:5Þ
The gas volume is described by the ideal gas law. This yields
the total gas volume and the components partial pressures using
the definition for the total amount of substance and the compo-
nent mole fractions.
n ¼
XNCOM
i ¼ 1
ni ð2:6Þ
xi ¼
ni
n
ð2:7Þ
Vgas ¼
n Á R Á T
p
ð2:8Þ
pi ¼ xi Á p ð2:9Þ
The initial conditions are chosen to be in the typical range for the
air based ethylene oxide process according to Table 2. In addition
to E, O2, CO2, and N2, some small amounts of EO, H2O and CH4 are
present. The upper and lower mole fraction of each component
used for the optimization are also given in Table 2. The inlet mole
fractions of E, O2, and CO2, as well as the inlet pressure, and the
inlet temperature are optimization variables in all calculations.
Due to numerical reasons during the optimization calculations a
lower limit on the ethylene and oxygen mole fraction has to be
assigned. The inlet mole fractions of EO and H2O are set to
zero and the inlet mole fraction of N2 is calculated from
the summation constraint. The total pressure is in the range of
10–20 bar and the temperature range is 550–630 K.
The space time yield is related to the catalytic channel and is
given by the following equation:
STY ¼
nEO,f ÀnEO,0
R tf
t0
Vgas
e
dt
¼ 0:27 mol=ðm3
sÞ ð2:10Þ
A general formulation of the selectivity considering ethylene
dosing is used as objective function:
S ¼
nEO,f ÀnEO,0
nE,0 þ
R tf
t0
jEdt
ÀnE,f
ð2:11Þ
2.2.3. Optimization problems of level 1
The full optimization problems which must be solved on level 1
are stated in (OP1). In order to compare the case stated above in a
comprehensive manner, the optimization functions, the optimization
Table 1
Model parameters for the reaction rates (Al-Saleh et al., 1988).
Parameter Reaction 1 Reaction 2
k0,j 6:275 Â 106
mol=ðkgp s Pa1:1
Þ 1:206 Â 107
mol=ðkgp s PaÞ
EA,j 74:9 kJ=mol 89:8 kJ=mol
nE,j 0.6 0.5
nO2 ,j 0.5 0.5
K0,j 1.985 Â 102
PaÀ1
1.08 Â 102
PaÀ1
Tads,j 2400 K 1530 K
Table 2
Limits for the mole fractions used for optimization, typical inlet conditions of the
air based process (Rebsdat and Mayer, 2007).
Component xL
xU
Typical inlet
conditions (% v/v)
E 10À5
0.1 2–10
O2 10À5
0.08 4–8
EO 0 0.1 % 0
CO2 0.05 0.1 5–10
H2O 0 0.1 % 0
N2 0 1 72–89
Table 3
Attributes and selectivity of each reaction concept investigated on level 1.
Case Attributes (DoF) si Selectivity (%)
Reference T, p si ¼ 0 8i A COM 79.10
Case 1 T, p, xi,0 si ¼ 0 8i A COM 81.35
Case 2 T(t), p, xi,0 si ¼ 0 8i A COM 81.48
Case 3 T(t), p, xi,0, jE(t) si ¼ 0 8i A COMfEg, sE ¼ 1 82.35
Case 4 T(t), p, xi,0, jO2
ðtÞ si ¼ 0 8i A COMfO2g, sO2
¼ 1 81.73
Case 5 T(t), p, xi,0, jCO2
ðtÞ si ¼ 0 8i A COMfCO2g, sCO2
¼ 1 81.99
Case 6 T(t), p, xi,0, si ¼ 0 8i A COMfE, O2g, 82.71
jEðtÞ, jO2
ðtÞ sE ¼ sO2
¼ 1
Case 7 T(t), p, xi,0, si ¼ 0 8i A COMfE, O2, CO2g, 82.84
jEðtÞ, jO2
ðtÞ, jCO2
ðtÞ sE ¼ sO2
¼ sCO2
¼ 1
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6457

6. parameters (together referred to as degree of freedom (DoF)), and the
integrated component fluxes are summarized in Table 3.
Obj ¼ max
DoF
S ðOP1Þ
s:t:
Component balances: Eq: ð2:1Þ
Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ
Constitutive equations: Eqs: ð2:6Þ2ð2:9Þ
Initial conditions: xiðt ¼ 0Þ ¼ xi,0
Residence time: t ¼ 30 s
STY : Eq: ð2:10Þ
Intrinsic bounds: TL
rT rTU
, pL
rprpU
, xL
i rxi rxU
i 8i A COM
Case selection: si according to Table 3
The bounds for temperature, pressure, and STY are stated in
the model description. The bounds for the composition are given
in Table 2 and the parameters of the reaction rates in Table 1.
0 10 20 30
560
570
580
590
600
610
620
T[K]
t [s]
case 2 (q)
case 3 (q, jE)
case 5 (q, jCO2
)
case 6 (q, jE, jO2
)
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
2
2.08
2.16
2.24
2.32
2.4
jE/nin10−3
[1/s]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
1.8
1.9
2
2.1
2.2
2.3
jO2
/nin10−3
[1/s]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
−23
−18
−13
−8
−3
2
jCO2
/nin10−3
[1/s]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
2
4
6
8
10
12
ji/nin10−3
[1/s]
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
t [s]
xi[−]
−1
−0.3
0.4
1.1
1.8
2.5
ji/nin10−3[1/s]
Fig. 5. (a) Temperature profiles. (b)–(h) Mole fractions and dosing profiles (—, E; – –, O2; Á Á Á Á, EO; - Á -, CO2; thin lines, mole fractions; thick lines, dosing profiles).
(a) Temperature profiles, (b) Case 1: isothermal. S¼81.35%, (c) Case 2: q. S¼81.48%, (d) Case 3: q, jE. S¼82.35%, (e) Case 4: q, jO2
. S¼81.73%, (f) Case 5: q, jCO2
. S¼81.99%, (g)
Case 6: q, jE, jO2
. S¼82.71%, and (h) Case 7: q, jE, jO2
, jCO2
. S¼82.84%.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696458

7. 2.2.4. Results
The results for all investigated reaction concepts of level 1 are
presented in Fig. 5 and the selectivities are summarized in Table 3.
The optimal temperature of the isothermal case with opti-
mized inlet composition (case 1) is 592.7 K. In this case, the
ethylene and oxygen inlet mole fractions are at their upper bound
(xE,0 ¼ xU
E ¼ 0:1, xO2,0 ¼ xU
O2
¼ 0:08) while the CO2 inlet mole frac-
tion is at its lower bound (xCO2,0 ¼ xL
CO2
¼ 0:05). The high inlet
fractions of ethylene and oxygen in combination with the low
inlet fraction of CO2 enable a lower temperature level compared
to the reference case. This reduced temperature level and the
direct influence of the higher ethylene mole fraction increase the
selectivity by 2.25% compared to the reference case.
In Fig. 5(a) the temperature profiles of selected cases where
the heat flux is optimized are shown. For the sake of clarity, the
temperature profile of case 4 ðq, jO2
Þ is not shown in Fig. 5(a) since
it is almost the same as in case 3 ðq, jEÞ. In case 7 ðq, jE, jO2
, jCO2
Þ the
optimal temperature profile is constant at 569.5 K and it is also
not shown for clarity.
From case 2–7, the mole fraction profiles and the flux profiles
of the manipulated components are presented in Fig. 5(c)–(h). In
addition, the selectivity of each case is given in the caption.
Referring to case 2, it can be concluded that an optimal
temperature profile exists for the conventional EO reactor, which
is not constant. The optimal temperature profile increases con-
tinuously starting from 570.1 K to 616.7 K. This temperature rise
counterbalances the lower reaction rates due the decreasing
ethylene and oxygen partial pressure and the increasing CO2
partial pressure with increasing residence time. However, the
selectivity increases only by 0.13% comparing case 1 with case 2.
Hence, it can be concluded that realizing a temperature profile in
the reactor is probably not worthwhile if additional investment
costs arise.
The optimal ethylene dosing profile with the resulting com-
ponent mole fraction profiles of case 3 are shown in Fig. 5(d). The
ethylene dosage makes up for the consumed ethylene so that the
ethylene mole fraction is always at its upper bound. The tem-
perature profile is similar to case 2 except that the temperature
level is lower. Besides the direct influence of ethylene on the
selectivity (compare reaction order of ethylene in r1 and r2), the
lower temperature level contributes to the selectivity increase of
3.25% compared to the reference case.
Referring to Fig. 5(e) belonging to case 4, the optimal oxygen
dosing strategy keeps oxygen at its upper boundary. The optimal
temperature profile is almost the same as in case 3. The selectivity
improvement is lower than in the ethylene dosing case. Hence,
dosing ethylene is more worthwhile to investigate than dosing
oxygen.
In case the produced CO2 can be removed from the reaction
mixture in situ (case 5), the selectivity can be increased by 2.89%.
Due to the inhibiting effect of CO2 on the catalyst, a lower CO2
partial pressure allows a temperature reduction while still match-
ing the required STY. As shown in Fig. 5(f), the optimal CO2
removal strategy starts with a high CO2 inlet mole fraction and
continuously removes CO2 from the reaction mixture until the
lower CO2 limit is reached. Afterwards, CO2 is kept at its lower
limit. This strategy is advantageous over starting with a CO2 inlet
mole fraction at its lower limit and keeping the CO2 mole fraction
constant at its lower limit since the former strategy keeps the
mole fractions of ethylene and oxygen high up to a residence time
of 3 s. This effect increases the selectivity more than always
staying at the lower limit for the CO2 mole fraction, which would
yield a selectivity 0.10% lower. Such a result can hardly be
obtained by intuition or heuristics, which demonstrates that our
model based approach is advantageous for the design of optimal
reactors.
If optimal ethylene, oxygen, and heat fluxes are provided (case
6), the ethylene mole fraction is always kept at its upper boundary
while the oxygen mole fraction is increased from xO2ðt0Þ ¼ 0:05 to
its upper boundary in the first 4 s (refer to Fig. 5(g)). Such a strategy
keeps the CO2 mole fraction low at the beginning and is advanta-
geous over a strategy where both the ethylene mole fraction and the
oxygen mole fraction are constant at their upper boundaries.
In case 7 ðq, jE, jO2
, jCO2
Þ the ethylene and oxygen mole fractions
are always kept at their upper bound, while the CO2 mole fraction
is kept at its lower bound (refer to Fig. 5(h)). Since these mole
fractions are all constant, the optimal temperature is also con-
stant (T¼596.5 K). In case the profiles for the temperature and all
influenced mole fractions are constant, these profiles could be
approximated by a completely back-mixed reactor. However, the
reactant mole fraction profiles must not be at their upper
boundaries for this case since the ethylene and oxygen concen-
tration in the feed are even higher, which is not allowed due to
the explosion hazard. In addition, the CO2 mole fraction profile
should not be at its lower boundary since the feed would have to
contain even less CO2 in this case and that contradicts the
assumed concentration bounds. The selectivity gain compared
to the reference case is 3.74%.
Case 7 has the highest potential for the selectivity increase, but
it is also most complicated to realize. The removal of CO2 could in
principle be realized by absorption with an amine solution or a
potassium carbonate solution. However, these solvents are not in
liquid state at reaction conditions, which makes an in situ
absorption impossible with these solvents. Hence, either inter-
mediate absorption must be applied or other solvents (such as
ionic liquids) allowing an in situ absorption at the reaction
conditions must be used. A detailed investigation which solute
may be used at the reaction temperature and designing a reactor
with integrated extraction of CO2 exceeds the scope of this paper
and hence all cases including CO2 removal are not further
investigated in this contribution. Nevertheless, the methodology
identifies the upper limit for the selectivity which can be obtained
by such a reaction concept.
From the results of level 1 it can be observed that the exact
knowledge of the explosive range is very important for the
optimal operation of the reactor. The optimal operation condi-
tions are often at the upper boundaries for the ethylene and
oxygen mole fractions defined by the explosion limit and this
indicates that higher ethylene and oxygen mole fractions will
further increase the overall EO selectivity. In addition, decreasing
the CO2 mole fraction will increase the selectivity for the same
STY and residence time. However, the rate law is not applicable at
CO2 levels below the chosen limit and hence the lower bound on
the CO2 mole fraction is necessary to obtain reliable results.
Summing up, it can be concluded that applying advanced con-
centration and temperature control strategies has high potential for
improving the EO selectivity. Hence, it is interesting to investigate if
this potential can also be exploited in case the optimal flux profiles
are approximated in a technical reactor. Taking the explosion limits
into account, ethylene and oxygen must be dosed using separate
channels. Hence, the dosing of only one of the components is much
easier. In this contribution, case 3 ðq, jEÞ will be further investigated
since it seems to be the most promising case from a selectivity and a
reaction engineering point of view.
2.3. Level 2: reactor concept, transport mechanisms, and control
variables
Based on the case studies of level 1, only the most promising
cases are further investigated. It is determined which transport
mechanisms and control variables are suited to make the desired
flux profiles attainable. For this purpose, a schematic reactor
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6459

8. design is proposed and control variables which can be changed by
the reactor design are identified. The losses due to the limited
mass and energy transport are quantified by comparing the
results of level 1 and level 2.
In principle, several mechanisms for the realization of heat and
component fluxes exist. Here, for the sake of simplicity the heat
flux is controlled by changing the cooling temperature along the
reaction coordinate and the distributed dosing of ethylene is
provided via a membrane. At constant pressure on the ethylene
supply side the permeance of the membrane is adjusted along the
reaction coordinate in order to control the ethylene flux in an
optimal manner. Alternatively, the exchange areas for both fluxes
or the pressure on the ethylene supply side could be controlled. It
should be noted that the cooling temperature and the permeance
of the membrane can still be ideally manipulated within the
specified bounds on the second level. The proposed schematic
reactor design is shown in Fig. 6.
At this point, the choice of the schematic reactor set-up
includes the choice of the catalyst support geometry. Here, a
fixed bed reactor with a randomly packed bed is chosen since it
offers the highest catalyst density compared to other catalyst
support concepts. A high catalyst density is required in order to
meet the required STY—a result which can already be obtained by
a sensitivity analysis on the first level.
The design is optimized in a wide design range for all optimiza-
tion variables, such as the inlet temperatures and exchanges areas.
Meaningful bounds for the inner and outer tube diameter (here:
0:5 cmrDi r3 cm, 4 cmrDo r10 cm) yield the bounds for the
specific exchange areas for heat and mass transfer. In case of a
fixed bed reactor with a conventional catalyst packing, an average
void fraction dependent on the inner and outer tube diameter and
on the catalyst particle diameter (here: 2 mmrDp r10 mm) must
be specified. If additional catalyst supports and reactor concepts,
for example coated wall catalyst and micro-reactors, should be
investigated, the void fraction correlation and the bounds for the
geometric design variables must be adjusted. However, including
different catalyst supports would only make sense in case higher
exchange areas to catalyst volume ratios are required.
In order to describe the heat transport, a suited heat transport
coefficient model for the one-dimensional heat transfer is used
(Li and Finlayson, 1977). Ethylene is provided via a porous stainless
steel membrane with a wide range of the permeance (here:
0 mol=ðPa s m2
ÞrPerr10À2
mol=ðPa s m2
ÞÞ. Such a membrane
was successfully used for EO reactors in a previous work reported
in the literature (Al-Juaied et al., 2001). It is assumed that the
ethylene flow through the porous membrane is purely convective
and hence driven by the pressure difference of both sides.
Criteria for external heat and mass transfer (Mears,
1971a, 1971b) as well as for internal heat transfer (Anderson,
1963) and mass transfer (expressed by the Thiele modulus
(Thiele, 1939) or the Damk¨ohler number (e.g. Emig and Klemm,
2005)) are considered. These criteria must be applied in order to
ensure a reasonable reactor design and to use the proposed
pseudo-homogeneous model with a high catalyst efficiency. In
addition, a maximal temperature difference between the cooling
media and the fluid and a maximum pressure difference between
the ethylene dosing side and the fluid are enforced. These criteria
limit the mechanical and thermal stress on the construction
material and hence are important for safety reasons.
The model equations are presented in Section 2.3.1 and the
arising optimization problem is stated in Section 2.3.2.
2.3.1. Model equations of level 2
In addition to the equations of level 1, the energy balance, the
momentum balance, the transport equations, and the equations
describing the specific exchange areas are required on this level.
The model is kept in the substantial formulation, however, the
local form of the balance equations can also be used if desired.
For the energy balance the same assumptions are taken as on
level 1, but heat exchange with the cooling side is considered.
Since it is not obvious that the desired heat flux is attainable, the
transport kinetics for the heat fluxes are included and the heat
exchange with the cooling side is controlled via the cooling
temperature profile.
n Á cp
Vgas
Á
dT
dt
¼ À Ao Á qo þrp Á
1Àe
e
Á
XNCOM
i ¼ 1
hi
XNR
j ¼ 1
ni,j Á rj
0
@
1
A
0
@
1
A ð2:13Þ
The specific exchange area based on the gas volume for the
exchange with the cooling channel and the exchange with the
ethylene supply channel are referred to as Ao and Ai, respectively.
Ao ¼
4
e
Á
Do
D2
oÀD2
i
ð2:14Þ
Ai ¼
4
e
Á
Di
D2
oÀD2
i
ð2:15Þ
The heat capacity of the mixture is modeled assuming a linear
mixing rule and individual heat capacities depending linearly
on the temperature according to Eq. (2.16). The coefficients Ai and
Bi are fitted to the higher polynomial functions (Shomate equa-
tion) from NIST chemical web book (Linstrom and Mallard, 2010),
since a linear function is by far easier to handle for the optimiza-
tion solver than higher order polynomials. In addition, the
component heat capacities depend almost perfectly in a linear
manner on the temperature in the investigated temperature
range.
cp,i ¼ Acp,i þBcp,i Á Tn
ð2:16Þ
The individual component enthalpies are calculated using
Eq. (2.17), where Fi is also fitted to the values given in the literature
(Linstrom and Mallard, 2010) ðTn
¼ T=1000, Tref ¼ 298:15 KÞ.
hiðTÞ ¼ Acp,i Á Tn
þ
Bcp,i
2
Á ðTn
Þ2
þFi ð2:17Þ
In case of a randomly packed fixed bed reactor with spherical
particles of uniform size, the Ergun equation with the parameters
stated in Eq. (2.19) approximates the pressure drop (Ergun, 1952).
Under steady state conditions, the operator equation (2.18) can be
used to convert the Ergun equation into the substantial formula-
tion as shown in Eq. (2.19).
dp
dt
¼ vi
@p
@z
ð2:18Þ
dp
dt
¼ À 150
mð1ÀeÞ2
D2
pe3
þ1:75
vsrð1ÀeÞ
Dpe3
!
v2
s
e
ð2:19Þ
vs ¼ vs,0
A0 Á p0 Á T Á n
A Á p Á T0 Á n0
ð2:20Þ
Ro
Ri
Ethylene supply
Coolant
Membrane
Reaction
Channel
z
Fig. 6. Reactor design for case 3 (q, jE) (longitudinal section view).
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696460

9. The inlet velocity vs,0 is chosen as degree of freedom. The dynamic
viscosity is assumed to be constant ðm ¼ 2:52 Â 10À5
kg=ðm sÞÞ.
This value was calculated for a typical inlet mixture at 500 K and
20 bar using Aspen Plus.
The heat transport between the reaction channel and the
cooling channel is described by an one-dimensional model
according to Eqs. (2.21) and (2.22) (Li and Finlayson, 1977). It is
assumed that the heat transfer resistance is completely on the
reaction channel side.
qo ¼ ao Á ðTÀTcÞ ð2:21Þ
ao ¼ 2:03 Á Re0:8
Dp
Á
l
Do
Á exp À
6 Á Dp
Do
with
20rReDp
¼
r Á vs Á Dp
m
r7600 ð2:22Þ
The heat conductivity of the mixture is assumed to be constant
for all calculations ðl ¼ 3:91 Â 10À2
W=ðm KÞÞ and was obtained
using Aspen Plus for a typical inlet composition at 500 K and
20 bar.
The mass transport through the membrane is assumed to be
purely convective and is described by Eq. (2.23), based on the
work of Lafarga and Varma (2000) and Al-Juaied et al. (2001), who
investigated the transport kinetics for the flux across the mem-
brane using the dusty gas model. Their results fit qualitatively to
the experimental observations, but a simpler model for the cross-
membrane transport yields similar results. This observation
justified the simple transport kinetics assumed in this work.
jE ¼ Ai Á Per Á ðpMÀpÞ ð2:23Þ
In case of an annular tube with randomly packed spherical
catalyst particles of uniform size, the average void fraction
depends on the outer tube diameter, on the inner tube diameter,
and on the particle diameter according to Eq. (2.24). Here, the
radius dependent void fraction eðrÞ is defined in Eq. (B.33).
e ¼
2
RRo
Ri
reðrÞdr
ðR2
oÀR2
i
Þ
ð2:24Þ
In order to ensure a reasonable reactor design and to be able to
use the proposed pseudo-homogeneous model with a high
catalyst efficiency, reactor design criteria on the external and
internal heat and mass transfer have to considered in the design
stage. All criteria are tracked over the whole reaction coordinate
and must be fulfilled at all times.
The criteria defined by Mears for external mass (Mears, 1971b)
and heat (Mears, 1971a) transfer are taken into account according
to the following equations, respectively:
0:3 Á bi Á ci,s
PNR
j ¼ 1ðÀni,j Á ni,j Á rjÞ Á rp Á Dp
Z1, i ¼ E,O2 ð2:25Þ
0:3 Á R Á T2
Á a
EA Á ð
PNR
j ¼ 1ðÀDHr,j Á rjÞ Á rp Á Dp
Z1 ð2:26Þ
The criteria for internal and external mass transfer must be
checked for both reactants, i.e. ethylene and oxygen. In Eq. (2.25),
ci,s is the surface concentration of the component calculated from
the ideal gas law and ni,j is the reaction order. To calculate the
transport coefficient bi the correlation from Thoenes and Kramers
(1958) for a packed bed of spherical particles is used according to:
bi ¼
1:9 Á Dm,i
Dp
vs Á Dp Á r
m
0:5
m
r Á Dm,i
0:33
ð2:27Þ
It is assumed that the molecular diffusion coefficients Dm,i can be
described by the method of Wilke and Lee (1955) assuming binary
diffusion in nitrogen. Here, mean values in the temperature range of
500–600 K at 20 bar are used ðDm,E ¼ 2:49 Â 10À6
m2
=s, Dm,O2
¼
3:15 Â 10À6
m2
=sÞ. The heat transfer coefficient a is related to the
mass transfer coefficient b by Eq. (2.28) (De Acetis and Thodos, 1960)
assuming a Lewis number close to unity. Here, the lower b value is
chosen in order to ensure a conservative design.
b
a
¼
0:7 Á M
r Á cp
Z1 m3
K=J with b ¼ min
i ¼ E,O2
bi ð2:28Þ
The Anderson criterion (Anderson, 1963) is considered to
ensure isothermal catalyst pellets (Eq. (2.29)). For the heat
transfer criteria the arithmetic mean value of both activation
energies EA is used.
3 Á R Á T2
Á lp
EA Á ð
PNR
j ¼ 1ðÀDHr,j Á rjÞ Á rp Á D2
p
Z1 ð2:29Þ
In order to ensure a catalyst efficiency above 95% the Damk¨oh-
ler criterion (e.g. Emig and Klemm, 2005) is applied according to
the following equation:
ffiffiffiffiffiffiffiffiffiffiffi
DaII,i
q
¼
PNR
j ¼ 1ðÀni,j Á rjÞ Á rp Á D2
p
4 Á Deff,i Á ci,s
!0:5
r1, i ¼ E,O2 ð2:30Þ
Here, the effective diffusion coefficient inside the pellet Deff,i is
calculated assuming no Knudsen diffusion influence, an inner
void fraction of 0.44, and a tortuosity of 3.
To limit the mechanical and thermal stress, criteria for a
maximum temperature difference ðDTmax ¼ 20 KÞ (Eq. (2.31))
between the cooling and the fluid side and for a maximum
pressure difference ðDpmax ¼ 1 barÞ between both sides of the
membrane (Eq. (2.32)) are applied.
TðtÞÀTcðtÞrDTmax ð2:31Þ
pM,0Àp0 rDpmax ð2:32Þ
2.3.2. Optimization problem of level 2
The full optimization problem which must be solved on level 2
is stated in (OP2). For temperature, pressure, composition, and STY
the same bounds apply as on level 1. The substance properties
required for the calculation of cp,i and hi are given in Table 4.
Obj ¼ max
TcðtÞ,PerðtÞ,xi,0,p,
Do,Di,Dp,T0,vs,0
S ðOP2Þ
s:t:
Component balances: Eq: ð2:1Þ
Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ
Energy balance: Eqs: ð2:13Þ2ð2:17Þ
Momentum balance: Eqs: ð2:18Þ2ð2:20Þ
Transport kinetics: Eqs: ð2:21Þ2ð2:23Þ
Catalyst support: Eq: ð2:24Þ
Design criteria: Eqs: ð2:25Þ2ð2:32Þ
Constitutive equations: Eqs: ð2:6Þ2ð2:9Þ
Table 4
Heat capacity coefficients.
Component Acp,i ðJ=ðmol KÞÞ Bcp,i ðJ=ðmol K2
ÞÞ Fi ðJ=molÞ
E 21.07 82.76 42.26
O2 27.73 7.68 À8.70
EO 20.14 110.45 À63.98
CO2 31.01 27.27 À404.13
H2O 29.74 10.95 À251.14
N2 27.08 5.65 À8.33
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6461

10. Initial conditions: xiðt ¼ 0Þ ¼ xi,0, Tðt ¼ 0Þ ¼ T0
Residence time: t ¼ 30 s
STY: Eq: ð2:10Þ
Intrinsic bounds: TL
rT rTU
, pL
rprpU
, xL
i rxi rxU
i 8i A COM
Design bounds: PerL
rPerrPerU
, DL
k rDk rDU
k for k ¼ i,o,p
Case selection: si ¼ 0 8i A COMfEg, sE ¼ 1
2.3.3. Results
In Fig. 7 the optimal permeance and optimal cooling profile as
well as the resulting temperature, reactor design criteria, ethylene
flux and mole fraction profiles are summarized. Referring to
Fig. 7(b) the temperature increases from T¼570.4 K to T¼597.6 K
which is qualitatively the same profile as in case 3 on level 1. The
temperature difference between the fluid and the cooling is always
below the maximum allowed temperature difference of 20 K. The
optimal outer tube diameter is Do¼4.07 cm.
Referring to Fig. 7(c) the ethylene and oxygen inlet mole
fractions are at their upper bound, while the CO2 inlet mole
fraction is at its lower bound. Due to the shown permeance profile
(refer to Fig. 7(a)) the ethylene mole fraction is always kept at its
upper boundary. The optimal ethylene pressure on the membrane
side is pE,M ¼ 20:78 bar and the inner tube diameter is at its lower
bound ðDi ¼ 0:5 cmÞ.
The particle diameter is at its lower bound ðDp ¼ 2 mmÞ, which
is the optimal trade-off between pressure drop, void fraction and
catalyst efficiency. The combination of the inner and outer tube as
well as particle diameter yields a void fraction of e ¼ 0:424, which
is slightly higher than the assumed void fraction on level 1. At the
inlet the velocity is 0.085 m/s, which gives rise to a small and
almost linear pressure drop of Dp ¼ 0:06 bar.
Referring to Table 5, the optimal reactor design never reaches
the bounds for the reactor design criteria. The minimal values for
the external heat and mass transfer as well for the internal heat
transfer are always above their minimal value of 1. The maximum
value for the internal mass transfer criterion is always below the
value where pore diffusion becomes limiting. Here, oxygen is the
more limiting component since it depletes along the reaction
coordinate. The profiles of the reactor design criteria, which are
close to their bounds, are shown in Fig. 7(d). The internal heat and
the external mass transfer are far away from being limiting, and
hence the according criteria are not shown.
The maximum selectivity of such a reactor set-up is 82.33%. In
case a degree of freedom is at its bound (e.g. the particle
diameter), a sensitivity analysis can be used to investigate
whether it is worthwhile to relax this bound. The relaxation of
such a bound refers to a different reactor concept or to the shift of
the design space, for example by advanced materials or miniatur-
ization. Since the difference in the selectivity between level 2 and
level 1 is merely 0.01%, it can be concluded that the cooling
temperature and the permeance of the membrane are suitable
control variables to obtain the desired fluxes and no design
bounds need to be shifted.
In addition, the chosen catalyst packing is optimal since it
features the highest catalyst density compared with wall coated
reactors, monolithic reactors, or foam like catalyst support struc-
tures, and no limitations on the overall heat and mass transport
occur. In case severe limitations of the heat and mass transport
occur, a reactor design with higher specific exchange areas is
0 10 20 30
3.45
3.5
3.55
3.6
Permeance.107[mol/(Pam2s)]
t [s]
0 10 20 30
550
560
570
580
590
600
T[K]
t [s]
Fluid
Coolant
0 10 20 30
0
0.02
0.04
0.06
0.08
0.1
xi[−]
t [s]
1.8
1.9
2
2.1
2.2
2.3
jE/nin⋅10−3
[1/s]
0 10 20 30
0
1
2
3
4
Massandheattransportcriteria[−]
t [s]
External heat
Internal mass E
Internal mass O2 external heat
transport criterion
internal mass
transport criteria
Fig. 7. Results level 2 ((c): —, E; – –, O2; Á Á Á Á, EO; - Á -, CO2; thin lines, mole fractions; thick line, dosing profile). (a) Permeance, (b) temperature, (c) mole fractions and
ethylene dosing, and (d) design criteria.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696462

11. required. However, such a reactor design will suffer from lower
catalyst densities and hence will yield lower selectivities for
the desired STY and residence time. Therefore, for the chosen
catalyst it can be concluded that alternative catalyst supports
and/or a micro-reactor concept is not necessary for the selective
production of EO.
2.4. Level 3: best technical reactor
2.4.1. Derivation of a technical approximation
On the third level, technical approximations based on the profiles
of the best control variables are developed. In case different ways
how to approximate the control variables are derived, the different
set-ups are compared using simple models before a chosen set-up is
further investigated using more detailed models.
In the investigated case, a technical approximation of the
control variable profiles based on existing apparatuses can
directly be proposed. The cooling temperature profile can be
approximated using a co-current heat exchanger. How the per-
meance profile can be approximated depends strongly on the
manufacturing technology of the porous membrane used. How-
ever, a membrane with a constant permeance is a reasonable
approximation since the optimal permeance profile on level
2 varies by less than 3.4%.
On this level, the change in the cooling temperature is
determined in a rigorous manner by solving the energy balance
for the coolant. On the ethylene supply side, the mass, energy, and
momentum balances are solved. The heat transport between the
reaction channel and the cooling channel is determined by the
heat transfer resistance on the gas side. In case of the heat
transfer between the ethylene supply side and reaction channel,
the heat transfer resistance of both sides are considered.
Due to the additional balance equations for the coolant and for
the ethylene supply side, the degrees of freedom of the optimiza-
tion problem are reduced. On this level, the flux profiles cannot be
optimized anymore. Now, only the design variables which are
listed with their according optimal values in Table 5 are degrees
of freedom. Hence, the solution of the optimization problem of
level 3 cannot be better—with respect to local optimality—than
the solution of (OP2). On level 3, the balance equations are
written in the local formulation and are listed in Appendix A.
The according optimization problem (OP3) is stated below:
Obj ¼ max
DoF refer to Table 5
_nEO,f À _nEO,0
_nE,0À _nE,f
ðOP3Þ
s:t:
Equations of changes: ðA:1Þ2ðA:3Þ
Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ
Transport kinetics: Eqs: ð2:21Þ2ð2:23Þ, ðA:16Þ; and ðA:17Þ
Catalyst support: Eq: ð2:24Þ
Design criteria: Eqs: ð2:25Þ2ð2:32Þ
Constitutive equations: Eqs: ð2:9Þ; ðA:4Þ2ðA:6Þ
Initial conditions: xiðt ¼ 0Þ ¼ xi,0, Tðt ¼ 0Þ ¼ T0, Tcðt ¼ 0Þ ¼ Tc,0,
TMðt ¼ 0Þ ¼ TM,0, _nMðt ¼ 0Þ ¼ _nM,0
Residence time: e
Z L
z ¼ 0
1
vs
dz ¼ 30 s
STY:
_nEO,f À _nEO,0
pðR2
oÀR2
i ÞL
¼ 0:27 mol=ðm3
sÞ
Intrinsic bounds: TL
rT rTU
, pL
rprpU
, xL
i rxi rxU
i 8iACOM
Design bounds: PerL
rPerrPerU
, DL
k rDk rDU
k for k ¼ i,o,p
Case selection: si ¼ 08iACOMfEg, sE ¼ 1
Coolant side: Eqs: ðA:7Þ and ðA:8Þ
Membrane side: Eqs: ðA:9Þ2ðA:15Þ
For temperature, pressure, composition, STY, and permeance
the same bounds apply as on the previous levels. The bounds for
the coolant constant and all additional required substance prop-
erties are stated in the model description. The heat capacity and
enthalpy of ethylene on the membrane side are calculated with
the same models and parameters as before.
The selectivity is 82.32% and it is still much higher than the
selectivity of the reference case. The difference in the objective
between level 2 and level 3 is caused by the non-ideal control
variable profiles. Since the difference between level 2 and level
3 is only 0.1%, the technical approximation is reasonable. In case
the losses due to the non-ideal control profiles are severe, a
different technical approximation should be developed. The
results of the derived technical approximation are shown in
Fig. 8(a)–(c).
In order to validate the results obtained by the one-dimen-
sional model on level 3, a more detailed reactor model accounting
Table 5
Results comparison between level 2 and level 3.
Decision variables Level 2 Level 3 (1D) Level 3 (2D)
Do (cm) 4.07 4.37 5.21
Di (cm) 0.5 0.84 2.94
Dp (mm) 2 2 2
L (m) – 5.88 4.58
T0 (K) 570.4 570.9 573.9
Tc,0 (K) 551.1 550.9 558.3
Kc ðK=WÞ – 4.0 Â 10À2
4.7 Â 10À2
TM,0 (K) – 550.9 550.0
_n0 ðmol=sÞ – 4.9 Â 10À2
3.9 Â 10À2
Per ðmol=ðPa m2
sÞÞ – 8.95 Â 10À7
5.42 Â 10À8
_nM,0 ðmol=sÞ – 0.1 0.1
Inlet composition (xE, xO2
, xCO2
, xN2
) 0.1, 0.08, 0.05, 0.77 0.1, 0.08, 0.05, 0.77 0.0973, 0.08, 0.05, 0.7727
Design criteria
External heat (min) 3.16 3.03 2.56
External mass (min) 132.8 140.5 41.5
Internal heat (min) 658.9 650.6 613.7
Internal mass (max) 0.83 0.81 0.57
Selectivity (%) 82.33 82.32 82.22
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6463

12. for radial temperature, concentration and porosity profiles is
derived and used for optimization in the next step.
2.4.2. Detailed investigation of the proposed reactor design
For the detailed investigation of the chosen reactor set-up a
two-dimensional, pseudo-homogeneous model is used (refer to
Appendix B). The model accounts for a radial temperature,
concentration, and porosity profile. In order to investigate if a
non-ideal temperature profile has a severe effect on the selectiv-
ity, the radial temperature distribution needs to be considered.
The radial porosity profile results in a radial distribution of the
axial velocity, which allows to investigate the effect of by-passing
of the dosed ethylene. Since the dosing of ethylene and the
radial distribution of the axial velocity gives rise to a non-plug
flow like profile of ethylene, also radial mole fraction profiles
need to be considered to ensure a non-explosive composi-
tion everywhere in the reactor. The optimization model has
1 2 3 4 5 6
550
560
570
580
590
600
T[K]
z [m]
Fluid
Coolant
1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
xi[−]
z [m]
1.6
1.7
1.8
1.9
2
2.1
jE⋅102[mol/(m2s)]
1 2 3 4 5 6
0
1
2
3
4
Massandheattransport
criteria[−]
z [m]
External heat
Internal mass E
Internal mass O2
external heat
transport criterion
internal mass
transport criteria
0 1 2 3 4 5
550
560
570
580
590
600
T[K]
z [m]
Coolant
Fluid T((Ro+Ri)/2)
Membrane
0 1 2 3 4 5
0
0.02
0.04
0.06
0.08
0.1
xi[−]
z [m]
1.52
1.53
1.54
1.55
1.56
1.57
jE⋅104
[mol/(m2
s)]
0
2
4
1.5
2
2.5
0.096
0.098
0.1
z [m]r [cm]
xE[−]
1.5 2 2.5
0
0.05
0.1
0.15
0.2
vs[m/s]
r [cm]
0 1 2 3 4 5
0
1
2
3
4
Massandheattransport
criteria[−]
z [m]
External heat
Internal mass E
Internal mass O2
external heat
transport criterion
internal mass
transport criteria
Fig. 8. Results level 3 ((b), (e): —, E; – –, O2; Á Á Á Á, EO; - Á -, CO2; thin lines, mole fractions; thick line, dosing profile). (a) 1D: temperature, (b) 1D: mole fractions and ethylene
dosing, (c) 1D: design criteria, (d) 2D: temperature, (e) 2D: mole fractions and ethylene dosing, (f) 2D: ethylene mole fraction, (g) 2D: inlet velocity profile, and
(h) 2D: design criteria.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696464

13. the same degrees of freedom as the 1D reactor model and is
stated in (OP4).
The equations describing the ethylene supply side and the
coolant side are kept one-dimensional since only the radial con-
centration and temperature gradients as well as the radial velocity
distribution in the reaction channel are important to validate the
results obtained with the one-dimensional model. All balance
equations of the reaction channel are written in cylindrical
coordinates:
Obj ¼ max
DoF refer to Table 5
S2D ðOP4Þ
s:t:
Component balances: Eqs: ðB:1Þ2ðB:10Þ
Reaction kinetics: Eqs: ð2:2Þ2ð2:4Þ
Energy balance: Eqs: ðB:11Þ2ðB:26Þ
Momentum balance: Eqs: ðB:27Þ2ðB:32Þ
Catalyst support: Eq: ðB:33Þ
Design criteria: Eqs: ð2:25Þ2ð2:32Þ
Constitutive equations: Eq: ð2:9Þ
Initial conditions: xiðt ¼ 0Þ ¼ xi,0, Tðt ¼ 0Þ ¼ T0, Tcðt ¼ 0Þ ¼ Tc,0,
TMðt ¼ 0Þ ¼ TM,0, _nMðt ¼ 0Þ ¼ _nM,0
Residence time: Eq: ðB:35Þ
STY: Eq: ðB:36Þ
Intrinsic bounds: TL
rT rTU
, pL
rprpU
, xL
i rxi rxU
i 8iACOM
Design bounds: PerL
rPerrPerU
, DL
k rDk rDU
k for k ¼ i,o,p
Coolant side: Eqs: ðA:7Þ and ðA:8Þ
Membrane side: Eqs: ðA:9Þ2ðA:15Þ
The equations for the case selections are directly implemented
in the mass balance of the reaction channel (refer to Eq. (B.3)),
and hence do not appear separately in (OP4). The equations of the
heat and mass transport kinetics are given together with the
component balances and energy balance, respectively.
Referring to the results presented in Table 5, it can be observed
that the inlet temperatures of the reaction, cooling, and mem-
brane channel are very close to the temperatures predicted by the
1D model. Also, the pellet diameter, the coolant constant, and the
ethylene flux of the membrane side are in accordance with the 1D
model. However, the diameters of the inner and outer tubes vary
significantly from the results obtained with the simpler model.
The different channel design results in different values for the
reactor length, the inlet mole flow and the permeance.
In Fig. 8(d) the profiles for the fluid temperature in the middle of
the reaction channel, the cooling temperature, and the membrane
temperature are shown. The profiles are similar to the 1D case and
the maximum radial hot spot in the fluid is DTradial,max ¼ 8:2 K.
Fig. 8(e) shows the average mole fraction profiles of all
components. These profiles are very similar to the results
obtained by the 1D model, however, the inlet mole fraction of
ethylene is only 9.73%. Due to the dosing of ethylene, the ethylene
mole fraction shows a strong radial distribution with maxima of
up to 10% at the inner tube wall (refer to Fig. 8(f)). This underlines
the necessity of more detailed models especially when safety
constraints have to be met.
The by-pass flow is significant as can be observed from the radial
velocity profile shown in Fig. 8(g). Similar as in the 1D model the
reactor design criteria are not critical and never exceeded along the
reaction channel (refer to Fig. 8(h) calculated with radially averaged
values and to Table 5 where the extreme values of the design criteria
are given considering the radial and axial profiles).
From a reaction engineering point of view, the difference in
selectivity between the one-dimensional and the two-dimensional
model is due to the non-ideal temperature and concentration
distribution as well as the non-ideal flow field. However, since both
models are different, a monotonic decrease in the objective function
cannot necessarily be expected from a mathematical point of view.
In case the non-ideal radial profiles cause a large decrease in
the objective function, a different catalyst packing which yields a
better radial heat and mass transport as well as a lower amount of
by-passing might improve the reactor performance. Such effects
can be realized for example by foam like catalyst packings.
Beside the radial hot spots and the high amount of by-passing,
the selectivity calculated with the optimized 2D model is still
82.25%, and hence an industrial realization of the proposed
reactor design seems to be worthwhile.
3. Numerical solution approach
For all calculations, the dynamic optimization problems are
transferred to large NLP problems using the simultaneous
approach. The problems were implemented in AMPL and solved
using CONOPT 3.14 G on a PC with an Intel(R)Core(TM)2 Duo CPU
E6850 with 3.00 GHz (calculation on a single CPU), a cache size of
4096 KB, a memory of 2 GB, and Ubuntu 10.04 as operating
system. For the one-dimensional problems orthogonal collocation
on finite elements (Logsdon and Biegler, 1989) is used as
discretization method. For the two-dimensional reactor model
on level 3, the axial coordinate is discretized using orthogonal
collocation on finite elements, while the radial coordinate is
discretized using the finite volume method with constant profiles
on each finite volume. This approach proved to be more robust in
the optimization especially with respect to mass balance errors.
In general, the computing times depend on the number of
equations, the degrees of freedom, the nonlinearity of the model
equations, the starting values, and the model formulation. Due to
these many influencing factors, the given numbers for the
computing times should only be considered for orientation
purposes. On the first level, the computing times are short since
the model formulation is efficient. However, the computing time
increases drastically for increasing degrees of freedom, for exam-
ple the CONOPT time for case 3 is 3 s, while the CONOPT time for
case 7 is 2824 s on the first level. The computing time for the 1D
model on level 3 is in the range of 92 s. In this case, the number of
equations is high due to the additional balance equations,
but only few degrees of freedom exist. In order to find feasible
starting points for the 2D reactor model, an advanced strategy
is necessary where single blocks of equality constraints defined
by the balance and flux equations are added to the optimiza-
tion problem step-by-step. In case of the two-dimensional
reactor model, the calculation times are in the range of 80 h,
where most of the time is required for the pre-processing to
obtain a feasible point. Here, a sensitivity analysis can also be
used in order to reduce the degree of freedom and help the
optimizer to converge.
This solution approach can only yield locally optimal solutions
as long as no global optimization solver such as BARON is used.
Hence, even better solutions might exist. In order to examine if
different local solutions can occur, each problem was solved
several times using different starting values. For all optimization
problems (OP1), (OP2), (OP3), and (OP4) no other local optima
apart from the stated solutions are obtained.
As can be seen from the computing times, the optimization
based on detailed reactor models takes much longer compared to
the screening stage. Hence, the computational effort is signifi-
cantly reduced by the proposed methodology compared to a brute
force approach where the results of detailed reactor models of
every reactor set-up are compared.
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6465

14. 4. Summary and conclusion
As a result of the current work, an improved reactor for the
ethylene oxide production, which can also be referred to as
optimal technical reactor, was designed. The proposed reactor
applies a co-current cooling strategy, which approximates the
optimal temperature profile quite well, and distributed ethylene
dosing via a membrane. The maximum selectivity of such a
reaction concept is calculated to be 82.35% (level 1). In addition,
it was shown by applying a two-dimensional reactor model that
the maximum achievable selectivity in a technical reactor is
82.23%. Hence, the losses due to the non-ideal temperature and
concentration distribution as well as the non-ideal flow field are
not severe and the proposed reactor design still has a large
selectivity optimization potential. The applied 1D model gave
good results with respect to the overall reactor performance
and the optimal control profiles. However, a more detailed
model—such as the used 2D reactor model—was required to
obtain validated results for the optimal design variables.
A comparison to the industrial used reactor set-up using the
same catalyst is done by the optimized reference case and it is
shown that a selectivity increase of 3.3% is possible. The selectiv-
ity values reported in the literature for the industrial air-based EO
process are approximately 80% (Rebsdat and Mayer, 2007). This
indicates that the used catalyst is comparable to the industrial
used catalysts and the increase in selectivity is realistic.
In comparison to other reactor configurations published in the
literature, the obtained selectivity with the derived reactor con-
cept is much higher. Lafarga and Varma (2000) investigated
different fixed bed membrane reactors. In agreement with the
current work, they found that dosing ethylene yields the highest
selectivity increase, however, the obtained selectivity is only 67%.
Kestenbaum et al. (2002) achieved a selectivity of up to 69% using
a micro-reactor set-up. The large discrepancy in the selectivities
between the work of Lafarga et al. (2000a) and Kestenbaum et al.
(2002) on the one hand and our work on the other hand is partly
due to the fact that the former works were performed with highly
active, but less selective catalysts. Zhou and Yuan (2005) opti-
mized a conventional fixed bed reactor with a highly selective
catalyst and obtained a selectivity up to 80.5%.
Summing up, the used reactor design methodology (Peschel
et al., 2010) is well suited to reliably predict the potential of
various integration and enhancement concepts as well as to
design advanced technical reactors.
The method proved to be able to design optimal reactors for
complex reaction systems. The approach is very useful to inves-
tigate the potential of dosing and removal of certain components
and yields the best suited integration and enhancement concept
for the investigated reaction system.
The methodology provides the optimal flux profiles and the
required relative mass and energy exchange areas. From the
exchange areas it can be concluded that a micro-reactor is not
required for the ethylene oxide production using the chosen
catalyst. In addition, it gives a strong catalog of requirements
which type of catalyst support is best suited for the reaction
system. Here, reactor design criteria ensure the validity of the
model and a robust reactor design.
Nomenclature
Latin symbols
A specific exchange area for flux, m2
=m3
gas
c concentration, mol/m3
cp heat capacity, J/(mol K)
D diameter, m
Dm molecular diffusion coefficient, m2
/s
Dr radial dispersion coefficient, m2
/s
EA activation energy, J/(mol K)
h specific enthalpy, J/mol
j component flux, mol/(m2
s), or molar flux into/out of
fluid element, mol/s
k heat transfer coefficient (2D model), W/(m2
K)
Kc coolant constant, K/W
L reactor length, m
_m mass flow, kg/s
M molecular weight, kg/mol
n molar amount in fluid element, mol
_n mole flow, mol/s
p pressure, Pa
Per permeance of porous membrane, mol/(m2
Pa s)
q heat flux, W/m2
rj reaction rate, mol/(kgp s)
R gas constant, J=ðmol KÞ, or radius, m
r radial coordinate, m
S selectivity, –
si selection variable indicating whether the component
flux is considered or not, –
STY space time yield, mol/(m3
s)
T temperature, K
t residence time, s
vi interstitial velocity, m/s
vs superficial velocity, m/s
vs average superficial velocity (2D model), m/s
V volume, m3
x mole fraction, –
z axial coordinate, m
Abbreviations, sets, and dimensionless numbers
COM set of components COM ¼ fE,O2,EO,CO2,H2O,N2g
Da Damk¨ohler number
DoF degree of freedom
E ethylene
EO ethylene oxide
NCOM number of components
NR number of reactions
Pe Peclet number
Pr Prandtl number
Re Reynolds number
Greek symbols
a heat transfer coefficient (1D model), W/(m2
K)
b mass transfer coefficient, m/s
e void fraction, –
e average void fraction (2D model), –
l heat conductivity, W/(m K)
L effective radial heat conductivity (2D model), W/(m K)
m dynamic viscosity, kg/(m s)
meff effective viscosity, kg/(m s)
ni,j stoichiometric coefficient (component i, reaction j), –
r density, kg/m3
t residence time, s
Suffices
c coolant side
gas gas phase (without catalyst fraction)
f final/outlet value
i reaction channel at r ¼ Ri, or component index
j reaction index
L lower bound
M membrane side
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696466

15. o reaction channel at r ¼ Ro
p catalyst particle
s surface conditions
U upper bound
0 inlet value
Acknowledgments
A.P. and F.K. thank the International Max Planck Research
School Magdeburg for financial support. The authors gratefully
acknowledge the discussion with Dr. R. Benfer, Dr. C. Großmann,
and Dr. G. Theis (all BASF SE, Ludwigshafen, Germany) as well as
the financial support of the BASF SE.
Appendix A. Model equations of level 3: one-dimensional case
Component balances:
@ _ni
@z
¼ 2pRisiji þpðR2
oÀR2
i Þ Á ð1ÀeÞ Á rp Á
XNR
j ¼ 1
ni,j Á rj ðA:1Þ
Energy balance:
XNCOM
i ¼ 1
ð _ni Á cp,iÞ
@T
@z
¼ À 2pðRoqo þRiqiÞþpðR2
oÀR2
i Þð1ÀeÞrp
XNCOM
i ¼ 1
hi
XNR
j ¼ 1
ni,j Á rj
0
@
1
A
0
@
1
A
ðA:2Þ
Momentum balance:
@p
@z
¼ À 150
mð1ÀeÞ2
D2
pe3
þ1:75
vsrð1ÀeÞ
Dpe3
!
vs ðA:3Þ
Constitutive equations:
vs ¼
_n Á R Á T
p Á pðR2
oÀR2
i
Þ
ðA:4Þ
_n ¼
XNCOM
i ¼ 1
_ni ðA:5Þ
xi ¼
_ni
_n
ðA:6Þ
Energy balance coolant side: The energy balance on the coolant
side (Index c) is written in temperature form and simplified so
that the change in the coolant temperature is proportional to the
heat flux. The introduced coolant constant is chosen within
meaningful bounds ðÀ1K=W rKc r1K=WÞ. A value of Kc smaller
than zero accounts for the possibility of counter-current cooling, a
value larger than zero for co-current cooling, and Kc¼0 accounts
for isothermal cooling, for example by an evaporating fluid.
dTc
dz
¼ Kc Á p Á Do Á qo ðA:7Þ
KL
c rKc rKU
c ðA:8Þ
Mass balance membrane side: The mass balance on the mem-
brane side (Index M) only has to account for an ethylene flow
through the membrane and is written in terms of the molar
ethylene flux. The membrane side consists of pure ethylene and
the inlet mole flow through the membrane channel is bounded
according to Eq. (A.10) with _nL
M,0 ¼ 0:01 mol=s and _nU
M,0 ¼ 1 mol=s.
d _nM
dz
¼ Àp Á Di Á Per Á ðpMÀpÞ ðA:9Þ
_nL
M,0 r _nM,0 r _nU
M,0 ðA:10Þ
Energy balance membrane side: The energy balance on the
membrane side is simplified so that it only accounts for the heat
exchange between ethylene supply channel and reaction channel.
Bounds on the temperature of the membrane side are given by
Eq. (A.12) with TL
M ¼ 490 K and TU
M ¼ 600 K.
_nMcp,M
dTM
dz
¼ p Á Di Á qi ðA:11Þ
TL
M rTM rTU
M ðA:12Þ
Momentum balance membrane side: For the momentum
balance on the ethylene supply side, the pressure drop correlation
for the flow through an empty pipe is used (Karst, 2006).
dpM
dz
¼ À
cw Á rM Á v2
M
2 Á Di
ðA:13Þ
cw ¼
0:3164
Re0:25
M
for 3000rReM ¼
rM Á vM Á Di
mM
r105
ðA:14Þ
Constitutive equations membrane side: The velocity on the
membrane side vM is calculated from the volume flow using the
ideal gas law and the cross sectional area of the ethylene supply
channel according to the following equation:
vM ¼
4 Á _nM Á R Á TM
pM Á p Á D2
i
ðA:15Þ
Heat transfer membrane side: The heat transfer between ethy-
lene supply channel and reaction channel depends on the heat
transfer resistance on both sides and is given by the following
equation:
qi ¼
ai Á aM
ai þaM
Á ðTÀTMÞ ðA:16Þ
On the ethylene side, the heat transfer coefficient is deter-
mined according to the turbulent flow through an empty pipe by
the following equation (Gnielinski, 2006):
aM ¼ 0:0214 Á
lM
Di
Á ðRe
4=5
M À110Þ Á Pr
1=4
M ðA:17Þ
The heat conductivity and the viscosity of ethylene
are assumed to be constant (lM ¼ 5:86 Â 10À2
W=ðm KÞ,
mM ¼ 1:75 Â 10À5
kg=ðm s)) and are obtained from Aspen Plus at
T¼550 K and p¼20 bar. The Prandtl number is also assumed to be
constant ðPrM ¼ 0:724Þ.
On the reaction side the heat transfer coefficient is calculated
using Eq. (2.22) with Di instead of Do.
Appendix B. Model equations of level 3: two-dimensional case
2D component mass balance: The component mass balances are
simplified by assuming no axial dispersion, no radial convection,
and steady state conditions.
@ðrivsÞ
@z
¼ À
1
r
@
@r
ðr Á Mi Á jr,iÞþrp Á ð1ÀeÞ Á Mi Á
XNR
j ¼ 1
ni,j Á rj ðB:1Þ
Mi Á jr,i ¼ ÀDr,i
@ri
@r
ðB:2Þ
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–6469 6467

16. jr,i9r ¼ Ri
¼
Per Á ðpMÀpÞ, i ¼ E
0 else
ðB:3Þ
jr,i9r ¼ Ro
¼ 0 ðB:4Þ
xi ¼
_mi
Mi
PNCOM
i ¼ 1
_mi
Mi
ðB:5Þ
The initial mass flow distribution ðri,0vs,0Þ is calculated by
solving Eq. (B.27) for the velocity distribution at the inlet. The
component densities ri,0 and the total density r0 at the inlet are
calculated from the inlet conditions (T0, p0) and the inlet compo-
sition (xi,0). The dispersion coefficients (Eqs. (B.6)–(B.9)) are
calculated in accordance to the models published in the literature
(e.g. Tsotsas, 2006a, 2006b). In order to estimate the dispersion
coefficient, the diffusion coefficient of ethylene in nitrogen is used
for all components in Eq. (B.7). To simplify the calculations, the
Peclet number is calculated with average inlet velocity.
Dr ¼ Dbed þK1 Á Pem Á Dm Á fDðrÞ Á
vs,0
vsðrÞ
ðB:6Þ
Dbed ¼ Dm Á ð1À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ÀeðrÞ
p
Þ ðB:7Þ
Pem ¼
vs,0 Á Dp
Dm
ðB:8Þ
K1 ¼
1
8
1þ
3
Pe0:5
m
!À1
ðB:9Þ
fDðrÞ ¼
RoÀr
0:44 Á Dp
2
if RoÀro0:44 Á Dp
1 if RoÀrZ0:44 Á Dp
8
:
ðB:10Þ
2D energy balance: The energy balance is simplified using
the assumptions of the LðrÞ-model resulting in the following
equations:
r Á cp Á vs
M
@T
@z
¼ Àrp Á ð1ÀeÞ
Á
XNCOM
i ¼ 1
hi
XNR
j ¼ 1
ni,j Á rj
0
@
1
Aþ
1
r
@
@r
r Á L Á
@T
@r
ðB:11Þ
Tðz ¼ 0Þ ¼ T0 ðB:12Þ
L Á
@T
@r

20. r ¼ Ri
¼
ki Á aM
ki þaM
ðTMÀTðRiÞÞ ðB:13Þ
L Á
@T
@r

24. r ¼ Ro
¼ ko Á ðTðRoÞÀTcÞ ðB:14Þ
The heat flux is calculated using the effective radial conduc-
tivity according to Eq. (B.15) (e.g. Tsotsas, 2006b). The average
velocity ðvs,0Þ and all properties depending on the fluid state are
calculated at the inlet conditions for the LðrÞ-model. The heat
conductivity of the particles ðlp ¼ 17:75 W=ðm KÞÞ is assumed to
be constant and calculated at 550 K (Yaws, 1999).
L ¼ lbed þ
1
8
Á Pe Á l Á fLðrÞ Á
vs,0
vsðrÞ
ðB:15Þ
lbed ¼ l Á kbed ðB:16Þ
kbed ¼ 1À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ÀeðrÞ
p
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ÀeðrÞ
p
Á kc ðB:17Þ
kc ¼
2
N
B
N2
kpÀ1
kp
ln
kp
B
À
Bþ1
2
À
BÀ1
N
ðB:18Þ
N ¼ 1À
B
kp
ðB:19Þ
kp ¼
lp
l
ðB:20Þ
B ¼ 1:25
1ÀeðrÞ
eðrÞ
10=9
ðB:21Þ
Pe ¼
vs,0 Á r Á Dp Á cp
l Á M
ðB:22Þ
fLðrÞ ¼
RoÀr
K2 Á Dp
2
if RoÀroK2 Á Dp
1 if RoÀrZK2 Á Dp
8
:
ðB:23Þ
K2 ¼ 0:44þ4 Á exp À
Re
70
ðB:24Þ
Re ¼
_m0 Á Dp
pðR2
oÀR2
i
Þm
ðB:25Þ
At the boundaries (r ¼ Ri and r ¼ Ro) the fluid heat conductivity
is used, which is in agreement with the LðrÞ-model. For the heat
transport coefficients on the reaction channel side the according
correlations for a 2D model must be used according to the
following equation (Li and Finlayson, 1977):
ki ¼ ko ¼ 0:17
l
Dp
Dp Á r Á vs,0
m
0:79
for
20rReDp
¼
r Á vs,0 Á Dp
m
r7600 ðB:26Þ
2D momentum balance: The Brinkman equation (B.27) is used
to calculate the radial profile of the axial velocity for the inlet
conditions. In order to reduce the model complexity so that it can
be handled by the optimization solver, the radial velocity dis-
tribution is only calculated for the inlet since the radial concen-
tration and temperature gradients do not strongly influence the
velocity distribution along the channel. The Reynolds number is
defined in Eq. (B.25) and the void fraction profile in Eq. (B.33). To
calculate the pressure drop from the Brinkman equation, the total
mass balance according to Eq. (B.32) is required. The pressure
gradient is assumed to be constant over the reactor length, which
is in good agreement with the results of the 1D model.
@p
@z

28. z ¼ 0
¼ Àf1 Á vs,0ðrÞÀf2 Á vs,0ðrÞ2
þ
meff
r
Á
@
@r
r
@vs,0ðrÞ
@r
ðB:27Þ
f1 ¼ 150 Á
1ÀeðrÞð Þ2
Á m
eðrÞ3
Á D2
p
ðB:28Þ
f2 ¼ 1:75 Á
ð1ÀeðrÞÞ Á r0
eðrÞ3
Á Dp
ðB:29Þ
pðz ¼ 0Þ ¼ p0, vs,0ðr ¼ RiÞ ¼ 0, vs,0ðr ¼ RoÞ ¼ 0 ðB:30Þ
meff ¼ 2 Á m Á expð2 Â 10À3
Á ReÞ ðB:31Þ
_m0 ¼ 2p
Z Ro
r ¼ Ri
vs,0ðrÞ Á r0 Á r dr ðB:32Þ
Radial void fraction profile: The radial void fraction for imper-
fect spherical particles is usually described using exponential
functions. Here, a modified correlation for the void fraction
distribution of an annular tube obtained from the literature
A. Peschel et al. / Chemical Engineering Science 66 (2011) 6453–64696468

29. Tsotsas (2006a) is used according to:
eðrÞ ¼
0:4 Á 1þ1:36 Á exp
À5 Á ðrÀRiÞ
Dp
, Ri rro
Ro þRi
2
0:4 Á 1þ1:36 Á exp
À5 Á ðRoÀrÞ
Dp
,
Ro þRi
2
ZrZRo
8
:
ðB:33Þ
Selectivity, residence time, and STY for the 2D case: The selectivity,
residence time, and STY for the 2D case are given by the following
equations:
S2D ¼
RRo
r ¼ Ri
ðvs,f rEO,f Àvs,0rEO,0Þr dr
RRo
r ¼ Ri
ðvs,0rE,0Àvs,f rE,f Þr drþRiME
RL
z ¼ 0 jEðr ¼ RiÞdz
Á
ME
MEO
ðB:34Þ
t2D ¼
Z L
z ¼ 0
ðR2
oÀR2
i Þ
R Ro
r ¼ Ri
vsr dr
dz ¼ 30 s ðB:35Þ
STY2D ¼
RRo
r ¼ Ri
ðvs,f rEO,f Àvs,0rEO,0Þr dr
ðR2
oÀR2
i ÞL Á MEO
¼ 0:27 mol=m3
s ðB:36Þ
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