Received: 9 December 2019
Revised: 3 February 2020
Accepted: 6 February 2020
DOI: 10.1002/er.5286
RESEARCH ARTICLE
Optimal design and operation of ammonia decomposition
reactors
Viorel Badescu1,2
1
Candida Oancea Institute, Polytechnic
University of Bucharest, Spl.
Independentei 313, Bucharest, Romania
2
Summary
The design and steady-state operation of a packed bed reactor with tubular
Romanian Academy, Calea Victoriei 125,
Bucharest, Romania
geometry is optimized. Direct optimal control methods are used. Two objective
functions are considered: (i) minimization of the ammonia mass fraction at
Correspondence
Viorel Badescu, Candida Oancea Institute,
Polytechnic University of Bucharest, Spl.
Independentei 313, Bucharest 060042,
Romania.
Email: badescu@theta.termo.pub.ro
reactor outlet and (ii) minimization of the heat flux necessary to reach a
predefined value of the ammonia mass fraction at reactor outlet. The optimization process is performed by using different controls, that is, the space distributions of (1) tube wall temperature Tw, (2) circular tube diameter Dtube, and
(3) diameter dp of the catalyst spherical particles. Results for the first objective
function are as follows. The optimal distribution of Tw along the reactor consists of a constant temperature or a U-shaped space temperature distribution,
respectively, depending on the allowed range of variation of Tw. The optimal
space distribution of Dtube (or, in other words, the shape of the reactor tube)
depends of Tw. For smaller values of Tw the tube is narrower at inlet and larger
at outlet while the reverse situation happens for larger values of Tw. For lower
Tw values, particles with smaller diameter dp are placed at reactor inlet while
when higher values of Tw are considered, particles with larger dp are placed at
reactor inlet. When both Dtube and dp are used as controls, the optimization
results are generally different from the results obtained from one-control optimization. Results for the second objective function are as follows. The optimal
space distribution of Tw starts with high values at reactor inlet. Next, the temperature decreases abruptly towards a minimum (which is lower for longer
tubes). Finally, the temperature increases smoothly towards a maximum near
the reactor outlet. The required heat flux slightly decreases by increasing the
tube length. The optimal Dtube ranges between its maximum allowed value
(at reactor inlet) and its minimum allowed value (at reactor outlet). The best
performance is obtained for catalyst particles of the smallest allowed diameter.
KEYWORDS
ammonia decomposition reactor, minimum heat consumption, optimal design, optimal
operation
Int J Energy Res. 2020;1–25.
wileyonlinelibrary.com/journal/er
© 2020 John Wiley & Sons Ltd
1
2
1 | INTRODUCTION
Energy may be transported by using reversible chemical
reactions from a high temperature heat source, such as a
solar collector array or a nuclear reactor, to a central
power plant or heat engine.1 A working fluid receives a
heat flux during an endothermic reaction at the high temperature heat source; it is further transported through
pipes to the end point where it releases the accumulated
thermal energy during an exothermic reaction. Several gas
phase reactions have been considered for storage and
energy transport such as methane reforming reactions and
ammonia decomposition reaction2 and the systems are
usually divided into two classes: nonseparating systems
such as sulphur trioxide dissociation/synthesis and two
phases separating systems such as ammonia decomposition/synthesis.1 The main advantages of using ammoniabased systems are an existing base of industrial knowledge
of ammonia and the fact that the nitrogen and hydrogen
gas storage mediums are both abundant resources. Also,
the reaction is free from complex side reactions and takes
place at temperatures relatively easily to obtain by using
solar or nuclear sources3 or by using hybrid and energy
integrated systems.4,5 Further details about the role of
ammonia in the hydrogen economy or for the storage and
transport of energy may be found in Section S0 of the Electronic Supplementary Material (ESM).
The endothermic decomposition of ammonia in thermochemical power plants is usually assisted by catalysts
(ruthenium, indium, nickel, Fe-Cr) and takes place at
temperatures in the range of 1123 to 1273 K while the
exothermic synthesis reaction requires pressures in the
range of 13 to 25 MPa and temperatures in the range of
523 to 873 K.6 Dissociation efficiencies of more than 90%
have been practically achieved by using solar cavity type
reactors.6 The usage of high pressure ammonia decomposition places extra restrictions on component design but
shifts the equilibrium towards higher temperatures and
helps to make the system feasible. Also, when the mixture of reactants (ie, various amounts of hydrogen, nitrogen and ammonia) is cooled to close to 300 K, the
majority of the ammonia component condenses and
spontaneously separates from the mixture. Therefore, the
reactant feedstock for both endothermic and exothermic
reactors can be stored in the same vessel and the composition of the decomposition and synthesis reactions can
be chosen independently during the system design.7 The
most usual separation method is by liquefaction. It can
be used for both synthesis and decomposition of ammonia.8 Operation of liquid/gas separation systems has been
analyzed in References 1, 9.
Solar thermochemical systems based on ammonia
decomposition and synthesis have been first proposed in
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1974 at the Australian National University and studied in
early 1980s at Colorado State University.6,10 More recent
experimental work has been performed by using the prototype solar ammonia receiver/reactors Mark I and Mark
II in Reference 11. In Reference 12, the design of a reference 10 MWe solar thermal plant is considered with liquid ammonia as a working fluid for energy production in
a Rankine cycle as well as a thermal storage medium.
Experimental results have been presented for solar
decomposition of ammonia in Reference 3. The program
SunShot of the United States Department of Energy was
focused on ammonia-based solar thermochemical energy
storage.13 Good literature reviews on the solar thermochemical system and experimental results may be found
in References 3, 7. Solar decomposition of ammonia has
been reviewed in Reference 10.
The catalytic ammonia decomposition was studied by
many researchers. Alumina supported nickel catalyst at
temperatures ranging from 673 to 873 K with two catalysts, 10.0% and 15.0% nickel content, respectively, has
been studied in Reference 14. Catalytic decomposition
and synthesis of ammonia over transition metals have
been analyzed in Reference 15 where reaction kinetics
models different from the classical Temkin-Pyzhev mechanism16 has been proposed. Ruthenium catalyst
supported on carbon nanotubes and promoted by potassium hydroxide was found in Reference 17 to be the most
effective catalyst for the thermal decomposition of ammonia. Three catalysts have been analyzed in Reference 18:
nickel, ruthenium, and iridium. It was found that ruthenium is the most active catalyst.
An important component of thermochemical systems
is the ammonia decomposition reactor. Different types of
reactors have been proposed, depending on their utilization. Packed-bed reactors are used in thermochemical
storage and transport systems, where component separators are placed at decomposition reactor exit.8 Tube type
and plate type ammonia cracker systems for the production of hydrogen have been compared in Reference 19
where it was shown that the plate type cracker has a
more uniform heat distribution.
The objective of this paper is to optimize the design
and operation of a packed bed decomposition ammonia
reactor to be used in thermochemical storage and power
plants. Thermochemical systems have been modeled and
their performance has been evaluated mainly in connection with the utilization of solar energy. A pseudohomogeneous steady-state model originating from Reference 20 and modified for ammonia decomposition and
synthesis in Reference 21 has been used in Reference 9.
Optimization of ammonia-based thermochemical energy
storage systems has been treated in several papers. The
objective was to maximize the work recovery efficiency of
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the exothermic process.1 The mass flow rate and the
operation pressure were considered optimization parameters in Reference 10. The main focus was on the most
critical process, that is, ammonia synthesis. The ammonia synthesis reactor has been analyzed in Reference 9. A
solar thermochemical power plant has been analyzed in
Reference 6 where the heat recovery from the synthesis
reactor has been maximized. Optimal energy recovery
from ammonia synthesis in a solar thermal power plant
has been studied in Reference 22. The synthesis reactor
of a Kellog-type ammonia plant has been optimized by
using genetic algorithms in Reference 23. The optimal
temperature profile for the synthesis reactor has been
analyzed in Reference 6 using the principles of variational calculus and optimal control.
Here, we focus on the optimization of the other important process, that is, ammonia decomposition in catalytic
packed bed reactors. Results obtained with a high-pressure
ammonia decomposition reactor that is intended to operate as part of a closed-loop thermochemical energy storage
experiment have been reported in Reference [7]. Here, the
operation of a high-pressure ammonia decomposition
reactor is optimized. Also, the design of the reactor
(in general) and its shape (in special) are optimized for the
first time here.
The processes inside the decomposition reactor have
been treated in several papers. For instance, the relationships among the power density profile on a solar
reactor, the reaction thermodynamics and kinetics, and
the heat transfer characteristics have been considered in
determining the absorber costs.24 Also, the heat transfer
behavior of fluid flow through packed bed catalytic reactors is analyzed in Reference 25. The mixed or layered
configuration of the catalyst bed has been shown to
have significant influence on reactor performance.26
These processes are taken into account here. The optimization is performed from both design and operational
perspectives. We use powerful direct optimal control
methods. The structure of the paper is as follows. The
ammonia decomposition reactor is described in
Section 2 while the model is presented in Section 3.
Details about the optimal control procedure are given in
Section 4. Results are presented in Section 5 and
Section 6 contains the conclusions.
2 | D E S C R I P T I O N OF AM M O N I A
DECOMPOSITION REACTOR
The ammonia decomposition reaction yields a mixture of
ammonia, hydrogen, and nitrogen. A conventional fixedbed plug-flow ammonia decomposition reactor has been
investigated experimentally in Reference 27. Decomposition
3
of ammonia was carried out on individual supports, such as
silica, alumina, HY, and H-ZSM-5. Details about the usage
of the ammonia decomposition reactors in combination
with fuel cells are given in Section S1 of the ESM.
Figure 1 shows a schematic of a thermochemical
power plant.10 The decomposition and synthesis of
ammonia take place in endothermic and exothermic
reactors, respectively. A storage tank is in between the
reactors and counterflow heat exchangers are attached
on each side of the storage tank. The ammonia decomposition reactor receives a heat flux which is used to dissociate ammonia into hydrogen and nitrogen. These hot
products of reaction transfer thermal energy to liquid
ammonia in a heat exchanger and later on enter the storage tank. The tank temperature is kept above the ambient temperature saturation pressure of ammonia.
Therefore, ammonia condenses at the bottom of the tank
while the syngas consisting of hydrogen and nitrogen
stays above the liquid layer. Subsequently, the syngas
enters a heat exchanger, where its temperature increases,
and finally reacts exothermally producing ammonia and
an amount of heat which is used to increase the temperature of the working fluid (steam, for example) in a power
generation cycle.
It has been shown that for ammonia, the optimal synthesis catalyst is not necessarily the optimal decomposition catalyst. However, synthesis catalysts are often used
for the decomposition process.28 Iron is the most often
used catalyst for ammonia production since it is effective,
abundant, and inexpensive.8 Iron-cobalt catalysts have
been used in Reference 10.
Catalysts activity depends on support. The ammonia
decomposition activity per metal site was found to be
greater for a silica support compared with alumina.
Ammonia decomposition studies on supported Ni, Ir, and
Ru catalysts are reported in table 1 of Reference 27. The
ammonia conversion at various temperatures shows the
following trend: Ru > Ir > Ni for the same nominal metal
loading (table 2 of Reference 27). The activity of
supported Ni catalysts is not significantly lower than the
supported Ir catalysts, making it attractive as an economical catalyst for ammonia decomposition.27 Typical catalysts include iron oxide, molybdenum, ruthenium, and
nickel. Ru catalysts perform better than Fe catalysts, particularly at lower temperatures but they are more extensive.28 Different catalysts have been considered for the
ammonia decomposition reaction such as Ni2O3 catalyst
with an α-Al2O3 support,29 Ni-Pt,28 gamma alumina with
Ni,7 and iron-cobalt.30 Different particle shapes have
been considered: cylinders, spheres, and Raschig rings.25
Several reactor geometries have been studied in the
literature. A simple tubular reactor with one feed stream
and one drain stream has been considered in Reference
4
FIGURE 1
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Simple scheme of a thermochemical power plant
10. The inlet stream consists of ammonia while the syngas product consists of nitrogen, hydrogen, and ammonia. An ammonia decomposition catalyst is placed inside
a tubular reactor in Reference 10. Catalysts’ pellets are
placed inside the co-current catalytic membrane reactor
in Reference 26. In order for the reaction to occur during
the decomposition stage, a packed bed reactor is used
with standard commercial catalyst material in Reference
10. Twenty packed bed catalytic reactor tubes arranged in
frustum (a truncated cone) inside the solar receiver cavity
have been considered in Reference 3. A cylindrical solar
received cavity has been analyzed in Reference 10 having
on its inner surface a bundle of ammonia-dissociated
tubular reactors. The distribution of the reactors is in
such a way to allow equal distribution of the solar flux on
them. A conventional catalytic reactor provided with a
nonporous tube for hydrogen transport has been compared in Reference 29 with a multifunctional catalytic
membrane reactor, which used a permeable palladium
membrane for hydrogen transport and removal from the
product stream via trans-membrane diffusion.
The system considered here for ammonia decomposition consists of a circular tubular catalytic packed bed reactor of length Ltube (see Figure 2). The inlet fluid consists of
_ m of ammonia at known tema known mass flow rate m
perature Tin. Heat is transferred from the tube wall to the
fluid inside and decomposition of ammonia into hydrogen and nitrogen takes place along the reactor. The outlet
fluid consists of a mixture of hydrogen, nitrogen, and
ammonia. The system works at constant pressure. Therefore, a compressor is placed before the reactor to control
and regulate the pressure, as previously proposed in Reference 10. The reactor tube wall has the temperature Tw
which is dependent on the abscissa z. The temperature of
the mixture inside the reactor, Tm, depends on z.
The reactor configuration is more general than usually considered in literature. Later, this will allow finding
the optimal reactor design. The reactor is circular tubular
but its diameter Dtube is unspecified function of the space
variable z. Depending on the number of catalysts, several
catalyst bed patterns such as single-catalyst pattern, wellmixed patterns, and spatially-layered patterns are used in
practice.26 Here, we consider a single catalyst whose particles, assumed of spherical shape, have the diameter dp.
This diameter is again unspecified function of z. The catalyst used here is Ni/Al2O3 as shown in Section 3.5.
The reactor will be optimized from two points of
view: design optimization and operation optimization.
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5
F I G U R E 2 Schematic
representation of the tubular circular
reactor for ammonia decomposition
Design optimization involves optimization of the reactor
shape (ie, finding the optimal variation of the tube diameter along the reactor) and optimization of packed bed
composition (ie, finding the optimal distribution of catalyst particles diameter along the reactor). Operation optimization involves finding the optimal distribution of the
tube wall temperature along the reactor.
The optimization is applied to a single tube decomposition reactor. This is similar with most experimental works,
which are performed for single tube reactors. Using results
for single tube reactors (obtained either by theory or by
experiments) in case of multi-tubular industrial reactors
requires additional assumptions. For instance, in Reference 10 the assumption is that each tube of a multi-tubular
solar reactor receives the same solar flux and the multitube reactor performance is estimated from results
obtained by modeling a single tube. A more involved
approach is to optimize the shape of the whole bundle of
tubes but in that case the results correspond to specific
configurations and are of limited general interest.
3 | M OD E L
Comments about models of ammonia decomposition are
presented in Section S2 of the ESM. A 1D model is used in
this paper. A comparison between 2D and 1D ammonia
decomposition models have been performed in Reference 6.
The relative errors for the outlet mixture temperature have
been found in the range 0.8%-1.5% for 1D models and 0.2%0.4% for 2D models. Therefore, 1D models perform generally well and are easier to implement than 2D models.
Assumptions usually adopted in literature are used
here. The catalyst bed is treated in Reference 9 as a continuum with thermal conductivity and diffusivity averaged on the reactor radius while the axial thermal
conduction and radial mixture velocity are neglected.
The hypotheses adopted in Reference 7 are that axial flow
dominates axial diffusion and temperature and reactionextent gradients are much higher than gradients of
specific heat, effective conductivity, and effective diffusivity. Axial and radial dispersion and heterogeneous effects
due to solid-gas inter-phase gradients are neglected in
Reference 6. The mixture components in the reactor have
been modeled as ideal gases in Reference 10. The equilibrium constant for ammonia decomposition is calculated
from the Gibbs free enthalpy of reaction as a function of
the conventional enthalpies and entropies of the participating species.10
The ammonia decomposition reaction occurs only
inside the catalyst bed. Steady state operation is assumed.
Similar hypotheses have been adopted in Reference 28.
However, three zones of constant temperature have been
considered in Reference 28 while here the reactor wall
temperature and the mixture temperature are variable
along the reactor.
Another hypothesis is that the mixture in the catalyst
bed can be fully described by bulk variables (temperature, concentrations, pressure). The fluid is assumed to
move as a plug through the reactor tube and the reaction
rate depends on local species concentration and temperature. Uniform temperature and concentration at the
radial cross-section are assumed. Another assumption
very often adopted is that the feed stream ideally consists
of ammonia only, and the syngas product contains three
species: nitrogen, hydrogen, and ammonia.10 This
assumption is used here (see Figure 2). Heat loss is considered to be zero.10
Pressure effects on ammonia decompositions have
been analyzed in many papers. The dependence of hydrogen recovery on the operation pressure in the range of
1.5 to 2.5 MPa has been modeled in Reference 29 for catalytic membrane reactors. The case when no hydrogen is
being removed via a membrane has been also considered.
That case is similar with the present approach. The
impact of pressure in the range of 5 to 7 MPa on the reaction rate is found to be negligible.10
Relationships to estimate the frictional pressure drops
along packed beds may be found in Reference 24. An
irreversible ammonia decomposition reaction taking
6
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place in a packed-bed tubular reactor at temperatures
793 to 853 K by using Ni-Pt catalysts is modeled in Reference 31. Since the inlet pressure is low (0.1-0.2 MPa), the
pressure drop along the reactor is taken into account by
using an Ergun equation. Pressure drops do not have significant effects when the system pressure is large. The
pressure drops in the reactor considered in Reference 6 is
less than 2% of the system pressure and was ignored. Plug
flow with negligible pressure drop is considered in Reference 28. A high system pressure is considered here and
the pressure drop across the reactor is neglected.
Several steps must be taken to implement local equilibrium models for the ammonia decomposition reactor.10 The first step is to define the control volume. In
case of 0D models the control volume is simply the reactor volume. In 1D models such as the present one, the
control volume is a narrow layer of thickness dz from the
reactor, located at abscissa z. The outlet quantities which
are usually of interest are the mixture temperature
(which is obtained from energy balance) and the content
of nitrogen, hydrogen, and ammonia (which are obtained
from mass balance for the three components).
Ammonia decomposition is endothermic.29 The
ammonia decomposition reaction is:
1
3
NH3 $ N2 + H2 :
2
2
ð1Þ
The conversion of ammonia may be as high as 98% to
99% at temperature as low as 700 K but the reaction kinetics
is slow.32 Therefore, catalysts such as Pt, Ru, Pd, and Ir are
used in practice to speed up the rate of decomposition.32
The temperature Tm of the mixture changes along the
reaction tube. The ammonia decomposition rate is found
to be almost zero below 720 K31 and the authors concluded that the adiabatic operation is impractical and an
external source of heat is necessary. A constant heat flux
is provided through the reactor walls by electrical heating
in Reference 31. From a mathematical point of view this
means using Neumann boundary conditions. Here, we
assume the tube wall temperature is controlled or is
fixed, which means using Dirichlet boundary conditions.
Boundary conditions based on reactor tube wall temperature have been used in Reference 9.
The steady-state energy balance equation for the mixture is as follows:
_ m cp,m
m
dT m
= hw Ptube ðT w −T m Þ− RNH3 ΔH r,NH3 Atube ,
dz
ð2Þ
where cp, m is the mixture specific heat capacity, hw is the
convection heat transfer coefficient at tube wall, Ptube is
tube perimeter, Tm and Tw are mixture and tube wall
temperatures, respectively, RNH3 is ammonia decomposition rate, ΔH r,NH3 is the enthalpy change of ammonia
decomposition reaction while Ptube and Atube are the tube
perimeter and the cross-sectional area of reaction tube
given by, respectively:
Ptube = πDtube ,
ð3Þ
πD2tube
:
4
ð4Þ
Atube =
The l.h.s. member of Equation (2) represents the rate
of mixture enthalpy change per unit length while the first
and second terms in the r.h.s. of Equation (2) are the heat
flux transferred from the tube wall to the mixture per unit
tube length and the energy rate needed per unit tube
length to perform ammonia decomposition, respectively.
The energy balance Equation (2) is similar with equation
(4) of Reference 7 except the fact that we neglect the radial
heat transfer considered in that 2D heat transfer model.
1D steady state equations for mixture composition temperature and pressure are found in equations (3.6) of Reference 6. However, those authors adopted the adiabatic
assumption (ie, the heat transfer through the reactor walls
is not taken into account in that paper). The ammonia
heat of formation depends on temperature and pressure.
For the pressure of 10 MPa used in this work, it ranges
between 52.04 kJ/mol at 573 K, 54.09 kJ/mol at 773 K,
and 55.39 kJ/mol at 973 K (see table 2 of Reference 8).
Here, the mixture temperature ranges in the reactor (from
inlet to outlet) between 500 K and 800 K. The ammonia
heat of formation increases by about 4% in this range of
temperature. This is not a strong increase and the enthalpy
of the decomposition reaction is considered constant here.
Also, its dependence on composition is neglected.
The steady state mass balance equation is as follows:
_m
m
dgNH3
= RNH3 M NH3 Atube ,
dz
ð5Þ
where gNH3 and M NH3 are the ammonia mass fraction
and molar mass, respectively. The l.h.s. member of Equation (5) is the ammonia mass change per unit length
while the r.h.s. member of Equation (5) is the rate of
ammonia mass decomposition per unit length. The differential mass balance Equation (5) for NH3 is similar with
that presented in equation (24) of Reference 26, where
the volume fraction of the catalyst has been used. Similar
balance equations for the species have been used in References 9, 28.
Since the mass is conserved along the reactor, the
mixture mass flow rate equals the known inlet ammonia
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7
_ m = const: The initial conditions
mass flow rate, that is, m
associated with Equations (2) and (5) are:
T m ðz = 0Þ = T in ,
gNH3 ðz = 0Þ = 1:
ð6Þ
ð7Þ
The inlet ammonia temperature Equation (6) is similar with the initial condition used in Reference 9 and the
initial condition equation (5) of Reference 7. Inlet value
for the content of the ammonia has been used as initial
condition in Reference 9.
The reactor may be optimally designed and operated
by taking into account different objective functions, as
described in Section 4.2. Details of the model follow.
r NH3 =
gNH3 =M NH3
g
= NH3 ,
nm
2 −gNH3
ð9aÞ
3 1 −gNH3 =M NH3 3 1 −gNH3
=
,
r H2 =
nm
2
2 2 −gNH3
ð9bÞ
1 1 −gNH3 =M NH3 1 1 −gNH3
=
:
r N2 =
nm
2
2 2 −gNH3
ð9cÞ
Here, Equation (8) has been used. Summing up the
three volume fractions r NH3 , r H2 , and r N2 , given by Equations (9a), (9b), and (9c), respectively, yield unity, as
expected.
Taking into account Equation (1), the hydrogen and
nitrogen mass fractions, gH2 and gN2 , respectively, are:
3.1 | Thermophysical properties of
mixture components
M H2
3
3
,
gH2 = nNH3 ,diss M H2 = 1 −gNH3
M NH3
2
2
ð10aÞ
The mixture in the reactor consists of ammonia, hydrogen, and nitrogen. The properties of the mixture components needed here are dynamic viscosity, thermal
conductivity, and specific heat at constant pressure.
These properties depend on mixture temperature and
pressure. Details are given in Appendix A.
M N2
1
1
gN2 = nNH3 ,diss M N2 = 1 −gNH3
:
M NH3
2
2
ð10bÞ
Notice that the molar mass of ammonia, hydrogen,
and nitrogen is 17 kg/kmol, 2 kg/kmol, and 28 kg/kmol,
respectively. Therefore, summing up gNH3 , gH2 (given by
Equation (10a)), and gN2 (given by Equation (10b)), yields
unity, as expected.
3.2 | Mass and volume/molar fractions
of mixture components
The value of the ammonia mass fraction gNH3 at reactor
outlet may be used as an indicator of the reactor effectiveness. It is useful to express the volume/molar and
mass fractions of the mixture components as functions of
gNH3 .
The ammonia, hydrogen, and nitrogen volume fractions, r NH3 , r H2 , and r N2 , respectively, are obtained as follows. At reactor inlet (z=0) the ammonia mass fraction is
1. At abscissa z the ammonia mass fraction is gNH3 . In
dissociated,
between, 1 −gNH3 kg of ammonia
has been
which correspond to nNH3 ,diss = 1 −gNH3 =M NH3 moles of
ammonia dissociated. Taking into account
Equation (1),
=M
and
they generated
ð
3=2
Þ
1
−g
NH
NH3
3
ð1=2Þ 1 − gNH3 =M NH3 moles of hydrogen and nitrogen,
respectively. The total number of moles nm in the mixture at abscissa z is:
nm =
gNH3
3 1 − gNH3 1 1 −gNH3 2 −gNH3
+
+
=
:
M NH3 2 M NH3
M NH3
2 M NH3
3.3 | Thermophysical properties of
mixture
The specific heat of the mixture at constant pressure is
computed by:
cp,m =
X
gi cp,i ,
ð11Þ
i = NH3 ,N2 ,H2
where cp, i is the specific heat at constant pressure of
component i.
The mole average method can lead to significant
errors in the computation of mixture viscosity due to the
presence of hydrogen. Wilke's method has been used in
this case to compute mixture viscosity.31 Here, the
dynamic viscosity of the mixture is computed by using
the Herning-Zipperer relationship33,34:
ð8Þ
The volume fraction of the mixture components are
as follows:
P
pffiffiffiffiffiffiffiffiffiffiffiffiffi
i = NH3 ,N2 ,H2 r i μi M i T c,i
pffiffiffiffiffiffiffiffiffiffiffiffiffi ,
μm = P
i = NH3 ,N2 ,H2 r i M i T c,i
ð12Þ
8
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where μi is the dynamic viscosity of the component
i while Tc,i is the critical temperature of the component
i (T c,NH3 = 404:5K , T c,H2 = 33:2K , T c,N2 = 126:0K ). The
root mean square error of Equation (12) for 34 binary
mixture including hydrogen has been found as 6.17%.35
The Wilke's rule is used to compute the mixture thermal conductivity36:
X
km =
P
i = NH3 ,N2 ,H2
r i ki
j = NH3 ,N2 ,H2 r i Φij
,
ð13Þ
where
1+
Φij =
0:5 0:25 2
μ
μj
Mj
Mi
:
i0:5
pffiffiffih
i
8 1+ M
Mj
ð14Þ
Equations (13), (14) may have errors up to 14%.37
3.4 | Convection heat transfer coefficient
The heat transfer in packed beds has been studied in several papers. More details may be found in Section S3 of
the ESM. In Reference 29 it has been found that the effective wall heat transfer coefficient is best modeled as a
function of the Reynolds number by the Li-Finlayson correlation.38 Li and Finlayson collected experimental data
from 16 papers (see their table 5) and used four analysis
methods. Their correlation is used here. Therefore, the
convection heat transfer coefficient at tube wall hw (units:
W/(m2 K)) to be used in Equation (2) is38:
_ m dp
km m
hw = 0:17
dp Atube μm
0:79
,
ð15Þ
where km (units: W/(mK)) and μm (units: Pa s) are the
radial thermal conductivity and the dynamic viscosity of
the mixture, respectively. Equation (15) applies for spherical pickings, 0.05 ≤ dp/Dtube ≤ 0.3, 20 ≤ Rep ≤ 7600, and
constant wall temperature. The average deviation from
experimental results is 14% while the correlation coefficient is R2 = 0.98. Only data which is accepted as being
free from length effects have been used in deriving Equation (15) and those data were checked by two additional
tests described in Reference 38. The experiments have
been done with air so no dependence on Prandtl number
is needed. A correction to take into account the Prandtl
number is suggested but it has not been tested and is not
considered here.
3.5 | Kinetic parameters
Different kinetic models have been proposed for ammonia decomposition (see Section S4 in the ESM). Several
authors concluded that the Temkin-Pyzhev model is able
to evaluate very well the reaction extent at high conversion rates.28 The Temkin-Pyzhev model has been slightly
adjusted in Reference 9 by using an effectiveness factor.
Here, a 1D model is developed based on the TemkinPyzhev kinetic model.
The reverse reaction in the Temkin-Pyzhev model is
often neglected in case of membrane reactors since the
decomposition reaction is far from equilibrium, taking
into account the low hydrogen partial pressures due to
hydrogen removal.29 At temperatures and pressures
between 810 and 1366 K and 1.8 to 3.5 MPa, respectively,
the magnitude of the reverse reaction was negligible compared to the forward reaction.29 Simplified versions of the
Temkin-Pyzhev model without the reverse reaction
have been used in several studies on ammonia
decomposition.6,32,39
Here, the Temkin-Pyzhev rate equation was used to
estimate the ammonia decomposition rate. Since lower
temperatures are used, both the forward and reverse
reactions are taken into account. The equation has the
following form when written in terms of ammonia
decomposition Equation (1)40:
2
RNH3 = k 4
f 2NH3
f 3H2
!β
−f N2 K 2eq
f 3H2
f 2NH3
!1 − β 3
5
units : mol= m3 reactors :
ð16Þ
The reaction rate depends on many parameters such
as catalyst loading and dispersion of the metal catalyst on
the binder. Therefore, the constants to define the expression of Temkin-Pyzhev should be adjusted for the catalyst
adopted.28 It is expected that the activation energy and
the exponential constant depend on the main metal while
the preexponential factor depends on the catalyst loading
on the binder.28 For instance, activation energies for
nickel-based catalysts operating at pressures up to
10 MPa are listed in Reference 7. For iron-based catalysts,
β = 0.25-0.60.29 β = 0.27 was found for aluminasupported Ru catalysts.28 In other cases, β = 0.5 to 0.724.8
A value β = 0.5 has been used in Reference 7 while
β = 0.674 is used in Reference 26.
Details about the parameters to be used in Equation (16) are given in Table 1. The values k0, Keq, and β
correspond to experiments performed at 1.6 MPa and
720 to 873 K in an ammonia decomposition reactor
(Dtube = 0.07 m, Ltube = 0.055 m) on a supported
BADESCU
9
TABLE 1
Quantities to be used in Equation (16). pm and Tm
are mixture pressure and temperature, respectively, while
R = 8314 J/(kmol K) is the universal gas constant
Relationship
+ E b pm
k = k 0 exp − Ea RT
m
Units
Ea = 2.187 × 105
mol
m3 s Pa −0:674
mol
m3 s Pa −0:674
J
mol
Eb = 1.16 × 10−3
J
molPa
k0 = 1.09 × 1020
−1
2250:3
T m − 0:8534
−4
=
log10 K eq ðatmÞ
−1:5105log10 T m −2:5898 × 10
+ 1:4896 × 10 − 7 T 2m
4.1 | Optimal control methods
-
Tm
β = 0.674
-
Ni/Al2O3 catalyst (average particle diameter 0.72 mm) in
Reference 40. To take into account pressure effects, the
values Ea and Eb from equation 3.5 of Reference 41 have
been used. Rather similar values have been used in Reference 26. However, the constant k does not depend on
pressure and a quadratic term in Tm is missing in Reference 26.
The fugacities fi (i = NH3, N2, H2) entering Equation (16) are computed by using:
f i = Φfug,i r i pm ,
the starting point. Hybrid methods, increasing the chance
of finding a global optimum, and improving the convergence speed have been also proposed.
The optimal control techniques used in this paper are
faster and more accurate than the stochastic methods
and are appropriate to perform parametric studies.
The optimal control problem consists of the constrained
extremization of an objective function. The constraints are
ordinary differential equations for the state variables and
control(s). Direct or indirect methods may be used to solve
optimal control problems. Here, a method based on direct
optimal control is used. A discretization is performed in the
space of the independent variable, applied to the state variables and control(s), as well as the constraints’ differential
equations. This way the optimal control problem is transformed into a nonlinear programming problem. Details
may be found in References 42, 43. Despite being less accurate than indirect methods, direct methods are widely used
in engineering applications since they are more robust with
respect to initialization and more straightforward to apply.
ð17Þ
4.2 | Optimal control model
where the fugacity factors Φfug,i for various species may
be found in Reference 6 as functions of pressure and temperature. Here, we assume that the reaction is ideal and
therefore the fugacity factors equal unity.
4 | OPTIMAL CONTROL
Controlling the heat transfer for increasing the ammonia
decomposition is not a new idea. For instance, a three temperature zones control strategy has been used during experiments performed in Reference 28. When mathematical
optimization is considered, methods based on stochastic
methods (such as genetic algorithms, simulated annealing,
and artificial life algorithms) and methods based on calculus (such as the calculus of variations and optimal control)
are usually envisaged. Stochastic methods allow global
searching and have the advantage that the solution is not
dependent on the initial configuration but they are very
slow in going towards the global solutions. Constraints on
parameter values cannot be always introduced and these
methods are not very useful when parametric studies for
many configurations should be performed. The methods
based on calculus are faster and more accurate than the stochastic methods and in most case are effective in finding
local solutions but their convergence strongly depends on
Different ways have been imagined in practice to control the
operation of ammonia decomposition reactors. The reactor
wall temperature was kept at a constant temperature by an
electrical heater in Reference 29. A bundle of solar ammonia
decomposition reactors have been organized in a way that
allows equal distribution of the solar flux.10 A three-zone
heater has been used in Reference 28 to maintain the catalyst
bed at desired temperature. The temperature of the wall is
controlled in such a way that the overall temperature along
the reactor axis is as uniform as possible. Different decomposition rates, average tube temperature, and temperature distributions along the reactor have been obtained by using an
electrically heated solar receiver of different geometries in
their optimal flow range.3 The distribution of wall temperature along the reactor may be controlled by using bundles of
electrical wires or pipes with superheated thermal agent distributed in a differential way along the reactor, as suggested
by the approach in Reference 38. A quasi-isothermal reactor
operation may be achieved by appropriate placement of hot
air injector nozzles alongside the reactor tubes, as done in
Reference 30. This method may be used to control the wall
tube temperature in a desirable way.
The ammonia decomposition reactor is optimized in
this paper from two different points of view: design and
operation. Two different objective functions are defined,
10
BADESCU
as shown next. In addition, for each objective function,
the optimization process may be performed by using different controls.
transformed into a Mayer problem in two steps. First, a
new dependent variable f is defined through the equation:
df
= hw PðT w −T m Þ,
dz
4.2.1
|
ð19Þ
First objective function
with the initial condition:
The purpose of the ammonia decomposition reactor is to
provide hydrogen. The mass fraction of hydrogen is a
maximum when the ammonia mass fraction in the syngas is a minimum. Therefore, the first objective function
consists of the minimization of the NH3 mass fraction at
reactor outlet. The optimal control problem is defined as
follows:
• independent variable: coordinate z;
• state variables: mixture temperature Tm and NH3 mass
fraction gNH3 ;
• objective function: outlet ammonia mass fraction
gNH3 ðz = Ltube Þ, which is to be minimized.
• in case of one-control optimization, the control is one
of the following quantities: the tube wall temperature
Tw(z), the catalyst particle diameter dp(z), the inner
tube diameter Dtube(z);
• in case of two-control optimization, the controls are
dp(z) and Dtube(z);
The objective function gNH3 ðz = Ltube Þ is minimized
under the constraints of the ordinary differential Equations (2) and (5), which are solved by using the initial
conditions Equations (6) and (7).
4.2.2
|
Second objective function
The ammonia decomposition reaction is endothermic.
The ammonia mass fraction at reactor outlet depends on
the heat flux received by the reactor. Generally, smaller
ammonia mass fraction corresponds to larger heat
amount received. As a compromise between reactor effectiveness and heat costs, in practice the expected outlet
ammonia fraction is significantly higher than zero. The
second objective function is defined as follows. For
expected value of the ammonia mass fraction at reactor
target
outlet, gNH3 ðz = Ltube Þ = gNH3 , the heat flux transferred to
the mixture in the reactor,
Q
ðL
Ptube hw ðT w −T m Þdz,
ð18Þ
0
should be minimized. The objective function Equation (18)
and the constraints’ Equations (2) and (5) constitute an
optimal control problem of Bolza type which is
f ðz = 0Þ = 0:
ð20Þ
Second, the objective function of the Mayer problem
is defined:
f ðz = Ltube Þ ! min:
ð21Þ
However, the constraint
target
gNH3 ðz = Ltube Þ = gNH3 ,
ð22Þ
should also be taken into account. Therefore, a new
0
objective function, f (z = Ltube) is defined:
h
i2
target
f 0 ðz = Ltube Þ f ðz = Ltube Þ + α gNH3 ðz = Ltube Þ −gNH3 ,
ð23Þ
h
i2
target
where α gNH3 ðz = Ltube Þ −gNH3
is the penalization
function while the value of the penalization factor
is α = 1011. Minimization of f 0 (z = Ltube) ensures both
the minimization of f(z = Ltube) and the condition
target
gNH3 ðz = Ltube Þ = gNH3 .
Therefore, the constrained Mayer optimal control
problem is defined as follows:
• independent variable: coordinate z;
• state variables: mixture temperature Tm and NH3 mass
fraction gNH3 ;
• objective function: f 0 (z = Ltube), which is to be
minimized.
• in case of one-control optimization, the control is one
of the following quantities: the tube wall temperature
Tw(z), the catalyst particle diameter dp(z), the inner
tube diameter Dtube(z);
• in case of two-control optimization, the controls are
dp(z) and Dtube(z);
The objective function Equation (23) is minimized
under the constraints of the ordinary differential Equations (2), (5), and (19), which are solved by using the initial conditions Equations (6) and (7) and the final
condition Equation (22).
BADESCU
11
4.3 | Optimal control implementation
Here, we are using a direct optimal control method based
on the BOCOP programming package.44 The user
describes the optimal control problem through several C
++ functions. The optimal control problem is transformed into a nonlinear programming problem.
BOCOP has several discretization methods. Here, we
use the method Midpoint (implicit, 1- stage, order 1). The
number of discretization steps for the independent variable is 500. This corresponds to a dimensionless space
step of 0.002. The maximum allowed number of iterations is 10 000 while the tolerance is 10−14.
4.3.1
|
Assumptions and input quantities
Ammonia decomposition is endothermic with an approximate standard enthalpy of reaction of 46.4 kJ/mol.29,31
This value is used here.
Thermochemical power plants consist of both decomposition and synthesis reactors. Choosing the pressure of
the decomposition process depends on the synthesis process, which is more effective at higher pressures. Ammonia is usually stored at moderate high pressure (>1 MPa)
since in this way at normal environment temperature it
results in liquid state. Operative pressures 0.1, 0.5, and
1 MPa have been considered in Reference 28. Producing
hydrogen at a higher pressure could save energy for the
secondary gas compression and might have several benefits on the overall system efficiency28 since it increases
the efficiency of the synthesis reactor.10 A pressure of
3.64 MPa has been used in Reference 26 and other
authors used 5 to 7 MPa,10 11.4 MPa,7 15 MPa,6 and
25 MPa.30 In Reference 24, it is recommended to operate
at pressures in the range of 10 to 30 MPa and a pressure
of 30 MPa has been used in Reference 8. Here, we are
using a moderately high operation pressure of 10 MPa.
For a solar decomposition reactor, an inlet ammonia
temperature of 523 K has been used10 and the authors
noticed that an increase in gas inlet temperature has a
minor effect on the efficiency. Therefore, a constant inlet
ammonia temperature of 500 K has been adopted here.
Some authors assume a known heat flux uniformly
distributed over the reactor length.10 Here, we assume a
controlled wall temperature. The ammonia decomposition reactor is expected to operate at tube wall temperatures of typically 1023 K.24 However, it is known that
excessive temperatures have destructive effects on the
catalyst.26 Also, the high temperature environment of
1073 to 1273 K results in rapid degradation of the materials.32 Usual fluid temperatures range between 645 and
720 K,26 773 K,32 and 675 to 853 K.31 Such fluid
temperatures may be achieved in the range of wall temperatures adopted here, that is, 600 to 900 K.
Here, we assume a Ni/Al2O3 catalyst in agreement
with the kinetics Equation (16) and Table 1.
When cylinder catalyst particles have been used, their
size was, for example, 4.5 mm diameter and 4.5 mm
height7 or 5.2 mm diameter and 5 mm height.10 In case
of catalyst particles of spherical shapes the range of variation is larger. For instance, small particles of diameter
0.35 mm31 and 0.6 mm28 have been used. Also, intermediate size particles of diameter 1.5 to 2 mm,26 2.23 mm
(with accuracy ±0.02 mm),7 or 5 mm29 were considered.
Large particles of diameter 29 mm, 38 mm, and 48 mm
were used in Reference 25. Notice that reduction of particle diameter from 8 mm to 1 mm would increase the
reaction rate five times.8 Here intermediate size spherical
catalyst particles of diameter 1.2 to 6.2 mm are
considered.
Short ammonia decomposition reactors of length
0.14 m,26 0.2 m, 0.31 m,31 and 0.5 m3 have been considered. Other authors studied longer reactors, of length
1.036 m,7 1.1 m,29 1.6 m,25 3 m,10,29 and 5 m.30 The reactor lengths covered in this paper range from very short to
long, that is, 0.15 to 3.0 m.
The inner diameter of the reactor tube is small in
some cases, for instance 4 mm8 or 10 mm.28 A tube with
external diameter of 7 mm has been used in Reference 3.
Intermediate size diameter such as 27.86 mm,30 41 mm,29
50 mm,31 and 61.66 mm10,29 have been considered. Also,
some authors studied larger tube diameters such as
158 mm7 and 217 mm.25 The tube diameters considered
in this paper are small and intermediate size:
10 to 24 mm.
A literature review shows a large variation range for
the ratio Dtube/dp: 14-28; 8-16; 3.9-51; 6-24; 3-5; 5.5-6.6;
4.5-7.5 (details about the appropriate references may be
found in Reference 25). The ratio Dtube/dp ranges
between 1.66 and 12.5 in six references quoted in Reference 38. The following ratios of Dtube/dp have been considered in Reference 25: <4, 5-12, and 7-27. All
computations in this paper used couple of values for the
diameters of the tube and catalysts particles in the range
3.33 < Dtube/dp < 20. These values ensure compatibility
with Equation (15) giving the heat transfer coefficient
and covers most cases considered in literature.
The mass flow rate of the inlet ammonia depends on
the size of the decomposition reactor. Small values such as
0.05 g/s and 0.0875 g/s,3 0.3175 g/s,30 and 0.447 g/s7 have
been used. Also, intermediate mass flow rates have been
used by some authors: 2.2 g/s,24 2.7 g/s,8 and 3.5 g/s.30
Large mass flow rates ranging between 20 and 40 g/s have
been adopted by other researchers.10 Small mass flow rates
are considered in this paper: 0.05-1 g/s.
12
BADESCU
The thermochemical storage system based on ammonia has the advantage that, when an arbitrary mixture of
reactants is cooled to ambient temperature, the ammonia
component condenses and spontaneously separates from
the mixture. Therefore, the reactant feedstock for both
endothermic and exothermic reactors can be stored in
the same vessel (see Figure 1).2 Therefore, the decomposition reaction needs not proceed to completion because
the effluent of each reactor separates spontaneously into
the basic reactants which may then be stored or
recirculated to the reactors in any desired proportions.1
Reaction extents of 100% are not needed and in general
are not obtained.3 The adopted reaction extent is 0.830 or
0.85.24 The mass flow control has been regulated in Reference 30 for a minimum of 80% of the ammonia feed
being dissociated. The likely range of exit reaction extents
expected for the ammonia-based system operating at
20 MPa is 0.6 to 0.8.45 A larger range of variation for the
reaction extent is considered in this paper: 0.0668-0.5157.
Table 2 shows the range of variation for most parameters used in this paper. A reference value is also stated for
each parameter. These reference values are used during
calculations except other values as explicitly mentioned.
4.3.2
|
Constraints
A constraint for both first and second objective functions
is that the wall temperature must exceed the mixture
temperature:
T w ðzÞ −T m ðzÞ≥0:
adopted for state variables and controls. Here we have
minimization problems. In this case, increasing the variation range for the state variables and controls usually
yield lower values of the objective function. However,
using larger upper or lower bounds is not always the
best solution since the local suboptimal solutions may
be lost, due to the finite subspace of approximation.
Finding appropriate bounds is a matter of experience
and trial.
Lower and upper bounds for the controls and state
variables are shown in Table 3 and Table 4, respectively,
for both first and second objective functions.
5 | RESULTS
Section 5.1 focuses on the minimization of the first
objective function while results concerning the minimization of the second objective function are presented in
Section 5.2. Other results may be found in Section S5 of
the ESM. Table 5 contains a summary of the boundary,
initial, and final conditions used in Sections 5.1
and 5.2.
T A B L E 3 Lower and upper bounds for controls, for both first
and second objective function
Lower
Bound
Upper
Bound
Tube wall
temperature, Tw (K)
500
800
One control
Tube diameter,
Dtube, (mm)
10
25
One control
Catalyst particle
diameter, dp (mm)
1
6
Case
Control
One control
ð24Þ
The solution and the convergence of the optimal
control methods depend on the lower and upper bounds
Two controls
TABLE 2
Range of variation for several parameters and
reference values
Parameter
Range of
Variation
Reference
Value
Reaction enthalpy, ΔHr (kJ/mol)
46.4
46.4
Mixture pressure, pm (MPa)
10
10
Inlet temperature, Tin (K)
500
500
Wall temperature, Tw (K)
600-900
800
Tube length, Ltube (m)
0.15-3.0
2.0
Tube diameter, Dtube (mm)
10-24
18
Catalyst particle diameter, dp (mm) 1.2-6.2
2.2
_ m (g/s)
Mass flow rate, m
0.05-1.0
0.4
Outlet ammonia mass
fraction, gNH3 ðz = Ltube Þ
0.0668-0.5157 0.18
Tube diameter, Dtube (mm)
10
25
Catalyst particle
diameter, dp (mm)
1
6
T A B L E 4 Lower and upper bounds for state variables, for both
first and second objective function
Objective Function State Variable
Lower Upper
Bound Bound
First and second
objective function
500
Mixture
temperature, Tm (K)
800
First and second
objective function
Outlet ammonia
mass fraction, gNH3
10−10
1
Second objective
function
f(z = Ltube) (W/m2)
0
-
BADESCU
13
T A B L E 5 Summary of boundary, initial, and final conditions
used in Sections 5.1 and 5.2
Boundary conditions
Objective function
Equation
number
Boundary condition
1 and 2
2, 5, and
19
Fixed or controlled
tube wall
temperature Tw
Objective function
Equation
number
Initial condition
1
2
Tm(z = 0) = 500 K
5
gNH3 ðz = 0Þ = 1
2
Tm(z = 0) = 500 K
5
gNH3 ðz = 0Þ = 1
19
f(z = 0) = 0
Equation
number
Final condition
5
gNH3 ðz = Ltube Þ = 0:18
Initial and final
conditions
2
5.1 | First objective function
The first objective function consists of minimizing the
outlet ammonia content. Several controls have been
envisaged. One of them (the space distribution of the
tube wall temperature) is related to operation optimization (see Section 5.1.1) while two other controls (optimal
space distribution of tube diameter and catalyst particle
diameter, respectively) are related to design optimization
(see Section 5.1.2).
5.1.1
|
Optimal operation
Here, we focus on operation optimization. The main
parameters are kept constant (see the reference values in
Table 2) while the space distribution of the tube wall
temperature Tw along the reactor is controlled by some
technique as described in the beginning of Section 4.2.
The optimal distribution of the tube wall temperature
depends on the range of variation of Tw, which is denoted
ΔTw (see Figure 3). When Tw and ΔTw are rather reduced
(500-600 K and 600-700 K) the optimal distribution consists of a constant temperature value along the reactor
(Figure 3A,B). That constant temperature equals the
maximum value in ΔTw and ensures a minimum outlet
value of gNH3 , for that variation range of Tw (see Table 6).
For higher values of the maximum allowed temperature Tw (ie, 800 K, 850 K, 900 K) a more interesting
optimal space distribution of tube wall temperature is
obtained (Figure 3A-D). In all those cases, the optimal
tube wall temperature at reactor inlet is significantly
higher than the fluid inlet temperature (which is 500 K).
This allows a strong rate of the ammonia decomposition
and a significant decrease of the ammonia mass fraction
gNH3 in the inlet region (see for instance Figure 4B). All
cases considered in Figure 3 have in common the fact
that the optimal temperature Tw at reactor outlet equals
the maximum allowed value. This is needed since a lower
temperature Tw would make the thermodynamic equilibrium to be changed and the consequence would be a
higher ammonia mass fraction at reactor outlet. Between
the reactor inlet and outlet the tube wall temperature is
lower and rather constant on a segment of the reactor
length, which is longer for higher values of Tw at the
reactor outlet (compare for instance Figure 3A and
Figure 3D). This ensures a low decomposition rate associated with a smooth decrease of gNH3 in the mid reactor
region (see Figure 4B).
The range of temperatures covered by Table 6 may be
compared with those of previous works. Mixture temperatures ranging between 675 and 875 K have been
reported for a solar reactor in Reference 3. The temperatures explored in Reference 29 ranged from 623 to 923 K.
The authors concluded that elevated temperatures reduce
the required reactor length. The ideal operation temperature found in Reference 29 is 923 K. The decomposition
reactor described in Reference 8 operates at 973 K, which
is higher than the range of temperatures considered here.
The outlet mixture composition in Reference 8 consists
almost exclusively of hydrogen and nitrogen. The results
shown in Table 6 are in good agreement with the values
of the outlet ammonia mass fraction shown in figure 3 of
Reference 8 for a reactor operating at 10 MPa and temperatures between 673 and 873 K.
To save space, we do not show graphical results for
the optimal variation of the mixture temperature Tm
along the reactor. Tm increases monotonously from the
inlet temperature, in a way similar with figure 2 of Reference 24. Inlet and outlet mixture temperatures around
500 K and 875 K, respectively, are shown in figure 3 of
Reference 7 for a pressure of 11.4 MPa and a mass flow
rate of 0.507 g/s. Those results are quite similar with
results obtained here. Notice that the space variation of
the mixture temperature in figure 7 of Reference 31 is
similar with that of present in Figure 3. However, the
reactor is heated electrically in Reference 31 and the tube
wall temperature is not a boundary condition. Also, the
space distribution of the heat flux in figure 7c of Reference 30 for a catalytic bed is similar in shape with the
optimal distribution of the tube wall temperature
obtained here (see Figure 3).
14
BADESCU
F I G U R E 3 Optimal space dependence of the tube wall temperature along the reactor for different variation ranges ΔTw of the tube
wall temperature, starting from (A) 500 K, (B) 600 K, (C) 700 K, and (D) 800 K. The first objective function has been considered and the
control is the tube wall temperature Tw. Tube length Ltube = 2 m, tube diameter Dtube = 18 mm, catalyst particle diameter dp = 2.2 mm,
_ m = 0:4g=s, and mixture pressure pm = 10 MPa [Colour figure can be viewed at wileyonlinelibrary.com]
mass flow rate m
T A B L E 6 Minimum outlet values of the ammonia mass
fraction gNH3 for the variation ranges of the tube wall temperature
considered in Figure 3
Variation range of
tube wall
temperature, Tw (K)
Minimum
outlet values of
the ammonia
mass fraction, gNH3
500-600
0.8253
500-700, 600-700
0.3192
500–800, 600-800, 700-800
0.1462
800-850
0.0988
500-900, 600-900, 700-900, 800-900
0.0680
The optimal distribution of the tube wall temperature
Tw along the reactor depends on the tube length Ltube
(see Figure 4A). However, the distribution shape is
similar in all cases. The wall temperature is higher at
reactor inlet and equals the maximum value of the variation range (which is 500-800 K, see the reference value in
Table 2) at reactor outlet. In the mid region of the reactor
the optimal temperature is lower, in agreement with
Figure 3. The length of the segment with lower temperature Tw increases by increasing the reactor length. Along
that segment of lower Tw values, the decrease of the
ammonia mass fraction gNH3 is small (see Figure 4B).
Notice that a negative ammonia conversion exists in the
inlet reactor region due to the reverse reaction creating
ammonia from hydrogen.29 The explanation is that the
reaction rate is an Arrhenius type relationship that varies
exponentially with temperature. The ammonia mass fraction reaches its outlet value well before the reactor exit
(see Figure 4B). This is in agreement with results shown
in figure 4 of Reference 7.
BADESCU
15
F I G U R E 4 Optimal space
dependence of (A) tube wall
temperature and (b) ammonia mass
fraction gNH3 along the reactor for
different values of the tube length
Ltube. The first objective function has
been considered and the control is
the tube wall temperature Tw. Tube
diameter Dtube = 18 mm, catalyst
particle diameter dp = 2.2 mm, mass
_ m = 0:4g=s, and mixture
flow rate m
pressure pm = 10 MPa [Colour figure
can be viewed at
wileyonlinelibrary.com]
A few qualitative comments about the energy consumption follow. Shorter and longer reactors yield the
same outlet value of gNH3 . At first sight, shorter reactors
should be preferred, since they are intuitively associated
with lower energy consumption. However, shorter reactors (for instance, Ltube = 1.1 m) are associated with
shorter segments of lower Tw values and a larger decrease
of gNH3 in the middle of the reactor, followed by a relatively long segment where Tw has the highest value. Longer reactors (eg, Ltube = 3.0 m) have longer segments of
lower Tw values in the reactor middle and a smaller
decrease of gNH3 near the outlet of the reactor. The
highest temperature Tw is reached on a relatively short
segment at reactor outlet. This suggests that the energy
needed to complete the ammonia decomposition is not
significantly different in short and long reactors. Further
comments about energy consumption may be found in
Section 5.2.
5.1.2
|
Optimal design
Here, we focus on design optimization. This involves a
controlled space distribution of the tube diameter Dtube
or/and catalyst particle diameter dp. The main parameters are kept constant (see the reference values in
Table 2) except those explicitly mentioned. The pressure
drop is about 2% of the operation pressure, when the tube
diameter and particle size are constant along the reactor.6
Therefore, pressure drops have been neglected in many
previous works.6,7,9,10,28 Here, we assume that the pressure drop is negligible even if the tube diameter and particle size are changing along the reactor length. This
assumption is justified in part by the low values of the
mass flow rate, which, in turn mean reduced pressure
drops, and by the fact that the dependence of the reaction
rate on pressure is negligible.10
One control
First, the space distribution of the tube diameter Dtube is
controlled in the range 0.010 m to 0.025 m, as shown in
Table 2. When lower values of the tube wall temperature
Tw are considered (ie, 600 K and 700 K) the optimal
space distribution of the tube diameter consists of two
regions (Figure 5A). There is a short inlet region where
the tube diameter has the smallest allowed value (ie,
0.010 m), followed by sudden jump to a longer region
where the tube diameter has the largest allowed value
(ie, 0.025 m). The control is of the bang-bang type. The
smaller tube diameter at reactor inlet ensures a higher
heat transfer coefficient (see Equation (15)) which in turn
increases the mixture temperature (see Equation (2)) and
changes the thermodynamic equilibrium towards smaller
values of the ammonia mass fraction. The ammonia mass
fraction gNH3 decreases slowly along the reactor length
for a tube wall temperature of 600 K while for Tw = 700 K
the decrease of gNH3 is more abrupt near the reactor inlet
(Figure 5B). Despite the space variation of the tube
16
BADESCU
F I G U R E 5 Optimal space
dependence of (A) tube diameter
Dtube and (B) ammonia mass fraction
gNH3 along the reactor for different
values of the constant tube wall
temperature Tw. The first objective
function has been considered and the
control is the tube diameter Dtube.
Tube length Ltube = 2 m, catalyst
particle diameter dp = 2.2 mm, mass
_ m = 0:4g=s, and mixture
flow rate m
pressure pm = 10 MPa [Colour figure
can be viewed at
wileyonlinelibrary.com]
diameter is similar for both values of Tw, the outlet gNH3
is different since the thermodynamic equilibrium at reactor outlet is different for the two temperatures.
For higher tube wall temperatures (750 K and 800 K),
the optimal distribution of Dtube consists of a larger inlet
diameter followed by a decreasing diameter interval
towards a region of constant diameter (Figure 5A). The
larger value of the inlet tube diameter is explained as follows. The high tube wall temperature ensures the necessary heat flux to perform the decomposition reaction and
there is no need to increase the heat transfer coefficient,
by using a smaller tube diameter. For intermediate tube
wall temperature (725 K), the optimal variation of the
tube diameter consists of a region where the diameter
increases followed by a region of constant diameter. In
this case, the contribution of the heat transfer coefficient
to the maximum extent of the decomposition reaction
decreases smoothly along the reactor inlet region. The
space variation of the ammonia content gNH3 is similar
for the high and intermediate wall temperature but the
outlet value is different, as expected (Figure 5B).
Second, the space distribution of the catalyst particle
diameter dp is controlled in the range 0.001 m to 0.006 m,
as Table 2 shows. When lower values of the tube wall
temperature Tw are considered (ie, 600 K and 700 K), the
optimal space distribution consists of catalyst particles of
the smallest allowed diameter (ie, 0.001 m) (Figure 6A).
The smallest particle diameter ensures a higher heat
transfer coefficient (see Equation (15)) and this allows
smaller values of the ammonia mass fraction to be
obtained, as explained in relation with the optimal distribution of the tube diameter. The ammonia mass fraction
gNH3 decreases slowly along the reactor for both tube wall
temperatures of 600 K and 700 K (Figure 6B).
For higher tube wall temperatures (750 K and
800 K), the optimal distribution consists of catalyst particles of larger diameter dp at reactor inlet. The diameter of the particles decreases gradually until a region of
particles of constant diameter is reached (Figure 6A).
The particles of larger diameter at reactor inlet are consequences of the fact that smaller particles are not
needed since the high tube wall temperature ensures
the necessary heat flux to perform the decomposition
reaction. For intermediate tube wall temperature
(725 K), the optimal distribution consists of three
regions: the inlet region, with particles having the
smallest allowed diameter, followed by an intermediate
region containing particles of increasing diameter, and
a rather long region consisting of particles of almost
constant diameter. In this case, the contribution of the
heat transfer coefficient to the maximum extent of the
decomposition reaction decreases smoothly along the
first two regions. The space variation of the ammonia
content gNH3 depends on the wall temperature, as
expected (Figure 6B).
Two controls
Here, both space distributions of the tube diameter Dtube
and catalyst particle diameter dp are controlled in the
ranges shown in Table 2. This is optimization with two
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controls. It is interesting to compare results obtained
from two controls optimization with previous results,
obtained from one-control optimization.
The optimal distribution of the tube diameter Dtube
obtained from two-control optimization is shown in
Figure 7A. This has to be compared with Figure 5A
showing results for one-control optimization. The results
F I G U R E 6 Optimal space
dependence of (A) catalyst particle
diameter dp and (B) ammonia mass
fraction gNH3 along the reactor for
different values of the constant wall
tube temperature Tw. The first
objective function has been
considered and the control is the
catalyst particle diameter dp. Tube
length Ltube = 2 m, tube diameter
Dtube = 18 mm, mass flow rate
_ m = 0:4g=s, and mixture pressure
m
pm = 10 MPa [Colour figure can be
viewed at wileyonlinelibrary.com]
F I G U R E 7 Optimal space
dependence of (A) tube diameter
Dtube, (B) catalyst particle diameter
dp, and (C) ammonia mass fraction
gNH3 along the reactor for different
values of the constant tube wall
temperature Tw. The first objective
function has been considered and the
controls are the tube diameter Dtube
and the catalyst particle diameter dp.
Tube length Ltube = 2 m, mass flow
_ m = 0:4g=s, and mixture
rate m
pressure pm = 10 MPa [Colour figure
can be viewed at
wileyonlinelibrary.com]
17
are quite similar in case of lower tube wall temperatures
(600 K and 700 K). However, the jump from the smallest
to the highest tube diameter in Figure 7A occurs at
slightly different distances from the reactor inlet for the
two values of the wall temperatures while in Figure 5A the
distance is the same for both temperatures. In case of the
high wall temperatures (750 K and 800 K) few differences
18
exist between one-control and two-control optimizations. For
instance, the segment of constant tube diameter starts at
0.6 m from the inlet in case of one-control while in case of
two-control it starts at 0.7 m. When the intermediate tube
wall temperature of 750 K is considered, the differences
between one- and two-control optimization are more significant. In the inlet region the tube diameter for one control
increases while in case of two controls it decreases. Also, the
region of constant diameter is longer for the one-control than
for the two-control and the constant tube diameter at reactor
outlet is smaller for the one-control than for the two-control.
The optimal distribution of the catalyst particle diameter dp obtained from two-control optimization is shown
in Figure 7B. This has to be compared with Figure 6A
showing results for one-control optimization. In case of
low tube wall temperatures (600 K and 700 K) and high
wall temperatures (750 K and 800 K) there are small differences between one- and two-control optimizations.
When the intermediate wall temperature 725 K is considered, obvious differences exist in the inlet region. The
optimal distribution for one-control consists of region of
small particle diameter followed by a region of particles
with increasing diameter. In case of two controls, particles with large diameter are optimal in the inlet region.
The optimal distribution of the ammonia mass fraction gNH3 obtained from two-control optimization is
shown in Figure 7C. This has to be compared with
Figure 5B and Figure 6B showing results for one-control
optimization. Differences between the two approaches
are less obvious and the outlet gNH3 values are the same,
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no matter how many controls are used, since they only
depend on the tube wall temperature.
Previous results concerning the dependence of the
reactor performance on the inlet mass flow rate are
shortly reminded now. The ammonia mass fraction at
reactor inlet increases almost linearly by increasing the
inlet ammonia mass flow rate (see figure 4.12 of Reference 6). The mixture temperature decreases by increasing
the mixture mass flow rate (see figure 5 of Reference 10).
For small mass flow rate values the mixture temperature
reaches 1200 K and the outlet ammonia mass fraction is
practically 0%. Notice that such high temperatures may
be obtained in Reference 10 since the heat flux through
the tube wall (not the tube wall temperature) is given.
For larger values of the mass flow rate lower mixture
temperatures are reached in Reference 10. The outlet
ammonia mass fraction ranges between 0.15 and 0.20 for
a temperature of 875 K. The exit reaction extent
decreases from 0.8 to 0.25 in Reference 7 for a mass flow
rate increasing 15 times (from 0.1 g/s to 1.5 g/s).
_ m is constant along the
The mixture mass flow rate m
reactor and equals the inlet ammonia mass flow rate.
Optimal distributions obtained from two-control optimi_ m as shown in Figure 8. A constant
zation depend on m
tube wall temperature of 800 K is considered (see the reference value in Table 2). The larger the mass flow rate is,
the larger are the optimal tube diameter (Figure 8A) and
the catalyst particle diameter (Figure 8B). A few comments follow about Figure 8B. The dependence of dp on
_ m is obvious in the middle of the
the mass flow rate m
F I G U R E 8 Optimal space
dependence of (A) tube diameter
Dtube, (B) catalyst particle diameter
dp, and (c) ammonia mass fraction
gNH3 along the reactor for different
values of the inlet ammonia mass
_ m . The first objective
flow rate m
function has been considered and the
controls are the tube diameter Dtube
and the catalyst particle diameter dp.
Tube length Ltube = 2 m, tube wall
temperature Tw = 800 K, and mixture
pressure pm = 10 MPa [Colour figure
can be viewed at
wileyonlinelibrary.com]
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reactor, where the mixture temperature Tm is close to the
wall temperature Tw and the ammonia mass fraction gNH3
is close to its outlet value. Since Tm cannot exceed Tw the
r.h.s. of Equation (2) should be positive and slightly
higher than zero. Therefore, the first and the second
terms in the r.h.s. member of Equation (2) should be of
the same order of magnitude. However, the second term
is almost constant while the first term depends on the
heat transfer coefficient hw, which in turn depends on
_ m and dp (see Equation (2)). Therefore, keeping a conm
_ m is increased, dp should
stant value hw asks that, when m
be increased, too. Similar arguments apply in the case of
Figure 8A, but they are complicated by the fact that both
terms in the r. h. s. of Equation (2) depend on Dtube.
The outlet value of the ammonia mass fraction does
_ m (Figure 8C) since the tube wall temnot depend on m
perature is the same for all values of the mass flow rate.
To conclude, in general one-control is not a special
case of two-control since the optimal design solutions in
the two cases are different. However, there are specific
situations when the two approaches yield similar optimal
solutions.
5.2 | Second objective function
The second objective consists of minimizing the heat flux
(see Equation (18)) necessary to obtain an expected outlet
target
(details are
ammonia content gNH3 ðz = Ltube Þ = gNH3
given in Section 4.2.2). Three different controls have been
considered: the tube wall temperature Tw, the tube diameter Dtube, and the catalyst particle diameter dp, respectively. Tw is related to operation optimization while Dtube
and dp are related to design optimization.
A few comments about the values of the outlet ammonia mass fraction reported in literature are useful. The reaction extent decreases by increasing the pressure (see figure
10 of Reference 7 where pressures between 4 MPa and
18 MPa have been considered). When the reverse reaction
is neglected, the outlet ammonia mass fraction is practically
zero for mixture temperatures higher that 700 K (see figure
3.8 and figure 4.1 of Reference 6). Equilibrium conversion
is about 99.5% for 800 K and 0.19 MPa, when the reverse
reaction is neglected.31 In case of Ni catalysts the ammonia
decomposition may range between 72% conversion at
773 K and 95% conversion at 873 K.32 The exit reaction
extent is not 100% for some values of the inlet ammonia
mass rate, even for the high operating temperature of
1023 K (see figure 6 of Reference 7). The effect of the activation energy on the outlet ammonia mass fraction is important. For instance, for mixture temperature around 873 K,
changing the activation energy from 1.15 × 105 J/mol to
3.87 × 105 J/mol, may change the outlet ammonia mass
19
fraction from 0.25 to 0.15 (see figure 7 of Reference 7). An
outlet ammonia mass fraction around 0.9 has been modeled
in Reference 7 (see their figure 4). A reaction extent equal
to 0.85 has been assumed in Reference 24.
The target outlet ammonia mass fraction considered
target
here is gNH3 = 0:18.
5.2.1 |
Optimal operation
The distribution of the tube wall temperature Tw along
the reactor is used as a control while other parameters
are kept constant (see the reference values in Table 2).
The range of variation for Tw is 500 to 800 K.
The optimal distribution of the tube wall temperature
depends on the tube length Ltube, as expected, but its shape
is similar in all cases (see Figure 9A). The tube wall temperature at reactor inlet is always 800 K. Next, the wall temperature decreases abruptly towards a minimum temperature
which is lower for longer tubes. A smooth temperature
increase follows towards a maximum temperature, which is
reached near the tube outlet. Then, the wall temperature
decreases to the value 773.98 K at reactor outlet. This is
associated with an outlet mixture temperature 773.98 K
ensuring the thermodynamic equilibrium needed to have
target
an outlet ammonia mass fraction gNH3 = 0:18 . The space
variation of the tube wall temperature in Figure 9A is
similar with the variation of the mixture temperature
along the reactor in figure 3.9 of Reference 6.
The optimal space distribution of the ammonia mass
fraction gNH3 is shown in Figure 9B. gNH3 decreases more
abruptly for shorter tubes, as expected. Notice that the
reactor segment where the mixture has reached the target
outlet ammonia content is very short, for all tube lengths.
This has to be compared with the longer reactor segments with constant outlet ammonia content obtained in
case of the first objective function (see for instance
Figure 4B). Therefore, more effective reactor utilization is
obtained by using the second objective function.
Table 7 shows the minimum heat flux values associated
with Figure 9. Generally, the needed heat flux decreases by
increasing the tube length. However, longer tubes are more
expensive than shorter tubes. Also, when longer tubes are
considered, the needed heat flux decreases slowly by
increasing the tube length. Therefore, tubes of intermediate
length are recommended in practice, as a compromise
between investments and operational costs.
5.2.2 |
Optimal design
The design optimization is considered next by controlling
the space distribution of the tube diameter Dtube in the
20
BADESCU
F I G U R E 9 Optimal space
dependence of (A) tube wall
temperature Tw and (B) ammonia
mass fraction gNH3 along the reactor
for different values of the tube length
Ltube. The second objective function
has been considered and the control
is the tube wall temperature Tw.
Tube diameter Dtube = 18 mm,
catalyst particle diameter
dp = 2.2 mm, mass flow rate
_ m = 0:4g=s, target outlet ammonia
m
target
mass fraction gNH3 = 0:18, and
mixture pressure pm = 10 MPa
[Colour figure can be viewed at
wileyonlinelibrary.com]
T A B L E 7 Minimum heat flux values needed to obtain an
target
outlet value of the ammonia mass fraction gNH3 = 0:18 for several
values of the tube length Ltube considered in Figure 9
Tube length,
Ltube (m)
Minimum value of the
heat flux needed to obtain
the outlet ammonia mass
target
fraction, gNH3 = 0:18 (W)
2.0
454.38
1.5
455.29
1.0
456.21
0.8
457.80
0.5
459.11
0.4
459.60
0.3
461.23
0.2
463.36
0.15
469.50
range 0.010 m to 0.025 m, as shown in Table 2. The other
parameters are kept constant (see the reference values in
Table 2). A constant wall tube temperature 800 K is
considered.
Generally, the optimal tube diameter at reactor inlet
has the maximum allowed value (0.025 m) while at
reactor outlet it has the minimum allowed value
(0.010 m) (see Figure 10A). This applies for all values of
the catalyst particle diameter considered here. The transition between the maximum and minimum diameters
occurs near the reactor inlet. The smaller the catalyst
particle diameter dp, the closer to the reactor inlet this
transition is. The optimal space distribution of the
ammonia mass fraction gNH3 is shown in Figure 10B. It is
quite similar for all values of the catalyst particle diametarget
ter. The target outlet ammonia mass fraction gNH3 = 0:18
is reached quite shortly after the reactor inlet. This shows
that the reactor length (2 m) is oversized in this case.
Notice that in case of larger values of dp (ie, 0.003 m and
0.004 m) the target outlet ammonia mass fraction is
reached before the mixture enters the reactor segment of
minimum diameter.
Table 8 shows the minimum heat flux values associated with Figure 10. Generally, the needed heat flux
slightly decreases by increasing the catalyst particle
diameter.
Next, the space distribution of the catalyst particle
diameter dp is controlled in the range 0.001 m to
0.006 m, as shown in Table 2. For all temperatures Tw
considered here, the optimal space distribution consists
of catalyst particles of the smallest allowed diameter
(ie, 0.001 m). This applies for different constant values
of the tube diameter, tube length, and inlet mass flow
rate. The explanation is that the smallest particle diameter ensures a higher heat transfer coefficient (see
Equation (15)).
BADESCU
21
F I G U R E 1 0 Optimal space
dependence of (A) tube diameter
Dtube and (B) ammonia mass fraction
gNH3 along the reactor for different
values of the catalyst particle
diameter dp. The second objective
function has been considered and the
control is the tube diameter Dtube.
Tube length Ltube = 2 m, mass flow
_ m = 0:4g=s, tube wall
rate m
temperature Tw = 800 K, target outlet
target
ammonia mass fraction gNH3 = 0:18,
and mixture pressure pm = 10 MPa
[Colour figure can be viewed at
wileyonlinelibrary.com]
T A B L E 8 Minimum heat flux values needed to obtain an
target
outlet value of the ammonia mass fraction gNH3 = 0:18 for several
values of the catalyst particle diameter dp, considered in Figure 10
Catalyst particle
diameter, dp (m)
Minimum value of the
heat flux needed to obtain
the outlet ammonia mass
target
fraction, gNH3 = 0:18 (W)
0.001
506.42
0.002
505.86
0.004
504.10
0.006
503.76
6 | C ON C L U S I ON S
Reversible chemical reactions may be used for energy storage
or energy transport from a heat source to a work extractor,
for instance a heat engine or a power plant. Such configurations are usually called thermochemical storage and/or
power systems. The reversible chemical reaction considered
in this paper is decomposition/synthesis of ammonia. Most
previous studies focus on the ammonia synthesis reactor.
The novelty here is that a packed bed ammonia decomposition reactor having tubular geometry is analyzed. Ammonia
decomposition is enhanced by using a Ni/Al2O3 catalyst.
The present paper brings two innovations. The first
innovation refers to the reactor operation: we suggest
that a spatially nonuniform heating of the ammonia
decomposition reactor may improve its performance. The
second innovation refers to the reactor design: for uniform space heating of the reactor, we suggest that its performance may be improved by changing the reactor
shape or/and by changing the size of the catalyst particles
along the reactor. Both the reactor operation and the
reactor design are optimized.
The optimization is performed by using direct optimal
control methods. The minimization of two objective functions is considered: (i) the ammonia mass fraction at reactor
outlet and (ii) the heat flux necessary to reach a given value
of the outlet ammonia mass fraction at reactor outlet. The
optimal control problem consists of the constrained
extremization of the objective functions. The constraints are
ordinary differential equations for the space variation of syngas temperature and the ammonia mass fraction along the
reactor, respectively. For each objective function, the optimization process is performed by using different controls, that
is, the tube wall temperature, the circular tube diameter,
and the diameter of the catalyst spherical particles.
A detailed list of conclusions may be found in
Section S6 of the ESM. Here, the main findings are
shortly reminded. First, results concerning the first objective function are listed.
1. The optimal space distribution of the reactor tube
temperature is U-shaped. The optimal variation of the
mixture temperature increases monotonously along the
reactor. The optimal distribution of the tube wall temperature depends on the tube length but the temperature
profile is similar in all cases.
2. For high tube wall temperatures, the optimal distribution of the tube diameter consists of a larger inlet
diameter followed by a decreasing diameter interval
towards a region of constant diameter.
3. For high values of the tube wall temperature, the
optimal catalyst particle diameter is larger at reactor inlet
22
and it gradually decreases until a rather long region of
particles of constant diameter is reached.
4. Two-control optimization yields results which, for
some particular cases, are different from the results
obtained from one-control optimization. Generally, the
larger the mass flow rate is, the larger are the optimal
tube diameter and the catalyst particle diameter.
Next, results concerning the second objective function
are reminded.
5. The optimal distribution of the tube wall temperature depends on the tube length. It is high at reactor
inlet, decreases abruptly towards a minimum temperature which is lower for longer tubes, and next increases
smoothly towards a maximum temperature, which is
reached near the reactor outlet.
6. The optimal value of the tube diameter ranges
between its maximum allowed value (at reactor inlet)
and its minimum allowed value (at reactor outlet).
7. For all values considered here for the tube wall
temperature, tube diameter, tube length, and inlet mass
flow rate, the optimal space distribution consists of catalyst particles of the smallest allowed diameter.
Our results open new perspectives on the design and
operation of ammonia decomposition reactors. It has been
shown here that the performance of these devices may be
improved by using rather unusual methods such as changing the reactor shape or changing the space distribution of
the catalyst particle size. However, further research work is
necessary from a theoretical point of view, by improving
the present model and relaxing some of the assumptions
adopted here. Also, experiments should adjust and consolidate the present and future theoretical results.
A C K N O WL E D G M E N T S
The author thanks Prof. M.J. Assael (Aristotle University,
Tessaloniki, Greece) for providing the database with
thermophysical properties of ammonia. The author thanks
the reviewers for useful comments and suggestions.
NO MEN CLATU RE
cross-sectional area of reactor tube (m2)
Atube
specific heat capacity (J/(K kg))
cp
g
component mass fraction (kg of component
/kg mixture)
convection heat transfer coefficient at reactor
hw
tube wall (W/(m2 K))
reactor tube length (m)
Ltube
M
molar mass (kmol/kg)
_m
m
mixture mass flow rate (kg mixture/s)
Ptube
reactor tube perimeter (m)
ammonia decomposition rate (mol NH3/(m3 s))
RNH3
r
component molar/volume fraction (kmol of
component/kmol of mixture)
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Tm
Tw
ΔH r,NH3
mixture temperature (K)
tube wall temperature (K)
enthalpy change of ammonia decomposition
reaction (J/mol NH3)
Subscripts
NH3 ammonia
hydrogen
H2
nitrogen
N2
m
mixture
ORCID
Viorel Badescu
https://orcid.org/0000-0002-7708-5108
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SU PP O R TI N G I N F O RMA TI O N
Additional supporting information may be found online
in the Supporting Information section at the end of this
article.
How to cite this article: Badescu V. Optimal
design and operation of ammonia decomposition
reactors. Int J Energy Res. 2020;1–25. https://doi.
org/10.1002/er.5286
24
BADESCU
A P P EN D I X A
Specific heats for hydrogen, nitrogen and ammonia are
expressed in [6] as functions of temperature in the range
500-800 K, only. The viscosity of the components may be
estimated by using measurements data or theoretical
models as shown in [31].
Details about the computation of the physical properties of the three components of the mixture follow.
The EES software [46] has been used to generate tabulated data of specific heat at constant pressure cp,
dynamic viscosity μ and thermal conductivity k for
hydrogen and nitrogen within the temperature interval
500 K to 1200 K (step of 100 K) and pressure interval
2 MPa to 11 MPa (step of 1 MPa). The physical properties
of ammonia require special care since measurements
above 700 K are not considered because spontaneous
decomposition of ammonia into hydrogen and nitrogen
was observed.47,48 The REFPROP database49 contains
properties of ammonia up to 1000 K. The high temperature part of these data has been obtained by modeling
(see Fig. 9 of [47] and Fig. 8 of [48]). The database has
been linearly extrapolated up to 1200 K.
Dimensionless values cp cp, =cp,ref , μ μ=μref and k k=kref have been generated by using the reference values
cp,ref, μref and kref defined in Table A1.
The dimensionless tabulated data were fitted by
using the TableCurve2D software
[50] against the
dimensionless temperature T m T m =T ref and pressure
pm ð pm =pref Þ, where the reference values are Tref = 1200 K
and pref = 11 MPa. Results are presented in Table A2 for
hydrogen and nitrogen and in Table A3 for ammonia.
Reference values cp,ref, μref and kref for hydrogen, nitrogen and ammonia
TABLE A1
Component
cp,ref (J/(kgK))
μref (Pa s)
kref (W/(mK))
Hydrogen
15372
0.00002446
0.47420
Nitrogen
1210
0.00004706
0.07574
Ammonia
5061.5
0.00002334
0.27212
Dimensionless properties of hydrogen and nitrogen as functions of mixture dimensionless pressure pm pm =pref and
temperature T m T m =T ref (temperature interval 500 K to 1200 K and pressure interval 2 MPa to 11 MPa)
TABLE A2
Relationship
Coefficients
Accuracy indicators
a = 0.4081590627615998
b = 0.3793674476036514
c = 0.5777866231107818
r2 = 0.9989059963998608
StdErr = 0.0006108313437706014
a = 0.7758750960400124
b = 0.3375686150045503
c = −0.1148260714987987
r2 = 0.9999567665541
StdErr = 0.001017864198757912
a = 0.02283112627762948
b = 1.592740989894128
c= −0.6171195663197747
r2 = 0.9998569463424971
StdErr = 0.001689674374307304
cp,N 2 ¼ a þ bpm þcT m
a = 0.8466588099991154
b = 0.01584005757592666
c = 0.1455038614031149
r2 = 0.9891495111777651
StdErr = 0.003756491435587898
0:5
a = −0.2305771513979486
b = 0.01246644227084282
c = 1.217447746701695
r2 = 0.9996972091834886
StdErr = 0.002497284263793677
0:5
a= −0.316492188448136
b = 0.02459847655509067
c = 1.289094046230622
r2 = 0.9988343455808792
StdErr = 0.005193859683053323
cp,H 2 ¼ a þ bT m þ cexp − T m
3
μH 2 ¼ a þ bT m þ Tc
m
2
k H 2 ¼ a þ bT m þ cT m
1:5
μN 2 ¼ a þ bpm þ cT m
k N 2 ¼ a þ bpm þ cT m
BADESCU
25
Dimensionless properties of ammonia as functions of mixture dimensionless pressure pm pm =pref and temperature
T m T m =T ref (temperature interval 500 K to 1200 K and pressure interval 2 MPa to 11 MPa)
TABLE A3
Relationship
cp,NH 3 ¼
ð
Þ
ð
Þ
2
3
aþbpm þcp2m þdlnT m þe lnT m þf lnT m
1þgpm þhp2m þilnT m þjðlnT m Þ2
μNH 3 ¼ a þ bpm þ
þgp3m þ
c
e
fp
þ dp2m þ 2 þ m
Tm
Tm Tm
h
ipm
3 þ 2 þ
Tm Tm
jp2m
Tm
aþbpm þcp2m þdlnT m þeðlnT m Þ þf ðlnT m Þ
2
k NH 3 ¼
1þgpm þhp2m þilnT m þjðlnT m Þ þk ðlnT m Þ
2
3
3
Coefficients
Accuracy indicators
a = 0.7078913187189729
b= −0.2248646373450243
c = 0.595108494896019
d = 1.845937929529775
e = 1.391558146130148
f = 0.1809565432557225
g = −0.3649092143337899
h = 0.787467280137581
i = 2.392679093103923
j = 1.502736212696683
r2 = 0.9737499037506497
StdErr = 0.01857863174873989
a = 1.938091639934885
b = −0.7071289228510781
c = 1.074158560129445
d = 0.9785531317259019
e = −1.702171588369741
f = 0.2184794728000117
g = −0.07536591665909596
h = 0.4112489323328676
i = 0.6916671645181923
j = −0.7460920764505295
r2 = 0.9966722001944903
StdErr = 0.02418291213560008
a = 0.7045538075430287
b = 0.1606764990062799
c = 0.02689985889649394
d = 2.20527941234809
e = 2.302501866855694
f = 0.8241133393245334
g = 0.1195284412465949
h = −0.0001156397360247473
i = 1.984566812877896
j = 0.9993129739025465
k = −0.0002206634338783622
r2 = 0.9979626290511955
StdErr = 0.007732362743086001