i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Available online at www.sciencedirect.com
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Modelling the hydrodynamics and kinetics of
methane decomposition in catalytic liquid metal
bubble reactors for hydrogen production
Lionel J.J. Catalan*, Ebrahim Rezaei
Department of Chemical Engineering, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, P7J 5E1 Canada
highlights
graphical abstract
Faster superficial gas velocity increases gas-liquid interfacial area
and gas holdup.
Larger diameter tubes lower the
total melt volume in multitubular
reactor designs.
Liquid metal volume increases
faster than
exponentially with
decreasing temperature.
14.8 m3 of Ni0$27Bi0.73 melt at
1050 C produces 10,000 Nm3.h1
of H2 at 80% CH4 conversion.
Molten CueBi alloy results in
smaller reactors (shorter tubes)
than NieBi alloy.
article info
abstract
Article history:
Bubble reactors using molten metal alloys (e.g, nickel-bismuth and copper-bismuth) with
Received 16 August 2021
strong catalytic activity for methane decomposition are an emerging technology to lower
Received in revised form
the carbon intensity of hydrogen production. Methane decomposition occurs non-
15 November 2021
catalytically inside the bubbles and catalytically at the gas-liquid interface. The reactor
Accepted 9 December 2021
performance is therefore affected by the hydrodynamics of bubble flow in molten metal,
Available online 11 January 2022
which determines the evolution of the bubble size distribution and of the gas holdup along
the reactor height. A reactor model is first developed to rigorously account for the coupling
Keywords:
of hydrodynamics with catalytic and non-catalytic reaction kinetics. The model is then
Hydrogen
validated with previously reported experimental data on methane decomposition at
Catalytic methane decomposition
several temperatures in bubble columns containing a molten nickel-bismuth alloy. Next,
Liquid metal
the model is applied to optimize the design of multitubular catalytic bubble reactors at
Bubble reactor
industrial scales. This involves minimizing the total liquid metal volume for various tube
Hydrodynamics
diameters, melt temperatures, and percent methane conversions at a specified hydrogen
Kinetics
production rate. For example, an optimized reactor consisting of 891 tubes, each measuring
0.10 m in diameter and 2.11 m in height, filled with molten Ni0$27Bi0.73 at 1050 C and fed
* Corresponding author.
E-mail address: lionel.catalan@lakeheadu.ca (L.J.J. Catalan).
https://doi.org/10.1016/j.ijhydene.2021.12.089
0360-3199/© 2021 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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with pure methane at 17.8 bar, may produce 10,000 Nm3.h1 of hydrogen with a methane
conversion of 80% and a pressure drop of 1.6 bar. The tubes could be heated in a fired
heater by burning either a fraction of the produced hydrogen, which would prevent CO2
generation, or other less expensive fuels.
© 2021 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction
Methane decomposition, also called methane pyrolysis (Eq.
(1)), is an alternative to the traditional hydrogen production
process. The latter combines steam methane reforming (SMR,
Eq. (2)) with the water gas shift reaction (WGS, Eq. (3)). The
main advantage of methane decomposition is the lack of
direct CO2 emissions and the production of solid carbon,
which may be sold or sequestrated. In contrast, the combination of SMR with WGS, (Eq. (4)), generates 1 mol of CO2 for
every 4 mol of H2. A downside of methane decomposition is
the low molar ratio of H2 production to CH4 consumption,
which is only 2:1 compared to 4:1 for SMR-WGS. This results in
a significant economic hurdle for methane decomposition
that may be overcome with the ability to sell the carbon byproduct, government incentives, large taxes on CO2 emissions that would make SMR-WGS less profitable, or a combination of the above. The carbon by-product should not be used
as a fuel since its combustion would produce CO2 and therefore negate the environmental benefit of methane
decomposition.
CH4 )/ 2H2 þ C(s) DH 298 ¼ 75 kJ mol1
(1)
CH4 þ H2O )/ 3H2 þ CO DH 298 ¼ 206 kJ mol1
(2)
CO þ H2O )/ H2 þ CO2 DH 298 ¼ -41 kJ mol1
(3)
CH4 þ 2H2O )/ 4H2 þ CO2 DH 298 ¼ 165 kJ mol1
(4)
Methane decomposition is one among many technological
approaches to reduce CO2 emissions in H2 production processes [1]. Nonetheless, a recent life-cycle based study
comparing the relative costs of CO2 mitigation for twelve
different H2 production technologies using fossil fuels, nuclear energy, and renewable resources found that methane
decomposition may be the most cost-effective abatement
solution in the short term [2].
The formation of elemental carbon during methane
decomposition creates a practical challenge for stable
continuous reactor operation. When the reaction is carried
out in the presence of solid catalyst pellets in a packed bed, the
pressure drop increases rapidly and the catalyst quickly deactivates as carbon deposits on its surface [1,3]. On the other
hand, non-catalytic methane decomposition in void tubular
reactors causes buildup of hard carbon deposits on the reactor
walls that eventually block the gas flow [4]. Fluidized bed reactors (FLBR) have been studied for their ability to mitigate
these operational problems [5,6]. Qian et al. [7] found that an
FLBR using 40 wt% Fe/Al2O3 catalyst maintained a low pressure drop and catalytic activity even though carbon was produced with a yield as high as 10.7 g carbon/g catalyst. Attrition
of the produced carbon on the Fe catalyst surface in FLBRs can
help maintain the conversion rate of methane over time [8].
Spent Fe/Al2O3 catalyst could be successfully regenerated
multiple times by CO2 oxidation, and the regenerated catalyst
not only required a reduced activation time but also improved
the methane conversion [7]. Fe based catalysts are advantageous for methane decomposition because of their low cost,
environmental friendliness, and easy activation. Recently,
several industrial wastes or by-products containing iron were
tested as catalysts for methane decomposition with encouraging results [9].
Liquid metal bubble reactors (LMBRs) provide an effective
solution to the carbon deposition problem in laboratory
studies. The gas injected at the bottom of the reactor rises as
bubbles through the molten metal. Methane decomposition
occurs inside the bubbles and at the gas-liquid interface if the
molten metal has catalytic activity. Because carbon is much
less dense than molten metal, it rises to the surface of the
melt. In Catalan and Rezaei [10], we included a review of
experimental studies of methane decomposition in LMBRs up
to 2019. All studies concurred that most of the carbon formed
a powder layer above the melt. Deposition of carbon on the
reactor walls appears to be limited to very thin films. Upham
et al. [11] observed that carbon deposited slowly on the walls
of the reactor via precipitation from the saturated melt;
however, the carbon-on-carbon deposition decreased with
time such that the thickness of the deposited carbon layer
remained less than 10 mm after one week of continuous
operation. These observations are consistent with those of
Geibler et al. [12] who reported that the carbon layer that had
deposited on the reactor wall was only around 10 mm thick
after 15 days of operation.
Various methods have been envisioned in the literature for
continuously removing the carbon floating on the surface of
the melt. One method involves skimming the carbon mechanically from the surface, which is similar to the common
practice for removing slag material from melts in metallurgical processes [11]. Alternatively, the fine powder of carbon
could be entrained in the gas stream exiting the reactor and
separated using cyclones, filters, or electrostatic precipitators
[11,13]. To the authors’ knowledge, none of these methods has
yet been demonstrated experimentally. Considering the high
cost of molten metals, the carbon/metal separation should be
very effective to limit the loss of metal and the economic
burden of metal make-up. The contamination of carbon by
metal may be substantially reduced by installing a molten salt
layer floating on the liquid metal [14].
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
In the past few years, methane decomposition in LMBRs
has been increasingly tested in catalytic molten metal alloys
and salts such as Ni-Bi [11,14,15], Cu-Bi [16], Ga [17], MnCl2eKCl [18], NaCleKCleFe [19], and various alkali halide salts
[20,21]. Additions of catalysts containing La, Ni, Co and Mn as
particle suspensions in molten salts have also been investigated [22]. The main advantage of a catalytic melt is the potential to reduce the reactor size and to operate at lower
temperatures ( < 1100 C) than with non-catalytic systems
(e.g, molten Sn or Fe), while still achieving industrially relevant H2 production rates (several thousand Nm.3 h1) at relatively high CH4 conversions (>60%). Lower temperatures are
desirable because they provide more flexibility for selecting
materials of construction [23] and they reduce the costs of
operation.
Our previous work [10] centered on the design of noncatalytic LMBRs where methane decomposition occurred in
the gas throughout the bubble but not specifically at the gasliquid interface. This required knowing the non-catalytic reaction rate in the gas phase, which in turn depends on reactant and product concentrations as well as on the gas volume
in the melt. The gas volume per unit volume of melt (i.e., the
gas holdup) is a complex function of the gas volumetric flow
rate, pressure, temperature, composition, and of the liquid
metal properties (density, viscosity, and surface tension). The
gas holdup increases along the reactor height not only
because the hydrostatic pressure decreases but also because
each CH4 molecule that reacts produces two molecules of H2
(Eq. (1)). Hence, the reaction kinetics and hydrodynamics of
LMBRs influence each other. We were the first to develop a
coupled hydrodynamic and kinetic model of LMBRs for noncatalytic decomposition of methane [10]. Prior to this, other
researchers had assumed an arbitrary gas holdup of 25%
either in the entire melt [11,24] or at the melt bottom [13,25].
In our earlier study [10], we predicted the gas holdup using
a drift flux correlation that had been experimentally corroborated in gas-molten metal mixtures [26,27], and we found
that relatively large superficial gas velocities (>0.10 m.s1)
were required to achieve gas holdups of about 10 vol percent
or more. We developed thermodynamically consistent kinetic
equations for non-catalytic methane decomposition based on
the experimental data of Keipi et al. [28] obtained in a tubular
reactor for the temperature range of 900e1450 K. Next, we
applied our coupled hydrodynamic and kinetic model to
optimize the diameter, height, and inlet pressure of a noncatalytic LMBR containing molten Sn by minimizing the volume of melt required to produce 200 kt.a1 of H2 at different
temperatures and CH4 conversions. Since methane decomposition is endothermic (DHo298 ¼ 75 kJ/mol), heat needs to be
provided to the molten metal, and we discussed various potential heating methods. Our LMBRs were designed with
isothermal conditions since the turbulent flow of gas rising at
high superficial velocities in the melt was assumed to cause
sufficient mixing and liquid recirculation to equalize the
temperature [10].
Unlike our earlier work, the present article focusses on
the design of catalytic LMBRs. Kinetic data for methane
decomposition in two catalytic molten alloys (Ni0$27Bi0.73 and
Cu0$45Bi0.55) have been recently published [11,16]. Coupling
kinetics and hydrodynamics in catalytic LMBRs poses
7549
significant challenges because the catalytic reaction occurs
at the interface between the gas bubbles and the liquid
metal; hence, the catalytic reaction rate is proportional to the
interfacial area, which in turn depends on the size distribution of bubbles. Previous attempts to model catalytic LMBRs
have made simple assumptions about bubble sizes. Upham
et al. [11] assumed an average bubble diameter of 1 cm
throughout the melt. In Farmer et al. [25], the bubbles were
assumed to be 1.5 cm in diameter at the bottom of the melt
and to change in size as they travelled upward due to the
decrease in hydrostatic pressure, the stoichiometry of the
methane decomposition reaction, and the diffusion of H2
across the gas-molten metal interfacea. Von Wald et al. [29]
assumed that the bubble size at any height was proportional
to its value at the bottom of the melt and to a factor accounting for the CH4 conversion and the stoichiometry of the
reaction. The method to calculate the bubble radius at the
bottom of the reactor was not reported. By contrast, the
present article relies on a dimensionless correlation developed and experimentally corroborated by Akita and Yoshida
[30] to calculate the specific gas-liquid interfacial area at any
height in LMBRs as a function of the superficial gas velocity,
the diameter of the column, and the properties (surface
tension, viscosity and density) of the molten metal. Changes
in CH4 conversion and hydrostatic pressure with height are
reflected in the superficial gas velocity. This approach leads
to a coupled hydrodynamic and kinetic model of catalytic
LMBRs that does not require making any assumption about
the bubble sizes.
The rest of this article has the following organization. The
Theory section (Section: Theory) derives the material balance
and pressure equations for the coupled model. It includes
subsections on the kinetics of catalytic and non-catalytic
methane decomposition in molten metals (Subsections: Kinetics of catalytic methane decomposition in molten
metalseExpansion of the differential material balance equation), the specific gas-liquid interfacial area (Subsection: Specific gas-liquid interfacial area), the gas holdup (Subsection:
Gas holdup in molten metals), and the reactive surface area
(Subsection: Reactive surface area in catalytic molten metals).
Next, the Applications section (Section: Applications) starts by
validating the coupled model with previously reported
experimental data obtained in laboratory-sized catalytic
LMBRs (Subsection: Experimental validation of the catalytic
liquid metal bubble reactor model). It then proceeds to scaleup of an industrially-sized multitubular reactor (Subsection:
Reactor scale-up) and to design optimization by minimizing
the melt volume for a given hydrogen production rate (Subsection: Reactor design optimization). The Discussion section
(Section: Discussion) begins by examining the effect of practical considerations (e.g. limits on maximum pressure) on
designs (Subsection: Variations from optimum designs). Next,
the designs of the present article are compared to previous
a
Farmer et al. [25] modelled an LMBR surrounded by a membrane that was selectively permeable to H2. This created the
conditions for net diffusion of H2 from the gas inside the bubbles
toward the molten metal. In the absence of such a membrane, the
net diffusion of H2 through the gas-liquid interface is nil when
the reactor operates at steady state.
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catalytic LMBR designs from literature (Subsection: Comparison to previous catalytic LMBR design from the literature) and
to steam methane reformer designs (Subsection: Comparison
to steam methane reformer).
In summary, this article differs from our previous work [10]
in that it accounts for catalytic methane decomposition at the
gas-molten metal interface. Furthermore, it improves on
previous attempts to model catalytic methane decomposition
in LMBRs [11,25,29] by developing a new model that calculates
the interfacial area and gas holdup as a function of the LMBR
operating conditions, so that methane conversion is successfully predicted without making ad hoc assumptions about
the bubble size.
Theory
Methane decomposition in catalytic LMBRs is determined by
the interplay between reaction kinetics and the hydrodynamics of gas rising in the melt. Because the catalytic reaction
occurs at the gas-liquid interface, the reactor model must be
able to predict the interfacial area. The model must also
calculate the gas holdup to account for non-catalytic methane
decomposition inside the gas bubbles. Moreover, both the
interfacial area and the gas holdup depend on methane conversion because the reaction generates 2 mol of hydrogen for
each mole of methane consumed (Eq. (1)). As gas rises in the
reactor, the increasing number of gas moles and the
decreasing hydrostatic pressure affect the gas holdup and the
size distribution of bubbles. In this section, we develop a
coupled hydrodynamic and kinetic model that will form the
basis for designing catalytic LMBRs that can achieve desired
H2 production rates and CH4 conversions.
Fig. 1 shows a simplified diagram of an LMBR. The material
balance equation for the gas phase in a differential volume dV
(m3) consisting of a horizontal slice of the melt with length dL
(m) is:
n_CH4 ;b dXCH4 ¼ ðRc þ Rn Þ a dV ¼ ðRc þ Rn Þ a
pD2
dL
4
(5)
where Rc and Rn , both having units of mol CH4.m3.s1, are
respectively the catalytic and non-catalytic reaction rates per
unit volume of gas, a is the gas holdup (m3 gas.m3 reactor), D
is the reactor inner diameter (m), n_CH4 ;b is the inlet mole flow
rate (mol.s1) of methane at the bottom of the reactor, and
dXCH4 is the change in methane conversion (dimensionless)
occurring in the differential volume.b
The pressure change (Pa) of the gas as it rises through the
differential volume element is:
dP ¼ rl ð1 aÞ þ rg a g dL ¼ ðrl aDrÞg dL
(6)
where rl is the liquid metal density (kg.m3), rg is the gas
density (kg.m3), Dr ¼ rl rg , and g is the gravitational
acceleration.
Eqs. (5) and (6) constitute a system of coupled differential
equations that must be solved numerically between the bottom (L ¼ 0, XCH4 ¼ 0, P ¼ Pb ) and the top of the melt (L ¼ Lt ) to
find the methane conversion and the pressure at the outlet of
the reactor. This requires relating the reaction rates Rc and Rn ,
as well as the gas holdup a, to XCH4 and P. These relationships
are developed below.
Kinetics of catalytic methane decomposition in molten
metals
The catalytic decomposition of methane is assumed to be
proportional to the interfacial area between the gas and the
liquid [11,25]. Hence, the catalytic reaction rate per unit volume of gas, Rc , is related to the catalytic reaction rate per unit
of interfacial area, (mol CH4.m2.s1), and the specific interfacial area, (m2 (interfacial area).m3 (gas)), by:
Rc ¼ ag rc
(7)
The value of ag depends on the bubble size distribution,
which is related to the hydrodynamics of the rising gas bubbles in the melt. The calculation of ag is addressed in Section:
Specific gas-liquid interfacial area. The rest of the present
section deals with rc .
The experiments of Upham et al. [11] and Palmer et al. [16]
in Ni0$27Bi0.73 and Cu0$45Bi0.55 melts, respectively, have shown
that the forward catalytic rate, rc;f (mol.m2.s1), is first-order
with respect to CH4:
rc;f ¼ kc;f CCH4
(8)
where kc;f is the forward rate coefficient of the catalytic reaction (m.s1) and CCH4 is the concentration of methane in the
b
Fig. 1 e Simplified graphical representation of an LMBR.
For convenience, all symbols and units are defined in the
Nomenclature section at the end of the article.
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gas (mol.m3). The net catalytic reaction rate is the difference
between rc;f and the backward catalytic rate, rc;b (mol.m2.s1):
kc;f
CH2 2
CH2 2 ¼ kc;f CCH4 1 rc ¼ rc;f rc;b ¼ kc;f CCH4 KC
CCH4 $KC
(9)
where CH2 is the concentration of hydrogen in the gas
(mol.m3) and Kc is the concentration-based equilibrium
constant (mol.m3). The backward reaction term in Eq. (9)
ensures that rc tends toward zero when the methane
decomposition reaction approaches thermodynamic equilibrium [31]. From Le Chatelier's principle, the CH4 conversion at
equilibrium increases at higher temperature because the reaction is endothermic and decreases at lower pressure
because methane decomposition causes a net increase in the
number of gas moles (Eq. (1)).
The Arrhenius equation for kc;f is:
kc;f ¼ koc;f exp Eac;f
!
(10)
RT
91204:6
K ¼ exp 13:2714 RT
(12)
Kinetics of non-catalytic methane decomposition
Non-catalytic methane decomposition occurs in the gas phase
inside the bubbles. The net non-catalytic reaction rate ; Rn
(mol CH4.m3.s1), per unit volume of gas is given by Ref. [10]:
Rn ¼Rn;f Rn;b ¼kn;f CCH4 n kn;f
CH2 2
CH2 2 CCH4 n1 ¼kn;f CCH4 n 1
KC
CCH4 $KC
(13)
where Rn;f and Rn;b are respectively the forward and backward
non-catalytic reaction rates, kn;f is the forward rate constant
(m3(n1).mol(1n).s1), and n is the order of the forward reaction with respect to methane. The Arrhenius equation for kn;f
is:
Ean;f
!
The experimental values of the forward pre-exponential
factor, koc;f (m.s1), and forward activation energy, Eac;f
kn;f ¼ kon;f exp (J.mol1), are given in Table 1 for Ni0$27Bi0.73 and Cu0$45Bi0.55
melts [11,16]. Experimental results reported in Parkinson et al.
[20] indicate that catalytic methane decomposition is very
likely reaction limited, not mass transfer limited, in the range
of relevant ag values. Hence, the net catalytic reaction rate (Eq.
(9)) does not need to be corrected by an effectiveness factor
that would otherwise account for mass transfer limitation.
This is explained in more detail in Section: 1. Evidence of reaction limited kinetics at the surface of bubbles in the Supplementary Material.
The concentration-based equilibrium constant KC is
related to K, the unitless reaction equilibrium constant, by
Ref. [31]:
Table 2 lists the numerical values of kon;f , Ean;f and n determined in Catalan and Rezaei [10] from experimental data reported in Keipi et al. [28].
o Pni
P iðgÞ
KC ¼ K
RT
(11)
where Po is the standard pressure (105 Pa), R is the universal
gas constant (8.314 m3.Pa.K1.mol1), T is the temperature (K),
P
and ni is the sum of the stoichiometric coefficients of the
Expansion of the differential material balance equation
Substituting the equations for the catalytic and non-catalytic
rates of methane decomposition, Eqs. (7), (9) and (13), in the
differential material balance equation, Eq. (5), yields:
CH2 2
CH2 2
þ kn;f CCH4 n 1 ag kc;f CCH4 1 CCH4 $KC
CCH4 $KC
pD2
a dL ¼ n_CH4 ;b dXCH4
4
iðgÞ
900e1200 C, K is approximated very accurately by Ref. [10]:
Table 1 e Kinetic parameters for catalytic methane
decomposition in Ni0·27Bi0.73 and Cu0·45Bi0.55 from Refs.
[11,16], respectively.
(15)
The concentrations of CH4 and H2 in the gas phase are
related to the methane conversion, XCH4 , as follows [31]:
CCH4 ¼
iðgÞ
gas components (CH4 and H2) taking part in methane
P
decomposition: ni ¼ 2 1 ¼ 1. In the temperature range of
(14)
RT
CH2 ¼
Tb
n_CH4 ;b ð1 XCH4 Þ P
T
V_ g;b ð1 þ εXCH4 Þ Pb
(16)
n_CH4 ;b qH2 þ 2XCH4
P
Tb
Pb
T
V_ g;b ð1 þ εXCH4 Þ
(17)
where Pb and Tb are the pressure (Pa) and temperature (K) at
the bottom of the melt, ε is the mole fraction of CH4 in the
reactor feed, qH2 is the mole ratio of H2 to CH4 in the reactor
feed, and V_ g;b is the volumetric flow rate of gas at the bottom of
the melt (m3.s1). With the ideal gas approximation, V_ g;b is
given by:
Parameter Units Value in Ni0$27Bi0.73 Value in Cu0$45Bi0.55
koc;f
Eac;f
a
m.s1
J.mol1
7.88 104
208,000
4.32 105a
222,000
Palmer et al. [16] calculated ko ¼ 3.7 108 ± 2.0 108 s1 in Rc ¼
ko expðEac;f =RTÞ CCH4
by assuming a bubble diameter
Table 2 e Kinetic parameters for non-catalytic methane
decomposition.
db ¼ 0.007 m. They kept the methane conversion below 10% to
minimize the backward reaction in their experiments. Therefore,
koc;f is related to their pre-exponential factor ko by koc;f ¼ ko = ag
Parameter
with ag ¼ 6=db :
n
kon;f
Ean;f
Units
3(n1)
m
.mol
J.mol1
unitless
Value
(1n)
.s
1
1.4676 1010
284,948
1.0809
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n_CH4 ;b þ n_H2 ;b þ n_I;b RTb
V_ g;b ¼
Pb
(18)
In Eq. (18), n_H2 ;b and n_I;b are the inlet flow rates (mol.s1) of
H2 and inert components (e.g., argon) at the bottom of the
reactor. Substitution of Eq. (18) in Eqs. (16) and (17) gives:
P
n_CH4 ;b ð1 XCH4 Þ
CCH4 ¼ n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ RT
CH2 ¼ n_CH4 ;b qH2 þ 2XCH4
P
n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ RT
(19)
(20)
Substituting Eqs. (19) and (20) in Eq. (15) and further rearrangement yields the expanded differential material balance
equation of the LMBR:
(
ag kc;f ð1 XCH4 Þ
P
n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ RT
#n )
ð1 XCH4 Þ
P
þ kn;f ,
n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ RT
(
)
2
n_CH4 ;b qH2 þ 2XCH4
P
1
n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þð1 XCH4 Þ,KC RT
dXCH4 a pD2
¼
dL
4
"
(21)
Before Eq. (21) can be solved, the terms ag and a must be
written in terms of XCH4 and P. This is done in the next two
sections.
Specific gas-liquid interfacial area
For a distribution of bubble sizes, the specific interfacial area
ag (m2 (interfacial area).m3 (gas)) is related to the volume-
Nevertheless, these measurement techniques and correlations for initial bubbles sizes (as they leave the sparger
orifice or nozzle) are not relevant for measuring or predicting
bubble sizes in the main part of bubble columns. This is
because bubbles interact with each other as they rise, especially at conditions of high superficial gas velocities relevant to
industrial operation. Consequently, the size of the majority of
bubbles throughout most of the column depends mostly on
the balance between the rates of bubble coalescence and
breakup. These rates, in turn, depend on the superficial gas
velocity and the liquid properties [30]. To the authors’
knowledge, no published study has reported bubble size
measurements in molten metal at conditions of high superficial gas velocities in regions where bubble coalescence and
break up are taking place.
The present study uses the experimentally-based correlation of Akita and Yoshida [30] to predict dvs in LMBRs (Eq. (23)).
These researchers used a photographic technique to measure
bubble sizes in square columns having cross sections of
7.7 cm 7.7 cm, 15 cm 15 cm, and 30 cm 30 cm. The experiments consisted of injecting air or oxygen in water,
30e100% glycol solutions, methanol, or carbon tetrachloride.
The bubble sizes were measured at a height of 150 cm from the
bottom of the column in the region of bubble coalescence and
breakup. They were found to be independent of the device
used to inject the gas in the column (perforated plate, porous
gas plate, or single orifice gas sparger).
dvs
¼ 26ðNBo Þ0:50 ðNGa Þ0:12 ðNFr Þ0:12
D
(23)
The dimensionless Bond number (NBo ), Galilei number
(NGa ), and Froude number (NFr ) are defined as follows:
surface mean bubble diameter dvs (m), also called the Sauter
mean diameter, by:
NBo ≡
gD2 rl
s
(24)
6
ag ¼
dvs
NGa ≡
gD3
n2l
(25)
(22)
In other words, dvs is the diameter of a bubble having the
same volume/surface area ratio as the entire distribution.
Optical and photographic techniques are not applicable
to the measurement of bubble sizes in opaque liquids such
as molten metals. Techniques based on differential pressure
[32] and acoustical [33,34] measurements have been reported for determining the size of bubbles as they detach
from a submerged orifice in various liquid metals (tin, lead,
copper, mercury, and silver). Sun et al. [32] found that helium bubble diameters experimentally measured in molten
tin at temperatures between 400 and 700 C were in relatively good agreement (average deviations of only 4.97%)
with the empirical correlation of Jamialahmadi et al. [35].
Interestingly, this correlation was based on experiments
carried out with water, wateralcohol, and waterglycerol
mixtures. This indicates that under the circumstances of
the experiments carried out by Sun et al. [32], the detaching
bubble sizes in molten metal could be acceptably predicted
by correlations developed with fluids having different
properties (particularly much lower density and surface
tension).
NFr ≡
jg
ðgDÞ1=2
(26)
where nl is the liquid kinematic viscosity (m2.s1), s is the
surface tension (N.m1), and jg is the superficial gas velocity
(m.s1).
Akita and Yoshida [30] found that their correlation was in
agreement with experimental data from an earlier study [36]
where wider variations of system properties, column size, and
superficial gas velocity were tested. Combining the results of
their two articles, the dvs correlation was experimentally
validated for the ranges NBo; exp ¼ 7.98 102 e 4.85 104,
NGa; exp ¼ 6.25 106 e 1.79 1012, and NFr ¼ 8 104 e
1.35 101.
The superficial gas velocity is given by:
jg ¼
4V_ g
pD2
(27)
where V_ g is the actual volume flow rate of the gas (m3.s1),
which depends on the local pressure and methane conversion
in the reactor:
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Pb
T
V_ g ¼ V_ g;b ð1 þ εXCH4 Þ
Tb
P
(28)
Substituting V_ g;b from Eq. (18) in Eq. (28) gives:
RT
V_ g ¼ n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ
P
(29)
Next, Eq. (29) is substituted in Eq. (27) to yield:
jg ¼
4 n_CH4 ;b þ n_H2 ;b þ n_I;b ð1 þ εXCH4 Þ RT
2
P
pD
(30)
Eq. (30), in combination with Eqs. (22)e(26), is used to relate ag
to XCH4 and P.
The dvs correlation of Eq. (23) applies to the parts of the
column where bubble sizes are determined by the balance
between the rates of bubble coalescence and breakup. This
correlation, however, does not apply to the initial bubbles in
the first few centimeters above the injector, where the local
volume-surface mean bubble diameter d*vs (m) depends on the
orifice diameter do (m) and the gas velocity through the orifice
uo (m.s1). For initial bubbles, Akita and Yoshida [30] found the
following correlation:
d*vs
uo
¼ 1:88 pffiffiffiffiffiffiffiffi
do
gdo
!1=3
(31)
Andreini et al. [33] also found that the size of the initial
bubbles in liquid tin, lead, and copper depended on do and uo .
Nevertheless, for reactor design, which is the focus of the
present article, Eq. (23) applies unless the reactor is very short.
Predicting dvs requires estimating liquid metal properties
(density, viscosity, surface tension, and liquidus temperature).
There is only a very limited amount of experimental data for
the physical properties of NieBi and CueBi alloys in the
7553
literature. Therefore, several models were utilized to estimate
the properties of these alloys as a function of temperature
from the properties of their constituent pure liquid metals.
This is explained in detail in Section: 2. Estimation of properties of liquid metal alloys in the Supplementary Material.
Fig. 2A shows dvs predicted by Eq. (23) as a function of jg for
a base case consisting of liquid Ni0$27Bi0.73 in a 0.18-m diameter column at 1000 C, and for variations on this base case.
Increasing jg causes dvs to progressively decrease. Reducing
the column diameter by a factor of 3 down to 0.06 m results in
dvs increasing by approximately 40%. Raising the temperature
to 1200 C has a minimal effect on dvs . Changing the liquid
alloy to Cu0$45Bi0.55 increases dvs by approximately 11%.
For simplicity, dvs will be called the mean bubble diameter
from now on in this article. It is important to remember,
however, that dvs is a measure of the specific gas-liquid
interfacial area through Eq. (22) rather than an actual bubble
diameter.
Gas holdup in molten metals
Kataoka and Ishii [26] developed a correlation for gas holdup
that was experimentally corroborated in molten metal baths
[27]. If the liquid metal is not flowing in and out of the reactor,
a is given by:
a¼
jþ
jg
g
¼
þ
þ
C0 jg þ Vgj C0 jg þ Vgj
(32)
where Vgj is the void-fraction weighted mean drift velocity
(m.s1) and C0 is the distribution parameter which depends on
the column geometry. For a round column,
qffiffiffiffiffiffiffiffiffiffiffi
C0 ¼ 1:2 0:2 rg rl
(33)
Fig. 2 e Effects of superficial gas velocity, reactor diameter, temperature, alloy type, pressure, and gas composition on (A) the
volume-surface mean bubble diameter and (B) the reactive surface area per unit volume of reactor. The base case
corresponds to D ¼ 0.18 m, T ¼ 1000 C, Alloy ¼ Ni0·27Bi0.73, P ¼ 15 bar, and Gas ¼ CH4. The legend indicates the parameters
that change from the base case.
7554
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
The dimensionless superficial gas velocity, jþ
g , and the
þ
, are defined by:
dimensionless drift velocity, Vgj
jþ
g ≡ jg
þ
≡ Vgj
(34)
1=4
sgDr r2l
Vgj
sgDr r2l
(35)
1=4
þ
depends on the viscosity
The method for evaluating Vgj
number, Nm , and the dimensionless hydraulic equivalent
diameter of the vessel, D*H , which are defined as follows:
Nm ≡ ml
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rl s s=ðgDrÞ
(36)
1=2
Cu0$45Bi0.55, this situation occurs when the column diameter is
larger than 65 mm, the temperature is above 800 C, the
pressure ranges from 1 to 80 bar, and the gas is a mixture of
CH4 and H2. In Catalan and Rezaei [10], we reported graphically how the superficial gas velocity, pressure, temperature,
and phase compositions affect the gas holdup in a molten
metal bath.
In order to use the above equations to calculate a in LMBRs,
jg and rg must be expressed as functions of XCH4 and P. This
has already been done for jg in Eq. (30). The equation relating
rg to XCH4 and P is developed next.
In accordance with the ideal gas approximationc, the gas
density is given by:
rg ¼
D
D*H ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s=ðgDrÞ
(37)
PMg
RT
(41)
where Mg (kg.mol1) is the molar mass of the gas:
CH2 MH2 þ CCH4 MCH4 þ CI MI
CH2 þ CCH4 þ CI
where ml is the liquid dynamic viscosity (Pa.s).
When jþ
g [0:5, a significant amount of liquid recirculation
Mg ¼
þ
occurs in the melt, and Vgj
is given by either Eqs. (38a), (38b), or
In Eq. (42), MH2 , MCH4 and MI are respectively the molar
masses of H2 (0.002016 kg.mol1), CH4 (0.01604 kg.mol1), and
an inert component (if present). The inert component concentration is given by Ref. [31]:
(38c):
þ
¼ 0:0019D*H
Vgj
0:809 rg rl
0:157
Nm 0:562 for Nm 2:2 103 and D*H
30
(38a)
þ
¼ 0:030 rg rl
Vgj
þ
¼ 0:92 rg rl
Vgj
0:157
0:157
(43)
(38b)
(38c)
Mg ¼
Conversely, when jþ
g ≪0:5, liquid recirculation is less and
the experimental data are adequately predicted by the drift
flux correlations of Ishii [37] for bubbly and churn-turbulent
flow:
pffiffiffi
þ
¼ 2ð1 aÞ1:75 for bubbly flow
Vgj
(39a)
pffiffiffi
þ
Vgj
¼ 2 for churn turbulent flow
(39b)
qH2 þ 2XCH4 MH2 þ ð1 XCH4 ÞMCH4 þ qI MI
1 þ qH2 þ XCH4 þ qI
(44)
Then, substituting Eq. (44) in Eq. (41) provides the desired
relationship between rg ; XCH4 , and P:
rg ¼
qH2 þ 2XCH4 MH2 þ ð1 XCH4 ÞMCH4 þ qI MI P
RT
1 þ qH2 þ XCH4 þ qI
(45)
Reactive surface area in catalytic molten metals
Eqs. (38) and (39) can be combined to account for the whole
range of jþ
g as follows [27]:
o
n
þ
þ
þ
þ Vgj
¼ Vgj
ðEq:39Þexp Ajþ
ðEq:38Þ 1 exp Ajþ
Vgj
g
g
P
n_CH4 ;b qI
n_CH4 ;b þ n_H2 ;b þ n_I;b Rð1 þ εXCH4 Þ RT
where qI is the mole ratio of inert component to methane in
the reactor feed. Substitution of Eqs. (19), (20) and (43) in Eq.
(42) gives:
Nm 0:562 for Nm 2:2 103 and D*H 30
for Nm 2:2 103 and D*H 30
CI ¼ (42)
(40)
þ
with A ¼ 1.39 so that expðAjþ
g Þ ¼ 0.5 at jg ¼ 0.5.
Hibiki et al. [27] used neutron radiography to visualize nitrogen bubbles and measure the gas holdup in a PbeBi melt
contained in a rectangular tank for a range of nitrogen flow
rates and liquid metal heights. Their measurements of gas
holdups in the N2ePbeBi system and those of Saito et al. [38] in
the N2eGa system were adequately predicted by Eq. (40). The
relative error on gas holdup was approximately ±30% for
102 jþ
g 2 and 0 a 0:32.
The reactive surface area per unit volume of reactor is the
product ag a, which has units of m2 (interfacial area).m3
(reactor). Fig. 2B shows that ag a increases considerably with jg
at low values of jg and more moderately at larger jg values.
Reducing the column diameter or pressure and increasing the
temperature cause ag a to decrease. Changing the gas from CH4
to H2 decreases ag a due to the lower density of H2. The reactive
surface area of Cu0$45Bi0.55 is lower than that of Ni0$27Bi0.73 by
approximately 15% when used in a bubble column under
same operating conditions.
The product ag a depends on longitudinal position along the
reactor because both ag and a depend on L. The reactive surface area for the entire column, Ar (m2), is the integral of ag a
from the bottom to the top of the melt:
þ
It is noteworthy that Vgj
becomes independent of the col-
umn diameter when D*H 30 (Eqs. (38b) and (38c)) because the
gas flow is no longer affected by the vessel walls except in
close proximity to the walls. For molten Ni0$27Bi0.73 or
c
The error introduced by the ideal gas approximation is
inconsequential compared to the relative error of ±30% on gas
holdup predictions.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Ar ¼
ð Vt
0
pD2
ag adV ¼
4
ð Lt
0
6
a dL
dvs
(46)
Applications
The system of Eqs. (5) and (6) was numerically solved in
MATLAB 2019b using the function ode23s with given reactor
dimensions and temperature, as well as inlet feed flowrate
and composition, to obtain the profiles of CH4 conversion, gas
holdup, pressure, superficial gas velocity, and mean bubble
diameter along the height of the reactor. Reactive surface
areas (Eq. (46)) were calculated using the trapezoidal numerical integration method in MATALB 2019b.
Experimental validation of the catalytic liquid metal bubble
reactor model
Upham et al. [11] reported methane decomposition experiments in a 0.03-m diameter bubble column filled with molten
Ni0$27Bi0.73 at 1040 and 1065 C. The depth of the melt above
the injection point was varied up to 1.15 m. The feed gas
consisted of an 80/20 mol% mixture of methane/argon with a
standard feed flow rate 10 cm3 (std).min1 and a pressure of
200 kPa. The composition of the reactor outlet was measured
with a mass spectrometer to calculate the methane conversion. The experimental methane conversions reported in
Ref. [11] are compared with the predictions of our catalytic
LMBR model in Fig. 3A.
The agreement between the catalytic LMBR model predictions and the experimental data is excellent at both temperatures. It is crucial to emphasize that this agreement was
obtained without the need to adjust any parameter. This is in
contrast to previous models in Refs. [11,25] where the bubble
size had to be adjusted so that the predicted methane conversions would match the experimental measurements. The
model presented here is a significant advance because it calculates the specific gas-liquid interfacial area and the gas
holdup from the superficial gas velocity and the properties of
the liquid and gas phases. This allows using the model to
design LMBRs at larger scales with an increased degree of
confidence (Sections: Reactor scale-up and Reactor design
optimization).
Fig. 3A also shows that when only the non-catalytic
methane decomposition reaction is included in the model,
the predicted CH4 conversions are much reduced at all reactor
heights. This demonstrates the benefit of catalytic molten
metals for decreasing the required reactor size for a given CH4
conversion.
Fig. 3B shows the profiles of the pressure and superficial
gas velocity along the catalytic melt at 1040 C. The pressure
decreases almost linearly from bottom to top, as expected
from Eq. (6). At any height, the pressure gradient is given by
ðrl aDrÞg. Since the gas holdup remains 1% (see
Fig. 3C) and the density of the liquid alloy is constant, the
pressure gradient is nearly constant along the reactor height.
The superficial gas velocity increases almost linearly with
respect to reactor height in the bottom half of the melt and
faster than linearly in the top half. This is consistent with Eq.
7555
(30) which shows that jg is proportional to ð1 þXCH4 Þ=P. As the
height increases, XCH4 increases (Fig. 3A) and P decreases
(Fig. 3B).
Fig. 3C shows the model predictions for the mean bubble
diameter, gas holdup, and reactive surface area versus reactor
height at 1040 C for the 1.15-m long bubble reactor in Ref. [11].
As bubbles rise, dvs decreases and a increases from the bottom
(dvs;b ¼ 0.809 cm, ab ¼ 0.00280) to the top (dvs;t ¼ 0.697 cm,
at ¼ 0.00917). This is because the superficial gas velocity jg
increases with height (Fig. 3B). The reactive surface area increases faster than linearly with respect to height. This is
consistent with Eq. (46) but contrasts with the assumption
made by Upham et al. [11] that the reactive surface area is a
linear function of height. According to Eq. (46), the slope of the
curve relating Ar to height (dAr =dLÞ is proportional to 6a=dvs .
As height increases, the slope increases because a increases
and dvs decreases (Fig. 3C).
Rahimi et al. [14] carried out methane decomposition experiments in a 1000-mm long, 22-mm inner diameter bubble
column reactor containing a 660-mm layer of molten
Ni0$27Bi0.73 under a 260-mm layer of molten salt (NaBr). The
purpose of the salt layer was to reduce metal losses and carbon contamination by the metal. The experiments investigated the effects of melt temperature and volume flow rate of
the inlet gas (pure methane) on CH4 conversion. Our coupled
catalytic LMBR model was also able to reproduce adequately
the experimental results of Rahimi et al. [14], as shown in
Fig. 4. In particular, the decreasing trend of CH4 conversion
with increasing inlet flow rate was reproduced by accounting
for the non-catalytic methane conversion in the injection tube
submerged in the melt. More details on the interpretation of
Fig. 4 are provided in Section: 3. Comparison with the experimental data of Rahimi et al. in the Supplementary Material.
Reactor scale-up
Industrial scale catalytic LMBRs should be able to produce
several thousand Nm3.h1 of hydrogen. A practical implementation may consist of several vertical tubes filled with
catalytic liquid metal and externally heated by burners in a
furnace (Fig. 5). The burners could be fed with natural gas or
a fraction of the produced hydrogen. Burners may be located
in the bottom of the furnace as illustrated in Fig. 5, or they
may be placed in rows on the sidewalls of the furnace. The
tops of the tubes connect to a production manifold where
the carbon layer could be continuously removed (e.g,
skimmed mechanically).
Fig. 6A shows the profiles of predicted CH4 conversion and
superficial gas velocity in a single tube measuring 0.10 m in
internal diameter and 2.11 m in length that could be part of a
multitubular reactor for industrial use. The tube is filled with
Ni0$27Bi0.73, and the conditions inside the tube are isothermal
at 1050 C. Pure CH4 is fed at the bottom of the tube at a rate of
0.0858 mol.s1 and a pressure of 17.8 bar. The CH4 conversion
reaches 80.0% at the top of the tube. The superficial gas velocity increases with height: it starts at 6.75 cm s1 at the
bottom of the tube and reaches 13.4 cm.s1 at the top. The
Froude number, which is proportional to the superficial gas
velocity for a given tube diameter (Eq. (26)), is purposefully
limited to 0.135 at the top of the tube, since this is the
7556
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Fig. 3 e Profiles of (A) predicted and experimental methane conversions at 1040 and 1065 C, (B) predicted superficial gas
velocity and pressure at 1040 C, and (C) predicted mean bubble diameter, gas holdup, and reactive surface area at 1040 C
versus reactor height in a bubble column filled with Ni0·27Bi0.73. The melt was 1.15 m in length and 0.03 m in diameter. The
feed was 10 cm3 (std).min¡1 of 80 mol% methane and 20 mol% argon mixture at 200 kPa. Markers are experimental data
from Upham et al. [11]. Lines represent model predictions.
maximum NFr value at which the correlation for dvs has been
experimentally validated [30,36].
Fig. 6B shows the predicted pressure and gas holdup for the
same tube. The pressure decreases almost linearly with height
down to 16.2 bar at the tube outlet, which corresponds to a
pressure drop of 1.6 bar. The gas holdup increases from 11.2%
to 13.1% from the bottom to the top of the tube, which is well
below the maximum a values of 30e32% for the dvs and a
correlations [27,30]. The gas holdups in the scaled up tube are
much larger than in the laboratory sized column where they
remained below 1% (Fig. 3C). This is due to the higher superficial gas velocities in the scaled up tube.
Fig. 6C shows the profiles of the predicted mean bubble
diameter and reactive surface area. The mean bubble diameter decreases from 3.16 mm at the bottom to 2.91 mm at the
top of the tube, which is considerably smaller than in the
laboratory sized column of Upham et al. [11] where dvs ranged
from 7 to 8 mm (Fig. 3C). This is due to a combination of a
larger tube diameter and higher superficial gas velocity in the
scaled-up case, and is consistent with the trends shown in
Fig. 2A. The reactive surface area increases slightly faster
than linearly with respect to height because the local reactive
surface area per unit volume of reactor, 6a=dvs , rises from
213 m2.m3 at the bottom to 270 m2.m3 at the top of the
tube.
The average time tr (s) taken by bubbles to rise from the
bottom of the melt to a height L is:
tr ¼
ðL
1
dL ¼
0 vg
ðL
a
0 jg
dL
(47)
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
7557
close to the reactor inlet. The average residence time of the
gas in the tube is tR ¼ 2.39 s.
Reactor design optimization
The optimization presented here is based on a H2 production
rate of 10,000 Nm3.h1, which corresponds to a small-scale
hydrogen plant by recent standards [39]. The feed is pure
methane and the selectivity to H2 is assumed to be 100% as this
value was nearly achieved under laboratory conditions [11].
The capital cost of the reactor depends greatly on the total
volume of the melt given by:
Fig. 4 e Predicted and experimental methane conversions
versus inlet gas flow rate and melt temperature in a bubble
column measuring 22 mm in inner diameter, 1000 mm in
length, and containing layers of 660 mm of Ni0·27Bi0.73 and
260 mm of NaBr. The feed was pure methane. Markers are
experimental data from Rahimi et al. [14]. Lines represent
model predictions.
Vmelt ¼
Nt pD2 Lt
4
(49)
where Nt is the number of tubes. The other important contributors to the capital cost are the melt composition, the
tube material of construction, and the pressure inside the
tubes. Molten Cu0$45Bi0.55 is cheaper than Ni0$27Bi0.73 as the
price of Cu is roughly half that of Ni [40]. The tube material of
construction is selected based on the melt composition,
temperature, and pressure. Higher pressures require a larger
tube wall thickness, thus increasing the cost of tubes.
_ H2 ðtotalÞ, was held
The total outlet flow rate of hydrogen, n
constant at 122.4 mol s1, which is equivalent to 10,000
Nm3.h1. Because methane decomposition produces 2 mol of
H2 for each mole of CH4 that reacts, the total feed flow rate of
_ CH4 ðtotalÞ, is given by:
methane, n
_ CH4 ðtotalÞ ¼
n
_ H2 ðtotalÞ
n
2 XCH4 ;t
(50)
where XCH4 ;t is the methane conversion at the top of the tubes.
The number of tubes and the inlet flow rate of methane in
each tube, n_CH4 ;b , are related by:
Nt ¼
_ CH4 ðtotalÞ
n
n_CH4 ;b
(51)
Combining Eqs. (49)e(51) yields:
Vmelt ¼
Fig. 5 e Conceptual design of a multi-tubular LMBR heated
by burners.
where vg ¼ jg =a represents the local average rise velocity
(m.s1) of the gas through the melt. The average residence
time of the gas in the tube is obtained by setting the upper
bound of the integral to L ¼ Lt in Eq. (47).
ð Lt
_ H2 ðtotalÞ pD2 Lt
n
8 n_CH4 ;b XCH4 ;t
(52)
Several optimization problems were solved for specified
tube diameters, melt temperatures, CH4 conversions, and
melt compositions.
-
Tube diameters: 0.075, 0.1, 0.125, 0.15, and 0.2 m
Melt temperatures: 950, 1000, 1050, 1100, and 1150 C
CH4 conversions: 60, 70, 80, and 90%
Melts: Ni0$27Bi0.73 and Cu0$45Bi0.55
(48)
For each problem, the optimization procedure consisted of
finding the tube length, Lt , the feed molar flow rate per tube,
_ CH4 ;b , and the feed pressure, Pb , that minimized Vmelt (Eq. (52))
n
Fig. 6D shows how vg and tr increase as a function of the
reactor height. The increase in gas rise velocity is more pronounced near the bottom of the tube because jg is proportional
to 1 þ XCH4 (Eq. (30)), and methane decomposition is faster
using the fmincon function with the sqp algorithm in MATLAB
2019b. The number of tubes was then calculated with Eq. (51).
The optimizations were subject to the following
constraints:
tR ¼
0
a
dL
jg
7558
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Fig. 6 e Predicted profiles of (A) CH4 conversion and superficial gas velocity, (B) gas holdup and pressure, (C) reactive surface
area and mean bubble diameter, and (D) average rise velocity and rise time of the gas in a single tube. Tube
diameter ¼ 0.10 m, length ¼ 2.11 m, melt ¼ Ni0·27Bi0.73, temperature ¼ 1050 C, feed ¼ 0.0858 mol s¡1 of pure CH4 at 17.8 bar.
102 jþ
g 2
(53a)
at 0:30
(53b)
NFr 0:135
(53c)
0:1 Lt ðmÞ 150
(53d)
_ CH4 ;b mol:s1 150
0:001 n
(53e)
1 Pb ðbarÞ 150
(53f)
The upper limits on dimensionless superficial gas velocity
(Eq. (53a)), gas holdup (Eq. (53b)), and Froude number (Eq. (53c))
correspond to the ranges of experimental validation of the
correlations for a and dvs . The allowable ranges for tube length
(Eq. (53d)), methane feed to each tube (Eq. (53e)), and bottom
pressure (Eq. (53f)) were chosen to provide a wide search space
for the optimization variables.
Table 3 summarizes the optimized designs that examine
the effects of tube diameter, temperature, methane conversion, and melt composition. These designs are easily adjustable for different rates of H2 production by only changing the
number of tubes. For example, optimum designs for 100,000
Nm3.h1 of H2 would require ten times the number of tubes
indicated in Table 3. All other parameters would stay the
same. In all designs, the Froude number is at the upper limit of
its allowed range (0.135) at the top of the tubes. This maximizes the superficial gas velocity, which in turn minimizes
7559
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Table 3 e Optimal reactor designs (minimum melt volumes) for producing 10,000 Nm3.h¡1 of H2 in a multitubular catalytic
LMBR fed by pure CH4.
Design
Melt composition
Methane conversion (%)
Temperature ( C)
Tube inner diameter (m)
Tube length (m)
Number of tubes
Melt volume (m3)a
Methane feed rate per tube (mmol.s1)
Inlet pressure (bar)
Outlet pressure (bar)
Bottom superficial gas velocity (cm.s1)
Top superficial gas velocity (cm.s1)
Bottom gas holdup (%)
Top gas holdup (%)
GHSV (Nm3 CH4.m3 catalyst.h1)b
Gas residence time (s)
Reactor duty (MW)
Average heat flux (kW.m2)
Bond number (x 103)
Galilei number (x 1010)
Inlet Froude number (x 102)
Outlet Froude number (x 102)
a
b
A
B
Ni0$27Bi0.73
80
80
1050 1050
0.075 0.10
2.15
2.11
1830 891
17.4
14.8
41.8
85.8
17.84 17.81
16.17 16.19
5.83
6.75
11.6
13.4
10.5
11.2
12.2
13.1
405
482
2.62
2.39
7.52
7.52
8.1
12.7
1.39
2.47
8.90
21.1
6.80
6.82
13.5
13.5
C
D
E
F
G
H
I
J
K
L
M
80
1050
0.125
2.07
510
13.0
150
17.79
16.20
7.57
15.0
11.9
13.9
554
2.22
7.52
18.1
3.85
41.2
6.84
13.5
80
1050
0.15
2.04
323
11.6
237
17.76
16.21
8.31
16.4
12.5
14.6
622
2.09
7.52
24.2
5.54
71.2
6.85
13.5
80
1050
0.20
1.98
157
9.8
487
17.72
16.23
9.78
18.9
13.4
15.9
750
1.90
7.52
38.5
9.86
169
6.98
13.5
80
950
0.10
9.42
1987
147
38.5
14.24
6.71
3.50
13.4
8.3
12.5
47.5
13.0
6.76
1.15
2.46
16.4
3.53
13.5
80
1000
0.10
4.49
1242
43.8
61.6
14.70
11.19
5.65
13.4
10.4
12.9
162
5.31
7.13
4.07
2.46
18.7
5.71
13.5
80
1100
0.10
1.02
675
5.39
113
22.97
22.19
7.18
13.4
11.6
13.2
1324
1.14
7.91
36.7
2.47
23.6
7.25
13.5
80
1150
0.10
0.51
526
2.11
146
29.90
29.51
7.50
13.4
11.8
13.4
3393
0.57
8.31
98.5
2.47
26.1
7.57
13.5
60
1050
0.10
1.03
333
2.70
306
52.03
51.26
8.23
13.4
14.0
15.9
3636
1.35
8.03
74.5
2.47
21.1
8.31
13.5
70
1050
0.10
1.46
518
5.94
169
31.13
30.02
7.58
13.4
12.7
14.5
1393
1.78
7.74
32.6
2.47
21.1
7.66
13.5
90
1050
0.10
3.25
2073
52.9
32.8
9.10
6.53
5.05
13.4
9.1
11.3
117
3.44
7.35
3.5
2.47
21.1
5.10
13.5
Cu0$45Bi0.55
80
1050
0.10
1.54
886
10.7
86.3
17.44
16.28
6.93
13.4
11.6
13.4
667
1.77
7.52
17.5
2.18
14.8
7.00
13.5
Includes the volume of gas bubbles dispersed in the liquid metal.
The catalyst volume is the liquid metal volume excluding the gas phase.
the mean bubble size and maximizes the gas holdup (while
keeping it below its allowed maximum of 30%). The net effect
is to maximize the reactive surface area, as shown by Eq. (46).
A potential source of error when predicting dvs in liquid
metals with Eq. (23) is that Akita and Yoshida [30] developed
this dimensionless number correlation with liquids having
significantly lower densities and surface tensions than those
of NieBi and CueBi alloys. Nevertheless, the catalytic LMBR
designs in Table 3 correspond to ranges of the Bond, Galilei,
and Froude numbers (NBo ¼ 1.39 103 e 9.86 103,
NGa ¼ 8.90 1010 e 1.69 1012 and NFr ¼ 3.53 102 e
1.35 101) that are contained within the experimental ranges
reported by Akita and Yoshida [30,36] (NBo; exp ¼ 7.98 102 e
4.85 104, NGa; exp ¼ 6.25 106 e 1.79 1012 and
NFr; exp ¼ 8 104 e 1.35 101).
Effect of tube diameter
The first set of optimizations (Designs A e E) was carried out
for five different tube diameters between 0.075 and 0.20 m
using Ni0$27Bi0.73 at a constant temperature of 1050 C and a
CH4 conversion of 80%. The melt volume (Eq. (52)), which includes all the tubes, decreases from 17.4 to 9.8 m3 as the tube
diameter increases from 0.075 to 0.20 m. Fig. 7A shows this
trend graphically. Meanwhile, the required number of tubes
decreases from 1830 to 157 and the tube length decreases from
2.15 to 1.98 m.
Fig. 7A provides additional insights on the effect of tube
diameter on reactor performance. As the tube diameter increases, the reactive surface area comprising all the tubes,
NtAr, where Ar is given by Eq. (46), stays nearly constant.
Therefore, the main reason why the melt volume grows with
decreasing tube diameter is that the ratio of reactive surface
area to melt volume increases. This ratio is expressed
mathematically as Nt Ar =Vmelt and is obtained by combining
Eqs. (46) and (49):
Nt Ar 1
¼
Vmelt Lt
ð Lt
0
6
a dL
dvs
(54)
Eq. (54) shows that Nt Ar =Vmelt is simply the average of the
ratio 6a=dvs taken over the length of a tube. Fig. 7B reveals
that the average mean bubble diameter decreases with
increasing tube diameter, while the average gas holdup increases. These trends, combined with Eq. (54), explain that
the ratio of reactive surface area to melt volume increases
with tube diameter.
Fig. 7C shows that the top pressure stays nearly constant at
16.2 bar as the tube diameter varies. In order for the catalytic
and non-catalytic reaction rates to remain above zero at the
top of the tubes, the outlet pressure must stay well below the
pressure at which the equilibrium conversion is 80%. This
pressure, which is denoted Pressure @XCH4, eq on Fig. 7C, is
20.7 bar at 1050 C. The bottom pressure is the sum of the top
pressure and the hydrostatic pressure exerted by the melt,
which is itself proportional to the tube length. Because the
tube length decreases slightly as the tube diameter increases,
the bottom pressure also decreases slightly from 17.8 to
17.7 bar.
The above results favor larger diameter tubes as they
reduce the melt volume, number of tubes, and pressure drop;
however, material and heat transfer limitations must also be
considered in the selection of tube diameter. The average heat
flux from the burners through the tube wall, qav (W.m2), depends on tube diameter and must be kept in a reasonable
range.
7560
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Fig. 7 e Effect of tube diameter on (A) melt volume, reactive surface area, and ratio of reactive surface area to melt volume, (B)
average mean bubble diameter and gas holdup, and (C) pressures and tube length. Melt ¼ Ni0·27Bi0.73, T ¼ 1050 C, H2
production ¼ 10,000 Nm3.h¡1, XCH4;t ¼ 0.80.
QR
qav ¼
pNt DL
(55)
In Eq. (55), QR is the reactor duty (W), which was determined with the process simulation software Aspen Plus. In
our calculations of QR and qav reported in Table 3, the temperature of the methane entering the bottom of the tube was
assumed to be 800 C and D was taken as the inner tube
diameter. Table 3 shows that qav based on the inner tube
surface increases with increasing tube diameter but stays
within the normal range of steam methane reformers (
150 kW.m2) [39,41] even for diameters as high as 0.20 m.
Effect of temperature
The second set of optimizations (Designs FeI) was carried out
for four additional temperatures between 950 C and 1150 C
with molten Ni0$27Bi0.73 in tubes measuring 0.10 m in diameter
at a CH4 conversion of 80% (Table 3). Fig. 8A shows that the
total reactive surface area decreases nearly exponentially
with increasing temperature. This is consistent with the
exponential increase of catalytic reaction rates with temperature implied by Arrhenius law. On the other hand, the melt
volume decreases somewhat faster than exponentially with
respect to temperature, from 147 m3 at 950 C to 2.11 m3 at
1150 C. This is because the ratio of reactive surface area to
melt volume, which is itself related to the ratio of gas holdup
to mean bubble diameter by Eq. (54), increases with temperature. Indeed, Fig. 8B shows that the average gas holdup increases and the average mean bubble diameter decreases with
increasing temperature.
Fig. 8C shows that the top pressure increases with temperature as the pressure at which the equilibrium conversion
is 80% (Pressure @XCH4, eq) increases from 10.5 bar at 950 C to
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
7561
Fig. 8 e Effect of temperature on (A) melt volume, reactive surface area, and ratio of reactive surface area to melt volume, (B)
average mean bubble diameter and gas holdups, and (C) pressures and tube length. Melt ¼ Ni0·27Bi0.73, tube
diameter ¼ 0.10 m, H2 production ¼ 10,000 Nm3.h¡1, XCH4;t ¼ 0.80.
37.1 bar at 1150 C. The increase in temperature causes the
optimum tube length to drop considerably from 9.4 to 0.51 m,
and the number of tubes to decrease from 1987 to 526 tubes.
The optimum bottom pressure, which is the sum of the top
pressure and the hydrostatic pressure exerted by the melt,
increases with increasing temperature. The pressure drop, i.e.
the difference between the bottom and top pressures, decreases with increasing temperature.
Effect of methane conversion
The third set of optimizations (Designs J e L) was carried out at
three additional CH4 conversions (60%, 70% and 90%) with
molten Ni0$27Bi0.73 in tubes measuring 0.10 m in diameter at a
constant temperature of 1050 C (Table 3). Higher CH4 conversions require larger reactive surface areas and, therefore,
larger melt volumes and smaller gas hourly space velocities
(GHSV) (Fig. 9A). The melt volume at 60% conversion is relatively small (2.70 m3) but increases to 52.9 m3 at 90%
conversion.
As conversion increases from 60 to 90%, the pressure that
corresponds to the equilibrium of the methane decomposition
reaction (denoted Pressure @XCH4, eq on Fig. 9B) decreases
from 65.4 to 8.6 bar. Hence, the optimum pressure at the top of
the tubes must also decrease in order to stay below the
equilibrium pressure and, thus, allow for a positive reaction
rate. The pressure drop in the tubes (difference between bottom and top pressures) depends on the height of the tubes
though the hydrostatic pressure exerted by the melt. Because
the tube length increases with methane conversion, the
pressure drop also increases.
The reactor duty decreases from 8.03 MW to 7.35 MW as the
CH4 conversion increases from 60% to 90%. This is because
less methane needs to be heated from 800 C to the reaction
temperature of 1050 C as conversion increases. The heat
consumed by the reaction is the same at all CH4 conversions
since the hydrogen production rate is fixed at 10,000 Nm3.h1.
The inner area of the tubes (i.e., the area for heat transfer to
the melt) is proportional to the number of tubes and their
7562
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Fig. 9 e Effect of CH4 conversion on (A) melt volume, reactive surface area, and GHSV, (B) pressures and tube length, and (C)
average heat flux and inner tube area. Melt ¼ Ni0·27Bi0.73, T ¼ 1050 C, tube diameter ¼ 0.10 m, H2 production ¼ 10,000
Nm3.h¡1.
length, both of which increase with methane conversion. As a
result, the average heat flux decreases from 74.5 kW.m2 to
3.5 kW.m2 as the CH4 conversion increases from 60% to 90%
(Fig. 9C).
Effect of catalytic molten metal alloy composition
The effect of catalyst composition can be determined by
comparing Designs B and M, which use different molten metal
alloys but the same temperature (1050 C), tube diameter
(0.10 m), and CH4 conversion (80%). The required melt volume
is smaller for Design M (10.7 m3 of Cu0$45Bi0.55) than for Design
B (14.8 m3 of Ni0$27Bi0.73), which represents a volume reduction
of 38%. This is consistent with the experimental observation of
Palmer et al. [16] that Cu0$45Bi0.55 is a somewhat more active
catalyst than Ni0$27Bi0.73. The lower melt volume for Cu0$45Bi0.55
vs. Ni0.27Bi0.73 results in shorter tubes (1.54 m vs 2.11 m) but
nearly the same number of tubes (886 tubes vs 891 tubes). The
inlet pressure is somewhat lower for Cu0$45Bi0.55 (17.4 bar vs
17.8 bar). Given that copper costs about half the price of nickel
per tonne of metal [40], Cu0$45Bi0.55 is expected to result in a
cheaper initial catalyst load for the reactor. Nevertheless, other
factors such as potential differences in the quality of the produced carbon, tendencies of Ni and Cu to leave the reactor with
the carbon product, and melt compatibility with tube construction materials should be investigated.
Discussion
Variations from optimum designs
The optimized reactor designs listed in Table 3 minimize the
melt volumes but sometimes lead to large internal pressures
that may cause excessive creep rates in metal tubes subjected
to very high temperatures. The combination of internal
pressure, tube diameter, and tube wall thickness determines
the hoop stress. At very high temperatures, excessive hoop
stresses lead to creep damage that limits the tube lifetime
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
[39,41,42]. Special alloys containing high concentrations of Ni
and Cr (e.g., HP-40 grade Centralloy G4852 Micro R [43]) are
used for steam methane reformer tubes exposed to temperatures up to 1050 C [44]. The lifetime of a tube exposed to a
given temperature and hoop stress can be estimated by means
of a Larson-Miller plot specific to the tube material of construction, as shown in Section: 4. Influence of creep on tube
life in the Supplementary Material.
Our analysis is based on a design life of 100,000 h, a
maximum tube temperature of 1050 C, and HP-40 grade
Centralloy G4852 Micro R as the tube material. When the tube
diameter is set at 0.10 m and the maximum internal pressure
is 52 bar (Design J), the tubes require a wall thickness of
55 mm. This is much thicker than tubes used in steam
methane reformer, which typically have wall thicknesses
between 10 and 25 mm [42]. The internal pressure should be
below 33 bar to keep the wall thickness under 25 mm. This
prediction does not account for how the lifetime of the tubes
may be affected by chemical compatibility issues between the
molten metal alloy and the tube material. This important
aspect is outside the scope of this article.
In the case of Design J (Ni0$27Bi0.73, T ¼ 1050 C, D ¼ 0.10 m,
CH4 conversion ¼ 60%), it is worth finding what penalty must
be paid in terms of increased melt volume if the maximum
internal pressure is limited to 30 bar in order to bring the tube
wall thickness to a reasonable value (21 mm). Hence, Design J
was re-run with the additional constraint Pb 30 bar, resulting in a new design named J’. The two designs are compared in
Table 4.
Table 4 e Effect of constraining the pressure to Pb 30 bar
on optimal reactor design (minimum melt volumes) for
producing 10,000 Nm3.h¡1 of hydrogen with a CH4
conversion of 60% in a multitubular reactor fed by pure
methane.
Design
Melt composition
Temperature ( C)
Methane conversion (%)
Tube inner diameter (m)
Tube length (m)
Number of tubes
Melt volume (m3)a
Methane feed rate per tube (mmol.s1)
Inlet pressure (bar)
Outlet pressure (bar)
Bottom superficial gas velocity (cm.s1)
Top superficial gas velocity (cm.s1)
Bottom gas holdup (%)
Top gas holdup (%)
GHSV (Nm3 CH4.m3 catalyst.h1)b
Gas residence time (s)
Reactor duty (MW)
Average heat flux (kW.m2)
Bond number (x 103)
Galilei number (x 1010)
Inlet Froude number (x 102)
Outlet Froude number (x 102)
a
b
J
Ni0$27Bi0.73
1050
60
0.10
1.03
333
2.70
306
52.03
51.26
8.23
13.4
14.0
15.9
3636
1.35
8.03
74.5
2.47
21.1
8.31
13.5
J0
0.81
582
3.70
175
30.00
29.39
8.19
13.4
13.1
14.9
2618
1.02
8.03
54.3
2.47
21.1
8.27
13.5
Includes the volume of gas bubbles dispersed in the liquid metal.
The catalyst volume is the liquid metal volume excluding the gas
phase.
7563
Unsurprisingly, the new optimum inlet pressure is the
maximum allowed, i.e., 30 bar. The melt volume increased
from 2.70 m3 (Design J) to 3.70 m3 (Design J’). The number of
tubes increased from 333 to 582, while the tube length
decreased from 1.03 to 0.81 m. The pressure drop decreased
from 0.77 to 0.61 bar. Hence, the main penalty for limiting the
maximum pressure is a higher reactor volume and catalyst
load (þ37% in this particular example).
Comparison to previous catalytic LMBR design from the
literature
Von Wald et al. [29] investigated the optimal dimensions of a
catalytic LMBR to produce 10.4 kt.a1 (i.e., 13,400 Nm3.h1) of
hydrogen as part of a methane pyrolysis energy system which
also included heat exchange and separation equipment. The
optimization minimized the H2 price required to achieve a net
present value of zero at a specified internal rate of return. The
reactor consisted of a single bubble column containing
Ni0$27Bi0.73. The reactor inlet pressure, temperature, and
methane conversion were fixed at 30 barg, 1100 C, and 90%,
respectively. The optimum column radius and height were
determined to be approximately 2.2 m and 1 m, respectively.
When the coupled catalytic LMBR model developed in the
present article is applied to the column dimensions determined by Von Wald et al., the resulting methane conversion is
only 78.7%, which is well below the specified 90%. In order to
reach 90% methane conversion with a column measuring 2.2m in radius, our model predicts that the height should extend
to 24.5 m, which is considerably larger than the height of 1 m
calculated by Von Wald et al. These discrepancies are
explained by differences in assumptions between the two approaches. Most importantly, our kinetic model is constrained
by thermodynamic equilibrium (Eqs. (9) and (13)). By contrast,
Von Wald et al. included no equilibrium constraint and stated
that the continuous removal of carbon from the system may
allow the maximum mole fraction of H2 to exceed its equilibrium value. In Section: 5. Chemical equilibrium equation of the
Supplementary Material, we show that the chemical equilibrium of methane decomposition is independent of the concentration of carbon and, therefore, is not affected by carbon
removal. Von Wald et al. also contended that improvements in
bubble column design made by Farmer et al. [25] can allow H2
mole fractions to exceed equilibrium values. Farmer et al.
modelled a reactor containing a membrane around the melt to
selectively remove nearly all H2 before the bubbles left the
melt. Selective H2 removal shifts the equilibrium of the CH4
decomposition reaction to the right to enable the CH4 conversion to reach nearly 100%. In contrast, non-membrane LMBRs,
such as the reactor modelled by Von Wald et al., do not
selectively remove the products of the reaction along their
length and therefore necessitate the inclusion of thermodynamic equilibrium limitations to be realistically sized.
Another reason for differences in predicted heights between our model and that of Von Wald et al. is related to the
reactor pressure drop. At 1100 C, the pressure corresponding
to 90% equilibrium conversion is 11.5 bar. Since the inlet
pressure at the bottom of the column is 31 bar (30 barg), the
CH4 conversion cannot reach 90% until such a height where
the pressure drops below 11.5 bar. Indeed, our model predicts
7564
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
that the outlet pressure is 11.2 bar at the height of 24.5 m
where the methane conversion reaches 90%d. By contrast, Von
Wald et al. assume that the pressure stays constant in the
reactor. Moreover, our model accounts for the rate of noncatalytic methane decomposition, unlike that of Von Wald
et al.
Besides the thermodynamic equilibrium constraint, the
reactor pressure drop, and accounting for the non-catalytic
methane decomposition, there are also significant differences between the two models regarding the treatment of
hydrodynamics. Von Wald et al. assume that the changes in
bubble diameter and gas holdup with height only depend on
CH4 conversion according to the following two equations:
dv ¼ dvb ð1 þ XCH4 Þ1=3
(56)
av ¼ avb ð1 þ XCH4 Þ
(57)
where dv and av are respectively the bubble diameter and the
gas holdup in their model, and the subscript b refers to the
bottom of the column. Eqs. (56) and (57) are tantamount to
assuming that the mean bubble volume and the gas holdup
are only affected by the stoichiometry of the methane
decomposition reaction, and they neglect the effect of
decreasing gas pressure with height. By contrast, our model
calculates the mean bubble diameter and gas holdup at any
height using respectively Eq. (23) and Eq. (32), which have been
validated experimentally. Fig. 10 shows that Eq. (56) in the
model of Von Wald et al. and Eq. (23) in our model predict
opposing trends for the change of bubble diameter versus
height. Moreover, Eq. (32) in our model predicts substantially
smaller increases in gas holdup with respect to height than Eq.
(57) from Von Wald et al. [29].
Comparison to steam methane reformer
In this section, the operational characteristics of a multitubular catalytic LMBR are compared to those of a steam
methane reformer (SMR). Both designs use a large number of
externally fired tubes. The SMR tubes are packed with porous
catalyst pellets. A typical modern reformer for hydrogen
production operates with a molar H2O/CH4 ratio of 1.8e2.5, a
pressure of 30 bar, and a syngas outlet temperature of 920 C.
The produced syngas contains 71.1 mol% H2, 17.6 mol% CO,
4.6 mol% CO2 and 6.7 mol% CH4 on a water free basis [44]. The
reformer is followed by a water gas shift (WGS) reaction system where nearly all the CO reacts with H2O to produce more
H2 and CO2 (Eq. (3)). As a result, a typical syngas composition
at the exit of the shift reactor is 75.4 mol% H2, 18.9 mol% CO2
and 5.7 mol% CH4 on a water free basis. By contrast, methane
decomposition produces no CO2, and the gas outlet of the
catalytic LMBR consists of a mixture of H2 and unreacted CH4.
When pure methane is fed to a catalytic LMBR, the stoichiometry of methane decomposition (Eq. (1)) indicates that the
mole fraction of H2 in the outlet gas, yH2 , is related to the
methane conversion by:
yH2 ¼
2XCH4;t
1 þ XCH4 ;t
(58)
Eq. (58) implies that a methane conversion of 60.5% is
required in the catalytic LMBR to obtain yH2 ¼ 0.754, i.e., the
same hydrogen mole fraction produced by a modern SMR
followed by WGS. If a higher hydrogen purity is required, it can
be increased to 99.999 mol% by using a pressure swing
adsorption (PSA) unit downstream of the LMBR or WGS. The
PSA off-gas may then be used as fuel in the burners of the
LMBR or SMR.
Table 5 e Comparison of operating conditions for
producing hydrogen by SMR in tubular catalytic
reformers and by methane decomposition in a catalytic
LMBR.
Design
H2 production rate (Nm3/h)
P (bar)
Outlet gas temperature ( C)
Tube diameter (m)
Tube heated length (m)
Number of tubes
Fig. 10 e Comparisons of mean bubble diameter and gas
holdup vs height calculated by our model and that of Von
Wald et al. [29]. Tube diameter ¼ 0.10 m, length ¼ 2.1 m,
melt ¼ Ni0·27Bi0.73, temperature ¼ 1050 C,
feed ¼ 0.0858 mol s¡1 of pure CH4 at 17.8 bar.
d
Although the column diameter of 4.4 m results in Bond and
Galilei numbers that are well above their validated ranges for
prediction of mean bubble diameters, our model still reliably
demonstrates the effect of thermodynamic equilibrium on the
required reactor height.
Reactor volume (m3)
GHSV (Nm3 CH4.m3
catalyst.h1)
Pressure drop (bar)
Average heat fluxc
(kW.m2)
a
b
c
Modern tubular
reformer
LMBR (Case
J0 )a
2500e300,000 [39]
30 [39]
700e950 [39,44,45]
0.07e0.20 [39,41]
6e13 [39,41]
12 to more than 800
[39,41,46]
30b [39]
1345b [39]
100,000
30
1050
0.10
0.81
5820
0.6e0.7 [46]
45e150 [39,41]
0.61
54.3
37
2618
The number of tubes and the reactor volume of Case J0 in Table 4
were multiplied by a factor of 10 to account for the tenfold increase in the hydrogen production rate.
Based on 100,000 Nm3/h hydrogen, reformer outlet gas at 875 C
and 31 bar, 0.10-m inner tube diameter, 13-m long tubes, and 294
tubes [39].
Based on m2 of inner tube surface.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
Table 5 compares the operating conditions of a catalytic
LMBR producing 100,000 Nm3.h1 hydrogen with 60% CH4
conversion to those of modern tubular steam methane reformers. The LMBR gas pressure, tube diameter, pressure
drop, and average heat flux fall within the range of SMR
operating conditions. In contrast, the LMBR uses a higher
outlet gas temperature, a shorter tube length, and a larger
number of tubes. Considering the high density of the molten
metal alloy, the lower tube length is beneficial for limiting the
pressure drop in the LMBR. For the given hydrogen production
rate, the volumes of the catalytic LMBR and tubular reformer
are remarkably similar (37 vs 30 m3). The GHSV of the catalytic
LMBR is nearly double that of the tubular reformer because
methane decomposition (Eq. (1)) consumes twice as much
methane compared to SMR þ WGS (Eq. (4)) to produce a given
amount of hydrogen.
7565
with higher temperature and lower methane conversion. The
pressure drop, which is nearly proportional to tube length,
decreased with increasing temperature and decreasing
methane conversion.
Changing the liquid metal composition from Ni0$27Bi0.73 to
Cu0$45Bi0.55 resulted in a smaller melt volume for a given
application because the kinetics of catalytic methane
decomposition is faster with the copper-based catalyst. This is
encouraging since the price of copper is roughly half that of
nickel on a mass basis. All the multitubular LMBR designs
presented in this article (Tables 3 and 4) can be easily adapted
to different H2 production rates by changing the number of
tubes proportionally. The optimum values of the tube length,
bottom pressure, and top pressure are independent of the H2
production rate.
Declaration of competing interest
Conclusions
The catalytic LMBR model developed in this article was successfully validated at various temperatures with previously
reported experimental data for methane decomposition in
laboratory-sized bubble columns filled with molten
Ni0$27Bi0.73. The model was applied to design industrial multitubular LMBRs producing 10,000 Nm3.h1 of hydrogen. The
designs were optimized by minimizing the volume of the
molten metal for various combinations of tube diameters,
melt temperatures, methane conversions and molten metal
compositions. The optimization always resulted in the Froude
number reaching its maximum allowed value of 0.135 at the
top of the tubes. For a given tube diameter, maximizing the
Froude number is equivalent to maximizing the superficial gas
velocity. Large superficial gas velocities increase the specific
gas-liquid interfacial area and the gas holdup, thus favouring
both the catalytic and non-catalytic reaction rates.
Mean bubble sizes were smaller and gas holdups were
higher in industrial applications compared to laboratory
LMBRs due to faster superficial gas velocities and larger tube
diameters. For a given melt temperature and methane conversion, the lowest melt volume was achieved for the largest
tube diameter (20 cm) because the ratio of reactive area to
melt volume increased with tube diameter. Larger tube diameters resulted in slightly shorter tube lengths and a much
lower number of tubes. Nevertheless, the maximum tube
diameter is limited by material and heat transfer considerations in practical applications.
The minimum melt volume dropped somewhat faster than
exponentially as a function of melt temperature for a given
tube diameter and methane conversion. Both the optimum
tube length and number of tubes decreased greatly with
increasing temperature.
Specifying a larger methane conversion increased the melt
volume considerably for a given tube diameter and melt
temperature. Both the optimum tube length and number of
tubes increased with increasing methane conversion.
The outlet pressure at the top of the tubes always stayed
below the pressure corresponding to the equilibrium CH4
conversion in order to keep the reaction rate positive. The
optimum inlet pressure at the bottom of the tubes increased
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank the Natural Sciences and
Engineering Research Council of Canada (Discovery Grant
Program) for providing funding for this work.
Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.ijhydene.2021.12.089.
Nomenclature
Ar
ag
Ci
C0
D
DH
do
dvs
d*vs
Eac;f
Ean;f
g
reactive surface area for the entire column or tube
(m2)
specific interfacial area per unit of gas volume (m2
(interface).m3 (gas))
concentration of component i in the gas phase
(mol.m3)
distribution parameter (unitless)
reactor or tube inner diameter (m)
dimensionless hydraulic equivalent diameter of the
reactor (m)
orifice diameter (m)
volume-surface mean bubble diameter in most of
the reactor (m)
volume-surface mean bubble diameter just above
the injector (m)
activation energy of the forward catalytic reaction
(J.mol1)
activation energy of the forward non-catalytic
reaction (J.mol1)
gravitational acceleration (9.81 m.s2)
7566
jg
kc;f
koc;f
kn;f
kon;f
K
KC
L
M
n
n_
NBo
NFr
NGa
Nm
Nt
P
qav
QR
rc
R
Rc
Rn
T
tr
tR
uo
V
Vgj
Vmelt
V_
vg
X
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 7 ( 2 0 2 2 ) 7 5 4 7 e7 5 6 8
superficial gas velocity (m.s1)
forward rate constant of the catalytic reaction
(m.s1)
forward pre-exponential factor of the catalytic
reaction (m.s1)
forward rate constant of the non-catalytic reaction
(m3(n1).mol(1n).s1)
forward pre-exponential factor of the non-catalytic
reaction (m3(n1).mol(1n).s1)
reaction equilibrium constant based on fugacity
ratios (unitless)
reaction equilibrium constant based on
concentrations (mol.m3)
length (m)
molar mass (kg.mol1)
order of the forward non-catalytic reaction with
respect to CH4 (unitless)
mole flow rate (mol.s1)
Bond number (dimensionless)
Froude number (dimensionless)
Galilei number (dimensionless)
viscosity number (dimensionless)
number of tubes in multitubular reactor
pressure (Pa)
average heat flux (W.m2)
reactor duty (W)
catalytic reaction rate per unit of interfacial area
(mol CH4.m2.s1)
universal gas constant (8.314 J.mol1.K1)
catalytic reaction rate per unit volume of gas (mol
CH4.m3.s1)
non-catalytic reaction rate per unit volume of gas
(mol CH4.m3.s1)
temperature (K)
average rise time of the gas to a height L in the melt
(s)
average residence time of the gas in the melt (s)
gas velocity through orifice (m.s1)
volume (m3)
void-fraction weighted mean drift velocity (m.s1)
total volume of melt (m3)
volumetric flow rate (m3.s1)
local average rise velocity of the gas (m.s1)
conversion (fraction)
Greek letters
a
gas holdup (fraction)
ε
mole fraction of CH4 in the reactor feed
mole ratio of component i (H2 or inert) to CH4 in the
qi
reactor feed
ml
dynamic viscosity of liquid metal (Pa.s or kg.m1.s1)
kinematic viscosity of liquid metal (m2.s1)
nl
stoichiometric coefficient of component i (unitless)
ni
Dr
density difference between liquid metal and gas
(kg.m3)
r
density (kg.m3)
s
surface tension of the liquid metal (N.m1)
sh
hoop stress (MPa)
Subscripts
b
bottom
c
catalytic
f
forward
g
gas
I
inert component
l
liquid metal
n
non-catalytic
t
top
w
wall
Superscripts
þ
dimensionless
o
standard
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