Post on 09-Apr-2023
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7
Avai lab le a t www.sc iencedi rec t .com
j ourna l homepage : www.e lsev ier . com/ loca te /he
Modeling and analysis of calcium bromide hydrolysis
Steven A. Lottes, Robert W. Lyczkowski*, Chandrakant B. Panchal, Richard D. Doctor
Energy Systems Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA
a r t i c l e i n f o
Article history:
Received 22 April 2008
Received in revised form
10 July 2008
Accepted 24 July 2008
Available online 30 October 2008
Keywords:
Computational fluid dynamics
Droplets
Bubbles
Water splitting
Surface reaction
* Corresponding author. Energy Systems DiviIL 60439, USA. Tel.: þ1 630 252 5923; fax: þ1
E-mail address: rlyczkowski@anl.gov (R.W0360-3199/$ – see front matter ª 2008 Interndoi:10.1016/j.ijhydene.2008.07.127
a b s t r a c t
The main focus of this paper is the modeling, simulation, and analysis of the calcium bromide
hydrolysis reactor stage in the calcium–bromine thermochemical water-splitting cycle for
nuclear hydrogen production. One reactor concept is to use a spray of calcium bromide into
steam, in which the heat of fusion supplies the heat of reaction. Droplet models were built up
in a series of steps incorporating various physical phenomena, including droplet flow, heat
transfer, phase change, and reaction, separately. Given the large heat reservoir contained in
a pool of molten calcium bromide that allows bubbles to rise easily, using a bubble column
reactor for the hydrolysis appears to be a feasible and promising alternative to the spray
reactor concept. The two limiting cases of bubble geometry, spherical and spherical-cap, are
considered in the modeling. Results for both droplet and bubble modeling with COMSOL
MULTIPHYSICS� are presented, with recommendations for the path forward.
ª 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights
reserved.
1. Introduction (Br2), in contrast to the UT-3 cycle, which employed two iron
The goal of the Nuclear Hydrogen Initiative [1] is to develop
economical carbon-free routes to the production of hydrogen
in connection with the Generation IV Nuclear Energy Systems
Initiative for the development of a proliferation-resistant,
sustainable, nuclear-based energy supply system that will
optimize the delivery of both heat and electricity to thermo-
chemical or electrolysis water-splitting cycles.
The thermodynamic basis for a three-stage calcium–
bromine (Ca–Br) water-splitting cycle being investigated at
Argonne National Laboratory builds upon pioneering work
done on the four-stage University of Tokyo UT-3 thermo-
chemical cycle [2–6]. However, the current Ca–Br cycle is
a marked departure in four ways:
� Molten calcium bromide (CaBr2) is employed, rather than
a solid monolith, to overcome the heat and mass transfer
limitations of the UT-3 process.
� Electrolysis (or possibly a plasma-chemical stage) will be
used for the recovery of HBr as hydrogen (H2) and bromine
sion, Argonne National La630 252 1342.. Lyczkowski).
ational Association for H
beds that swung semicontinuously between the oxide and
bromide states.
� All the steps in the hybrid Ca–Br cycle will be continuous.
� The process is now a ‘‘hybrid’’ in that it requires electricity.
The hydrogen will be produced at much lower temperatures
than those required by the UT-3 cycle and at much higher
molar concentrations.
The efficiency of the UT-3 process for hydrogen production
has been analyzed ‘‘.to be less than 13%, and is likely to be
even lower when additional process uncertainties are
accounted for.’’ [7]. A later study that identified problems in
the UT-3 cycle estimated the efficiency to be 15% [8].
The Ca–Br cycle being investigated is a hybrid cycle for
hydrogen production employing both heat and electricity. It
is particularly attractive because nearly one-half the required
thermodynamic energy for water splitting is delivered as Gener-
ation IV nuclear reactor heat at around 1023 K (750 �C), and it is
envisioned that this temperature will facilitate the engineering
of materials when compared to other, higher-temperature
boratory, Building 362, Room C349, 9700 S. Cass Avenue, Argonne,
ydrogen Energy. Published by Elsevier Ltd. All rights reserved.
Nomenclature
Roman symbols
a spherical-cap base radius, m
cp specific heat, J/(kg K)
cpg gas specific heat, J/(kg K)
cpi specific heat of species i
cpl liquid specific heat¼ 459 J/(kg K)
cps solid specific heat¼ 445 J/(kg K)
Cj Reynolds-averaged molar concentration of species
j, mol/m3
Cj molar concentration of species j, mol/m3
Cd drag coefficient
C31 k-3 model constant¼ 1.44
C32 k-3 model constant¼ 1.92
Cm k-3 model constant¼ 0.09
d diameter, m
Deffj effective diffusivity for species j, m2/s
Dj molecular binary diffusivity for species
j¼ 0.26� 10�4 m2/s for H2O and HBr and
0.2� 10�8 m2/s for CaO and CaBr2
E activation energy, J/mol
F body force, Pa/m
g gravity vector, m/s2
g acceleration due to gravity, m2/s
DH heat of reaction, kJ/mol
Hs smoothed Heaviside function flc1hs in COMSOL�I identity tensor
k molecular thermal conductivity, W/(m K)
k turbulent kinetic energy, m2/s2
keff effective thermal conductivity, W/(m K)
kg gas thermal conductivity, W/(m K)]
kl liquid thermal conductivity, W/(m K)
ks solid thermal conductivity, W/(m K)
k* first-order reaction rate coefficient, 1/s
K reaction rate coefficient in Eq. (26), m4=ðs molH2OÞn unit normal vector on a boundary
p pressure, Pa
pd droplet pressure, Pa
pg gas domain pressure, Pa
p0 exit pressure, Pa
q00 heat flux vector, W/m2
_q00g heat flux from the gas into the interface, W/m2
_q00r ¼ � _r00CaBr2DH heat flux sink due to reaction at the
gas–liquid interface, W/m2
_r00CaBr2reaction rate of calcium bromide, mol/(m2 s)
R universal gas constant¼ 8314 J/(mol K)
Rb (Equivalent) bubble radius, m
Rc spherical-cap radius of curvature, m
Rd droplet radius, m
Re Reynolds number, Re¼ rud/m
Reb Reynolds number based on equivalent radius Rb,
Reb¼ 2ruRb/m
sH half size of smoothing interval for Hs, K
t time, s
t unit tangential vector along a boundary
T Reynolds-averaged temperature, K
T temperature, K
Tm melting point temperature, K
T0 uniform inflow temperature, K
Tsurf droplet–gas boundary surface temperature, K
u Reynolds-averaged velocity vector, m/s
u velocity vector, m/s
u0 uniform inflow velocity vector, m/s
uf fluid velocity vector, m/s
ug gas velocity vector, m/s
ul Reynolds-averaged liquid velocity vector, m/s
u velocity magnitude, m/s
uN bubble terminal rise velocity, m/s
us friction velocity in turbulent wall function in Eq.
(30)¼ (sw/r)0.5, m/s
X_CaO mole fraction of CaO in liquid
X_HBr mole fraction of HBr in bubble
Greek symbols
dTm magnitude of the temperature interval that
incorporates the heat of fusion into the specific
heat, K
dw wall distance, m
dz unit vector in the axial (z) direction
3 turbulent dissipation rate, m2/s3
q spherical-cap half angle, deg
k von Karman constant¼ 0.4
lfs heat of fusion, J/mol
m viscosity, Pa s
mf fluid viscosity, Pa s
mg gas viscosity, Pa s
mt eddy (turbulent) viscosity, Pa s
n kinematic viscosity, m2/s
nj stoichiometric coefficient of species j
nT turbulent kinematic viscosity, m2/s
r density, kg/m3
rf fluid density, kg/m3
rg gas density, kg/m3
s surface tension, N/m
sw wall shear, Pa
s3 k-3 model constant¼ 1.3
sk k-3 model constant¼ 1.0
sD turbulent Schmidt number¼ 1.0
sT turbulent Prandtl number¼ 1.0
V gradient
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74156
cycles. The work at Argonne National Laboratory focuses on
two special aspects of the hybrid Ca–Br cycle:
� Determining the feasibility of H2 generation from HBr.
� Determining the feasibility of CaBr2 hydrolysis in a contin-
uous mode, recognizing that there is a eutectic phase with
calcium oxide (CaO) [9].
The three reactions in the hybrid Ca–Br cycle are given by:
1) CaBr2 hydrolysis with HBr formation (z1013–1050 K)
CaBr2 þ H2O / CaO þ 2HBr (1)
2) CaBr2 regeneration with oxygen (O2) formation (z850–
1050 K)
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4157
CaO þ Br2 / CaBr2 þ 1=2O2 (2)
3) Br2 regeneration – PEM electrochemical (z333 K)
2HBr / Br2 þ H2 (3)
Eq. (1), the hydrolysis stage, is endothermic, 6H¼ 182.8 kJ/
mol at 1015 K (742 �C), with an equilibrium constant of
5.39� 10�6 [10]. Therefore, means of supplying sufficient heat
in the reaction process and shifting the reaction to the right to
obtain adequate conversion must be identified. The purpose
of this analysis is to present computer model options to
accomplish CaBr2 hydrolysis in a continuous flow system.
Two options are considered for bringing the reactants
together continuously. In the first option, molten CaBr2 is
sprayed with or into a high-temperature steam environment.
The CaBr2 droplets act as heat carriers for the reaction, as well
as being one of the reactants. In the second option, steam
bubbles are sparged through a pool of molten CaBr2. Given the
much greater heat-carrying capacity of a pool of molten
CaBr2, the second option for the CaBr2 reactor appears to be
the more promising of the two and is being pursued as the
primary candidate in the system design. Both of these
candidate systems consist of a continuous fluid medium as
one reactant and a dispersed fluid in the form of droplets or
bubbles as the other reactant. Modeling of these systems is,
therefore, similar but with the roles of the gas and liquid
phases reversed.
2. Droplet models
Droplet models were built up in a series of steps incorporating
various physical phenomena, including droplet flow, heat
transfer, phase change, and reaction, separately. The goal of
droplet modeling is to develop an understanding of droplet
processes in terms of time scales, boundary layer thicknesses,
and extents of droplet wakes as they move through a flow field
of steam and possible reaction products. Results from droplet
modeling can be used to guide reactor design in determining
requirements for spray characteristics. The results can also be
used to develop higher level models for a spray reactor that
combines the effects of tens to hundreds of thousands of
droplets.
Fig. 1 – Mesh and boundary types for CaBr2 droplet flow in
steam.
2.1. Droplet flow
The initial COMSOL� [11] model built was for a water droplet
moving through air. Once a flow computation with this model
was successful and results compared with known solutions
for flow around a sphere [12], the material properties were
changed to model a molten CaBr2 droplet in an isothermal
flow through steam at 1073 K (800 �C). Material properties,
including thermodynamic and transport properties, were
obtained from literature sources reporting experimental
results where possible [10], and the data set was completed
using the Aspen process simulation software material prop-
erties library and estimation module [13].
To simplify the problem of solving for the flow field around
a droplet, the domain is defined such that the droplet is
stationary and the gas flows past it. As a further simplification,
the gas flow approaching the droplet is taken to have
a uniform velocity, u0. The flow field can be assumed to be
modeled as axisymmetric through the center of the droplet in
the direction of u0 even when droplet motion is not aligned
with the gravitational field.
With these assumptions, the domain is defined to be two-
dimensional, axisymmetric through the droplet center, with
the other boundaries to the side, front, and bottom of the
droplet. If the side boundary is sufficiently far from the droplet
center, conditions there have little effect on the flow in the
vicinity of the droplet. Alternatively, the droplet can be taken
to be one of an array of droplets equally spaced in a plane of
hexagonal cells. An array of non-interacting droplets may be
modeled using a single droplet with a slip-symmetry
boundary condition at the midpoint between droplet centers.
The model domain with a typical finite-element mesh used in
the computations is shown in Fig. 1. A high density of mesh
elements is generated near the droplet surface in order to
resolve flow structures in that region. Boundary types are also
indicated in the figure.
The governing equations for a droplet moving through
a gas flow are the same for both the droplet domain, d, and the
gas domain, g, for an observer at the center of the droplet. The
continuity equations (conservation of mass) are given by
VðriuiÞ ¼ 0 (4)
where u is the velocity vector and r is the density. The
subscript i denotes the droplet, d, or gas domain, g. The Navier–
Stokes equations (conservation of momentum) are given by
riðuiVÞui ¼ Vh� piIþ mi
�Vui þ ðVuiÞT
�iþ rig; (5)
where m is the viscosity, p is the pressure, g is the gravity vector,
and the superscript T signifies transpose. The boundary
conditions for the external gas flow field and droplet–gas
interface are specified as follows. At the inflow boundary, the
Fig. 2 – Isothermal 1073 K (800 8C) steam flow field around
a CaBr2 droplet: Left: 1 mm diameter, Re [ 15 at 3 m/s;
Right: 2 mm diameter, Re [ 511 at 50 m/s.
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74158
gas velocity is taken to be uniform and parallel to the droplet
centerline: ug¼ ug0dz where ug0 is the specified gas velocity and
dz is a unit vector in the axial (z) direction. At the outflow
boundary, the gas velocity is assumed to be straight out of the
unit cell: tug¼ 0. The total gas stress normal to the exit plane is
set equal to the exit pressure, p0:
nh� pgIþ mg
�Vug þ
�Vug
�T�i
n ¼ �p0; (6)
where t and n are unit tangential and normal vectors, respec-
tively, at a point on the boundary, and I is the identity tensor.
At symmetric boundaries, the normal component of the gas
velocity is zero, nug¼ 0, and the shear stress is zero:
th� pgIþ mg
�Vug þ
�Vug
�T�i
n ¼ 0: (7)
In addition to the symmetry conditions above, along the
axisymmetric boundary through the center of the droplet, the
radial coordinate is zero: r¼ 0.
At the droplet–gas boundary, the velocity is parallel to the
boundary, nu¼ 0, and continuous, tud¼ tug. In addition, the
shear stress is also continuous at the boundary:
th� pgIþ mg
�Vug þ
�Vug
�T�i
n ¼ th� pdIþ md
�Vud þ ðVudÞT
�i:
(8)
A series of simulations was run with droplet diameters
ranging from 100 mm to 2 mm and steam inflow velocities
ranging from 3 m/s to 100 m/s. The droplet Reynolds numbers
for these conditions ranged from 1.5 to 511. In the low Rey-
nolds number range, a strong recirculation zone in the wake
of the droplet would not be expected, as shown in the liter-
ature [12]. Fig. 2 shows isothermal 1073 K (800 �C) velocity
magnitude color plots in m/s around and within a molten
CaBr2 droplet at two different diameters and steam inlet
velocities. For Re¼ 15 on the left, a recirculation zone is not
seen in the wake. For Re¼ 511, on the right, a recirculation
zone is seen extending about 2 diameters downstream in the
droplet wake. In a flow with reaction at the droplet surface,
reaction products will tend to be swept into the wake.
Recirculation zones will tend to hold reaction products in
droplet wakes. The size of the wake also gives some idea of
required droplet spacing if droplet–droplet interactions are
undesirable.
1 Laboratory work at Argonne found the melting point for CaBr2
to be in the range 728–732 �C [10], but these results do not changethe conclusions of the modeling.
2.2. Heat transfer to a droplet in an external steamflow field
A next step in the development of the full reacting flow droplet
model is to add heat transfer from the external gas flow
domain to the droplet domain. Energy equations to be solved
are added to the equations in Section 2.1:
V�riuicpiTi � kiVTi
�¼ 0; (9)
where T is the temperature, cp is the specific heat, and k is the
thermal conductivity. The inflow boundary condition is taken
to be a uniform gas temperature, Tg0.
At the outflow boundary, thermal conduction of the gas in
the stream-wise direction normal to the exit plane is assumed
to be negligible:
n��kgVTg
�¼ 0: (10)
In this case the heat flux of the gas, q00g, at the exit boundary
is due to convection only:
nq00g ¼ n�
rgcpgugTg
�: (11)
At symmetric boundaries, the heat fluxes for both gas and
droplet domains are zero:
n�ricpiuiTi � kiVTi
�¼ 0: (12)
At the droplet–gas boundary, the temperature is taken to be
a known surface temperature, Tsurf. The effect of cooling of the
external steam flow field by a droplet surface that is at the
freezing point of CaBr2, Tsurf¼ 1033 K (760 �C)1, at a center-to-
center droplet spacing of four and eight diameters is shown in
Fig. 3. At smaller droplet spacing, the temperature of steam
leaving the domain is about 20 K lower than the incoming
flow, compared to the eight-diameter spacing where the
steam temperature drops less than 10 K. These results give an
indication of the droplet spacing where droplet–droplet
interactions become important in terms of their effect on the
surrounding flow field.
2.3. Droplet freezing
A state change (liquid to solid) model was developed for the
freezing of a CaBr2 droplet. Internal circulation is neglected;
Fig. 3 – Heat transfer to 200-mm CaBr2 droplets with 4- and
8-diameter center-to-center spacing at 1015 K (742 8C) with
an incoming 1073 K (800 8C) steam flow at 3 m/s, Re [ 1.5.
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4159
only conduction and phase change are considered. The energy
equation is given by:
rcpvTvt¼ VðkVTÞ: (13)
An assumed uniform internal droplet temperature,
T(r,0)¼ T0, is used for the initial condition. The boundary
condition at the droplet–gas boundary, Rd, is T(Rd,t)¼ Tsurf.
To avoid having to track the liquid–solid interface explicitly
in the solution space, the heat of fusion for CaBr2 was treated
via a temperature-dependent specific heat function, cp(T ),
over a small temperature range around the freezing point,
Tm¼ 1033 K (760 �C). Other material property differences
between liquid and solid were also accounted for via step
functions.
Using this approach, functions for the CaBr2 thermal
conductivity, k, and specific heat, cp, covering both the solid, s,
and liquid, l, states are given as:
kðTÞ ¼�
ks; T < Tm;kl; T � Tm;
(14)
cp ¼
8<:
cps T < Tm � dTm;
cpl þ lfsdTm; Tm � dTm < T � Tm;
cpl Tm < T:(15)
where subscript s refers to the solid, subscript l refers to the
liquid, lfs¼ 17.5 kJ/mol is the heat of fusion for calcium
bromide, and dTm¼ 1.0 K is the magnitude of the temperature
interval that incorporates the heat of fusion into the specific
heat.
The specific heat formulation given by Eq. (15) does not
have continuous derivatives at the bounds of the freezing
interval. Numerical stability in COMSOL� is improved by
using a built-in smoothed Heaviside function, Hs, having
a continuous first derivative. The specific heat formulation is
given by:
cpðTÞ ¼ cp;s þlfs
dTm
HsðT� Tm þ dTm ; sHÞ þ�
cp;f � cp;s �lfs
dTm
�
�HsðT� Tm; sHÞ;(16)
which has a smoothed transition in the interval �sH< 0< sH,
where 2sH¼ 0.1 K is the interval over which the temperature is
ramped. The simulations performed with droplet state change
did not couple the internal droplet state with the external gas
thermal field. A fixed-temperature boundary condition at the
droplet surface was used instead. For 100 mm droplets and
a surface temperature 40 K below the freezing point, the time
for the phase change was found to be around 1 ms, depending
on the initial temperature of the droplet. A more rigorous
approach would be to couple the droplet phase change to the
external gas field.
The droplet freezing models were run over a range of
different time steps. Variations of other solver options, such
as convergence tolerances, were also tested. The use of the
smoothed Heaviside functions having a continuous first
derivative allowed the solver to converge in most cases.
However, physically unrealistic undershoots and overshoots
in the temperature history at some radial positions as the
freezing front passed occurred for larger time steps. For a 100-
mm-diameter droplet with surface temperature conditions
that would yield a solidification time of approximately 1 ms,
a time step of around 1 ms was required to obtain a physically
realistic, smooth, monotonic temperature development
within the freezing droplet. Much larger time steps produced
unrealistic temperature overshoots and unrealistic tempera-
ture evolution curves.
Fig. 4 (top) shows the temperature evolution curves over
time for a 100-mm droplet with an initial uniform tempera-
ture 60 K above the freezing point and a surface temperature
40 K below the freezing point. Also plotted (bottom) is the
smoothed heat capacity for calcium bromide given by
Eq. (16). The time for the phase change front to reach the
center of the droplet is seen to be just over 1.0 ms. The figure
clearly shows that the temperature in the droplet interior
cannot drop below the freezing point until the freezing front
passes. As a consequence of the temperature being pinned at
the freezing point, higher temperature gradients result in the
solid between the liquid core and the surface producing
higher heat transfer rates than would result if pure
conduction without freezing occurred in the droplet interior.
This higher heat transfer rate through the solid zone
includes the heat of fusion. The effect is seen in the sharp
changes in slope of the temperature curves at the freezing
point.
3. Limitations of CaBr2 droplets as heatcarriers
Freezing of CaBr2 droplets (heat of fusion, lfs¼ 17.5 kJ/mol) can
supply only about 100(lfs/6H )¼ 9.6% of the endothermic heat
of reaction, 6H¼ 182.6 kJ/mol, for the hydrolysis stage. With
a CaBr2 liquid specific heat of 0.091 kJ/(mol K) and a range of
about 50 K between melting and boiling point, superheating
the CaBr2 liquid could add another 2% of the energy required
Fig. 4 – Top: Radial variation of temperature in a freezing CaBr2 droplet initially 60 K above a freezing point of 1033 K (760 8C)
with a droplet surface temperature of 993 K (720 8C) at various times. Bottom: plot of Eq. (16).
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74160
for the reaction. This can be compared with using the sensible
enthalpy of steam at 1073 K (800 �C) with a 100 K temperature
drop to supply heat for the reaction. The supplied heat would
be about 4.2 kJ/mol or only about 2.3% of the required heat.
This indicates why the earlier work with this hydrolysis
system, where only steam provided the heat of reaction in the
UT-3 cycle, proceeded at such a slow pace [2–8,10].
4. Bubble column reactor analysis andmodeling
The analysis in Section 3 indicates that running the hydrolysis
reaction in a molten CaBr2 bath, which would act as a much
larger heat reservoir, could be beneficial in supplying the
requiredheat of reaction. At Argonne, qualitative experimental
observations of molten CaBr2 in a quartz container with inert
helium bubbled through it revealed a behavior similar to that of
water or some other low-viscosity liquid at room temperature
[10]. The viscosity of CaBr2 at 1023 K (750 �C) is about
1.2� 10�3 Pa s [13], close to that of water at room temperature
(0.9� 10�3 Pa s). This low viscosity opened the possibility of
using a molten pool of CaBr2 as the heat reservoir for the
reaction, Eq. (1), together with steam sparging.
Using a bubble column for the hydrolysis reaction appears
to be a feasible and promising alternative to the spray reactor
concept. To identify the engineering challenges of this system,
bubble dynamics in the system must be understood, along
with the processes of heat and mass transfer at the bubble
surface where the reaction takes place.
The bubble rise velocity determines the bubble residence
time in a given height of a liquid-filled reactor. Bubbles are first
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4161
considered in isolation. They riseas a consequence of buoyancy.
Ultimately, buoyancy is balanced by drag forces, and the bubble
moves at a constant terminal velocity, uN. Many researchers
have investigated bubble dynamics [14–18]. For small bubbles at
low Reynolds numbers, bubbles remain spherical and the
terminal velocity is close to that of Stokes’s solution, which
corresponds to a drag coefficient on a bubble of Cd¼ 24/Re. This
relation is valid for solid spheres where the sphere is assumed to
be a rigid body, and the velocity of the liquid is zero with respect
to the surface of the sphere, the no-slip condition.
For gas bubbles rising in a liquid, the gas in the bubble
offers little resistance to tangential flow at the gas–liquid
interface, mg� ml, and consequently the drag is primarily
pressure drag (negligible skin friction). For laminar flow, the
bounds of free-slip and no-slip at the bubble surface place the
drag coefficient in the range 16/Re� Cd� 24/Re. Contami-
nants, which tend to accumulate at the rear surface of the
bubble, change the surface tension, and in addition, thermal
and other gradients along the surface can affect the force
balance, yielding some shear forces at the surface, a condition
between free-slip and no-slip [12]. The surface tension force
maintains the spherical shape of the bubble. As the bubble
size increases, the buoyancy force increases, and the
balancing pressure drag increases. Bubbles deform as the
inertial force approaches the surface tension force as char-
acterized by the Weber number, We ¼ 2rfu2NRb=s. Bubble
deformation is complex and unsteady, but roughly speaking,
it goes through an ellipsoidal to a spherical-cap geometry for
large bubbles at We z 20 [14]. As a bubble deforms, its cross-
sectional area normal to the flow direction increases, thereby
increasing the form or pressure drag. Consequently, the
terminal velocity of the bubble decreases.
The two limiting cases of bubble geometry, spherical and
spherical-cap, are considered for the bubble column
modeling. The optimal bubble size, which should maximize
HBr yield without adverse side effects, may be in or close to
the spherical bubble regime, where the reacting surface-to-
volume ratio is quite large. Brennen [14] indicates that the
bubble wake transitions from laminar to turbulent at
Re z 360; therefore, turbulence in the liquid must be modeled
above this transition Reynolds number.
To conserve computer resources and make simulation
feasible, a coordinate system at rest with respect to the bubble
is used for bubble models and simulation. This approach
allows the computational domain for bubble simulations to be
divided into two sub-domains: one for the liquid, and one for
the gas in the interior of the bubble. The geometry of the
bubble–liquid interface is predetermined and built into the
computational domain as an interior boundary. The spher-
ical-cap has a radius of curvature, Rc, and a half angle of q. An
equivalent bubble radius, Rb, is defined to be that of a sphere
having the same volume as the spherical-cap. Wegener and
Parlange [18] indicate that q / 50� for Reb> 100, with data
scatter in the interval 40� < q< 60�. For modeling purposes,
q¼ 50� is reasonable for Reb> 100.
4.1. Bubble terminal velocity
The bubble terminal velocity, uN, is determined by a balance
between buoyancy and drag forces. Determination of drag is
complicated by the effects of contaminants and gradients at
the liquid–gas interface, affecting the velocity of the interface
between gas and liquid. Deformation of a bubble at higher
velocities also affects pressure drag. Wallis [17] presents
correlations for the terminal velocity of isolated bubbles from
the work of Peebles and Garber [16]. Four of these correlations
and an additional data fit for air bubbles in water from Clift
et al. [12] are plotted in Fig. 5, which shows a comparison of
plots using both the properties of water (dashed lines) and
CaBr2 (solid lines).
The material properties of the gas are not significant in
determining the bubble rise velocity. The plot for bubbles with
a radius Rb< 0.01 cm (bubble Reynolds number, Reb< 2) is not
shown. Bubbles up to a radius of about 0.1 cm are still in the
spherical range and have terminal velocities that range over
about an order of magnitude, offering the greatest opportunity
to control residence time with bubble size. After the peak in
terminal velocity, bubbles enter a regime in which they
deform, becoming ellipsoidal and finally spherical-caps, and
terminal velocities decline in the bubble radius range from
about 0.1 to 0.5 cm. The horizontal lines in the range of Rb of
approximately 0.2–0.4 cm represents the uncertainty range in
the correlation, which is independent of bubble radius [16].
For larger spherical-cap bubbles, the terminal velocity
again increases with radius. Primarily because the liquid
density of CaBr2 at 1013 K (740 �C) is about three times that of
water at 298 K (25 �C), the curves for CaBr2 are shifted to the
left, giving higher terminal velocities (lower residence times)
for spherical bubbles in CaBr2. To maximize residence time for
the CaBr2-steam reaction and to maximize the bubble–gas
interface area, where the reaction takes place, operating with
the smallest practical bubble size in the spherical regime
between 100 mm and about 600 mm would appear to be
advantageous. In this operating range bubble wakes will be
laminar. Modeling of bubble flow with heat transfer and
reaction that characterizes the distribution of reaction prod-
ucts is needed to determine if there are possible advantages to
operating in the spherical-cap regime, where bubble wakes
will be turbulent.
4.2. Models for spherical bubbles in laminar flow andspherical-cap bubbles with turbulent wakes
For small bubbles at low Reynolds number, the flow of liquid
around the bubble and the gas flow within the bubble are
laminar. Under these conditions, the flow field in both
domains is governed by the incompressible non-isothermal
continuity and Navier–Stokes equations, given by Eqs. (4) and
(5). The same unit cell approach is applied to defining
a computational domain as was done for the droplet models.
In this case the exterior sub-domain is liquid, and the interior
sub-domain is the gas bubble. Fig. 6 shows typical two-
dimensional domains for a spherical and a spherical-cap
bubble with meshes and labeled boundaries. For rising
bubbles in a stationary domain, the flow enters from the top at
the terminal rise velocity, uN. For a 400-mm-diameter bubble,
uN¼ 0.06 m/s, giving an inlet boundary condition of u¼ uNdz.
Outflow, symmetric, and bubble–liquid interface boundaries
are the same as those given in Section 2.1 except that in this
case the liquid is the fluid in the external sub-domain.
Fig. 5 – Terminal velocity of bubbles: water at 298 K (25 8C); calcium bromide at 1013 K (740 8C).
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74162
For larger bubbles that become spherical-caps as they
reach terminal velocity, the flow in the wake of the bubble
becomes turbulent, and a turbulence model is required. For
spherical-cap bubbles, the standard k-3 model of Launder and
Fig. 6 – Typical mesh and boundary types for spherical and sph
are the surfaces of bubbles.
Spalding [19] is used with Reynolds-averaged equations of
motion. In the k-3 model, the equation of momentum includes
an eddy viscosity defined in terms of turbulent kinetic energy
and its dissipation rate. The eddy viscosity, mt, accounts for the
erical-cap bubble domains. The curved interior boundaries
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4163
rapid transport and mixing of flow field constituents across
mean flow streamlines. The averaged continuity and
momentum equations, respectively, for the velocity field, u,
are given by
Vu ¼ 0 (17)
and
rðuVÞu� V½ðmþ mtÞVu ¼ �Vpþ F: (18)
The eddy viscosity, mt, is given by
mt ¼ Cmrk2=3: (19)
where Cm is a model constant, k is the turbulent kinetic energy,
and 3 is its dissipation rate.
Transport equations for the turbulent kinetic energy, k, and
dissipation rate, 3, respectively, are given by
rðuVÞk� V½ðmþ mt=skÞVk ¼ mt
�vui
vxjþ vuj
vxi
�vui
vxj� r3 (20)
and
rðuVÞ3� V½ðmþ mt=s3ÞV3 ¼ C31mt3
k
�vui
vxjþ vuj
vxi
�vui
vxj� C32
r32
k: (21)
To include the hydrolysis reaction at the bubble surface, the
effects of reaction on the velocity field are assumed to be
small, so that the steady-state solution of the momentum
equations can be used to solve transient equations for species
and heat transfer in the interior of the bubble and in the
exterior liquid. Turbulent energy and species transport
equations must be provided for the liquid phase. The Rey-
nolds-averaged energy equation for the liquid phase is given
by
rlcplvTl
vtþ rlcplulVTl � V
�keffVTl
�¼ 0; (22)
where keff is the effective thermal conductivity, given by
keff ¼ rlcplnt
sTþ kl: (23)
The kinematic turbulent viscosity is given by
nt ¼ Cm
k2
3; (24)
and kl is the molecular thermal conductivity of the liquid.
The source term for the energy equation is zero because
the chemical reaction is taken to occur at the interface
between the liquid and the gas bubble; therefore, the
energy sink due to endothermic reaction is incorporated
into the boundary conditions coupling the gas and liquid
sub-domains. Along the gas–liquid interface for the liquid
phase, the boundary condition for the energy equation in
the liquid is
n�rlcplulTl � keffVTl
�¼ _q00r � _q00g: (25)
where _q00r ¼ � _r00CaBr2DH is the heat flux sink due to reaction at the
gas–liquid interface, and _q00g is the heat flux from the gas into
the interface. As mentioned in the introduction, the heat of
reaction, DH, 182.8 kJ/mol at 1015 K (742 �C) [20], is negative for
the endothermic reaction given by Eq. (1).
The forward reaction for the hydrolysis reaction of calcium
bromide given by Eq. (1) is assumed to be first-order in both
calcium bromide and steam, given by the expression:
_r00CaBr2¼ KCCaBr2
CH2O; (26)
where _r00CaBr2is the reaction rate of calcium bromide and K is the
forward kinetic rate constant. CCaBr2and CH2O are the Reynolds-
averaged molar concentration of calcium bromide and the
molar concentration of steam, having initial values of
15.6� 103 and 11.9 mol/m3, respectively. Since the reaction is
assumed to be irreversible, the backward reaction rate is zero.
The Reynolds-averaged transport equations for species in
the liquid-phase domain are given by
vCj
vtþ ulVCj � V
�DeffjVCj
�¼ 0; (27)
where Deffj is the effective binary diffusivity for species Cj
determined by
Deffj ¼nt
sDþ Dj: (28)
The molecular binary diffusivity for species j is Dj, and sD is
the turbulent Schmidt number. The source term for the
species transport energy equation is zero because the chem-
ical reaction is taken to occur only on the bubble surface, and
therefore, the reaction rate term is incorporated into the
boundary conditions coupling the gas and liquid sub-
domains. The boundary condition for the species j transport
equation at the gas–liquid interface is
n�Cjul � DeffjVCj
�¼ nj _r
00CaBr2
; (29)
where nj is the stoichiometric coefficient of species j, taken to
be positive for reactants and negative for products in Eq. (1).
While the liquid flow is modeled as turbulent, the gas flow
inside the bubble remains laminar, and Eqs. (4) and (5) apply.
As in the case of a spherical droplet in Section 2.1, the domain is
divided into two sub-domains with an internal boundary at the
gas–liquid interface. The boundary conditions for the k-3
turbulence model in the liquid sub-domain are similar to those
for the spherical bubble model, with the addition of conditions
on turbulence kinetic energy, k, and its dissipation rate, 3.
The concentration of molten CaBr2 at the inlet flow
boundary is 15.6� 103 mol/m3 at 1013 K (740 �C) and u ¼ uNdz,
where u ¼ 0:2 m=s for a spherical-cap bubble with equivalent
radius Rb¼ 2 mm. The gradients of k and 3 are taken to be zero
at symmetric, slip, and outflow boundaries. At the inlet,
k¼ 1.8� 10�4 m2/s2 and 3¼ 4.4� 10�6 m2/s3.
For the interior boundary at the gas–liquid interface, the
boundary conditions are approximated by assuming that
the bubble maintains its shape and radius. Along the cap
on the liquid side the boundary is taken to be free-slip,
assuming that the gas inside the bubble yields negligible
resistance to tangential flow. At the interface between the
two domains, the liquid and gas velocities are set equal to
each other. Along the bottom surface, a turbulent wake will
have eddies with flow in random directions. The mean
velocity of this random direction flow is taken to be zero.
Here, knowledge of the presence of the turbulent wake is
incorporated into the boundary condition along the bottom
interface of the cap. The interfacial boundary conditions on
the liquid side of the top of the cap are also zero gradient.
Along the bottom of the spherical-cap on the liquid side,
the boundary conditions are given as nu ¼ 0 and wall
functions from COMSOL�, given as
Fig. 7a – Left: Laminar flow field around a 400-mm diameter spherical steam bubble rising at uN [ 0.06 m/s. Right: Mean flow
around a 4-mm equivalent sphere diameter spherical-cap bubble rising at uN [ 20 cm/s in molten calcium bromide at 1013
K (740 8C).
Fig. 7b – Left: CaO and HBr mole fractions X_CaO and X_HBr in liquid and bubble, respectively, for a reacting 4-mm
equivalent sphere diameter spherical-cap bubble rising at 20 cm/s in molten calcium bromide. Reaction rate coefficient,
K [ 10L8 m4/(s 3 molH2O), time [ 1.0 s. Right: Temperature field for a reacting 4-mm equivalent sphere diameter spherical-
cap bubble rising at 20 cm/s in molten calcium bromide. Reaction rate coefficient, K [ 10L8 m4/(s3molH2O), time [ 1.0 s.
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74164
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4165
tu=us ¼ ð1=kÞlnðdwus=nÞ þ C; (30)
3 ¼ u3s=kdw; (31)
and
nV
�ðvþ nT=skÞVlnðkÞ þ 2
3ðvþ nT=skÞVlnð3Þ
¼ 0; (32)
where C¼ 5.5.
Chemical reaction, reactant consumption, and product
generation at the gas–liquid interface are assumed to have
negligibly small effects on the flow field. Therefore, the
steady-state solution for the flow field at the initial chemical
composition is used in the solution of transient heat and mass
transport during reaction of steam in the bubble with the
surrounding liquid mixture of CaBr2 and CaO.
Transient laminar heat and mass transport from the gas in
the bubble to and from the reaction interface are governed by
rgcpgvTg
vtþ rgcpgugVTg � V
�keVTg
�¼ 0 (33)
and
vCj
vtþ ugVCj � V
�DjVCj
�¼ 0; (34)
respectively. The boundary conditions at the bubble surfaces
for these equations are the usual symmetry condition at the
bubble centerline, equal temperatures of liquid and gas, and
n�Cju� DjVCj
�¼ nj _r
00CaBr2
¼ njKCCaBr2CH2O: (35)
4.3. Modeling results and discussion
Examples of simulation results for the cases of laminar flow
around a spherical bubble and for turbulent flow around
a spherical-cap bubble are shown in Fig. 7a on the left and
right, respectively. The velocity in the axial direction is used to
color the plots. The velocity scales just to the right of each plot
are for the interior of the bubbles and to their right, for the
external liquid flow fields. The plots also show streamlines
and velocity vectors. No recirculation zone appears in the
wake; therefore, the model indicates that reaction products
would not be retained near the bottom surface of the bubble in
its wake at significant concentrations. The flow pattern in the
interior of the bubble shows that steam from the bubble
interior near the centerline circulates to its surface, where the
reaction front would be for steam reacting with liquid CaBr2.
The equivalent sphere diameter of the spherical-cap
bubble is 4 mm, with a Reynolds number of about 2500. The k-3
turbulence model disguises the true nature of the wake, which
in reality will consist of a chaotic mixture of vortices shedding
from the bottom of the bubble. However, the model can be
expected to capture the effect of entrainment of reaction
products in the wake at the bottom of the bubble, which may
affect reaction rates there. The interior of the cap also shows
circulation that would bring steam from the interior to the
reaction front at the bubble surface. Brennen [14], Clift et al.
[12], and Wallis [17] have noted that contaminants in the
liquid and surface tension and thermal gradients along the
surface can act to slow this internal circulation.
A transient solution for a reacting spherical-cap bubble was
performed using a reaction rate coefficient, K¼ 10�8,
producing a 0.47 HBr mole fraction in 1.0 s, the assumed resi-
dence time of a bubble in a typical bubble column. The initial
temperature is 1023 K (750 �C). Plots of the HBr and CaO mole
fractions and temperature field are shown in Fig. 7b, respec-
tively, at 1.0 s. The temperature scales to the right of the
temperature plot are inside the bubble and in the external flow
field, respectively. The temperature decrease due to the
endothermic surface reaction was found to be negligible,
producing an essentially isothermal temperature field within
and in the near vicinity of the bubble. This was anticipated
because a bubble of steam with an equivalent sphere diameter
of 4 mm can only extract heat from a very small volume of
liquid CaBr2. The concentration of CaO in the bubble wake is
also found to be negligible. These results justify the steady
flow field assumption.
The relationship of this value of reaction rate constant,
K¼ 10�8, which is used in Eq. (26), to that analyzed from the
Robinson et al. [20] data was determined. Robinson et al.
conducted hydrolysis experiments at four temperatures by
flowing steam over a porcelain boat placed in a silica tube
heated electrically and containing 0.5 g samples of solid
calcium bromide crystals for 30 min. The weights of calcium
bromide decomposed were reported, but no analysis was
performed. Assuming excess steam was used in these exper-
iments, a first-order reaction with calcium bromide was
assumed to obtain reaction rate coefficients, k*, for each
temperature. The slope of the curve plotting the log of the
resulting reaction rate coefficients vs. the reciprocal of abso-
lute temperature, T, was used to determine the activation
energy, E, in the Arrhenius equation, given by
k ¼ Aexpð�E=RTÞ: (36)
Then the pre-exponential constant, A, was evaluated. Using
these values, a reaction rate constant of 0.05 min�1 at 1023 K
was determined.
In order to convert the reaction rate coefficient k* (min�1) to
K ðm4=s molH2OÞ, the following expression was used:
K ¼ kðV=SÞ=�CH2O;i � 60
�; (37)
where CH2O;i is the initial molar concentration of steam
(11.9 mol/m3) and the volume to surface area ratio of the
equivalent spherical bubble V/S¼ r/3. Using an equivalent
sphere radius, r¼ 2 mm, yields K¼ 4.7� 10�8, which is of the
same order of magnitude as the value of 10�8 used above.
Using this value, the mole fraction of HBr at 1.0 s¼ 0.9. Anal-
ysis of the initial reaction rate data of Avogadro et al. [9] in
kinetics experiments for the HBr production and CaBr2
consumption rates at 1023 K (750 �C) results in a reaction rate,
K, in the range of approximately 5� 10�9 to 10�8. The HBr
yields with values of K ranging from 10�7 to 4.7� 10�9 were
computed for hydrolysis in molten calcium bromide. Table 1
summarizes the results of this parametric study. As before,
the temperature fields were essentially isothermal.
Based on these modeling results, it appeared that
laboratory experiments involving the reaction of steam
with molten calcium bromide in a sparging reactor would
be successful, producing appreciable conversion. Proof of
principle experiments have been carried out at Argonne
showing that molten CaBr2 hydrolysis is feasible for
producing reaction rates up to an order of magnitude
Table 1 – HBr production and steam conversion at 1.0 s asa function of reaction rate coefficient, K, at 1023 K (750 8C)for a spherical-cap bubble model.
Reaction ratecoefficient,K m4=ðs molH2OÞ
Mole fractionHBr
Mole fractionsteam
10�7 0.99þ 0.01�4.7� 10�8 0.90 0.10
10�8 0.47 0.53
4.7� 10�9 0.28 0.72
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 74166
higher than previous experiments [9,20], as described in
a companion paper [21].
The models developed in this paper, combined with
further experiments to determine the reaction rate, will help
to size the molten CaBr2 hydrolysis reactor and to identify the
optimal operating range for bubble size to maximize conver-
sion of steam to HBr in the reactor. The reaction might
produce a eutectic and/or solid CaO products, which would be
convected away from the bubble surface reaction front. Some
of these reaction products may end up in the interfacial
boundary layer of bubbles. Effects of products in the mixture
on the local reaction rate and physical properties need
investigation. The bubble models developed will account for
the effects of internal circulation, mixing, and diffusion of
steam and HBr to and from the reaction front at the surface.
Results of this modeling will aid in reactor design. Additional
modeling may be required to aid in sparger tube or bubble
distributor design. Based on results of bubble scale modeling,
sub-models will be developed to characterize reaction prog-
ress and interfacial heat, mass, and momentum transfer
between bubbles and the surrounding molten CaBr2 pool.
These sub-models can then be incorporated into larger
reactor-scale models.
5. Conclusions
Sparging steam into molten CaBr2 appears to be a prom-
ising candidate for a calcium bromide hydrolysis reactor in
a hybrid thermochemical water-splitting cycle. For a single
2-mm-diameter bubble, a negligible amount of CaO will be
formed, go into solution, and be drawn into the wake of the
spherical-cap bubble at an essentially isothermal condition.
The analysis indicates that good conversion of steam to HBr
can be obtained over a broad range of kinetic rate coeffi-
cient, K, near that determined from published experimental
data. Experiments with molten calcium bromide are needed
to determine K more precisely. A preliminary reaction
kinetics analysis has been performed [22]. In a pilot-scale
unit, many more bubbles are required to achieve the
desired production rate of HBr.
Acknowledgements
This effort was sponsored by the U.S. Department of
Energy’s (DOE’s) Nuclear Energy Research Initiative. The
investigations described herein were conducted by
Argonne National Laboratory under DOE Contract No. DE-
AC02-06CH11357.
r e f e r e n c e s
[1] Nuclear hydrogen initiative: ten year program plan, U.S.Department of Energy (DOE) Office of Nuclear Energy,Science and Technology, Office of Advanced NuclearResearch; March 2005.
[2] Kameyama H, Tomino Y, Sato T, Amir R, Orihara A,Aihara M, et al. Process simulation of mascot plant using theUT-3 thermochemical cycle for hydrogen production. Int JHydrogen Energy 1988;14(5):323–30.
[3] Aihara M, Umida H, Tsutsumi A, Yoshida K. Kinetic study ofUT-3 thermochemical hydrogen production process. Int JHydrogen Energy 1990;15(1):7–11.
[4] Yoshida K, Kameyama H, Aochi T, Nobue M, Aihara M,Amir R, et al. A simulation study of the UT-3 thermochemicalhydrogen production process. Int J Hydrogen Energy 1990;15(3):171–8.
[5] Sakurai M, Tsutsumi A, Yoshida K. Improvement of Ca-pelletreactivity in the UT-3 thermochemical hydrogen productioncycle. Int J Hydrogen Energy 1995;20(4):297–301.
[6] Sakurai M, Bilgen E, Tsutsumi A, Yoshida K. Adiabatic UT-3thermochemical process for hydrogen production. Int JHydrogen Energy 1996;21(10):865–70.
[7] Teo ED, Brandon NP, Vos E, Kramer GJ. A critical pathwayenergy efficiency analysis of the thermochemical UT-3 cycle.Int J Hydrogen Energy 2005;30(5):559–64.
[8] Lemont F, Lafon C, Dedryvere R, Gonbeu D. Physicochemicaland thermodynamic investigation of the UT-3 hydrogenproduction cycle: a new technological assessment. Int JHydrogen Energy 2006;31(7):906–18.
[9] Avogadro A, Gotz L, Hoffman A, Soldanl G. Hydrogenproduction from water using nuclear heat. Progress ReportNo. 3, Joint Nuclear Research Centre, Ispra Establishment,Italy, EUR 5059e; 1974.
[10] Doctor RD, Henry M, Lin YP, Lottes SA, Lyczkowski RW,Panchal CB. Nuclear hydrogen initiative calcium–brominecycle CH16TC31: feasibility of a calcium–brominethermochemical cycle for nuclear hydrogen production.Argonne National Laboratory report, Available from: Part 1 of4, <http://66.250.193.139/uploads/deliverables/rdoctor_6009/7_6_2006%2011_21_54%20AM/NHI%20Ca-Br%20CH16TC31%202006-06-30%20Part%201.pdf>; 2006. Part 2 of 4, <http://66.250.193.139/uploads/deliverables/rdoctor_6010/7_6_2006%2011_37_40%20AM/NHI%20Ca-Br%20CH16TC31%202006-06-30%20Part%202.pdf>; 2006. Part 3 of 4, <http://66.250.193.139/uploads/deliverables/rdoctor_6011/7_6_2006%2011_43_31%20AM/NHI%20Ca-Br%20CH16TC31%202006-06-30%20Part%203.pdf>; 2006. Part 4 of 4, <http://66.250.193.139/uploads/deliverables/rdoctor_6012/7_6_2006%2011_47_09%20AM/NHI%20Ca-Br%20CH16TC31%202006-06-30%20Part%204.pdf>; 2006.
[11] COMSOL, Inc. COMSOL version 3.2a user’s guide; 2004.[12] Clift R, Grace J, Weber ME. Bubbles, drops, and particles.
Academic Press, Inc.; 1978.[13] Aspen Technology, Inc. Aspen properties application;
2005.[14] Brennen CE. Fundamentals of multiphase flow. Cambridge
University Press; 2005.[15] Harmathy TZ. Velocity of large drops and bubbles in media of
infinite or restricted extent. AIChE J 1960;6:281–8.[16] Peebles FN, Garber HJ. Studies on the motion of gas bubbles
in liquid. Chem Eng Prog 1953;49:88–97.
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 3 4 ( 2 0 0 9 ) 4 1 5 5 – 4 1 6 7 4167
[17] Wallis GB. One-dimensional two-phase flow. McGraw-Hill;1969.
[18] Wegener PP, Parlange J-Y. Spherical-cap bubbles. Ann RevFluid Mech 1973;5:79–100.
[19] Launder BE, Spalding DB. The numerical computation ofturbulent flows. Comp Methods Appl Mech Eng 1974;3:269–89.
[20] Robinson PL, Smith HC, Briscoe HVA. The hydrolytic actionof low-pressure superheated steam on salts of the alkalineearth metals. J Chem Soc 1926;129:836–9.
[21] Yang J, Panchal CB, Doctor RD. Experimental investigation ofmolten-phase calcium hydroxide hydrolysis. Int J HydrogenEnergy [submitted for publication].
[22] Panchal CB, Lyczkowski RW, Lottes SA, Yang J, DoctorRD. Analysis of kinetics of CaBr2 hydrolysis in a directsteam sparging contactor. Paper 532f, presented in theSession Developments in Thermochemical andElectrolytic Routes to Hydrogen. American Institute ofChemical Engineers Annual Meeting, Salt Lake City;November 4–9, 2007.