Modeling and analysis of calcium bromide hydrolysis

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

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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: [email protected] (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.

mailto:[email protected]

http://www.elsevier.com/locate/he

http://www.sciencedirect.com

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


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.


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.


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).


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


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).


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


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.



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


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.

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Modeling and analysis of calcium bromide hydrolysis

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