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Accepted Manuscript
Diamagnetically enhanced electrolysis and phase separation in
low-gravity
Álvaro Romero-Calvo ∗and Hanspeter Schaub †
Department of Aerospace Engineering Sciences, University of Colorado Boulder, CO, 80303, United States
Gabriel Cano-Gómez‡
Departamento de Física Aplicada III, Universidad de Sevilla, Sevilla, 41092, Spain
The management of fluids in space is complicated by the absence of relevant buoyancy
forces. This raises significant technical issues for two-phase flow applications. Different ap-
proaches have been proposed and tested to induce phase separation in low-gravity; however,
further efforts are still required to develop efficient, reliable, and safe devices. The employment
of diamagnetic buoyancy is proposed as a complement or substitution of current methods, and
as a way to induce the early detachment of gas bubbles from their nucleation surfaces. The
governing magnetohydrodynamic equations describing two-phase flows in low-gravity are pre-
sented with a focus on bubble dynamics. Numerical simulations are employed to demonstrate
the reachability of current magnets under different configurations, compare diamagnetic and
Lorentz forces on alkaline electrolytes, and suggest scaling up procedures. The results support
the employment of new-generation centimeter-scale neodymium magnets for electrolysis, boil-
ing and phase separation technologies in space, that would benefit from reduced complexity,
mass, and power requirements.
Nomenclature
𝐴= effective surface of the electrode, m2
𝑩= magnetic flux density, T
𝜒mass = mass magnetic susceptibility, m3/kg
𝜒mol = mole magnetic susceptibility, m3/mol
𝜒vol = volume magnetic susceptibility
𝑫= electric displacement field, C/m2
𝐷0= contact line diameter, m
∗
Graduate Research Assistant, Department of Aerospace Engineering Sciences, University of Colorado Boulder,
alvaro.romerocalvo@
colorado.edu, AIAA Student Member.
†
Professor, Glenn L. Murphy Chair in Engineering, Department of Aerospace Engineering Sciences, University of Colorado Boulder,
hanspeter.schaub@colorado.edu, AIAA Member.
‡-Associate Professor, Departamento de Física Aplicada III, Universidad de Sevilla, gabriel@us.es.
The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021
Accepted Manuscript
𝑑0= bubble break-off diameter, m
𝑑𝑏= electrolytic bubble break-off diameter, m
Δ𝜒vol = differential volume magnetic susceptibility
𝑬= electric field, V/m
𝒆𝑖= Cartesian frame
𝜖0= permittivity of free space, F/m
𝜂= dynamic coefficient of viscosity, Pa s
𝒇𝑆= surface force density, N/m2
𝒇𝑉= body force density, N/m3
𝑭= Force, N
𝒈= inertial acceleration, m/s2
𝛾= volume coefficient of viscosity, Pa s
𝑯= magnetic field, A/m
𝑯∗= virtual magnetic field, A/m
𝑯0=applied magnetic field, A/m
H= arithmetic mean interface curvature, 1/m
𝑰= identity matrix
𝐼= electrode current intensity, A
𝑱𝑒= electric current density, A/m2
𝒌= normal vector perpendicular to the bubble pit
𝑲𝑠= surface electric current density, A/m
𝜆= combined coefficient of viscosity, Pa s
L= Lorentz force per unit volume, N/m3
𝑴= magnetization field, A/m
𝑚𝑏= virtual bubble mass, kg
𝜇0= permeability of free space, H/m
M= molar mass, kg/mol
𝒏= external normal vector
𝜈= specific volume, m3/kg
𝑷= electric polarization field, C/m2
𝑝∗= composite pressure, Pa
𝑝= hydrostatic pressure, Pa
2
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Accepted Manuscript
𝑝𝑖= mass fraction
𝑅𝑏= bubble radius, m
𝑅𝑒 = Reynolds number
𝜌= fluid density, kg/m3
𝜌𝑣= free charge density, C/m3
𝜎= surface tension, N/m
𝜎𝑠= surface free charge density, C/m2
𝑇= temperature, K
𝒕= stress vector, Pa
𝒕𝑓 𝑔 = tangent unit vector in the meridian plane
T= Maxwell stress tensor, Pa
𝜃= apparent contact angle, rad
𝒗= fluid velocity, m/s
𝒗𝑡= terminal velocity, m/s
𝒙= position vector, m
Subscripts
e = electric term
f = liquid environment
g = gas environment
in = inertial
m = magnetic term
me = magnetic environment
n = normal component
𝜈= viscous term
p = pressure term
𝜎= surface tension term
t = tangential component
I. Introduction
The term water electrolysis refers to the electrochemical decomposition of water into hydrogen and oxygen. The
reaction was first performed by Troostwijk and Deiman in 1789 [
1
,
2
] and was already considered for space applications
in the early 1960s [
3
]. A wide range of environmental control and life support systems [
4
], space propulsion technologies
3
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Accepted Manuscript
[
5
–
7
], or energy conversion and storage mechanisms [
8
,
9
] rely on this process. Furthermore, future interplanetary
missions are likely to employ water as a commodity acquired and processed by In Situ Resource Utilization (ISRU)
methodologies to produce propellants, thereby reducing vehicle launch mass [10, 11].
Water electrolysis technologies can be classified according to the nature of the electrolyte. Three chemistries are
considered for space applications: alkaline, proton exchange membrane (PEM), and solid oxide ceramics. Of these,
the low temperature alkaline and PEM electrolytes require phase separation at the electrode. The liquid alkaline
technology employs two metallic electrodes separated by a porous material and immersed in a conductive aqueous
solution, usually prepared with
𝐾𝑂 𝐻
or
𝑁𝑎𝑂 𝐻
. The cell separator allows the exchange of the
𝑂𝐻−
groups and prevents
the recombination of
𝐻2
and
𝑂2
into water. PEM cells, on the contrary, are fed with pure water and make use of a
proton-conducting polymer electrolyte. PEM cells allow high current densities, prevent the recombination of oxygen
and hydrogen (and so, are safer), and produce high-purity gases. However, they lack the long-term heritage of alkaline
cells and are sensitive to water impurities [12].
The operation of alkaline and PEM cells in low-gravity is severely complicated by the absence of strong buoyancy
forces, resulting in increased complexity, mass, and power consumption. Dedicated microgravity experiments have
shown how the weak buoyancy force gives rise to a layer of gas bubbles over the electrodes, shielding the active surface
and limiting mass transport [
13
–
15
]. Gas bubbles tend to be larger than in normal-gravity conditions due to the longer
residence time on the electrodes and the absence of bubble departure. Besides, and unlike in normal-gravity, the bubble
departure diameter increases with increasing current intensity [16]. A forced water flow can be employed to flush this
structure, but this approach increases the complexity of the system and has a limited efficiency [
17
]. Most types of
electrolytic cells also require a liquid/gas phase separation stage. Rotary [
18
,
19
] and membrane-type [
17
,
20
] devices are
nowadays employed. Passive approaches that make use of surface tension by means wedge geometries [
21
,
22
], springs
[
23
], or eccentric annuli [
24
] have also been proposed and tested. As an alternative, the generation of an equivalent
dielectric buoyancy force by means of strong electric fields has been considered since the early 1960s [
25
] and has been
studied and successfully tested for low-gravity boiling [
26
,
27
] and two-phase flow [
28
,
29
] applications. However, the
drawbacks of these approaches are numerous: centrifuges add to system power loads and may represent a safety hazard,
membranes have limited lifetime and tend to clog in the presence of water impurities [
5
,
30
], surface tension-based
approaches require careful geometrical design and are sensitive to moderate departures from the operational design point
[
21
], and electric fields consume power and may represent a safety hazard for both human and autonomous spaceflight
due to the large required potential differences.
Both alkaline and PEM technologies have flown to space and dealt with the phase separation problem in different
ways. The Russian Elektron module, first operated at Mir and then at the ISS, makes use of a circulating alkaline
electrolyte (
25 %
wt
𝐾𝑂 𝐻
) and a fluid circuit with gas/liquid static separators and heat exchangers [
31
]. The operation
of the system has been compromised in the past by notorious malfunction events [
32
–
34
]. NASA’s Oxygen Generation
4
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Accepted Manuscript
System (OGS), installed at the ISS in July 2006, makes use of a cathode-feed PEM and a rotary phase separator and
absorber modules to produce dry oxygen. Unlike anode-feed PEMs, cathode feeding avoids the humidification of
𝑂2
due to proton-induced electro-osmosis [
5
]. Technical problems associated with the management of two-phase flows in
the OGS in microgravity have also been reported [
18
,
35
]. JAXA has recently developed a cathode-feed PEM cell for
𝑂2
generation. The system relies on the removal of the electrode gas cover by means of forced convection. The separation
of gas hydrogen and liquid water is performed by means of a membrane-type phase separator [
17
,
20
,
36
]. Subsequent
versions of the cathode-feed cell rely on an internal water/gas separation function that makes water circulation and phase
separator unnecessary, creating a simple, energy-efficient, and lightweight system. However, difficulties were found to
reach a stable phase separation process [
37
–
39
]. As a way to remove the water purification and phase separation stages,
substantial efforts have been devoted to the development of Static Water Feed (SWF) electrolytic cells, which avoid
the phase separation stage by means of a second PEM. Technological demonstrators by Life Systems were tested on
the STS-69 Endeavor (1995) and the STS-84 Atlantis (1997) for NASA [
40
–
45
], being followed by relatively modern
systems [
5
,
46
]. In spite of its inherent advantages, this approach requires larger cells to deliver a specific gas output due
to the presence of a second membrane, that increases the water gradient, and the adoption of a cathode-feed configuration
for the second membrane [5, 36].
This scientific and technological review unveils the numerous challenges associated with the low-gravity gas/liquid
separation process and shows important limitations in current and foreseen technologies. As a complement or substitution
of the previous methods, the inherent magnetic properties of water and water-based electrolytes may be employed to
induce the natural detachment and collection of gas bubbles. Inhomogeneous magnetic fields induce a weak body force
in continuous media [
47
] that, due to the differential magnetic properties between phases, results in a net buoyancy
force. This phenomenon is known as magnetic buoyancy and has been applied to terrestrial boiling experiments with
ferrofluids [
48
,
49
]. Previous works on low-gravity magnetohydrodynamics have explored the diamagnetic manipulation
of air bubbles in water [
50
,
51
], the positioning of diamagnetic materials [
52
], air-water separation [
53
], protein crystal
growth [
54
], magnetic positive positioning [
55
–
59
], magnetic liquid sloshing [
60
,
61
], or combustion enhancement
[
51
], among others. The application of Lorentz’s force on liquid electrolytes has also been studied as a way to enhance
hydrogen production [
62
–
72
]. However, the use of magnetic buoyancy in phase separation, electrolysis, and boiling in
low-gravity remains largely unexplored.
The development of low-gravity magnetic phase separators may lead to reliable, lightweight, and passive devices. If
the magnetic force was applied directly over the electrodes, the bubble departure diameter would be reduced and a
convective flow would be induced on the layer of bubbles, enlarging the effective electrode surface and minimizing the
mass transport limitations and cell voltage. The same benefits would be obtained for low-gravity boiling devices, with
the boiling surfaces being equivalent to the electrodes.
In this paper, the applications of magnetic buoyancy in low-gravity electrolysis are first explored in Sec. II. A set of
5
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(a) Diamagnetic (b) Para/Ferromagnetic
Fig. 1 Conceptual representation of a magnetically enhanced electrolysis cell. Blue arrows represent the
liquid/gas flow, while red arrows denote the magnetization vector.
general theoretical tools that enable a comprehensive analysis of the problem are derived Sec. III, while an analysis of
bubble dynamics is given in Sec. IV. Fundamental results and technical solutions are shown in Sec. V, and conclusions
are finally drawn in Sec. VI.
II. Magnetic buoyancy applications to low-gravity water electrolysis
The application of magnetic buoyancy may be beneficial for at least two common components of low-gravity water
electrolysis systems: the electrolytic cell and the gas/liquid phase separator. These elements are present in both alkaline
and PEM cells, with the exception of SFW configurations. As shown in Sec. III.D, magnetic buoyancy can be induced in
virtually all liquids of technical interest, with pure water being the least magnetically susceptible of them. Although this
discussion focuses on water electrolysis, the concepts here introduced have a direct equivalent in applications involving
pool boiling or bubble growth, such as water boiling or other types of electrolysis, recombination, or combustion
reactions.
A. Electrolytic cells
The application of strong inhomogeneous magnetic fields over the electrodes would potentially induce a convective
flow on the layer of bubbles, reducing the break-off diameter, enlarging the effective surface, and reducing the cell voltage
while effectively separating the phases. Some of these effects have been observed in terrestial boiling experiments with
ferrofluids, where a significant influence of the magnetic field on the boiling plate bubble coverage and heat transfer
coefficient is reported [48, 49].
Depending on the type of reactant, the diamagnetic or para/ferromagnetic configurations depicted in Fig. 1 should
be considered. While in the former a diamagnetic liquid (such as water) is repelled from the magnet, in the latter a
paramagnetic or ferromagnetic substance is employed, producing the opposite effect. In virtue of Archimedes’ principle,
gas bubbles experience a magnetic buoyancy force that attracts (diamagnetic) or expels (para/ferromagnetic) them from
6
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Accepted Manuscript
(a) Standalone
(b) Combined
Fig. 2 Conceptual representation of a magnetic standalone and surface-tension enhanced diamagnetic phase
separator. Blue arrows represent the liquid/gas flow, while red arrows denote the magnetization vector.
the magnet.
This distinction may sound contrived to the reader, as water and its associated electrolytes are diamagnetic materials
and applications where paramagnetic or ferromagnetic substances are employed cannot be easily found. However, the
renewed interest in nanofluids-enhanced [
73
] and magnetically-enhanced [
48
,
49
] heat transfer may open new interesting
avenues of research. Highly susceptible liquids, such as ferrofluids, may be employed to boost the productivity of the
electrodes and boiling surfaces, both on Earth and in space. In spite of the numerous technical challenges that such
technologies would face (e.g. thermal stability or particle deposition [
74
]), the possibility of using ferrofluids is covered
by this work.
B. Phase separators
Although the magnetic electrolytic cell concept presents intrinsic advantages, modularity is commonly sought in an
industry characterized by a complicated and expensive design flow. From a systems engineering perspective, magnetic
phase separators could replace the functionality of existing phase separation technologies without forcing qualitative
modifications at the cell level, and with applications to any combination of phases. The magnetic buoyancy force could
also be employed in combination with existing systems, such as surface-tension enabled phase separators.
This is conceptually represented in Fig. 2, where magnetic and combined surface tension/magnetic phase separators
are shown for a dielectric liquid. The magnetic phase separator consists on a channel surrounding a magnet that attracts
the bubbles from an incoming two-phase flow. The combined phase separator consists on a wedge-shaped channel
that pushes large bubbles to the open end as they evolve towards their configuration of minimum energy (spherical
geometry). This approach was tested in the Capillary Channel Flow experiment, conducted at the ISS in April 2010
7
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Accepted Manuscript
[
21
]. Magnetic buoyancy may be particularly useful to attract small bubbles that are unlikely to contact the free surface
and hence remain within the liquid flow.
III. Magnetic buoyancy in two-phase flows
The conceptual solutions here presented should be studied in the framework of the magnetohydrodynamic theory.
This section introduces a series of fundamental theoretical tools together with simplified expressions that enable
preliminary analyses. Due to their potential interest in future applications, the use of ferrofluids is considered. No
assumptions are made regarding the constitutive relation of the material.
A. Governing equations for polarizable, viscous, compressible fluids
The magneto/electrohydrodynamic mass and momentum conservation equations are given by [75]
𝜕𝜌
𝜕𝑡 + ∇ · (𝜌𝒗)=0,(1a)
𝜌𝐷𝒗
𝐷𝑡 =𝜌𝒈+ ∇ · T,(1b)
subject to appropriate boundary conditions, with
𝒗
being the velocity field,
𝜌
the fluid density,
𝒈
the inertial acceleration,
𝐷/𝐷𝑡
the material derivative, and Tthe Maxwell stress tensor that includes pressure, viscous, magnetic, and electric
terms.
The stress tensor Tis defined in terms of the magnetic and electric fields acting on the system. Those are computed
from Maxwell equations
∇ · 𝑫=𝜌𝑣,(2a)
∇ · 𝑩=0,(2b)
∇ × 𝑬=−𝜕𝑩
𝜕𝑡 ,(2c)
∇ × 𝑯=𝑱𝑒+𝜕𝑫
𝜕𝑡 ,(2d)
where
𝑬
,
𝑫=𝜖0𝑬+𝑷
, and
𝑷
are the electric, electric displacement, and polarization fields, and
𝑩
,
𝑯=(𝑩/𝜇0)−𝑴
,
and
𝑴
are the flux density, magnetic, and magnetization fields, respectively;
𝜖0
and
𝜇0
are the permittivity and
permeability of vacuum,
𝜌𝑣
is the free charge density, and
𝑱𝑒
is the electric current density. For soft magnetic materials,
the magnetization field is aligned with the magnetic field and follows the relation
𝑴=𝜒vol(𝐻)𝑯
, with
𝜒vol(𝐻)
being
the volume magnetic susceptibility. Dia/paramagnetic materials exhibit a constant and small
𝜒vol
. The interfacial
8
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electric and magnetic boundary conditions are
𝒏· (𝑫2−𝑫1)=𝜎𝑠,(3a)
𝒏× (𝑬2−𝑬1)=0,(3b)
𝒏· (𝑩2−𝑩1)=0,(3c)
𝒏× (𝑯2−𝑯1)=𝑲𝑠,(3d)
with
𝜎𝑠
and
𝑲𝑠
being the surface free charge and electric current densities, respectively. Therefore, the normal
component of
𝑩
and, in the absence of surface currents, the tangential component of
𝑯
, are continuous across the
interface.
For most applications covered in this work, the dielectric force can be neglected (see the comparison between
diamagnetic and dielectric terms in Sec. V.A) and the stationary Maxwell equations in the absence of electric charges
are employed due to the consideration of a neutrally charged medium. The theoretical framework can be further
simplified when considering dia / paramagnetic materials, whose small magnetization field justifies the approximation
𝑯≈𝑯0
, with
𝑯0
being the applied magnetic field. From a practical perspective, this implies that
𝑯0
can be calculated
independently of the state of the system under analysis. A simplified set of governing equations for incompressible,
neutral fluids subject to static magnetic fields is given in Appendix A.
B. The viscous Maxwell stress tensor
The magneto/electrodynamic state of a continuous medium can be described by means of the viscous Maxwell
stress tensor, which has been formulated in the classical literature as [47, 75–77]
T=T𝑝+T𝜈+T𝑚+T𝑒,(4)
with the pressure, viscous, magnetic, and electric terms being given by
T𝑝=−𝑝∗𝑰,(5a)
T𝜈=𝜂∇𝒗+ (∇𝒗)𝑇+𝜆(∇ · 𝒗)𝑰,(5b)
T𝑚=−𝜇0
2𝐻2𝑰+𝑩𝑯,(5c)
T𝑒=−𝜖0
2𝐸2𝑰+𝑫𝑬,(5d)
9
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and where
𝑝∗=𝑝(𝜈, 𝑇 ) + 𝜇0∫𝐻
0
𝜕
𝜕𝜈 [𝜈 𝑀]𝑑𝐻0+∫𝐸
0
𝜕
𝜕𝜈 [𝜈 𝑃]𝑑𝐸 0(6)
is the composite pressure, that includes hydrostatic
𝑝(𝜈, 𝑇 )
and magneto/electropolarization terms. In the previous
expressions,
𝑰=𝛿𝑖 𝑗 𝒆𝑖𝒆𝑗
is the unit dyadic in the Cartesian
𝒆𝑖
reference system, and
𝜈=𝜌−1
is the specific volume of
the medium. In the viscous tensor,
𝜆=(𝛾−
2
𝜂)/
3, and
𝜂
and
𝛾
are the dynamic and volume coefficients of viscosity.
When considering ferrofluids,
𝜂
shows a nonlinear dependence with the magnetic field [
78
]. Applications involving
unequilibrated ferrofluid solutions (i.e. those for which
𝑴×𝑯≠
0), should incorporate the effects resulting from
particle rotation in a viscous carrier liquid. An additional term should be added to the viscous stress tensor T
𝜈
, and the
angular momentum and magnetic relaxation equations should be considered [75, 77].
The structure of the magnetic tensor reflects the complete analogy between magnetostatics and electrostatics [
47
].
The electric interaction has been widely studied in the context of two-phase flows [
28
] and does not have a significant
incidence for the applications discussed in this work. Even though the small differential electrode potential (
≈
1
.
5
V) produces an electric field, the associated electric force is several orders of magnitude weaker than its magnetic
counterpart, as shown in Sec. V.A. Consequently, this discussion focuses on the magnetohydrodynamic effect by
assuming an electrically neutral medium that remains in thermodynamic equilibrium with constant density, temperature
and chemical potentials. A separate analyses is presented in Sec. V.B for charge-carrying electrolytes, where the Lorentz
force becomes dominant. After dropping the electric terms, the stress tensor becomes
T=T𝑝+T𝜈+T𝑚.(7)
This formulation does not implement any assumption regarding the constitutive relation of the material. In some cases,
however, it may be useful to particularize the analysis to linear media, as presented in Ref. [
47
]. Linear electric results
are presented in Ref. [
28
] and can be easily adapted to the dia / paramagnetic case by making use of the analogy between
magnetostatics and electrostatics.
1. Body force distributions
The forces per unit volume exerted on the medium in the absence of electric fields can be computed as the divergence
of the stress tensor given by Eq. 7, resulting in [75]
𝒇𝑉=∇ · T=𝒇𝑉
𝑝+𝒇𝑉
𝜈+𝒇𝑉
𝑚(8)
with
𝒇𝑉
𝑝=∇ · T𝑝=−∇𝑝∗,(9a)
10
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𝒇𝑉
𝜈=∇ · T𝜈=∇ · 𝜂∇𝒗+ (∇𝒗)𝑇+𝜆(∇ · 𝒗)𝑰,(9b)
𝒇𝑉
𝑚=∇ · T𝑚=𝜇0𝑀∇𝐻. (9c)
If the viscosity coefficients 𝜂and 𝜆, are considered constant, the viscous term reduces to
𝒇𝑉
𝜈=𝜂∇2𝒗+(𝜂+𝜆)∇(∇ · 𝒗).
2. Boundary conditions
Surface forces appear in the interface between immiscible media as a consequence of the discontinuity in the stress
tensor. Those forces are balanced according to the condition of stress equilibrium, leading to [75]
𝒕𝑛,2−𝒕𝑛, 1=2𝜎H𝒏1,(10)
where
𝒕𝑛,𝑖 =𝒏𝑖·
T
𝑚,𝑖
is the stress vector,
𝒏𝑖
is the external normal of the medium
𝑖
, the right term is the capillary
pressure,
𝜎
is the surface tension, and
H
is the arithmetic mean curvature of the interface. Implementing the magnetic,
viscous stress tensor given by Eq. 7, the stress vector is expressed as
𝒕𝑛=−𝑝∗𝒏+𝜂2𝜕𝑣 𝑛
𝜕𝑥𝑛
𝒏+𝜕𝑣 𝑛
𝜕𝑥𝑡
+𝜕𝑣 𝑡
𝜕𝑥𝑛𝒕+𝜆(∇ · 𝒗)𝒏−𝜇0
2𝐻2𝒏+𝐵𝑛𝑯,(11)
where
𝑣𝑛
and
𝑣𝑡
are the normal and tangential velocity components, and
𝑥𝑛
and
𝑥𝑡
the distances along the normal and
tangential directions, respectively. Computing the balance at the interface, considering Gauss’ and Ampère’s laws, and
expressing the result in the normal (
𝑛
) and tangential (
𝑡
) directions, the ferrohydrodynamic (FHD) viscous boundary
condition is obtained [75]
𝑛:𝑝∗−2𝜂𝜕𝑣 𝑛
𝜕𝑥𝑛
−𝜆(∇ · 𝒗) + 𝑝𝑛+2𝜎H=0,(12a)
𝑡:𝜂𝜕𝑣 𝑛
𝜕𝑥𝑡
+𝜕𝑣 𝑡
𝜕𝑥𝑛 =0,(12b)
with
𝑝𝑛,𝑖 =𝜇0𝑀2
𝑛,𝑖 /
2being a pressure-like term named magnetic normal traction, and the brackets denoting a difference
across the interface. If the second medium is nonmagnetic and viscosity is neglected, the normal balance reduces to the
inviscid boundary condition between magnetizable and nonmagnetizable media obtained in Ref. [78], as it should.
C. Effective total forces
As shown in Ref. [
79
], different equivalent formulations can be employed to compute the total magnetic force
experienced by a body. One of the most common procedures consists on integrating the volume and surface magnetic
11
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force densities as
𝑭𝑚=∫𝑉
d𝑉𝒇𝑉
𝑚+∮𝜕𝑉
d𝑆𝒇𝑆
𝑚,(13)
where
𝑉
and
𝜕𝑉
denote the volume and surface of a magnetized medium, respectively. The surface force distribution is
generated by the discontinuity of the imanation field, and the volume force distribution is given by the well-known
Kelvin force expression given by Eq. 9c. In the case of a magnetic body (
𝑏
) surrounded by a magnetic environment
(
𝑚𝑒
), Eq. 13 may be reformulated by integrating the environmental pressure on the interface. After considering the
quasi-static momentum balance arising from Eq. 1b, the effective total magnetic force results to be
𝑭eff
𝑚=∫𝑉
d𝑉𝒇𝑉 ,eff
𝑚+∮𝜕𝑉
d𝑆𝒇𝑆
𝑚,(14)
where the surface force distribution in 𝜕𝑉 is only due to the discontinuity of the imanation field
𝒇𝑆
𝑚=𝜇0
2𝑀2
𝑛,𝑏 −𝑀2
𝑛,𝑚𝑒 𝒏(15)
with 𝒏being the external normal of the body surface 𝜕𝑉 . The effective volume force distribution in 𝑉is
𝒇𝑉 ,eff
𝑚=𝜇0𝜒vol
𝑏𝐻∇𝐻−𝜒vol
𝑚𝑒 𝐻∗∇𝐻∗,(16)
where
𝐻∗
is the virtual magnetic field that would be present if the volume
𝑉
was occupied by the environment. The
same expression can be obtained by applying the Archimedes’ principle.
If the system is in thermodynamic equilibrium, the total force can be also computed by integrating the magnetic
stress force in the external contour
𝜕𝑉+
[
79
]. Taking again into account the Archimedes’ principle, the effective magnetic
force acting on the magnetic medium 𝑏can be formulated as
𝑭eff
𝑚=∮𝜕𝑉
d𝑆𝒏·T+
𝑚− (T∗
𝑚)+,(17)
where T
+
𝑚
is the magnetic stress tensor in the external contour
𝜕𝑉+
when the volume
𝑉
is occupied by the medium
𝑏
,
and (T∗
𝑚)+is the magnetic stress tensor at the same points computed as if the volume 𝑉was part of the environment.
A third equivalent formulation of the effective magnetic force can be obtained by applying the Principle of Virtual
Works to the free energy variation of a magnetizable medium caused by changes in the applied magnetic field
𝑯0
. The
result is a well-known expression [47, 76, 79] that modified as before results in
𝑭eff
𝑚=𝜇0∫𝑉
d𝑉(𝜒vol
𝑏𝑯−𝜒vol
𝑚𝑒 𝑯∗) · ∇𝑯0.(18)
12
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The previous expressions constitute the formulation of the Archimedes’ principle for the magnetic component of the
external fields. The effective electric and inertial forces can be obtained by following the same procedure, with the latter
being given by
𝑭eff
in =∫𝑉
d𝑉(𝜌𝑏−𝜌𝑚𝑒)𝒈.(19)
D. Magnetic susceptibility
The magnetic susceptibility is an intrinsic property of the medium that defines the relation between the fields
𝑴
and
𝑯
, which are aligned in soft magnetic materials. Diamagnetic and paramagnetic substances generally have small and
constant volume susceptibility values. Ferrofluids, on the contrary, are characterized by large susceptibilities and a
non-linear dependence between
𝑴
and
𝑯
. Magnetic susceptibilities are commonly expressed per unit volume (
𝜒vol
),
mass (𝜒mass), or mole (𝜒mol ) in the international or CGS systems [80].
Since
𝐾𝑂 𝐻
and
𝑁𝑎𝑂 𝐻
solutions are widely employed in water electrolysis technologies, a brief analysis of their
magnetic susceptibility is here presented. Assuming that dipole-dipole interactions are negligible, Wiedemann’s
additivity law states that
𝜒mass
sol =
𝑁
Õ
𝑖=1
𝑝𝑖𝜒mass
𝑖,(20)
where
𝜒𝑚𝑎𝑠 𝑠
sol
is the mass susceptibility of the solution, and
𝑝𝑖
is the mass fraction of each substance. Equivalent
expressions are found for volume and molar susceptibilities [
80
]. The magnetic susceptibility of diluted salts can be
computed as [81]
𝜒mass
salt =𝜒mol
cation +𝜒mol
anion
Msalt
,(21)
with Msalt being the molar mass of the salt. The susceptibilities of the ions are expressed per unit mole, as commonly
reported in the literature. Values for 𝐾𝑂 𝐻 and 𝑁 𝑎𝑂𝐻 solutions are given in Table 1.
The approximate evolution of the magnetic susceptibility of
𝐾𝑂 𝐻
and
𝑁𝑎𝑂 𝐻
solutions with the solute mass fraction
is reported in Fig. 3, where the solubility of the solutions is taken from Ref. [
83
] at
25°
C and a constant solution
volume is assumed. Since the magnetic force is directly proportional to the magnetic susceptibility, this result implies
that liquid electrolytes are particularly well suited for magnetic buoyancy applications, with increases of magnetic
susceptibility of up to an 80%. Since PEM cells employ deionized water in contact with the electrodes, the magnetic
Table 1 Relevant magnetic parameters of alkaline electrolytes expressed in the CGS system [81, 82].
Solute Msolute
[g/mol]
𝜒mol
cation/10−6
[cm3/mol]
𝜒mol
𝑂𝐻 −/10−6
[cm3/mol]
𝜒mass
𝐻2𝑂/10−6
[cm3/g]
NaOH 39.9971 -6.8 -12 -0.720
KOH 56.1056 -14.9
13
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Fig. 3 Volume magnetic susceptibility of KOH and NaOH solutions as a function of the mass fraction of solute
neglecting dipole interaction
susceptibility of water should be employed in the calculations.
IV. Bubble dynamics
The magnetic buoyancy force can produce significant effects in the generation and evolution of gas bubbles over
electrodes or boiling surfaces. Such effects have been observed in experiments involving electric fields [
27
] and
ferrofluids subject to magnetic fields [
48
,
49
]. In consequence, understanding this process is of major importance for
future applications.
Fig. 4 Conceptual stages of single bubble evolution when subject to an inhomogeneous magnetic field in
microgravity. Detachment occurs when the vertical momentum balance is no longer satisfied, inducing a
microconvection flow in the surrounding liquid. The bubble subsequently accelerates until viscous drag 𝑭𝜈
compensates the magnetic buoyancy force, reaching the terminal velocity.
14
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The evolution of an isolated gas bubble subject to an inhomogeneous magnetic field in microgravity can be studied
as a four-step process, represented in Fig. 4: nucleation, growth, detachment, and transport. The magnetic force should
not produce significant effects in the nucleation phase, but may impact the rest. Although microgravity experiments
show that the actual electrolysis reaction is significantly more complicated due to the formation of a layer of bubbles and
their coalescence [
13
–
16
], the tools here introduced are still useful to draw fundamental conclusions. A comprehensive
chemical analysis of the bubble nucleation process can be found in Ref. [84].
A. Growth
The quasi-static momentum balance is one of the fundamental and most widely extended tools to study bubble
growth. Let’s consider a liquid environment with density
𝜌𝑓
and a body consisting on a single gas bubble with volume
𝑉
, density
𝜌𝑔
, and surface tension
𝜎
. The bubble is sitting on an horizontal electrode with apparent contact angle
𝜃
while subject to an inertial acceleration
𝒈
. In the absence of dynamic forces, the momentum balance can be obtained as
done in Ref. [28] for the electric interaction, resulting in
∫𝑉
d𝑉 𝜌𝑔𝒈+∫𝐶𝐿
d𝐿𝜎 𝒕𝑓 𝑔 +∮𝜕𝑉
d𝑆𝒏·T+
𝑝+∮𝜕𝑉
d𝑆𝒏·T+
𝑚=0,(22)
where
𝐶𝐿
denotes the circular contact line of diameter
𝐷0
, and
𝒕𝑓 𝑔
is the tangent unit vector in the meridian plane.
It should be noted that
𝜕𝑉
, that can be decomposed as a surface
𝜕𝑆
on the liquid face of the gas-liquid interface and
surface
𝐴
delimited by CL in the gas region, denotes a complete surface enclosing the pinned bubble volume
𝑉
. The
pressure term can be expanded as
∮𝜕𝑉
d𝑆𝒏·T+
𝑝=−∮𝜕𝑉
d𝑆 𝑝∗
𝑓𝒏+∫𝐴
d𝑆(𝑝∗
𝑓−𝑝𝑔)𝒏,(23)
where
𝑝∗
𝑓
is the virtual composite pressure applied to the magnetic fluid if it occupied the bubble volume
𝑉
. The term
(𝑝∗
𝑓−𝑝𝑔)
is the virtual fluid overpressure with respect to the gas flow pressure evaluated at the plane
𝐴
. In quasi-static
conditions, the first term in the right equals the inertial and magnetic flotability forces acting on the bubble, and Eq. 22
can be reformulated as
∫𝐶𝐿
d𝐿𝜎 𝒕𝑓 𝑔 +∫𝐴
d𝑆(𝑝∗
𝑓−𝑝𝑔)𝒏+𝑭eff
in +𝑭eff
𝑚=0,(24)
where
𝑭eff
𝑚
is given by Eq. 17 or, equivalently, Eqs. 14 or 18, and
𝑭eff
in
is defined by Eq. 19. For practical purposes, it is
useful to project Eq. 24 on an axis 𝒌perpendicular to 𝐴, which results in
𝐹𝑏+𝐹𝑝+𝐹𝜎+𝐹𝑚=0,(25)
15
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with the buoyancy, internal overpressure, surface tension, and magnetic forces being given by
𝐹𝑏=𝒌·𝑭eff
in ≈𝑉𝜌𝑔−𝜌𝑓𝒌·𝒈,(26)
𝐹𝑝=𝜋𝐷 2
0
4𝑝𝑔−𝑝∗
𝑓,(27)
𝐹𝜎=∫𝐶𝐿
d𝐿𝜎 𝒌·𝒕𝑓 𝑔 ≈ −𝜋𝐷0𝜎sin 𝜃, (28)
𝐹𝑚=𝒌·𝑭eff
𝑚,(29)
and where uniform fluid density and overpressure on
𝐴
have been assumed. For water-gas solutions, with susceptibilities
of the order of
|𝜒vol| ≈
10
−6
, the magnetic fields in Eq. 18 can be approximated as
𝑯,𝑯∗≈𝑯0
. The total force exerted
on a small, spherical, gas bubble is then
𝑭eff
𝑚≈2
3𝜋𝑅3
𝑏𝜇0Δ𝜒vol∇𝐻2
0,(30)
where
𝑅𝑏
is the radius of the bubble and with
Δ𝜒vol =𝜒vol
𝑏−𝜒vol
𝑒
denoting the differential magnetic susceptibility
between gas and the water environment. This approach has been employed in previous works on dielectric manipulation
in low-gravity [53, 54]. For the quasi-axisymmetric case, Eq. 29 can be then approximated by
𝐹𝑚≈2
3𝜋𝑅3
𝑏𝜇0Δ𝜒vol 𝜕𝐻2
0
𝜕𝑧 .(31)
The momentum balance may consider a forced viscous shear flow by including the viscous stress tensor and its associated
lift and drag expressions [85].
B. Detachment
The detachment of the bubble is produced when the balance of vertical forces cannot longer be satisfied with
increasing volume [
27
]. In this context, the magnetic force
𝐹𝑚
can be employed to accelerate the detachment process or,
equivalently, reduce the critical bubble volume.
Alternative simplified expressions can be developed to estimate the bubble detachment radius. In boiling and heat
transfer research, the maximum break-of diameter of a bubble on an upward facing surface is usually estimated form
Fritz’s equation [86]
𝑑0=1.2𝜃r𝜎
𝑔𝜌𝑓−𝜌𝑔.(32)
If the bubble is sufficiently small, the magnetic force may be approximated by a constant, uniform field. The magnetic
16
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Fritz equation would then be rewritten as
𝑑0=1.2𝜃r𝜎
𝑓𝑚+𝑔𝜌𝑓−𝜌𝑔,(33)
with
𝑓𝑚=𝐹𝑚/𝑉
being the overall magnetic body force density (in N/m
3
). The departure diameter may deviate from this
result due to the microconvection flow associated to the detachment process [
86
] and the interactions between adjacent
bubbles [
13
,
14
,
16
]. Furthermore, in electrolysis applications the break-of diameter also depends on the surface current
density through an expression of the form [87]
𝑑𝑏
𝑑0
=1+𝑘1𝐼/𝐴
[𝐴/𝑚−2]−𝑘2
,(34)
where 𝑘𝑖are fitting parameters, 𝐼is the electrode current intensity, and 𝐴is the effective surface of the electrode.
C. Displacement
The movement of a spherical bubble within a liquid can be described by the balance between buoyancy and viscous
forces
𝑚0
𝑏
𝑑2𝒙
𝑑𝑡2=𝑭eff
𝑚+𝑭eff
in +𝑭𝑅,(35)
with
𝑚0
𝑏=(
4
/
3
)𝜋𝑅3
𝑏(𝜌𝑔+
0
.
5
𝜌𝑓)
being the virtual mass of the bubble [
88
], and
𝑭𝑅=−
6
𝜋𝑅𝑏𝜂(𝑑𝒙/𝑑𝑡 )
the viscous
drag according to Stokes’ law (
𝑅𝑒
1). By making use of the simplified total force expression given by Eq. 30, the
momentum balance is reduced to
𝜌𝑔+1
2𝜌𝑓𝑑2𝒙
𝑑𝑡2≈𝜇0
2Δ𝜒vol∇𝐻2
0+𝜌𝑔−𝜌𝑓𝒈−9𝜂
2𝑅2
𝑏
𝑑𝒙
𝑑𝑡 .(36)
Small gas bubbles experience large accelerations due to their low density, rapidly reaching a steady-state dynamic
regime. This justifies the employment of the terminal velocity, defined as the steady-state velocity of the bubble, as a
physically meaningful parameter. The terminal velocity can be derived from Eq. 36, resulting in
𝒗𝑡≈2𝑅2
𝑏
9𝜂h𝜇0
2Δ𝜒vol∇𝐻2
0+ (𝜌𝑔−𝜌𝑓)𝒈i(37)
The validity of this expression is limited to small bubbles and low-susceptibility gases and liquids. Similar formulations
can be found in the literature [51, 53].
17
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V. Numerical analysis
A series of numerical results are here presented to better understand low-gravity magnetic buoyancy and its
applications in electrolysis and phase separation. This preliminary analysis and system sizing is made based on the
previously introduced expressions. For this purpose, an N52 neodymium magnet, one of the strongest categories
commercially available, is considered. The magnet is characterized by a magnetization of 1150 kA/m. Relevant
physicochemical properties of water, gas hydrogen, and gas oxygen at 25°C and 1 atm are given in Table 2.
A. Electrically neutral media
The effects of magnetic buoyancy on electrically neutral media, which are the main subject of this work, are first
addressed. This includes pure water in contact with the external face of PEM electrodes and alkaline electrolytes outside
the
𝑂𝐻−
transport region. Although in the second case the presence of charged electrodes leads to a local distribution
of charge, the Debye length [
89
] of such distribution becomes about 0.1 nm for
𝑁𝑎𝑂 𝐻
and
𝐾𝑂 𝐻
solutions in water in
standard conditions. That is, the alkaline electrolyte outside the
𝑂𝐻−
transport region can be considered electrically
neutral, and hence unaffected by Lorentz’s electric and magnetic force terms.
To illustrate the magnetic buoyancy concept, the volume force density
𝑓𝑉 ,eff
𝑚
(Eq. 16) is first computed by means of
finite-element simulations in Comsol Multiphysics. The equations and boundary conditions of the magnetic model
are similar to the ones employed in Ref. [
60
]. Figure 5 represents the radial cross-section of the volume force density
field induced by a cylindrical magnet with 1 cm radius and 0.5 cm height in a
𝑂2
bubble. Due to the small magnetic
susceptibility of water, values of 1 nN/mm
3
, corresponding to an inertial acceleration of
≈
1 mm/s
2
, are reached at 2 cm
from the surface of the magnet. In contrast, an hypothetical square PEM cell with an electrode surface of 2 cm2and a
potential difference of 1.2 V exerts a dielectric force of 10
−5
to 10
−1
nN/mm
3
on a gas bubble sitting on the electrode. It
is then justified to neglect the dielectric force for the applications here considered.
Figure 6 shows the terminal velocity field (Eq. 37) of a 1 mm radius
𝑂2
bubble immersed in water and subject to the
influence of a permanent neodymium magnet in microgravity (
𝑔≈
0). The red arrows, solid lines, and dashed lines
correspond to the non-scaled velocity vector, the constant velocity contours, and the magnetic flux lines, respectively.
Three different cylindrical magnets magnetized along the axis are studied, the first (a) with 10 mm radius and 5 mm
Table 2 Relevant physicochemical properties of water, gas hydrogen, and gas oxygen at 25°C and 1 atm [83].
Material M
[g/mol]
𝜌
[kg/m3]𝜒vol 𝜂
[Pa·s]
𝐻2𝑂(l) 18.015 997 −9.1·10−60.0009
𝐻2(g) 2.016 0.082 1·10−10 -
𝑂2(g) 31.999 1.308 3.73 ·10−7-
18
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0.1
1
10
100
1000
1000
Magnetic flux lines
Force contour
Force direction
Fig. 5 Radial cross-section of the magnetic force density induced by a cylindrical magnet in an 𝑂2gas bubble.
The red arrows, solid lines, and dashed lines represent the non-scaled force vector, the constant force contours,
and the magnetic flux lines, respectively.
height, the second (b) with 20 mm radius and 5 mm height, and the third (c) with 10 mm radius and 20 mm height.
The velocity vectors point towards the magnets, which adopt the role of a bubble sink. This effect can be employed to
induce phase separation and the detachment of gas bubbles from an electrode or boiling surface in microgravity. The
performance of the magnets is hampered by the rapid magnetic field decay, leading to terminal velocities of the order of
1 mm/s at approximately 15 mm from their surface. Larger velocities are experienced in the corners of the magnets,
where the magnetic field gradient is maximum.
The magnetic body force is proportional to the gradient of the magnetic field
𝑯
and its module. When a quasi-uniform
field is generated, as observed near the axis of Fig. 6b, the magnetic forces and terminal velocities are reduced. It is
then convenient to select a magnetic configuration that maximizes the force exerted on the bubbles. Similar problems
appear in biomedical applications dealing with magnetic drug delivery and targeting [
90
–
93
] or magnetic resonance
imaging [
94
,
95
], and have been faced by means of Halbach magnet arrays. A Halbach magnet array is an arrangement
of permanent magnets that reinforces the magnetic field on one side of the array and cancels it on the other [
96
]. These
characteristics are convenient for space applications, where the performance of the magnet should be maximized, and its
electromagnetic interference and mass should be minimized.
Figure 7 represents a linear array of five 1
×
1
×
0
.
5cm
3
neodymium magnets configured considering (a) aligned
magnetizations, and (b) Halbach-oriented magnetizations. As in Fig. 6, the terminal velocity map computed with Eq.
37 is represented. It can be observed how the Halbach configuration produces an asymmetrical magnetic field and a
19
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0 1 2 3 4
r (cm)
-3
-2
-1
0
1
2
3
z (cm)
0.01
0.1
1
10
(a) Base configuration
0.1
1
10
(b) Radially extended
0.1
1
10
10
Magnetic flux lines
Velocity contour
Velocity direction
(c) Axially extended
Fig. 6 Radial cross-section of the microgravity terminal velocity 𝑣𝑡induced by a cylindrical magnet in an 𝑂2
gas bubble with 1 mm radius in water. The red arrows, solid lines, and dashed lines represent the non-scaled
velocity vector, the constant velocity contours, and the magnetic flux lines, respectively.
more homogeneous terminal velocity distribution, with the 1 mm/s contour line staying at approximately 2 cm from the
magnets along the x axis. However, the terminal velocity is shown to decay faster than in the linear configuration, as
exemplified by the 0.1 mm/s line. This characteristic may guide the design of future phase separators. For instance,
the linear configuration may be more suitable for the gas collection process due to the convergence of the velocity
vectors towards the extremes of the magnet, while the Halbach array may produce a more homogeneous magnetic force
distribution over the electrodes.
These results can be easily extended to the
𝐾𝑂 𝐻
or
𝑁𝑎𝑂 𝐻
solutions studied in Sec. III.D by noting the linear
dependence of the terminal velocity with the volume magnetic susceptibility
𝜒vol
. Because this parameter is a 60-80%
larger than that of pure water, the performance of the system would be greatly improved. Similar effects would be
observed in applications involving ferrofluids, whose magnetic susceptibility can be of the order 10. Without considering
the many technical difficulties associated with their operation, such technologies could easily reach magnetic force
values equal or larger than the acceleration of gravity. This may lead to large improvements in the productivity of the
cell both on Earth and in space.
A second effect on interest arising from the application of an inhomogeneous magnetic field to a nucleation surface
is the potential reduction of the break-of diameter. This is explored in Fig. 8 for an isolated bubble by making use of
Eq. 33. A 10 mm radius, 5 mm height cylindrical magnet is considered in microgravity, assuming a contact angle
of
𝜃=5°
. The magnetic Fritz equation predicts a reduction of the break-of diameter from 10 cm to few millimeters
as the bubble approaches the magnet. Without considering the variations in contact angle and surface tension, the
20
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(a) Uniform magnetization
(b) Halbach array
Fig. 7 Two-dimensional simulation with 1 cm depth of the microgravity terminal velocity 𝑣𝑡induced by an
array of magnets in an 𝑂2gas bubble with 1 mm radius in water. The black arrows, red arrows, solid lines, and
dashed lines represent the magnetization direction, non-scaled velocity vector, the constant velocity contours,
and the magnetic flux lines, respectively.
employment of saturated
𝐾𝑂 𝐻
/
𝑁𝑎𝑂 𝐻
solutions would reduce the diameter by a 25% due to the increase in magnetic
susceptibility. On the other hand, no significant differences are observed between
𝑂2
or
𝐻2
gas bubbles due to their
small magnetic susceptibility. These predictions should however be taken with care, as the magnetic Fritz equation
assumes an homogeneous magnetic force in the bubble volume, and this assumption is being violated in a significant
portion of the solution domain. Even if this was not the case, the Fritz equation describes the detachment of an isolated
bubble. Experimental observations have shown that the break-of diameter in microgravity is actually much smaller due
to the interaction between bubbles located in the first layer over the electrodes [
13
,
14
,
16
]. Numerical simulations
based on the framework of analysis presented in Sec. IV.A and experimental results are then required to shed light on
this problem.
21
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1
1
10
100
Magnetic flux lines
Break-of diameter contour
(a) Break-of diameter distribution (b) Break-of diameter in axis of symmetry
Fig. 8 (a) Radial cross-section of the break-of diameter 𝑑0induced by a 10 mm radius, 5 mm height cylindrical
magnet in an 𝑂2gas bubble in water with 𝜃=5°. The solid and dashed lines represent the constant break-of
diameter contours and the magnetic flux lines, respectively. (b) Break-of diameter in the axis of symmetry for
different gas-liquid combinations.
Non-magnetic cell components have been considered throughout this discussion. However, electrodes and bipolar
plates are made of diamagnetic (carbon), paramagnetic (titanium), or ferromagnetic (nickel, ferritic stainless steel)
materials. Those from the third group, with relative permeabilities up to 2000, can be strongly magnetized by external
fields and modify significantly their local magnetic force distributions. If not taken into consideration, these disturbances
may lead to the undesired accumulation of bubbles at the surface of the electrodes. Although the local effect needs to
be evaluated in a case-by-case basis, it can become important for massive, ferromagnetic electrodes subject to strong
magnetic fields. In particular, corner geometries will tend to generate magnetic singularities, leading to the generation
of bubble sinks (as it happens in the well-known lightning rod effect [97]).
B. Effect of magnetic field in unbalanced electrolyte
Lorentz’s force must be considered when an electromagnetic field is applied to unbalanced electrolyte solutions,
adopting the form
L=𝜌𝑉𝑬+𝑱𝑒×𝑩.(38)
As with the diamagnetic force, a buoyancy effect is induced on the gas bubbles. In PEM electrolysis the only volume
where there is a charge unbalance is the membrane itself, where a highly acidic medium is created in the presence of
22
The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021
Accepted Manuscript
Fig. 9 Alkaline cell where the charge unbalance in the 𝑂𝐻−transport region leads to a magnetic Lorentz
buoyancy effect in the presence of an out-of-plane magnetic field 𝑩.
water. Assuming a current density of 1 A/cm
2
, a magnetic field of 1 T, a potential difference between electrodes of 1.2
V, and a membrane thickness of 100
𝜇
m in an acidic solution with pH 1, the electric term dominates over the magnetic
term by a factor 10
7
. This factor increases for more acidic solutions, so it can be concluded that the imposed magnetic
field has virtually no effect in the solid electrolyte.
As for alkaline electrolysis, previous works have reported the effects of an external magnetic field in the productivity
of alkaline cells when such field is applied to the
𝑂𝐻−
transport region [
62
,
63
,
65
,
66
,
69
]. For instance, the setup
depicted in Fig. 9 employs two parallel flat electrodes immersed in an alkaline electrolyte to which a constant magnetic
field is imposed. The magnetic field is applied parallel to the plane of the electrode, and since the mean electric current
density vector
𝑱𝑒
is perpendicular to such electrode, a vertical force is induced by the magnetic term in Eq. 38. With a
current density of 0.5 A/cm
2
and a characteristic magnetic field of 1 T, a Lorentz buoyancy force of 5000 nN/mm
3
would be generated. This term is several orders of magnitude larger than the diamagnetic force studied in Sec. V.A and
could lead to interesting low-gravity applications. However, the need to generate gas bubbles between the electrodes
may raise safety concerns in space applications, where the recombination of products represents a critical safety hazard.
Such bubbles would also modify the local current flow, leading to more complex microfluidic interactions arising from a
non-uniform Lorentz force distribution [98, 99].
C. Scale-up process
Many times innovations at the sub-cell or cell levels do not survive the ‘scale-up’ process from a single cell to
full-size stack. It is then convenient to give some hints on how such process should be carried out for the diamagnetic
cell architectures here introduced.
23
The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021
Accepted Manuscript
Fig. 10 Two-dimensional simulation with 10 cm depth of the microgravity terminal velocity 𝑣𝑡induced by an
array of magnets in an 𝑂2gas bubble with 1 mm radius in water. The hypothetical location of the Membrane
Electrode Assemblies is represented by light gray areas. Black arrows, red arrows, and solid lines represent the
magnetization direction, non-scaled velocity vector, and constant velocity contours, respectively.
Two main scale-up strategies may be followed: either a continuous magnetic sheet with Halbach-like arrays (like the
one represented in Fig. 7(b)) is located in parallel to the electrodes, or a series of magnets are strategically positioned
to collect the bubbles. In both cases, the magnetic system can be adapted to any cell surface. However, the second
approach leads to important mass savings. This is shown in Fig. 10, where a 1 kg array of twelve 1
×
1
×
10 cm
3
magnets
is employed to induce diamagnetic buoyancy at the surface of three 100 cm
2
PEMs. The bubble velocity vectors point
toward the magnets, that can be used as gas collection points. This design can be largely improved by optimizing the
distribution of magnets in the
𝑧
axis, or by employing anode- and cathode-feed PEM architectures where only one side
of the membrane requires phase separation.
In addition to selecting an efficient magnetic architecture, the movement of the bubbles should be constrained by
means of an optimized wall (or bipolar plate) profile. Such profile would be adapted to the magnetic force potential to
push the bubbles toward specific collection points, where the gas is finally extracted. Hydrophobic and hydrophilic
surfaces may be employed to induce the accumulation and coalescence of bubbles.
VI. Conclusions
The applications of diamagnetic buoyancy in low-gravity electrolysis, boiling, and phase separation have been
introduced. The diamagnetic force can be employed to induce the early detachment of gas bubbles from the electrodes,
increasing the effective surface area and effectively separating the phases. A comprehensive theoretical analysis of the
problem has been presented together with simplified expressions that ease preliminary studies.
24
The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021
Accepted Manuscript
Numerical simulations are employed to show how modern neodymium magnets induce a significant magnetic force
in gas-water flows at distances of the order of 2 cm. The reachability of the system is increased by an 80% when
saturated
𝑁𝑎𝑂 𝐻
and
𝐾𝑂 𝐻
electrolytes are considered. For unbalanced alkaline electrolytes, the magnetic term of the
Lorentz force can lead to strong magnetic buoyancy forces, an effect that may open interesting avenues for research.
Finally, potential approaches to scale up the diamagnetic electrolysis architecture have been suggested.
There are several scientific and technical questions of interest that need to be solved before diamagnetic buoyancy
is employed in low-gravity technologies. A non extensive list includes the characterization of the bubble collection
process, the experimental and numerical study of magnetically-induced bubble detachment, the development of reliable
gas collectors, or the analysis of applications employing ferrofluids, whose larger magnetic susceptibilities and a strong
magnetic response would lead to significant increases of the magnetic force, also enabling terrestrial applications.
Appendix A: Governing equations for incompressible fluids subject to static magnetic fields
The magnetic phase separation and bubble detachment concept discussed in this paper can be applied to different
electrolysis and boiling technologies. However, most of them share four important characteristics: (i) the fluids involved
are treated as incompressible, (ii) steady magnetic fields are imposed, (iii) para/diamagnetic substances are employed,
and (iv) viscous coefficients are considered constant. It is then useful to particularize Eqs. 1 and 2 to this case.
Under the previous assumptions, the magnetohydrodynamic mass and momentum conservation equations become
∇ · 𝒗=0,(39a)
𝜌𝐷𝒗
𝐷𝑡 =𝜌𝒈− ∇ 𝑝+𝜂∇2𝒗+𝜇0𝑀∇𝐻 , (39b)
subject to the boundary conditions
𝑛:𝑝∗−2𝜂𝜕𝑣 𝑛
𝜕𝑥𝑛
+𝑝𝑛+2𝜎H=0,(40a)
𝑡:𝜂𝜕𝑣 𝑛
𝜕𝑥𝑡
+𝜕𝑣 𝑡
𝜕𝑥𝑛 =0,(40b)
The steady state Maxwell’s equations in the absence of surface currents are employed to compute the magnetic fields,
giving
∇ · 𝑩=0,(41a)
∇ × 𝑯=𝑱𝑒,(41b)
25
The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021
Accepted Manuscript
due to the absence of electric fields. The simplified electric and magnetic boundary conditions at the interface are
𝒏· (𝑩2−𝑩1)=0,(42a)
𝒏× (𝑯2−𝑯1)=0.(42b)
Acknowledgments
The authors acknowledge the financial support offered by the la Caixa Foundation (ID 100010434) under agreement
LCF/BQ/AA18/11680099 to support the PhD studies of A.R.C. The assistance of Isabel Romero Calvo and Morphology
Visuals in the formatting of the figures presented in this manuscript is gratefully acknowledged.
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