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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

The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021

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[

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

The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021

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

The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021

Accepted Manuscript

(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

The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021

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

The final version of this paper can be found in http://arc.aiaa.org/doi/abs/10.2514/1.A35021

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[

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)

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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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Accepted Manuscript

(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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Accepted Manuscript

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

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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

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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

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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

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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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