Content uploaded by Álvaro Romero-Calvo

Author content

All content in this area was uploaded by Álvaro Romero-Calvo on Feb 06, 2022

Content may be subject to copyright.

AAS 091

MICROGRAVITY RESTART OF LIQUID ROCKET ENGINE WITH

LOW PROPELLANT RESIDUALS

´

Alvaro Romero-Calvo∗

, Vladislav Urbansky †

, Vadim Yudintsev ‡

, Hanspeter

Schaub§

, and Valeriy Trushlyakov¶

The active deorbiting and passivation of launch vehicles has become key for the implemen-

tation of modern space debris mitigation guidelines. Appropriate engine restart conditions

must be provided as part of this process. Ullage motors have been traditionally employed

to induce active settling and ensure a gas-free propellant supply to the engines. Although

robust and reliable, ullage rockets are also heavy, which motivates the study of alternative

approaches to the problem. This paper explores for the ﬁrst time several high-risk-high-

return propellant settling strategies that may result in signiﬁcant beneﬁts for future space

systems. In particular, three distinct Magnetic Positive Positioning concepts, a hydrogen-

peroxide-based Propellant Gasiﬁcation System, and a hybrid device that combines both ap-

proaches are introduced. The preliminary feasibility analysis indicates that the successful

development of these technologies may lead to mass savings of hundreds of kilograms and

economic gains of several hundred thousand dollars per launch. However, the robustness

of some of these methods may be compromised by complex ﬂuid-structure interactions that

require a careful numerical and/or experimental analysis.

INTRODUCTION

The exploration and commercialization of space has led to the increasing contamination of the Low Earth

Orbit (LEO) by non-functional man-made objects. Space debris represents a serious safety hazard for current

and future satellites due to the risk of in-orbit collisions, and a concern for the general population during

uncontrolled re-entry events. The minimization of debris release during normal operations has consequently

become a major goal for the international space community.1

Launch vehicles (LVs) represent more than 42% of the abandoned intact objects in orbit and account for

57% of the abandoned mass.2Recent studies have identiﬁed the most dangerous elements in an attempt

to guide future active debris removal efforts,2–4 resulting in a list that includes 290 second stages of the

Soviet/Russian “Cosmos-3M” LV, 7 of which have been considered among the 50 most concerning space

debris objects,4110 third stages of the Soviet/Russian “Cyclone-3” LV, 54 units of the American ﬁrst and

second stages of the “Delta” LV, as well as 38 third stages of the Chinese “CZ-4” and “CZ-2D” LVs. Further

concerns arise due the presence of propellant residuals in the tanks, which account for up to ∼3% of the initial

fuel mass.5During the long stay of a stage in orbit, the remaining fuel evaporates under the inﬂuence of solar

radiation, which leads to an increased risk of explosion in orbit and, therefore, to the generation of space

debris.1The uncontrolled descent of ﬁrst LV stages in sensitive drop areas can also lead to environmental

∗Graduate Research Assistant, Department of Aerospace Engineering Sciences, University of Colorado, 431 UCB, Colorado Center for

Astrodynamics Research, Boulder, CO 80309-0431. Contact: alvaro.romerocalvo@colorado.edu

†Graduate Research Assistant,Aircraft and Rocket Building Department, Omsk State Technical University, pr. Mira 11, 644050 Omsk,

Russian Federation. Contact: vladurba95@gmail.com

‡PhD, Aircraft and Rocket Building Department, Omsk State Technical University, pr. Mira 11, 644050 Omsk, Russian Federation.

Contact: yudintsev@gmail.com

§Professor, Glenn L. Murphy Chair, Department of Aerospace Engineering Sciences, University of Colorado, 431 UCB, Colorado Center

for Astrodynamics Research, Boulder, CO 80309-0431. AAS Fellow. Contact: hanspeter.schaub@colorado.edu

¶Professor, Aircraft and Rocket Building Department, Omsk State Technical University, pr. Mira 11, 644050 Omsk, Russian Federation.

Contact: vatrushlyakov@yandex.ru

1

pollution caused by the depressurization of toxic fuels, ﬁres in the drop sites, and the contamination of water

bodies. This problem is more relevant for Russian LVs like “Soyuz”, “Proton”, and “Angara”, where most of

the drop zones are located on land.6, 7

Modern launch vehicle operations are subjected to strict space debris mitigation policies.1When graveyard

orbits are not attainable, the orbital lifetime is limited and systems are passivated by removing all energy

sources. Active deorbiting represents a highly attractive alternative to those strategies, but it is not exempt

from risks and technical challenges.8Among them, proper engine restarting conditions must be provided

once the stage is separated from the rest of the vehicle in order to ensure a safe reorbiting or reentry. This

decoupling induces strong disturbances on the propellant residuals and leads to highly non-linear sloshing

dynamics, compromising the operation of the engine feed system.9

Ullage engines have been traditionally employed to settle the propellant residuals during insertion, orbital

coast, or on-orbit operations in an approach known as active settling.10 These independent rockets induce

accelerations that can be as weak as 10−4to 10−3m/s2and involve solid, mono-propellant, bi-propellant, or

cold gas technologies, sometimes fed by vaporized propellant vented from the main tanks.11 Some examples

include the Saturn IV-B’s hypergolic liquid bi-propellant Auxiliary Propulsion System (APS),12,13 SpaceX’s

Falcon 9 nitrogen cold gas thrusters for coast attitude control,14 or the two Sistema Obespecheniya Zapuska

(SOZ) ullage engines of the Blok DM-2 Proton upper stage. This last unit has raised concerns in the space

debris community after being responsible for up to 50 on-orbit explosions between 1984 and 2019.15

The technical speciﬁcations of ullage engines are not usually accessible to the general public, which ham-

pers any effort to perform an “external” evaluation of these systems. However, numerous reports from the

Apollo era can still be consulted. The two Saturn IV-B APSs were usually ﬁred in three consecutive ullaging

burns for a total of ∼245 s, consuming ∼13.5 kg of propellant (∼23.5% of the total propellant mass of each

APS).16 The dry mass of the APS is unknown to the authors but seems of the order of several hundred kg

judging by the volume of the system. The dry mass of Saturn IV-B was about 13.5 t. On the other hand, each

one of the two SOZ units of the Block DM-2 upper stage had a dry mass of ∼106 kg and a total propellant

mass of up to 114 kg, while the stage itself weighted 2.1 t. Although determined by the characteristics of the

vehicle and its mission proﬁle, the total mass of the ullage rocket system should be around ∼200 kg.17 With

a Falcon 9’s launch cost to LEO of ∼2700 $/kg,18 an economic penalty per launch and stage of ∼500.000

USD may be estimated. This value is doubled for GEO orbits, and multiplied by an even larger factor in a

Mars mission.

Ullage engines are a robust and well-established solution to deal with the restart of rocket engines in

microgravity conditions. However, that should not preclude the study of alternative approaches to the problem

with lower mass budgets and/or enhanced reliability. In this paper, the feasibility and performance of a

Magnetic Positive Positioning (MP2),19 an on-board Propellant Gasiﬁcation System (PGS),20 and a hybrid

device are explored for the ﬁrst time. The historical background of each system is presented together with a

preliminary technical analysis. The ultimate goal is to initiate an open discussion on these technologies and

inform the design of new-generation propellant settling systems.

LAUNCHER CHARACTERISTICS

Overview

Although applicable to multiple low-gravity propellant settling systems, the discussion that follows focuses

on the ﬁrst and second launch stages of a Falcon-9-like LV. The basic parameters of the vehicle are reported in

Table 1, with the geometrical deﬁnitions being depicted in Fig. 1. Some of these values are found in SpaceX’s

Falcon User’s Guide,14 while others can only be estimated from unofﬁcial sources∗.

Propellant behavior during stage separation

The acceleration proﬁle experienced by each stage during separation is key to understand the dynamic

behavior of the propellant. A simple mechanical model is introduced in the Appendix A and employed to

∗See www.spacelaunchreport.com/falcon9ft.html. Consulted on 13/01/2022.

2

Table 1: Geometrical and inertial parameters of the launch vehicle.

Parameter First Stage Second Stage

Propellant type LOX + RP1 LOX + RP1

Empty mass [t] 22 4.5

Propellant mass [t] 411 111.5

Oxygen tank capacity [t] 287.4 78

Kerosene tank capacity [t] 123.5 33.5

Total mass [t] 433 116

Propellant mass after stages separation [t] 13 3.5

Residual propellant mass after landing [t] 1 0.3

Thrust (stage total) [kN] 7686 981

Throttle capability [kN] 4381 to 7686 626 to 981

Number of engines 9 1

Diameter [m] 3.66 3.66

LOX tank height L0[m] 22.5 8.7

Length of 1 stage h0[m] 29 11.2

Length of fuel tank hg[m] 19.2 7.4

First stage mass center xc[m] 14.9 5.7

Moment of inertia [kg·m2]2.68 ·1063·104

Length to PGS nozzles hn[m] 39 15

Figure 1: Geometrical parameters of a launch vehicle stage.

obtain the acceleration curves reported in Fig. 2 using representative values. Peak accelerations of ∼1m/s2

are applied to the system and sustained for less than 1 s. Additional effects that may impact the propellant

behavior include the release of strain energy from the walls of the tank, the ﬂow movement induced by engine

suction, or thermal convection.11

Modeling this problem is far from trivial, and experimental data is not easily available because of its

consideration as Export-Controlled material. However, a partial recording of Falcon 9’s ﬁrst and second

stage liquid oxygen (LOX) tanks during the CRS 5 mission is publicly available†, allowing for a qualitative

analysis of the problem. Figure 3 shows the sequence of video frames for (a) the instant before second-stage

engine cut off (SECO), (b) the lateral sloshing wave caused by the structural relaxation after SECO, (c) the

cloud of LOX bubbles generated after separation, and (d) the state of the cloud 45 s after separation. It can be

readily concluded that (i) the SECO induces a mild lateral sloshing wave, but does not signiﬁcantly disturb

the liquid, (ii) the stage separation atomizes the residual LOX in a myriad of droplets that reach the top of

the tank in less than 40 s (i.e. the droplets move at least at ∼20 cm/s), and (iii) the droplets keep moving for

at least 6 minutes while coalescing with each other. This behavior is also (brieﬂy) observed in the ﬁrst stage,

where the droplets seem to move at about 0.5 m/s. This value has been employed in the derivation of the

†The interested reader is referred to https://youtu.be/p7x-SumbynI. Consulted on 13/01/2022.

3

2nd Stage

1st Stage

a [m/s2]

−1

−0.8

−0.6

−0.4

−0.2

0

t [s]

0 0.2 0.4 0.6 0.8

Figure 2: Estimated acceleration proﬁle of the 1st and 2nd stages after separation.

(a) Pre-SECO (b) Post-SECO (c) Post-stage-separation (d) 45 s after stage separation

Figure 3: Top view of the second-stage LOX tank of SpaceX’s Falcon 9 during the CRS-5 mission. Source:

www.youtube.com/watch?v=mVAGoWJuDKk

acceleration proﬁles shown in Fig. 2.

Engine restart conditions

The propellant must be settled over the fuel outlet to ensure a safe engine restart. Although this usually

implies bringing all the liquid back to the bottom of the tank before ignition, the requirement may be instead

reformulated by noting that the engines will also accelerate the stage. The goal is thus to have enough

propellant at the outlet so that, when the thrust-induced settling brings all the liquid to the bottom of the tank,

no gas bubbles have made their way into the engines.

The ﬁrst factor to consider is mass ﬂow rate: higher thrust will settle the propellant droplets faster, but will

also require a larger initial mass. The relation between thrust Tand mass ﬂow rate ˙mpis given by

T=Ispg0˙mp,(1)

where Isp is the speciﬁc impulse and g0= 9.81 m/s2is the standard gravity acceleration. For a LOX-RP1

chemical engine with a mass ratio of ∼2.3 the speciﬁc impulse should be around 285 s at sea level.21 In fact,

the old SpaceX website attributed to Falcon 9’s Merlin engines the values of 282 s at sea level and 311 s in

vacuum‡. The second factor is the propellant settling time, that can be divided into two phases. On the ﬁrst,

the propellant residuals return to the intake device, while on the second, gas bubbles are removed from the

liquid. The total settling time can thus be expressed as

ts=tI+tII .(2)

For a tank of length L, the duration of the ﬁrst phase is bounded by the kinematic result

tI=r2Lm0

T,(3)

‡See web.archive.org/web/20130501002858/http://www.spacex.com/falcon9.php. Consulted on:

13/01/2022.

4

with m0being the dry mass of the stage. The duration of the second phase, assuming a constant terminal

velocity of the bubbles in the liquid, is22

tII =l8

3

rbT

Cdmt1−ρg

ρl−1/2

(4)

where rbis the bubble radius, Cdis the drag coefﬁcient, ρgis the gas density, ρlis the liquid density, mtis

the total mass of the stage after settling, and lis the height of the longest liquid column. Consequently, the

initial mass of propellant required to complete the maneuver for a giving thrust level is

mp= ˙mpts=T

Ispg0(r2Lm0

T+l8

3

rbT

Cdmt1−ρg

ρl−1/2),(5)

which increases with √T, showing that small thrust values are convenient to minimize the mass of propellant

required to restart the engines. Table 2 reports the stage acceleration, settling time, and initial oxidizer and

fuel masses for different thrust conﬁgurations of Falcon 9’s ﬁrst and second stages. The values Cd= 0.47

(sphere), rb= 5 mm, ρg= 0.1785 kg/m3(He), ρl= 1141 kg/m3(LOX), and l=mr/(πR2ρl), with Rbeing

the tank radius and mrthe residual LOX mass, are employed in combination with those presented in Table 1

using the larger LOX tank as a reference. The masses reported in Table 2 are the minimum absolute values

required near the fuel outlet to initiate the restart maneuver. Unofﬁcial sources claim that Falcon 9’s ﬁrst stage

restart acceleration is less than 50 m/s2by employing reverse engineered telemetry data§, which indicates that

either the central engines (maximum thrust) or three outer engines (minimum thrust) are actually employed

in this process. However, the authors were not able to verify this information.

First Stage Second Stage

g

[m/s2]

ts

[s]

LOX

[kg]

RP1

[kg]

g

[m/s2]

ts

[s]

LOX

[kg]

RP1

[kg]

Maximum Thrust 350 0.66 1287 401 218 0.39 88 38

Minimum Thrust 200 0.88 971 303 140 0.49 70 30

Single Engine 22 2.63 324 101 140 0.49 70 30

Table 2: Stage acceleration, settling time, and minimum initial oxidizer and fuel masses for different restart

conﬁgurations of Falcon 9’s ﬁrst and second stages.

MAGNETIC POSITIVE POSITIONING

Concept and overview

The ability of controlling the position of susceptible liquids by means of magnetic ﬁelds in microgravity

leads to several potential space applications. Those include, but are not limited to, mass transfer,23–25 thermo-

magnetic convection,26,27 or micropropulsion.28, 29 The volume force density that enables these technologies

is induced by inhomogeneous magnetic ﬁelds on susceptible liquids, and adopts the form

fm=µ0M∇H, (6)

with µ0being the permeability of free space, and Mand Hdenoting the magnetization and magnetic ﬁelds,

respectively. In addition, the magnetic normal traction

pm=µ0

M2

n

2,(7)

should be considered at the liquid interface, where Mnis the normal magnetization component.30 This

pressure-like term is usually neglected for natural liquids, such as LOX, but becomes relevant for highly

§See https://github.com/shahar603/SpaceXtract for a remarkable example of reverse engineering. Consulted on

17/01/2022

5

Figure 4: Magnetic Positive Positioning

susceptible materials like high-density ferroﬂuids.31 Since both Hand its gradient decay with the distance to

the source, the magnetic force vanishes relatively quickly. Therefore, powerful magnets or coils are needed

for most applications.

The Magnetic Positive Positioning approach, sketched in Fig. 4 for the system under study, seeks to induce

a magnetic acceleration that holds, collects, and/or traps the liquid near the fuel outlets. The concept was

ﬁrst proposed in 1963 by Steve Papell in the same patent where he invented ferroﬂuids.32 The idea was

abandoned until 2001 when, motivated by the advent of stronger permanent magnets and high-temperature

superconductors, the NASA Magnetically Actuated Propellant Orientation (MAPO) experiment explored

the positioning of ferroﬂuid solutions in a series of parabolic ﬂights.33 Such ferroﬂuids were selected to

approximate the magnetization curve of LOX for different magnetic ﬁeld intensities. It should be noted

that LOX is the most susceptible natural paramagnetic liquid,34 making it particularly appropriate for this

application. Subsequent publications by Marchetta and coworkers presented reﬁned numerical models and

results of technical relevance for the development of liquid oxygen magnetic positioning devices.35–43 Recent

works have also explored the free surface oscillations of ferroﬂuids in microgravity, which may be relevant for

slosh control and the development of novel propellant management devices (PMDs).44–49 A comprehensive

review of the ﬁeld can be found in Ref. 19.

Signiﬁcant advances have been made in the modeling and fundamental understanding of magnetic positive

positioning devices during the last two decades. However, none of the aforementioned works explored the

feasibility of this approach as part of the operation of LVs. Although limited by the lack of reliable technical

information, this study aims at covering this knowledge gap by exploring the application of magnetic positive

positioning to the restart of Falcon 9’s ﬁrst and second stages.

Passive retention strategy

The ﬁrst and most intuitive approach to magnetic positive positioning is the liquid retention strategy, where

a magnet or coil is used to hold the paramagnetic liquid in the presence of adverse accelerations that tend to

destabilize the free liquid surface. In the classical literature, the critical Bond number

Bo∗=ρg∗R2

σ(8)

is employed to compute the critical acceleration load g∗for which surface tension cannot longer stabilize

the meniscus. Myshkis and coworkers provide a best-case Bo∗=−3.32 for cylindrical tanks at a contact

angle of 90◦,50 which results in g∗= 2.9·10−6m/s2for the LOX tank considered in this work. In other

words, surface tension does little or nothing to prevent the atomization of the residual LOX volume observed

in Fig. 3 under the action of the acceleration loads estimated in Fig. 2. An obvious questions is whether the

magnetic force can hold the liquid against adverse accelerations of ∼1m/s2.

Marchetta and coworkers explore the problem of magnetic LOX retention in a 12 cm diameter 24 cm

height cylindrical tank under the inﬂuence of a point dipole of 1.4 cm diameter and an inertial acceleration

along its major axis.42 The dipole strength required to hold ∼30 ml of liquid is shown to be about 10 Am2

for g∗= 1m/s2. Similarly, in Ref. 19 an analytical model is developed to study the stability of magnetic

interfaces and applied to a 10 cm diameter cylindrical tank, showing that a 60 g magnet can increase the

critical load by 31.5%. None of these low-gravity studies can be easily extended to Falcon 9’s 3.66 m

diameter LOX tank, where the liquid outlet has a diameter of about 90 cm and the maximum estimated

6

Figure 5: Magnetic acceleration contours induced on LOX by a 35 cm diameter coil operating at 1 At.

acceleration is about 1 m/s2. The problem, however, can be easily addressed by plotting the axisymmetric

magnetic acceleration contours induced by a 1 A cylindrical coil on the LOX tank volume as done in Fig. 5.

The ﬁgure depicts the magnetic acceleration levels in a logarithmic scale and its direction using black arrows.

A coil mean diameter of 35 cm is chosen to ensure that the liquid gets attracted toward the PMD located at the

the tank outlet. The minimum mass to be retained is 70 kg (second stage), which translates to a LOX sphere

of 25 cm radius. The magnetic acceleration at this distance is about 10−11 m/s2. Since the acceleration scales

with the square of the current intensity,19 values of ∼106At (i.e. coils current intensity times number of coil

turns) would be required to retain the oxidizer against accelerations of 1 to 10 m/s2. Further computations are

not required to conclude that the mass and/or power requirements of this approach are well beyond reason

with existing technologies, particularly for the ﬁrst stage.

Recovery strategy

The passive magnetic retention strategy sets an upper limit for the magnetic ﬁeld strength. Since this limit

is hard to reach with existing technologies, alternative strategies must be explored. The ﬁrst of them is here

introduced and seeks to collect the LOX droplets after they are atomized rather than holding part of the liquid

at the bottom of the tank¶. In order to evaluate this idea, the time required to settle a LOX droplet is ﬁrst

derived with a simpliﬁed framework of analysis.

Magnetic settling time: If the settling of the propellant is induced by the magnetic interaction and not by a

uniform acceleration g, the derivation of tsis complicated by the presence of an inhomogeneous acceleration

ﬁeld. A strict approach to the problem would require solving the Navier-Stokes equations with a magnetic

force source term. Although less computationally expensive than the fully coupled ﬂuid-magnetic simulations

that are necessary for highly susceptible ferroﬂuids,19 this approach is still prohibitive for a preliminary

study. Instead, the movement of a perfectly spherical droplet along the symmetry axis of an axisymmetric

coil or magnet is analyzed. The droplet is small in comparison with the variation of the magnetic ﬁeld

and exhibits linear magnetization with susceptibility χ1. It is further assumed that external, internal,

and magnetization ﬁelds are collinear, that residual liquid volumes do not contribute to the magnetic ﬁeld,

¶Following the popular saying, “if you can’t beat them, join them”.

7

and that magnetic surface force terms are negligible. In this simpliﬁed framework, the total magnetic force

induced on the liquid droplet by a circular coil with nturns, radius R, and current intensity Iat a distance z

along the symmetry axis ezisk

Fm≈ −3µ0χ(nI)2R4

4

z

(R2+z2)4ez.(9)

This expression can also be applied to axially magnetized cylindrical magnets with magnetization Mm, radius

Rand height lmby considering an equivalent circular coil with the same radius and current intensity nI =

Mmlm. Its main advantage is that it allows the derivation of a quasi-analytical expressions for tI. After

considering Newton’s second law and solving the resulting second-order differential equation with initial

position z(0) = Land initial velocity ˙z(0) = 0, the duration of the ﬁrst phase becomes

tI(L) = s4πρl

µ0χ(nI)2R4ZL

01

(z2+R2)3−1

(L2+R2)3−1/2

dz, (10)

where it should be noted that tIis inversely proportional to nI (or, if a magnet is employed, to Mmlm) and

R2. Of these, only the current intensity can be considered a design parameter, because Ris bounded by the

fuel intake radius (see previous subsection).

The time required to debubble the multiphase mixture near the fuel outlet using the paramagnetic force

can be derived in a similar way, as done in Ref. 52 after adopting the Stokes’ law for viscous drag (which is

valid for Re 1) instead of the drag coefﬁcient. However, the magnetic debubbling process is much faster

than the ﬁrst phase because the liquid is closer to the magnetic source, and hence it is further assumed that

ts≈tI.

Even though the assumptions employed in the derivation of Eq. 9 are not appropriate for highly susceptible

ferroﬂuids, the volume magnetic susceptibility of the liquids employed in this work is bounded by that of

LOX (χLOX = 0.0034 at 90 K and 1 atm34). It would not make sense to use high density ferroﬂuids in the

fuel tank when the most demanding requirements are associated with the LOX tank.

Performance analysis: The time of ﬂight of the droplets for a coil diameter of 35 cm is represented in

Fig. 6 as a function of the initial droplet distance to the coil and the applied current intensity. A LOX settling

time of tmax ≈6minutes, estimated in the analysis of the second stage in Fig. 3, is superposed and treated as

kEquation 9 can also be found in the literature divided by the term (1+χ)2or (1 + Dχ)2, with D= 1/3being the demagnetization

factor of a sphere.19,31, 51 Each choice denotes a different modeling of the internal magnetic ﬁeld inside the droplet. Since in the problem

here considered χ1, the simplest approach, that assumes H0≈H, is adopted.

Max. T.

Min. T.

Max. T.

S.E.

Min. T.

tmax

1st Stage

2nd Stage

102 A·turn

103 A·turn

104 A·turn

105 A·turn

106 A·turn

Time of flight [s]

10−2

10−1

100

101

102

103

104

Initial droplet distance to fuel outlet [m]

0 1 2 3 4 5

Figure 6: Time required by a LOX droplet to reach the bottom of the tank as a function of its distance to the

coil and the applied current intensity. The minimum tank settling length required by the single engine (S.E.),

minimum thrust (Min. T.) and maximum thrust (Max. T.) conﬁgurations is superposed.

8

a deadline for the collection process. However, unofﬁcial telemetry data shows that the ﬁrst stage restarts 2

to 3 minutes after stage separation (see footnote at the end of page 5). As it will be seen, this does not change

the qualitative results of the analysis. It is arbitrarily assumed that, after atomization, the LOX droplets are

uniformly distributed in the tank volume. In this framework, the vertical lines represent the tank length that

needs to be settled for each one of the conﬁgurations detailed in Table 2 before the LOX droplets stop moving

(i.e. get attached to the walls of the tank).

A qualitative difference is ﬁrst observed between ﬁrst and second stages. The LOX mass required to restart

the engines, listed in Table 2, drops by an order of magnitude in the second stage, and thus the length of the

tank that needs to be settled is much smaller under the uniform droplet distribution assumption. The second

factor that should be considered is the density of residual LOX per unit tank length. Although the lower

LOX tank has a length of 22.5 m and the upper of just 8.7 m, the density of residual LOX per unit length

is relatively similar, decreasing from 383.5 kg/m to 267 kg/m, respectively. Figure 6 shows that a coil with

a conﬁguration of 103A·turn can satisfy the requirements of the second stage. However, 104to 105A·turn

are needed to settle the ﬁrst stage using the single engine restart conﬁguration, with 106to 107A·turn being

required for the rest. In other words, the liquid recovery strategy can potentially reduce the coil strength

requirements by one order of magnitude in the ﬁrst stage and three orders of magnitude in the second. These

conclusions do not change if tmax drops to 120-180 s for the ﬁrst stage, as indicated by unofﬁcial telemetry

data.

These results should be taken with care due to the number of assumptions employed in the derivation of

Eq. 10. In particular, ﬂuid-structure interactions have been completely neglected, but Fig. 3 shows that, after

a few minutes, the liquid droplets tend to get stuck to the walls of the tank. This is a natural consequence

of the presence of corner geometries in the interface between PMDs and the walls. The robustness of the

liquid recovery strategy may thus be compromised by this effect, which should be evaluated with ﬂight data

that is not available to the authors. Possible mitigation strategies include the elimination of gaps and corner

geometries or the application of a LOX-phobic treatment to the internal surfaces.

Active retention strategy

The third and ﬁnal approach seeks to soften the requirements imposed on the magnetic system by storing

the restart liquid in a smaller auxiliary tank during the stage separation process. The propellant is later re-

leased near the fuel outlet, where it is held against disturbing accelerations by means of a permanent magnet

or coil. The restart liquid remains thus unaffected by large stage separation accelerations. The main draw-

backs of this method are the addition of ancillary components and the limitation of the number of restart

events.

A representative magnetic conﬁguration is displayed in Fig. 7, which depicts the magnetic acceleration

ﬁeld on LOX of a 5 cm radius 10 cm height cylindrical magnet magnetized at 1300 kA/m. The magnet

geometry differs from that of Fig. 5 in the adoption of a smaller magnet radius, which increases the magnetic

force close to the source.19 The magnet volume is chosen to impose an acceleration of 10−4m/s2(one

order of magnitude larger than microgravity disturbances11) at ∼40 cm from the magnet. This leads to

the approximate LOX volume that needs to be retained in the single-engine ﬁrst stage scenario reported in

Table 2. Although the mass of the magnet is just ∼5.5 kg, ancillary components like the tanks or PMDs must

also be taken into account.

Four 26 cm radius spherical containers are ﬁrst considered to store the LOX volume in the ﬁrst stage. The

four tanks are located around the fuel outlet and outside the main tank space, as shown in Fig. 8a. Although

this choice leaves the internal conﬁguration virtually unaffected, an alternative internal positioning would

minimize the pressure difference supported by the walls, and thus the mass requirements. The total mass of

this system is hard to estimate without access to further technical details, but it would not be surprising to

discover that ancillary components account for most of it. For instance, emptying the tanks in the microgravity

environment would require a bladder12 or outlet magnet45 to ensure a gas-free expulsion and a small pressure

difference applied, for instance, by means of the same helium tank employed to pressurize the LOX vessel. An

alternative system that minimizes the number of ancillary components at the expenses of providing a single

9

Figure 7: Magnetic acceleration ﬁeld induced by a 5 cm radius 10 cm height praseodymium magnet mag-

netized at 1300 kA/m.

engine restart opportunity is shown in Fig. 8b. This conﬁguration employs one ∼40 cm radius internal LOX

tank with a top gas valve and an open-ended bottom. After the stage separation, the gas valve is opened and

the LOX ﬂows under the inﬂuence of the magnet to the minimum-energy conﬁguration given by the constant

acceleration lines in Fig. 7. The total mass of this system would be close to 20 kg after considering a 1

mm thick Aluminum LOX tank. The major risk of this approach is the potential onset of Rayleigh-Taylor

(a) External (b) Internal

Figure 8: Conceptual active liquid retention conﬁgurations for Falcon 9’s ﬁrst stage LOX tank.

10

instabilities (see Ref. 53) at the open bottom end if the LOX interface is accidentally exposed to the gas. A

careful ﬂuid dynamic analysis of the problem is thus needed to ensure a robust operation.19,40, 41

The same system can be applied to the second stage and would require four 16 cm radius tanks or a single

25 cm radius vessel with a magnet of just 0.5 kg. The mass of the single tank aluminum walls would account

for ∼2.2 kg. That is, the active liquid retention strategy may potentially achieve mass savings of one to two

orders of magnitude with respect to existing ullage engines.

Magnetic ﬁeld generation

With the nI parameter already sized for the different engine restart conﬁgurations, the next logical step

is to determine how to produce the required magnetic ﬁelds. Three technologies are subsequently studied:

copper/aluminum coils, rare earth permanent magnets, and superconducting coils.

The magnetic ﬁeld generated by a coil is linearly dependent on the nI parameter, that can be increased by

adding more wire turns or employing higher currents. Total coil mass and power dissipation are the driving

factors of the design. The mass of the coil can be estimated as

m= 2πRnSρw,(11)

where Sis the cross-section of the wire and ρwits density. The heat dissipated by the coil can be derived

from Ohm’s law, resulting in

P= 2πRInρe

I

S,(12)

with ρebeing the resistivity of the material. In a worst-case scenario this heat is stored in the coil instead of

being dissipated, causing a temperature increase of

∆T=P tmax

mCp

=ρetmax

ρsCpI

S2

,(13)

where Cpis the heat capacity of the wire. In order to constrain the design, the heat dissipated by the coil is

limited by considering two worst-case scenarios: in the ﬁrst, the heat is fully transferred to the residual LOX.

The maximum power is arbitrarily set to the one that vaporizes 1 kg of LOX during the 6 minutes operation of

the coils. The latent heat of vaporization of LOX is 6.82 kJ/mol (or 213.13 kJ/kg) at atmospheric pressure,34

which results in a maximum coil power of 592 W. Because nI is ﬁxed, the ratio I/S is determined by Eq. 12.

In the second scenario, the heat is stored in the coil, causing a temperature increase that is limited by choice

to 10 K. Therefore, from Eq. 13 the I/S parameter is extracted. The most restrictive constraint is chosen for

each design so that the thermal runaway of the material and the vaporization of the residual LOX volume are

avoided. Then, the mass is computed for the I /S value from Eq. 11. The second requirement concerns the

voltage of the coil, set to 24 V to ease integration with Falcon 9’s power subsystem. After inserting the I/S

value in Eq. 12 and employing Ohm’s law, the current intensity of the coil and its resistance are computed.

At the boiling temperature of LOX (90 K) the resisitivities of copper and aluminum are 3.5·10−9Ωm and

4.5·10−9Ωm, respectively.54 Although copper is slightly more conductive than aluminum, its density and

heat capacity are 8960 kg/m3and 0.385 kJ/kgK, while aluminum has a density of 2700 kg/m3and a heat

capacity of 0.89 kJ/kgK. Therefore, aluminum is chosen to minimize the mass of the design.

The second approach focuses on employing rare earth permanent magnets to generate a constant, unpow-

ered magnetic ﬁeld. Neodymium (Nd2Fe14B) is the most popular rare earth material, has a density of 7008

kg/m3,34 and exhibits a remanent magnetization of Mm≈1200 kA/m. It is classiﬁed as a “hard material”,

implying that it can be used to manufacture magnets of any shape.55 As previously noted, the sizing parame-

ter nI of a cylindrical coil can be translated to the length lmof an equivalent cylindrical magnet with the same

diameter by means of the expression In =Mmlm. Magnet tessellation strategies such as Halbach arrays can

be employed to boost the paramagnetic force on one side, while partially canceling it on the other.56 Halbach

arrays have already been proposed for space applications52 and would be particularly well suited to the LOX

settling problem for two reasons: the reach of the magnet is increased, and the interaction between the LOX

magnet and the droplets generated in the fuel tank is reduced.

11

Neodymium magnets experience a slight increase of their magnetic ﬂux as temperature decreases. At

around 135 K, a transition point is reached and the magnet undergoes spin reorientation (i.e. a change in the

preferred direction of the magnetization vector) that decreases the ﬂux by no more than a 14%. This process

is reverted as soon as the temperature increases.57 If needed, the transition point could be avoided by isolating

the magnet in the LOX tank and actively controlling its temperature. A more elegant solution is, however, to

employ praseodymium magnets to avoid the spin reorientation. Praseodymium magnets do not suffer from

spin reorientation and have been shown to reach a remanent magnetization of ∼1300 kA/m at 85 K,58,59

which makes them ideal for LOX control applications.

The design points of the aluminum coil and praseodymium magnet are shown in Table 3 as a function of the

nI parameter. In all cases but 106At, the design of the coil is driven by the thermal requirement (maximum

temperature increase of 10 K). Magnets are orders of magnitude lighter for all nI values, incurring in a

– still reasonable– mass penalty of 52 kg at 105At. nI values beyond 105At seem unreachable without

incurring in large mass penalties, and it is in this context where high-temperature superconductors (HTC)

can become a game-changing alternative. A HTC wire exhibits zero resistance in a certain operational range,

resulting in no heat loss and a potential reduction in mass and power requirements. This happens when (i)

it is operated below its critical temperature Tc–greater, by deﬁnition, than the boiling point of nitrogen (77

K)–, (ii) it is subjected to a magnetic ﬁeld below the critical ﬁeld Bc, and (iii) the critical current Icis not

exceeded. The simultaneous satisfaction of these three requirements is far from trivial; in fact, Icdecreases

continuously with increasing temperature and magnetic ﬁeld.60 For example, Bi2223 (Bi2Sr2Ca2Cu3O10+δ)

has a critical temperature of 110 K, but its critical current drops to zero when the material is exposed to a

ﬁeld of less than 1 T. RE-123 ((RE)Ba2Cu3O7, where RE stands for Rare Earth element) superconductors

(also known as REBCO), on the contrary, can resist up to 10 T, but only well below a critical temperature of

around 90 K.61 It is nowadays feasible to generate very strong magnetic ﬁelds at the boiling point of Helium

(4.22 K), the best example being the 32 T superconducting magnet62 of the National High Magnetic Field

Laboratory∗∗. Reaching similar values at higher temperatures seems, unfortunately, still beyond our technical

capabilities. In the application here discussed the superconductor would be immersed in LOX, which would

act as a cooling agent only if Tc90 K, and the maximum magnetic ﬁeld imposed near the coils would be

∼10 T at In = 106At. The results presented in Ref. 60 for different commercial REBCO conductors seem

to indicate that such operation point cannot be reached with current technologies. However, the Icvalue of

4 mm wide superconductors is shown to be 450–1000 A/mm at 12 T and 77 K, a value that jumps up to 60

kA/mm2at 18 T and 4.2 K. This indicates that cooling mechanisms need to be put in place to reach the 106

At conﬁguration with HTCs, which may open an opportunity for multiple-use of the helium tanks employed

for tank pressurization. This possibility, although attractive, would require a deeper technical analysis that is

beyond the scope of this paper.

∗∗See https://nationalmaglab.org/magnet- development/magnet-science- technology/

magnet-projects/32- tesla-scm. Consulted on: 26/12/2021

Table 3: Mass and power budget for different magnetic conﬁgurations.

Aluminum coil1Praseodymium magnet2

Current ·Turns

[At]

I

[A]

P

[W]

n

[# turns]

d

[mm]

m

[kg]

h

[mm]

m

[kg]

1022.51·10−30.06 4 1.62·10−12.44 0.08 0.052

1032.51·10−20.60 40 5.12·10−124.4 0.77 0.52

1042.51·10−16.03 405 1.62 244 7.7 5.2

1052.51 60.26 4054 5.12 2438 77 52

10624.67 592 40541 16.2 24814 769 519

1Coil of 35 cm diameter operating at 24 V and 90 K.

2Cylindrical magnet of 35 cm diameter magnetized at 1300 kA/m at 90 K.

12

Fuel tank

From the magnetic actuation perspective, LOX determines the design envelope of the system. On one

hand, the LOX tank is more than two times larger than the fuel tank, and therefore a given magnetic source

will reach a larger portion of the latter. On the other, LOX is a paramagnetic substance with volume magnetic

susceptibility χLOX = 0.0034, while kerosene is a diamagnetic with χKe ≈ −8·10−6.63 In order to apply the

magnetic retention and settling approaches to the fuel tank, it must be transformed into a para/ferromagnetic

by adding magnetic nanoparticles and creating a kerosene-based ferroﬂuid. With this approach, the suscepti-

bility of the solution is bounded by the concentration of magnetic nanoparticles.

Kerosene has been employed as a carrier liquid since the invention of ferroﬂuids in 196332 and kerosene-

based ferroﬂuids are synthetized and used in numerous ﬁelds.64–67 Commercial solutions like Ferrotec’s

EMG-905†† are now widely available at a relatively low cost. The initial susceptibility of a monodisperse,

colloidal ferroﬂuid can be estimated as30

χini = 8φλ, (14)

where φis the volume fraction of magnetic solids and λis the coupling coefﬁcient, given by

λ=µ0M2

dV

24kT ,(15)

with Mdbeing the saturation moment of the bulk magnetic solid, Vthe nanoparticle volume, kthe Boltzmann

constant, and Tthe absolute temperature. Assuming an iron oxide nanoparticle radius of 5 nm, an absolute

temperature of 293 K, and a saturation moment of 446 kA/m,68 the approximate volume fraction required to

match the magnetic susceptibility of LOX starting from the value of kerosene would be just φ≈3.2·10−4.

Not surprisingly, this value is within the range tested by Martin and Holt in the NASA MAPO experiment.33

If the whole kerosene tank volume is magnetized, the magnetic nanoparticles add ∼40 kg and ∼11 kg to

the ﬁrst and second stages, respectively, in addition to negligible variations in density and speciﬁc impulse.19

The very low ferroﬂuid concentration should prevent damage to the engines. Although simple, this approach

is expensive and inefﬁcient in comparison with the active liquid retention strategy, where only the kerosene

employed to restart the engine would need to be enhanced with ferromagnetic nanoparticles. In this case, the

mass penalty associated with such nanoparticles would be of just 40 to 140 g based on the total RP-1 masses

presented in Table 2. The same approach could be applied to the passive liquid retention and liquid recovery

strategies if a concentrated ferroﬂuid volume is mixed with the RP-1 residuals shortly before MECO/SECO.

Operation

The discussion provided in this section focuses on moving the propellant toward the bottom of the tank. A

logical follow-up question is what to do once that happens. Unless some sort of valve or screen mesh is put

in place, it must be assumed that the propellant stored in the tubes leading to the engines is partially released

to the tank. Thus, once the liquid is attracted back to the tank outlet, a gas bubble is generated in the tubes

leading to the engines. This would represent an unacceptable safety risk.

There are several ways to face this problem, and all of them require a deep knowledge of Falcon 9’s tank

outlet design. A possibility would be to employ a combination of PMDs to hold the liquid at the entrance

of the engine against accelerations of 1 m/s2. This should effectively prevent the development of a Rayleigh

Taylor instability and the entrance of gas bubbles. Active strategies may involve purge valves and membranes

located in strategic positions along the liquid path. Unfortunately, a more detailed analysis requires technical

information that is not available to the authors.

††See https://ferrofluid.ferrotec.com/products/ferrofluid- emg/oil/emg-905/. Consulted on:

28/12/2021

13

Figure 9: Propellant Gasiﬁcation System

PROPELLANT GASIFICATION SYSTEM

Concept and overview

The injection of hot gases into the tanks for chemical pressurization through propellant evaporation and

combustion has long been known and used in the “Proton”, “Rokot” and “Dnepr” launch vehicles.69 These

systems do not require heat exchangers, which are used to heat cold helium gas from 90 K to 300 K in current

pressurization systems, and are instead based on feeding a nitric acid-based oxidizer and fuel into the fuel and

oxidizer tanks, respectively. The components ignite and heat is released, causing the fuel to vaporize, which

increases the pressure in the tanks. Referring to Sutton, while this type of pressurization system is small

and lightweight, it has generally not yielded reproducible tank pressures due to the difﬁculties to stabilize the

combustion reaction. For instance, fuel sloshing caused by vehicle maneuvers results in sudden cooling of the

hot pressurizing gas and leads to erratic tank pressure changes. This problem can be avoided by physically

separating the hot gas from the liquid propellant. If the hot gas is generated from a solid propellant reaction

or from mono-propellant decomposition instead of a high-pressure gas supply, a signiﬁcant reduction in the

mass of the pressurizing system can be achieved.21

The Propellant Gasiﬁcation System concept was ﬁrst proposed in the early 2010s as a method to vaporize

the propellant residual of 2nd launch vehicle stages and provide attitude and orbit control capabilities by

means of dedicated vapor-fed thrusters.70–72 The original idea was to inject the combustion products of two-

component propellants (AA and NDMH) in the tank to move the stage from its initial circular orbit to an

elliptical orbit. Such orbit would ensure a successful deorbiting in the time frame of 25 years. In 2015, the

use of solid fuel instead of a two-component propellant was investigated to simplify the design and improve

the energy performance of the PGS. Further analyses on the Soyuz 2.1v launch vehicle showed that the PGS

could also lead to launch vehicle characteristic speed enhancements of up to 5%.73 The PGS baseline design

has currently evolved to reduce its mass and environmental impact using a green mono-propellant (hydrogen

peroxide) that adds the possibilities of i) controlling the movement of the stage to reach a given drop area, ii)

providing conditions for LRE restart by executing the ﬂip around and propellant settling maneuvers, and iii)

passivating the propellant after a normal or emergency cutoff of the LRE.74 Highly concentrated hydrogen

peroxide (85%) has already been employed as a green mono-propellant in substitution of hydrazine on the

“Soyuz” launch vehicle for the operation of turbo-pump units.75

The PGS considered in this work vaporizes the propellant residuals in the oxidizer tank of the launch

vehicle using the catalytic decomposition of hydrogen peroxide, which is placed in an auxiliary tank and

used as a heat source. The mono-propellant is passed through a catalyst chamber that leads to the formation

of up to 823K hot oxygen and steam. The vapor-gas mixture is then transferred into the oxidizer tank, which

leads to vaporization of the liquid phase and a pressure increase. The gas, consisting of vaporized propellant

and pressurizing agent (helium), is used to feed a set of gas thrusters that are employed for attitude control and

tank settling, as sketched in Fig. 9. This approach can be regarded as the active equivalent of the hydrogen

venting strategy employed in the Apollo era.11

14

It is important to remark that the risk of combustion or explosion is virtually non-existent in the proposed

PGS approach. Since only hot hydrogen peroxide decomposition byproducts (water vapor and oxygen) are

injected into the oxidizer tank and a reducing agent at auto-ignition temperatures is not present, the combus-

tion reaction cannot be produced. Such a hazard could only occur if the PGS was used in the propellant tank.

However, this approach has already been applied to the aforementioned chemical pressurization systems, and

the reaction was safely controlled by tuning the pressurizer gas ﬂow rate.

The following main subsystems compose the PGS: (i) a hot gas generator that includes a bladder-controlled

hydrogen peroxide tank and a catalyst chamber where the exothermic decomposition of hydrogen peroxide

happens, producing a high-temperature vapor-gas mixture (VGM) with a 34% H2Oand 66% O2composi-

tion, (ii) a system of nozzles installed after the catalyst chamber that injects the VGM into the LOX tank

minimizing tank wall heating, and (iii) a system of gas nozzles used to discharge the VGM from the tank and

produce the required thrust. The PGS provides control over the tank discharge valves, hydrogen peroxide

feeding, and gas nozzles. Cold helium gas, which is stored in balloons at the bottom of the oxidizer tank, can

also be used to reduce the temperature of the VGM (see next section).

Concept of Operations:

Figure 10 shows the ConOps for the PGS approach. Three seconds after MECO, the ﬁrst stage separates

from the second stage using a pneumatic pusher (or equivalent). The control system of the 1st stage starts

the PGS to increase the pressure in the oxygen tank. Nine seconds after MECO, the PGS opens the attitude

control nozzles with a total thrust of 1 kN to start the ﬂip around maneuver, which involves an acceleration

phase, a constant angular velocity phase, and a deceleration phase. About 55 seconds after MECO, a set of

nozzles provide 5 kN of axial acceleration before engine re-start.

The PGS can be operated in three distinct modes in combination with the helium pressurization system:

1. Standalone PGS operation: In this case, the main task of the PGS is to vaporize the liquid residuals.

The highest gas temperature is employed (823 K).

2. Alternate operation: The residual cold helium pressurizer gas (90 K) is fed into the tank in order to

reduce the VGM temperature. This reduces or stops the evaporation of liquid propellant and ensures

that the proper quantity of liquid is vaporized to operate the gas thrusters.

3. Combined operation: In this case, the VGM does not vaporize the propellant residuals, acting instead

as a heat exchanger that warms up the pressurizer gas up to an accepted operational temperature (300

K), replacing the engine heat exchanger.76

Flip around maneuver:

The ﬂip around maneuver to the required ∆φangle consists of three phases that are evaluated in Fig. 11.

The acceleration phase is produced with constant angular acceleration until the speed ωis reached, and lasts

t=ω

,(16)

Flip around maneuver

Propellant settling

+9s

+3 s

+54.7s

+95 s

+18.7s

+45s

1st burn

Starting PGS

MECO

LOX evaporation

0 s

5 kN

1 kN

1 kN

Figure 10: PGS Operation cycle

15

Ang. Vel. [m/s]

0

1

2

3

4

5

t [s]

10 20 30 40 50

Ang. Acc. [deg/s2]

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

t [s]

10 20 30 40 50

Figure 11: Angular velocity and angular acceleration proﬁles of the 1st stage during ﬂip around maneuver

using the attitude control nozzles with a total thrust of 1 kN

where zero initial speed and angle are assumed. The second phase is produced with constant angular velocity

ωand duration tω. Finally, the deceleration phase happens with the same angular deceleration. The total

turnaround time tris given by

tr=ω

+∆ψ

ω.(17)

The magnitude of angular acceleration is determined by the torque produced by the attitude control nozzles

relative to the center of mass and by the lateral moment of inertia of the stage. For instance, around the z-axis

the acceleration becomes

=F(hn−xc)

Jz

,(18)

with Fbeing the total thrust of the nozzles, hn−xcthe thrust arm, and xcthe distance from the bottom

section of the stage to the center of mass. The parameters of the ﬂip around maneuver are given in Table 4.

Oxygen vaporization model

Vaporization in a tank can occur according to three mechanisms: evaporation without boiling, ﬁlm boiling,

and nucleate boiling. The criterion that determines whether vaporization belongs to one of three types is the

temperature of the liquid. The mass rate of vaporization during bubble and ﬁlm boiling is calculated following

a thermodynamic approach that assumes that all the heat supplied is employed to vaporize the liquid.

The mathematical model of the oxygen vaporization process is based on the ﬁrst law of thermodynamics,

and leads to74

dp

dt=k−1

VHu˙mhc +ihc ˙mhc +iev ˙mev −iout ˙mout −k

k−1pdV

dt ,(19a)

dρ

dt=1

V˙mhc + ˙mev −˙mout −ρdV

dt ,(19b)

Table 4: Parameters of the ﬂip around maneuver

Parameter Value

Angular velocity, ω[deg/s] 5

Total attitude nozzle thrust force, F[N] 1000

Acceleration and deceleration phases, t[s] 9.7+9.7

Constant angular velocity phase, tω[s] 26.3

Total ﬂip around time, tr[s] 54.7

16

dTw

dt=qmix-w

rad +qmix-w

con +qw-lox

rad +qw-lox

con −qw

rad +qext

cwmw

(19c)

dTmix

dt=−qmix-w

con −qmix-w

rad +qw-lox

rad +qmix-lox

con −qev +Hu˙mhc

cmixmmix

,(19d)

dTlox

dt=qmix-lox

rad −qmix-lox

con +qw-lox

rad +qw-lox

con −qev

cloxmlox

.(19e)

The system given by Eqs. 19a-e describes the change of the pressure in the tank p, density of the vapor-

gas mixture ρ, and the temperatures of the vapor-gas mixture Tmix, LOX Tlox, and tank walls Tw. The

temperature of the vapor-gas mixture depends on the radiative and convective heat ﬂux to the wall of the tank

(qmix-w

con ,qmix-w

rad ), radiative and convective heat ﬂux to LOX (qw-lox

rad ,qmix-lox

con ), heat of vaporization of LOX qev,

and the heat generated during decomposition reaction of H2O2with mass of mhc,Hu˙mhc. The temperature

of the walls of the tank depends on the external heat ﬂux from the atmospheric heating qext, heat ﬂux from the

vapor-gas mixture in the tank qmix-w

rad and qmix-w

con , heat ﬂux to LOX (qw-lox

rad ,qw-lox

con ), and radiative heat ﬂux qw

rad.

The temperature of LOX is determined by the heat ﬂux from the wall, vapor-gas mixture in the tank and heat

of vaporization of LOX.

The key heat ﬂuxes that determine the vaporization process of LOX are now considered. The radiant heat

ﬂux from the vapor-gas mixture to the LOX surface is

qmix-lox

rad =σmix Flox T4

mix −T4

lox,(20)

where the area of the surface of vaporization of LOX, Flox, depends on the mechanical condition in the tank.

If LOX is accumulated at the bottom of the tank, then Flox equals the cross section area of the tank. If the

liquid oxygen is in the drop state, then Flox is the total surface of all the drops. The convective heat ﬂux from

the vapor-gas mixture to the oxygen surface is

qmix-lox

con =αmixFlox (Tmix −Tlox ),(21)

with

αmix =λmixN umix

l(22)

being the heat-exchange coefﬁcient of the vapor-gas mixture to the surface of evaporation of the oxygen in the

tank, and where λmix is the thermal conductivity of the vapor-gas mixture in the tank, and lis the characteristic

dimension that is equal to the diameter of the oxygen tank. αmix depends also on the Nusselt number N umix,

which is determined by the vaporization type of LOX: evaporation from the free surface, nucleate boiling or

ﬁlm boiling. qcon and qrad , among other contributors to the heat exchange process, are deﬁned similarly to

Eqs. 21 and 22.

If free surface vaporization is the dominant vaporization mechanism, the Nusselt number can be expressed

as77

Nuev =C1Re0.8

ev P r0.43

ev (23)

where Reev is the Reynolds number, P rev is the Prandtl number for the vapor-gas mixture, and C1is equal to

0.037.

In the case of nucleate boiling, the Nusselt number becomes77

Nunuc =αd0

λ= 75(P e ·F o)0.7P r −0.2.(24)

For the nucleate boiling case, the characteristic dimension is the average diameter of the bubble departure

diameter d0, that can be considered independent on the heat ﬂux q. Consequently, the Peclet number for heat

exchange during nucleate boiling is

P e =qd0

r∗ρva,(25)

17

with an average diameter of the bubble departure diameter deﬁned by Fritz’s equation78

d0= 1.2θrσ

g(ρl−ρv),(26)

where θis the contact angle in rad, σis the surface tension coefﬁcient, gis the inertial acceleration, ρland ρv

are the densities of the liquid and gas oxygen, qis the heat ﬂux density, r∗is the speciﬁc heat of evaporation,

and ais the temperature conductivity coefﬁcient. The Fourier number for the nucleate boiling of oxygen is

F o =a

d2

0f,(27)

with

fpd0= 0.56 ρl−ρv

ρv

g1/2

(28)

being the frequency of formation of the vapor bubbles.

Finally, in a ﬁlm boiling scenario with a large liquid volume on vertical surfaces as well as on horizontal

cylinders and spheres, it is recommended to use the similarity equation

Nuf= 0.18Ra1/3,(29)

where

Ra =gl3

0

νvav

ρl−ρv

ρv

(30)

is the Rayleigh number that characterizes the behavior of a ﬂuid with a temperature gradient, and where

l0=σ

g(ρl−ρv)(31)

is the capillarity constant, selected as a characteristic length.

Performance analysis

The alternate PGS mode, that reduces the average tank temperature in comparison with a standalone PGS

operation, is employed in the analysis. A combined operation with the tank pressurization system reduces

oxygen evaporation and preserves it for the operation of the LRE. The ratio of hydrogen peroxide and helium

mass ﬂow is not optimized in the results presented in this work, where one of the possible combinations was

chosen. However, even this sub-optimal set point reﬂects the efﬁciency of the alternate operation of the PGS

and helium pressurization systems.

Figure 12 shows the time evolution of different variables of the PGS for the nozzle thrust values reported

in Table 4. After stage separation, the supply of hydrogen peroxide to the LOX tank leads to an operating

pressure of ∼4 atm in the oxygen tank. Figure 12a shows the time evolution of this variable. During the

ﬁrst six seconds after separation of the stage and supply of heat to the tank, the pressure increases from 2.7

to 4 atm. As noted above, the given pressure level in the tank is maintained by the joint operation of the

PGS and the supply of helium to the LOX tank, that are set manually and may be optimized in future works.

From 9 to 95 s, the stage performs the ﬂip-around and propellant settling maneuvers as described in Fig. 10.

Figures 12b and 12c show the mass ﬂow rates of hydrogen peroxide and helium during this process. Figures

12f and 12e show thrust plots and mass ﬂow rates of the PGS nozzles. During the entire operation, the gas

temperature in the tank increases up to 275 K, which does not violate the standard thermal stability limits of

300-350 K.

Table 5 shows the results of the ﬂip around and propellant settling maneuvers separately. A total of 327.9

kg of gas are employed by the nozzles during the operation of the PGS, with 354 kg of oxygen, 95 kg

of hydrogen peroxide, and 0.7 kg of helium being consumed. After the PGS operation, the gas residuals

(composed of vaporized oxygen, water vapor, and helium) are needed to maintain the tank pressure of 4 atm

18

Pressure [atm]

2.5

2.75

3

3.25

3.5

3.75

4

t [s]

0 20 40 60 80 100

(a) Pressure in the oxygen tank

H2O2 mass flow rate [kg/s]

0

2

4

6

8

t [s]

0 20 40 60 80 100

(b) Hydrogen peroxide mass ﬂow rate

He mass flow rate [kg/s]

0

0.01

0.02

0.03

t [s]

0 20 40 60 80 100

(c) Helium mass ﬂow rate

Thrust [N]

0

1000

2000

3000

4000

5000

t [s]

0 20 40 60 80 100

(d) Nozzle thrust in turning and fuel shrinking modes

Gas mass flow rate [kg/s]

0

2

4

6

8

t [s]

0 20 40 60 80 100

(e) Gas mass ﬂow rate through gas jet nozzles

Tank gas temperature [K]

150

175

200

225

250

275

300

t [s]

0 20 40 60 80 100

(f) Gas temperature in oxidizer tank

Mass evaporation rate [kg/s]

0

5

10

15

20

t [s]

0 20 40 60 80 100

(g) Mass oxygen evaporation rate

Figure 12: Performance of the PGS.

19

Table 5: Mass from different components required for the ﬂip around and settling maneuvers.

System Flip around maneuver Propellant settling

Gas through nozzles [kg] 22.9 305

Vaporized oxygen [kg] 53.9 304

PGS [kg] 40 -

Catalyst [kg] 0.5 -

H2O2balloon [kg] 4.6 5.8

H2O2[kg] 42 53

Helium [kg] 0.069 0.668

as a working body for the gas thrusters, that are in charge of stage orientation and stabilization during the

ﬂight. The PGS mass budget of 40 kg includes the gas generator (5 kg), nozzles (10 kg), valves, pipes and

other hardware components. The total mass of the nozzles depends on the selected conﬁguration: 8 nozzles

are arbitrarily selected in this work, with 2 of them having a higher thrust. The mass values are strongly

dependent on the technical decisions of the designers, who will have their own constrains and engineering

vision of the PGS, including the length of the pipes, number of valves, etc. Active catalysts (e.g. potassium

permanganate grains, silver mesh, copper, or other metals79) are used for hydrogen peroxide decomposition.

The catalyst does not lose its properties when exposed to the low temperature of the cryogenic tank or the high

temperatures of the combustion reaction. The temperature of the hydrogen peroxide decomposition products

can be controlled by tuning the hydrogen peroxide concentration,80 which allows the mass evaporation rate

to be modiﬁed when the PGS is used in combination with the pressurizer gas helium.

In order to minimize propellant residuals in the gas phase at the moment of PGS operation termination,

it is necessary to optimize the helium and hydrogen peroxide input cyclogram (e.g. using the Pontryagin

maximum principle or dynamic programming81). An optimum ﬂow rate combination reduces the residual

gas both in the balloons and in the tank after vaporization, and also the ﬁnal PGS mass.

Computations for the second stage would follow the same process as for the ﬁrst stage and will be addressed

in future works.

HYBRID MAGNETIC GASIFICATION

A combination of the MP2and PGS technologies can potentially enhance the robustness and performance

of the overall propellant settling system. This hybrid approach involves a permanent magnet located at the

fuel outlet and a smaller PGS aimed at carrying out the ﬂip-around phase and a shorter propellant settling

maneuver. The initial acceleration induced on the liquid residuals during the settling phase starts a slow

movement toward the bottom of the tank, where the magnetic force is stronger and thus able to efﬁciently

collect the liquid droplets.

Figure 13 depicts the time required by an LOX droplet to reach the bottom of the tank when subject to the

magnetic ﬁeld generated by a 104At magnet as a function of its initial velocity and distance to the tank outlet.

Based on Table 3, such magnet would have a mass of ∼5.2 kg. The curves are computed from an extended

version of Eq. 10, and prove that an initial inertial kick can signiﬁcantly extend the reach of the MP2system.

Initial droplet velocities of just 5 to 10 mm/s allow the magnet to collect the necessary residual propellant

mass under all engine restart conﬁgurations listed in Table 2 for both stages assuming that the droplets are

uniformly distributed in the tank volume. The 5 kN PGS nozzles, that induce accelerations of ∼0.23 m/s2

in the ﬁrst stage, would theoretically need to operate for less than 0.05 s to induce these droplet velocities,

reducing the propellant settling window in about 40 s. The operation window of the nozzles would need to

be extended to account for transient effects and ﬂuid-structure interactions, but this would only increase the

performance of the system. Based on Table 5, the associated PGS mass savings would be close to 60 kg,

resulting in a total hybrid system mass of ∼92 kg.

20

Max. T.

Min. T.

Max. T.

S.E.

Min. T.

tmax

1st Stage

2nd Stage

0 mm/s

5 mm/s

10 mm/s

15 mm/s

20 mm/s

Time of flight [s]

100

101

102

103

Initial droplet distance to fuel outlet [m]

0 1 2 3 4 5

Figure 13: Time required by a LOX droplet to reach the bottom of the tank as a function of its initial position

and velocity when subject to a 104At coil located at the tank outlet. The minimum tank settling length

required by the single engine (S.E.), minimum thrust (Min. T.) and maximum thrust (Max. T.) conﬁgurations

is superposed.

SUMMARY AND CONCLUSIONS

Five novel high-risk-high-return propellant settling approaches have been explored in this work: passive

magnetic retention, magnetic recovery, active magnetic retention, propellant gasiﬁcation, and hybrid mag-

netic gasiﬁcation. The general advantages and disadvantages of each of them have been discussed, and

preliminary mass budgets have been derived. Table 6 summarizes and extends the main results from the

analysis.

While the passive magnetic retention strategy exceeds any reasonable mass budget, the active magnetic

Table 6: Comparison of different propellant settling approaches and mass budgets for ﬁrst and second stages.

Advantages Disadvantages Mass, 1st

[kg]

Mass, 2nd

[kg]

Passive

Magnetic

Retention

- Simple

- Robust

- Thoroughly studied

- Beyond current technological capabilities

- Massive unless stage operation is adapted

- Limited control → ∞ >1000

Magnetic

Recovery

- Lightweight

- Simple

- Sensitive to ﬂuid-structure interactions

- Slow

- Requires tank outlet redesign

- Very low TRL

104 1-10

Active

Magnetic

Retention

- Lightweight

- Simple

- Limited reusability

- Requires careful tank design

- Requires tank outlet redesign

- Very low TRL

40 6

Propellant

Gasiﬁcation

System

- Robust

- Provides settling and attitude control

- Fast to operate

- More traditional design

- Complex

- Sensitive to liquid movement

- Very low TRL

147 -

Hybrid

Magnetic

Gasiﬁcation

- Lightweight

- More robust than MP2systems

- Provides settling and attitude control

- Fast to operate

- Boosts magnet performance

- Complex

- Sensitive to liquid movement

- Requires tank outlet redesign

- Very low TRL

92 -

21

retention approach can potentially reduce the mass of existing propellant settling systems by one to two

orders of magnitude, leading to more than half a million dollar savings per launch and stage. This comes

at the cost of higher complexity and potential robustness and reliability issues. Although less efﬁcient, the

magnetic recovery system is also very competitive with respect to current technologies, particularly for upper

stages. Because this approach depends on the availability of uniformly distributed free-ﬂoating propellant

droplets, ﬂuid-structure interactions can easily undermine its performance. The PGS, which represents a

relatively more conventional approximation to the problem, can also lead to moderate mass savings that are

signiﬁcantly increased when operated in combination with a magnetic retention system. As with any other

low-TRL technology, numerous technical challenges remain that can only be addressed with a more detailed

numerical and experimental analysis.

Ullage engines have been employed since the beginning of the space era and are nowadays regarded as a

robust active settling solution. However, publicly available data indicates that they also involve signiﬁcant

mass and economic penalties that may be reduced with novel approaches. Such approaches must demon-

strate the same level of reliability and robustness in order to become competitive. Although the magnetic

positive positioning and propellant gasiﬁcation systems introduced in this work are still in a very early stage

of development, the analysis here presented offers reasons to persevere in their development.

APPENDIX A: STAGE SEPARATION MODEL

The accelerations experienced by each stage during separation are modeled after assuming that the process

is carried out by spring pushers with total initial force Fb, ﬁnal force Fe, and stroke h. Thus, the total force

acting on the stage is

F(x12) = Fb−cx12 if x12 ≤h, 0otherwise,(32)

with c= (Fb−Fe)/h being the total stiffness of the springs, and where the geometrical parameters of the

problem are deﬁned in Fig. 14. After applying Newton’s second law and solving the resulting differential

equation, the duration of the maneuver becomes

th=rm12

carccos Fe

Fb

,(33)

where m12 =m1m2/(m1+m2). The acceleration of the stage with respect to the orbital frame is

a1=−Fb

m1

cos rc

m12

tif t≤th,0otherwise.(34)

The values Fe= 0.3Fb, and h= 0.2m are adopted in this work, with Fb= 12.1kN and Fb= 4.5kN in the

ﬁrst and second stages, respectively. Empty and total masses are considered for the deployed and remaining

stages with a payload mass of 15 t. The resulting propellant dispersion velocities range between 0.45 and 0.5

m/s, a range that seems to agree with observations from SpaceX’s CRS-5 mission.

Figure 14: Stage separation model

22

REFERENCES

[1] Inter-Agency Space Debris Coordination Committee, “IACD Space Debris Mitigation Guidelines,”

Tech. Rep. IADC-02-01, IADC, 2020.

[2] L. Anselmo and C. Pardini, “Ranking upper stages in low Earth orbit for active removal,” Acta

Astronautica, Vol. 122, 2016, pp. 19–27, 10.1016/j.actaastro.2016.01.019.

[3] J. C. Liou, “An active debris removal parametric study for LEO environment remediation,” Advances in

Space Research, Vol. 47, No. 11, 2011, pp. 1865–1876, 10.1016/j.asr.2011.02.003.

[4] D. McKnight, R. Witner, F. Letizia, S. Lemmens, L. Anselmo, C. Pardini, A. Rossi, C. Kunstadter,

S. Kawamoto, V. Aslanov, J.-C. Dolado Perez, V. Ruch, H. Lewis, M. Nicolls, L. Jing, S. Dan, W. Dong-

fang, A. Baranov, and D. Grishko, “Identifying the 50 statistically-most-concerning derelict objects in

LEO,” Acta Astronautica, Vol. 181, 2021, pp. 282–291, 10.1016/j.actaastro.2021.01.021.

[5] V. Trushlyakov and Y. Shatrov, “Improving of technical characteristics of launch vehicles with liquid

rocket engines using active onboard de-orbiting systems,” Acta Astronautica, Vol. 138, 2017, pp. 19–27,

10.1016/j.actaastro.2017.05.018.

[6] S. Lednev, T. Koroleva, P. Krechetov, A. Sharapova, I. Semenkov, and A. Karpachevskiy, “Revegetation

of areas disturbed by rocket impact in Central Kazakhstan,” Ecoscience, Vol. 25, No. 1, 2018, pp. 25–38,

10.1080/11956860.2017.1396100.

[7] T. V. Koroleva, P. P. Krechetov, I. N. Semenkov, A. V. Sharapova, S. A. Lednev, A. M. Karpachevskiy,

A. D. Kondratyev, and N. S. Kasimov, “The environmental impact of space transport,” Transportation

Research Part D: Transport and Environment, Vol. 58, 2018, pp. 54–69, 10.1016/j.trd.2017.10.013.

[8] R. P. Patera, K. R. Bohman, M. A. Landa, C. D. Pao, R. T. Urbano, M. D. Weaver, and D. C. White,

“DMSP-17 Upper Stage Controlled Reentry Disposal,” Tech. Rep. ATR-2007(8083)-1, The Aerospace

Corporation, 2006.

[9] V. I. Trushlyakov, V. V. Shalay, and Y. T. Shatrov, “Reduction of the technogenic impact of launch

vehicles on the liquid toxic components of rocket fuel on the environment,” tech. rep., Omsk State

Technical University, 2004.

[10] F. Dodge, The New Dynamic Behavior of Liquids in Moving Containers. Southwest Research Institute,

2000.

[11] G. K. Platt, “Space vehicle low gravity ﬂuid mechanics problems and the feasibility of their experimen-

tal investigation,” Tech. Rep. TM X-53589, NASA, 1967.

[12] W. L. Browning, “S-4B/5 Auxiliary Propulsion System 90-day Recycle Capability Test Report, Module

1,” Tech. Rep. DAC-56728, McDonnell Douglas Astronautics Company, 1969.

[13] K. Coates and E. Donald, “Investigation of SA-501 S-4B Auxiliary Propulsion System Flight Anoma-

lies,” Tech. Rep. TN D-5207, NASA, 1969.

[14] Space Exploration Technologies Corporation, “Falcon User’s Guide,” September 2021.

[15] P. Anz-Meador, “Root Cause Classiﬁcation of Breakup Events 1961-2018,” First International Orbital

Debris Conference, Houston, TX, 2019.

[16] Saturn Flight Evaluation Working Group, Marshall Space Flight Center, “Saturn 5 launch vehicle ﬂight

evaluation report-AS-511 Apollo 16 mission,” Tech. Rep. TM-X-69535, NASA, 1972.

[17] A. P. Adzhan, E. L. Akim, and O. M. Alifanov, “Rocket and space technology,” Engineering.

Encyclopedia, Vol. 4, 2012, p. 925.

[18] H. Jones, “The Recent Large Reduction in Space Launch Cost,” 48th International Conference on

Environmental Systems, Albuquerque, NM, 2018.

[19] A. Romero-Calvo, F. Maggi, and H. Schaub, “Magnetic Positive Positioning: Toward the application in

space propulsion,” Acta Astronautica, Vol. 187, 2021, pp. 348–361, 10.1016/j.actaastro.2021.06.045.

[20] V. I. Trushlyakov, V. A. Urbansky, and V. V. Yudintsev, “Reducing Environmental Damage After Emer-

gency Engine Cutoff of the Launch Vehicle,” Journal of Spacecraft and Rockets, Vol. 58, No. 3, 2021,

pp. 685–696, 10.2514/1.A34912.

[21] G. Sutton and O. Biblarz, Rocket Propulsion Elements. John Wiley & Sons, 8 ed., 2010.

[22] A. P. Vasilev, V. M. Kudryavtsev, V. A. Kuznetsov, V. D. Kurpatenkov, and A. M. Obelnitskii,

Fundamentals of the theory and calculation of liquid propellant rocket engines, Vol. 2. Kudryavtsev

M.V., 4 ed., 1993.

[23] J. C. Boulware, H. Ban, S. Jensen, and S. Wassom, “Experimental studies of the pressures generated

by a liquid oxygen slug in a magnetic ﬁeld,” Journal of Magnetism and Magnetic Materials, Vol. 322,

No. 13, 2010, pp. 1752 – 1757, 10.1016/j.jmmm.2009.12.022.

[24] K. Kinefuchi and H. Kobayashi, “Theoretical and experimental study of the active control of bubble

point pressure using a magnetic ﬁeld and its applications,” Physics of Fluids, Vol. 30, No. 6, 2018,

p. 062101, 10.1063/1.5034222.

23

[25] A. Causevica, P. Sahli, F. Hild, K. Grunwald, M. Ehresmann, and G. Herdrich, “PAPELL: Interac-

tion Study of Ferroﬂuid with Electromagnets of an Experiment on the International Space Station,”

Proceedings of the 69th International Astronautical Congress, 2018, pp. 1–5.

[26] D. Ludovisi, S. S. Cha, N. Ramachandran, and W. M. Worek, “Heat transfer of thermocapillary con-

vection in a two-layered ﬂuid system under the inﬂuence of magnetic ﬁeld,” Acta Astronautica, Vol. 64,

No. 11, 2009, pp. 1066 – 1079, 10.1016/j.actaastro.2009.01.018.

[27] A. Bozhko and G. Putin, “Thermomagnetic Convection as a Tool for Heat and Mass Transfer Control

in Nanosize Materials Under Microgravity Conditions,” Microgravity Science and Technology, Vol. 21,

Jan 2009, pp. 89–93, 10.1007/s12217-008-9059-7.

[28] B. A. Jackson, K. J. Terhune, and L. B. King, “Ionic liquid ferroﬂuid interface deformation and

spray onset under electric and magnetic stresses,” Physics of Fluids, Vol. 29, No. 6, 2017, p. 064105,

10.1063/1.4985141.

[29] K. Lemmer, “Propulsion for CubeSats,” Acta Astronautica, Vol. 134, 2017, pp. 231 – 243,

10.1016/j.actaastro.2017.01.048.

[30] R. E. Rosensweig, Ferrohydrodynamics. Dover Publications, 1997.

[31] A. Romero-Calvo, G. Cano-G ´

omez, T. H. Hermans, L. Parrilla Ben´

ıtez, M. Herrada, and E. Castro-

Hern´

andez, “Total magnetic force on a ferroﬂuid droplet in microgravity,” Experimental Thermal and

Fluid Science, Vol. 117, 2020, p. 110124, 10.1016/j.expthermﬂusci.2020.110124.

[32] S. Papell, “Low viscosity magnetic ﬂuid obtained by the colloidal suspension of magnetic particles,”

1963. US Patent 3215572.

[33] J. Martin and J. Holt, “Magnetically Actuated Propellant Orientation Experiment, Controlling ﬂuid

Motion With Magnetic Fields in a Low-Gravity Environment,” Tech. Rep. TM-2000-210129, NASA,

2000.

[34] D. R. Lide, CRC Handbook of Chemistry and Physics: 84th Edition. CRC Press, 2003.

[35] J. Hochstein, J. R. Warren, J. George Schmidt, J. Hochstein, J. R. Warren, and J. George Schmidt,

“Magnetically Actuated Propellant Orientation (MAPO) Experiment - Pre-ﬂight ﬂow ﬁeld predictions,”

Proceedings of the 35th Aerospace Sciences Meeting and Exhibit, 1997, pp. 1–11. AIAA Paper 1997-

570, 10.2514/6.1997-570.

[36] J. Marchetta and J. Hochstein, “Fluid capture by a permanent ring magnet in reduced gravity,”

Proceedings of the 37th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 1999, pp. 1–14.

AIAA Paper 1999-845, 10.2514/6.1999-845.

[37] J. Marchetta and J. Hochstein, “Simulation and dimensionless modeling of magnetically induced reori-

entation,” Proceedings of the 38th Aerospace Sciences Meeting and Exhibit, Renno, NV, USA, 2000,

pp. 1–13. AIAA Paper 2000-700, 10.2514/6.2000-700.

[38] J. Marchetta, J. Hochstein, D. Sauter, and B. Simmons, “Modeling and prediction of magnetic storage

and reorientation of LOX in reduced gravity,” 40th AIAA Aerospace Sciences Meeting & Exhibit, 2002,

pp. 1–19. AIAA Paper 2002-1005, 10.2514/6.2002-1005.

[39] J. G. Marchetta, “Simulation of LOX reorientation using magnetic positive positioning,” Microgravity

- Science and Technology, Vol. 18, Mar 2006, p. 31, 10.1007/BF02908417.

[40] J. Marchetta and K. Roos, “A Three-Dimensional Computational Simulation of Magnetic Positive Posi-

tioning,” 45th AIAA Aerospace Sciences Meeting and Exhibit, 2007, pp. 1–11. AIAA Paper 2007-956,

10.2514/6.2007-956.

[41] J. Marchetta and K. Roos, “Simulating Magnetic Positive Positioning of Liquids in a Transient Ac-

celeration Field,” 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008, pp. 1–11. AIAA Paper

2008-820, 10.2514/6.2008-820.

[42] J. G. Marchetta, B. D. Simmons, and J. I. Hochstein, “Magnetic retention of LO2 in an accelerating

environment,” Acta Astronautica, Vol. 62, No. 8, 2008, pp. 478 – 490, 10.1016/j.actaastro.2008.01.016.

[43] J. Marchetta and A. Winter, “Simulation of magnetic positive positioning for space based ﬂuid man-

agement systems,” Mathematical and Computer Modelling, Vol. 51, No. 9, 2010, pp. 1202 – 1212,

10.1016/j.mcm.2010.01.002.

[44] A. Romero-Calvo, T. H. Hermans, G. C. G´

omez, L. P. Ben´

ıtez, M. H. Guti´

errez, and E. Castro-

Hern´

andez, “Ferroﬂuid Dynamics in Microgravity Conditions,” Proceedings of the 2nd Symposion on

Space Educational Activities, Budapest, Hungary, 2018, pp. 1–5.

[45] A. Romero-Calvo, G. Cano G´

omez, E. Castro-Hern´

andez, and F. Maggi, “Free and Forced Oscillations

of Magnetic Liquids Under Low-Gravity Conditions,” Journal of Applied Mechanics, Vol. 87, 12 2020.

021010, 10.1115/1.4045620.

[46] A. Romero-Calvo, A. Garc´

ıa-Salcedo, F. Garrone, I. Rivoalen, G. Cano-G´

omez, E. Castro-Hern´

andez,

M. H. Guti´

errez], and F. Maggi, “StELIUM: A student experiment to investigate the slosh-

ing of magnetic liquids in microgravity,” Acta Astronautica, Vol. 173, 2020, pp. 344 – 355,

10.1016/j.actaastro.2020.04.013.

24

[47] A. Romero-Calvo, M. A. Herrada, T. H. Hermans, L. P. Ben´

ıtez, G. Cano-G ´

omez, and E. Castro-

Hern´

andez, “Axisymmetric ferroﬂuid oscillations in a cylindrical tank in microgravity,” Microgravity

Science and Technology, Vol. 33, No. 50, 2021, 10.1007/s12217-021-09894-4.

[48] A. Romero-Calvo, F. Garrone, A. Garc´

ıa-Salcedo, I. Rivoalen, G. Cano-G´

omez, E. Castro-Hern´

andez,

and F. Maggi, “Free surface reconstruction of opaque liquids in microgravity. Part 2: Drop tower cam-

paign,” Acta Astronautica, Vol. 189, 2021, pp. 269–277, 10.1016/j.actaastro.2021.07.020.

[49] A. Romero-Calvo, A. Garc´

ıa-Salcedo, F. Garrone, I. Rivoalen, and F. Maggi, “Lateral and axisymmetric

ferroﬂuid oscillations in a cylindrical tank in microgravity,” AIAA Journal, 2022. in press.

[50] A. Myshkis and R. Wadhwa, Low-gravity ﬂuid mechanics: mathematical theory of capillary

phenomena. Springer, 1987.

[51] B. Pugh, D. Kramer, and C. Chen, “Demagnetizing Factors for Various Geometries Precisely Deter-

mined Using 3-D Electromagnetic Field Simulation,” IEEE Transactions on Magnetics, Vol. 47, Oct

2011, pp. 4100–4103, 10.1109/TMAG.2011.2157994.

[52] A. Romero-Calvo, G. Cano-G ´

omez, and H. Schaub, “Diamagnetically Enhanced Electrolysis and Phase

Separation in Low Gravity,” AIAA Journal of Spacecraft and Rockets, 2021, pp. 1–13. in press,

10.2514/1.A35021.

[53] D. Sharp, “An overview of Rayleigh-Taylor instability,” Physica D: Nonlinear Phenomena, Vol. 12,

No. 1, 1984, pp. 3–18, 10.1016/0167-2789(84)90510-4.

[54] J. Campbell, E. Eldridge, and J. Thompson, “Handbook on Materials for Superconducting Machinery,”

Tech. Rep. MCIC-HB-04, ARPA, 1974.

[55] A. Trench and J. P. Sykes, “Rare Earth Permanent Magnets and Their Place in the Future Economy,”

Engineering, Vol. 6, No. 2, 2020, pp. 115–118, 10.1016/j.eng.2019.12.007.

[56] K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nuclear

Instruments and Methods, Vol. 169, No. 1, 1980, pp. 1 – 10, 10.1016/0029-554X(80)90094-4.

[57] S. R. Trout and S. Constantinides, “Using Permanent Magnets at Low Temperature,” Tech. Rep. TN

0302, Arnold Magnetic Technologies, 2021.

[58] K. Uestuener, M. Katter, R. Blank, D. Benedikt, J. Bahrdt, A. Gaupp, B. Klemke, F. Gr¨

uner, and

R. Weingartner, “Sintered (Pr,Nd)-Fe-B permanent magnets with (BH)max of 520.kJ/m3at 85 K for

cryogenic applications,” 20th International Workshop on Rare-Earth and Future Permanent Magnets

and their Applications, 2008.

[59] J.-C. Huang, H. Kitamura, C.-S. Yang, C.-K. Yang, S. Mizumoto, C.-H. Chang, C.-H. Chang, and

C.-S. Hwang, “Development of cryogenic permanent magnet undulators at NSRRC,” AIP Conference

Proceedings, Vol. 2054, No. 1, 2019, p. 030022, 10.1063/1.5084585.

[60] K. Tsuchiya, X. Wang, S. Fujita, A. Ichinose, K. Yamada, A. Terashima, and A. Kikuchi, “Su-

perconducting properties of commercial REBCO-coated conductors with artiﬁcial pinning centers,”

Supercond. Sci. Technol, Vol. 34, 2021, p. 105005, 10.1088/1361-6668/ac1e65.

[61] W. Fietz, R. Heller, S. Schlachter, and W. Goldacker, “Application of high temperature superconductors

for fusion,” Fusion Engineering and Design, Vol. 86, No. 6, 2011, pp. 1365–1368. Proceedings of the

26th Symposium of Fusion Technology (SOFT-26), 10.1016/j.fusengdes.2010.11.018.

[62] H. Weijers, W. Markiewicz, D. Abraimov, H. Bai, D. Hilton, A. Gavrilin, D. Larbalestier, J. Lu, T. P.

Murphy, P. N. a. J. Voran, and NHMFL, “Testing of prototype coils for the NHMFL 32 T superconduct-

ing user magnet,” Applied Superconductivity Conference, Charlotte, NC, 2014.

[63] O. P. Ivakhnenko and D. K. Potter, “Magnetic susceptibility of petroleum reservoir ﬂuids,” Physics and

Chemistry of the Earth, Parts A/B/C, Vol. 29, No. 13, 2004, pp. 899–907. Paleo, Rock and Environ-

mental Magnetism, 10.1016/j.pce.2004.06.001.

[64] C.-Y. Hong, I. J. Jang, H. E. Horng, C. J. Hsu, Y. D. Yao, and H. C. Yang, “Ordered structures in

Fe3O4 kerosene-based ferroﬂuids,” Journal of Applied Physics, Vol. 81, No. 8, 1997, pp. 4275–4277,

10.1063/1.364800.

[65] E. Ghasemi, A. Mirhabibi, M. Edrissi, R. Aghababazadeh, and R. M. Brydson, “Study on the

Magnetorheological Properties of Maghemite-Kerosene Ferroﬂuid,” Journal of Nanoscience and

Nanotechnology, Vol. 9, No. 7, 2009, pp. 4273–4278, doi:10.1166/jnn.2009.M45.

[66] V. I. Zubko, Y. I. Dikanskii, D. V. Zubko, S. A. Kunikin, and G. I. Sitsko, “Electrical and Magnetic Prop-

erties of a Kerosene-Based Magnetic Fluid Subjected to the Action of Electric and Magnetic Fields,”

Journal of Engineering Physics and Thermophysics, Vol. 91, May 2018, pp. 806–811, 10.1007/s10891-

018-1803-2.

[67] D. Susan-Resiga, I. Malaescu, O. Marinica, and C. Marin, “Magnetorheological properties of a

kerosene-based ferroﬂuid with magnetite particles hydrophobized in the absence of the dispersion

medium,” Physica B: Condensed Matter, Vol. 587, 2020, p. 412150, 10.1016/j.physb.2020.412150.

25

[68] L. Maldonado-Camargo, M. Unni, and C. Rinaldi, “Magnetic Characterization of Iron Oxide Nanopar-

ticles for Biomedical Applications,” Methods in molecular biology (Clifton, N.J.), Vol. 1570, 2017,

pp. 47–71, 10.1007/978-1-4939-6840-4˙4.

[69] M. L. Voloshin, S. A. Kuda, A. I. Logvinenko, A. N. Mashchenko, and E. I. Shevtsov, “Experi-

ence of Development and Use of Generator Pressurization System for Tanks of Launch Vehicles

on High-Temperature Propellants,” Space Technology. Missile Armaments, Vol. 1, 2019, pp. 45–53,

10.33136/stma2019.01.045.

[70] V. Trushlyakov, V. Shalay, J. Shatrov, M. Jakovlev, and A. Kostantino, “Active de-orbiting onboard

system from LEO of upper stages of launchers,” 5th European Conference on Space Debris, Darmstadt,

2009.

[71] E. Yutkin, V. Trushlyakov, F. Maggi, L. Galfetti, and L. T. De Luca, “Active onboard deorbiting system

for the second stage of Cosmos 3M: a preliminary study,” 4th European Conference for Aerospace

Sciences (EUCASS), 2011, pp. 1–9.

[72] F. Maggi, L. Galfetti, L. De Luca, V. I. Trushlyakov, V. Y. Kudentsov, and D. B. Lempert, “Thermo-

chemical Considerations in Support of ADOS Propulsion,” Space Debris Mitigation Workshop, 2010,

p. Presentation.

[73] V. I. Trushlyakov, D. B. Lempert, and M. E. Bel’kova, “Possibility of using gas-generating compositions

for increasing the rocket propulsion efﬁciency,” Combustion, Explosion, and Shock Waves, Vol. 51,

No. 3, 2015, pp. 326–332.

[74] V. I. Trushlyakov, V. A. Urbansky, and N. V. Pustovoy, “Study of the unusable liquid propellant

residues evaporation processes parameters in the tanks of the launch vehicle expended stage in mi-

crogravity,” Journal of Physics: Conference Series, Vol. 1441, jan 2020, p. 012121, 10.1088/1742-

6596/1441/1/012121.

[75] Arianespace, “Soyuz User’s Manual,” March 2012. Issue 2, Revision 0.

[76] G. Dussollier and A. Teissier, “Ariane 5 main stage oxygen tank pressurization,” 29th Joint Propulsion

Conference and Exhibit, AIAA-93-1969, Monterey, CA, June 1993, pp. 1–10, 10.2514/6.1993-1969.

[77] S. S. Kutateladze, Fundamentals of heat transfer. Academic Press, New York, 1964.

[78] K. Stephan, Physical Fundamentals of Vapor Bubble Formation, pp. 126–139. 1992, 10.1007/978-3-

642-52457-8˙10.

[79] S. L. Guseinov, S. Fedorov, V. Kosykh, and P. Storozhenko, “Hydrogen Peroxide Decomposition Cat-

alysts Used in Rocket Engines,” Russian Journal of Applied Chemistry, Vol. 93, 2020, pp. 467–487,

10.1134/S1070427220040011.

[80] C. Schumb, C. Satterﬁeld, and R. Wentworth, “Hydrogen Peroxide Monograph,” Journal of

the American Pharmaceutical Association (Scientiﬁc Ed.), Vol. 45, No. 2, 1956, p. 128,

10.1002/jps.3030450224.

[81] I. M. Ross, A primer on Pontryagin’s principle in optimal control. Collegiate publishers, 2015.

26