# Experimental analysis of heat and mass transfer in non-isothermal sloshing
using a model-based inverse method

Pedro A. Marques  Alessia Simonini  Laura Peveroni  Miguel A. Mendez  von
Karman Institute for Fluid Dynamics, Waterloosesteenweg 72, 1640 Sint-
Genesius-Rode, Belgium  Université Libre de Bruxelles, Av. Franklin Roosevelt
50, 1050 Bruxelles, Belgium

###### Abstract

Nonisothermal liquid sloshing in partially filled reservoirs can significantly
enhance heat and mass transfer between liquid and ullage gasses. This can
result in large temperature and pressure fluctuations, producing thrust
oscillations in spacecraft and challenging thermal management control systems.
This work presents an experimental characterization of the thermodynamic
evolution of a cylindrical reservoir undergoing sloshing-induced thermal de-
stratification. We use a 0D model-based inverse method to retrieve the heat
and mass transfer coefficients in planar and swirl sloshing conditions from
the temperature and pressure measurements in the liquid and the ullage gas.
The experiments were carried out in the SHAKESPEARE shaking table of the von
Karman Institute in a cuboid quartz cell with a cylindrical cut-out of 80 mm
diameter in the centre, filled up to 60mm with the cryogenic replacement fluid
HFE-7200. A thermal stratification with $\Delta T=25$ K difference between the
ullage gas and liquid was set as the initial conditions. A pressure drop of
90% in the ullage gas was documented in swirling conditions. Despite its
simplicity, the model could predict the system’s thermodynamic evolution once
the proper transfer coefficients were derived.

###### keywords:

Sloshing, non-isothermal, pressure drop, thermal destratification, 0D model,
inverse method, data assimilation

††journal: Applied Thermal Engineering

##  1 Introduction

Cryogenic fuels such as LH2, LCH4 and LNG are gaseous in ambient conditions
but must often be transported and stored in the liquid state to have cost-
competitive volumetric energy densities [1]. This requires challenging
cryogenic conditions, especially for hydrogen, which can only exist as a
liquid (LH2) under 33 K. While these fuels are already employed in aerospace
propulsion [2], significant technological advances are still required to make
them economically viable for other applications on a global scale [3, 4, 5].

The efficient storage of cryogenic liquids requires complex thermal management
systems. Since no insulating system can entirely prevent heat exchange with
the surrounding [6], it is impossible to prevent some of the fuel from boiling
off and thus the tank pressure from rising over time [7, 8, 9]. The simplest
way to control the tank pressure is venting, but this brings a high cost of
energy and mass loss. Additional liquefaction steps can be used to recover the
vented fuel, but this further increase the operational costs.

Thermal losses increase significantly in fuel tanks installed on vehicles,
whether these travel on sea, land, air, or space because external
accelerations induce sloshing. Besides challenging the vehicle’s stability and
manoeuvrability [10], sloshing can significantly increase heat and mass
transfer rate between liquid and ullage gasses, leading to significant
variations of the tank’s pressure [11, 12, 13].

In typical operating conditions, the ullage gasses tend to be warmer than the
liquid. This is due to the pressurization, typically up to $2-12$ bar [1, 14],
and the heat fluxes with the environment [15, 16]. As a result, thermal
stratification is produced in the tank if this is left in quiescent
conditions. Sloshing produces a thermal mixing that can level the
stratification and suddenly reduce the temperature at the gas-liquid interface
[17]. Since the interface naturally sets the saturation condition of the tank,
a decrease in its temperature produces a decrease in the saturation pressure,
hence condensation [18]. This is known in the literature as the ‘pressure drop
effect’ [17, 19].

The opposite mechanism can also occur when heat exchange across the wall
increases the interface temperature near the contact line and thus enhances
evaporation [20], increasing the tank’s pressure [21]. The pressure
fluctuations due to sloshing-induced evaporation or condensation can produce
thrust oscillation in a spacecraft and force thermal management systems to
react with venting or re-pressurization; both operations bring a cost that
must be minimized.

The literature on the sloshing phenomenon is abundant and has been
significantly fostered by the aerospace industry during the 1960s [22, 23,
24]. These works focused on the (isothermal) wave dynamics and the damping
rates of several fluids under many excitation conditions and tank shapes. More
recently, the interest in the pressure drop effect has shifted towards
characterizing the non-isothermal consequences of sloshing [11, 17, 18, 21,
25, 26, 27]. These works have focused on the thermodynamic response of
partially filled tanks under different sloshing regimes, pressurization
techniques, pressurant gasses and tank designs (e.g. with or without baffles).
The data retrieved in these experiments allowed several authors to develop and
validate simplified models to predict the thermodynamic changes in fuel tanks
due to sloshing. The simplest and yet most popular approach is a quasi-
dimensional (0D) formulation [1, 28, 29, 30] in which the fuel tank system is
composed of three control volumes (CVs): the ullage gas, the liquid, and the
solid walls. Under this lumped formulation, the model equations are derived
from each volume’s energy and mass balances, using mass-averaged thermodynamic
properties for each volume. This results in a coupled system of Ordinary
Differential Equations (ODEs) which can be solved for a given set of inputs
and initial states to predict evaporation and condensation rate and hence the
thermodynamic evolution of the fuel tank. Since these models are simple to
derive and computationally inexpensive to evaluate, they can be easily
integrated with other sub-models to perform system simulations [31] in real
time.

However, these models must rely on closure laws for heat and mass transfer
coefficients to close the system of equations. While several empirical
correlations have been proposed for stationary/long-term storage conditions
[14, 28, 29], their estimation during sloshing events remains an open problem.
Moreover, no established methodology has been proposed in the literature for
driving these closure laws in dynamic conditions.

A natural approach is to ‘tune’ these coefficients such that numerical
prediction match well with experimental data (see [1] and [30]). This can
provide useful orders of magnitude, but it can hardly account for the time
variation of these coefficients. To the authors’ knowledge, the only
correlation currently available for modelling the sloshing-enhanced thermal
mixing has been proposed by Ludwig & Dreyer [32]. These authors have proposed
an empirical law for the sloshing-induced ‘effective diffusivity’, which
depends on the system’s initial condition and the maximum pressure variation
measured in the tank. Their correlation, expressed in terms of a sloshing-
based Nusselt number, proved valid for their experimental data and the
experiments of Das & Hopfinger [33] and Moran et. al [11]. However, the
agreement was limited to the initial stages of the sloshing excitation.

This work presents an experimental characterization of the sloshing-induced
heat and mass transfer and proposes a model-based inverse method to derive
empirical correlations from the measurements of pressure and temperatures in
the tank. The inverse method sets an optimization problem in which the closure
terms are automatically tuned to minimize the discrepancy between experimental
data and a model’s prediction. This data-driven approach, common in data-
assimilation literature [34], has been used in the calibration of constitutive
laws [35], modelling of dynamic menisci [36], thermal turbulence closure [37],
flow control [38] and modelling closure for stagnation line flow in hypersonic
reentry [39].

This work gives a promising proof of concept on using such a model-based
inverse method to derive empirical laws for sloshing-induced heat and mass
transfer from pressure and temperature measurements in the tank. We consider a
reduced-scale upright cylindrical tank using the non-cryogenic replacement
fluid HFE-7200 [40]. The experiments were carried out under different lateral
sloshing excitation in the presence of an axial thermal stratification. The
underlying thermodynamic model is a classic 0D formulation of the energy and
mass balance between three control volumes (liquid, ullage gas and solid),
while the optimization was carried out with the Basinhopping algorithm,
complemented with a Monte Carlo approach for estimating model uncertainties.

The rest of the article is structured as follows. Section 2 presents an
overview of the physical problem alongside all modelling assumptions and
approximations. Section 3 summarizes the experimental test cases and the
theoretical aspects used to reproduce different sloshing conditions in the
laboratory. Section 4 reports on the methodology, including a detailed
descriptions of the experimental setup and procedure, the 0D model, and the
inverse method. The results are shown and discussed in Section 5. Conclusions
and future outlook are presented in Section 6.

##  2 Problem Statement and Objectives

The problem considered in this work is schematically illustrated Figure 1. A
cylindrical tank of radius $R$ and height $H$, with a flat top and bottom
surfaces, is filled with liquid until a height $H_{l}$. The tank is closed and
pressurized until the ullage gas is at a pressure $p_{g}$. The ullage is
composed of a mixture of two species: an inert gas at partial pressure $p_{a}$
and the liquid’s vapour at partial pressure $p_{v}$.

The tank walls are adiabatic with mass $m_{w}$ and specific heat capacity
$c_{w}$. Because of the tank’s pressurization, the system tends towards a
thermally stratified configuration in which the liquid is colder than the gas.
More specifically, to a first approximation, one might expect an axisymmetric
temperature field $T(r,z)$, with $r$ the radial coordinate and $z$ the
vertical one, as sketched in the right-hand side of Figure 1. Subscripts $l$,
$g$ $w$ denote variables linked to the liquid, the gas and the walls. The gas-
liquid interface is at the saturation temperature $T_{\text{sat}}(p_{v})$
corresponding to the gas pressure, and a positive temperature gradient
$\partial_{z}T$ develops at the interface (cf. Figure 1). No mass transfer
occurs between liquid and gas if the ullage gas is saturated with the liquid
vapour.

Figure 1: Schematic of the the problem set. Fuel tank with radius $R$, total
height $H$, solid wall thickness $s$, and filled with liquid up to $H_{l}$
(left). Axial temperature distribution after pressurizing the tank at an
arbitrary radial coordinate $r$ (right). Because the tank is pressurized up to
a certain pressure $p_{g}$, the gas is initially warmer than the liquid.

In this work, we focus on the perturbation of this state due to lateral
sloshing of the tank. We consider an harmonic displacement along $x$ with
amplitude $A_{0}$ and angular frequency $\Omega$:

$X(t)=A_{0}\sin(\Omega t).$ (1)

Sloshing induces the mixing of the liquid interface, thus a destratification
of the thermal field. This reduces the (average) interface temperature
$T_{i}$, which in turn leads to an increase in the heat transfer and mass
transfer at the interface. Both mechanisms lead to a pressure drop. We here
seek to analyze experimentally whether a simple 0D model of the heat and mass
transfer can describe the thermodynamic evolution of the ullage gas after a
sloshing event. Furthermore, let

$\small
h_{li}(t)=\frac{\dot{Q}_{li}(t)}{S(\overline{T}_{l}(t)-T_{i}(t))}\,\,,\,\,h_{gi}(t)=\frac{\dot{Q}_{gi}(t)}{S(\overline{T}_{g}(t)-T_{i}(t))}$
(2)

$\small
h_{wl}(t)=\frac{\dot{Q}_{wl}(t)}{S_{wl}(\overline{T}_{w}(t)-T_{l}(t))}\,\,,\,\,h_{wg}(t)=\frac{\dot{Q}_{wg}(t)}{S_{wg}(\overline{T}_{w}(t)-T_{g}(t))}$
(3)

denote the instantaneous global heat transfer coefficients between the fluid
phases and interface, and between the fluids and the tank walls, respectively,
with $\dot{Q}$ the heat exchanged between the different regions, identified by
their $l$ (liquid), $g$ (gas), $i$ (interface), $w$ (walls) subscripts. In
these expressions, $S$ is the tank’s cross section area, $S_{wl}$ and $S_{wg}$
the tank’s wetted areas in contact with the liquid and gas at rest,
$\overline{T}_{l}(t)$, $\overline{T}_{g}(t)$, $\overline{T}_{w}(t)$ the
(instantaneous) volume averaged temperature in the liquid, gas, and solid
walls, $T_{i}(t)$ the instantaneous spatial average of the interface
temperature. Let

$h_{m}(t)=\frac{\dot{m}_{i}(t)}{S(\rho_{v,\text{sat}}(t)-\overline{\rho}_{v}(t))}$
(4)

denote the instantaneous global mass transfer coefficient between liquid and
gas, with $\dot{m}_{i}$ the exchanged vapour mass flow rate, $\rho_{v,sat}$
the vapour density at the saturation temperature and $\overline{\rho}_{v}$ the
volume average of the vapour density.

We seek to analyze whether a model-based inverse method can be used to
retrieve $h_{li}$, $h_{gi}$ and $h_{m}$ from real-time measurements of
pressure and temperature in the tank.

##  3 Sloshing conditions

This section describes the excitation conditions applied to the fuel tank to
produce liquid sloshing. The system was excited in the vicinity of the natural
frequency corresponding to the first asymmetrical lateral sloshing mode. The
natural frequencies in a partly-filled upright cylindrical reservoir subject
to these conditions are given by:

$\omega_{mn}^{2}=\left(\frac{g\xi_{mn}}{R}+\frac{\sigma}{\rho_{l}}\frac{\xi_{mn}^{3}}{R^{3}}\right)\tanh\left(\frac{\xi_{mn}H_{l}}{R}\right),$
(5)

where $m,n$ are the indices defining the sloshing mode, $\xi_{mn}$ is the
$n^{\text{th}}$ zero of the first derivative of the $m^{\text{th}}$ order
Bessel function, $\sigma$ is the surface tension and $\rho_{l}$ is the
liquid’s density [41]. The lowest wave mode that can be excited in these
conditions is the first asymmetrical one $m=n=1$. Investigations on several
flights of the Arianespace launchers show that the $(1,1)$ mode is the primary
excitation during the propelled phase [18, 42]. Therefore, this work focuses
on excitations close to $\omega_{11}$.

Three different wave responses may be observed in these conditions: (1) planar
waves, (2) chaotic, and (3) swirl sloshing [43]. The planar wave regime is
observed for low amplitudes and frequencies and is characterized by harmonic
motion. The chaotic regime is observed for excitation frequencies near
resonance conditions. In this regime, the sloshing dynamics quickly depart
from a linear condition, leading to unpredictable interface dynamics. Finally,
the swirling regime is observed for excitation frequencies slightly above
resonance. Swirling waves can be classified as stable if the rotational
velocity remains constant or unstable if it periodically changes direction.

The boundaries between these regimes have been analytically derived by Miles
[43], and experimentally verified by Royon-Lebeaud [44], who proposed the
following relation:

$\frac{A_{0}}{R}=\frac{1}{1.684}\left(\frac{\left(\Omega/\omega_{11}\right)^{2}-1}{\mathcal{B}_{i}}\right)^{3/2},$
(6)

where $A_{0}$ and $\Omega$ were defined in Section 2 and $\mathcal{B}_{i}$, is
the frequency offset parameter that separates the sloshing regimes in the
vicinity of $\omega_{11}$. These are $\mathcal{B}_{2}=-0.36$,
$\mathcal{B}_{3}=-1.55$, and $\mathcal{B}_{4}=0.735$.

Figure 2: Sloshing regimes diagram from Miles’ weakly nonlinear theory [43].
The tested experimental cases are identified and labelled with numbers ranging
from $1-5$. The excitation conditions are further detailed in Table 2.

Figure 2 shows Mile’s diagram together with boundaries between regimes
according to (6) and the six conditions investigated in this work. These are
further detailed in Table 2 and described in Section 4.2 together with the
experimental procedure. To complement the aforementioned kinematic
description, the dynamics of the sloshing regime and its mixing efficiency are
linked to the sloshing Reynolds number $\text{Re}_{s}$ proposed by Ludwig &
Dreyer [32], defined as:

$\text{Re}_{s}=\frac{\Omega}{\omega_{11}}\left(\frac{b}{R}\right)^{2}\frac{\left(gR^{3}\right)^{1/2}}{\nu}\sqrt{\xi_{11}},$
(7)

where $b$ is an estimate of the wave amplitude expected during the sloshing
motion. We take $b/A_{0}\approx
2[(\Omega/\omega_{11})^{2}/(1-(\Omega/\omega_{11})^{2})]$ in the planar
regime, $b\approx 0.54R$ in the chaotic regime and $b\approx 0.7R$ in the
swirl sloshing regime, as indicated by [32]. According to these authors, the
critical Reynolds number for the mixing is $(\text{Re}_{s})_{c}=4.0\cdot
10^{3}\pm 20\%$. Below this value, sloshing does not significantly increase
the heat or mass transfer in the tank.

##  4 Methodology

###  4.1 Experimental setup

Table 1: Location of the thermocouples (sorted by increasing height) with respect to a cylindrical coordinate system with origin centered in the tank’s lower inner surface. The numbering of the thermocouples and orientation of the axes is illustrated in Figure 4. Thermocouple no. [-] | 1 | 8 | 7 | 2 | 6 | 3 | 5 | 11 | 4 | 12 | 10 | 13 | 9
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$z$ [mm] | 9 | 19 | 34 | 39 | 44 | 46 | 49 | 50 | 52 | 64 | 69 | 84 | 94
$r$ [mm] | 37.5 | 37.5 | 37.5 | 37.5 | 37.5 | 37.5 | 37.5 | 1.5 | 37.5 | 15.1 | 15.1 | 30.0 | 30.0
$\theta$ [∘] | 10 | 170 | 155 | 25 | 140 | 40 | 120 | 90 | 60 | 5.7 | 174.3 | 2.9 | 177.1

The experimental campaign was carried out in the SHAKESPEARE (SHaking Aparatus
for Kinetic Experiments of Sloshing Projects with EArthquake Reproduction)
facility in the von Karman Institute. The table has dimensions of 1.5 m
$\times$ 1.5 m and allows to mount different experimental setups. This
facility is composed of three sliding modules, one for each axis, which can
recreate complex three-dimensional excitation with controlled amplitudes and
frequencies. A schematic of the table’s three sliding modules is shown in
Figure 3. Since this work focuses on lateral sloshing, only excitation in the
$x$ direction was considered. The displacement of this sliding module was
measured with an Optical Displacement Sensor (ODS30) [45]. This sensor
measures displacements in the range of $28-32$ mm.

Figure 3: 3D view of the three sliding modules of the SHAKESPEARE (SHaking
Aparatus for Kinetic Experiments of Sloshing Projects with EArthquake
Reproduction) shaking table facility.

The sloshing cell used in this work is a quartz rectangular cuboid with
dimensions 100 mm $\times$ 100 mm $\times$ 134 mm in which a cylindrical hole
with diameter $D=80$ mm was drilled down to a depth of 124 mm. The top cover
of the cell consisted of a 12 mm acrylic glass plate with a cylindrical
extrusion in the centre with a 12mm thickness and a diameter slightly smaller
than 80 mm. This assembly was instrumented with thirteen 0.25 mm grounded
mineral insulated type K thermocouples, a static pressure tap, and two heating
elements. Figure 4 shows a top view of the cell, illustrating the position of
the pressure tap and the thermocouples in radial and angular coordinates. The
thermocouples are numbered counter-clockwise, and their location is given in
Table 1 in a cylindrical reference system centred on the bottom wall, at the
origin of axes in Figure 1. Eight thermocouples were placed in one half of the
cell’s cross-section along its periphery at a diameter $D=75$ mm, and five
were located along the centerline. These thermocouples have a response time of
$7.5$ ms when plunged into boiling water from air at 20 ∘C, and their signals
were acquired through two NI9212 8 Channel National Instruments cards with a
sampling rate of 95 Hz per channel.

Figure 4: Schematic of the top view of the instrumented sloshing cell.
Thirteen 0.25mm type K thermocouples, one static pressure tap and two
thermofoil heating elements were mounted on the top-cover of the sloshing
cell.

The top-mounted pressure tap was connected to a Validyne with an M38 diaphragm
diameter and maximum admissible pressure of 0.55 bar. The Validyne CD15
carrier demodulator was used to convert the analogue pressure measurement into
an output voltage between $\pm 10$ V. The response time of the pressure line
was estimated to be around 156 ms.

Two Minco Polyimide Thermofoil HK6907 heaters [46] were selected to impose the
thermally stratified field during the initialization stage. The heating was
applied from the top to reproduce an axial thermal stratification similar to
[17, 18, 32]. These heaters have dimensions of 25.4 mm $\times$ 50.8 mm
$\times$ 1.14 mm, and minimum and maximum working temperatures of -32 and 100
∘C, respectively. Their maximum output power is 16.4 W. A 0.25 mm type K
thermocouple was attached to one of these elements with thermal paste to
monitor their temperature during the heating stage of the experiments. This
thermocouple was then connected to a Tempatron PID330 controller to regulate
the power provided by the TENMA 72-8700 [47] power supply, ensuring the
maximum limit of $100^{\circ}$C was never crossed.

The facility was insulated with a 50 mm layer of Rockwool 850 material [48].
This minimized the environmental effect and improved the experiments’
repeatability. Figure 5(a) shows a cut-out view of the experimental setup
composed of the quartz tank, top-cover and bottom-cover assemblies,
temperature sensing instrumentation, and insulation layer.

(a)

(b)

Figure 5: Instrumented experimental setup. a) Sectional view of the different
elements which compose the experimental setup: (1) plexiglas top-cover, (2)
bakelite insulating plate, (3) copper shield which houses both HK6907 Minco
heating elements, (4) quartz tank, (5) steel threads that allow for screwing
down the top-cover assembly, (6) Rockwool 850 insulating layer, (7) plexiglas
bottom-cover, (8) temperature sensing assembly, (9) double-grooved aluminium
interface plate with two O-rings. b) Pictures of the set-up without (left) and
with (right) insulation.

The facility was equipped for high speed visualization (HSV) in back-lighting
conditions and for Particle Image Velocimetry (PIV). Both were performed
without insulation, which prevents optical access. Nevertheless, the
importance of these tests was twofold, as further described in Section 5.1.
The HSV allowed a qualitative confirmation of the expected sloshing regime
while the PIV was performed to analyze the flow field in the liquid phase
_before_ the sloshing event and thus ensure that each experiment started from
the same (quiescent) conditions. For the PIV measurements, the liquid was
seeded with fluorescent polymer microspheres with a density of 1300 kg/m3 and
diameter 1-5 $\mu$m. The image acquisition was performed with a 12 bit Imager
SX 4M camera [49] with a resolution of 2360 × 1776 pixels. The region of
interest was illuminated by the Litron Bernoulli Laser [50]. This pulsed
Nd:YAG laser has a maximum repetition rate of 15 Hz, an output power of 200 mJ
per cavity, and a $3-7$ ns pulse duration at 532 nm. The laser beam was
focused through the spherical lens and turned into a sheet with a cylindrical
lens. This sheet was then deflected through a prism underneath the sloshing
cell, thus illuminating its mid-plane from the bottom.

###  4.2 Experimental procedure and test matrix

Table 2: Summary of the initial thermodynamic conditions at the start of the pressurization and sloshing stages, and the investigated lateral sloshing excitation. Superscripts ‘$Q$’ and ‘$T$’ refer to the start of the constant heat-flux and temperature heating regimes, respectively. The superscript ‘$S$’ is associated with the start of sloshing. Average gas and liquid temperatures $\tilde{T}_{g}$ and $\tilde{T}_{L}$, respectively, are presented at the start of the heating stage $t^{Q}$, and at the start of sloshing $t=0$. In this table, $A_{0}$ is the amplitude of the lateral motion, $R$ is the tank radius, $\Omega$ is the imposed angular frequency, $\omega_{11}$ is the natural frequency of the $(1,1)$ sloshing mode, $f_{e}$ is the excitation frequency in Hertz, and Res is the sloshing-based Reynolds number [32]. Case |  $t^{Q}$ [s] |  $t^{T}$ [s] |  $p_{g}^{Q}$ [kPa] |  $\tilde{T}_{l}^{Q}$ [K] |  $\tilde{T}_{g}^{Q}$ [K] |  $p_{g,0}^{S}$ [kPa] |  $\tilde{T}_{l,0}^{S}$ [K] |  $\tilde{T}_{g,0}^{S}$ [K] | $A_{0}/R$ | $\Omega/\omega_{11}$ |  $A_{0}$ [mm] |  $f_{e}$ [Hz] |  Res [-]
---|---|---|---|---|---|---|---|---|---|---|---|---|---
s0 |  $-$3602 |  $-$3129 | 100.0 | 291.5 | 291.4 | 103.5 | 295.1 | 313.9 | - | - | - | - | -
s1 |  $-$3613 |  $-$3179 | 99.4 | 297.7 | 296.9 | 102.5 | 298.4 | 315.6 | 0.025 | 0.80 | 1.0 | 2.70 | 5.1e3
s2 |  $-$3622 |  $-$3198 | 100.0 | 297.4 | 296.9 | 103.5 | 298.7 | 315.8 | 0.025 | 0.89 | 1.0 | 3.00 | 1.2e4
s3 |  $-$3610 |  $-$3178 | 99.3 | 296.8 | 296.8 | 102.6 | 298.3 | 315.3 | 0.045 | 0.84 | 1.8 | 2.83 | 1.3e4
s4 |  $-$3607 |  $-$3167 | 99.9 | 296.6 | 296.7 | 103.7 | 298.7 | 315.7 | 0.045 | 0.86 | 1.8 | 2.90 | 1.6e4
s5 |  $-$3610 |  $-$3185 | 99.8 | 297.0 | 296.6 | 103.3 | 298.4 | 315.6 | 0.025 | 1.04 | 1.0 | 3.5 | 5.1e4

The experimental procedure and the test matrix considered in this work are
described in this section. Each experiment consists of four phases, three of
which are preparatory. In the first phase, the quartz cell is filled with
$302\pm 2$ ml of HFE-7200. This results in a liquid level of $60\pm 0.4$ mm
measured from the bottom of the cell. Then, the top cover is closed and
screwed into position. Temperature-resistant grease is used to improve the
connection between the O-rings, the aluminium interface plate, the quartz cell
and the top cover. The setup remains closed for several hours until stable
temperature and pressure readings are established inside the cell. This
duration is long enough to allow the liquid’s vapour to saturate the ullage
gas.

Once this condition was met, the second phase begins with the acquisition of
the pressure and temperature signals. During this second phase, the heating
elements are switched on and produce a constant heat flux until the operating
temperature of 100 ∘C is reached. Then, the third phase begins with the PID
controller actively enforcing a constant temperature on the heater surface.
During this third phase, the system is warmed at a slower rate from the
constant temperature surface at the top of the cell.

The pressure evolution and temperature distribution in the cell were
continuously measured throughout this initialization stage, which lasted one
hour and generated an axial thermal stratification in the tank. Figure 6 shows
a typical reading for the pressure sensor in the ullage gas and the
thermocouples during the second, third and fourth phases. The fourth phase
consists of the actual sloshing experiment: the heater was switched off, and
the lateral excitation was initiated from the controller screen of the
SHAKESPEARE sloshing table. The time label $t=0$ is associated with the start
of the sloshing, which is the focus of this work. Each excitation lasted 10
minutes, after which the data acquisition system was terminated, and the
shaking table was turned off.

(a)

(b)

Figure 6: Thermodynamic evolution of the system during heating and sloshing in
a sample experimental test case (Case s2 in Table 2). a) Gauge pressure
evolution in the quartz cell and b) thermal profiles of the cell during
heating (left) and sloshing (right). The system’s heating starts at $t=t^{Q}$
with a constant heat flux until the heating elements’ temperature reaches 100
∘C ($t=t^{T}$). It then proceeds at a constant temperature until $t=t^{S}=0$,
when the heating is switched off, and the sloshing starts. The liquid
interface is at $z=60\pm 0.4$mm.

Figure 6(a) shows the impact of the different phases on the pressure. The
heating during the preparatory phase leads to an increase in the ullage
pressure. This rise is initially steeper both because of the warm-up and the
evaporation of the liquid. The temperature profiles, shown in Figure 6(a),
clearly reveal the thermal stratification reached before sloshing. Sloshing
mitigates the thermal stratification and reduces the temperature at the liquid
interface by approximately 3 ∘C in the illustrated example. This drastically
reduces the saturation pressure in the ullage gas and induces condensation.
The resulting pressure drop is visible in Figure 6(a) for $t>0$s.

The initial thermodynamic conditions for each of the performed test cases are
collected in Table 2. Case s0 was included as a reference condition to
investigate the pressure and temperature evolution of the initialized system
without sloshing. The positioning of these conditions in the Mile’s diagram is
shown in Figure 2. To compute the absolute pressure and spatially averaged
temperatures required by the model, the measurement was complemented with a
reading of the atmospheric pressure from the VKI’s meteorological station
while volume averaged temperatures were computed numerically at each time-
instant through adaptive quadrature in Scipy [51] using the instantaneous
readings in each thermocouple.

###  4.3 Modeling of heat and mass transfer

The quasi-dimensional (0D) model used to determine the heat and mass transfer
coefficients relies on the assumption that the propellant tank’s wall are
adiabatic and the system is closed. The mass and energy conservation thus
give:

$\frac{dm_{g}}{dt}+\frac{dm_{l}}{dt}=0$ (8)
$\frac{dU_{g}}{dt}+\frac{dU_{l}}{dt}+\frac{dU_{w}}{dt}=0,$ (9)

where $m_{g},m_{l}$ and $U_{g},U_{l}$ are the masses and the internal energies
of the ullage gas (subscript $g$) and the liquid (subscript $l$), and $U_{w}$
is the internal energy of the walls. The mass in the ullage gas is the sum of
the inert gas ($m_{a}$, in our case air) and the liquid’s vapour ($m_{v}$)
masses, thus $m_{g}=m_{a}+m_{v}$. Assuming that the system is closed, i.e.
$dm_{a}/dt=0$, the mass and the energy balances in each subsystem yield

$\frac{dm_{v}}{dt}=-\frac{dm_{l}}{dt}=-\dot{m}_{i}$ (10)

$\frac{dU_{l}}{dt}=-\dot{Q}_{li}-\dot{m}_{li}h_{l}^{\text{s}}+\dot{Q}_{wl}$
(11)

$\frac{dU_{g}}{dt}=-\dot{Q}_{gi}+\dot{m}_{i}h_{v}^{\text{s}}+\dot{Q}_{wg}$
(12)

$\frac{dU_{w}}{dt}=-\dot{Q}_{wl}-\dot{Q}_{wg}$ (13)

where $\dot{m}_{i}$ is the net mass flux from the gas to the liquid,
$h_{l}^{\text{s}}$ and $h_{v}^{\text{s}}$ are the saturated liquid and vapour
enthalpies, and $\dot{Q}$ represents the heat fluxes linked to the heat
transfer coefficients defined in Equations (2) and (3).

The interface temperature in (2) is assumed to be equal to the saturation
temperature at the partial vapour pressure in the ullage gas, denoted as
$p_{v}$. This can be estimated using the Clausius-Clapeyron relation [52]:

$T_{i}=\left(\frac{1}{T_{\text{sat}}^{\text{ref}}}-\frac{R_{v}}{\mathcal{L}_{v}}\ln\left(\frac{p_{v}}{p_{v,\text{sat}}^{\text{ref}}}\right)\right)^{-1}\,,$
(14)

where $p_{v,\text{sat}}=14.53$ kPa and $T_{\text{sat}}^{\text{ref}}=298.15$ K
are the reference vapour pressure and corresponding saturation temperature and
$\mathcal{L}_{v}=125.6$ kJ/kg is the latent heat of vaporization [40]. The net
mass transfer due to the phase change is linked to the heat fluxes through the
energy balance at the interface:

$\dot{m}_{i}=-\frac{\dot{Q}_{gi}+\dot{Q}_{li}}{\mathcal{L}_{v}}\,.$ (15)

Equations (8)-(13) can be combined into the rate of changes for the volume-
averaged temperatures in the liquid, gas and tank walls:

$\frac{d\overline{T}_{l}}{dt}=\frac{-\dot{m}_{i}\left(c_{p,l}\left(\overline{T}_{l}-T_{i}\right)-{p_{v}}/{\rho_{l}}\right)-\dot{Q}_{li}+\dot{Q}_{wl}}{m_{l}c_{p,l}}$
(16)

$\frac{d\overline{T}_{g}}{dt}=\frac{\dot{m}_{i}\left(c_{p,v}\left(\overline{T}_{g}-T_{i}\right)-{p_{v}}/{\rho_{v}}\right)-\dot{Q}_{gi}+\dot{Q}_{wg}}{m_{a}c_{v,a}+m_{v}c_{v,v}}$
(17)

$\frac{d\overline{T}_{w}}{dt}=-\frac{\dot{Q}_{wl}+\dot{Q}_{wg}}{m_{w}c_{w}}$
(18)

where $c_{v}$ and $c_{p}$ are the specific heat capacities at constant volume
and constant pressure, respectively. Equations (16) - (18) were derived
considering that $dh\approx c_{p}dT$ and $du\approx c_{v}dT$ in the ullage,
whereas $du\approx c_{p}dT$ for the liquid region as proposed in [53].

Given the lack of real gas equations of state for the HFE-7200 used in this
work, the ideal gas law was used to model vapor’s properties in the ullage. In
a first approximation, this appears reasonable considering that the reduced
pressure is in the range $p_{R}=p_{v}/p_{c}=0.01-0.02$ and the reduced
temperature is in the range $T_{R}=T/T_{c}=0.60-0.65$, with critical
conditions $T_{c}=483.15$ K, and $p_{c}=2.01$ MPa [40]. At such low reduced
pressure, the compressibility factor $Z={p_{v}}/{\rho_{v}R_{v}T_{g}}$ can be
expected to be close to unity for most common fluids [52] and it this thus
believed that the impact of this simplifying assumption is within the
uncertainties of the inverse method.

Therefore, the pressure evolution in the ullage can be determined by taking
the time derivative of the ideal gas law and expanding it with the chain rule.
This gives

$\frac{dp_{g}}{dt}=\frac{m_{a}R_{a}+m_{v}R_{v}}{V_{g}}\left(\frac{d\overline{T}_{g}}{dt}-\frac{dV_{g}}{dt}\frac{\overline{T}_{g}}{V_{g}}\right)+\frac{R_{v}\overline{T}_{g}}{V_{g}}\frac{dm_{g}}{dt}.$
(19)

Equation (19) highlights the three fundamental mechanisms responsible for
pressure variations in closed reservoirs. These are time variations in the (1)
ullage gas temperature, (2) ullage volume, and (3) vapour mass. The rate-of-
change of the ullage volume was derived from the volume variation on the
liquid side. Since the liquid is treated as incompressible, its density is
only a function of the temperature and thus

$\frac{dV_{l}}{dt}=-\frac{dV_{g}}{dt}=\frac{1}{\rho_{l}}\frac{dm_{l}}{dt}-\frac{m_{l}}{\rho_{l}^{2}}\left.\frac{\partial\rho_{l}}{\partial
T}\right|_{p}\frac{d\overline{T}_{l}}{dt}$ (20)

where $\left.\partial\rho_{l}/\partial T_{l}\right|_{p}$ is the liquid’s
isobaric compressibility.

Under the assumption that the vapour and the inert in the ullage gas are at
the same temperature $\overline{T}_{g}$ and the interface is always at
saturation conditions (equation (14)), the thermal evolution of the problem
must be constrained to the condition that $\overline{T}_{g}>{T}_{i}$. However,
the energy balance underlying (16)-(18) does not guarantee this condition to
be satisfied. To enforce this constraint, the system was complemented with the
equations governing the evolution of the pressure and average temperatures of
gas and liquid along the saturation line. These read:

$\frac{dp_{g}}{dt}=\left.\frac{\partial
p_{g}}{\partial\rho}\right|_{\text{sat}}\left(\frac{1}{V_{g}}\frac{dm_{v}}{dt}-\frac{m_{g}}{V_{g}^{2}}\frac{dV_{g}}{dt}\right)$
(21)

$\frac{d\overline{T}_{g}}{dt}=\left.\frac{\partial\overline{T}_{g}}{\partial\rho}\right|_{\text{sat}}\left(\frac{1}{V_{g}}\frac{dm_{v}}{dt}-\frac{m_{g}}{V_{g}^{2}}\frac{dV_{g}}{dt}\right)$
(22)

$\frac{d\overline{T}_{l}}{dt}=\frac{Q_{wg}+Q_{wl}-(m_{a}c_{v,a}+m_{v}c_{v,v})\frac{d\overline{T}_{g}}{dt}-c_{v,v}\overline{T}_{g}\frac{dm_{v}}{dt}}{m_{l}c_{p,l}}\,,$
(23)

where $\left.\partial p_{g}/\partial\rho\right|_{\text{sat}}$ and
$\left.\partial\overline{T}_{g}/\partial\rho\right|_{\text{sat}}$ were
obtained by differentiating the Clasius-Clapeyron relation coupled with the
ideal gas law. These equations were solved in parallel to (16)-(18) but
activated only at the limiting case $\overline{T}_{g}=T_{i}$ to prevent
$\overline{T}_{g}$ to drop below $T_{i}$.

Equations (16)-(23) constitute a system of ordinary differential equations in
which eight variables describe the state of the system. These are:

$\mathbf{x}=\begin{bmatrix}\overline{T}_{g}&\overline{T}_{l}&\overline{T}_{w}&m_{v}&m_{l}&V_{g}&V_{l}&p_{g}\end{bmatrix}\,.$
(24)

The evolution of this system from a set of initial conditions $\mathbf{x}^{0}$
depends on the fluid properties and tank geometry, as well as four heat
transfer coefficients $h_{li},h_{gi}$, $h_{wl},h_{wg}$ (see eq. (2)).

Special treatment was given to the averaged wall temperature
$\overline{T}_{w}$, because this was not measured during the experiments and
cannot be linked to the other quantities unless its initial value
$\overline{T}_{w}(t=0)$ is given. Therefore, the model predictions were run
considering three scenarios for $\overline{T}_{w}(t=0)$. These are: (1)
$\overline{T}_{w}(0)=\overline{T}_{g}(0)$, (2)
$\overline{T}_{w}(0)=(\overline{T}_{l}(0)+\overline{T}_{g}(0))/2$ and (3)
$\overline{T}_{w}(0)=\overline{T}_{l}(0)$. These cover the full range of
possible wall temperatures and allow for analyzing the impact of this variable
on the closure derivation.

Concerning the heat transfer coefficients $h_{li}$ and $h_{gi}$, experimental
observation suggested that these could be modelled as decaying exponentials

$h_{gi}(t)=h^{0}_{gi}e^{-t/\tau}+h^{\infty}_{gi}$ (25)

$h_{li}(t)=h^{0}_{li}e^{-t/\tau}+h^{\infty}_{li}\,,$ (26)

which define $h^{0}_{gi},h^{0}_{li},h^{\infty}_{li}$, $h^{\infty}_{li}$ and
$\tau$. The time constant $\tau$ is related to the transient heat transfer due
to the sloshing: within $t\ll\tau$, the heat exchange is strongly dominated by
the sloshing-induced thermal de-stratification, while at $t\gg\tau$, the heat
transfer is mostly due to conduction and hence less dependent on the sloshing
regime.

The model prediction in the proposed inverse method thus depends on seven
parameters

$\mathbf{h}=\begin{bmatrix}h_{gi}^{0}&h_{li}^{0}&h_{gi}^{\infty}&h_{li}^{\infty}&\tau&h_{wg}&h_{wl}\end{bmatrix},$
(27)

which must be estimated from the available data. The global mass transfer
coefficient defined in Equation (4) can be computed a posteriori to evaluate
the evaporation/condensation rates at the interface.

In summary, the thermodynamic model described in this section can be cast in
the form of a parametric initial value problem:

$\systeme{\dot{\mathbf{x}}=\mathbf{f}{(\mathbf{x},t;\mathbf{h})},\mathbf{x}{(0)}=\mathbf{x}^{0}}$
(28)

in which $\dot{\mathbf{x}}$ is the time-derivative of the state in (24), and
$\mathbf{f}{(\mathbf{x},t;\mathbf{h})}$ collect the right-hand-side of the
system of equations (16)-(23). In what follows, however, the problem set was
further simplified by assuming that no heat exchange occurs between the gas
and the wall (hence forcing $h_{wg}=0$). This simplification was motivated
a-posteriori because the optimization detailed in the following section
systematically zeroed this term in all the investigated experimental
conditions.

###  4.4 Model-based inverse method

The inverse method consists in identifying the set of parameters
$\mathbf{h}^{*}$ which minimizes the error between experimental data
$\tilde{\mathbf{x}}_{e}{(\mathbf{t})}$ and the corresponding model outputs
$\tilde{\mathbf{x}}(\mathbf{t})$. Here $\mathbf{t}\in\mathbb{R}^{n_{t}}$
denotes a set of randomly sampled points in time and the tilde denotes
variable normalization. This scaling is necessary because the variables in
$\mathbf{x}$ include different physical quantities with largely different
numerical values and physical units. The scaling is performed by mean shifting
and standardization (i.e. to unitary standard deviation).

The optimal parameters are thus those that minimize the objective function
$\mathcal{J}{(\mathbf{x},\mathbf{h})}$ with respect to $\mathbf{h}$:

$\underset{\mathbf{h}}{\mathrm{argmin}}\ \
\mathcal{J}{(\mathbf{x}(\mathbf{t});\mathbf{h})}=||\tilde{\mathbf{x}}_{e}{(\mathbf{t})}-\tilde{\mathbf{x}}{(\mathbf{x}^{0},t;\mathbf{h})}||_{2}^{2}=||\mathbf{R}(\mathbf{h})||^{2}_{2},$
(29)

where $||\bullet||_{2}$ denotes the induced $l_{2}$ norm of a matrix. For a
given set of times $\mathbf{t}\in\mathbb{R}{n_{t}}$, the residual are stored
into a matrix $\mathbf{R}(\mathbf{h})\in\mathbb{R}^{8\times n_{t}}$.

The optimization was performed with the Basinhopping algorithm [54]. This
alternates a quasi-Newton method with a random search for re-initialization.
The random search is carried out using simulated annealing: a new candidate
solution $\mathbf{h}^{\text{new}}$ is generated by perturbing the current
guess, and this is accepted as a new solution with a piece-wise probability
density of the form

$\mathcal{P}=\left\\{\begin{matrix}1&\mathrm{if}\ \ \
\mathcal{J}{(\mathbf{h^{\text{new}}})}<\mathcal{J}{(\mathbf{h^{\text{old}}})}\\\\[5.0pt]
\exp{\left(-\frac{\mathcal{J}{(\mathbf{h^{\text{new}}})}-\mathcal{J}{(\mathbf{h^{\text{old}}})}}{\lambda}\right)}&\mathrm{if}\
\ \
\mathcal{J}{(\mathbf{h^{\text{new}}})}\geq\mathcal{J}{(\mathbf{h^{\text{old}}})}\end{matrix}\right.$
(30)

The parameter $\lambda$ plays the role of the temperature in simulated
annealing, where it is usually made iteration-dependent. In this work, this
was taken as $\lambda=0.01$. The new solution is then given as an initial
guess for the quasi-Newton L-BFGS-B algorithm (see [55]), which operates using
finite differences for the gradient calculation. Once the stopping criterion
for the BFGS is reached, the Basinhopping algorithm repeats the random search
unless the optimization is terminated. The perturbation is made large enough
to pick a new guess outside the basin of attraction of the identified local
minima but small enough not to exceed the pre-defined boundaries of the
parameter space. This was achieved by placing the new points halfway between
the identified local minima. The termination occurs if the optimal solution
does not change by more than $0.1\%$ in the last $500$ iterations.

The uncertainty in determining the optimal coefficients was estimated via a
Monte Carlo method. Given that the measured signals contain noise, the input
data in $\tilde{\mathbf{x}}_{e}{(\mathbf{t})}$ was randomly sampled from the
available experiments such that only half of the points were used for training
the model. In each of these sampling, the first entry in time is used as the
initial condition for the numerical simulation, and the model calibration was
performed $N=100$ times.

The initial conditions $\mathbf{x^{0}}$ supplied to the model were given as
the starting values of the training set $\tilde{\mathbf{x}}_{e}(0)$.
Therefore, the model calibration was performed through $N=100$ optimizations
by randomly selecting different training sets and initial conditions each
time. This resulted in a matrix of optimal solutions
$\mathbf{H}^{*}\in\mathbb{R}^{100\times 5}$ in which the i-th row collects the
optimized parameters at the i-th run. The variability in this matrix reflects
the uncertainties in the model’s training; thus, the standard deviation in
each column is associated with the uncertainty in each parameter. Finally,
these uncertainties were propagated to the numerical predictions of
$p_{g}(t)$, $\overline{T}_{g}(t)$ and $\overline{T}_{l}(t)$ by evaluating the
model for each set of coefficients in $\mathbf{H}^{*}$.

In each of these evaluations, the initial conditions were perturbed with
Gaussian distributions with mean and standard deviation taken from the
experimental data of $p_{g}$, $\overline{T}_{g}$ and $\overline{T}_{l}$ in the
quasi-stationary conditions before the sloshing.

##  5 Results and discussion

This section is dedicated to presenting and discussing the main results of
this research campaign. Section 5.1 experimentally validates the assumptions
made in Section 2 on the boundary and initial conditions for the sloshing
experiment. Then, Section 5.2 describes the wave dynamics of the tested
sloshing excitation by comparing the transient interface displacement of cases
s1 (subcritical planar waves), s3 (supercritical planar waves) and s5 (swirl
waves). The consequent sloshing-induced pressure drop and thermal mixing are
documented in Section 5.3. The impact of different excitations is assessed,
and the results are compared to reference case s0 in quiescent conditions.
Finally, Section 5.4 reports on the model calibration through the inverse
method described in Section 4.4.

###  5.1 Verification of initial and boundary conditions

A preliminary experimental campaign was carried out to assess the adiabaticity
of the cell. These experiments analyzed the temperature profile of the empty
cell (i.e. filled with air) after one heating cycle (i.e., heating up the cell
for one hour and allowing it to cool down passively for another hour). The
tests were repeated with the insulated and the non-insulated facilities (see
Section 5(b)).

Figure 7: Thermal profiles in the non-insulated (black) and insulated (blue)
setups filled only with air at $t=3600$ s. The heating procedure was initiated
at $t\approx-3600$ s, and the heaters were turned off at $t=0$ s. The dotted
lines are the optimal linear fit of each profile.

During these tests, the thermal profile was analyzed with the available
thermocouples (cf. Section 4.1 and Table 1). Under the assumption of fully
adiabatic and axisymmetric conditions, and in the absence of buoyancy-driven
flows, the conduction-dominated thermal field in the tank should lead to a
linear temperature profile along its $z$ axis once steady-state conditions are
reached. The results of this analysis are reported in Figure 7 for both the
non-insulated and the insulated facilities. The linearity of the temperature
profile proves that the assumption of adiabatic conditions is reasonable for
the case of the insulated facility, which is the one analyzed in the following
subsections.

A second campaign was also carried out, with the aim of characterizing the
velocity field in the liquid during the heating phase. In this phase, before
sloshing, the liquid should be quiescent to ensure repeatability. However, the
temperature gradients in the cell inevitably produce buoyant flows that could
undermine this assumption. Therefore, Particle Image Velocimetry (PIV) was
used to analyze the potential impact of buoyancy in setting the initial
conditions for the sloshing experiments. Although the thermal profiles are
different in the absence of the insulation, Figure 7 shows that the $\Delta T$
between the top and the bottom surfaces of the cell are comparable. Therefore,
it is believed that the PIV analysis in non-insulated conditions is also
representative for the insulated setup.

Figure 8: Velocity profiles captured at the start of the constant-temperature
heating stage $t=t^{T}$ (left), and right before the start of sloshing
$t=t^{S}$ (right).

The cell was filled up to $H=60$mm, and the velocity field was captured at the
start of the constant temperature phase (i.e. $t\in[t^{T},t^{S}]$, see Figure
6 and Section 4.2. The PIV was sampled at $10$ Hz, and time-averaged velocity
fields were captured every 500 images. The first (at $t\approx t^{T}$) and the
last (at $t\approx t^{S}$) time-averaged velocity fields are shown in Figure
8.

(a)

(b)

Figure 9: a) Snapshots of the sloshing motion triggered by the lateral
excitation in cases s1, s3 and s5. Cases s1 and s3 are characterized by
lateral interface displacement, which remains approximately flat and follows a
harmonic motion. On the other hand, case 6 displays a stable rotational
velocity which induced a high-amplitude swirl wave. b) Interface displacement
measured at a point located in the mid-plane of the quartz tank at a radial
distance $R/4$ from the side wall during the first 80 seconds of sloshing.

As shown in Figure 8 on the left, upward convective currents are observed at
$t\approx t^{T}$, with velocities of the order of $\approx 1$mm/s. However,
these decrease during the course of the heating stage and nearly vanish at
$t=t^{S}$ as shown in Figure 8 on the right. In conclusion, these results show
that the impact of buoyancy-driven currents on the liquid is negligible, and
the assumption of quiescent initial conditions for the sloshing phase is
reasonable.

###  5.2 Lateral sloshing dynamics

We here report on the response of the liquid interface for the experimental
conditions detailed in Table 2. The observed responses were in line with
Mile’s regime classification (Figure 2) except for test case s4, which was
expected to be in the chaotic regime but produced a planar wave response like
cases s2 and s3. This discrepancy might be explained by the proximity of the
point to the boundaries between these two regimes (see Figure 2). However, as
conditions s3 are equally close to this boundary but on the planar wave regime
side, it appears that the frequency offset parameter $\mathcal{B}_{3}$ might
be slightly under-predicted by the theory.

Concerning the test cases in the planar waves regimes, case s1 is in
subcritical conditions, Res $<$ (Res)c in (7), while cases s2, s3, s4 are in
supercritical conditions, Res $>$ (Res)c in (7). Finally, case s5 is in the
swirl wave regime as expected. The lack of points in the chaotic regime in the
reported campaign stems from the difficulties in performing the measurements
in this regime. Several conditions in chaotic sloshing conditions were
reproduced during the campaign, but the violent response of the interface
resulted in severe wetting/de-wetting of thermocouples and pressure taps,
which led to unreliable measurements.

Figure 9(a) illustrates a snapshot of the liquid interface in the three
investigated conditions, namely planar subcritical sloshing (s1), planar
supercritical sloshing (s2) and swirl sloshing (s5). For each of these, the
figures in the bottom reports on the time evolution of the vertical
displacement $\eta(t)$ at a point located in the mid-plane of the tank at a
distance $R/4$ away from the lateral wall. This displacement is computed with
respect to the original height of the interface at $t=0$. This measurement was
obtained via image processing of the video sequence, by identifying regions of
sharp changes in the image’s intensity along each ‘pixel-column’. The edge
detection approach is similar to the approaches in [56, 57].

As expected, the interface response for each of the sloshing modes in the
planar test cases is the sum of the harmonic response to the perturbation and
exponentially vanishing transients of the form
$\exp(-\gamma_{n,j}\omega_{n,j}t)$. The decay rates $\gamma_{n,j}$ of the
transients are linked to the viscous damping and can be computed analytically
using the linear theory in [41], in which the flow velocity is assumed to be
the sum of a potential contribution and a rotational one due to viscosity.

Focusing on the first mode $(n,j)=(1,1)$, by far the most energetic, the decay
rate $\gamma_{1,1}$ is derived in [41] and reads

$\gamma_{1,1}=\frac{\delta_{1,1}}{\sqrt{4\pi^{2}+\delta_{1,1}^{2}}}$ (31)

where the logarithmic decrement, $\delta_{1,1}$, is

$\delta_{1,1}=2\pi
K\frac{\nu^{\frac{1}{2}}\left(1+2\left(1-\frac{h}{R}\right)\right)}{R^{\frac{3}{4}}g^{\frac{1}{4}}\tanh^{\frac{1}{4}}\left(\xi_{1,1}\frac{h}{R}\right)\sinh\left(2\xi_{1,1}\frac{h}{R}\right)}$
(32)

for upright cylinders without baffles, where $K\approx 0.59$ is a parameter
that accounts for the geometry of the tank [18]. This derivation, also
verified experimentally in [58], considers that the frictional losses between
the oscillating flow field and the solid walls are limited to the Stokes
boundary layer [59].

The exponentially decaying envelope of the transient response using (31) is
shown with dashed lines in Figure 9(b) for cases s1 and s3. As the transient
vanishes, the interface motion tends towards its harmonic response. The
agreement between experimental data and theory is remarkable. Moreover, the
interface’s response is also characterized by the ‘beating phenomena’ during
the transient, as the excitation frequency ($2.7$ Hz in s1 and $2.83$Hz in s3)
is close to the natural frequency ($3.375$ Hz). This results in the periodic
modulation at $f_{b}=0.67$ Hz for s1 and $f_{b}=0.54$ Hz for s3. Finally,
concerning the swirling case (case s5), the linear theory illustrated for the
previous test cases does not hold. The sloshing initially begins with lateral
waves (at the associated beating) but gradually gains a rotational component
until a stable swirl wave is produced at $t>30$s. The impact of these sloshing
excitation on the thermodynamic state of the system is described in the
following section.

###  5.3 Pressure and temperature transducer measurements

The data acquired from the instrumentation mounted on the top cover of the
sloshing cell is analyzed in this section. The measurements were non-
dimensionalized to better compare the different cases. The non-dimensional
temperature on the gas and liquid side are defined, respectively, as:

$\Theta_{g}(t)=\frac{T(t)-T_{\text{sat,0}}}{T_{\text{t,0}}-T_{\text{sat,0}}}\,,$
(33)

$\Theta_{l}(t)=\frac{T(t)-T_{\text{b,0}}}{T_{\text{sat,0}}-T_{\text{b,0}}}\,,$
(34)

where $T_{\text{t,0}}$ is the temperature measured at $t=0$ by the
thermocouple closest to the top of the tank, $T_{\text{b,0}}$ is the
temperature measured the same instant by the thermocouple closest to the
bottom, and $T_{\text{sat,0}}$ is the corresponding saturation temperature.

The non-dimensional pressure is defined as the ratio between the instantaneous
gauge pressure in the cell $\Delta p_{g}(t)$ and its initial value $\Delta
p_{g,0}$:

$\Delta\hat{p}_{g}=\frac{\Delta p_{g}(t)}{\Delta p_{g,0}}.$ (35)

We focus on the gauge pressure instead of the absolute pressure because the
objective was to assess the pressure change in the ullage due to sloshing.

Figure 10 shows the normalized gauge pressure evolution in all test cases
after the system pressurization (phase three in Section 4.2) is concluded and
the heating is turned off, that is $t\in[0,10]$ min. This figure should be
analyzed together with Figure 11, which shows the temperature evolution in the
five cases, as measured by the thermocouples in the ullage volume (top row of
plots) and the ones immersed in the liquid (bottom row). The height of the
thermocouple location (see Table 1) is recalled at the bottom of each line of
plots.

Figure 10: Normalized gauge-pressure in the quartz for experimental test cases
s0-s5 after the pressurization/heating stage was concluded $t>0$ s. The
excitation of cases s1-s5 started at $t=0$ s and lasted for 10 minutes. The
legend underneath the figures indicates the height of each sensor with respect
to the bottom of the tank.

(a)

(b)

Figure 11: Non-dimensional ullage (a) and liquid (b) temperatures of cases s0,
s1, s3, and s5, as a function of time, measured by the thermocouples immersed
in the liquid phase after the pressurization/heating stage was concluded $t>0$
s.

We begin with case s0, which is the baseline test without sloshing. During the
10 minutes of observation, the gauge pressure decreased by roughly 45%. This
is due to the cooling of the ullage to the lateral walls and the liquid under
it, as visible from Fig 11(b). Noteworthy is the continuous warm-up of the
liquid due to the thermal energy stored in the thick quartz wall of the tank.

Moving to case s1, which is in planar sloshing conditions, its pressure
evolution in Figure 10 is nearly identical to the baseline in quiescent
conditions. The analysis of the temperature evolution reveals a moderate heat
exchange between liquid and gas. On the gas side, this is noticeable only in
the thermocouple closest to the interface (at approximately $4$ mm) near the
interface. On the liquid side, a sloshing-induced temperature drop is observed
down to the second thermocouple (about 40mm underneath the interface).
However, soon after the sloshing event, the temperature evolution restores the
linearly increasing trend observed for case s0. The moderate impact of
sloshing on the pressure evolution in this test case is compatible with its
low sloshing-based Reynolds number (see (7)), which is roughly 5.1e3 and thus
within the $\pm 20\%$ range of the critical value $(\text{Re}_{s})_{c}$ below
which no significant heat or mass transfer is expected [32].

Cases s2, s3, and s4 exhibited similar pressure and temperature evolutions.
According to their sloshing-based Reynolds numbers, which were one order of
magnitude higher than case s1, it was expected that these experiments should
produce increasingly larger sloshing-induced pressure drops (i.e., s2 $<$ s3
$<$ s4). However, case s4 produced a pressure evolution higher than s2, but
lower than s3. This might be explained by slightly higher initial temperatures
in the quartz walls and the liquid bulk (Table 2). Nevertheless, the
normalized pressure evolutions are within the uncertainty bars of each other,
which was expected given that the sloshing-based Reynolds number was in the
same order of magnitude across all three tests (Table 2). Because of this
similarity, only the temperature distribution measured in case s3 is shown in
Figure 11(b). These cases are characterized by a sharp drop in the ullage
pressure at the sloshing onset, followed by a strong thermal mixing in the
vicinity of the interface. The sloshing event does not influence the
temperature at the bottom of the tank (at about $51$ mm from the interface).

Finally, case s5, in swirling conditions, results in a significant pressure
drop and thermal destratification. The gauge pressure (see Figure 10) reduces
by $90\%$ within the first 70 seconds, and the temperature in the liquid (see
Figure 11(b)) becomes nearly uniform at $t\approx 100$ s. The temperature drop
in the gas is also significant and becomes nearly constant near the interface.
At $t>150$s, after the mixing is complete, the temperature in the liquid
increases again because of the exchanges with the wall. As this occurs, the
pressure rises again, possibly due to evaporation as the interface warms up.
This test case showcases the tremendous impact that sloshing-induced heat and
mass transfer can have on the pressure and temperature of the ullage gas.

Table 3: Optimal set of closure terms obtained after completing one hundred global optimizations in each test case. The uncertainties associated with each mean value are presented next to it, and were computed as two times the standard deviation. Case |  $\overline{T}_{w}^{0}$ [K] |  $h_{gi}^{0}$ [W/m2K] |  $h_{li}^{0}$ [W/m2K] |  $h_{wl}$ [W/m2K] |  $h_{gi}^{\infty}$ [W/m2K] |  $h_{li}^{\infty}$ [W/m2K] |  $1/\tau$ [ms-1]
---|---|---|---|---|---|---|---
s0 | $\overline{T}_{l}^{0}$ |  |  | 25.82 ± 9.6 | 0.15 ± 0.01 | 0.1 ± 0 |
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ |  |  | 4.06 ± 0.05 | 0.14 ± 0.01 | 0.12 ± 0 |
$\overline{T}_{g}^{0}$ |  |  | 2.01 ± 0.03 | 0.14 ± 0 | 0.12 ± 0 |
s1 | $\overline{T}_{l}^{0}$ | 0.14 ± 0.03 | 0.25 ± 0.03 | 31.13 ± 11.5 | 0.11 ± 0.02 | 0.07 ± 0 | 13.4 ± 1.2
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ | 0.13 ± 0.02 | 0.26 ± 0.03 | 2.56 ± 0.21 | 0.12 ± 0.01 | 0.09 ± 0 | 14.2 ± 1
$\overline{T}_{g}^{0}$ | 0.13 ± 0.02 | 0.26 ± 0.04 | 1.26 ± 0.08 | 0.12 ± 0.01 | 0.09 ± 0 | 14.5 ± 1.4
s2 | $\overline{T}_{l}^{0}$ | 0.71 ± 0.2 | 2.97 ± 0.61 | 12.56 ± 7.93 | 0.23 ± 1.96 | 0.03 ± 0 | 34.6 ± 3.4
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ | 0.71 ± 0.15 | 2.98 ± 0.5 | 3.34 ± 0.87 | 0.12 ± 0.02 | 0.04 ± 0 | 34.1 ± 3.3
$\overline{T}_{g}^{0}$ | 0.7 ± 0.12 | 3.02 ± 0.45 | 1.65 ± 0.46 | 0.13 ± 0.01 | 0.04 ± 0 | 34.5 ± 3
s3 | $\overline{T}_{l}^{0}$ | 0.95 ± 0.14 | 2.88 ± 0.49 | 23.38 ± 10.27 | 0.14 ± 0.02 | 0.04 ± 0 | 35.9 ± 3.4
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ | 0.92 ± 0.13 | 2.84 ± 0.43 | 3.02 ± 0.66 | 0.14 ± 0.02 | 0.06 ± 0 | 35.5 ± 3.5
$\overline{T}_{g}^{0}$ | 0.86 ± 0.58 | 2.69 ± 0.9 | 1.59 ± 0.71 | 1.12 ± 5.92 | 0.06 ± 0 | 35.1 ± 4.2
s4 | $\overline{T}_{l}^{0}$ | 1.09 ± 0.13 | 3.95 ± 0.28 | 12.53 ± 7.44 | 0.1 ± 0.01 | 0.02 ± 0 | 41.1 ± 1.7
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ | 1.08 ± 0.14 | 3.95 ± 0.27 | 2.76 ± 0.79 | 0.1 ± 0.01 | 0.04 ± 0 | 40.8 ± 2
$\overline{T}_{g}^{0}$ | 1.05 ± 0.2 | 4.57 ± 3.63 | 1.46 ± 0.74 | 0.09 ± 0.05 | 0.1 ± 0.36 | 40.2 ± 5.2
s5 | $\overline{T}_{l}^{0}$ | 1.78 ± 1.58 | 6.36 ± 0.77 | 108.1 ± 37.14 | 3.84 ± 1.72 | 0 ± 0 | 44 ± 5.8
$(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ | 2.64 ± 1.8 | 6.12 ± 0.69 | 25.41 ± 10.65 | 2.09 ± 1.91 | 0.1 ± 0.99 | 0.03 ± 0
$\overline{T}_{g}^{0}$ | 1.44 ± 1.03 | 5.92 ± 0.42 | 9.9 ± 0.62 | 3.1 ± 1.07 | 0 ± 0 | 34.3 ± 2.7

###  5.4 Data-driven discovery of heat transfer coefficients

This section reports on the identification of the heat transfer coefficients
through the inverse method and the accuracy of the resulting model’s
prediction. Table 3 collects the main results for the terms in (25) and (26)
identified for the five test cases s0-s5 and the three scenarios for
$\overline{T}_{w}(t=0)=\overline{T}_{w}^{0}$ described in Section 4.3. For
each quantity, the table provides the mean value and the confidence interval,
defined as twice the standard deviation, obtained from the columns of the
$\mathbf{H}^{*}$ matrix in the Monte-Carlo evaluation (see Section 4.4).

Figure 12 complements this analysis by showing the distribution of the optimal
$h_{gi}^{0}$, $h_{li}^{0}$ coefficients for test cases s1-s5. Case s0 was
omitted because only $h_{gi}^{\infty}$, $h_{li}^{\infty}$ and $h_{wl}$ were
calibrated in the absence of sloshing.

The results in Table 3 demonstrate that the different scenarios for
$\overline{T}_{w}^{0}$ have minimal impact on the coefficients discovered
through the inverse method, except for $h_{wl}$. This parameter is maximum
when $\overline{T}_{w}^{0}=\overline{T}_{l}^{0}$, and minimum when
$\overline{T}_{w}^{0}=\overline{T}_{g}^{0}$. Interestingly, the inverse method
adapts the optimal $h_{wl}$ to the wall temperature to provide the same heat
exchange $\dot{Q}_{wl}(t)$. This could be expected from the fact that $h_{wl}$
and $\overline{T}_{w}(t)$ appear in the system of ODEs through the
$\dot{Q}_{wl}(t)$ term. This is confirmed in Table 4, which shows the time-
averaged value of this heat flux for all experimental test cases and the three
different wall temperature scenarios. The impact of the initial wall
temperature on the predicted heat flux $\dot{Q}_{wl}$ is within 2.8%,
confirming its negligible influence on the regression of the other
coefficients.

(a)

(b)

(c)

(d)

Figure 12: Swarm plots with the distribution of the optimal transient heat transfer coefficients (a) $h_{gi}^{0}$ with $\overline{T}_{w}^{0}=\overline{T}_{l}^{0}$, (b) $h_{gi}^{0}$ with $\overline{T}_{w}^{0}=(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$, (c) $h_{li}^{0}$ with $\overline{T}_{w}^{0}=\overline{T}_{l}^{0}$, (b) $h_{li}^{0}$ with $\overline{T}_{w}^{0}=(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$ for experimental test cases s1-s5. Table 4: Average heat flux exchanged between the liquid volume and the tank walls $\dot{Q}_{wl}$ for test cases s0-s5, and for the three different wall temperature scenarios at $t=0$ s. Case |  Time-averaged $\dot{Q}_{wl}$ [W]
---|---
$\overline{T}_{w}^{0}=\overline{T}_{l}^{0}$ | $\overline{T}_{w}^{0}=\frac{\overline{T}_{l}^{0}+\overline{T}_{g}^{0}}{2}$ | $\overline{T}_{w}^{0}=\overline{T}_{g}^{0}$
s0 | 0.72 | 0.72 | 0.74
s1 | 0.42 | 0.42 | 0.43
s2 | 0.58 | 0.50 | 0.54
s3 | 0.49 | 0.50 | 0.53
s4 | 0.43 | 0.45 | 0.50
s5 | 2.84 | 2.82 | 2.89

From Table 3 and Figure 12, it is evident that $h_{li}^{0}$, $h_{gi}^{0}$, and
$h_{wl}$ increase with the sloshing-based Reynolds number. This is consistent
with the pressure and temperature evolutions reported in the previous section.
As expected, the 0D model correlates the sudden pressure and the temperature
drops observed in Figures 10-11 for cases s2-s5 with larger values of
$h_{li}^{0}$ and $h_{gi}^{0}$ than those identified in cases s1 and s0. On the
other hand, he identified $h_{gi}^{\infty}$ is nearly the same in all cases
s0-s4. This suggests that the heat exchanges between the gas and the
interface, at large times, are independent of the sloshing conditions. These
results agree with the experimental measurements presented in Section 5.3,
which show that planar sloshing triggers the initial thermal destratification,
but the long-term evolution is driven by conduction because the convective
fluxes cannot reach the bulk fluid.

This result also aligns with the identified decay time $\tau$, reported in the
last column, Table 3. As highlighted in Section 5.2, the transient sloshing
dynamic is characterized by a decay rate $\omega_{1,1}\gamma_{1,1}=48.7$ ms-1,
which is comparable to the time scale $\tau$ for the cases s3-s5, where the
pressure drop was particularly evident. This suggests that the time scale of
the de-stratification is comparable to the time scale of the sloshing’s
transient dynamics. Moreover, it is possible that the derived coefficients are
not only linked to the sloshing regime per se (planar or swirl), but also the
transient sloshing dynamics.

Figure 12 shows that the distribution of identified coefficients is narrow and
unimodal for all cases except for case s5. This test case produced the
steepest pressure drop, leading to homogeneous temperature fields in the
liquid after 30 excitation periods. Accordingly, the inverse method derives
the highest $h_{gi}^{0}$ and $h_{li}^{0}$ coefficients. Moreover, it predicts
the largest amount of heat exchanged with the walls (see Table 4), which
appears much larger than the one exchanged with the interface ( $\dot{Q}_{li}$
). Therefore, the model suggests that the rise in the liquid temperature in
Figure 11(b) is primarily due to the heat transfer with the solid walls.

(a) Case s0: system relaxation in quiescent conditions

(b) Case s1: subcritical planar sloshing

(c) Case s3: supercritical planar sloshing

(d) Case s5: swirl sloshing

Figure 13: Results of the 0D model calibration applied to test cases s0, s1,
s3 and s5, considering the initial wall temperature
$\overline{T}_{w}^{0}=(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$. The data
was generated using the mean closure terms from Table 3, and the uncertainty
region was computed via a Monte Carlo approach (see Section 4.4).

The results of the 0D model are compared against the experimental measurements
for cases s0, s1, s3 and s5 in Figure 13, which was generated for the scenario
$\overline{T}_{w}^{0}=(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$. The
dashed lines represent the pressure and temperature evolutions generated using
the mean $\mathbf{\bar{h}^{*}}$ coefficients from Table 3, and with the
initial conditions of each test case $\mathbf{X}^{0}$. The shaded area
represents the uncertainty of the prediction, estimated as described in
Section 4.4.

The results of the calibrated model showed good agreement with the
experimental data, which was generally within the uncertainty bands of the
prediction. Figures 13(a) and 13(b) correspond to tests s0 and s1,
respectively. While the pressure and gas temperature evolutions were nearly
identical in these tests, steeper initial drops were registered for s1. This
observation aligns with the much lower value of $h_{gi}^{0}$ in case s0
compared to s1. This difference is likely due to to the sudden excitation
applied to case s1, which reduced the response time of the system compared to
s0. Since the optimal coefficients in s0 were determined without the transient
terms on the interface, the temperature drop in the ullage is overpredicted by
11%, whereas the numerical pressure drop is 5% more than the experimental one.
On the other hand, the matching of case s1 is remarkable, with the mean
numerical evolutions overlapping the experimental ones. The supercritical
planar sloshing tests are represented by case s3 in Figure 13(c). This figure
highlights the model’s ability to capture the system’s transient pressure and
temperature variations. Similar to s1, there is nearly perfect agreement
between the numerical and experimental curves, which is remarkable given the
steep initial variation in pressure and temperature.

Finally, Figure 13(d) concerns the swirl test case s5. While the transient
evolution of the system was well captured by the model, the following pressure
rise was not well reproduced. Moreover, the rise in liquid temperature was
overestimated by nearly two degrees, while the decrease in ullage temperature
was steeper in the simulations. Nevertheless, the model showed promising
agreement with the experiments, especially given its simple formulation and
negligible computational cost.

Lastly, the model was used to assess the relative importance of the different
terms which drive the pressure evolution according to Equation (19). Figure 14
shows the different contributing terms in $dp_{g}/dt$, computed with
$\overline{T}_{w}^{0}=(\overline{T}_{l}^{0}+\overline{T}_{g}^{0})/2$, for
$t\in[0,200]$ s. Solid lines represent the overall pressure change-rate,
dotted lines correspond to the variation due to thermal expansion/contraction,
dashed lines represent the term accounting for changes in volume, and dash-
dotted lines indicate the contribution of adding or removing vapour mass due
to phase change through the interface. This last term depends on $dm_{v}/dt$,
which depends on the mass transfer coefficient defined in Equation (4). These
figures show that the volumetric term can be neglected in all test cases.

(a)

(b)

(c)

(d)

Figure 14: Pressure drop rate due to variations of the ullage (1) temperature,
(2) volume, and (3) mass identified in Equation (19) for cases (a) s0, (b) s1,
(c) s3, and (d) s5. These individual contributions are also compared to the
overall rate-of-change of the ullage pressure, drawn in solid black lines.

On the other hand, the relative weight of the other two terms changes
significantly with the sloshing conditions. The pressure drop in cases s0 and
s1 is dominated by the thermal (cooling) contribution. Mass transfer and
cooling have comparable weight in case s3, while case s5 is largely dominated
by the mass transfer (condensation) effect until $t\approx 40$s. Approximately
85% of the initial pressure drop in s5 is due to condensation. Afterwards, the
pressure evolution is again driven by the thermal effect, and a change of sign
is observed at $t>60$ s. In this last phase, the liquid’s temperature
homogenized (see Figure 11(b)) and started warming up because of heat
exchanges with the walls. This triggers evaporation, hence a rise in the
tank’s pressure.

##  6 Conclusions

We presented an experimental characterization of non-isothermal sloshing in an
upright cylindrical tank under lateral harmonic excitation and proposed a
model-based inverse method to infer heat and mass transfer coefficients.

The experiments were carried out on the Karman Institute’s SHAKESPEARE
sloshing table in a reduced-scale quartz tank filled with HFE-7200 and
instrumented with an array of thermocouples and a pressure tap. In all
sloshing experiments, the tank is first pressurized and thermally stratified
using heating elements on the ullage gas side. Temperature distribution and
pressure were monitored both in the preparatory phase and the sloshing
experiments, and Particle Image Velocimetry (PIV) was used to prove that
buoyant flows have a negligible impact on the liquid. Hence, the initial
conditions for the sloshing are those of a quiescent and thermally stratified.
Four main scenarios were analyzed: (1) absence of sloshing, (2) subcritical
planar waves (2) subcritical planar waves, (3) supercritical planar waves, and
(4) swirl waves.

The temperature and pressure measurements were used to infer heat and mass
transfer coefficients in the tank using a model-based inverse method. These
are identified via an optimization, carried out with the Basinhopping
algorithm, as the one that minimizes the discrepancy between data and model
prediction. The model is a simple 0D formulation of heat and mass balance
between liquid, gas and walls.

In line with previous experimental works, it was observed that the subcritical
planar waves (Res $<$ (Res)c) do not appreciably increase the heat and mass
transfer from the quiescent conditions. Instead, supercritical planar waves
promote thermal mixing in the vicinity of the gas-liquid interface and produce
an appreciable pressure drop in the tank. The mixing and the associated
pressure drop are considerably larger in swirling conditions, which destroy
the thermal stratification.

In all cases of sloshing enhanced heat and mass transfer rate, the model
predicts that the pressure drop is primarily due to condensation from the
ullage gas due to a change in temperature of the gas-liquid interface.

Considering the important simplifications underpinning the 0D model used in
the inverse method, the results are particularly promising. If correctly
calibrated, the model could reproduce the experimental data accurately in all
the investigated conditions at a negligible computational cost. These results
thus open the path to using model-based inverse derivations of closure laws to
describe the thermodynamic evolution of non-isothermal sloshing. The same
approach could be extended to all critical phases of cryogenic fuel storage
(tank filling, pressurization, and long-term storage). Besides allowing for
finding correlations from real-time data (for example, by searching for a link
between the closure coefficients and states of the system), these could also
be used for model predictive control and anomaly detection in thermal
management systems. Both applications are currently under investigation.

## Acknowledgments

This work was supported by the European Space Agency (ESA) in the framework of
the GSTP-SLOSHII project with reference number 4000129315/19/NL/MG. The
authors gratefully acknowledge the financial support of the ‘Fonds de la
Recherche Scientifique (F.R.S. -FNRS)’ for the FRIA grant with reference
40009348 supporting the PhD of Mr. Marques.

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