P Fraundorf

Traveler-point dynamics: A 3-vector bridge to curved spacetimes

Locally-defined "flat-patch" parameters, chosen to be minimally frame-variant, can be useful for describing the perspective of a single observer in accelerated frames and in curved spacetimes. In particular the metric-equation's scalar proper-time, and the 3-vectors proper-velocity and properacceleration, are useful because they don't rely on extended arrays of synchronized clocks, which may be hard to find. This use of an observer's "proper reference frame" is where diverse metric definitions of extended-simultaneity converge to agree on a set of locally-approximate "geometric" (i.e. connection-coefficient) forces, which in turn further prepare intro-physics students for "flatpatch" engineering in curved spacetimes. Newton's approximation to gravity is the oldest example of this. The strategy also opens the door to engineering nomograms that allow one to track arbitrary local and extended radar time trajectories from the perspective of accelerated and curved-spacetime travelers, as well as real-time single-traveler trajectories that are exact at any speed in flat spacetime, and approximate around gravitating masses. CONTENTS I. introduction 1 II. proper time 2 III. proper velocity 2 IV. proper acceleration 3 V. causes of motion 4 VI. engineering nomograms 7 VII. discussion 7 VIII. conclusion

I. INTRODUCTION

In an excellent 1968 article 1 , Robert W. Brehme discussed the advantage of teaching relativity with fourvectors. This is not done for intro-physics courses. One reason may be that spacetime's 4-vector symmetry is broken locally into (3+1)D, so that e.g. engineers working in curved spacetimes 2 (including that here on earth) will naturally want to describe time-intervals in seconds and space-intervals e.g. in meters. In this note, we therefore explore in depth an idea about use of minimally frame-variant scalars and 3-vectors, suggested by many but perhaps first hinted at in Percy W. Bridgman's 1928 ruminations [START_REF] Bridgman | The Logic of Modern Physics[END_REF] about the promise of defining 3-vector velocity using traveler proper time instead of an "extended" time system.

The problem with the former, of course, is that minimally frame-variant quantities like proper time, 3-vector proper velocity [START_REF] Bondi | Relativity and Common Sense (Doubleday/Dover[END_REF] , and 3-vector proper acceleration [START_REF] Einstein | The particle problem in the general theory of relativity[END_REF] are inherently local to one traveler's perspective, hence we refer to them as "traveler-point" quantities. The advantage is that they are either frame-invariant (in the case of proper time and the magnitude of proper-acceleration) or synchrony-free [START_REF] Shurcliff | Special relativity: The central ideas[END_REF] (i.e. do not require an extended network of synchronized clocks which is often unavailable in curved space time and accelerated frames).

In a larger context, general-relativity revealed a century ago why we can get by with using Newton's laws in spacetime so curvy that it's tough to jump higher than a meter. It's because those laws work locally in all frames of reference [START_REF] Misner | Gravitation[END_REF][START_REF] Hartle | Gravity[END_REF] , provided we recognize that motion is generally affected by both proper and "geometric" (i.e. accelerometer-invisible connection-coefficient) forces. The emergent "metric-first" approach to exploring curved spacetimes with calculus 9 can choose to restrict all measurement to locally-flat patches of spacetime in this context, which can then be sewn together by a global metric [START_REF] Wheeler | Exploring Black Holes: Introduction to General Relativity[END_REF][START_REF] Derbes | Exploring black holes: Book review[END_REF] . This opens the door to engineering use of spatial 3vectors and temporal scalars at any speed, and locally in curved spacetimes, for engineers who have little interest in ignoring the local (3+1)D break in our global 4-vector symmetry which gives rise to fundamental differences in the way that space and time are experienced. The metric equation also contents itself with a single definition of extended simultaneity (cf. Fig. 1) i.e. that associated with the set of book-keeper (or map) coordinates with which one chooses to describe one's local patch (for measurement) and one's global spacetime metric (e.g. for formally defining extended simultaneity). Avoiding the Lorentz transform's need for two frames with synchronized clocks in special relativity [START_REF] French | Special Relativity[END_REF] of course opens the door to simple treatments of accelerated motion in flat spacetime as well. When simultaneity is defined by a flat-spacetime bookkeeper coordinate frame, time-elapsed on traveling clocks is always "dilated" (e.g. spread out), since from the metric equation the differential-aging or Lorentz factor γ ≡ dt/dτ ≥ 1. Expressions for this in terms of velocity will be discussed below.

We won't discuss the utility of proper-time τ in detail here because it is widely used now even by introductory textbook authors [START_REF] Halliday | Fundamentals of Physics[END_REF] . However it's worth pointing out that early treatments of special relativity were so focused on the equivalence between frames, each with their own extended set of synchronized clocks, that proper-time (if discussed [START_REF] Einstein | Relativity: The special and the general theory, a popular exposition[END_REF] ) was generally a late-comer to the discussion, and of course used only for "rest frame" (i.e. unaccelerated) travelers [START_REF] French | Special Relativity[END_REF] .

Moving yardsticks also have a "proper-length" associated with them, which can be useful too. However this is a "non-local" quantity not defined by the metric equation, the concept of "rigid body" is only an approximation in spacetime, and length contraction itself involves three separate events as distinct from the two involved in FIG. 2. Proper-velocity addition for a sci-fi puzzler, in which your starfleet battle-cruiser drops in from hyperspace heading away from a nearby star, only to find the enemy ship heading off in another direction.

time-dilation. Hence we don't include proper-length in a collection of those traveler-point parameters which show promise of wide-ranging usefulness in accelerated frames and curved spacetime.

III. PROPER VELOCITY

Proper-velocity 15 w ≡ d x/dτ is the rate at which bookkeeper or map 3-vector position x is changing per unittime elapsed locally on traveler clocks. Because it is proportional to momentum (i.e. p = m w), unlike coordinate velocity v ≡ d x/dt it has no upper limit. Also unlike coordinate velocity, its measurement does not depend on map-clock readings (synchronized or not) along that world line. Proper-velocity is also represented by the space-like components of a traveling object's velocity 4-vector.

An interesting thought problem for future engineers (as well as teachers) might be to ask if speed-limit signs, in a "Mr. Tompkins" world [START_REF] Gamow | Tompkins in paperback[END_REF] with a much slower lightspeed spacetime constant, would use coordinate-speed or proper-speed. Four criteria to consider might involve the measure's connection to momentum, kinetic-energy, and to reaction-times (after the "warning photons" arrive) for both the driver, and a pedestrian who is tempted to cross the street.

The time-like component of that 4-vector is cγ, where γ ≡ dt/dτ is the also-useful (but not synchrony free) Lorentz differential-aging or "speed of map-time" factor mentioned above. In flat spacetime, from the met-ric equation it is easy to show that γ = w/v = 1 + (w/c) 2 ≥ 1. This works better at high speeds than the version in terms of coordinate speed because there is no difference in the denominator between numbers very near to 1.

This same proper velocity form also emerges directly for circular orbits around a spherically symmetric mass, as the exact azimutal motion factor in the Schwarzchild time-dilation equation γ ≡ dt/dτ = γ r 1 + (w φ /c) [START_REF] David | Relativistic timekeeping, motion, and gravity in distributed systems[END_REF] , where the gravitational time-dilation factor is γ r ≡ 1/ 1 -(r s /r) ≥ 1 and r ≥ r s is distance from the center of a mass M object whose Schwarzschild radius r s = 2GM/c 2 . Global positioning system satellites have to consider both of these factors in compensating for the rate of time's passage in their orbits around earth.

For unidirectional velocity addition, the advantage of a collider over an accelerator is easily seen because w AC = γ AB γ BC (v AB +v BC ). In other words, the Lorentz-factors multiply even though the coordinate-velocites (which add) never exceed c. Proper velocity's direct connection to momentum makes it more relevant to speed-limits e.g. in worlds with a reduced value for lightspeed, and its direct connection to traveler time makes it more relevant to passengers and crew when planning long high-speed trips.

More generally, unlike coordinate-velocities, proper velocities add vectorially provided that one rescales (in magnitude only) the "out of frame" component. In other words w AC = γ AC v AC = ( w AB ) C + w BC . Here C's view of the out-of-frame proper-velocity ( w AB ) C is in the same direction as w AB but rescaled in magnitude by a factor of (γ BC + (γ AB -1) w BC • w AB /w 2 AB ) ≥ 0. Hence vector diagrams, and even the original low-speed equation for velocity-addition (almost), survive intact when propervelocity is used.

This would open the door to new settings for those relative motion problems (that students dislike so much). For example, suppose that a starfleet battle-cruiser drops out of hyperspace in the orbital plane of a ringworld, traveling at a proper-velocity of 1[ lightyear/traveler-year] radially away from the ringworld's star. An enemy cruiser drops out of hyperspace nearby at the same time, traveling 1[ly/ty] in the rotation-direction of the ringworld's orbit, and in a direction perpendicular to the starfleet cruiser's radial-trajectory. What is the proper-velocity (magnitude in [ly/ty] and direction) of the enemy cruiser relative to the starfleet ship? The vector solution to this is illustrated in the Fig. 2.

IV. PROPER ACCELERATION

The felt or proper-acceleration 3-vector α is defined as the acceleration felt by accelerometers (broadly defined) that are moving with an object 17 relative to the tangent free-float (or geodesic as a generalization of "inertial") frame. In flat spacetime its magnitude is the frame-invariant magnitude of the net-acceleration 4-FIG. 3. Accelerometer data from a phone dropped and caught three times. During the free-fall segments, the accelerometer reading drops to zero because gravity is a geometric force (like centrifugal) which acts on every ounce of the phone's structure, and is hence not detected. The positive spikes occur when the fall is arrested as the falling phone is caught (also by hand) before it hit the floor.

vector A λ ≡ δU λ δτ ≡ δ 2 X λ δτ 2 . This four-vector has a null time-component in the frame of the accelerated traveler, from whose "traveler-point" perspective its 3-vector direction is defined.

In flat spacetime, this 3-vector is simply related to the rate of proper-velocity change through α = δ w δt || w + γ δ w δt ⊥ w , which in the unidirectional case reduces to α = δ w δt . This link between proper acceleration and proper velocity will be important when we look at traveler-point dynamics in the next section.

To illustrate what happens in curved spacetime (and accelerated frames), try running an accelerometer app on your phone and performing a hold, drop, catch sequence as shown in Fig. 3. Because accelerometers only detect proper-accelerations, like that due to the upward force of your hand when holding the phone in place before the drop, nothing is detected when the phone is released even through the net acceleration 4-vector in that case has a vertically downward component of g.

Thus in curved spacetimes and accelerated frames, understanding motion requires a recognition of geometric (connection-coefficient) effects which thankfully (following Newton) can often be approximated locally as gravitational and inertial forces. The bottom line is that in flat spacetime motion is explained by the forces that cause proper acceleration, but as soon as we move to curved spacetimes and accelerated frames (even here on earth) a second kind of cause comes into play. As we'll see, these geometric effects act on every ounce of their target, vanish locally from the perspective of a comoving freefloat frame, and are therefore undetected by on-board accelerometers. Returning to flat spacetime for the moment, let's also consider the equations of constant proper-acceleration (cf. Fig. 4). In flat spacetime, it is proper and not coordinate acceleration that can be (and normally is) held constant.

If we stick to (1+1)D i.e. unidirectional motion, constant proper acceleration yields some wonderfully simple integrals of motion, namely α = ∆w/∆t = c∆η/∆τ = c 2 ∆γ/∆x, where "hyperbolic velocity angle" or rapidity η = sinh -1 [w/c] = tanh -1 [v/c]. These are approximated by the familiar introl-physics relationships a = ∆v/∆t = 1 2 ∆(v 2 )/∆x at low speed, but also allow beginning students to explore interstellar constant proper-acceleration round-trip problems almost as easily as one does low-speed constant acceleration problems.

In (3+1)D the integrals are also analytical, but messy and perhaps best expressed 18 in terms of a characteristic time τ o equal to (γ o + 1)/2c/α, which connects to a hyperbolic velocity angle τ /τ o . Here γ o is the "aging factor at turnaround", which for unidirectional motion yields the (1+1)D rapidity discussed above.

V. CAUSES OF MOTION

We've discussed a robust strategy for describing motion (kinematics) at any speed plus locally in curved spacetime & accelerated frames, using (3+1)D parameters which are either frame-invariant or synchrony free. The causes of motion (dynamics) on the other hand are traditionally described by a connection between net 3vector force as the cause, and the second time derivative of position where e.g. at low speed we normally write Σ F = mδ 2 x/δt 2 .

To extend this (3+1)-vector analysis to high speeds and curved spacetimes we simply follow the tradition already established with gravity, but add some new notation. For instance, we might denote the proper forces (with frame-invariant magnitudes which are felt by onboard accelerometers) with the greek letter xi ( ξ). For inertial frames in flat spacetime, then, Σ ξ ≡ m α = mδ 2 x/δτ 2 .

In the "local patch" of curved spacetimes and accelerated frames, we can often approximate the effects of metric connection coefficients on the equation of motion by imagining geometric forces which: (a) are unfelt by on-board accelerometers, (b) act on every ounce of an object's being, and (c) disappear when seen from the vantage point of a local "free-float frame". We denote these geometric forces (like gravity and inertial forces such as centrifugal) with the greek letter zeta ( ζ). Locally then we can write Σ ξ + Σ ζ ≃ mδ 2 x/δτ 2 .

Although these force expressions work well for tracking map position versus time, energies, accelerometer readings and differential aging, etc., in the special case of high speed non-unidirectional motion a kinematic correction is needed when looking at momentum transfers (e.g. in collisions). This is because rates of proper velocity change (proportional to momentum change) are frame variant The idea that rates of momentum change depend on one's frame of reference may seem odd, but rates of energy change (δE/δt = Σ ξ • v → Σ f • v) are frame-variant even at low speeds. This in spite of the fact that when I asked an AAPT audience in 1998 for a show of hands by those who thought that classical rates of an object's energy change are frame-variant, the only hand to go up in the audience was that of Edwin F. Taylor (that year's Oersted Medal winner).

As mentioned above, in curved spacetimes the move from 4-vector to local (3+1)D dynamics is normally done by the "geometric force" approximation, in which motion is seen as caused by adding geometric to proper forces. These relations are inspired by the general-relativistic 4-vector equation of motion, written in force-units as mDU λ /dτ -mΓ λ µν U µ U ν = mdU λ /dτ , which may be described as "net proper 4-vector force" + "net geometric 4-vector force" = mass times the "net 4-vector acceleration". As usual here m is frame-invariant mass, upper case D denotes the covariant derivative of 4-velocity component U λ , Γ λ µν denotes one of 4 × 4 × 4 = 64 connection coefficients defined in terms of metric tensor derivatives, and repeated indices in products like Γ λ µν U µ U ν are im-plicitly summed over all (spatial and temporal) values of those indices.

The traveler-point approach expresses this in terms of a locally-useful set of 3-vector and scalar relations, like Σ ξ + Σ ζ ≃ mδ 2 x/δτ 2 and δE ≃ (Σ ξ + Σ ζ) • δ x. The scalar part connects to energy conservation, while the vector part connects to momentum conservation locally through the flat spacetime connection to frame-variant forces f discussed above. Both the "geometric force" approximation which gives rise to the ζ' vectors, and the connection of "felt plus geometric" forces to rates of momentum change, rely on the general assertions: (a) that spacetime is "locally flat" even in curved spacetimes and accelerated frames, and (b) that the proper acceleration 4-vector is purely spacelike (i.e. a.spatial 3-vector) from the perspective of the traveling object. The locality requirement, of course, may be more severe as spacetime curvature increases.

For example, if we imagine a stationary charge q held in place by some combination of electromagnetic forces in a Schwarzschild (non-spinning spherical mass) potential, the net proper force is the Lorentz 4-vector qF λ β U β = qγ{ E • v/c, E + v × B}. In traveler-point form [START_REF] David | Classical Electrodynamics[END_REF] for charge q this becomes Σ ξ = q E ′ = q( E || w +γ E ⊥ w + w× B), where E and B are the bookkeeper-frame field vectors, and E ′ is only the electric field seen by charge q. FIG. 6. This compares tangent free-float-frame and radar-time extended simultaneity models for constant proper acceleration. The dashed green line corresponds to the world line of a stationary object, moving forward as if it had been dropped by our accelerated observer (red line) at the turnaround point.

The magnetic field B ′ seen by the charge has no effect on its motion, so that in traveler-point terms such problems become purely electrostatic even in curved spacetime! As an aside in this context, the above relations may be used to show how frame-variant proper forces can in general be decomposed into "static" and "kinetic" components, f s = γ 3 m a = ξ || w + γ ξ ⊥ w and f k = ( 1 γ -γ) ξ ⊥ w , which are analog to the electrostatic and magnetic forces (respectively) associated with electromagnetic fields.

The net geometric force, on the other hand, is obtained by summing the 13 out of 64 non-zero connection coefficients (8 of 40 with independent values) for a Schwarzschild object of mass M and "far-coordinate" radius r, to get Σ ζ = -GM mr/r 2 . If we wish to support the object in place so that net acceleration 3-vector α is zero, then we require an E-field directed in the radial direction for which upward proper force qE ′ = GM m/r 2 . One nice thing about discussing this force-balance in traveler-point terms is that all observers, those using fartime in a Schwarzschild potential as well as e.g. those passing by in a star ship accelerating at 1 gee, will be talking about the same thing in terms of time-intervals and forces experienced by the traveler, albeit with position coordinates of their own choosing.

The net proper-force that our cell-phones measure is frame-invariant in magnitude as is proper time, with a 3-vector direction presumably defined in traveler terms. This makes it useful [START_REF] Fraundorf | The proper-force 3 vector[END_REF] for comparing electromagnetic forces in different frames (e.g. to see how magnetic attraction becomes purely electrostatic), and makes curved spacetime (like that we experience on earth's surface as illustrated in Fig. 5) a bit simpler to understand than we might have imagined in a world where time passes differently according to your location as well as your rate of travel. Thus Einstein's general relativity, far from invalidating Newton, revealed that the classical laws work locally in all frames (including accelerated-frames in curvedspacetime) provided that, in addition to proper-forces, we recognize geometric (i.e. connection-coefficient) forces like gravity and inertial-forces (acceleration "gees", centrifugal, etc.) that "act on every ounce" of an object's being.

These highly frame-dependent geometric forces also have effective potentials ∆U linked to the differentialaging (generalized Lorentz) factor γ ≡ dt/dτ (equal to 1/g 00 for motionless objects even when the metric is non-diagonal) via L τ -L t = (γ-1)mc 2 where Lagrangian L ≡ K -U , e.g. in rotating habitats, accelerating spaceships [START_REF] Rindler | Relativity: Special, General, and Cosmological[END_REF] , and gravity on earth [START_REF] Hartle | Gravity[END_REF] . Potential energies associated with proper forces instead involve delocalized interactions linked to the total rest mass-energy m of a composite system, and do not involve differential aging.

VI. ENGINEERING NOMOGRAMS

Engineering nomograms which map the extended "proper coordinates" for an accelerated or curved-space time traveler allow one to visualize where traveler-point or flat-patch dynamics does and does not work, as well as to examine how extended definitions of simultaneity will apply to "arbitrary object trajectories" in terms of both bookkeeper and proper coordinates. The propercoordinate overlay was e.g. discussed by Dolby and Gull [START_REF] Dolby | On radar time and the twin paradox[END_REF] for an accelerated traveler, while arbitrary trajectories (e.g. of a dropped object) generally involve integration of coordinate-time and/or proper-time Lagrange equations. In (1+1)D quite accurate nomograms on flat screens and/or paper are thereby possible.

In order to illustrate how "flat-patch" or "travelerpoint" dynamics works for accelerated frames and curved spacetimes, let's first consider views of an object dropped by an accelerated observer. Then we consider an object dropped by a shell-frame observer at fixed radius above a non-rotating gravitational mass.

Figure 6 offers x-ct diagrams for an observer undergoing constant proper acceleration α ≃ "one-gee" who drops an object at the "turnaround point" with respect to our flat spacetime bookkeeper frame, using two different definitions of extended simultaneity (i.e. models for the time and place of events non-local to our observer). The first (tangent free-float-frame simultaneity) only works in flat spacetimes as clocks can't otherwise be synchronized, while the latter (radar-time simultaneity [START_REF] Dolby | On radar time and the twin paradox[END_REF] ) depends on future as well as past elements of our observer's trajectory.

Our accelerated observer's x,t coordinates relative to the map frame are described, in terms of proper time τ on the observer's clocks, by t = (c/α) sinh[ατ /c] and x = (c 2 /α)(cosh[ατ /c]-1). Our dropped object's motion (which is at fixed x object the map frame) using the tangent free-float-frame definition then looks like:

ρ object ≃ - 1 2 ατ 2 - 7α 3 24c 2 τ 4 + O[τ ] 5 , (1)
while using radar time simultaneity it looks like:

ρ object ≃ - 1 2 ατ 2 - 10α 3 24c 2 τ 4 + O[τ ] 5 . (2)
The figures also show isocontours for τ and ρ in case you want to examine other particle trajectories as well. Both highlight the "Rindler" event-horizon, which like the Schwarzchild event-horizon illustrated below prevents photons beneath it from ever catching our observer.

Regardless, the flat-patch or traveler-point description of our dropped object's motion is simply ρ object ≃ -(1/2)ατ 2 , which is a good local approximation to both expressions above (and likely other definitions of extended simultaneity as well). The range of validity of this local approximation is denoted schematically by yellow shading along the worldline of our accelerated traveler.

Figure 7 shows a Schwarzschild "far-coordinate" r-ct diagram of trajectory models for a particle dropped from a "one-gee" shell frame outside a non-spinning supermassive (145 billion solar mass) black hole, whose mass has been chosen simply to make the event horizon radius a significant fraction of the shell radius as well. The figure shows isocontours for τ and ρ only for radar-time simultaneity (since tangent-fff simultaneity doesn't work in curved spacetimes), again in case you want to examine other particle trajectories as well.

This plot here shows Schwarzschild geodesic trajectory of our dropped particle (dashed), which is asymptotic to (i.e. never crosses) the event horizon, as well as versions of the flat-patch or traveler-point trajectory r object ≃ -(1/2)ατ 2 (top dotted line) and the Newtonian approximation ρ object ≃ -(1/2)ατ 2 (bottom dotted line) where α = GM/r 2 .

As you can see, the flat-path or traveler-point trajectory is a much better approximation than the Newtonian one, even though its validity too is best in the shaded region close to our accelerated observer's world line.

VII. DISCUSSION

The foregoing, summarized in Fig. 4, is targeted toward university physics teachers, but uses concepts which may be unfamiliar to some. In that context, we are compiling example problems work with dimensioned variables using a single map-frame which involve high speed acceleration not within the scope of a "Newton-first" introductory course, andworking on a opening "one-class" intro to Newtonian variables as useful approximations [START_REF] Wentzel-Long | A class period on spacetime-smart 3-vectors with familiar approximates[END_REF] instead.

The traveler-point proper coordinate perspective on the importance of specifying "which clock", and on the distinction between proper and geometric forces with help from your cell-phone, can be helpful for students at the very beginning of their engineering physics education. It also provides "Newtonian-like" tools for the 3-vector addition of proper-velocities and momenta, as well as for proper-force/acceleration analysis at any speed using simultaneity defined by a free-float bookkeeper frame. In curved spacetimes the best bet for this reference frame may be the tangent free-float 9 frame (or Fermi-normal 8 ) coordinate system, like for instance the "shell frame" that we use for our inertial reference here on earth.

This traveler-point perspective perhaps does not do much for rigorous simplification of "geometric-force" analysis in curved spacetime, because e.g. the geodesic equation (i.e.

without proper-forces) for arbitrary coordinate-systems is not reducible to purely 3-vector form. However, the electromagnetic example discussed above would make it quite straightforward e.g. to define a "gravitational proper-force" of the form mG λ µν U µ U ν . This proper-force approximation for including gravitomagnetic effects at any speed might, for example, come in handy for making 3-vector simulators of interstellar travel, e.g. using the celestiaproject.net database, which would work pretty well in the vicinity of main-sequence stars and planets, as well as between stars.

In that case, interesting extreme-gravity effects might have to be patched into the algorithms after the fact, e.g. with help from a mean-field estimate of gravitational dt/dτ . Regardless of whether one uses geometric-force gravity [START_REF] Bell | A numerical solution of the relativistic Kepler problem[END_REF] or its proper-force approximation, metric-first simulations of accelerated motion between and around stars using simultaneity defined by bookkeeper far-time, FIG. 9. A (2+1)D acceleration puzzler schematic of a starfleet battlecruiser traveling upward which, on seeing light from an enemy spaceship dropping out of hyperspace (dotted red circle), accelerates rightward in order to intercept. The enemy trajectory is in gray, while the starfleet cruiser trajectory is in black. Proper-time intervals are marked on both trajectories in 2 traveler-month intervals. Units for distance are lightyears, for proper-velocity vectors (green) are lightyears/traveler-year, and for proper-acceleration (red) is in lightyears/year. The battlecruiser then reverses acceleration direction in order to (eventually) recover its original trajectory, amazingly about four months of ship time ahead of schedule following a detour that took 4 years. A detour that saved time, at least for the traveler! but which are predicted on actions taken by a single traveler on their proper-time clock (cf. Fig. 8), will likely become quite robust in the days ahead.

VIII. CONCLUSION

Thus "flat-patch" or "traveler-point" dynamics opens the door to "single map-frame" word problems, like those in introductory physics (cf. Fig. 9), which work globally at any speed and locally in curved spacetimes and accelerated frames. It also uses minimally frame-variant and synchrony-free variables expressed as scalars and 3vectors for the convenience of workers who experience time and space in terms of very different units. It allows for exact descriptions of accelerated motion in terms specific to one-traveler in flat spacetime, as well as locally accurate descriptions of motion in curved spacetimes as well. The local and single-traveler foci are limits, but one thereby also avoids the ambiguity of multiple ways to define extended simultaneity. Extended simultaneity in practice (e.g. when discussing conventions for "galactic mean time") will likely be determined by convention in practical situations regardless.

As a result introductory physics texts downstream might simply introduce Newton's laws as approximations to "traveler-point" equations which work at any speed and (with help of local "geometric-force" approximations) in curved spacetimes and in accelerated frames. This would allow teachers at the outset to deconstruct Newton's implicit assumption of global time, and give students a simple path on their own to explore more extreme physics situations without having to spend time on those situations in the class itself. It would also then make it natural to mention the roots of inertial forces and gravity (and "geometric forces" generally) in terms of differential aging, and of magnetism in terms of electrostatic charge combined with length contraction, thus whetting potential interest in learning more while keeping the focus on the Newtonian approaches at hand.

FIG. 1 .

 1 FIG.1. Even one extended-frame with synchronized clocks may be more than we have in curved spacetime.

FIG. 4 .

 4 FIG. 4. Metric-1st equations of (1+1)D motion. The dot over the equals sign means this works only for unidirectional motion. The Lorentz transform section at far right is the only place which requires more than one map (or bookkeeper) reference frame of yardsticks and synchronized clocks.

FIG. 5 .

 5 FIG. 5. Two views of proper (red) and geometric (dark blue or brown) forces in some everyday settings.

FIG. 7 .

 7 FIG. 7. Schwarzschild "far-time" r-ct view of a particle trajectory (green dashed) dropped from a 1 gee shell frame (red line) around a large supermassive (145 billion solar mass) nonrotating black hole with event horizon in dot-dashed blue.

FIG. 8 .

 8 FIG. 8. Years elapsed in proper time (blue) and map time (red) on a 16+12+20 = 48 lightyear 1-gee acceleration round trip lasting about 17.2 traveler years and 53.7 map years.

ACKNOWLEDGMENTS

Thanks to: Matt Wentzel-Long for his ideas, Edwin F. Taylor and the late Bill Shurcliff (1909Shurcliff ( -2006) ) for their counsel on "minimally-variant" approaches, Phil Asquith for calling the Bridgman, Bondi & Brehme papers to our attention, plus D. G. Messerschmitt & D. Neustadter for their interest.


Raúl De

La Fuente

Thomas Watteyne
email: thomas.watteyne@inria.fr

Hands-on Evaluation of Kinéis Satellite IoT Technology

Keywords: Satellite, IoT, hands-on evaluation, open-source, Kinéis

Satellite technology offers exciting new opportunities for IoT applications. This poster shows the performance one may expect using Kinéis -a leading operator -as a representative example. We use an implementation we provide as open-source for repeatability, and measure an end-to-end latency of 45 min, a battery lifetime of 1,000 packet when using a pair of AA batteries, and an end-to-end reliability of 23%/54%/99% when using 1/10/18 repetitions.

INTRODUCTION

The Industrial IoT is the pinnacle of low-power wireless technology, with standardized solutions that offer wire-like reliability, a decade of battery lifetime and certified security [START_REF] Bajic | Industry 4.0 Implementation Challenges and Opportunities: A Managerial Perspective[END_REF]. Yet, today's local and cellular networks are not perfectly suited for deployments in remote areas with limited or no network infrastructure.

The "new space" trend is enabled by the development of affordable, durable and miniaturized Low Earth Orbit (LEO) satellites [START_REF] Akyildiz | The Internet of Space Things/CubeSats[END_REF][START_REF] Golkar | Definition of New Space-Expert Survey Results and Key Technology Trends[END_REF]. The availability of Low Power Global Area Networks (LPGANs) wireless technologies enables devices on the ground to send their data directly to a satellite passing above [START_REF] Centenaro | A Survey on Technologies, Standards and Open Challenges in Satellite IoT[END_REF]. Kinéis (https://www.kineis.com/en/), a startup company based in France, is one of the sensor-to-satellite leaders. They operate the popular historic ARGOS constellation (7 polar orbiting satellites launched between 1998 and 2019), and are preparing to launch a new 25-satellite constellation by May 2024. They offer a complete solution to end users, in particular the postage stamp-sized KIM1 module that can be soldered onto a board and that contains the radio to send data to the satellite.

The contributions of this poster abstract are threefold:

• We provide a snapshot of the state-of-the-art in satellite connectivity, with the down-to-Earth angle of an end user; • We develop and provide a full set of source code to drive the KIM1 module, published under an open-source license; • We measure the key performance indicators of this technology: end-to-end reliability, end-to-end latency, battery lifetime. While we describe it in the context of Kinéis's solution, all providers use the same general architecture. IoT devices are deployed somewhere on Earth. One big advantage of satellite connectivity is that it works even when the devices are in remote locations, such as deserts or in the middle of the ocean. A constellation of LEO satellites continuously orbit the Earth in about 128 min. In the general case, a device is in radio range of a satellite for 10-12 min. Since there are a limited number of satellites, the device needs to wait for one to pass overhead before transmitting. Once it has received a frame from a device, the satellite waits to pass above one of Kinéis's ground stations, and transfers the device's data to it. The ground station then forward the data to the company's processing and distribution center, which sends it to the end user using tradition Internet protocols such as MQTT. Today, Kinéis operates seven LEO satellites. These form the wellknown ARGOS constellation, used since 1978 for animal tracking and other environmental monitoring. The company will be launching a 25-satellite constellation in groups of five by May 2024.

SATELLITE FOR IOT

Satellite technologies such as Kinéis enable a wide range of applications, including transport and logistics, network and infrastructure, smart agriculture, environmental monitoring, maritime activities, and smart water management. To simplify evaluation, Kinéis has integrated the KIM1 module in an Arduino shield that we connect to an Arduino board (2) On it, we implement the AT command-based interface of the KIM1.

OPEN-SOURCE IMPLEMENTATION

The KIM1 doesn't know when a satellite is overhead; Kinéis provides C libraries to compute it. While one can use it on an embedded micro-controller, we chose to run our own Python library on a Raspberry Pi (1).

EVALUATION SETUP AND RESULTS

We deploy the setup in Fig. 2 for a week, having the KIM1 send a 13 byte data frame every 90 s when a satellite is overhead.

Part of the payload we send is a sequence number, allowing us to measure the end to end reliability. As show in Fig. 3 the KIM1 module sent 1479 packets in total. 346 of these packets (23%) were successfully received by the user application. For this setup, we would need to retransmit the same packet 18 times to yield an end-to-end reliability above 99%.

The notification received from the Kinéis processing center contains several timestamps, allowing us to break down the end-to-end latency, as shown in Table 1. The largest contributions to the latency waiting for a satellite to pass over 35 min sending frame to satellite 5.103 s satellite travels and sends to ground station 9 min 23 s transfer from ground station to application 31 ms total 45 min Table 1: End-to-end latency of a representative packet.

DISCUSSION

Satellite IoT technology opens up new exciting opportunities. It is not meant to replace low-power wireless mesh technologies such as 6TiSCH in industrial applications (which require wire-like reliability), nor cellular technologies such as NB-IoT for smart city applications (which are readily available). Rather, it fills a technology gap for nodes deployed in remote areas without infrastructure. Ideal applications include pipeline monitoring, smart agriculture or wildfire detection. It is fantastic to see companies such as Kinéis providing ready-to-use solutions. The goal of this poster is to show the attendance the performance they can expect from such solutions, Kinéis being representative of other providers.

Figure 1 :

 1 Figure 1: The "technology chain" of satellite IoT application, using the representative Kinéis terminology.

Fig. 1

 1 Fig.1depicts the "technology chain" in sensor-to-satellite solutions. While we describe it in the context of Kinéis's solution, all providers use the same general architecture. IoT devices are deployed somewhere on Earth. One big advantage of satellite connectivity is that it works even when the devices are in remote locations, such as deserts or in the middle of the ocean. A constellation of LEO satellites continuously orbit the Earth in about 128 min. In the general case, a device is in radio range of a satellite for 10-12 min. Since there are a limited number of satellites, the device needs to wait for one to pass overhead before transmitting. Once it has received a frame from a device, the satellite waits to pass above one of Kinéis's ground stations, and transfers the device's data to it. The ground station then forward the data to the company's processing and distribution center, which sends it to the end user using tradition Internet protocols such as MQTT.Today, Kinéis operates seven LEO satellites. These form the wellknown ARGOS constellation, used since 1978 for animal tracking and other environmental monitoring. The company will be launching a 25-satellite constellation in groups of five by May 2024.Satellite technologies such as Kinéis enable a wide range of applications, including transport and logistics, network and infrastructure, smart agriculture, environmental monitoring, maritime activities, and smart water management.

Figure 2 :

 2 Figure 2: We use a Raspberry Pi and an Arduino to drive the KIM1 module.

Figure 3 :

 3 Figure 3: End-to-end reliability measured over one week.

Fig. 2

 2 Fig. 2 shows the setup used. As an online addition to this paper, we publish all source code used under an open-source license at https:// github.com/openwsn-berkeley/starwars, ensuring full replicability of our results. The setup is composed of three devices. The KIM1 module (3) is the main element we are testing. It features a low-power radio and implements the ARGOS-2 physical layer and protocol. It is certified by the French Space Agency (CNES). On top of being able to send data to ARGOS/Kinéis polar satellites, it allow for GPS-free Doppler localization of the sending IoT device, with a localization accuracy of 150 m.To simplify evaluation, Kinéis has integrated the KIM1 module in an Arduino shield that we connect to an Arduino board (2) On it, we implement the AT command-based interface of the KIM1.The KIM1 doesn't know when a satellite is overhead; Kinéis provides C libraries to compute it. While one can use it on an embedded micro-controller, we chose to run our own Python library on a Raspberry Pi (1).

Figure 4 :

 4 Figure 4: Current draw at 5 V of the KIM1 module when sending 3 frames (left). The right plot zooms onto one frame.

* This document is issued within the frame and for the purpose of the OpenSwarm project. This project has received funding from the European Union's Horizon Europe Framework Programme under Grant Agreement No. 101093046.


Tanathep Leungtongkum

Onrawee Laguerre

Denis Flick

Alain Denis

Steven Duret

Nattawut Chaomuang
email: nattawut.ch@kmitl.ac.th

Experimental investigation of airflow and heat transfer by natural convection in an insulated box with a Phase Change Material using a Particle Image Velocimetry technique

Keywords: Tanathep Leungtongkum Conceptualization, Methodology, Investigation, Validation, Formal analysis, Software, Writing -Original Draft Preparation, Visualization. Onrawee Laguerre Validation, Formal analysis, Writing -Review & Editing, Supervision, Project Administration, Funding acquisition Denis Flick Methodology, Validation, Formal analysis, Nattawut Chaomuang Conceptualization, Methodology, Investigation, Validation, Formal analysis, Software, Writing -Original Draft Preparation, Visualization, Funding acquisition

Airflow and heat transfer via natural convection in an insulated box with Phase Change Material (PCM) were experimentally investigated using Particle Image Velocimetry (PIV) and temperature measurements. The effects of PCM positions (side wall and lid) on flow pattern and temperature distribution were studied under empty and loaded conditions. Two loads were considered to study the obstacle effect and the influence of heat exchange with air. When PCM was either at the side wall or at the lid of the box, laminar flow was observed and the corresponding Rayleigh number was about 10 7 . Upward flow was always observed near the side walls of the box. When PCM was on the side, downward flow occurred along the PCM; in the empty case, flow was almost 2D but became 3D when the load was added. When PCM was on the lid, the air cooled in contact with PCM, detached from it and flowed downwards. In

𝛽

 et al., 2017), fruit and vegetables (Zhao et al., 2019), and pharmaceutical products (Robertson et al., 2017). The advantages of insulated boxes with PCM include low investment and operating costs, flexibility in storage temperatures and volumes, and ease of maintenance (Zhao et al., 2020). However, two drawbacks emerge: the temperature in an insulated box is difficult to control, and temperature heterogeneity is often observed, with a low temperature in the vicinity of the PCM and a high temperature further away from the PCM (Laguerre et al., 2008a;[START_REF] Margeirsson | Numerical modelling of temperature fluctuations in superchilled fish loins packaged in expanded polystyrene and stored at dynamic temperature conditions[END_REF][START_REF] Navaranjan | Thermal insulation requirements and new cardboard packaging for chilled seafood exports[END_REF]. This problem can result in the degradation of product quality, for instance chilling/freezing injury (a temperature that is too low leads to cell damage) and pose a safety risk (a temperature that is too high leads to bacterial growth) [START_REF] Laguerre | How to predict product temperature changes during transport in an insulated box equipped with an ice pack: Experimental versus 1-D and 3-D modelling approaches[END_REF]. As reviewed by [START_REF] Leungtongkum | Insulated box and refrigerated equipment with PCM for food preservation: State of the art[END_REF], numerous experimental and numerical studies have been conducted to evaluate the thermal performance of insulated boxes equipped with PCM. [START_REF] Choi | Practical mathematical model to predict the performance of insulating packages[END_REF] developed simplified heat transfer models based on an ice melting test for the estimation of heat flow resistance (R-value) which is a factor generally used to determine the insulation performance of the box. Later, Singh et al. (2008) applied this method and reported the R-values of insulated boxes made of various insulating materials with different wall thicknesses and box dimensions. Various 2D and 3D

Computational Fluid Dynamics (CFD) models of insulated boxes with PCM were developed to study the influences of different factors on the evolution of air/product temperatures [START_REF] Du | Cooling performance of a thermal energy storage-based portable box for cold chain applications[END_REF][START_REF] Laguerre | How to predict product temperature changes during transport in an insulated box equipped with an ice pack: Experimental versus 1-D and 3-D modelling approaches[END_REF][START_REF] Laguerre | Modelling of fish refrigeration using flake ice[END_REF][START_REF] Margeirsson | Numerical modelling of temperature fluctuations in superchilled fish loins packaged in expanded polystyrene and stored at dynamic temperature conditions[END_REF]Paquette et al., 2017;Xiaofeng and Xuelai, 2021). These factors included box characteristics (dimensions, shape of inner corners, types and thicknesses of insulating materials, internal surface emissivity), PCMs (types, mass, and position in a box), products (types, initial temperature, mass, and arrangement in a box), and operating conditions (ambient temperature and transport J o u r n a l P r e -p r o o f duration) [START_REF] Leungtongkum | Insulated box and refrigerated equipment with PCM for food preservation: State of the art[END_REF]. To reduce computational time, analytical and zonal models were also developed as a complementary approach [START_REF] East | A method for quantitative risk assessment of temperature control in insulated boxes[END_REF][START_REF] Laguerre | How to predict product temperature changes during transport in an insulated box equipped with an ice pack: Experimental versus 1-D and 3-D modelling approaches[END_REF][START_REF] Laguerre | Modelling of fish refrigeration using flake ice[END_REF]. These models enable useful information (e.g., PCM melting duration and product temperature change during shipment) to be acquired as a function of the box design and usage conditions. Xiaofeng and Xuelai (2021) developed an insulated box with multiple partitions for delivery of various products, which require different preservation temperatures (ambient, chilled, and frozen temperatures) in the same box. Despite these efforts, there is still no universal optimal condition that can be applied to control the air/product temperatures in an insulated box.

All heat transfer modes can occur simultaneously in an insulated box with PCM: conduction (inside the product, PCM and the walls of the box), natural convection (between air and product/PCM in the box), natural/forced convection (between external air and the box), and radiation (between the walls and product/PCM in the box) [START_REF] Leungtongkum | Insulated box and refrigerated equipment with PCM for food preservation: State of the art[END_REF]. For model simplification, most numerical studies consider heat transfer by conduction, and sometimes also by radiation, while natural convection inside the box tends to be neglected [START_REF] Du | Cooling performance of a thermal energy storage-based portable box for cold chain applications[END_REF]. In fact, all heat transfer modes can be of the same order of magnitude, and natural convection can exert significant impacts, especially on temperature heterogeneity in the insulated box [START_REF] Laguerre | Temperature prediction in domestic refrigerators: Deterministic and stochastic approaches[END_REF].

Natural convection in closed cavities occurs in a wide range of engineering applications and has been extensively studied with both experimental and numerical approaches [START_REF] Miroshnichenko | Turbulent natural convection heat transfer in rectangular enclosures using experimental and numerical approaches: A review[END_REF][START_REF] Pandey | An exhaustive review of studies on natural convection in enclosures with and without internal bodies of various shapes[END_REF]. The configurations commonly studied are air or water in a rectangular cavity of which two opposite (horizontal/vertical) walls are maintained at different constant and uniform temperatures (hot and cold) while the other walls are perfectly insulated (adiabatic). Many studies also introduce porous medium or solid J o u r n a l P r e -p r o o f objects (flat plate, cylinder and sphere) in the cavity to observe their effects on heat transfer and airflow [START_REF] Ataei-Dadavi | Flow and heat transfer measurements in natural convection in coarse-grained porous media[END_REF][START_REF] Lee | Three-dimensional natural convection around an inner circular cylinder located in a cubic enclosure with sinusoidal thermal boundary condition[END_REF]. The knowledge acquired using a cavity filled with porous media can be applied to the case of small products such as cereal grains (ratio between characteristic lengths of product and cavity ≤ 0.02). However, the models are limited in the case of larger products (e.g., meat, fruit and vegetables) where the ratio is much higher (> 0.1) (Laguerre et al., 2008b). Moreover, the configuration of a product-loaded insulated box with PCM is more complex because all walls are subjected to heat loss and the wall equipped with PCM (cold wall) has a non-uniform temperature as PCM melts more rapidly at some positions (Jevnikar and Siddiqui, 2019). Several experimental and numerical studies were conducted to investigate heat transfer and airflow in air-filled cavities with PCM at one wall. [START_REF] Aitlahbib | Numerical study of heat transfer inside a Keeping Warm System (KWS) incorporating phase change material[END_REF] developed a two-dimensional numerical model to evaluate the thermal performance of an insulated container with one vertical wall equipped with PCM, the opposite one kept at higher temperature while other walls were well-insulated. The results demonstrated that the container with PCM wall achieved superior thermal performance than the one without. The same cavity configuration was numerically studied by [START_REF] Labihi | Effect of phase change material wall on natural convection heat transfer inside an air filled enclosure[END_REF].

The effect of volume expansion of the PCM during solidification was considered in the model.

Better predictions were obtained when this effect was taken into account. [START_REF] Moreno | Numerical and experimental study of heat transfer in a cubic cavity with a PCM in a vertical heated wall[END_REF] developed would be useful to suggest the optimal practices for handling insulated boxes used for food transport. The experimental results obtained in this study enable deep knowledge in the exchange phenomena. They will be used to develop CFD and simplified thermal models, which will be presented in the future. These numerical approaches enable the prediction of product temperature change with time at different positions in an insulated box exposed to variable ambient temperatures as in a real shipment.

Materials and methods

Experimental device

J o u r n a l P r e -p r o o f

Two insulated boxes were used. Box A was a commercially manufactured box which was used for the thermal study (temperature measurement by thermocouples). Box B was the same box in which two walls were replaced with triple-glazed windows ensuring almost the same insulation, and it was used for the momentum study (air velocity measurement by a PIV system). As shown in Fig. 1a, the wall structure of Box A is composed of an expanded polystyrene foam layer (thickness: 25 mm, 𝜆 = 0.029 W•m -1 •K -1 ) sandwiched between two polypropylene plastic layers (thickness: 3.5 mm, 𝜆 = 0.12 W•m -1 •K -1 , 𝜀 = 0.97) (Cengel and Ghajar, 2020). The inner gap between the polypropylene layers is 35 mm so that an air layer

(thickness: 5 mm, 𝜆 = 0.025 W•m -1 •K -1
) is also present. The internal dimensions of the box were 310 mm (W) × 500 mm (L) × 300 mm (H), corresponding to about 0.05 m 3 in volume.

Box B had the same dimensions and wall structure, but two vertical walls were replaced by triple-glazed windows. They were composed of three glass layers (𝜆 = 1.4 W•m -1 •K -1 ) each with a thickness of 4 mm and two 10-mm argon gaps (𝜆 = 0.018 W•m -1 •K -1 ), as shown in Fig.

1b. In addition, the panes were coated with a low-emissivity material (𝜀 = 0.03, manufacture data) to avoid the transmission of infrared radiation. In this manner, the overall heat transfer coefficient of the glass wall was almost identical to that of the unmodified commercially manufactured box: To facilitate interpretation of the results, the boxes (for both the thermal and momentum studies)

𝑈 𝑔 ≅ 𝑈 𝑤 = 0.9 W•m -2 •K -1 .
were placed on small supports (with a height of 50 mm) placed on a table (with a height of 700 mm), thereby ensuring homogeneous airflow around the box (including underneath it). The experimental device was placed in a test chamber (dimensions: 340 cm × 340 cm × 250 cm) in which the ambient temperature was controlled at 20.0 ± 1.0°C throughout the experiments. The humidity in the room was not controlled. The relative humidity measured by using a hygrometer (Testo 174H, accuracy ±3%rh) lied in the range of 45-65%, corresponding to the humidity ratio of 0.006-0.009 kg of water vapor/kg dry air.

Thermal study

J o u r n a l P r e -p r o o f

Temperature measurements were performed in Box A using calibrated T-type thermocouples (200 μm diameter, ± 0.2°C accuracy) connected to a data logger (Agilent 34972A). The protocols for the measurements under empty and loaded conditions are described hereafter.

Experimental protocol under empty conditions

The temperature measurements were conducted on a middle plane (x = 250 mm) of the box as shown in Fig. 2a. Twelve thermocouples were installed on a portable stand at intervals varying from 5 mm (near the top and bottom) to 35 mm (mid-height) and were used to measure the air temperatures across the cavity. Another twelve thermocouples were fixed on the surfaces of the box walls and the PCM in order to measure their surface temperatures. Three thermocouples were also installed at three positions inside the PCM.

To establish the temperature profiles, several temperature measurements were carried out by moving the stand from one position to another across the plane. The stand was positioned at a distance of 5 mm from the walls for the initial measurement. The measurement began at least 90 min. after the box closure (steady state had been reached) and lasted for a duration of 5 min.

with recording intervals of 15 s. The box was re-opened in order to move the stand to the next position and then was closed rapidly to limit the effects of external air ingress (total duration of this operation: < 1 min). Then, temperature measurement was undertaken 15 min. after closure (steady state had been reached again). To address the temperature profile in the boundary layer, fine incremental steps of 5 mm were applied for the first five positions near the wall, then coarser incremental steps (up to 50 mm) were used (eighteen y-positions). The temperature at each position was averaged over 5 min. and the reported temperature profiles were based on these average values.

It needs to be emphasized that the phase change process of the PCM is a transient phenomenon and the real steady state never exists. The "pseudo" steady state stipulated in this study indicate J o u r n a l P r e -p r o o f the duration in which the air temperatures at different positions in the cavity were relatively constant. Preliminary experiments showed that "pseudo" steady state was reached after 90 min.

Then, the standard deviations over 3 h of the air temperatures at eight positions in the box never exceeded 0.3°C.

Experimental protocol under loaded conditions

The stack of TYL packs was carefully placed in the center of the box, which corresponded to a loading percentage (including the PCM) of almost 50% by volume. In this case, all air and load temperatures (core and surface) were simultaneously measured throughout the experiment without intermittent openings. Fig. 2b shows the positions of the thermocouples used to measure the air temperatures on the middle (x = 250 mm) and the lateral (x = 15 mm) planes of the box as well as the core and surface temperatures of four TYL packs. Thermocouples were also installed at three positions inside the PCM. Measurements were undertaken 90 min. after the closure of the box and lasted until the PCM was completely melted (i.e. all measured PCM temperatures started to increase). A recording interval of 30 s was set for the experiment. The air and product (core and surface) temperatures at each point were averaged over 200 min.

during which the PCM was melting. These time-averaged values were used to establish the temperature field in the loaded box.

Momentum study

Fig. 3 shows the overall view of experimental setup for the air velocity measurements in the insulated box (Box B) using a PIV system which requires tracer particles (oil smoke in our case) for the measurement. The box was connected to a smoke container using a flexible duct in which four small PCM packs were used to precool the smoke before entering the box. A small fan was used to assist the introduction of precooled smoke (~10°C) into the box and its flow rate was controlled by a valve on the connecting duct.

J o u r n a l P r e -p r o o f

PIV system

A 2D-PIV system (LaVision, FlowMaster 2D) was used to visualize the flow pattern and to measure air velocity in the box. The system is composed of three main components: a doublepulsed Nd:YLF laser (527 nm wavelength, 10 mJ pulse energy), a high-speed 12-bit CMOS video camera (Photron, FASTCAM SA3; 1024 ×1024 pixels in resolution) mounted with a lens (Sigma; 105 mm, f/1:2.8), and a programmable timing unit (PTU-X) for the synchronization of the device. For light scattering, a smoke machine (Antari, F-80Z) was used to generate oil particles (Levenly, Smoke Standard; mean diameter of 0.3 μm). Image acquisition and postprocessing were performed using DaVis 10.2 interfaced software. The camera and the laser were installed on a three-dimensional displacement system (displacement precision ± 1 mm)

and aligned in such a manner that the field of the camera view was perpendicular to the light sheet (thickness of 1 mm).

Image acquisition

The PIV measurements under both unloaded and loaded conditions were performed on the same plane as the temperature measurement under loaded conditions: the middle plane (x = 250 mm) and the lateral plane (x = 15 mm). Based on the image calibration using a ruler and the DaVis software, the magnification factor of 0.113 mm/pixel was determined and it corresponded to an image size of approximately 115 mm × 115 mm. For each measurement plane, several measured windows with a partial overlap between them were used to cover the entire area of the plane.

Fig. 4 shows the position and its corresponding number of measured windows on the measurement plane for different experimental conditions (unloaded/loaded and PCM on the side/lid). The position of the measured windows was changed by using a displacement system.

For each measured window, 500 pairs of images were recorded every 20 ms with a time interval (∆𝑡) between two paired images (two pulsed laser illuminations) of 900 μs, and the total J o u r n a l P r e -p r o o f measurement duration was 10 s. Based on the preliminary experiment, the measurement duration should not exceed 10 s to avoid heat generation by the laser which caused an increase in wall temperature, thereby affecting airflow. This time interval was considered as an optimal value for a velocity estimation in our case, because it allowed a mean particle displacement of less than a quarter of the smallest width of interrogation window [START_REF] Keane | Theory of cross-correlation analysis of PIV images[END_REF].

Image post-processing

A multi-pass correlation algorithm was used to process instantaneous vector calculation. The cross-correlation between individual paired images was performed with decreasing interrogation window sizes: 64 × 64 pixels with 50% overlap for the first passes and 32 × 32 pixels with 75% overlap for the final passes. Given the interrogation dimensions of the final pass, the spatial resolution of the vector field (distance between two vectors) was 8 pixels (about 0.9 mm) in both vertical and horizontal directions. After 500 instantaneous vector fields were obtained, the mean velocity field (𝑣: 2D velocity magnitude) was then calculated as follows

𝑣 = 1 𝑁 ∑ √𝑣 𝑦,𝑖 2 + 𝑣 𝑧,𝑖 2 𝑁 𝑖=1 (1)
where 𝑁 is the total number of measured windows (𝑁 = 500 in our case), 𝑣 𝑦 and 𝑣 𝑧 are the horizontal and vertical velocity components expressed in m•s -1 , respectively.

The mean velocity fields of all measured windows were then connected to establish the velocity field of the entire measurement plane. It should be emphasized that the out-of-plane regions and the regions near high reflection surfaces in the images (e.g. the surfaces of PCM, wall, and load) were deleted and excluded prior to the vector calculation.

The DaVis software uses the correlation statistics method (a-posteriori approach) to quantify the uncertainty of the PIV measurement. Based on the statistical analysis (Wieneke, 2015), the J o u r n a l P r e -p r o o f uncertainty of the mean air velocity reported in this study was less than 5% for 𝑣 > 0.04 m•s -1 and less than 10% for 0.02 < 𝑣 < 0.04 m•s -1 .

Experimental protocol

Air velocity measurements were carried out under steady state conditions which were achieved 90 min. after the PCM was introduced into the box. At steady state, the precooled smoke was introduced into the box until its concentration was sufficient. To ensure flow stabilization, the PIV measurement was performed about 30 min. after the smoke introduction. Nine and five measured windows of 500 paired images were captured for the experiments under empty and loaded conditions, respectively. Table 2 summarizes all experimental conditions for the temperature and the velocity measurements.

Results and discussion

Air velocity and temperature profiles under the empty condition

The experiment was firstly performed under empty conditions to gain an understanding of underlying momentum and energy transport phenomena inside the insulated box equipped with PCM. The influence of PCM positions (side wall and lid) on these phenomena were investigated and the results are presented as follows.

PCM on a side wall of the box

When the frozen PCM was placed on a side wall of the box, the apparent width (𝑊′) of the box was reduced to 260 mm and accordingly the aspect ratio (𝐴 = 𝐻 𝑊′ ⁄ ) of the box was about 1.15. Based on the temperature difference between the inner surfaces of the PCM and the walls (∆𝑇 = 𝑇 ℎ -𝑇 𝑐 = 7.9°C), the Rayleigh (𝑅𝑎) number based on the box height as defined in Eq.

J o u r n a l P r e -p r o o f 2 was approximately 2.8×10 7 . Thus, the air flow in the box was in the laminar regime (< 10 9 ) (Saury et al., 2011).

𝑅𝑎 = 𝐺𝑟 • 𝑃𝑟 = 𝑔𝛽(𝑇 ℎ -𝑇 𝑐 )𝐻 3 𝜐 2 • 𝑐 𝑝 𝜇  (2)
The thermal properties of air were calculated at the mean air temperature in the box (𝑇 𝑚𝑒𝑎𝑛 = 7.7°C) by using the correlations proposed by [START_REF] Mcquillan | Properties of dry air at one atmosphere, Microelectronics Heat Transfer Lab[END_REF]. The calculated values are summarized in Table 3.

Fig. 5 shows the air velocity profiles on the middle (x = 250 mm) and the lateral (x = 15 mm) planes of the box equipped with the PCM on the side wall. For comparison purposes, the color scale was limited to 0.13 m•s -1 over which the maximum value recorded among all experimental conditions never exceeded. Large (red) arrows were drawn over the velocity field in order to provide better visualization of flow patterns represented by velocity vectors originally generated from the PIV software.

As shown in Fig. 5a, the airflow on the middle plane exhibits a flow pattern similar to the simple case, largely documented in the literature, of a rectangular cavity with opposite isothermally hot and cold vertical walls and well-insulated horizontal walls [START_REF] Lee | Three-dimensional natural convection of air in an inclined cubic cavity[END_REF]:

upward and downward flow streams adjacent to the vertical surfaces of the side wall and the PCM, respectively. These flow streams moved horizontally along the bottom and the top walls, respectively, thus forming a recirculation cell. However, due to the technical limitations of the experimental device, the visible height was limited to z = 260 mm; thus, the complete flow recirculating cell could not be visualized in our study (dashed-line arrow in Fig. 5a).

A secondary flow was also detected (see arrow A on Fig. 5a): the air at a distance of about 30 mm from the box wall changes its direction from upward to downward and flows toward the cold wall. This could result from the heat transfer through the bottom wall of the box. The J o u r n a l P r e -p r o o f airflow near the 'warm' bottom wall has a slightly ascending slope (this is not the case for an adiabatic bottom wall). The air located about 30 mm from the walls is entrained by the main recirculation flow, but it is relatively colder than the air nearer to the wall; thus, at a given point, its trajectory leaves the main cell and becomes downwards.

As illustrated in Fig. 5a decrease, conceivably because the flow 'turned' at the corner. The flow pattern on the lateral plane (Fig. 5b) was almost identical to that on the middle plane, but the maximal velocity magnitude was higher (0.12 m•s -1 ). Note that the velocity field on the lateral plane (x = 15 mm) was extended to the region of the PCM because the PCM was symmetrically placed in the box, thereby allowing the PIV measurement in the gap between the lateral wall and the PCM (x > 20 mm).

On both planes, there is a zone of stagnant air in the core region of the box. However, this region on the lateral plane was smaller than that on the middle plane. Three-dimensional flow due to additional heat gain through the side wall could explain this difference.

PCM on the lid of the box

The apparent height was 𝐻' = 250 mm; thus, the aspect ratio of the box with PCM on the lid (𝐴 = 𝐻'/𝑊) was about 0.81 and the 𝑅𝑎 number was 2.4×10 7 (Δ𝑇 = 6.8°C). As in the case of PCM on the side wall, the airflow was laminar. The values of thermal properties of air are summarized in Table 3 (𝑇 𝑚𝑒𝑎𝑛 = 6.2°C).

Fig. 7 shows the air velocity profiles on the middle (x = 250 mm) and the lateral (x = 15 mm) planes of the box with PCM on the lid. It was found from Fig. 7a that there were two almost symmetric, counterrotating air-flow cells. This result qualitatively agrees with the numerical solution obtained by [START_REF] Corcione | Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above[END_REF] who also observed such a two-cell flow pattern in a rectangular cavity with one cold top wall, one hot bottom wall and two hot side walls (𝐴 = 0.5 and 𝑅𝑎 = 10 6 in his study). As shown in Fig. 7a', the positive vertical velocity components were detected along the side walls (y < 50 mm and y > 260 mm) while the negative ones were mostly in the core region. On the middle plane, the absolute values of the vertical velocity components never exceeded 0.08 m•s -1 . The upward flow along the side walls was induced by the relatively high air temperature in these regions as a result of heat conduction through the box walls. Accordingly, these flow streams converged on the top where air was cooled down via the PCM. Becoming heavier, the air then flowed downward in the center region.

J o u r n a l P r e -p r o o f

Near the lateral wall (x = 15 mm, Fig. 7b andb'), the flow is almost everywhere upwards as is the case near the side wall (𝑣 𝑧 > 0 for y = 15 mm in Fig. 7a'). In fact, one would expect a similar 2D cell flow pattern in x-z plane (𝑦 = 𝑊/2) as that observed in the y-z plane (𝑥 = 𝐿/2, Fig. 7a). Overall, air flows upwards along the lateral and side walls, whereas it flows downward in the central region (3D flow). Near the corners (junction of lateral and side walls e.g. x = 15 mm, y = 15 mm) the heat flow by conduction through the walls is the highest, and this explains why the vertical velocity is also the highest in these positions: 0.12 m•s -1 .

It should be borne in mind that the presented velocity fields are time-averaged over 10 s and are composed of 9 windows recorded at different times (typically at 2 min. intervals taking into account the time needed to move the camera and save the recorded data). Direct observation of smoke in the middle plane showed that the flow was not stable (it was unsteady) in the central region: the downward flow oscillated in the y direction. This explains why the velocity observed in Fig. 7a is not strictly symmetric and that there are some 'jumps' between the 3 parts (3 windows in the y direction) of the profiles in Fig. 7a'. This type of instability has been observed also for free convection in domestic refrigerators [START_REF] Laguerre | Experimental study of heat transfer by natural convection in a closed cavity: application in a domestic refrigerator[END_REF].

Fig. 8 shows the air temperature field on the middle plane of the box. As expected, the cold region was in the center where downward flow was observed, while the warm region was near the side walls where upward flow was observed. The maximum air temperature was about 9°C near the top of the side walls and the minimum air temperature was observed just below the PCM (top of central region); the surface of the PCM container was at a temperature of around 1-2°C.

Comparison between PCM on the lid or on the side

In comparison with the case of PCM on the side wall, the box with PCM on the lid exhibited a lower maximal temperature: 9.3°C (lid)/10.5°C (side). This high temperature was observed near J o u r n a l P r e -p r o o f the walls. In fact, the product should not touch the walls and should even be placed outside the boundary layers whose thickness was around 30 mm. If we exclude the boundary layer zone, the mean temperature (in the middle plane) was lower for PCM on the lid: 𝑇 𝑚𝑒𝑎𝑛 = 6.2°C (lid) and 7.7°C (side) and the temperature distribution was also more homogeneous for PCM on the lid: 𝑇 𝑚𝑎𝑥 -𝑇 𝑚𝑖𝑛 = 1.7°C (lid) and 2.7°C (side).

As mentioned previously, flow fluctuations were visually observed in the central region where the PCM was placed on the lid. To a lesser extent, fluctuations were also observed near the bottom of the PCM when it was placed on the (right) side. Fig. 9 presents the instantaneous velocity evolution during 10 s at two positions (near the center and near the bottom/right corner). This confirms that the airflow was more stable in the case of PCM on the side. Velocity variations of up to 0.10 m•s -1 were detected at the center of the box with PCM on the lid (Fig. 9b).

The convective heat transfer coefficients can be estimated from the measured temperature profiles. Very near to the wall, the air velocity is close to zero. So, the heat flux (W•m -²) along y-direction can be given by

𝜆 𝑎 𝜕𝑇 𝜕𝑦 = ℎ 𝑧 (𝑇 𝑤 -𝑇 ∞ ) (3)
where 𝑇 𝑤 and 𝑇 ∞ are the temperatures of the wall and the air outside the boundary layer (free stream), respectively, 𝜆 𝑎 is the thermal conductivity of the air, and ℎ 𝑧 is the local convective heat transfer coefficient at a given height (z) which can be approximately estimated from

ℎ 𝑧 = 𝜆 𝑎 (𝜕𝑇 𝜕𝑦 ⁄ ) 𝑇 𝑤 -𝑇 ∞ (4)
For example, when the PCM was at the side wall, at the mid height (z = 160 mm), the temperature at the PCM wall, at 5 mm from the wall and outside the boundary layer were 0.8°C, 5.5°C and 7.2°C, respectively. The slopes (𝜕𝑇 𝜕𝑦 ⁄ ) of the tangent line to the temperature profile near the PCM wall was thus approximately 1°C/mm. Accordingly, the local convective heat transfer coefficient at PCM wall could be estimated around 4 W•m -2 •K -1 . In the same way, the J o u r n a l P r e -p r o o f

heat transfer coefficient at the vertical internal box walls (warm walls) could be estimated between 2 and 3 W•m -2 •K -1 . It is to emphasize that the temperature fields were measured only during the 'pseudo' steady state period during which the air temperature was almost invariant with time. Therefore, the estimation of ℎ 𝑧 was given for this period only.

Despite low heat transfer coefficients, convection cannot be neglected compared to conduction in air because the maximum air velocities observed in the box were around 0.1 m•s -1 , corresponding to the Peclet number (𝑃𝑒) of more than 100.

The Peclet number (𝑃𝑒) is defined as

𝑃𝑒 = 𝑣 𝑎 𝐿 𝑐 𝛼 𝑎 (5)
For the empty box, the value was approximately 1500, given that the characteristic length (𝐿 𝑐 ) was 0.3 m (height of the box), the air velocity (𝑣 𝑎 ) was 0.1 m•s -1 , and the air thermal diffusivity (𝛼 𝑎 ) was about 2×10 -5 m 2 •s -1 .

Air velocity and temperature profiles under loaded conditions

The experiments were performed under loaded conditions. The experiment was firstly conducted with inert blocks (XPS) that made it possible to study the obstacle-effect alone on the flow pattern. Then, an experiment with the test packs (TYL) was performed to study both the obstacle-effect and the influence of heat exchange with air. Due to the presence of the obstacles, the light sheet was restricted; thus, the velocity field behind the load was not available.

PCM on the side wall of the box

As shown in Fig. 10, the airflow in the loaded box, regardless of obstacle types, exhibited similarities with that in the empty box: upward and downward flow streams close to the vertical J o u r n a l P r e -p r o o f surfaces of the side wall and the PCM, respectively. However, under loaded conditions, the upward flow on the middle plane did not result from a two-dimensional recirculation cell (located in this plane) but instead from a three-dimensional flow pattern as illustrated in Fig.

11.

At State 1, the air flowed downward (-z direction) in the space between the PCM and the load. Once approaching the bottom wall (State 2), the air flowed rather horizontally (+/-x direction) toward the lateral walls of the box. At the edge of the load (State 3), the air turned and flowed between the lateral wall and the load (-y direction). This is confirmed in Figs. 10b and 10d (lateral plane, x = 15 mm) where a strong flow from the right to the left was observed in the bottom region. When it reached the bottom of the side wall (State 4), the air moved both horizontally to occupy the entire gap between the side wall and the load and upwards because it became warmer and warmer (heat exchange with the walls). Finally, starting from State 5, it recirculated to the PCM.

The velocity fields obtained with inert blocks (XPS) and with test products (TYL) were very similar. This is because thermal steady state was practically reached in both cases (the test products were introduced practically at the equilibrium temperature and the measurement began after 2 h). In this manner, thermal inertia became negligible. For XPS (with very low thermal conductivity) the air temperature was expected to be relatively homogeneous in the gaps between the load and the lateral wall or the PCM. For TYL, due to conduction, the load surface temperature was lower than the air temperature in the wall-side gap but higher in the PCM-side gap (as shown hereafter in Fig. 12a). This could diminish free convection, but the results showed a minor effect (𝑃𝑒 ~150, 𝐿 𝑐 (gap) = 30 mm). This means that (steady state) flow characterization can be carried out with inert blocks (XPS), which is much simpler. Certainly, if warm products were introduced in the box initially, the flow pattern would have been substantially altered.

J o u r n a l P r e -p r o o f Fig. 12 shows the temperature field on the middle (x = 250 mm) and the lateral (x = 15 mm) planes of the box with PCM on the side wall and TYL load. The temperature field was coherent with the flow pattern: air was cooled down along the PCM and warmed up along all the box walls. It is obvious that the cold air near the PCM resulted in relatively low load temperatures on this (right) side. Conversely, the warmer air near the wall resulted in relatively high load temperatures on the opposite (left) side. Conduction in the load was not sufficient to homogenize the load temperature. The highest load temperature, 7.8°C, was reached near the side wall (opposite the PCM location). The average load temperature was 6.0°C and the maximum difference was 3.9°C. Air temperature stratification was observed on the lateral plane of the box (Fig. 12b).

PCM on the lid of the box

A complex flow pattern was observed in the loaded box with PCM on the lid as shown in Fig.

13.

On the middle plane (x = 250 mm, Figs. 13a and13c), regardless of load types, cold air coming from the top (near the PCM) flowed downward in the center of the (left) gap between the side wall and the load. Then, air flowed upwards along the side wall and the load surface.

An explanation for the upward flow along the side wall is that the air near this wall is warmed up by conduction through the wall (as in the empty case). The flow direction along the load surface is logically upwards if the surface temperature is higher than the average air temperature in the gap. This was the case for the XPS load because radiation from the wall to the load surface tended to increase the load surface temperature. For the TYL, in addition to radiation, conduction also occurred inside the load. This could explain why upward flow occurred all along the XPS load but occurred only along the upper part of the TYL load. This difference appeared also on the lateral plane (x = 15 mm): there were more upward flow regions in the This is related to the non-linear term in the flow equations (Navier-Stokes) which is responsible for a break in symmetry (even before turbulence). This dissymmetry was also confirmed by Dissymmetry was also observed: the right gap had a lower average air temperature than the left gap which in turn affected the load temperature. This also seemed to be due to a stronger cold air stream from the PCM toward the right gap (see the red arrow in Fig. 14). For temperature measurements, the dissymmetry cannot be imputed to a difference in insulation (Box A was J o u r n a l P r e -p r o o f used). This configuration did not allow stratification since the cold PCM container was placed at the top.

In the case of PCM on the lid, the highest load temperature, 7.3°C (7.8°C for PCM on the side) was reached near the bottom (opposite the PCM location). The average load temperature was 5.7°C (6.0°C for PCM on the side), and the maximum difference was 3.0°C (3.9°C for PCM on the side). So, there was little apparent difference between the two configurations. These findings suggested that the PCM can be placed either at the side wall or at the lid without compromising the quality and safety of food products if spaces between the PCM and the load and between the side walls and the load are reserved. According to our estimation, the insulated box with ice as a PCM is feasible for the transport of food products in the temperature range of 4-8°C for about 10 h. The experimental results obtained by this study will be used for the development of CFD and simplified thermal models to predict product temperature evolution along a logistic chain. This evolution makes possible the prediction of product quality thanks to a relation with the product temperature. In this manner, the product shelf life under different logistic scenarios can be predicted. The thermal and quality modelling would help the supply chain management to optimize the logistic conditions to reduce food loss and waste.

Conclusions

The present study was carried out to characterize the airflow and the heat transfer due to natural convection in an insulated box equipped with PCM by using PIV technique and temperature measurement. The influence of the PCM position on the flow pattern and temperature distribution was investigated. The study was conducted in a progressively more complex manner: empty, loaded with extruded polystyrene (low conductivity and almost no thermal inertia), and loaded with tylose (thermal properties close to those of food). The key findings are summarized as follows:

J o u r n a l P r e -p r o o f

Whatever the configuration, the highest observed air velocities were around 0.1 m•s -1 ; therefore, convection cannot be neglected compared to conduction in air (Peclet number > 100).

Numerical simulations with either CFD or simplified models should include free convection.

When the PCM is on the side wall, the flow pattern is simple to predict. Air flows downwards along the PCM surface and upwards along the side walls. In the empty case, the flow pattern can be approximated by a 2D recirculation cell, but the presence of the load leads to a 3D flow pattern. When the PCM is on the lid, after cooling down in contact with the PCM, a cold air stream detaches from the PCM surface and flows downwards. This cold air stream is unstable in the empty case and shows preferential pathways (symmetry breaking) in the loaded case. The flow pattern is less predictable.

In all cases, after cooling down near the PCM, the air temperature increases progressively along the trajectories until returning close to the PCM. The product surface temperatures are close to the temperature of the adjacent air. At steady state, conduction in the load has a minor effect on the flow pattern which can be approximated by replacing the real load with an obstacle of low conductivity.

From a practical point of view, it is recommended to leave a space between the PCM and the load (to promote free convection) and between the side walls and the load (to allow evacuation of heat from the ambient via conduction through the walls). The gap should be at least of the order of the boundary layer thicknesses: 2-3 cm. Further experiments without such gaps would be useful.

At thermal steady state, there was no significant difference in terms of maximum product temperature and heterogeneity between the PCM on the lid or at the side. The study showed the coldest and warmest zones in both cases, suggesting the best location for products that are sensitive to bacterial growth or chilling injury.

J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f J o u r n a l P r e -p r o o f • Effect of PCM positions on flow and temperature fields was investigated.

• The highest observed air velocities in the box were around 0.1 m•s -1 .

• PCM position exerted no significant effect on maximum product temperature.

• Gaps should be left between the product and the box walls or PCM.

J o u r n a l P r e -p r o o f

  Abbreviations

  a transient 2D-CFD model to describe the airflow and the heat transfer in an airfilled cavity with a PCM wall. The numerical results revealed flow patterns similar to ones observed in experiment during melting process. The temperature stratification was intensified after the liquid fraction of the PCM started to predominate (> 50%).[START_REF] Orozco | The effect of a segmented wall filled with Phase Change Material on heat transfer and airflow in a closed cavity[END_REF] further performed a numerical study with the same cavity configuration but this time the PCM wall was segmented into small volumes. The results showed that the segmentation of the PCM wall had no significant effect on the flow patterns and the temperature distributions in the air-filled cavity. Both flow topology and temperature stratification in the cavity with the segmented PCM J o u r n a l P r e -p r o o f wall were similar to those in the cavity with the non-segmented PCM wall. However, for the same requirement of thermal retention duration, the amount of the PCM could be reduced by dividing the PCM container into segments. These mentioned studies essentially dedicated to the use of PCM to enhance the energy efficiency of buildings. Only the temperature measurements were used for the model validation (no air velocity measurement). To our best knowledge, no experimental studies combining temperature and air velocity measurements inside an empty and loaded insulated box with PCM have been conducted. The present study attempted to bridge this research gap. The objective was to investigate experimentally the airflow and heat transfer due to natural convection in an insulated box with PCM. The influence of the PCM position and that of the presence of a load on flow patterns and temperature distribution in the insulated box were examined in the study. A non-intrusive technique, Particle Image Velocimetry (PIV), was implemented to characterize the air velocity field. The originality of this study lies in a) the implementation of PIV measurement for low air velocity and b) the analysis of both velocity and temperature fields in different configurations of an insulated box with PCM under a real use condition of food transport i.e., all walls of the box subjected to heat losses and the presence of loads in the box. The obtained knowledge

  ', the vertical velocity component (𝑣 𝑧 ) near the box wall tended to increase from bottom to mid-height (z = 120 mm) reaching almost 0.09 m•s -1 . Beyond this height (z > 120 mm), the velocity started to decease. A similar trend was observed near the PCM, but in the opposite manner. The vertical velocity component (|𝑣 𝑧 |) steadily increased from the top until roughly z = 90 mm reaching almost 0.08 m•s -1 , then the velocity began to

Fig. 6a shows

  Fig.6ashows the air temperature field on the middle plane (x = 250 mm) of the box with PCM

J

  o u r n a l P r e -p r o o f case of the XPS load. However, the easier experiments with the XPS load gave a good approximation of the flow pattern in the presence of load in the box.Due to the limitations of the PIV technique, velocity measurement was not possible in the (right) gap. A similar flow pattern could be expected due to symmetry. However, in fact, flow was not symmetrical above the load: it seemed that a stronger cold air stream flowed down in the right gap than in the left gap. Such dissymmetry was already observed in the empty case (PCM on the lid) where instability was induced by oscillations of downward flow. It seemed that in the loaded case, the flow 'chose' one or another preferential pathway (through the left or right gap).

Figs. 13b

 13b Figs. 13b and 13d (lateral plane, x =15 mm) where flow from the right to the left was observed

Fig. 14

 14 Fig. 14 presents the temperature field on the middle (x = 250 mm) and the lateral (x = 15 mm)

  XPS is a stack of four extruded polystyrene blocks (block dimensions = 200 mm × 400 mm × 50 mm); TYL is a stack of 16 Tylose packs (pack dimensions = 200 mm × 100 mm × 50 mm); the window number is referred to that in Fig.4.

Fig. 1

 1 Fig. 1 Insulated boxes: (a) commercially manufactured box for temperature measurement

Fig. 2

 2 Fig. 2 Diagram showing the experimental setup for temperature measurement for PCM

Fig. 3

 3 Fig. 3 Diagram (a) and photograph (b) showing the PIV setup.

Fig. 4

 4 Fig. 4 Position of measured windows for the PIV measurement: (a) empty box/PCM on the

Fig. 5

 5 Fig. 5 Air velocity fields on (a) the middle (x = 250 mm) and (b) the lateral (x = 15 mm)

Fig. 6

 6 Fig. 6 (a) Air temperature field on the middle plane (x = 250 mm) of the box with PCM on

Fig. 7

 7 Fig. 7 Air velocity fields at (a) the middle (x = 250 mm) and (b) the lateral (x = 25 mm)

Fig. 8

 8 Fig. 8 Air temperature field on the middle plane (x = 250 mm) of the box with PCM on the

Fig.

  Fig. Air velocity fields at (a and c) the middle (x = 250 mm) and (b and d) the lateral (x

Fig.

  Fig. Illustration of three-dimentional flow in a box with PCM on the side wall. Numbers

  J o u r n a l P r e -p r o o f

Fig. 12

 12 Fig. 12 Air temperature field on (a) the middle (x = 250 mm) and (b) the lateral (x = 15

Fig. 13

 13 Fig. 13 Air velocity fields on (a and c) the middle (x = 250 mm) and (b and d) the lateral (x

Fig. 14

 14 Fig. 14 Air temperature field on (a) the middle (x = 250 mm) and (b) the lateral (x = 15

  Paquette, J.-C., Mercier, S., Marcos, B., & Morasse, S. (2017). Modeling the thermal performance of a multilayer box for the transportation of perishable food. Food and Bioproducts Processing 105, 77-85. Robertson, J., Franzel, L., & Maire, D. (2017). Innovations in cold chain equipment for immunization supply chains. Vaccine 35(17), 2252-2259.Saury, D., Rouger, N., Djanna, F., & Penot, F. (2011). Natural convection in an air-filled cavity:

	Experimental results at large Rayleigh numbers. International Communications in Heat
	and Mass Transfer 38(6), 679-687.
	Singh, S., Burgess, G., & Singh, J. (2008). Performance comparison of thermal insulated
	packaging boxes, bags and refrigerants for single-parcel shipments. Packaging
	Technology and Science: An International Journal 21(1), 25-35.
	Wieneke, B. (2015). PIV uncertainty quantification from correlation statistics. Measurement
	Science and Technology 26(7), 074002.
	Xiaofeng, X., & Xuelai, Z. (2021). Simulation and experimental investigation of a multi-
	temperature insulation box with phase change materials for cold storage. Journal of Food
	Engineering 292, 110286.
	Zhao, X., Xia, M., Wei, X., Xu, C., Luo, Z., & Mao, L. (2019). Consolidated cold and modified
	atmosphere package system for fresh strawberry supply chains. LWT 109, 207-215.
	Zhao, Y., Zhang, X., & Xu, X. (2020). Application and research progress of cold storage
	technology in cold chain transportation and distribution. Journal of Thermal Analysis and
	Calorimetry 139(2), 1419-1434.

Table 1

 1 Thermophysical properties of materials used in the study

	Material	Density [kg•m -3 ]	Specific heat [J•kg -1 •°C -1 ]	Thermal [W•m -1 •K -1 ] conductivity	Reference
	Extruded polystyrene	35	1210	0.029	Cengel and Ghajar (2020)
	Polypropylene	910	1925	0.120	Cengel and Ghajar (2020)
	Tylose	1070	3372	0.510	Icier and Ilicali (2005)
	Water (liquid)	1000	4217	0.561	Cengel and Ghajar (2020)
	Water (solid)	920	2040	1.880	Cengel and Ghajar (2020)

Table 2

 2 Experimental conditions for thermal (temperature measurement) and momentum (air velocity measurement) studies.

Table 3

 3 Thermophysical properties* of air used for the 𝑅𝑎 estimation. calculated at the average air temperature (𝑇 𝑎 ) from the correlations proposed by[START_REF] Mcquillan | Properties of dry air at one atmosphere, Microelectronics Heat Transfer Lab[END_REF]

	Parameter	Unit	𝑇 𝑎 = 6.2°C	Value	𝑇 𝑎 = 7.7°C
	𝜌	kg•m -3	1.264		1.258
	𝜐	m 2 •s -1	1.387×10 -5		1.400×10 -5
	𝑐 𝑝	J•kg -1 •°C -1	1006		1006
	𝜆	W•m -1 •K -1	0.0245		0.0247
	𝛽	K -1	0.0036		0.0036

*

Acknowledgements

The authors gratefully acknowledge King Mongkut's Institute of Technology Ladkrabang, Thailand (contract no. KREF156402), the French Embassy in Thailand, and the National Research Institute for Agriculture, Food and Environment, France, for their financial support.

The first author, T. Leungtongkum, would also like to thank the Office of the Civil Service Commission of Thailand and Chulalongkorn University, Thailand, for the awarding of his PhD scholarship. The authors thank to LaVision for PIV technical support.

Further studies are planned in order to compare these results with CFD simulations and to develop a simplified model that enables prediction of temperature evolution (at different locations) as a function of the box, the product and PCM properties, along with ambient temperature changes.

Conflict of Interest and Authorship Conformation Form

Please check the following as appropriate: