Real-time identification of parametric sloshing-induced heat and mass transfer in a horizontally oriented cylindrical tank

This paper experimentally investigates how vertical harmonic excitation induces parametric sloshing in a horizontally oriented cylindrical tank, using an Augmented-state Extended Kalman Filter to provide real-time identification of the resulting rapid increases in heat and mass transfer and the subsequent thermal destratification.

The Gist

Imagine you are traveling in a high-tech, futuristic airplane or a massive cargo ship powered by super-cold liquid fuels (like liquid hydrogen). To keep these fuels stable, they are stored in giant tanks. But there is a problem: if the vehicle vibrates or bounces up and down, the liquid inside doesn't just sit there—it starts to "slosh."

This paper investigates a specific, tricky kind of sloshing that happens when a tank is shaken vertically (up and down), and how that movement can cause a sudden, dangerous "thermodynamic meltdown" inside the tank.

Here is the breakdown of what the scientists discovered, using some everyday analogies.

1. The "Layer Cake" Problem (Thermal Stratification)

Normally, a tank of cold fuel is like a perfectly layered cake. The liquid at the bottom is very cold, and the vapor (gas) at the top is slightly warmer. Because they are layered, they stay that way. This "layering" is good—it keeps the pressure in the tank steady and predictable.

2. The "Blender Effect" (Parametric Sloshing)

When the tank is shaken up and down at just the right rhythm, something magical and scary happens. It’s not like pushing a swing; it’s more like vibrating a bowl of Jell-O.

The researchers found that once the shaking hits a certain "sweet spot" (called resonance), the liquid doesn't just ripple—it erupts. A massive jet of liquid shoots up from the center, hitting the ceiling of the tank. This is the "Blender Effect." It takes that neat, layered "cake" and instantly whips it into a messy, lukewarm smoothie.

3. The "Pressure Pop" (The Thermodynamic Consequence)

When the liquid and gas are suddenly mixed by this "blender," the temperature changes instantly. The warm gas hits the cold liquid and turns into liquid (condensation).

Think of it like opening a shaken soda bottle. Because the gas is rapidly turning into liquid, the pressure inside the tank suddenly "collapses" or drops. The researchers used a high-tech mathematical tool (called a Kalman Filter) to act like a "digital detective," looking at noisy sensor data to figure out exactly how much heat was being moved and how fast the pressure was changing in real-time.

4. The "Two-Mode Dance" (Mode Competition)

At certain levels, the liquid gets "confused." Instead of just doing one type of motion, it starts a weird dance. One moment it’s a central jet shooting upward (the symmetric mode), and the next, it’s a wave sliding side-to-side (the asymmetric mode). It’s like a dancer who can’t decide whether to spin or slide, switching back and forth rhythmically.

Why does this matter?

If you are designing a hydrogen-powered airplane, you can't just assume the fuel will stay calm. If the engine or the wind causes the tank to vibrate at that specific "sweet spot," the pressure could drop so fast that it messes up the fuel system.

In short: This paper provides the "instruction manual" for how vertical shaking turns a calm, layered tank into a chaotic, pressure-dropping blender, helping engineers build safer, more stable vehicles for the future of green energy.

Technical Summary

Technical Summary: Real-time identification of parametric sloshing-induced heat and mass transfer in a horizontally oriented cylindrical tank

1. Problem Statement

The research addresses a critical gap in the thermodynamic characterization of cryogenic fuel storage. While much is known about lateral sloshing in upright tanks, the behavior of vertically forced (parametric) sloshing in horizontally oriented cylindrical tanks—a geometry highly relevant for future aeronautical and ground transportation applications—remains largely uncharacterized.

The core problem is that vertical vibrations periodically modulate the effective gravity, triggering parametric resonance (Faraday-like instability). Under non-isothermal conditions, this sloshing disrupts the natural thermal stratification between the liquid and vapor phases. This disruption intensifies heat and mass transfer, leading to rapid pressure fluctuations (specifically pressure drops driven by condensation) that can compromise the stability of cryogenic storage systems.

2. Methodology

The authors employ a decoupled experimental and computational approach to isolate hydrodynamic effects from thermodynamic responses:

• Experimental Setup:
  • Isothermal Campaign: Conducted in a transparent PMMA tank to visualize free-surface dynamics using high-speed videography and Proper Orthogonal Decomposition (POD).
  • Non-Isothermal Campaign: Conducted in a stainless-steel tank using 3M™ Novec™ HFE-7000 as a surrogate for cryogenic fluids. The setup includes a pressurizing tank, surface heaters for thermal stratification, and an array of thermocouples and pressure transducers.
• Hydrodynamic Analysis: Natural frequencies were determined experimentally through free-decay tests. POD and Continuous Wavelet Transform (CWT) were used to identify dominant sloshing modes and their frequency evolution.
• Thermodynamic Modeling & Inference:
  • A lumped-parameter thermodynamic model was developed, partitioning the system into liquid, vapor, and wall control volumes.
  • To overcome the difficulty of measuring heat transfer coefficients directly, the authors implemented an Augmented-state Extended Kalman Filter (AEKF). This allows for the real-time, time-resolved inference of unknown Nusselt numbers ($Nu$) by assimilating noisy experimental data (pressure and temperature) into the dynamic model.

3. Key Contributions

• First Experimental Characterization: Provides the first experimental data on the relationship between vertical parametric sloshing and pressure drop in a 3D horizontal cylindrical geometry.
• Novel Inference Framework: Demonstrates the use of an AEKF to treat heat and mass transfer coefficients as "hidden" time-varying states, allowing for adaptive estimation during transient regimes.
• Scaling Framework: Establishes a comprehensive set of dimensionless groups (e.g., modified Fourier, Biot, and Jakob numbers) to relate laboratory results to full-scale cryogenic applications.
• Mode Interaction Discovery: Identifies complex subharmonic mode interactions (e.g., the competition between the $(2,0)$ and $(1,0)$ modes) in horizontal geometries.

4. Key Results

• Critical Threshold & Destratification: A clear forcing threshold exists. Below it, the fluid remains quiescent and stratified. Above it, parametric resonance drives a "middle-jet" that causes complete thermal destratification.
• Nusselt Number Jump: Upon the onset of sloshing, the inferred Nusselt numbers increase by several orders of magnitude, marking the transition from natural convection/conduction to intense forced convection.
• Pressure Dynamics: The pressure drop is primarily driven by condensation (accounting for $\sim 80\%$ of the pressure rate change), rather than simple ullage cooling. A characteristic "pressure burst" (due to transient evaporation) is observed immediately before the rapid pressure collapse.
• Wall Response: The thermal response of the tank wall is found to be governed by its own intrinsic diffusive timescale (the modified Fourier number $f_{Fo} \approx 1$), making it relatively independent of the specific forcing amplitude once mixing has begun.
• Mode Competition: At 50% fill levels, the dominant symmetric $(2,0)$ mode intermittently alternates with a planar $(1,0)$ mode, indicating non-linear energy transfer between modes.

5. Significance

This work provides a fundamental benchmark for the design of next-generation cryogenic storage systems (e.g., liquid hydrogen tanks for aircraft). By quantifying how vertical vibrations trigger rapid pressure changes, the study enables more accurate predictive modeling for vehicle stability and thermal management. The AEKF methodology offers a robust tool for real-time monitoring and state estimation in complex, multi-phase industrial processes.