Oscillatory bursting of gel fuel droplets in a reacting environment

Understanding the combustion behavior of gel fuel droplets is pivotal for enhancing burn rates, lowering ignition delay and improving the operational performance of next-generation propulsion systems. Vapor jetting in burning gel fuel droplets is a crucial process that enables an effective transport (convectively) of unreacted fuel from the droplet domain to the flame zone and accelerates the gas-phase mixing process. Here, first we show that the combusting ethanol gel droplets (organic gellant laden) exhibit a new oscillatory jetting mode due to aperiodic bursting of the droplet shell. Second, we show how the initial gellant loading rate (GLR) leads to a distinct shell formation which self-tunes temporally to burst the droplet at different frequencies. Particularly, a weak-flexible shell is formed at low GLR that undergoes successive rupture cascades occurring in same region of the droplet. This region weakens due to repeated ruptures and causes droplet bursting at progressively higher frequencies. Contrarily, high GLRs facilitate a strong-rigid shell formation where consecutive cascades occur at scattered locations across the droplet surface. This leads to droplet bursting at random frequencies. This method of modulating jetting frequency would enable an effective control of droplet trajectory and local fuel-oxidizer ratio in any gel-spray based energy formulation.

. Dynamic response of envelope diffusion flame (for a 3 wt.% ethanol gel droplet) during an oscillatory rupture cascade with 6 jetting cycles. This shows that cascade constitutes a series of aperiodically occurring jetting events that lead to both flame envelope distortion and possible localized extinction depending on the jetting intensity (jet speed =Δ Δ ⁄). For the image sequence (from top to bottom and left to right) the images are time instants: Cycle 1: t = 0ms, 1ms. Cycle2: 13, 14 ms.  during an oscillatory cascade with 6 jetting cycles. Physically, the distorted flame structure can be characterized as mild distorted or severely distorted based on the magnitude of , / . Fig. 2 indicates that despite irregular jetting disruptions, the overall oscillatory flame response resulting from a cascade is clearly reflected by harmonic oscillations in . Furthermore, the fact that rupture holes during successive cycles occur in the same region and overlap to a large extent is also evident at the flame scale, since jets during all cycles are observed to emanate along the same angular direction. There is negligible variation (within ~13°) in the angular position of jets.

Shell bursting Dynamics:
Stages exhibiting a sequential transition from bubble growth/formation to shell rupture and jetting is shown schematically in Fig. 3 and the critical length scales used in the analysis are shown marked. The droplet volume ( ) before and after the bubble formation is given by: Before bubble formation (Stage 1): 1 = 1 + 1 (1) After bubble formation (Stage 2) : 2 = + ( 1 − , ) + 2 (2) Here, , and are respectively the volumes of liquid fuel, organic gellant and the bubble while , is the quantity of liquid fuel that gasified to form the bubble. During the swelling period the evaporative mass flux at the droplet surface is zero. Therefore, fuel mass conservation at anytime instant during the bubble growth period can be expressed as: Further, as the organic gellant is non-volatile the gellant volume remains constant ( 1 = 2 ). From equations (1), (2) and (3) the change in droplet volume ∆ is then given by: Assuming that the droplet temperature is uniform and constant and equal to the boiling point of high volatility species (ethanol: T=78.5°C at 0.1 MPa), the density ratio ≪ 1 ( ≈ 1.6 Kg/m 3 and ≈ 725 Kg/m 3 ). Thus, the change in internal liquid volume is negligible compared to the change in vapour volume or in other words the change in droplet volume is essentially due to bubble formation∆ ≈ . Based on this analysis, as the droplet swells the gellant mass conservation demands that: where, 1 and 1 are respectively the droplet radius and shell thickness initially (prior to bubble formation) while 2 and 2 are the corresponding values after swelling.
Next, as the droplet expands, the gellant shell thins downs further and ruptures when the tensile stress applied to the spherical shell reaches the rupture stress (~yield point of the material). The critical hoop (or circumferential) stress at rupture (Stage 3: Fig. 3) is given by: From the experimental data, droplet radius at first rupture is ~ (1mm). As reported by He et. al [28] the initial shell thickness for millimeter sized droplets is 0~ (5 µm). Given that at rupture 0 ⁄~ 1.25 and using Eq. (5) the shell thickness at rupture is of the order of ~ O (3 µm). Assuming a quasi-steady balance between the pressure difference across the shell and tension in the shell at rupture ∆ can be estimated using the expression: Using the representative values from present experiments and as reported by Cho et. al [33] the jet speed following rupture ~ O (1 m/s), vapour density ~ 1.6 Kg/m 3 . The pressure difference across shell is calculated to be ∆~ O (1 Pa). Through this order of magnitude analysis and using Eq. (6) the tensile stress at rupture is ~ (1 kPa). This is very close to the magnitude of 2 kPa as reported by He et. al [28].
where, ( ), ( ) and ( ) are the temporally varying area of the jet hole, the jet speed and the bubble radius respectively. Fig. 1 of the manuscript shows that in a single jetting cycle the jet hole area expands and then contracts till its complete recovery. Consequently, this would cause a variation in fuel mass flux. However, for a first order analysis we assume that the jet hole area and the jet speed remain constant during the active jetting period.
Using Eq. (7) = √2( − 0 )/ and given the initial condition that at rupture onset (time t = 0) the bubble is at its maximum radius , , eq. (8) above can be used to estimate the variation in bubble radius during the jetting time period: ( ) = ( , 3 − 3 4 √ 2( − 0 ) . ) 1 3 Finally, note that the temporal variation of bubble radius is intrinsically a function of rheophysical properties of the shell as the pressure difference is related to shell thickness and tensile stress at rupture through eq. (6).