Atomisation of liquid sheets

If a liquid sheet issues from a nozzle, its later development is governed by the initial sheet velocity as well as physical properties of the gas and liquid. Fraser et al. (1963) identified three distinct modes of sheet disintegration namely: rim, perforated and wave disintegration regimes. In the rim mode, surface tension forces contract the free edge of the sheet into a thick rim which disintegrates further due to jet break-up. This mode is dominant when the viscosity and surface tension of the sheet are high and produces large droplets along with numerous satellite droplets. In the perforated mode, irregularly shaped holes appear on the liquid sheet. These holes grow rapidly in size until the liquid portions of adjacent holes coalesce and form a thick rim. This rim will then disintegrate into droplets of various sizes. Disintegration can also happen as a result of wave propagation on the liquid sheet. In this case, areas of the sheet proportional to half of the oscillation wavelength are detached from the sheet and reorganise themselves into spherical droplet due to surface tension forces.

27

According to the studies of York et al. (1953) and Dombrowski & Johns (1963) the main mechanism of sheet break-up is the generation of wave-induced instabilities at the interface of liquid and gas phases. As a result of fluctuations in gas and liquid local pressure and velocity, infinitesimal disturbances in the form of waves are generated on the liquid sheet surface. These waves may be either damped, in which case they die away, or, alternatively, may overcome the surface tension forces and grow further in amplitude. The most unstable disturbance has the largest growth rate and, hence, is responsible for sheet disintegration. Eventually sheet disintegration leads to separation of droplets from the liquid bulk. York et al. (1953) conducted a theoretical study of sheet disintegration considering a two-dimensional infinite sheet with a finite thickness. Using calculated local velocities around the disturbed sheet, it was observed that under certain conditions, the disturbance amplitude exponentially grows. It was then concluded that such disturbances are the most dominant ones causing sheet disintegration.

Squire (1953) performed an instability analysis of an inviscid moving liquid film through a stagnant medium. Wave induced instabilities on the liquid surface were considered to be responsible for sheet breakup. The wavelength that is responsible for sheet disintegration (so called optimum wavelength) is the one with maximum growth rate. For 𝑊𝑒𝑔 ≫ 1 the

wave number 𝐾𝑠, corresponding to the maximum growth rate is obtained by equation ‎2-12:

𝐾_𝑠 = 𝜌𝑔𝑉𝑟𝑒𝑙

2

2𝜎 ‎2-12

In which 𝑉𝑟𝑒𝑙 is the relative velocity between moving sheet and surrounding gas. Hence

optimum wavelength can be calculated using equation ‎2-13: 𝜆_𝑜𝑝𝑡 =2𝜋 𝐾_𝑠 ‎2-13 And 𝑊𝑒𝑔 = 𝜌𝑔𝑉𝑟𝑒𝑙2 ℎ 𝜎 ‎2-14

28

Where ℎ is the thickness of the sheet. The maximum growth rate can be worked out using the relation below:

𝜔𝑚𝑎𝑥 =

𝜌𝑔𝑉𝑟𝑒𝑙2

(𝜎𝜌𝑙ℎ)0.5

‎2-15

Fraser et al. (1963) extended Squire’s theory to predict the droplet size produced by a low viscosity fan spray sheet. The work assumes the existence of an intermediate stage of unstable ligament formation prior to droplet generation. The wave with maximum growth rate is assumed to detach from the front edge in a form of two cylindrical ligaments per wavelength. The wave number corresponding to the fastest growing wave is calculated from equation ‎2-12. The diameter of the cylindrical ligament, 𝑑𝑙𝑖𝑔 is:

𝑑_𝑙𝑖𝑔 = √4ℎ

𝐾_𝑠 ‎2-16

Taking advantage of Rayleigh’s analysis (Rayleigh, 1878) the diameter of the droplets generated by the collapse of the ligament can be worked out by using equation ‎2-11 where the jet diameter is replaced by the ligament diameter:

𝐷_𝑑 = 1.89 𝑑_𝑙𝑖𝑔 ‎2-17

Mechanism of sheet disintegration by impaction of liquid sheet by a high velocity air flow was also investigated by Fraser et al. (1963). In their study of flat liquid sheets formed by means of a spinning cup, the atomisation air was introduced through an annular gap located axially symmetrically to the cup. It was observed that wave-induced instabilities formed at the point of air-to-sheet impaction. The existence of an intermittent stage of unstable ligament formation prior to droplet generation was also confirmed.

The influence of the initial sheet thickness on spray characteristics was studied by Rizk and Lefebvre (1980) who used air blast atomisers that were capable of forming flat liquid sheets across the centreline of a two-dimensional air duct. The liquid sheets were exposed to high velocity air from both sides. It was concluded that high values of flow rates and liquid viscosity resulted in the formation of thicker film. It was further reported that thin liquid

29

films break down to droplets, following the relation of 𝑆𝑀𝐷 ∝ 𝑑𝑠𝑜0.4, where SMD is Sauter

mean diameter and subscript 𝑆𝑂, denotes spray orifice. This relationship was also confirmed by a separate study by (El-Shanawany & Lefebvre, 1980).

Rizk and Lefebvre (1984) used high-speed flash photography with time increments of 0.2 µs to further investigate sheet disintegration and subsequent droplet formation. A similar mechanism to that proposed by York et al. (1953) and Dombrowski & Johns (1963) was observed in experiments using water as the working fluid and air velocity of 55 m/s. This includes the formation of wave-like instabilities due to interaction of water and air, followed by the formation of unstable liquid ligaments and lastly the formation of droplets. It was also seen that the process of liquid sheet disintegration accelerates and ligaments form closer to the atomiser lip by increasing the air velocity to 91 m/s.

The breakup length of liquid sheets was studied by Arai and Hashimoto (1986), by injecting liquid sheets into co-flowing air. The break-up length is determined based on an average of many observations. It was observed that for a constant sheet thickness, the breakup length decreases by increasing relative velocity between liquid and air. The breakup length showed inverse relation with the liquid viscosity.

In a recent review, Ashgriz (2011) and co-workers provide a comprehensive picture regarding the mechanism of sheet breakup. This review includes the linear and nonlinear instability of a viscous and inviscid liquid sheet, showing the effect of the aerodynamic forces on the growth rate of the initially small perturbations. It appears that nothing new has emerged to significantly change the perception of sheet breakup mechanism and more recent works just developed much more fine details. Nevertheless the materials in section ‎2.3.1.3 are appropriate context for the current work.