Hybrid atomisation

4.3.2.3.1 Onset of different atomisation regimes

Determination of the onset of atomisation regime is based on spatial comparison of liquid layer void fraction, 𝛼𝑙𝑎𝑦 (equation ‎3-42 which used with subscript 𝑙𝑎𝑦) against critical void

fraction, 𝛼𝑐. Critical void fraction is strongly dependent on bubble nuclei topological

arrangement inside the liquid layer. In the absence of any visual evidence, an arbitrary assumption has to be made in order to determine droplet size scale. Therefore the simplest assumption would be a primitive cubic system (PCS) arrangement of nuclei inside the liquid layer. This assumption is also used in previous works (Sher & Elata, 1977; Senda et al., 1994;

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Kawano et al., 2006). Such nuclei geometric arrangement suggests that the critical void fraction occurs when growth is sufficient for adjacent bubbles to touch each other. In a PCS arrangement the corresponding void fraction is 𝛼𝑐 ≅ 0.52.

If 𝛼𝑙𝑎𝑦 < 𝛼𝑐 everywhere along the spray nozzle, aerodynamic atomisation based on LISA

model prevails at the spray orifice exit plane. However, if at any point along spray nozzle 𝛼𝑙𝑎𝑦 ≥ 𝛼𝑐, bubbles are assumed to touch each other, the liquid layer fragments and

liquid blobs (also termed as parent droplet) are formed. Here flash-boiling atomisation prevails inside the spray nozzle. The liquid blobs are then exposed to aerodynamic force of the vapour core and disintegrate by one of the assumed secondary atomisation mechanism, into number of fine child droplets.

4.3.2.3.2 Parent droplet formation by flashing

Once the void fraction criterion is satisfied, bubble bursting leads to liquid layer disintegration. The arrangement of the initial bubble nuclei within the liquid segment has a direct impact on the size of the product liquid droplets. In this model two possible arrangements are considered for the bubbles based on the liquid layer thickness namely two-dimensional (2D) and three-dimensional (3D) PCS arrangements. Homogeneous distribution of nuclei leads to identical average distance, 𝑙𝑎𝑣𝑒 between any two adjacent

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Figure ‎4.4 2D nuclei arrangement in liquid layer

Figure ‎4.5 3D nuclei arrangement in liquid layer

Parent droplet Unit cell Parent droplet Unit cell ℎ ℎ

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If this average distance is larger than the layer thickness, then there is no space for the nuclei to be placed on top of each other, along the thickness of the layer. Therefore, one bubble per unit cell is appropriate as shown in Figure ‎4.4. Thus a 2D arrangement is the topological nuclei arrangement and average distance between two bubbles can be calculated as follows:

𝑙𝑎𝑣𝑒 = √

𝐴_𝑙𝑎𝑦 𝑛𝑏

‎4-12

Where 𝐴𝑙𝑎𝑦 is the surface area of the liquid layer (= 𝜋𝐷𝑠𝑜× 𝐿𝑙𝑎𝑦) and 𝑛𝑏 is the absolute

number of bubbles in liquid layer calculated by equation ‎3-40. As a result, number of parent droplets (shown by circle in Figure ‎4.4), 𝑛𝑑 is twice the number of bubble nuclei (per bubble

nucleus one droplet is formed from the top segment surface and one from bottom segment surface).

On the other hand, if the average distance between two bubbles is smaller than the liquid layer thickness, a 3D arrangement of nuclei lattice would occur. In this case, there is a possibility for the nuclei to be situated on top of each other along the layer thickness (as shown in Figure ‎4.5). In this situation the average distance is estimated using equation ‎4-13:

𝑙_𝑎𝑣𝑒 = √𝑉𝑜𝑙𝑙𝑎𝑦 𝑛_𝑏

3

‎4-13

Where 𝑉𝑜𝑙𝑙𝑎𝑦 is the liquid layer volume and is calculated using the expression below:

𝑉𝑜𝑙𝑙𝑎𝑦= 𝜋𝐷𝑠𝑜× 𝐿𝑙𝑎𝑦× ℎ ‎4-14

Where 𝐿𝑙𝑎𝑦 is the length of liquid layer segment (= 𝑉. Δ𝑡 where 𝑉 is local flow velocity and

Δ𝑡 is time step over which liquid flows into spray orifice). For this case, the residual liquid surrounding each of the bubbles is used for one droplet formation. Thus, the number of droplets is equal to the number of bubbles in the liquid segment. At the point where condition 𝛼𝑙𝑎𝑦≥ 𝛼𝑐 is satisfied, the size of the residual droplets is calculated based on the

129 𝐷𝑝 = √ 6𝑚_𝑙(1 − 𝛼_𝑐)𝑣_𝑙,𝑐 𝑛_𝑑𝜋 3 ‎4-15

Where the 𝑚𝑙 is the mass of the liquid layer flowing into the spray orifice.

4.3.2.3.3 Child droplet formation

A parent droplet undergoes further disintegration due to being exposed to high velocity HFA gas phase, flowing inside spray nozzle. This interaction produces the final fine droplets. These final droplets are also termed child droplets. In accordance with Faeth et al. (1995), determination of the appropriate secondary atomisation regime depends on the prevailing droplet Weber number (𝑊𝑒𝑔 = 𝜌𝑔𝑉2𝐷𝑝/𝜎). Flow calculations suggest that the transitional

Weber number should be in the range of 20 < 𝑊𝑒𝑔 < 80. As noted by Faeth et al. (1995)

multimode secondary breakup occurs with in this range, resembling a combination of bag and sheet-thinning breakup modes (Lefebvre, 1988; Ashgriz, 2011; Guildenbecher et al., 2009). Bag and sheet-thinning modes are best represented by TAB and KH secondary atomisation models (Zeoli & Gu, 2006), respectively. Realistically, the transition between TAB and KH is a continuous function of Weber number. There is no guideline in the literature to establish a single transitional Weber number from TAB to KH. However, modelling is simplified by assuming that each model is activated over distinct regions of gas Weber number (Faeth et al., 1995). Therefore, we use the average Weber number of 50 (average of 20 and 80) as the transition criterion. For gas Weber numbers below 50, TAB model is activated. For Weber numbers above 50, KH model is activated.

4.3.2.3.3.1 TAB model

According to O’Rourke & Amsden (1987), the non-dimensionalised equation of motion of the oscillating droplet is:

𝑑2_𝑦́ 𝑑𝑡2 = 2 3 𝜌_𝑔 𝜌𝑙 𝑉2 𝑟𝑝2− 8 𝜌𝑙 𝜎 𝑟_𝑝3𝑦́ − 5𝜇𝑙 𝜌𝑙𝑟𝑝2 𝑑𝑦́ 𝑑𝑡 ‎4-16

Where 𝑦́ is non-dimensionalised distortion parameter defined as 𝑦 =́ 2𝑥́/𝑟𝑝 (𝑥́ is the

displacement of droplet equator from its equilibrium position). In equation ‎4-16, 𝑟𝑝 is the

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Assuming that the droplet-ambient relative velocity is constant and is equal to the local prevailing velocity in the spray orifice and then equation ‎4-16 can be analytically integrated and solved for each liquid blob which is produced by the procedure described in ‎4.3.2.3.2. The solution reads as follows:

𝑦́(𝑡) =𝑊𝑒𝑔 12 + 𝑒−𝑡/𝑡𝑑[(𝑦́(0) − 𝑊𝑒_𝑔 12 ) cos(𝜔𝑡) + ( 𝑑𝑦́/𝑑𝑡 𝜔 + 𝑦́(0) − 𝑊𝑒_𝑔/12 𝜔𝑡_𝑑 sin (𝜔𝑡))] ‎4-17 Where: 1 𝑡_𝑑= 5𝜇_𝑙 2𝜌_𝑙𝑟𝑝2 ‎4-18 And: 𝜔2 ₌ 8𝜎 𝜌_𝑙𝑟_𝑝3− ( 1 𝑡𝑑) 2 ‎4-19

If 𝑦́ > 1 at some point during the oscillation, then child droplet is generated from the blob and its size is calculated as follows (O’Rourke & Amsden, 1987)

𝐷_𝑑 = 𝐷𝑝

1 +8𝐾𝑦́_{20 +}2 𝜌𝑙𝑟𝑝3(𝑑𝑦 ́/𝑑𝑡)_𝜎 2(6𝐾 − 5_{120 )} ‎4-20

Where 𝐾 is equal to 10/3 (O’Rourke & Amsden, 1987).

4.3.2.3.3.2 KH model

As derived by Reitz (1987), the wave length  of the disturbance with maximum growth rate, which is responsible for pinching off child droplets from parent droplet, is calculated from equation ‎4-21:

Λ 𝐷_𝑝 =

9.02(1 + 0.45𝑂ℎ0.5_{)(1 + 0.4𝑇𝑎}0.7₎

(1 + 0.87𝑊𝑒_𝑔1.67₎0.6 ‎4-21

131 Ω (𝜌𝑙𝐷𝑝 3 𝜎 ) 0.5 = (0.34 + 0.38𝑊𝑒𝑔 1.5₎ (1 + 𝑂ℎ)(1 + 1.4𝑇𝑎0.6₎ ‎4-22

Where 𝐷𝑝 is the diameter of the parent droplet, 𝑂ℎ is the Ohnesorge number as defined by

equation ‎4-11 with the length scale of 𝐷𝑝 and eventually 𝑇𝑎 is the Taylor number defined by

the following expression:

𝑇𝑎 = 𝑂ℎ√𝑊𝑒𝑙 ‎4-23

In which 𝑊𝑒𝑙 is the Weber number defined based on liquid phase density. The resulting

child droplets are linearly proportional to the wavelength of instability:

𝐷𝑑 = 2𝐵0Λ ‎4-24

Where B0 is a model constant set equal to 0.61 based on the work of Reitz (1987).