Operation Study of Miniature Air-blast Atomizer under Very Low Liquid Pressures

Igor Gaissinski; Yeshayahou Levy; Daniel Kutikov; Valery Sherbaum; Vladimir Rovenski

doi:10.1515/tjj-2016-0073

Article

Operation Study of Miniature Air-blast Atomizer under Very Low Liquid Pressures

Igor Gaissinski , Yeshayahou Levy , Daniel Kutikov , Valery Sherbaum and Vladimir Rovenski

Published/Copyright: April 12, 2017

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From the journal International Journal of Turbo & Jet-Engines Volume 36 Issue 4

Abstract

Miniature air blast atomizers are intended for use in small jet engines. The typical miniature atomizer with rotating air outside the pressure swirl nozzle was investigated by theoretically and experimentally. The equations which describe the flow of rotating hollow liquid jet in a gaseous environment, where there is difference between outer and inner pressures of the liquid film were developed. The effects of surface tension, gravity forces, and inner-outer pressure difference on liquid film modes and its disintegration were considered. A set of the relevant equations was solved numerically. The initial conditions for the equations: pressure distribution inside and outside of the ‘onion’ as function of outer rotating air and liquid parameters, were obtained experimentally. The numerical results were verified by the experimental study. It was shown that exists two shapes of liquid film nearby nozzle exit – ‘onion’ and open film. Shape of the liquid film for the given nozzle pressure drop depends mainly on air pressure difference outside and inside the film. Two resulting spray modes are distinguished in droplet characteristics drastically. The results can be applied for estimation of the operational parameters of air-blast atomizers that operate with very low liquid pressure drop.

Keywords: rotational liquid film; stress tensor; numerical solution; experimental study; CFD study

PACS: 47.60Kz; 47.70Pq

Funding statement: The authors are grateful to ISRAEL SCIENCE FOUNDATION (ISF) for its support (ISF Grant No 661/11).

Appendix 1

In the case of ε≡Re−1=0,Eu=0,Fr→∞ the problem (5) – (7) leads to the following system of differential equations:

(43)dVτds=Vθ2VτRsinψ,dVθds=−VθRsinψ,dψds=cosψR⋅Vθ2/Vτ−We−1RVτ−We−1R,dRds=sinψ,dxds=cosψ

with the boundary conditions

(44)x|s=0=0,R|s=0=1,Vτ|s=0=1,Vθ|s=0=VΩ,ψ|s=0=ψ0

Substitution the fourth equation into the first and the second ones gives

VτdVτds=Vθ2RdRds,1VθdVθds=−1RdRdsdψds=cosψR⋅Vθ2/Vτ−We−1RVτ−We−1R,dRds=sinψ,dxds=cosψ

Hence

Vτ=−12C2/R4+C1,Vθ=C/Rdψds=cosψR⋅Vθ2/Vτ−We−1RVτ−We−1R,dRds=sinψ,dxds=cosψ

Using the boundary conditions, one can obtain the following:

(45)Vτ=12VΩ2(1−R−4)+1,Vθ=VΩ/Rdψds=cosψR⋅(VΩ2/R2)/[12VΩ2(1−R−4)+1]−We−1R12VΩ2(1−R−4)+1−We−1R, dRds=sinψ,dxds=cosψ

Proposition 1

In the case when R=const,ψ=const,Vτ=const, and Vθ=const is the solution to the problem (45) – (44) it is necessary and sufficient to have the following boundary values: ψ0=0, VΩ=We−1/2.

Proof: 1

Necessity: If R=const,ψ=const, then follow initial conditions (44) R≡1, ψ=0, and so the system (45) has the following simplified form:

R=const=1,ψ=const=0,Vτ=const=1,Vθ=const=VΩdψds=−cosψVΩ2−We−11−We−1=0

The result VΩ=We−1/2 proves the necessity.

2. Sufficiency: Let VΩ and ψ0 be equal VΩ=We−1/2,ψ0=0, and we assume that R≠const, Vτ≠const, Vθ≠const, and ψ≠const is the solution to the problem (45) – (44). Note that the constant functions R=1,ψ=0,Vτ=1, and Vθ=We−1/2 also satisfy the problem above as it was proven in the previous point (1). Hence, because of the uniqueness theorem for the Cauchy problem, only the single solution namely R=1,ψ=0,Vτ=1,Vθ=We−1/2 satisfies the problem (45) – (44). The last completes the proof.

Appendix 2. Representation of the measured air pressure distribution in the form of an analytic function

Due to the axisymmetric nature of swirl flow, the empirical results are expected to be axisymmetric. Visual inspection of the results revealed a distinctive pattern of a sub-pressure pit on the symmetry axis at some particular distance from the atomizer. Consequently, the following function was selected for further fitting procedure:

(46)F(r,x)=Pmin1−[1+exp{−a2(r−b2)}]−11+exp{−a1(r−b1)} ×1−[1+exp{−a4(x−b4)}]−11+exp{−a3(x−b3)}

The suggested function is formed based on an elementary component, named the “logistic function”:

(47)Sx=1+exp−ax−b−1

In graphical representation, logistic function has the pattern shown in Figure 32. Consequently, the suggested function F, composed of basic logistic-function elements, attains the following pattern:

– The leading coefficient Pmin corresponds to the lowest pressure value within the field – the “pit” pressure;
– The “pivot point” coefficients b_i are evaluated by means of visual examination of the results.

Figure 32:

The logistic function.

Since b_i numerical values are associated with the particular swirler used in the experiments, the “pivot point” coefficients are also defined by means of the swirler diameter, Dmean=Dout+Din/2. This means that the diameter dimension is believed to be the “pivot point” governing factor. In order to define the steepness coefficients a_i, a procedure of “least square” approximation was applied:

(48)∑i∑jFri,xj−Pri,xj2=minfa1,a2,a3,a4

where P represents the empirical pressure field.

The resulting analytical function has the form of function F (eq. (46)) with the coefficients placed in Table 8:

To simplify the function and reduce number of coefficients the coefficients ai were fitted separately for each flow rate and then averaged, Table 9. The leading coefficient Pmin, which represents the lowest pressure in the distribution and was presented by the following function

(49)Pminm˙a=−33.0m˙a2+6.97m˙a+0.364Pa

Table 9:

Fitted analytical function F, b_i coefficients (Dmean=10mm, Figure 8).

b₁	b₂	b₃	b₄
−0.4Dmean	0.4Dmean	−0.1Dmean	1.1Dmean

where m˙a air mass flow rate, g/s.

Finally, the following analytical formula for the pressure distribution, from (46) and Tables 9 and 10, is obtained:

(50)FR,xˉ=Pmin1−11+exp−a˜2λR−0.41+exp−a˜1λR+0.4⋅1−11+exp−a˜4λxˉ−1.11+exp−a˜3λxˉ+0.1

Table 10:

ai coefficients.

a1	a2	a3	a4
17.9/Dmean	28.6/Dmean	15.4/Dmean	4.4/Dmean

where λ=R0/Dmean,a˜i=aiDmean,R=r/R0,xˉ=x/R0 are our dimensionless variables.

Comparison of measured and analytical approximating function of negative pressure distribution (relatively to ambient) along radius for different air mass flow rates is shown in Figure 33.

Figure 33:

Comparison of measured and analytical approximating function of negative pressure distribution (relatively to ambient) along radius for different air mass flow rates downstream distances x₁= 2 mm, x₂= 5 mm, x₃= 8 mm, top-down.

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Received: 2016-11-21

Accepted: 2016-11-22

Published Online: 2017-04-12

Published in Print: 2019-11-18

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Articles in the same Issue

https://doi.org/10.1515/tjj-2016-0073

Keywords for this article

rotational liquid film; stress tensor; numerical solution; experimental study; CFD study