# Application of maximum entropy method for droplet size distribution prediction using instability analysis of liquid sheet

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## Abstract

This paper describes the implementation of the instability analysis of wave growth on liquid jet surface, and maximum entropy principle (MEP) for prediction of droplet diameter distribution in primary breakup region. The early stage of the primary breakup, which contains the growth of wave on liquid–gas interface, is deterministic; whereas the droplet formation stage at the end of primary breakup is random and stochastic. The stage of droplet formation after the liquid bulk breakup can be modeled by statistical means based on the maximum entropy principle. The MEP provides a formulation that predicts the atomization process while satisfying constraint equations based on conservations of mass, momentum and energy. The deterministic aspect considers the instability of wave motion on jet surface before the liquid bulk breakup using the linear instability analysis, which provides information of the maximum growth rate and corresponding wavelength of instabilities in breakup zone. The two sub-models are coupled together using momentum source term and mean diameter of droplets. This model is also capable of considering drag force on droplets through gas–liquid interaction. The predicted results compared favorably with the experimentally measured droplet size distributions for hollow-cone sprays.

## Keywords

Droplet Size Nozzle Exit Droplet Formation Droplet Size Distribution Liquid Sheet## List of symbols

*A*_{l,o}Vortex strength (m

^{2}/s or l/s)*A*_{cross}Jet cross section area

*A*Droplet cross section area

*C*_{f}Drag coefficient over the liquid sheet

*C*_{D}Drag coefficient on a droplet

*d*_{nozz}Nozzle diameter

*d*_{L}Ligament diameter

*d*_{Drop}Droplet diameter

*D*_{i}Diameter of ith droplet

*D*_{30}Mass mean diameter

- \( \dot{E}_{0} \)
Energy flow rate get into the C.V

*g*Gas-to-liquid density ratio

*h*Ratio of inner and outer radius

*h*_{S}Liquid sheet thickness

- H
Shape factor

- \( \dot{J}_{0} \)
Momentum flow rate get into the C.V

*I*_{n}nth order modified Bessel function of first kind

*K*_{n}nth order modified Bessel function of second kind

*k*= 1/*λ*Axial wave number (l/m)

- k
Boltzmann constant

*L*_{b}Breakup length

- \( \dot{m}_{\text{o}} \)
Mass flow rate get into the C.V

*n*Circumferential wave number (Rad)

- \( \dot{n} \)
Total number of droplets being produced per unit time

- N
Normalized cumulative droplet number

*p*_{i}Probability of occurrence of state i

*P*Mean pressure (N/m

^{2})- \(p^{\prime }\)
Disturbance pressure (N/m

^{2})g*R*_{a}Inner diameter of liquid sheet (m)

*R*_{b}Outer diameter of liquid sheet (m)

*Re*\( {{\rho_{\rm l} U^{2} h} \mathord{\left/ {\vphantom {{\rho_{\rm l} U^{2} h} {\mu_{\rm l} }}} \right. \kern-\nulldelimiterspace} {\mu_{\rm l} }} \)

*S*_{m}Dimensionless mass source term

- \( S_{\text{mu}} \)
Dimensionless momentum source

*S*_{e}Energy source term

*U*Mean axial velocity (m/s)

*u*Disturbance axial velocity (m/s)

- \( \bar{u}_{\text{o}} \)
Mean velocity of jet in nozzle outlet

*U*_{m}Droplets mean velocity

*u*Droplet velocity

*V*Mean radial velocity (m/s)

*V*_{m}Mean volume of droplet

*V*_{i}Volume of ith droplet

*v*Disturbance radial velocity (m/s)

*W*Mean tangential velocity (m/s)

*We*Weber number (\( {{\rho_{\rm l} U^{2} R_{\text{b}} } \mathord{\left/ {\vphantom {{\rho_{l} U^{2} R_{\text{b}} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \))

*We*_{g}Weber number (\( {{\rho_{\text{g}} U^{2} h} \mathord{\left/ {\vphantom {{\rho_{\text{g}} U^{2} h} \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \))

*w*Disturbance tangential velocity (m/s)

*η*Displacement disturbance (m)

*σ*Surface tension (kg/s

^{2})*ω*Temporal growth rate (l/s)

*λ*_{i}Lagrange coefficient

## Notes

### Acknowledgments

Work presented in this paper was performed while the lead author (E. M.) was on leave at the University of Alabama in Huntsville. Supports from Tarbiat Modares University and Chemical and Material department in UAHuntsville are acknowledged.

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