Skip to main content
SpringerLink
Log in

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

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

A l,o :

Vortex strength (m2/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/m2)

\(p^{\prime }\) :

Disturbance pressure (N/m2)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/s2)

ω :

Temporal growth rate (l/s)

λ i :

Lagrange coefficient

References

  1. Jones WP, Sheen DH (1999) A probability density function method for modeling liquid fuel sprays. Flow Turbul Combust 63:379–394

    Article  Google Scholar 

  2. Fritsching U (2004) Spray simulation. Cambridge University Press, Cambridge

  3. Babinsky E, Sojka PE (2002) Modeling droplet size distributions. Progr Energy Combust Sci 28:303–329

    Article  Google Scholar 

  4. Lefebvre AH (1989) Atomization and sprays. Hemisphere Publishing, Washington

  5. Sellens RW, Brzustowski TA (1985) A prediction of drop-size distribution in a spray from first principles. At Spray Technol 1:89–102

    Google Scholar 

  6. Li X, Tankin RS (1987) Derivation of droplet size distribution in sprays by using information theory. Combust Sci Technol 60:345–357

    Google Scholar 

  7. Sellens RW (1989) Prediction of the drop size and velocity distribution in a spray based on the maximum entropy formalism. Part Part Syst Charact 6:17–27

    Article  Google Scholar 

  8. Ahmadi M, Sellens RW (1993) A simplified, maximum entropy based drop size distribution. At Sprays 3:291–310

    Google Scholar 

  9. Dumouchel C (2006) A new formulation of the maximum entropy formalism to model liquid spray drop-size distribution. Part Part Syst Charact 23:468–479

    Article  Google Scholar 

  10. Sirignano WA, Mehring C (2000) Comments on energy conservation in liquid-stream disintegration. In: Proceedings of ICLASS, Pasadena

  11. Ibrahim AA, Jog MA (2008) Nonlinear instability of an annular liquid sheet exposed to gas flow. Int J Multiphase Flow 34:647–664

    Article  Google Scholar 

  12. Chu CC, Chou SF, Lin H, Liann YH (2007) Theoretical analysis of heat and mass transfer in swirl atomizers. Heat Mass Transf 43:1213–1224

    Article  Google Scholar 

  13. Rayleigh L (1878) On the stability of jets. Proc Lond Math Soc 10:4–13

    Article  Google Scholar 

  14. Sirignano WA, Mehring C (2000) Review of theory of distortion and disintegration of liquid streams. Prog Energy Combust Sci 26:609–655

    Article  Google Scholar 

  15. Lin SP (2003) Breakup of liquid sheets and jets. Cambridge University Press, Cambridge

  16. Mitra SK, Li X (1999) A predictive model for droplet size distribution in sprays. At Sprays 9:29–50

    Google Scholar 

  17. Kim WT, Mitra SK, Li X (2003) A predictive model for the initial droplet size and velocity distributions in sprays and comparison with experiments. Part Part Syst Charact 20:135–149

    Article  Google Scholar 

  18. Chu CC, Chou SF, Lin H, Liann YH (2008) An experimental investigation of swirl atomizer sprays. Heat Mass Transf 45:11–22

    Google Scholar 

  19. Ommi F, Nekofar K, Movahednejad E (2009) Designing and experimental investigation of characteristics of a double-base swirl injector in a liquid rocket propellant engine. J Appl Sci Res 5(8):955–968

    Google Scholar 

  20. Liao Y, Jeng SM, Jog MA, Benjamin MA (2001) Advanced sub-model for airblast atomizers. J Prop Power 17:411–417

    Article  Google Scholar 

  21. Jeandel X, Dumouchel C (1999) Influence of the viscosity on the linear stability of an annular liquid sheet. Int J Heat Fluid Flow 20:499–506

    Article  Google Scholar 

  22. Gordillo JM, Saborid MP (2005) Aerodynamic effects in the break-up of liquid jets: on the first wind-induced break-up regime. J Fluid Mech 541:1–20

    Article  MATH  Google Scholar 

  23. Ommi F, Mahdavi SA, Movahednejad E, Hosseinalipour SM (2009) Estimation of the breakup length for the annular and the round liquid jet using linear instability analysis. ASME Fluids Engineering Summer Meeting, Colorado

  24. Dombrowski N, Johns WR (1963) The aerodynamic instability and disintegration of vicious liquid sheets. Chem Eng Sci 18:203–214

    Article  Google Scholar 

  25. Shen J, Li X (1996) Instability of an annular viscous liquid jet. Acta Mech 114:167–183

    Article  MATH  Google Scholar 

  26. Bruce CA (1976) Dependence of ink jet dynamics on fluid characteristics. IBM J Res Dev 1:258–270

    Article  Google Scholar 

  27. Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Urbana

    MATH  Google Scholar 

  28. Movahednejad E, Ommi F, Hosseinalipour SM (2010) Prediction of droplet size and velocity distribution in droplet formation region of liquid spray. Entropy 12:1484–1498

    Article  Google Scholar 

  29. Li X, Chin LP, Tankin RS, Jackson T et al (1991) Comparison between experiments and predictions based on maximum entropy for sprays from a pressure atomizer. Combust Flame 86:73–89

    Article  Google Scholar 

  30. White FM (1991) Viscous fluid flow, 2nd edn. McGrow Hill, New York

    Google Scholar 

  31. Sallam KA, Dai Z, Faeth GM (2002) Liquid breakup at the surface of turbulent round liquid jets in still gases. Int J Multiphase Flow 28:427–449

    Article  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Tarbiat Modares University, 14115-143, Tehran, Iran

    E. Movahednejad, F. Ommi & S. A. Mahdavi

  2. Iran University of Science and Technology, Tehran, Iran

    S. M. Hosseinalipour

  3. University of Alabama in Huntsville, Huntsville, AL, 35899, USA

    E. Movahednejad & C. P. Chen

Authors
  1. E. Movahednejad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Ommi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. M. Hosseinalipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. P. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. A. Mahdavi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to E. Movahednejad or F. Ommi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Movahednejad, E., Ommi, F., Hosseinalipour, S.M. et al. Application of maximum entropy method for droplet size distribution prediction using instability analysis of liquid sheet. Heat Mass Transfer 47, 1591–1600 (2011). https://doi.org/10.1007/s00231-011-0797-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-011-0797-5

Keywords

Search

Navigation