# Prediction of Droplet Size and Velocity Distribution in Droplet Formation Region of Liquid Spray

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Governing Equations

_{ij}, which is the probability of finding a droplet with volume V

_{i}and velocity u

_{j}. Hence, the mass, momentum and energy conservation equation can be restated as:

_{m}, S

_{mu}and S

_{e}are the source terms for mass, momentum and energy equations, respectively.

_{i}terms are considered in the energy equation. To obtain a more proper form of these equations, it is possible to normalize the equation with ${\dot{m}}_{o}$ ،${\dot{J}}_{O}$ ،${\dot{E}}_{O}$. Utilizing the definition of averaged velocity (U

_{0}) and droplet-averaged volume (V

_{m}) in the spray, mass, momentum and energy equations can be rewritten.

_{ij}which satisfy equations (1) to (4); therefore, the most appropriate distribution is the one in which Shannon entropy is maximized [13]:

_{i}is a collection of arbitrary Lagrange multipliers which must be evaluated for each particular solution. To obtain the coefficient λ

_{i}, equations (1) to (4) and (6) should be solved simultaneously. According to Li and Tankin model [8] it is also feasible to convert the analytical domain from volume and velocity of droplets to their diameter and velocity. Hence, the formulation can be written according to the probability of finding droplets which their diameters are between ${\overline{D}}_{n-1}$ and ${\overline{D}}_{n}$ and their velocities are between ${\overline{u}}_{m-1}$ and ${\overline{u}}_{m}$. Although as discussed by Dumouchel [14,15], such a procedure is inconsistent with the MEF mathematical manipulation and must be prohibited without taking the appropriate precaution to ensure entropy invariance.

_{i}) in PDF (f), it is necessary to solve the following normalized set of equations [8,12]:

_{i}in equations (7) which can be computed from solving the equation set (8) simultaneously. In this paper, to solve this set of equation, the Newton-Raphson method was used. At first, some initial value for the λ

_{0}, λ

_{1}, λ

_{2}, λ

_{3}was assumed. Then, using these values and the Newton-Raphson procedure, new value for λ

_{0}and then λ

_{1}, λ

_{2}, λ

_{3}was computed and this procedure continued until final answer was obtained. It is noted that functions in equation set (8) and their derivatives are integral functions. Therefore, double integrals function should be solved numerically for all iterations. Another important point is that integral functions and the terms in these integrals are exponential; hence, if the selection of an initial guessed of λ

_{i}turns out to be not enough close to the answer, the solution will not converge to the answer.

## 3. Modeling

**Table 1.**Spray characteristics [12].

Liquid Density | Surface Tension | Ambient Pressure | Gas Density | Flow rate | Liquid Average Velocity | D_{30} |
---|---|---|---|---|---|---|

998.2 (Kg/m^{3}) | 0.0736 (N/m) | 1 atm | 1.22 (Kg/m^{3}) | 2.809*10^{-3} (Kg/s) | 40.8 (m/s) | 1.37*10^{-5} (m) |

_{f}is the drag coefficient for flow over a flat liquid plate with contact area of A, which has different values for laminar and turbulent flows.

_{b}and h are breakup length and thickness of the liquid sheet and A

_{cross}shows it’s cross section area. Considering a laminar boundary layer flow passing on a flat plate, C

_{f}is computed; so the momentum source term can be evaluated as shown in Table 2. The Reynolds number is based on the jet velocity at the outlet of injector [12].

Weber Number | Reynolds Number | Drag Force | Momentum flow rate to Control volume | Non-dimensional Momentum source term |
---|---|---|---|---|

W_{e} | Re | F (N) | ${\dot{J}}_{O}$ | ${\overline{S}}_{mu}$ |

311 | 18200 | 1.953*10^{-3} | 0.1147 | -0.01702 |

**Figure 2.**Comparison of theoretical (solid line) and experimental (dashed line) [12] droplet size distribution.

Experiment | Calculation | ||||||
---|---|---|---|---|---|---|---|

U_{0} (m/s) | Ug (m/s) | Breakup Length (mm) | D_{30}(micron) | Re_{g} | Drag Coefficient (C_{f}) | S_{mu} | S_{m} |

4.6 | 56 | 4 | 45 | 11420 | 0.0124 | 0.047 | 0.0 |

**Figure 3.**Comparison between the theoretical (MEP) and experimental (Mitra [18]) droplet size distribution for PWC nozzle.

## 4. Modeling of Droplets Formation Region

_{r}is the relative velocity and C

_{D}is the drag coefficient evaluated from following relation [21].

**Figure 4.**Atomization regions of the spray from the hollow cone nozzle [22].

_{u}and n

_{d}show the number of intervals on velocity and size domains respectively. The modification of the momentum source term continues until the results converge to the unique answer, as seen in Figure 5. The final droplet’s size and velocity distributions belong to the droplets formed at downstream of jet breakup regime or final stage of primary breakup. Before this stage the instabilities on liquid jet and aerodynamic forces have the most important effect on the droplet formation process. After it, in the next stage (secondary breakup) turbulence plays a more significant role in droplet’s breakup. Without considering any turbulence effect (as in the present model), the drag force on droplets is only capable of modeling droplet formation to certain drop characteristics, where a steady condition established on droplet formation process, so the distribution of droplet characteristics converges to certain solution.

## 5. Results and Discussion

Parameter | Step 1 | Step 2 | Step 3 | Step 4 |
---|---|---|---|---|

λ_{1} | -1.4491 | -0.0393 | 0.0547 | 0.0625 |

λ_{2} | 62.5777 | 11.1397 | 9.3288 | 9.1889 |

λ_{3} | -128.727 | -22.5275 | -18.7530 | -18.4611 |

λ_{4} | 65.4985 | 11.7133 | 9.7962 | 9.6479 |

D_{30} | 0.929286 | 0.741426 | 0.72169 | 0.72 |

U_{m} | 0.285355 | 0.244805 | 0.2468 | 0.247 |

S_{mu}(droplets) | 0 | -0.01858 | -0.0036 | -0.00033 |

S_{mu}(drop + sheet) | -0.0172 | -0.0 3558 | -0.0391 | -0.0395 |

**Figure 7.**Droplet size distribution of PWC nozzle for two different positions in downstream direction [18].

**Figure 8.**Prediction of droplet size distribution of PWC nozzle using MEP model for two different positions in downstream direction.

## 6. Conclusions

_{i}and by using a wrong initial value, and the solution diverged immediately.

## Acknowledgements

## List of notations

$\dot{n}$ | Total number of droplets being produced per unit time |

S_{m} | dimensionless mass source term |

S_{mu} | dimensionless momentum source |

S_{e} | Energy source term |

${\dot{m}}_{o}$ | mass flow rate |

${\dot{J}}_{O}$ | Momentum flow rate |

${\dot{E}}_{O}$ | Energy flow rate |

λ_{i} | Lagrange coefficient |

C_{D} | Droplets drag coefficient |

C_{f} | Drag coefficient over the liquid sheet |

D_{30} | Mass mean diameter |

D_{i} | diameter of i_{th} droplet |

f | probability density function |

H | Shape factor for velocity profile |

K | Boltzmann constant |

N | normalized cumulative droplet number |

p_{i} | probability of occurrence of state i |

u | Droplet velocity |

U_{0} | Mean velocity of jet in nozzle outlet |

U_{m} | Droplets mean velocity |

V_{i} | Volume of i_{th} droplet |

V_{m} | Mean volume of droplet |

We | Weber number |

## References and Notes

- Jones, W.P.; Sheen, D.H. A probability density function method for modelling liquid fuel sprays. Flow Turbul. Combust.
**1999**, 63, 379–394. [Google Scholar] [CrossRef] - Fritsching, U. Spray Simulation; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Chen, C.P.; Shang, H.M.; Jiang, Y. An efficient pressure-velocity coupling method for two-phase gas-droplet flows. Int. J. Numer. Method Fluids
**1992**, 15, 233–245. [Google Scholar] [CrossRef] - Babinsky, E.; Sojka, P.E. Modeling droplet size distributions. Prog. Energ. Combust. Sci.
**2002**, 28, 303–329. [Google Scholar] [CrossRef] - Lefebvre, A.H. Atomization and Sprays; Taylor & Francis: New York, NY, USA, 1989. [Google Scholar]
- Liu, H.F.; Gong, X.; Li, W.F.; Wang, F.C.; Yu, Z.H. Prediction of droplet size distribution in spray of prefilming air-blast atomizers. Chem. Eng. Sci.
**2006**, 61, 1741–1747. [Google Scholar] [CrossRef] - Sellens, R.W.; Brzustowski, T.A. A prediction of drop-size distribution in a spray from first principles. Atomization Spray Tech.
**1985**, 1, 89–102. [Google Scholar] - Li, X.; Tankin, R.S. Derivation of droplet size distribution in sprays by using information theory. Combust. Sci. Technol.
**1987**, 60, 345–357. [Google Scholar] - Ahmadi, M.; Sellens, R.W. A simplified maximum entropy based drop size distribution. Atomization Sprays
**1993**, 3, 292–310. [Google Scholar] [CrossRef] - Cousin, J.; Yoon, S.J.; Dumouchel, C. Coupling of classical linear theory and maximum entropy formalism for prediction of drop size distribution in sprays. Atomization Sprays
**1996**, 6, 601–622. [Google Scholar] [CrossRef] - Sellens, R.W; Brzustowski, T.A. A simplified prediction of droplet velocity distribution in a spray. Combust. Flame
**1986**, 65, 273–279. [Google Scholar] [CrossRef] - Li, X.; Chin, L.P.; Tankin, R.S.; Jackson, T.; Stutrud, J.; Switzer, G. Comparison between experiments and predictions based on maximum entropy for sprays from a pressure atomizer. Combust. Flame
**1991**, 86, 73–89. [Google Scholar] [CrossRef] - Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Dumouchel, C. The maximum entropy formalism and the prediction of liquid spray drop-size distribution. Entropy
**2009**, 11, 713–747. [Google Scholar] [CrossRef] - Dumouchel, C. A new formulation of the maximum entropy formalism to model liquid spray drop-size distribution. Part. Part. Syst. Charact.
**2006**, 23, 468–479. [Google Scholar] [CrossRef] - Movahednejad, E. Prediction of Size and Velocity Distribution of Droplets in Spray by Maximum Entropy Principle and Using Wave Instability and Turbulence Analysis. PhD Thesis, Tarbiat Modares University, Tehran, Iran, 2010. [Google Scholar]
- White, F.M. Viscous Fluid Flow, 2nd ed.; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Mitra, S.K. Breakup Process of Plane Liquid Sheets and Prediction of Initial Droplet Size and Velocity Distributions. PhD Thesis, University of Waterloo, Waterloo, ON, Canada, 2001. [Google Scholar]
- Kim, W.T.; Mitra, S.K.; Li, X. A predictive model for the initial droplet size and velocity distributions in sprays and comparison with experiments. Part. Part. Syst. Charact.
**2003**, 20, 135–149. [Google Scholar] [CrossRef] - Sirignano, W.A.; Mehring, C. Comments on Energy Conservation in Liquid-Stream Disintegration. In Proceedings of ICLASS, Pasadena, CA, USA, July 2000.
- Clift, R.; Grace, J.R.; Weber, M.E. Bubbles, Drops and Particles; Academic Press: New York, NY, USA, 1978. [Google Scholar]
- Eberhart, C.J.; Lineberry, D.M.; Moser, M.D. Experimental Cold Flow Characterization of a Swirl Coaxial Injector Element. In Proceedings of 45th AIAA/ASME/SAE Joint Propulsion Conference, AIAA 2009-5140, Denver, CO, USA, August 2009.
- Ommi, F.; Movahednejad, E.; Hosseinalipour, S.M.; Chen, C.P. Prediction of Droplet Size and Velocity Distribution in Spray Using Maximum Entropy Method. In proceedings of the ASME Fluids Engineering Division, FEDSM2009-78535, Vail, CO, USA, August 2009.

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Movahednejad, E.; Ommi, F.; Hosseinalipour, S.M.
Prediction of Droplet Size and Velocity Distribution in Droplet Formation Region of Liquid Spray. *Entropy* **2010**, *12*, 1484-1498.
https://doi.org/10.3390/e12061484

**AMA Style**

Movahednejad E, Ommi F, Hosseinalipour SM.
Prediction of Droplet Size and Velocity Distribution in Droplet Formation Region of Liquid Spray. *Entropy*. 2010; 12(6):1484-1498.
https://doi.org/10.3390/e12061484

**Chicago/Turabian Style**

Movahednejad, Ehsan, Fathollah Ommi, and S. Mostafa Hosseinalipour.
2010. "Prediction of Droplet Size and Velocity Distribution in Droplet Formation Region of Liquid Spray" *Entropy* 12, no. 6: 1484-1498.
https://doi.org/10.3390/e12061484