Modeling of splat particle splashing data during thermal spraying with the Burr distribution

Document Type : Research Article


1 Department of Mathematics and Statistics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

2 Department of Mechanical Engineering, Payame Noor University (PNU), Tehran, Iran


Splashing of splat particles is one of the most important phenomena in industrial processes such as thermal spray coating. The data relative to the degree of splashing of splats sprayed with a normal angle are commonly characterized by the Weibull distribution function. In this present study, an effort has been made to show that the Burr distribution is better than the Weibull distribution for presenting the distribution of the degree of splashing. For this purpose, the Burr Type XII distribution and Weibull distribution are compared using different criteria. Furthermore, because of the great importance of statistical prediction of censored data in reducing costs and improving quality of the coating process, we consider different predictors of this data based on a progressively censored sample. For computing the prediction values we obtain the maximum likelihood estimates using the Expectation-Maximization (EM) algorithm. An important implication of the present study is that the Burr Type XII distribution more appropriately described the degree of splashing data. Therefore, the Burr Type XII can be used as an alternative distribution that adequately describes the splashing data and thereby predicts the censored data.


  • Inference about the splat particle splashing data which sprayed with a normal angle.
  • Perform the number of model selection tests to determine the appropriate probability model under complete and progressive censored sample.
  • Study of different methods for predicting the missing splat particle splashing data.


[1] S.D. Aziz, S. Chandra, Impact, recoil and splashing of molten metal droplets, Int. J. Heat Mass Tran. 43 (2000) 2841-2857.
[2] A.M. Worthington, The splash of a drop, Society for Promoting Christian Knowledge, London, 1895.
[3] A. Worthington, On the forms assumed by drops of liquids falling vertically on a horizontal plate, Proc. R. Soc. Lond., 25 (1876) 261-272.
[4] O.G. Engel,Waterdrop collisions with solid surfaces, J. Res. NBS, 5 (1955) 281-298.
[5] C. Stow, M. Hadfield, An experimental investigation of fluid flow resulting from the impact of a water drop with an unyielding dry surface, Proc. R. Soc. Lond. A 373 (1981) 419-441.
[6] G. Montavon, S. Sampath, C. Berndt, H. Herman, C. Coddet, Effects of the spray angle on splat morphology during thermal spraying, Surf. Coat. Technol. 91 (1997) 107-115.
[7] S. Thoroddsen, J. Sakakibara, Evolution of the fingering pattern of an impacting drop, Phys. Fluids 10 (1998) 1359.
[8] Y. Hardalupas, A. Taylor, J. Wilkins, Experimental investigation of sub-millimetre droplet impingement on to spherical surfaces, Int. J. Heat Fluid Fl. 20 (1999) 477-485.
[9] S. Asadi, M. Passandideh-Fard, M. Moghiman, Numerical and analytical model of the inclined impact of a droplet on a solid surface in a thermal spray coating process, Iran. J. Surf. Eng. 4 (2008) 1-14.
[10] J. Liu, H. Vu, S.S. Yoon, R.A. Jepsen, G. Aguilar, Splashing phenomena during liquid droplet impact, Atomization Spray. 20 (2010) 297-310.
[11] S. Asadi, Simulation of nanodroplet impact on a solid surface, Inter. J. Nano Dim. 3 (2012) 19-26.
[12] H. Li, S. Mei, L. Wang, Y. Gao, J. Liu, Splashing phenomena of room temperature liquid metal droplet striking on the pool of the same liquid under ambient air environment, Int. J. Heat Fluid Fl. 47 (2014) 1-8.
[13] G. Liang, Y. Guo, Y. Yang, N. Zhen, S. Shen, Spreading and splashing during a single drop impact on an inclined wetted surface, Acta Mech. 224 (2013) 2993-3004.
[14] G. Liang, Y. Yang, Y. Guo, N. Zhen, S. Shen, Rebound and spreading during a drop impact on wetted cylinders, Exp. Therm. Fluid Sci. 52 (2014) 97-103.
[15] H. Akaike, Information theory and an extension of the maximum likelihood principle, Proceeding of the Second International Symposium on Information Theory, Akademinai Kiado, Budapest, 1973, pp. 267- 281.
[16] G. Schwarz, Estimating the dimension of a model, Ann. Stat. 6 (1978) 461-464.
[17] B. Pradhan, D. Kundu, On progressively censored generalized exponential distribution, Test 18 (2009) 497-515.
[18] A. Asgharzadeh, R. Valiollahi, Point Prediction for the Proportional Hazards Family under Progressive Type-II Censoring,J. Iran. Stat. Soc. 9 (2010) 127-148.
[19] R.R.A. Awwad, M.Z. Raqab, I.M. Al-Mudahakha, Statistical inference based on progressively type II censored data from Weibull model, Commun. Stat. Simulat. 44 (2015) 2654-2670.
[20] H. Ng, Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples, IEEE T. Reliab. 54 (2005) 374-380.
[21] I.W. Burr, Cumulative frequency functions, Ann. Math. Stat. 13 (1942) 215-232.
[22] M.K. Rastogi, Y.M. Tripathi, Estimating the parameters of a Burr distribution under progressive type II censoring, Stat. Methodol. 9 (2012) 381-391.
[23] H. Panahi, A. Sayyareh, Parameter estimation and prediction of order statistics for the Burr Type XII distribution with Type II censoring, J. Appl. Stat. 41 (2014) 215-232.
[24] A. Abd-Elfattah, A.S. Hassan, S. Nassr, Estimation in step-stress partially accelerated life tests for the Burr type XII distribution using type I censoring, Stat. Methodol. 5 (2008) 502-514.
[25] B. Abbasi, S.Z. Hosseinifard, D.W. Coit, A neural network applied to estimate Burr XII distribution parameters, Reliab. Eng. Syst. Safe. 95 (2010) 647- 654.
[26] N.L.Johnson, S. Kotz, N.Balakrishnan,Continuous Univariate Distributions, Wiley Ser. Prob. Stat., vol. 1, 1994.
[27] R Development Core Team, R: A Language and Environment for Statistical Computing. Vienna, Austria : The R Foundation for Statistical Computing, (2011).
[28] M.V. Aarset, How to identify a bathtub hazard rate, IEEE T. Reliab. 36 (1987) 106-108.
[29] A. Asgharzadeh, R. Valiollahi, Point Prediction for the Proportional Hazards Family under Progressive Type-II Censoring, J. Iran. Stat. Soc. 9 (2010) 127- 148.
[30] M.Z. Raqab, H.N. Nagaraja, On some predictors of future order statistics, Metron, 53 (1995) 185-204.
[31] D. Kundu, B. Pradhan, Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring, Commun. Stat.-Theor. M. 38 (2009) 2030-2041.