An analytical model for laser drilling incorporating effects of exothermic reaction, pulse width and hole geometry G.K.L. Ng a, * , P.L. Crouse b , L. Li b a TWI Technology Centre (Yorkshire), P.O. Box 3314, Sheffield S13 9WZ, UK b Laser Processing Research Centre, School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK Received 21 June 2005 Available online 23 November 2005 Abstract An analytical model is presented which incorporates the effects of using O 2 as assist gas. The contribution of the enthalpy of oxidation used in the model was determined experimentally by capturing the ejected melt and measuring the volume percentage of oxidation. The formulation of recoil pressure used in the model takes into account hole diameter and depth, and the associated pressure variation. The model presented also considers pulse width which is shown to affect the drilling velocity. The model enables the prediction of the velocity of melt ejection, and the drilling rate, as well as the contributions of melt ejection and vapourisation to the overall drilling rate. The calculated drilling rates are in close agreement with the experimental results. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Laser drilling; Analytical model; Melt ejection; Drilling rate; Laser oxidation; Exothermic reaction 1. Introduction Laser drilling involves a number of physical processes, typically melting, vapourisation, heat transfer by radiation, convection and conduction, vapour and melt flow, droplet formation and condensation of the vapour, absorption and reflection of electromagnetic radiation. Numerous models related to laser drilling have emerged over the last 40 years [1–33]. Before the current availability of inexpensive com- puting power, the emphasis was on finding solutions of appropriately simplified models. Early work by various researchers was mostly analytical in nature and laser dril- ling was modelled as a one-dimensional heat conduction problem [1,3,6]. This was adequate for low extraction effi- ciencies and low repetition rates, which allowed drilling of only shallow craters. Nevertheless, most of the one- dimensional models matched experimental drilling rates reasonably well. The capabilities of modern lasers have increased dramatically, and holes with aspect ratios beyond 10 can be easily drilled. The facts that analytical solutions are not available for moving-boundary phase-change prob- lems, that actual geometries can be very complex, that the physical properties are invariably temperature-dependent, and that computer power has become very inexpensive have shifted current emphasis to numerical solutions. In recent years, there continues a growing number of one- dimensional models, with an ever-increasing development of higher-dimensional numerical models published in the open literature [8,9,12,30]. These models include both steady state and transient conditions and evaluate the effects of varying laser parameters on the temperature pro- file [21,22], removal rate and drilling speed [18,27–29,31] and the shape of the hole profile [11,12,20,28]. The model developed by Semak and Matsunawa [23] and later adapted to include the effects of using an O 2 assist gas for laser drilling by Low et al. [33], are both integral, steady-state, analytical models using mass and energy con- servation as basis. The main thrust of the Semak and 0017-9310/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2005.10.002 * Corresponding author. Tel.: +44 114 2699046; fax: +44 114 2699781. E-mail address: gary.ng@twi.co.uk (G.K.L. Ng). www.elsevier.com/locate/ijhmt International Journal of Heat and Mass Transfer 49 (2006) 1358–1374