Dimensional Analysis applied to predict depth of penetration

The second portion of my thesis was to find an equation for predicting depth of penetration for a given welding parameter for a given material.

The formation of weld bead in EBW is controlled by many factors and to predict bead formation by rigorous mathematical analysis was not possible.While width of weld in EBW was not a significant parameter, depth of penetration is. Many research papers published at that time depended on fitting equations to the experimental data. These curves are applicable in limited field. To get a generalaised equation covering all metals and all welding parameter needed a different approach.

I had some experience in Dimensional Analysis and thought of applying this technique to predict depth of penetration.

The various factors that control depth of penetration are: beam current, beam voltage,beam diameter, welding speed and material properties such as thermal conductivity,melting point,specific heat,and latent heat of fusion. An advanced method of Dimensional Analysis was applied to solve the resulting equations. Three non-dimensional parameters were obtained. The first parameter related to the ratio of heat lost by conduction to the beam power, the second, the heat required to melt a given volume to the beam power and the last was the heat required to heat a given volume to the melting temperature to the latent heat of fusion. Plotting these parameters one against the other we got graphs which predicted the depth of penetration.The theoretical depth of penetration closely followed our Experimental results and the data obtained from other workers.These equations can used to predict depth of penetration in EBW for any metal at any given voltage,current and speed.

This was the first time any one had applied dimensional analysis to solve a welding problem.