General temperature rise solution for a moving plane heat source problem in surface grinding

In the present study, the general solutions for a transient state as well as for the temperature rise formed everywhere in the workpiece due to a rectangular-shaped moving plane heat source arising at the grinding zone are derived. The present analysis starts from a point heat source solution by applying the method of separation of variables to a three-dimensional heat conduction problem. Because the workpiece moving velocity is quite small, the convective term related to the workpiece velocity is first excluded from the heat conduction equation. This workpiece velocity effect will be included in the model by slightly modifying the coordinate variable in the sliding direction shown in the solution of the point heat source. Therefore, the general three-dimensional solution of the stationary temperature rise can be expressed in an integral form as a function of the product value of the unknown initial condition and the particular solution of temperature rise. The unknown initial temperature rise in the solution can be replaced by the point heat source due to frictional that multiplying the product of the Dirac delta functions defined for three directions. Using the definition of the Dirac delta function, the temperature rise solution for a point heat source can thus be obtained. This solution is further extended to obtain the moving and uniform heat sources arising in a rectangular grinding zone. A comparison among the experimental result and the theoretical results predicted by the present model and Jaeger's model [Jaeger JC (1942) Proc Roy Soc, NSW 76:203-224.] show that the present model is quite accurate and is generally superior to Jaeger's model; it can be applied to predict the three-dimensional temperature rise distributions in the workpiece.