Recursive solution of an inverse heat transfer problem in rapid thermal processing systems

Rapid thermal processing (RTP) is a new technique for performing various wafer fabrication operations such as annealing, oxidation and chemical vapor deposition in a single chamber. The success of the RTP depends on the precise control of wafer temperature by adjusting wall heat flux. In the present investigation, an efficient recursive method is developed to solve the inverse heat transfer problem of estimating the wall heat flux on the wafer from the measurements of wafer temperature. The present recursive method is based on the Kalman filtering technique and the Karhunen–Loève Galerkin procedure. Although the direct implementation of the Kalman filter on heat conduction equation is never feasible due to the tremendous requirement of computer time and memory, a practical method of recursive estimation is devised in the present investigation by reducing the partial differential equation to a minimal set of ordinary differential equations by means of the Karhunen–Loève Galerkin procedure.