An approximate method for computing the temperature distribution over material thickness during hot flat rolling

An analytic solution for computing the temperature distribution over material thickness during hot flat rolling is so complex that finding the exact answer takes too much time. We approximate the analytic solution to cut significantly the run time required in calculating the temperature distribution. The proposed approach finds a finite number of eigenvalues instead of an infinite number of eigenvalues via minimizing the natural metric of definite real-valued functions. The initial condition of the cooling conditions altering sequentially is approximated using a sixth-order polynomial function so that we can perform the integral inherent in the analytic solution without numerical analysis.To substantiate the usefulness of the proposed method, we applied it to the hot rolling process where cooling conditions such as convection and radiation are activated repetitively. The run time is reduced drastically to less than 3% of that required by the analytic solution. The accuracy of the proposed method is as good as those by the finite difference method and analytic method.