Thermophysical Properties Estimation in Annealing Process Using the Iterative Dynamic Programming Method and Gradient Method

Abstract

In annealing, steel coils should be heated and consequently cooled according to the technological prescription defined for the annealed type of steel. It is appropriate to develop the systems and methods for estimation of the steel coil inner temperature for that reason. The proposal for such a system of indirect measurement of inner temperature is described in this study. This system, in the form of the mathematical model, is developed based on the theory of heat transfer and needs thermophysical parameters as inputs. In many cases, the thermophysical properties are difficult to access or unknown for the specific composition of the material being processed. In this paper, two optimization methods were applied to estimate two thermophysical properties. The application of the iterative dynamic programming method is aimed to estimate optimal thermal diffusivity. The verification of this method was performed on 11 laboratory measurements. The algorithm of the gradient method was used for estimating thermal conductivity and was verified on seven operational measurements. Results show that the optimized values of thermophysical properties increased the accuracy of the steel coil inner temperature estimation in the locations nearer to the steal coil central axis.

Soft-Sensing in Batch Annealing Based on Finite Differential Method and Support Vector Regression

“…The temperature at nodes is unambiguously defined by the following indices: Then T i, j, k represents the temperature at the node that lies on the intersection of the i-th parallel line with the x-th coordinate and the j-th parallel line with y coordinate at time τ = k • ∆τ. For a small step grid (i.e., with low values of Δx, Δy, Δτ), the following is valid: and analogously in direction of the y-axis (10) By substituting equations (6), (9) and (10) into equation 3and after algebraic modification, the final model (11) can be obtained. (11) This equation is used to calculate temperatures at the nodal points of the coil at any time step k. On the basis of equation 11, the temperature at any node point in the next time step can be calculated (i.e., if the temperatures at the nodal points by which the point is surrounded in the previous time step are known).…”

“…The parameter i represents the time step of the simulation (i = 1, 2, ..., τ k ) and τ k is the total number of T meas samples. The core of the adaptation of selected thermophysical properties is the optimization algorithm, which is based on the principle of iterative dynamic programming (IDP) and the gradient method as discussed in the paper [9]. Several simulations were performed from experimental measurements on a laboratory coil (see Table 1) in the electric furnace.…”