Kolbasin Sergei

The problem of identifying the coefficient of heat in the zone of secondary cooling machines continuous casting billets. Unsteady processes of internal heat describes parabolic nonlinear partial differential equations. Boundary conditions include radiation and convection heat transfer components and take into account the complex mechanism of heat from water-cooling air. We propose a method of solution of the problem with the method of least squares. There are analyzed and the results of calculations.

To reduce the calculations, give a simplified mathematical model in rectangular coordinates. Consider some station continuously moving steel ingot in the system of coordinates, tied to the construction of continuous casting machines.

The equation for a two-dimensional model of heat and mass transfer in a rectangle (0, l) x (0, m) is as follows:

where v (t) - speed environment, T (t, x, y) - temperature, c (T) - specific heat, p (T) - density, and lyambda (T) - thermal conductivity continuum.

Perform initial:

and boundary conditions:

here a (x) - coefficient of heat convection, sigma - given the rate of radiant heat, T - the temperature inside the ingot, T | y = l - the temperature on the surface of ingot, To.s. -- Ambient temperature.

To simplify the task in three parts of the boundary review rectangle raised the heat flow as zero. On the border, an appropriate surface cooled ingot, set the boundary conditions 3 - the first kind. In general, the heat flow in ZSC has two components: convection (Newton's law-Rihmana) and radiant (Stefan-Boltzmann Law).

Since the surface of ingot in ZSC is in the range of temperatures, in which a large proportion in the total belongs to radiant heat flow component, the boundary conditions (4) take into account both sorts of heat. You need to define the rate of heat loss a (x). As additional information were measurements of temperature on the surface of ingot.

Such tasks are called boundary inverse problem [3.4]. They are incorrect in the classical sense. Correctness in the classical sense (or even speak on Adamaru correctness) means the existence of a solution of the task, its uniqueness and sustainability (ie continuous dependence on the input data). In our case, not running third condition - a condition of sustainability.

The method of direct treatment. To get an idea of the degree of volatility solutions problem, use direct treatment. To do this from (4) express a (x)

The measurements of temperature on the surface of ingot known, so the task falls into two parts: the definition of temperature field inside the ingot and then the definition of the coefficient of heat a (x).

The first task is to challenge boundary with boundary conditions 1 - the first kind (Dirichlet conditions) and 2 - the first kind (conditions Neumann). It used finite differences. In a rectangle before us introduce finite-difference grid wq, p, uniform in each direction with the steps q, p = const, q = l / N, p = m / M, where N, M - the number of segments along the partitioning and spatial coordinates x y respectively (Figure 2).

As a result of the decision-course challenges, we can see the difference field temperatures before us plot. To calculate the derivative nodes in the grid imposed, it certainly replace the analogue-difference.

From (5) and (6) receive an expression for calculating a (x):

The coefficient of heat found in a manner suitable for little practical use. Summand corresponding radiant heat transfer increases the error until the fourth order. In addition, the formula is present numerical differentiation, which in itself is an incorrect challenge. Ultimately, the relatively small deviations (errors in measurements) correspond fairly high temperature deviations in the coefficient of heat. The results of calculations by direct treatment are presented in Fig. 4.

To apply the method of least squares take into account the following information.

The coefficient of a (x) has a special distribution along the surface of ingot. It is known that at the site, covering a torch jets, it could bring parabolic function, which is the maximum value at a point corresponding to coordinate nozzle jets, while the remaining stations - a constant. Because the jets in the same section before us, they give the same water-air torch, hence the rate of heat loss - the same parabola, move along the axis abscissa (Fig.3).

Here are all the sites under the jets to top coordinate such a way as to peak over the parabola was the beginning of coordinates. The value of h is determined polushirinoy capture plume jets. Consequently, we have to identify only two options - A and a book. As has been said, a c = const, but at sites subject to forced cooling, will seek a (x) in the form of:

Consider the first sites at which a (x) = a c = const. Let many knots xi, which we believe KT permanent, K. lot of other nodes, where KT is distributed according to a parabolic law denote V. From (7) get a formula for the residual heat flow at the border:

Denote

A necessary condition for the existence of extremum S (a):

Hence, we find a

Each node xi from the set in place in line yi point on the interval [-h, h] in a manner that | yi | equal distance from the coordinates to the nearest xi jets. From (7) and (8) receive residual

We find A, in which

Of the necessary conditions for the existence of extremum

A find

It should also be noted that in certain values with the help of MNCs and with a sufficient condition for the existence of running a minimum function S. Easy to verify that the second order partial derivatives S for each of these parameters strictly greater than zero.

Thus, we find some spline approximation distributed in the space of the coefficient of heat at the surface moving ingot, which gives us the minimum rms deviation between the surface temperature measured and calculated on the model solutions as a result of direct challenges.

Both the above methods were carried out numerical calculations. As thermophysical parameters for the models were chosen data process continuous casting steel (for stamps art 40), the width of 1 m slabs, slabs polutolschiny l = 0,1 m and speed ingot v = 1 m / min. These calculations are presented in Fig. 4. Here one can see that the decision obtained by direct treatment, is unsustainable and unsuitable for practical use. The second curve is spline approximation, which is the result of the decision of the same tasks of the method of least squares.

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