Estimation of heat transfer coefficients and heat flux on the billet surface by an integrated approach
Introduction
The continuous casting is the main solidification process of molten steel. This is a process of heat transfer between the metal and cooling zones. The cooling methods have significant influence on the formation of internal and surface defects of billets. In order to guarantee defect-free products, a well-designed and operator spray cooling system is considered. Secondary cooling control model based on the heat transfer equation has been widely used. In 1993, Louhenkilpi et al. [1] introduced a real-time heat transfer model and considered the model accuracy. Hardin et al. [2], in 2003, a two-dimensional heat-transfer model is presented to realize the control of continuous steel slab caster. Petrus etc [3] established one-dimensional finite-difference model with a decentralized controller configuration in 2011. In 2013, Ke etc [4] proposed real-time slab quality diagnosis and analysis system which was based on heat transfer model. Hence accurate heat transfer model is crucial to continuous casting steel production. Heat transfer coefficients are essential to the accurate heat transfer model. In many research works, heat transfer coefficients were determined by the lab trials [5], [6], [7], [8], [9]. These results may include many unsatisfied factors. Firstly, the slab surface temperature, the cooling water volume, spray coverage area and nozzle type may influence the results. Secondly, a certain discrepancy for the heat transfer coefficients may be existence, owing to long cycle term and the existence of the deviation between experimental situation and actual casting process. This discrepancy can bring serious impact on the calculation accuracy. Thirdly, the necessary instruments in the laboratory may cost too much to finish measuring the heat transfer coefficient. Recently, some research works obtained heat transfer coefficients by using the surface temperature measurements. In 2006, Carlos etc [10] quantified heat transfer coefficients based on the solution of the inverse heat transfer problem by using industrial measured billet surface temperatures. In 2014, Yang etc [11] identified the physical parameters and heat transfer coefficients by chaos particle swarm optimization algorithm. At the same time, Wang etc [12] determined the heat transfer coefficients by solving the inverse heat transfer problem and validated effectiveness of this method by using the pin-shooting experiment. However, these research works did not consider the effect of surface temperature measurement error on the inverse results. Therefore, this paper presents an integrated approach of overcoming error disturbances which assembles the online temperature measurement and heat transfer model to identify heat transfer coefficients by solving the nonlinear inverse heat transfer problem.
In practice, nonlinear inverse heat transfer problem [13], [14] is ill-posed. That is to say, the surface temperature measurements with small error can create large disturbance in the results. The classical Levenberg–Marquardt (LM) [15], [16], [17], [18] algorithm is a valid method for solving this nonlinear inverse heat transfer problem. Recently, many research works [19], [20], [21] focus on improving this algorithm with non-exact line search. These works are seldom used in industry engineering because the non-exact line search needs to calculate the inequality continually until the condition is satisfied, which may increase the amount of calculation, especially for heat transfer model in continuous casting. In order to overcome this problem, this paper presents a modified Levenberg–Marquardt (MLM) algorithm with the exact linear search to estimate the heat transfer coefficients. The convergence of MLM algorithm is analyzed. By the actual measured data, the heat transfer coefficients are obtained and the results match with the measured data very well. At last, the measured data with error are used to estimate the heat transfer coefficients. Simulation experiment illustrates this integrated approach of overcoming error disturbances can overcome the error effectively.
Section snippets
Mathematical model of solidification heat transfer
In process of the continuous casting, the three-dimensional steady state heat transfer model is obtained from literature [22]:
Assuming that [23], [24], [25], [26]: (1) the discussion focuses on two-dimensional heat transfer model in literature [26]; (2) A pseudo-steady temperature field which is occurred during the undisturbed operational cycle of the casting device (Fig. 1) is considered; (3) since the release of heat is small compared to that in the cross
Identification of heat transfer coefficients based on the MLM algorithm
We define that is the calculated value of the billet surface temperature, is the experiment value of the billet surface temperature with error level δ. Because is dependent on the heat transfer coefficients h, so the process of solving the heat transfer coefficients h can be transformed into solving the following nonlinear operator:where is the time when the billet in continuous casting machine. However, solving the heat transfer
Difference calculation of solidification heat process model
The parameters and in Eq. (2) are nonlinear, thus the finite difference method is used to calculate this solidification heat process model. The Eq. (2) is approximated by explicit difference method and equivalent specific heat capacity is applied. According to literature [28], [29], we have in steady-state. So the heat transfer Eq. (2) can be written as by using explicit difference method in literature [25]:where i is the
Experimental procedure and results discussion
In order to illustrate the effectiveness of the integrated method, we use the industrial data to estimate the heat transfer coefficients and heat flux.
Conclusions
- (1)
An integrated and effective method of overcoming error disturbances is presented to estimate the heat transfer coefficients. This method combines the online temperature detection and heat transfer model, which is expected to describe heat transfer during the process of continuous casting precisely. The heat transfer coefficients can be used to calibrate the heat transfer model. This corrected heat transfer model can be applied to predict liquid pool length, the growth of shell and the billet
Conflict of interest
None declared.
Acknowledgments
This work was supported by National Natural Science Foundation of China (60974091, 61333006, 61473074), the Fundamental Research Funds for the Central Universities (N120708001).
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