Flow and Heat Transfer of Basalt Melt in the Feeder of the Smelter Furnace

The flow and heat transfer of the basalt melt in the boundary layer on a flat plate is considered. The conditions of formation of the layer and the intensity of heat transfer are determined. A self-similar analysis using the symmetry method was used. A system of ordinary differential equations in self-similar form is obtained. The fluid flow and heat transfer of molten basalt at a laminar steady-state flow in the feeder furnaces are numerically researched. The term “protective layer” on the interface “basalt melt-lining” is introduced. The dependences for the calculation of dimensionless shear stresses and the Nusselt number on the lining surface are obtained. The conditions of rational organization of the technological process of basalt melt feeding in the furnace feeder are formulated.


Introduction
It is known that the technology of production of high-performance thermal insulation based on staple basalt fiber has the potential for modernization [1]. In particular, due to the rational organization of hydrodynamic and heat transfer processes in the elements of technological equipment. According to current concepts, basalt melt is a multiphase heterogeneous system containing liquid, crystalline and gaseous phases. The structure and action of this system are largely determined by the temperature and pressure at which it is located [2] [3]. The peculiarity of the basalt melt is its ability to restore in time its complex crystal structure from the liquid state when found in a certain temperature range-be-tween the lower and upper limits of crystallization. In relation to the basalts of Ukrainian deposits [4], the critical temperature range-between the lower and upper crystallization limits-is between 900˚C and 1270˚C. The occurrence in the smelter furnace of conditions under which the basalt melt acquires a critical temperature leads to the process of secondary crystallization of the melt. At the point of contact between the melt and the furnace lining, secondary crystallization contributes to the formation of a protective layer that impedes the destruction of the latter-the advantage-guaranteeing the long-term operation of the lining. On the other hand, the occurrence of conditions under which it is possible to recrystallize the basalt melt at the place of forming coarse fibers is a problem that leads to a decrease in the quality of fibers, a violation of the technological process, a decrease in plant productivity and, as a consequence, the increased production price of the final product. Studying the flow and heat transfer of basalt melt on the flat surface gives qualitative and quantitative characteristics of the process, and opens up the possibility of developing technical proposals for process control.
The theme of fluid flow and heat transfer on a flat plate-both in the approximation of the boundary layer, analytically, and in its full formulation, numerically-is widely discussed in numerous publications and monographs [5] [6]. The peculiarity of the formulation under consideration is that the classical phenomenological approach is applied to a deliberately heterogeneous system, with a significant dependence of viscosity on temperature-basalt melt. As will

Mathematical Model
The Equations (1) : To solve the system of differential equations with partial derivatives (1)-(3), we convert them to ordinary differential equations. For this purpose, we use symmetry analysis (analysis of Lie groups) [8]. The algorithm for symmetry analysis for heat and mass transfer and hydrodynamics problems is given in [9].
The transformation of the system of Equations (1)-(3) into a system of ordinary differential equations is considered in the work [10]. The following system of ordinary differential equations in self-similar form was obtained The self-similar variable was defined by the expression The velocity u in self-similar variables are represented in the form The velocity v in self-similar variables are represented in the form Self-similar function for temperature is introduced as The dependence of the dynamic viscosity coefficient of the liquid on the temperature is described by the equation In the self-similar form equation (12) is presented as where µ ∞ is liquid viscosity outside the boundary layer.
The system of Equations (6), (7) was solved under the following boundary

Results
Based on the system of Equations (6), (7) with boundary conditions (14), (15), numerical simulation of fluid motion and heat transfer in the boundary layer over a flat surface in a wide range of parameters μ and Θ is performed. Numerical solutions were obtained using the 4th order Runge-Kutta method [11].
For the numerical solution of system (6), (7), we replace the variables and perform its transformation .
The system takes the form The system was closed by the following boundary conditions The results of the calculations are presented in the graphical form. The increased viscosity of the medium and its dependence on the temperature on the surface of the plate significantly affects the velocity distribution in the boundary layer. In Figure 2      Nu Re where The average heat flux density on the wall q ave is determined by integrating an expression for the local heat flux density (27) Figure 7 shows the dependence of the average heat flux density q ave on the temperature on the wall T w .
As it can be noted, the q ave dependence on the T w temperature is nonlinear.    The resulting nonlinearity is explained by the change in the heat transfer mechanism in the layer. Heat transfer changes from predominantly convective (molar), at a temperature on the surface of the wall close to the temperature of the undisturbed flow, to predominantly conductive (molecular) transfer, at surface temperatures below 1100˚C.
Its maximum value of 813 W/m 2 q ave reaches at a temperature on the surface T w = 1007˚C.
Of practical interest are the values of q ave corresponding to the temperature range on the wall surface T w = 1300˚C -1400˚C. They constitute, respectively, q ave = 532 -218 W/m 2 .

Conclusions
The concept of "protective layer" in relation to the sedentary area of the melt on the interface "basalt melt-lining" is formulated. The thermophysical conditions determining the formation and size of the protective layer on the interface "basalt melt-lining" are identified. The temperature range on the wall surface is 1300˚C -1400˚C. The corresponding range of heat flux densities is 532 -218 W/m 2 .
The dependences for the calculation of dimensionless shear stresses and the Nusselt number on the interface "basalt melt-lining" are obtained.