It is known that the technology of production of high-performance thermal insulation based on staple basalt fiber has the potential for modernization . 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  . 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—between the lower and upper limits of crystallization. In relation to the basalts of Ukrainian deposits , 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  . 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 be shown below, the viscosity of the basalt melt in the temperature range under consideration varies by 14 orders of magnitude. The purpose of the study was to quantify the processes of flow and heat transfer of the basalt melt and obtain the characteristics of resistance, heat transfer intensity, heat flux values, and sizes of the zone of secondary crystallization of the melt. The task was solved analytically and numerically.
2. Study of the Flow and Heat Transfer of Basalt Melt on a Flat Plate
2.1. Physical Model
The influence of basalt melt viscosity on the flow and heat transfer in the laminar boundary layer on a flat surface under the condition of the formation of a sedentary liquid layer near the wall is considered. The viscosity of the basalt melt is a nonlinear function that depends significantly on the temperature  (Figure 1).
2.2. Mathematical Model
The stationary problem of flow and heat transfer in the boundary layer of a liquid on a flat surface is considered. For this problem a system of equations of hydrodynamics and heat transfer in a boundary layer on a flat plate in a two-dimensional approximation  is considered
Figure 1. Dependence of the coefficient of dynamic viscosity on temperature for basalt from the Ivanova Dolina deposit.
Equations (1)-(3) are analyzed under the following boundary conditions
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) . The algorithm for symmetry analysis for heat and mass transfer and hydrodynamics problems is given in . The transformation of the system of Equations (1)-(3) into a system of ordinary differential equations is considered in the work . 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 is liquid viscosity outside the boundary layer.
The system of Equations (6), (7) was solved under the following boundary conditions
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 .
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 quantities and were determined by the shooting method , with the aim of satisfying conditions (15).
The grid spacing along η was taken equal to .
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, the comparison of the classical solution of Blasius for a self-similar velocity function with the solution of the system (6), (7) for different temperatures on the wall surface is displayed. The deformation of the velocity profiles within the temperature boundary layer at decrease of Tw is clearly visible. Curve 1 is the classic Blasius solution obtained for isothermal flow on the plate surface. Curves 2-4 are the solutions obtained for the nonisothermal flow of the melt on the plate at different surface temperatures—1300˚C, 1100˚C, 900˚C, respectively. The melt temperature and the Prandtl number in the free flow are T∞ = 1450˚C; Pr∞ = 5180. Accepted conventions are also relevant to Figure 3.
To calculate the shear stress on the wall, the dependence of the second derivative of the self-similar velocity function on the wall on the wall temperature Tw is obtained (Figure 3).
The dimensionless shear stress on the wall is determined by the expression
Tabulated values is presented in Table 1.
Figure 2. Distribution of velocity profiles in the boundary layer at: 1—Blasius solution; 2—Tw = 1300˚C; 3—Tw = 1100˚C; 4—Tw = 900˚C; T∞ = 1450˚C; Pr∞ = 5180.
Figure 3. Dependence of the second derivative of the self-similar function on the wall temperature Tw (˚C): T∞ = 1450˚C; Pr∞ = 5180.
Table 1. Dependence of the second derivative of the self-similar function on the wall temperature.
Figure 4 shows the dependence of the self-similar temperature function on the self-similar coordinate. The classical solution obtained by Levek is compared with the solution of the system (6), (7) for different temperatures on the wall surface. As the temperature on the wall decreases, the temperature distribution in the temperature boundary layer approaches the linear one, which is typical for non-movable layers.
As already noted, near a flat surface with a temperature Tw < T∞ basalt melt forms a sedentary layer that protects the surface from destruction. We formalize the concept of “sedentary layer”. We will term the melt layer near the surface, in which the velocity of the medium does not exceed 1% of the velocity in the undisturbed melt flow a protective layer. Using the dependence of the self-similar velocity function on the self-similar coordinate Figure 2, we calculate the value of the protective layer formed on a flat surface for different temperatures Tw. The calculated area is selected taking into account the actual geometric dimensions of the melting plants. The results of calculations are presented in graphical form in Figure 5. It should be noted that in the temperature range Tw = 900˚C - 1100˚C the thickness of the protective layer exceeds the size of the feeder in the melting plant. Based on this, it was concluded that it was not advisable to maintain such temperatures. On the other hand, in the temperature range Tw = 1300˚C - 1400˚C the size of the protective layer of the same order with the geometric dimensions of the feeder in the melting plant and is sufficient to fulfill its technological purpose. The specified temperature range is advisable to choose as the determining, when calculating the value of the heat flux density on the surface of the lining.
The dependence of the first derivative of the self-similar temperature function at η = 0 on the temperature on the wall surface, in order to calculate the intensity of heat exchange on the plate is shown in Figure 6. Traditionally, the heat transfer intensity on a flat surface is described by the dimensionless Nusselt number. In relation to the problem under consideration, the Nusselt number is determined by the expression
where is obtained from the solution of the system (6), (7)
The average heat flux density on the wall qave is determined by integrating an expression for the local heat flux density
Figure 7 shows the dependence of the average heat flux density qave on the temperature on the wall Tw.
As it can be noted, the qave dependence on the Tw temperature is nonlinear.
Figure 4. Distribution of temperature in the boundary layer at: 1—Leveque solution; 2—Tw = 1300˚C; 3—Tw = 1100˚C; 4—Tw = 900˚C; T∞ = 1450˚C; Pr∞ = 5180.
Figure 5. Dependence of the thickness of the protective layer δ (m) on the x (m) coordinate (for different temperatures on the wall Tw (˚C)): T∞ = 1450˚C; Pr∞ = 5180.
Figure 6. Dependence of the first derivative of the self-similar function on the temperature on the wall Tw (˚C): T∞ = 1450˚C; Pr∞ = 5180.
Figure 7. Dependence of the average heat flux density qave (W/m2) on the temperature on the wall Tw (˚C): T∞ = 1450˚C; Pr∞ = 5180.
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/m2 qave reaches at a temperature on the surface Tw = 1007˚C.
Of practical interest are the values of qave corresponding to the temperature range on the wall surface Tw = 1300˚C - 1400˚C. They constitute, respectively, qave = 532 - 218 W/m2.
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/m2.
The dependences for the calculation of dimensionless shear stresses and the Nusselt number on the interface “basalt melt-lining” are obtained.
We express our words of appreciation to the directorate and to the plant’s employees LLC CE “Chernivtsi Plant of Heat-Insulating Materials” (Chernivtsi, Ukraine) on the basis of which all experimental-industrial research has been conducted.
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