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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.

It is known that the technology of production of high-performance thermal insulation based on staple basalt fiber has the potential for modernization [

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 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 [

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 [

∂ u ∂ x + ∂ v ∂ y = 0 , (1)

ρ ( u ∂ u ∂ x + v ∂ u ∂ y ) = ∂ ∂ y ( μ ∂ u ∂ y ) , (2)

ρ c p ( u ∂ T ∂ x + v ∂ T ∂ y ) = ∂ ∂ y ( λ ∂ T ∂ y ) . (3)

Equations (1)-(3) are analyzed under the following boundary conditions

y = 0 : u = v = 0 ; T = T w (4)

y → ∞ : u = U ∞ ; v = 0 ; T = T ∞ (5)

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) [

1 2 ⋅ f ( η ) ⋅ f ″ ( η ) + M ′ [ Θ ( η ) ] ⋅ Θ ′ ( η ) ⋅ f ″ ( η ) + M [ Θ ( η ) ] ⋅ f ( 3 ) ( η ) = 0 , (6)

1 2 ⋅ P r ∞ ⋅ f ( η ) ⋅ Θ ′ ( η ) + Θ ″ ( η ) = 0 . (7)

The self-similar variable was defined by the expression

η = y U ∞ ρ μ ∞ x . (8)

The velocity u in self-similar variables are represented in the form

u = U ∞ f ′ ( η ) . (9)

The velocity v in self-similar variables are represented in the form

v = 1 2 U ∞ ⋅ μ ∞ ρ ⋅ x ( f ′ ( η ) ⋅ η − f ( η ) ) . (10)

Self-similar function for temperature is introduced as

T = T w + ( T ∞ − T w ) ⋅ Θ ( η ) . (11)

The dependence of the dynamic viscosity coefficient of the liquid on the temperature is described by the equation

μ ( T ) = 3.83 × 10 − 3 ⋅ e 2.41 × 10 6 ( T 4 ) 7 . (12)

In the self-similar form equation (12) is presented as

μ ( Θ ( η ) ) = μ ∞ ⋅ M [ Θ ( η ) ] , (13)

where μ ∞ is liquid viscosity outside the boundary layer.

The system of Equations (6), (7) was solved under the following boundary conditions

η = 0 : f ( 0 ) = 0 ; f ′ ( 0 ) = 0 ; Θ ( 0 ) = 0 ; (14)

η → ∞ : f ′ ( ∞ ) = 1 , Θ ( ∞ ) = 1. (15)

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

f ( η ) = y 1 , f ′ ( η ) = d y 1 d η = y 2 , f ″ ( η ) = d y 2 d η = y 3 , Θ ( η ) = t 1 , Θ ′ ( η ) = d t 1 d η = t 2 . (16)

The system takes the form

d y 1 d η = y 2 , (17)

d y 2 d η = y 3 , (18)

d y 3 d η = − 1 2 M [ t 1 ] ⋅ y 1 ⋅ y 3 − M ′ [ t 1 ] M [ t 1 ] ⋅ t 2 ⋅ y 3 , (19)

d t 1 d η = t 2 , (20)

d t 2 d η = − 1 2 ⋅ P r ∞ ⋅ y 1 ⋅ t 2 . (21)

The system was closed by the following boundary conditions

η = 0 , f ( 0 ) = y 1 ( 0 ) = 0 , f ′ ( 0 ) = y 2 ( 0 ) = 0 , Θ ( 0 ) = t 1 ( 0 ) = 0 ; (22)

the quantities f ″ ( 0 ) = y 3 ( 0 ) and Θ ′ ( 0 ) = t 2 ( 0 ) were determined by the shooting method [

The grid spacing along η was taken equal to h = 1.0 e − 06 .

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 _{w} 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

To calculate the shear stress on the wall, the dependence of the second derivative of the self-similar velocity function on the wall f ″ ( η ) η = 0 on the wall temperature T_{w} is obtained (

The dimensionless shear stress on the wall is determined by the expression

1 2 c f = f ″ ( η ) η = 0 R e x . (23)

Tabulated values f ″ ( η ) η = 0 is presented in

f ″ ( η ) η = 0 | 3.285E-05 | 5.108E-04 | 4.095E-03 | 2.071E-02 | 0.0750 | 0.2123 | 0.3291 |
---|---|---|---|---|---|---|---|

T_{w}, ˚C | 900 | 1000 | 1100 | 1200 | 1300 | 1400 | 1449 |

As already noted, near a flat surface with a temperature T_{w} < 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 _{w}. 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 _{w} = 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 T_{w} = 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

N u x = ( d Θ d η ) η = 0 ⋅ ( R e x ) 1 / 2 , (24)

where ( d Θ d η ) η = 0 is obtained from the solution of the system (6), (7)

( d Θ d η ) η = 0 = 0.218 ⋅ e 2.269 × 10 − 3 T w . (25)

The average heat flux density on the wall q_{ave} is determined by integrating an expression for the local heat flux density

q x = N u x ⋅ ( λ ∞ x ) ⋅ ( T ∞ − T w ) , (26)

q a v e = 1 L ∫ 0 L q x d x . (27)

_{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}.

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Kremnev, V., Basok, B., Davydenko, B., Timoshchenko, A. and Timoshchenko, A. (2019) Flow and Heat Transfer of Basalt Melt in the Feeder of the Smelter Furnace. Journal of Applied Mathematics and Physics, 7, 2555-2563. https://doi.org/10.4236/jamp.2019.711174