The connection between air flow regime and heat transfer in the boundary layer of the brake disc cooling unit

. The boundary layer of the air flow blowing the surfaces of the brake mechanism prevents heat transfer from heated surfaces. The obtained analytical dependences of the velocities` and temperatures` distribution in the boundary layer represent the field of the boundary layer as a multilayer structural model, the laws of change in the velocities and temperatures of the air flow blowing depend on various sub layers of the boundary layer. According to these analytical dependencies, a thermal model "ventilated brake disc - external condition" was developed. The thicknesses of the boundary layers of the air flow blowing on the various surfaces of the brake disc are determined by various processes. For the working surfaces of the brake disc, the determining criterion is the thickness of the diffusion layer, while for the surfaces of the ventilation apparatus, the criterion for estimating the thickness of the boundary layer is thermal processes. In future, the developed thermal model "ventilated brake disc – external condition" should be the basis for thermal calculations in designing disc-shoe type brake mechanisms.


Introduction
The paper [1] considers the magnitude of aerodynamic losses that can be caused by the brake discs of the railway transport. By analyzing the described measurement data and modeling, the dependences of aerodynamic losses on the rotation speed and linear speed of the rolling stock for radial and curvilinear channels, using a segmental ventilation apparatus and solid disks, were obtained. The paper [2] compares the heat dissipation performance of two interchangeable ventilated brake discs, solid hub and perforated hub versions. The perforated hub design has shown excellent convective heat dissipation, depending on rotational speed and crossflow angle, as well as disc temperature. The increase in the heat transfer coefficient ranged from 3.5% to 20%. The hub brake disc design is 8.5% lighter than solid hub design. The aim of the study [3] was to investigate the behavior of the airflow inside a ventilated brake disc to improve heat dissipation. The influence of disk geometry and rotation speed on the average flow, turbulence intensity in the channel and mass flow was determined and it was found that the aerodynamic characteristics of the flow depend on the geometry of the ventilation apparatus. In the framework of [4], using the CFD model, it was studied the effect of changing the geometry of the first row of cylindrical segments on the aerothermodynamics' properties of a brake disc with a ventilated apparatus. A validated CFD model showed that reducing the thickness of the first row of pins by 10% improved the air mass flow through the brake disc vent by 14% and the heat transfer rate by 6%. The authors of the study [5] developed a procedure for maximum dissipation of convective heat. Ultimately, the specific power dissipation as the product of the average convective heat transfer coefficient and the wetting area of the disk provided the only quantitative measure of the effectiveness of the disk convective cooling. The developed design showed an increase in convective heat release by more than 10% compared to the existing disk. The study [6] substantiates the influence of the exit angle of the ventilation apparatus ribs on the thermo aerodynamic characteristics of brake discs with curved channels using numerical models. It was found that increasing the outlet angle increases the circumferential pressure difference, which enhances the secondary flow perpendicular to the main flow, and therefore obviously enhances local heat transfer on the disc inner surface. The paper [7] compares the thermal characteristics of three types of ventilation apparatus with radial, inclined and curvilinear channels under braking conditions. It was established that a brake disc with curved channels has the smallest distributions of temperature and thermal stresses under braking conditions.
From the analysis of special literary sources, it can be concluded that the connection between the air flow regime and heat transfer from the heated surfaces of the brake mechanism to the external condition was previously considered as a homogeneous system. Nevertheless, studies [8,9] show that the boundary layer near different surfaces of the brake disc in the turbulent flow regime in Figure 1 has different properties and calculation methods. Accordingly, the turbulent boundary layer has three sub layers: first is a viscous sub layer, second is a transition field and third is a field of completely turbulent flow. The viscous sub layer is characterized by molecular transport processes. The transition field connects the viscous sub layer with the field of completely turbulent flow. This area can be considered a laminar sub layer, as it is great influence of the surface on the aerodynamic and thermal parameters. The field of completely turbulent flow characterizes the entire turbulent boundary layer.
The total shear stress in the turbulent boundary layer will be determined from the equation of motion written in polar coordinates (l, φ): where μ is a dynamic viscosity, Pa s; υl is the velocity in the longitudinal direction of the boundary layer, m/s; ρ is the density of the environment, kg/m3; ϕ is a universal constant.
To describe the total shear stress in the boundary layer and the stress in the turbulent boundary layer of the air flow, it is necessary to write the equation of motion in a dimensionless form: Under the boundary conditions for the viscous sub layer 0 ≤ lcosφ ≤ l0cosφ, 0≤τs≤τsmax, it follows that ς0=η0. Consider the transitional field of the turbulent boundary layer (τt>τl) (τl are shear stresses in the laminar sub layer). According to studies [10], the total tangential stress in the boundary layer will be constant and equal to the tangential frictional stress of the flow on the surface. Therefore, the lower equation of system (2) will have the following form: (3) where ς1 and η1 are the dimensionless values of the velocity and stagnation velocity for the air flow at the boundary of the laminar sub layer and the turbulent flow field. Based on the boundary conditions of the laminar sub layer 0≤lcosφ≤l1cosφ, τs>> τl, the velocity ratio will correspond to the ratio of the deceleration rate to the kinematic viscosity in the boundary layer ς1=η1.
As the laminar sub layer is an idealized model, the transitional sub layer can have the properties of turbulent and laminar sections of the flow. Therefore, it is advisable to use the power law of velocity distribution, which is valid for both smooth and rough surfaces. Figure 2 shows the distribution curves of the air flow velocities at п=10. The power law of velocity distribution is better approximated than the law of velocity distribution for a laminar sublayer. For comparison, the power law of the distribution of air flow velocities for smooth and rough surfaces is shown in Figure 3.
It is similar to curved velocity distributions in the boundary layer, and it is possible to represent the temperature distribution in semilogarithmic coordinates.
Using the Fourier Law for the transition sub layer of the air flow as the main law of heat transfer, we obtain the heat balance equation between the free flow and the surface of the brake mechanism. St = − ( ) . (5) where St is the Stanton criterion; ϑ = Тs -Tff is temperature change between the surface of the brake mechanism and free air flow, K; λ is the thermal conductivity of the environment boundary layer, W/(m K). It is similarly to the power law of velocity distribution (4), we write the power law of temperature distribution: where Pr is the Prandtle criterion;

St
-is the law of temperature distribution in the transition sub layer of the air flow near the surfaces of the braking mechanism; is the ratio of the deceleration rate in the boundary layer to the kinematic viscosity in the transition sub layer of the air flow near the surfaces of the braking mechanism; Сf is the friction coefficient of the air flow on the surface of the brake disc.
As it can be seen from dependence (6), the temperature distributions in the boundary layer will depend on the deceleration rate of the external medium flow in the boundary layer and the Prandtle criterion  Considering the third field of the turbulent boundary layer, it can be assumed that the heat flux in the field of the turbulent air flow will be constant and will be equal to the heat flux from the surface of the braking mechanism. It is written the heat flux in dimensionless form as: The law of temperature distribution in the turbulent boundary layer will be the law of velocity distribution, consist of two zones in Figure 4.
In the first field, namely, in the viscous sub layer (ηt≤5), the temperature distribution law will change linearly with respect to the sub layer thickness: In the field of the transition sub layer of the air flow (5≤ηt≤10 4 ), the temperature distribution law will have the following form: In the third field of the turbulent boundary layer, the distribution law is presented in the equation form (7). This is possible if ϕ of the Prandtle criterion is constant. If a similar relationship is applied to the working surfaces of the brake mechanism, then, in addition to the boundary and thermal turbulent layers, it is necessary to take into account the diffusion turbulent boundary layer.
is the law of the elements` distribution of a multicomponent suspension in the turbulent boundary layer of the air flow near the working surfaces of the brake mechanism.
After considering the laws of distribution of velocities, temperatures, and components of the air flow in a turbulent boundary layer, it is considered the laws of heat transfer from the heated surfaces of the brake mechanism.
It is used the general law of heat transfer for a turbulent boundary layer for an impermeable surface [11]: where W is a dimensionless coefficient; Ret is the Reynolds criterion in the thermal boundary layer. It is substituted the equation of thermal energy losses in the boundary layer of the air flow of the brake mechanism surface into the heat transfer law, and integrate over the boundary conditions of the relative parameter and obtain: (12) where L1 is the parameter counted from the boundary of the laminar sub layer and the turbulent boundary layer, m; = St 0 is the relative law of heat transfer of the air flow inside the boundary layer under the condition = idem. Local heat transfer for a turbulent boundary layer will be determined from the system of equations: where μs, μff are dynamic viscosities in the boundary layer of the air flow near the disk surface and at the free flow boundary, Pa s.
The temperature change in the turbulent boundary layer of the air flow will be determined by the formula: This dependence is necessary to determine the heat transfer from heated surfaces, provided that the turbulent boundary layer will be considered from the working and nonworking surfaces of the brake mechanism to the boundary with free flow (L= L1). .
The obtained law of heat transfer for a local section of the turbulent boundary layer makes it possible to calculate the heat transfer from the heated surfaces of the braking mechanism. When the relative parameter L is replaced in the boundary conditions by the parameter Lsi, it is possible to calculate the heat transfer from the i-th surface area in the turbulent boundary layer of the air flow.

Results and discussion
Analytical dependences (8), (9), obtained for the temperature distribution in the boundary layer of the blowing air flow, make it possible to develop a thermal model of a ventilated brake disc in Figure 5. According to the Reynolds analogy, the process of heat transfer in the boundary layer, as well as the process of aerodynamic resistance, depends on the longitudinal velocity of the air flow in the ventilation channel of the brake disc. With an increase in friction losses and aerodynamic resistance of the segments or ribs of the ventilation apparatus, heat transfer from the heated surfaces of the brake disc increases [12]. The theory of the boundary layer introduces its own corrections into the air flow around the surfaces of workers and the ventilation apparatus, that is turning the systems "working surfaces -external condition" and "ventilation apparatus -external condition" into a multilayer system. Due to the fact that the air flow around the working surfaces of the brake disc and entering the ventilation apparatus is in the turbulent flow mode, therefore, the boundary layer will also work in the turbulent flow mode. Based on this statement, the authors will develop thermal models of multilayer systems "working surfaces -external condition" and "ventilation apparatus -external condition".

Conclusion
The main criteria characterizing the processes occurring in the boundary layers of the surrounding air flow are the working surfaces of the friction elements, the thickness of the dynamic, thermal and diffusion layers. Each of these quantities is responsible for certain processes occurring in the boundary layer. The dynamic layer characterizes the thickness of air mass changes in the boundary layer. Thermal characterizes the heat exchange processes occurring in the boundary layer, the magnitude and direction of the heat flow. The diffusion layer determines the chemical reactions occurring in the boundary layer, depending on the mass fraction of compounds in it. All these criteria quantify the obstacle to air exchange and heat exchange processes near the working heated surfaces of the friction elements. Without understanding the air exchange processes within the boundary layer, it is impossible to develop a strategy for intensifying cooling from the friction units of the brake mechanisms. As a result, the thermal models of metal brake elements are a multilayer model of heat transfer depending on the thickness of its layers. Consequently, it is difficult to analytically and empirically determine the separation boundary between the viscous and transitional thermal boundary layers, it is advisable to combine them into a single boundary layer. For working surfaces, the boundary layer will be determined primarily by diffusion, which allows equating the boundary layer to diffusion (δb≈δd). Whereas on non-working surfaces the diffusion processes of mass transfer will be less than on the working surfaces of the brake mechanism, the boundary layer will be entirely determined by the thermal boundary layer (δb≈δt). The developed multilayer thermal model "ventilated brake disc -external condition" will be a thermal calculation of disc-shoe type brake mechanisms.