**Abstract:** Taking the cylindrical roller bearing as the research object, the local heat transfer model of the bearing outer ring raceway is established, and the axial temperature distribution of the bearing outer ring is analyzed by the heat flow network method. The influence of different working conditions on the axial temperature distribution of the outer ring is analyzed by a numerical example. The results show that the temperature of the outer ring of the bearing increases with the increase of the speed and the load, and the surface temperature of the outer ring has little difference along the axial direction.

Most bearings work far below the rated speed and rated load, so there is no need to consider the problem of overheating. However, when the speed is close to the rated speed and the load is close to the rated load for a long time, the heat will be very large. If the heat cannot be dissipated in time, the temperature of the bearing will increase abnormally, causing the viscosity of the lubricant to decrease, the wear of the roller and the raceway will be accelerated, and even the material of the contact surface will be tempered and softened, resulting in premature bearing fatigue failure. death, the consequences are very serious. Therefore, it is necessary to consider the external cooling system to reduce the temperature of the bearing system and ensure the normal operation of the bearing within a reasonable temperature range.

Slotting the coolant between the bearing outer ring surface and the bearing seat is an effective cooling method, but the number, size, location and distribution of the grooves depends on the temperature distribution of the bearing outer ring surface. This example takes the cylindrical roller bearing as the research object, analyzes the temperature distribution of the outer ring surface under different working conditions, and provides a theoretical basis for the design of the external cooling system of the bearing.

**1 Calculation of bearing calorific value**

During the operation of the bearing, energy loss will be caused by the friction between the moving parts, which is manifested as a temperature rise in the bearing system. Therefore, the calculation of the calorific value of the bearing is the calculation of the friction torque first.

Bearing friction torque is a very complex problem, involving contact mechanics, tribology and other disciplines. Various factors interact and interfere with each other, which causes certain difficulties in the calculation of bearing calorific value. Literature [1] proposed that to accurately calculate the friction torque of the bearing, it is necessary to consider four causes of friction

where M is the total friction torque; Mrr is the rolling friction torque; Msl is the sliding friction torque; Mseal is the friction torque of the seal; Mdrag is the friction torque due to drag loss, eddy current and splash.

The method analyzes the root cause of friction and combines various factors to calculate the total friction torque.

In the formula: Dpw is the diameter of the pitch circle of the roller group, mm; Fr is the radial load, N; ν is the kinematic viscosity of the lubricant, mm2/s; n is the rotational speed, r/min; R1, S1 and S2 are the friction torque Fa is the axial load, N; μsl is the sliding friction factor; Ks1, β and Ks2 are constants depending on the bearing type and seal type, respectively; ds is the diameter of the bearing shoulder; VM is the drag loss variable ; Kroll is a constant for roller bearings; B is the width of the bearing inner ring, mm.

In order to calculate more accurately, the new SKF model takes into account the influence of factors such as cut-in heat generation and lean oil backfill effect, where: φish is the cut-in heat reduction coefficient; φrs is the lean oil backfill reduction coefficient; Krs is the lean oil backfill constant; Kz is the geometric constant corresponding to the bearing type; d and D are the inner diameter and outer diameter of the bearing, mm.

**2 Establishment of heat transfer model**

For a bearing working state of a given speed and load, its final steady-state temperature distribution will be affected by various complex heat transfer processes, including heat transfer, fluid mechanics and other disciplines, while convection in the heat transfer process. The heat transfer problem is even more difficult to determine. Therefore, it is difficult to calculate the final temperature distribution of the bearing accurately. To simulate the heating and heat transfer process of the bearing and obtain the local temperature distribution, it is necessary to make appropriate assumptions to simplify the model.

Figure 1 is a schematic diagram of the contact area between the cylindrical roller and the outer ring raceway. When the bearing is working, the contact position between the cylindrical roller and the raceway of the ring changes continuously. The contact and separation of the rollers from the contact area occurs intermittently. When the rollers pass through the contact area as they roll in the raceway, the contact area actually becomes a heat source due to frictional heating. The temperature of the contact area and the surrounding material increases, and after the roller and the contact area are separated at the next moment, they are cooled by the surrounding medium, and the temperature of the contact area and the surrounding material decreases. Therefore, the rollers in the raceway are experiencing constant heating and cooling cycles when they pass through each surface of the contact belt surface, and finally reach a steady-state temperature field [2].

For high-speed cylindrical roller bearings, the cooling time interval of the contact area is very short, and the heat generated each time is transferred to the surface and interior of the roller and raceway through the contact area, and then reaches a stable state, so the contact area can be regarded as a heat source . But for high-speed bearings, each contact area can be heated hundreds of times per second. In order to simplify the calculation, the contact area can be regarded as a constant heat source for high-speed bearings, and the cross section of any ring in the raceway is taken as the research object. , to analyze the heat transfer and the steady-state temperature distribution of the section, which can greatly reduce the amount of calculation. Figure 2 shows the cross-sectional schematic diagram of the inner ring, outer ring and rollers as an example of an inner ring guided cylindrical roller bearing.

Since the physical properties (thermal conductivity, specific heat capacity, density) of the bearing parts are the same or similar, the heat generated by friction is distributed by 1:1 on the contact surfaces of the rollers and raceways involved in contact [3]. Assuming that the bearing is in operation, the outer ring is fixed and the inner ring is rotating. Due to the friction between the roller and the raceways of the inner and outer rings at the same time, the relative angular velocity of the inner and outer rings is the same, and the linear velocity of the inner ring relative to the roller is greater than that of the outer ring. The number of times is greater than the number of friction with the outer ring. In addition, the contact stress between the roller and the inner raceway is higher than the contact stress between the roller and the outer raceway, so the heat generation of the inner raceway is higher than that of the outer raceway. The outer ring of the bearing not only absorbs the heat generated by friction with the rollers, but also absorbs the frictional heat of the inner race conducted through the rollers.

The purpose of this example is to study the temperature distribution on the outer surface of the outer ring when the bearing operates to achieve steady heat transfer, so the heat generation and conduction of the bearing inner ring, cage, etc. will not be described in detail here. Considering only steady-state heat transfer, the heat generated by bearing friction is conducted to the outer ring through the contact area shown in Figure 3.

In Figure 3, the contact surface between the roller and the raceway is heat conduction, and the raceway surface is subjected to the convection heat transfer of the lubricant. Since the propagation of heat into the friction body does not depend on radiation, the influence of radiation heat dissipation on the temperature distribution of the outer ring surface is ignored in the calculation, and only the heat conduction on the raceway surface and the heat conduction through the outer ring, and the final temperature distribution on the AB surface are considered. .

The heat flow network method selects some temperature nodes in the analysis system. Different temperature nodes are connected to each other with different thermal resistances to form a thermal network, and the heat flow is established by using the principle of equal inflow and outflow heat flow at each node in steady heat transfer. Equation system [4], in order to solve the bearing temperature distribution. The cross-section shown in Figure 3 is left-right symmetrical, and the heat transfer and heat exchange surfaces are also left-right symmetrical. Therefore, the left half is taken as the research object to establish a heat transfer model as shown in Figure 4, and its temperature distribution is solved by the heat flow network method.

Figure 4 establishes five temperature nodes at different positions of the bearing outer ring raceway under the condition of steady heat transfer: among them, ① is the center temperature T1 of the contact area between the roller and the raceway; ② is the axial center position temperature of the outer surface T2; ③ is the edge temperature of the outer surface T3; ④ is the temperature of the non-contact area between the raceway and the roller T4; ⑤ is the oil temperature T5; A is the ambient temperature. Among them, ① has heat input; ①, ④ and ⑤ have convective heat transfer; ②, ③ and ⑤ have convective heat transfer with the ambient temperature.

The heat transfer relationship between the five nodes is shown in Table 1.

The frictional heat generation is calculated by the formula (9), and the friction torque is obtained by the formula (6). Heat conduction calculation formula, in which: S is the area of the vertical heat flow direction between two points; d is the distance between these two points; k is the thermal conductivity of the material.

In the formula: hv is the convective heat transfer coefficient, which is a function of the temperature of the solid surface and the fluid, the thermal conductivity of the fluid, the fluid velocity near the solid surface, the surface size, and the fluid viscosity and density. It can be calculated by the approximate formula given by Harris [4].

In the formula: λ is the thermal conductivity of the lubricating oil; Pr is the Prandtl number of the lubricating oil; Re is the Reynolds number of the lubricating oil; Dpw is the pitch circle diameter of the roller group; Cp, ρ are the specific heat capacity and density of the lubricating oil; u and l are the characteristic velocity and characteristic length of the fluid, respectively.

In steady state heat transfer, the incoming heat flow at each node should be equal to the outgoing heat flow. Thus, the sum of the heat fluxes through each temperature node equals zero. Taking node 1 as an example, the sum of the inflow and outflow heat flow is zero

**3 Calculation examples and result analysis**

The NU313 single-row cylindrical roller bearing is selected as an example for heat transfer calculation, and the material of the inner and outer rings and rollers of the bearing is 8Cr4Mo4V. Table 2 shows the basic structural dimensions of NU313 bearings.

Substitute the corresponding data into the heat flow equations of the heat transfer system, where the ambient temperature is 30°C, the kinematic viscosity of the lubricating oil is 20mm2/s, and the thermal conductivity of the bearing material is 42W/(m·k). The temperature distribution of each node is shown in Figure 5 to Figure 8.

As shown, the highest temperature occurs at node 1 and the lowest temperature occurs at node 3. Nodes 2 and 3 are the highest and lowest temperature points on the outer surface, respectively.

**4 Conclusion**

Through the heat transfer analysis of the bearing outer ring section, it can be known that: (1) the temperature of the outer surface of the bearing outer ring increases with the increase of the bearing load, and increases with the increase of the speed; (2) the temperature difference between the outer surface of the bearing outer ring No more than 1% at most, and can be treated as uniform temperature distribution along the axial direction.

Based on this, if you want to make grooves between the bearing outer ring and the bearing seat to cool the bearing, you can choose to open several grooves of equal size and distribute them evenly. As for the opening position and geometric parameters of the groove, the circumferential temperature distribution of the outer surface of the bearing outer ring needs to be additionally considered.