Transient temperature characteristics of friction clutch disc considering thermal contact conductance under sliding conditions

High temperatures are generated due to the sliding contacts between the rubbing surfaces of the friction clutch system. In this work, by considering the effective thermal contact conductance under sliding conditions, a simulation model of a two-dimensional transient temperature field of the clutch disc was developed. A numerical solution to obtain the surface temperature at different radii was presented based on the finite difference method. Compared with the experimental data, the proposed model for estimating the surface temperature is more accurate than the conventional prediction method. The results showed that the errors of the calculated temperatures at radii of 114 and 106 mm have obviously reduced by 12.98% and 12.60%, respectively. In addition, the influences of pressure and relative speed on the surface temperature were investigated. The temperature increases with the increase of the relative speed and pressure during the sliding period, and there is an interaction effect between pressure and speed on the surface temperature rise.


Introduction
Wet multi-plate clutches are critical components in the fields of automobile, ship, aerospace, and heavy machinery. The functional behavior and shifting comfort of the clutch system are essentially based on the frictional behavior between two rotating elements in sliding contacts. High surface temperatures appear as a result of the frictional heat generated when the friction clutch starts to work [1,2]. Overheating and bad cooling during the slipping phase are responsible for the problems of the increasing wear [3], surface cracks and thermal instability [4,5] of engagement process. Thermal failure is accompanied by non-uniform temperature distribution of the clutch friction pair. By comparing the simulation results with the experimental data, Li et al. [6] found that the maximum radial temperature difference is twice the circumferential temperature difference. Thus, the radial temperature difference is the main reason causing the clutch failure. Yu et al. [7] investigated the thermodynamic differences of a multi-disc clutch before and after the elastic deformation of separate plate through numerical simulation and experiments. The result shows that the critical circumferential temperature gradient for the elastic deformation of the plate is 2-3 °C, which can be used as a theoretical basis to evaluate the real-time status of friction components. Li et al. [8] developed a thermal model to investigate the temperature field of carbon fabric wet clutch, and found that lower skeleton density, lower specific heat capacity, and higher thermal conductivity could reduce the bonding temperature.
Yang and Tang [9] constructed a dynamic model through MSC/NASTRAN to describe the transient contact and heat transfer rules of the disks. The radial temperature difference makes the heat conduction direction of the steel disc gradually change from the initial axial direction to the radial direction. Zhao et al. [10] conducted a transient temperature displacement coupling analysis on wet clutch through numerical simulation. The simulation results showed that the transverse thermal conductivity has a more significant effect on the maximum temperature, which compared with the radial thermal conductivity. Abdullah and Schlattmann [11] established a finite element model to study the temperature field of the clutch disc under different assumptions of uniform pressure and uniform wear, respectively. Based on a uniform pressure assumption, the surface temperature varies linearly with disc radius during the sliding period. While the surface temperature values are approximately distributed assuming uniform wear, which exists errors in the results. Recently, they developed a 3D model of a dry clutch system and found that the contact pressure decreases with a reduction of the structural stiffness (modules of the elasticity) of the friction facing, whereas the displacements are increased [12]. Wu et al. [13] proposed a numerical model of clutch temperature field considering the time-varying characteristics of contact stress distribution and cooling flow field distribution. The findings suggested that the temperature distribution is consistent with the contact stress distribution in the radial direction of the clutch friction pair. In addition, in terms of different testing technologies, such as optical fiber sensors and infrared thermal imaging technology, a series of experiments were carried out to accurately measure the surface temperature trends during the sliding period [14][15][16]. Most studies [17][18][19][20] have focus their attention on the influences of the contact pressure distribution, the sliding speed, and the cooling flow field on the temperature of the clutch.
As is well known, the real contact area takes a quite small fraction (1%-2% for metallic contact) of the nominal contact area, and most of the heat flowing through the actual contact spots is restricted by an effective thermal contact resistance (which is the reciprocal of thermal contact conductance) between the surfaces, which causes the 'bulk temperature' at the interface or on a plane slightly below the interface in the two contacting bodies to differ [21]. Moreover, the frictional heating is located at these contact spots, leading to high local flash temperature. Therefore, an imperfect contact should be considered in pursing more accurate surface temperature of the clutch system. In particular, an effective thermal contact conductance/resistance under sliding conditions should be added in the calculation process. There exist a wide range of factors that may affect thermal contact resistance or thermal contact conductance such as material properties, contact pressure, roughness, and surface oxidation [22][23][24][25][26][27]. In a way, the motion is a form of convective heat transfer. The authors [28,29] proposed a way to investigate the effective thermal contact conductance for non-static contacts considering not only the contact pressure, surface roughness, but also the sliding speed. In addition, combing with this effective thermal contact conductance, it is possible to create a macroscopic numerical model to solve the heat transfer problem where the whole individual asperities are not easily to be established.
At present, few studies have been conducted on the temperature characteristics of wet clutch disc, taking account of the thermal contact conductance under sliding conditions. The objective of this paper is to develop a numerical model of clutch friction pair to find the temperature distribution during the sliding period. An effective thermal contact conductance of non-static contacts is added for obtaining surface temperature with high accuracy. Compared with the experimental data, the modified model is more accurate than the traditional one. The influences of the pressure and relative speed on the surface temperature characteristics are also investigated.

Calculation model
The wet clutch disc system consists of steel discs and friction plates. A single friction pair of the clutch without grooves is taken as the research object (shown in Fig. 1). The materials of the steel disc and the friction plate are respectively 65Mn and the copper matrix composites. The geometric parameters and material parameters are listed in Table 1. www.Springer.com/journal/40544 | Friction  It can be assumed that the thermal model of the friction pair satisfies the axisymmetric geometry, uniform contact pressure, axisymmetric boundary conditions, and constraints [30]. Therefore, the distribution of heat flow will not change in circumferential direction, which implies that the transient heat conduction problem reduces to an axisymmetric two-dimensional heat conduction equation: where in R and out R are the inner and outer radii, respectively;  is the half thickness of the steel disc;  is the thermal diffusivity ( c     );  is the thermal conductivity;  is the density; c is the heat capacity; and T is the temperature.
Also, it is possible to simulate only the half of the steel disc (upper half) or the friction plate (lower half) to reduce the calculation process. As the clutch starts to engage, the heat generated on the frictional facing during sliding contact, which will cause raises in the values of the temperature locally. Lubricating oil flows from the center to the outer circumference and the convection heat transfers occur on the boundaries. Radiation is neglected due to the short launching time. Schematic of heat transfer model is shown in Fig. 2. For the steel disc, the initial and boundary conditions are established as Eq. (2): where q is the total heat exchange at the contacting surface of the steel disc; oil T is the temperature of the lubricating oil and 0 T is the initial temperature; and in h and out h correspond to the heat transfer coefficients.
The heat conduction equation is discretized by finite difference method. As shown in Fig. 3, for the internal nodes of the steel disc, the central difference form is used in the space step, and the forward difference form is used in the time step. The explicit finite difference formula is as Eq. (3): where x  and y  are the spatial step lengths on the x-axis and y-axis, respectively; t  is the time step length; m and n are the serial numbers of nodes on the x-axis and y-axis, respectively, 1 1 mm , and the model of the half-steel disc has 6,416 nodes in total. The above difference formula for the internal node is arranged based on Fourier's method as Eqs. (4) and (5): where Fo is the Fourier number. And the stability criterion requires that the coefficient of , p m n T should not be smaller than 0. For Eq. (4), there is

Convective heat transfers on the inner and outer radii of disc
During the working process, the heat transfer between the internal (or external) surface and the lubricating oil can be considered as the laminar forced convective heat transfer in a rectangular channel. And the correlative formula for calculating the average convective heat transfer coefficient con h is given as where Nu is Nusselt number; oil  is the thermal conductivity of the lubricating oil; e d is the equivalent diameter of the channel, which is defined as the characteristic length: where c A is the flow section of the channel; L is the wetted perimeter; and the value of the characteristic length is e 0.0015 m d  . In the practical engineering, Sieder-Tate formula is often used to calculate the average Nusselt number of the laminar forced convection heat transfer in a rectangular channel, as Eqs. (9) where Re is Reynolds number; Pr is Prandtl number; l is the length of the channel;  is the oil dynamic viscosity; f  is the oil dynamic viscosity at the qualitative temperature; w  is the oil dynamic viscosity at the wall average temperature of the channel; v is the relative velocity of oil;  is the kinematic viscosity of oil; and p c is specific heat at constant pressure.

Convective heat transfers between the local oil film and the contact surfaces
The interface between the friction pair of the wet clutch is composed of the lubrication regions and the asperity contact areas, plotted in Fig. 4  where the radial distribution index of the temperature is 0 1 m  .

Frictional heat
The rate of the heat generated by the friction force during the sliding contacts is given as Eq. (13): where p and s V are the contact pressure and sliding speed; k is the coefficient of friction which is assumed to be barely affected by the operating parameters, such as pressure and velocity. The friction heat flows into the steel disc and friction plate in relation to their thermal properties, and the heat partition factor can be expressed as [32]: Consequently, the heat flux f q and f q obtained from the steel disc and the friction plate are as Eqs. (15) and (16):   Notice that these results are appropriate when the sliding speeds are high. At very slow sliding speeds, the steady heat transfer will be obtained and the frictional heat will be partitioned in the ratio of conductivities 1 2   [33].

Thermal contact conductance under sliding conditions
According to the principle of energy conservation, the mean heat flux y q in the interface can be obtained as the product of the thermal contact conductance c h and the temperature difference T  , which is given as Eq. (17) The thermal contact conductance c h has already been defined as a function of the sliding speed, the mean contact pressure, and the surface roughness, which is [28]: ; i R is the summit radius of the asperity; i  is the summit height standard deviation, and E  is the composite elastic modulus of the friction pair. Equation (18) shows that the thermal contact conductance is in direct proportion to 1 2 s V . This is based on the assumption that during sliding, the individual asperity interactions are in a very short duration of time and hence fail to approach a thermal steady state [28].

Numerical procedure
Friction heat generation, local oil film heat dissipation, and heat conduction occur simultaneously on the interface of the friction pair. Therefore, the total heat q obtained by the upper surface of the steel disc is the sum of the frictional heat f q , the convective heat c q , and the heat conduction quality y q due to the temperature difference of the contacting surfaces. There is , , , where n is the serial number on the y-axis of the friction plate on contact interface, which is corresponding to the serial number m on the x-axis. Thus, the finite difference expressions for the nodal temperatures on the boundaries in Fig. 3 can be written as where m q is the total heat flux input of the node whose radius corresponds to the serial number m; Bi is Biot number. Similarly, the finite difference equation of four vertices can be derived.
The initial temperature of the friction pair and the oil temperature are 45 and 20 °C, respectively. The oil temperature is assumed to be constant. After 14 s sliding, the sectional temperature distribution of the steel disc at contact pressure ( p ) of 1.4 MPa and relative speed ( n  ) of 400 r/min is shown in Fig. 5. Temperature variation in the height (thickness) direction is very small, while a significant change can be seen in the radial direction.
The surface temperature distribution of the steel disc is drawn by Fig. 6. The steel disc has a large temperature gradient in the region with small radius. The surface temperature first increases and then decreases with the increase of the radius. The maximum temperature appears at the radius of 118 mm, as illustrated in Fig. 7. This is due to the convective heat transfer between the outer edge of the steel disc and the lubricating oil. The high temperature zone of the steel disc exists between the middle diameter and the outer diameter.

Comparison and test verification
The calculated surface temperatures with or without considering the effective heat contact conductance are both compared with the experimental data in Ref. [34] under the same working conditions. In the test process, the friction pair is controlled to rotate, and the values of surface temperatures at different radii are measured by the temperature sensing fibbers. The calculation surface temperatures at radii of 114 and 106 mm are separately compared with the experimental data during the sliding period, plotted in Figs. 8 and 9. It can be seen that the modified simulation results with considering the effective thermal contact conductance are more accurate, and have better consistency with the experimental results than the traditional ones. It should be noted that the oil temperature is assumed to be constant in the simulation process, however, the temperature of the lubricating oil will rise slightly in the test, resulting from the heat dissipation from the friction pair. Therefore, the calculated temperatures will take less  time to reach a steady state than the measured temperatures.
The maximum surface temperature error of the modified model at radius of 114 mm appears in 4.5 s, and its value is 14.12%. At radius of 106 mm, the maximum error occurs in 5.5 s, and the error is 8.87%. The torque transmission of the test bench is unstable at the start-up stage, and the actual boundary conditions are different from the assumptions. Consequently, within a certain error range of 15%, it can be considered that the modified numerical model is able to reflect the surface temperature variation of the steel disc. In addition, the numerical temperatures obtained at 14 s are compared with the experimental data, listed in Table 2. With considering the thermal contact conductance, the errors of the simulated temperatures at radii of 114 and 106 mm are obviously reduced by 12.98% and 12.60%, respectively.

Temperature characteristic analysis of the friction pair
The variation of the cross-section temperature with sliding time of the friction pair is shown in Fig. 10. The upper section is the friction plate and the lower one is the steel disc. It can be noted that the surface temperature increases rapidly in initial contact. With the same thickness, the temperatures of the friction plate or the steel disc increase firstly and then slightly decrease in the radial direction during the sliding period. For each thickness, the maximum value appears near the radius of 118 mm toward the outer radius. The results are in consistent with the conclusions illustrated by Fig. 7. In addition, with the same radius, the temperature in the cross section has a small decline in the thickness direction when it is away from the contact surface. The varying tendency of the surface temperatures of the friction pair with sliding time at radius 106 mm is plotted in Fig. 11. The convective heat transfer on the inner and outer surface, local film heat dissipation, and contact heat transfer on the contact interface lead to the decrease of temperature increment with sliding time. Based on Table 1, the heat partition factor is 0.5948   , which is bigger than 0.5. More frictional flows into the steel disc, resulting in that the surface temperature of the steel disc is higher than that of the friction plate. The value of the temperature difference reaches 3.13 °C after sliding 14 s.

Influences of pressure and relative speed on the surface temperature
Taking the steel disc as an example, the influence of the pressure on the surface temperature at a fixed speed 500 r/min is illustrated in Fig. 12   The influence of the relative speed on the surface temperature when the pressure keeps 1.4 MPa is shown in Fig. 13. The increases in speed enhance the frictional heat, convective heat transfer, and contact heat transfer, among which the frictional heat increases more dramatically during the sliding period. As a  result, the final temperature rises with the growing speed, and the temperature increment per unit time is also elevated.
Five uniform simulation levels of the speed and pressure are designed respectively, as shown in Table 3. A total of 25 groups of orthogonal level combinations are simulated. The contour map of the temperature rise at radius 114 mm is plotted in Fig. 14. It shows that the temperature rise increases with the increasing of the speed and pressure, and there is an interaction effect between pressure and relative speed on the surface temperature rise. The difference in temperature rises between 0.6 and 1.0 MPa increases with the increasing of speed. Similar variation trend of the temperature rise differential has been found for each 0.4 MPa increase in pressure. While, when the speed is constant, for every additional 0.4 MPa increase in pressure, the temperature rise difference remains essentially unchanged. Moreover, the differences of the temperature rise between 200 and 350 r/min increase as pressure increasing. The change of the temperature rise differential has the same tendency for each additional 150 r/min in speed. However, with every 150 r/min rise in speed, the temperature rise difference decreases gradually with the increase of speed when the pressure is constant. This is because there is a growing temperature gap between the steel disc and the environment, as well as the friction plate, which leads to the increase of convective heat transfer and contact heat transfer. Therefore, when the clutch system works under high speed, controlling the contact pressure can effectively reduce the temperature rise of the friction pair and avoid the thermal failure.

Conclusions
In this paper, a two-dimensional transient temperature model of the clutch disc was developed by considering the thermal contact conductance under sliding conditions. A numerical solution to obtain the surface temperature at different radii was presented based on the finite difference method. The influences of the pressure and relative speed on the surface temperature were investigated under different work conditions. This study highlights the error existing in the temperature distribution without considering the effective thermal contact conductance. The main conclusions are summarized.
(1) The results obtained from uniform pressure show that the values of the temperature are not uniform over the frictional facing, which increase firstly and then decrease with disc radius. The maximum radial temperature occurs at the radius of 118 mm.
(2) The modified numerical model with considering the effective thermal contact conductance is more accurate, and has better consistency with the experimental data than the conventional one. By considering the thermal contact conductance, the errors of the simulated temperatures at radii of 114 and 106 | https://mc03.manuscriptcentral.com/friction mm have obviously reduced by 12.98% and 12.60%, respectively. Moreover, the surface temperature of the steel disc is higher than that of the friction plate. The value of the temperature difference reaches 3.13 °C after sliding 14 s.
(3) The temperature increases with the increase of the relative speed and pressure, and there is an interaction effect between pressure and speed on the surface temperature rise. When the pressure is constant, the temperature rise rate decreases with the raising of the speed.