The reliable operation of mechanical power transmission systems heavily depends on the performance of their critical components, among which spur gears are one of the most prevalent. The longevity and efficiency of a spur gear pair are fundamentally governed by the lubrication conditions at the tooth contact interface. During the initial operating period, known as the running-in process, asperities on the contacting tooth surfaces interact, leading to the generation of wear debris particles. While nano-scale particles may act as beneficial additives, micron-sized particles, comparable in dimension to the elastohydrodynamic lubrication (EHL) film thickness, can significantly alter the lubrication regime. These solid contaminants, suspended in the lubricant, form a solid-liquid two-phase flow whose influence on the thermal EHL performance of spur gears is not yet fully understood. This work establishes a comprehensive transient thermal EHL model for a spur gear pair that explicitly accounts for the presence and dynamics of a solid particle within the contact, analyzing its effects on pressure, film thickness, and temperature distributions.
The analysis focuses on a single, spherical, rigid particle traversing the conjunction zone of a meshing spur gear pair. The particle size is assumed to be smaller than the minimum film thickness to prevent entrapment and catastrophic failure, typically in the range of 0.1 to 0.3 µm. The model divides the contact into three distinct regions along the direction of motion (x-direction): Region 1 (inlet before the particle), Region 2 (the region containing the particle), and Region 3 (outlet after the particle). The core modification to the classical EHL formulation lies in the derivation of a modified Reynolds equation for Region 2, where the flow is partitioned by the solid particle.
Considering the high pressures and thus high effective viscosity in the EHL contact, the drag force exerted by the lubricant on the particle is substantial. For a spherical particle, this dynamic drag force can be expressed as $F_{dyn} = 3.4 \eta R U$, where $\eta$ is the local dynamic viscosity, $R$ is the particle radius, and $U$ is the relative flow velocity. Given this strong coupling and assuming negligible particle inertia, the particle’s translational velocity $u_p$ is approximated to be equal to the local lubricant velocity at its center. Furthermore, the model considers the possibility of particle spin, either clockwise or counterclockwise, which induces a local vortex-like flow perturbation around the particle.
Mathematical Model and Governing Equations
The foundation of the analysis is the modified Reynolds equation incorporating thermal effects. For Regions 1 and 3 (lubricant-only zones), the generalized Reynolds equation is used:
$$
\frac{\partial}{\partial x}\left[ \left(\frac{\rho}{\eta}\right)_e h^3 \frac{\partial p}{\partial x} \right] = 12u \frac{\partial}{\partial x}(\rho^* h) + 12\frac{\partial}{\partial t}(\rho_e h)
$$
For Region 2, which contains the solid particle with a semi-length in the z-direction denoted as $z_0$, a new equation is derived from first principles by applying the continuity equation to the combined flow in sub-regions above and below the particle. The resulting modified thermal Reynolds equation for Region 2 is:
$$
\frac{\partial}{\partial x}\left[ \left(\frac{\rho}{\eta}\right)^*_e (h – 2z_0)^3 \frac{\partial p}{\partial x} \right] = 48\frac{\partial}{\partial x}\left[ \rho^* (h – 2z_0) \frac{u_1 + u_2 + 2u_p}{4} \right] + 48\frac{\partial}{\partial t}[\rho_e (h – 2z_0)]
$$
In these equations, $p$ is pressure, $h$ is film thickness, $u_1$ and $u_2$ are the surface velocities of the pinion and gear teeth, $u=(u_1+u_2)/2$ is the entrainment velocity, $u_p$ is the particle velocity, and $t$ is time. The terms $\rho^*$, $(\rho/\eta)_e$, and $(\rho/\eta)^*_e$ are equivalent parameters accounting for viscosity and density variation across the film, defined as:
$$
\begin{aligned}
\left(\frac{\rho}{\eta}\right)_e &= 12\left(\frac{\eta_e \rho’_e}{\eta’_e} – \rho”_e\right), \\
\left(\frac{\rho}{\eta}\right)^*_e &= 48\left(\frac{\eta_e \rho’_e}{\eta’_e} – \rho”_e\right), \\
\rho^* &= [\rho_e \eta_e (u_2 – u_1) + \rho_e u_1] / u, \\
\rho_e &= \frac{1}{h} \int_0^h \rho \, dz, \quad
\rho’_e = \frac{1}{h^2} \int_0^h \rho \int_0^z \frac{1}{\eta} \, dz’ dz, \quad
\rho”_e = \frac{1}{h^3} \int_0^h \rho \int_0^z \frac{z’}{\eta} \, dz’ dz, \\
\eta_e &= h / \int_0^h \frac{1}{\eta} \, dz, \quad
\eta’_e = h^2 / \int_0^h \frac{z}{\eta} \, dz.
\end{aligned}
$$
The film thickness equation for the spur gear contact includes the geometric gap, elastic deformation of the teeth, and the central film thickness $h_0(t)$:
$$
h(x,t) = h_0(t) + \frac{x^2}{2R(t)} + v(x,t)
$$
where $R(t)$ is the time-varying equivalent radius of curvature along the path of contact, and $v(x,t)$ is the elastic deformation calculated using the Boussinesq integral:
$$
v(x,t) = -\frac{2}{\pi E’} \int_{x_{in}}^{x_{out}} p(s,t) \ln|x-s| \, ds
$$
The energy equation for the lubricant film, which is crucial for capturing thermal effects in spur gear contacts, is given by:
$$
\rho c \left( u \frac{\partial T}{\partial x} + w \frac{\partial T}{\partial z} \right) = k \frac{\partial^2 T}{\partial z^2} + \eta \left( \frac{\partial u}{\partial z} \right)^2 + \frac{T}{\rho} \frac{\partial \rho}{\partial T} \left( u \frac{\partial p}{\partial x} + w \frac{\partial p}{\partial z} \right)
$$
where $c$ is specific heat, $k$ is thermal conductivity, $T$ is temperature, and $w$ is the lubricant velocity component in the z-direction. The Roelands viscosity-pressure-temperature relationship and a common density-pressure-temperature relationship are employed:
$$
\eta(p, T) = \eta_0 \exp\left\{ (\ln \eta_0 + 9.67) \left[ -1 + (1 + 5.1 \times 10^{-9}p)^{Z_0} \left( \frac{T – 138}{T_0 – 138} \right)^{-S_0} \right] \right\}
$$
$$
\rho(p, T) = \rho_0 \left[ 1 + \frac{A_1 p}{1 + A_2 p} – A_3 (T – T_0) \right]
$$
The model is solved under the following boundary conditions: Reynolds boundary condition ($p = \partial p/\partial x = 0$) at the outlet, ambient pressure at the inlet, and specified temperature at the inlet and solid boundaries (gear tooth surfaces).
The equations are normalized for numerical stability and efficiency. The dimensionless parameters are defined as follows:
| Dimensionless Variable | Definition | Description |
|---|---|---|
| $X$ | $x/b$ | Coordinate |
| $H$ | $hR_0/b^2$ | Film thickness |
| $P$ | $p/p_H$ | Pressure |
| $\bar{\eta}$ | $\eta/\eta_0$ | Viscosity |
| $\bar{\rho}$ | $\rho/\rho_0$ | Density |
| $U_0$ | $\eta_0 u_0/(E’ R_0)$ | Speed parameter |
| $G$ | $\alpha E’$ | Material parameter |
| $W_0$ | $w_0/(E’ R_0)$ | Load parameter |
| $\bar{z_0}$ | $z_0 R_0 / b^2$ | Particle semi-length |
Numerical Solution Method
The system of coupled equations is solved using a robust numerical procedure. The transient analysis is performed along the entire path of contact of the spur gear, discretized into 120 time steps from the start to the end of a single-tooth engagement. The pressure distribution is obtained using the multigrid method with a Gauss-Seidel iterative solver, which efficiently handles the strong nonlinearity. A W-cycle with six grid levels is implemented, with the finest grid containing 961 nodes. The elastic deformation is calculated using the multigrid multi-level integration technique to accelerate the convolution process. The energy equation is solved using a column-by-column scanning method, with temperature and pressure solutions iteratively coupled until convergence. The convergence criteria are set to a relative error of less than $10^{-3}$ for dimensionless pressure and load, and less than $10^{-4}$ for dimensionless temperature at each time step. The initial condition for the first time step (near the start of engagement) is derived from a steady-state EHL solution.
Results and Discussion
The analysis is conducted for a standard spur gear pair with parameters listed in the table below. The transmitted power is 12 kW, and the pinion rotates at 1000 rpm. The dimensionless particle radius $\bar{z_0}$ is set to 0.1.
| Parameter | Value |
|---|---|
| Ambient Oil Viscosity, $\eta_0$ | 0.075 Pa·s |
| Pressure-Viscosity Coefficient, $\alpha$ | $2.19 \times 10^{-8}$ Pa⁻¹ |
| Density at Ambient Condition, $\rho_0$ | 870 kg/m³ |
| Number of Teeth (Pinion/Gear) | 35 / 140 |
| Module, $m$ | 2 mm |
| Pressure Angle, $\phi$ | 20° |
| Environmental Temperature, $T_0$ | 313 K |
| Young’s Modulus, $E’$ | $2.06 \times 10^{11}$ Pa |
Particle Velocity and Position Effects
The particle’s translational velocity $u_p$ is not constant but varies with its position in the contact. When the particle is in the inlet region (far from the high-pressure zone), the velocity profile across the film (z-direction) is more Couette-like, leading to a significant variation in $u_p$ depending on its vertical position $Z_c$. However, as the particle enters the central contact region of the spur gear, the extreme pressure causes the lubricant to behave more like a plastic solid, flattening the velocity profile. Consequently, $u_p$ becomes nearly constant across the film and closely matches the surface velocities $u_1$ and $u_2$.
The position of the solid particle significantly influences the minimum film thickness ($h_{min}$) and the maximum contact temperature ($T_{max}$) in the spur gear conjunction. When the particle is located near the contact center or at the inlet to the Hertzian pressure zone ($X_c \approx -1.29$), it causes a noticeable reduction in $h_{min}$. This is most pronounced when the particle is at the Hertzian inlet. Simultaneously, $T_{max}$ rises considerably when the particle is near the contact center, indicating a severe adverse effect on thermal performance. Conversely, when the particle is positioned very close to either the driving or driven spur gear tooth surface ($Z_c \rightarrow 0$ or $1$), both $h_{min}$ and $T_{max}$ show a decrease. The proximity to the solid boundary likely enhances heat conduction away from the contact, mitigating the temperature rise.
The numerical model’s validity for the spur gear application was checked by comparing the calculated minimum film thickness at several mesh positions (under pure EHL conditions, i.e., no particle) against predictions from the well-established Dowson-Higginson formula: $h_{min} = 2.65 \alpha^{0.54} (\eta_0 u)^{0.7} E’^{0.03} R^{0.43} w^{-0.13}$. The relative errors were all below 9%, confirming the accuracy of the numerical methodology.
| Meshing Position | Numerical $h_{min}$ (µm) | Empirical $h_{min}$ (µm) | Relative Error |
|---|---|---|---|
| 1 (Start) | 0.3721 | 0.3971 | 6.3% |
| 2 | 0.4795 | 0.4962 | 3.4% |
| 3 (Mid) | 0.4665 | 0.4866 | 4.1% |
| 4 | 0.5069 | 0.5451 | 7.0% |
| 5 (End) | 0.6894 | 0.7546 | 8.6% |
Transient Temperature Rise Due to Particle Presence
The localized heating effect of the particle is vividly captured in the transient temperature profiles. When the particle is in the far inlet region, its effect on the overall temperature field is minimal. However, as it traverses the high-pressure, high-shear central region of the spur gear contact, it generates a significant local temperature spike in its immediate vicinity. This “hot spot” moves through the contact alongside the particle. The most severe transient temperature rise occurs when the particle is near the contact center, aligning with the location of maximum pressure and shear. This phenomenon underscores the critical importance of effective filtration and lubricant cooling during the running-in process of spur gears to remove such debris and dissipate excess heat, preventing localized scoring or thermal degradation.
Influence of Particle Spin
Under severe operating conditions common in spur gear applications—characterized by high speed, heavy load, and significant sliding—the flow field becomes complex, and particles may undergo spin. The model accounts for both clockwise (negative angular velocity $\omega$) and counterclockwise (positive $\omega$) spin. Spin introduces a local perturbation in the flow field, akin to a micro-vortex, which modifies the effective entrainment and shear conditions.
Clockwise spin of the particle has a pronounced detrimental effect on the spur gear contact, leading to a marked decrease in $h_{min}$ and a significant increase in $T_{max}$ compared to the no-spin or counterclockwise spin cases. Counterclockwise spin shows a lesser impact on $h_{min}$ but still elevates $T_{max}$ relative to the no-spin condition. The physical mechanism is that particle spin exerts a drag on the adjacent lubricant layers. Clockwise spin, in the geometry of the contact, likely acts against the main entraining flow, thereby impeding film formation and increasing shear heating. As the magnitude of the spin angular velocity $|\omega|$ increases, these effects are amplified: $h_{min}$ further diminishes, and $T_{max}$ rises more sharply.
The transient temperature maps reveal distinct patterns based on spin direction. For a clockwise-spinning particle, the most intense heating is localized in the region surrounding the particle itself. For a counterclockwise-spinning particle, while the particle region is still hot, a more pronounced temperature rise is also observed in the exit region of the spur gear contact. In all spinning cases, the local temperature in the particle’s vicinity rises dramatically, and this rise scales with $|\omega|$. The enhanced drag and vortex formation impede efficient heat diffusion, causing heat to accumulate locally.
Conclusion
This investigation into the thermal elastohydrodynamic lubrication of spur gears contaminated by solid particles during running-in yields several critical conclusions. The presence of micron-sized particles in the lubricant can significantly degrade the lubrication performance of a spur gear pair. The particle’s position within the contact is a key factor: particles near the Hertzian contact center or inlet cause the greatest reduction in minimum film thickness and the largest increase in maximum temperature, creating dangerous local hot spots. The translational velocity of the particle is strongly dependent on its position, stabilizing near the contact center due to the non-Newtonian behavior of the lubricant under high pressure.
Furthermore, particle spin, a plausible dynamic in harsh spur gear environments, exacerbates these issues. Clockwise spin is particularly detrimental, substantially thinning the lubricant film and elevating temperatures. Increasing spin angular velocity intensifies all adverse effects. The analysis confirms that effective contaminant control through filtration and thermal management through cooling are essential practices, especially during the running-in period of spur gear systems, to mitigate the risks posed by solid debris and ensure long-term operational reliability.