Research on the Static Thermal Degradation Law of Lubricating Grease for Wind Power Bearings

Abstract

This research addresses the issue of lubricant performance degradation in the main shaft bearings of wind turbines. Through multi-temperature accelerated aging tests, the static thermal degradation patterns were elucidated, and an aging model was developed. Initially, 176 samples were prepared at temperatures of 80 °C, 100 °C, 120 °C, and 140 °C using the static thermal degradation method, with 44 samples at each temperature point. Subsequently, key parameters such as the quality change rate, penetration, oil separation rate, and evaporation amount of the lubricant were systematically measured. Ultimately, the mathematical aging model of the lubricant was derived by fitting the aging kinetics model. The results indicate that as aging time and temperature increase, the degradation characteristics of the lubricant, such as quality change rate, penetration, oil separation rate, and evaporation amount, exhibit discernible patterns. The mathematical aging model was successfully fitted, with the maximum deviation generally within 20% of the error margin, meeting the established criteria. This research provides a theoretical foundation for the establishment of a lubricant condition monitoring system in wind farms. Predicting the performance inflection point of the lubricant can effectively prevent unplanned bearing shutdowns resulting from lubrication failures, thereby offering significant engineering value in enhancing the operational reliability of wind turbine units.

1. Introduction

With the increasing development of domestic wind power technology, wind turbines are evolving towards larger capacities, lower costs, and higher reliability. In wind turbines, the main shaft bearing serves as a crucial transmission support component [1,2,3]. The cumulative installed capacity of wind turbines is steadily rising; however, the lubrication conditions for these systems are quite severe [4,5]. Numerous tests and practical applications have demonstrated that the performance of grease-lubricated bearings—including failure rates and lubrication lifespan—significantly depends on the type of grease used within the bearing [5,6,7,8,9]. Lubricating grease serves as the lubricating medium for the main shaft bearings of wind turbines, and its condition directly influences the performance of these bearings. The lubricating grease in the main shaft bearings of wind turbines primarily deteriorates due to adverse operational conditions characterized by low speed and heavy load. Chemical deterioration represents the predominant form of this degradation. Therefore, exploring the thermal deterioration mechanisms of lubricating grease is crucial in enhancing the reliability of wind turbine bearings. To reduce maintenance costs and prevent premature failure of critical wind turbine components, researchers both domestically and internationally have conducted studies on the mechanisms of grease deterioration [10,11].

Frolov, Kilyakova, Drangai, and Dorr [12] investigated the properties of grease fillers, including molybdenum disulfide, tungsten disulfide, and graphite, using a friction testing machine. They assessed the impact of these fillers on the tribological properties of grease. Yan, Wang, Du, Su, Zheng, and Li [13] employed a rotating rheometer to investigate the rheological behavior of grease at various temperatures, ultimately deriving the composition equation for grease. Zhou, Huang, Wang, Zhang et al. [14] examined the impact of temperature on the thixotropic ring area, storage modulus, loss modulus, strain amplitude, and apparent viscosity of lithium-based greases. Their findings indicate that as the temperature increases, the structure of lithium grease becomes unstable, with significant degradation occurring when temperatures exceed 100 °C. Hodapp, Conrad, Hochstein, Jacob et al. [15] investigated the viscosity and yield stress of grease across a temperature range of −40 °C to 20 °C, concluding that grease can be classified into the NLGI category based on temperature. Paszkowsk, Stelmaszek, and Krzak [16] assessed the lubrication performance of two greases by comparing the rheological and tribological properties of both new and contaminated greases using a rotary rheometer and a ball-disk friction testing machine. Sun and Wang [17] tested the rheological properties of seven types of lithium-based greases using a rotating rheometer. They obtained a calculation formula for apparent viscosity and analyzed the rheological properties of the greases. Garshin, Porfiryev, Zaychenko, Shuvalov et al. [18] examined the influence of base oil composition on the low-temperature performance of greases. They determined how the fraction and group partition ratio of these base oils affect both the low-temperature performance and friction performance of the greases. Wang, Zhang, Lin, and Gao [19] investigated the rheological properties of lithium-based grease and polyurea grease at varying temperatures and consistencies using a rotational rheometer. They also assessed the friction and wear properties of these two greases with a high-temperature friction and wear testing machine. Japar, Aziz, and Hamid [20] utilized waste transformer oil as the base oil to formulate grease and studied its performance, concluding that transformer oil can be effectively configured with grease to achieve significant lubricating performance. Johnson [21] proposed a technology that minimizes the amount of grease required to provide viscosity information, addressing the limitations of traditional methods for detecting grease consistency and cone penetration, which are not applicable for condition monitoring. Lin and Meehan [22] conducted oxidation measurements on the base oil of grease for various deterioration periods to study the degradation of spherical roller bearing grease. They discovered that the grease deterioration process was inconsistent, with the deteriorated grease exhibiting reduced oil separation ability and base oil fluidity. Due to the absence of suitable parameters to characterize grease deterioration, Lin did not establish a unified deterioration process for grease. In a separate study, Li, Liu, Tian, Zhang et al. [23] evaluated the aging performance of a specific type of automotive grease and found that the number of large particles in aged grease was significantly higher than that in fresh grease. Osara and Bryant [24] conducted a comprehensive thermodynamic analysis of the deterioration of lubricating grease. Utilizing thermodynamic principles, they developed a deterioration model for lubricating grease, which was subsequently validated using experimental methods. This model introduced novel material and process parameters for characterizing the performance of lubricating grease, facilitating system optimization. Camousseigt, Galfré, Couenne, Oumahi et al. [25] proposed a dynamic oil seepage model derived from a static oil seepage device. The system is characterized by mass and momentum balance equations. By employing parameter estimation methods, the resulting dynamic simulation structure was compared with experimental results to determine the permeability values. Peng, Li, Shangguan, Zhang et al. [6] investigated the viscoelasticity, fluidity, and temperature-dependent rheological properties of grease across varying temperatures. By fitting the Herschel–Bulkley (H-B) model, they derived a constitutive equation for grease that effectively predicts its rheological behavior under different thermal conditions. Shetty, Meijer, and Lugt [26] proposed a vaporization model that quantifies the influence of base oil volatility and sealing quality on vapor loss. Rezasoltani and Khonsari [27] employed principles of irreversible thermodynamics to predict the viscosity reduction of grease under mechanical shear action and verified this prediction with experiments conducted on three types of greases at varying shear rates and temperatures. Yin, Li, Wang, and Dong [28] proposed a physics-based degradation trajectory modeling method for prediction of the remaining useful life of rolling bearings, wherein physical knowledge is embedded in input preparation, model construction, and output specification. The results indicate that this method outperforms alternative approaches in terms of accuracy and robustness. This data-driven approach is advantageous for advancing research on the static thermal aging behavior of lubricating grease and enhancing the accuracy and applicability of mathematical models.

Most studies in the aforementioned papers concentrated on analyzing specific properties of deteriorated grease and the novel substances generated during the degradation process [29,30,31,32]. Most studies on the mathematical models of grease deterioration laws focus on individual deterioration laws, resulting in a lack of systematic research on the static thermal deterioration law of grease. This paper conducts a comprehensive analysis of a type of grease commonly used in the main shaft bearings of wind turbines. The performance deterioration law of grease is examined under thermal deterioration testing, and a model is established, thereby laying the groundwork for future studies on grease deterioration laws.

This paper focuses on a specific grease used in wind power bearings as the research subject. It employs the static thermal aging method within a constant-temperature drying oven. The study examines the variations in the deterioration behavior under different temperature and time conditions. The characteristics of the evaporation rate, cone penetration, and oil separation rate of the grease during different deterioration periods were analyzed using a cone penetration meter and a steel-net oil separation tester. Relationships between these characteristics, deterioration time, and temperature were established to formulate a mathematical model of grease deterioration. This research aims to elucidate the static thermal degradation mechanisms of grease in wind turbine spindle bearings, thereby laying a foundation for future studies on grease condition prediction.

2. Static Thermal Deterioration Test of Lubricating Grease

2.1. Test Sample

This study focuses on a specific type of lubricating grease used in wind power bearings, conducting static thermal degradation tests. The parameters for this research are presented in

Table 1. Characteristics of grease used in a wind power bearing.

NLGI Rating	1.5
Thickener type	Composite lithium base
Color, visual	red (color)
Penetration, 25 °C, GB/T 269-2023 [33]	305
melting point (of lubricating oil), °C	255
Viscosity, cSt @ 40 °C	460
Timken OK load bearing, GB/T 11144-2007 [34]	55
Four-ball sintering, SH/T 0202-1992 [35], load bearing, kgf	250
water abrasion, SH/T 0109-2004 [36], 79 °C loss. %wt	10
Rust resistance, Distilled water	0.0

below, which outlines the characteristics of the selected lubricating grease for wind power applications.

2.2. Test Equipment and Test Methods

2.2.1. Blast-Type Constant-Temperature Drying Oven

The blast-type constant-temperature drying oven has a temperature control range of room temperature +10 to 200 °C, with a temperature fluctuation of ±1 °C. The blast-type constant-temperature drying oven and the grease placed in the thermostat are illustrated in Figure 1.

2.2.2. HD269 Penetration Tester

To assess the penetration of grease, approximately 5 g of the grease sample should be placed in the test container. The surface must be smoothed according to the prescribed usage method before measuring the penetration of the grease sample. It is advisable to take the average of two measurements for each sample, as illustrated in Figure 2.

2.2.3. Silver-Mesh Oil Separator

To evaluate the grease separation rate, approximately 8 to 10 g of grease sample should be filled into the mesh, ensuring that the level reaches that indicated in Figure 3a while avoiding bubble formation. The assembled silver-mesh oil separator apparatus should then be placed in a constant-temperature drying oven for the specified duration to measure the grease sample’s separation rate. The average of two measurements for each sample should be recorded, as illustrated in Figure 3.

2.2.4. FA2004B Electronic Balance

An FA2004B electronic balance was utilized to test the rate of change in grease quality and the oil separation rate. For each test, the mass of the silver-screen oil separator was measured twice, ensuring that the difference between the two measurements did not exceed 0.05 g. The final result was calculated as the average of these measurements, as illustrated in Figure 4.

2.3. Thermal Degradation Test Conditions

The static thermal deterioration method was employed to investigate the deterioration patterns of grease for various temperatures and durations. For wind power bearings, the temperature performance range of the grease is −30 °C to 150 °C. The operating temperature of wind power bearings is generally above 60 °C. Therefore, the selected test temperatures for this study are 80 °C, 100 °C, 120 °C, and 140 °C. The deterioration test temperatures and durations are presented in

Table 2. Experimental temperature and number of days.

	Time/Day	2	4	6	8	…	80	…	88
Temperature/°C
80 °C		√	√	√	√	√	√	√	√
100 °C		√	√	√	√	√	√	√	√
120 °C		√	√	√	√	√	√	√	√
140 °C		√	√	√	√	√	√	√	√

below.

2.4. Test Procedures

(1) Prepare several glassware containers and apply a layer of lubricating grease.

(2) Sample collection: During the testing period, extract samples every 48 h (every 2 days), with each sample weighing approximately 120 g. After allowing the samples to cool naturally, utilize 60 g for the grease performance test while retaining the remaining 60 g. Note that samples will not be returned after testing.

(3) Test study analysis: Evaluate the 1/4 working conicity, oil separation rate, and mass change rate of the lubricating grease at various test temperatures and durations of deterioration.

A picture of grease under different deterioration temperature and time conditions is presented in Figure 5 below. In the figure, from left to right, the grease deteriorates for 10, 20, 30, 40 and 44 days respectively. From top to bottom, the black wire frame is 80 °C, the yellow wire frame is 100 °C, the green wire frame is 120 °C, and the blue wire frame is 140 °C.

3. Test Results and Analysis

3.1. Mass Change Rate of Grease

The lubricating grease was placed in the steel-mesh oil separator, and subsequently, it was inserted into the thermostats set to test temperatures of 80 °C, 100 °C, 120 °C, and 140 °C to initiate the heat deterioration test. The mass change rate of the lubricating grease, denoted as Y₁ [% (mass fraction)], is presented in Equation (1) below.

$$Y_1=100\times {\displaystyle \frac{m_1}{m_2}}$$

(1)

where Y₁ is the mass change rate of grease, m₁ is the mass reduction of grease (g), and m₂ is the initial mass of grease (g).

When the test interval is set to two days, the mass change rate of grease remains relatively constant. Therefore, the following table presents the mass change rate of grease observed when the test interval was extended to four days. The mass change rates of lubricating grease for different test temperatures and numbers of deterioration days are presented in

Table 3. Mass change rate of lubricating grease under different test temperatures.

τ/Day	Mass Change Rate of Grease (Y1%)
	80 °C	100 °C	120 °C	140 °C
2	0.0095	0.0083	0.0528	0.1642
6	0.00015	0.0122	0.0688	0.1828
10	0.017	0.0204	0.0590	0.2021
14	0.0213	0.0233	0.0882	0.2165
18	0.0198	0.0256	0.0803	0.2593
22	0.0212	0.0267	0.0891	0.2781
26	0.0216	0.0268	0.0906	0.2876
30	0.0231	0.0329	0.0961	0.3068
34	0.0244	0.0321	0.1094	0.3219
38	0.0228	0.0335	0.13	0.35
42	0.0244	0.0355	0.1256	0.3720
46	0.0228	0.0363	0.1334	0.4025
50	0.0271	0.0351	0.1466	0.4597
54	0.0257	0.0356	0.1516	0.4828
58	0.0220	0.0419	0.1512	0.4995
62	0.0259	0.0436	0.1539	0.4996
66	0.0254	0.0539	0.1569	0.5245
70	0.0250	0.0518	0.1588	0.5305
74	0.0225	0.0524	0.1592	0.5399

below.

Figure 6 below illustrates the relationship between the grease mass change rate and deterioration time across different test temperatures.

As illustrated in Figure 6, at test temperatures of 80 °C, 100 °C, 120 °C, and 140 °C, the mass change rate of grease increases with the duration of deterioration. Notably, higher temperatures correspond to a more rapid increase in the mass change rate. This increase becomes particularly pronounced after 10 days of deterioration. As the lubricating grease deteriorates, two primary factors contribute to this process. First, from a chemical perspective, the oxidation of both the base lubricating oil and the thickener disrupts the soap fiber structure of the grease. This disruption weakens the thickener’s capacity to bind to the base oil, resulting in softening and loss of consistency. Additionally, the reaction with oxygen produces organic acids and low-molecular-weight gels, which further alters the quality of the grease. Second, from a physical standpoint, thermal conditions lead to the evaporation of the base oil, causing hardening and caking phenomena. Consequently, over time, the mass change rate of the grease experiences a significant increase.

The lifespan of grease can be assessed by analyzing the performance data of degraded grease. Initially, the aging dynamics model is employed to simulate and align the trend of the mass change rate of grease with the duration of deterioration. The formula is expressed as follows:

$$1-Y=A{e}^{-K{\tau }^{\alpha }}$$

(2)

where $\alpha$ is a constant, τ is the aging time, K is the aging rate, A is the fitting constant, and Y is the grease aging rate.

The mass change rate and degradation time of the grease at various test temperatures were analyzed using Equation (4), resulting in the fitting equation presented in

Table 4. Fitting equation of mass change rate of grease with deterioration time.

Test Temperature	Fitted Equation	R2	p-Values
80 °C	$1-Y_1=0.02354e^{-0.08162\tau }+0.9751$	0.7615	1.044 × 10−5
100 °C	$1-Y_1=0.1444e^{-0.004366\tau }+0.844$	0.9319	5.119 × 10−10
120 °C	$1-Y_1=0.2199e^{-0.01066\tau }+0.7334$	0.9632	2.584 × 10−12
140 °C	$1-Y_1=16.37e^{-0.0003521\tau }-15.52$	0.9855	7.548 × 10−16

below.

The R² value, or coefficient of determination, quantifies the explanatory power of the model concerning the variation in data. Specifically, it indicates the percentage of variation in the dependent variable (Y) that can be accounted for by the independent variable (X) through the model. The value ranges from 0 to 1, with higher values typically indicating better model performance; however, it is essential to consider this in the context of the actual situation. The p-value is employed to assess whether the relationship between a specific independent variable (X) and the dependent variable (Y) is statistically significant. A p-value of 0.05 is generally regarded as indicative of statistical significance.

Based on the observed trends in the data points on the test curve, the relationship between the mass change rate of grease (Y₁) and the deterioration time (τ) can be expressed in the following functional form:

$$1-Y_1=A{e}^{B\tau }-C$$

(3)

Table 4 demonstrates that the values of A, B, and C vary under different test temperature conditions. Additionally, since the test temperature influences the mass change rate of the grease, the values of A, B, and C also differ accordingly. This indicates that A, B, and C are related to the test temperature (T).

Furthermore, analysis of Table 4 reveals an approximate exponential relationship between coefficients A, B, and C and the test temperature (T):

$$A=A_0{e}^{A_{1}T}$$

(4)

$$B=B_0{e}^{B_{1}T}$$

(5)

$$C=C_0{e}^{C_{1}T}+C_2$$

(6)

A regression analysis was conducted, and the resulting values are presented in

Table 5. Fits of the values of parameters in the formula.

Parameter	Parameter Values in Fitting Formula		R2	p-Values
A	A0	1.46 × 10−12	0.9999	0.00728
	A1	0.2146
B	B0	−2830	0.9759	0.04845
	B1	−0.1307
C	C0	−2.058 × 10−72	0.9999	0.01489
	C1	1.199
	C2	0.85

.

A static thermal degradation test of grease was performed at temperatures of 80, 100, 120, and 140 °C, leading to the acquisition of a curve depicting the mass change rate of grease for various degradation times and four different test temperatures. By fitting the data from these four test groups, we derived the calculation formula for the mass change rate of grease at the specified temperatures, along with fitting coefficients A, B, and C, as shown in Table 4. Coefficients A, B, and C in the calculation formula were determined using Equations (4), (5), and (6), respectively (refer to Table 5), resulting in the formulation of the mass change rate for a specific grease:

$$1-Y_1=A{e}^{B\tau }-C$$

(7)

where T is the test temperature, τ is the deterioration number of days, and Y₁ is the mass change rate of the grease.

$$A=1.456\times {10}^{-12}{e}^{0.2146T}$$

(8)

$$B=-2830\times e^{-0.1307T}$$

(9)

$$C=-2.058\times {10}^{-72}{e}^{1.199T}+0.85$$

(10)

Table 4 and Table 5 indicate that the R² values of the fitted equations exceed 0.3, and all p-values are below 0.05, thereby satisfying the criteria for a fitting curve. The calculated values can be derived by substituting the test temperature and the duration of deterioration into Equation (7). The calculated values can be obtained by comparing them with the measured values, revealing a maximum difference of 0.05898. The maximum residual mean square is 2.382 × 10⁻⁴. Figure 7 presents the comparison curve between the measured and predicted values. From the analysis of the graph and the preceding discussion, it is evident that the established calculation formula can accurately predict the mass change rate of grease within the test range and meets the required standards.

3.2. Change Rule of Grease Cone Penetration

The non-working coning degree of one-fourth of the four groups of deteriorated grease samples at various test temperatures was determined. The coning degree of the grease at different deterioration times is presented in

Table 6. Cone penetration of grease for different degradation times at four test temperatures.

τ/Day	Cone Penetration of Grease/mm				τ/Day	Cone Penetration of Grease/mm
	80 °C	100 °C	120 °C	140 °C		80 °C	100 °C	120 °C	140 °C
0	7.36	7.36	7.36	7.36	46	4.68	4.78	5.215	5.095
2	5.69	5.32	5.17	5.79	48	5.36	5.12	4.93	5.355
4	5.51	4.99	4.99	5.345	50	5.28	5.69	5.3	5.91
6	5.61	5.84	6.41	5.2	52	5	4.65	4.915	5.04
8	5.33	5.2	6.835	5.615	54	5.08	4.58	4.96	4.885
10	5.91	4.42	6	5.81	56	4.66	5	5.135	5.045
12	5.55	5.12	5.405	5.46	58	5.1	4.88	5.015	5.23
14	5.3	4.79	4.975	5.08	60	5.37	5.45	5.22	6.32
16	5.5	5.14	4.95	5.72	62	4.43	5.38	5.28	5.65
18	4.9	4.74	4.805	5.22	64	4.62	5.24	5.15	5.5
20	5.32	4.27	5.03	5.5	66	4.52	5.53	5.36	5.435
22	4.59	5.82	5.08	4.97	68	4.87	5.22	5.19	4.82
24	4.89	5.33	5.96	6.12	70	5.34	6.28	5.22	6.23
26	4.91	4.95	6.305	5.02	72	4.77	4.82	4.935	5.435
28	4.96	5.5	5.005	5.94	74	4.95	4.94	5.055	5.335
30	5.2	5.28	5.13	6.2	76	4.72	4.55	4.935	5.92
32	4.72	4.47	5.405	5.255	78	5.17	4.61	5	6
34	4.77	5.58	5.01	5.315	80	5.39	5.74	5.36	6.13
36	4.9	4.87	5.305	5.64	82	4.46	4.98	5.005	6.2
38	4.98	5.35	5.16	5.465	84	4.49	4.94	5.135	6.22
40	4.82	5.52	5.66	6.02	86	4.54	5.09	5.28	6.13
42	4.47	5.12	5.05	5.595	88	4.14	5.05	5.355	6
44	4.9	5.35	4.94	5.11

below.

The changes in grease cone penetration over deterioration time at various test temperatures are illustrated in Figure 8 below.

Figure 8 indicates that at test temperatures of 80 °C and above, the coning degree of the lubricating grease initially decreases, then increases and subsequently decreases again as deterioration time progresses. Overall, the trend shows a decline. This behavior can be attributed to the oxidation decomposition and evaporation of the grease’s base oil at elevated temperatures, which lead to the hardening of the grease and a reduction in coning degree.

The taper of the grease gradually increases with the test temperature, suggesting that elevated temperatures may have compromised the structure of the grease, resulting in a relatively softer consistency.

Equation (6) was employed to model the cone penetration and deterioration time of a specific grease at varying test temperatures, leading to the derivation of the fitting equation presented in

Table 7. Fitting equation of cone penetration of grease changing with deterioration time.

Test Temperature	Fitted Equation	R2	p-Values
80 °C	$1-Y_2=-1.866e^{-0.1263\tau }-3.86$	0.5936	3.92 × 10−9
100 °C	$1-Y_2=-2.241e^{-1.2305\tau }-4.12$	0.3934	2.147 × 10−5
120 °C	$1-Y_2=-1.372e^{-0.1163\tau }-4.158$	0.3360	1.501 × 10−4
140 °C	$1-Y_2=-1.852e^{-1.0651\tau }-4.512$	0.3437	3.346 × 10−4

below.

Based on the variation trend of data points in the test curve, the relationship between grease cone penetration (Y₂) and deterioration days (τ) can be expressed in the following functional form:

$$1-Y_2=D{e}^{E\tau }-F$$

(11)

Table 7 illustrates that the values of D, E, and F vary under different test temperature conditions. Furthermore, variations in test temperature lead to differences in the coning degree of the mass change rate of the grease, resulting in corresponding changes in the values of D, E, and F. This indicates that D, E, and F are dependent on the test temperature (T).

An analysis of Table 7 reveals a specific relationship between coefficients D, E, and F and the test temperature (T):

$$D=D_0\sin \left(D_{1}T+D_2\right)+D_3$$

(12)

$$E=E_0\sin \left(E_{1}T+E_2\right)+E_3$$

(13)

$$F=F_0{T}^{F_1}$$

(14)

A regression analysis was carried out, and the obtained values are presented in

Table 8. Fits of the values of parameters in the formula.

Parameter	Parameter Values in Fitting Formula		R2	p-Values
D	D0	−0.5039	0.9998	0.01135
	D1	0.2099
	D2	−1.167
	D3	−1.824
E	E0	−0.5775	0.9882	0.00005362
	E1	0.1371
	E2	6.732
	E3	−0.6532
F	F0	−1.24877	0.9998	2.79 × 10−4
	F1	0.25699

.

Through a static thermal degradation test of grease at temperatures of 80, 100, 120, and 140 °C, we obtained the change curves of the cone penetration of the grease over time under four distinct testing conditions. By fitting the data from these four tests, we derived a formula for calculating the cone penetration of the grease at the specified temperatures, along with the fitting values for coefficients D, E, and F presented in Table 7. Equations (12)–(14) were employed to compute the coefficients used in the calculation of D, E, and F, as shown in Table 8. This process ultimately led to the formulation of the cone penetration calculation for a specific grease:

$$1-Y_2=D{e}^{E\tau }-F$$

(15)

where T is the test temperature, τ is the number of deterioration days, and Y₂ is the cone penetration of grease.

$$D=-0.5039\sin \left(0.21T-1.167\right)-1.824$$

(16)

$$E=-0.578\sin \left(0.137T+6.732\right)-0.653$$

(17)

$$F=-1.24877T^{0.25699}$$

(18)

Table 7 and Table 8 demonstrate that the R² values of the fitted equations exceed 0.3, and their corresponding p-values are all below 0.05, indicating that the fitting curve requirements are satisfied. Some R² values in the mathematical model are relatively low, at approximately 0.3. This can be attributed to numerous influencing factors affecting grease penetration. At a temperature of 100 °C, grease deteriorates rapidly; however, in the experiment, samples were taken every two days, which may have led to inconsistencies in the data regarding changes in grease penetration. Consequently, the model exhibits relatively low R² values. Therefore, there are inherent limitations in the grease penetration aspect of this model. Future research should consider reducing the sampling time interval for grease.

The calculated values can be derived by substituting the test temperature and the deterioration days into Equation (15). A comparison of the calculated values with the measured values reveals that the relative error is generally less than 16%, with the maximum error not exceeding 20%. Figure 9 illustrates the comparison curve between the measured and predicted values. From the graph and the preceding analysis, it is evident that the established calculation formula can accurately predict the cone penetration of lubricating grease within the test range and meets the necessary requirements.

3.3. Variation Rule of Oil Separation Rate of Grease

The oil separation rates of four groups of deteriorated grease samples were determined at various test temperatures. The oil separation rates corresponding to different deterioration times are presented in

Table 9. Grease separation rates at different test temperatures and numbers of deterioration days.

τ/Day	Oil Separation Rate of Grease%				τ/Day	Oil Separation Rate of Grease%
	80 °C	100 °C	120 °C	140 °C		80 °C	100 °C	120 °C	140 °C
0	0.18%	0.18%	0.18%	0.18%	46	0.61%	0.52%	0.85%	0.52%
2	0.18%	0.28%	0.38%	0.57%	48	0.69%	0.32%	1.09%	0.08%
4	0.41%	0.17%	0.31%	0.81%	50	0.57%	0.27%	0.27%	0.09%
6	0.33%	0.30%	1.40%	1.14%	52	0.54%	0.15%	0.65%	0.39%
8	0.36%	0.55%	2.99%	1.25%	54	0.61%	0.27%	0.35%	0.39%
10	0.334%	0.27%	0.66%	0.96%	56	0.37%	0.11%	0.71%	0.14%
12	0.39%	0.32%	0.65%	0.96%	58	0.41%	0.24%	0.57%	0.13%
14	0.33%	0.29%	0.65%	1.16%	60	0.29%	0.32%	0.63%	0.10%
16	0.28%	0.21%	0.69%	1.74%	62	0.15%	1.11%	0.90%	0.45%
18	0.25%	0.25%	0.66%	1.67%	64	0.14%	1.02%	0.99%	0.50%
20	0.20%	0.11%	0.53%	1.80%	66	0.12%	1.51%	0.84%	0.17%
22	0.23%	0.74%	0.58%	2.21%	68	0.31%	0.88%	0.85%	0.42%
24	0.13%	0.56%	0.61%	1.56%	70	0.22%	0.87%	1.28%	0.53%
26	0.11%	0.55%	0.71%	2.58%	72	0.14%	0.99%	0.943%	0.67%
28	0.15%	0.61%	0.67%	2.56%	74	0.15%	1.03%	0.62%	0.69%
30	0.27%	0.17%	0.66%	3.08%	76	0.19%	0.90%	0.57%	0.83%
32	0.13%	0.14%	0.69%	2.92%	78	0.20%	0.67%	0.73%	0.79%
34	0.13%	0.37%	0.52%	3.07%	80	0.22%	0.68%	0.69%	0.83%
36	0.25%	0.35%	0.81%	2.30%	82	0.26%	0.55%	0.72%	0.8%
38	0.20%	0.29%	0.84%	0.59%	84	0.26%	0.58%	0.68%	0.76%
40	0.17%	0.13%	0.43%	1.20%	86	0.25%	0.81%	0.85%	0.83%
42	0.33%	0.71%	0.75%	1.02%	88	0.27%	0.61%	0.92%	0.84%
44	0.50%	0.96%	0.76%	0.96%

.

Under varying test temperatures, the oil separation rate of grease changes over the duration of deterioration, as illustrated in Figure 10 and Figure 11 below.

As shown in Figure 10, at a test temperature of 80 °C, the oil content of the grease initially increases, followed by a decrease, then another increase and, finally, another decrease, resulting in fluctuating oil content rates. At 100 °C, the oil content of the grease also increases initially, then decreases before increasing again; however, the overall trend shows a slight increase. At 120 °C, the oil content of the grease rises initially, then decreases rapidly before stabilizing. At 140 °C, the oil separation rate of the grease gradually increases, but after 40 days of deterioration, the oil separation rate begins to gradually decrease before slowly increasing again.

Equation (8) was employed to model the oil separation rate and deterioration time of grease at various test temperatures, resulting in the fitting equation presented in

Table 10. The fitting equation of oil separation rate of grease with deterioration time.

Test Temperature	Fitted Equation	R2	p-Values
80 °C	$1-Y_3=-0.4357e^{-{\displaystyle \frac{{\left(x-49.5888\right)}^2}{46.91}}}+0.774$	0.7002	6.425 × 10−10
100 °C	$1-Y_3=-0.7533e^{-{\displaystyle \frac{{\left(x-70.05\right)}^2}{95.22}}}+0.635$	0.524	1.159 × 10−5
120 °C	$1-Y_3=-11.3212e^{-{\displaystyle \frac{{\left(x-7.1369\right)}^2}{0.468}}}+0.3153$	0.749	3.489 × 10−4
140 °C	$1-Y_3=-2.233e^{-{\displaystyle \frac{{\left(x-28.23\right)}^2}{132.68205}}}+0.4567$	0.7979	7.766 × 10−12

below.

The relationship between the grease oil separation rate (Y₃) and the number of deterioration days (τ) can be described in the following functional form, according to the variation trend of data points in the test curve:

$$1-Y_3=G{e}^{-{\displaystyle \frac{{\left(\tau -I\right)}^2}{H}}}-J$$

(19)

Table 10 illustrates that the values of G, H, I, and J vary under different test temperature conditions. Furthermore, as the test temperature changes, the degree of coning in the mass change rate of the grease also varies, leading to corresponding changes in the values of G, H, I, and J. This indicates a relationship between G, H, I, and J and the test temperature (T).

An analysis of Table 10 reveals the specific relationships between coefficients G, H, I, and J and the test temperature (T):

$$G=G_0+G_1\cos \omega x+G_2\sin \omega x$$

(20)

$$H=H_0+H_1\cos \omega x+H_2\sin \omega x$$

(21)

$$I=I_0+I_1\cos \omega x+I_2\sin \omega x$$

(22)

$$J=J_0+J_1\cos \omega x+J_2\sin \omega x$$

(23)

A regression analysis was carried out, and the obtained values are presented in

Table 11. Fits of the values of parameters in the formula.

Parameter	Parameter Values in Fitting Formula		R2	p-Values
G	G0	−0.4375	0.99	0.00263
	G1	−6.274
	G2	3.09
	$\omega$	0.09993
H	H0	58.59	0.99	0.00356
	H1	75.38
	H2	38.56
	$\omega$	0.1416
I	I0	38.71	0.99	0.000632
	I1	−33.66
	I2	18.84
	$\omega$	0.09537
J	J0	0.545	0.99	0.000283
	J1	0.2276
	J2	0.09431
	$\omega$	0.7873

.

Through a static thermal degradation test of grease at temperatures of 80, 100, 120, and 140 °C, the variation curves of the oil fraction of grease with degradation time were obtained for four different test temperatures. By fitting the data from these four groups of tests, we derived the calculation formula for the oil separation rate of grease at these temperatures, along with the fitting values of coefficients G, H, I, and J, as presented in Table 10. Equations (20)–(23) are employed to calculate coefficients G, H, I, and J (refer to Table 11), which ultimately leads to the formulation of the mass evaporation rate of grease:

$$1-Y_3=G{e}^{-{\displaystyle \frac{{\left(\tau -I\right)}^2}{H}}}-J$$

(24)

where T is the test temperature, τ is the number of deterioration days, and Y₃ is the oil separation rate of grease.

$$G=-4.375-6.274\cos \left(0.09993x\right)+3.09\sin \left(0.09993x\right)$$

(25)

$$H=58.59+75.38\cos \left(0.1416x\right)+38.56\sin \left(0.1416x\right)$$

(26)

$$I=38.71-33.66\cos \left(0.09537x\right)+18.84\sin \left(0.09537x\right)$$

(27)

$$J=0.545+0.2276\cos \left(0.07873x\right)+0.09431\sin \left(0.07873x\right)$$

(28)

Table 9 and Table 10 demonstrate that the R² values of the fitted equations exceed 0.3, and all p-values are below 0.05, indicating that the fitting curves meet the necessary criteria. The calculated values can be derived by substituting the test temperature and number of deterioration days into Equation (24). The calculated values can be obtained by comparing them with the measured values, revealing a maximum difference of 0.1434. The maximum residual mean square is 1.806 × 10⁻⁵. Figure 11 illustrates the comparison curve between the measured and predicted values. From the analysis of the graph and the preceding discussion, it is evident that the established calculation formula can reliably predict the oil separation rate of grease within the test range with high accuracy, thereby fulfilling the specified requirements.

4. Conclusions

This paper investigates the relationship between the performance of deteriorated grease and the factors of deterioration time and temperature, as well as the static thermal deterioration behavior of grease. Additionally, a mathematical model is developed to elucidate the deterioration law of grease. The main findings are summarized as follows:

(1) As the deterioration time increases, the rate of mass change in the grease gradually rises. The cone penetration of the grease initially increases before subsequently decreasing, leading to an overall downward trend. Additionally, the oil content of the grease exhibits a pattern of first increasing, then decreasing.

(2) With the increase in deterioration temperature, the mass change rate of grease, the degree of coning, and the oil separation rate of grease gradually increase. When the test temperature is at or below 100 °C, the oil separation rate initially increases, then decreases, followed by another increase and subsequent decrease. Overall, there is a higher trend of increase, although the magnitude of this increase is not significant. When the test temperature is higher than 100 °C, the oil separation rate of the grease increases rapidly at first, then decreases and finally stabilizes. When the deterioration time exceeds 60 days, the oil separation rate of the grease gradually decreases, then slowly increases.

(3) The relationship between the evaporation, coning degree, oil separation rate of grease, temperature, and deterioration time was analyzed and fitted to establish a mathematical model of grease deterioration. The R² values of the models were all greater than 0.3, and the p-values were all less than 0.05, indicating that the models met the necessary statistical requirements. Some of the models concerning the penetration index of grease exhibit relatively low R² values of approximately 0.3. This can be attributed to numerous influencing factors affecting grease penetration. At a temperature of 100 °C, grease deteriorates rapidly; however, in the experiment, samples were taken every two days, which may have led to inconsistencies in the data regarding changes in grease penetration. Consequently, the model exhibits relatively low R² values. Therefore, there are inherent limitations in the grease penetration aspect of this model. Future research should consider reducing the sampling time interval for grease. By substituting the test temperature and the number of deterioration days into the mathematical model, we can obtain the calculated values. A comparison of these calculated values with the measured values reveals that the maximum difference corresponds to the highest error, which does not exceed 20%. The maximum residual mean square is 2.382 × 10⁻⁴. Based on the comparison curve between the measured and predicted values, as well as the aforementioned analysis, it can be concluded that the established calculation formula is capable of predicting the performance of deteriorated grease within the tested range.

This mathematical model predicts the deterioration trend of lubricating grease under high temperatures and enables abnormal warnings by monitoring key performance parameters in real time. This model is applicable to lithium-based grease, specifically under conditions of low speed and heavy load. The relevant bearing parameters correspond to those used in wind power bearings. The rate of change in the quality of the grease exceeding the fitted value by 15% may indicate seal failure and the intrusion of oxygen. If the rate of change in quality sharply increases within a short period, one should be alert to the risk of oil leakage from the bearing box. In response to abnormal oil separation rates, the system can automatically activate the supplementary oil injection device upon detecting that the oil separation rate falls below the safety threshold, thereby ensuring the integrity of the lubricating film. The bearing temperature exceeding 120 °C causes a sharp increase in the oil separation rate and accelerates the loss of base oil, significantly shortening the service life of the grease. Therefore, by strictly controlling the bearing temperature to remain below 120 °C, the amount of retained base oil can be effectively maintained, thereby prolonging the overall service life of the grease.