The influence of water-based nanolubrication on mill load and friction during hot rolling of 304 stainless steel

Using pure water in comparison to water-based lubricant containing 4% TiO2 nanoparticles (NPs), the hot rolling tests of 304 stainless steel were carried out at a rolling temperature of 1050 °C under varying rolling reductions and speeds. The effects of lubrication on rolling force, torque, power and contact friction were systematically investigated. The coefficient of friction (COF) during steady-state hot steel rolling was inversely calculated using a developed flow stress model. The COF models including the effects of rolling reduction and speed were proposed via multiple linear regression. The results indicated that the use of the nanolubricant enabled a reduction of rolling force up to 6.1% and decreases in rolling torque and power up to 21.6%, compared to that of water condition. The results obtained from the linear regression agreed well with those from the inverse calculation, suggesting the developed COF models had high accuracy. The lubrication mechanisms were derived from a boundary lubrication regime, owing to ball bearing and mending effects of TiO2 NPs, and formation of thin lubricant film under high rolling pressure.


Introduction
Hot rolling is a metalworking process extensively applied in steel manufacturing to produce a final product or a basic material for downstream processing [1]. In hot steel rolling, rolling temperatures normally range from 800 to 1050 °C, depending on the material being rolled [2]. Stainless steels, in particular, are usually hot rolled at temperatures above 1000 °C, which aims to enhance the resistance to sticking on work roll surface [3,4]. Such high rolling temperatures significantly expedite the degradation of work rolls in the form of oxidation and wear. There also exists a challenge to enhance the capacity of rolling mill and lower its energy consumption by reducing the mill load, especially for rolling heavy plates with high rolling reductions. Hot rolling with high reductions is also one of the most commonly used techniques to produce ultra-fine-grained steels. Therefore, a coolant and/or lubricant is needed to wet/lubricate the work roll surface during hot rolling process. Over the past decades, conventional oil lubricants such as neat oils and oil-in-water (O/W) emulsions have been widely used in hot steel rolling for reduced mill load [5], reduced work roll wear [6], thinned oxide scale and improved strip surface quality [7], controlled strip thickness [8], ameliorated flatness [9], refined grains [10], and optimised recrystallization texture [11]. Nevertheless, oil lubricants usually contain a high concentration of oil to ensure sufficient lubricity which is in fact weakened when oil is burning at high rolling temperatures. Burning of oil lubricants causes environmental issues. Their disposal has the same concern owing to the non-biodegradable nature and inherent toxicity of oil [12]. To enhance the lubricity of oil lubricants but reduce the oil usage for environmental protection, nanoadditives are added into the O/W emulsions that have a low oil content [13][14][15]. By doing this, encouraging results have been obtained in improving the lubrication performance and mitigating the pollution problems compared with conventional oil lubricants. The use of oil-containing lubricants, however, unavoidably has adverse effects on the environment and the regular maintenance of nozzles and pipes. It is thus essential to develop eco-friendly and highperformance lubricants that can be employed in hot steel rolling even at temperatures above 1000 °C.
In recent years, water-based nanolubricants have attracted increasing attention and interest in many engineering fields such as metal machining [16][17][18], micro rolling [19] and other tribological applications [20][21][22]. These lubricants containing TiO 2 NPs [23][24][25][26], Eu-doped CaWO 4 NPs [27] or MoS 2 -Al 2 O 3 nanocomposites [28,29] applied in hot steel rolling have become increasingly prevalent due to their good environmental compatibility, high recyclability, superb cooling capacity and excellent lubricity at high temperatures. The nanoadditives for synthesising water-based lubricants include pure metals, metal and non-metal oxides, metal sulphides, carbides, nitrides, carbon-based materials and composites [30]. Among all the candidates of aqueous nanoadditives, nano-TiO 2 has gained popularity in the syntheses of water-based nanolubricants on account of their low cost and exceptional dispersion stability in water [31][32][33]. Applying such lubricants containing TiO 2 NPs in hot steel rolling has demonstrated great potential in reducing rolling force, alleviating roll wear, decreasing oxide scale thickness, improving strip surface quality, and refining grains in rolled steels [34]. A recent study also showed that such water-based nanolubricants could be synthesised in a facile process with a super low operating cost, which sheds some light on industrial-scale hot steel rolling [35]. However, the effects of rolling parameters on the mill load using waterbased nanolubricants have been scarcely reported.
When developing the water-based nanolubricants for hot steel rolling, it is imperative to investigate the contact friction between the work roll and the workpiece. The main reason behind this is that friction is of paramount importance for accurate modelling, optimal design and control of industrial rolling processes [36]. In some previous studies, the friction in hot rolling was investigated through characterising the relations between coefficient of friction (COF) and rolling parameters under dry [37,38] or oil lubrication conditions [5,39]. Nevertheless, the friction associated with hot rolling of steels using water-based nanolubrication remains not well understood likely due to the difficulty in measuring the COF.
This study aimed to investigate the effects of different rolling parameters on the mill load and friction during hot rolling of 304 stainless steel. The optimal water-based nanolubricant ascertained in our previous study [35] was utilised to have a comparison with pure water in this matter. The lubrication mechanisms associated with the hot rolling process were unveiled based on the lubrication regime determined through a lubricant film thickness model. New COF models of hot steel rolling were developed and validated using the rolling test results.

Materials
Water-based nanolubricants used in this study consist of low-cost rutile TiO 2 NPs (~ 300 nm in diameter), and chemical additives including glycerol, sodium dodecyl benzene sulfonate (SDBS) and Snailcool. Rutile is the most stable polymorph of TiO 2 at temperatures up to its melting point of 1843 °C, and it can be irreversibly converted from metastable phases of anatase or brookite upon being heated above temperatures ranging from 600 to 800 °C [40,41]. Glycerol is a colourless, odourless and viscous liquid that assists in enhancing viscosity and wettability of the lubricants [25]. SDBS is an organic dispersant that has a linear structure with a hydrophilic group, which helps improve the wettability, dispersion stability and viscosity of the water-based lubricants [23]. Snailcool is a novel water-soluble extreme pressure agent that not only enhances the extreme pressure property of the lubricants, but also improves the lubrication performance [34]. All the chemical additives are nontoxic and biodegradable to environment. The preparation of the water-based nanolubricants is a facile process that only involves mechanical stirring without using extra ultrasonic treatment. This process helps reduce the synthesis cost of the water-based nanolubricants, which is of particular significance for industrial application. According to the results obtained in our previous study [35], all the water-based nanolubricants with varying constituents exhibited excellent dispersion stability within 48 h. In this study, only the lubricant with the best lubrication performance was selected to compare with pure water which is a commonly used coolant for industrial-scale work rolls. The specific chemical composition of the optimal lubricant in weight percentage includes TiO 2 4.0%, glycerol 10.0%, SDBS 0.2% and Snailcool 1.0%.
A 304 austenitic stainless steel was hot rolled in this study. Its main chemical composition in weight percentage includes C 0.047%, Si 0.51%, Mn 1.13%, Cr 18.09%, Ni 8.45%, Cu 0.046%, V 0.12%, N 0.027 and P 0.027%. The steel workpieces were machined to 300 ± 0.5 mm in length, 48 ± 0.3 mm in width and 10.8 ± 0.1 mm in thickness with tapered edges for ease of biting. Both sides of the workpieces were ground to achieve a consistent surface roughness (R a ) of around 0.5 μm. The workpieces were then cleaned with acetone to remove any residues retained from machining.

Hot rolling test
Hot rolling tests were carried out on a 2-high Hille 100 experimental rolling mill. The work roll diameter and the roll barrel length were 225 and 254 mm, respectively. The initial surface roughness of work rolls was 0.9 μm in R a . During testing, the steel workpieces were first reheated in a high-temperature electric resistance furnace at 1100 ℃ and then soaked for 40 min in a nitrogen atmosphere. The actual rolling temperature was about 50 ℃ lower than the reheating temperature. As shown in Table 1, the rolling reduction ranged from ~ 10 to ~ 30%, and the rolling speed was in the range of 10-50 rpm, i.e. 0.12-0.6 m•s −1 . The hot rolled strips were immediately put into a cooling box inflated with nitrogen to restrain further oxidation. As reported elsewhere [26], prior to each rolling test, the lubricant was continuously sprayed onto the pre-cleaned rotating work rolls until supersaturated coverage was achieved. The hot steel rolling under the same testing condition was repeated twice to attain average data of rolling force, torque and power.

Flow stress model for inverse calculation of COF
The COF during steady-state hot rolling of 304 stainless steel was acquired using a numerical model developed by Alexander based on the Orowan rolling model [42,43]. The Alexander model with detailed features and theory can be found elsewhere [44,45]. In the Alexander model, the flow stress of the steel used for COF calculation is expressed as [42]: where A = 0 ⋅ e −aT ; 0 is the base yield stress (MPa); T is the temperature (K); B , D , a , n 1 and n 2 are constants; and ̇ denote the strain and strain rate (s −1 ), respectively. Assuming B = D = 1000 ≫ 1 , A , n 1 and n 2 in Eq. (1) can be determined using the multiple regression of the stress-strain curves obtained from hot compression tests. A MATLAB programme was developed based on the Alexander model to conduct the inverse calculation of COF. When running the MATLAB programme, the COF value was input and adjusted until the calculated rolling force was less than 1% error with the measured one.
The flow stress of 304 stainless steel was achieved by hot compression tests conducted on a Gleeble-3500 Thermomechanical Simulator in vacuum under 1.0 × 10 −2 torr pressure at 1050 ℃. The compressive strain was limited up to 0.5. The strain rates being selected were 7.6 and 10.0 s −1 , falling into the range of strain rate in the hot rolling tests. The strain rate of hot steel rolling was calculated using the Ford-Alexander equation [46]: where ̇ is the strain rate (s −1 ); N is the rotation speed (rpm); r is the work roll radius (mm); h 0 is the entry thickness (mm); and ′ is the rolling reduction.

Measurement and characterisation
Rolling force data during hot steel rolling were acquired through two individual transducers located at both the drive and operation sides over the bearing blocks of the top work roll. Two torque transducers were placed in the drive spindles to measure the rolling torque. The rolling speed was monitored using a tachometer, and the roll gap was precisely controlled using a rotary knob mounted on a control panel.
A pyrometer was applied to measure the entry temperature. All the data were collected using a programme developed on the basis of MATLAB xPC technology [35]. Surface morphologies of rolled steels were examined and analysed under a high-resolution field emission scanning electron microscope equipped with an energy-disperse spectrometer (EDS) to investigate the role of NPs during rolling process.
Dynamic viscosity values of the applied lubricants were measured at 80 ℃ under high pressures of 0, 50, 100 and 150 bar, i.e. 0, 5, 10 and 15 MPa, using an Anton Paar MCR 302 Rheometer. The testing temperature was chosen to avoid substantial loss of water caused by rapid evaporation at a temperature above 80 ℃. By doing this, the lubricant concentration would not be affected significantly during the viscosity measurement. The shear rate used for the viscosity measurement ranged from 1 to 1000 s −1 . Each measurement was repeated three times, and the averaged values were obtained.

Rolling force, torque and power
Rolling force values are plotted as a function of the rolling reduction varying from ~ 10 to ~ 30% for all the rolling speeds being used, as shown in Fig. 1. The rolling force increases with the increased rolling reduction for both water and 4% TiO 2 lubricant. The rolling force ( F , kN) is linearly related to the rolling reduction ( ′ ) with correlation coefficients (R 2 ) above 0.98, which can be expressed below.
(3) F = 11.6 + 713.83 � R 2 = 0.987 for water In comparison to water, 4% TiO 2 lubricant enables a linear function with a smaller slope, indicating that the rolling force changes less significantly with the reduction when using 4% TiO 2 lubricant instead of water. In the hot rolling of 304 stainless steel at 1050 ℃, Eqs. (3) and (4) can be used to predict the rolling force values under certain rolling reductions due to their high linear correlation coefficients. Figure 2 shows the rolling force that changes with the rolling speed under rolling reductions of 10-30% using water compared with 4% TiO 2 lubricant at the rolling temperature of 1050 ℃. Under each reduction, the rolling force obtained using water or 4% TiO 2 lubricant increases with the increased rolling speed. Compared to water, 4% TiO 2 lubricant brings about much lower rolling force, especially under relatively low rolling reduction and speed. In particular, the rolling force is decreased up to 6.1% when using 4% TiO 2 lubricant under 10% rolling reduction at a rolling speed less than 0.36 m•s −1 . The lubrication effect on decreasing the rolling force appears to be less significant under higher rolling reduction and speed.
According to Eq. (2) and the rolling parameters listed in Table 1, the strain rate calculated in this study ranges from 1.1 to 10.1 s −1 . The rolling force plotted as a function of the strain rate using water and 4% TiO 2 lubricant is shown in Fig. 3. It can be seen that the rolling force changes insignificantly with the strain rate under each rolling reduction for both water and the lubricant. Besides, the linear fitting results with 4% TiO 2 lubricant are slightly better than those with water.
The rolling torque values are plotted against the rolling speed under different rolling reductions in Fig. 4. The results show a similar variation trend to those in Fig. 2. The use of 4% TiO 2 lubricant can decrease the rolling torque obtained with water by around 5-21%, under all the rolling conditions. A slower rolling test combined with a smaller rolling reduction causes a better lubrication effect on decreasing the rolling torque.
The power applied to the rolling mill results from the torque applied to the work rolls. Rolling power is a quantity that represents the energy consumption of a rolling mill, which is calculated as: where P is the rolling power (kW); M is the rolling torque (kN•m); and is the rolling speed (rpm). Figure 5 shows that the rolling power rises with the increased rolling speed under each reduction. The results also show that the rolling power falls significantly by up to 21% using 4% TiO 2 lubricant. The extent of power reduction presents a descending trend with the increased rolling speed and rolling reduction, but it is generally over 5%. Figure 6 shows the true stress-strain curves obtained from the hot compression tests at 1050 ℃ with the strain rates of 7.6 and 10.0 s −1 . It is evident that the flow stress rises dramatically with the increase in strain, and a higher strain rate leads to a higher flow stress. The parameters in Eq. (1) are obtained from the regression result of the stress-strain curves:

Coefficient of friction
where is the stress (MPa); and ̇ represent the strain and strain rate (s −1 ), respectively.  Table 2. It is found that the use of 4% TiO 2 lubricant results in a lower COF compared to that of water under each rolling condition. Of particular interest is that the COF can be lowered to a greater extent under the rolling reduction of ~ 10% combined with a rolling speed of 0.12 m•s −1 .
According to [47], the COF ( ) during hot steel rolling is linearly related to workpiece temperature ( T , ℃) and rolling speed ( v , m•s −1 ), depending on different work roll materials, expressed as:  These equations predict that the COF decreases with the increased workpiece temperature or rolling speed. It is noted that the effect of rolling reduction is not included in Eqs. (7)-(9), nor is that of lubrication. In this work, the improved COF models for both water and 4% TiO 2 lubricant were obtained via linear regression using the data listed in Table 2, written as: where is the COF during steady-state hot rolling of 304 stainless steel; ′ is the rolling reduction, and v is the rolling speed (m•s −1 ). As the rolling temperature remains unchanged at 1050 ℃, its effect is not included in Eqs. (10) and (11). The correlation coefficients (R 2 ) of these equations for pure water and 4% TiO 2 lubricant are 0.926 and 0.962, respectively. The improved COF model appears more accurate with the lubricant than pure water. Figure 7 a and b compare the calculated COF values with those predicted by the improved COF models for water and the lubricant, respectively. In both cases, the results obtained from the COF models agree well with those obtained from the calculation, indicating high accuracy and reliability of the models. In addition to this, the agreement for lubricant case appears better than pure water.

Effects of rolling reduction, speed and lubrication on mill load and friction
When using water-based nanolubricants for hot steel rolling, the key factors that affect mill load, i.e. force and torque, include rolling reduction, temperature, speed and lubrication, among which rolling reduction has the most significant effect, and lubrication has the least [48]. As can be seen in Figs. 1 and 2, when the rolling reduction is increased, the rolling force increases significantly. Two competing mechanisms are associated with the rolling process. The increase in rolling reduction raises the strain and the strain rate, which in turn raises the resistance of the steel to deformation, thus leading to an increased rolling force. On the other hand, the increasing reduction results in more plastic work during rolling. This generates more deformation heat and hence a higher temperature on the steel surface, which produces a lower rolling force. Overall, the effect of increased deformation resistance overwhelms that of increased deformation heat, thereby causing a rise in rolling force. In this study, the entry temperature (1050 ℃) remains unchanged for all the rolling tests. Therefore, rolling speed becomes the second important factor that affects mill load. There exist several competing mechanisms that can interpret the speed effect. First, the strain rate rises markedly with the increased rolling speed (see Fig. 3), leading to a higher resistance from the workpiece to deformation and thus an increase in rolling force. Second, rolling speed affects the effective quantity of lubricants in the contact zone. In a faster rolling test, more lubricants can be dragged into the contact zone by the moving surfaces of work rolls. This lowers the COF between the work roll and the workpiece, which contributes to a decreased rolling force. Third, a higher rolling speed corresponds to a less temperature drop during rolling, which should help decrease the rolling force to some extent. Figures 2 and 3 show that an increased speed results in an increased force, suggesting that the strain rate effect overwhelms those caused by lubricant and workpiece temperature. Previous research also supports that the effect of strain rate on steel's resistance to deformation is more significant, particularly at higher rolling temperatures [49], such as the case of 1050 ℃ in our study. The 4% TiO 2 lubricant produces much lower mill loads than water, which is particularly true when relatively low rolling reductions and relatively low rolling speeds are employed, as shown in Figs. 2 and 4. This may be because with a higher reduction TiO 2 NPs are more difficult to enter the contact zone as the extreme pressure property of the lubricant becomes poorer. Moreover, the NPs tend to agglomerate with the increase in reduction, which further weakens the lubrication effect. The results shown in Fig. 8 support these derivations. As can be seen in Fig. 8a3, b3, c3, the EDS mappings demonstrate that the effective quantity of TiO 2 NPs remained on the rolled steel surface is inclined to decrease with the increased reduction. Correspondingly, the TiO 2 NPs presented in Fig. 8a4, b4, c4 appear coarser. The ball-bearing effect of the NPs is exemplified in Fig. 8a2, where individual TiO 2 NPs spread over the steel surface rolled with ~ 10% reduction. In contrast, as shown in Fig. 8c2, when the reduction is increased to ~ 30%, TiO 2 clusters are found on the rolled steel surface, indicating agglomeration of NPs. More and smaller TiO 2 NPs in the contact zone actually help lower mill load to a greater extent [26]. When rolling at relatively high speeds, the increased speed significantly enhances the resistance to material deformation, thus leading to a higher mill load for both water and lubricant. As a result, the load difference caused by water and the lubricant becomes less distinct. The NPs can also affect rolling through mending the surface defects, as shown in Fig. 8a1, b1, c1.
Introducing lubricants into the contact zone between the work roll and the workpiece increases the complexity of friction. As discussed earlier, the increase in rolling speed allows more lubricants to enter the contact zone, and hence a drop in COF. As can be derived from Eq. (11), an increasing reduction also leads to a drop in COF. When the reduction is increased, the experienced pressure of the lubricant increases, so does the viscosity [50]. The increased viscosity prompts a decreased COF. On the other hand, the risen interfacial temperature is always associated with the increased reduction, which somewhat lowers the viscosity of lubricant and thus leads to a higher COF. Figure 8 demonstrates that fewer and larger NPs are present in the contact zone when rolled under a higher reduction, which therefore aggravates the friction between the work roll and the workpiece. In this study, pressure appears to have more dominant effect on reducing the COF than other factors.

Lubrication mechanism
To understand the possible lubrication mechanisms involved in the hot rolling of 304 stainless steel, it is necessary to first determine the lubrication regime. According to the wellknown Stribeck curve, there are three types of lubrication regimes, including boundary lubrication, mixed lubrication  Fig. 7 The COF values obtained from the inverse calculation verses those from the linear regression for a water and b 4% TiO 2 lubricant and hydrodynamic lubrication [51]. It is possible to determine the lubrication regime during rolling by comparing the thickness of the lubricant film ( h min ) and the combined asperity heights ( R ′ q ) of the work roll and the workpiece [52]. The ratio ( ) and R ′ q are defined as: where R q1 and R q2 are the surface roughness values ( R q ) of the work roll and workpiece, respectively. As reported elsewhere [53], boundary lubrication is present when is less than 1; mixed lubrication prevails when 1 ≤ ≤ 3 , while hydrodynamic lubrication occurs at a value over 3.
The lubricant film thickness ( h s ) at the inlet of rolling mill can be calculated using the formula derived by Wilson and Walowit [54], expressed as: where 0 is the dynamic viscosity of lubricant under ambient pressure (Pa·s); is the pressure-viscosity coefficient (Pa −1 ) which links the variation of viscosity with a change in pressure; v entry and v roll are the strip entry speed and roll surface speed, respectively, both in m•s −1 . They are assumed to be the same in this study. l in Eq. (14) is the contact length, defined as l = √ r ⋅ Δh , where r is the radius of work roll, and Δh is the reduction of thickness. y is the yield strength of the strip at ambient temperature. r ′ is the deformed roll radius that can be calculated with the knowledge of the Poisson's ratio and Young's modulus of the work roll [55]. It should be noted that Eq. (14) was proposed for cold rolling of metals. Some parameters such as 0 and y in this equation should be corrected to high-pressure dynamic viscosity of the lubricant and high-temperature yield strength of the strip, respectively. For the hot steel rolling conditions, the actual yield strength of the strip at 1050 ℃ under varying strain rates can be estimated using Eq. (6), in which the strain is selected at 0.2%. In fact, the viscosity of lubricant during hot steel rolling is not constant, but rather a function of temperature and pressure [56]. Figure 9 shows the results of dynamic viscosity measurement at 80 ℃ under increasing pressures up to 15 MPa. It can (b3) and (c3), respectively, at 1000 × magnification; (a2), (b2) and (c2) corresponds to (a4), (b4) and (c4), respectively, at 10,000 × magnification. The rolling speed is fixed at 0.36 m•s −1 be seen in Fig. 9a that the viscosity of lubricant decreases continuously within 600 s −1 and then remains stable as the strain rate is increased to 1000 s −1 , indicating that the lubricant is a non-Newtonian fluid. For calculating the lubricant film thickness in this work, the viscosity values at the stable stages are averaged, as shown in Fig. 9b. It is found that the viscosity rises with the increased pressure. The pressure-viscosity coefficient ( ) can be determined using the data in Fig. 9b, which aims to predict the viscosities of a lubricant under unmeasured high contact pressures generated in hot steel rolling. The pressure-viscosity coefficient ( ) is defined as: where a p is the pressure shift factor; p and p ref are lubricant viscosities under a selected pressure ( P ) and a reference (15) , respectively, according to the viscositypressure data in Fig. 9b. The calculation result shows that the pressure-viscosity coefficient ( ) is 0.0153 MPa −1 .
In hot steel rolling, the contact pressure gradually increases from the entry to the neutral point [57]. The average contact pressure ( P ) during rolling can be used to approximately represent the pressure applied on the lubricant. P is calculated as: where F is the rolling force; B is the average width of the workpiece before and after rolling; r is the radius of work roll and Δh is the thickness reduction. From Eqs. in the range of 0.003-0.024, indicating a boundary lubrication regime occurred in the hot rolling of 304 stainless steel. This implies that the lubricant film is insufficiently thick to prevent the work roll and the workpiece from direct contact, while the TiO 2 NPs indeed play a more dominant role in lubrication effectiveness. In view of this, the lubrication mechanisms involved are ascribed to three components. First, there exist substantial metal to metal contacts via the surface asperities of work roll and workpiece, which exacerbates the friction in between. Second, the TiO 2 NPs act as ball bearings that can roll in the contact zone, contributing to the decrease in friction. Meanwhile, the NPs can be pressed into the surface defects of the strip, playing a mending effect. Third, some thin lubricant films are likely to be entrapped in the surface pockets, and therefore relieve the direct metal to metal contact to some extent. These lubrication mechanisms are summarised illustratively in Fig. 10.

Conclusions
In this study, the hot rolling of 304 stainless steel was carried out at 1050 ℃ under rolling reductions of 10-30% and rolling speeds of 0.12-0.6 m•s −1 using both water and waterbased lubricant containing 4% TiO 2 NPs. The influence of water-based nanolubrication on rolling force, torque, power and contact friction was systematically investigated, and the lubrication mechanisms were proposed. The main conclusions are drawn as follows: • The rolling force has a linear relationship with the rolling reduction or strain rate. The rolling force with 4% TiO 2 nanolubricant is 6.1% lower than that with pure water lubrication for the ~ 10% reduction with the rolling speed less than 0.36 m•s −1 . • The rolling torque and power obtained using pure water lubrication can be decreased by over 5% in all the rolling tests when using 4% TiO 2 nanolubricant. • The lubrication becomes less effective at relatively high rolling reduction and speed, in terms of load reduction. • Improved COF models are developed with the inclusion of rolling reduction and speed. The models exhibit satisfactory accuracy in the prediction of the COF in the steady-state hot rolling of 304 stainless steel. • A boundary lubrication regime is formed during the hot steel rolling, and the excellent lubricity of the nanolubricant is ascribed to the ball bearing and mending effects of TiO 2 NPs, as well as the formation of thin lubricant films.