Heat transfer in cutting and grinding processes: mechanisms, models, and applications

Abstract

The heat transfer process is a critical topic in the field of cutting and grinding machining, playing a vital role in reducing machining temperatures and improving machining quality. In these operations, heat transfer is generally characterized by specific parameters, including the energy distribution coefficient and the convective heat transfer coefficient. These parameters affect the magnitude and direction of energy flow in the heat transfer process, directly impacting cutting and grinding temperatures. However, comprehensive reviews summarizing current studies on heat transfer processes are lacking. To effectively control heat transfer, thereby managing cutting/grinding temperatures while improving workpiece surface quality, it is essential that we understand how machining parameters, material properties, and cooling methods influence the heat transfer, and this knowledge is critical for guiding the practical production. This paper analyzes and summarizes heat transfer process parameter models in cutting and grinding operations. First, the study examines energy flow and distribution ratios during cutting and grinding processes, classifying and summarizing various energy partition coefficient models based on different research methods. Second, convective cooling mechanisms in cutting and grinding machining are analyzed, summarizing models of convective heat transfer coefficients. The paper then reviews the practical application of these models, highlighting the influence law of each factor on the models. Finally, the most widely recognized and accurate models of heat transfer process parameters are summarized and identified, analyzing the mechanisms by which different factors alter these parameters. Based on current challenges in heat transfer process parameter research, the paper proposes possible future research directions. The goal is to provide theoretical guidance and technical support for advancing research in heat transfer and improving thermal control in machining.

Graphical abstract

Keywords

cutting / grinding / energy distribution coefficient / convective heat transfer coefficient / heat transfer theory

1 Introduction

Thermal damage during machining is one of the main factors limiting the productivity of the workpiece and the quality of the finished product [1]. In industries such as aerospace and toolmaking, machining accuracy and workpiece performance are critical [2–4]. However, operations such as turning, drilling, milling, and grinding generate heat due to the interaction of various parts. In particular, grinding produces a substantial amount of heat that accumulates in the machining area [5–7]. Excessive heat buildup and rising temperature negatively affect the surface quality and performance of the workpiece and may even reduce the lifespan of the cutting tool or grinding wheel [8–10].

A portion of the energy input into the machining area is converted into heat through the interaction between the tool and the workpiece. This heat is transferred between different parts in the machining area via the contact interface. During this process, the amount of heat transferred to each part continuously varies due to multiple influencing factors [11,12]. Aiming to better understand and investigate the mechanisms and variations in heat transfer, specific parameters are used to represent the process. In cutting and grinding machining, most of the generated heat is mainly transferred to the workpiece. In non-dry machining processes, the coolant also absorbs a considerable portion of the heat due to its effective cooling properties [13,14]. Therefore, two key heat transfer process parameters are used: the energy distribution coefficient (EDC) for measuring the amount of heat transferred into the workpiece and the convective heat transfer coefficient (CHTC) for measuring the heat transfer capacity of the fluid. The EDC quantifies the proportion of energy transferred into the workpiece relative to the total energy input into the machining area [15–17]. Aiming to control the surface temperature of the workpiece during machining and ensure quality, the heat transferred to the workpiece should be minimized (that is, the EDC should be reduced) [18]. Through the construction of a model, the influencing factors of the EDC can be identified and adjusted to reduce its value effectively [19,20]. The CHTC characterizes the efficiency of heat exchange between a fluid and a solid surface. In non-dry machining, coolant removes heat from the machining area through thermal convection with the workpiece, tool, or grinding wheel. In addition, high-speed rotation of the workpiece or machining tool can enhance convective heat transfer due to airflow effects. The solution of CHTC is also notably complex due to the influence of machining parameters, workpiece material, coolant type, and flow rate, affecting the efficiency of convective heat transfer [21,22]. Simultaneously, excessive use of coolant raises environment and health concerns [23–25]. Aiming to address these concerns, researchers have explored alternative lubrication methods, such as the minimum quantity lubrication (MQL) [26–28] and nanofluid minimum quantity lubrication (NMQL) [27,29,30], which further complicate the modeling and calculation of CHTC. However, constructing accurate CHTC models enables effective evaluation of convective heat transfer within the machining zone and helps identify strategies to enhance CHTC, thereby reducing heat accumulation, particularly in the grinding zone [31–33].

The study of heat transfer process parameters is conducive to an in-depth understanding of the laws of heat transfer in the machining area and control of the influencing factors, which is of great significance for the control of temperature in the machining process. The EDC reflects the proportion of heat transferred to the workpiece, and the CHTC reflects the heat carried away by the cutting fluid, all of which have a direct influence on the temperature change in the machining zone [34–36].

Domestic and international researchers have extensively studied the heat transfer process parameters, as shown in Fig.1. In the machining process, the main energy source is supplied by the machining tool or grinding wheel. The contact area between the tool and the workpiece becomes a major heat source, generating a substantial amount of heat that accumulates in the machining area [37–39]. Therefore, when studying the heat transfer process, establishing a heat source model within the machining area, commonly represented as either a triangular or rectangular heat source, is necessary [40–42]. Heat generated during machining is distributed among several destinations, including the workpiece, the tool or grinding wheel, the coolant, and the chips. The amount of heat dissipated through other methods is relatively small. Therefore, most researches typically assume that the heat generated in the machining zone is transferred only to the following components: the workpiece, the grinding wheel or tool, the coolant, and the debris [43].

The energy transferred to the workpiece is a primary focus of study, and its proportion relative to the total energy input into the machining area is defined as the EDC. Two main approaches to determining this coefficient are as follows: theoretical analysis or a combined approach with experiments. Models established through theoretical analysis typically do not require experimental data. These models include analytical methods that consider factors such as abrasive grain–workpiece contact [44,45] and the physical properties of materials [46]. The required parameters for these models are inherent to the workpiece or machining tool, along with standard machining parameters, which can be used for simulation and model computation. By contrast, models that incorporate experimental data require not only inherent material and process parameters but also certain values that must be obtained through experimental measurements [47,48]. For instance, models that consider surface temperature rise of the workpiece or the CHTC depend on experimentally measured surface temperature data and CHTC values during machining to construct the model [49,50]. CHTC modeling aims to facilitate energy transfer to the cooling fluid. The difficulty of model construction lies in the complexity of fluid changes during processing and the variety of cooling methods with technological advancements [51–53]. The model is constructed based on different ideas. One category focuses on fluid flow, considering factors such as hydrodynamics, different Péclet numbers, particle contact conditions, and other models. The other category considers the boiling state of the fluid, because varying levels of cam cause the fluid to transition between different boiling states, namely, no boiling, nucleate boiling, transition boiling, and film boiling, which notably impact model construction [28]. Additionally, with the emergence of new cooling methods such as MQL, parameters such as the number of cooling droplets [54,55] and the diffusion area [27,56,57] must be considered.

The research framework is shown in Fig.2, focusing on the study of heat transfer process parameters in machining, specifically, the EDC and CHTC. The two coefficients are investigated across different machining modes, including cutting (turning, milling, and drilling) and grinding. Among these modes, grinding typically generates more heat due to its larger contact area, making it more prone to thermal damage [58,59]. Therefore, EDC and CHTC have been studied more extensively and carefully in the context of grinding. Aiming to investigate heat transfer process parameters across different machining modes, researchers have adopted different ideas and proposed a variety of models. This study compiles and summarizes these models and their applications, aiming to address current problems in heat transfer research and provide valuable references for future studies.

In response to existing challenges, this study presents future research directions for the heat transfer process in cutting and grinding machining. By summarizing the EDC and CHTC models and their applications, the influence of grinding parameters, grinding wheel parameters, workpiece physical parameters, cooling methods, and other factors on heat transfer process parameters, as well as the underlying influence mechanisms, are revealed. The classification emphasizes high-precision models developed under various conditions and analyzes how each factor affects the heat transfer process. The goal is to provide theoretical guidance and technical support for advancing the understanding of heat transfer mechanisms, ultimately contributing to improved workpiece quality and processing efficiency.

2 Energy distribution mechanisms and models

In the machining process, excessive heat is generated due to material deformation, contact friction and other factors, and is transferred to various parts of the processing area. The fraction of heat transferred to the workpiece substantially affects its temperature, making this proportion a key focus in the study of heat transfer process parameters. This proportion is referred to as the EDC R_w. The study of R_w is a prominent research topic domestically and internationally, with many scholars exploring it using different methods.

The EDC is a complex variable to determine; however, its fundamental definition is the ratio of the energy transferred to the workpiece to the total energy input in the machining area. Therefore, the EDC of the workpiece is expressed as follows:

(1)R_{\text{w}} =\frac{q_{\text{w}}}{q_{\text{t}}},

where q_w represents the heat transferred to the workpiece and q_t is the total heat input to the machining area, which is closely related to factors such as machining power. The following section provides an overview of EDC values for different machining methods.

2.1 Turning

Turning is a common metal cutting process widely used in machinery manufacturing, automotive industry, and other fields. This process plays a particularly important role in the production of precision parts [60]. During turning, most of the heat is generated by friction between contact surfaces, while plastic deformation of the workpiece material also contributes to heat generation [61–63]. The high temperatures and heat distribution within the cutting zone are key factors influencing tool wear and overall machining quality [64]. The heat distribution among the cutting tool, chip, and workpiece is primarily determined by the relative thermophysical properties of the tool and workpiece materials [65,66]. The following section presents and summarizes reported EDC models for the cutting process.

2.1.1 Multi-system calculation based on analytical methods

In 2005, Quan and He [67], using an analytical method, assumed that all the cutting deformation work is converted into heat, with no heat lost to the external environment, and that the heat at the contact surfaces is uniformly distributed. As shown in Fig.3, let R₁ represent the proportion of heat flowing to the chip from the shear surface, R₂ the proportion of heat flowing to the workpiece from the tool–chip interface, and R₃ the proportion of heat flowing to the workpiece from the tool–workpiece interface.

For orthogonal free dry cutting, the EDCs for energy distribution are as follows:

(2)R_{\text{1}} =\frac{1}{1+1.328\sqrt{{\displaystyle \frac{a_1{\mathrm{\Lambda }}_{\text{h}}}{v_{\text{c}}{h}_{\text{D}}}}}},

(3)R_2=\frac{q_2(l_{\text{f}}\overline{A}/\phantom{l_{\text{f}}\overline{A}{k}_{\text{t}}}{k}_{\text{t}})-{\overline{\theta }}_{\text{s}}+{\theta }_0^{\mathrm{\prime }}}{q_{\text{2}}\left[{\displaystyle \frac{l_{\text{f}}\overline{A}}{k_{\text{t}}}}+{\displaystyle \frac{0.377l_{\text{f}}}{k_{\text{w}}\sqrt{v_{\text{ch}}{l}_{\text{f}}/\phantom{v_{\text{ch}}{l}_{\text{f}}(4a_2)}(4a_2)}}}\right]},

where a₁ and a₂ are the thermal diffusivity of the workpiece material at different temperatures, Λ_h is the cutting deformation coefficient, v_c is the cutting speed, h_D is the cutting thickness, q₂ is the cutting heat generated per unit of time at the tool–chip interface, l_f is the tool–chip contact length, {\theta }_0^{\prime } is the initial temperature of the tool, {\overline{\theta }}_{\text{s}} is the average temperature of the shear surface, k_t is the thermal conductivity of cutting tools, k_w is the thermal conductivity of the workpiece material, A_t is the tool–chip contact area coefficient, and v_ch is the chip flow rate.

Determining the values of parameters such as heat flow density and friction on the rear blade face is difficult; thus, these values are often ignored in practical calculations.

2.1.2 Based on infinitesimal element

In 2021, Zhao et al. [68] introduced the concept of tool–chip contact infinitesimal element and developed a time-varying tool–chip EDC prediction model for coated tools. They clarified the time-varying effect of tool–chip heat distribution coefficients and revealed the coupled effects of tool coating thickness, tool coating, tool substrate, and the thermophysical properties of the workpiece. Additionally, they examined the influence of intermittent milling and machining speed on heat distribution at the tool–chip interface.

As shown in Fig.4, the infinitesimal element model of the cutting process using a coated tool is taken as a semi-infinite body, based on the principles of energy conservation and the Block criterion. At a given moment t, the tool–chip contact infinitesimal element is also considered a semi-infinite body. The coated tool is assumed to maintain perfect thermal contact with the chip, indicating no additional thermal resistance at the interface. Furthermore, the materials are considered isotropic with constant physical properties, and no internal heat generation occurs within either the coated tool or the chip. Under these assumptions, the time-varying distribution coefficient of the tool–chip interface for the coated tool can be calculated as follows:

(4)R_{\text{chip}}( t)=\frac{A(x_{\text{1}},k_{\text{c}},{\lambda }_{\text{c}},k_{\text{s}},{\lambda }_{\text{s}},t)}{A(x_1,k_{\text{c}},{\lambda }_{\text{c}},k_{\text{s}},{\lambda }_{\text{s}},t)+3{\left[2B(k_{\text{w}},{\lambda }_{\text{w}},t)\right]}^{-1}},

where

\begin{array}{rl}& A(x_1,k_{\text{c}},{\lambda }_{\text{c}},k_{\text{s}},{\lambda }_{\text{s}},t)=\frac{4\sqrt{{\lambda }_{\text{c}}}}{k_{\text{c}}}\cdot \sum _{m=0}^{\mathrm{\infty }}{\left|C\right|}^{n+1}\\ & \cdot \sqrt{\frac{t}{\text{π}}}\{\exp [-\frac{{(n+1)}^2{x}_1^2}{t{\lambda }_{\text{c}}}]\\ & -\text{erfc}[-\frac{(n+1)x_1}{\sqrt{t\cdot {\lambda }_{\text{c}}}}]\}+\frac{2}{k_{\text{c}}}\sqrt{\frac{t{\lambda }_{\text{c}}}{\text{π}}}\end{array}

is the time-varying coefficient considering the coating thickness x₁, the coating thermal conductivity k_c, the coating thermal conductivity rate λ_c, the substrate thermal conductivity k_s, and the substrate thermal conductivity rate λ_s, while

B(k_{\text{w}} ,{\lambda }_{\text{w}},t) =\frac{2}{k_{\text{w}}}\sqrt{\frac{t{\lambda }_{\text{w}}}{\text{π}}}

is the time-varying coefficient considering the workpiece thermal conductivity k_w and the workpiece thermal conductivity rate λ_w.

2.2 Milling

Milling is mainly used to remove material and shape the workpiece to the desired dimensions. The heat generated during the milling process mainly originates from the friction between the tool and the workpiece surface. This heat directly affects tool wear and durability, as well as the machining accuracy and surface quality of the workpiece [69].

Komanduri and Hou [70] presented an analytical model of orthogonal cutting to determine energy distribution. They also confirmed the relationship between energy distribution in the main shear zone in orthogonal cutting for different materials and cutting conditions and the heat number N_th.n = h_u·v_c·α⁻¹, where h_u is the undeformed chip thickness and α is the temperature diffusion coefficient of the workpiece material. To this day, research on the EDC in milling processes has mainly focused on the development of empirical models for determining energy distribution during cutting.

In 2012, Sölter and Gulpak [71] discretized the grinding arc into four domains V_i and determined the heat flux from the workpiece by iteratively applying an inverse process. They assumed that the associated cutting power remained approximately constant within each domain.

As shown in Fig.5, assuming that 100% of the spindle’s effective power is converted into mechanical energy, the proportion of energy R_w dissipated to the workpiece within the domain Ω_i can be calculated as follows:

(5)R_{\text{w,}i}= \frac{q_{\text{w}}{A}_i}{P_{\text{el}.\text{meas}.i}},

where A_i is the area of region Ω_i and P_el.meas.i is the effective spindle power.

2.3 Drilling

Drilling is mainly used for creating holes in workpieces. In the drilling process, when the drill bit contacts the workpiece material, friction generates a substantial amount of heat, especially at high drilling speeds [72,73]. During the stripping and movement of chips, the friction between the chips, drill bit, and workpiece also generates heat [74]. Furthermore, the heat flowing into the tool is unfavorable for surface processing in plough-slip [75].

In 2016, Cuesta et al. [76] proposed a model based on a new experimental approach to determine the difference in heat transfer to the workpiece between dry and lubricated drilled holes. The temperature of the sidewall of a drilled hole depends on the heat distribution between the chip and the workpiece. The heat flow in the drilled hole can be considered as an instantaneous transfer to the surface drilled, which is then propagated by conduction. As shown in Fig.6, the heat in the workpiece can be approximated as a cylindrical annulus. Therefore, the heat transferred to the workpiece is as follows:

(6)q_{\text{w}} =mcT= L\rho (\text{π}/4)(d_2^2- d_1^2)cT,

where c is the specific heat capacity, ΔT is the temperature rise, L is the depth of the borehole, d₁ is the borehole diameter, d₂ is the diameter of the heated cylinder, and ρ is the density.

Assuming that all work is converted to heat, the heat produced during drilling (q_t) is calculated as shown below.

(7)q_{\text{t}} =W=( \text{π}/\phantom{\text{π}4}4)\cdot d_{\text{drill}}^2\cdot L{K}_{\text{s}} ,

where K_s is the ratio of cutting force to cutting cross-sectional area, d_drill is the diameter of the drill bit, and the proportion of energy entering the workpiece can be calculated from the definition. Subsequently,

(8)R_{\text{w}} =\frac{q_{\text{w}}}{q_{\text{t}}}=\frac{\rho (d_2^2-d_1^2)c\mathrm{\Delta }{T}_{max}{f}_{z}z}{8M_z},

where f_z is the feed per tooth, z is the number of teeth, and M_z is the torque.

Studies on modeling the EDC coefficient of proportionality in planning and boring machining are limited, and no relevant models currently exist; thus, they will not be discussed further in this study. Tab.1 provides a summary of the factors influencing the EDC model in cutting machining.

2.4 Grinding

Grinding is a machining process suitable for finishing high-hardness materials and is generally the final process for important components. Therefore, ensuring that components of acceptable quality are produced is crucial [77,78]. For some materials, thermal damage is a major limiting factor in improving removal rates while maintaining surface quality [79,80].

Simultaneously, during grinding, material is removed by a large number of abrasive particles with a substantially shallow depth of cut. Negative cutting angles result in higher energy consumption per unit volume of material removed compared to other cutting processes, such as milling and turning [81–83]. A schematic of the heat sources generating heat around the abrasive grits is shown in Fig.7. Heat is generated at the following three locations [84]:

(a) Grain wear flat–workpiece interface, AB

(b) Shear surface, BC

(c) Abrasive grain and Grinding debris contact interface, BD

The EDC for the grinding process has been studied by many scholars domestically and internationally. In their investigations, some scholars have derived models based on the grinding wheel’s abrasive grains, such as the single-grain model and the abrasive grains–workpiece contact model [85], among others. Other scholars combine experimental research, consider the physical properties of the processed material, or measure grinding temperatures through grinding experiments to derive energy distribution models. Additionally, some researchers have explored convection heat transfer, heat-source inclination, and other factors to conduct an in-depth study of the model. Below are the EDC models that have been reported or introduced in the grinding process.

2.4.1 Based on a single grain of grinding wheel

The EDC model for grinding, based on the contact between a single abrasive grain and the workpiece, was first proposed by Rowe [84] to measure the thermal attributes of a single abrasive grain through grinding tests. The proportional model of heat distribution for a single abrasive grain is applicable to normal and slow-feed grinding processes, regardless of the grinding process parameters and cooling methods.

GUO et al. [86] constructed a numerical calculation model for the profile grinding temperature, using the distribution of a triangular heat source within the grinding arc and simplifying the one-dimensional heat-conducting process. This model was also based on the theoretical analysis of the continuous grinding temperature of a plane. Therefore, the abrasive particle was modeled as a truncated cone.

On this basis, Guo [87] conducted a theoretical discussion and experimental analysis of the profile grinding temperature. Using this model, the proportion of energy entering the workpiece material in the ordinary flat grinding processing zone can be calculated, as shown in Fig.8. The conical abrasive grain moves along the surface of the workpiece with a velocity of v_s. The grinding heat is conducted along the contact surface A_0=\text{π}{r}_0^2 between the abrasive grain and the workpiece, and is transferred to the grain, the workpiece, and the coolant. The proportion of heat transferred to the workpiece at the initial moment is R_dry, while the remaining heat is transferred to the abrasive grain. The maximum temperature rise at the abrasive grain–workpiece contact surface is T_gmax.

(9)\{\begin{array}{c}{T}_{\text{gmax}}=1.13\frac{(1-R_{\text{dry}})q_{\text{t}}}{(k\rho c{)}_{\text{g}}^{1/2}}{\left(\frac{l_{\text{c}}}{{\nu }_{\text{s}}}\right)}^{1/2}\frac{1}{f(\zeta )A},\\ \zeta \equiv {\left(\frac{{\gamma }^2\text{π}{\alpha }_{\text{g}}{l}_{\text{c}}}{2A_0{\nu }_{\text{s}}}\right)}^{1/2},\end{array}

where the subscript g refers to the abrasive grain, and α_g represents the thermal diffusivity of the abrasive grain. The function γ is the geometry coefficient of a single grain, defined as γ = d_rg/d_z. A is the ratio of the total wear area of the protruding abrasive grains on the grinding wheel surface to the total area of the grinding wheel. Additionally, r_g denotes the radius of the top of the conical abrasive grains.

When the coolant enters the grinding wheel aperture, its running speed matches that of the grinding wheel. At this time, the contact surface between the workpiece, as well as the highest temperature rise of the machined surface, are T_fmax and T_wmax, respectively.

(10)T_{\text{fmax}}=1.06 \frac{(R_{\text{dry}}-R_{\text{w}})q_{\text{t}}}{(k\rho c{)}_{\text{f}}^{1/2}}{\left(\frac{l_{\text{c}}}{v_{\text{s}}}\right)}^{1/2}\frac{1}{1-A} ,

(11)T_{\text{wmax}}=1.06 \frac{R_{\text{w}}{q}_{\text{t}}}{(k\rho c{)}_{\text{w}}^{1/2}}{\left(\frac{l_{\text{c}}}{v_{\text{w}}}\right)}^{1/2},

where the subscripts f and w denote the coolant and the workpiece material, respectively.

Assuming that the maximum temperature rise at the contact surface between the workpiece and the abrasive grain, the contact surface between the workpiece and the coolant, and the machined surface of the workpiece are all equal (i.e., T_gmax = T_fmax = T_wmax), the heat distribution ratio of the heat transferred into the inner part of the workpiece material, R_w, can be calculated as follows [86,88]:

(12)R_{\text{w}} ={\left[1+\mathrm{\Omega }{\left(\frac{v_{\text{s}}}{v_{\text{w}}}\right)}^{1/2}\right]}^{-1},

(13)\mathrm{\Omega }\equiv 0.94\frac{(k\rho c{)}_{\text{g}}^{1/2}}{(k\rho c{)}_{\text{w}}^{1/2}}Af( \zeta )+ \frac{(k\rho c{)}_{\text{f}}^{1/2}}{(k\rho c{)}_{\text{w}}^{1/2}}(1-A).

When the temperature of the grinding arc exceeds the boiling point temperature of the coolant, the coolant enters the bubble nucleation boiling state, enhancing the convective heat transfer capacity. However, when the temperature of the grinding arc exceeds 130 ℃, the coolant in the arc area is transitions into a film boiling state, notably reducing its heat transfer capacity. At this point, the coolant loses its ability to transfer heat and cool the system. In typical grinding processes, the grinding temperature in the contact arc zone can exceed 130 ℃, thereby causing (kρc)_f = 0. The single grain heat distribution ratio model does not consider the energy carried away by the chips. The melting energy of steel-like metal materials is approximately 6 J/mm³, which is equivalent to 45% of the chip energy. Therefore, the proportion of heat removed by the chip, R_ch, can be calculated as follows:

(14)R_{\text{ch}}=\frac{0.45e_{\text{sch}}}{e_{\text{s}}}\approx \frac{6}{e_{\text{s}}},

where e_sch is the specific chip formation energy and e_s is the specific grinding energy.

Considering the heat carried away by the chips and subtracting the proportion of this heat, the model for the heat distribution ratio of a single grain gives the proportion of energy entering the workpiece, R_w, for ordinary flat grinding as follows [89]:

(15)\begin{array}{rl}& R_{\text{w(AB)}}=R_{\text{w(EF)}}\\ =& \left(1-\frac{6}{e_{\text{s}}}\right){\left[1+0.94\frac{(k\rho c{)}_{\text{g}}^{1/2}}{(k\rho c{)}_{\text{w}}^{1/2}}Af(\zeta ){\left(\frac{v_{\text{s}}}{v_{\text{w}}}\right)}^{1/2}\right]}^{-1}.\end{array}

In 2010, Xu et al. [90] improved the traditional heat-source model by integrating the heat transfer process, the end surface temperature of single wear grains, and a one-dimensional heat transfer model to analyze the dynamic effects of grinding on heat transfer. They then studied the energy distribution ratio. As shown in Fig.9, the wear grains are assumed to be cylindrical, and thermal convection between the sides of the grains and the surrounding air is neglected. Under these conditions, the heat transfer state of the wear grains can be modeled as a semi-infinite cylindrical unsteady state with one-dimensional heat transfer, subject to the first boundary condition.

In time t, the heat transfer is q_a per square meter, and then the heat transferred to the workpiece is calculated as follows:

(16)q_{\text{w}} =q_{\text{t}}-q_{\text{a}}\text{π}{R}^2=F_{\text{t}} v_{\text{s}} {\tau }_0-2b \sqrt{\frac{{\tau }_0}{\text{π}}}({\theta }_{\text{a}}-{\theta }_0)\text{π}{R}^2,

where F_t is the tangential grinding force, b is the heat storage coefficient, θ₀ is the average temperature of the abrasive grain, and θ_a is the surface temperature of the workpiece. Introducing the empirical formula for the grinding force F_t, the EDC is given by the following:

(17)R_{\text{w}} =1-\frac{2b({\theta }_{\text{a}}-{\theta }_0)\sqrt{\text{π}}{R}^2}{C_1{v}_{\text{s}}^{-x+1/2}{v}_{\text{w}}^y{a}_{\text{p}}^{z+1/4}{d}_{\text{s}}^{1/4}}.

Based on Eq. (17), R_w is influenced by the thermal properties of the wear grain and the coating material, as well as by the mechanical properties of the coating material and the grinding parameters.

2.4.2 Considering abrasive grain–workpiece contact

In 2001, Rowe [84] studied the heat-source modeling using an inclined heat-source model, as shown in Fig.10. In this model, the total heat directed toward the contact zone is divided into four directions, with the proportions of heat in each direction summing up to the total heat.

(18)q_{\text{t}} =q_{\text{w}}+q_{\text{s}}+q_{\text{ch}}+q_{\text{f}},

(19)1=R_{\text{w}}+R_{\text{s}}+R_{\text{ch}}+R_{\text{f}},

where q_s is the heat flow to the abrasive grain, q_ch is the heat flow to the chip, and q_f is the heat flow to the coolant. R_w = q_w/q_t, and so on.

After rearrangement, the heat absorbed by the workpiece and the grain can be determined.

(20)q_{\text{ws}}=q_{\text{w}} +q_{\text{s}}=q_{\text{t}}-q_{\text{ch}}-q_{\text{f}}.

The heat of the abrasive chips, q_ch, and the heat of the fluid, q_f, can be estimated based on the processing conditions. The temperature of the abrasive chips is assumed to be slightly below the melting temperature, and the fluid is divided into boiling and non-boiling cases. For the non-boiling case, the temperature in the contact region is approximately two-thirds of the maximum temperature. Therefore, based on the model of Hahn [84], the workpiece–grain particle energy distribution ratio, R_ws, can be obtained as follows:

(21)R_{\text{ws}}=\frac{q_{\text{w}}}{q_{\text{w}}+q_{\text{s}}}={\left[1+\frac{k_{\text{g}}}{\sqrt{r_0{v}_{\text{s}}{(k\rho c)}_{\text{w}}}}\right]}^{-1}.

R_ws depends on the thermal conductivity of the abrasive particles k_g, the thermal properties of the workpiece {\beta }_{\text{w}} =\sqrt{{(k\rho c)}_{\text{w}}}, the rotational speed of the grinding wheel v_s, and the radius of the particles to wear r₀. The total wear area increases with the wear of the grinding wheel. Therefore, the radius of wear will also increase as the grinding wheel is used.

In 2018, Guo [87] analyzed the temperature in profile grinding theoretically and experimentally. As shown in Fig.11, during the grinding process with an integral grinding wheel, the closed area between the grinding wheel and the workpiece (i.e., between B and E) makes it difficult for the coolant to penetrate, preventing effectively cooling. Therefore, most of the grinding heat is carried away by the workpiece and the grinding wheel. The high temperature in the grinding contact arc area causes the coolant to enter a film boiling state, losing its cooling capability. Therefore, the proportion of grinding heat removed by the coolant is assumed to be R_f = 0.

Subtracting the heat carried away by the chips, the ratio of energy transferred to the workpiece is as follows [84,87]:

(22)\begin{array}{rl}{R}_{\text{w}}=& R_{\text{wg}}(1-R_{\text{ch}})\\ \approx & \left(1-\frac{6}{e_z}\right){\left[1+\frac{k_{\text{g}}}{{(r_{\text{0}}{\nu }_{\text{c}})}^{1/2}(k\rho c{)}_{\text{w}}^{1/2}}\right]}^{-1}.\end{array}

Rowe [84] proposed an energy distribution ratio model for surface grinding based on the contact between the workpiece and the abrasive grits, combined with the heat transfer coefficient. The schematic of the abrasive grain grinding process is shown in Fig.12. During the grinding process, the wear planes of the abrasive grains generate heat, which is partly transferred to the workpiece and partly to the abrasive grains. Rowe correlated the theoretical model for the predicted distribution ratio with experimental results, which led to the development of an effective model applicable to cubic boron nitride (CBN) and aluminum oxide abrasives. Based on the solution provided by the circular heat-source model for a semi-infinite body, as shown in Fig.13(a), the heat transfer coefficient of the workpiece, h_w, was obtained.

(23)h_{\text{w}} =\frac{(k\rho c{)}_{\text{w}}^{1/2}{v}_{\text{s}}^{1/2}}{0.974r_0^{1/2}}.

As shown in Fig.13(b), the abrasive grain is assumed to be a stationary object that is continuously subjected to a steady circular heat flux during grinding. The abrasive grain is treated as an infinitesimal protrusion on a semi-infinite object. Based on this model, the heat transfer coefficient h_g of the abrasive grain can be determined.

(24)h_{\text{g}} =\frac{k_{\text{g}}}{2r_0\tau }\left[\frac{1}{{\text{π}}^{1/2}}-\text{ierfc}\left(\frac{1}{2\tau }\right)\right],

where the time constant \tau =({\alpha }_{\text{g}}{l}_{\text{c}}/\phantom{{\alpha }_{\text{g}}{l}_{\text{c}}{r}_0^2{v}_{\text{s}}}{r}_0^2{v}_{\text{s}}{)}^{1/2}.

By incorporating the two heat transfer coefficients into the transient EDC model, the proportion of energy transferred to the workpiece material at the modified transient moment can be obtained as follows:

(25)\begin{array}{rl}{R}_{\text{w}}=& (1-R_{\text{ch}})\frac{1}{1+q_{\text{g}}/\phantom{q_{\text{g}}{q}_{\text{wg}}}{q}_{\text{wg}}}\\ =& (1-R_{\text{ch}})\frac{1}{1+h_{\text{g}}/\phantom{h_{\text{g}}{h}_{\text{wg}}}{h}_{\text{wg}}}\\ \approx & \left(1-\frac{6}{e_{\text{s}}}\right){\left[1+\frac{0.974k_{\text{g}}}{{(r_{\text{0}}{\nu }_{\text{s}})}^{1/2}(k\rho c{)}_{\text{w}}^{1/2}}\frac{1}{\Phi (\tau )}\right]}^{-1}.\end{array}

(26)\begin{array}{rl}\varphi (\tau )=& 1+\frac{1}{6{\tau }^2}+\frac{4\tau }{3\sqrt{\text{π}}}\\ & \cdot \left\{1-{\left[\exp \left(\frac{1}{4{\tau }^2}\right)\right]}^{-1}\left(1+\frac{1}{4{\tau }^2}\right)\right\}\\ & -\text{erf}\left(\frac{1}{2\tau }\right)\left(1+\frac{1}{6{\tau }^2}\right).\end{array}

The heat transfer coefficient of the abrasive grain, h_g = k_g/r₀, and the time constant τ approaches infinity when the heat transfer reaches a steady state. At this point, in the steady state, the proportion of energy transferred to the workpiece material in the closed region between point B and point E is as follows:

(27)R_{\text{w(BE)}}=\left(1-\frac{6}{e_{\text{s}}}\right)\cdot {\left[1+\frac{0.974k_g}{{(r_0{v}_{\text{s}})}^{1/2}(k\rho c{)}_{\text{w}}^{1/2}}\right]}^{-1}.

In their model of abrasive grain and workpiece contact, as shown in Fig.14, Li and Zhao [91] indicated that the nominal contact area between the grinding wheel and the workpiece is denoted by A, while the actual contact area is A_R. According to the linearized model, the energy E_w entering the workpiece per unit of time due to thermal convection is as follows:

(28)E_{\text{w}} =\frac{{(c_{\text{p}})}_{\text{w}}{\theta }_{\text{m}}}{2}\cdot (v_{\text{w}}{b}_{\text{s}}{\overline{a}}_{\text{p}}{)}_{\text{w}}=\frac{{\theta }_{\text{m}}{b}_{\text{s}}}{2}\cdot {\left[2{(k\rho c_{\text{p}})}_{\text{w}}{v}_{\text{w}}{l}_{\text{c}}\right]}^{1/2},

where c_p denotes the mass constant pressure specific heat capacity, θ_m is the temperature, b_s is the grinding width, a_p is the grinding depth, and (kρc_p)_w is related to the thermal properties of the workpiece.

The energy E_s entering the grinding wheel per unit time due to thermal convection is as follows:

(29)\begin{array}{rl}{E}_{\text{s}}=& \frac{{(c_{\text{p}})}_{\text{s}}{\theta }_{\text{m}}}{2}\cdot \left(v_{\text{s}}\frac{A_{\text{R}}}{A}\cdot b\cdot {\overline{a}}_{\text{p}}\right)\\ =& \frac{1}{2}{\theta }_{\text{m}}b{\left[2{(k\rho c_{\rho })}_{\text{s}}\cdot v_{\text{s}}\cdot \frac{A_{\text{R}}}{A}l\right]}^{1/2}.\end{array}

The resulting EDC coefficients are presented as

(30)R=\frac{E_{\text{w}}}{E_{\text{w}}+E_{\text{s}}}=\frac{1}{{\left[1+{\displaystyle \frac{{(k\rho c_{\text{p}})}_{\text{s}}}{{(k\rho c_{\text{p}})}_{\text{w}}}}{\displaystyle \frac{v_{\text{s}}}{v_{\text{w}}}}{\left({\displaystyle \frac{A_{\text{R}}}{A}}\right)}_{\text{s}}\right]}^{1/2}}.

For flat grinding, (v_w/v_s)(A_R/A)_s ≈ 1. Thus, the EDC of energy distribution under flat grinding conditions is computed as follows:

(31)R=\frac{1}{{\left[1+{(k\rho c_{\text{p}})}_s/\phantom{{(k\rho c_{\text{p}})}_{\text{s}}{(k\rho c_{\text{p}})}_{\text{w}}}{(k\rho c_{\text{p}})}_{\text{w}}\right]}^{1/2}}.

When the effect of the coolant is considered, the energy transferred to the grinding wheel is equal to the sum of the energy imparted to the abrasive particles and the energy transferred to the coolant on the surface of the wheel. Therefore, under wet grinding conditions, the proportion of energy transferred to the grinding wheel is as follows [92]:

(32)\begin{array}{rl}R=& \frac{R_{\text{w}}}{R_{\text{w}}+R_{\text{s}}}=\frac{E_{\text{w}}}{E_{\text{w}}+E_{\text{s}}}\\ =& {\left[1+\sqrt{\frac{{(k\rho c_{\text{p}})}_{\text{s}}}{{(k\rho c_{\text{p}})}_{\text{w}}}\frac{v_{\text{s}}}{v_{\text{w}}}{\left(\frac{A_{\text{R}}}{A}\right)}_{\text{s}}}+\sqrt{\frac{{(k\rho c_{\text{p}})}_{\text{f}}}{{(k\rho c_{\text{p}})}_{\text{w}}}\frac{v_{\text{s}}}{v_{\text{w}}}}\right]}^{-1},\end{array}

where (kρc_p)_f is related to the coolant and (kρc_p)_s is related to the abrasive grains on the surface of the grinding wheel.

2.4.3 Considering the surface temperature of the workpiece

In the grinding process, the surface temperature rise of the workpiece can be measured to calculate the amount of heat absorbed by the workpiece [93,94]. This measurement serves as the basis for calculating the proportion of energy distribution during the grinding process.

For example, Rowe [84] obtained the energy distribution ratio by relating the maximum background temperature to the heat entering the workpiece based on the Jaeger solution for a sliding heat source and corrections for different flux distributions.

(33)R_{\text{w}} \text{=}\frac{\text{π}}{C}b\cdot \sqrt{{\left(k\rho \text{c}\right)}_{\text{w}}{v}_{\text{w}}{l}_{\text{e}}}\cdot \frac{{\theta }_{\text{m}}}{q_{\text{t}}},

where θ_m is the maximum background temperature. The value of C depends on the shape of the heat source and the value of the Péclet number.

When the Péclet number is high, the C value is shown in Tab.2.

In 2011, Hadad et al. [95] investigated the temperature and energy distribution during MQL grinding. The experiments were conducted using CBN and Al₂O₃ grinding wheels. As shown in Fig.15, the experiment treats the contact zone as a strip heat source of length l_c, which moves along the surface of the workpiece with a velocity v_ft to calculate the temperature of the contact zone.

An assumption is that the contact zone heat is transferred to the workpiece, grinding wheel, chips and coolant during the grinding process (Fig.16). Based on the assumption of a triangular heat-source distribution in the x–z plane, the temperature rise of the workpiece is as follows:

(34)\begin{array}{rl}{T}_0(x,z)=& \frac{q_{\text{w}}}{\text{π}{k}_{\text{w}}}{\int }_{-l_{\text{c}}/\phantom{l_{\text{c}}2}2}^{l_{\text{c}}/\phantom{l_{\text{c}}2}2}\exp \left[-V_{\text{ft}}(x-a)/\phantom{-V_{\text{ft}}(x-a)(2x)}(2x)\right]\\ & \cdot K_0\left\{\frac{V_{\text{ft}}}{2{\alpha }_{\text{w}}}{\left[{(x-a)}^2+z^2\right]}^{1/2}\right\}f(a)\text{d}a,\end{array}

where f(a) = 1 + 2a/l_c is the triangular heat-source distribution and K₀ is the Type II zero-order modified Bessel function.

The maximum temperature rise T_max of the workpiece during grinding was measured in the experiment, and the total energy input can be calculated by measuring the tangential grinding force as q_t as shown below:

(35)q_{\text{t}} =\frac{F_{\text{t}}{V}_{\text{c}}}{b{l}_{\text{c}}},

where F_t is the tangential grinding force, b is the workpiece width, and l_c is the actual grinding contact length.

Therefore, calculating the energy distribution of the workpiece by evaluating the maximum temperature rise T_max is possible.

(36)T_{max}=\frac{\beta q_{\text{w}}{\alpha }_{\text{w}}^{1/2}{l}_{\text{c}}^{\text{1/2}}}{k_{\text{w}}{V}_{\text{ft}}^{1/2}},

where k_w denotes the thermal conductivity of the workpiece, α_w is the heat diffusivity of the workpiece, and β denotes a constant related to the shape of the heat source (β = 1.06 for triangularly distributed heat sources).

In 2021, Peng et al. [96] investigated the height of the energy distribution ratio and its changing law by high-speed grinding of nickel-based high-temperature alloys. As shown in Fig.17, for high-speed grinding, an inclined triangular heat-source is more appro-priate, and the angle Φ between the line of motion and the plane of the belt source can also more accurately represent the amount of moving heat, the maximum grinding temperature rise T_max on the contact workpiece surface based on the one-dimensional conduction sliding heat-source theory can be expressed as [37,97]

(37)T_{max}=C_0\cdot \frac{q_{\text{w}}}{{\beta }_{\text{w}}}\cdot \sqrt{\frac{l_{\text{c}}}{{\nu }_{\text{w}}}},

where C₀ is the parameter factor in the circular moving heat-source model. {\beta }_{\text{w}}=\sqrt{{(k\rho c)}_{\text{w}}}. For the value of C₀ [97], the shape of the heat flow density distribution during the grinding process is approximately triangular. The sliding heat-source solution gives a value of C₀ approximately equal to 1.

Therefore, the energy distribution ratio R_w to the workpiece can be simply calculated using the measured grinding surface temperature T_s:

(38)\{\begin{array}{c}{R}_{\text{w}}=\frac{q_{\text{w}}}{q_{\text{t}}}=\frac{T_{max}{h}_{\text{w}}}{q_{\text{t}}}=\frac{(T_{\text{s}}-T_0)\cdot h_0}{q_{\text{t}}},\\ h_{\text{w}}=\frac{{\beta }_{\text{w}}}{C}\sqrt{\frac{{\nu }_{\text{w}}}{l_{\text{c}}}},\end{array}

where T₀ is the ambient temperature. T₀ = 20 ℃, and h₀ is defined as the heat transfer factor, which is determined by the thermal properties of the workpiece, the speed of movement of the heat-source, and the contact geometry.

In 2018, Ding et al. [98] investigated the temperature and energy distribution in the cylindrical grinding process under various process and thermophysical parameter conditions. They used a physical model-based approach, combined with finite element simulation and experimental validation. The research emphasized the effect of workpiece speed on energy distribution.

Temperature modeling was conducted using a moving tilted triangular strip source temperature model, based on the Jaeger model:

(39)\begin{array}{rl}T=& \frac{8{\alpha }_{\text{w}}^{2}q}{\text{π}k{l}_{\text{c}}{\nu }_{\text{w}}^2}\cdot {\int }_{x{\nu }_{\text{w}}/\phantom{x{\nu }_{\text{w}}(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}^{(x-l_{\text{c}})\nu /\phantom{(x-l_{\text{c}})\nu (2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}u\cdot \exp (-u)\cdot k_0(u)\text{d}u\\ & -\frac{4{\alpha }_{\text{w}}qx}{\text{π}k{l}_{\text{c}}{\nu }_{\text{w}}}\cdot {\int }_{x{\nu }_{\text{w}}/\phantom{x{\nu }_{\text{w}}(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}^{(x-l_{\text{c}})\nu /\phantom{(x-l_{\text{c}})\nu (2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}\exp (-u)\cdot k_0(u)\text{d}u.\end{array}

The model for calculating the energy distribution of the workpiece, based on the maximum temperature of the workpiece, can be expressed as follows:

(40)\begin{array}{rl}{R}_{\text{w}}=& T_{max}\cdot \{P_{\text{net}}^{\mathrm{\prime }}[k_1{\int }_{xv/\phantom{xv(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}^{(x-l_{\text{c}})v/\phantom{(x-l_{\text{c}})v(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}{u}^2\cdot \exp (-u)\cdot k_0(u)\text{d}u\\ & +k_2{\int }_{xv/\phantom{xv(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}^{(x-l_{\text{c}})v/\phantom{(x-l_{\text{c}})v(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}u\cdot \exp (-u)\cdot k_0(u)\text{d}u\\ & +k_3{\int }_{xv/\phantom{xv(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}^{(x-l_{\text{c}})v/\phantom{(x-l_{\text{c}})v(2{\alpha }_{\text{w}})}(2{\alpha }_{\text{w}})}\exp (-u)\cdot k_0(u)\text{d}u]\}.\end{array}

In 2003, Li and Zhao [91] stated that the heat transferred to the workpiece during the grinding process notably affects the temperature of the grinding zone and referred to the energy distribution model. In 2019, Zhang et al. [99] applied this model to the MQL grinding of Ti–6Al–4V. The density of heat flow into the workpiece during grinding is as follows:

(41)q_{\text{w}} =\frac{q_{\text{w}}}{q_t}\cdot q_t=\frac{{\nu }_{\text{s}}{F}_{\text{t}}}{b{l}_{\text{c}}}\cdot R= \frac{{\nu }_{\text{s}}{F}_{\text{t}}}{b\sqrt{d_{\text{s}}{a}_{\text{p}}}}\cdot R.

The energy scaling factor can be expressed as shown below.

(42)R=\frac{k_{\text{w}}{\nu }_{\text{w}}^{1/2}}{q_{\text{t}}\beta {\alpha }_{\text{w}}^{1/2}{a}_{\text{p}}^{1/4}{d}_{\text{s}}^{1/4}}{\theta }_{max},

where θ_max is the peak rise of temperature.

2.4.4 Based on material physical parameters

Mao et al. [100] considered various factors to re-evaluate the heat distribution ratio into the workpiece. On this basis, Rowe [97] conducted a thermal analysis of efficient deep grinding to explore optimal working conditions for the process.

In this analysis, the heat in the grinding zone is assumed to flow in the following four directions: workpiece, grinding wheel, chips, and fluid. The total heat flow is as follows:

(43)q_{\text{t}} =q_{\text{w}}+q_{\text{S}}+q_{\text{ch}}+q_{\text{f}}.

The heat flow to the workpiece, grinding wheel, fluid, and chips can be expressed in terms of convection/conduction factors [101]. This heat flow is related to the maximum contact temperature T_max-c and the fluid burnout temperature T_b, while the heat of the chips q_ch, is related to the average temperature of the workpiece \overline{T}, but not to the melting temperature of the workpiece material T_w.

(44)\begin{array}{rl}& q_{\text{w}}=h_{\text{w}}{T}_{\text{max}}{}_{-\text{c}},\\ & q_{\text{s}}=h_{\text{s}}{T}_{\text{max}-\text{c}},\\ & q_{\text{f}}={h_{\text{f}}{T}_{\text{max}-\text{c}}|}_{T_{\text{max-c}}⩽T_{\text{b}}},\\ & q_{\text{ch}}=h_{\text{ch}}\overline{T}.\end{array}

An assumption is that the heat flowing into the chip enables it to reach the critical temperature for melting; thus,

(45)q_{\text{ch}}={\rho }_{\text{w}} c_{\text{w}} T_{\text{w}}\frac{a_{\text{p}}{v}_{\text{w}}}{l_c}.

The heat flow into the coolant q_f requires suitable experimental data. Two main types of coolant are available: pure oil and oil-in-water emulsion. For the oil-in-water emulsion, q_f = 0 is assumed under boiling conditions [97]. During the grinding process, the temperature in the grinding zone is typically high and can easily reach the boiling point of the coolant. At this time, the coolant and the surface of the workpiece form an insulating layer due to vaporization, notably reducing the heat transfer between the coolant and the workpiece surface [102]. Therefore, the heat flow density transferred to the coolant is assumed to be zero, denoted as q_f = 0.

Based on Hana’s distribution model, the grinding wheel and the workpiece are regarded as a subsystem, and the total heat of the grinding wheel and the workpiece is q_ws = q_w + q_s. Thus, the “workpiece–grinding wheel” distribution ratio R_ws is defined as shown below.

(46)R_{\text{ws}}=\frac{q_{\text{w}}}{q_{\text{w}}+q_{\text{s}}}={\left[1+\frac{0.974k_{\text{g}}}{\sqrt{{(k\rho c)}_{\text{w}}{r}_0{v}_{\text{s}}}}\right]}^{-1}.

The following represents the heat flow into the workpiece during grinding.

(47)\begin{array}{rl}{q}_{\text{w}}=& R_{\text{ws}}\left(q_{\text{t}}-q_{\text{ch}}-q_{\text{f}}\right)\\ =& {\left[1+\frac{0.974k_{\text{g}}}{\sqrt{{(k\rho c)}_{\text{w}}{r}_0{v}_{\text{s}}}}\right]}^{-1}\cdot \left(\frac{F_{\text{t}}{\nu }_{\text{s}}}{l_{\text{c}}b}-{\rho }_{\text{w}}{c}_{\text{w}}{T}_{\mathrm{w}}\frac{a_{\text{p}}{\nu }_{\text{w}}}{l_{\mathrm{c}}}\right).\end{array}

By substituting the above heat model of the workpiece, the proportion of energy flowing into the workpiece can be obtained.

(48)R_{\text{w}} ={\left[1+\frac{0.974k_{\text{g}}}{\sqrt{{(k\rho c)}_{\text{w}}{r}_0{v}_{\text{s}}}}\right]}^{-1}\left(1-\frac{{\rho }_{\text{w}}{c}_{\text{w}}{T}_{\text{w}}{\displaystyle \frac{a_{\text{p}}{\nu }_{\text{w}}}{\sqrt{a_{\text{p}}{d}_{\text{s}}}}}}{{\displaystyle \frac{F_{\text{t}}{\nu }_{\text{s}}}{\sqrt{a_{\text{p}}{d}_{\text{s}}}b}}}\right).

Based on the equation, the model fully considers several factors. By incorporating these effects, the model extends its range of applicability and has become a commonly used method for calculation.

Ramanath and Shaw [103] proposed a method to calculate the energy distribution ratio between the workpiece and the grinding wheel. As shown in Fig.18, during a surface grinding operation, they assumed that the grinding wheel behaves as a perfect adiabatic body, with all the heat flowing into the workpiece. By applying the Jaeger solution to Fig.18, the average working temperature over the grinding wheel–workpiece contact arc can be easily obtained.

Assuming that the average surface temperatures of the workpiece and the grinding wheel in the grinding zone are equal, the EDC between the workpiece and the grinding wheel is as follows:

(49)R_{\text{ws}}={\left[1+\sqrt{\frac{{(k\rho c)}_{\text{g}}}{{(k\rho c)}_{\text{w}}}}\right]}^{-1}.

However, the model applies only to the energy distribution between the grinding wheel and the workpiece. The model does not involve the speed and grinding parameters of the grinding wheel, nor does it consider changes in the temperature gradient or the grain shape of the wheel. Therefore, this model can only provide an approximate energy distribution ratio.

Kohli et al. [104] investigated the energy distribution of workpieces during grinding with aluminum oxide and CBN grinding wheels. The grinding temperature is typically calculated using either a rectangular (uniform) heat flow density distribution or a triangular heat flow density distribution. While results from both models show only a small difference, the internal temperature distribution remains largely the same. As shown in Fig.19, the triangular heat-source mode was found to be highly relevant after the study because it provides better correlation between the measured and calculated temperatures.

The energy input to the workpiece was calculated analytically by matching the calculated temperature with the measured temperature. The energy distribution was then obtained by dividing the input workpiece energy by the measured net grinding energy. Additionally, the effective number of grains was incorporated into the grinding energy distribution model. By equalizing the maximum temperatures of the workpiece and the abrasive grains at their interfaces, the energy distribution of the workpiece for the triangular heat-source distribution was obtained as follows:

(50)R_{\text{w}} ={\left[1+1.06\sqrt{\frac{\text{π}(k\rho c{\text{)}}_{\text{g}}{v}_{\text{s}}}{2{(k\rho c)}_{\text{w}}{v}_{\text{w}}}}f(\zeta )A_0{G}_{\text{a}}\right]}^{-1},

where G_a is the number of active abrasive grains per unit area on the surface of the grinding wheel, A₀ is the average single-grain workpiece contact area corresponding to the cross-sectional area of the cone, and the value of A₀ corresponding to the measured value of R_w can be obtained by calculation. The function f(ζ) considers that the cross-sectional area of the abrasive grain is larger than the contact area between the abrasive grain and the workpiece.

2.4.5 Slicing multiple energy distribution subsystems

In 2006, Jin and Stephenson [105] proposed a comprehensive method to analyze the energy distribution ratio R_w of the workpiece, establishing the energy distribution relationship between the workpiece, abrasive, abrasive fluid, and grinding chip as follows:

(51)R_{\text{w}} =\frac{q_{\text{w}}}{q_{\text{t}}}=\frac{R_{\text{ws}}\cdot R_{\text{wch}}}{R_{\text{ws}}+R_{\text{wch}}-R_{\text{ws}}\cdot R_{\text{wch}}\left(1-\frac{h_{\text{f}}}{h_{\text{w}}}\right)} ,

where h_f is the convection coefficient of the coolant.

(52)\begin{array}{rl}{R}_{\text{ws}}=& \frac{q_{\text{w}}}{q_{\text{w}}+q_{\text{s}}}=\frac{h_{\text{w}}}{h_{\text{w}}+h_{\text{s}}},\\ R_{\text{wch}}=& \frac{q_{\text{w}}}{q_{\text{w}}+q_{\text{ch}}}=\frac{h_{\text{w}}}{h_{\text{w}}+h_{\text{ch}}},\end{array}

where h_ch is defined as the heat transfer coefficient of the chip.

In 2012, Hadad and Sadeghi [106] investigated the thermal analysis of the MQL grinding process. They introduced a new method for calculating the grinding temperature and workpiece energy distribution during the MQL grinding process by examining the grinding wheel and workpiece as a system.

In the model (Fig.20), heat is assumed to be generated at the grinding wheel–workpiece interface (global scale) and within the workpiece. The two mechanisms of heat generation are friction and plastic deformation. Three heat-source locations are available in the grinding process: the contact surface between the abrasive grit and the workpiece, the contact surface between the abrasive grit and the chip, and the shear surface between the workpiece and the chip.

As shown in Fig.21, the total heat flux q_t generated by grinding is instantaneously transferred to the chip and to the system of the workpiece and grinding wheel.

(53)q_{\text{t}} =q_{\text{c}}+q_{\text{w-g}}.

Within the system of workpieces and grinding wheels, heat flows into the workpiece and the grinding wheel, where q_w-g = q_w + q_g. The heat inflow into the workpiece, removed by the coolant, is subtracted, leaving the actual heat transferred to the workpiece. For the circular contact, the speed between grinding wheel grain and workpiece is assumed to be consistent with the grinding wheel speed. By considering the heat distribution ratio, the proportion of heat transferred to the workpiece can be obtained as follows:

(54)R_{\text{w}} ={\left[1+\frac{0.974k_{\text{g}}}{\sqrt{{(k\rho c)}_{\text{w}}\cdot (l_{\text{wf}}/\phantom{l_{\text{wf}}2}2)\cdot V_{\text{c}}}}\frac{1}{F}\right]}^{-1},

where F is the transient function, and Black gives an approximate computational model F. The τ is modeled as a dimensionless time and l_wf is the average diameter of the wear plane.

(55)\{\begin{array}{c}F=1-\exp \left(-\frac{\tau }{2}\right),\\ \tau ={\left(\frac{4{\alpha }_{\text{g}}{l}_{\text{c}}}{l_{\text{wf}}^2{V}_{\text{c}}}\right)}^{1/2}.\end{array}

In 2012, Zhu et al. [107] indicated that existing grinding temperature models generally assume that the effective contact radius of the abrasive grain is a constant, resulting in poor prediction accuracy. Aiming to address the modeling accuracy, an enhanced model that considers the geometry and distribution of abrasive grains is proposed. As shown in Fig.22, a single abrasive grain is regarded as a cone, providing the heat diffusion coefficient of the grain as shown below.

(56){\alpha }_{\text{g}} =\frac{k_{\text{g}}}{{\rho }_{\text{g}}{c}_{\text{g}}}.

The effective contact radius of the abrasive grains varies considerably depending on the depth of cut and the cone angle of the abrasive grains. Therefore, treating it as a constant for estimation is unreasonable and introduces some error. Based on the model shown in Fig.22, the calculation model for the effective contour radius, considering different depths of cut and variable taper angles, is r₀ = a_p·tan(θ/2), which, when incorporated into the existing EDC, yields a more reasonable model of R_w.

(57)R_{\text{ws}}={\left[1+\frac{0.974k_{\text{g}}}{{\beta }_{\text{w}}\sqrt{a_{\text{p}}\cdot \tan (\theta /\phantom{\theta 2}2)\cdot {\nu }_{\text{s}}}}\cdot \frac{1}{F}\right]}^{-1}.

In 2016, Jin et al. [108] divided the heat distribution in the grinding zone into multiple systems, derived the EDC for each system based on the existing models. They then derived the EDC model for the workpiece based on existing models. Previous grinding models mainly focused on plane grinding, with the complexity of gear grinding, the point-source temperature rise solution was further developed. As shown in Fig.23, this model accounts for the non-uniform heat sources occurring within the grinding zone for three-dimensional involute shapes. Thus, the model allows for the calculation of grinding temperatures and an analysis of how the energy distribution ratio of the workpiece changes along the gear tooth profiles under different grinding conditions.

The energy distribution ratio R_ws between the workpiece and the grinding wheel was obtained based on Hahn’s model, and the energy distribution ratio R_wch between the workpiece and the chips was obtained based on Stephenson and Jin’s model.

(58)R_{\text{wch}}( {\xi }_{\text{i}} )=\frac{h_{\text{w}}({\xi }_{\text{i}})}{h_{\text{w}}({\xi }_{\text{i}})+h_{\text{ch}}({\xi }_{\text{i}})}= \frac{1}{1+0.753\sqrt{{\displaystyle \frac{{\nu }_{\text{s}}({\xi }_{\text{i}})t_{\text{a}}({\xi }_{\text{i}})}{{\alpha }_{\text{w}}\gamma }}}},

where α_w = k_w/(ρ_wc_w) is the thermal diffusivity of the abrasive chips, γ is the shear strain in the chip formation zone, and t_a(ξ_i) is the undeformed chip thickness.

Combining the above sub-models, the EDC for the workpiece can be derived as follows:

(59)\begin{array}{rl}& R_{\text{w}}\left({\xi }_{\text{i}}\right)=\frac{q_{\text{w}}}{q_{\text{t}}}=R_{\text{ws}}({\xi }_{\text{i}})R_{\text{wch}}({\xi }_{\text{i}})\\ & \cdot \{R_{\text{ws}}({\xi }_{\text{i}})+R_{\text{wch}}({\xi }_{\text{i}})-R_{\text{ws}}({\xi }_{\text{i}})R_{\text{wch}}({\xi }_{\text{i}})\cdot \left[1-\frac{h_{\text{f}}({\xi }_{\text{i}})}{h_{\text{w}}({\xi }_{\text{i}})}\right]{\}}^{-1},\end{array}

where ξ is the roll angle.

2.4.6 Considering convective heat transfer

In 2002, Jin et al. [101] investigated the combustion threshold for efficient deep grinding. Based on thermal modeling of deep grinding conditions, they derived the theoretical expression for the workpiece burning threshold by comprehensively considering the grinding parameters, grinding wheel conditions, thermal properties of the workpiece and abrasive grains, as well as the convective heat dissipation of the grinding coolant and chips.

On this basis, Yang et al. [109], in their study of nanoparticle spray cooling, found that nanofluids play a major role in heat dissipation within the grinding zone, with the convective heat transfer of the nanofluid carrying away most of the heat. They developed a mathematical model for R_w by calculating the CHTC of the grinding wheel, workpiece, coolant, and chips, without the need to directly measure the grinding zone temperature.

Rowe [110] improved the thermal model by considering the effect of convective heat transfer in different states of the coolant on the energy distribution. He developed a division expression that distinguishes between the upper and lower boundaries, enhancing the modeling of heat flow distribution between the grinding wheel and the workpiece. In wet grinding, if the liquid film boiling temperature is exceeded, then the coolant no longer serves as an effective heat sink in the grinding zone.

In the calculation process, as shown in Fig.24, the energy distribution ratio between the grinding wheel and the workpiece is calculated based on a simple two-slider model. However, after further consideration, the chips and the coolant, as heat dissipation components, have no substantial effects on the calculation of R.

The grinding wheel and the workpiece can be regarded as two sliders. After considering the density and nature of the heat flow in the grinding wheel and the workpiece, an assumption is that both are affected by the sliding heat source. Considering these factors, the energy ratio of the workpiece can be determined.

(60)R_{\text{ws}}\text{=}\frac{\text{1}}{\text{1 +}\sqrt{{\displaystyle \frac{v_{\text{s}}}{v_{\text{w}}}}{\displaystyle \frac{{(k\rho \text{c})}_{\text{s}}}{{(k\rho \text{c})}_{\text{w}}}}}}.

If the energy from the convection of chips and coolant is considered, then the predicted distribution ratio is reduced and the EDC can be corrected as shown below.

(61)R=R^{\prime }\left(1-\frac{e_{\text{cc}}+e_{\text{cf}}}{e_{\text{c}}}\right) .

Typical values of e_cc are 6 J/mm³, and the value of e_cf is 0 when fluid boiling occurs, leading to an increasingly substantial effect of chips at relatively low specific energies.

(62)R=R^{\prime }\left(1-\frac{6}{e_{\text{c}}}\right) .

In 2022, Zheng et al. [111] considered the role of coolant in the grinding process and established an improved external wet grinding temperature model that accounts for the lubricating effect of the coolant, building on previous models. The model determines the EDC of the workpiece through the relationship between the EDC of different subsystems. As shown in Fig.25, the schematic illustrates cylindrical grinding and the heat distribution within the grinding zone.

The energy ratio R_w flowing into the workpiece can be calculated by the following relation [108].

(63)R_{\text{w}} ={\left[\frac{1}{R_{\text{ws}}}+\frac{1}{R_{\text{wch}}}-\left(1-\frac{h_{\text{f}}}{h_{\text{w}}}\right)\right]}^{-1},

where h_f is the CHTC of the coolant, h_w is the CHTC of the workpiece, and R_ws and R_wch are the energy distribution ratios of the workpiece–grinding wheel and the workpiece–chip, respectively.

(64)R_{\text{ws}}=\frac{1}{1+{\displaystyle \frac{0.974k_{\text{s}}}{{\beta }_{\text{w}}\sqrt{r_0{\nu }_{\text{s}}}}}{\left[1-\text{exp}\left(-\frac{\tau }{2}\right)\right]}^{-1}},

(65)R_{\text{wch}}=\frac{1.328\sqrt{{\alpha }_{\text{w}}\gamma /\phantom{{\alpha }_{\text{w}}\gamma ({\nu }_{\text{s}}{a}_{\text{gmax}})}({\nu }_{\text{s}}{a}_{\text{gmax}})}}{1+1.328\sqrt{{\alpha }_{\text{w}}\gamma /\phantom{{\alpha }_{\text{w}}\gamma ({\nu }_{\text{s}}{a}_{\text{gmax}})}({\nu }_{\text{s}}{a}_{\text{gmax}})}},

where the dimensionless number τ is given by \tau ={\left[k_{\text{s}}^{1/2}{l}_{\text{c}}{({\rho }_{\text{s}}{c}_{\text{s}})}^{-1/2}/(r_0^2{\nu }_{\text{s}})\right]}^{1/2}, k_s is the thermal conductivity of the grinding wheel, ρ_s is the density of the grinding wheel, and c_s is the specific heat of the grinding wheel. The shear strain is defined as \gamma =\cos \theta / [\sin \phi \cdot \cos (\phi +\theta )], where \theta and \phi are the half of the grinding tip angle and the shear angle, respectively, and a_gmax is the maximum undeformed chip thickness.

2.4.7 Grinding wheel and fluid composite as a whole

In 2000, Guo and Malkin [88] proposed a quantitative method for calculating grinding temperature and workpiece energy distribution. Aiming to analyze heat transfer during the grinding process, they approximated the heat transfer problem in two dimensions by considering straight-face plunge grinding, as shown in Fig.26. The contact between the grinding wheel, fluid, and workpiece was simplified as two planar contact surfaces, and the heat source was modeled as a suitable rectangular heat source.

This method of estimating energy distribution is based on matching the temperature in the grinding area. Therefore, the grinding wheel and fluid are considered as a composite material. Assuming that the energy distributed to the workpiece along the grinding zone remains constant, the energy distribution in the workpiece can be calculated as follows:

(66)R={\left[1+\sqrt{\frac{{(k\rho c\nu )}_{\text{c}}}{{(k\rho c\nu )}_{\text{w}}}}\right]}^{-1},

where k is the thermal conductivity and ρ_c is the volume-specific heat. The subscript c represents the composite and w represents the workpiece.

The thermal properties of the wheel–liquid composites are unknown and the location of the maximum temperature varies. Thus, this relatively simple model can only be used for qualitative analysis of the energy distribution.

Lavine [112] presented a simple analytical model for heat transfer between the grinding wheel, workpiece surface, and coolant, predicting the proportion of energy entering the workpiece. As shown in Fig.27, the grinding zone is modeled as a two-dimensional region. Assuming that the coolant fills the thermal boundary layer pores and remains stationary relative to the grinding wheel, the grinding wheel is considered as a composite material comprising coolant and abrasive grains, with their combined properties determining the overall properties of the wheel.

The total heat generated is divided into two parts: the heat conducted to the workpiece and the heat entering the complex comprising the grinding wheel and the fluid. The proportion of energy transferred to the workpiece is then given by the following:

(67)R={\left[1+\frac{\sqrt{{(k\rho c_{\text{P}}V)}_{\text{c}}}}{\sqrt{{(k\rho c_{\text{P}}V)}_{\text{w}}}}\right]}^{-1}.

The ratio of the actual contact area of the grinding wheel to the nominal contact area of the workpiece, namely A_R/A, is much less than 1 for the grinding wheel and equal to 1 for the liquid film. The proportion of energy allocated to the workpiece is then calculated as follows:

(68)R_{\text{w}} ={\left[1+\sqrt{\frac{{(k\rho c)}_{\text{g}}{\nu }_{\text{s}}{A}_{\text{R}}}{{(k\rho c)}_{\text{w}}{\nu }_{\text{w}}A}}+\sqrt{\frac{{(k\rho c)}_{\text{f}}{\nu }_{\text{s}}}{{(k\rho c)}_{\text{w}}{\nu }_{\text{w}}}}\right]}^{-1}.

Kim et al. [113] investigated the heat flow density distribution and energy distribution in the workpiece during creep feed grinding. On this basis, they measured the surface temperature of the workpiece and used inverse analysis to determine the energy distribution value. These results were the compared with those calculated using the moving heat-source theory. The analysis was conducted separately under different working conditions for down grinding and up grinding (Fig.28).

The energy distribution in the workpiece can be easily obtained by modeling the grinding zone as a band heat source moving at the wheel speed along the workpiece surface. This approach involves calculating the maximum surface temperature at the interface on one side of the composite and assuming a sufficiently large Péclet number, such that heat conduction in the direction of motion can be neglected.

(69)\epsilon ={\left[1+\sqrt{\frac{{(k\rho c)}_{\mathrm{c}}{v}_{\text{s}}}{{(k\rho c)}_{\text{w}}{v}_{\text{w}}}}\right]}^{-1}.

Takazawa [114] considered the thermal parameters of the grinding wheel and the workpiece, treating them as a composite heat conductor. The grinding process was modeled as two contact heat conductors moving at different speeds. A one-dimensional heat conduction model was then used to calculate the percentage of heat transferred to the workpiece as follows:

(70)

In 2017, Yin and Marinescu [115] modeled the grinding zone as a heat source, assuming that most of the heat is transferred to the workpiece, while the remaining energy is carried away by the grinding wheel, abrasive fluid, and chips. Based on Lavin’s model [116], they assumed that the grinding wheel is a composite of coolant and wheel, with the heat entering the wheel denoted as q_s.

(71)q_{\text{s}} =\frac{T_{\text{m}}b}{2}\sqrt{2{(k\rho c)}_{\text{s}}\cdot v_{\text{s}}\frac{A_{\text{R}}}{A}{l}_{\text{c}}},

where T_w is the melting point of the workpiece and A_R is the contact area of the abrasive grains.

Based on Malkin’s study, the energy imparted to the abrasive chips can be calculated using the specific energy of the abrasive chips.

(72)q_{\text{c}} =\frac{e_{\text{c}}\cdot MRR}{b\cdot l_{\text{c}}}=\frac{{\rho }_{\text{w}}{c}_{\text{w}}{T}_{\text{m}}b{a}_{\text{p}}{v}_{\text{w}}}{b\cdot \sqrt{a_{\text{p}}{d}_{\text{s}}}}=\frac{{\rho }_{\text{w}}{c}_{\text{w}}{T}_{\text{m}}{a}_{\text{p}}{v}_{\text{w}}}{\sqrt{a_{\text{p}}{d}_{\text{s}}}},

where d_s is the diameter of the grinding wheel. MRR is the material removal rate.

An assumption is that most of the heat is transferred to the workpiece and the coolant. Therefore, the heat involved in the heat transfer process comprises the heat flow density q_w of the workpiece and the convective heat flow density q_f of the coolant. The energy distribution can then be expressed as follows:

(73)\begin{array}{rl}{R}_{\text{wf}}=& \frac{q_{\text{w}}+q_{\text{f}}}{q_{\text{t}}}=1-\frac{q_{\text{c}}+q_{\text{s}}}{q_{\text{t}}}=1-\frac{1}{{\displaystyle \frac{F_{\text{t}}{V}_{\text{s}}}{b\sqrt{a_{\text{p}}{d}_{\text{s}}}}}}\\ & \cdot \left[{\displaystyle \frac{{\rho }_{\text{w}}{c}_{\text{w}}{T}_{\text{m}}{a}_{\text{p}}{V}_{\text{w}}}{\sqrt{a_{\text{p}}{d}_{\text{s}}}}}+\frac{T_{\text{m}}b}{2}\sqrt{2{(k\rho c)}_{\text{s}}{V}_{\text{s}}\beta \sqrt{a_{\text{p}}{d}_{\text{s}}}}\right].\end{array}

This energy distribution equation considers the effects of workpiece feed rate, depth of cut, equivalent grinding wheel diameter, tangential grinding force, and the physical and thermal properties of the grinding wheel and the workpiece on R_wf. This equation also considers the cooling effect of the coolant, and the results for the tangential grinding force F_t reflect the effect of the coolant.

2.4.8 Considering heat source tilt angle

In 2001, Jin et al. [117] conducted a thermal study of high-efficiency deep grinding (HEDG) investigated the relationship between workpiece energy distribution and grinding parameters. They developed an analytical model for deep grinding that considered the influence of the inclination angle of the heat-source plane on its direction of motion. Under HEDG conditions with a large Péclet number, the inclination angle substantially affects the temperature of the grinding zone, which cannot be ignored. Therefore, they developed an analytical thermal model for deep grinding conditions that incorporates the effect of the inclination angle.

Using the tilt-shift heat-source planar model, the average heat flow of the workpiece can then be obtained as shown below.

(74)q_{\text{w}} =\frac{{\theta }_{\text{max}}{(c\rho )}_{\text{w}}{\nu }_{\text{w}}\sin \phi }{2\text{erf}\left[{\nu }_{\text{w}}\sin \phi \cdot {(t_{\text{o}}/\phantom{t_{\text{o}}4\alpha }4\alpha )}^{1/2}\right]} .

The average heat flow of the composite solid is q_{\text{c}}=( {\theta }_{\text{max}}/\phantom{{\theta }_{\text{max}}1.128}1.128)\sqrt{{\nu }_{\text{s}}{l}^{-1}{(k\rho c)}_{\text{c}}}. Based on the defini-tion, the EDC considering the inclination of the heat source is given as

(75)R={\left\{1+\frac{2(k\rho c{)}_{\text{c}}^{1/2}{({\nu }_{\text{s}}/\phantom{{\nu }_{\text{s}}{l}_{\text{c}}}{l}_{\text{c}})}^{1/2}}{1.128{(c\rho )}_{\text{w}}{\nu }_{\text{w}}\sin \phi }\cdot \text{erf}\left[{\nu }_{\text{w}}\sin \phi \cdot \left(\frac{t_{\text{o}}}{4{\alpha }_{\text{w}}}\right)\right]\right\}}^{-1}.

Tab.3 further summarizes the factors influencing the EDC model in grinding processing.

3 Convection heat transfer mechanism and modeling

Convection heat transfer is a complex process that involves thermal conduction and convection. This process occurs when a fluid is in direct contact with a solid surface, a relative motion exists between the fluid and the wall, and a temperature difference emerges. Owing to fluid viscosity, in the flow through the solid surface, by the influence of frictional resistance will be formed on the solid surface with a large velocity gradient of the boundary layer. Several factors affect convective heat transfer, including the properties of the fluid, the nature of fluid flow, and the characteristics of the heat transfer surface [118,119].

The cooling effect provided by the coolant plays a crucial role in the machining process. Therefore, accurately determining the CHTC is essential for calculating surface temperature in non-dry machining processes [120]. However, predicting or solving for the CHTC is challenging due to variability in processing parameters, uncertainties in the physical parameters of materials, and differences in fluid properties [121,122].

3.1 Turning

In the turning process, cutting fluid cooling is one of the most commonly used methods for temperature control. Additionally, the high-speed rotation of the workpiece or tool generates strong convection heat transfer with the surrounding air, which plays a crucial role in effectively reducing the workpiece temperature.

In 2023, Yin et al. [123] constructed a model of the CHTC during the turning of slender shafts, aiming to improve the machining accuracy. During the cutting process, the end face and surface of the slender shaft experience relative motion with the surrounding air, leading to convective heat transfer.

For the slender shaft end face, the CHTC h₁ is calculated as follows:

(76)h_1=28\left(1+\sqrt{0.45u}\right),

where u is the linear velocity of the ambient air around the circumferential end face, which can be calculated using the spindle speed n and the diameter d of the slender shaft end face. u = πdn/60000.

For elongated axial circumferential surfaces, the forced CHTC h₂ can be calculated using the Nusselt equation, which is given by:

(77)h_2=1000\cdot Nu\cdot k_{\text{a}}/\phantom{Nu\cdot k_{\text{a}}d}d,

where k_a is the thermal conductivity of air. The Nusselt number Nu is calculated as follows:

(78)

where Re is the Reynolds number and Pr is the Prandtl number of the flowing air.

3.2 Milling

Milling is an intermittent process; therefore, air convection also facilitates effective cooling of the tool between machining intervals [124].

3.2.1 Based on fluid mechanics

In 2022, Chen et al. [125] used an analytic approach to model convective heat transfer in tools. Milling, as an intermittent cutting process, allows the tool to cooled through convective heat transfer with the surrounding air when the cutting edge is not in contact with the workpiece [126].

(79)h_0=\frac{Nu\cdot k_{\text{a}}}{D},

where D is the diameter of the milling cutter.

The average Nussle number Nu is

(80)Nu=0.318R e_{\text{r}}^{0.571},

where Re_r is the dimensionless rotational Reynolds number,

(81)R{e}_{\text{r}}=\frac{\mathrm{\Omega }{D}^2{\rho }_{\text{a}}}{2\mu },

where Ω is the cylinder angle, ρ_a is the air density, and μ is the dynamic viscosity of air, and the physical parameters of air are shown in Tab.4.

3.2.2 Considering air thermal circulation

In 2004, Wang et al. [127] analyzed the CHTC on the surface of a wave-shaped flute milling cutter blade during face milling by integrating principles from heat transfer, fluid mechanics, and cutting theory. As shown in Fig.29, during face milling, the rotating cutter induces high-speed tangential motion of the surrounding air along the outer edge of the cutter disk, resulting in a phenomenon known as thermal circulation. Considering the complexity of the heat-source flow problem, the CHTC is assumed to be uniform along each convection boundary of the milling cutter. Additionally, the air adjacent to the milling cutter disk is considered an incompressible Newtonian fluid with constant physical properties, no internal heat source, and negligible heat dissipation from viscous friction. Moreover, another assumption is that the air at the wall of the milling cutter disk is in the exuberant turbulence zone.

The layer of air adhering to the wall of the rotating disk moves with a linear velocity v along the outer edge, undergoing uniform circular motion. This flow condition closely resembles to the fluid flow around a single tube. Therefore, the Reynolds number of the air can be expressed as follows:

(82)Re =\frac{\text{π}{d}^{2}n}{60\upsilon }.

The air becomes turbulent at normal operating speeds near the knife plate against the wall; thus, airflow is assumed to remain consistently in a turbulent state. The CHTC at the knife plate against the wall is given as follows:

(83)h=C\frac{k_{\text{f}}}{d}{\left(\frac{\text{π}{D}^{2}n}{60v}\right)}^m,

where D is the diameter of the cutter, k_f is the thermal conductivity of the fluid, and C and m are coefficients to be determined based on the Re criterion.

3.3 Drilling

The drilling process typically occurs deeper within the workpiece, making it more difficult for the cutting fluid to reach the machining area, which notably affects the cooling efficiency [128,129]. Drilling is generally a fast process and is therefore suitable for analysis under transient situations. Moreover, different regions of the drill experience varying cooling effects [128].

3.3.1 System transient heat transfer

In 2012, Kurgin et al. [130] investigated the CHTC under MQL lubrication during hole reaming. The reaming and drilling processes is typically remarkably rapid, and the system is transient. The capacitance method can be used to solve for the transient CHTC. Assuming no temperature gradient exists within the object, the capacitance equation is as follows:

(84)\frac{T_t-T_{\mathrm{\infty }}}{T_{\text{a}}-T_{\mathrm{\infty }}}=\text{exp}\left(-\frac{h{A}_{\text{f}}}{\rho cV}t\right),

where T_t is the temperature at moment t, T_a is the initial temperature of the object, T_∞ is the convective ambient temperature, A_f is the convective surface area, and V is the volume.

The Biot number must be calculated to determine the presence of internal temperature gradients. The results of the total capacity analysis are valid when the Biot number is less than 0.1. The Biot number is calculated as follows:

(85)

where k is the thermal conductivity of the object.

3.3.2 Zoned heat transfer analysis in drilling tools

In 2017, Pontes et al. [131] investigated the CHTC of the cutting fluid over the auger complex to effectively model the temperature at the tool tip. As shown in Fig.30 and Fig.31, heat exchange can be idealized into four regions during the drilling process. The maximum heat removal occurs in the second region, where the drilling tool and cutting fluid are in contact. Under steady-state conditions, where all heat transferred to the drill is expected to be exchanged with the cutting fluid, the average temperature of the drill surface in contact with the cutting fluid is calculated as follows:

(86)T_{\text{d}} =\frac{1}{A_{\text{f}}}{\int }_{A_{\text{f}}}T\text{d}A .

Once the average drill surface temperature T_d is identified, the average CHTC is calculated as follows:

(87)h=\frac{q_{\text{f}}}{A_{\text{f}}(T_{\text{d}}-T_{\text{f}})} ,

where T_f is the temperature of the cutting fluid.

Studies focused on modeling CHTC in planning and boring machining are few, and no relevant models have been established. Therefore, these topics will not be discussed further herein. Tab.5 provides a summary of the factors that influence the modeling of CHTC in cutting operations.

3.4 Grinding

Compared to cutting, grinding generates more frictional heat due to the intense friction between the abrasive grains and the workpiece. This heat tends to accumulate in the grinding zone, potentially causing burns on the workpiece and damaging the grinding wheel. Therefore, the cooling effect provided by the coolant is critical in the grinding process [132]. The convective heat transfer of the coolant is a primary mechanism for dissipating heat in the grinding zone, especially during coolant flooding, where most of the heat is carried away through the coolant [133]. Therefore, in non-dry grinding, determining the CHTC is a key factor in predicting the workpiece surface temperature and controlling machining quality [134].

However, excessive use of coolant has many disadvantages [135], and in high-speed grinding, breaking through the air barrier layer and effectively reaching the grinding zone for cooling is difficult for the coolant [136,137]. Therefore, researchers have developed a variety of cooling methods such as MQL [138,139] and NMQL [140,141]. Owing to the diversity of cooling methods and the variability of machining conditions, determining the CHTC becomes a complex problem [142]. Researchers have used a variety of approaches to derive different CHTC models, including those that consider factors such as boiling states of the fluid, particle contact conditions, different grinding zones, and those based on hydrodynamics, statistical theory, energy distribution.

3.4.1 Considering the fluid boiling state

In 2019, Zhang et al. [99] investigated the grinding process of low-temperature air nanofluid MQL lubrication and established a CHTC model based on the theory of boiling heat transfer and conduction. Notably, the boiling heat transfer process in the grinding zone can be divided into four stages: natural convective heat transfer, nucleate boiling heat transfer, transition boiling heat transfer, and film boiling heat transfer. The effect of grinding temperature on the heat transfer coefficient is shown in Fig.32, demonstrating the four boiling stages of the cooling medium.

In a natural convection state, the coolant does not boil, and nanofluid convection heat transfer becomes the main heat transfer mode. Yang et al. [143] investigated the CHTC under MQL conditions without considering the boiling convection heat transfer coefficient. When the nanofluid is injected into the grinding zone, the ejected nanofluid is discretized into N_l droplets, each with a volume V_l per unit time as follows:

(88)\{\begin{array}{rl}& \begin{array}{rl}{V}_{\text{i}}=& r_{\text{surf}}^3\cdot \{\frac{\text{π}}{2}\frac{1}{\tan {\theta }_{\text{n}}}\left(\frac{1}{\cos {\theta }_{\text{n}}}-1\right)\\ & {\cdot \left[1+\frac{1}{3}\frac{1}{{\tan }^2{\theta }_{\text{n}}}{\left(\frac{1}{\cos {\theta }_{\text{n}}}-1\right)}^2\right]\}}^{-1},\end{array}\\ & V_{\text{l}}=\frac{\text{π}{d}_0^3}{6},\\ & N_{\text{l}}=\frac{Q^{\prime }t}{V_{\text{l}}},\\ & t=\frac{a_{\text{w}}}{v_{\text{w}}},\end{array}

where Q^{\prime } is the nanofluid supply during the grinding time, r_surf is the diffusion radius of a single droplet, N_l is the droplet volume, t is the total grinding time, a_w is the length of the workpiece, V_l is the volume of a single droplet, d₀ is the diameter of the sphere of a single droplet, and θ_n is the contact angle.

Therefore, the CHTC for the natural convection state is introduced as follows:

(89)h_{n1}=\frac{N_{\text{l}}{c}_{\text{l}}{\rho }_{\text{l}}{V}_{\text{l}}}{\text{π}{r}_{\text{surf}}^{\text{2}}\cdot t} +h_{\text{a}}^{\mathrm{\prime }},

where c_l is the droplet specific heat capacity, ρ_l is the nanofluid density, T_s is the saturation temperature, and h_{\text{a}}^{\mathrm{\prime }} is the room temperature air CHTC.

When the boiling state of the coolant is considered, as shown in Fig.32, the CHTC reaches its maximum value at the endpoint of nucleated boiling, which marks the starting point of transitional boiling heat transfer. At this point, the CHTC can be calculated as follows:

(90)h_{\text{n2}}=\frac{\left[h_{\text{fa}}+c_{\text{l}}(T_{\text{s}}-T_{\text{l}})\right]Q^{\prime }{\rho }_{\text{l}}}{\text{π}{r}_{\text{surf}}^{\text{2}}(T_{\text{n2}}-T_{\text{l}})}+h_{\text{a}}^{\prime },

where h_fa is the latent heat of vaporization and T_l is the nanofluid temperature.

When the nanofluid enters the film boiling state, the endpoint of transitional boiling marks the starting point of film boiling. At this stage, the CHTC is calculated as follows:

(91)\{\begin{array}{rl}& \begin{array}{rl}{h}_{\text{n}3}=& \frac{N_{\text{l}}{Q}^{\prime }{\rho }_{\text{l}}\left[h_{{\text{f}}_{\text{d}}}+c_{\text{l}}(T_{\text{s}}-T_{\text{l}})\right]}{b{l}_{\text{c}}(T_{\text{n}3}-T_{\text{l}})}\\ & \cdot \{0.027\exp [\frac{0.08\sqrt{\ln (We/\phantom{We35+1}35+1)}}{B^{1.5}}]\\ & +0.21k_{\text{d}}B\cdot \text{exp}(-\frac{90}{We+1})\}+h_{\text{a}}^{\mathrm{\prime }},\end{array}\\ & We=\frac{{\rho }_{\text{l}}{d}_0{\nu }_{\text{n}}^2}{\sigma },\\ & {\nu }_{\text{n}}=\sqrt{{\displaystyle \frac{{\displaystyle \frac{p_0-p_0}{{\rho }_1}}+{\displaystyle \frac{16{Q^{\prime }}^2}{{\text{π}}^2{d}_0^2}}}{1+\epsilon }}}\cdot \cos \theta ,\\ & B=\frac{c_{\text{v}}(T_{\text{m}}-T_{\text{s}})}{h_{\text{fa}}},\\ & l_{\text{c}}=\sqrt{a_{\text{p}}{d}_{\text{s}}},\\ & k_{\text{d}}=\frac{k_{\text{v}}}{c_{\text{v}}{\mu }_{\text{v}}},\end{array}

where p_a is the pressure inside the nozzle, p₀ is the atmospheric pressure, v_l is the droplet jet velocity along the nozzle direction, v_n is the droplet vertical impact velocity on the heat transfer surface, k_v is the thermal conductivity of the steam, μ_v is the steam kinetic viscosity, c_v is the mass heat capacity of the steam, θ is the droplet jet and the horizontal direction of the angle, and σ is the surface tension.

In 2023, Liu et al. [144] analyzed the heat transfer behavior of the liquid film at the interface of the low-temperature cold air MQL-lubricated grinding wheel–workpiece. They further established a CHTC model based on the liquid film flow heat transfer process within the grinding zone. As shown in Fig.33, the total liquid film heat transfer q_t in the grinding zone comprises the liquid film flow heat transfer q_f and the heat transfer from the boiling bubble group q_b.

For the total heat of the liquid film, the energy input into the grinding zone is assumed to be completely converted into the heat generated during the interaction between the grinding wheel and the workpiece.

(92)Q_{\text{t}} =F_x\cdot l_{\text{c}} =F_x\cdot \sqrt{d_{\text{s}}{a}_{\text{p}}}.

For the boiling bubble heat exchange, the liquid film in the nucleate boiling state is assumed to form spherical bubbles on the workpiece surface. Applying the Fritz formula, all the bubbles, from the nucleation of the gas to their growth and detachment, contribute to the total heat absorbed, denoted as Q_b.

(93)Q_{\text{b}} =N_{\text{b}}w{l}_c{\zeta }_{\text{b}}{\rho }_{\text{a}}\frac{\text{π}{d}_b^3}{6},

where ζ_b is the latent heat of the bubble, taken as 384.3 kJ/kg. d_b is the diameter of a single bubble, ρ_a is the density of the air, and N_b is the number of nucleation sites (per unit area) on the superheated surface.

The CHTC of the flowing liquid film can be modeled by substituting the parameters into the relevant calculation equation as follows:

(94)h_{\text{f}} =\frac{(Q_{\text{t}}-Q_{\text{b}})\cdot h_{\text{a}}}{N{u}_{\text{f}}\cdot w{l}_c(T_{\text{w}}-T_{\text{f}})} .

In 2014, Mao et al. [145] established a mathematical model for the heat transfer coefficient in the MQL grinding zone, based on the atomization mechanism and the varying heat transfer characteristics exhibited of droplets at different wall temperatures. Aiming to simplify the calculation, an assumption is that a constant heat flow is applied to a small unit of the grinding surface within a unit time step. The droplets are further assumed to penetrate the air barrier and vapor layer surrounding the grinding wheel, entering the small contact area for effective heat transfer. The heat transfer on the surface is assumed to occur through convection for the air, while the droplets transfer heat through heating and boiling. Given the small diameter of the droplets, they are assumed to reach the wall temperature instantaneously upon contact with the heat transfer surface.

No boiling heat exchange. The heat transfer in this region is divided into two components: convective heat transfer between the air and the wall, and heating heat transfer of the liquid droplets. By applying the Nusselt number calculation equation, the CHTC for the air to the wall is as follows:

(95)h_{\text{a}} =0.906R{e}^{1/2}P{r}^{1/3}{k}_{\text{a}} l^{-1}.

The heat transfer coefficient h_l for droplet heating is as follows:

(96)h_{\mathrm{l}}=\frac{c_1{m}_0(t_{\text{w}}-t_{\mathrm{l}})}{t_{\text{w}}-t_{\mathrm{l}}},

where m₀ is the mass of the droplets involved in the heat transfer and t₁ is the initial temperature of the droplets.

The total heat transfer coefficients are presented as follows:

(97)h=h_{\text{a}}+h_{\text{l}}.

In the non-boiling zone, the heat carried away by MQL cooling and the wall temperature are approximately linear. Therefore, the heat transfer coefficient in this region can be approximated as a constant value.

Nucleate boiling heat transfer and transition boiling heat transfer. In the nucleate boiling zone and the transition boiling heat transfer zone, the change in heat transfer is highly complex, and the relationship between wall temperature becomes non-linear. However, the spray heat transfer and wall temperature follow a parabolic distribution, reaching a maximum at the critical heat flow density at the maximum value. The critical heat flow density is as follows:

(98)\begin{array}{c}{q}_{\text{max}}=N_{\text{l}}\left[h_{\text{fa}}+c_{\text{l}}(t_{\text{s}}-t_{\text{l}})\right]\phi +q_{\text{a}}^{\mathrm{\prime }},\\ q_{\text{a}}^{\mathrm{\prime }}=0.906R{e}^{1/2}P{r}^{1/3}\frac{\lambda }{l}(t_{\text{w}}-t_0),\end{array}

where q_{\text{a}}^{\prime } is the air convection heat transfer and φ is the liquid mass flow rate.

Assuming that all droplets sprayed onto the heat transfer surface are evaporated and that N_l is 1, it can be simplified as follows:

(99)q_{max}=\left[\begin{array}{c}{h}_{\text{fa}}+c_{\mathrm{l}}(t_{\text{s}}-t_{\text{l}})\end{array}\right]\phi +q_{\text{a}}^{\prime }.

Therefore, the heat transfer coefficient at the critical heat flow density is calculated as follows:

(100)h_{max}=\frac{q_{max}}{t_{\text{w}}-t_{\text{l}}}.

Stabilized film boiling heat transfer. The heat transfer in this area includes the following three components: heat transfer between liquid droplets and the hot surface, convective heat transfer between the airflow and the hot surface, and the heat transfer of a single droplet hitting the high-temperature wall, which can be expressed as follows:

(101)\begin{array}{rl}{q}_{\text{d}}=& \frac{\text{π}{d}^3}{6}{\rho }_{\text{l}}\left[h_{\text{fa}}+c_{\text{l}}(t_{\text{s}}-t_{\text{l}})\right]\\ & \cdot \{0.027\cdot \exp [\frac{0.08\sqrt{\ln (We/\phantom{We35}35+1)}}{B^{1.5}}]\\ & +0.21k_{\text{d}}B\cdot \exp (\frac{-90}{We+1})\},\end{array}

where We is the Weber number; We={\rho }_{1}d{v}_{\text{n}}^{\text{3}}/\phantom{{\rho }_{1}d{v}_{\text{n}}^{\text{3}}\sigma }\sigma. Also, we have B=c_{\text{v}}(t_{\text{w}}-t_{\text{s}})/\phantom{c_{\text{v}}(t_{\text{w}}-t_{\text{s}})h_{\text{fa}}}{h}_{\text{fa}} and k_{\text{d}} =k_{\text{v}}/\phantom{k_{\text{v}}(c_{\text{v}}{\mu }_{\text{v}})}(c_{\text{v}}{\mu }_{\text{v}}).

High-temperature surface heat transfer due to droplet collision per unit area per unit time:

(102)q_1^{″}=q_{\text{d}} N_{\text{l}} =q_{\text{d}}\frac{6Q_1}{S\text{π}{d_0}^3},

where S is the area covered by the nozzle jet and d₀ is the droplet diameter.

The convective heat exchange between air flow and wall surface is calculated as follows:

(103)q_{\text{a}}^{″}=0.906R e^{1/2}P{r}^{1/3} \frac{\lambda }{l}(t_{\text{w}}-t_0).

Consequently, the CHTC in the boiling zone of the stable film state is presented below.

(104)h_{\text{b}} =\frac{q_1^{\mathrm{\prime }\mathrm{\prime }}+q_{\text{a}}^{\mathrm{\prime }\mathrm{\prime }}}{t_{\text{w}}-t_0}.

3.4.2 Based on fluid mechanics

In 2024, Gupta and Yadav [22] employed computational fluid dynamics to develop a mathematical model for the CHTC of the coolant. They analyzed the total heat generated in the grinding zone and its distribution among the grinding wheel, workpiece, fluid, and chips. The heat flow to the fluid, denoted as q_f, is expressed as follows:

(105)q_{\text{f}} =h_{\text{f}}\cdot T_{max}= q_{\text{t}} -(q_{\text{w}} +q_{\text{s}}+q_{\text{ch}}),

where the total heat q_t depends on the speed ratio, grinding depth, grinding wheel, workpiece material, and diameter.

\begin{array}{rl}& q_{\text{t}}=\frac{{\nu }_{\text{w}}}{{\nu }_{\text{t}}}{a}_{\text{p}}^{3/2}{d}_{\text{s}}^{1/2}{F}_{\text{vm}},\\ & q_{\text{w}}=\frac{{\beta }_{\text{w}}}{\text{C}}\sqrt{\frac{{\nu }_{\text{p}}}{l_{\text{c}}}}\cdot T_{max},\\ & q_{\text{s}}=q_{\text{w}}\frac{k_{\text{g}}}{{\beta }_{\text{w}}\sqrt{r_{\text{a}}{\nu }_{\text{t}}}},\end{array}

(106)q_{\text{ch}} =\frac{pca{v}_{\text{p}}}{l_{\text{c}}}\cdot T_{\text{ch}} ,

where F_vm is the specific main grinding resistance.

In 2017, Yang et al. [146] investigated micro-scale cranial grinding under various cooling conditions and developed a CHTC model based on fluid dynamics.

(107)h=0.023k_{\text{nf}}(2u_0{b}_0{)}^{0.8}{\eta }^{-0.8}P{r}^{0.4}/\phantom{P{r}^{0.4}(2b_0)}(2b_0) ,

where k_nf is the thermal conductivity of the fluid, u₀ is the exit velocity, b₀ is the half-width of the nozzle exit cross-section, and η is the kinematic viscosity of the fluid. After transformation, h can be modified as follows:

(108)h=105.6k_{\text{nf}}^{0.6}{\rho }_{\text{nf}}^{0.8}{c}_{\text{nf}}^{0.4}/\phantom{c_{\text{nf}}^{0.4}{\mu }^{0.4}}{\mu }^{0.4},

where ρ_nf is the fluid density, c_nf is the specific heat capacity of the fluid, and μ_f is the dynamic viscosity of the fluid.

In 2003, Jin et al. [147] estimated the CHTC using a fluid dynamics and thermal model fit. When the coolant enters the grinding area, the temperature distribution of the cooling medium within the thermal boundary layer is shown in Fig.34(a). Assuming the surface temperature of the workpiece is uniform and constant at T_w, the coolant temperature outside the thermal boundary layer is also set as T_w. As shown in Fig.34(b), δ_t denotes the thickness of the thermal boundary layer at position x. An assumption is that the flow rate of the coolant is equal to the peripheral speed of the grinding wheel. At the surface of the workpiece, the heat flow q is expressed as shown below.

(109)q=-k{\frac{\mathrm{\partial }T}{\mathrm{\partial }y}|}_{\text{w}}=h_{\text{c}} (T_{\text{w}}-T_{\mathrm{\infty }}).

Solving for the CHTC, h_c, requires a coolant temperature gradient at the workpiece surface, while the internal temperature distribution of the coolant, T_w and T_∞, is related as follows:

(110)\frac{T-T_{\text{w}}}{T_{\mathrm{\infty }}-T_{\text{w}}}=\frac{3}{2}\frac{y}{{\delta }_{\text{t}}}- \frac{1}{2}{\left(\frac{y}{{\delta }_{\text{t}}}\right)}^3,

where δ is the thickness of the coolant film in the grinding zone.

By solving the integral equation with appropriate boundary conditions, the fluid mass flow is determined, from which the pressure gradient is derived. Then, through energy balance, the thermal boundary thickness δ_t is obtained as follows:

(111)A_4{\delta }_{\text{t}}^4+A_3{\delta }_{\text{t}}^3+A_2{\delta }_{\text{t}}^2+A_1{\delta }_{\text{t}}+A_0=0 ,

where A_0=-\frac{3}{2}\alpha ( T_{\mathrm{\infty }}- T_{\text{w}} ), A_1=0, A_2=\frac{\mu }{\rho c}[\frac{1}{2}{\left(\frac{v_{\text{s}}}{\delta }\right)}^2-\frac{v_{\text{s}}{k}_1}{2\mu }+\frac{k_1^2}{8{\mu }^2}]+\left[T_{\text{w}}+\frac{4}{5}(T_{\mathrm{\infty }}-T_{\text{w}})\right]\cdot \left(\frac{v_{\text{s}}}{\delta }-\frac{k_1\delta }{2\mu }\right)\cdot (T_{\mathrm{\infty }} -T_{\text{w}}), A_3=\frac{k_1}{\rho c}\left(\frac{v_{\text{s}}}{4\delta }-\frac{k_1\delta }{8\mu }\right)+\frac{k_1}{2\mu }\left[T_{\text{w}}+\frac{7}{8}(T_{\mathrm{\infty }}-T_{\text{w}})\right] \cdot (T_{\mathrm{\infty }}-T_{\text{w}} ), and A_4=\frac{k_1^2}{24\mu \rho c}.

This condition leads to four solutions, wherein only one fits the thermal boundary thickness range from which the CHTC h_c can be derived.

In 2009, Lin et al. [148,149] investigated the CHTC of a fluid using fluid dynamics theory and an analytical model to describe heat transfer in the grinding zone under different grinding conditions. They developed an analytical model for calculating the CHTC by coupling fluid dynamics with heat transfer theory. As shown in Fig.35, a schematic illustrates the heat transfer between the grinding wheel and the workpiece under typical surface grinding conditions.

First, semi-empirical equations based on the theory of heat conduction in solids yield the following:

(112)Nu=0.664R{e}^{1/2}P{r}^{1/\phantom{13}3},

where Re={\rho }_{\text{f}}{\nu }_{\text{av}}{l}_{\text{c}}/{\mu }_{\text{f}}, Pr={\mu }_{\text{f}}{C}_{\text{f}}/k_{\text{f}}, and Nu=h_{\text{f}}{l}_{\text{c}}/k_{\text{f}}.

Combining the above equations, the CHTC h_f is then derived as follows:

(113)h_{\text{f}} =G\sqrt{\frac{{\nu }_{\text{a}\nu }}{l_{\text{c}}}},

where v_av is the average velocity of the fluid through the grinding zone. G is a constant related to density, temperature, and the friction coefficient.

(114)\begin{array}{c}G=0.664{\rho }^{1/2}{C}_{\text{p}}^{1/3}{\mu }^{-1/6},\\ l_{\text{c}}={\left[8R_{\text{r}}^2{F}_{\text{n}}^{\mathrm{\prime }}(k_{\text{s}}+k_{\text{w}})d_{\text{s}}+a{d}_{\text{s}}\right]}^{1/2},\end{array}

where R_r is the roughness factor.

3.4.3 Adoption of the backpropagation method

In 2022, Jin et al. [150] established a backpropagation calculation method to determine the distribution of fluid CHTC in the grinding zone under deep-cut grinding conditions. This method is based on measured grinding temperature signals and a circular contact moving heat-source model.

By considering the energy distribution of heat flow in the grinding zone—among the workpiece, grinding wheel, grinding chips, and grinding fluid—the temperature distribution obtained through actual measurement can be used for inverse analysis to deduce the heat flux density distribution across these subsystems. This process leads to the introduction of the CHTC in the grinding zone, as illustrated in Fig.36.

(115)h_{\text{f}} (x_i)=h_{\text{w}} (x_i)\left[\frac{1}{R_{\text{w}}(x_i)}-\frac{1}{R_{\text{ws}}}-\frac{1}{R_{\text{wch}}(x_i)}+1\right].

Gao et al. [151] improved the model for calculating the CHTC in vertical axis surface grinding, based on the research of Hideo Fukui in Japan. They analyzed the grinding temperature using an analytical method and obtained the temperature distribution along the depth direction of the surface by measuring temperatures at various depths within the workpiece surface layer. This data was then used to derive a formula for calculating the grinding CHTC.

The amount of heat convected outward from a surface is proportional to the temperature difference between the surface and the surrounding fluid, as well as the heat transfer area. For a one-dimensional heat conduction problem, the heat flux through a surface can be determined from the basic definition of heat conduction, and applying the law of conservation of energy yields the following expression:

(116)h=k{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=0}(T_{\text{w}}-T_{\text{f}}{)}^{-1}.

The temperature gradient near the surface and its surface temperature are difficult to determine directly. However, they can be measured by measuring the temperature of the surface layer of the workpiece at different depths, allowing for the determination of the temperature distribution along the depth direction of the object, and subsequently, the required parameters. Let the temperature distribution of the surface layer of the object follow a parabolic distribution.

(117)T(z)=A_0+ A_{1}z+A_2{z}^2,

where A₀, A₁, and A₂ are coefficients to be determined.

(118)\begin{array}{rl}& T_{\text{w}}={T(z)|}_{z=0}=A_{\text{0}},\\ & {\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=0}=A_{\text{f}}.\end{array}

Considering Eq. (117), we have

(119)h=k{A}_{\text{1}}(A_{\text{0}}-T_{\text{f}}{)}^{-1}.

When the three-point method is used to measure the surface temperature of the workpiece today, the coefficients A₀ and A₁ and are determined using the following equation:

(120)\{\begin{array}{rl}{A}_0=& \frac{z_1\cdot z_3}{(z_3-z_1)(z_2-z_1)}{T}_1-\frac{z_1\cdot z_3}{(z_2-z_1)(z_3-z_2)}{T}_{\text{2}}\\ & +\frac{z_1\cdot z_2}{(z_3-z_1)(z_3-z_2)}{T}_{\text{3}},\\ A_1=& -\frac{z_3+z_1}{(z_2-z_1)(z_3-z_1)}{T}_1+\frac{z_3+z_1}{(z_3-z_2)(z_2-z_1)}{T}_2\\ & -\frac{z_2+z_1}{(z_3-z_2)(z_3-z_1)}{T}_{\text{3}}.\end{array}

3.4.4 Based on the Péclet number

Lavine and Jen [152] assumed that the heat flow in the grinding zone is uniform and that the coolant fills the intergranular pores. They further assumed that the coolant is stationary relative to the grinding wheel and can be treated as a moving “semi-infinite solid”. By neglecting heat transfer in the direction of motion, the fluid is assumed to be in an unboiled liquid state with a depth greater than the thermal boundary layer thickness. Based on the large Péclet number of the coolant (typically of the order of 10⁵ or greater), the heat transfer coefficient of the fluid is as follows:

(121)h_{\text{f}} (x)=\sqrt{\text{π}(k\rho c_{\text{p}}{\text{)}}_{\text{f}}{v}_{\text{s}}/(4x)},

where x is the distance from the grinding zone.

In 2015, Vinay and Srinivasa Rao [153] briefly analyzed the temperature destination in the grinding zone, as shown in Fig.37. They also calculated the CHTC of the grinding fluid based on the motion characteristics during the grinding process, as shown below.

(122)h_{\text{wp}}=\frac{{\beta }_{\text{wp}}}{C}\sqrt{\frac{{\nu }_{\text{s}}}{l_{\text{c}}}},

where C is the correction factor and depends on the dimensionless Péclet number Pe, according to Rowe [97]:

(123)

3.4.5 Differences in fluids in different areas of the grinding zone

In 2008, Shen [154] estimated the CHTC during the grinding process using the finite difference method for grinding heat modeling. Considering the complexity of the actual boundary conditions in the grinding process, different regions were analyzed and discussed in detail. The planar grinding configuration is shown in Fig.38. The actual depth of cut in the grinding zone decreases from the leading edge to the trailing edge, resulting in uneven convective cooling, the leading edge, contact zone, and trailing edge of the workpiece are considered separately. The grinding zone CHTC h_contact is assumed to be a linear function, as presented in Fig.38:

(124)h_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}\left(x\right)=h_1+\frac{h_2-h_1}{l_{\text{c}}}\left(x+\frac{l_{\text{c}}}{2}\right),

where h₁ = max[h_contact(x)] and h₂ = max[h_contact(x)]. x is the local coordinate of the origin at the center of the grinding zone.

Assuming that the CHTC at the trailing edge is uniform, the CHTC between the trailing edge and the end of the contact zone does not vary. Therefore, at the trailing edge: h_trailing = h₁.

The cooldown effect of the leading edge is negligible; therefore, h_leading = 0.

3.4.6 Based on particle contact conditions

In 2013, Zhu et al. [107] proposed an improved temperature calculation model that considers the geometry and distribution of abrasive grains to improve model accuracy. Considering the effects of abrasive grain geometry, size, and concentration, the CHTC model was modified to propose a calculation model based on cross-arrays of grains. Assuming that the abrasive grains on the surface of the grinding wheel are arranged in an orderly manner according to a cross-array pattern, as shown in Fig.22 and Fig.39, and that the abrasive grains are modeled as cylinders with the same diameter, a new calculation model for the CHTC of the grinding surface of the workpiece is derived based on the multirow bundle heat transfer model for the outward flow of the fluid, as follows:

(125)h_{\text{f}} =\frac{Nu\cdot k_{\text{f}}}{L_{\text{c}}}=\frac{Nu\cdot k_{\text{f}}}{d_{\text{g}}N}=\frac{Nu\cdot k_{\text{f}}}{d_{\text{g}}{l}_{\text{c}}{L}_{\text{g}}^{-1}},

where L_c is the characteristic length, d_g is the equivalent mean abrasive grain diameter, and L_g is the mean particle spacing.

If the coefficients are integrated into the model by taking the average grain spacing to be approximately 1.5 times the grain spacing in the grinding wheel, and using the semi-empirical equations for the forced convection theory, the model for calculating the CHTC can be then rewritten as follows:

(126)h_{\text{f}} =\frac{0.664v_{\text{s}}^{1/2}{L}_{\text{g}}^{1/2}{k}_{\text{f}}}{v_{\text{f}}^{1/6}{\alpha }_{\text{f}}^{1/3}{d}_{\text{g}}^{1/2}{l}_{\text{c}}^{1/2}},

where α_f represents the thermal diffusion coefficient of the fluid.

3.4.7 Considering the fluid flow state

Zhang and Rowe [155,156] studied the role of fluid convection in cryogenic grinding around 2020. Convective heat transfer models typically assume that the fluid flows over the workpiece surface in a laminar flow state within the grinding zone. However, in reality, the fluid is mechanically agitated due to the constraints of the abrasive material, making the assumption of steady laminar flow problematic. Therefore, the CHTC in various cases is investigated based on the different flow states of the fluid.

Laminar flow modeling. When the fluid state in the grinding zone is determined to be laminar flow, the CHTC is obtained based on the assumption that the fluid flows through a smooth surface, as follows:

(127)h_{\text{f}} =\frac{4}{9}{\rho }_{\text{f}}^{\text{1/2}}{C}_{\text{f}}^{\text{1/3}}{\eta }^{-1/6}{k}_{\text{f}}^{2/3}{\nu }_{\text{s}}^{1/2}{l}_{\text{c}}^{-1/2}.

Turbulent flow modeling. The results revealed that the fluid in the grinding zone behaves turbulently. This assumption is highly realistic because the fluid is vigorously agitated by the high-speed particles in the limited pore space between the grinding wheel particles. Turbulence helps bring the cooling fluid closer to the working surface by reducing the laminar boundary layer. As shown in Fig.40, the dashed line shows the typical fluid temperature distribution. Therefore, the calculation of CHTC, considering turbulent flow, is modeled as follows:

(128)h_{\text{f}} \approx \frac{1}{40}{\rho }_{\text{f}}^{4/5}{C}_{\text{f}}^{1/3}{\eta }^{-7/15}{k}_{\text{f}}^{2/3}{\nu }_{\text{s}}^{4/5}{l}_{\text{c}}^{-1/5}.

Fluid–wheel model. The boundary layer of the fluid is completely eliminated, and the fluid is assumed to maintain a zero-temperature rise to the workpiece surface, at which point the maximum CHTC will be produced, indicating an impossible limiting assumption. The following assumptions are also presented: the fluid entering the grinding contact zone is at zero-temperature rise above room temperature, the fluid passes through the workpiece at wheel speed, and a thermal boundary layer is established in a solid with heat transfer characteristics of the fluid. Thus, the CHTC is as follows:

(129)h_{\text{f}} =k_{\text{f}}^{1/2}{\rho }_{\text{f}}^{1/2}{c}_{\text{f}}^{1/2}{\nu }_{\text{s}}^{1/2}{l}_{\text{c}}^{-1/2}.

In 2016, Wang [157] expressed the convective heat transfer problem in terms of a set of convective heat transfer differential equations and definite solution conditions. As shown in Fig.41, when the grinding fluid first enters the grinding zone, it can be approximated as a laminar flow with velocity v_s over the workpiece surface. However, as the flow distance x increases, the flow transitions toward turbulence.

In the laminar flow state, the boundary layer integral equation system is used to solve for the CHTC. The average CHTC can then be obtained by averaging the local CHTC integrals over the length l_c under the condition of Pr ≥ 1:

(130){\overline{h}}_{\text{f}}=0.664\frac{k_{\text{f}}}{l_{\text{c}}}R{e}^{1/2}P{r}^{1/3}.

The temperature distribution is inhomogeneous over the entire length of the grinding arc; thus, the average CHTC is two-thirds localized CHTC.

(131)\{\begin{array}{c}{\overline{h}}_{\text{f}}=0.4427\frac{k_{\text{f}}}{l_{\text{c}}}R{e}^{1/2}P{r}^{1/3},\\ Re=\frac{{\rho }_{\text{f}}{\nu }_{\text{s}}{l}_{\text{c}}}{{\eta }_{\text{f}}},\\ Pr=\frac{C_{\text{f}}\eta }{k_{\text{f}}}.\end{array}

The same method is applied in the case of turbulent flow. When the laminar flow section is notably short due to high fluid velocity, approximating the boundary layer as being fully turbulent, the average heat transfer coefficient is calculated as follows:

(132){\overline{h}}_{\text{f}}=0.0247\frac{k_{\text{f}}}{l_{\text{c}}}R{e}^{4/5}P{r}^{1/3}.

If the laminar flow section is not negligible and the combined laminar and turbulent flows are considered, then the average heat transfer coefficient along the grinding arc length for the mixed boundary layer is given as follows:

(133){\overline{h}}_{\text{f}}=\frac{2k_{\text{f}}}{3l_{\text{c}}}\left[0.664R{e}^{1/2}+0.037(R{e}_{\mathrm{l}\mathrm{c}}^{4/5}-R{e}_{\mathrm{c}}^{4/5})\right]P{r}^{1/3}.

The critical Reynolds number R{e}_{\mathrm{l}\mathrm{c}} is influenced by the degree of disturbance in the incoming flow and the roughness of the wall surface. During the grinding process, factors such as the jet pressure of the grinding fluid, the jet speed, the particle size of the grinding wheel, and the surface roughness of the workpiece all affect the value of the critical Reynolds number.

3.4.8 Based on statistical theory

During MQL grinding, surface heat transfer is highly nonlinear due to the large temperature gradient on the grinding surface, the complex kinematic characteristics of the spray, and the random distribution of droplet sizes [158,159].

In 2021, Yang et al. [143,160] investigated the atomization mechanism of nanofluid aerosol cooling and statistically analyzed the probability density of droplet sizes in the grinding zone. They used mathematical and statistical methods to analyze the effective heat transfer droplets within the nanofluid aerosol jet during grinding. Based on this analysis, a theoretical model for the CHTC of nanodroplets was established by calculating the heat transfer coefficients of the nanofluid jet and the accompanying high-pressure gas jet.

As shown in Fig.42, when an aerosol droplet impinges on the grinding zone, it can rebound, diffuse, and splash. The rebound and crown droplet splash do not effectively contribute to heat transfer. The most effective heat transfer occurs when the droplet impinges on the surface of the heat source and diffuses along the surface, forming a liquid film. The critical Weber number that determines the effective droplet impact pattern is given by 2.0 × 10⁴ × La^−0.2 ≤ We ≤ 2.0 × 10⁴ × La^−1.4, where La is the Laplace number.

Assuming that the influence of the airflow around the grinding tool is overlooked, the velocity of the droplets hitting the solid surface can be taken as the velocity of the spray outlet. Based on the critical Weber number, the effective droplet diffusion size can be derived, that is, D_min ≤ D ≤ D_max, and the ratio of effective heat transfer droplets can be calculated as shown below.

(134)P_{\text{d}} ={\int }_{D_{\text{max}}}^{D_{\text{min}}}\frac{1}{6{\overline{D}}^4}{D}^3\text{exp}\left(-\frac{D}{\overline{D}}\right)\text{d}D.

For a single cooling droplet, the heat transfer coefficient h_s is presented below.

(135)\{\begin{array}{c}J=c_{\text{f}}{m}_{\text{d}}\mathrm{\Delta }T,\\ h_{\text{s}}=\frac{q_{\text{d}}}{\mathrm{\Delta }T},\\ J=q_{\text{s}}{A}^{\prime }{t}_{\text{s}},\end{array}

where J is the amount of heat transfer for a single droplet, ∆T is the heat transfer temperature difference, q_d is the heat flow density of a single droplet heat transfer, t_s is the heat transfer time, m_d is the droplet mass, and A^{\prime } is the diffusion area of the droplet.

By integrating over the effective region, the heat transfer coefficient for all effective heat transfer droplets is given by the following:

(136)h_{\text{n}} =N_z{\int }_{D_{\text{max}}}^{D_{\text{min}}}\frac{h_{\text{s}}}{6{\overline{D}}^4}{D}^3\text{exp}\left(-\frac{D}{\overline{D}}\right)\text{d}D,

where N_z is the number of all droplets.

The CHTC of the gas introduced to high pressure h_a = k_a·Nu/b_c, and the integrated CHTC for MQL cooling is as follows:

(137)h_{\text{f}} =h_{\text{n}}+h_{\text{a}}.

Tab.6 further summarizes the factors influencing the EDC in grinding operations.

4 Application of heat transfer process parameters modeling

4.1 Energy distribution coefficient

Rowe [84] investigated the energy distribution and temperature prediction in the grinding process using CBN and alumina grinding as examples. By comparing two different models and demonstrating the sensitivity of important parameters, the abrasive grain model aligns more closely with the underlying reasoning.

The experimental results show that the thermal contact coefficients, derived indirectly from the grinding experiments, are notably lower than the performance values obtained from direct measurements. Therefore, the effective values obtained by correlating the theoretical model with the grinding experiments are lower, while the abrasive grain model demonstrates better agreement between the theoretical and experimental values of thermal performance.

Qiu et al. [89] studied the grinding heat transfer process in thread profile grinding and analyzed the process using two EDC models for two distinct grinding regions. One model uses a single-particle energy distribution, while the other uses an energy distribution model at the abrasive grain–workpiece interface. At a_p = 0.005 mm, the energy distribution to the ground workpiece reaches 76.5% and 84.4% for the AB and BC surfaces (Fig.11), respectively. The thermal partition R_w(AB) is approximately 8%–9% lower than R_w(BE), which is due to the better heat dissipation conditions in the AB plane compared to the BE plane.

In 1999, Guo et al. [86] developed a numerical calculation model for profile grinding temperature based on the distribution of heat sources in right-angled triangles within the grinding arc and the simplification of the one-dimensional heat conduction process. This model was further supported by the theoretical analysis of continuous grinding temperature on a plane. After conducting numerous grinding tests with conventional aluminum oxide abrasives and CBN grinding wheels, and using different fluids (including soluble oil, cold air, and ester oil), the temperature rise of the workpieces under conventional and creep-feed conditions was measured. The energy distribution was obtained by measuring the temperature, integrating the positive heat flow density, and dividing by the grinding power. Notably, the results indicate that the energy distribution is influenced by grinding conditions, the type of abrasive, and the application of the fluid, all of which can be quantitatively considered by using a single-grain thermal model.

Ding et al. [98] investigated the temperature and energy distribution in the cylindrical grinding process under various process parameter and thermophysical conditions through finite element simulation and experimental validation. The results revealed that the effect of workpiece rotational speed on the energy distribution to the chip is more pronounced than that of other grinding parameters. The rational selection of grinding parameters can notably reduce the energy distribution into the workpiece, from 80% to a range of 30%–50%, or even lower. The variation in the energy distribution ratio under different process parameters is shown in Fig.43.

Wang et al. [161] determined the heat flow density of the workpiece using the finite difference method, based on measurements of the grinding force and grinding temperature, which allowed them to obtain the energy distribution. As shown in Fig.44, the results indicate that the feed rate has a more substantial effect on grinding temperature and energy distribution compared to other factors. An increase in feed rate reduces the machining temperature but results in a considerably higher energy distribution. Studies have revealed that the energy distribution in dry grinding is approximately 5% when grinding the titanium alloy material TC4 with a vitrified bond CBN grinding wheel.

Hadad et al. [95] investigated the temperature and energy distribution in the MQL grinding process. The thermal performance of MQL grinding was analyzed and compared with dry grinding and fluid grinding. The results, shown in Tab.7 and Tab.8, indicate that the energy distribution for MQL grinding with Al₂O₃ grinding wheel ranges from 73% to 77%, whereas the energy distribution for dry grinding is approximately 82%. In fluid grinding with Al₂O₃ grinding wheels, the cooling effect of the fluid in the contact zone reduces the energy distribution to less than 36%. By contrast, grinding with CBN grinding wheels results in a notably lower energy distribution to the workpiece due to the thermal conductivity of the CBN material, which is 46% in combination with MQL, 52% in dry grinding, and 14% in fluid grinding.

Zhan and Xu [162] conducted an experimental study on the temperature and energy distribution in grinding carbide with vacuum-brazed diamond grinding wheels. The effects of grinding conditions such as grinding wheel speed, feed rate, and depth of cut on the grinding temperature and energy distribu-tion are shown in Fig.45. The results indicate that the grinding temperature measured under different grinding conditions ranged from 10 to 100 °C. The energy distribution of dry grinding to the workpiece was between 35% and 70%.

Li et al. [163] used nanofluids prepared from different vegetable oils as base oils for high-temperature nickel-based alloy MQL grinding experiments and compared the energy proportionality coefficients of seven vegetable oils for MQL grinding. The results are shown in Fig.46, where castor oil-based MQL grinding demonstrates the highest energy proportion factor, while palm oil-based MQL grinding exhibits the lowest energy proportion factor. The results for the other five vegetable oils were similar, with values in between the above two. Palm oil is the best base oil for MQL grinding with an EDC of 42.7%.

Li et al. [164] conducted experiments on the efficient deep grinding of nickel-based high-temperature alloys using porous metal-bonded CBN supergrinding wheels to study the energy distribution to the workpiece during grinding. The results show that the energy distribution values for porous metal-bonded CBN grinding wheels generally range from 2% to 6%, which is lower than that of conventional vitrified CBN grinding wheels and alumina grinding wheels. In comparison, the energy distribution to the workpiece for conventional vitrified CBN grinding wheels ranges from 4% to 8%, while for conventional alumina grinding wheels, this distribution ranges from 25% to 65%.

Hou and Yao [165] studied the grinding process using fluted grinding wheel grindings and modeled the temperature distribution of the workpiece under conditions of periodic heat-source variation. As shown in Fig.47, the experiment shows that R_w increases with the feed rate, while R_f decreases as the feed rate increases. The R_w for slotted grinding wheels is higher than that of non-slotted grinding wheels at feed rates ranging from 400 to 2000 mm/min. Additionally, the R_f for slotted grinding wheels is lower than that for non-slotted wheels. The heat flow density transferred to the coolant when the grinding wheel is slotted is not greater than that when the grinding wheel is not slotted.

Jin et al. [108] analyzed the heat transfer mechanism in involute gear form grinding, considering and analyzing in detail the variation in tooth contact geometry. The heat effect on the involute tooth form was investigated using a transient, non-uniform heat-source moving along a three-dimensional curved shape. The distribution of the energy distribution ratio of the workpiece along the involute contact profile under dry grinding conditions, with different grinding depths and table speeds, is shown in Fig.48.

Overall, the energy distribution along the involute contact profile, from root to tip, gradually decreases under dry grinding conditions and varies with the grinding parameters. Increasing the grinding depth and table speed helps reduce the energy distribution.

Pang et al. [166] investigated thermal models in the grinding zone by analyzing dry and wet cylindrical plunge grinding. In studying the EDC coefficient of energy distribution, the effect of CHTC coefficient was considered. The analysis was conducted using two energy distribution models: R_ws and R_wch for workpiece–grinding wheel and the workpiece–chip, respectively. The results show that v_f and a_p have substantial effects on R_w, while v_s has minor effects on R_w. Specifically, R_w decreases with increasing v_f and increases with rising a_p.

Zhu et al. [107] proposed an improved temperature calculation model that considers the geometry and distribution of abrasive grains, as well as the workpiece–wheel energy distribution ratio, considering the effect of the cone angle during the machining process. Experimentally, the effective contact radius of abrasive grains was found to substantially influence the workpiece–wheel distribution ratio. As shown in Fig.49(a), when r₀ < 15 μm, the predictions of the two models are similar; however, as r₀ continues to increase, the two models show considerable differences. Regarding the effect of abrasive geometry, the cone angle also has a substantial influence on the workpiece–wheel distribution ratio. As shown in Fig.49(b), at a depth of cut of 5 μm, the R_ws value increases almost linearly with the taper angle in the range of 90°–170°. As the depth of cut increases, especially beyond 25 μm, the sensitivity of R_ws to depth decreases.

Peng et al. [96] investigated the height of the grinding chip energy distribution ratio and its variation in the high-speed grinding process of nickel-based alloys, analyzing changes in the energy distribution mechanism under wet and dry grinding conditions were studied. As shown in Fig.50, which presents the ratio of grinding temperature and energy distribution to the workpiece, as well as the heat flow and heat transfer coefficients under identical grinding conditions, the grinding temperature at v_s = 80 m/s is substantially higher than at higher wheel speeds. This finding indicates that the energy distribution mechanism at v_s = 80 m/s differs markedly from that at higher wheel speeds, possibly due to the presence of burnout of the grinding fluid or film boiling occurring once the threshold temperature is exceeded.

The conclusions show that under wet grinding conditions, high-speed grinding increases the CHTC and reduces the energy distribution indications. However, when the temperature exceeds the combustion threshold temperature associated with membrane boiling, the fluid has difficulty in wetting and cooling the heated surfaces in the grinding region. This behavior is similar to dry grinding condition, leading to notable increase in the EDC of the workpiece. Under “dry” grinding conditions, approximately 30%–40% of the grinding heat is carried away by the grinding debris.

Zheng et al. [111] established an improved external wet grinding temperature model that accounts for the lubrication effect of the grinding fluid and solved it numerically. They studied and compared the temperature variation of alloy steel during cylindrical wet grinding under different grinding process parameters, considering the lubrication effect. Experimental results show that the value of q_w with lubrication is larger than that without lubrication, and the difference between the two gradually increases with v_s. The value of q_w is directly proportional to the total grinding heat flow q_t of the workpiece and the energy distribution ratio R_w, as shown in Fig.51.

As shown in Fig.52, the energy distribution ratio R_w is higher for a lubricated workpiece compared to one without lubrication. This finding indicates that, under lubricated conditions, a greater portion of the grinding heat flow q_w enters the workpiece, resulting in higher grinding temperatures.

Lavine et al. [116] analyzed the grinding wheel surface in the grinding zone by approximating it as a thermal composite material comprising abrasive grains and grinding fluid within its pores. The energy distribution was calculated by modeling the grinding zone as a rectangular band of heat sources moving at the working speed along the wheel surface. The study concluded that the thermal properties of this composite solid are difficult to specify and are highly sensitive to the porosity of the grinding wheel near the workpiece surface.

Therefore, Kim et al. [113] investigated the heat flow density and energy distributions of the workpiece during creep-feed grinding

Ren et al. [167] analyzed and modeled the dynamic energy distribution in belt grinding, considering the dynamic characteristics of belt grinding in terms of grinding effect and heat. The thermal characteristics of the system were obtained using a combination of the finite element method and an optimization algorithm. The dynamic energy distribution during continuous grinding was then calculated using an iterative method that considers the heat accumulation effect in the workpiece.

Validation was conducted using SUS304 and AA6061-T6 workpieces, and the results demonstrate the effectiveness of the proposed method for calculating dynamic energy distribution. For the SUS304 workpiece, the validation results are shown in Fig.53(a) and Fig.53(b), where the maximum percen-tage error in the calculated q_w is 14.3%, and the maximum error between the calculated and measured temperatures is less than 32.4 ℃. In comparison, for the AA6061-T6 workpiece, the validation results are presented in Fig.53(c) and Fig.53(d), with the calculated grinding temperature closely following the trend of the measured temperature and a maximum error of 17.18 ℃.

The dynamic energy distribution for the SUS304 workpiece is shown in Fig.53(e), while the results for the AA6061-T6 workpiece are shown in Fig.53(f). The results indicate that, unlike traditional grinding with wheel grinding, the mechanical properties of the workpiece in belt grinding substantially influence the energy distribution and the material removal mechanism. As grinding proceeds, the grinding temperature softens the workpiece, making material removal easier. Simultaneously, heat accumulation in the abrasive grits reduces the cooling capacity of the belt.

Mohamed et al. [168] improved the estimation of energy distribution during grinding using surface roughness measurements. Their experimental findings demonstrated that surface roughness can serve as an effective indicator for estimating particle radius. The average difference between the energy distribution predicted using this model and the conventional method was 7.5%. The results, showing the relationship between energy distribution and surface roughness under three different dressing conditions, are plotted in Fig.54. Additionally, the energy distribution values calculated using this method are compared to those derived from the traditional trial-and-error method.

Tab.9 further summarizes the factors influencing the EDC in cutting and grinding machining.

4.2 Convective heat transfer coefficient

Liu et al. [144] analyzed the flow liquid film heat transfer law at the interface of grinding wheel–workpiece with low-temperature cold air MQL, based on the evolution law of the physical properties of the lubricant and the liquid film heat transfer mechanism in the grinding zone. They also conducted verification experiments on the flow liquid film CHTC and the heat transfer performance of low-temperature cold air MQL for grinding titanium alloy. The results show that, under the same lubricant flow rate, the flow liquid film CHTC h_f increases as the cold air temperature decreases. Under different cold air temperatures, the theoretical calculated value of the heat transfer coefficient h_f is generally consistent with the actual measured value, with the model error ranging from 8.5% to 11.6%.

The theoretical value of CHTC coincides with the measured value, with an error of 8.5% when the cold air temperature is −10 ℃. Additionally, the experi-mental surface temperature value of the workpiece coincides with the trend of the theoretical value, with an error of 7.7% when the grinding depth is 30 μm and the cold air temperature is −40 ℃.

Gupta and Yadav [22] used the research method of fluid dynamics to establish a mathematical model for the convective CHTC of grinding fluids. Through experimental and simulation studies under different grinding conditions, they derived the trends of grinding parameters and the influence of the grinding fluid’s density, specific heat, conductivity, and viscosity on the changes in CHTC. Experiments were conducted using two grinding fluids at different grinding wheel speeds and grinding depths, and the results indicated that the CHTC of the grinding fluids increases with the speed of the grinding wheel and decreases with the depth of cut. Additional experiments were conducted under the same conditions by separately controlling the differences in the density, specific heat, conductivity, and viscosity of the grinding fluid. The results showed that changes in specific heat had a small effect on the coefficient, while changes in viscosity and conductivity had a substantial effect on the coefficient. The CHTC coefficient increased with the concentration, specific heat, and conductivity of the grinding fluid, and decreased with the viscosity of the grinding fluid.

Lin et al. [148,149] described the heat transfer in the grinding region under different grinding conditions based on fluid dynamics and heat transfer theory. Grinding experiments were conducted on two material powders separately, and the results are shown in the table. Notably, the CHTC values can be remarkably high, substantially exceeding those previously reported. As shown in Fig.55, the experimental results indicated that the values of CHTC varied with process parameters, such as the actual depth of cut and wheel speed, and decreased as the contact length increased.

Jin et al. [147] used fluid dynamics and thermal simulations to investigate and estimate the CHTC in the grinding zone. They concluded that the CHTC is determined by the grinding wheel speed and the liquid film thickness in the contact zone, which is, in turn, affected by factors such as the grinding wheel speed, porosity, particle size, coolant type, flow rate, and nozzle size.

The grinding temperatures for deep grinding and shallow grinding were calculated based on the estimated CHTC values and compared with the actual grinding temperatures.

Jin et al. [150] used the inverse method to investigate the fluid CHTC in the grinding zone under deep-cut grinding conditions by measuring the temperature in the grinding zone with the help of the EDC. The experiment showed that the distribution pattern of CHTC in the grinding zone under slow-in deep-cut mode is notably different from that of conventional shallow grinding. The distribution of CHTC in the grinding zone is divided into two curves: in the small interval at the front end of the grinding zone, the grinding fluid CHTC exhibits an obvious upward section, while in the main convective heat transfer zone, the CHTC change is relatively flat. The contact arc length and feed rate have minimal influence on the CHTC of the main convective heat transfer zone, with grinding speed being the main factor.

Gao et al. [151] experimentally determined the CHTC and its distribution in the contact zone between the grinding wheel and workpiece during vertical axis surface grinding. Based on heat conduction theory and a large number of experimental results, the grinding temperature follows a parabolic distribution along the depth direction. The CHTC was determined by inverting the model and using the three-point method to measure the surface temperature of the workpiece.

The CHTC within the contact area between the vertical axis plane grinding wheel and the workpiece is unevenly distributed. In dry grinding, the coefficient demonstrates minimal variations, whereas in wet grinding, substantial changes occur at different locations on the contact surface of the grinding wheel. The installation height of the cooling nozzle has a greater impact on the cooling effect. Additionally, the coolant flow rate notable affects h. When the flow rate is small, the effect is more pronounced. However, once the flow reaches a certain value, the cooling heat transfer coefficient increases slowly; that is, the coolant flow rate is too small, the cooling effect is poor, while excessive flow rate leads to waste.

Shen [154] estimated the CHTC using a grinding heat model based on the finite difference method. The accuracy of the model was verified by comparison with the traditional semi-infinite workpiece grinding heat model. The results showed that the effects of free convection and leading-edge cooling are negligible, while trailing-edge cooling only affects the trailing-edge region. The model was further applied to analyze convective cooling in grinding experiments. The estimated average CHTC in the grinding contact zone was approximately 4.2\times {10}^5\mathrm{W} /({\mathrm{m}}^2\cdot \mathrm{K}) for wet grinding and approximately 2.5\times {10}^4\mathrm{W} /({\mathrm{m}}^2\cdot \mathrm{K}) for MQL grinding, whereas the estimated CHTC at the trailing edge was notably lower.

By matching the measured temperature distri-butions with those calculated by the FDM heat transfer model, the CHTC within the grinding zone (h_contact) and at the trailing edge (h_trailing) can be obtained. The CHTC in the contact zone is higher than at the trailing edge due to the high flow rate. Additionally, the CHTC in wet grinding is notably higher than in MQL grinding for the contact zone and trailing edge. The results for the trailing edge in wet grinding show good agreement, whereas the results for MQL grinding at the trailing edge do not match well due to the low flow rate.

Zhu et al. [107] proposed an improved temperature calculation model considering the abrasive grain geometry and distribution. Assuming that the abrasive grains on the surface of the grinding wheel are arranged in an orderly manner in a cross-array, an experimental study was conducted to accurately assess the effect of abrasive grain geometry (cone angle, grain size, and grain concentration) on the CHTC.

Notably, increasing the cone angle leads to a reduction in the CHTC, as shown in Fig.56(a), which can be illustrated by the increasing Reynolds number with the rising cone angle. As shown in Fig.56(b), a large abrasive grain size leads to a small CHTC. This phenomenon is due to the large grinding wheel size, which results in a smaller corresponding grinding wheel pitch than the finer grinding wheel, making it difficult for the coolant to penetrate into the gap between the grinding wheel and the workpiece. This condition results in an increase in the Reynolds number and a decrease in the CHTC. As shown in Fig.56(c), a high abrasive grain concentration leads to a low CHTC. This finding is equivalent to Fig.56(b), where a small abrasive spacing results in a high Reynolds number, leading to a small CHTC.

Zhang and Rowe [155] investigated the modeling of CHTC for different fluid states and compared the experimental values with the predicted values. They compared the predictions of the laminar flow model with the experimental results, as well as those of the fluid wheel model. The CHTC predictions from the turbulent and laminar flow models were compared to a series of earlier experimental results over a range of low grinding temperature values. The turbulent flow model predictions were closer to the actual results. The turbulent flow model predicted drastically higher h_f values than the laminar flow model. However, below the boiling range, the TFM predictions were still lower than the measured results. At notably low grinding temperatures, the predicted values from both models are excessively low, indicating that as the grinding temperature decreases, the heat transfer coefficient diminishes, leading to reduced heat dissipation during the grinding process.

Tab.10 further summarizes the factors influencing the CHTC coefficient in cutting and grinding machining.

5 Model synthesis evaluation and parametric impact analysis

Several different models of heat transfer process parameters are available, and the effect of influencing factors on the parameters is full of variations. Models with high accuracy can be effectively identified by summarizing and evaluating the existing models. By analyzing the mechanism of the influence of each factor on the parameters, the change trend of heat transfer process parameters under the influence of each factor can also be effectively understood.

5.1 Evaluation of heat transfer process parameters models

5.1.1 EDC model

In the study of the heat transfer process, a reasonable choice of different EDC models can effectively reduce error projection results. Owing to differences in model construction process, each model adopts different approaches or methodologies, with varying emphasis on influencing factors. Therefore, under different processing conditions, selecting the appropriate type of EDC model based on the importance of influencing factors can help minimize modeling errors. The different models are categorized as shown in Fig.57.

EDC models can generally be divided into two main categories based on whether experimental involvement is required. The first category does not require experimental data; instead, the models are constructed using known parameters and pre-determined processing parameters. For example, models associated with abrasive grain–workpiece contact are developed by employing intrinsic parameters of the grinding wheel and workpiece (thermal conductivity, specific heat capacity, abrasive grain radius), along with a preset parameters such as rotational speed of the grinding wheel. By modeling the grinding wheel and the fluid as a composite system, a relatively simple predictive model can be constructed based on the thermal parameters of the grinding wheel and the coolant. However, due to unclear overall thermal characteristics of the composite, such models are mainly used for qualitative analysis of the energy distribution during the grinding process. This category also includes partial models based on single particles, material physical parameters, slicing multiple energy distribution subsystems, and considerations such as heat-source tilt angle. A comprehensive model in this group is the one developed from the studies of Mao et al. [100] and Rowe [97], which fully incorporates the physical parameters of the grinding wheel and the workpiece, effective contact radius of the abrasive, the speed of the workpiece and the grinding wheel, coolant properties, effective grinding force per unit width, and wheel accuracy. Owing to its broad adaptability and capability to capture the influence of multiple factors, this model has become a widely adopted method for EDC calculation. The EDC model proposed by Yin and Marinescu [115], based on the assumption that the grinding wheel and the grinding fluid are compounded as a whole, also considers the effects of several factors, including the workpiece feed rate, depth of cut, equivalent grinding wheel diameter, tangential grinding force, physical and thermal properties of the grinding wheel and the workpiece, and the cooling effect of the grinding fluid. This model has the best accuracy among such models, with errors ranging from 6.94% to 14.87%. The second category requires experimental participation, and the required parameters in the model calculation must be measured experimentally. For example, the turning energy distribution model based on the analytical method of multisystem construction combines the physical and thermal parameters of the workpiece and tool materials, as well as the measured temperature during the turning process and the calculation of the energy distribution of the subsystems, to obtain the energy distribution of the workpiece. Moreover, the grinding energy distribution model, considering the surface temperature of the workpiece, calculates the density of heat flow into the workpiece by combining its temperature increase, which leads to the calculation results. The accuracy of such models is roughly in the range of 10% to 20%, and the accuracy of the energy distribution model derived by Hadad et al. [95], from the maximum temperature increase of the work-in during grinding, can be achieved with an error in the range of 5.4% to 15.4%.

5.1.2 CHTC model

CHTC plays a crucial role in the study of heat transfer during machining. Aiming to control the temperature of the machining zone, the use of coolant is the most common method. As the coolant flow throughs the machining zone, convective heat transfer between the continuously flowing coolant and the surface of the machining zone removes a substantial amount of heat, effectively lowering the temperature of the machining zone. Therefore, an appropriate CHTC model enables more accurate calculation of the heat carried away by the coolant, thereby minimizing the errors in predicting the temperature of the machining zone. Moreover, a well-established convective heat transfer model helps identify the key factors affecting the heat transfer efficiency. This understanding allows for targeted improvements in the cooling process, effectively reducing thermal damage caused by untimely heat dissipation in the machining process.

As shown in Fig.58, during the machining process, convective heat transfer includes gases and coolants. In terms of gas-based convective heat transfer, one category arises due to natural convective heat transfer between the tool or workpiece and the surrounding air. For example, Yin et al. [123] modeled the CHTC between the workpiece and the air, considering the heat exchange with the air during the continuous rotation of the workpiece. Another category involves the input of cold air into the machining area. This approach, known as low-temperature cold air cooling technology, facilitates strong convective heat transfer between the cold air and the grinding wheel or workpiece, effectively removing the heat generated in the machining area.

In terms of convective heat transfer in coolants, several categories are presented.

The first category focuses on a model that considers the flow state of the coolant, highlighting the physical properties of the coolant itself and the state of the flow process. Owing to variations in type, density, viscosity, and other physical parameters, different coolants exhibit different heat transfer efficiencies. As the coolant flows through the machining zone, it is agitated by the rotating tool, which leads to changes in its flow state and, consequently, its convective heat transfer efficiency. With the emergence of advanced cooling methods such as MQL, factors such as droplet number and droplet diffusion area have gained importance. Relevant models include those based on fluid mechanics, those incorporating the Péclet number, and those that consider the difference of fluid in different areas of the grinding zone and the fluid flow state. Among such models, the convective heat transfer model derived by Wang [157], considering the flow state and using a system of differential equations, has a good accuracy with an error of 2%–7%.

The second category is a model that considers the boiling state of the coolant. The coolant will change its boiling state after heat absorption in the processing zone. Different boiling states will have varying effects on the convective heat transfer efficiency. Liu et al. [144] constructed a convective heat transfer model with an accuracy error of 8.5%–11.6% in low-temperature cold-air MQL machining, considering the effects of liquid film boiling.

The third category focuses on a model that considers other processing parameters. The machining process is an interconnected and complex process, with various factors affecting each other. The speed of the grinding wheel, particle distribution, contact length, and surface temperature will influence the heat transfer efficiency of the coolant, for example, the model based on the particle contact conditions, and the model using the backpropagation method. Zhu et al. [107] considered the optimization of abrasive particle geometry, and distribution of CHTC calculation model accuracy can achieve errors ranging from 7.5% to 9.5%.

The various types of models are not completely independent, and a cross-mixture often exists. In this paper, the categorization is based primarily on the main influencing factors.

5.2 Parameter effects and mechanistic analysis

5.2.1 Parameters related to EDC

The main parameters affecting the variation of the EDC during machining include the depth of grind (depth of cut), grinding wheel speed, feed rate, coolant, workpiece material, grinding wheel type, grit distribution, and geometry.

The grinding depth refers to the depth of the grinding wheel cutting feed during each grinding process. When the grinding depth is greater, the amount of material removed in a single pass increases, the grinding force increases, and additional heat is generated by friction and cutting. Thus, when the grinding depth is increased, additional energy is consumed in other areas, and the proportion of energy absorbed by the workpiece is relatively reduced [108,162]. As for the grinding wheel speed, at high grinding speeds, higher speeds result in higher cutting temperatures and more heat generated by friction, which leads to the transfer of additional heat to the workpiece surface. Therefore, as the grinding wheel speed increases, the EDC of the workpiece generally increases. The effect of feed rate on the EDC is similar to that of the grind depth in that an increase in feed rate raises the grinding forces and generates additional heat. Therefore, the EDC typically increases with rising feed rate [161,162,165]. However, a larger feed rate helps to spread the generated heat quickly over a larger area, reducing the risk of localized high temperatures and leading to a reduction in the amount of heat absorbed by the workpiece. Consequently, a situation in which the EDC decreases with an increase in the feed rate also occurs. The coolant increases the heat exchange efficiency between the workpiece and the grinding wheel. Thus, minimal heat is absorbed on the surface of the workpiece, and additional heat is carried away. Therefore, coolant application is usually effective in reducing the energy distribution ratio of the workpiece [95,111,163]. Properties such as thermal conductivity, hardness, and heat capacity of different workpiece materials can substantially affect the absorption and conduction of heat energy. Harder workpiece materials generally absorb more thermal energy, while workpieces with higher thermal conductivity can dissipate heat to other areas more quickly, thus changing the energy distribution ratio of the workpiece [167]. Changes in the properties of different grinding wheels can directly influence their cutting performance and heat generation. A grinding wheel with effective thermal conductivity facilitates the transfer of additional heat to the wheel, reducing the energy distribution of the workpiece [84,164]. With the development of technology, the use of new grinding wheels such as heat pipe grinding wheels has increased [39,169,170]. The high thermal conductivity of new grinding wheels can effectively change the proportion of heat transferred to the workpiece [171]. Furthermore, the distribution and geometry of grinding wheel grits affect the energy distribution. The coarseness and fineness of the abrasive grains, the size of the cone angle, and the unevenness of the distribution all affect the overall energy distribution [107].

5.2.2 Parameters related to CHTC

The main parameters affecting the variation of CHTC during machining include the following: fluid properties (density, specific heat, viscosity, and thermal conductivity), flow rate, speed of the grinding wheel, depth of grinding, parameters of the grinding wheel (taper angle, size of the abrasive grit, and porosity), and temperature difference.

Fluid properties mainly affect convective heat transfer [22]. Dense fluids generally have high heat capacity and efficient heat transfer characteristics. The greater the specific heat, the greater the ability of the fluid to absorb heat, absorbing the heat between the grinding wheel and the workpiece more effectively and improving the heat transfer efficiency. Fluids with high thermal conductivity can efficiently conduct heat. Therefore, when the three aforementioned factors increase, the CHTC will also increase. The greater the viscosity of the fluid, the greater the resistance to fluid flow, making turbulence less likely to occur and reducing heat transfer efficiency of the fluid. The flow rate of the fluid also affects its flow state during the process. A large flow rate of fluid has a greater capability to carry away heat, with higher shear and turbulence intensity on the contact surface, which disrupts the thermal boundary layer and improves heat transfer efficiency [151,155]. Cooling methods, such as the more environmentally friendly MQL, have emerged, but high-flow cast cooling usually provides better cooling results in comparison. Although the increase in the speed of the grinding wheel leads to the generation of additional heat, the high-speed rotation of the grinding wheel helps to increase the disturbance of the fluid, creating stronger turbulence, which, in turn, enhances the convective heat transfer capability of the fluid. The influence of the parameters of the grinding wheel grit on the heat transfer efficiency lies in the size of the heat transfer area. A large cone angle results in a large abrasive grain size, a small abrasive grain spacing, a small porosity, and a reduced heat exchange area with the fluid, thus minimizing the CHTC [107]. Temperature difference is a key factor affecting convective heat transfer efficiency. A larger temperature difference indicates a stronger heat transfer driving force. The larger the temperature difference, the more pronounced the convective effect of the fluid and the higher the heat exchange efficiency. Therefore, low-temperature cold-air cooling technology, which increases the temperature difference by reducing the air temperature, has a good cooling effect.

6 Conclusions and prospects

6.1 Conclusions

This paper provides a comprehensive review of EDC and CHTC models for cutting and grinding machining, including their specific applications and scope. In addition, potential directions for grinding thermal research are introduced to address current issues related to grinding thermal modeling.

(1) Aiming to solve the EDC model of the grinding process, the uniform heat-source model or the triangular heat-source model can be used. The triangular heat source closely algins with the actual working conditions, and the EDC model proposed by Yin, based on the contact between the abrasive particles and the workpiece, has a high accuracy, revealing approximately 6.94%–14.87%.

(2) The study of CHTC requires consideration of process parameters and fluid characteristics, including flow velocity, flow rate, boiling conditions, flow state, fluid type, physical parameters, and injection mode. Wang proposed a CHTC model based on the convective heat transfer differential equation system and fixed conditions, which demonstrated a high accuracy of approximately 2%–2.7%.

(3) Normally, the higher the speed of the grinding wheel, the higher the feed rate, and the harder the workpiece material, the higher the EDC of the workpiece. Conversely, a better cooling effect and a higher thermal conductivity of the grinding wheel result in a smaller workpiece EDC.

(4) The greater the grinding wheel speed, fluid specific heat, and fluid flow rate, the greater the CHTC. A high depth of cut, feed rate, fluid viscosity, and grain size lead to a small CHTC. Simultaneously, the nozzle position, flow rate, and fluid type also have substantial effects on the coefficient of variation.

6.2 Prospects

(1) Current modeling studies on EDC and CHTC are mainly conducted for grinding processes, with a lack of widely recognized, practical, and reliable models in the cutting process. Therefore, further research and exploration of thermal modeling in the cutting process is urgently needed.

(2) The energy distribution ratio plays a crucial role in studying workpiece temperature during machining. However, this coefficient is a complex variable that cannot be directly measured and is affected by many factors such as machining parameters, grain shape, grinding wheel parameters, coolant, and workpiece properties. While various models have been developed using different methods, a convincing, unified theory is lacking.

(3) The determination of CHTC is crucial for studying the cooling effect in the machining zone. However, due to uncertainties in the physical properties of the coolant, workpiece, grinding wheel, and other physical parameters, as well as the difficulty in fully controlling and simulating the real working conditions in the experiments, the change in boundary conditions will affect the measurement results. Additionally, with the introduction of nanofluids and advanced materials, existing models may not effectively describe their heat transfer characteristics. Therefore, further in-depth research and exploration on this area is still needed.

References

[1] Guo C S, Chen Y. Thermal modeling and optimization of interrupted grinding. CIRP Annals, 2018, 67: 321–324

[2] Gong P, Zhang Y B, Wang C J, Cui X, Li R Z, Sharma S, Liu M Z, Gao T, Zhou Z M, Wang X M, Dambatta Y S, Li C H. Residual stress generation in grinding: mechanism and modeling. Journal of Materials Processing Technology, 2024, 324: 118262

[3] Xian C, Shi Y Y, Lin X J, Liu D. Experimental study on energy partition of polishing aero-engine blades with abrasive cloth wheel. The International Journal of Advanced Manufacturing Technology, 2020, 106: 1839–1853

[4] Yao C F, Wang T, Xiao W, Huang X C, Ren J X. Experimental study on grinding force and grinding temperature of Aermet 100 steel in surface grinding. Journal of Materials Processing Technology, 2014, 214(11): 2191–2199

[5] SunJ G, Yang M, YangY Y, ZhangY B, Dambatta Y S, ZhouZ M, LiC H. Ultrasonic micro-grinding biological bone non-uniform heat source temperature field model and experimental verification. Journal of Mechanical Engineering, 2024, in press (in Chinese)

[6] Brinksmeier E, Aurich J C, Govekar E, Heinzel C, Hoffmeister H W, Klocke F, Peters J, Rentsch R, Stephenson D J, Uhlmann E, Weinert K, Wittmann M. Advances in modeling and simulation of grinding processes. CIRP Annals, 2006, 55(2): 667–696

[7] Ma X F, Yao B, Cai Z Q, Li Z M Q, Li Z S, Chen G F, Liu W S. Temperature field responses to face gear grinding conditions based on a comprehensive force–thermal model. Precision Engineering, 2023, 83: 22–41

[8] Dai C W, Ding W F, Zhu Y J, Xu J H, Yu H W. Grinding temperature and power consumption in high speed grinding of Inconel 718 nickel-based superalloy with a vitrified CBN wheel. Precision Engineering, 2018, 52: 192–200

[9] Dai S J, Wang X J, Zhang H B, Wen B R. Research on variation of grinding temperature of wind turbine blade robotic grinding. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2021, 235(3): 367–377

[10] Langenhorst L, Sölter J, Heinzel C. Partitioning of primary shear zone heat in face milling. CIRP Annals, 2022, 71(1): 53–56

[11] Brosse A, Naisson P, Hamdi H, Bergheau J M. Temperature measurement and heat flux characterization in grinding using thermography. Journal of Materials Processing Technology, 2008, 201(1–3): 590–595

[12] Zhang Z Y, Shang W, Ding H H, Guo J, Wang H Y, Liu Q Y, Wang W J. Thermal model and temperature field in rail grinding process based on a moving heat source. Applied Thermal Engineering, 2016, 106: 855–864

[13] Mohamed A O, Bauer R, Warkentin A. Uncut chip thickness and coolant delivery effects on the performance of circumferentially grooved grinding wheels. The International Journal of Advanced Manufacturing Technology, 2016, 85: 1429–1438

[14] Kizaki T, Zheng Q R, Shu L M, Tanaka J, Katsuma T. On reduction of energy flow into workpiece in continuous generating grinding. CIRP Annals, 2022, 71(1): 277–280

[15] Jamshidi H, Budak E. Grinding temperature modeling based on a time dependent heat source. Procedia CIRP, 2018, 77: 299–302

[16] Fang C F, Lin Y F, Xu X P. A thermal model for the calculation of energy partitions in sawing with a segmented blade. High Temperatures – High Pres-sures, 2016, 45(1): 35–56

[17] Gao T, Xu P M, Wang W, Zhang Y B, Xu W H, Wang Y Q, An Q L, Li C H. Force model of ultrasonic empowered minimum quantity lubrication grinding CFRP. International Journal of Mechanical Sciences, 2024, 280: 109522

[18] Shen J Y, Zeng W M, Huang H, Xu X P. Thermal aspects in the face grinding of ceramics. Journal of Materials Processing Technology, 2002, 129(1–3): 212–216

[19] Zhang J, Lv M, Wang S Y. The influence of side contact area on the grinding heat partition in narrow narrow deep groove. Machinery Design & Manufacture, 2019, 166–169

[20] Yang M, Hao J C, Wu W T, Li Z H, Ma Y Q, Zhou Z M, Gao T, Liu M Z, Cui X, Zhang Y B, Li B K, Ma X, Dambatta Y S, Li C H. Critical cutting thickness model considering subsurface damage of zirconia grinding and friction–wear performance evaluation applied in simulated oral environment. Tribology International, 2024, 198: 109881

[21] Mao M Z, Liu S B, Jiang J L, Sun S F, Wang D X. Study on flow field and convective heat transfer characteristics in grinding zone of large spiral angle flow disturbance grooved wheel. The International Journal of Advanced Manufacturing Technology, 2023, 129(1-2): 39–63

[22] Gupta P K, Yadav N P. Numerical investigation into convective heat transfer coefficient of the grinding fluid used in a deep grinding process. Transactions of Famena, 2024, 48(2): 129–144

[23] Song Y X, Li C H, Zhou Z M, Liu B, Sharma S, Dambatta Y S, Zhang Y B, Yang M, Gao T, Liu M Z, Cui X, Wang X M, Xu W H, Li R Z, Wang D Z. Nanobiolubricant grinding: a comprehensive review. Advances in Manufacturing, 2025, 13(1): 1–42

[24] He Q S, Fu Y C, Chen J J, Zhang W. Investigation on heat transfer performance of heat pipe grinding wheel in dry grinding. Journal of Manufacturing Science & Engineering, 2016, 138(11): 111009

[25] He Q S, Fu Y C, Chen J J, Zhang W, Cui Z M. Experimental investigation of cooling characteristics in wet grinding using heat pipe grinding wheel. The International Journal of Advanced Manufacturing Technology, 2018, 97(1–4): 621–627

[26] GuG Q, Wang D Z, WuS J, ZhouS, ZhangB X. Research status and prospect of ultrasonic vibration and minimum quantity lubrication processing of nickel-based alloys. Intelligent and Sustainable Manufacturing, 2024, 1(1): 10006

[27] LiC H. Thermodynamic Mechanism of MQL Grinding with Nano Bio-lubricant. Singapore: Springer Nature Singapore Pte. Ltd., 2024, 117–140

[28] Mao C, Zou H F, Huang Y, Li Y F, Zhou Z X. Analysis of heat transfer coefficient on workpiece surface during minimum quantity lubricant grinding. The International Journal of Advanced Manufacturing Technology, 2013, 66(1): 363–370

[29] SongY X, Xu Z L, LiC H, ZhouZ M, LiuB, ZhangY B, Dambatta Y S, WangD Z. Research progress on the grinding performance of nanobiolubricant minimum quantity lubrication. Surface Technology, 2023, 52(12): 1–19 (in Chinese)

[30] Sinha M K, Madarkar R, Ghosh S, Rao P V. Application of eco-friendly nanofluids during grinding of Inconel 718 through small quantity lubrication. Journal of Cleaner Production, 2017, 141: 1359–1375

[31] Cui X, Li C H, Zhang Y B, Ding W F, An Q L, Liu B, Li H N, Said Z, Sharma S, Li R Z, Debnath S. Comparative assessment of force, temperature, and wheel wear in sustainable grinding aerospace alloy using biolubricant. Frontiers of Mechanical Engineering, 2023, 18(1): 3

[32] Sharma P, Said Z, Kumar A, Nižetić S, Pandey A, Hoang A T, Huang Z H, Afzal A, Li C H, Le A T, Nguyen X P, Tran V D. Recent advances in machine learning research for nanofluid-based heat transfer in renewable energy system. Energy & Fuels, 2022, 36(13): 6626–6658

[33] Liu J Y, Srebric J, Yu N Y. Numerical simulation of convective heat transfer coefficients at the external surfaces of building arrays immersed in a turbulent boundary layer. International Journal of Heat and Mass Transfer, 2013, 61: 209–225

[34] Zhu D H, Li B Z, Ding H. A combined model with optimal solution for prediction of grinding temperature. International Journal of Materials & Product Technology, 2014, 49(4): 203–223

[35] García E, Méresse D, Pombo I, Harmand S, Sánchez J A. Identification of heat partition in grinding related to process parameters, using the inverse heat flux conduction model. Applied Thermal Engineering, 2014, 66(1–2): 122–130

[36] Zhao B, Huang Q, Cao Y, Ding W F, Xu J H, Liu G L. Thermal analysis of ultrasonic vibration-assisted grinding with moment-triangle heat sources. International Journal of Heat and Mass Transfer, 2023, 216: 124552

[37] Rowe W B, Jin T. Temperatures in high efficiency deep grinding (HEDG). CIRP Annals, 2001, 50(1): 205–208

[38] Wang Y L, Zhang Y B, Cui X, Liang X L, Li R Z, Wang R X, Sharma S, Liu M Z, Gao T, Zhou Z M, Wang X M, Dambatta Y S, Li C H. High-speed grinding: from mechanism to machine tool. Advances in Manufacturing, 2025, 13(1): 105–154

[39] Zhang L Y, Chen J J, Jiang H F, Xu Y L, Qian N, Fu Y C, Chen Y, Dai C W. Analysis for green grinding of Ti-6Al-4V titanium alloys with profile rotating heat pipe-grinding wheel. The International Journal of Advanced Manufacturing Technology, 2024, 131: 2537–2549

[40] Jiang J L, Ge P Q, Sun S F, Wang D X, Wang Y L, Yang Y. From the microscopic interaction mechanism to the grinding temperature field: an integrated modelling on the grinding process. International Journal of Machine Tools & Manufacture, 2016, 110: 27–42

[41] WangY, Xie J H, XiongW, YangL, ZhangS. Research on simulation of heat source model based on finite element method in surface grinding. Journal of System Simulation, 2016, 28(11): 9 (in Chinses)

[42] ZhangL, Ge P Q, MengJ F, ChengJ H, WangM. Newheat flux model in surface grinding. Materials Science Forum, 2004, 471–472: 298–301

[43] Lefebvre A, Vieville P, Lipinski P, Lescalier C. Numerical analysis of grinding temperature measurement by the foil/workpiece thermocouple method. International Journal of Machine Tools & Manufacture, 2005, 46(14): 1716–1726

[44] Zhang Y B, Gong P, Tang L Z, Cui X, Jia D Z, Gao T, Dambatta Y S, Li C H. Topography modeling of surface grinding based on random abrasives and performance evaluation. Chinese Journal of Mechanical Engineering, 2024, 37(5): 93

[45] Wang D X, Ge P Q, Bi W B, Jiang J L. Grain trajectory and grain workpiece contact analyses for modeling of grinding force and energy partition. The International Journal of Advanced Manufacturing Technology, 2014, 70: 2111–2123

[46] Ren X K, Huang X K, Feng H J, Chai Z, He Y B, Chen H B, Chen X Q. A novel energy partition model for belt grinding of Inconel 718. Journal of Manufacturing Processes, 2021, 64: 1296–1306

[47] Li Z, Ding W F, Liu C J, Su H H. Prediction of grinding temperature of PTMCs based on the varied coefficients of friction in conventional-speed and high-speed surface grinding. The International Journal of Advanced Manufacturing Technology, 2016, 90: 2335–2344

[48] Duan Z J, Wang S S, Wang Z H, Li C H, Li Y H, Song J L, Liu J Y, Liu X. Tool wear mechanisms in cold plasma and nano-lubricant multi-energy field coupled micro-milling of Al-Li alloy. Tribology International, 2024, 192: 109337

[49] Chen J Y, Xu X P. Temperature and energy partition in high-speed grinding of alumina with a brazed diamond wheel. Machining Science and Technology, 2010, 14(4): 440–454

[50] Lefebvre A, Sinot O, Torrance A A, Lipinski P. Determination of the partition coefficient for the grinding of a hard WC-Co-Cr coating with a diamond wheel. Machining Science and Technology, 2014, 18(4): 585–602

[51] Jin T, Stephenson D J, Xie G Z, Sheng X M. Investigation on cooling efficiency of grinding fluids in deep grinding. Cirp Annals, 2011, 60(1): 343–346

[52] Zhang Y B, Li L Y, Cui X, An Q L, Xu P M, Wang W, Jia D Z, Liu M Z, Dambatta Y S, Li C H. Lubricant activity enhanced technologies for sustainable machining: mechanisms and processability. Chinese Journal of Aeronautics, 2025, 38(6): 103203

[53] Duan Z J, Wang S S, Li C H, Wang Z H, Bian P, Sun J, Song J L, Liu X. Cold plasma and different nano-lubricants multi-energy field coupling-assisted micro-milling of Al-Li alloy 2195–T8 and flow rate optimization. Journal of Manufacturing Processes, 2024, 127: 218–237

[54] Zhang J Z, Tan X M, Liu B, Zhu X D. Investigation for convective heat transfer on grinding work-piece surface subjected to an impinging jet. Applied Thermal Engineering, 2013, 51(1–2): 653–661

[55] Duan Z J, Wang S S, Li C H, Wang Z H, Bian P, Song J L, Liu X. Performance evaluation of cold plasma and h-BN nano-lubricant multi-field coupling assisted micro-milling of aluminum alloy 6061–T651. Alexandria Engineering Journal, 2024, 109: 610–625

[56] Li B K, Li C H, Zhang Y B, Wang Y G, Yang M, Jia D Z, Zhang N Q, Wu Q D. Effect of the physical properties of different vegetable oil-based nanofluids on MQLC grinding temperature of Ni-based alloy. The International Journal of Advanced Manufacturing Technology, 2017, 89(9–12): 3459–3474

[57] Said Z, Sundar L S, Rezk H, Nassef A M, Chakraborty S, Li C H. Thermophysical properties using ND/water nanofluids: an experimental study, ANFIS-based model and optimization. Journal of Molecular Liquids, 2021, 330: 115659

[58] Tahvilian A M, Liu Z H, Champliaud H, Hazel B. Experimental and finite element analysis of temperature and energy partition to the workpiece while grinding with a flexible robot. Journal of Materials Processing Technology, 2013, 213(12): 2292–2303

[59] Yi J, Jin T, Deng Z H. The temperature field study on the three-dimensional surface moving heat source model in involute gear form grinding. The International Journal of Advanced Manufacturing Technology, 2019, 103(5–8): 3097–3108

[60] YangY. Li J L, ZhouL. Research on cutting thermal modeling and simulation of turning AISI 1045. Modular Machine Tool & Automatic Manufacturing Technique, 2018, 5: 34–36 (in Chinese)

[61] DingZ H, Huang L, WangX C, YuanJ T. Simulation study on cutting heat distribution of cemented carbide coated tools for titanium alloy cutting. Tool Engineering, 2023, 57: 77–82 (in Chinese)

[62] LiuW Y, Zhang Y, LiJ, LiuX K. Simulation of heat distribution with FEA in high-speed cutting. Tool Engineering, 2009, 43(8): 14–17 (in Chinese)

[63] Huang S W, Tao B, Li J D, Fan Y J, Yin Z P. Estimation of the time and space-dependent heat flux distribution at the tool-chip interface during turning using an inverse method and thin film thermocouples measurement. The International Journal of Advanced Manufacturing Technology, 2018, 99(5–8): 1531–1543

[64] QuanY M, He Z W, DouY. Cutting heat dissipation in high-speed machining of carbon steel based on the calorimetric method. Frontiers of Mechanical Engi-neering in China. 2008, 3(2): 175–179

[65] Ma L J, Chen J Q, Deng H, Zhou Y G, Cai C Y, Sun Z C. Theoretical model of heat conduction of brittle material in the process of turning fluorophlogopite ceramics. Science Technology and Engineering, 2020, 20(11): 4345–4350

[66] Bartoszuk M. Temperature and heat partition testing in the cutting zone for turning AISI 321 steel. Strojniški vestnik - Journal of Mechanical Engineering, 2020, 66(11): 629–641

[67] Quan Y M, He Z W. Cutting heat distribution in turning of carbon steel. China Mechanical Engineering, 2006, 17: 2155–2158

[68] Zhao J F, Zheng L R, Li W Q, Liu Z Q, Li L L, Wang B, Cai Y K, Ren X P, Liang X L. Effects of PVD CrAlN/(CrAlB)N/CrAlN coating on pin–disc friction properties of Ti2AlNb alloys compared to WC/Co carbide at evaluated temperatures. Metals, 2024, 14(6): 662

[69] XiaoM H, He N, LiL, LuA H. Study on cutting heat in high speed milling of nickel-based superalloy. Journal of Harbin Institute of Technology, 2011, 43: 105–109 (in Chinese)

[70] Komanduri R, Hou Z B. Thermal modeling of the metal cutting process. International Journal of Mechanical Sciences, 2000, 42(9): 1715–1752

[71] Sölter J, Gulpak M. Heat partitioning in dry milling of steel. CIRP Annals, 2012, 61(1): 87–90

[72] Hu Y H, Ding H, Shi Y T, Zhang H Y, Zheng Q C. A predictive model for cortical bone temperature distribution during drilling. Physical and Engineering Sciences in Medicine, 2021, 44(1): 147–156

[73] Patne H S, Kumar A, Karagadde S, Joshi S S. Modeling of temperature distribution in drilling of titanium. International Journal of Mechanical Sciences, 2017, 133: 598–610

[74] Liu Y, Pan Z T, Zhou H G, Jing X W, Li G C. Review and prospect of drilling heat for fiber reinforced composite. Acta Materiae Compositae Sinica, 2023, 40(8): 4416–4439

[75] Feng K K, Zhang F P, Wang W H. The micro-drilling process fine modeling with the strain characteristics of undeformed chip considered. Journal of Mechanical Science and Technology, 2022, 36(2): 883–895

[76] Cuesta M, Aristimuño P, Garay A, Arrazola P J. Heat transferred to the workpiece based on temperature measurements by IR technique in dry and lubricated drilling of Inconel 718. Applied Thermal Engineering, 2016, 104: 309–318

[77] Thanedar A, Dongre G G, Joshi S S. Analytical modelling of temperature in cylindrical grinding to predict grinding burns. International Journal of Precision Engineering and Manufacturing, 2019, 20(1): 13–25

[78] Zhang B, Lu S X, Rabiey M, Axinte D, Bleicher F. Grinding of composite materials. Cirp Annals, 2023, 72(2): 645–671

[79] Wang L, Ge P Q, Qin Y, Liu Z C, Sun J G, Gao W. Analysis of wet-grinding temperature field based on finite element method. Journal of Mechanical Engineering, 2002, 38(9): 155–158

[80] Su J X, Zhang Y Z, Deng X Z. Analysis and experimental study of cycloid gear form grinding temperature field. The International Journal of Advanced Manufacturing Technology, 2020, 110: 949–965

[81] Aurich J C, Herzenstiel P, Sudermann H, Magg T. High-performance dry grinding using a grinding wheel with a defined grain pattern. Cirp Annals, 2008, 57(1): 357–362

[82] Biermann D, Holtermann R, Menzel A, Schumann S. Modelling and simulation of thermal effects in internal traverse grinding of hardened bearing steel. Cirp Annals, 2016, 65(1): 321–324

[83] Fritsche A, Bleicher F. Experimental investigation of the heat flux distribution in grinding of titanium alloy. Procedia Engineering, 2015, 100: 987–993

[84] Rowe W B. Temperature case studies in grinding including an inclined heat source model. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2001, 215(4): 473–491

[85] HouZ B, Komanduri R. On the mechanics of the grinding process, Part II - thermal analysis of fine grinding. International Journal of Machine Tools & Manufacture, 2004, 4444: 247–270

[86] Guo C, Wu Y, Varghese V, Malkin S. Temperatures and energy partition for grinding with vitrified CBN wheels. CIRP Annals, 1999, 48(1): 247–250

[87] Guo G Q, An Q L, Lin L F, Yang C Q, Chen M. Analytical and experimental investigation of tempera-ture in form grinding. Journal of Mechanical Engi-neering, 2018, 54(3): 203–215

[88] Guo C S, Malkin S. Energy partition and cooling during grinding. Journal of Manufacturing Processes, 2000, 2(3): 151–157

[89] Qiu K X, Guo G Q, Yang C Q, Dong D P, Chen M. Investigation on temperature distribution in form grinding of 9Mn2V thread gauge. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2018, 232: 1342–1350

[90] Xu K Z, Wei C J, Hu D J, Xu L M, Xu M M. Temperature investigation of coated workpieces in intermittent grinding with a cup wheel. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2012, 226(2): 239–246

[91] LiB M, Zhao B. Modern Grinding Technology. Beijing: China Machine Press, 2003, 44–64

[92] Su J X, Ke Q X, Deng X Z, Ren X Z. Numerical simulation and experimental analysis of temperature field of gear form grinding. The International Journal of Advanced Manufacturing Technology, 2018, 97: 2351–2367

[93] Kizaki T, Takahashi K, Katsuma T, Shu L M, Sugita N. Prospects of dry continuous generating grinding based on specific energy requirement. Journal of Manufacturing Processes, 2021, 61: 190–207

[94] Zhang P F, Zhang W L, Yuan Y J, Fan X Q, Zhu M H. Probing the effect of grinding-heat on material removal mechanism of rail grinding. Tribology International, 2020, 147: 105942

[95] HadadM J, Tawakoli T, SadeghiM H, SadeghiB. Temperature and energy partition in minimum quantity lubrication-MQL grinding process. International Journal of Machine Tools & Manufacture, 2012, 54–55: 10–17

[96] Peng K L, Lu P, Lin F H, Jin T, Bao W C, Xie G Z, Shang Z T. Convective cooling and heat partitioning to grinding chips in high speed grinding of a nickel based superalloy. Journal of Mechanical Science and Technology, 2021, 35: 2755–2767

[97] Rowe W B. Thermal analysis of high efficiency deep grinding. International Journal of Machine Tools & Manufacture, 2001, 41(1): 1–19

[98] Ding Z S, Jiang X H, Guo M X, Liang S Y. Investigation of the grinding temperature and energy partition during cylindrical grinding. The International Journal of Advanced Manufacturing Technology, 2018, 97(5–8): 1767–1778

[99] ZhangJ C, Wu W T, LiC H, YangM, ZhangY B, JiaD Z, Hou Y L, LiR Z, CaoH J, AliH M. Convective heat transfer coefficient model under nanofluid minimum quantity lubrication coupled with cryogenic air grinding Ti–6Al–4V. International Journal of Precision Engineering and Manufacturing-Green Technology, 202202, 8(4): 1113–1135

[100] MaoC, ZhouZ X, ZhouD W, Gu D Y. The properties and the influence factors of the white layer in the surface grinding. Advanced Materials Research, 2008, 53–54: 285–292

[101] Jin T, Stephenson D J, Corbett J. Burn threshold of high-carbon steel in high-efficiency deep grinding. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2002, 216(3): 357–364

[102] Yang M, Kong M, Li C H, Long Y Z, Zhang Y B, Sharma S, Li R Z, Gao T, Liu M Z, Cui X, Wang X M, Ma X, Yang Y Y. Temperature field model in surface grinding: a comparative assessment. International Journal of Extreme Manufacturing, 2023, 5(4): 042011

[103] Ramanath S, Shaw M C. Abrasive grain temperature at the beginning of a cut in fine grinding. Journal of Manufacturing Science and Engineering, 1988, 110(1): 15–18

[104] Kohli S, Guo C, Malkin S. Energy partition to the workpiece for grinding with aluminum oxide and CBN abrasive wheels. Journal of Engineering for Industry, 1995, 117(2): 160–168

[105] Jin T, Stephenson D J. Analysis of grinding chip temperature and energy partitioning in high-efficiency deep grinding. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2006, 220(5): 615–625

[106] Hadad M, Sadeghi B. Thermal analysis of minimum quantity lubrication-MQL grinding process. International Journal of Machine Tools & Manufacture, 2012, 63: 1–15

[107] Zhu D H, Li B Z, Ding H. An improved grinding temperature model considering grain geometry and distribution. The International Journal of Advanced Manufacturing Technology, 2013, 67: 1393–1406

[108] Jin T, Yi J, Li P. Temperature distributions in form grinding of involute gears. The International Journal of Advanced Manufacturing Technology, 2017, 88: 2609–2620

[109] Yang M, Li C H, Said Z, Zhang Y B, Li R Z, Debnath S, Ali H M, Gao T, Long Y Z. Semiempirical heat flux model of hard-brittle bone material in ductile microgrinding. Journal of Manufacturing Processes, 2021, 71: 501–514

[110] Rowe W B. Temperatures in grinding––a review. Journal of Manufacturing Science and Engineering, 2017, 139(12): 121001

[111] Zheng Y, Wang C Q, Zhang Y F, Meng F M. Study on temperature of cylindrical wet grinding considering lubrication effect of grinding fluid. The International Journal of Advanced Manufacturing Technology, 2022, 121(9–10): 6095–6109

[112] Lavine A S. On the linear stability of mixed and free convection between inclined parallel plates with fixed heat flux boundary conditions. International Journal of Heat and Mass Transfer, 1993, 36(5): 1373–1381

[113] Kim N K, Guo C, Malkon S. Heat flux distribution and energy partition in creep-feed grinding. CIRP Annals, 1997, 46(1): 227–232

[114] TakazawaK. Theory and measuring method of the temperature distribution in ground surface layer. Journal of the Japan Society of Precision Engineering, 1964, 30(358): 851–857

[115] Yin G X, Marinescu I D. A heat transfer model of grinding process based on energy partition analysis and grinding fluid cooling application. Journal of Manufacturing Science and Engineering, 2017, 139(12): 121015

[116] Li Y X, Eldredge J D, Lavine A S, Fisher T S, Drolen B L. A conjugate heat transfer model of oscillating heat pipe dynamics, performance, and dryout. International Journal of Heat and Mass Transfer, 2024, 227: 125530

[117] Jin T, Cai G Q, Jeong H D, Kim N K. Study on heat transfer in super-high-speed grinding: energy partition to the workpiece in HEDG. Journal of Materials Processing Technology, 2001, 111(1–3): 261–264

[118] WeiW, RenJ L. Finite element analysis of cutting force and cutting temperature for stainless steel. Tool Engineering, 2009, 43(2): 67–70 (in Chinese)

[119] Thomas J A, DeVincentis B, Janz E, Turner B. A general approach for predicting convective heat transfer coefficients in turbulent systems. International Journal of Heat and Mass Transfer, 2024, 220: 124989

[120] Gong P, Zhang Y B, Cui X, Xu S Q, Yang M, Jia D Z, Li C H. Lubricant transportation mechanism and wear resistance of different arrangement textured turning tools. Tribology International, 2024, 196: 109704

[121] Pervaiz S, Deiab I, Wahba E, Rashid A, Nicolescu M. A numerical and experimental study to investigate convective heat transfer and associated cutting temperature distribution in single point turning. The International Journal of Advanced Manufacturing Technology, 2018, 94: 897–910

[122] Zhang D K, Li C H, Zhang Y B, Jia D Z, Zhang X W. Experimental research on the energy ratio coefficient and specific grinding energy in nanoparticle jet MQL grinding. The International Journal of Advanced Manufacturing Technology, 2015, 78: 1275–1288

[123] YinJ, ZhouX H, ZhangL, Yang J J, LiuW M, ZhaoH J. Deformation analysis of twin-cutter turning slender shaft under thermal-mechanical coupling. Ordnance Industry Automation, 2023, 42: 92–96 (in Chinese)

[124] Yi J, Zhou W, Deng Z H. Experimental study and numerical simulation of the intermittent feed high-speed grinding of TC4 titanium alloy. Metals, 2019, 9(7): 802

[125] Chen G, Gao Q, Yang X P, Liu J, Su Y X, Ren C Z. Investigation of heat partition and instantaneous temperature in milling of Ti-6Al-4V alloy. Journal of Manufacturing Processes, 2022, 80: 302–319

[126] Özerdem B. Measurement of convective heat transfer coefficient for a horizontal cylinder rotating in quiescent air. International Communications in Heat and Mass Transfer, 2000, 27(3): 389–395

[127] WangY B, Li Z J, TanG Y, LiuL. Study on boundary condition of temperature field for waved-edge milling insert. Tool Engineering, 2004, 11: 16–19 (in Chinese)

[128] Shariar F, Karagüzel U, Karpat Y. Heat partition evaluation during dry drilling of thick CFRP laminates with polycrystalline diamond drills. Manufacturing Letters, 2024, 41: 483–493

[129] Farahani S D, Tahmasebi V, Toghraie D. Estimation of heat flux entering the bone during the drilling process using the inverse heat transfer method. International Communications in Heat and Mass Transfer, 2024, 154: 107444

[130] Kurgin S, Dasch J M, Simon D L, Barber G C, Zou Q. Evaluation of the convective heat transfer coefficient for minimum quantity lubrication (MQL). Industrial Lubrication and Tribology, 2012, 64(6): 376–386

[131] Pontes D W, Crichigno Filho J M, Zdanski P S B. Prediction of the convective heat transfer coefficient of cutting fluid in helical drills. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2017, 39(12): 5195–5204

[132] Jin T, Stephenson D J. A study of the convection heat transfer coefficients of grinding fluids. CIRP Annals, 2008, 57(1): 367–370

[133] Kizaki T, Takahashi K, Katsuma T, Tanaka J, Shu L M, Sugita N. Effect of grinding fluid supply on workpiece temperature in continuous generating grinding. Journal of Manufacturing Processes, 2020, 60: 410–417

[134] Jin T, Stephenson D J. Three dimensional finite element simulation of transient heat transfer in high efficiency deep grinding. CIRP Annals, 2004, 53(1): 259–262

[135] WeberD, Kirsch B, da SilvaE J, AurichJ C. 3D-FEM-Wärmeübertragungssimulation des Schleifens. Zeitschrift für wirtschaftlichen Fabrikbetrieb, 2022, 117(7–8): 484–488 (in German)

[136] Xie G Z, Huang H. An experimental investigation of temperature in high speed deep grinding of partially stabilized zirconia. International Journal of Machine Tools & Manufacture, 2008, 48(14): 1562–1568

[137] Xiao X W, Li B, Peng R T, Hao X Q, Huang X F, Li Y Y, Zhao L F, Chen M L, Gao J X. Analysis of the material removal and heat transfer during internal cooling grinding of GH4169. Journal of Materials Processing Technology, 2023, 322: 118199

[138] Shao Y M, Li B Z, Chiang K N, Liang S Y. Physics-based analysis of minimum quantity lubrication grinding. The International Journal of Advanced Manufacturing Technology, 2015, 79(9–12): 1659–1670

[139] He Z Y, Jia D Z, Zhang Y B, Qu D, Lv Z L, Zeng E J. Investigation into the heat transfer behavior of electrostatic atomization minimum quantity lubrica-tion (EMQL) during grinding. Lubricants, 2024, 12(5): 158

[140] Huang W T, Liu W S, Wu D H. Investigations into lubrication in grinding processes using MWCNTs nanofluids with ultrasonic-assisted dispersion. Journal of Cleaner Production, 2016, 137: 1553–1559

[141] Yang M, Li C H, Zhang Y B, Jia D Z, Zhang X P, Hou Y L, Shen B, Li R Z. Microscale bone grinding temperature by dynamic heat flux in nanoparticle jet mist cooling with different particle sizes. Materials and Manufacturing Processes, 2018, 33(1): 58–68

[142] SunL Y, Zhang Y B, CuiX, AnQ L, ChenY, JiaD Z, Gong P, LiuM Z, DambattaY S, LiC H. Magnetic lubricants: preparation, physical mechanism, and application. Friction, 2024

[143] YangM, Li C H, ZhangY B, WangY G, LiB K, LiR Z. Theoretical analysis and experimental research on temperature field of microscale bone grinding under nanoparticle jet mist cooling. Journal of Mechanical Engineering, 2018, 54(18): 194–203 (in Chinese)

[144] LiuM Z, Li C H, ZhangY B, YangM, CuiX, LiB K, Gao T, WangD Z, AnQ L. Heat transfer mechanism and convective heat transfer coefficient model of cryogenic air minimum quantity lubrication grinding titanium alloy. Journal of Mechanical Engineering, 2023, 59(23): 343–357 (in Chinese)

[145] MaoC, ZouH F, HuangY, Zhou Z X. Research on heat transfer mechanism in grinding zone for MQL surface grinding. China Mechanical Engineering, 2014, 25(6): 826–831 (in Chinese)

[146] Yang M, Li C H, Zhang Y B, Wang Y G, Li B K, Jia D Z, Hou Y L, Li R Z. Research on microscale skull grinding temperature field under different cooling conditions. Applied Thermal Engineering, 2017, 126: 525–537

[147] Jin T, Stephenson D J, Rowe W B. Estimation of the convection heat transfer coefficient of coolant within the grinding zone. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2003, 217(3): 397–407

[148] Lin B, Morgan M N, Chen X W, Wang Y K. Study on the convection heat transfer coefficient of coolant and the maximum temperature in the grinding process. The International Journal of Advanced Manufacturing Technology, 2009, 42(11–12): 1175–1186

[149] Lin B, Chen X W, Morgan M N, Wu H. Estimation of the convection heat transfer coefficient of coolant and prediction of the maximum temperature within the grinding zone. International Journal of Abrasive Technology, 2009, 2(2): 130–153

[150] Jin T, Ma X. Hu H, Qu M N, Shang Z T. Inverse approach to derive the distribution of convection heat transfer coefficient of grinding fluid within grinding zone for deep grinding of nickel based super alloy. Journal of Mechanical Engineering, 2022, 58(15): 55–62

[151] GaoH, SongZ W, ZhenH W. Experimental determination of cooling CHTC for vertical spindle surface grinding. Diamond & Abrasives Engineering, 19911991: 16–20 (in Chinese)

[152] Lavine A S, Jen T C. Thermal aspects of grinding: heat transfer to workpiece, wheel, and fluid. Journal of Heat Transfer, 1991, 113(2): 296–303

[153] Vinay P V, Srinivasa Rao C. Temperature assessment in surface grinding of tool steels. Journal of Mechanical Science and Technology, 2015, 29(11): 4923–4932

[154] ShenB. Minimum quantity lubrication grinding using nanofluids. Dissertation for the Doctoral Degree. Ann Arbor: University of Michigan, 2008, 107–135

[155] Zhang L, Rowe W B. Fluid convection models for low-temperature grinding and effect of fluid warming. Journal of Manufacturing Science and Engineering, 2021, 143(2): 021010

[156] Zhang L, Rowe W B. Study of convective heat transfer in grinding applied to tool carbide. Journal of Manufacturing Science and Engineering, 2020, 142(2): 021001

[157] WangY C. Study on effective supply of grinding fluid and convective heat transfer in grinding zone. Thesis for the Master’s Degree. Jinan: Shandong University, 2016, 31–40 (in Chinese)

[158] Yang M, Ma H, Hao J C, Li Z H, Li R Z, Zhou Z M, Gao T, Liu M Z, Cui X, Wang X M, Zhang Y B, Dambatta Y S, Long Y Z, Li C H. Droplet size distribution model of needle electrode electrostatic atomization and milling nickel-based alloy performance evaluation. Journal of Manufacturing Processes, 2024, 119: 682–698

[159] Yang M, Ma H, Li Z H, Hao J C, Liu M Z, Cui X, Zhang Y B, Zhou Z M, Long Y Z, Li C H. Force model in electrostatic atomization minimum quantity lubrication milling GH4169 and performance evaluation. Frontiers of Mechanical Engineering, 2024, 19: 28

[160] Yang M, Li C H, Luo L, Li R Z, Long Y Z. Predictive model of convective heat transfer coefficient in bone micro-grinding using nanofluid aerosol cooling. International Communications in Heat and Mass Transfer, 2021, 125: 105317

[161] Wang X Z, Yu T B, Sun X, Shi Y, Wang W S. Study of 3D grinding temperature field based on finite difference method: considering machining parameters and energy partition. The International Journal of Advanced Manufacturing Technology, 2016, 84: 915–927

[162] Zhan Y J, Xu X P. An experimental investigation of temperatures and energy partition in grinding of cemented carbide with a brazed diamond wheel. The International Journal of Advanced Manufacturing Technology, 2011, 61: 117–125

[163] Li B K, Li C H, Zhang Y B, Wang Y G, Jia D Z, Yang M. Grinding temperature and energy ratio coefficient in MQL grinding of high-temperature nickel-base alloy by using different vegetable oils as base oil. Chinese Journal of Aeronautics, 2016, 29(4): 1084–1095

[164] Li Z, Ding W F, Ma C Y, Xu J H. Grinding temperature and wheel wear of porous metal-bonded cubic boron nitride superabrasive wheels in high-efficiency deep grinding. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2017, 231(11): 1961–1971

[165] Hou Z B, Yao Z Q. Investigation on grinding temperature and heat flux distribution with grooved grinding wheels. The International Journal of Advanced Manufacturing Technology, 2023, 124(10): 3471–3487

[166] Pang J Z, Wu C J, Shen Y M, Liu S Q, Wang Q X, Li B Z. Heat flux distribution and temperature prediction model for dry and wet cylindrical plunge grinding. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2019, 233(10): 2047–2060

[167] Ren X K, Huang X K, Chai Z, Li L F, Chen H B, He Y B, Chen X Q. A study of dynamic energy partition in belt grinding based on grinding effects and temperature dependent mechanical properties. Journal of Materials Processing Technology, 2021, 294: 117112

[168] Mohamed A M O, Warkentin A, Bauer R. Use of surface roughness measurements to improve the estimation of the heat partition in grinding. Journal of Materials Processing Technology, 2011, 211(4): 566–572

[169] ChenJ J, Liu S Y, YangY, YuanD D, ZhangL Y, FuY C, Qian N. Evaluation of heat transfer performance of nanofluid heat pipe grinding wheel forming and grinding titanium alloy. Journal of Mechanical Engineering, 2024, 6060: 407–419 (in Chinese)

[170] Chen J J, Fu Y C, Qian N, Ching C Y, Ewing D, He Q S. A study on thermal performance of revolving heat pipe grinding wheel. Applied Thermal Engineering, 2021, 182: 116065

[171] He Q S, Liu J W, Wang X, Fu Y C. Thermal performance analysis of phase change heat storage grinding wheel in high-efficiency dry grinding. The International Journal of Advanced Manufacturing Technology, 2024, 134(7–8): 3599–3609

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