Temperature field model in surface grinding: a comparative assessment

Grinding is a crucial process in machining workpieces because it plays a vital role in achieving the desired precision and surface quality. However, a significant technical challenge in grinding is the potential increase in temperature due to high specific energy, which can lead to surface thermal damage. Therefore, ensuring control over the surface integrity of workpieces during grinding becomes a critical concern. This necessitates the development of temperature field models that consider various parameters, such as workpiece materials, grinding wheels, grinding parameters, cooling methods, and media, to guide industrial production. This study thoroughly analyzes and summarizes grinding temperature field models. First, the theory of the grinding temperature field is investigated, classifying it into traditional models based on a continuous belt heat source and those based on a discrete heat source, depending on whether the heat source is uniform and continuous. Through this examination, a more accurate grinding temperature model that closely aligns with practical grinding conditions is derived. Subsequently, various grinding thermal models are summarized, including models for the heat source distribution, energy distribution proportional coefficient, and convective heat transfer coefficient. Through comprehensive research, the most widely recognized, utilized, and accurate model for each category is identified. The application of these grinding thermal models is reviewed, shedding light on the governing laws that dictate the influence of the heat source distribution, heat distribution, and convective heat transfer in the grinding arc zone on the grinding temperature field. Finally, considering the current issues in the field of grinding temperature, potential future research directions are proposed. The aim of this study is to provide theoretical guidance and technical support for predicting workpiece temperature and improving surface integrity.

Intermediate quantity Ac Average effective area of contact between single abrasive and workpiece Ag Truncated grain tips, rg is grain radius An Nominal contact area between grinding wheel and workpiece Ar Actual contact area between grinding wheel and workpiece As The ratio of the real contact area between the abrasive grain and the workpiece to the geometric contact area between grinding wheel and workpiece A ′ Spreading area of droplet b Distance from the position of triangular heat source distribution peak to the center of the grinding wheel bc Contact width between micro abrasive tool and workpiece bg Width Thermal conductivity, density and specific heat capacity of coolant fluid (kρc)g Thermal conductivity, density and specific heat capacity of abrasive grain (kρc)n Thermal conductivity, density and specific heat capacity of nanoparticles (kρc)s Thermal conductivity, density and specific heat capacity of grinding wheel (kρc)w Thermal conductivity, density and specific heat capacity of workpiece

Introduction
Grinding is a machining method that offers several advantages over other techniques, including high machining accuracy, versatile applications, and superior surface quality [1][2][3][4]. However, it requires a substantial amount of energy input, specifically in the form of grinding specific energy, to remove material per unit volume [5][6][7]. This energy is largely converted into heat, which accumulates in the grinding zone, resulting in a rapid temperature increase and a significant temperature gradient in the surface layer [8][9][10]. The thin metal layer being cut during grinding, compared to cutting processes, leads to higher specific grinding forces and energies, generating significant heat. Most of this heat is transferred to the workpiece, with only a small portion being removed by the grinding chips [11][12][13]. During grinding, heat tends to accumulate in the workpiece's surface layer, as it is not quickly dissipated into the depth of the material. This localized heat accumulation results in high surface temperatures, often exceeding 1000 • C, and a substantial temperature gradient in the surface layer (up to 600-1000 • C mm −1 ). The thermal effects of grinding greatly influence the performance and surface quality of the workpiece, particularly when the grinding temperature surpasses a critical value at the interface, leading to thermal damage on the workpiece surface [14][15][16]. This thermal damage can manifest as burns, surface oxidation, residual stress, and cracks, all of which can reduce wear resistance, increase stress corrosion susceptibility, and decrease fatigue resistance of the components [17][18][19][20][21]. Moreover, the cumulative temperature rise of the workpiece during the grinding cycle can result in size and shape errors [22][23][24]. Additionally, the heat generated in the grinding zone can affect the service life of the grinding wheel [25][26][27][28][29]. Therefore, controlling the grinding temperature is of utmost importance for enhancing the surface quality and performance of workpieces, extending the lifespan of components, and improving the service life of grinding wheels.
Grinding heat is primarily generated as a result of grinding power consumption [30,31]. The specific energy required for grinding is typically high, and only a small portion of this energy is utilized as surface energy for forming a new surface, strain energy in the ground surface layer, and kinetic energy for chip expulsion. The majority of the energy, however, is transformed into heat, which enters both the workpiece and the grinding wheel, leading to an increase in the temperature of the grinding area [32,33]. In engineering research, grinding temperature can be segregated into the overall average temperature of the workpiece, the temperature of the grinding wheel-workpiece contact surface, the temperature of the workpiece grinding surface, and the temperature of the abrasive grinding point [34,35]. The overall average temperature of the workpiece refers to the average temperature increase in the workpiece material caused by the transfer of grinding heat. The temperature of the grinding wheel-workpiece contact surface refers to the temperature at the interface where the grinding wheel and workpiece make contact during the grinding process. The temperature of the workpiece grinding surface represents the temperature of the processed surface of the workpiece. Last, the temperature of the abrasive grain's grinding point refers to the temperature of the small section of the abrasive blade that interacts with the workpiece, serving as both the source of grinding heat and the highest temperature point. When the literature discusses grinding temperature and grinding temperature field, it generally refers to the temperature at the grinding wheel-workpiece contact surface. This temperature value has the most significant impact on the surface quality of the ground workpiece [36].
Studying the mechanism of grinding heat generation through the analysis of grinding temperature is an important aspect of grinding technology [58][59][60][61][62]. Due to the inherent complexity of the grinding process compared to other machining methods, the grinding temperature is influenced by numerous parameters, including grinding parameters, physical properties of workpiece materials, cooling mode and associated parameters, and the structure and properties of the grinding wheel [63][64][65][66][67][68][69][70][71][72][73][74][75]. As a result, developing a predictive model for the grinding temperature field poses a significant and challenging task. This is especially true when grinding hard-tomachine materials such as nickel-based superalloys [35,76,77] and titanium alloys [78][79][80][81][82][83], for which a temperature field prediction model is of utmost importance.
Analytical methods establish mathematical models of the temperature field and obtain solutions in a functional form. The calculation process allows for clear logical reasoning and understanding of physical concepts, and the final solution effectively represents the impact of various factors in the grinding zone on the temperature distribution and heat conduction processes. However, analytical methods may become difficult or even impossible to solve when there are slight changes in the grinding zone conditions, often requiring simplifications of the original problem. Researchers often make assumptions when using analytical methods, such as simplifying the heat transfer state on the surface of the heat conductor, the shape of the parts, and the distribution state of the heat source on the workpiece surface. These simplifications can affect the accuracy of the solution.
On the other hand, numerical methods rely on discrete mathematics and utilize computers as computational tools. Although lacking a rigorous theoretical foundation, numerical methods are better suited for solving the actual grinding temperature field [110,111] and are more widely employed. Numerical solutions encompass FDMs and FEMs, with the FEM commonly used to deduce the grinding temperature field. The FEM approach involves establishing a geometric model, dividing it into units, and applying convection and heat flux loadings. FEM offers advantages such as fast computation, ease of operation, suitability for constant grinding forces, stable input of heat flux density in the grinding zone, and consideration of convective heat transfer with cooling media under stable conditions [112][113][114][115]. However, when the input parameters of the grinding temperature field become complex, and the grinding force and cooling media are involved in convective heat transfer within the grinding zone, the FEM may no longer be applicable. In such cases, the FDM [45,110,[116][117][118][119][120], another numerical method, serves as an effective approach for calculating the grinding temperature field. The FDM is based on actual grinding conditions, and theoretical models are developed for the boundary conditions (heat flux, heat distribution ratio, convective heat transfer coefficient, etc) of the temperature field, enabling accurate temperature field calculations.
Extensive research conducted by numerous scholars has made significant contributions to the advancement of the grinding processing industry, particularly in understanding the grinding temperature field using analytical and numerical methods. However, the current mathematical modeling and simulation techniques for the grinding temperature field are based on the common assumption of a uniform and continuous distribution of heat sources within the grinding wheelworkpiece contact area. In reality, grinding heat is generated through the interaction between abrasive particles and the workpiece, with the positioning of abrasive particles on the grinding wheel surface being completely random. Each abrasive particle in contact with the workpiece can be seen as a moving heat source, and the heat generated by each individual heat source varies. Recognizing this, this paper reviews prediction models of the distribution of the grinding temperature field, considering discrete heat sources based on the randomized positioning of grits on the grinding wheel surface. Furthermore, the review takes into account the micro-scale interference effects between the effective grinding grains and the workpiece. In this approach, the temperature field produced by a single diamond grit is determined at different stages and eventually combined to obtain the temperature field in the grinding zone through superposition of discrete temperature fields.
Existing review papers on grinding temperature tend to focus primarily on the uniform and continuous grinding temperature field, lacking a comprehensive analysis of the nonuniform and discontinuous grinding temperature field, as well as the corresponding grinding thermal models. Additionally, these reviews do not extensively cover the laws and application scope of the temperature field and grinding thermal theoretical models. Given that surface grinding is one of the most common grinding methods, this study aims to provide a comprehensive review of temperature field models and related thermal models specifically for surface grinding.
The research idea is presented in figure 2. The initial focus is on analyzing the methods used to solve the grinding temperature field. Depending on the uniformity and continuity of the heat source, the grinding temperature field is categorized into two groups: the uniform and continuous temperature field based on the continuous strip heat source, and the grinding temperature field based on the random discrete heat source, considering the position and orientation of abrasive particles on the wheel surface.
Regardless of the method used to solve the grinding temperature field, the key scientific challenge lies in establishing a grinding thermal model. This includes models for heat source distribution, energy distribution proportional coefficients, and heat transfer coefficients. This study provides a comprehensive summary of these models, their applications, and their scopes. It aims to address the current issues in temperature field modeling and grinding thermal models.
Furthermore, the study proposes future research directions for the grinding temperature field, taking into account the existing challenges. By summarizing the grinding temperature field models and grinding thermal models, this study establishes a multi-parameter mapping relationship between the grinding temperature field, surface integrity of the grinding workpiece, and various parameters such as grinding parameters, physical parameters of the workpiece, grinding wheel parameters, cooling modes, and their corresponding parameters. The aim is to provide theoretical guidance and technical support for improving workpiece surface integrity in surface grinding processes.

Uniform continuous temperature field
In the context of grinding, a uniform continuous temperature field assumes that the heat source in the grinding area is considered a moving heat source with a continuous distribution, and the temperature field is determined based on this assumption. Many researchers have developed theoretical models for grinding temperature using the concept of moving heat sources, originally proposed by Jaeger [121]. The main idea behind these models involves the superposition of heatsource temperature fields.
To explain further, the grinding interface is simplified as a surface heat source consisting of numerous line heat sources. Each line heat source can be further simplified as a combination of multiple microelement line heat sources [31,[122][123][124]. In turn, each microelement line heat source can be approximated as the combined effect of multiple point heat sources. Therefore, the superposition principle is employed to calculate the temperature field of the heat source.
Essentially, the temperature field resulting from the moving heat source theory is determined by solving the temperature field of an instantaneous point heat source in an infinitely large object at any given time after a certain amount of heat is instantaneously released [125][126][127][128][129]. By considering the grinding process as a continuous distribution of heat sources, this approach enables the calculation of the temperature distribution across the interface between the workpiece and the grinding wheel.

Temperature field based on an instantaneous point heat
source. As shown in figure 3, suppose that a certain heat source is located in an infinitely large object at the origin of the coordinates. At the initial time (t = 0 s), the temperature (T) is 0 • C, and heat is emitted instantaneously from the point heat source at the origin of the coordinates and immediately stops heating. According to the theory of heat transfer, the temperature rise of point M (x, y, z) at any position in the coordinate system is where c w is the specific heat of the conductive medium, Q is the instantaneous calorific value of the point heat source, α is the thermal diffusivity of the conductive medium, ρ w is the density of the heat conduction medium, t is the time after instantaneous heating of the heat source, and R is the distance between M and the origin of the coordinates. When the table moves, the workpiece moves at the same speed, and when it passes the grinding wheel, a band heat source is formed under the interaction between the workpiece and the grinding wheel. This band heat source, which moves at the same speed as the workpiece, passes over the surface of the workpiece, causing a temperature rise on the working surface, and this action is called the thermal action of the grinding process. Therefore, in the calculation of the surface temperature rise, the grinding contact interface is a surface heat source, and the combination of the surface heat source as an innumerable strip heat source, this part of the strip heat source along the x axis movement speed for the workpiece speed v w [130,131]. Based on this theory, Takasawa and Kawamura derived a calculation formula for the temperature field in the contact arc region of grinding according to the uniform and triangular heat source models, respectively [132][133][134]: where q w is the heat flux flowing into the workpiece, k w is the thermal conductivity of the workpiece material, K 0 (u) is the zero-order second-class modified Bessel function (K 0 (u) = and l c is the contact arc length l c = √ a p · d s , a p is the cutting depth, d s is the grinding wheel diameter).
The aforementioned analytical method is used to calculate the grinding temperature field. The superposition of the temperature field of this type of heat source successfully derived the theoretical solution for the grinding interface temperature field under ordinary continuous grinding.

Temperature field based on the FDM.
The object is divided into finite grid cells, the difference equations are obtained by transforming differential equations, and the temperature at the node of each grid microcell is solved numerically. As shown in figure 4, the object is divided into rectangular grids along the x-and y-directions according to the spacing between ∆x and ∆y in the 2D thermal conductivity problem. A node is defined as the intersection point of each grid line, and p (i, j) is used to represent the position of each node. i is the sequence number of nodes along the x direction, and j is the sequence number of nodes along the y direction. The basic principle of this method is to replace the micro commercial finite difference quotient and then transform the original differential equation into a difference equation [116].
In the heat conduction problem of grinding, Fourier's law is the most basic heat conduction equation, that is, the heat passing through the microelement isothermal surface dA in a finite time interval dt is dQ, which is proportional to the temperature gradient ∂T ∂n and is opposite to the direction of the temperature gradient: where n is the unit vector in the normal direction.  For heat flux density: where q x is the heat flux in the x x-direction, and ∂T ∂x is the temperature gradient in the x direction.
The Fourier and energy conservation laws can be used to obtain the thermal conductivity differential equation, assuming that there is no internal heat source in the body. Based on the above assumptions, the microelement body dV = dxdydz is divided from the object undergoing heat conduction, as shown in figure 5. The three edges of the microelement body are parallel to the x, y and z axes. According to the law of conservation of energy, the net heat of incoming and outgoing micro elements in time dt should be equal to the increase in energy in the micro element, namely, the net heat of incoming and outgoing elements (I) = increase in internal energy of elements (II) [135]. I can be obtained by adding the net heat introduced and exported in the x, y, and z directions: According to Fourier's law: Substituting equation (7) into equation (6), the following can be obtained: Increment of energy in the cell in dt time: After finishing, the following can be obtained: Equation (10) can be simplified as: The workpiece is assumed to be a rectangular plane and discretized into a planar grid structure. The space step of equal length ∆x = ∆z = ∆s is considered as two groups of parallel lines with equal intervals, and the rectangular sample is divided. The equation for parallel lines [116] is as follows: where x i and z j are the coordinate values of the ith horizontal line in the x-direction and the jth vertical line in the z direction of the difference grid, respectively; l w and b w are the length and height of the workpiece, respectively; and M and N are natural numbers. After subdivision, the grid area for differential calculation is obtained. A system of difference equations is established based on the second-order difference quotient: The difference equations of each node of the internal grid can be obtained: The grinding temperature can be determined by solving equation (14). Furthermore, equation (14) indicates that the temperature field calculated using the continuous heat source model is smooth and continuously differentiable.
Another significant numerical method for solving the grinding temperature field is the FEM. Many researchers have employed the FEM to analyze the grinding temperature field and have achieved valuable research outcomes. However, as the theoretical analysis models of temperature fields are typically embedded within FEM software, this paper does not discuss them in detail.

Nonuniform discontinuous grinding temperature field
Currently, most grinding temperature field theories are based on Jaeger's theory of a moving heat source, assuming that the heat source is uniformly and continuously distributed in the contact area between the grinding wheel and workpiece. However, in reality, grinding heat is generated through the interaction between abrasive grains and the workpiece, and the position and orientation of these abrasive grains on the grinding wheel's surface are completely random. Each abrasive grain participating in cutting can be considered a moving heat source, and the heat generated by each moving heat source varies. Therefore, the heat flux is not uniformly distributed along the width direction of the grinding wheel or continuously distributed along the feed direction of the workpiece. Instead, it represents a superposition of random vibration signals.
Recognizing this, Li and Axinte [136] proposed a stochastic grain-discretized temperature model (SGDTM) by considering the contact zone between the wheel and workpiece as a moving heat source. However, this assumption alone is not sufficient to fully capture the complexity of the grinding process. The model first establishes a surface topography model of the randomized grinding wheel, which takes into account the random distribution of abrasive grains. Then, based on the relationship between the maximum undeformed chip thickness of an individual abrasive grain and its critical depth in the stages of friction, ploughing, and cutting (as illustrated in figure 6), the model quantitatively analyzes the determination mechanism of effective abrasive grains within the grinding area.
Building upon the key geometric characteristics of effective abrasive particles and the distribution laws of their positions and orientations, an analysis was conducted on the microinteraction state between abrasive particles and the workpiece, as well as the heat distribution mechanism of abrasive grains at different stages. Assuming that the heat generated by a single abrasive grain per unit time is proportional to the volume of material removed per unit time, the heat allocated to each abrasive grain can be determined by identifying the grinding stage of the abrasive grain. This allows for the calculation of the proportion of heat distribution entering the workpiece, as depicted in figure 7.
The temperature domain induced by a moving infinitely small heat source is obtained by the discrete integration of Jaeger's moving heat source temperature domain equation. The discrete temperature field distribution can be obtained by solving equation (15) and is shown in figure 8 where k is the grain k location; m is the total number of moving heat sources, that is, the cutting grains in the computational domain; t 0 is the start time of the heat source movement; c w and ρ w are the specific heat capacity and density of the workpiece material, respectively; v s is the grinding wheel peripheral speed; and v sx , v sy , and v sz are the velocities of a heat source moving along the x, y, and z axes, respectively. Liu et al [137] further expanded the study by taking into account the random geometric shape distribution of grinding wheel grains. They proposed the law of heat superposition attenuation in the grinding area and developed a nonuniform discontinuous grinding temperature field model. In their research, as shown in figure 9, the surface of the grinding wheel predominantly consists of conical, spherical, conical, and pyramid-shaped abrasive grains.
While existing models of grinding wheel surface topography mainly consider conical or spherical grains, the observation results of actual cubic boron nitride (CBN) grinding wheels indicate that most of the grains are irregular quadrilateral pyramids. Therefore, in their model, the shape of the abrasive particles is assumed to be quadrilateral pyramids, which accurately represents the surface characteristics of CBN grinding wheels.
During the grinding process, the heat generated at any point on the workpiece surface by the point heat source is rapidly dissipated, resulting in the instantaneous unloading of the heat source. However, the temperature field generated by the heat source does not immediately drop to zero. Instead, it gradually decreases over time, following a specific attenuation coefficient. Additionally, the thermal residual effect of the heat source continues to influence the temperature changes in neighboring locations. This phenomenon is also applicable to the heat source area within the grinding arc region.   The grinding temperature field is characterized by dynamic alternation, where the thermal effects of the heat source persist even after the heat source has been removed. This means that the temperature distribution during grinding is not static but rather undergoes dynamic changes. As a result, a discontinuous nonuniform grinding temperature field is obtained, as illustrated in figure 10. This model captures the dynamic nature of the temperature distribution during grinding, taking into account the residual effects of the heat source and the gradual attenuation of the temperature field over time.
After analyzing various temperature field models, it can be concluded that the choice of heat source distribution model depends on whether a continuous or discrete heat source distribution is considered when solving the grinding temperature field. In the case of a continuous heat source distribution, the heat source distribution model needs to be determined. On the other hand, in the case of a discrete heat source distribution, this step is not necessary.
Regardless of the method used to solve the grinding temperature field, certain aspects of the grinding thermal model need to be determined. First, the proportion of heat transferred into the workpiece in the total heat needs to be established. This is represented by the grinding heat distribution coefficient (R w ) [138][139][140][141][142][143]. The grinding heat distribution coefficient quantifies the distribution of heat among the workpiece, grinding wheel, abrasive grain debris, and cooling medium due to heat conduction.
Second, since most grinding processes involve the use of a cooling medium, the influence of the cooling medium on the temperature needs to be considered. The convective heat transfer coefficient (h f ), which encompasses all factors related to convective heat transfer, serves as the parameter that directly characterizes the heat transfer capacity of the cooling medium [8,[144][145][146][147]. The heat source distribution model, grinding heat distribution coefficient model, and convective heat transfer coefficient model together form the grinding thermal model. However, it is important to note that both the grinding temperature field model and the grinding thermal model have limitations. When establishing these models, certain simplifications and assumptions are made regarding the influencing factors, which can introduce inherent errors. Additionally, the models are constrained by the knowledge and understanding of the model builder, which may limit their ability to capture all nuances of the grinding process accurately.

Experimental verification of temperature field models
To validate the grinding temperature field models, researchers have conducted different grinding experiments, as shown in table 1.

Grinding thermal model
Grinding thermal models, which consist of the heat source distribution model, grinding heat distribution coefficient model, and convective heat transfer coefficient model, play a crucial role in solving the grinding temperature and providing essential loading conditions for temperature calculations. The following is an overview of thermal grinding models.

Heat source distribution model
During the grinding process, heat is primarily generated in the grinding area where each grinding particle acts as a heat source. Considering the large number of grains involved in grinding within a small distance, it is reasonable to treat them as a surface heat source applied to the workpiece surface [152][153][154]. As the grinding progresses, the grinding contact area moves forward along the feed direction on the workpiece at a velocity v w . Consequently, the surface heat source also moves along the feed direction at the same velocity v w on the workpiece surface. However, as the workpiece temperature increases, various defects such as oxidation, burns, cracks, and residual stresses can occur on the surface, significantly reducing the corrosion resistance and fatigue strength of the components, which affects the service life and reliability [155][156][157]. To achieve a more accurate prediction of the grinding temperature field, researchers have further explored the concept of a moving heat source model for the grinding process.
In conventional grinding temperature studies, the influence of grinding depth on temperature is often neglected due to small cutting depths. Therefore, the machined surface and the un-machined surface of the workpiece are treated as the same surface, and the angle between the grinding arc area and the workpiece's feed velocity direction is assumed to be zero. In other words, the temperature problem in ordinary deep cutting grinding can be approximated as a planar strip heat source moving on the surface of a semi-infinite object. Building upon this assumption, Jaeger [121] proposed the theory of heat source movement to calculate the grinding temperature. This theory considers the grinding process as a rectangular strip heat source moving along the feed direction with a velocity v w , while assuming the workpiece to be a semi-infinite heat conductor. When the strip heat source reaches a specific point, the temperature accumulation of the heat source at any point in the workpiece can be indirectly obtained by integrating the temperature fluctuation generated by the temperature field at that point.

Shallow cutting grinding.
After the introduction of the rectangular strip heat source by Jaeger [121] in 1942 (as shown in figure 11(a)), further advancements were made in 1964 by Bei [158] and Takazaw [132]. In their investigation of the temperature field during the grinding process, they observed dynamic changes in the grinding thickness of the abrasive particles within the contact area. This varying thickness directly influenced the generation of heat, resulting in uneven heating power across the contact arc region. To address this issue, they proposed a right-triangle heat source model, as illustrated in figure 11(b). Experimental studies confirmed that the right-triangle heat source model closely matched the heat flux of the grinding area during actual processing [159].
In 1995, Zhang and Mahdi [160] introduced a common triangle heat source model, which deviates from the righttriangle shape. This model is determined by the value of l a (l a = ξ a /l c ′ , l c ′ = l c /2, where ξ a is the value of ξ at the peak value of the ordinary triangular heat source distribution and ξ is the coordinate of the moving heat source), as shown in figure 11(c). To investigate the temperature field, researchers utilized the rectangular heat source distribution model, the right triangle heat source distribution model, and various Ti-6Al-4V Dry Thermocouples (1) The grinding temperature field is not uniform and continuous, namely, the temperature in the grinding zone is not equivalently distributed along the length/width direction of the workpiece.
(2) The high temperature point on the surface of the workpiece is positively correlated with the cutting depth.
The minimum error of 64 sets of data is 4.9%, and the proportion of errors less than 10% reaches 86% Figure 11. Heat source distribution model for shallow cutting grinding: (a) rectangular heat source, (b) triangular heat source, (c) ordinary triangular heat source, (d) parabolic heat source, (e) comprehensive distribution model of heat source, (f) inclined rectangular heat source, (g) inclined triangular heat source, (h) inclined parabolic heat source, (i) inclined trapezoidal heat source, (j) inclined Gaussian heat source distribution, (k) inclined common triangle heat source distribution.
common triangle heat source distribution models. Through calculations, they analyzed and compared the effects of these different heat source distribution models on the resulting temperature field. In 2008, Mao et al [161] made an important discovery regarding the heat flow density into the workpiece during grinding. It was found that this heat flow density was influenced by the thickness of the undeformed swarf present in the grinding zone, which did not exhibit uniform or triangular characteristics. Consequently, the heat flow density into the workpiece was determined based on the swarf thickness ranging from zero to its maximum value. The heat flow into the workpiece varied gradually as the swarf thickness changed from zero to its maximum. Taking into account the actual contact pattern between the grinding wheel and the workpiece, it was observed that the undeformed abrasive chip had a rounded shape. Therefore, the circular arc surface formed by the wheelworkpiece contact was considered the heat source surface of the workpiece. As a result, a parabolic distribution of the heat flow intensity into the workpiece was proposed, as illustrated in figure 11(d).
During the grinding process, abrasive particles interact with the workpiece in three ways: sliding, ploughing, and cutting. Each interaction type has a distinct effect on the heat generation. When the interaction is sliding or ploughing, the primary effect is friction, resulting in approximately equal heat flux intensities. This leads to a rectangular heat source distribution. On the other hand, when the interaction is cutting, the tangential and normal forces gradually increase with increasing abrasive input, leading to an increment in the heat flux intensity. This results in a triangular heat source distribution.
It is important to consider that during the grinding process, abrasive particles protrude from the surface of the grinding wheel at different heights. Consequently, at any given position and time, both sliding and cutting actions coexist. Therefore, a calculation of the grinding temperature field based solely on rectangular or triangular heat source distributions would be inadequate. In 2006, Zhang et al [162] developed a comprehensive model for the heat source distribution, as depicted in figure 11(e). This model takes into account the combined effects of sliding, ploughing, and cutting actions during the grinding process. The origin of the coordinate system O is set at the entry point between the abrasive particles and the workpiece at the bottom of the grinding wheel. The form function equation of the comprehensive distribution model of the grinding heat source is represented by the ordinate s(x), and a Cartesian coordinate system is established with the s(x) axis as the vertical direction.
The distribution of the grinding heat source can be categorized into two components: a rectangular heat source distribution and a triangular heat source distribution. The rectangular heat source is generated by the effect of grinding particles on the workpiece materials and ploughing, characterized by a heat flux intensity of ξ h q 0 (where q 0 represents the average heat flux) and a length of a. On the other hand, the triangular heat source distribution is formed due to the cutting action of abrasive particles on the material. It has a peak heat flux of uq 0 , a distance b from the origin of the coordinates, and a grinding arc length of l c .
The values of a, ξ h , and the sharpness of the grinding wheel are influenced by the lubrication performance of the grinding fluid. On the other hand, the value of b is related to the grinding mode, such as forward and reverse grinding. In cases where a = b = l c and ξ h = u = 1, the abrasive particles are relatively blunt, with sliding and ploughing playing the dominant roles, while the cutting action can be neglected. In such scenarios, the comprehensive heat-source distribution model simplifies into a rectangular heat-source distribution model.
When a = 0, b = l c , ξ h = 0, and u = 2, the abrasive particles are sharp, and the cutting action plays a dominant role while the effects of sliding and ploughing can be ignored. In the front part of the grinding arc, where less material is removed, the heat source intensity is low. Conversely, in the back part of the grinding arc, where the cutting depth is significant and material is being cut out, the heat source intensity is high. This transformation results in the comprehensive heat source distribution model being simplified into a right-triangular heat source distribution model.
When a = 0, b = l c /2, ξ h = 0, and u = 2, the cutting effect of abrasive particles in the middle of the grinding arc is strong, leading to a high heat source intensity. In this case, the comprehensive heat source distribution model is transformed into an isosceles triangle heat source distribution model.
When b = l c , the comprehensive model of the heat source distribution is transformed into a model that exhibits a rectangular heat source distribution at the front of the grinding arc and a triangular heat source distribution at the back. This configuration is known as the moment triangle heat source distribution model. As the grinding grain and workpiece come into contact, the smaller depth at which the grinding grain cuts into the workpiece results in grinding between the grain and workpiece having the effect of a slip brush, thus forming a rectangular heat source distribution. With an increase in the cutting depth of the abrasive, the cutting action becomes more prominent, resulting in a triangular heat source distribution. Consequently, the overall process exhibits a moment of triangle heat source distribution models.
When a = 0 and b = l c , the comprehensive heat source distribution model is transformed into a trapezoidal heat source distribution model. The trapezoidal heat source distribution can be seen as a combination of rectangular and triangular heat source distributions. Due to the varying protrusion height of the abrasive particles during the grinding process, both sliding and ploughing effects, as well as cutting effects, are present. In the front of the grinding arc, the cutting depth of the abrasive particles and the heat source intensity are small. Conversely, at the back of the grinding arc, the cutting depth of the abrasive particles increases, resulting in a higher heat source intensity.
In 2015, Wang et al [163] introduced a heat source distribution model specifically for the arc region of quartic polynomial grinding. This model was based on probability statistics derived from an analysis of the trajectory and contact of abrasive particles. By analyzing the heat distribution relationship in the grinding arc area, the researchers were able to determine the heat distribution shape without assuming that the total heat distribution shape and heat distribution ratio were identical. Finite element simulation of the grinding temperature field, as well as experimental analysis, were conducted to validate the model. The results demonstrated a strong consistency between the temperature values predicted by the fourth-order polynomial simulation calculation of the grinding arc heat source distribution model and the experimental measurements. The highest temperature in the grinding arc area showed an error range between 2.24% and 15.3% when comparing the predicted and measured values.
In 2019, He et al [164] made assumptions about the shape of the abrasive particles, considering them to be spherical, with particle size and protrusion height following a Rayleigh distribution. By applying the Taylor formula to expand the expression for the contact radius of the abrasive particles, a highorder functional curve heat source model was established for the grinding arc region. Fitting errors of third-to sixth-order function heat source models were analyzed, and it was found that the fifth-order function yielded an error value below 2.5%, with a gentle decrease in error as the order increased. Thus, a new fifth-order heat source model was developed.
He et al [165] proposed six inclined heat source models and conducted a simulation analysis using the FEM. The models included an inclined rectangular heat source, right triangle, common triangle, parabolic distribution, trapezoidal distribution, and Gaussian distribution, as shown in figures 11(f)-(k). These models were used to simulate the grinding temperature field and were compared with experimental results. The analysis revealed that the right triangle heat source model exhibited the smallest difference between the calculated results and the experimentally measured values.
In the context of ultra-precision grinding, where machining accuracy is below 0.1 µm and surface roughness R a is below 0.1 µm, certain factors that are often neglected in the heat source models used for conventional shallow cutting grinding or large cutting deep grinding must be taken into account. One such factor is the variation in grinding forces under different grinding conditions. The simplistic homogenization model and triangle model fail to accurately describe the intricacies of the real grinding process in ultra-precision grinding. Therefore, a more complex thermal model is necessary to predict the theoretical temperature distribution in ultraprecision grinding.
In 2009, Tian et al [166] proposed an inclined heat source model specifically for ultra-precision grinding. This model encompassed various heat source distribution shapes, including inclined rectangles, triangles, parabolas, trapezoids, Gaussian distributions, and isosceles triangles, as illustrated in figures 12(a)-(e).

Deep cutting grinding.
In contrast to ordinary shallow grinding, ultra-precision grinding techniques exhibit different characteristics. In ultra-precision grinding, the grinding depth can be neglected, and the ground surface is considered the same as the unground surface. Consequently, the heat source plane is parallel to the direction of motion.
Large cut-depth grinding includes creep feed grinding [92,100,[167][168][169] and high-efficiency deep grinding (HEDG) [170,171]. HEDG, in particular, is characterized by high wheel speed, high workpiece speed, and large cut depth. It presents distinct grinding conditions compared to conventional shallow grinding and slow feed grinding [172]. In large cut-depth grinding, the grinding depth can reach up to 30 mm, and the angle formed between the grinding arc area and the machined surface of the workpiece can exceed 20 • . Ignoring the influence of cutting depth and inclination angle on the grinding temperature may lead to significant errors. To address this issue, in 2001, Jin and Cai [173] proposed an analytical thermal model of a planar heat source moving along a semi-infinite solid surface, as depicted in figures 13(a) and (b). This model focused on studying the temperature distribution in the grinding area under deep cutting conditions. It was found that the angle between the heat source plane and the direction of motion played a crucial role in the temperature rise and distribution in the grinding area during HEDG. In 2003, Zhao et al [174] further extended Jin's work and proposed an arc heat source model. As shown in figure 13(c), the contact area between the grinding wheel and workpiece is represented by an arc plane, and the heat source is formed by an accumulation of infinite moving-line heat sources distributed around the contact arc.
The equations for each model and the applicable grinding methods are summarized in table 2.

Energy distribution coefficient model
In the establishment of the grinding temperature model, one crucial issue is determining the proportion of grinding heat generated during the grinding process that is transferred into the workpiece. This proportion is referred to as the heat distribution ratio R w [34]. Both domestic and international theoretical studies have focused on determining R w based on different approaches. Some reports propose that the average temperature of the grinding wheel surface at the wheelworkpiece interface is equal to the average temperature of the workpiece surface. Others suggest that the maximum temperature of the grinding wheel surface corresponds to the maximum temperature of the workpiece surface. These approaches are based on the assumption that the highest temperature occurs at the interface between the abrasive grains and the workpiece, and that the same relationship applies to the highest temperature on the workpiece surface [4,175]. However, it is important to note that the heat distribution ratio is a variable that depends on various factors, including the properties of the grinding wheel and workpiece, grinding depth, workpiece feed speed, and grinding wheel linear velocity. These factors collectively influence the transfer of heat from the grinding process to the workpiece.
Under the nondry grinding condition, as shown in figure 14, the total heat generated by the grinding zone is where b g is the width of the grinding wheel, q g is the heat passing into the abrasive particles of the abrasive tool, q c is the heat removed by the abrasive debris, and q f is the heat coming out of the cooling medium. Therefore, the heat distribution coefficient of the workpiece is: Therefore, calculating the heat-distribution ratio is a very complicated problem. Thus far, there is still no convincing theory to explain the factors that determine the heat distribution ratio. The following are the theoretical models of the heat partition coefficient that have been reported and introduced. Outwater [176] identified that the heat produced during the grinding process primarily originates from three key areas: (1) the wear plane of abrasive particles interacting with the workpiece's surface, exemplified by the AB surface depicted in figure 15, (2) the shear slip surface of the debris, represented by the BC surface, and (3) the contact surface between abrasive particles and debris, or the BD surface. The heat generated from the plastic deformation of the material, induced by the abrasive particles, is subsequently transferred to the abrasive particles and the workpiece via these three surfaces.
Building on the Outwater model, Hahn [177] enhanced the heat allocation model. This new version disregards the cutting force on the shear plane and assumes that the workpiece surface is smooth, with abrasive particles gliding over this polished surface. This concept is referred to as the 'abrasive particle sliding hypothesis' model. In this model, grinding heat is assumed to be generated at the grinding interface, specifically at the plane of abrasive wear. A portion of this grinding heat is channeled into the workpiece, while the rest flows into the abrasive particles. On this basis, Hahn simplified the movement of abrasive particles along the workpiece surface (at velocity 'vs') to that of conical shapes. Since the thermal conductivity of the abrasive particles surpasses that of the cooling medium, it is assumed that the cooling medium does not absorb the heat transferred to the abrasives. Instead, all of this heat is transferred to them, leading to a uniform temperature along the radius of the abrasive particles. Bei [158] and Takazaw [132] Plane right triangle heat source Plane ordinary triangular heat source The maximum error is controlled within 6.5% Zhang et al [162] Plane comprehensive heat source  The error between the predicted and measured maximum temperature ranges from 2.24% to 15.3% He et al [164] Fifth order heat source qw =

60.35
The error of the heat source is less than 2.5% (Continued.)

Ultra precision grinding
Tian et al [166] Inclined rectangle heat source     Under these assumptions, the proportion of grinding heat that enters the workpiece can be determined as follows: where the subscripts g and w represent abrasive particles and workpiece, respectively. Based on the analysis of the energy distribution at the abrasive/workpiece wear interface (shown in figure 16), Rowe [150,178,179] modified equation (18), the wheel-workpiece heat distribution is defined as follows: where R s is the proportion of heat flowing into the grinding wheel, k g is the thermal conductivity of the abrasive, and r 0 is the effective contact radius of grinding wheel grains.

Model considering material physical properties.
Mao et al [180] re-evaluated the heat distribution ratio channeled into the workpiece, this time accounting for various factors. These included the physical property parameters of the grinding wheel and workpiece, the effective contact radius of the grinding wheel grains, the grinding parameters, the grinding fluid, the grinding force, and variations in these elements at different grinding temperatures.
(1) Heat flow into debris Assuming that the chip reaches its melting temperature during grinding, heat flows into the chip.
where T mp is the melting point of the material, a p is the actual cutting depth, and l g is the geometric contact arc length. (2) Heat flow into coolant In Mao's research, during the grinding process, it is challenging for the grinding fluid to penetrate the grinding zone. Consequently, the amount of heat carried away by the grinding fluid in this zone is negligible. When the grinding fluid in the grinding area reaches boiling point, a vaporization phenomenon occurs between the fluid and the workpiece surface, forming an insulating layer. This substantially reduces the heat transfer between the grinding fluid and the workpiece surface. Therefore, it is assumed that the heat flux density transferred into the grinding fluid is zero, denoted as q f = 0. (3) Heat flows into the workpiece and the grinding wheel.
The heat entering the workpiece and grinding wheel can be expressed as: Then, the heat distribution ratio between the incoming workpiece and the grinding wheel is: where (kρc) w denotes the thermal contact coefficient of the workpiece material, r 0 denotes the effective contact radius of the grinding wheel grit, and k g denotes the thermal conductivity of the abrasive particles. (4) Heat flow into workpiece The heat flow into the workpiece: where F t is the tangential component of the grinding force.
The proportion of heat flowing into the workpiece can be obtained by substituting the above types of heat into the workpiece.
From equation (24), it is evident that the model's distribution thoroughly considers a multitude of factors, including the physical performance parameters of the grinding wheel and workpiece, the effective contact radius of the abrasive, the velocities of both the workpiece and the grinding wheel, the coolant, the effective diameter of the grinding force per unit width, and the precision of the grinding wheel. The model's consideration of these influences has led to an enhancement in its application and it is now a widely used method of calculation.
In 1999, Guo et al [181] used an inverse analysis method to investigate the energy distribution ratio during the grinding process of a ceramic binder CBN wheel. This was achieved by fitting the measured workpiece temperature to the analytical calculated value. It was discovered that for the swing grinding of steel, the energy distribution range of the alumina grinding wheel increased from 60% to 75% using the inverse analysis method. This is slightly less than the energy distribution proportion coefficient value obtained using the thermal method [182,183]. When using a resin-bonded CBN grinding wheel, the energy distribution ratio can be reduced by 20%. However, using alumina grinding wheels for slow-feed grinding results in a significantly smaller energy distribution of only 2%-4.5%, as determined by inverse analysis. This low energy distribution ratio is attributed to the cooling effect of the fluid in the grinding zone. Grinding experiments were conducted with a CBN wheel, and the energy proportional coefficient values of different grinding parameters were obtained through inverse analysis, as shown in figure 17(a).
Taking into account the heat transfer to the abrasive particles and grinding fluid, an energy proportional coefficient model is established, as illustrated in figure 17(b). Each effective abrasive is treated as a flat-headed cone moving along the workpiece surface at velocity 'v s '. It is assumed that all grinding energy is expended within the contact area 'Ac' (A c = πr 0 2 ) between the abrasive and the workpiece, and that the maximum temperature at the workpiece/grinding fluid interface, the workpiece/abrasive interface, and the workpiece surface are equal. Consequently, the heat distribution ratio transferred to the workpiece is given as follows: where Ω ≡ 0.94  , γ = drg dz , A g is the truncated grain tip, r g is the grain radius, and γ is the geometric grain shape factor.
In 2002, Lan [184] conducted a study on heat distribution during the grinding process using an electroplated CBN wheel. Owing to the high thermal conductivity of the electroplated CBN grinding wheel and its metal matrix, a portion of the heat generated in the grinding zone is transferred to the workpiece during the grinding process, while another portion is passed to the metal grinding wheel matrix via CBN abrasive particles. This greatly contrasts with the assumption in traditional grinding heat transfer mechanisms, where the grinding wheel does not absorb heat. Therefore, the heat transfer model of a traditional grinding wheel does not fit the scenario of ultrahighspeed grinding with electroplated CBN wheels.
Analyzing the heat transfer mechanism of grinding with both traditional and CBN wheels, Lan introduced the equivalent coefficient Φ e and provided a calculation formula for the heat distribution ratio R w when using an electroplated CBN wheel: The Φ e values of the different grinding parameters were obtained by fitting the experimental measurement results to the theoretical calculation results, as listed in table 3. The results show that when the electroplated CBN wheel is ground, only approximately 30% of the heat is transferred to the workpiece, which is much smaller than that of an ordinary wheel, which is approximately 60%-75%.
Ramanath and Shaw [185] proposed a method to calculate the energy distribution ratio between the workpiece and the grinding wheel. He established a heat distribution coefficient model specifically for flat, dry grinding, as illustrated in figure 18. In this model, when the grinding depth is relatively small, the chip carries away only a minor amount of heat. Therefore, the heat source model during grinding is presumed to be a uniform heat source moving between the abrasive cutting surface and the workpiece surface.
Under the assumption that the average surface temperatures of the workpiece and the grinding wheel at the grinding contact interface are equal, the expression for the proportionality coefficient of heat distribution between the workpiece and the grinding wheel can be described as follows: The model considers the inherent thermal characteristics of the material, but does not consider the influence of the grinding state and parameters. Moreover, it does not factor in the grain of the grinding material or the structure of the grinding wheel. As a result, it can only yield an approximate heat distribution ratio.
In 1992, Kohli et al [183] incorporated an effective abrasive count into the grinding heat distribution model, taking into account the fact that the contact area between the grinding wheel grains and the workpiece did not equal the total grinding contact area. Under the assumption that the highest grinding temperature of the workpiece in the grinding arc area is the same as that of the grinding wheel, the grinding heat distribution ratio of the workpiece under a triangular heat source is given as: where C a is the number of effective abrasives per unit area of the grinding wheel, A 0 is the average effective area of contact between a single abrasive particle and workpiece, and f (ζ) is the shape function of the abrasive particles.

Model considering abrasive/grinding fluid composite.
Lavine [186,187] proposed that the grinding wheel can be viewed as a complex composed of grinding fluid and abrasive particles, with the characteristics of the grinding wheel and grinding fluid together defining the attributes of this complex. In this model, the grinding fluid is considered part of the surface of the grinding wheel, rather than a medium for convective heat transfer; therefore, there is no need to determine a convective heat transfer coefficient. The heat flow between the complex and the workpiece is partitioned, with one part transferred to the workpiece and the remainder channeled to the complex. The heat flow directed to the workpiece moves along its surface at the same rate as the feed speed of the workpiece, while the heat flow directed to the complex moves along its surface in accordance with the linear velocity vs of the grinding wheel.
Based on Jaeger's formula for calculating the linear specific grinding temperature and under wet grinding conditions, it is assumed that the liquid film adheres to the wheel surface. The ratio of the actual contact area of the wheel to the workpiece, compared to the nominal contact area, A r /A n , is much less than 1 for the abrasive grain, and A r /A n equals 1 for the liquid film. The proportion of heat distribution channeled into the workpiece is calculated as follows: Moreover, according to research by Zhang et al [188] and Wang et al [189], the energy ratio coefficient for the incoming workpiece using flood and MQL methods can be determined.
As NJMQL grinding has gradually become a hot topic in the field of machining in recent years, researchers [146,[190][191][192][193][194] have explored the R w model for NJMQL. For the NMQL cooling technique, solid nanoparticles are added to the grinding fluid to enhance its thermal conductivity coefficient, given that nanoparticles are key heat-conducting materials [195][196][197]. This improvement in the grinding fluid's heat transfer capacity will also result in increased heat input into the grinding fluid. The heat directed into the grinding fluid includes heat absorbed by the base fluid of the grinding fluid and heat absorbed by the nanoparticles. After adding the nanoparticles, the energy distribution coefficient in the workpiece can be represented as follows: where φ n is the concentration of nanofluids, and (kρc) n is the thermal conductivity, density, and specific heat capacity of the nanoparticles. The heat produced within the grinding zone permeates into the workpiece, grinding wheel, coolant, or the material chips. This total heat is bifurcated into two segments: the heat directed into the workpiece and the heat absorbed by the composite mass composed of the grinding wheel and the coolant (without considering the chips explicitly). The heat flux entering the workpiece and the composite mass (denoted as q w and q c , respectively) is considered uniformly distributed within the grinding zone. By treating q w and q c as known variables, the heat transfer issue within the workpiece and the composite mass is segregated. As reviewed in previous literature, the heat conduction issue within a workpiece with specified convective coefficients on its surface has been addressed previously. The current analysis addresses the problem of heat transfer within composite masses by determining the convective coefficient. The proposed model is depicted in figure 19. This heat transfer model for composite masses is also applicable to cutting and cutting grinding processes. In this research, the heat transfer within the composite mass was coupled with the heat transfer within the workpiece.  Based on this, Rowe [198] conducted further research and determined the heat distribution ratio between the grinding wheel and workpiece:

Model considering grinding wheel/workpiece as a system.
Hadad and Sadeghi [199] developed a model for the heat distribution coefficient of the grinding wheel/workpiece system. He proposed that the heat generated during the grinding process, resulting from the abrasive particles' interaction with the workpiece materials, comprises three components: (1) heat produced by the frictional action between the wear plane of the abrasive particles and the workpiece, as well as the interface of the abrasive particles and the debris, and (2) heat produced by the plastic deformation of the shear surface of the abrasive particle and the workpiece.
As shown in figure 20, it is assumed that heat is instantaneously transferred to the chip and grinding wheel/workpiece system The heat q w-s transferred to the grinding wheel/workpiece system is further transferred to the workpiece and grinding wheel as follows: The heat into the workpiece eventually flows into the workpiece ground and cooling medium:

Model considering macro and micro heat transfer processes.
Drawing from the heat transfer processes involved in both macro and micro grinding operations, and taking into account the real contact area between the grain and workpiece, as well as the geometry of the grinding wheel and workpiece, Zhang [162] developed a mathematical model to calculate the heat distribution ratio among the workpiece, grinding fluid, and grinding wheel. The contact area ratio, A s , and the effect of the abrasive contact radius on the abrasive surface temperature were key considerations in this process.
From a macroscopic perspective, grinding is perceived as a grinding wheel working on a workpiece, with heat being transferred to the grinding wheel, workpiece, and grinding fluid. The heat transfer relationship among these components is illustrated in figure 21(a). A microscopic analysis reveals the abrasive's role in sliding and cutting the workpiece surface and generating heat. Because the heat flow intensity generated by the interaction between the abrasive grains and workpiece varies, the heat flow intensity absorbed by each abrasive grain also differs. This heat flux transfer relationship is represented in figure 21(b). When the grinding fluid is applied, it serves a cooling and lubricating function in the gap between the grinding wheel and workpiece. The computation of the convective heat transfer coefficient is complex due to the difficulty in determining the flow state of the grinding fluid [182]. Thus, the convective heat transfer coefficient is disregarded, and it is assumed that the grinding fluid is distributed in the pores between the abrasive particles in the grinding zone, becoming part of the grinding wheel and rotating along with it.
Assuming that the temperature of the workpiece, and the contact surface of the abrasive and grinding fluid are equal, and that the heat absorbed by the chip can be disregarded due to its negligible amount, the heat distribution ratios of the workpiece, grinding fluid, and grinding wheel can be calculated as follows:

Model based on grinding wheel/workpiece composite heat conductor.
Takazawa [200] regarded the grinding wheel and workpiece as two contacting heat conductors moving at different speeds and calculated the percentage of heat transferred to the workpiece using the one-dimensional heat conduction model: where f is a coefficient depending on Pe.

Model considering convective heat transfer in the grind-
ing zone. In light of the above analysis, researchers typically neglect the heat outflow ratio (R f ) of the cooling medium when calculating the heat distribution of the grinding area, or they estimate the heat transfer coefficient (h f ) to ascertain R f and subsequently derive the heat distribution ratio of the sample introduced. However, in nanoparticle jet mist cooling, the nanofluid is the primary medium for heat dissipation in the grinding zone. While Rowe's model for the heat distribution ratio (Rw) takes into account the convective heat transfer coefficient of the cooling medium, it requires knowing the temperature of the grinding zone to compute Rw and does not allow prediction of R w prior to grinding. To address this issue, Yang et al [201] established a mathematical model for R w by calculating the convective heat transfer coefficients of the grinding wheel, workpiece, grinding fluid, and debris.
The heat flux in the incoming workpiece, grinding wheel, grinding fluid, and debris is related to the maximum contact temperature T max , T b (boiling point of the grinding fluid), and T m (melting point of the workpiece) (T max ⩽ T b ) [178]: where h w , h g , h f and h d represent the heat-transfer coefficients of the workpiece material, grinding wheel, grinding fluid, and debris, respectively.
The heat distribution ratio into the workpiece: The heat transfer coefficient is defined as the amount of heat transferred through a unit area in a unit time. In the grinding process, the heat distribution coefficient of the workpiece represents the ratio of the heat transferred to the workpiece material relative to the total heat generated. Consequently, the material heat distribution coefficient, abrasive particle heat distribution coefficient (R g ), abrasive debris heat distribution coefficient (R c ), and cooling medium heat distribution coefficient (R f ) characterize the capacity of each heat transfer medium to absorb grinding heat. Simultaneously, the heat transfer coefficient of debris (h d ) is related to the melting point of the workpiece material. However, the grinding temperature of the raw material cannot reach the melting point of the workpiece material or the boiling point (T b ) of the cooling medium, even under dry grinding conditions. Based on these considerations, the heat distribution coefficient of the workpiece can be assumed as follows: In particular, for the nanoparticle jet MQL cooling method, h f is calculated using the convective heat transfer coefficient model established in the next section. For h w , h g and h d [150,178]: The applicability of each energy distribution coefficient model and the model accuracy are summarized in table 4.

Heat transfer coefficient model
In the context of grinding, the cooling effect provided by the grinding fluid plays a significant role. Therefore, when considering non-dry grinding processes, determining the convective heat transfer coefficient becomes a crucial factor in calculating the temperature of the grinding surface [100,143,151,202,203]. However, due to the inherent uncertainty associated with physical parameters such as the coolant, workpiece, and grinding wheel, quantifying the convective heat transfer coefficient (h) accurately based on scientific principles is challenging.
In 1985, Susumu and Furukawa [204] u conducted a study on heat transfer in the grinding process using an electrically heated mock workpiece. They found that the convection coefficient of the coolant (h) ranged from 1 × 10 4 -1 × 10 5 W m −2 K −1 . These values were utilized to determine the dimensionless heat transfer coefficient (h f ). Their findings revealed that in conventional grinding, with a workpiece speed of approximately 10 m·min −1 , h f was approximately 0.05, resulting in an expected temperature reduction of only 10%. Conversely, in creep feed grinding, where the workpiece speed is reduced by a factor of 1000, h f increased up to 50, indicating that the temperature rise can be expected to be only approximately 5% compared to grinding without cooling (h f = 0), representing a significant reduction of approximately 95%.
To determine the convection coefficient around the workpiece, the researchers conducted experiments by measuring the temperature change in a small copper cylinder heated and subsequently placed in the flow of the coolant. This approach allowed them to obtain the convection heat transfer coefficients at five different locations, as illustrated in figure 22.

Estimation of convective heat transfer coefficient.
In 1997, Kim [205] estimated the values for downfeed creep grinding to be 2 × 10 4 W m −2 K −1 ahead and 1.5 × 10 4 W m −2 K −1 behind the grinding zone. For upgrinding, h f was projected to be 5000 W m −2 K −1 ahead and 2 × 10 4 W m −2 K −1 behind the grinding zone. In the case of dry grinding, the convective heat-transfer coefficient was assumed to be zero. Table 5 provides a detailed breakdown of the convective heat transfer coefficients and boiling temperatures for various cooling methods and media.
In 2001, Rowe [150] proposed three methods for estimating the convective heat transfer coefficient: (1) Based on extensive experimental data, the convective heat transfer coefficients for various fluids are determined under non-boiling and boiling conditions. The two primary cooling media studied are pure oil and oil-in-water emulsions. For an oil-in-water emulsion, h f is assumed to be 1 × 10 4 W m −2 K −1 under non-boiling conditions when T max ⩽ T b . It is assumed that h f is 0 under boiling conditions or estimated to be below 1 × 10 4 W m −2 K −1 under specific circumstances. For pure oil, the convective coefficient is assumed to be 4000 W m −2 K −1 . (2) Fluid convection is proposed not to be related to the maximum temperature of the contact surface, but rather, to depend on the average temperature. For a parabolic heat source distribution, the average temperature is 2/3 of the maximum temperature, and for a triangular heat source distribution, the average temperature is 1/2 of the maximum temperature. For water-based fluids, h f is 6700 W m −2 K −1 , and the convection coefficient assumed is 2700 W m −2 K −1 for pure oils. (3) Another method to estimate the cooling effect of a cooling medium involves 'fluid wheel' hypothesis [206]. This approach assumes that a layer of fluid flowing at the speed of the grinding wheel covers all contact areas. The hypothesis includes the use of a triangular heat source and a fluid wheel [181]: where β f is the thermal property for transient heat conduction in a coolant β f = √ (kρc) f . As an example for water, taking k f 0.61 W m −1 K −1 , c f 4200 J kg −1 K −1 and ρ f 1000 kg m −3 with v s 30 m s −1 and l c 10 mm, h f is 2600 W m −2 K −1 . For the same conditions using neat oil, with k f 0.14 W m −1 K −1 , c f 2100 J kg −1 K −1 and ρ f 900 kg m −3 , h f is 840 W m −2 K −1 .

Model based on Peclet number.
In 1991, Lavine [207] assumed that the cooling medium completely filled the space around the abrasive particles and that the depth was greater than the thickness of the thermal boundary layer. Assuming that the fluid is stationary relative to the grinding wheel, the fluid flows through the workpiece at a uniform speed v s . based on the large Peclet number (typically, Pe is on the order of 10 5 or greater), and conduction in the direction of motion is neglected. Assuming that the fluid remains liquid (non-boiling), the heat transfer coefficient for the fluid is In 2015, Vinay and Srinivasa Rao [208] calculated the convective heat-transfer coefficient of grinding as where β w is the thermophysical parameter of the workpiece (β w = √ (kρc) w ), C depends on Pe and is approximated by Rowe [150]: where Pe is the relation between the forced convection of a system and its conduction [209] and can be obtained by Rowe [150]:

Model based on fluid dynamics.
Most researchers assume that the heat transfer coefficient in the grinding contact zone is equivalent to the convective heat transfer coefficient of the cooling medium. In 2003, Jin [210,211] estimated these parameters using fluid dynamics and thermal simulations. Figure 23 illustrates the temperature distribution of the cooling medium in the thermal boundary layer when the coolant enters the grinding area. The temperature of the workpiece surface is designated as T w , and the temperature of the coolant outside the thermal boundary layer is T ∞ . δ t represents the thickness of the thermal boundary layer at position x, and it is presumed that the flow velocity of the majority of the coolant is equal to the speed of the edge of the grinding wheel, denoted as v s .
On the workpiece surface, the heat flow q w is : where h c is the convective heat transfer coefficient of the coolant in the grinding zone.
The relationship between T w and T ∞ : The hypothetical thermal boundary layer thickness, δ t is less than the hydrodynamic boundary layer thickness, δ h . From the energy balance, the thickness of the thermal boundary can be obtained as follows: , µ is the dynamic viscosity of the grinding fluid, k 1 = ∂p ∂x = − 2µ vs δ 2 , δ is the coolant film thickness within the grinding zone, p is the coolant pressure, and For a grinding wheel with a grain size of 60 ∼ 70#, the coolant film thickness is 0.15 mm during slow-feed grinding and efficient deep grinding. In cases of shallow cutting grinding, where both the coolant and wheel speeds are low, the coolant film thickness is assumed to be between 0.05 mm and 0.1 mm. Figure 24 illustrates the predicted convective heat transfer coefficient of the coolant. From the figure, it can be observed that the convective heat transfer coefficient of the water-based coolant is higher than that of the oil-based coolant. Furthermore, the two primary factors determining the  convective coefficient (h f ) in the grinding zone are the wheel speed and the liquid film thickness in the contact zone. The thickness of the oil film can be determined by considering various factors such as the speed of the grinding wheel, porosity, grain size, coolant type, flow rate, and nozzle size.
Furthermore, through theoretical analysis and experimental research, Jin found that the relationship between h f and h c is h f = 0.67 h c . When the coolant reaches the ignition point (130 • C), the convective heat transfer coefficient of the waterbased coolant is a constant value (2.9 × 10 5 W m −2 K −1 ).
In 2009, Lin [212] developed a model for the convective heat transfer coefficient, grounded in fluid dynamics and heat conduction theory. Figure 25 depicts the contact conditions and temperature conduction between the grinding wheel and the workpiece. The following assumptions were made in the development of this model: (1) fluid motion is represented by average velocity, (2) convection is the primary mode of heat transfer, and (3) the fluid flow is predominantly laminar.
The semi-empirical equation based on the theory of heat conduction in solids is as follows: where Re is the Reynolds number (Re = ρ f ualc µ ), Pr is the Prandtl number (Pr = µc f k f ), Nu is the Nusselt number (Nu = h f lc k f ), u a is the average fluid velocity, and µ is the dynamic viscosity.  h f can be obtained as: where G is a comprehensive parameter related to the density, specific heat capacity, and thermal conductivity. Grinding experiments of cast iron and M50 were carried out using a water-based coolant. v s was 36 m s −1 , v w was 270 mm s −1 , and a p was 0.01-0.06 mm. The calculated and measured h f values are listed in tables 6 and 7, respectively. Simultaneously, the effect of a p on h f was studied, as shown in figure 26. The results show that h f decreased with an increase in a p .

Nonuniform coefficient in different regions of the
grinding zone. In 2002, Wang [189] calculated the average heat-transfer coefficient in the grinding arc range. There is a complete formula for calculating the convective heat transfer coefficient in the laminar flow state of the plate. The local heat transfer coefficient is By introducing the Reynolds and Prandtl criteria into equation (51), the average convective heat transfer coefficient in the grinding arc range can be obtained as follows: where u ∞ is the velocity of free flow. Figure 27 shows the trend of the convective heat transfer coefficient variation in the grinding arc length at different temperatures and positions. It can be seen from the figure that the temperature at x = 0.157 mm (cut-in) is 160 • C, and the convective heat transfer coefficient is 3.56 × 10 5 W m −2 K −1 . At x = 1.57 mm (cut source), the workpiece wall temperature is approximately 20 • C, and the convective heat transfer coefficient is 2.63 × 10 4 W m −2 K −1 , which is nearly ten times different.
In 2008, Shen [116] conducted a detailed study on the convective heat transfer coefficient in the grinding zone. As the chip thickness in the grinding arc area gradually increases, this results in nonuniform convective cooling, as shown in figure 28. The convective heat transfer area is divided into three parts: the grinding leading edge, contact area, and grinding trailing edge. The convective heat transfer coefficient h contact in the grinding contact area is assumed to be a linear function as follows: where h 2 = max {h contact (x), h 1 = min (h contact (x)}, and x is the local coordinate, where the origin is located at   The convective heat transfer coefficient of the grinding leading edge is ignored and the convective heat transfer coefficient of the grinding trailing edge is assumed to be uniform,  [116]. Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works. Reproduced with permission from [116].
MQL (soybean oil) 3.9 × 10 4 2.5 × 10 4 1 × 10 4 Flood (5 vol.% Cimtech 500 synthetic grinding fluid) 7.4 × 10 5 4.2 × 10 5 9.5 × 10 4 Table 9. MQL parameters and coolant-lubricants.   figure 29, the average convective heat-transfer coefficient in the grinding contact zone (h contact ) and trailing edge (h trailing ) can be obtained, as shown in table 8. As shown in the table, because the fluid velocity in the contact area is much higher, the convection heat transfer coefficient in the contact area is much higher than that in the trailing edge area. The convective heat transfer coefficient in the contact area and trailing edge of the pouring type is much higher than that of the MQL method.
In 2012, Hadad and Sadeghi [199] analyzed the convective heat-transfer coefficient model proposed by Shen. For dry grinding, the MQL of different jet parameters and pouring cooling conditions, h contact and h trailing were calculated, as listed in tables 9 and 10. To further verify Shen's law, namely, that the convective heat transfer coefficient of the contact zone is much higher than that of the trailing edge, in the contact zone and trailing edge, the convective heat transfer  coefficient of pouring grinding is much higher than that of MQL grinding.

Model considering particle contact condition.
In 2013, Zhu et al [213] proposed a model for the convective heat transfer coefficient that took into account the conditions of the abrasive, considering the influence of the abrasive's geometry, size, and distribution density on the heat transfer capabilities of the grinding fluid. The shape of the abrasive particles was assumed to be approximately conical, and the abrasive particles were organized in a systematic manner according to a cross-array, as depicted in figure 30.
Assuming that the convective heat transfer coefficients of the coolant on the workpiece and grinding wheel surfaces are equal, the calculation model for the convective heat transfer coefficient of the workpiece grinding surface can be derived based on the heat transfer model of multiple rows of tube bundles with outwardly flowing fluid, as shown below: where L c is the characteristic length, d g is the equivalent average grain diameter, and L g is the average particle spacing.

Model for low-temperature grinding considering different fluid states.
Previous models for the convective heat transfer coefficient assumed fluid flowing over the surface of the workpiece in the grinding arc zone. However, in the grinding arc, the fluid is restricted by rough abrasive particles and cannot flow freely over the workpiece surface, implying that the fluid is mechanically stirred within the grinding arc zone. Taking this into account, Zhang et al [214,215] proposed different convection heat-transfer coefficient models according to various fluid states, as depicted in figure 31.
(1) Laminar-flow model In the laminar flow model (LFM), the assumption is that the fluid state in the grinding zone is laminar and that the fluid is flowing over a smooth surface. An assumed constant average temperature increase ∆T av on the work surface is considered to be two-thirds of the maximum temperature increase. The highest velocity of the boundary layer in laminar flow generates the lowest convection heat transfer coefficient, as illustrated in figure 31(b). The final convective heat transfer coefficient, derived based on the LFM, is as follows: where C f ′ is the sliding heat source shape factor.
(2) Turbulent-flow model Turbulence is considered a more realistic assumption for laminar flow because abrasive particles are very rough compared to the depth of the fluid. These abrasive particles cut into the workpiece surface, scraping off any potential fluid boundary layers in the abrasive/workpiece contact area. Additionally, if the fluid is not entering the grinding zone, it can be assumed that it has established a stable flow in that area. The fluid is stirred in the closed pores between the grinding wheel particles and is agitated intensely by the high-speed moving particles. As the laminar boundary layer decreases, turbulence brings the cooler fluid closer to the working surface. However, a thin boundary layer persists near the stationary surface. As a result, the convective heat transfer coefficient for turbulence is higher than that for laminar flow. The fluid convection factor for turbulent flow is as follows: (3) Fluid-wheel model The fluid is assumed to pass through the workpiece at the circumferential speed of the grinding wheel, and a thermal boundary layer is established in the solid, characterized by the thermal conductivity of the fluid. It is further assumed that the temperature of the fluid entering the grinding zone is higher than room temperature. With these assumptions, the convection factor can be calculated as follows: 3.3.7. Model based on statistical theory. For the MQL and NMQL cooling methods, the probability density distribution of atomized droplet sizes significantly influences heat transfer performance [197,[216][217][218]. In 2021, Yang et al [219] studied the atomization mechanism and characteristics of micro-lubrication. He conducted a probability density statistical analysis and calculation of atomized droplet size density, establishing a mathematical model for the convective heat transfer coefficient under atomization cooling conditions. As the Weber number (We) of incident droplets gradually increases, the droplets exhibit three sequential behaviors: rebound, spreading, and splashing. When the initial energy of the droplet is low, it will rebound. When the droplet collides with the heat source surface at high energy, it will fly out from the crown's edge and decompose into many smaller droplets. In both cases, droplets cannot effectively participate in heat transfer. Effective heat transfer occurs when a droplet spreads, i.e. when a droplet hits the heat source surface and spreads along that surface to form a liquid film [220]. The critical Weber number (We) of the spreading droplet is where La is the Laplace number. According to equation (58), the range of droplet size D with spread, that is, the effective heat transfer, can be calculated as D min ⩽ D ⩽ D max , as shown in figure 32. Therefore, the proportion of effective heat transfer droplets is as follows: For single cooling droplets, the heat transfer coefficient h s [217]: where J is the heat exchange of a single droplet, ∆T is the heat exchange temperature difference, q d is the heat flux of singledroplet heat transfer, t s is the heat exchange time, m d is the droplet mass, and A ′ is the spreading area of the droplet.
Heat transfer coefficient of all effective heat transfer droplets: where N z is the number of all droplets. The convective heat transfer coefficient of a high-pressure gas jet impinging on the heat source surface is: where b c is the contact width between the micro abrasive tool and workpiece. Comprehensive convective heat transfer coefficient of MQL cooling: To validate the theoretical model, a system to measure the convective heat transfer coefficient was designed and assembled, as depicted in figure 33. An atomization cooling experiment was conducted using nanoparticle jet spray cooling with hydroxyapatite (HA), SiO 2 , Fe 2 O 3 , Al 2 O 3 , and carbon nanotube (CNT) nanoparticles, with pure normal saline coolant serving as a comparison. Figure 33 presents the convective heat transfer coefficient values computed theoretically and measured experimentally. When compared to the pure saline spray (1.62 × 10 −2 W mm −2 K −1 ), the measured convective heat transfer coefficients for nanoparticle jet spray cooling with HA, SiO 2 , Fe 2 O 3 , Al 2 O 3 , and CNT nanoparticles increased by 141.98%, 137.65%, 130.25%, 141.36%, and 145.06%, respectively.
The applicability of each heat transfer coefficient model and the model accuracy are summarized in table 11.

Influence of the heat source distribution model on the temperature field
Li et al [222] investigated the impact of varying models on the temperature field through FEM, concluding that different heat source models significantly influence the workpiece surface's temperature distribution. Under the triangular and rectangular heat-source models, the temperature distribution of the workpiece became uniform with an increase in depth. According to Meng and Li [221], heat transfer in the grinding process can be classified into four heat conduction models based on different heat source intensity distributions and heat conduction directions: (1) One-dimensional heat conduction model with a uniform heat source intensity distribution. The surface grinding temperature of the GH33A superalloy was calculated under identical grinding parameters and heattransfer conditions for all models, as depicted in figure 34(a). The results showed significant variations in the grinding temperature when the law governing the heat source intensity distribution differed. The grinding temperature calculations for the one-dimensional and two-dimensional heat conduction models were similar when under the same heat source intensity distribution. The grinding temperature field's distribution is primarily dependent on the heat source intensity distribution in the grinding zone, with the heat conduction direction having a negligible influence.
Wang proposed a heat source shape coefficient, n h , for the grinding zone, suggesting that the value of n h fluctuates depending on different grinding environments. Under conditions of good lubrication, n h tends toward a triangle (n h = 0), while in situations of high friction or when the abrasive particles are blunt, n h leans toward a rectangle (n h = 1). Applying different n h values to the workpiece in the same grinding environment results in diverse temperature profiles. As demonstrated in figure 34(b), during inverse grinding, as n h approaches 1, the peak temperature in the grinding zone shifts toward the entrance of the grinding arc (X/L = 0). Conversely,  if the grinding wheel is dressed sharply and the abrasive grain primarily performs the cutting role, material removal leads to a smaller sliding and ploughing effect at the front of the contact zone, resulting in lower heat source intensity. At the rear end of the contact zone, the larger cutting depth and greater material removal result in a high heat source intensity, and n h is approximately 0. This makes the peak temperature in the grinding zone move towards the center of the grinding arc (X/L = 0.5).
As shown in the figure, changes in the n h value only slightly modify the maximum temperature but significantly influence the distribution of this maximum temperature within the grinding zone. Additionally, the temperature in the contact zone first increases then decreases at varying rates depending on the heat source model. The temperature rises faster and falls slower under the triangular heat source distribution model compared to that under the rectangular heat source distribution model. Zhang et al [162] used several heat source distribution models, including rectangular, right-triangle, and various common triangular models, to calculate the temperature field. The results revealed significant differences in workpiece surface temperature distributions depending on the heat source distribution model used. The lowest maximum grinding temperature value in the contact zone was produced by the rectangular model, whereas the triangular models generated higher grinding temperatures. The peak grinding temperature was found to be 0.25 away from the contact zone's center when the triangular heat source model was employed. The position of the highest grinding temperature on the surface varied, with the right-angle triangle model typically displaying the peak temperature near the contact area's entrance. The only exception was the rectangular heat source model, which showed a maximum grinding temperature biased toward the contact area's outlet. The temperature of the contact zone also rose and fell at different rates depending on the heat source model; the triangular model's temperature increased faster and decreased slower than the rectangular model's. On the ground surface, the temperature values from all models were very similar.
In 2004, Zhang et al [162] expanded his analysis to include the isosceles triangle and moment triangle heat source distribution models. As shown in figure 34(c), these models led to distinct surface temperature distributions. The isosceles triangle model yielded the highest temperature, followed by the right triangle, rectangular, and finally the moment triangle model. The location of these highest temperatures also varied, with the right triangle model's peak near the grinding arc's exit area, the rectangular and moment triangle models' peaks near the entrance, and the isosceles triangle model's peak in the middle of the arc.
Wang et al [163] employed FEM to load a right-angle triangle heat source and a quartic function curve heat source as the moving heat source. The results, validated through experimental grinding, showed that the maximum temperature generated by the right-angle triangle heat source was slightly higher than that generated by the quartic function curve heat source. The maximum temperature generated by the quartic function curve heat source was closer to the front of the grinding arc area, and its temperature distribution more closely mirrored the experimental results ( figure 34(d)).
He et al [164] compared simulated workpiece surface temperatures using rectangular, triangular, and fifth-order function heat source models with experimentally measured grinding temperatures (as depicted in figure 34(e)). The results showed that the grinding temperature calculated using the fifth-order heat source model more closely matched the measured temperature, both in value and location of peak temperature.
In the context of high-efficiency deep grinding, Xu et al [223] proposed a positive correlation between the grinding zone temperature and the constant C, which is related to the contact angle and Pe. They studied the influence of the contact angle and Pe on the C value using both oblique plane heat source and arc surface heat source models ( figure 34(f)).
He et al [165] used several inclined heat source models, including rectangles, triangles, parabolas, trapezoids, Gaussian distributions, and isosceles triangles, to simulate the ultraprecision grinding temperature of the titanium alloy TC4. As shown in figures 34(g) and (h), while the temperature curves of different models exhibited similar trends, the temperatures calculated using the inclined triangle heat source model were closest to the experimentally measured temperatures.
The influence law of the heat-source distribution model on the temperature field is summarized in table 12.

Influencing factors of the proportion coefficient of heat distribution
Lan [184] studied the influence of grinding parameters on the heat distribution ratio R w , as demonstrated in figures 35(a) and (b). The results showed that an increase in the grinding wheel speed (v s ) caused a decrease in R w , while an increase in workpiece speed (v w ) produced the opposite effect, increasing the energy distribution rate R w .
Zhang et al [162] systematically explored the impact of grinding wheel parameters and grinding parameters on the heat distribution ratio. As illustrated in figure 35(c), the study utilized grinding wheels with different grain sizes under dry grinding conditions to calculate the heat distribution ratio as per Rowe and Lavine's formulas. The results revealed that the heat distribution ratio decreases as the grinding wheel grit size increases.
The investigation also examined changes in the workpiece heat distribution ratio concerning different grinding parameters in Zhang's research [162]. As presented in figures 36(a)-(c), increasing the grinding depth caused a slight rise in the workpiece's heat distribution ratio, although the effect was relatively small. Additionally, as the grinding wheel speed increased, the workpiece's heat distribution ratio increased. A minor increase in the workpiece's heat distribution ratio was also observed with an increase in feed speed, although the effect was less significant than that of the grinding wheel speed. Figure 36(d) shows the influence of different cooling methods on the workpiece's heat distribution ratio. Applying coolant resulted in a substantial decrease in the workpiece's (2) One-dimensional heat conduction model with triangular distribution of heat source intensity; (3) Two-dimensional heat conduction model with uniform heat source intensity distribution; (4) Two-dimensional heat conduction model with triangular distribution of heat source intensity. The distribution law of grinding temperature field mainly depends on the distribution law of heat source intensity in the grinding area, and the direction of heat conduction has little influence on it.
Wang et al [ Dry grinding Right angle triangle heat source, quartic function curve heat source (1) The maximum temperature generated by the right angle triangle heat source is slightly higher than that generated by the quartic function curve heat source; (2) The maximum temperature generated by the quartic function curve heat source is closer to the front of the grinding arc area; (3) The temperature distribution obtained by loading the quartic function curve heat source is closer to the temperature distribution measured in the experiment.
(Continued.) but the position of the highest grinding temperature is different. The energy center of the fifth-order heat source is closer to the entrance of the grinding arc area than that of the triangular heat source and the rectangular heat source.
(2) Experimental results show that the fifth-order heat source is closer to the measured temperature than the rectangular heat source or the triangular heat source, and the position of its highest temperature is also closer to the measured situation, and only the fifth-order heat source meets the hysteresis characteristics of the measured temperature when the temperature rises sharply.
Xu et al [223] 0.  (4), inclined Gaussian distribution (5) and inclined common triangle heat source (6)  (1) The temperature calculation results of all heat source models have similar trends, with the calculation results of triangular distribution heat sources being the closest to the experimental measurements. In addition, as the feed speed of the workpiece increases, the grinding temperature decreases.
(2) The Heat-affected zone of heat source 1 is triangle, and others are arc. The heat influence band of heat source 4 is the widest and the temperature is the lowest. Heat source 5 and 6 have narrow Heat-affected zone and high temperature. heat distribution ratio, with water-based grinding fluid providing better cooling effects than oil-based fluid. Additionally, the study analyzed the workpiece's heat distribution ratio under different heat-source distribution forms, as illustrated in figure 36(e). The heat distribution ratio computed with a triangular heat source distribution was 0.7% higher than that calculated with a rectangular distribution, but overall, the heat source distribution had a negligible impact on the heat distribution ratio calculation. Finally, as shown in figure 36(f), the choice of grinding method (forward or reverse grinding) had little influence on the calculation results of the heat distribution ratio. Wu et al [224] conducted experiments to assess the influence of the heat distribution ratio on grinding parameters. As indicated in figures 36(g)-(i), increasing the grinding wheel speed (v s ) raised R w from 12.1% to 24.3%, R c from 5.5% to 11.3%, and reduced R s from 82.4% to 64.4%. Additionally, increasing the grinding depth (a p ) resulted in an increase in R w from 17.3% to 30.3%, R c from 6.6% to 15.8%, and a decrease in R s from 76.1% to 53.9%. Increasing the workpiece feed speed (v w ) led to a decrease in R w from 23.1% to 11.7%, an increase in R s from 71.1% to 78.3%, and an increase in R c from 5.8% to 10%.
Xu [225] studied changes in the heat distribution ratio of a workpiece under varying grinding parameters. As shown in figures 36(j)-(l), increasing the grinding wheel linear speed (v s ) from 30 to 150 m s −1 led to an increase in R w from 0.14 to 0.57. As illustrated in the figure, R w displayed a decreasing trend with increases in table speed and grinding cutting depth. Specifically, as the table speed rose from 700 to 3500 mm min −1 , R w fell from 0.53 to 0.28. Similarly, as the cutting depth increased from 5 to 40 µm, R w fell from 0.43 to 0.2.
Shen et al [226] conducted a study to explore the impact of various factors, such as the coolant and its application mode, on the heat distribution ratio. This research specifically examined the effects of cooling conditions and the coolant application method during face and surface grinding of zirconia ceramics, using water as the coolant. Under dry grinding conditions, a decrease in the heat distribution ratio (R w value) was observed to different extents. This suggests that when coolant penetrates the grinding arc area, it efficiently  removes a substantial amount of heat during the grinding process. As a result, it reduces heat transfer at the interface between the diamond grinding wheel and the ceramic workpiece, thus effectively lowering the temperature in the grinding arc area. The researchers fit the temperature curve of the ceramic surface grinding process to the theoretical analytical temperature curve to derive the heat distribution ratio. To further examine the effect of different coolant application methods on the heat distribution ratio, a vertical constant pressure grinding device was used. The constant pressure grinding temperature was recorded online under various processing parameters by applying coolant from the outside and inside of the grinding wheel. After fitting the experimental and theoretical analysis temperature curves, the R w value for the heat distribution ratio was obtained. The results showed that when the coolant was applied from the inside of the grinding wheel (R w = 0.72), the heat distribution ratio was lower compared to when the coolant was applied from the outside (R w = 0.85). In other words, coolant applied from the inside of the grinding wheel was more effective at removing grinding heat through the grinding arc area, thereby reducing the temperature of the grinding arc area.
Rowe et al [227,228] investigated the effect of the contact arc length of the grinding wheel/workpiece on the energy distribution proportional coefficient, as shown in figure 37. Here, l c represents the actual contact length and l g denotes the geometric contact length. For a rectangular heat source distribution, the highest temperature is seen at the position x = l g starting from the point of contact between the grinding wheel/workpiece. For a triangular heat source distribution, the highest temperature occurs at the midpoint of the contact area, i.e. x = 0.5l c . Therefore, if a triangular heat source distribution is employed, the value of l c is approximately twice that of l g . It can also be observed that the triangular heat source distribution can yield higher R w values, especially at smaller cutting depths.
The influencing factors of the energy proportional coefficient are further summarized in table 13.

Influence law of convective heat transfer coefficient on temperature
Hadad and Sadeghi [199] studied the influence of different h trailing and h contact conditions on the rise in grinding temperature. Various convection heat transfer coefficients were applied to the trailing edge, and the resulting temperature distribution on the workpiece surface is depicted in figure 38(a). The results indicated that while h trailing significantly affects the temperature distribution, its influence is limited only to the trailing edge and does not extend to the contact zone or the leading edge. The peak temperatures calculated for different h trailing values are approximately the same. Different h contact values were employed to calculate the temperature in the grinding zone, and the temperature distribution on the workpiece surface that reached a steady state is shown in figure 38(b). Effective cooling in the grinding zone was found to significantly influence both the peak temperature and the trailing-edge temperature. These results indicate that cooling in the contact zone is the most critical factor in grinding.
Feng [229] studied the influence of the convective heat transfer coefficient on the theoretical temperature of the workpiece grinding surface under external circular grinding conditions. The results are summarized in table 14. The table demonstrates that when the convective heat transfer coefficient values vary within the ranges of 1 × 10 4 ∼ 9 × 10 4 W m −2 K −1 , 1.1 × 10 5 -1.9 × 10 5 W m −2 K −1 , and 2.1 × 10 5 -2.9 × 10 5 W m −2 K −1 , the theoretical results of the grinding surface temperature of the workpiece do not show significant change. However, when the convective heat transfer coefficient changes by different orders of magnitude, the theoretical temperature difference becomes substantial. For instance, when the convection heat transfer coefficient is 1 × 10 4 W m −2 K −1 and 2.9 × 10 5 W m −2 K −1 , the theoretical temperatures are 933 • C and 704 • C, respectively, indicating a clear difference.
Jin et al [210] calculated the convective heat transfer coefficient value and the grinding temperature under the conditions of both deep cut (0.5-1 mm, encompassing creep feed grinding and HEDG) and shallow cut (0.005, 0.01, 0.03 mm). These calculations were based on the models and process parameters established by various researchers, as depicted in tables 15 and 16.
Shen [116] measured the in-tube flow convection coefficient of a nanofluid, as demonstrated in figure 39. The experiment involved using Al 2 O 3 nanoparticles with varying volume fractions, and pure water was used as a point of comparison. The results indicated that the in-tube flow convection coefficient for the Al 2 O 3 nanofluid was not significantly different from that of water.
Zhang and Rowe [144,214,215] performed research to scrutinize the influence of different heat transfer coefficient    models on grinding temperature. In particular, they studied the heat transfer coefficient models built upon the LFM, turbulent flow model (TFM), and fluid wheel models. Figure 40      temperature of approximately 400 • C. Nonetheless, when the grinding temperature surpassed 400 • C, h f showed an upward trajectory. In figure 40(b), the heat transfer coefficients predicted by the TFM and LFM are compared with the experimentally determined values. The TFM predicted value was found to be more congruent with the experimental measurement value. It is worth noting that the heat transfer coefficient projected by the turbulence model was substantially greater than that predicted by the laminar model. Furthermore, the experimental results indicated a marked downward trend when the grinding temperature fell below the boiling point, implying that the heat transfer coefficient decreases as the grinding temperature drops, leading to diminished heat dissipation during the grinding process. The influence law of the convective heat transfer coefficient on the temperature is summarized in table 17.

Conclusion
(1) This study delivers a comprehensive overview of grinding temperature field models and grinding thermal models, including their specific applications and scopes. Additionally, it proposes potential future directions for research in grinding temperature fields to tackle prevailing issues related to temperature fields and grinding thermal models. (2) Grinding temperature fields can be divided into two categories: uniform continuous temperature fields and nonuniform discontinuous grinding temperature fields, contingent upon the uniformity and continuity of the heat source. When working with grinding wheels that contain randomly arranged abrasive particles, utilizing the nonuniform discontinuous temperature field calculation is more congruent with real-life working conditions. The temperature field model proposed by Liu, predicated on the random distribution of grinding wheel particles, delivers high precision with a minimum error of 4.9%. (3) Based on the grinding depth, the heat source distribution model can be broadly separated into three types: heat source distribution models for general shallow cutting grinding, deep cutting grinding, and ultra-precision grinding. Nevertheless, the grinding depth is not the sole determinant for the heat source distribution model. It is essential to consider an aggregate of grinding parameters and grinding methods, among other factors, to determine the most suitable heat source distribution model. (4) The energy distribution proportion models, such as the one formulated by Mao [180], consider numerous factors, encompassing material physical property parameters of the grinding wheel and workpiece, effective contact radius of grinding particles, workpiece speed, grinding wheel speed, coolant, grinding force per unit width, and the effective diameter of the grinding wheel. Consequently, the accuracy of this model has markedly improved, with a maximum error of 10%. This calculation method is widely favored among current researchers.
(5) In the context of convective heat transfer coefficient models, Jin [210] took into account the thermal boundary layer of the workpiece in the grinding area and devised a model for the convective heat transfer coefficient (h c ) and the convection coefficient of the cooling medium (h f ). Jin's model, which has been broadly accepted by researchers, defines h f as 2.9 × 10 5 W m −2 K −1 for water-based emulsions and 1 × 10 5 -1.6 × 10 5 W m −2 K −1 for oil-based lubricants. The relationship between h f and h c is expressed as h f = 0.67 h c .

Prospects
(1) Current theoretical models for the temperature field in grinding are primarily predicated on the heat conduction theory for isotropic and homogeneous materials. However, the rise in the use of anisotropic and heterogeneous materials such as carbon fiber composites, polymer composites, and optical crystal materials necessitates accurate control over grinding temperatures. Regrettably, the prevailing theory of the grinding temperature field is not applicable to these materials, thereby underscoring an immediate need to establish a specific heat conduction theory for the grinding of anisotropic and heterogeneous materials. (2) The heat distribution ratio during the grinding process is a variable subject to numerous factors including grinding wheel parameters, grain morphology, structure, coolant, and the properties of the grinding wheel and workpiece. Calculating the heat distribution ratio has consistently posed a complex problem, and thus far, there is a paucity of compelling theories explaining the key factors influencing this ratio. Additional research is necessary to elucidate the underlying mechanisms and furnish a comprehensive understanding of this phenomenon. (3) The cooling process, facilitated by grinding fluids, plays a pivotal role in the grinding process. In the context of non-dry grinding, the accurate calculation of the convective heat transfer coefficient is vital for determining the grinding surface temperature of the workpiece. However, due to uncertainties related to the coolant, workpiece, grinding wheel, and other physical parameters, determining the convective heat transfer coefficient remains a formidable challenge. Therefore, additional in-depth studies are required to scientifically quantify this coefficient and augment our understanding of convective heat transfer in grinding.