Thermal Analysis in Grinding: Modelling Heat Partition and Surface Integrity

1.1. Physical Definition of Energy Dissipation and Total Heat Flux

The grinding process is fundamentally an intense process of energy dissipation.The shear deformation and friction that occur as abrasive grains scratch the workpiece surface convert nearly all (90% to 100%) of the input mechanical energy into instantaneous thermal energy. Therefore, the first step in controlling grinding temperature is to quantify the Total Heat Flux (q_t), which represents the total energy generated per unit time at the grinding zone.

When the total power (P), determined by the tangential grinding force (F_t) and the wheel peripheral speed (v_s), is concentrated within the contact area (width b, length l_c), the heat flux is defined as follows:

This heat flux induces a localized temperature rise at the machining point, acting as a decisive physical parameter that determines the Surface Integrity of the workpiece.

1.2. Heat Distribution Mechanism and Energy Partitioning

The generated total heat is not dissipated through a single path but is distributed among four major elements constituting the machining system. According to the law of conservation of energy, the total heat flux (q_t) is equal to the sum of the heat fluxes transferred to the workpiece (q_w), the grinding wheel (q_s), the grinding chips (q_chip), and the grinding fluid (q_f).

The decisive variable for preventing thermal damage to the workpiece is the Heat Partition Ratio (R_w). This represents the proportion of the total heat flux that actually penetrates into the workpiece, defined by the following equation:

1.3. Variation of R_w and Its Thermodynamic Impact

The value of R_w varies significantly depending on the grain characteristics of the grinding wheel and cooling conditions. In dry grinding with conventional Aluminum Oxide (Al₂O₃) wheels, a large portion of the generated heat is conducted into the workpiece due to the low thermal conductivity of the abrasive. In such cases, R_w can reach approximately 0.6 to 0.8, a dangerous level that can cause severe thermal damage.

Conversely, the situation changes dramatically when applying CBN (Cubic Boron Nitride) wheels, which possess excellent thermal conductivity. CBN grains rapidly dissipate generated heat into the wheel structure, effectively blocking the heat flow into the workpiece. In this case, R_w is significantly reduced to a level of approximately 0.2 to 0.5.

This imbalance in energy flow induces a rapid thermal cycle on the surface when the workpiece is viewed as a “moving semi-infinite body.” The instantaneous high-temperature heating and subsequent cooling go beyond simple physical expansion, serving as a critical factor in forming microstructural transformations and residual stresses within the material.

2.1. Jaeger’s Moving Heat Source Theory

Quantitatively predicting the temperature rise at the grinding zone is an essential process for optimizing machining conditions. To understand how the workpiece heat flux (q_w), defined in Part 1, is distributed spatially and temporally within the material, we employ an analytical approach that assumes the workpiece to be a Moving Semi-infinite Body.

The most widely cited model is the theory proposed by Jaeger, which calculates the steady-state temperature rise (ΔT) when a band-shaped heat source (q_w) is applied to a workpiece surface moving at a constant velocity (v_w). The maximum temperature rise (ΔT_max) at the grinding surface can be approximated by the following equation:

where k is thermal conductivity, ρ is density, C_p is specific heat, and l_c is the grinding contact length. This formula demonstrates that the maximum temperature rise is proportional to the heat flux (q_w) and the square root of the contact time (√(l_c / v_w)). Physically, this provides the evidence that increasing the feed speed (v_w) effectively lowers the surface temperature by reducing the time the heat source dwells on a specific point.

2.2. Temperature Decay with Depth and the Peclet Number (Pe)

Heat is generated at the surface and conducted inward. The temperature at a depth (z) below the surface decreases exponentially, and this decay characteristic is determined by the Peclet Number (Pe). The Peclet number is a dimensionless value representing the ratio of heat transfer by advection (movement) to heat transfer by conduction.

where α is the thermal diffusivity (k / ρC_p). This implies that at high grinding speeds, the heat source moves faster than the rate of thermal diffusion into the bulk material, effectively “trapping” the heat at the surface.

The grinding process typically involves high Peclet numbers (Pe > 10), meaning the workpiece moves away before heat can conduct deeply into the interior. Consequently, grinding thermal damage is characterized by its concentration within an extremely shallow layer , typically ranging from tens to hundreds of micrometers (μm) from the surface.

2.3. Micro-heat Source Mechanism at the Grain Level

Beyond the macro-perspective of the Jaeger model, an analysis at the individual grinding grain level is required. The total heat flux (q_w) is the summation of localized heat sources generated as countless microscopic grains sequentially scratch across the workpiece.

The temperature at the grain tip is significantly higher than the macro-average temperature, a phenomenon known as the “Flash Temperature.” While this instantaneous high temperature has a positive aspect—inducing localized softening of the material to assist chip formation—it mostly acts as the fundamental mechanism for microstructural changes in the workpiece.

3.1. Occurrence Conditions of Grinding Burn and Critical Heat Flux

The ultimate goal of grinding temperature modeling is to identify and evade the “thermal threshold” that compromises the mechanical properties of the material. When the maximum temperature (ΔT_max) calculated in Chapter 2 exceeds the material’s intrinsic critical temperature (T_c), irreversible microstructural changes and Grinding Burn occur on the workpiece surface.

In ferrous metals (Steel), grinding burn primarily manifests as surface re-hardening or tempering. To prevent this, researchers utilize the concept of Critical Heat Flux (q_bc). The heat flux at the onset of grinding burn is modeled by the following energy threshold equation:

According to this equation, as the workpiece velocity (v_w) increases, the allowable critical heat flux also rises. In other words, high-speed grinding serves not only to increase productivity but also acts as a sophisticated control mechanism that physically inhibits the risk of grinding burn by blocking the time required for heat to penetrate into the material.

3.2. Reversal of Residual Stress Due to Heat Influx

In addition to visible carbonization (burn), temperature rise alters the state of microscopic residual stress. While “compressive residual stress,” which is beneficial to the surface, is formed when mechanical loads dominate, it transitions into detrimental “tensile residual stress” through rapid thermal expansion and contraction once the thermal load exceeds the threshold.

Tensile residual stress drastically shortens the fatigue life of components and serves as a starting point for micro-cracks. Therefore, modern precision machining emphasizes “deterministic process control,” which monitors R_w and temperature in real-time via temperature sensors or Acoustic Emission (AE) sensors to ensure the stress state does not transition into the tensile regime.

The heat distribution and temperature modeling discussed in this report are core tools that elevate the grinding process from the realm of simple experience to the realm of science. To summarize, the following physical strategies are required for high-quality machining:

• Partition Control: Minimize the workpiece partition ratio (R_w) by using highly conductive abrasives such as CBN.
• Moving Heat Source Optimization: Shorten the heat dwell time on the surface by considering the Peclet number.
• Threshold Monitoring: Establish a real-time monitoring system to ensure the process is performed within the critical heat flux (q_bc).

These thermodynamic insights guarantee the reliability of precision parts and form the foundation of high-value-added manufacturing technology capable of withstanding extreme environments.

1. Efficiency of Energy Conversion: Why Does Mechanical Power (P) Become Heat (Q)?

Physically, power (Watt) and the rate of thermal energy generation (J/s) share the same units. The mechanical power (F_t · v_s) input during the grinding process is consumed by the shear deformation and friction of the metal. According to the law of conservation of energy in metal machining, more than 95% of this consumed energy is instantaneously converted into thermal energy.

Since only a negligible amount (less than 5%) is stored as residual energy within the metal lattice, engineering models employ the ‘Adiabatic Dissipation’ assumption, treating all input power as dissipated heat. This provides the physical rationale for why the value obtained by dividing power by the contact area directly equates to the heat flux.

2. The Heat Equation and the Physical Significance of the Square Root (√)

The reason a square root is included in the surface temperature rise equation is due to the diffusion characteristics of heat. Solving the 1D unsteady state heat conduction equation:

reveals that the distance heat penetrates into the interior is not linearly proportional to time (t); instead, it scales with the square root of time (√t) due to the stochastic nature of thermal diffusion.

This mathematically proves that as the heat source moves faster across the workpiece surface (increasing v_w), the time window for heat to penetrate decreases sharply according to the square root function. Consequently, high-speed grinding is not a strategy to reduce the amount of heat generated itself, but a ‘temporal isolation mechanism’ that finishes the machining before the heat can destroy the internal microstructure.

3. Origin of Constant 1.13: The Geometric Shape Factor

The constant 1.13 appearing in the Jaeger model is an approximation of 2 / √π derived during the mathematical derivation. It quantifies the geometric distribution characteristics of how heat diffuses internally when a constant heat flux is applied to the surface of an infinitely wide plane (workpiece).

This constant serves as a mathematical guarantee that our model mirrors a ‘Moving Semi-infinite Body’ similar to real-world grinding. It allows us to calculate complex 3D heat transfer phenomena using practical 1D equations.

This geometric factor is the physical reason why the constant 1.13 is consistently applied in our previous calculations for ΔT_max and q_bc.

References for Thermodynamic Modeling:

• Malkin, S., & Guo, C. (2008). Grinding Technology: Theory and Applications of Machining with Abrasives. Industrial Press Inc.
• Jaeger, J. C. (1942).
  “Moving sources of heat and the temperature at sliding contacts.” Proc. Roy. Soc. NSW.
• Rowe, W. B. (2014).
  Principles of Modern Grinding Technology. William Andrew.
• Marinescu, I. D., et al. (2006).
  Handbook of Machining with Grinding Wheels. CRC Press.