## 2 Transient Temperature Field Model of Wear Band on the Flank of End Mills
### Temperature Drop Model of the Wear Band Affected by Dissipating Heat Source  
Tool-workpiece contact is intermittent during the peripheral milling process, when the tool is out of contact with the workpiece, the heat sources in the PSZ, RCZ and FWZ will disappear, and then the dissipating heat sources will appear. Because the processing method in this paper is dry milling, the dissipating heat source at this time is caused by wear land on the flank of end mills and the natural convection cooling of air. Because air is between the wear land on the flank of end mills and the machined surface, the former is regarded as a dissipating heat source. The dissipating heat source is a fixed heat source relative to any point on the flank face, so the temperature drop model is analogous with the temperature rise model of the tool/workpiece heat source established by [PERSON] et al. [32]. It is assumed that the length of the dissipating heat source is the length of the wear band on the flank face and the width of dissipating heat source is the cutting width, as shown in Figure 9. Temperature drop in wear land on the flank of end mills is affected by both native dissipating heat source and mirror dissipating heat source.  
The temperature drop \(\Delta T_{\textit{flank-air}}\) (\(x^{\prime}\), 0, 0) at any point \(P_{air}\) (\(x^{\prime\prime}\), 0, 0) along \(X^{\prime\prime}\) direction can be calculated by Equations (35)-(37).  
\[\Delta T_{air}(x^{\prime\prime},0,0)=\frac{q_{air}(x^{\prime\prime})}{2\pi \lambda_{t}}\int_{0}^{VB_{1}}dx_{i-4}^{\prime\prime}\int_{-\frac{\
u^{\prime} }{2}}^{\frac{\
u^{\prime}}{2}}\frac{1}{R_{i-4}}dy_{i-4}^{\prime\prime} \tag{35}\]  
\[\Delta T_{air}(x^{\prime\prime},0,0)=\frac{q_{air}(x^{\prime\prime})}{2\pi \lambda_{t}}\int_{VB_{1}}^{VB_{1}[1+\cos(2\gamma_{n})]}\frac{1}{\cos(2\gamma_{ n})}dx_{i-4}^{\prime\prime}\int_{-\frac{\
u^{\prime}}{2}}^{\frac{\
u^{\prime}}{2}} \frac{1}{R_{i-4}}dy_{i-4}^{\prime\prime} \tag{36}\]  
\[\Delta T_{\textit{flank-air}}(x^{\prime\prime},0,0)=\Delta T_{air}(x^{\prime \prime},0,0)+\Delta T_{air^{\prime}}(x^{\prime\prime},0,0) \tag{37}\]  
From Equation (35) to (37), \(\Delta T_{air}\) (\(x^{\prime\prime},0,0\)) and \(\Delta T_{air}\) (\(x^{\prime\prime}\), 0, 0) are the temperature drop caused by native dissipating heat source and mirror dissipating heat source, respectively. The distance from any  
Figure 9: Dissipating heat source.  
point \(P_{air}\) (\(x^{\prime\prime}\), 0, 0) along \(X^{\prime\prime}\) direction to the points of native dissipating and mirror dissipating heat sources can be measured by Equation (38).  
\[\left\{\begin{array}{l}R_{i-4}=\sqrt{\left(x^{\prime\prime}-x_{i-4}^{\prime \prime}\right)^{2}+\left(y_{i-4}^{\prime\prime}\right)^{2}}\\ R_{i-4^{\prime}}=\sqrt{\left(x^{\prime\prime}-x_{i-4^{\prime}}^{\prime\prime} \right)^{2}+\left(y_{i-4^{\prime}}^{\prime\prime}\right)^{2}+\left(\frac{x_{i -4^{\prime}}^{\prime\prime}-VB_{1}}{\cos(2\gamma_{n})}\right)^{2}}\\ w^{\prime}=\frac{w}{\cos\lambda_{s}}\end{array}\right. \tag{38}\]  
According to [PERSON]'s cooling law, the dissipating heat intensity is calculated by Equation (39).  
\[q_{air}(x^{\prime\prime})=\overline{h}\left[T_{fank}(x^{\prime\prime},0,0)-T_{ e}\right] \tag{39}\]  
In Equation (39), \(\overline{h}\) is the average air convective heat transfer coefficient, \(T_{fank}\) (\(x^{\prime\prime}\), 0, 0) is the temperature at any point in the \(X^{\prime\prime}\) direction of the flank wear band of end mills, and \(T_{e}\) is the ambient temperature.