## 3 Numerical Analysis of the Thermal Performance
### Thermal Resistance Analysis
#### 3.3.1 Tool-Holder and Internal Fluid Channel  
The proposed independent internal cooling structure, composed of connected cooling fluid channels, can be regarded as a novel form of cylindrical cooling systems (CCS). CCS is often theoretically characterized by temperature gradients in radial directions. As shown in Figure 7a, the cylindrical wall separates two fluids of different temperatures. According to [PERSON]'s law of heat transfer, the thermal resistance for radial conduction in a cylindrical wall is expressed as [16]:  
\[R=\frac{\ln(r_{2}/r_{1})}{2\pi Lk} \tag{6}\]  
where \(r_{1}\), \(r_{2}\) are the radius of the inner surface and outer surface respectively; \(L\) is the length of the channel; and \(k\) is thermal conductivity.  
We divide the interface surface between the insert and the tool-holder into different elements, as shown in Figure 7b. The connected internal fluid channels are located within the tool-holder and deployed according to the geometric shape of the tool-holder such that they are very closed to the interface marked by the blue dash lines in Figure 7b. The thermal resistance between the unit \((i,j)\) can be regarded as the parallel connection of every internal fluid channel unit. Therefore, the thermal resistance of unit \((i,j)\) is presented written as:  
\[R_{(i,j)}=\frac{\prod\limits_{k=1}^{n}R_{k,cylinder}}{\sum\limits_{k=1}^{n}R_{k,cylinder}} \tag{7}\]  
where \(R_{k,cylinder}\) is the thermal resistance between the \(k\)th internal fluid channel unit and unit \((i,j)\) of the interface between the insert and the tool-holder, and \(n\) is the number of the unit.  
Figure 7: Schematic diagram of thermal resistance: (**a**) the cylindrical wall; (**b**) interface unit division.  
From the above results, the thermal resistance distribution can be calculated to validate the enhancement of the heat transfer process. Figure 8 shows the thermal resistance at different locations on the interface in this research. A total of 840 points that are evenly spaced are sampled from the interface. Figure 8a is the coordinate definition diagram, with the origin located in the tool nose. The minimum thermal resistance located on the center of the interface indicates good heat dissipation performance in this interface.  
The Reynolds number is defined as the ratio between the inertia force and the viscous force [17; 18], utilized to characterize different flow regimes within a similar fluid such as laminar or turbulent flow [19]. The Reynolds number is defined as:  
\[R_{e}=\frac{UL}{v}=\frac{\rho vd}{\eta} \tag{8}\]  
where \(U\) is a scale of variation of velocity in a length scale \(L\); \(v\) is the velocity of flow; \(d\) is the diameter of the channel and \(\eta\) is the liquid viscosity. Therefore, the convection coefficient can be written as:  
\[h_{s}=\lambda\frac{Nu}{l}=\lambda\frac{CR_{e}{}^{m}P_{r}{}^{1/3}}{l} \tag{9}\]  
where \(\lambda\) represents the thermal conductivity of the fluid; \(Nu=CR_{e}{}^{m}P_{r}{}^{1/3}\) is the Nusselt number [20]; the constants \(C\) and \(m\) can be chosen from references [20; 21]; \(P_{r}\) is the Prandtl number; and \(l\) is the characteristic length.  
The thermal circuit can be constructed on the basis of the fact that the thermal resistance is associated with the heat conduction between the insert and the tool-holder. The equivalent thermal resistance between the tool-holder and the internal fluid channel can be acquired as:  
\[R_{ci}=\frac{1}{h_{s}A}+R_{cylind} \tag{10}\]  
Therefore, on the interface between tool-holder and internal fluid channel, the heat flow can be expressed as:  
\[q_{tl}=\frac{T_{insert}-T_{o}}{R_{ci}} \tag{11}\]