## 3 Analysis of the Cooling Mechanism of Injection Mold
### Heat Transfer Analysis of the Cooling Channel of the Injection Molding  
#### 3.3.1 Cooling Channel Heat Transfer Mathematical Model Establishment  
During the cooling process of plastic parts, the coolant flow in the channel is three-dimensional but can be simplified to a one-dimensional steady-state flow process when analyzing heat transfer behavior. Therefore, it is not necessary to study the precise details of the coolant flow. In the modeling process, parameters such as coolant flow rate and pressure can be regarded as remaining constant along the length of the cooling channel but may vary with changes in the channel geometry or coolant properties.  
Based on the analysis above, it is clear that convective heat transfer is responsible for heat transfer between the mold and cooling channels. The third boundary condition is applied in the boundary analysis model for cooling channels, which requires calculation of the convective heat transfer coefficient of the coolant. This coefficient is closely related to the flow conditions of the coolant. When the coolant flows uniformly into the channel with a Reynolds number \(Re\) greater than a critical value, the boundary layer inside the channels will gradually rise to a turbulent boundary layer and will subsequently flow into the center of the channels and become a fully turbulent zone.  
At a uniform flow rate, when the coolant is forced into a circular tube with a Reynolds number (\(Re\)) surpassing the critical value, gradual development of a turbulent boundary layer occurs at the pipe's boundary. This turbulent boundary layer then converges toward the center of the tube, eventually establishing a fully developed turbulence zone.  
Consider that the coolant is located in channels of length \(L\) flowing. The inner wall temperature \(T_{W}\) of the channels is kept constant. The inlet and outlet temperatures of the cooling channels are \(T_{in}\) and \(T_{out}\), respectively, and the coolant flow rate is \(\upsilon\). Figure 4 represents a schematic diagram of the heat balance of the coolant in a micro-element control body of length \(\mathrm{d}x\). Assuming that the average temperature of the coolant at the cross-section from the inlet \(x\) is \(T_{a}\), the heat flow \(Q_{1}\) obtained by the micro-element is:  
\[Q_{1}=\rho c\upsilon TA \tag{10}\]  
where \(\rho\) is the coolant density, \(c\) is the specific heat capacity, \(A\) is the cross-sectional area of the channels, and  
\[A=\frac{\pi}{4}d^{2} \tag{11}\]  
Figure 3: A visualization of turbulent and laminar flow. In fully developed laminar channel flow, the fluid flows in parallel layers with small differences in velocity, and fluid mixing between these layers is not present.  
Similarly, the heat flow \(Q_{2}\) out of the micro-element and the heat flow \(Q_{3}\) from the inner wall of the channels to the micro-element can be obtained as:  
\[Q_{2}=\frac{\pi}{4}d^{2}\rho cv(T+\frac{\text{d}T}{\text{d}x}) \tag{12}\]  
\[Q_{3}=h(\pi d\dot{x})(T_{W}-T) \tag{13}\]  
where \(h\) is the convective heat transfer coefficient that can be computed as:  
\[h=0.027(\frac{v\text{d}\rho}{\mu})^{0.8}(\frac{c\mu}{\lambda})^{\frac{1}{3}}( \frac{\mu_{f}}{\mu_{W}})^{0.14}\frac{\lambda}{d} \tag{14}\]  
where \(\mu_{W}\) is the viscosity of the cooling liquid when the wall temperature is \(T_{W}\), and other parameters are determined by the average temperature of the coolant.