## 2 Geometric Model, Mesh Division, and Boundary Conditions  
### Geometric Model  
The multi-branch cold plate is a crucial liquid cooling method in the radar cooling field, with extensive applications in both military and civilian industries. Investigating the flow and heat transfer characteristics of the 6 \(\times\) 5 parallel multi-branch cold plate enhances its cooling efficiency and broadens its application scope. Figure 0(a) illustrates the 3D geometric model of the 6 \(\times\) 5 parallel multi-branch liquid cold plate. The model comprises 12 layers, including a main liquid supply pipe, a main liquid return pipe, and 13 supply and return branches, specifically seven supply branches and six return branches. The model is a three-dimensional geometric model that is modeled in _Ansys 2022 R1_[11]. In Figure 0(a), the blue dot is the inlet of the coolant, and the red dot is the outlet. The integrated liquid cooling framework is primarily designed for the T/R components, comprising a total of 104 units. Each row of the model contains T/R components, with six components in rows 1, 2, 11, and 12, and ten components in all other rows. The four corners of the model are occupied by distribution components. The spacing between each branch pipe in the model is 0.192 m. Both the inlet and outlet are circular, with a diameter of 0.0059 m. The main supply and return channels have rectangular cross-sections measuring 0.04 m \(\times\) 0.02 m, while the supply and return branches have square cross-sections of 0.02 m \(\times\) 0.02 m. Each T/R component consists of a connecting base and a connecting pipe. The cooling liquid channels of two adjacent T/R components form a pair, creating a total of 52 pairs in the liquid cooling framework. Figure 0(a) is the overall flow channel frame of the large cold plate. Figure 0(b) shows the meshing results of the entire computational domain. Figure 0(a) Geometric model and mesh division. Figure 2 shows the size of the geometric model and marks it in the diagram, which can more intuitively see the size of the geometric model. The flow and heat transfer of the medium in the pipes involve the mass conservation equation, the energy conservation equation, and the momentum conservation equation:  
Mass conservation equation: The increase in the mass of a fluid micro-element per unit time is equal to the net mass flowing into the micro-element during the same period [12]. \[\
abla\overset{\rightarrow}{U}=\frac{\partial u}{\partial x}+\frac{\partial v }{\partial y}+\frac{\partial w}{\partial z}=0 \tag{1}\]  
In the equation, u, v, and w are the velocity components in the x, y, and z directions, respectively. Momentum conservation equation:  
The physical meaning of the momentum equation is to control the change rate of momentum in the body = surface force (stress) + volume force (gravity, etc.). \[\rho\left(\partial u/\partial t+u\bullet\
abla u\right)=-\
abla p+\mu\
abla^{2 }u+f \tag{2}\]  
where \(\partial\) is the Nabla operator \(\left(\partial/\partial x,\partial/\partial y,\partial/\partial z\right)\) and \(\partial\) 2 is the Laplacian operator \(\left(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}+\partial^{2}/ \partial z^{2}\right)\). At low Re, the viscous force is very large relative to the inertial force \(\left(\rho u\cdot\partial\ u\right)\). The influence of the inertia term is relatively small, and the balance of viscous force and pressure gradient mainly drives the flow. The N-S equation is simplified as  
\[\rho\left.\partial u/\partial t\right.\approx-\
abla p+\mu\
abla^{2}u \tag{3}\]  
When Re is large, the convective inertia term \(\left(\rho\ u\cdot\
abla u\right)\) plays a dominant role. The viscous force \(\left(\mu\
abla^{2}u\right)\) is relatively small in most regions (large scale), but it is crucial at the smallest scale (dissipative scale) for dissipating kinetic energy.