## 2 Model  
### Physical Model  
Figure 1 shows the hot plate which is discussed in this study. Its basic structure is the same as the hot plates discussed in previous studies [37; 38; 39]. The hot plate has a square plate structure with a side length of L and a thickness of d. The actual working area where work is possible is a square area with a side length of L-2g which excludes the hot plate perimeter region with a width of g. On the hot plate's sides, there are 2 rows of staggered circular holes with a diameter D in the length (z) direction, one row above the other. The circular holes are arranged horizontally at intervals of S, and one of the core advantages of the hot plate is its extensibility, by which it can easily be extended to a large area by increasing the number of holes. As shown in Figure 1, a heat source is provided in heating mode by inserting circular cartridge heaters in the lower circular holes, and the upper circular holes are used for cooling. The main dimensions of the hot plate used in this study are as follows.  
Figure 1: _Cont_.  
L = 240 mm, D = 6 mm, g = 10 mm, d = 20 mm, S = 12 mm. Using this as a reference, 20 cartridge heaters were installed in the lower part and 19 cooling holes were made in the upper part. Each cartridge heater has a heating capacity of 300 W. The heating plate material is stainless steel STS304, and the density is 8030 kg/m\({}^{3}\). The thermal conductivity is 15.1 W/m K. The thermal diffusion coefficient is \(3.75\times 10^{-6}\) m\({}^{2}\)/s, which is considerably smaller than the thermal diffusion coefficient of the aluminum used in previous studies, which is \(7.10\times 10^{-5}\) m\({}^{2}\)/s. This study considered two hot plate models with different methods of using the cooling holes. Model I is a model which uses the cooling holes as flow channels for refrigerant as in an existing study [36]. The coolant flows in cooling mode only, and the hot plate is cooled directly. Model II is a model with a heat pipe attached to the cooling holes. It was considered as a method for improving the temperature uniformity problems which can occur in Model I. In this model, a heat pipe that is longer than the hot plate is used, and the heat pipe is indirectly cooled by cooling the end of the heat pipe which protrudes outside of the hot plate. The outer diameter  
Figure 1: Schematic illustration of the hot plate and simulation model. (**a**) Top view and the front view. (**b**) The enlarged front view near the temperature sensing point. (**c**) Different configurations of cooling holes and cartridge parallel. (**d**) Mesh structure of simulation model.  
of the heat pipe is 6 mm, and the length is 340 mm. Of the 100 mm part that protrudes outside of the hot plate, the 30 mm at the end is cooled.  
Feedback control is performed on the cartridge heater's heat generation to allow the hot plate temperature to optimally approach the set temperature and stabilize. As done in a previous study [33; 34], 6 PID controllers are used to control the 20 heaters. Six temperature monitoring points are arranged as shown in Figure 1a. The two heaters near each perimeter monitoring point are combined into single groups, and the 4 heaters near the internal temperature monitoring points are combined into single groups to achieve independent temperature control for a total of 6 groups.