# Water Jacket Systems for Temperature Control of Petri Dish Cell Culture Chambers
## 3 Results and Discussion  
### Thermal Analysis of the Water Jacket under Static Reference Conditions  
First, we investigate the thermal performance of the water jacket system under steady-state reference conditions. The reference condition was defined as water temperature at the inlet port of the water jacket, \(T_{water@inlet}\) = 39 \({}^{\circ}\)C; water flow rate through the water jacket, \(Q_{water}\) = 60 mL/min; free convection coefficient, \(h_{air}\) = 10 W/m\({}^{2}\)K; ambient temperature, \(T_{air}\) = 20 \({}^{\circ}\)C; Petri dish external diameter, \(D_{Petri\;dish}\) = 35 mm; target sample temperature, \(T_{sample}\) = 37 \({}^{\circ}\)C; and volume ratio of the sample inside the Petri dish, \(V_{sample}\) = 50%. Furthermore, the Petri dish was made of polystyrene, and a rubber washer was inserted between the water jacket and the Petri dish, as shown in Figure 1.  
Figure 2a shows the flow streamlines of the water circulating through the water jacket chamber, colored by the magnitude of the flow velocity. The results were obtained at the reference water flow rate of 60 mL/min, as described above. The Reynolds number for recirculating water was \(\sim\)50, reflecting the laminar nature of the recirculating flow at this rate. The flow streamlines were highly ordered  
across the water jacket due to the laminar flow characteristics, except at the inlet region where flow was not fully developed. A total pressure drop of 4.5 Pa was obtained along the water jacket cylinder (excluding the connecting tubes). Figure 2b presents the temperature contours across the system. The simulations revealed an almost uniform temperature distribution across the water jacket surface, which was due to the dominance of forced convection. In contrast, a large temperature gradient was obtained along the radial axis of the Petri dish (from the side walls toward the middle), which was due to the dominance of conduction  
Figure 2: Analyzing the thermal performance of the water jacket system under steady state reference conditions. (**a**) Flow streamlines along the water jacket chamber are colored by the magnitude of the velocity. (**b**) Temperature contours across different surfaces of the system. (**c**) Temperature contour at the top free surface of the water jacket system obtained by numerical simulations. (**d**) Temperature contour at the top free surface of the water jacket system obtained by thermal imaging camera. (**e**) Variations in temperature along the radial axis of the Petri dish at its top free surface obtained by experimental, numerical and theoretical analyses. Inset shows the temperature contours at the middle cross section of the Petri dish. and the free convection heat loss across the free surfaces of the Petri dish, as defined in Equation (13). This free convection heat loss caused a temperature gradient along the height of the Petri dish. The filled level of the sample inside the Petri dish is shown by a dashed line. Experimental measurements were also conducted to further investigate the temperature gradient along the radial axis of the Petri dish. We took advantage of a thermal imaging camera to monitor the temperature distribution across the top free surface of the system. Figure 2c,d present the temperature contours at the top free surface of the system obtained by numerical simulations and thermal imaging, respectively. The results confirmed the existence of a temperature gradient along the radial axis of the Petri dish. Interestingly, the lowest temperature was not located at the center of the Petri dish, and instead drifted toward the colder side of the water jacket (toward the outlet port), which is attributed to the temperature drop of the water while passing through the water jacket. The numerical simulations predicted this drift as 0.4 mm, whereas the thermal imaging recorded this drift as 1.9 mm. Figure 2e compares the temperature profiles along the radial axis of the Petri dish (parallel with the inlet/outlet ports axis) obtained from experimental, numerical and theoretical analyses.