# Experimental Study on Cooling Performance of a Hybrid Microchannel and Jet Impingement Heat Sink
## 3 Data Reduction and Uncertainty Analysis  
The temperature of the heating wire is a function of its resistance, and resistances at different temperatures are measured in an oven with a controlled temperature. The correlation between temperature (40-240 \({}^{\circ}\)C) and resistance is fitted by using the experimental data.  
\[T=93.732R_{e}-1506.043 \tag{1}\]  
The thermal conductivities of the AlN substrate and tungsten copper used in the measuring device are measured by NETZSCH LFA447. Based on the measured data, the temperature-dependent thermal conductivity of the AlN substrate can be determined by Equation (2).  
\[\lambda_{AIN}=-0.304T+178.330 \tag{2}\]  
The temperature-dependent thermal conductivity of tungsten copper can be determined by Equation (3).  
\[\lambda_{WCu}=248.860T^{-0.045} \tag{3}\]  
The heat flux can be calculated by fitting the temperature gradient according to the thermocouple measurements.  
\[q_{i}=\lambda_{WCu}\frac{\mathrm{d}T}{dx} \tag{4}\]  
The heat flux through the interface is the average value of heat fluxes through the upper and lower blocks. The thermal resistances of the AlN substrate and silicone grease are calculated by an average of five tests. The thermal resistance is measured under a weight of 1.28 kg, which is the same as the spring plate to fix the heater in the experiment. The detailed results are discussed in Section 4.  
\[R_{th}=3.056\times 10^{-5}-3.083\times 10^{-4}T^{-1}+2.933\times 10^{-3}T^{-2} \tag{5}\]  
The heating power of the AlN heater on the heat sink is determined by the measurements of voltage and current.  
\[Q=UI \tag{6}\]  
The heating area is 1.4 cm \(\times\) 1.5 cm, and then the heat flux is determined by Equation (7).  
\[q^{\prime\prime}=\frac{Q}{A} \tag{7}\]  
Figure 3: Schematic of the core structure in the measuring device.  
The wall temperature on the cooling surface of the heat sink is determined by Equation (8).  
\[T_{i}=T_{i-1}-\frac{qx_{i}}{\lambda_{i}} \tag{8}\]  
The outlet vapor quality is calculated according to the inlet and outlet conditions.  
\[x=\frac{qA-\dot{m}(h_{sat}-h_{in})}{\dot{m}h_{lv}} \tag{9}\]  
The thermal resistance is used to evaluate the cooling performance of the heat sink, which is defined by Equation (10).  
\[R_{h}=\frac{(T_{ow}-T_{in})}{qA} \tag{10}\]  
where \(T_{ow}\) is the outer wall temperature of heat sinks, and \(T_{in}\) is the inlet fluid temperature.  
The uncertainties of calculated data are determined by Equation (11) according to the reference [36]. Uncertainties of thermocouples, differential pressure sensor, flowmeter, voltage and current measurements are \(\pm\)0.2 K, \(\pm\)0.5%, \(\pm\)2.5%, \(\pm\)0.15% and \(\pm\)0.2%, respectively. The uncertainty of heat flux is 0.25%. The uncertainty of temperature for the heating wire is 1.8-8.5%. The uncertainty of the outlet vapor quality is 4.7-12.5%. The uncertainty of thermal resistance is 0.8-13%. It is necessary to check the repeatability of the experiments. Two different runs are adopted with a volumetric flow rate of 0.86 L/min under different heat fluxes. The comparison of the two different runs is shown in Figure 4. It can be seen that the difference between the two runs is quite small with a maximum temperature difference of 3.7 K over a heat flux range from 50 W/cm\({}^{2}\) to 350 W/cm\({}^{2}\).  
\[\frac{\delta_{y}}{y}=\frac{1}{y}\sqrt{\sum_{i=1}^{n}\left(\frac{\partial f}{ \partial x_{i}}\right)(\delta_{x_{i}})^{2}} \tag{11}\]