# Method for Measuring Cooling Efficiency of Water Droplets Imping onto Hot Metal Discs
## 4 Determining Cooling Efficiency
### Water Droplet Cooling Efficiency  
Data from the two reference measurements without applying water droplets, sandwiching the water droplet experiment, were used to calculate the average reference cooling curve. The heat losses were in these cases mainly associated with heat radiation and convective heat losses, with some minor heat loss through the four thermocouples suspending the disc. Between these reference measurements, the droplet-cooling test was done with water droplets applied to the hot metal discs at a constant rate. Typical temperature versus time curves are shown in Figure 4. The respective cooling rates (\(dT/dt\)) versus temperature are shown in Figure 5. It is clearly seen that at certain temperatures, the cooling rate is more significant when water droplets were applied to cool the disc. Based on the differences in cooling rates, disc mass, and specific heat, the heat flow to the impinging water droplets was calculated.  
Figure 4: Temperatures of a representative aluminum disc (**blue**) with 2.4-mm diameter droplets and an application rate of 0.02 g/s, with the reference temperatures recorded prior to and after the droplet measurements (**black**).  
Subtracting the temperature versus time derivatives presented in Figure 5 gives the net water droplets' disc-cooling rate, i.e., \(\left\{\frac{dT}{dt}\right\}_{\text{Net}}\) (K/s). Knowing the mass, \(m\) (kg), of the disc, which is in this case 52.9 g, and the specific heat of aluminum as a function of temperature, \(C_{\text{P}}(T)\) (J/kg K), the water droplet cooling heat flow can be calculated by:  
\[\dot{Q}_{\text{Drops}}(T)=m_{\text{Disc}}\cdot C_{\text{P}}(T)\cdot\left\{ \frac{dT}{dt}\right\}_{\text{Net}}\ (\text{W}), \tag{1}\]  
where the specific heat as a function of temperature is given by the NIST-JANAF Thermochemical Tables [29] as:  
\[C_{\text{P}}(T)=A+B\cdot\tau+C\cdot\tau^{2}+D\cdot\tau^{3}+E/\tau^{2}\ \ \ (\text{J/kg K}), \tag{2}\]  
where \(A=28.08920\), \(B=-5.414849\), \(C=8.560423\), \(D=3.427370\), \(E=-0.277375\) and \(\tau=T(\text{K})/1000\). Ignoring the heat flow that is needed to heat the water droplets to 100 \({}^{\circ}\)C and possibly heating the steam above 100 \({}^{\circ}\)C, the heat flow that is needed to evaporate the droplets at a rate of \(\dot{m}_{\text{Drops}}\) (kg/s) is given by:  
\[\dot{Q}_{\text{Max}}=\dot{m}_{\text{Drops}}\cdot\Delta H_{\text{vap}}\ \ \ (\text{W}), \tag{3}\]  
where the heat of vaporization at 100 \({}^{\circ}\)C, \(\Delta H_{\text{vap}}\), is 2571 kJ/kg.  
The droplet cooling rate as a function of temperature and the droplet cooling potential, \(\dot{Q}_{Max}\), for the test presented in Figure 4 are shown in Figure 6. The temperature of the highest droplet cooling rate, i.e., 124 \({}^{\circ}\)C, which corresponds to the critical heat flux, is marked in the figure. It is clearly seen that the cooling capacity does not reach \(\dot{Q}_{Max}\). This was simply caused by some droplets being lost from the disc before evaporating completely.  
The relative droplet cooling efficiency, i.e., \(\dot{Q}_{\text{Drops}}(T)/\dot{Q}_{\text{Max}}\), is shown in Figure 7. The cooling efficiency relative to heating the steam to the disc temperature is also shown in the figure.  
It should be noted that the cooling efficiency at disc temperatures below 100 \({}^{\circ}\)C is quite high. This is a result of droplet wetting, i.e., the remains of previous droplets spread out to make a quite substantial total wet surface area for film evaporation. This was also observed by [PERSON] et al. [14] in their study of droplets impinging onto metal surfaces at temperatures between 50-120 \({}^{\circ}\)C.  
Figure 5: Cooling rate, \(dT/dt\), for droplet application (**blue**) and reference recordings (**black**) based on temperature data from Figure 4.