## 6 Application in Hot Wire Calibration
### HotWire Calibration
The root mean square error \(\sigma_{V}\) and the relative error \(\sigma_{V}/V_{\text{avg}}\) of the 100 velocity values can be further calculated by using the data processing formulas Equations (7) and (8) in Section 5.1. In fact, the root mean square error \(\sigma_{V}\) and the relative error \(\sigma_{V}/V_{\text{avg}}\) obtained here from the calibration data have little difference from those listed in Tables 4 and 5 obtained in the adjustment test of velocity control system (see Section 5.1). At the same time the output of voltage value \(E\) from the hot-wire anemometer was synchronously acquired at the same acquisition frequency of 1000 Hz for 6 s with a total of 6000 times. The voltage values were averaged every 60 ms to obtain 100 mean voltage data. These 100 mean voltage values \(E_{i}\) (\(i=1\) to 100) were averaged again to obtain average voltage \(E_{\text{avg}}\) corresponding to the average velocity \(V_{\text{avg}}\). The root mean square error \(\sigma_{E}\) and the relative error \(\sigma_{E}/E_{\text{avg}}\) of 100 voltage values can be further calculated as  
\[E_{\text{avg}}=\frac{\sum\limits_{i=1}^{n}E_{i}}{n}, \tag{19}\]  
\[\sigma_{E}=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(E_{i}-E_{\text{avg}}\right) ^{2}}{n-1}}, \tag{20}\]  
The 10 average velocity values (\(V_{\text{avg}}\)) and the 10 corresponding voltage values (\(E_{\text{avg}}\)) form a discrete functional relationship \(E\) vs. \(U\) (\(U\) represents the velocity \(V\)). The relationship can be drawn into a curve, i.e., fitted to an analytical continuous curve. The best known and most commonly used fitting function is the modified King's law, i.e.,  
\[E^{2}=a+bU^{n}, \tag{21}\]  
where \(a\), \(b\), and \(n\) are constants. A calibration can also be regarded as finding the value of the constants \(a\), \(b\), and \(n\) for each fitting curve under different temperature and humidity values. Figure 18 presents the calibrated velocity points and corresponding fitting curves fitted by modified King's law formula at 3 different humidity values \(H=20\%\)RH, 60%RH, and 80%RH, respectively, under the same temperature \(T=30\ ^{\circ}\)C. Figure 19 shows the calibrated velocity points and the corresponding fitting curves at 3 different humidity values \(H=20\%\)RH, 60%RH, and 80%RH, respectively, under the same temperature \(T=40\ ^{\circ}\)C. Figure 20 gives the velocity points and fitting curve under temperature \(T=60\ ^{\circ}\)C and humidity \(H=19\%\)RH. The humidity \(H=19\%\)RH is a value that naturally evolved from the  
Figure 17: The hot-wire anemometer, its probe, and the probe fixed on the strut in the enclosed chamber. ambient atmospheric humidity value as the temperature was changed from atmospheric value to 60 \({}^{\circ}\)C without being humidified or dehumidified. The figures reveal that the fitting curves fit the calibrated velocity points very smoothly. Table 10 summarizes the values of constants \(a\), \(b\), and \(n\) of the fitting curves at different humidity values under three temperatures. The values of \(n\) are between 0.5578 and 0.7074,  
Figure 19: Calibrated velocity points and fitting curves for \(H\) = 20%RH, 60%RH, and 80%RH under \(T\) = 40 \({}^{\circ}\)C.  
Figure 20: Calibrated velocity points and fitting curve under \(T\) = 60 \({}^{\circ}\)C, \(H\) = 19%RH. which agrees with the statement of Ref. [13] that \"if a calibration is restricted to the range U \(<\) 1 m/s, values of \(n\) in the range from 0.5 to 1 would be more appropriate\".