# On the Variation of Cup Anemometer Performance Due to Changes in the Air Density
## 1 Introduction
\[Q_{f}=B_{0}+B_{1}\omega+B_{2}\omega^{2}, \tag{2}\]0 Percentage difference between the rotational frequencies in air, \(f_{r,a}\), and water, \(f_{r,a}\), at the same fluid speed, \(V\), in relation to the rotational frequency in air, \(f_{r,a}\). These data were graphically extracted from the corresponding figure in the Deutsche WindGuard GmbH report [28]; see also \[Q_{f}=B_{0}+B_{1}\omega+B_{2}\omega^{2}, \tag{2}\]1  
If the data in Table 1 are assumed to correspond to a Thies Claima 4.3350 cup anemometer (this detail is not clear from references [28; 29], and could not be corroborated by the staff of Deutsche WindGuard GmbH. However, the reasoning stated hereafter can be applied to other First Class cup anemometer models such as any one of the Vector Instruments A100 series); the above-mentioned average calibration data of this instrument can be taken and, bearing in mind that these data were obtained with an air density \(\rho\sim 1.1\) kg\(\cdot\)m\({}^{-3}\) (corresponding to average meteorological conditions in the city of Madrid (Spain)), it is possible to estimate the output frequencies of the instrument corresponding to the mentioned Table 1, see \[Q_{f}=B_{0}+B_{1}\omega+B_{2}\omega^{2}, \tag{2}\]2  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\rho\) [kg\(\cdot\)m\({}^{-3}\)]} & \multicolumn{6}{c}{\(V\) [m\(\cdot\)s\({}^{-1}\)]} \\ \cline{2-9}  & **15.5** & **14** & **12** & **10** & **8** & **6** & **4** \\ \hline
0.855 & 0.98058 & 0.98643 & 0.98923 & 0.9909 & 0.99207 & 0.99286 & 0.99336 \\
0.916 & 0.98641 & 0.99009 & 0.99188 & 0.99305 & 0.99372 & 0.99422 & 0.99452 \\
0.976 & 0.98903 & 0.99224 & 0.99379 & 0.99479 & 0.9955 & 0.99596 & 0.99617 \\
1.038 & 0.99198 & 0.99436 & 0.99536 & 0.99607 & 0.99661 & 0.99674 & 0.99691 \\
1.100 & 0.99689 & 0.99748 & 0.99777 & 0.9981 & 0.99814 & 0.99831 & 0.99823 \\
1.161 & 0.99938 & 0.99942 & 0.99959 & 0.99955 & 0.99946 & 0.99951 & 0.99938 \\
1.227 & 1.00033 & 1.00017 & 1.00012 & 1.00021 & 1.00025 & 1.00021 & 1.00021 \\
1.294 & 1.00354 & 1.00228 & 1.00182 & 1.00161 & 1.00115 & 1.00116 & 1.00107 \\
1.352 & 1.00323 & 1.00252 & 1.00206 & 1.00168 & 1.00147 & 1.00139 & 1 From here, it is possible to estimate the variation of the calibration constants \(A\) and \(B\) with the variation of the density. These variations have been plotted in the graph in Figure 7 for the density values studied. It can be observed in this graph that the values of both constants have been fitted to a fourth-order polynomial curve. Based on these fits, the value of both constants can be extrapolated to study the variation of the velocity measured by the cup anemometer as a function of the measured output frequency. Figure 8 shows, as a function of the anemometer output frequency, \(f\), the relationship between the wind speed that should be measured with respect to that measured with the calibration performed with a density \(\rho\) = 1.1 kg\(\cdot\)m\({}^{-3}\), \(V/V_{\rho=1.1}\), for three lower air densities (\(\rho\) = 0.95 kg\(\cdot\)m\({}^{-3}\), \(\rho\) = 0.8 kg\(\cdot\)m\({}^{-3}\) y \(\rho\) = 0.7 kg\(\cdot\)m\({}^{-3}\)). It can be observed that for high frequencies the behavior is different than for low frequencies. For high frequencies, a decrease in density causes a decrease in the measured velocity, while for low frequencies, this tendency is reversed. We have chosen not to extrapolate the results using the settings shown in Figure 8 since for densities less than \(\rho\) = 0.7 kg\(\cdot\)m\({}^{-3}\), a greater decrease would imply zero or even negative values of the calibration constant \(B\) (Equation (3)).