# On the Variation of Cup Anemometer Performance Due to Changes in the Air Density
## 1 Introduction
If we take the average results measured in the wind tunnel of a high precision cup anemometer such as the Thies Clima 4.3350: \(A=0.0483\) m y \(B=0.248\) m\(\cdot\)s\({}^{-1}\), and taking into account the annual average air density at the LAC-IDR/UPM laboratory where these calibrations were performed, \(\rho=1.096\) kg\(\cdot\)m\({}^{-3}\), and the number of pulses per revolution of that anemometer, \(N_{p}=37\), it is possible to represent the calibration curve in the same terms as those used in the graph in Figure 2. Figure 3 shows a comparison between the calibration curve represented by the constants indicated above and the results of the air flow meter tests obtained by [PERSON] and [PERSON]. The comparison has been made with the data of revolutions per foot dimensioned with the number of turns per foot for \(V\rho^{0.5}=12\) (expressed in United States customary units). This graph shows the great similarity between the two curves. Figure 2: **(Top)** Current meter tested with water and air flow by [PERSON] and [PERSON] [27]. (**Bottom**) Some of the results of the testing campaign. Revolutions per foot, \(f_{r}/V\), in relation to the flow speed, \(V\), multiplied by the square ratio of the flow density, \(\rho\). Reprinted with permission from the National Institute of Standards and Technology, U.S. Department of Commerce. Not copyrightable in the United States. Once it has been established that the calibration curve is essentially composed of the points graphed by [PERSON] and [PERSON] (Figure 2) [27], it is possible to transfer the points of the graph of this figure to some graphs of what is today a cup anemometer calibration curve. Figure 4 shows these calibration curves (fluid velocity in m-s\({}^{-1}\) represented in relation to the frequency of rotation of the instrument, \(f_{r}\), measured in both air and water). In both graphs, the equations of the linear regression fitted to the data have been included. It is quite striking that the slopes of both curves are so similar. This means that the rotational speed of this instrument's rotor at high speeds will be practically the same regardless of whether the fluid is air or water, whose density is much higher. However, it can be seen in the graph that, at low fluid velocities, the response of the instrument changes, becoming much lower in the case of the lower-density fluid; see Figure 5 for the percentage difference between the rotational frequencies in air, \(f_{r,a}\), and water, \(f_{r,w}\), at the same fluid speed, \(V\), in relation to the rotational frequency in air, \(f_{r,a}\). This is mentioned in the work of [PERSON] and [PERSON]. Furthermore, these authors warn that this effect will be more relevant the lower the density of the fluids being compared. As described in the first section of this article, thanks to the work of Deutsche Wind-Guard GmbH in 2018 [28], and later published in 2023 [29], information is available on the performance of a cup anemometer operating with air of different densities at a constant temperature. Table 1 shows the plot data relating the performances of a cup anemometer as a function of air density [28,29]. The performances are analyzed with the variable \(k\), defined as the rotational frequency divided by the fluid velocity divided by the same ratio measured for an air temperature \(T^{*}=20\)\({}^{\circ}\)C and an atmospheric pressure \(P^{*}=1000\) hPa:  
\[k=\frac{f/V|_{T;P}}{f/V|_{T^{*};P*}}. \tag{5}\]  
Figure 3: Non-dimensional rotation rate per length unit, \(f_{r}/V^{*}\), of the Thies Clima 4.3350 cup anemometer and the flow meter tested by [PERSON] and [PERSON] [27], in relation to the product of wind speed and the square ratio of the air density, \(V_{P}\)\({}^{0.5}\). Figure 4: Data points from the graph included in Figure 2 (**Top**: Air tests; **Bottom**: Water rating after air tests), expressed in terms of fluid velocity, \(V\), in relation to the instrument’s rotation frequency, \(f_{r}\).