# On the Variation of Cup Anemometer Performance Due to Changes in the Air Density
## 1 Introduction  
The cup anemometer (see Figure 1) is the most widely used wind measurement instrument used today in the meteorological and wind energy sectors. It was invented by [PERSON] in the mid-19 th century [1; 2; 3], although its design seems to have been suggested earlier [4]. This is an instrument that combines advantages in price, robustness, and economy in maintenance and calibration processes, which make it very competitive with other more modern and, in principle, more accurate measurement systems such as the LIDAR, SODAR, or the sonic anemometer. One of the main advantages is the experience accumulated in its use, and the research carried out on its characteristics. Leaving aside the quite large amount of research carried out on this instrument in the past [5], it can be said that research focused on cup anemometers has decreased today. However, some very interesting works have been recently published on this instrument, most of them related to its accuracy [6; 7; 8; 9; 10]. The fundamental equation that defines the motion of the rotor of a cup anemometer is the following:  
\[I\frac{\mathrm{d}\omega}{\mathrm{d}t}=Q_{A}-Q_{f}, \tag{1}\]  
where \(\omega\) is the angular velocity of rotation of the anemometer's rotor, and \(I\) is the moment of inertia of the rotor. On the other hand, \(Q_{A}\) is the torque produced by aerodynamic forces,and \(Q_{f}\) is the torque produced by the friction on the shaft that holds the cup anemometer rotor [11]. The aerodynamic torque, \(Q_{A}\), depends on the following:  
* The dynamic pressure, \(0.5\rho V^{2}\), which depends on both the air density, \(\rho\), and the air velocity with respect to the cup anemometer, \(V\);
* The ratio \(R_{c}/R_{rc}\), between the radius of the cups, \(R_{c}\), and the radius of rotation of the center of the cup, \(R_{rc}\);
* The cup's normal force coefficient, \(c_{N}\), [12]. There are different models to relate the aerodynamic moment, \(Q_{A}\), to the variables mentioned above [13; 14; 15]; however, these models have some limitations, such as that the cups' normal force coefficients, \(c_{N}\), are provided based on measurements performed on static cups [16] (that is, they are not rotating), and precisely because the cups of an anemometer are rotating means that the point of application of the aerodynamic force on them is not the same as in a static situation. However, for the study of the problem at hand, it is reasonable to assume a linear relationship between the air density, \(\rho\), and \(Q_{A}\). On the other hand, the friction torque, \(Q_{f}\), depends on the temperature, \(T\), and the rotational speed of the anemometer rotor, \(\omega\), by means of the following expression:  
\[Q_{f}=B_{0}+B_{1}\omega+B_{2}\omega^{2}, \tag{2}\]  
where \(B_{0}\), \(B_{1}\), and \(B_{2}\) are coefficients that depend on the temperature, \(T\)[17]. The effect of temperature on cup anemometers has been studied with interest by several researchers. However, emphasis has almost always been placed on the effect of friction [11; 18; 19] and on the accumulation of ice on the cups when this instrument operates in extremely cold conditions [20; 21; 22; 23; 24; 25; 26]. The effect of air density variation on cup anemometer performance has received less attention from the scientific community, probably because this is a second-order effect when this instrument is used at altitudes not far from the one where it was calibrated. Changes in air density taken into account independently from air temperature seem to be possible only in extreme weather conditions or specific geographic locations. Figure 1: Cup anemometer installed over a wind vane. The cups and the rotor of this instrument are indicated in the figure. Photo courtesy of the _Agencia Estatal de Meteorologia_ (AEMET), Spain. [[http://hdl.handle.net/20.500.11765/11425](http://hdl.handle.net/20.500.11765/11425)]([http://hdl.handle.net/20.500.11765/11425](http://hdl.handle.net/20.500.11765/11425)), accessed on 22 December 2023.