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The influence of anemometer rotor shape parameters, such as the cups' front area or their center rotation radius on the anemometer's performance was analyzed. This analysis was based on calibrations performed on two different anemometers (one based on magnet system output signal, and the other one based on an opto-electronic system output signal), tested with 21 different rotors. The results were compared to the ones resulting from classical analytical models. The results clearly showed a linear dependency of both calibration constants, the slope and the offset, on the cups' center rotation radius, the influence of the front area of the cups also being observed. The analytical model of Kondo

Rotation anemometers, such as cup and propeller anemometers, are the most commonly used instruments for wind speed measurements. Thanks to their linearity and accuracy they are optimal for a large number of applications in the wind energy sector, from routine observations to field measurements. Cup anemometers have been widely studied since the first half of the twentieth century, with the early works devoted to studying the optimal number of cups and arm length [

In a previous study at the IDR/UPM Institute [_{r}_{r}_{p}

This research, performed on more than 20 models of commercial cup anemometers, showed a linear correlation between the coefficients A_{r}_{rc}_{r}_{rc}_{rc}_{r}_{rc}^{2} = 0.753 instead of ^{2} = 0.485, in the previous fitting) leaving aside some anemometers, those that have very different shape from the others, when calculating the linear fitting. Finally, a brief calculation using the 2-cup analytical method showed a very close result to the mentioned fitting, A_{r}_{rc}_{rc}_{rc}_{c}

This simple model was used in the past to study the aerodynamics of the cup anemometers' rotor (see a sketch of the model in the _{A}_{f} is the frictional torque. The frictional torque, _{f}, can be neglected as it is usually (for wind speeds larger than 1 m·s^{−1}) very small in comparison to the aerodynamic torque, _{A}_{1} = _{2}, where _{1} and _{2} are the aerodynamic forces on the cups (see _{rc}_{c}_{d}_{1} and _{d}_{2} are the drag coefficients of the cups, respectively at 0° and 180° regarding the wind direction (the wind direction angle with respect to the cup is indicated in

The ratio between the wind speed, _{rc}

From an early study by Patterson, this factor was found to be between 2.5 and 3.5 [_{d}_{2}/_{d}_{1}.

Ramachandran [_{N}_{r}_{r}

Ramachandran made two important assumptions in his calculations:

Taking into account the aforementioned ratios between the wind speed and the rotation speed (that is, the anemometer factor,

The anemometer's behavior in steady state can be obtained from _{m}_{m}

As mentioned, in order to calculate the last expression Ramachandran suggested the use of Brevoort and Joyner results [_{N}_{N}_{r}_{rc}_{rc}_{r}_{rc}

Kondo _{rc}, α

Kondo _{N}_{r}_{rc}_{rc}

The aim of the present work is to analyze the correlation between the cups' rotor dynamics in steady state (that is, the anemometer's transfer function) and the rotor's shape (more specifically, the cups center rotation radius, _{rc}_{c}

Two anemometers, Climatronics 100075 (also known as F460 model, by Climatronics Corp.: Bohemia, New York, USA), and Ornytion 107A (Ornytion: Bergondo, A Coruña, Spain) were used in the testing campaign (see _{c}_{rc}

The calibrations were carried out at the IDR/UPM Institute, in the S4 wind tunnel. This facility is an open-circuit wind tunnel with a closed test section measuring 0.9 by 0.9 m. It is served by four 7.5 kW fans with a flow uniformity under 0.2% in the testing area. More details concerning the facility and the calibration process are included in references [^{−1} wind speed).

The aerodynamic forces on the cups were measured in the wind tunnel of the Department of Mechanical Engineering of the Vrije Universiteit Brussel (Belgium). This facility is also an open-circuit wind tunnel with a 2 by 1 m closed test section. The facility is served by a 55 kW centrifugal fan. The testing section is equipped with a 6-component balance made by TEM Engineering Limited (Sussex, UK). Three larger-scale cups (0.2 m diameter) were manufactured to measure the aerodynamic forces. These cups were also made in a 3D printer of ABS plastic, and they are an exact scale-replica of the rotors' ones. As stated in the introduction, two different tests were carried out. In the first one, the forces on an isolated cup were measured by varying the wind direction, whereas in the second one the cup was surrounded by the other two in order to better simulate the anemometer's rotor (see ^{−1} wind speed. 12,000 samples were taken at 50 Hz in each measurement. The forces were made non dimensional with the dynamic pressure directly measured by a BnC-Lambrecht 630a (Goettingen, Germany) pitot tube located at the ceiling of the testing chamber, upstream to the point where the models are allocated, and connected to a SETRA Model 239 (Boxborough, MA, USA) differential pressure sensor.

The calibration results, together with the characteristics of each tested rotor, are included in _{r}_{rc}_{r}^{2}. In _{r}_{r}_{rc}_{rc}_{r}_{0}, are different from one case to another. However, once divided by the cups' front area, _{c}_{r}

The same analysis can be performed on the anemometers' transfer function offset, B. Let's assume that this coefficient has a linear behavior with _{rc}

In _{rc}_{r}_{rc}_{0}, are quite different from one fitting to the other one. Nonetheless, it is possible to find some fitting to describe them as a function of the other important parameter of the rotors' shape, the cups' front area, _{c}

The previous analysis illustrates the relation between both calibration constants, A_{r}_{rc}_{c}^{−1}, is shown in _{r}

To compare these results with the ones from the classical models it must be taken into account that the anemometer's factor depends on both calibration constants:

However, it can be accurate enough to leave aside the second term of this expression, assuming an average deviation up to 8.6% (based on the calibration results in ^{−1} wind speed). The results are then _{r}_{rc}_{r}_{rc}_{r}_{rc}_{r}_{rc}_{r}_{rc}

The results of the normal aerodynamic force coefficient measured on a cup are included in _{rc}_{mz}_{c}_{rc}

The use of these experimental results in combination with the described analytical models gives more accurate values of _{r}_{rc}

In _{N}_{r}_{rc}

In the present study, the influence of the rotor's shape (cups' front area, _{c}_{rc}

The major conclusions resulting from this work are:

Both calibration constants, A_{r}_{rc}_{r}_{r}_{rc}_{r}_{0}, seems to depend only on the cups' front area, _{c}_{r}_{rc}_{r}_{0}, seem to depend only on the cups' front area.

The slope of an anemometer's transfer function, that is, the calibration constant A_{r}

The authors are indebted to Enrique Vega, Alejandro Martínez, Pedro López, Luis García and Eduardo Cortés, from the IDR/UPM Institute for their friendly help and support in the present research. This work was done in collaboration with the Department of Mechanical Engineering of the Vrije Universiteit Brussel. In this sense, the authors are truly indebted to Chris Lacor and Alain Wery, for the opportunity to carry out part of the testing campaign in Brussels, and also for their friendly support. The authors would also like to thank Angel Sanz for sharing his ideas and giving his encouraging support. The authors are indebted to Alfonso Rosende from Ornytion for his constant and kind support for the authors' research. Finally, the authors are grateful to Brian Elder and Tania Tate for their kind help in improving the style of the text.

2-cup anemometer model: anemometer factor (see Section 1.1), _{d}_{1}, and at 180° wind angle, _{d}_{2}.

Normal aerodynamic force coefficient, _{N}

Difference between the local wind angle with respect to the cup (commonly known as angle of attack),

Normal aerodynamic force coefficient, _{N}

Ornytion 107A (

Cups sets correspondent to the 50/60 (

Both cup configurations measured in the wind tunnel of the Vrije Universiteit Brussel: isolated cup (

Calibration coefficients, A_{r}_{rc}

Calibration coefficients, A_{r}_{rc}

Offset of the linear fittings from _{r}_{0}/_{c}_{c}

Calibration coefficients, B, as a function of the cups' center rotation radius, _{rc}

Slope and offset (divided by the squared front area of the cups) of the linear fittings from _{c}

Normal aerodynamic force coefficient, _{N}_{mz}_{c}_{rc}_{c}_{rc}

Standard deviation of the normal force on the cup divided by the mean value of this force, _{N}

Results of the calibration performed on the Climatronics 100075 and Ornytion 107A anemometers. The calibration constants, A, B and A_{r}_{c}_{rc}

| |||||
---|---|---|---|---|---|

_{c} |
_{rc} |
_{r} | |||

| |||||

40/40 | 40 | 40 | 0.0310 | 0.2593 | 0.9295 |

40/50 | 40 | 50 | 0.0420 | 0.3010 | 1.2592 |

40/60 | 40 | 60 | 0.0518 | 0.3562 | 1.5526 |

50/40 | 50 | 40 | 0.0293 | 0.2867 | 0.8777 |

50/60 | 50 | 60 | 0.0495 | 0.2567 | 1.4850 |

50/80 | 50 | 80 | 0.0697 | 0.3387 | 2.0909 |

50/100 | 50 | 100 | 0.0890 | 0.5447 | 2.6692 |

60/40 | 60 | 40 | 0.0279 | 0.1900 | 0.8361 |

60/60 | 60 | 60 | 0.0481 | 0.1559 | 1.4425 |

60/80 | 60 | 80 | 0.0682 | 0.2167 | 2.0464 |

60/100 | 60 | 100 | 0.0866 | 0.3731 | 2.5991 |

60/120 | 60 | 120 | 0.1064 | 0.4731 | 3.1922 |

70/60 | 70 | 60 | 0.0461 | 0.1642 | 1.3833 |

70/80 | 70 | 80 | 0.0674 | 0.1754 | 2.0210 |

70/100 | 70 | 100 | 0.0873 | 0.2003 | 2.6200 |

70/120 | 70 | 120 | 0.1067 | 0.2997 | 3.1997 |

80/60 | 80 | 60 | 0.0454 | 0.1376 | 1.3633 |

80/80 | 80 | 80 | 0.0653 | 0.1864 | 1.9601 |

80/100 | 80 | 100 | 0.0859 | 0.2221 | 2.5781 |

80/120 | 80 | 120 | 0.1052 | 0.2539 | 3.1557 |

80/140 | 80 | 140 | 0.1248 | 0.3040 | 3.7455 |

| |||||

| |||||

_{c} |
_{rc} |
_{r} | |||

| |||||

40/40 | 40 | 40 | 0.4809 | 0.2739 | 0.9617 |

40/50 | 40 | 50 | 0.6396 | 0.3707 | 1.2792 |

40/60 | 40 | 60 | 0.7827 | 0.5387 | 1.5654 |

50/40 | 50 | 40 | 0.4477 | 0.1479 | 0.8954 |

50/60 | 50 | 60 | 0.7584 | 0.3513 | 1.5168 |

50/80 | 50 | 80 | 1.0571 | 0.5058 | 2.1141 |

50/100 | 50 | 100 | 1.3489 | 0.6509 | 2.6978 |

60/40 | 60 | 40 | 0.4313 | 0.0702 | 0.8625 |

60/60 | 60 | 60 | 0.7363 | 0.1824 | 1.4727 |

60/80 | 60 | 80 | 1.0397 | 0.2957 | 2.0795 |

60/100 | 60 | 100 | 1.3143 | 0.4646 | 2.6285 |

60/120 | 60 | 120 | 1.6139 | 0.5543 | 3.2278 |

70/60 | 70 | 60 | 0.7074 | 0.1848 | 1.4148 |

70/80 | 70 | 80 | 1.0275 | 0.2796 | 2.0549 |

70/100 | 70 | 100 | 1.3173 | 0.2182 | 2.6347 |

70/120 | 70 | 120 | 1.6022 | 0.3444 | 3.2044 |

80/60 | 80 | 60 | 0.6794 | 0.1500 | 1.3588 |

80/80 | 80 | 80 | 0.9936 | 0.1902 | 1.9873 |

80/100 | 80 | 100 | 1.3001 | 0.2174 | 2.6003 |

80/120 | 80 | 120 | 1.5906 | 0.2359 | 3.1812 |

80/140 | 80 | 140 | 1.8830 | 0.3191 | 3.7659 |

Linear fittings (slope, _{r}_{rc}_{r}_{0}, and correlation coefficient, ^{2}) of calibration constants A_{r}_{rc}_{c}_{c}

| |||||
---|---|---|---|---|---|

_{c} |
_{c}^{2}] |
_{r}_{rc} |
_{r}_{0} |
^{2} | |

| |||||

40/40, 40/50, 40/60 | 40 | 1,256.6 | 3.116E–02 | –3.107E–01 | 0.99888 |

50/40, 50/60, 50/80, 50/100 | 50 | 1,963.5 | 2.990E–02 | –3.125E–01 | 0.99986 |

60/40, 60/60, 60/80, 60/100, 60/120 | 60 | 2,827.4 | 2.934E–02 | –3.243E–01 | 0.99974 |

70/60, 70/80, 70/100, 70/120 | 70 | 3,848.5 | 3.024E–02 | –4.157E–01 | 0.99953 |

80/60, 80/80, 80/100, 80/120, 80/140 | 80 | 5,026.5 | 2.980E–02 | –4.194E–01 | 0.99989 |

| |||||

| |||||

_{c} |
_{rc} |
_{r}_{rc} |
_{r}_{0} |
^{2} | |

| |||||

40/40, 40/50, 40/60 | 40 | 1,256.6 | 3.019E–02 | –2.406E–01 | 0.99911 |

50/40, 50/60, 50/80, 50/100 | 50 | 1,963.5 | 3.002E–02 | –2.955E–01 | 0.99980 |

60/40, 60/60, 60/80, 60/100, 60/120 | 60 | 2,827.4 | 2.943E–02 | –3.003E–01 | 0.99968 |

70/60, 70/80, 70/100, 70/120 | 70 | 3,848.5 | 2.974E–02 | –3.497E–01 | 0.99923 |

80/60, 80/80, 80/100, 80/120, 80/140 | 80 | 5,026.5 | 3.004E–02 | –4.254E–01 | 0.99970 |

Linear fittings (slope, _{rc}_{0}, and correlation coefficient, ^{2}) of calibration constants B as a function of the cups center rotation radius, _{rc}_{c}_{c}

| |||||
---|---|---|---|---|---|

_{c} |
_{c}^{2}] |
_{rc} |
_{0} |
^{2} | |

| |||||

40/40, 40/50, 40/60 | 40 | 1,256.6 | 4.8445E–03 | 6.3273E–02 | 0.99370 |

50/40, 50/60, 50/80, 50/100 | 50 | 1,963.5 | 4.2794E–03 | 5.7171E–02 | 0.72444 |

60/40, 60/60, 60/80, 60/100, 60/120 | 60 | 2,827.4 | 3.9178E–03 | 2.0341E–02 | 0.83575 |

70/60, 70/80, 70/100, 70/120 | 70 | 3,848.5 | 2.1567E–03 | 1.5786E–02 | 0.81394 |

80/60, 80/80, 80/100, 80/120, 80/140 | 80 | 5,026.5 | 2.0019E–03 | 2.0607E–02 | 0.99374 |

| |||||

| |||||

_{c} |
_{rc} |
_{rc} |
_{0} |
^{2} | |

| |||||

40/40, 40/50, 40/60 | 40 | 1,256.6 | 1.3239E–02 | –2.6754E–01 | 0.97653 |

50/40, 50/60, 50/80, 50/100 | 50 | 1,963.5 | 8.3167E–03 | –1.6821E-01 | 0.99334 |

60/40, 60/60, 60/80, 60/100, 60/120 | 60 | 2,827.4 | 6.2523E–03 | –1.8674E–01 | 0.99262 |

70/60, 70/80, 70/100, 70/120 | 70 | 3,848.5 | 2.0861E–03 | 6.9016E-02 | 0.58547 |

80/60, 80/80, 80/100, 80/120, 80/140 | 80 | 5,026.5 | 1.9197E–03 | 3.0577E–02 | 0.93054 |

Percentage variations of the anemometer transfer function slope and offset, A_{r}^{−1} wind speed are also included in the table.

| ||||||
---|---|---|---|---|---|---|

_{r} |
_{r} |
|||||

| ||||||

40/40 | −3.47% | 4.93% | −3.16% | −0.83% | 92.54% | 2.82% |

40/50 | −4.92% | 7.59% | −4.38% | −1.99% | 77.80% | 2.24% |

40/60 | −3.57% | 5.49% | −3.11% | −0.75% | 46.84% | 2.91% |

50/40 | −0.72% | −29.27% | −1.89% | 1.46% | 132.63% | 4.23% |

50/60 | −0.92% | 10.13% | −0.51% | −0.55% | 46.92% | 1.83% |

50/80 | −0.93% | 7.08% | −0.55% | −0.27% | 36.06% | 2.36% |

50/100 | 0.08% | −18.74% | −1.38% | 0.40% | 32.16% | 3.35% |

60/40 | 1.21% | −16.50% | 0.73% | 0.31% | 211.19% | 2.43% |

60/60 | 0.25% | 42.76% | 1.20% | −0.50% | 79.73% | 1.59% |

60/80 | −0.01% | 32.19% | 0.98% | −0.68% | 47.77% | 1.36% |

60/100 | 1.81% | −6.06% | 1.39% | 1.40% | 17.58% | 2.47% |

60/120 | 1.69% | −12.41% | 0.74% | 1.16% | 18.25% | 2.51% |

70/60 | 2.72% | 9.90% | 2.89% | 0.61% | 2.86% | 0.67% |

70/80 | 0.00% | 32.78% | 0.82% | −1.53% | −9.35% | −1.84% |

70/100 | 0.03% | 42.43% | 1.25% | −0.43% | 45.23% | 1.00% |

70/120 | 0.66% | 12.71% | 1.18% | 0.59% | 10.40% | 1.08% |

80/60 | 2.35% | 8.49% | 2.47% | 1.76% | −43.43% | 0.80% |

80/80 | 1.80% | 3.59% | 1.85% | −0.23% | −40.53% | −1.32% |

80/100 | 0.67% | 6.60% | 0.86% | −0.67% | −34.94% | −1.74% |

80/120 | 1.26% | 10.49% | 1.59% | 0.05% | −28.06% | −0.90% |

80/140 | 1.33% | 6.65% | 1.56% | 0.45% | −37.96% | −1.30% |

Anemometer factor, _{r}_{r}_{rc}

| ||
---|---|---|

_{r}_{rc} | ||

| ||

2-cup model [ |
3.88 | 0.0244 |

Ramachandran [ |
2.64 | 0.0166 |

Kondo |
3.5 | 0.022 |

| ||

| ||

_{r}_{rc} | ||

| ||

2-cup model | 8.39 | 0.0527 |

Ramachandran | 3.37 | 0.0212 |

Kondo |
4.8 | 0.0302 |

| ||

| ||

_{r}_{rc} | ||

| ||

2-cup model | 9.51 | 0.0598 |

Ramachandran | 3.46 | 0.0218 |

Kondo |
4.98 | 0.0313 |

Kondo |
4.75 | 0.0298 |

| ||

| ||

_{r}_{rc} | ||

| ||

Based on calibrations on commercial anemometers [ |
3.02 | 0.019 |

Present research | 4.77 | 0.03 |