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=== <br>'''Angle of blade and the resulting forces to spin the blades versus surface area exposed''' ===
=== <br>'''Angle of blade and the resulting forces to spin the blades versus surface area exposed''' ===
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<br> <br> The angle that the windmill blades are tilted compared to the stream of fluid will determine how much energy can be converted into rotational motion and then be captured by the system for meaningful work. The amount of force is calculated by finding the wind pressure.
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The angle that the windmill blades are tilted compared to the stream of fluid will determine how much energy can be converted into rotational motion and then be captured by the system for meaningful work. The amount of force is calculated by finding the wind pressure.
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<br> The wind pressure exerted by the wind is given by: <span class="texhtml">''P'' = 1 / 2(1 + ''c'') * ρ * ''v''<sup>2</sup></span>
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The wind pressure exerted by the wind is given by:
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*where c is a constant and equals 1.0 for long flat plates.
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<math> P = 1/2 (1 + c) * \rho * v^2 </math>
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*where c is a constant and equals 1.0 for long flat plates.
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<br>
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The force of the wind against the windmill blade is based on the wind pressure multiplied by the area of the blade facing the oncoming flow. In the event that the blade is tilted at an angle to the oncoming airstream, then the area of the blade exposed to the fluid is reduce by a factor of <span class="texhtml">''s'''i'''n''θ</span>. As such, the wind pressure calculation is multiplied by <span class="texhtml">''A'' * ''s'''i'''n''θ</span> to obtain the force of the wind on the blades
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<br> In addition, the force of the wind converted into rotational motion is related to the angle of the blade in relationship to the oncoming fluid flow. This relationship is given by a factor of <span class="texhtml">''c'''o'''s''θ</span>.
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<br> Furthermore, the blades will encounter a drag coefficient related to the angle of the blades as they rotate in their own axis perpendicular to the oncoming flow of fluid. This drag coefficient will be represented by <span class="texhtml">''D'' * ''c'''o'''s''θ</span>.
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<br> Therefore, the combined calculation to determine the force balance on the blades is:
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<span class="texhtml">''F'' = ρ * ''v''<sup>2</sup> * ''A'' * ''s''</span><span class="texhtml">'''''i'''n''θ * ''c'''o'''s''θ * ''D'' * ''c'''o'''s''θ</span>
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<br> An important relationship to note is that between force and <span class="texhtml">θ</span>. The combined force balance indicates a relationship between force and <span class="texhtml">''s'''i'''n''θ * ''c'''o'''s''θ * ''c'''o'''s''θ</span>.
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As a result, the optimal tilt of the blades would provide an angle to the airflow such that <span class="texhtml">''s'''i'''n''θ * ''c'''o'''s''θ * ''c'''o'''s''θ</span> is a maximum. This value has been presented in the graph below to show how the value changes as <span class="texhtml">θ</span> is adjusted. <br>
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<br>
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The force of the wind against the windmill blade is based on the wind pressure multiplied by the area of the blade facing the oncoming flow. In the event that the blade is tilted at an angle to the oncoming airstream, then the area of the blade exposed to the fluid is reduce by a factor of <math>sin \theta </math>. As such, the wind pressure calculation is multiplied by <math>A * sin \theta </math> to obtain the force of the wind on the blades
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[[Image:Blade Angle cos cos sin.jpg|left|575x340px]]
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In addition, the force of the wind converted into rotational motion is related to the angle of the blade in relationship to the oncoming fluid flow. This relationship is given by a factor of <math>cos \theta </math>.
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<br>
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<br>
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Furthermore, the blades will encounter a drag coefficient related to the angle of the blades as they rotate in their own axis perpendicular to the oncoming flow of fluid. This drag coefficient will be represented by <math>D * cos \theta </math>.
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<br>
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Therefore, the combined calculation to determine the force balance on the blades is:
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<br>
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<math> F = \rho * v^2 * A * sin \theta * cos \theta * D * cos \theta </math>
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<br>
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An important relationship to note is that between force and <math> \theta </math>. The combined force balance indicates a relationship between force and <math>sin \theta * cos \theta * cos \theta</math>.
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As a result, the optimal tilt of the blades would provide an angle to the airflow such that <math>sin \theta * cos \theta * cos \theta</math> is a maximum. This value has been presented in the graph below to show how the value changes as <math>\theta </math> is adjusted.
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The angle is adjusted in radians and seems to indicate a maximum value at approximately 0.62 radians, or roughly 35.5 degrees. This translates in a maximum conversion of 38.5% of the wind force into rotational motion. Therefore, the blades should be tilted at an angle of roughly 35.5 degrees from the oncoming air stream to obtain the optimal amount of energy using flat blade windmills.
= '''-- Regional Considerations --''' =
= '''-- Regional Considerations --''' =