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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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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">''sin''θ</span>. As such, the wind pressure calculation is multiplied by <span class="texhtml">''A'' * ''sin''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> 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">''cos''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> 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'' * ''cos''θ</span>.
<br> Therefore, the combined calculation to determine the force balance on the blades is:
<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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<span class="texhtml">''F'' = ρ * ''v''<sup>2</sup> * ''A'' * ''sin''θ * ''cos''θ * ''D'' * ''cos''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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<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">''sin''θ * ''cos''θ * ''cos''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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As a result, the optimal tilt of the blades would provide an angle to the airflow such that <span class="texhtml">''sin''θ * ''cos''θ * ''cos''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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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.
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.