# How Does Wind Turbine Blade Length Affect Power Output?

• shk
In summary, a do-it-yourself company is advertising a wind turbine with a power output of 1 kW and a swept area of 2.4 m2. At a wind speed of 12.5 m s–1, the blades have a length of 0.9 m and an angular velocity of 63 rad s–1. The theoretical power available is 3 kW, but the actual power is less due to factors such as air hitting at an angle and energy being lost to heat and sound. To prevent damage to a chimney stack, the turbine is designed to cut out at a wind speed of 14 m s–1, which would result in a force of 610 N on the blades.
shk
[Note: Thread moved to homework forum by a moderator. shk, please post all homework questions in the homework forums in the future and be sure to use the template provided when creating a new thread therein.]

A do-it-yourself company is advertising a wind turbine that they state can deliver a power of 1 kW. Their specification provides the following data: • area swept out by the blades in one revolution = 2.4 m2 • power output = 1 kW at a wind speed of 12.5 m s–1 • typical operating speed of blades = 600 revolutions per minute (a) (i) Show that the length of each blade is approximately 0.9 m.
(ii) Show that the angular velocity of the blades at the typical operating speed is approximately 63 rad s–1.
(iii) Calculate the speed at which the tips of the blades will then be travelling.
(b) The theoretical power available from a wind turbine is given by p = ½ ρAv3 where ρ = density of air = 1.3 kg m–3 A = area swept out by blades per revolution v = wind speed (i) Show that when the wind speed is 12.5 m s–1, the theoretical power from the advertised turbine is about 3 kW.
(ii) Suggest two reasons why the actual power is less than the theoretical power.
(c) The manufacturer has to ensure that when the turbine is attached to a chimney stack, the force exerted on the chimney does not cause it to collapse. The turbine is designed to cut out at a wind speed of 14 m s–1. (i) Calculate the mass of air hitting the blades each second when the wind speed is 14 m s–1.
(ii) Hence calculate the maximum force that the wind could exert on the blades
(a) (i) Use of A = πr 2 leading to 0.87 (m)
(ii) Correct use of ω = 2π/t leading to 62.8 (rad s–1)
(iii) Correct use of v = rω = 55 m s–1 [allow use of show that value]
( b) (i) Substitution into p = ½ ρAv3 (1) 3047 (W)
(ii) Air is hitting at an angle/all air not stopped by blades ,Energy changes to heat and sound
(c) (i) Attempts to find volume per second (A × v) (1) 44 kg s–1
(ii) Use of F = ∆mv/∆t (1) F = 610 N (1)

my problem is part (C)
c)i) my answer is V=Axv= 2.4x14=33.6
c)ii) F=∆mv/∆t = 33.6x14=470.4 N
as I think the density of air is 1 therefore m=V(volume)
my answeres for this part are very different from the mark scheme !
I need help wih this.
thanks

shk said:
I think the density of air is 1
shk said:
where ρ = density of air = 1.3 kg m–3
470 × 1.3 = 611.

shk
kuruman said:
Here is your problem.470 × 1.3 = 611.
many thanks Kuruman . That really helped

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