V2 Derivation of wind turbine formula A derivation of the function R in Exercise 69, based on three equations from physics, is out- lined here. Consider again the figure given in Exercise 69, where vị equals the upstream velocity of the wind just before the wind stream encounters the wind turbine, and v, equals the downstream velocity of the wind just after the wind stream passes through the area swept out by the turbine blades. An equation for the power extracted by the rotor blades, based on conservation of momen- tum, is P = v*pA(, – v2), where v is the velocity of the wind (in m/s) as it passes through the turbine blades, p is the density of air (in kg/m³), and A is the area (in m²) of the circular region (in swept out by the rotor blades. a. Another expression for the power extracted by the rotor blades, based on conservation of energy, is P = pVA(v,² – v,²). 2 Equate the two power equations and solve for v. pA b. Show that P = " (v, + v,)(v,² – v,²). c. If the wind were to pass through the same area A without being disturbed by rotor blades, the amount of available power would pAv,3 be P, = -. Letr = 2 and simplify the ratio to obtain Po the function R(r) given in Exercise 69. (Source: Journal of Applied Physics, 105, 2009)
V2 Derivation of wind turbine formula A derivation of the function R in Exercise 69, based on three equations from physics, is out- lined here. Consider again the figure given in Exercise 69, where vị equals the upstream velocity of the wind just before the wind stream encounters the wind turbine, and v, equals the downstream velocity of the wind just after the wind stream passes through the area swept out by the turbine blades. An equation for the power extracted by the rotor blades, based on conservation of momen- tum, is P = v*pA(, – v2), where v is the velocity of the wind (in m/s) as it passes through the turbine blades, p is the density of air (in kg/m³), and A is the area (in m²) of the circular region (in swept out by the rotor blades. a. Another expression for the power extracted by the rotor blades, based on conservation of energy, is P = pVA(v,² – v,²). 2 Equate the two power equations and solve for v. pA b. Show that P = " (v, + v,)(v,² – v,²). c. If the wind were to pass through the same area A without being disturbed by rotor blades, the amount of available power would pAv,3 be P, = -. Letr = 2 and simplify the ratio to obtain Po the function R(r) given in Exercise 69. (Source: Journal of Applied Physics, 105, 2009)
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Chapter1: Units, Trigonometry. And Vectors
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Transcribed Image Text:V2
Transcribed Image Text:Derivation of wind turbine formula A derivation of the function
R in Exercise 69, based on three equations from physics, is out-
lined here. Consider again the figure given in Exercise 69, where
vị equals the upstream velocity of the wind just before the wind
stream encounters the wind turbine, and v, equals the downstream
velocity of the wind just after the wind stream passes through the
area swept out by the turbine blades. An equation for the power
extracted by the rotor blades, based on conservation of momen-
tum, is P = v*pA(, – v2), where v is the velocity of the wind
(in m/s) as it passes through the turbine blades, p is the density
of air (in kg/m³), and A is the area (in m²) of the circular region
(in
swept out by the rotor blades.
a. Another expression for the power extracted by the rotor blades,
based on conservation of energy, is P = pVA(v,² – v,²).
2
Equate the two power equations and solve for v.
pA
b. Show that P = " (v, + v,)(v,² – v,²).
c. If the wind were to pass through the same area A without being
disturbed by rotor blades, the amount of available power would
pAv,3
be P, =
-. Letr = 2 and simplify the ratio to obtain
Po
the function R(r) given in Exercise 69. (Source: Journal of
Applied Physics, 105, 2009)
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