In Chapter 3 we learned that the force of air drag on an object moving rapidly through the air is proportional to v2, where v is the speed of the ball relative to the air. The same relation applies to the force of the wind on the blades of a wind turbine (Figure 9.4), where v is the speed of the wind. The angular velocity of the turbine is also proportional to the wind speed.
(a) Show that the mechanical power exerted by the wind on a turbine varies as Pvá and find the value of á .
P is proportional to V to the power of alpha". In otherwords P~V^a, which means that P = K * V^a, where K is some number that we don't know, but it does not depend on velocity.
(b) At some promising sites for wind energy, the wind speed averages about 51 mph. If a wind turbine at this site produces 9 MW with a wind speed of 51 mph, what power would it produce when the wind speed is 49 mph? Your answer should show why it is important to place wind turbines at places where the wind speed is largest.
... I am completely lost and have no idea where to even start.. Please help
To solve this problem, we need to use the given information that the force of air drag on an object moving rapidly through the air is proportional to v^2, and the same relation applies to the force of the wind on the blades of a wind turbine.
(a) To show that mechanical power exerted by the wind on a turbine varies as Pv^a, we need to find the value of α.
First, let's define some variables:
P = mechanical power exerted by the wind on the turbine.
V = speed of the wind.
According to the given information, we know that P is proportional to V^α. So, we can write it as:
P = K * V^α, where K is a constant.
Now, let's consider the relationship between the force of air drag and the speed of the ball relative to the air. We know that the force of air drag is proportional to v^2, where v is the speed of the ball relative to the air.
The force of air drag can be expressed as:
F_drag = -C * v^2, where C is a constant.
However, the force exerted by the wind on the turbine is in the opposite direction to the motion of the blades, so we can write it as:
F_wind = -C * V^2.
The mechanical power exerted by the wind on the turbine can be calculated as the product of force and velocity:
P = F_wind * V = -C * V^2 * V = -C * V^3.
Comparing this to the expression P = K * V^α, we can see that α = 3.
Therefore, the mechanical power exerted by the wind on a turbine varies as P = K * V^3.
(b) To find the power produced by the wind turbine when the wind speed is 49 mph, we can use the power produced when the wind speed is 51 mph as a reference.
Given:
Power at 51 mph (P1) = 9 MW
Wind speed at 51 mph (V1) = 51 mph
Wind speed at 49 mph (V2) = 49 mph
Using the proportionality equation P = K * V^3, we can write:
P1 = K * V1^3
9 MW = K * (51 mph)^3
Now, we can find the value of K:
K = 9 MW / (51 mph)^3
Finally, we can calculate the power at a wind speed of 49 mph (P2):
P2 = K * V2^3
P2 = (9 MW / (51 mph)^3) * (49 mph)^3
Evaluating this expression will give us the power produced when the wind speed is 49 mph.
It is important to place wind turbines at places where the wind speed is largest because the power generated by the wind turbine increases significantly with the cube of the wind speed. Therefore, even a small increase in wind speed can result in a substantial increase in the power output of the wind turbine.
Don't worry, I'm here to help you understand and solve this problem. Let's break it down step-by-step:
(a) We are given that the force of the wind on the blades of a wind turbine is proportional to the wind speed (v) raised to the power of α. We need to show that the mechanical power (P) exerted by the wind on the turbine varies as P = Kv^α, where K is a constant.
To do this, we need to relate the force of the wind on the turbine to the mechanical power it generates. Power is defined as the rate at which work is done or energy is transferred. In this case, the mechanical power generated by the turbine is equal to the work done by the wind on the turbine's blades per unit time.
The work done by the wind on the turbine is given by the product of the force (F) exerted by the wind and the distance (d) over which it acts. Since the force of the wind on the blades is proportional to v^α, we can express this as:
work = F * d
= (K * v^α) * d
Now, let's consider the definition of power. Power (P) is equal to the work done per unit time. So, we divide the work by the time (t):
P = (K * v^α * d) / t
However, in this case, wind speed (v) is the distance covered by the blades per unit time (i.e., v = d/t). Substituting this into the equation above, we get:
P = K * (v^α * v)
= K * v^(α+1)
So, we can see that the mechanical power (P) exerted by the wind on the turbine varies as P = Kv^(α+1). Comparing this with P = Kv^α, we find that α + 1 is the exponent for the power equation. Therefore, we can conclude that α + 1 = á.
(b) To find the value of á, we need to compare two situations: when the wind speed is 51 mph (let's call this v₁) and when the wind speed is 49 mph (let's call this v₂). We know that the power (P) is proportional to v^á, where v is the wind speed.
Using the given information, we have:
P₁ = 9 MW (at a wind speed of 51 mph)
P₂ = ? (at a wind speed of 49 mph)
Now, let's use the proportional relationship P = Kv^á to find the value of á. Divide the two power equations:
P₁ / P₂ = (v₁^á) / (v₂^á)
Substituting the given values, we have:
9 MW / P₂ = (51 mph)^á / (49 mph)^á
Simplifying this equation, we can cancel out the exponents by taking the logarithm (base 10) of both sides:
log(9 MW / P₂) = á * log(51 mph / 49 mph)
We can find the value of á by solving this equation for á, using logarithmic rules and elementary algebra.
Once we determine the value of á, we can use the power equation P = Kv^á to calculate the power (P₂) produced by the wind turbine when the wind speed is 49 mph.
Finally, to understand why it is important to place wind turbines at locations with higher wind speeds, we observe that the power generated by a wind turbine is directly proportional to the wind speed raised to the power of á. Hence, as the wind speed increases, the power output of the turbine increases significantly (since the power equation is not linear). Placing wind turbines at locations with higher wind speeds maximizes their energy production potential.