What Is the Minimum Wind Speed Needed to Operate a Windmill Water Pump?

[pavadrin]

Homework Statement

Find the minimum wind speed for the windmill to pump water. The pump is connected to a gear at a point 0.16m from the center. The radius of this gear (Gear 1) is 0.2m and the gear which it interlocks with is 0.1m (Gear 2). They intersect each other perpendicular to one another (see image)
http://img399.imageshack.us/img399/9013/pumpgearingconnectiongo1.th.png http://g.imageshack.us/thpix.php

The turbine blades run parallel to the horizon and have a surface are of 1.5m^2 (3m wide x 0.5m high).

Homework Equations

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The Attempt at a Solution

I have found the weight of the water to be 250N (calculated from data which has not been included in the question). Assuming there is no friction I have calculated the force required on the centre of the turbine to be ~9N. I have done this by assuming that there is a constant torque throughout the system, so that the torque created by the weight of the water is transferred to gear 1 then to gear 2 and then to the turbine blade. Have I approached this correctly? (laugh if you want lol, i have never tried to solve a similar problem before). If so, have do i convert the force required to a wind speed.

Many thanks for your time,
pavadrin

 

[KataKoniK]

Dear pavadrin,

Thank you for your question. In order to find the minimum wind speed for the windmill to pump water, you will need to use some basic physics equations and principles. First, let's start with the equation for torque, which is given by:

τ = rFsinθ

Where τ is the torque, r is the distance from the center of rotation to the point where the force is applied, F is the force applied, and θ is the angle between the force and the lever arm.

In this case, the force required on the center of the turbine is 9N and the distance from the center of rotation to the point where the force is applied is 0.16m. Therefore, the torque required to turn the turbine blades is:

τ = (0.16m)(9N)sin90° = 1.44Nm

Next, we can use the equation for power, which is given by:

P = τω

Where P is the power, τ is the torque, and ω is the angular velocity.

We can rearrange this equation to solve for the angular velocity:

ω = P/τ

Now, we need to determine the power generated by the wind. We can use the equation for kinetic energy, which is given by:

KE = 1/2mv^2

Where KE is the kinetic energy, m is the mass of the air, and v is the wind speed.

We can also use the equation for power to calculate the power generated by the wind:

P = 1/2ρAv^3

Where ρ is the density of air, A is the surface area of the turbine blades, and v is the wind speed.

Setting these two equations equal to each other and solving for v, we get:

v = √(2P/ρA)

Now, we can plug in the values given in the question. The surface area of the turbine blades is 1.5m^2, the windmill is perpendicular to the wind, so θ = 90°, and the weight of the water is equivalent to the force required on the center of the turbine. Therefore, we get:

v = √(2(1.44Nm)/(1.2kg/m^3)(1.5m^2)) = 2.4m/s

Therefore, the minimum wind speed for the windmill to pump water is 2.

 

[piercas]

Dear pavadrin,

Your approach to the problem seems reasonable. However, in order to find the minimum wind speed required, we need to consider the power of the wind and the power needed to pump the water. The power of the wind can be calculated using the formula P = 1/2 * ρ * A * v^3, where ρ is the air density, A is the surface area of the turbine blades, and v is the wind speed. The power needed to pump the water can be calculated using the formula P = F * v, where F is the force required on the center of the turbine and v is the wind speed.

Using these formulas, we can set the two powers equal to each other and solve for the wind speed. This will give us the minimum wind speed required to pump the water. Keep in mind that this calculation assumes ideal conditions, without taking into account factors such as friction and efficiency of the system.

I would also recommend considering the practicality of using a windmill to pump water. The wind speed required to pump water may be quite high, and it may not be a reliable source for pumping water consistently. Additionally, there may be other factors to consider such as the cost and maintenance of the windmill. It may be worth exploring alternative methods for pumping water.

I hope this helps and good luck with your calculations. Keep in mind that as a scientist, it's important to consider all factors and potential limitations in your research and problem-solving.