Homework 5

Due at 11:59pm on Thursday, May 14th. Submit to https://canvas.uw.edu/courses/1883403/assignments/11424522. It is fine to submit scanned or a picture of handwritten answers, but we must be able to read it. Indicate your final answer clearly, show your work. Each problem is 10 points (all subproblems weighted equally).

Problem 1

The lift force exerted on each blade of a three-blade wind turbine is \(1000 \text{ N}\). The center of mass of the blade is \(20 \text{ m}\) from the hub. Find the total torque generated by the turbine and the total power generated if the blades are turning at \(35 \text{ rpm}\) (revolutions per minute).

Problem 2

The wind speed at a site is uncertain. Suppose we can predict the mean speed to be \(10 \text{ m/s}\) and the deviation to be \(\pm 10 \%\). That is, wind speed is in the range of \([9,11] \text{ m/s}\). Using this information, how well can be predict wind power? That is, what is the % deviation from the mean?

Problem 3

Complete the following steps. Suppose for a given area, the wind speed before the turbine is \(w_u\) and the wind speed after the turbine is \(w_d\).

  1. To compute the power that is extracted by the turbine, we need to know the wind speed at the turbine. Making a continuity assumption, suppose that the speed is at the turbine is the average of the upwind and the downwind speed. That is, the wind speed at the turbine, \(w\), is \(w=\frac{1}{2} (w_u+w_d)\).

  2. The power extracted by the turbine is also given by the difference in power in upwind airmass and the power in the downwind airmass.

  3. We assume air is incompressible, that is, the same amount or air is moving upwind, through the turbine, and downwind.

  4. Combining the last two terms, it is now possible to write the power extracted by the turbine as

    \[P_{\text{turbine}}=P_u \cdot C_p,\]

    where \(C_p\) is in terms of the ratio \(\gamma=\frac{w_d}{w_u}\).

  5. Now maximize \(C_p\) by varying \(\gamma\). This leads the to Betz ratio, which states \(C_p \leq 0.593\).

Note, this is not a trivial derivation and you may use the Internet or AI tools if needed. Although it’s worthwhile to give it a try.

Problem 4

Each U-235 fission releases approximately \(3.20 \times 10^{-11} \text{ J}\) of energy. The heat of combustion of natural gas (primarily methane) is approximately \(55 \text{ MJ/kg}\).

Find the amount of energy \(10 \text{ kg}\)$ of \(\mbox{U}^{235}\) can produce. How much natural gas is needed to produce the same amount of energy (assuming perfect efficiency)? Hint: the molar mass of \(\mbox{U}^{235}\) is \(M = 235 \text{ g/mol}\) and Avogadro’s number is \(N_A = 6.022 \times 10^{23} \text{ mol}^{-1}\).

Problem 5

We use three phase power for transmission systems. In a few short sentences, explain why we do not use four phase or higher number of phases?

Problem 6

A balanced Y-connected three-phase source is connected to a balanced three phase load. At the source, the rms voltage and current per phase are \(160 \angle 50^\circ \text{ V}\) \(85 \angle 70^\circ \text{ A}\), respectively.

Find:

a) The (rms) line-to-line voltage.

b) Total power delievered by the source.

Problem 7

A Y-connected balanced three-phase source is feeding a balanced three-phase load. The voltage and current on phase \(a\) of the source are:

\[v(t) = 240 \cos(100\pi t + 2\pi/9) \text{ V}\] \[i(t) = 300 \cos(100\pi t + 3\pi/18) \text{ A}\]

Find:

a) The impedance of the load if it is Y-connected.

b) The impedance of the load if it is \(\Delta\)-connected.