Energy and Environment II

HW#9

Due Friday, March 8, 2002

 

1.      Off the southern coast of Australia, the annual average wave power (per crest length) is about 75 kw/m.  Assuming these waves have a wavelength of 100 m, find the following:

a.      The period of the waves (T, seconds).

b.      The wave speed (V, m/s)

c.      The height of the waves (H = 2a, m)

d.      The maximum speed of the water particles (u, m/s)

Clearly show all steps in your work.

 

2.      Box 5.8 on page 209 of the text contains some important equations for hydropower.  Start with the equation:

P (kw) x 1000 (w/kw) = h x r (kg/m3) x g (m/s2) x Q (m3/s) x H (m)

Assume H is the effective (or working) head, that is, the head after the effect of the friction in the penstock as been taken into account.  The mechanical efficiency of the turbine is h.  The power output of the turbine is P.

  1. Derive the first equation in the box, ie, the first equation for P/H2H1/2.  Note vw = uj (the jet velocity, m/s), and r = rj (jet radius, m).  There is only one jet.  The density of water is 1000 kg/m2 and the acceleration of gravity is 9.81 m/s2.  DO NOT ASSUME g = 10 m/s2 AS THE TEXT HAS DONE.  ALSO NOTE THERE IS AN ERROR IN THE TEXT EQUATION.
  2. Now derive the second equation in the box, ie, the second equation for P/H2H1/2.  Retain the efficiency h in your equation.  The constant in your equation should reflect the correct value for the acceleration of gravity.
  3. Now derive the third equation in the box, ie, the equation for Ns.  Retain the efficiency h in your equation.  The constant in your equation should reflect the correct value for g.  Note n is the revolutions per minute of the turbine, vB is the tip speed of the turbine, and R is the radius of the turbine.  Note this equation holds for all types of water turbines.
  4. For a particular application: the total head is 3 m, the flow rate is 1.0 m3/s, and mechanical power output of the turbine is 20 kw.  Find the velocity of the jet, the radius of the jet, and the mechanical efficiency of the turbine.  State any assumptions used.
  5. Would you use a Pelton wheel turbine for this application?  Explain.  What would be the rotational speed of your turbine (n, rev/min)? 

    

3.      We have studied some methods of obtaining energy from water.  Which of the methods studied do you believe offers the best ratio of “benefit to environmental-impact”?  Explain.  Type your answer double-spaced on one page.