A hydroelectric dam holds back a lake of surface area 3.0×106 m2 that has vertic

Question

A hydroelectric dam holds back a lake of surface area 3.0×106 m2 that has vertical sides below the water level. The water level in the lake is a height 140 m above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted into electrical energy with 88 % efficiency. a)If gravitational potential energy is taken to be zero at the base of the dam, how much energy is stored in the top meter of the water in the lake? The density of water is 1000 kg/m3 . Express your answer using two significant figures. J

b) What volume of water must pass through the dam to produce an amount of electrical energy totalling 1000 kilowatthours of electrical energy? Express your answer using two significant figures. m3

c)What distance does the level of water in the lake fall when this much water passes through the dam? Express your answer using two significant figures. m

Explanation / Answer

   Given

   surface area of lake is 3*10^6 m2, height h = 140 m

and When the water passes through turbines at the base of the dam, its mechanical energy is converted into electrical energy with 88 % efficiency

as the water is at 140m height above the ground the energy will be stored in the form of gravitational potential energy

a)   gravitational potential energy U = mgh

density of water is d = 1000 kg/m3, we know that density d= m/V ==> m = d*V

   energy is stored in the top meter of the water in the lake is

   U = (1000*3*10^6)(9.8)(140-1) J = 4.0866*10^12 J

b)
   Given electrical energy is 1000 kWh= 1000*10^3*3600 J = 3.6*10^9 J

energy efficiency is 89% so the energy is 3.6*10^9/8.9 = 4.04494382*10^8 J

  
to produce 1000 kWh of energy the water should decrease is

           3.6*10^9 J = 1000*3*10^6*h
           h = 1.2 m

the volume of water must pass through is v = m/d = 1000*3*10^6/1000 = 3*10^6 m3 of water should pass

c)

   the distance of the level of the water in the lake fall when water produce the energy is h = 1.2 m

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