In the water behind a high power dam (110 m high) the temperature difference between surface and bottom may be 10ºC. Compare the possible energy extraction from the thermal energy of a gram of water with that generated by allowing the water to flow over the dam through turbines in the conventional way.
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Sol . The efficiency of a perfect engine is
η = 1 – T low /T high .
The energy extracted from one gram of water is then
W = η Q = 
.C v (T high – T low ),
where Q is the heat extracted from one gram of water, C v is the specific heat of one gram of water. Thus
W = C v (T high – T low ) 2 /T high .
If T high can be taken as the room temperature, then
W = 1 × 10 2 /300 = 0.3 cal.
The energy generated by allowing the water to flow over the dam is
W' = mgh = 1 × 980 × 100 × 10 2 = 10
7 erg = 0.24 cal.
We can see that under ideal conditions W' < W. However, the efficiency of an actual engine is much less than that of a perfect engine. Therefore, the method by which we generated energy from the water height difference is still more efficient.
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