Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 4.4.
A different kind of proposed power plant uses the thermal gradient of the ocean drive a heat engine. If the temperature of the water at the surface of the ocean in some (presumably fairly tropical; it never reaches that temperature here in Scotland) location is and at the bottom is , then the maximum efficiency of an engine operating between these two reservoirs is
This is a pretty dire efficiency. If we want to produce 1 GW of electricity, so the plant is comparable to land-based power plants, how much surface water would we need to process to extract the required heat ? We have
The amount of heat to be extracted per second is therefore
The heat expelled to the cold reservoir is
To find the minimum volume of water that must be processed per second, we can try the following argument. [Disclaimer: I’m not 100% convinced this argument is correct, but it seems to make some sense. Comments welcome.]
The amount of water needed clearly depends on how much of a temperature drop the extraction of heat produces; the smaller the volume of water, the larger the temperature drop produced. The same is true (in reverse) when expelling heat into the cold layer; the smaller the volume of water, the larger the temperature increase.
Now, we can’t decrease the temperature of the upper layer to a final temperature that is lower than the final temperature of the cold layer, so the maximum temperature drop of the upper layer is to a value of that is equal to the final temperature of the lower layer after is dumped into it. If we take the masses (I’ll use mass instead of volume, since heat capacities are expressed in terms of mass) of water processed to be the same in both layers, then we have
where is the mass of water processed in each layer and is the specific heat capacity. We can add these two equations to get
Plugging in the numbers we get
This corresponds to a volume (since of water has a mass of 1000 kg) of
The total volume of water processed (hot + cold) is twice this, or