HW#3
Due
Friday, October 13, 2000
In tropical regions, ocean surface temperatures can be as high as 30 degrees C (303 K), whereas at a depth of about 500 meters, the water temperature can be as cool as 10 degrees C (283 K).
Use the surface water as the heat source, and use the water at 500 meters depth as the heat sink.
Suppose you build (or have built) a heat engine to operate between these two heat reservoirs.
A. What would be the maximum efficiency of a heat engine in this application?
Answer:
The heat source is at 30 degrees C = 303 K. The heat sink is at 10 degrees C (283 K). The maximum possible efficiency is the Carnot efficiency, which for this application is 1 - TL/TH = 1 - 283/303 = 0.066 or 6.6%.
B. The objective is to generate 1 million kw-hr of electricity over the year using the ocean heat reservoirs. Suppose the working fluid in the real engine has a maximum temperature of 27 degrees and a minimum temperature of 13 degrees. That is, the hot water is cooled from 30 to 27 degrees as it transfers heat to the engine, and the cold water is heated from 10 to 13 degrees C as it receives heat rejected from the engine. Suppose the efficiency of your real heat engine (with electrical generator) is one-half the maximum efficiency found in part A. How many cubic meters of ocean water would be required over the year? Noting there are 264 US gallons in one cubic meter, what would be the flow rate of water as gallons per minute? The density of water is 1000 kg per cubic meter and the specific heat of water is 4.2 kJ/kg-K [kilojoules per kilogram per degree K (or C)].
Answer:
For the real engine (with electrical generator), efficiency = 0.066/2 = 0.033 = WNET/ QADD. Since WNET = 1,000,000 kw-hr = 3,600,000,000 kJ, QADD = 109,100,000,000 kJ and QREJECT = 105,500,000,000 kJ. The QADD must equal the mass of the hot water times its specific heat times its temperature drop. Thus, the mass of hot water required is m = 109,100,000,000 kJ divided by the quantity (4.2 kJ/kg-C x 3 degrees C) = 8,659,000,000 kg. Since the density of water is 1000 kg per cubic meter, the volume of hot water required over the year is 8,659,000 cubic meters. The amount of cold water is 1055/1091 = 0.967 (one minus efficiency) times the hot water amount, or 8,373,000 cubic meters. The total of the hot and cold water is 17,032,000 cubic meters. In US gallons this amount is 4,495,000,000 gallons. Since there are 525,600 minutes in a year, the flow rate of the sum of the hot and cold water is 8555 gallons per minute.