A battle has erupted over a local power plant owner\\\'s decision to raise the p

Question

A battle has erupted over a local power plant owner's decision to raise the power delivered by the plant from 144 MW to 411 MW (The delivered power accounts for energy losses). The local population is increasing, so the area needs more power. However, the plant expends its heated waste water* into the Frustration River; local citizens are concerned that the heated water will raise water temperatures to the point where the endangered river smelt will die from oxygen depletion. Local biologists called into the fray report that the smelt can only withstand an average increase of 3.44°C. The river is 202 ft wide at the point where the waste water is dumped, but the depth and cross section is not well known. Farther downstream, however, the water flows through a narrow man-made concrete channel of rectangular cross-section, which has a width of 80.7 ft and depth of 32.2 ft. The water flows through this channel at 10.4 mph. The power plant claims a planned efficiency of 36.10% for its new power plant design. Based on this information, what is the maximum power that the power plant can output without endangering the river smelt?

Explanation / Answer

Given, area of cross section of river through concrete man made channel A = 80.7*32.2 ft^2 = 2598.54 ft^2 = 241.41 m^2
river speed thorough this point, v = 10.4 mph = 4.649 m/s
so voilume flow rate = A*v
mass flow rate = rho*Av ( where rho is density of water = 1000 kg/m^3)
mass flow rate, m' = 1000*A*v = 1122315.09 kg/s
now, eneequired to raise temperature of 1 kg of water by 1 C = heat capacity of weater = 4184 J

so, for m' = 1122315.09 kg/s
power requirement for raising temperature by 3.44 C = 1122315.09 kg/s * 4184 * 3.44 = 16153.4361977664 MW

for 36.1 percent efficiency power output of power plant be P
then power wasted = (1-0.361)of total power = 16153.4361977664 MW
total power = 25279.24287 MW
power genertated by power plant, capacity = 0.361*25279.24 = 9125.8066 MW

so the power plant can increase its power output by 9125.806 MW and still will not increase the temperature of the river by more than 3.44 C

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