A power plant, having a Carnot efficiency, produces electric...

A power plant, having a Carnot efficiency, produces electric power $P$ from turbines that take in energy from steam at temperature $T_{h}$ and discharge energy at temperature $T_{c}$ through a heat exchanger into a flowing river. The water downstream is warmer by $\Delta$ T due to the output of the power plant. Determine the flow rate of the river.

Key Concepts

Carnot Efficiency

Carnot efficiency is the theoretical maximum efficiency that a heat engine operating between two temperature reservoirs can achieve. It is defined by the temperatures of the hot source and cold sink (expressed in absolute units), and it sets an upper limit on how efficiently thermal energy can be converted into work, independent of the working substance or detailed processes. This concept is fundamental in understanding idealized thermodynamic cycles and serves as a benchmark for real engines.

Energy Balance

The energy balance concept involves applying the conservation of energy principle to systems such as power plants. It relates the input energy in the form of heat to the engine, to the useful work output and the energy rejected to the cold reservoir. In problems involving power plants and heat exchangers, energy balance is key to determining the repartition of energy flows between work production and the warming of an output fluid like river water.

Specific Heat Capacity and Flow Rate Calculation

Specific heat capacity is a material property that quantifies the amount of energy required to raise the temperature of a unit mass of a substance by one degree. When a known amount of heat is transferred to a fluid, the resulting temperature change is used along with the fluid’s specific heat capacity to calculate its mass flow rate. This relationship, expressed as Q = m·c·?T, is crucial in determining how much fluid must flow to absorb or relinquish a given amount of energy in systems like heat exchangers.

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