A power plant operates at a 32.0 % efficiency during the...

A power plant operates at a $32.0 \%$ efficiency during the summer when the seawater used for cooling is at $20.0^{\circ} \mathrm{C}$. The plant uses $350^{\circ} \mathrm{C}$ steam to drive turbines. If the plant's efficiency changes in the same proportion as the ideal efficiency, what would be the plant's efficiency in the winter, when the seawater is at $10.0^{\circ} \mathrm{C}$ ?

Key Concepts

Proportional Change in Efficiency

When a real engine’s efficiency is said to change in proportion to the ideal (Carnot) efficiency, it implies that the ratio of the actual efficiency to the theoretical maximum remains constant even as operating conditions, such as the cold reservoir temperature, change. This proportionality allows one to estimate the new actual efficiency based on changes in the ideal efficiency.

Heat Engine Efficiency

Heat engine efficiency is the ratio of work output to heat input. This efficiency reflects how well the engine converts heat energy into useful work. In real engines, this efficiency is typically lower than the Carnot efficiency (the ideal limit) due to practical limitations and irreversible processes.

Carnot Efficiency

Carnot efficiency is a theoretical limit on the efficiency of a heat engine operating between two temperatures. It is given by the formula ? = 1 - (T_cold/T_hot), where T_hot and T_cold must be in an absolute scale. This concept defines the maximum possible efficiency achievable by any heat engine and serves as a benchmark for real-world engines.

Temperature Conversion to Kelvin

In thermodynamics, temperatures must be expressed on an absolute scale, typically Kelvin, when used in efficiency calculations. Converting Celsius to Kelvin (by adding 273.15) ensures that temperature ratios are correctly computed, which is critical for accurately determining efficiencies using formulas like that of Carnot's.

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