One alternative energy source is the temperature difference...

One alternative energy source is the temperature difference between surface and deep water in the tropical oceans. Suppose a heat engine uses $25^{\circ} \mathrm{C}$ surface water as its hot reservoir and $4^{\circ} \mathrm{C}$ deep water as its cold reservoir. (a) Find the maximum efficiency of such an engine. (b) How much heat would have to be removed from the surface water each day in order to produce energy at the rate of a typical large power plant, about $1000 \mathrm{MW} ?$

One alternative energy source is the temperature difference between surface and deep water in the tropical oceans. Suppose a heat engine uses $25^{\circ} \mathrm{C}$ surface water as its hot reservoir and $4^{\circ} \mathrm{C}$ deep water as its cold reservoir. (a) Find the maximum efficiency of such an engine.
(b) How much heat would have to be removed from the surface water each day in order to produce energy at the rate of a typical large power plant, about $1000 \mathrm{MW} ?$

Key Concepts

Carnot Efficiency

Carnot efficiency is the maximum theoretical efficiency that any heat engine operating between two thermal reservoirs can achieve. It is determined by the temperatures of the hot and cold reservoirs (measured on an absolute scale) using the formula ? = 1 - (Tc/Th), where Th is the absolute temperature of the hot reservoir and Tc is the absolute temperature of the cold reservoir. This concept establishes the upper limit of energy conversion efficiency for heat engines regardless of the working substance or cycle details.

Heat Engine

A heat engine is a device that converts thermal energy into work by exploiting temperature differences between a high-temperature (hot) reservoir and a low-temperature (cold) reservoir. The performance of a heat engine is limited by the second law of thermodynamics, which dictates that not all the absorbed heat can be transformed into work, emphasizing the importance of understanding efficiency limits like those given by the Carnot cycle.

Absolute Temperature Scale

The absolute temperature scale, typically measured in Kelvin, is essential in thermodynamics because efficiency formulas such as the Carnot efficiency require temperatures to be expressed in absolute terms. Converting temperatures from Celsius to Kelvin by adding 273.15 is a crucial step when applying thermodynamic equations, ensuring that the ratios of temperatures are physically meaningful.

Power and Energy Unit Conversion

Understanding the relationship between power and energy is vital in evaluating energy systems. Power, measured in watts (or megawatts for large-scale applications), indicates the rate at which energy is produced or consumed, while energy is the total work performed over time. Converting between these units, such as determining the total heat removal required over a day for a given power output, is a foundational skill in energy engineering and thermodynamics.

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