Question

An oil-burning electric power plant uses steam at \(773 \mathrm{K}\) to drive a turbine, after which the steam is expelled at \(373 \mathrm{K} .\) The engine has an efficiency of \(0.40 .\) What is the theoretical maximum efficiency possible at those temperatures?

Short answer

The theoretical maximum efficiency is approximately 51.73%.

Step-by-step solution

Understand the Concept

The theoretical maximum efficiency of a heat engine operating between two temperatures is given by the Carnot efficiency, which depends solely on the temperatures of the hot and cold reservoirs. It's calculated using the formula: \[ \eta_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}} \] where \( T_{cold} \) and \( T_{hot} \) are the absolute temperatures of the cold and hot reservoirs respectively.

Apply the Formula

Substitute the given temperatures into the Carnot efficiency equation. The temperature of the hot reservoir \( T_{hot} = 773 \mathrm{K} \) and the cold reservoir \( T_{cold} = 373 \mathrm{K} \).\[ \eta_{Carnot} = 1 - \frac{373}{773} \]

Calculate the Carnot Efficiency

Perform the calculation:\[ \eta_{Carnot} = 1 - \frac{373}{773} = 1 - 0.4827 \approx 0.5173 \] Thus, the Carnot efficiency is approximately \(0.5173\) or 51.73%.

Key concepts

Heat Engine

A heat engine is a fascinating device that plays a crucial role in converting thermal energy into mechanical work. It operates on the basic principle of absorbing heat from a hot reservoir and expelling a portion of it to a cold reservoir. The work done by the engine is the difference between the heat absorbed and the heat expelled. A heat engine is an integral part of many power plants, automobiles, and various industrial systems.

Here's how a heat engine works:

• It receives heat from a high-temperature source or hot reservoir, such as steam at 773 K in our exercise.
• It converts part of this heat into work - this could be seen as mechanical energy like turning a turbine.
• The remaining heat is expelled to a low-temperature sink or cold reservoir, such as into steam expelled at 373 K.

In essence, a heat engine makes use of the natural flow of heat from hot to cold to generate work. The efficiency of this heat conversion process is of great interest and varies depending on the engine design and conditions of operation.

Thermodynamic Efficiency

Thermodynamic efficiency is a measure of how effectively a heat engine converts heat into work. It factors heavily into the performance and usefulness of an engine in practical applications. While real engines never achieve perfect efficiency due to unavoidable losses and imperfections, understanding the theoretical maximum helps in improving designs.

The efficiency of a heat engine, particularly for idealized models like the Carnot engine, is calculated using the expression for Carnot efficiency:\[\eta_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}}\]

• Where \( T_{cold} \) is the absolute temperature of the cold reservoir.
• And \( T_{hot} \) is the absolute temperature of the hot reservoir.

The theoretical maximum efficiency, or Carnot efficiency, provides a benchmark. Even though our particular engine has an efficiency of 40%, the Carnot efficiency calculated at 51.73% offers insight into the limitations imposed by the laws of thermodynamics. This efficiency dictates the upper limit, guiding engineers in evaluating how close real engines are to this idealized limit.

Temperature Reservoirs

Temperature reservoirs are fundamental concepts in studying heat engines, forming the backbone of understanding how these engines function. They serve as the thermal energy sources from which engines draw heat and the sinks to which they expel it.

There are two primary reservoirs for any heat engine:

• The **hot reservoir:** This is where the engine gets its heat. In our example, the steam at 773 K acts as the hot reservoir.
• The **cold reservoir:** This acts as the destination for expelled heat. For the given exercise, this is the steam at 373 K.

The difference in temperature between these reservoirs is what allows the heat engine to convert some of the thermal energy into work. The greater this temperature difference, the more potential there is for work, given an ideal process. Understanding the role of these reservoirs is key to designing efficient systems and maximizing the work output given certain constraints.