Question

Obtain a relation for the second-law efficiency of a heat engine that receives heat \(Q_{H}\) from a source at temperature \(T_{H}\) and rejects heat \(Q_{L}\) to a sink at \(T_{L},\) which is higher than \(T_{0}\) (the temperature of the surroundings), while producing work in the amount of \(W\)

Short answer

Answer: The second-law efficiency of the heat engine can be expressed as \(\eta_{II} = \frac{1 - \frac{Q_L}{Q_H}}{1 - \frac{T_L}{T_H}}\).

Step-by-step solution

Calculate the efficiency of a reversible heat engine

The efficiency of a reversible heat engine is given by the formula: $$ \eta_{rev} = 1 - \frac{T_L}{T_H} $$ This is based on the Carnot principle, which states that the efficiency of a reversible heat engine operating between two temperatures is independent of the working substance and the specific process.

Calculate the actual efficiency of the heat engine

The actual efficiency of the heat engine is determined by the ratio of work output to heat input. From the given parameters, we know the work output (\(W\)) and the heat input (\(Q_H\)). Therefore, the actual efficiency is given by: $$ \eta_{actual} = \frac{W}{Q_H} $$

Relate work output and heat transfer

We can relate the work output (\(W\)) to the heat transfers: $$ W = Q_H - Q_L $$ This equation is based on the first law of thermodynamics, which states that the difference between heat input and heat output will produce work.

Calculate the second-law efficiency

The second-law efficiency is defined as the ratio of the actual efficiency to the maximum possible efficiency (reversible efficiency). We can use the expressions we derived for \(\eta_{actual}\) and \(\eta_{rev}\) to compute the second-law efficiency: $$ \eta_{II} = \frac{\eta_{actual}}{\eta_{rev}} \\ \eta_{II} = \frac{\frac{W}{Q_H}}{1 - \frac{T_L}{T_H}} $$

Substitute the expression for work output

Now, replace the work output (\(W\)) in the second-law efficiency formula with the relation we derived in step 3. $$ \eta_{II} = \frac{\frac{Q_H - Q_L}{Q_H}}{1 - \frac{T_L}{T_H}} $$

Simplify the expression for second-law efficiency

To get the final expression for second-law efficiency (or Exergetic efficiency), we can simplify the previous equation: $$ \eta_{II} = \frac{1 - \frac{Q_L}{Q_H}}{1 - \frac{T_L}{T_H}} $$ This is the relation for the second-law efficiency of a heat engine that receives heat \(Q_{H}\) from a source at temperature \(T_{H}\) and rejects heat \(Q_{L}\) to a sink at \(T_{L},\) which is higher than \(T_{0}\) (the temperature of the surroundings), while producing work in the amount of \(W\).

Key concepts

Carnot Principle

The Carnot principle is fundamental to understanding heat engine efficiency. Named after French engineer Sadi Carnot, it stipulates an upper limit on efficiency that any engine operating between two heat reserves can achieve—no matter what substances or processes it uses. This principle can be concisely described by the equation:
\[\begin{equation}\eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}\end{equation}\]
where \(\eta_{\text{Carnot}}\) represents the Carnot efficiency, \(T_H\) is the absolute temperature of the hot reservoir, and \(T_L\) is the absolute temperature of the cold reservoir.

• The principle assumes that the process is reversible, meaning that no entropy is produced in the cycle.
• It implies that as the temperature difference between the hot and cold reservoirs increases, so too does the potential efficiency of the engine.
• It is also a benchmark for comparing actual engines to an idealized, reversible one.

Carnot's insight shapes how we evaluate the performance of all heat engines, including those found in power plants and automobiles.

First Law of Thermodynamics

The first law of thermodynamics, often termed the law of energy conservation, reveals how energy is balanced within physical systems. It establishes that energy cannot be created or destroyed, only transformed from one form to another. When applied to heat engines, the first law is expressed as:\[\begin{equation}W = Q_H - Q_L\end{equation}\]
where \(W\) is the work done by the engine, \(Q_H\) is the heat absorbed from the hot source, and \(Q_L\) is the heat rejected to the cold sink. Understanding this law is crucial for the following reasons:

• It allows us to track how much of the input energy is converted into productive work versus lost as waste heat.
• It stands as the basis for evaluating how energy efficient a machine or process is.
• It provides a fundamental equation that relates heat transfer to mechanical work, essential in engineering and science.

In essence, the first law provides the guidelines for the energy input-output balance, from which we can derive a machine's actual operational efficiency.

Heat Engine Efficiency

Heat engine efficiency is a measure of how well an engine converts heat into work. The actual efficiency, denoted as \(\eta_{\text{actual}}\), of a heat engine is found by comparing the work output to heat input. This ratio can be quantified by the equation:\[\begin{equation}\eta_{\text{actual}} = \frac{W}{Q_H}\end{equation}\]
where \(W\) represents the work produced, and \(Q_H\) is the heat absorbed from the hot reservoir. The following points elaborate on its importance:

• It reflects the real-world performance of engines, which is always less than the idealized efficiency suggested by the Carnot principle.
• It considers losses due to irreversibilities such as friction, heat transfer limitations, and other factors reducing the system's efficiency.
• Differentiating between actual and theoretical (Carnot) efficiency helps engineers and scientists identify inefficiencies and potential areas for improvement in engine design.

By assessing the real-world efficiency, we're better positioned to enhance the energy conversion process, saving costs and reducing environmental impacts.

Exergetic Efficiency

Exergetic efficiency, often termed second-law efficiency, examines the quality of energy conversion processes through the lens of the second law of thermodynamics. This law introduces the concept of entropy and acknowledges that some energy will always be degraded through processes. The second-law efficiency is calculated with the ratio:\[\begin{equation}\eta_{\text{II}} = \frac{1 - \frac{Q_L}{Q_H}}{1 - \frac{T_L}{T_H}}\end{equation}\]
where \(\eta_{\text{II}}\) is the second-law (exergetic) efficiency, \(Q_H\) and \(Q_L\) are the heat transfers from the high and low temperature reservoirs, and \(T_H\) and \(T_L\) are the absolute temperatures of those reservoirs. Key insights into exergetic efficiency include:

• It accounts for the inevitable increase of entropy and gives a truthful measure of how effectively a heat engine utilizes the potential of the heat supplied.
• This efficiency compares actual performance against the maximum possible performance permitted by the second law, rather than the first law.
• It is a critical criterion for optimization because it identifies the potential for improving energy conversion processes beyond just conserving energy, by also managing the entropy production or energy quality.

Taking second-law efficiency into account is vital for driving advancements in the performance and sustainability of thermal systems within industries and technologies.