Question

A heat engine that receives heat from a furnace at \(1200^{\circ} \mathrm{C}\) and rejects waste heat to a river at \(20^{\circ} \mathrm{C}\) has a thermal efficiency of 40 percent. Determine the second-law efficiency of this power plant.

Short answer

Answer: The second-law efficiency of the heat engine is approximately 49.94%.

Step-by-step solution

Convert temperatures to Kelvin

Before calculating the Carnot efficiency, both source and sink temperatures must be converted to Kelvin. To do this, we add 273.15 to the Celsius temperature. \(T_H = 1200 + 273.15 = 1473.15 \, K\) \(T_C = 20 + 273.15 = 293.15 \, K\)

Calculate maximum efficiency (Carnot efficiency)

Now that the temperatures are in Kelvin, we can use the Carnot efficiency formula, which is given by: Carnot efficiency, \(\eta_C = 1 - \frac{T_C}{T_H}\) Plugging in the values for \(T_H\) and \(T_C\), \(\eta_C = 1 - \frac{293.15}{1473.15} = 1 - 0.1989 \approx 0.8011\) Carnot efficiency is approximately 0.8011 or 80.11%

Calculate second-law efficiency

We have the actual thermal efficiency, which is 40% (\(0.4\) as a decimal), and we have determined the maximum efficiency from the previous step, which is 80.11% (\(0.8011\) as a decimal). Second-law efficiency is the ratio of the actual efficiency to the maximum efficiency, given as: Second-law efficiency, \(\eta_{SL} = \frac{\eta}{\eta_C}\) Plugging in the actual thermal efficiency and the Carnot efficiency: \(\eta_{SL} = \frac{0.4}{0.8011} \approx 0.4994\) Second-law efficiency is approximately 0.4994 or 49.94%. Thus, the second-law efficiency of this power plant is approximately 49.94%.

Key concepts

Carnot Efficiency

Carnot efficiency is a theoretical maximum efficiency that a heat engine can achieve based on the temperatures of its heat source and heat sink. It represents the upper limit on the efficiency of any heat engine due to the Second Law of Thermodynamics. The concept is named after the French engineer Sadi Carnot who first described it in the 1820s.

To calculate the Carnot efficiency \( \eta_C \), we use the formula:
\[ \eta_C = 1 - \frac{T_C}{T_H} \]
where \( T_H \) is the absolute temperature of the hot reservoir (heat source) and \( T_C \) is the absolute temperature of the cold reservoir (heat sink), both measured in Kelvin. The Kelvin scale addresses the absolute zero temperature as the point where all thermal motion ceases, thus providing a true baseline for thermal efficiency calculations.

In the context of the example provided, by converting the temperatures to Kelvin and applying them to the formula, we derived the Carnot efficiency as approximately 80.11%. This figure represents the ideal efficiency, highlighting the potential maximum work obtainable from a heat engine under the given temperature conditions, if it were a perfect, reversible engine without any losses.

Thermal Efficiency

Thermal efficiency of a heat engine is the fraction of heat converted to useful work. It is a real-world measure of how well the engine performs compared to the ideal efficiency. Unlike Carnot efficiency, which is a theoretical limit, thermal efficiency takes into account the actual losses that occur during the process, such as friction and heat escaping to the environment.

The formula for thermal efficiency \( \eta \) is:
\[ \eta = \frac{\text{Work output}}{\text{Heat input}} \]
In practical terms, it tells us what percentage of the heat energy the engine is actually turning into work. The example in the exercise states that the heat engine has a thermal efficiency of 40 percent, meaning that only 40% of the input heat is being converted into useful work, and the remaining 60% is lost (mostly as waste heat).

Heat Engine

A heat engine is a system that converts heat energy into mechanical work. Heat engines operate on the basis of thermal cycles, which typically involve the transfer of heat from a hot source, doing work through expansion or other means, and the rejection of leftover heat to a cooler sink. The operation of heat engines is a cornerstone of classical thermodynamics and covers a vast range of applications, from automobile engines to power plants.

The performance of a heat engine is often assessed by its efficiency, which can be calculated using the second-law efficiency or thermal efficiency. Real heat engines can never achieve the maximum efficiency predicted by Carnot, due to inherent irreversibilities in the processes they perform. Still, they're pivotal in converting thermal energy into usable power.

Kelvin Temperature Conversion

Kelvin temperature conversion is fundamental in thermodynamics because it allows various thermal processes to be compared on an absolute scale. For scientific work, the Kelvin scale is preferred because it starts at absolute zero, the point where no thermal motion exists in particles, and proceeds with increments identical to those of the Celsius scale.

To convert from Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature:
\[ K = \degree C + 273.15 \]

For example, applying this conversion to the temperatures in the exercise, the furnace at \(1200^\circ C\) becomes 1473.15 K and the river at \(20^\circ C\) becomes 293.15 K. These conversions are critical when using thermal efficiency equations, as they rely on absolute temperatures for accuracy.