Question

An inventor claims to have developed a device that executes a power cycle while operating between reservoirs at 800 and \(350 \mathrm{~K}\) that has a thermal efficiency of (a) \(56 \%\), (b) \(40 \%\). Evaluate the claim for each case.

Short answer

The claims are possible because both 56% and 40% are less than or equal to the Carnot efficiency of 56.25%.

Step-by-step solution

- Understand the Problem

An inventor's device operates between two reservoirs. The temperatures are given as 800 K for the hot reservoir and 350 K for the cold reservoir. The claim involves evaluating the device's thermal efficiency for both 56% and 40%.

- Recall the Formula for Carnot Efficiency

The maximum possible efficiency for any heat engine operating between two temperatures is given by the Carnot efficiency formula: \[ \eta_{carnot} = 1 - \frac{T_c}{T_h} \], where \( T_c \) is the temperature of the cold reservoir and \( T_h \) is the temperature of the hot reservoir.

- Calculate Carnot Efficiency

Using the given temperatures, calculate Carnot efficiency: \[ \eta_{carnot} = 1 - \frac{350}{800} = 1 - 0.4375 = 0.5625 \] or 56.25%.

- Evaluate the Claim for 56%

Compare the given efficiency of 56% to the Carnot efficiency of 56.25%. Since 56% is less than 56.25%, it is physically possible.

- Evaluate the Claim for 40%

Compare the given efficiency of 40% to the Carnot efficiency of 56.25%. Since 40% is significantly less than 56.25%, this claim is also physically possible.

Key concepts

thermal efficiency

Thermal efficiency is a measure of how well a heat engine converts heat into work. It's important because it tells us how much useful work we can get from a certain amount of heat energy. The higher the thermal efficiency, the better. For a heat engine, thermal efficiency \(\text{η}\) is given by the formula \( \text{η} = \frac{W_{net}}{Q_{in}} \), where \( W_{net} \) is the net work output and \( Q_{in} \) is the heat input.

• The efficiency ranges from 0 to 1 (or 0% to 100%).
• Perfect efficiency (100%) is not possible due to inherent energy losses such as friction and heat dissipation.

Therefore, thermal efficiency helps us understand the effectiveness of a heat engine, and serves as a benchmark when comparing different engines or cycles.

power cycle

A power cycle is a process where a working fluid (like steam or gas) goes through a series of state changes, transforming heat energy into mechanical work. This cycle is repeated over and over.
For example:

• The Carnot cycle is a theoretical model of an ideal power cycle.
• Real-life cycles include the Rankine cycle for steam engines and the Otto cycle for internal combustion engines.

The efficiency and effectiveness of a power cycle depend on factors like the type of working fluid, the temperature range, and the design of the engine. In our example, the power cycle involves reservoirs at 800 K and 350 K.

Carnot efficiency formula

The Carnot efficiency formula allows us to calculate the maximum possible efficiency a heat engine can achieve, operating between two temperature reservoirs. It's given by:
\[ \text{η}_{carnot} = 1 - \frac{T_c}{T_h} \]
where:

• \( T_c \) is the temperature of the cold reservoir (in Kelvin).
• \( T_h \) is the temperature of the hot reservoir (in Kelvin).

The Carnot efficiency sets the upper limit for the efficiency of any real engine. Any actual engine will have a lower efficiency than the Carnot efficiency because real processes have losses such as friction and non-reversible processes.

reservoir temperatures

Reservoir temperatures are crucial in determining the Carnot efficiency and thus the overall thermal efficiency of a heat engine. In thermodynamics, reservoirs are large thermal bodies that maintain a constant temperature even when heat is added or extracted.
In our example, the hot reservoir is at 800 K and the cold reservoir is at 350 K.
Higher temperature differences between the hot and cold reservoirs generally result in higher efficiencies:

• Higher \( T_h \) relative to \( T_c \) means more potential work output.
• Lower \( T_c \) relative to \( T_h \) also increases efficiency.

Therefore, to achieve higher efficiencies, heat engines strive to operate between a very high temperature source and a very low temperature sink.

heat engine

A heat engine is a device that converts thermal energy into mechanical work. It does this by exploiting the differences in temperatures between two reservoirs. The main components of heat engines include a working fluid, a source of heat (hot reservoir), and a heat sink (cold reservoir).
In this example:

• The working fluid absorbs heat from the hot reservoir (800 K).
• It then does work (like moving a piston or turning a turbine).
• Finally, it releases unused heat to the cold reservoir (350 K).

The effectiveness of a heat engine is measured by its thermal efficiency. By understanding and applying the Carnot efficiency formula, we can determine how close a given heat engine's performance is to the theoretical maximum efficiency.