Question

To increase the thermal efficiency of a reversible power cycle operating between thermal reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{C}\), would it be better to increase \(T_{\mathrm{H}}\) or decrease \(T_{\mathrm{C}}\) by equal amounts?

Short answer

It is better to decrease \(T_{\text{C}}\) by equal amounts to increase the thermal efficiency more effectively.

Step-by-step solution

Understand the Problem

The task is to determine which action—either increasing the high-temperature reservoir, \(T_{\text{H}}\), or decreasing the low-temperature reservoir, \(T_{\text{C}}\) by equal amounts—would more effectively increase the thermal efficiency of a reversible power cycle.

Thermal Efficiency Formula

The thermal efficiency \( \eta \) of a reversible power cycle (like a Carnot engine) is given by: \[ \eta = 1 - \frac{T_{\text{C}}}{T_{\text{H}}} \]

Determine Change in Efficiency for Increasing \(T_{\text{H}}\)

Consider increasing \(T_{\text{H}}\) by a small amount \( \Delta T \). The new efficiency will be: \[ \eta' = 1 - \frac{T_{\text{C}}}{T_{\text{H}} + \Delta T} \]

Determine Change in Efficiency for Decreasing \(T_{\text{C}}\)

Consider decreasing \(T_{\text{C}}\) by a small amount \( \Delta T \). The new efficiency will be: \[ \eta'' = 1 - \frac{T_{\text{C}} - \Delta T}{T_{\text{H}}} \]

Compare the Two Scenarios

Compare the new efficiencies: For an increase in \(T_{\text{H}}:\) \[ \eta' = 1 - \frac{T_{\text{C}}}{T_{\text{H}} + \Delta T} \] For a decrease in \(T_{\text{C}}:\) \[ \eta'' = 1 - \frac{T_{\text{C}} - \Delta T}{T_{\text{H}}} \]

Evaluate Efficiency Gains

Write the approximate changes in efficiency using the binomial expansion for small \( \Delta T \): For increasing \(T_{\text{H}}:\) \[ \eta' \approx 1 - \frac{T_{\text{C}}}{T_{\text{H}}} + \frac{T_{\text{C}} \Delta T}{T_{\text{H}}^2} \] For decreasing \(T_{\text{C}}:\) \[ \eta'' \approx 1 - \frac{T_{\text{C}}}{T_{\text{H}}} + \frac{\Delta T}{T_{\text{H}}} \] The first approximation shows that the term increases more significantly in the second case (bigger increment in efficiency).

Conclusion

From the approximations, one can see that the efficiency gain is more substantial when decreasing \(T_{\text{C}}\) by a small amount \( \Delta T \).

Key concepts

reversible power cycle

To fully grasp the problem, let's first understand the concept of a reversible power cycle. In thermodynamics, a power cycle is a series of processes that convert heat energy into work. A reversible power cycle is ideal and theoretical, meaning it can be reversed without any loss of energy. This reversibility implies there is no entropy change, making it as efficient as possible.
These cycles are crucial in studying thermodynamic efficiency because they set the upper bound of what's possible. The Carnot cycle, composed of isothermal and adiabatic processes, is a prime example of a reversible power cycle.
The core takeaway is that the more reversible a cycle, the more efficient it is. Practically, though, achieving total reversibility is impossible due to real-world factors like friction and heat losses. However, studying reversible cycles helps engineers aim for higher efficiency in real engines.

thermal reservoirs

In our exercise, the power cycle operates between two thermal reservoirs. A thermal reservoir is a large body that maintains a constant temperature even when heat is added or removed. Think of it as an infinite heat sink.
The two reservoirs in our problem are at temperatures, \( T_{\text{H}} \) (hot) and \( T_{\text{C}} \) (cold). The challenge is to determine if increasing the hot temperature \( T_{\text{H}} \) or decreasing the cold temperature \( T_{\text{C}} \) would more effectively increase thermal efficiency by the same amount.
The role of these reservoirs is foundational: heat is absorbed from the hot reservoir to do work, and the remaining heat is expelled to the cold reservoir. Think of them as source and sink for heat in the cycle, and their constant temperatures help to define the cycle's efficiency.

Carnot engine

The Carnot engine is a specific example of a reversible power cycle often used as a benchmark for maximum efficiency. It operates between two thermal reservoirs and consists of two isothermal processes and two adiabatic processes. The thermal efficiency \( \eta \) of a Carnot engine is given by
\[ \eta = 1 - \frac{T_{\text{C}}}{T_{\text{H}}} \]
This formula tells us that the efficiency depends solely on the temperatures of the reservoirs. Thus, if you want to increase \( \eta \), you can either increase \( T_{\text{H}} \) or decrease \( T_{\text{C}} \).
The beauty of the Carnot cycle is its simplicity and elegance in defining the theoretical limits of efficiency. By comparing your practical cycles to the Carnot cycle, you can gauge how optimal your engine design is.

temperature difference

The crux of the problem lies in understanding how changes in the temperature difference affect efficiency. According to the formula for thermal efficiency
\[ \eta = 1 - \frac{T_{\text{C}}}{T_{\text{H}}} \]
we can increase \( \eta \) by either increasing \( T_{\text{H}} \) or decreasing \( T_{\text{C}} \).
If we increase \( T_{\text{H}} \) by \( \Delta T \), the new efficiency will be:
\[ \eta' = 1 - \frac{T_{\text{C}}}{T_{\text{H}} + \Delta T} \]
If we decrease \( T_{\text{C}} \) by \( \Delta T \), the new efficiency will be:
\[ \eta'' = 1 - \frac{T_{\text{C}} - \Delta T}{T_{\text{H}}} \]
Both changes improve efficiency, but to what extent?

binomial expansion

To evaluate which scenario gives a better efficiency increase, we use the binomial expansion, a mathematical method to approximate the changes. For small \( \Delta T \), we expand the efficiency formulas to:
\[ \eta' \approx 1 - \frac{T_{\text{C}}}{T_{\text{H}}} + \frac{T_{\text{C}} \Delta T}{T_{\text{H}}^2} \] for an increase in \( T_{\text{H}} \).

And:
\[ \eta'' \approx 1 - \frac{T_{\text{C}}}{T_{\text{H}}} + \frac{ \Delta T}{T_{\text{H}}} \] for a decrease in \( T_{\text{C}} \).
Comparing these, we notice that the second term (which implies a higher contribution to efficiency change) is generally larger when decreasing \( T_{\text{C}} \) than when increasing \( T_{\text{H}} \), assuming small \( \Delta T \).