Problem 15 - An Introduction to Thermal Physics

Question

In an absorption refrigerator, the energy driving the process is supplied not as work, but as heat from a gas flame. (Such refrigerators commonly use propane as fuel, and are used in locations where electricity is unavailable. \(^{*}\) ) Let us define the following symbols, all taken to be positive by definition: \(Q_{f}=\) heat input from flame \(Q_{c}=\) heat extracted from inside refrigerator \(Q_{r}=\) waste heat expelled to room \(T_{f}=\) temperature of flame \(T_{c}=\) temperature inside refrigerator \(T_{r}=\) room temperature (a) Explain why the "coefficient of performance" (COP) for an absorption refrigerator should be defined as \(Q_{c} / Q_{f}\) (b) What relation among \(Q_{f}, Q_{c},\) and \(Q_{r}\) is implied by energy conservation alone? Will energy conservation permit the COP to be greater than \(1 ?\) (c) Use the second law of thermodynamics to derive an upper limit on the \(\mathrm{COP},\) in terms of the temperatures \(T_{f}, T_{c},\) and \(T_{r}\) alone.

Step-by-Step Solution

Answer

COP is maximized at \( \text{COP}_{max} = \frac{T_c}{T_f - T_r} \).

Step 1: Understanding COP

The Coefficient of Performance (COP) for an absorption refrigerator is defined as \( \frac{Q_c}{Q_f} \). This definition makes sense because \( Q_c \), the heat extracted from the interior, is the useful energy transfer we want. The energy that we put in is \( Q_f \), the heat absorbed from the flame. Therefore, \( \frac{Q_c}{Q_f} \) measures the effectiveness of using the absorbed heat to extract heat from the refrigerator.

Step 2: Applying Energy Conservation

According to the energy conservation principle, the total energy input into the system must equal the total energy output. Therefore, by energy conservation, we have:\[ Q_f = Q_c + Q_r \]This implies that the input heat \( Q_f \) is used in part to extract heat \( Q_c \) and the remaining \( Q_r \) is expelled as waste heat. This equation suggests that energy conservation alone might allow a COP greater than 1 because \( Q_c \) could be larger than \( Q_f \).

Step 3: Using the Second Law of Thermodynamics

The second law of thermodynamics gives restrictions on energy transfer processes. For an absorption refrigerator operating between three reservoirs at temperatures \( T_f \), \( T_c \), and \( T_r \), this creates a limitation for the COP as follows:The entropy change due to the process can be written as:\[ \frac{Q_f}{T_f} - \frac{Q_c}{T_c} - \frac{Q_r}{T_r} \geq 0 \]Reorganizing using \( Q_f = Q_c + Q_r \), and solving for \( \frac{Q_c}{Q_f} \), we can find the maximum COP. Solving these equations gives the upper limit:\[ \text{COP}_{max} = \frac{T_c}{T_f - T_r} \]

Key Concepts

Coefficient of Performance (COP)

In the world of refrigeration, the Coefficient of Performance, often abbreviated as COP, is a crucial metric. It gives us an idea of how effectively our refrigerator, especially in the case of an absorption refrigerator, is performing.

The COP is defined as the ratio of the heat extracted from inside the refrigerator (\( Q_c \)) to the heat input from an external source, like a flame (\( Q_f \)). So, the formula for COP in this context is:

• \( \text{COP} = \frac{Q_c}{Q_f} \)

This ratio tells us the efficiency of using the supplied heat to achieve cooling. Higher COP values indicate better performance, as more heat is extracted from the fridge per unit of heat supplied from the flame.

Understanding COP is like looking at the miles per gallon of a car; it helps compare different refrigerators' efficiency using the same input, encouraging smart energy use.

Energy Conservation

Energy conservation is a fundamental concept that states that energy cannot be created or destroyed; it can only change forms. In the context of an absorption refrigerator, this principle helps us analyze how energy is distributed across the system.

Here, the heat input from the flame (\( Q_f \)) is the total energy supply, and according to energy conservation, it splits into two parts:

• Heat extracted from inside the refrigerator (\( Q_c \))
• Waste heat expelled to the environment (\( Q_r \))

Which gives us this equation:

• \( Q_f = Q_c + Q_r \)

This equation implies that all of the energy supplied (\( Q_f \)) ensures the fridge operates effectively, and the leftover energy not used for cooling is expelled as waste heat (\( Q_r \)).

With this understanding, one might wonder if having the COP greater than 1 is possible. The equation alone suggests yes, because the heat extracted (\( Q_c \)) can indeed be higher than the heat supplied (\( Q_f \)), especially in systems that recycle energy efficiently.

Second Law of Thermodynamics

The Second Law of Thermodynamics plays a critical role in determining the constraints of energy transfers in an absorption refrigerator, as it introduces the concept of entropy.

This law essentially places a limit on efficiency regarding energy transfers and heat flow among reservoirs at different temperatures—here, the flame (\( T_f \)), the refrigerator interior (\( T_c \)), and the environment (\( T_r \)). The law can be simplified into an inequality that governs the operation:

• \( \frac{Q_f}{T_f} - \frac{Q_c}{T_c} - \frac{Q_r}{T_r} \geq 0 \)

Using the energy conservation equation (\( Q_f = Q_c + Q_r \)) and simplifying, we can solve for the COP's upper limit, which is specified by:

• \( \text{COP}_{max} = \frac{T_c}{T_f - T_r} \)

This equation illustrates that the maximum achievable COP depends on the temperatures of the involved heat reservoirs. The further apart these temperatures are, the less efficient the system becomes. This understanding is crucial for optimizing absorption refrigeration systems for better performance.