Question

An absorption refrigeration system receives heat from a source at \(120^{\circ} \mathrm{C}\) and maintains the refrigerated space at \(0^{\circ} \mathrm{C}\) If the temperature of the environment is \(25^{\circ} \mathrm{C}\), what is the maximum COP this absorption refrigeration system can have?

Short answer

Answer: The maximum COP for the given absorption refrigeration system is 2.276.

Step-by-step solution

Convert given temperatures to Kelvin

We need to work with absolute temperatures (in Kelvin) for this exercise. Convert the given temperatures from Celsius to Kelvin using the formula K = °C + 273.15. Heat source temperature: \(T_H = 120^{\circ}C + 273.15 = 393.15\,K\) Refrigerated space temperature: \(T_L = 0^{\circ}C + 273.15 = 273.15\,K\) Environment temperature (not needed for this calculation, but given for context): \(T_E = 25^{\circ}C + 273.15 = 298.15\,K\)

Apply the formula for the COP of a reversible Carnot refrigeration cycle

The formula for the COP of a reversible Carnot refrigeration cycle is: \(\text{COP}_\text{max} = \frac{T_L}{T_H - T_L}\) Plug in the values of \(T_H\) and \(T_L\) obtained in Step 1 into this formula to find the maximum COP: \(\text{COP}_\text{max}=\frac{273.15\,K}{393.15\,K - 273.15\,K}\)

Calculate the maximum COP

Perform the calculation to find the maximum COP: \(\text{COP}_\text{max}=\frac{273.15}{393.15 - 273.15} = \frac{273.15}{120}=2.276\) Therefore, the maximum COP this absorption refrigeration system can have is 2.276.

Key concepts

Carnot Refrigeration Cycle

The Carnot refrigeration cycle is the most efficient refrigeration cycle operating between two temperatures, and it forms the benchmark for assessing the efficiency of actual refrigeration systems. In the realm of thermodynamics, the Carnot cycle is revered for its theoretical maximum efficiency. It is a reversible cycle, consisting of two isothermal processes and two adiabatic processes.

During the isothermal expansion stage, the refrigerant absorbs heat from the cold reservoir, which is the space we wish to cool down. It then undergoes an adiabatic expansion, where its temperature and pressure drop without heat exchange. Next, during the isothermal compression stage, the refrigerant rejects heat to the hot reservoir. This is followed by an adiabatic compression that brings the refrigerant back to its initial state, completing the cycle.

This cycle sets the gold standard for efficiency, indicating the least amount of work needed for a given amount of heat removal. In practice, no real refrigeration system can achieve the Carnot cycle's efficiency due to irreversible processes such as friction and non-ideal gas behavior. Nonetheless, it serves as an essential tool for engineers to understand the limits of thermodynamic cycle efficiency and to aim for improvement in real-world systems.

Temperature Conversion

Understanding temperature conversion is crucial when working with thermodynamic equations, which typically require temperature measurements to be in Kelvin. Kelvin is the base unit of temperature in the International System of Units (SI), and it is imperative for accurately reflecting the thermodynamic properties of substances.

Temperature conversion is straightforward: to convert Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature. Thus, the conversion formula is:
\( T(K) = T(^\text{{\textdegree}}C) + 273.15 \)

This step is essential when calculating properties such as entropy or when utilizing equations such as those governing the Carnot cycle, where temperature must be represented in terms of absolute scale (Kelvin) to ensure accuracy. In essence, a sound understanding of temperature conversion ensures the credibility of thermodynamic computations and real-world applications like refrigeration.

Absorption Refrigeration

Absorption refrigeration is an alternative refrigeration technology that differs from the standard vapor-compression refrigeration systems. Instead of using a mechanical compressor to increase the refrigerant's pressure, absorption refrigeration systems use a heat source to drive the refrigeration cycle. This heat source can be waste heat from an industrial process, solar energy, or even natural gas.

The process revolves around the absorption of a refrigerant by a transport medium, often a solution of water and lithium bromide or ammonia and water. The solution circulates through the system, alternating between absorber, where it absorbs refrigerant vapor, and generator, where heat is applied to separate the refrigerant vapor from the solution. The cycle then proceeds similarly to conventional refrigeration cycles, with the refrigerant being condensed, expanded, and evaporated to remove heat from the space to be cooled.

One of the key advantages of absorption refrigeration is its ability to utilize heat sources that might otherwise be wasted, making it an attractive option for combined heating and cooling applications or in contexts where electricity is scarce or expensive.

Thermodynamic Efficiency

Thermodynamic efficiency is a measure of how well a system converts energy from one form to another, especially in the context of heat engines or refrigeration cycles. It's a dimensionless quantity that compares the work output of a cycle with the energy input necessary to run it. For cooling systems, this efficiency is often expressed in terms of Coefficient of Performance (COP), which is the ratio of heat removed from the cold space to the work input required.

The maximum COP of any refrigeration system is the COP of a Carnot cycle operating between the same two temperatures. As thermodynamic efficiency is bound by the second law of thermodynamics, actual systems invariably operate at a lower efficiency due to real-world factors such as friction, non-ideal fluid dynamics, and material limitations. Nevertheless, analyzing thermodynamic efficiency helps engineers design more effective and sustainable systems by minimizing energy losses and optimizing performance.