Practical Examples

Practical examples of refrigerators and heat pumps: system sizing, energy consumption calculation and comparison with theoretical limits

Conceptual context

Practical examples of refrigerators and heat pumps connect theoretical thermodynamic principles with the design and operation of real systems. The central problem is determining how much electrical power a compressor needs to maintain a space at a desired temperature, how much heat the system can extract or deliver, and how its measured efficiency compares with the Carnot theoretical limit. This connection between theory and practice is essential for system sizing, energy cost estimation and design quality evaluation.

🟢 Essential level

A domestic refrigerator extracts heat $Q_C$ from the cold interior and delivers it to the warm environment by consuming electrical power $W_{comp}$ in the compressor. The $COP_{real}$ indicates how many units of heat are extracted per unit of power consumed. A typical refrigerator has a $COP_{real}$ between 2 and 4, meaning that for each watt of electricity consumed, 2 to 4 watts of heat are extracted from the interior.

A heat pump does the same but with the opposite objective: it extracts heat from the cold exterior and delivers it to a building interior. Its $COP_{real}$ measures how many units of heat are delivered to the heated space per unit of power. Geothermal heat pumps achieve $COP_{real}$ of 3.5 to 5, converting each watt of electricity into 3.5 to 5 watts of heating.

The $COP_{Carnot}$ establishes the absolute efficiency ceiling. No real system reaches it. The Carnot fraction $\eta_{\\text{Carnot}}$ measures what percentage of the theoretical limit the system achieves, typically 30 to 60 %.

🔵 Formal level

Basic formulas

The definition of real refrigeration COP is:

Real refrigeration COP

For a heat pump, the real COP definition is:

Real heat pump COP

The reference Carnot COP for refrigeration is calculated as:

Reference Carnot COP

The Carnot fraction quantifies the relative efficiency of the real system:

Carnot fraction

Real student questions

Why does a refrigerator consume 150 W but extract 375 W of heat? Because the refrigerator does not generate heat but transfers it. The compressor performs work to move heat from the cold interior to the warm exterior. A COP of 2.5 means each watt of electricity moves 2.5 watts of heat, which does not violate any law.

How do I know if my system is well designed? By calculating the Carnot fraction. If the system achieves 45 to 55 % of the Carnot COP, it is in the normal range. Below 30 %, there is significant room for improvement.

Final synthesis

Practical examples demonstrate that thermodynamic principles are not abstractions but predictive tools. The $COP_{Carnot}$ sets the ceiling, the $COP_{real}$ measures reality, and the Carnot fraction quantifies the distance between them. The daily energy consumption $E_{dia}$ translates these coefficients into real economic impact: multiplying $W_{comp}$ by the operating hours gives the total energy consumed, which converted to kWh allows directly estimating the electricity cost. Mastering the sizing of these systems requires integrating theoretical thermodynamics with the practical constraints of real components, including estimation of $E_{dia}$ to evaluate economic viability. In industrial applications, a small improvement of 5 to 10 percent in $COP_{real}$ can translate to savings of thousands of euros annually, making these calculations essential for engineering practice. The systematic comparison between real performance and the Carnot limit through the Carnot fraction provides the universal diagnostic tool for assessing the quality of any refrigeration or heat pump installation.