6.2 Refrigerators, heat pumps, and coefficients of performance

Refrigerator and Heat Pump Principles

Refrigerators and heat pumps both move heat from a cold region to a hot region, which doesn't happen spontaneously. This requires work input, and understanding how and how well these devices perform that task is a core application of the Second Law.

Vapor-Compression Refrigeration Cycle

The vapor-compression cycle is the workhorse behind nearly all practical refrigerators and heat pumps. It transfers heat from a low-temperature reservoir to a high-temperature reservoir through cyclic compression and expansion of a refrigerant.

The cycle has four main components, and each plays a specific role:

1. Compressor — Increases the pressure and temperature of the refrigerant vapor. The refrigerant enters as a low-pressure gas and leaves as a high-pressure, high-temperature gas. This is where the work input $W$ goes.
2. Condenser — The hot, high-pressure vapor releases heat $Q_H$ to the warm surroundings and condenses into a high-pressure liquid.
3. Expansion valve — The high-pressure liquid passes through a throttling device that drops its pressure sharply. The refrigerant partially evaporates and cools significantly. No work is done here; it's an irreversible pressure drop.
4. Evaporator — The cold, low-pressure mixture absorbs heat $Q_L$ from the cold space (like the inside of a fridge), and the remaining liquid evaporates completely before returning to the compressor.

The net effect: heat is pulled out of a cold space and dumped into a warm space, at the cost of compressor work.

Applications of Refrigeration Principles

Refrigerators maintain a cold space by removing heat from the interior and rejecting it to the surroundings. Examples include household refrigerators, walk-in coolers, and industrial refrigeration systems.

Heat pumps do the reverse job: they extract heat from a low-temperature source (outdoor air, ground, or water) and deliver it to a high-temperature sink (the indoor space you want to heat). The same cycle applies; the difference is which reservoir you care about. Common types include air-source, ground-source, and water-source heat pumps.

Coefficient of Performance

Definition and Calculation

The coefficient of performance (COP) measures how effectively a refrigerator or heat pump uses work input. It's defined as the ratio of the desired heat transfer to the work input required.

The key distinction is what counts as "desired":

• Refrigerator: You want to remove heat from the cold space, so:

$COP_{ref} = \frac{Q_L}{W}$

• Heat pump: You want to deliver heat to the warm space, so:

$COP_{hp} = \frac{Q_H}{W}$

Since energy is conserved around the cycle ($Q_H = Q_L + W$), these two COPs are related:

$COP_{hp} = COP_{ref} + 1$

This means a heat pump's COP is always at least 1 greater than the corresponding refrigerator COP operating between the same reservoirs.

Theoretical Maximum COP

The best possible performance comes from a Carnot refrigerator or heat pump, which operates as a reversed Carnot cycle between two thermal reservoirs. These set the upper bound on COP for any device operating between temperatures $T_H$ and $T_C$ (both in absolute units, i.e., Kelvin):

$COP_{ref,\text{Carnot}} = \frac{T_C}{T_H - T_C}$

$COP_{hp,\text{Carnot}} = \frac{T_H}{T_H - T_C}$

Note: these expressions come directly from the Carnot cycle analysis, not from the ideal gas constant $R$. You just need the two reservoir temperatures in Kelvin.

Example: A refrigerator keeps food at 3°C (276 K) while the kitchen is at 27°C (300 K).

$COP_{ref,\text{Carnot}} = \frac{276}{300 - 276} = \frac{276}{24} = 11.5$

That's the theoretical maximum. Real refrigerators fall well below this due to irreversibilities like friction in the compressor, pressure drops in piping, heat transfer across finite temperature differences, and non-ideal refrigerant behavior.

Factors Affecting Performance

Temperature Difference and Compressor Efficiency

Temperature difference is the single biggest factor. Look at the Carnot COP formulas: as $T_H - T_C$ grows, COP drops. More work is needed to "push" heat across a larger temperature gap.

• A refrigerator in a hot kitchen (large $T_H - T_C$) will have a lower COP than one in a cool pantry.
• A heat pump in a mild climate performs much better than one in a very cold climate, because the outdoor source temperature $T_C$ is higher.

Compressor efficiency also matters significantly. Mechanical friction, internal heat losses, and leakage all reduce the fraction of input energy that actually compresses the refrigerant. Scroll compressors, for instance, tend to have higher efficiencies than reciprocating compressors because they have fewer moving parts and less internal leakage.

Heat Exchanger Effectiveness and Refrigerant Selection

The condenser and evaporator are heat exchangers, and their effectiveness directly impacts COP. Factors include surface area, material thermal conductivity, and fluid flow characteristics. Increasing the surface area of condenser fins, for example, improves heat rejection and raises COP.

Refrigerant choice also affects performance. Different refrigerants have different boiling points, latent heats of vaporization, and specific heat capacities, all of which influence how efficiently the cycle operates. R-134a is common in household refrigerators, while R-410A is widely used in heat pumps for its favorable heat transfer properties.

Proper maintenance matters too. Dirty heat exchanger surfaces reduce heat transfer, and low refrigerant charge forces the compressor to work harder, both of which degrade COP over time.

Efficiency vs Other Systems

Comparison with Electric Resistance Heaters and Air Conditioners

The reason heat pumps get so much attention is that they move heat rather than generate it. This lets them deliver more thermal energy than the electrical energy they consume.

• A heat pump with a COP of 3 delivers 3 kJ of heat to your home for every 1 kJ of electrical energy the compressor uses. The extra 2 kJ comes from the outdoor environment.
• An electric resistance heater converts electrical energy directly into heat with a COP that maxes out at 1. Every 1 kJ of electricity gives you exactly 1 kJ of heat.

So a heat pump with COP = 3 is effectively three times as efficient as a resistance heater for space heating.

Air conditioners are functionally refrigerators designed to cool indoor spaces. They typically have lower COPs than heat pumps used for heating, because in cooling mode the device often works against a larger temperature difference (hot summer outdoor air vs. desired cool indoor air).

Advanced Technologies for Improved Efficiency

Several technologies push real-world COPs closer to theoretical limits:

• Variable-speed compressors adjust their output to match the actual heating or cooling demand, avoiding the energy waste of cycling on and off at full power.
• Multi-stage compression uses two or more compressors in series, reducing the pressure ratio each one handles and improving overall efficiency.
• Advanced refrigerants with lower global warming potential (GWP) are being developed to reduce environmental impact without sacrificing thermodynamic performance.

When comparing heating and cooling systems in practice, local climate, electricity prices, and the specific application all matter. A ground-source heat pump might be highly cost-effective in a moderate climate but less so where electricity is expensive and winters are extreme.