13.5 Gas Refrigeration, Heat Pump Cycles & Cryogenics

The Reversed Carnot Cycle: The Absolute Limit

Every refrigeration system is a heat engine run backwards: work is supplied to move heat from a cold reservoir at $T_L$ to a hot reservoir at $T_H$. The best possible performance is set by the reversed Carnot cycle.

Coefficient of performance is defined as the useful effect divided by the work input, and the definition of useful effect differs between the two applications:

$\text{Refrigerator: } COP_R = \frac{Q_L}{W} = \frac{T_L}{T_H - T_L}$

$\text{Heat pump: } COP_{HP} = \frac{Q_H}{W} = \frac{T_H}{T_H - T_L}$

Since $Q_H = Q_L + W$, the two are related by

$\boxed{COP_{HP} = COP_R + 1}$

This relation holds for any refrigeration system, not merely the Carnot one, and it is examined constantly. A heat pump is always "better" than a refrigerator by exactly one — not because it is a superior machine, but because the work input itself ends up as useful heat in the hot space.

Note also that COP is not an efficiency and routinely exceeds one. Quoting a COP as a percentage is a conceptual error.

Worked example

A refrigerator maintains $-10$ degrees Celsius while rejecting heat at 40 degrees Celsius. The maximum possible COP is

$COP_R = \frac{263}{313 - 263} = \frac{263}{50} = 5.26$

A heat pump between the same reservoirs would have $COP_{HP} = 6.26$. Note that temperatures must be absolute; using Celsius here would give a meaningless answer, which is the single commonest error in the topic.

The COP falls sharply as the temperature difference widens. Cooling to $-30$ degrees Celsius while rejecting at the same 40 gives $COP_R = 243/70 = 3.47$ — a third lower. This is why cold-storage design pays such attention to minimising the temperature lift.

Units of Refrigeration

One tonne of refrigeration is the rate of heat removal required to freeze one short ton of water at 0 degrees Celsius into ice at 0 degrees Celsius in 24 hours:

$1 \text{ TR} = 3.5 \text{ kW} \approx 211 \text{ kJ/min}$

The Bell-Coleman (Reversed Brayton) Cycle

Where vapour compression uses a fluid that changes phase, the gas refrigeration cycle keeps the working fluid — normally air — in the gaseous phase throughout. It is the Brayton cycle run in reverse and comprises four processes:

The critical difference from vapour compression is that expansion occurs in an expander that recovers work, not through a throttle valve. Throttling a gas produces almost no cooling — as the Joule-Thomson analysis showed, an ideal gas does not cool on throttling at all — so an expansion machine is essential.

With $r_p$ the pressure ratio, the COP of the ideal cycle is

$COP = \frac{1}{r_p^{(\gamma-1)/\gamma} - 1}$

Equivalently, in terms of temperatures,

$COP = \frac{T_1 - T_4}{(T_2 - T_3) - (T_1 - T_4)}$

Comparison with vapour compression

The COP is poor because sensible heat capacity is far smaller than latent heat, so a large mass flow of air is needed for a given cooling duty. Yet the cycle survives in aircraft, where the working fluid is available free from the engine compressor, no refrigerant charge or leakage is possible, and the weight per tonne of refrigeration is far lower than any vapour-compression plant.

Aircraft Refrigeration Systems

Aircraft cabins need cooling even at altitude because ram air compression, avionics and solar gain generate substantial heat, while bleed air from the engine compressor arrives very hot. Standard configurations, in increasing sophistication:

The bootstrap arrangement is the most common on commercial aircraft, because using the turbine output to drive a second compressor rather than wasting it improves the overall pressure ratio available for expansion.

Heat Pumps

A heat pump is thermodynamically identical to a refrigerator; only the useful effect differs. Where the ambient is the source and a building is the sink, the pump delivers $Q_H$ units of heat for every $W$ units of work, with $COP_{HP} = Q_H/W$ typically 3 to 4 in temperate conditions.

The engineering significance is that direct electric heating delivers one unit of heat per unit of electricity, while a heat pump delivers three or four. Heat pumps are therefore the most effective route to decarbonised space heating wherever the temperature lift is modest.

Performance falls as source temperature drops, because the lift $T_H - T_L$ widens. Ground-source heat pumps exploit the near-constant ground temperature below a few metres to keep the lift small year-round, at the cost of borehole or trench installation.

A reversible heat pump switches the roles of the two heat exchangers with a four-way valve, giving cooling in summer and heating in winter from one machine — the basis of the reverse-cycle air conditioner.

Cryogenics and Gas Liquefaction

Cryogenics concerns temperatures below about 123 K. Producing them requires a different approach, because no single cycle spans from ambient to a few kelvin.

The Linde-Hampson process

Gas is compressed, cooled to ambient, passed through a counter-current heat exchanger, then throttled. The throttling produces cooling only if the gas is below its inversion temperature, as established in the thermodynamic-relations section. The cooled gas returns through the heat exchanger, pre-cooling the incoming stream, so the system cascades down to progressively lower temperatures until liquefaction begins.

Air and nitrogen work directly because they sit well below their inversion temperatures at ambient. Hydrogen must be pre-cooled with liquid nitrogen and helium with liquid hydrogen before throttling produces any cooling at all.

The Claude process

An improvement in which part of the stream is diverted through an expansion engine rather than a throttle. Because the expander extracts work, the temperature drop is much greater than in throttling alone, and the process is thermodynamically far more efficient. Most industrial air-separation plants use this principle.

Industrial relevance

Cryogenic air separation produces the oxygen and nitrogen used in steelmaking and in inerting; liquefied natural gas transport depends on cryogenic containment; and in coal operations, nitrogen is used for inertisation of sealed-off districts to suppress spontaneous heating and mine fires — a direct link between this topic and mine safety practice.