Heat Pumps: More Heat Than You Pay For

Moving heat instead of making it

Junior level — plain language, no maths

An electric heater is a perfect machine, and that is exactly its problem. Every joule of electricity you feed it becomes a joule of heat in the room - one for one, and there is no way to do better, because you cannot get more than everything. Yet a heat pump, fed the same joule, delivers three or four joules into the house. It looks like something for nothing.

It is not, because the heat pump is not making heat. It is moving it - carrying warmth that already exists outdoors into your living room, the way a fridge carries warmth out of the salad drawer and dumps it behind the cabinet. A heat pump is just a fridge pointed the other way, with your house as the bit that gets warm.

The obvious objection is that it is cold out there. But cold is only cold compared to us. Air at −5 °C is still 268 degrees above absolute zero and is packed with jiggling molecules; there is a colossal amount of heat in it. All you need is something even colder to soak it up - and that is the clever bit. Inside the pipes runs a liquid that boils at around −25 °C. To that liquid, a freezing night feels like a warm bath, so it boils, and boiling is how it swallows heat.

Then the machine goes round in a circle. The cold vapour gets squeezed by a compressor, and squeezing a gas makes it hot - feel the barrel of a bicycle pump after you inflate a tyre. Now hot, it flows through a coil indoors, where it gives its heat to your room and turns back into a liquid. Finally it is squirted through a tiny nozzle, the pressure collapses, the temperature plummets, and it goes outside to do it all again. The only electricity you pay for is the squeeze. And here is the practical punchline: the smaller the gap between the outdoor cold and the temperature you demand indoors, the less squeezing is needed. Feeding underfloor pipes at 35 °C is dramatically cheaper to run than feeding old radiators at 55 °C - which is why a heat pump installation is really an argument about your pipes, not about the box outside.

COP, the temperature lift, and why 400% is not a violation

Student level — the core equations

The number that matters is the coefficient of performance, \(\mathrm{COP} = Q_h/W\): heat delivered divided by electricity consumed. A good air-source machine feeding underfloor heating runs at 4 to 5. It is not called an efficiency deliberately, because efficiency compares an output to the same kind of input, and here the two are different. Energy is conserved exactly: \(Q_h = Q_c + W\). Four units arrive in the room, one came from the socket, three were dragged in from the cold outside. Nothing has been created.

What limits it is the second law. Heat does not flow uphill on its own, so moving it from cold to hot costs work, and the steeper the hill the more it costs. Run the Carnot cycle backwards and you get the ceiling: \(\mathrm{COP}_{\max} = \frac{T_h}{T_h-T_c}\), with both temperatures in kelvin. Everything hangs on that denominator - the temperature lift. Delivering 35 °C water from 7 °C air is a lift of 28 K and an ideal COP near 11. Delivering 55 °C water from −7 °C air is a lift of 62 K and an ideal COP near 5.3. Same machine, less than half the performance, purely from geometry on the temperature axis.

Real machines land at roughly 40-50% of the Carnot value, because heat exchangers need a temperature difference to push heat across, compressors are not isentropic, and the throttling valve destroys work on purpose. That factor is remarkably stable, which is why you can predict a datasheet from thermodynamics alone: at A7/W35 expect a COP near 4.5, at A−7/W55 expect close to 2.2. Around 0 to 5 °C in damp air there is a further nuisance - moisture freezes on the outdoor coil and the machine must periodically run backwards to melt it, costing perhaps 10% in that band.

Which explains the whole design brief for a heat-pump-ready house. Lower the flow temperature - bigger radiators, or underfloor pipes - and the COP rises for free. And the comparison with gas is not close: a condensing boiler delivers about 0.9 units of heat per unit of gas, while a heat pump at COP 3.5 running on electricity from a 50%-efficient gas turbine still delivers 1.75 units per unit of gas. On a grid with any renewables at all, it is not a contest.

The reversed cycle, where the Carnot fraction goes, and the refrigerant question

Scholar level — full mathematical depth

01Why work is required at all

Clausius' statement of the second law is precisely the design brief: no process can have as its sole result the transfer of heat from a colder to a hotter body. The word doing the work is sole. Transfer heat uphill and something else must change, and in a heat pump that something is the shaft work paid to the compressor, which leaves the system as part of the heat rejected indoors. Entropy accounting makes it exact: rejecting \(Q_h\) at \(T_h\) while absorbing \(Q_c\) at \(T_c\) requires \(Q_h/T_h \geq Q_c/T_c\), and combining that with \(Q_h = Q_c + W\) gives the ceiling directly - no cycle details needed.

02The ceiling, and the cycle that chases it

A reversed Carnot cycle between reservoirs gives \(\mathrm{COP} = T_h/(T_h-T_c)\), which diverges as the lift goes to zero - the correct and slightly counterintuitive statement that moving heat across no temperature difference is free. No real machine uses that cycle, because compressing a two-phase mixture is mechanically hostile. Instead the vapour-compression cycle rides the saturation dome: isobaric evaporation with a few kelvin of superheat, near-isentropic compression to the condensing pressure, isobaric desuperheating and condensation, then an isenthalpic throttle back to the evaporating pressure. On the p-h diagram, \(\mathrm{COP} = (h_2-h_3)/(h_2-h_1)\), which is a ratio of horizontal distances you can read off with a ruler.

03Where the missing half goes

Second-law efficiency \(\eta_{II} = \mathrm{COP}/\mathrm{COP}_{\text{Carnot}}\) sits at 0.40-0.50 for good equipment, and the losses are well ordered. Both heat exchangers need finite approach temperatures - typically 5 to 10 K - so the cycle's internal lift is always larger than the lift between the room and the outdoor air, and this alone accounts for much of the gap. Compressor isentropic efficiency runs 0.65-0.80. The throttling valve is deliberately irreversible: expanding a liquid through an expander would recover a few per cent, but the hardware costs more than it returns except in large plant. Then cycling losses, defrost, fan and pump parasitics, and pressure drops each take their slice.

04Frost, and the worst day of the year

An air-source evaporator runs several kelvin below ambient, so whenever ambient is roughly 0-7 °C and humid, the coil is below the dew point and below freezing: water condenses and then freezes on the fins, choking the airflow and the heat transfer. The remedy is a reverse-cycle defrost - the machine briefly becomes an air conditioner and melts its own ice with heat taken from the house - costing perhaps 5-12% of seasonal output in a maritime climate. Notice the awkwardness this creates for sizing: COP falls precisely when demand peaks, so the electrical draw at design conditions is much worse than the seasonal average suggests, and system planners care about that coincidence far more than about SCOP.

05The refrigerant question

The working fluid must have a saturation curve in the right place, a large latent heat, chemical stability, tolerable compressor discharge temperatures - and now, a low global warming potential, since some of it always leaks. The industry has walked from R22 (ozone-depleting, banned) to R410A (GWP 2088) to R32 (675), and is arriving at propane R290 (GWP 3, but flammable, so charge limits and outdoor siting) and transcritical CO2 R744, whose peculiar gliding gas cooler suits high flow temperatures and domestic hot water. It is an unusual engineering field where the environmental regulation, not the thermodynamics, sets the direction of travel.

06What it does to the system

Electrifying heat moves an enormous seasonal energy demand from gas pipes onto wires. A million heat pumps in a cold snap is a large, weather-correlated, simultaneous load arriving exactly when solar output is lowest - so the marginal question is never the annual average but the coincident winter peak. That is also the opportunity, because a building has thermal inertia: with a hot-water buffer or simply the fabric of the house, a heat pump is one of the most flexible loads on the grid, able to shift hours without anyone noticing. The heat pump is not merely a cheaper boiler; it is a demand-side asset that happens to keep you warm.