Thermal systemsEnergy

Heat pump

A vapor-compression heat pump on the refrigerant cycle — the P-h diagram, compressor and heat exchangers, the COP and its Carnot limit, how COP falls with temperature lift, and the seasonal SCOP that beats resistance heat fourfold.

DjiniousLabModelling & simulation

A vapor-compression cycle drawn on a pressure-enthalpy diagram, the four states traced around the two-phase saturation dome — compression, condensation, expansion, evaporation

COP at 0 °C → 45 °C: 4.27COP at 0 °C → 45 °C
of the Carnot limit: 60%of the Carnot limit
seasonal SCOP: 3.70seasonal SCOP
less electricity vs resistance: 3.7×less electricity vs resistance
balance point: −2 °Cbalance point
design notebooks: 10design notebooks

One unit of electricity, four of heat.

A heat pump doesn't make heat — it moves it, and that's why it can deliver three to four units of warmth per unit of electricity. The trick is a refrigerant boiling and condensing around a loop. This program models the vapor-compression cycle on the pressure-enthalpy diagram, follows it through the compressor and heat exchangers, and answers the questions that decide a system: how high a COP, how far it falls in the cold, and what it actually averages over a heating season.

Boil low, condense high.

The refrigerant evaporates at low pressure, pulling heat from the cold outdoor air; the compressor squeezes it to high pressure; it condenses indoors, dumping that heat plus the compressor work; and an expansion valve drops it back to low pressure to start again. Drawn on the pressure-enthalpy diagram, the cycle is a rectangle riding the two-phase saturation dome, and every performance number — heat delivered, work in, coefficient of performance — is a width on that chart. At a 0 °C source and a 45 °C sink the cycle returns a COP of 4.27, about 60% of the reversible Carnot bound.

The vapor-compression cycle traced on the pressure-enthalpy diagram with the two-phase dome and the four state points labelled — The cycle on the P-h diagram: compression up the right, condensation across the top, expansion down the left through the saturation dome, evaporation along the bottom. The heat delivered is the top span, the work is the compression rise, and their ratio is the COP — the whole machine on one chart.

COP falls exactly when you need it most.

A heat pump's efficiency depends on the temperature lift — how far it has to pump heat uphill. On a mild day, lifting from 5 °C to a 45 °C flow, COP is near five. On a cold one, lifting from −15 °C, it drops to three, because the same compressor faces a far larger pressure ratio. That's the central design tension: demand is highest when efficiency is lowest. The program traces COP across the whole source-and-sink envelope, finds the balance point where capacity meets the building's heat loss (around −2 °C here), and shows where a backup is needed — the analysis that sizes a real installation.

COP_heat falling with outdoor source temperature for several supply temperatures, from about 6.7 at +15 °C to 3.1 at −15 °C — COP versus the cold side: heating COP falls from 6.7 with a mild +15 °C source to 3.1 at −15 °C (45 °C supply), and a lower supply temperature lifts the whole family of curves. Underfloor heat at 35 °C beats radiators at 55 °C for exactly this reason.

A heatmap of heating COP over source temperature and sink temperature, brightest at low lift — The full operating map: heating COP over every source-and-sink pair. The bright low-lift corner is where a heat pump shines; the dim high-lift corner is where it struggles — and where lowering the supply temperature or raising the source (ground loops, waste heat) buys the most.

Heat-pump capacity falling with outdoor temperature, crossing the building heat-loss line at the balance point, with an auxiliary-heat region — Capacity versus demand: the heat pump's output falls as it gets colder while the building's heat loss rises, and they cross at the balance point (~−2 °C). Below it the shaded region needs auxiliary heat — the sizing decision that defrost and cold-climate performance turn on.

Seasonal heat delivered per unit of input energy — the heat pump at SCOP 3.70 versus resistance heat at 1.0 and a boiler — What it averages over a season: weighting COP by the hours at each outdoor temperature gives a seasonal SCOP of 3.70 — 3.7× less electricity than a resistance heater for the same warmth, and well ahead of a condensing boiler. The single number that decides the bill.

Every number is one you can re-run.

The sign-off notebook re-derives each requirement from the same cycle the program builds.

Result

COP @ 0 °C / 45 °C: 4.27
Second-law efficiency: 60.4%
COP @ −15 °C source: 3.06
Seasonal SCOP: 3.70
Requirements verified: 6 / 6

Requirement

COP @ 0 °C / 45 °C: ≥ 4.0
Second-law efficiency: ≥ 50%
COP @ −15 °C source: ≥ 3.0
Seasonal SCOP: ≥ 3.5
Requirements verified: PASS

Idealized refrigerant, real cycle.

The refrigerant is an idealized R290 — Clausius-Clapeyron saturation pressure with constant latent heat and specific heats, not a CoolProp/REFPROP equation of state — which keeps every number reproducible and the physics teachable. The cycle is steady thermodynamics, not a dynamic component model; the compressor is an isentropic-efficiency block; the SCOP is a temperature-bin method, not a full EN 14825 rating. That is exactly the fidelity a heat-pump concept and sizing study needs first — choosing the refrigerant and pressures, mapping COP against climate, finding the balance point, and trading supply temperature against efficiency — on your own numbers, before a manufacturer's detailed model.