Heat Pump Math

How to Calculate COP, Refrigerant Charge, and Thermal Storage for Industrial Heat Pumps

The five core formulas behind industrial heat pumps, worked end to end with real inputs, units, and where each number comes from.

Every industrial heat pump metric traces back to coefficient of performance: COP equals useful heat output divided by electrical work input, both in kilowatts. A unit delivering 500 kW of process heat while drawing 125 kW electrical runs at COP 4.0. Reversed for cooling, you use the evaporator duty instead of condenser duty, so the same machine might show a cooling COP of 3.0 and a heating COP of 4.0 simultaneously. Keep the two separate. The COP Payback tool needs the heating figure for a boiler displacement case and the cooling figure when you are also offsetting a chiller, and mixing them inflates savings by 25 to 33 percent.

To sanity-check a quoted COP, compute the Carnot ceiling first. Carnot COP for heating equals the hot sink temperature divided by the temperature lift, both in Kelvin. Delivering 80 C water (353 K) while sourcing 10 C waste heat (283 K) gives a 70 K lift and a Carnot COP of 353 divided by 70, or 5.04. Real screw and reciprocating systems land at 45 to 55 percent of Carnot, so expect a field COP near 2.5 at that lift. If a vendor claims 4.5 across a 70 K lift, that implies 89 percent of Carnot and is not credible.

Heat exchanger effectiveness governs how much of the theoretical duty you actually capture. Effectiveness equals actual heat transferred divided by the maximum possible, where Qmax equals the minimum heat capacity rate Cmin times the inlet temperature difference. With a source stream at Cmin of 12 kW per Kelvin and a 40 K inlet difference, Qmax is 480 kW. Measure 384 kW of real transfer and effectiveness is 0.80. The Heat Exchanger Yield calculator uses this to convert a clean-design duty into the derated duty you will see after fouling, which typically costs 5 to 12 percent within the first year.

Refrigerant charge is a mass balance, not a guess. Total charge equals the sum of each component internal volume times the refrigerant density in that section. Liquid line volume times liquid density (roughly 1,150 kg per cubic meter for R-134a at condensing conditions) dominates, while suction line gas contributes only 20 to 40 kg per cubic meter. A system with 0.045 cubic meters of liquid-phase volume holds about 52 kg in that section alone. The Refrigerant Charge Cost tool multiplies that mass by price per kilogram, so a 10 percent volume error on a 90 kg charge moves the number by 9 kg, often 200 to 400 dollars.

Thermal storage sizing for sensible water tanks uses Q equals mass times specific heat times temperature swing. A 10,000 liter tank (10,000 kg) with a 30 K usable swing and water specific heat of 4.18 kJ per kg per Kelvin stores 1.254 GJ, which is 348 kWh. Divide by your discharge duty to get run time: at 150 kW that tank covers 2.32 hours. The Thermal Storage Sizing calculator inverts this to find the volume you need for a target hold time. Watch the usable swing, since stratification and approach temperatures often shrink a nameplate 40 K swing to a real 28 to 32 K.

Defrost energy per cycle is the sum of sensible warming and latent melting of accumulated frost. Melting 4 kg of ice needs 4 times 334 kJ per kg, or 1,336 kJ of latent load, plus sensible energy to raise coil metal and ice from minus 8 C to 2 C. Total often runs 1.8 to 2.5 MJ per defrost. Multiply by cycles per day (commonly 4 to 8) and by days to get annual defrost draw. The Defrost Energy Cost calculator combines this with electricity price, and on a poorly controlled coil defrost can consume 6 to 10 percent of total heat pump energy.

Compressor capacity from a test stand comes from mass flow times enthalpy difference. Refrigerant mass flow of 0.85 kg per second across an evaporator enthalpy rise of 165 kJ per kg yields 140.25 kW of cooling duty. The Compressor Test Capacity calculator pairs this with swept volume and volumetric efficiency, where flow equals displacement times speed times suction density times volumetric efficiency. A 180 cubic centimeter compressor at 50 revolutions per second with 0.90 volumetric efficiency and 15 kg per cubic meter suction density moves about 0.122 kg per second, which you cross-check against the measured enthalpy method.

Pull inputs from the right instruments so the math holds. Take COP electrical input from a true-power meter at the motor terminals, not nameplate amps, since power factor and part-load derating can swing readings 8 to 15 percent. Read inlet and outlet temperatures with calibrated RTDs within 0.2 K, because a 1 K error on a 40 K difference is a 2.5 percent duty error that compounds through effectiveness and storage calculations. Log flow with an ultrasonic or Coriolis meter rather than pump curves. Every downstream number, from Refrigerant Charge Cost to Thermal Storage Sizing, is only as good as these four field measurements.

Published 2026-07-02.