A radiator gives about a third of its rated output at 45 °C flow instead of 70 °C. Emitter size, not outdoor temperature, is what stops a heat pump working.
Output falls with roughly the 1.3 power of the water-to-air temperature difference, and radiators are rated at 50 K above room temperature. A heat pump at 45 °C sits around 21 K above the room, so the same radiator delivers a third of its label. Installers who cannot enlarge the emitters raise the flow temperature instead, and every extra degree costs roughly 2.5% of the efficiency the heat pump was bought for. Scandinavia runs on heat pumps at temperatures Britain never sees — the cold was never the problem.
Not in a house with underfloor heating, which is already a low-temperature emitter and will run happily at 35 °C flow. And in a genuinely uninsulated solid-wall house the fabric loss can exceed what any realistic emitter can put back — there the cold really is the problem.
At 45 °C the rule says 0.5 and the exact answer is 0.4061 — 23% high. At the point most people use it the shortcut is already outside its own 10% tolerance. It only holds between 56.4 °C and 75 °C, which is not where it gets used.
A radiator is rated at 70 °C flow, and output does not scale with the temperature difference — it scales with that difference to about the power 1.3. Guess linearly and a 45 °C heat-pump flow looks like half the heat; it is nearer 40%. That gap is the entire reason a heat pump needs bigger emitters.
The rule against the exact answer, computed across the range. Inside the shaded band the shortcut is close enough to use; outside it, reach for the calculator.
Estimate with the rule, then check it against the calculator that models it properly.
Open Heat Loss & Heat Pump Sizing →A radiator gives about a third of its rated output at 45 °C flow instead of 70 °C. Emitter size, not outdoor temperature, is what stops a heat pump working. Output falls with roughly the 1.3 power of the water-to-air temperature difference, and radiators are rated at 50 K above room temperature.