The dT Power-Law Formula
Radiators transfer heat primarily by free convection. The rate is roughly proportional to dTn, where dT is the difference between mean water temperature and room temperature, and n is a radiator-type-specific exponent. The formula for actual output at any condition is BTU_actual = BTU_rated × (dT_actual / dT_rated)n. The rated dT depends on the test standard — US fin-tube baseboard uses 115 °F rated dT (180 °F water, 65 °F room), cast iron IBR uses 100 °F, and European panel radiators (EN 442) use 50 K ≈ 90 °F.
Typical Exponents
n = 1.30 for panel radiators (EN 442) and copper fin-tube baseboard. The 1.30 comes from the empirical behavior of convection-dominated heaters. n = 1.45 for cast iron radiators because radiation (which scales as T4 and thus dT~1.5 at linearization) is a significant fraction of total output. Some manufacturers publish exact exponents from their test data — use those when available. 1.33 is a safe general-purpose default.
Why Low-Temperature Hydronics Matter
Traditional oil and gas boilers delivered 180-200 °F water. Modern condensing boilers run 140-160 °F for best efficiency. Heat pumps deliver 95-120 °F. For the same radiator, this shift means dramatically less output: a panel radiator rated for 180 °F water at 70 °F room (dT 110) drops to about 38% of rated at 120 °F water (dT 50). So a retrofit from an oil boiler to an air-source heat pump usually requires either larger radiators, added radiant floor, or accepting longer run times.
Sizing for Retrofits
For a heat pump retrofit, pick radiators with 2-3× the calculated heating load at the target low water temperature. A room with 5,000 BTU/hr heat loss on 120 °F water needs a radiator with 13,000 BTU/hr rated output at 180 °F. That sounds extreme until you see that at 120 °F water its real output is 5,000 BTU/hr. Low-temp hydronic design always works backwards from the operating condition, not the rating.
Radiator Capacity Ratio at Common Water Temperatures
Output as a fraction of nameplate rating at different mean water temperatures, 70 °F room, panel radiator (n = 1.30) rated at 180 °F water (dT = 110 °F):
| Mean water temp | dT (water − room) | Capacity ratio | Use case |
|---|---|---|---|
| 180 °F (82 °C) | 110 °F | 100% | Old oil/gas boiler (rating cond) |
| 160 °F (71 °C) | 90 °F | 77% | Standard condensing boiler |
| 140 °F (60 °C) | 70 °F | 56% | Low-temp condensing design |
| 130 °F (54 °C) | 60 °F | 46% | Air-to-water heat pump warm |
| 120 °F (49 °C) | 50 °F | 37% | Typical heat pump design |
| 110 °F (43 °C) | 40 °F | 28% | Heat pump cold weather |
| 95 °F (35 °C) | 25 °F | 16% | Radiant floor only |
Heat pump retrofit math: a radiator rated 10,000 BTU/hr at 180 °F delivers only 3,700 BTU/hr at 120 °F water. To heat the same room with a heat pump you need ~2.7× the nameplate capacity. Options: bigger radiators, more radiators, or add radiant floor loops to distribute low-temp heat over more area.
Frequently Asked Questions
What is mean water temperature?
The average of supply and return temperatures. For 180 °F supply and 160 °F return, mean = 170 °F.
Can I use a thermostatic radiator valve (TRV)?
Yes — TRVs throttle flow to maintain room temperature. Output varies with both dT and flow, but the dT formula is usually accurate enough for sizing.
How do I size radiators for a heat pump?
Use the calculator with your heat pump's design supply temperature (110-120 °F) to find the required rated capacity at standard conditions.
Why do European radiators use 50K rated dT?
EN 442 chose 50K because 70 °C water and 20 °C room (typical European conditions) give exactly 50 K dT — standardizing comparison.
What is the difference between IBR and SBI ratings?
IBR is Institute of Boiler and Radiator (now HYDI) — the US standard with 170 °F water and 70 °F room. SBI is the Canadian equivalent. Both use n ≈ 1.45 for cast iron.
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