Heat crossing empty space, needing no medium at all. Near room temperature it carries about as much as free convection in still air; the fourth-power law makes it dominant in a furnace, and the curve beside the stage shows where it pulls ahead.
Radiation — the Stefan–Boltzmann Law
This browser cannot draw the 3D view. The diagram works exactly the same.
Heat rate Q
18.21 kW
Thermal resistance R
0.02745 K/W
Heat flux q
18,213 W/m²
Driving ΔT
500 K
Radiative h_r
36.43 W/m²·K
Emitted at T₁ (εσT⁴)
18,581 W/m²
Mode
Radiation
Formula
Q = εσFA(T₁⁴ − T₂⁴)
No warnings — every assumption behind these numbers holds for this set-up.
T₁⁴ − T₂⁴ = 4.10e+11 − 8.10e+9 = 4.02e+11 K⁴
h_r = εσF(T₁+T₂)(T₁²+T₂²) = 36.43 W/m²·K — exact, not a linearisation
R = 1 / (h_r·A) = 0.02745 K/W
Q = ΔT / R = 500 / 0.02745 = 18213.2 W
check: εσFA(T₁⁴−T₂⁴) = 18213.2 W — the same number
Pick a preset to load a real case, or set your own with the controls below.
The heat crossing a slab is proportional to its area and to the temperature difference, and inversely proportional to its thickness. The constant k is a property of the material, which is what makes it tabulable.
The instrument is now set to a carbon steel wall 100 mm thick over 2 m², between 300 °C and 50 °C. That gives 302,500 W, with a resistance of 0.0008 K/W and a gradient of 2,500 K/m.
Thickness works against you; area and temperature difference work for you.
Change nothing but the material. The same 100 mm wall over the same 2 m² with the same 250 K across it passes 302,500 W in carbon steel and 190.0 W in fibreglass — a factor of about 1,592.
No geometry changed. Nothing was made thicker. This is why insulation is a material problem long before it is a thickness problem, and why the material list here deliberately runs from copper all the way down to still air.
A change of material is worth more than any plausible change of thickness.
Rewrite Fourier’s Law as Q = ΔT/R with R = L/(kA) and conduction becomes an Ohm’s law. Resistances in series add, which is how a real wall — plaster, brick, insulation, render — is actually solved.
The reason this matters here is that all three modes take the same form. Convection gives R = 1/(hA) and radiation gives R = 1/(hrA). The readout shows R in every mode for exactly that reason, and it is the bridge to the overall coefficient U in a heat exchanger, where 1/U = ΣR.
Every mode is a resistance. That is the one idea this whole tool is built on.
Take the same steel wall down from 100 mm to 5 mm and the heat rises from 302,500 W to 6,050,000 W — twenty times, exactly as R ∝ L says it must.
This is why the metal wall of a heat exchanger tube contributes almost nothing to its total resistance, and why a heat exchanger is really a contest between the two convection films on either side of it. The metal is there for strength and corrosion, not for insulation.
In a metal wall the resistance is usually negligible; the films either side are not.
The heat leaving a surface into a fluid is hA times the temperature difference. Simple to write — and the entire difficulty is hidden inside h, which is not a material property and cannot be looked up.
The stage is now a 1.5 m vertical panel at 60 °C in still 20 °C air. The correlation gives h = 4.7 W/m²·K from Ra = 1.09e+10 and Pr = 0.71. Watch the working: no number was typed in.
h is a result of the fluid, the shape, the speed and ΔT — never an input.
Same panel, same temperatures, now with 8 m/s of air across it. h goes from 4.7 to 15.1 W/m²·K — about 3.2 times — for nothing but moving the air.
The honest test of which regime you are in is the ratio Gr/Re². Well below one, forced flow dominates and buoyancy can be ignored. Well above one, it is really free convection. Near one it is mixed and neither correlation is trustworthy — and this tool says so instead of quietly picking a side.
Forced convection beats free convection by roughly an order of magnitude.
A 50 mm pipe at 1.5 m/s. In water the correlation gives h = 6,240 W/m²·K; in air at the same speed it gives 18.9. A factor of about 330.
Water is denser, has a far higher specific heat and conducts better, and all three feed the Reynolds and Prandtl numbers. It is why engines are water-cooled and why an air-cooled heat exchanger needs enormous fins to keep up.
The fluid is worth more than the speed. Water beats air by two orders of magnitude.
Fluid touching a wall does not move — the no-slip condition — so heat must conduct across a thin film before anything can carry it away. The thickness of that film essentially is 1/h, and every trick for improving convection works by thinning it.
The Prandtl number sets how the thermal layer compares with the velocity layer. In air (Pr ≈ 0.71) the thermal layer is the thicker of the two. In engine oil (Pr ≈ 1,214) it is dramatically thinner — 19.0 mm of velocity layer against 1.78 mm of thermal layer.
Prandtl number is the ratio of the two boundary layers, and it decides the shape of everything.
Everything above absolute zero radiates, and the emissive power goes as the fourth power of absolute temperature. σ is 5.670374419 × 10⁻⁸ W/m²·K⁴ — since 2019 an exactly defined constant rather than a measured one.
Note what the formula is not. It is not εσA(ΔT)⁴. With T₁ = 800 K and T₂ = 300 K the difference of fourth powers is about six times the fourth power of the difference, and the reference tool this page was built against prints the wrong one on its readout chip.
T₁⁴ − T₂⁴ is not (T₁ − T₂)⁴, and the two are not close.
Against a 20 °C background at ε = 0.8, one square metre at 350 K radiates about 360 W — a light breeze would beat that easily. Take the same surface to 1200 K and it radiates about 93,744 W, which no practical air flow can match.
The temperature rose by a factor of 3.4; the heat rose by a factor of about 260. The curve in the right-hand pane is that fourth power, with your operating point marked on it — which is the fastest way to see where the crossover sits.
Below about 600 °C radiation is a correction; above it, radiation is the answer.
Same geometry, same temperatures, only the finish changes. At ε = 0.04 — polished aluminium — the surface radiates 4,687 W. At ε = 0.98 — matt black paint — it radiates 114,836 W. A factor of 25, from a coating.
Polished aluminium is 0.04 and anodised aluminium is 0.82: the same metal, twenty times apart, separated by a few microns of oxide. Low-emissivity window coatings, loft radiant barriers and the silvering in a vacuum flask are all this idea used deliberately.
Visible shine is a poor guide: white paint is almost black in the infrared.
Two surfaces exchange radiation only insofar as they can see each other. The view factor F is the fraction of radiation leaving one that lands on the other, and setting it to 1 assumes each sees only the other and nothing else.
That assumption is exactly right for a small body inside a large enclosure — a component in a furnace, a thermometer in a duct — and wrong for almost everything else. Halving F halves the net exchange, because the geometry simply removes that fraction of the paths.
Two surfaces that cannot see each other exchange nothing, however hot they are.
Across a vacuum. Modest when cool, unbeatable when hot, because of T⁴. The law is Q = εσFA(T₁⁴ − T₂⁴), and as a thermal resistance it reads R = 1 / (h_r·A), so the heat rate is always Q = ΔT / R.
At the instrument's opening set-up it gives a heat rate of 18.21 kW through a resistance of 0.02745 K/W, a flux of 18,213 W/m² and a radiative h_r of 36.43 W/m²·K. Open The arithmetic, with your numbers in it above to see each of those worked through with the values on the sliders.
Every object above absolute zero radiates. The Stefan–Boltzmann law says the emissive power of a blackbody goes as the fourth power of absolute temperature, and for a real surface exchanging with another the net flow is Q = εσF A (T₁⁴ − T₂⁴), where ε is the emissivity, σ is 5.670374419 × 10⁻⁸ W/m²·K⁴ and F is the view factor describing how much of one surface the other can actually see.
The fourth power is the whole character of the mode. Against a 20 °C background, a surface at 350 K radiates about 360 W per square metre at ε = 0.8; take the same surface to 1200 K and it radiates about 93,744 W. The temperature went up by a factor of 3.4 and the heat went up by a factor of roughly 260. This is why radiation is a rounding error in a centrally heated room and the dominant term inside a furnace.
Two absolute temperatures are required, in kelvin or degrees Rankine. Feeding celsius into a fourth power is not a small error — it is the single most common mistake in the subject, and it does not merely shift the answer, it changes it by orders of magnitude.
Radiation is the only mode that crosses a vacuum, which makes it the entire thermal design problem for a spacecraft and the reason a vacuum flask works. It is also why a clear night can put frost on a car while the air temperature never drops below zero: the roof is radiating to a sky that behaves as though it were at about 230 K, and it loses more that way than convection can replace.
Emissivity is a surface property, not a bulk one, and it is where most of the engineering leverage sits. Polished aluminium sits near 0.04 and anodised aluminium near 0.82 — the same metal, a factor of twenty apart, decided by a few microns of oxide. Low-emissivity window coatings, radiant barriers in lofts and the silvering inside a flask are all the same idea applied deliberately.