Heat, Temperature and the Three Modes
Energy and power, specific heat, conduction, convection and radiation, thermal resistance and the numbers to remember.
Heat, temperature and energy
Temperature says how hot something is; heat is the energy that flows because of a temperature difference. It always flows from hot to cold, and the bigger the difference, the faster it flows. The energy needed to change a body’s temperature is
Q = m c ΔT
Worked example: warming an aluminum block
- A 2 kg aluminum block (c = 896 J/kg·K) is heated from 20 °C to 200 °C: Q = 2 × 896 × 180 = 322.6 kJ.
- A 1 500 W heater, with no losses, needs t = Q ÷ P = 215 s, about 3.6 minutes. With losses it takes longer: see Transient Heating and Cooling.
- Water has an unusually high specific heat (4 180 J/kg·K, five times that of aluminum per kilogram), which is why it is the usual coolant.
The heat needed to melt or boil a substance with no temperature change is the latent heat: 334 kJ/kg to melt ice, 2 257 kJ/kg to boil water at atmospheric pressure. It is why steam carries so much energy and why evaporating even a little water cools a surface strongly.
The three modes
- Conduction is heat flow through a stationary material, by vibrating atoms and free electrons. It is why a metal spoon in hot soup gets hot, and why metals (with free electrons) conduct best. See Conduction: Walls, Pipes and Insulation.
- Convection is heat carried away by a moving fluid at a surface: the fluid next to a hot surface warms, and its motion (stirred by a fan, a pump or buoyancy) carries the heat off. See Convection and Heat Transfer Coefficients.
- Radiation is heat carried by electromagnetic waves, which needs no medium: the warmth of the sun, a fire or a hot furnace wall. Every surface above absolute zero radiates. See Thermal Radiation.
The rate equations
| Mode | Rate equation | Key property |
|---|---|---|
| Conduction | Q̇ = −k A dT/dx (Fourier’s law); plane wall Q̇ = k A ΔT ÷ L | thermal conductivity k (W/m·K) |
| Convection | Q̇ = h A (Ts − T∞) (Newton’s law of cooling) | heat transfer coefficient h (W/m²·K) |
| Radiation | Q̇ = ε σ A (Ts⁴ − Tsurr⁴) (Stefan–Boltzmann) | emissivity ε (0 to 1); σ = 5.67 × 10⁻⁸ W/m²·K⁴; kelvin |
Conduction and convection are both proportional to a temperature difference. Radiation goes with the difference of fourth powers of absolute temperature, so it is negligible near room temperature compared with convection, and dominant at several hundred degrees.
Thermal resistance
Because each mode is proportional to a temperature difference, we can use an electrical analogy: temperature difference is the voltage, heat flow is the current, and the thermal resistance R = ΔT ÷ Q̇ (K/W) is the resistance. Resistances in series add, in parallel combine as 1 ÷ R = Σ 1 ÷ Ri, exactly like a circuit.
| Mode | Resistance (K/W) |
|---|---|
| Conduction through a plane wall | R = L ÷ (k A) |
| Conduction through a cylinder | R = ln(r₂ ÷ r₁) ÷ (2 π k L) |
| Convection | R = 1 ÷ (h A) |
| Radiation (linearized) | R = 1 ÷ (hr A), hr = ε σ (Ts + Tsurr)(Ts² + Tsurr²) |
Worked example: a single-pane window
- A 6 mm glass pane (k = 1.0 W/m·K): conduction resistance per square meter R″ = L ÷ k = 0.006 m²·K/W. With 20 K across the glass itself the flux would be 3 333 W/m².
- But the glass is not at the full 20 K difference: the air films on both sides add resistance. The real overall coefficient of a single pane is about U = 5.7 W/m²·K and of a sealed double glazed unit about 2.8 W/m²·K. At 20 K the loss is 114 and 56 W per square meter.
- Nearly all the resistance is in the still air, not the glass: the 6 mm glass is only 3.4 percent of the single-pane resistance. That is the lesson of thermal resistance networks: find where the resistance is.
Numbers to remember
| Material | Conductivity k (W/m·K) | Specific heat c (J/kg·K) |
|---|---|---|
| Copper | 390 | 385 |
| Aluminum | 205 to 237 | 896 |
| Carbon steel | 45 to 55 | 490 |
| Stainless steel 304 | 15 | 500 |
| Concrete | 1.4 | 880 |
| Brick | 0.7 | 840 |
| Glass | 1.0 | 840 |
| Water (liquid) | 0.6 | 4 180 |
| Wood (pine) | 0.12 | 1 700 |
| Mineral wool, glass wool | 0.035 to 0.045 | 840 |
| Polyurethane or PIR foam | 0.022 to 0.028 | 1 400 |
| Still air | 0.026 | 1 007 |
Metals conduct hundreds of times better than insulators, and still air is among the best insulators of all: good insulation is mostly trapped air or gas in small cells that cannot circulate. Typical convection coefficients are on the page Convection and Heat Transfer Coefficients.
Putting the modes together
Worked example: a bare steel pipe in still air
- A bare steel pipe, 60.3 mm outside diameter (DN50), at 80 °C in air at 20 °C, emissivity 0.9 for rusty or painted steel.
- Natural convection (a standard fit for a horizontal cylinder in air, h ≈ 1.32 (ΔT ÷ D)0.25): hc = 7.4 W/m²·K.
- Radiation: hr = ε σ (Ts + Ta)(Ts² + Ta²) = 6.9 W/m²·K. Radiation is 48 percent of the total: not negligible even at 80 °C.
- Heat loss per meter: q′ = (hc + hr) π D ΔT = 14.4 × π × 0.0603 × 60 = 163 W/m (84 convection + 79 radiation). A 100 m run loses 16.3 kW. Insulating it is the next step.
Units and R-values
Heat flow is in watts (1 W = 1 J/s = 3.412 Btu/h); 1 kWh = 3.6 MJ = 3 412 Btu. Building and insulation work uses the U-value (W/m²·K, the inverse of the total resistance) in metric countries and the R-value (ft²·h·°F/Btu) in the US. They are related by U = 5.678 ÷ RUS, so an R-19 wall (US) has RSI = 3.35 m²·K/W and U = 0.30 W/m²·K. Note that the metric R-value (m²·K/W) is not the same unit as the US one. The Energy and Power converters help with the rest.
Related
- Next: Conduction: Walls, Pipes and Insulation.
- Heat Sink and Thermal Resistance Calculator.
- Fluid Mechanics and Pipe Flow: the fluid side of convection.
- Energy Converter and Temperature Converter.
- Heat Transfer: the whole guide.
FAQ
What is the difference between heat and temperature?
Temperature measures the average energy of the particles; heat is energy that moves because of a temperature difference. A bath of warm water holds more heat than a spark that is far hotter, because it has so much more mass.
Why does metal feel colder than wood at the same temperature?
Because metal conducts heat away from your hand far faster, so the surface of your skin drops towards the metal’s temperature. The thing you feel is the heat flow, not the temperature.
Do I use °C or kelvin?
For temperature differences either, as the size of the degree is the same. For radiation and for the gas law, always absolute temperature in kelvin (K = °C + 273.15).
Is heat transfer a vector or a scalar?
Heat flux (W/m²) has a direction, from hot to cold, and is a vector in three dimensions; for one-dimensional problems it is just a number with a sign.
References
- Incropera, F. P., et al., Fundamentals of Heat and Mass Transfer, Wiley.
- Çengel, Y. A. and Ghajar, A. J., Heat and Mass Transfer: Fundamentals and Applications, McGraw-Hill.
- ASHRAE Handbook – Fundamentals (thermal properties, heat transfer).
- CODATA 2018 for the Stefan–Boltzmann constant.
Last reviewed: October 2026. Property values are typical.