Conduction, Convection and Radiation: Heat Transfer with Numbers
Thermodynamics tells you how much energy must move; heat transfer tells you how fast it moves and what size of surface you need. Heat travels in three ways: by conduction through solids, by convection between a surface and a moving fluid, and by radiation as electromagnetic waves. Real problems such as a hot pipe, an electronics heat sink or a sun-baked wall usually involve all three. This article gives the governing law for each mode, solves a numerical example for each, and then combines them using the thermal resistance method.
The Three Governing Laws
Conduction (Fourier): Q = k * A * (T1 - T2) / L
Convection (Newton): Q = h * A * (Ts - Tf)
Radiation (Stefan-Boltzmann, small body in large surroundings):
Q = eps * sigma * A * (Ts^4 - Tsurr^4)
k = thermal conductivity, W/m K
h = convective heat transfer coefficient, W/m^2 K
eps = emissivity (0 to 1)
sigma = 5.67 x 10^-8 W/m^2 K^4
Radiation temperatures must be in kelvin.Conductivity is a material property. The convection coefficient is not; it depends on fluid, velocity, geometry and orientation, and is usually found from correlations or experiments. The ranges below show how much it varies.
| Situation | Typical h (W/m^2 K) |
|---|---|
| Free convection, gases | 2 to 25 |
| Free convection, liquids | 50 to 1000 |
| Forced convection, gases | 25 to 250 |
| Forced convection, liquids | 100 to 20,000 |
| Boiling and condensation | 2500 to 100,000 |
One Example for Each Mode
Conduction through a brick wall
A 230 mm brick wall (the common nine-inch wall) has an area of 10 m^2. Its inner surface is at 25 degrees C and outer surface at 35 degrees C. Take k = 0.7 W/m K for brick.
Q = k A dT / L = 0.7 * 10 * 10 / 0.23 = 304 W
Convection from a hot plate
A horizontal plate 0.5 m by 0.5 m is at 80 degrees C in still air at 30 degrees C. Taking h = 10 W/m^2 K for free convection:
Q = h A dT = 10 * 0.25 * 50 = 125 W
Radiation from the same plate
If the plate is painted (emissivity 0.9) and the room walls are also at 30 degrees C:
Ts = 353.15 K, Tsurr = 303.15 K Q = 0.9 * 5.67e-8 * 0.25 * (353.15^4 - 303.15^4) = 0.9 * 5.67e-8 * 0.25 * 7.108e9 = 90.7 W
Radiation carries about 42 percent of the total 216 W lost from the plate. Students often ignore radiation at low temperatures, but in free convection it is frequently as important as convection. A polished aluminium plate with emissivity near 0.05 would lose only about 5 W by radiation, which is why reflective surfaces are used on insulation jackets.
Combining Modes with Thermal Resistance
Heat flow behaves like electric current. Temperature difference plays the role of voltage and each layer or surface film acts as a resistance. Resistances in series simply add.
| Element | Resistance per unit area (m^2 K/W) |
|---|---|
| Plane conduction layer | L / k |
| Convective surface film | 1 / h |
| Radiation (linearised) | 1 / h_r, where h_r = eps sigma (Ts + Tsurr)(Ts^2 + Tsurr^2) |
Worked example: insulating a hot wall
The 230 mm brick wall of a room faces 45 degrees C outdoor air while the room is kept at 25 degrees C. Take inside film coefficient 10 W/m^2 K and outside 25 W/m^2 K. Compare the heat gain through 10 m^2 with and without a 50 mm layer of glass wool (k = 0.04 W/m K).
Inside film 1/10 = 0.100 Brick 0.23/0.7 = 0.329 Outside film 1/25 = 0.040 Total (bare) = 0.469 m^2 K/W q = 20 / 0.469 = 42.7 W/m^2 -> 427 W for 10 m^2 Add glass wool 0.05/0.04 = 1.250 Total (insulated) = 1.719 m^2 K/W q = 20 / 1.719 = 11.6 W/m^2 -> 116 W for 10 m^2
Just 50 mm of insulation cuts the heat gain by about 73 percent, which directly reduces the air-conditioner load. The insulation dominates because its resistance is nearly four times that of the brick.
Practical Tips and Common Mistakes
- Use kelvin for radiation. Celsius gives completely wrong answers because of the fourth power.
- Do not treat h as a fixed property; a fan can raise it ten times compared to still air.
- Check which resistance is largest. Improving a small resistance barely changes the heat flow.
- Remember contact resistance between layers, which can be significant in electronics and bolted joints.
- Account for both convection and radiation when estimating heat loss from hot surfaces in still air.
Summary
Conduction follows Fourier's law, convection follows Newton's law of cooling and radiation follows the Stefan-Boltzmann law. Each can be expressed as a thermal resistance, and resistances in series add, which turns multi-layer problems into simple arithmetic. With these tools you can estimate heat losses, choose insulation thickness and judge whether a heat sink needs a fan.