Heat Transfer: Conduction, Convection, and Radiation

Heat — the random kinetic energy of molecular motion — always flows from hot things to cold things. That direction is fixed by the second law of thermodynamics and is responsible for why ice melts in warm rooms, why hot coffee cools down, and why you can feel a fire from across a room without touching it. But the mechanism by which heat flows depends on the situation, and there are exactly three fundamental mechanisms:

1. Conduction — heat transfer through a stationary medium (solid, liquid, or gas) by molecular collision and electron drift
2. Convection — heat transfer by the bulk motion of a fluid (liquid or gas) that carries hot matter from one place to another
3. Radiation — heat transfer by electromagnetic waves, with no medium required at all

These three don't just describe different situations — they have wildly different physics, obey different mathematical laws, and dominate in different regimes. Engineers designing insulation, heat sinks, buildings, spacecraft, and ovens have to think about all three simultaneously, because any real surface can lose (or gain) heat by any combination of the three.

Let me unpack each one in depth, then compute a worked example where all three matter at once.

Conduction — heat through a solid

Imagine holding a metal spoon in a hot cup of tea. After a few seconds, the handle of the spoon feels warm even though you never put it in the tea. The heat traveled through the metal, not via any moving fluid. That's conduction.

Microscopically, conduction happens because hot atoms are vibrating more than cold ones, and through the electromagnetic forces that hold the solid together, they push on their neighbors, passing kinetic energy along. In a metal, free electrons add a second, faster conduction channel: hot electrons drift preferentially toward the cold side, carrying energy with them. This electronic contribution is why metals are much better conductors than insulators — electrons move much faster than atomic vibrations.

Fourier's law describes conduction mathematically. In steady state, through a flat slab of thickness $L$, cross-section area $A$, made of a material with thermal conductivity $k$, with temperatures $T_{\text{hot}}$ and $T_{\text{cold}}$ on either side:

$q_{\text{cond}} = k\,A\,\dfrac{\Delta T}{L}$

where $\Delta T = T_{\text{hot}} - T_{\text{cold}}$ is the temperature difference, and $q$ is the heat flow rate in watts.

The thermal conductivity $k$ is a material property with units of W/(m·K). It varies enormously across materials:

• Diamond (the all-time champion): $k \approx 2000\;\text{W/(m·K)}$
• Copper: $k \approx 400$
• Aluminum: $k \approx 240$
• Iron / steel: $k \approx 50$–$80$
• Granite / concrete: $k \approx 1$–$3$
• Glass: $k \approx 0.8$
• Wood: $k \approx 0.1$–$0.2$
• Fiberglass insulation: $k \approx 0.04$
• Still air: $k \approx 0.025$

The range from diamond to still air is 5 orders of magnitude. Good conductors are nearly all metals (free electrons). Good insulators are mostly things that trap air in small pockets (fiberglass, foam, fur) — the air itself is the real insulator, and the solid matrix just holds it still. If the air could move, convection would transport heat much faster.

Notice that Fourier's law is linear in $\Delta T$ — doubling the temperature difference doubles the heat flow rate. This is very different from radiation, which will have a $T^{4}$ dependence and is extremely nonlinear.

Convection — heat carried by moving fluid

Put your hand above a hot stove (not on it!) and feel the heat rising. That's convection. The air right next to the stove gets hot (by conduction, briefly), expands slightly so it becomes less dense, rises under buoyancy, and is replaced by cooler air from below. The rising hot air carries its extra energy upward, where it eventually mixes with the rest of the room.

Convection is fundamentally different from conduction in that it involves bulk motion of the fluid. The fluid itself flows from hot to cold, transporting energy by carrying warm molecules from one place to another. This is much more efficient than conduction alone for large temperature differences and large distances.

Convection comes in two flavors:

• Natural (or free) convection — driven by buoyancy alone. Hot fluid rises, cold fluid sinks, creating a slow circulation. Examples: the air above a radiator, hot water rising in a pot, ocean currents driven by temperature differences.
• Forced convection — driven by some external mechanism (pump, fan, wind). Examples: a hair dryer blowing hot air, a car radiator with coolant pumped through, a CPU with a fan blowing over the heat sink.

Quantitatively, convection is described by Newton's law of cooling:

$q_{\text{conv}} = h\,A\,\Delta T$

where $h$ is the convective heat transfer coefficient — a property of the fluid flow, geometry, and temperature, not just the fluid. $h$ varies over several orders of magnitude:

• Natural convection in still air: $h \approx 5$–$25\;\text{W/(m}^{2}\cdot\text{K)}$
• Forced air (fan blowing on a surface): $h \approx 10$–$200$
• Natural water convection: $h \approx 50$–$1000$
• Forced water (pumped through a pipe): $h \approx 500$–$10{,}000$
• Boiling water: $h \approx 2500$–$35{,}000$

Water is vastly better than air for convective heat transfer — roughly 100× better — which is why water cooling is used for CPU heatsinks in serious overclocking setups, and why you can burn your hand in 60°C water but not in 60°C air.

Convection is tricky because $h$ is hard to predict from first principles. It depends on the fluid's viscosity, thermal conductivity, specific heat, density, flow velocity, the surface geometry, and even the temperature (because all of the above vary with temperature). Engineers usually use empirical correlations or computational fluid dynamics to estimate it.

Radiation — heat from electromagnetic waves

Stand outside on a sunny day. The sun is 150 million kilometers away — there's no solid object connecting you to it, and there's essentially no air for 100 km above your head. Yet you can feel the heat on your skin. How?

Thermal radiation. Every object at a temperature above absolute zero emits electromagnetic waves. The waves carry energy, and when they're absorbed by another object, they heat it up. At room temperature, most of the radiation is in the mid-infrared (wavelengths around 10 μm) — you can't see it, but thermal imaging cameras can. At higher temperatures, the radiation moves to shorter wavelengths: ~800 K peaks in the near-infrared, ~1000 K starts visibly glowing red ("red hot"), ~5000 K peaks in visible light (like the sun), and hotter stars peak in the ultraviolet.

The key formula is the Stefan-Boltzmann law:

$q_{\text{rad}} = \sigma\,\varepsilon\,A\,\bigl(T^{4} - T_{\text{surr}}^{4}\bigr)$

where:

• $\sigma = 5.67 \times 10^{-8}\;\text{W/(m}^{2}\cdot\text{K}^{4})$ is the Stefan-Boltzmann constant
• $\varepsilon$ is the emissivity — a dimensionless number from 0 to 1 indicating how efficiently the surface emits radiation compared to a "perfect blackbody"
• $A$ is the surface area
• $T$ is the surface temperature of the emitting object (must be in kelvin!)
• $T_{\text{surr}}$ is the temperature of the surroundings the object is radiating into

Three critical features distinguish radiation from the other two mechanisms:

1. No medium required. Radiation works in vacuum. That's how the sun warms the Earth across 150 million km of empty space, how thermal imaging cameras see across rooms, and how spacecraft lose heat (only radiation is available in orbit).

2. $T^{4}$ dependence. Doubling the absolute temperature (in kelvin!) multiplies the radiation rate by 16. This is why radiation dominates at high temperatures — a light bulb filament at 3000 K radiates ~$16^{2} = 256$ times as much as room-temperature surfaces. It's also why the Celsius-vs-kelvin distinction is absolutely critical for radiation: the fourth power of 20°C (treated as 20) is 160,000, but the fourth power of 293 K is 7.4 × 10⁹. Completely different numbers.

3. Surroundings temperature matters, not the adjacent object. Radiation flows to the surroundings at temperature $T_{\text{surr}}$, not to the specific adjacent surface. This is why an object can simultaneously lose heat by radiation to the sky (cold) and gain heat by conduction from the warm ground, and why a car parked in direct sunlight can be dramatically hotter inside than the air around it (greenhouse effect).

Worked example: all three mechanisms on a steel wall

Let's tackle the setup from the problem: a steel wall 10 cm thick, with 500°C on the inside (hot side) and 20°C on the outside (cold side). We'll compute the rate of heat transfer by each mechanism, assuming the wall has 1 m² of surface area.

Given:

• $k_{\text{steel}} \approx 60\;\text{W/(m·K)}$
• $A = 1\;\text{m}^{2}$
• $L = 0.1\;\text{m}$
• $T_{\text{hot}} = 500°\text{C} = 773\;\text{K}$
• $T_{\text{cold}} = 20°\text{C} = 293\;\text{K}$
• $\Delta T = 500 - 20 = 480\;\text{K}$

Method 1: Conduction through the wall

$q_{\text{cond}} = k\,A\,\dfrac{\Delta T}{L} = (60)(1)\dfrac{480}{0.1} = 288{,}000\;\text{W} = 288\;\text{kW}$

Enormous! Nearly 300 kilowatts of heat pours through a 1 m² steel wall with a 480 K temperature difference. This is why steel is a terrible choice for insulating anything — it's much too good a conductor. If you were designing a furnace, you'd use a ceramic refractory lining (k ~ 1) to reduce this by a factor of 60.

Method 2: Convection on the cool (outside) surface

The outside surface is at 20°C = 293 K. If the ambient air is at, say, 20°C as well, then $\Delta T_{\text{surface-air}} \approx 0$ and convection is zero. Let's instead assume the outside surface is somewhat hotter than the ambient air — say 70°C (343 K) — because the enormous conductive heat flow from inside has warmed it above ambient. Then $\Delta T = 50\;\text{K}$.

For natural air convection: $q_{\text{conv}} = h\,A\,\Delta T = (10)(1)(50) = 500\;\text{W}$

For forced air (say a fan blowing across the wall): $q_{\text{conv, forced}} = (100)(1)(50) = 5000\;\text{W}$

Even with a powerful fan, convection is much smaller than the conductive heat flow through the wall. In steady state, these numbers have to balance — if the wall were really conducting 288 kW in from the inside but only 500 W out to the air, the outside surface would rapidly heat up until the conduction from the inside matches the loss to the outside. So the real steady-state temperatures would be different from the assumed 20°C outside surface.

Method 3: Radiation from the hot (inside) surface

The hot side is at 773 K. Suppose it's radiating to the inside of a furnace chamber at (say) 500 K on average. Steel has emissivity about 0.8. Then:

$q_{\text{rad}} = \sigma\,\varepsilon\,A\,\bigl(T^{4} - T_{\text{surr}}^{4}\bigr)$

Compute the fourth powers:

• $773^{4} = 773^{2} \cdot 773^{2} = 597{,}529 \times 597{,}529 \approx 3.57 \times 10^{11}$
• $500^{4} = 500^{2} \cdot 500^{2} = 250{,}000 \times 250{,}000 = 6.25 \times 10^{10}$

Difference: $T^{4} - T_{\text{surr}}^{4} = 3.57 \times 10^{11} - 6.25 \times 10^{10} \approx 2.95 \times 10^{11}\;\text{K}^{4}$

Multiply by $\sigma \cdot \varepsilon \cdot A$: $q_{\text{rad}} = (5.67 \times 10^{-8})(0.8)(1)(2.95 \times 10^{11}) \approx 13{,}400\;\text{W} = 13.4\;\text{kW}$

Significant — about 13 kilowatts of radiation from the hot wall to its surroundings, far larger than natural convection but still much less than conduction through the steel.

Comparison

At these temperatures (hot side of 773 K), the three mechanisms give:

• Conduction (through steel): ~288 kW
• Radiation (from hot surface): ~13 kW
• Convection (natural air on cool side): ~0.5 kW

Conduction dominates because steel is such a good conductor. Radiation is significant because of the high temperature ($T^{4}$ is huge at 773 K). Convection is weakest because natural air is a poor heat carrier compared to a solid conductor.

If you change the materials — say replace the steel wall with a fiberglass panel ($k \approx 0.04$) — conduction drops by a factor of 1500, and then convection and radiation become comparable or even dominant. This is how insulation works: block conduction, and then the other two mechanisms (which scale less aggressively with temperature difference) can't keep up.

Why $T^{4}$ is the signature of radiation

The fourth-power temperature dependence in Stefan-Boltzmann is perhaps the most important feature of radiation heat transfer. Let me illustrate why it matters:

• Doubling the absolute temperature (say from 300 K to 600 K) multiplies radiation by $2^{4} = 16$.
• Going from 300 K to 3000 K multiplies it by $10^{4} = 10{,}000$.
• Going from 300 K (room temperature) to 5800 K (sun's surface) multiplies it by about $(5800/300)^{4} = 19.3^{4} \approx 139{,}000$.

That 5800 K factor is why the sun delivers so much energy despite being 150 million km away. A 1 m² piece of sun at 5800 K emits about 64 million watts. The Earth at 300 K emits only about 460 W from the same area. Without radiation, we'd never feel the sun at all.

The $T^{4}$ dependence also explains why:

• Hot things dominate radiation at the expense of everything else. A light bulb filament at 3000 K is radiating at a rate far beyond what conduction or convection could ever remove — it MUST be radiating, which is the whole point.
• Cold objects barely radiate at all. A 100 K object emits about $10^{-4}$ as much as a 300 K object. This is why cryogenic experiments (detectors for astrophysics, quantum computers) use multi-stage cooling: each stage has to deal with only the small fraction of heat that leaked through from the warmer stage.
• Temperature regulation by radiation is tough. If your body is at 310 K and the room is at 293 K, the radiation difference is only $310^{4}/293^{4} - 1 \approx 26\%$. Small changes in room temperature have modest effects on your radiative heat loss, which is why your body relies more on convection, evaporation (sweating), and blood flow regulation to control temperature.

Real-world applications

• Home insulation. Fiberglass batting in a wall cavity traps still air, blocking conduction. Since still air has very low $k$ (0.025), this is highly effective. The trick is making sure the air doesn't circulate (so convection doesn't start up) — the fiberglass fibers break the air into tiny pockets that can't sustain bulk flow. Modern spray foam and aerogel do the same thing more efficiently.
• Thermos flasks. A vacuum gap between two walls blocks conduction (no medium) and convection (no fluid to move). The inner wall is silvered to reflect thermal radiation back. All three mechanisms are defeated simultaneously. A well-made thermos keeps coffee hot for 12+ hours.
• CPU cooling. Modern computer CPUs dissipate 50–200 W into tiny areas. Conduction brings heat from the chip to a metal heat sink (aluminum or copper, often with heat pipes filled with phase-change fluid). Forced air convection from a fan carries the heat from the heat sink fins into the room. Some gaming PCs use water cooling, where forced-convection water has much higher $h$ than air.
• Spacecraft thermal management. No air in space, so no convection. Conduction moves heat between components through the structure. Radiators on the outside of the spacecraft emit heat to deep space ($T_{\text{surr}} \approx 3$ K, the cosmic microwave background) via radiation. The Space Shuttle had radiator panels on the inside of the cargo bay doors that had to be opened to space for cooling to work.
• Thermal imaging. Passive infrared cameras detect the long-wave infrared radiation emitted by objects at room temperature. Every object in the scene is acting as a thermal emitter at its own temperature, and the camera measures the radiation to map temperature. Used in firefighting (seeing through smoke), medicine (detecting fevers or tumors), surveillance, and industrial inspection.
• Building energy efficiency. Modern low-E (low-emissivity) windows have thin metallic coatings that reflect thermal infrared radiation back into the room in winter (keeping heat in) and reject solar radiation in summer (keeping heat out). The coating doesn't affect visible light much, so you still get clear views. Conduction and convection through the glass are handled separately with multiple panes and gas fills.
• Greenhouses. Visible sunlight passes through the glass and warms the soil and plants inside. The warm interior then emits infrared radiation — but the glass is opaque to long-wave infrared, so the radiation is trapped. Combined with blocking convection (sealed air space), the greenhouse warms up. This is the origin of the term "greenhouse effect" for climate-related CO₂ trapping.

Common mistakes

• Using Celsius for radiation. The Stefan-Boltzmann law absolutely requires kelvin because it's a fourth-power law. The difference $773^{4} - 293^{4}$ is approximately $3.5 \times 10^{11}$, but if you mistakenly used 500°C and 20°C as is, you'd get $500^{4} - 20^{4} = 6.25 \times 10^{10} - 1.6 \times 10^{5} \approx 6.25 \times 10^{10}$, which is an order of magnitude smaller than the correct answer. Always convert to kelvin for radiation.
• Mixing up $k$ and $h$. Thermal conductivity $k$ has units W/(m·K) and is a bulk material property. Convective coefficient $h$ has units W/(m²·K) and is a property of the fluid flow. They differ by a factor of meters. Never mix them up.
• Assuming all three mechanisms are equally important. Usually one dominates by a factor of 10–100 or more, depending on the situation. Part of good engineering is identifying which one is dominant before you try to reduce total heat loss.
• Forgetting that convection requires a fluid. You can't have convection in a solid. It only happens in liquids and gases.
• Forgetting that radiation works in vacuum. The other two don't. This is why space thermal management is all about radiation.
• Ignoring emissivity. The Stefan-Boltzmann formula uses $\varepsilon$, which ranges from ~0.01 (polished silver) to ~0.98 (flat black paint). A highly polished metal surface radiates much less than a rough black one at the same temperature. Engineers tune emissivity intentionally (white thermal paint for spacecraft radiators, black paint for solar absorbers).
• Treating $T_{\text{surr}}$ as the temperature of an adjacent surface. In radiation, "surroundings" means whatever the object sees in its field of view — often the sky, or the inside of a room. For a hot pipe in open air, $T_{\text{surr}}$ is roughly the sky temperature, which on a clear night can be far below the air temperature (the reason frost forms on car windshields even when the air is above freezing).

Try it

• Pick "All three" mechanisms and drag the temperature difference up. Watch conduction scale linearly, convection scale linearly (but with different slope), and radiation scale like $T^{4}$. The radiation bar grows dramatically faster than the others.
• Switch to "Conduction only" and pick different materials. Diamond and copper conduct thousands of times better than fiberglass or still air. Notice how choice of material matters more than anything else for conduction.
• Switch to "Radiation only" and raise $T_{\text{hot}}$ to 1500 K (characteristic of molten lava). The radiation rate jumps by roughly $5^{4} = 625$ compared to room temperature. This is why lava and hot metal feel "glowing" — they're radiating huge amounts of infrared even before they get visibly red.
• Try a vacuum scenario. Set convection coefficient $h$ to near zero (simulating no fluid) and watch conduction still work through the wall, but convection vanish. This is why spacecraft have to use radiation for all their heat rejection.
• Compare steel vs. fiberglass as the wall material at the same $\Delta T$. Conduction drops by roughly a factor of 1500 when you switch from steel to fiberglass — which is the whole idea behind insulation.
• Try "Show comparison bar" to see all three heat flows plotted as bars. The dominant mechanism in your current configuration will be obvious immediately.