Heat Transfer — Conduction, Convection & Radiation
Three Modes Compared — Formulas, Differences, Applications & Solved Problems
Last Updated: March 2026
📌 Key Takeaways
- Conduction: Heat flows through a material by molecular vibration. Governed by Fourier’s Law: Q = −kA(dT/dx).
- Convection: Heat transferred by fluid motion (natural or forced). Governed by Newton’s Law of Cooling: Q = hA(Ts − T∞).
- Radiation: Heat transferred via electromagnetic waves — no medium required. Governed by Stefan-Boltzmann Law: Q = εσAT&sup4;.
- Conduction requires physical contact. Convection requires fluid motion. Radiation works through vacuum.
- Most real problems involve combined modes — e.g., a hot pipe loses heat by convection to air and radiation to surroundings simultaneously.
1. Overview — Why Heat Transfers
Heat transfer is the movement of thermal energy from a region of higher temperature to a region of lower temperature. This is a direct consequence of the Second Law of Thermodynamics — heat always flows spontaneously from hot to cold, never the reverse.
Nature uses three distinct mechanisms to accomplish this transfer, each with its own physics, governing equation, and engineering applications. Understanding when each mode dominates is essential for designing insulation systems, heat exchangers, cooling systems, furnaces, electronic cooling, and building HVAC systems.
2. Conduction — Fourier’s Law
Conduction is heat transfer through a material without any bulk movement of the material itself. Energy passes from molecule to molecule through vibrations, collisions, and (in metals) free electron movement. It is the dominant mode of heat transfer in solids.
Fourier’s Law of Conduction
Q = −kA(dT/dx)
For a flat wall of thickness L with steady-state, one-dimensional conduction:
Q = kA(T&sub1; − T&sub2;) / L
Where:
- Q = rate of heat transfer (W)
- k = thermal conductivity of the material (W/m·K)
- A = cross-sectional area perpendicular to heat flow (m²)
- dT/dx = temperature gradient (K/m)
- L = wall thickness (m)
- T&sub1;, T&sub2; = temperatures on each side (K or °C)
The negative sign indicates heat flows in the direction of decreasing temperature. Thermal conductivity k is a material property — metals have high k (good conductors), while materials like wood, foam, and air have low k (good insulators).
Thermal Resistance (Conduction)
Rcond = L / (kA) (K/W)
Q = ΔT / Rcond — analogous to Ohm’s law (I = V/R)
3. Convection — Newton’s Law of Cooling
Convection is heat transfer between a surface and a moving fluid. The bulk movement of the fluid carries thermal energy away from (or towards) the surface much faster than conduction alone could. It is the dominant mode for heat transfer in liquids and gases.
Newton’s Law of Cooling
Q = hA(Ts − T∞)
Where:
- Q = rate of convective heat transfer (W)
- h = convection heat transfer coefficient (W/m²·K)
- A = surface area exposed to the fluid (m²)
- Ts = surface temperature (K or °C)
- T∞ = fluid temperature far from the surface (K or °C)
Two types of convection:
| Type | Driving Force | h Range (W/m²·K) | Examples |
|---|---|---|---|
| Natural (free) convection | Buoyancy — hot fluid rises, cold fluid sinks | 5–25 (air), 20–100 (water) | Hot radiator warming a room, cooling of electronic components |
| Forced convection | External force — fan, pump, wind | 25–250 (air), 100–20,000 (water) | Car radiator with fan, CPU cooler, industrial heat exchangers |
The convection coefficient h is not a material property — it depends on the fluid, flow velocity, geometry, and whether the flow is laminar or turbulent. It is typically determined from correlations using dimensionless numbers (Nusselt, Reynolds, Prandtl).
4. Radiation — Stefan-Boltzmann Law
Radiation is heat transfer via electromagnetic waves (primarily infrared). Unlike conduction and convection, radiation does not require a medium — it can travel through a perfect vacuum. Every object above absolute zero emits thermal radiation.
Stefan-Boltzmann Law
Qemitted = εσAT&sup4;
Net radiation heat exchange between a small body and large surroundings:
Qnet = εσA(Ts&sup4; − Tsurr&sup4;)
Where:
- ε = emissivity of the surface (0 to 1; 1 for a perfect blackbody)
- σ = Stefan-Boltzmann constant = 5.67 × 10−8 W/(m²·K&sup4;)
- A = surface area (m²)
- T = absolute temperature (K) — must be Kelvin
Key properties of thermal radiation:
- Proportional to T&sup4; — doubling temperature increases radiation by 16 times.
- Dominant at high temperatures (furnaces, combustion, re-entry heating, sun).
- A blackbody (ε = 1) absorbs and emits maximum radiation. Real surfaces have ε < 1.
- Shiny, polished surfaces have low emissivity (poor radiators); dark, rough surfaces have high emissivity (good radiators).
5. Side-by-Side Comparison
| Feature | Conduction | Convection | Radiation |
|---|---|---|---|
| Mechanism | Molecular vibration and electron diffusion | Bulk fluid motion | Electromagnetic waves |
| Medium required? | Yes (solid, liquid, or gas) | Yes (liquid or gas) | No — works in vacuum |
| Governing law | Fourier’s Law | Newton’s Law of Cooling | Stefan-Boltzmann Law |
| Key material property | Thermal conductivity (k) | Convection coefficient (h) | Emissivity (ε) |
| Temperature dependence | Proportional to ΔT | Proportional to ΔT | Proportional to ΔT&sup4; |
| Dominant in | Solids | Fluids (liquids & gases) | High temperatures, vacuum |
| Speed | Slowest | Moderate | Speed of light |
| Example | Heat through a brick wall | Wind chill, boiling water | Sun warming Earth, campfire heat |
6. Thermal Conductivity — Material Values
| Material | k (W/m·K) | Category |
|---|---|---|
| Copper | 385 | Excellent conductor |
| Aluminium | 205 | Good conductor |
| Steel (mild) | 50 | Moderate conductor |
| Stainless Steel | 15 | Poor metal conductor |
| Glass | 0.8 | Insulator |
| Brick | 0.6–1.0 | Insulator |
| Wood | 0.12–0.17 | Good insulator |
| Fiberglass insulation | 0.04 | Excellent insulator |
| Air (still) | 0.026 | Excellent insulator |
| Aerogel | 0.013 | Best practical insulator |
7. Worked Numerical Examples
Example 1: Conduction Through a Wall
Problem: A brick wall is 0.25 m thick with k = 0.7 W/m·K and area 10 m². Inside temperature is 25°C, outside is 5°C. Find the heat loss.
Solution
Q = kA(T&sub1; − T&sub2;)/L = 0.7 × 10 × (25 − 5) / 0.25 = 0.7 × 10 × 20 / 0.25
Q = 560 W
Example 2: Convective Cooling
Problem: A heated plate at 80°C is cooled by air at 20°C blowing over it. h = 50 W/m²·K, plate area = 0.5 m². Find the convective heat loss.
Solution
Q = hA(Ts − T∞) = 50 × 0.5 × (80 − 20) = 50 × 0.5 × 60
Q = 1,500 W = 1.5 kW
Example 3: Radiation from a Hot Surface
Problem: A blackbody (ε = 1) with surface area 2 m² is at 500°C. The surroundings are at 25°C. Find the net radiation heat loss.
Solution
Ts = 500 + 273 = 773 K, Tsurr = 25 + 273 = 298 K
Q = εσA(Ts&sup4; − Tsurr&sup4;) = 1 × 5.67×10−8 × 2 × (773&sup4; − 298&sup4;)
773&sup4; = 3.57 × 1011, 298&sup4; = 7.89 × 109
Q = 5.67×10−8 × 2 × (3.57×1011 − 7.89×109) = 5.67×10−8 × 2 × 3.49×1011
Q ≈ 39,590 W ≈ 39.6 kW
At 773 K, radiation dominates — the T&sup4; dependence makes it enormous at high temperatures.
8. Common Mistakes Students Make
- Using Celsius in radiation formulas: Stefan-Boltzmann law uses T&sup4; — you MUST use Kelvin. The difference between (500)&sup4; and (773)&sup4; is enormous.
- Thinking convection coefficient h is a material property: Unlike thermal conductivity k, the convection coefficient h depends on flow conditions, geometry, fluid properties, and whether the flow is laminar or turbulent. It is not a fixed material constant.
- Forgetting radiation at moderate temperatures: Radiation is always present — not just at high temperatures. At room temperature, radiation contributes 30–50% of the total heat loss from a human body. Students often assume radiation is negligible below furnace temperatures.
- Neglecting the negative sign in Fourier’s law: The negative sign means heat flows from high T to low T (in the direction of decreasing temperature). The magnitude of Q is positive; the sign indicates direction.
- Confusing heat flux with heat rate: Heat flux q = Q/A is heat transfer per unit area (W/m²). Heat rate Q is total heat transfer (W). Fourier’s law can be written for either, and mixing them up leads to dimensionally incorrect answers.
9. Frequently Asked Questions
What are the three modes of heat transfer?
The three modes are conduction (through materials by molecular vibration), convection (by fluid motion), and radiation (by electromagnetic waves through any medium or vacuum). In most engineering situations, multiple modes act simultaneously — for instance, a hot pipe loses heat by conduction through the pipe wall, convection to the surrounding air, and radiation to the walls of the room.
What is the difference between conduction and convection?
Conduction transfers heat through a medium without moving the medium itself — energy passes from molecule to molecule by vibration and collision. Convection involves the actual movement of fluid (liquid or gas) carrying thermal energy from one place to another. Conduction is the dominant mechanism in solids; convection dominates in fluids.
Can radiation occur through a vacuum?
Yes — radiation is the only heat transfer mode that requires no material medium. It travels as electromagnetic waves at the speed of light through a complete vacuum. This is how solar energy reaches Earth across 150 million kilometres of space. Conduction and convection both require matter to operate.