You step out on a still, clear night in early spring. The air is a few degrees above freezing, yet the car roof is furred with frost. The next night is cloudy and the same car is dry. Your laptop, meanwhile, has a grille on the side and a fan that spins up when you open too many tabs. And the metal rail by your front door feels colder than the wooden step beside it, though both have sat out all night at the same temperature.
Those three puzzles have one answer: heat moves in exactly three ways, each with its own rule, and an engineer who has taken a heat transfer course sees which one is in charge. Thermodynamics tells you heat will flow from hot to cold. Heat transfer tells you how fast.
- Who takes it
- Mechanical, aerospace, chemical and nuclear engineers, usually in year 3.
- What it is for
- Predicting how quickly heat moves through, off and between things, so you can stop a house, an engine or a chip from getting too hot or too cold.
- Comes after
- Thermodynamics (34/14), Fluid Mechanics (34/15) and Differential Equations (34/03).
- Leads to
- No later subject in this curriculum lists it as a prerequisite, but it is behind turbine-blade cooling in Propulsion (34/21), the thermal design in Machine Design (34/19), and almost every capstone project (34/22) with a motor or a battery.
- What this piece gives you
- You will not be able to design a heat exchanger. You will see why a thick coat works, why a fin helps, and why a hot steel ball and a hot potato cool differently. Easier first read: The Physics of the Greenhouse Effect (13/15), which is radiation at planet scale.
Conduction: heat creeping through stuff
Conduction is heat passing through a solid (or still fluid) by jostling neighbours. Hot atoms vibrate harder and pass the shaking along. No bulk motion, just a relay. The rule is Fourier's law, set out by Joseph Fourier in his 1822 Théorie analytique de la chaleur: the heat flowing is proportional to how steep the temperature gradient is, and to a property of the material called its thermal conductivity, k.
Thicker walls slow heat; bigger temperature differences speed it up; the material sets the scale. And the scale is enormous. Copper has k of about 400 W/m·K, aluminium about 240, steel about 45, glass about 1, brick about 0.7, wood about 0.15, fibreglass insulation about 0.04 and still air about 0.026. Copper conducts ten thousand times better than fibreglass.
That also solves the cold rail. Metal and wood are the same temperature, but your hand is warmer than both. Metal drains heat out of your skin much faster, so it feels colder. Your nerves sense the rate of heat loss, not the temperature. The rate is what conduction sets.
Typical rounded conductivities near room temperature. Real values vary with alloy, density, moisture and temperature.
Try copper at 5 cm and 50 K: 401 kW through every square metre. Switch to fibreglass: 40 W. The ratio is the whole reason insulation exists. And notice still air: it is a superb insulator, which is why most insulation is just a way of trapping air so it cannot move. The moment air moves, we are in the next mode.
Thermal resistance: a wall is a circuit
Rearrange Fourier's law and it looks like Ohm's law. Temperature difference plays voltage, heat rate plays current, and the rest is a thermal resistance:
So a wall of brick, insulation and plasterboard is three resistors in a row, plus a thin film of nearly-still air on each face. The temperature drops across the layers like voltage across resistors, and most of the drop lands where the resistance is biggest. Engineers usually work per square metre and quote the U-value, which is 1 divided by the total resistance. Lower is better.
k: brick 0.72, fibreglass 0.04, plasterboard 0.17 W/m·K. Air-film resistances 0.13 inside and 0.04 outside m²·K/W, the standard values for a vertical wall. Ignores studs, which leak extra heat.
The default wall loses about 870 W through 100 m² when it is 25 K warmer inside. Slide the insulation to zero and the loss jumps to about 6,490 W, seven and a half times more. Ten centimetres of an insulator 18 times less conductive than brick beats the brick completely. The hidden lesson: resistance adds, so the biggest resistor dominates, and nothing you do to a small one matters much.
Convection: a moving fluid carries the heat away
Where air or water touches a hot surface, it warms, then moves away and is replaced by cooler fluid. That is convection: conduction into the fluid, plus the carrying. If the fluid moves by its own buoyancy (hot air rises) it is natural convection. If a fan, pump or wind drives it, it is forced. Newton's law of cooling bundles all that messy fluid behaviour into one number:
The catch: h is not a property of the material. It depends on the flow, the shape and the fluid, which is why an entire course exists to estimate it. Typical textbook ranges are roughly 2 to 25 for natural convection in air, 25 to 250 for forced air, and 100 to 20,000 for forced liquids. A thin, nearly stuck layer of fluid clings to the surface and acts like a blanket. Anything that thins it (wind, a fan, stirring) raises h.
h values are illustrative round numbers inside the textbook ranges, not a prediction for a particular flow.
Set the difference to 15 K. Still air takes 1.5 W off your 10 cm square of skin; moving water takes 150 W. That is why 15 °C water saps you far faster than 15 °C air, and why wind chill is just convection with a higher h. It is also why a fan cools a chip without making the air any colder: it raises h.
Radiation: the sky is cold
The third mode needs no fluid and no solid. Everything above absolute zero emits electromagnetic radiation, mostly infrared at everyday temperatures, and the power climbs with the fourth power of the absolute temperature. Josef Stefan found the T⁴ rule from measurements in 1879, and Ludwig Boltzmann derived it from theory in 1884. Two surfaces exchange heat:
Emissivity ε says how well a surface radiates (and absorbs). Matte paint, skin, wood and water are around 0.9. Polished metal can be a few percent, which is why a space blanket works and why a thermos is silvered inside. The Sun warms the Earth across empty space this way, and a spacecraft's radiator is the only way to get rid of its waste heat.
Now the car roof. At night the surface looks up at the sky, which is far colder than the air: much of the sky's radiation comes from very cold upper layers, and dry air passes the roof's infrared straight through to space. So the roof radiates away more than it gets back. It cools until warmth from the surrounding air, by convection, balances the loss. A cloudy night is a blanket of warm, wet air that radiates back down, so the roof stays near air temperature.
A thin, insulated surface that gets no heat from below, like a car roof's outer skin or a blade of grass. The effective sky temperature is not one fixed number: it depends on humidity, clouds and altitude, so the 20 K, 10 K and 3 K drops are illustrative assumptions. Real surfaces usually dip less because heat also leaks up from below.
With 3 °C air, a clear calm sky and paint-like emissivity, the model's skin settles near −2.4 °C: frost, with the air above freezing. Add a light breeze or a bit of cloud and it stays near 0 °C or above. That matches what you see: frost needs clear, calm nights. Emissivity matters too. Cover the roof with shiny foil (ε near 0.05) and it radiates little, so it barely cools.
Fins and heat sinks: buy surface area
Look at Newton's law again: Q̇ = h A ΔT. If h is stuck (air gives you what it gives you) and ΔT is capped (a chip must stay under about 100 °C), the only dial left is A. A fin is a thin strip of metal that sticks out from a hot surface to add area. A heat sink is a bunch of fins. The laptop grille hides one, and a car radiator is a heat sink with water inside.
But heat must travel along the fin by conduction, losing heat to the air as it goes, so the tip is cooler than the base. A long skinny fin of poor metal is mostly wasted: the far end barely helps. The standard result for a fin with an insulated tip is:
A straight rectangular fin, 2 mm thick and 50 mm wide, base 50 K above the air, insulated tip. Metal k: copper 401, aluminium 237, steel 45 W/m·K.
Drag the length and watch the curve flatten: the first centimetres carry most of the heat. Switch to steel and the curve saturates sooner and lower, because heat cannot reach the far end. Raise h (blow on it) and the fin gets less efficient, because it dumps heat before it travels far, but the fin moves more heat in total. That trade-off, between area, conductivity and airflow, is the whole design of a heat sink.
Transient cooling: the time constant and the Biot number
Everything so far held still. But a hot cup, a quenched blade or a switched-off chip cools over time. The simplest model treats the object as one lump at a single temperature, shedding heat through its surface. The lump has a thermal "capacitance" (how much energy it stores per kelvin: density × specific heat × volume) and a thermal "resistance" to the surroundings (1 ÷ hA). It is an RC circuit, and so it cools like one:
The model has a catch. It assumes the inside is the same temperature as the skin, so heat flows out easily compared with how fast the outside sheds it. The Biot number, Bi = h Lc ÷ k (with Lc = volume ÷ surface area), compares the two resistances: heat having to get out of the object versus heat leaving its surface. If Bi is below about 0.1, the lump model is fine. Above that, the inside lags behind the skin.
Starts at 100 °C in 20 °C air. ρ, c, k: copper 8960, 385, 401; steel 7850, 434, 60; water-like 1000, 4180, 0.6 (SI units). Ignores radiation.
A 10 mm steel ball in still air takes about 7.6 minutes per time constant, so about 23 minutes to come within 5% of room temperature. Biot number: 0.0014, so the lump model is excellent. Now choose "water-like" at 25 mm: τ is about 23 minutes but the Biot number is 0.35. Real potatoes stay hot in the middle long after the skin is cool, so the lump model underestimates how long they stay hot inside. Copper's huge k means any copper ball is almost always a lump, even in water. Metals pass the test; food does not.
What a graduate just sees
- Heat has three roads: through solids (conduction), by carried fluid (convection) and across space (radiation).
- A thing feels cold because it takes heat away quickly, not because it is colder.
- Resistances in series add, and the biggest one decides everything. Fix the big one.
- Insulation is mostly trapped still air. The moment the air can move, it stops working.
- Water cools you far faster than air at the same temperature, because h is a hundred times bigger.
- Radiation scales with temperature to the fourth power, so it dominates when things get hot or when nothing else can carry heat, as in space.
- When area is the only dial left, add fins. They have diminishing returns past a certain length.
- Everything cools exponentially, with a time constant that is the product of a thermal resistance and a thermal capacity, as long as Bi is small.
Where it shows up next
- Propulsion (34/21): a modern turbine runs hotter than its metal would survive, so blades are riddled with cooling passages and wrapped in a film of cool air, all heat-transfer design.
- Machine Design (34/19) and Materials Science & Manufacturing (34/13): a bearing, a brake disc or a motor winding has a thermal limit as real as its strength limit, and k is a material property.
- Control Systems (34/20): a thermostat loop is a time constant being chased, using the same exponential as the cooling ball.
- In the world: house insulation and windows, CPUs and phones, batteries and EV cooling, cooking, spacecraft radiators and the Earth's own energy balance.
If you want to go deeper
- 📘 Textbook. Theodore Bergman, Adrienne Lavine, Frank Incropera and David DeWitt, Fundamentals of Heat and Mass Transfer (Wiley, 8th edition). The standard text, thorough and full of tables.
- 📘 Textbook. Yunus Çengel and Afshin Ghajar, Heat and Mass Transfer: Fundamentals and Applications (McGraw Hill, 6th edition in 2020). Chattier, with many worked examples.
- 📘 Free to read alone. John Lienhard V and John Lienhard IV, A Heat Transfer Textbook (6th edition, free to download for personal use).
- 🌐 Free course. MIT OpenCourseWare 2.051 Introduction to Heat Transfer (Fall 2015, undergraduate). A shorter route is the heat-transfer part of 16.050 Thermal Energy.
Heat moves by conduction (Fourier: k A ΔT ÷ L), convection (Newton: h A ΔT) and radiation (Stefan–Boltzmann: ε σ T⁴). Treat a wall like a circuit, where resistances in series add and the biggest wins. Add area with fins when the other dials are stuck. And check the Biot number before treating an object as a single lump with one temperature and one time constant.
Try this: on a cold, clear night, feel the roof of a parked car and the roof of a car parked under a tree or a carport. Then check which one has frost. The difference is a clear sky versus a ceiling that radiates back.
Sources. The three modes, thermal resistance, fins and the lumped-capacitance model are standard textbook heat transfer (Incropera; Çengel & Ghajar; Lienhard). Fourier's 1822 Théorie analytique de la chaleur; Stefan 1879 and Boltzmann 1884 for the T⁴ law. σ = 2π⁵k⁴ ÷ 15h³c² = 5.670374 × 10⁻⁸ W/m²·K⁴ from the exact SI values of k, h, c. Thermal conductivities: rounded typical values (Incropera appendix; Engineering Toolbox: window glass 0.96, plasterboard 0.17, fibreglass 0.04, still air 0.026). Convection ranges: Incropera Table 1.1 (free convection in gases 2 to 25, forced in gases 25 to 250, forced in liquids 100 to 20,000 W/m²·K). Air-film resistances 0.13 and 0.04 m²·K/W: ISO 6946 values for horizontal heat flow. Frost above freezing on clear nights: radiative cooling to a sky colder than the air; the effective sky temperature is a convenient and approximate concept and is set by assumption in the interactive. Every number was computed in node: the 100 m² wall loses 866 W with 10 cm of fibreglass (U = 0.347 W/m²·K, R = 2.885) and 6,487 W without (R = 0.385); copper 5 cm slab at 50 K carries 401 kW/m²; the roof skin settles at −2.4 °C (clear, calm, h = 10), 0.4 °C (h = 25), 0.1 °C (10 K sky), 2.1 °C (3 K sky); aluminium fin, 50 mm, h = 25, carries 5.96 W (efficiency 92%); steel ball r = 10 mm, h = 25: τ = 454 s, Bi = 0.0014; water-like ball r = 25 mm: τ = 1,393 s, Bi = 0.347.
- 🌐 Lienhard, A Heat Transfer Textbook (free download)
- 🌐 MIT OCW 2.051 Introduction to Heat Transfer
- 📘 Bergman, Lavine, Incropera & DeWitt, Fundamentals of Heat and Mass Transfer