Heat flux is the rate at which heat energy flows through a surface, divided by the area of that surface. If you hold your hand near a campfire, the warmth you feel on your palm is a heat flux: so many joules of energy arriving on each square metre of skin every second. Engineers use heat flux to size insulation, cool computer chips, design boilers and protect spacecraft. Climate scientists use it to track the energy arriving from the Sun and the heat leaking out of Earth's interior.
The idea is simple, but it is precise in a way that "temperature" or "heat" alone is not. Temperature tells you how hot something is. Heat flux tells you how fast energy is moving, in which direction, and through how much area. This article explains the definition, the main physical laws that govern heat flux (Fourier's law for conduction, the heat transfer coefficient for convection and the Stefan–Boltzmann law for radiation), how heat flux is measured, and where it shows up in everyday life and science.
Definition and units
The Wikipedia article on heat flux defines it as the flow of energy per unit area per unit time. It is also called heat flux density, heat-flow density or heat-flow rate intensity. In the International System of Units (SI) it is measured in watts per square metre (W/m²). Since a watt is a joule per second, that is joules per second per square metre.
A few things follow from this definition.
It is a rate, not an amount. Heat flux is about how fast energy moves, not how much has moved in total. A small heat flux sustained for a long time can deliver a lot of energy; a huge flux for an instant may deliver little.
It is per unit area. The total rate of heat flow through a surface, measured in watts, is the heat flux multiplied by the area (if the flux is uniform). A 10-square-metre wall with 20 W/m² passing through it loses 200 watts.
It is a vector. Heat flux has a direction as well as a size. The Wikipedia article notes it is usually written as φ_q with an arrow over it to show it is a vector, where the subscript q distinguishes heat flux from other kinds of flux, such as mass or momentum flux. At any point inside a material, the heat flux vector points in the direction heat is flowing, and its length tells you how intense the flow is.
It is defined at a point. Formally, the heat flux at a point is the limit of the heat flow through a small surface divided by its area, as the surface shrinks towards zero. That lets heat flux vary from place to place: the flux through the middle of a window pane can differ from the flux at its frame.
The physical dimensions of heat flux are mass × time⁻³ (M·T⁻³), which you can verify: a watt is kg·m²·s⁻³, and dividing by m² leaves kg·s⁻³.
The three ways heat moves
Heat travels by three mechanisms, and each has its own law for heat flux.
- Conduction: energy passes from molecule to molecule (or via free electrons in metals) through a material without the material itself moving. This is how heat travels through a saucepan handle or a brick wall.
- Convection: a moving fluid, such as air or water, carries heat with it. This is how a radiator warms a room or how wind chills your face.
- Radiation: energy travels as electromagnetic waves, which need no medium at all. This is how sunlight reaches Earth across empty space.
In real situations, two or three of these usually act at once.
Conduction and Fourier's law
For most solids under normal conditions, conduction is the main way heat moves, and it is described by Fourier's law, named after the French mathematician Joseph Fourier. In one dimension, the Wikipedia heat flux article gives it as:
φ_q = −k (dT/dx)
Here φ_q is the heat flux, k is the material's thermal conductivity, and dT/dx is the temperature gradient, how quickly the temperature changes with distance x. In three dimensions:
φ_q = −k ∇T
where ∇T, the gradient of temperature, is a vector pointing in the direction in which temperature increases fastest.
Why the minus sign?
The gradient points "uphill", towards hotter regions. Heat flows the other way, from hot to cold. The minus sign takes care of that. As the Wikipedia article puts it, "The negative sign shows that heat flux moves from higher temperature regions to lower temperature regions." This is a direct expression of the second law of thermodynamics: heat never flows spontaneously from cold to hot.
Thermal conductivity
The constant k measures how readily a material conducts heat. Its unit is watts per metre per kelvin, W/(m·K). According to the Wikipedia article on thermal conduction, it depends mainly on the material's phase, temperature, density and molecular bonding. Metals conduct heat well because they contain free-moving electrons; copper, gold and platinum are good conductors. Liquids and gases are poor conductors, and electrical insulators carry heat mainly through vibrations of their atomic lattice, called phonons. The thermal conduction article notes that most gases conduct heat about as well as nitrogen does, with hydrogen and helium as exceptions.
A worked example
Consider a flat wall 0.2 metres thick. Suppose, for illustration, its material has a thermal conductivity of 0.5 W/(m·K). The inside surface is at 20 °C and the outside surface at 0 °C. The temperature gradient through the wall is (0 − 20) / 0.2 = −100 kelvin per metre. Fourier's law gives:
φ_q = −0.5 × (−100) = 50 W/m²
The positive answer means heat flows in the positive x-direction, from inside to outside. If the wall has an area of 12 square metres, it loses 50 × 12 = 600 watts. Double the thickness and the gradient halves, so the heat flux halves to 25 W/m². Halve the conductivity, by choosing a better insulator, and the flux also halves. That is the whole logic of building insulation in two lines of arithmetic.
Thermal resistance: the electrical analogy
Engineers often rearrange Fourier's law into a form that looks like Ohm's law for electric circuits. The Wikipedia thermal conduction article gives the thermal resistance of a simple slab as:
R = L / (kA)
where L is the thickness and A the area. Then the temperature difference across the slab equals the thermal resistance times the heat flow rate: ΔT = R × Q̇, just as voltage equals resistance times current. Thermal resistance is measured in kelvins per watt (K/W).
The analogy is powerful because resistances add up in series, as they do in circuits. A wall made of brick, an air gap and plasterboard has a total resistance equal to the sum of the three. Engineers can then find the heat flow through a complex layered structure with simple addition.
A layered-wall example
Here is an illustrative calculation using made-up but plausible round numbers. Imagine one square metre of wall made of two layers. The outer layer has a thermal resistance of 0.1 K/W for that square metre, and an insulating inner layer has 0.4 K/W. In series, the total is 0.1 + 0.4 = 0.5 K/W. With 20 °C inside and 0 °C outside, the heat flow is ΔT / R = 20 / 0.5 = 40 watts, so the heat flux is 40 W/m².
Now look at what happens inside the wall. The same 40 watts must pass through both layers (heat cannot pile up in a steady state), so the temperature drop across each layer is proportional to its resistance. Across the outer layer it is 40 × 0.1 = 4 kelvin; across the insulation it is 40 × 0.4 = 16 kelvin. The insulation does most of the work of holding back the heat, and the temperature at the boundary between the layers is 4 °C. Calculations like this tell builders where condensation might form inside a wall, since the point where the temperature falls below the dew point of the indoor air is where moisture can collect.
The Wikipedia heat flux article draws the same analogy in reverse when describing measurement: inferring heat flux from the temperature drop across a known material is like measuring an electric current from the voltage drop across a known resistor.
Convection and the heat transfer coefficient
When a fluid flows past a surface at a different temperature, the heat flux between them is usually described with a heat transfer coefficient, h, measured in W/(m²·K):
φ_q = h (T_surface − T_fluid)
This relationship is a form of Newton's law of cooling. The thermal conduction article describes Newton's law of cooling as a discrete analogue of Fourier's law, and notes that it applies well when the Biot number (a ratio comparing heat transfer at a body's surface with conduction inside it) is below about 0.1, meaning the body's interior is nearly uniform in temperature. In that case the body's temperature decays exponentially towards that of its surroundings.
The coefficient h is not a property of a material. It depends on the fluid, its speed, the shape of the surface and whether the flow is smooth or turbulent. A breeze raises h compared with still air, which is why wind makes a cold day feel colder: the temperature difference is the same, but h, and therefore the heat flux leaving your skin, is larger.
Radiation and the Stefan–Boltzmann law
All objects emit thermal radiation. For an ideal black body, the radiated heat flux depends only on its absolute temperature, according to the Stefan–Boltzmann law described in its Wikipedia article:
φ_q = σT⁴
where σ ≈ 5.670374419 × 10⁻⁸ W/(m²·K⁴). For a real surface, multiply by its emissivity ε (between 0 and 1): φ_q = εσT⁴.
The fourth power makes radiation dominant at high temperatures. A black surface at 300 kelvin (about room temperature) emits about 459 W/m². At 600 kelvin it emits sixteen times as much, about 7,350 W/m². In furnaces and flames, radiation usually carries most of the heat.
The net radiative flux between two surfaces depends on the difference of their fourth-power temperatures. A person in a room is constantly emitting and absorbing radiation from the walls; the net flux is what matters for comfort, which is why a room with cold windows can feel chilly even when the air is warm.
The energy balance: why heat flux matters to physics
Heat flux is the natural language for energy conservation, the first law of thermodynamics. Draw an imaginary closed surface around any system, such as a coffee cup, a building or a planet. If heat transfer is the only way energy enters or leaves, then the net rate at which the system's energy changes equals the total heat flux crossing its surface. The Wikipedia heat flux article writes this as a surface integral:
∮_S φ_q · dS
which simply means: add up the flux through every tiny patch of the surface, taking direction into account, so that heat leaving counts one way and heat entering counts the other.
In practice, nobody knows the exact flux at every point on a real surface. The Wikipedia article notes that approximation methods are used, giving Monte Carlo integration as an example: sample the flux at a set of points and estimate the total from them.
Measuring heat flux
Measuring heat flux directly is harder than measuring temperature. The Wikipedia heat flux article describes two broad approaches.
Method 1: from a known conductivity
If you know a material's thermal conductivity and thickness, you can measure the temperatures on both sides and calculate the flux from Fourier's law. The Wikipedia article notes this is often impractical, because the thermal resistance of the material being studied is usually not known accurately.
Method 2: a heat flux sensor
The more common approach is a dedicated heat flux sensor. It works by building a thin layer of known properties into a small device and measuring the temperature difference across it. Because the layer's properties are known, the device can convert that tiny temperature difference into a flux.
According to the Wikipedia article on heat flux sensors, most designs are flat plates containing many thermocouples wired in series, known as a thermopile. A thermocouple produces a small voltage proportional to the temperature difference between its two junctions (the Seebeck effect). Wiring many in series adds their voltages, so a thermopile turns a very small temperature difference across a thin plate into a measurable signal. The heat flux sensor article notes that thermopiles are stable, have low electrical resistance and give zero output with zero input, but their sensitivity is low, so the signals are typically microvolts.
Each sensor must be calibrated. The output voltage is divided by a sensor-specific calibration constant, which the article calls the sensitivity, to give heat flux in W/m². Calibration involves applying a known heat flux and recording the voltage. The heat flux sensor article recommends:
- a voltmeter or data logger with a resolution of about ±2 microvolts or better;
- filling air gaps between layers, for example with thermally conductive gel;
- calibrating at several temperatures, under conditions close to how the sensor will be used.
It cites the standard ASTM C1130-17 for calibrating thin heat flux transducers. Some modern sensors are self-calibrating, with a built-in heater that applies a known flux so the sensor can be recalibrated in place.
Specialised gauges
For very high fluxes, such as in fires, special designs are used. The heat flux sensor article describes:
- The Gardon gauge, a circular-foil gauge that measures combined radiative and convective flux. It must face the heat source directly to be accurate and is used in fire and flame research.
- The Schmidt–Boelter gauge, a wire-wound design mounted on a water-cooled body, used in fire resistance testing.
Common measurement errors
The heat flux sensor article lists several pitfalls:
- Slow response. Results assume steady conditions. The response time of a sensor grows with the square of its thickness, and reaching 95 percent of a step change can take several minutes. Over long averaging periods with steady heat flow, these errors tend to cancel.
- Temperature dependence. Thermocouple output varies with temperature, making sensitivity temperature-dependent; designs can compensate.
- Disturbing the flow. A sensor is itself a layer of material. If its thermal resistance differs from what it is attached to, it changes the very heat flow it is trying to measure. Thin sensors minimise this.
- Lateral flows and non-uniform sensitivity, which are hard to quantify.
Heat flux in the real world
Buildings
Heat flux sensors attached to walls and roofs measure heat loss and gain in real buildings and help determine their insulating performance (R-value and U-value), according to the heat flux sensor article. This matters because insulation as installed often performs differently from laboratory figures.
The human body and clothing
The heat flux sensor article notes that sensors are used to measure human heat exchange when testing clothing, immersion suits and sleeping bags. Very thin sensors are needed because skin acts as a heat sink and a thick sensor would distort the measurement.
Soil and agriculture
In agricultural meteorology, sensors are buried around 4 centimetres deep to track how much energy is going into or out of the soil, according to the same article, though water movement and poor sensor contact make reliable measurement difficult.
Industry and aerospace
The article lists aluminium smelting, boilers, blast furnaces and wind tunnels as industrial uses. Fast sensors called atomic layer thermopiles offer response times of microseconds for aerodynamic testing.
Sunlight: the solar constant
The best-known heat flux on Earth is sunlight. The total solar irradiance, sometimes called the solar constant, is the flux of solar energy at Earth's average distance from the Sun, measured above the atmosphere. According to the Wikipedia article on the solar constant, its mean value is about 1361 W/m² (1.3608 ± 0.0005 kW/m²), and the International Astronomical Union defines the nominal value as exactly 1361 W/m². It varies by only about 0.1 percent between solar minimum and maximum.
Much less reaches the ground. The same article says that at most about 75 percent of solar energy reaches the surface; with the Sun directly overhead, surface irradiance ranges from about 550 W/m² under cirrus cloud to about 1025 W/m² under a clear sky. Historical estimates varied widely: Claude Pouillet obtained 1.228 kW/m² in 1838, while Samuel Langley's 1884 value of 2.903 kW/m² was much too large.
Earth's internal heat
At the other extreme is the tiny heat flux leaking from Earth's hot interior. According to the Wikipedia article on Earth's internal heat budget, the total geothermal heat flow is about 47 terawatts (estimates range from 43 to 49 TW). Averaged over the planet's surface, that is a heat flux of about 91.6 milliwatts per square metre, based on more than 38,000 measurements. The mean is 70.9 mW/m² through continental crust and 105.4 mW/m² through oceanic crust, with the highest values at mid-ocean ridges where young, hot rock rises.
Compare the two: sunlight at the top of the atmosphere delivers roughly 15,000 times more heat flux than geothermal heat flow (1361 ÷ 0.0916 ≈ 14,900). That is why Earth's climate is driven by the Sun, even though its interior is thousands of degrees hot.
Heat flux versus related quantities
It is easy to confuse heat flux with similar terms. Some distinctions:
- Heat (joules) is an amount of energy transferred.
- Heat flow rate or thermal power (watts) is the total rate of transfer through a whole surface.
- Heat flux (watts per square metre) is that rate per unit area.
- Radiant flux refers specifically to energy carried by electromagnetic radiation; the Wikipedia article lists it as a related topic.
- Latent heat flux is energy carried by evaporation and condensation of water, important in weather and climate.
- Insolation is the solar radiation received on a given surface.
Frequently asked questions
What are the units of heat flux? Watts per square metre (W/m²) in SI units, which is joules per second per square metre.
Is heat flux the same as temperature? No. Temperature describes how hot something is. Heat flux describes how fast heat energy crosses a surface. Heat flux depends on temperature differences, not on temperature alone.
Why is there a minus sign in Fourier's law? Because heat flows from hot to cold, opposite to the temperature gradient, which points towards hotter regions.
How is heat flux measured? Usually with a thin heat flux sensor containing a thermopile, a chain of thermocouples that turns a small temperature difference across a layer of known properties into a voltage. The sensor is calibrated to convert its voltage into W/m².
What is a typical heat flux from sunlight? About 1361 W/m² above the atmosphere, and up to around 1000 W/m² at the ground under a clear sky with the Sun overhead.
How much heat comes out of the Earth? On average about 91.6 milliwatts per square metre, totalling roughly 47 terawatts.
Summary
Heat flux is the rate of heat flow per unit area, measured in watts per square metre, with both a size and a direction. Conduction is described by Fourier's law, φ = −k∇T, convection by a heat transfer coefficient, and radiation by the Stefan–Boltzmann law. Writing heat transfer as a flux turns the first law of thermodynamics into a simple accounting rule: the energy change of any system equals the net flux through its boundary. Heat flux can be calculated from known material properties or measured directly with calibrated thermopile sensors, which are used in buildings, clothing tests, soils, industry and fire research. From the 1361 W/m² of sunlight arriving at Earth to the 0.09 W/m² seeping up from its interior, heat flux is one of the most useful single numbers in physics and engineering.
Sources and further reading
- Wikipedia, "Heat flux": https://en.wikipedia.org/wiki/Heat_flux
- Wikipedia, "Thermal conduction": https://en.wikipedia.org/wiki/Thermal_conduction
- Wikipedia, "Heat flux sensor": https://en.wikipedia.org/wiki/Heat_flux_sensor
- Wikipedia, "Stefan–Boltzmann law": https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law
- Wikipedia, "Solar constant": https://en.wikipedia.org/wiki/Solar_constant
- Wikipedia, "Earth's internal heat budget": https://en.wikipedia.org/wiki/Earth%27s_internal_heat_budget