# Thermal conduction **Thermal conduction** is the transport of [[Thermal_energy|thermal energy]] through matter by microscopic carriers — vibrating atoms, [[Phonon|phonons]] and, in metals, free [[Electron|electrons]] — without any bulk movement of the material itself. It is one of the three modes of [[Heat_transfer|heat transfer]], the one that works through solids and through still fluids, and the one that operates whenever [[Convection|convection]] is suppressed and [[Thermal_radiation|radiation]] is weak. Its governing statement is Fourier's law: the heat flux is proportional to the negative of the [[Temperature|temperature]] gradient, q = −k·dT/dx, with the [[Thermal_conductivity_and_resistivity|thermal conductivity]] k as the material's entire contribution. In the microsim below the reader picks a material — copper, steel, glass, wood or still air, each supplying its own k and [[Thermal_diffusivity|thermal diffusivity]] α — and then sets the rod's length and the two end temperatures. The steady answer comes from Fourier's law; the interesting part is the approach to it, which the sim steps live from the transient equation ∂T/∂t = α·∂²T/∂x² using explicit finite differences under the stability condition Δt ≤ Δx²/(2·α).[^vuik-heat] Watch the temperature profile bend from its initial shape toward the straight line that steady state demands, in a time that scales as L²/α, and read out the steady [[Heat|heat]] flux, that settling time, and — on a second panel — the R-value of a wall built from layers in series, R = Σ Lᵢ/kᵢ. Push the step size past the stability limit and the numerical solution explodes into a sawtooth, which is the most honest lesson in the sim. On the [[Energy]] flagship this article is the child of Part VII — Energy transfer, section *Conduction* (row E57), one half of the shared C15 pair whose other half is [[Thermal_radiation|radiation]]. ## Types Conduction problems divide by what the temperature field is doing in time and by which carrier is doing the work. The four cases below run from the everyday to the exotic, and only the first two appear in the microsim. ### Steady-state conduction In steady state nothing stored inside the material changes, so whatever heat enters one face leaves the other. For a slab with no internal generation the temperature profile is a straight line and the flux is constant everywhere along the path. This is the regime in which conduction can be treated as a resistance in a circuit, and in which the answer to "how much heat?" needs only the geometry, the conductivity and the two surface temperatures. A [[Thermal_insulation|wall]] that has been at the same indoor and outdoor temperature for a day is in steady state; the sim reaches it when the profile stops moving. ### Transient conduction Before steady state, the material is charging or discharging its own [[Heat_capacity|heat capacity]]. The governing equation is the instationary heat equation, ρ·c·∂T/∂t = λ·∂²T/∂x² + Q, whose ratio of conduction to storage, α = k/(ρ·c), is the thermal diffusivity and the only material property the transient needs.[^vuik-heat] Diffusivity, not conductivity, sets the clock: the characteristic time to traverse a distance L is L²/α, so doubling the thickness quadruples the wait. A one-metre copper rod with α ≈ 1.1×10⁻⁴ m²/s settles in roughly 9×10³ s, about two and a half hours, while the same rod in glass, α ≈ 5×10⁻⁷ m²/s, would need about three weeks (derived).[^kvals] ### Relativistic conduction The classical heat equation is parabolic, and a parabolic equation propagates a disturbance to every point instantly: put a match to one end of an infinite rod and the far end warms, imperceptibly but immediately. That is a harmless fiction at laboratory scales and an embarrassment in principle, because nothing may outrun light. The repair is to give Fourier's law a finite relaxation time, so that the flux responds to a gradient with a lag rather than instantaneously. The resulting equation is hyperbolic and carries heat as a strongly damped wave of finite speed, reducing to the ordinary diffusion equation whenever the relaxation time is short compared with the timescale of interest — which, outside of ultrafast laser heating and cryogenic solids, it always is. ### Quantum conduction The carriers are quantised. In an electrical [[Insulator_(electricity)|insulator]] heat is carried by phonons, the quanta of lattice vibration whose population and spectrum are described by the [[Debye_model|Debye model]], and conductivity is limited by how far a phonon travels before it scatters off another phonon, an impurity or a boundary.[^likharev-ch2] In a metal the conduction [[Electron|electrons]] carry most of the heat as well as the charge, which is why the ratio of thermal to electrical conductivity at a given temperature is nearly the same for all metals — the [[Wiedemann–Franz_law|Wiedemann–Franz law]], a result the free-electron [[Drude_model|Drude model]] already gets approximately right.[^openstax-v3-ch9] When a sample is made smaller than the carrier mean free path, conduction stops being diffusive altogether and the conductance approaches a quantum limit set by the number of available modes; [[Graphene|graphene]] and [[Carbon_nanotube|carbon nanotubes]] are the standard laboratory examples. ## Fourier's law Fourier's law is the constitutive statement that heat flows down a temperature gradient at a rate proportional to it. Joseph Fourier published it, together with the analytical machinery invented to solve it, in the *Théorie analytique de la chaleur* of 1822 — a book that gave physics both a heat equation and [[Fourier_analysis|Fourier series]].[^fourier1822] Its structure is identical to [[Fick's_laws_of_diffusion|Fick's law]] for matter, to [[Electrical_resistivity_and_conductivity|Ohm's law]] for charge and to Darcy's law for water in soil, and the analogy is not a metaphor: all four are linear response laws with a driving gradient, a transported quantity and a material coefficient, and their series and parallel arithmetic is the same.[^ochsner-rh] ### Differential form The local statement is q = −k·∇T, or in the one dimension the microsim uses, q = −k·dT/dx, with q the flux in W/m² and k in W/(m·K). The minus sign encodes the [[Second_law_of_thermodynamics|second law]]: heat runs from hot to cold, never the other way, so the flux is opposite in sign to the gradient. Combining the differential form with conservation of energy in a volume element gives the heat equation quoted above, and it is the differential form that the sim actually integrates, cell by cell, along the rod. ### Integral form Integrated across a slab of area A, thickness L and conductivity k held between temperatures T₁ and T₂, the law gives the total heat rate Q = k·A·(T₁ − T₂)/L. The numbers separate materials brutally. Take a rod of cross-section 1 cm² and length 1 m with a 100 K difference across it: in copper, k ≈ 400 W/(m·K), the rod carries 4.0 W; in steel, k ≈ 45, it carries 0.45 W; in glass, k ≈ 1, it carries 10 mW; in wood across the grain, k ≈ 0.15, 1.5 mW; and in still air, k ≈ 0.026, 0.26 mW (derived).[^kvals] Four orders of magnitude separate the metal from the gas, which is why every good insulator is a way of trapping air, and why [[Aerogel|aerogel]] does better still: its pores are smaller than the mean free path of an air molecule, which at room temperature and atmospheric pressure is of order 10⁻⁷ m, so the trapped gas cannot carry heat by ordinary collisions at all.[^openstax-mfp] ## Conductance Because the integral form is linear, conduction can be written as a circuit. Define the thermal resistance of a layer as R = L/(k·A), and the heat rate is the temperature difference divided by the resistance, exactly as current is voltage over resistance. Layers stacked in series add their resistances, Q = ΔT/(R₁ + R₂ + …), which is the same algebra Ochsner uses for water crossing two soil horizons, where the hydraulic resistance of a layer is L/K and the layer resistances add.[^ochsner-rh] Paths side by side add their conductances instead, which is why a steel stud through an insulated wall — a thermal bridge — carries far more than its share. ### Intensive-property representation Dropping the area gives the per-unit-area resistance R = L/k in m²·K/W, the quantity the building trade calls the R-value, whose reciprocal U = 1/ΣR is the U-value. The sim's second panel builds a wall from layers and reports both. A wall of 100 mm of wood and 10 mm of glass has R = 0.100/0.15 + 0.010/1.0 = 0.677 m²·K/W and so U = 1.48 W/(m²·K); adding a still-air cavity of 20 mm, k ≈ 0.026, adds 0.77 and nearly halves U to 0.69 (derived).[^kvals] The arithmetic is the whole of steady-state insulation design, and it shows at a glance why the thin layer of trapped gas usually matters more than the thick layer of solid. Seasonal loss follows by multiplying U·A by heating degree-days, HDD = Σ(65 °F − T_avg) over the days below that threshold: Kerlin's table gives 6,104 degree-days for Hartford against 149 for Miami, a forty-fold difference in the same wall, and puts space heating at about 50 million Btu a year out of a US home's ≈100 million.[^kerlin-hdd] ### Cylindrical shells A pipe is not a slab: the area through which the heat passes grows with radius, so the temperature profile is logarithmic rather than linear and the heat rate through a shell of length L between radii r₁ and r₂ is Q = 2π·k·L·ΔT/ln(r₂/r₁). One consequence surprises people. Adding insulation to a thin pipe increases the conduction resistance but also increases the outer surface available to the surroundings, so below a critical radius, lagging a small pipe can raise its heat loss rather than lower it. ### Spherical For a spherical shell the area grows as r², and integrating gives Q = 4π·k·ΔT/(1/r₁ − 1/r₂). As r₂ → ∞ the resistance does not diverge: a sphere buried in an infinite medium loses heat at a finite rate, 4π·k·r₁·ΔT. That finite limit is why a buried tank, a radioactive waste canister or a magma body reaches a steady loss into surrounding rock, and it is the geometry behind the geothermal heat-in-rock calculations on the [[Geothermal_energy|geothermal]] shelf.[^kerlin-rock] ## Transient thermal conduction The transient problem is the one the sim steps. Discretising ∂T/∂t = α·∂²T/∂x² on a grid of spacing Δx with a forward Euler step and the central second difference −(T_{j−1} − 2·T_j + T_{j+1})/Δx² gives an explicit scheme that is simple, fast and conditionally stable: it works only while Δt ≤ Δx²/(2·α), and halving the grid spacing quarters the allowable time step.[^vuik-heat] The implicit (backward Euler) alternative is unconditionally stable and costs a linear solve per step.[^vuik-heat] For copper with Δx = 1 cm the explicit limit is Δt ≤ 0.45 s (derived); cross it in the sim and the profile does not merely lose accuracy, it oscillates with growing amplitude until it overflows, which is a different failure from being wrong. Two dimensionless numbers organise the rest. The Fourier number α·t/L² is elapsed time measured in units of the diffusion time, and a value near one means the transient is essentially over. The Biot number h·L/k compares the resistance inside the body with the resistance at its surface; when it is small, the body's interior is nearly isothermal and the whole object can be treated as a single lumped [[Heat_capacity|heat capacity]] cooling exponentially, which is [[Newton's_law_of_cooling|Newton's law of cooling]]. When it is large — a thick, poorly conducting body in a vigorous [[Convection|convective]] flow — the interior gradient dominates and the full equation is unavoidable. ### Interface heat transfer Two solids pressed together do not make perfect thermal contact. Real surfaces touch only at asperities, and the gaps between them are filled with a gas of low conductivity, so the interface behaves as an extra resistance in series and a temperature discontinuity appears across a joint that is geometrically continuous. The remedy is to increase the true contact area or replace the gas: pressure, a soft interlayer, or a thermal paste of higher k. In a layered wall the same effect means the measured U-value is usually a little worse than the sum of the layer R-values predicts. ## Applications Conduction is used deliberately in two opposite ways: to move heat out of something quickly, and to stop heat moving at all. Both are the L²/α clock read in different directions. ### Splat cooling Splat cooling drives a molten droplet onto a cold, highly conducting substrate so that it spreads into a film microns thick and freezes before its atoms can arrange themselves into a lattice. The rate follows from the diffusion time: a 20 μm layer on copper has L²/α ≈ (2×10⁻⁵)²/1.1×10⁻⁴ ≈ 3.6 μs, so a 500 K fall happens in microseconds, a cooling rate above 10⁸ K/s (derived).[^kvals] That is fast enough to freeze a metal into the glassy state, and it is the classical route to [[Amorphous_metal|amorphous metals]] and to extended [[Solid_solution|solid solutions]] that equilibrium [[Phase_transition|phase diagrams]] forbid. ### Metal quenching [[Quenching|Quenching]] is the same idea at industrial scale and lower rates: a hot steel part is plunged into water, oil or brine so that conduction out of the part, and convection into the fluid, outrun the diffusion of carbon inside it. What makes quenching hard is that the two resistances are comparable, so the Biot number is of order one and the surface cools far ahead of the core; the resulting gradients produce distortion and cracking as well as the desired hardness, and they are why section thickness is a design variable in [[Heat_treating|heat treating]] rather than an afterthought. ## Zeroth law of thermodynamics Conduction is the mechanism behind the law that makes [[Temperature|temperature]] meaningful. The [[Zeroth_law_of_thermodynamics|zeroth law]] says that if two bodies are each in thermal equilibrium with a third, they are in equilibrium with each other, which is what licenses a thermometer: the instrument is the third body, and its reading is transferable.[^yan-ch1] Equilibrium here means that no net heat flows, and the process that brings two touching bodies to that state, given time and no fluid motion, is conduction. The law is logically prior to the first and second laws and was named last because it had been assumed silently for a century. ## Instruments Because the heat carried by conduction depends on the conductivity of whatever fills the gap, conduction can be run backwards as a measurement: hold the geometry and the temperatures fixed and the heat flow reports the material. ### Thermal conductivity analyzer A thermal conductivity analyser, or katharometer, holds a thin wire at a temperature above its surroundings and measures the electrical power needed to keep it there. With the geometry fixed, that power is proportional to the conductivity of the surrounding gas. The detector is differential in practice — a sample cell and a reference cell in a bridge — which cancels drift in the supply and in the ambient temperature. ### Gas sensor The same device is a universal detector in [[Gas_chromatography|gas chromatography]], where it responds to any species whose conductivity differs from the carrier gas. Helium and hydrogen carriers are chosen precisely because their conductivities are far from those of ordinary analytes, which maximises the signal; the trade is sensitivity for universality, since a katharometer detects everything but identifies nothing.[^kvals] The same physics, packaged as a heated bead, is the basis of cheap [[Hydrogen_storage|hydrogen]] leak sensors. ## See also - [[Thermal_conductivity_and_resistivity]] - [[Thermal_insulation]] - [[Thermal_diffusivity]] - [[Heat_transfer]] - [[Convection]] - [[Thermal_radiation]] - [[Fick's_laws_of_diffusion]] - [[Newton's_law_of_cooling]] ## References [^vuik-heat]: Vuik, Kees; Vermolen, Fred; van Gijzen, Martin (2023). *Numerical Methods for Ordinary Differential Equations*. Chapter 8, "The instationary heat equation", pp. 127–135 — the heat equation in the form ρ·c·T_t = λ·T_xx + Q at pp. 127–128, the explicit forward-Euler stability limit Δt ≤ Δx²/2 at unit diffusivity at p. 129, and the unconditional stability of backward Euler at p. 130. The central second difference −(w_{j−1} − 2·w_j + w_{j+1})/Δx² and its O(Δx²) accuracy are at Chapter 7, "The finite-difference method", pp. 111–126, with the oscillation criterion for the central scheme at p. 122. Portal Book 030, https://open.umn.edu/opentextbooks/textbooks/numerical-methods-for-ordinary-differential-equations [^ochsner-rh]: Ochsner, Tyson (2019). *Rain or Shine*. Chapter 4, "Soil Water Flow", p. 127 (flow across layers as q = Δψ_t/(R_h1 + R_h2) with R_h = L/K, the hydraulic resistance of a layer; resistances in series add), and Chapter 12, "Surface Energy Balance and Evapotranspiration", p. 294 (in soil, conduction dominates heat transfer). Chapter 13, "Soil Temperature", pp. 297–317, treats soil conduction directly (page to pin). Portal Book 119, https://open.umn.edu/opentextbooks/textbooks/rain-or-shine [^kerlin-rock]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*. Chapter 7, pp. 249–266 (heat transfer in rock for geothermal extraction; page to pin). Portal Book 048, https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^kerlin-hdd]: Kerlin (2013), *Future Energy: Opportunities & Challenges*, pp. 176–179 — heating degree-days as HDD = Σ(65 °F − T_avg) over days below 65 °F (pp. 178–179); Table 6-1 values including Hartford 6,104 and Miami 149 (p. 179); a US home's annual use of ≈100 million Btu with space heating ≈50 million Btu/yr (pp. 176, 179). Portal Book 048. [^fourier1822]: Fourier, Joseph (1822). *Théorie analytique de la chaleur*. Paris: Firmin Didot, père et fils. [^likharev-ch2]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 2, "Principles of physical statistics", pp. 29–72 — harmonic-oscillator statistics, the Planck distribution and the Debye theory of the specific heat of solids, the basis for treating lattice vibrations as a phonon gas (page to pin). Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^openstax-v3-ch9]: Sanny, Jeff; Ling, Samuel J., et al. (2016). *University Physics Volume 3*, OpenStax. Chapter 9, "Condensed Matter Physics", pp. 393–440, §9.4 "Free Electron Model of Metals" (page to pin). Portal Book 079, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^openstax-mfp]: Sanny, Jeff; Ling, Samuel J., et al. (2016). *University Physics Volume 2*, OpenStax. Chapter 2, "The Kinetic Theory of Gases", pp. 75–114, with the mean free path λ = k_B·T/(√2·4π·r²·p) and the mean free time τ = λ/v_rms at pp. 94–95 (the equation displays were lost in extraction and the standard forms are supplied). Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^yan-ch1]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 1, "Basic Concepts and Definitions", pp. 31–58 (thermal equilibrium, the zeroth law and the definition of temperature; page to pin). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^kvals]: ILLUSTRATIVE. The conductivities used on this page and as the microsim's material presets — copper ≈ 400, steel ≈ 45, glass ≈ 1, wood across the grain ≈ 0.15 and still air ≈ 0.026 W/(m·K), with diffusivities of about 1.1×10⁻⁴ m²/s for copper and 5×10⁻⁷ m²/s for glass — are round textbook magnitudes given to one or two significant figures, not measured values, and every rod, wall, splat-cooling and katharometer number derived from them on this page is labelled "derived". No Portal Book on the Energy shelf carries a conductivity table; the standard open source is Sanny, Jeff; Ling, Samuel J., et al. (2016), *University Physics Volume 2*, OpenStax, Chapter 1 "Temperature and Heat", pp. 17–74, §1.6 "Mechanisms of Heat Transfer" (page to pin), Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 — the individual printed values have not been read and are marked [citation needed] until they are. ## Further reading - Vuik, Vermolen and van Gijzen, *Numerical Methods for Ordinary Differential Equations* (2023), Chapters 7 and 8 — finite differences and the instationary heat equation, with the stability limit the sim enforces. Portal Book 030. - Ochsner, *Rain or Shine* (2019), Chapter 13 "Soil Temperature", pp. 297–317, and Chapter 4 "Soil Water Flow" for the series-resistance arithmetic. Portal Book 119. - Yan, *Introduction to Engineering Thermodynamics* (2022), Chapter 1 "Basic Concepts and Definitions" and Chapter 4 "The First Law of Thermodynamics for Closed Systems", pp. 127–186. Portal Book 115. - Kerlin, *Future Energy: Opportunities & Challenges* (2013), Chapter 7, pp. 249–266 — conduction through rock as the limit on geothermal extraction. Portal Book 048. - Sanny and Ling et al., *University Physics Volume 2* (2016), Chapter 1 "Temperature and Heat", pp. 17–74 — the conductivity table and the three transfer mechanisms side by side. Portal Book 078. ## External links - [Numerical Methods for Ordinary Differential Equations](https://open.umn.edu/opentextbooks/textbooks/numerical-methods-for-ordinary-differential-equations), Vuik, Vermolen and van Gijzen, open textbook (Portal Book 030) - [Rain or Shine](https://open.umn.edu/opentextbooks/textbooks/rain-or-shine), Tyson Ochsner, open textbook (Portal Book 119) - The Wikipedia pair's external links list further open resources <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Thermal_conduction.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Thermal conduction* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Thermal_conduction.html" data-title="Thermal conduction"></div> *Built from `MICROSIM_GUIDE/specs/sims/Thermal_conduction.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Thermal_conduction) : [Wikitube](https://en.wikitube.io/wiki/Thermal_conduction) · pinned revision [1373951778](https://en.wikipedia.org/w/index.php?oldid=1373951778) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E57 · sim pending (matter/Thermal_conduction).*