Heat transfer

Heat transfer asks how quickly heat crosses a boundary and how the temperature varies along its path. Its principal mechanisms are thermal conduction, convection and thermal radiation. A warm wall can conduct heat through its interior while exchanging heat with moving air and emitting thermal radiation at its surfaces. Identifying these paths lets us estimate heating requirements and cooling times.

Distinguish the accumulated heat $Q$ from its rate $\dot{Q}=dQ/dt$, a power measured in $\mathrm{W}$. The heat flux $q$ is the rate per unit area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$: a uniform normal heat flux across area $A$ gives $\dot{Q}=Aq$. The steady bar in thermal conduction carries $q=400\mathrm{W}{\mathrm{m}}^{-2}$ through $A=0.01{\mathrm{m}}^2$, hence $\dot{Q}=4\mathrm{W}$; maintained for $60\mathrm{s}$, it transfers $Q=240\mathrm{J}$. This distinction makes a surface's intensity and a whole object's heating load easy to compare.

A measure of how rapidly a material exchanges heat with another body in transient contact. Here k is thermal conductivity, $\rho$ is mass density and $c_p$ is specific heat capacity. Perceived coldness at equal actual temperature can depend on this contact heat transfer rather than on an electronic Fermi temperature.

Transfer of energy by thermal radiation. A diffuse grey surface of area $A$ and emissivity $ϵ$ completely enclosed by isothermal black surroundings has net outward power $ϵ\sigma A(T_s^4-T_e^4)$. This subtracts absorbed environmental radiation from emitted radiation. Other geometries require exchange factors; a gross emitted power is not a net cooling rate.

Energy transport by bulk fluid motion together with local thermal conduction. A prescribed adiabatic process profile constrains temperature but does not require zero convergence of this energy transport. A cloud held at a specified temperature can therefore have nonzero net radiative heating balanced by convection.

Transfer of thermal energy between regions. In a constant-property thermal Boussinesq approximation, the advective heat flux per unit area is ${\rho }_0{c}_{p}T\mathbf{u}$ and transport across a section is its normal component integrated over that section. With zero net volume transport, adding a constant reference temperature does not change that integral. Otherwise the reference and the return-flow budget must be specified. Atmospheric energy transport also includes latent heat and gravitational potential energy.

The component of heat transport crossing a latitude. A poleward transport is northward in the Northern Hemisphere and southward in the Southern Hemisphere. Equal and opposite volume transports can carry net heat when the poleward branch is warmer. For eddies, the corresponding transport is determined by the correlation of meridional velocity and temperature anomalies, closely related to eddy buoyancy flux in a thermal Boussinesq approximation.

thermal insulation reduces unwanted heat exchange. Conduction through supports, convection in gas and thermal radiation require different treatments. vacuum suppresses gas transport but leaves solid conduction and radiation; lowering thermal conductance also increases the temperature rise caused by a fixed internal heat load.

A thermal shield reduces coupling to unwanted environmental temperature changes. Low-conductance supports and thermal insulation reduce heat transport, low-emissivity surfaces reduce thermal radiation, and a conducting inner enclosure can reduce temperature gradients. Combined with heat capacity these give a thermal low-pass response. Passive shielding cannot indefinitely reject a slow change in the entire environment and may increase the temperature rise from internal dissipation; active regulation addresses a different frequency range.

A lumped sample of heat capacity $C$ coupled by thermal conductance $G$ to a bath has the displayed linearized energy balance. For constant internal power, the bath-to-sample temperature response is $1/(1+i\omega C/G)$. Insulation and heat capacity suppress rapid bath fluctuations but have unity static bath-to-sample gain. A steady power change shifts the sample by $\delta P/G$, so improved isolation increases self-heating sensitivity and equilibration time. The lumped approximation requires internal equilibration faster than the variations being modeled.

A first-order lumped thermal system relaxes exponentially with time constant $C/G$ after a small bath-temperature or heating change. More complex systems have multiple thermal modes and need not have a single relaxation time.

A heat current is a rate of heat transfer across a specified boundary. Its sign requires a chosen direction; unlike heat flux, it is not a rate per unit area.

A heat-current relaxation time measures the decay of a departure from equilibrium carrying heat. It need not equal the momentum relaxation time or the time between arbitrary scattering events. In an isotropic degenerate-electron estimate, $K_e=c_e{v}_F^2{\tau }_Q/3$. Small-angle inelastic electron-phonon scattering can remove an excitation energy of order $k_{B}T$ while barely changing the electron's velocity direction, thus relaxing heat more efficiently than charge flow.

A heat exchanger transfers heat between a fluid and another fluid or its surroundings. In measuring the Joule-Thomson effect with a porous plug, an upstream coil restores the inlet temperature before the measurement throttle; it belongs outside the approximately insulated measurement control volume.

A heat exchanger in which the two streams move in opposite directions. Returning cold helium can cool incoming compressed helium before its expansion in helium liquefaction.

The Nusselt number is the ratio of total heat transfer to conductive heat transfer across the same length scale: $\mathrm{Nu}=qh/(k_T\Delta T)$, where $q$ is heat flux and $k_T$ is thermal conductivity.

For a homogeneous material, thermal inertia is $\Gamma =\sqrt{k\rho c_p}$, where $k$ is thermal conductivity, $\rho$ is density and $c_p$ is specific heat capacity. It measures the resistance of the surface temperature to periodic heating and controls the phase lag relevant to the Yarkovsky effect.

A body loses internal energy by emitting thermal radiation faster than it absorbs radiation or receives other heating. In a gravitating gas this can remove pressure support and allow further contraction.

Hot electrons can transfer energy to cosmic microwave background photons through Compton scattering. For a fixed electron fraction $x_e$ and electron temperature well above the radiation temperature, the optically thin gas cooling time scales approximately as $(1+z{)}^{-4}/x_e$. Its volume loss scales with electron density, not with $n_H^2$, because the radiation density is independent of local gas density. The process tends toward the radiation temperature; it heats rather than cools electrons colder than that radiation.

Dust grains emit thermal radiation after absorbing energy from radiation or collisions. Collisions can transfer thermal energy from dense gas to grains, whose radiation then cools the gas. This channel requires dust and does not operate in strictly pristine hydrogen-helium material without prior enrichment.

Thermal electrons can transfer energy to lower-energy radiation photons through Compton scattering. In the nonrelativistic Thomson regime, the net cooling power per electron is $4{\sigma }_{T}c{U}_{\gamma }{k}_B(T_e-T_{\gamma })/(m_e{c}^2)$ for a blackbody photon bath. A sufficiently hot electron population therefore loses energy; a colder population is heated instead. The cosmic microwave background radiation energy density grows as $(1+z{)}^4$, making this channel more important for high-redshift diffuse ionized gas.

An astrophysical cooling function packages radiative losses of a plasma into a temperature-dependent coefficient. In the hydrogen-density-squared convention, the volume energy loss is $n_H^2\Lambda (T)$, so the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. Its dependence on ionization state, chemical abundances and radiation field must be specified; a low-density collisional-equilibrium curve is not universal.

In an optically thin plasma, one convention defines the radiative cooling power per volume by $\mathcal{C}=n_e{n}_H{\Lambda }_{eH}(T)$, where $n_H$ counts hydrogen nuclei. A convention $\mathcal{C}=n_H^2{\Lambda }_{HH}$ instead has ${\Lambda }_{HH}=(n_e/n_H){\Lambda }_{eH}$. Thus the tabulated functions differ even for the same gas. With total particle density $n$, a monatomic-gas radiative gas cooling time is $3n{k}_{B}T/(2\mathcal{C})$ when heating is negligible.^[ref]

Radiation from rotational and vibrational transitions of molecules removes gas thermal energy. In primordial gas below the atomic threshold, molecular hydrogen is a principal coolant; hydrogen deuteride can matter in suitable chemical conditions. Molecular formation, photodissociation, optical depth and density-dependent level populations control the rate. This allows some small systems below the atomic virial temperature threshold to form stars.

Photoionization changes the ion population, often removing bound-state coolants at temperatures where collisional equilibrium would retain them. It can therefore suppress primordial line-cooling peaks. It also supplies photoionization heating. The net loss is cooling minus heating and depends on radiation spectrum, intensity, shielding and gas density. A universal density-independent function of temperature alone is not generally sufficient for irradiated gas.

For a nonrelativistic ideal gas with $n_{\mathrm{tot}}$ thermal particles per volume, divide its thermal energy $3n_{\mathrm{tot}}{k}_{B}T/2$ by the optically thin volume radiation loss. In the $n_H^2$ cooling convention this becomes $3\chi k_{B}T/(2n_H\Lambda )$, where $\chi =n_{\mathrm{tot}}/n_H$. Density scaling and ionization-dependent particle count must be kept consistent.

Equating the optically thin cooling time to the uniform-sphere free-fall time gives a critical gas density proportional to $T^2/{\Lambda }^2$. With hydrogen mass fraction $X$, particle ratio $\chi$ and gravitating gas fraction $f_g$, it is $[32G{m}_p/(3\pi X{f}_g)][3\chi k_{B}T/(2\Lambda ){]}^2$. At fixed temperature, gas above this density cools faster than free fall. A smaller gravitating gas fraction moves the boundary upward because gravity is faster at a fixed gas density.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.

Atomic line cooling removes gas kinetic energy by collisional excitation followed by photon escape. It is strongest where atoms or ions have bound electrons and thermally accessible transitions. If photons are trapped, or if the relevant atoms are fully ionized, the simple optically thin line-cooling rate no longer gives the same loss.

Heavy elements provide many ionic transitions at temperatures where hydrogen and helium have lost their most effective line emitters. Metal-line cooling can therefore increase the plasma cooling function substantially and broaden its temperature features. Low-energy fine-structure transitions also permit cooling below the hydrogen atomic threshold when the relevant ions and electrons are present.