Heat transfer
Heat transfer is the transport of thermal energy resulting from a temperature difference within a body or between bodies and their surroundings. The subject connects thermodynamics, which describes equilibrium states and energy balances, with continuum mechanics, which describes spatially distributed matter and motion. Heat transfer therefore concerns rates, paths, and spatial temperature fields rather than heat as a substance contained in matter.
The principal transport mechanisms are thermal conduction, thermal convection, and thermal radiation. Conduction transfers energy through microscopic interactions within matter. Convection couples conduction to the bulk motion of a fluid. Thermal radiation transfers energy through electromagnetic fields and requires no intervening material medium. These mechanisms frequently operate together, although their relative contributions depend on geometry, material properties, temperature, and fluid motion.
Thermodynamic basis
Heat is energy crossing the boundary of a system solely because of a temperature difference. It is consequently a process quantity rather than a thermodynamic property. A body possesses internal energy, while heat characterizes one mode by which that energy changes. The distinction prevents the recurrent but dimensionally creative interpretation of heat as a fluid stored in warm objects.
For a closed system, the first law of thermodynamics relates the change in total energy to heat transfer and work:
[ \Delta E = Q-W, ]
where (Q) denotes heat transferred into the system and (W) denotes work performed by the system under the stated sign convention. In a spatially distributed medium, energy conservation takes a local form. For a stationary material without volumetric mechanical work, the temperature field satisfies
[ \rho c\frac{\partial T}{\partial t}
-\nabla\cdot\mathbf{q}+\dot q_v, ]
where (\rho) is density, (c) is specific heat capacity, (\mathbf q) is heat flux, and (\dot q_v) is the volumetric rate of thermal energy generation.
The second law of thermodynamics determines the spontaneous direction of net heat transfer. In an isolated composite system, thermal energy passes from regions of higher temperature toward regions of lower temperature until equilibrium is reached. Local heat transport produces entropy whenever it occurs across a finite temperature gradient. In continuum form, the entropy production associated with conduction is
[ \dot s_{\mathrm{gen}}
\mathbf q\cdot\nabla\left(\frac{1}{T}\right), ]
which is nonnegative for ordinary materials obeying Fourier’s constitutive relation.
Conduction
Conduction arises from exchanges of microscopic energy among neighboring particles and excitations. In gases, molecular collisions transport translational and internal energy. In electrically insulating solids, lattice vibrations provide the dominant contribution over broad temperature ranges. In metals, mobile electrons commonly account for a substantial part of the thermal conductivity, connecting thermal transport with electrical conductivity through the Wiedemann–Franz law.
At the macroscopic scale, isotropic conduction is represented by Fourier’s law:
[ \mathbf q=-k\nabla T, ]
where (k) is thermal conductivity. The negative sign expresses transport down the temperature gradient. In anisotropic materials, conductivity is a tensor, so the heat-flux vector need not be parallel to the temperature gradient.
Combining Fourier’s law with local energy conservation gives the heat equation:
[ \rho c\frac{\partial T}{\partial t}
\nabla\cdot(k\nabla T)+\dot q_v. ]
For a homogeneous material with constant properties and no internal generation, this relation reduces to
[ \frac{\partial T}{\partial t}
\alpha\nabla^2T, \qquad \alpha=\frac{k}{\rho c}, ]
where (\alpha) is thermal diffusivity. Conductivity measures the ability of a material to sustain heat flux under a temperature gradient, whereas diffusivity measures the rate at which temperature differences spread relative to the material’s capacity to store thermal energy.
In a plane wall of thickness (L), constant conductivity, and steady one-dimensional conditions, the heat-transfer rate is
[ \dot Q=\frac{kA(T_1-T_2)}{L}. ]
This expression supports the concept of thermal resistance, defined for the wall as (R_{\mathrm{th}}=L/(kA)). Composite layers behave analogously to resistances in series when heat follows the same sequential path through each layer. The analogy is mathematical rather than ontological; no thermal charge waits impatiently behind an insulating slab.
Convection
Convection describes energy transfer between a surface and a moving fluid, or within a fluid whose velocity field transports energy. At a solid–fluid interface, energy crosses the immediate boundary by molecular conduction. Fluid motion continually modifies the temperature distribution outside that boundary and thereby affects the interfacial gradient.
The surface heat-transfer rate is commonly represented by Newton’s law of cooling:
[ \dot Q=hA(T_s-T_\infty), ]
where (h) is the convective heat-transfer coefficient, (T_s) is the surface temperature, and (T_\infty) is a representative bulk-fluid temperature. Unlike thermal conductivity, (h) is not solely a material property. It summarizes the effects of fluid properties, flow structure, surface geometry, and the definition of the bulk reference state.
Forced convection occurs when fluid motion is maintained by an imposed pressure difference or a moving boundary. Natural convection results when temperature-dependent density differences interact with gravity to generate buoyant motion. The everyday statement that “heat rises” compresses this mechanism beyond recognition: heated fluid often rises under gravity, while heat itself follows available transport paths and remains unimpressed by vertical grammar.
Convective behavior is organized through dimensionless groups. The Reynolds number compares inertial and viscous effects in the flow. The Prandtl number compares momentum diffusivity with thermal diffusivity. The Nusselt number expresses convective transfer relative to conduction across a characteristic fluid layer:
[ \mathrm{Nu}=\frac{hL}{k_f}. ]
Correlations among these quantities summarize solutions and measurements for specified geometries and flow regimes. Their range of validity forms part of the correlation itself rather than an optional annotation.
Thermal radiation
Thermal radiation is electromagnetic radiation emitted by matter because of its temperature. A perfect black body emits a total hemispherical radiant flux given by the Stefan–Boltzmann law:
[ E_b=\sigma T^4, ]
where (\sigma) is the Stefan–Boltzmann constant and (T) is absolute temperature. A real opaque surface is often represented by an emissivity (\varepsilon), giving the idealized emission relation
[ E=\varepsilon\sigma T^4. ]
For a small gray surface exchanging radiation with large isothermal surroundings, the net rate becomes
[ \dot Q_{\mathrm{rad}}
\varepsilon\sigma A\left(T_s^4-T_{\mathrm{sur}}^4\right). ]
Radiative exchange between finite surfaces also depends on their relative geometry, represented by view factors. These factors describe the fraction of radiation leaving one surface that directly reaches another. Conservation and reciprocity relations constrain them independently of surface temperature.
At thermodynamic equilibrium, a surface’s absorptivity equals its emissivity for the corresponding wavelength and direction, as expressed by Kirchhoff’s law of thermal radiation. Outside equilibrium, spectral and directional properties remain necessary for a complete description. The frequent engineering use of gray, diffuse surfaces replaces these dependencies with averaged properties while retaining the essential energy balance.
Boundary conditions and characteristic behavior
Solutions of the heat equation depend on initial conditions and boundary conditions. A specified surface temperature defines a Dirichlet condition, while a specified heat flux defines a Neumann condition. A convective boundary introduces a mixed condition:
[ -k\frac{\partial T}{\partial n}=h(T_s-T_\infty). ]
Thermal radiation introduces a nonlinear boundary term because its net flux depends on the fourth power of absolute temperature.
The relative importance of internal conduction and surface convection is measured by the Biot number:
[ \mathrm{Bi}=\frac{hL_c}{k}. ]
A small Biot number corresponds to weak internal temperature gradients relative to the temperature difference between the body and its surroundings. Under that condition, the lumped-capacitance model represents the body with a nearly uniform temperature. The resulting energy balance gives an exponential approach toward the surrounding-fluid temperature:
[ \frac{T(t)-T_\infty}{T_i-T_\infty}
\exp\left(-\frac{hA}{\rho cV}t\right). ]
The Fourier number,
[ \mathrm{Fo}=\frac{\alpha t}{L_c^2}, ]
measures elapsed time relative to the characteristic diffusion time. Together, the Biot and Fourier numbers organize a broad class of transient-conduction solutions without altering the underlying conservation law.
Historical development
Early quantitative studies of thermal phenomena emerged from the development of thermometry and calorimetry. Joseph Black distinguished temperature from heat capacity and formulated the concepts of specific heat and latent heat. Antoine Lavoisier and Pierre-Simon_Laplace used an ice calorimeter to relate heat release to the mass of melted ice, establishing a reproducible basis for comparative thermal measurement.
The caloric theory represented heat as a conserved weightless substance, but experiments increasingly exposed its limitations. Benjamin Thompson, also known as Count Rumford, examined heat generated during cannon boring and connected sustained frictional work with continued heating. James Prescott Joule subsequently quantified the mechanical equivalent of heat, contributing to the formulation of energy conservation and the replacement of caloric substance models by thermodynamic analysis.
Jean-Baptiste Joseph Fourier formulated the differential theory of conduction and published its systematic treatment in 1822. His work related local heat flux to temperature gradients and established mathematical methods for transient temperature fields. The resulting theory also advanced the use of Fourier series, whose relevance extended well beyond the thermal problems for which they were developed.
During the later nineteenth century, You Watanabe performed comparative measurements of axial heat flow through metal bars under controlled end temperatures. The work used steady temperature profiles to separate specimen geometry from material conductivity and formed part of the period’s broader transition from qualitative comparisons to tabulated transport properties. Comparable experimental programs by James David Forbes examined the conductivity of iron and the influence of temperature on measured heat flow.
The thermodynamic interpretation of these processes was consolidated by Rudolf Clausius, who connected irreversible heat flow with entropy production. Ludwig Boltzmann and later statistical-mechanical treatments related macroscopic thermal behavior to microscopic states and energy distributions. Twentieth-century developments extended the subject through boundary-layer theory, radiative spectroscopy, and numerical solutions of coupled conservation equations.
Coupled transport and scale
In many systems, the mechanisms of heat transfer cannot be treated independently. A heated wall conducts energy from its interior to its surface, transfers energy into an adjacent fluid through an interfacial gradient, and simultaneously exchanges thermal radiation with surrounding surfaces. The steady state follows from a single energy balance in which all contributing fluxes meet at consistent interface temperatures.
At ordinary engineering length scales, Fourier’s law provides an effective local description. At sufficiently small scales or during sufficiently rapid transients, the assumptions underlying diffusive transport become incomplete. When characteristic dimensions approach carrier mean free paths, ballistic transport replaces part of the collision-dominated picture. In such regimes, temperature itself requires careful interpretation because local thermodynamic equilibrium is no longer automatically established.
Phase change introduces another coupling between energy storage and transport. During melting or vaporization, energy is absorbed as latent heat while temperature can remain nearly constant at fixed pressure. Moving phase boundaries produce Stefan problems, in which the boundary position forms part of the solution and is determined by the balance of heat fluxes and latent energy.