Heat Equation

(redirected from Particle diffusion)

heat equation

[′hēt i‚kwā·zhən]
A parabolic second-order differential equation for the temperature of a substance in a region where no heat source exists: ∂ t /∂τ = (kc)(∂2 t /∂ x 2+ ∂2 t /∂ y 2+ ∂ t 2/∂ z 2), where x, y, and z are space coordinates, τ is the time, t (x,y,z, τ) is the temperature, k is the thermal conductivity of the body, ρ is its density, and c is its specific heat; this equation is fundamental to the study of heat flow in bodies. Also known as Fourier heat equation; heat flow equation.

Heat Equation


a parabolic partial differential equation that describes the process of propagation of heat in a continuous medium, such as a gas, liquid, or solid. It is the basic equation in the mathematical theory of thermal conductivity.

The heat equation expresses the heat balance for a small element of volume of the medium; heat gains from sources and heat losses through the surface of the element are taken into account for heat transport by conduction. The equation has the following form for an isotropic nonhomogeneous medium:

Here, ρ is the density of the medium; cv is the specific heat of the medium at constant volume; t is time; x, y, and z are space coordinates; T = T(x, y, z, t) is the temperature, which is calculated by means of the heat equation; λ is the coefficient of thermal conductivity; and F = F(x, y, z, t) is the specified density of the heat sources. The magnitudes of ρ cv, and λ depend on the coordinates and, generally speaking, on the temperature. For an anisotropic medium, the heat equation contains in place of λ the thermal conductivity tensor λik, where i, k = 1,2,3.

In the case of an istropic homogeneous medium, the heat equation assumes the form

where Δ is the Laplace operator, a2 = λ/(ρcv) is the coefficient of thermal diffusivity, and f = F/(ρcv). In a stationary state, where the temperature does not vary with time, the heat equation becomes the Poisson equation or, when there are no heat sources, Laplace’s equation ΔT = 0. The basic problems for the heat equation are the Cauchy problem and the mixed boundary value problem (seeBOUNDARY VALUE PROBLEMS).

The heat equation was first studied by J. Fourier in 1822 and S. Poisson in 1835. Important results in the study of the heat equation were obtained by I. G. Petrovskii, A. N. Tikhonov, and S. L. Sobolev.


Carslaw, W. S. Teoriia teploprovodnosti. Moscow-Leningrad, 1947.
Vladimirov, V. S. Uravneniia matematicheskoi fiziki. Moscow, 1967.
Tikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.


References in periodicals archive ?
Among the possible transport mechanisms deserves attention the particle diffusion, driven by a gradient law originated by a non-equilibrium situation; as it has been shown in a previous paper [14], this law is strictly connected with the global entropy increase of an isolated thermodynamic system, the diffusion medium plus the diffusing species both tending to the equilibrium configuration in the state of maximum disorder.
1980), it is likely that two particle diffusion mechanisms were involved (Sivasubramaniam and Talibudeen 1972).
Accordingly, shell progressive film diffusion and shell progressive particle diffusion equations (Hodges and Johnson 1987; Jung et al.
For reaction retarded particle diffusion, the appropriate equation is:
t should thus be linear if reaction-retarded particle diffusion controls the reaction rate (Hodges and Johnson 1987; Jung et al.
5 [micro]m) This result indicates that particle deposition on the top of moving object is influenced by particle diffusion while the fluid flow convection plays less effect on the particle deposition.
The change of these profiles is obvious, namely, in the case of no collision the axial velocity profile becomes narrower that is related to the reduction of the particle diffusion in the radial direction caused by the interparticle collisions [4, 5].
In addition, particle diffusion and uptake promoted by thermal capillary waves might play a role in particle transport through membranes.
Patterns of particle diffusion and deposition differ between plumes and puffs (Gifford, 1968; Hanna et al.
Normally, particle diffusion occurs from a region of high concentration of particles toward one of low concentration.
1992) indicated that changes in arrangement or in the number of exhaust openings do not have a significant effect on the entire flow field; however, such changes often have a large influence on the particle diffusion field (for 0.
c) particle diffusion due to interparticle collisions [17]: