The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. In the special case of a heat propagation in an isotropic and homogeneous medium, this equation is
where:
u(t,x,y,z) is temperature as a function of time and space
ut is the rate of change of temperature at a point over time
uxx, uyy, and uzz are the second spatial derivatives (thermal conductions) of temperature in the x, y, and z directions, respectively
k is a material-specific constant (thermal diffusivity)
To solve the heat equation, we also need to specify boundary conditions for u.
Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.
Heat conduction in non-homogeneous anisotropic media
In general, the study of heat conduction is based on several principles. Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space.
The time rate of heat flow into a region V is given by a time-dependent quantity qt(V). We assume q has a density, so that
Heat flow is a time-dependent vector function H(x) characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area dS and with unit normal vector n is
Thus the rate of heat flow into V is also given by the surface integral
where n(x) is the outward pointing normal vector at x.
The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient
where A(x) is a 3 x 3 real matrix, which in fact is symmetric and non-negative.
By Green's theorem, the previous surface integral for heat flow into V can be transformed into the volume intergal
The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ
Putting these equations together gives the general equation of heat flow:
Remarks.
The constant κ(x) is the inverse of specific heat of the substance at x × density of the substance at x.
In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity.
This approach, however, yields a diffusion coefficient tensor that characterizes a particular experiment and cannot, in general, be applied to different scenarios.
Although this equation neglects adsorption effects, it is otherwise a complete equilibrium thermodynamic description of nonreactive diffusive transport of charged species in concentrated electrolytes.
In those cases where the self-diffusion coefficients of all the diffusing species are nearly identical, the diffusion potential will be nearly zero, and the apparent bulk diffusion coefficient will be nearly equal to the agglomerated diffusion coefficient.
Diffusion fluxes can cause changes in the concentration distribution of the diffusing molecules, and to deal with these we need the full differential equation for diffusion, which can be written
Let's assume that the atmosphere is well-mixed by wind and turbulance at a height of 100 meters, and that diffusion is called upon to move CO from 100 meters to the ground.
In the steady state, we assume that all of the molecules reaching the surface of the cell at radius a are adsorbed, resulting in C=0 at r=a, and a constant rate, Q, of uptake of nutrient molecules.
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