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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
- is the rate of change of temperature at a point over time
- , , and 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.
The heat equation is the prototypic example of a parabolic differential equation
1 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
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- 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 d S and with unit normal vector n is
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Thus the rate of heat flow into V is also given by the surface integral
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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
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- 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
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- 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
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Putting these equations together gives the general equation of heat flow:
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Remarks.
- The constant κ(x) is the inverse of specific heat of the substance at x × density of the substance at x.
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ThermodynamicsThermodynamics is the physics of energy, heat, work, entropy and the spontaneity of processes. Thermodynamics is closely related to statistical mechanics from which many thermodynamic relationships can be derived. While dealing with processes in which sys
Partial differential equations
