The law of heat conduction is also known as Fourier's law; equation, assumptions and features of Fourier's Law of heat conduction. where the vector is the heat flux (W/m2) in the positive x-direction, dT/dx is the Fourier's Law, as the basic rate equation of the conduction process, when. Jump to Solving the heat equation using Fourier series - The heat equation is a consequence of Fourier's law of conduction (see heat conduction). 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.‎Statement of the equation · ‎The physical problem and · ‎Heat conduction in non.


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The image to the right is animated and describes the way heat changes in time along a metal bar.

What is Fourier's Law of Heat Conduction?

One of the interesting properties of the heat fourier heat conduction equation is the maximum principle that says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern.

This is essentially saying that temperature comes either from some source or from earlier in time because heat permeates but is not created from nothingness. This is a property of parabolic partial differential equations and is not difficult to prove mathematically see below.

For example, if a bar fourier heat conduction equation metal has temperature 0 and another has temperature and they are stuck together end to end, then very quickly the temperature at the point of connection will become 50 and the graph of the temperature will run smoothly from 0 to The heat equation is used in probability and describes random walks.

It is also applied in financial mathematics for this reason. It is also important in Riemannian geometry and thus topology: A famous example is shown in A Christmas Story fourier heat conduction equation, where Ralphie dares his friend Flick to lick a frozen flagpole, and the latter subsequently gets his tongue stuck to it.

He found that heat flux is proportional to the magnitude of a temperature gradient.

Heat equation - Wikipedia

His equation is called Fourier's Law. The symbol q is the heat flux, which is the heat per unit area, and it is a vector. Q is the heat rate. The minus sign is to show that the flow of heat is from hotter to colder.

To quantify the ease with which a particular medium conducts, engineers employ the thermal conductivityalso known as the conductivity constant or conduction coefficient, k.

Thermal conductivity is a material property that is primarily dependent on the medium's phasetemperature, density, and molecular bonding.

Thermal effusivity is a quantity derived from conductivity, which is a measure of its ability to exchange thermal energy with its surroundings.

Steady-state conduction[ edit ] Steady state conduction is the form of conduction that happens when the temperature difference s fourier heat conduction equation the conduction are constant, so that after an equilibration timethe spatial distribution of temperatures temperature field in the conducting object does not change any further.

Thus, all partial derivatives of temperature with respect to space may either be zero or have nonzero values, but all derivatives of temperature at any point with respect to time are uniformly zero. In steady state conduction, the amount of heat entering any region of an object is equal to amount of heat coming out if this were not so, the temperature would be rising or falling, as thermal energy was tapped or trapped in fourier heat conduction equation region.

Heat equation

For example, a bar may be cold at one end and hot at the other, but after fourier heat conduction equation state of steady state conduction is reached, the spatial gradient of temperatures along the bar does not change any further, as time proceeds.

Instead, the temperature at any given section of the rod remains constant, and this temperature varies linearly in space, along the direction of heat transfer.

In steady state conduction, all the laws of direct current electrical conduction can be applied fourier heat conduction equation "heat currents". In such cases, it is possible to take "thermal resistances" as the analog to electrical resistances.

In such cases, temperature plays the role of voltage, and heat transferred per unit time heat power is the analog of electric current. Steady state systems can be modelled by networks of such thermal resistances in series and in parallel, in exact analogy to electrical networks of resistors.


See purely resistive thermal circuits for an example of such a network. Heat equation During any period in which temperatures changes in time at any place within an object, the mode of thermal energy flow is termed transient conduction.