Heat Transfer: Conduction

Fourier Law of Heat Conduction 
When there exists a temperature gradient within
a body, heat energy will flow from the region of high temperature to
the region of low temperature. This phenomenon is known as
conduction heat transfer, and is described by Fourier's Law (named after the French
physicist Joseph Fourier),
This equation determines the heat flux vector q for a given temperature profile T and thermal conductivity k. The minus sign ensures that heat flows down the temperature gradient. 
Heat Equation (Temperature Determination) 
The temperature profile within a body depends
upon the rate of its internallygenerated heat, its capacity to
store some of this heat, and its rate of thermal conduction to its
boundaries (where the heat is transfered to the surrounding
environment). Mathematically this is stated by the Heat Equation,
along with its boundary conditions, equations that prescribe either the temperature T on, or the heat flux q through, all of the body boundaries W,
In the Heat Equation, the power generated per unit volume is expressed by q_{gen}. The thermal diffusivity a is related to the thermal conductivity k, the specific heat c, and the density r by,
For Steady State problems, the Heat Equation simplifies to,

Derivation of the Heat Equation 
The heat equation follows from the conservation of energy for a small element within the body, 
heat conducted in  +  heat generated within  =  heat conducted out  +  change in energy stored within 
Mathematically, this equation is expressed as,
The change in internal energy e is related to the body's ability to store heat by raising its temperature, given by,
One can substitute for q using Fourier's Law of heat conduction from above to arrive at the Heat Equation,

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