Write the heat equation for each of the following cases: a. A wall, steady state, stationary, one-dimensional, incompressible and no energy generation.
b. A wall, transient, stationary, one-dimensional, incompressible, constant k with energy generation.
c. A cylinder, steady state, stationary, two-dimensional (radial and axial), constant k, incompressible, with no energy generation.
d. A wire moving through a furnace with constant velocity, steady state, one-dimensional (axial), incompressible, constant k and no energy generation.
e. A sphere, transient, stationary, one-dimensional (radial), incompressible, constant k with energy generation.

Explanation:

a) the steady-state, 1-D incompressible and no energy generation equation can be expressed as follows:

b) For a transient, 1-D, constant with energy generation

suppose T = f(x)

Then; the equation can be expressed as:

where;

= heat generated per unit volume

= Thermal diffusivity

c) The heat equation for a cylinder steady-state with 2-D constant and no compressible energy generation is:

where;

The radial directional term = and the axial directional term is

d) The heat equation for a wire going through a furnace is:

since;

the steady-state is zero, Then:

'

e) The heat equation for a sphere that is transient, 1-D, and incompressible with energy generation is:

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explanation:

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explanation:

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