Derivation of the heat equation can be explained in one dimension by considering an infinitesimal rod. The heat equation is a parabolic partial differential equation, describing the distribution of heat in a given space over time. The mathematical form is given as:
Heat equation derivation in 1D
- The amount of heat energy required to raise the temperature of a body by dT degrees is sm.dT and it is known as the specific heat of the body where,
s: positive physical constant determined by the body
m: a mass of the body
- The rate at which heat energy crosses a surface is proportional to the surface area and the temperature gradient at the surface and this constant of proportionality is known as thermal conductivity which is denoted by 𝛋
Consider an infinitesimal rod with cross-sectional area A and mass density ⍴.
Temperature gradient is given as:
Rate at which the heat energy crosses the right hand is given as:
Rate at which the heat energy crosses in the left hand is given as:
For the temperature gradients to be positive on both sides, temperature must increase.
As the heat flows from the hot region to a cold region, heat energy should enter from the right end of the rod to the left end of the rod.
Where, dt: infinitesimal time interval
Temperature change in the rod is given as:
Mass of the rod is given as: ⍴Adx
Dividing both sides by dx and dt and taking limits
Hence, the above is the heat equation.
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Frequently Asked Questions – FAQs
What are the different modes of heat transfer?
The different modes of heat transfer are:
What is the mathematical form of the Heat Equation?
What is thermal diffusivity?
k is the thermal conductivity
cp is the specific heat capacity
ρ is density
ρcp is the volumetric heat capacity