Derivation Of Heat Equation

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:

\(\frac{\partial u}{\partial t}-\alpha (\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})=0\)

Heat equation derivation in 1D

Assumptions:

  • 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: 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: \(\frac{\partial T}{\partial x}(x+dx,t)\)

Rate at which the heat energy crosses in right hand is given as: \(\kappa A\frac{\partial T}{\partial x}(x+dx,t)\)

Rate at which the heat energy crosses in left hand is given as: \(\kappa A\frac{\partial T}{\partial x}(x,t)\)

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.

Therefore, \(\kappa A\frac{\partial T}{\partial x}(x+dx,t)-\kappa A\frac{\partial T}{\partial x}(x,t)dt\)

Where, dt: infinitesimal time interval

Temperature change in the rod is given as: \(\frac{\partial T}{\partial t}(x,t)dt\)

Mass of the rod is given as: ⍴Adx

\(s\rho Adx\frac{\partial T}{\partial t}(x,t)dt=\kappa A[\frac{\partial T}{\partial x}(x+dx,t)-\frac{\partial T}{\partial x}(x,t)]dt\)

Dividing both sides by dx and dt and taking limits \(dx,dt\rightarrow 0\) \(s\rho A\frac{\partial T}{\partial t}(x,t)=\kappa A\frac{\partial^2 T}{\partial x^2}(x,t)\) \(\frac{\partial T}{\partial t}(x,t)=\alpha ^{2}\frac{\partial^2 T}{\partial x^2}(x,t)\)

Where,

\(\alpha ^{2}=\frac{\kappa }{s\rho }\) is the thermal diffusivity.

Hence, above is the heat equation.

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