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:

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

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: 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:

\(\begin{array}{l}\frac{\partial T}{\partial x}(x+dx,t)\end{array} \)

Rate at which the heat energy crosses the right hand is given as:

\(\begin{array}{l}\kappa A\frac{\partial T}{\partial x}(x+dx,t)\end{array} \)

Rate at which the heat energy crosses in the left hand is given as:

\(\begin{array}{l}\kappa A\frac{\partial T}{\partial x}(x,t)\end{array} \)

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,

\(\begin{array}{l}\kappa A\frac{\partial T}{\partial x}(x+dx,t)-\kappa A\frac{\partial T}{\partial x}(x,t)dt\end{array} \)

Where, dt: infinitesimal time interval

Temperature change in the rod is given as:

\(\begin{array}{l}\frac{\partial T}{\partial t}(x,t)dt\end{array} \)

Mass of the rod is given as: ⍴Adx

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

Dividing both sides by dx and dt and taking limits

\(\begin{array}{l}dx,dt\rightarrow 0\end{array} \)
\(\begin{array}{l}s\rho A\frac{\partial T}{\partial t}(x,t)=\kappa A\frac{\partial^2 T}{\partial x^2}(x,t)\end{array} \)
\(\begin{array}{l}\frac{\partial T}{\partial t}(x,t)=\alpha ^{2}\frac{\partial^2 T}{\partial x^2}(x,t)\end{array} \)

Where,

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

Hence, the above is the heat equation.

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Frequently Asked Questions – FAQs

Q1

What are the different modes of heat transfer?

The different modes of heat transfer are:

  • Conduction
  • Convection
  • Radiation
Q2

What is the mathematical form of the Heat Equation?

The mathematical form of the Heat Equation is given as:
\(\begin{array}{l}\frac{\partial u}{\partial t}-\alpha (\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})=0\end{array} \)
Q3

What is thermal diffusivity?

Thermal diffusivity is defined as the rate of temperature spread through a material. It is the measurement of heat transfer in a medium. It measures the heat transfer from the hot material to the cold. Thermal diffusivity is denoted by the letter D or α (alpha). SI unit of thermal diffusivity is m²/s. The thermal diffusivity of a material is given by the thermal conductivity divided by the product of its density and specific heat capacity where the pressure is held constant.
\(\begin{array}{l}\alpha = \frac{k}{\rho c_p} \end{array} \)
where,
k is the thermal conductivity
cp is the specific heat capacity
ρ is density
ρcp is the volumetric heat capacity
Q4

What is the movement of molecules in fluids from higher temperature regions to lower temperature regions known as?

It is known as convection.
Q5

What is the SI unit of heat?

SI unit of heat is Joules.

What is heat? Why do we experience it? How does it travel?

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