# 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.

Stay tuned with BYJU’S to learn more on other Physics related articles.

Related Physics Articles:

## 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?

Test your knowledge on Heat equation derivation