# Fourier's Law of Heat Conduction, Formula and Derivation

The physical law governing the transfer of heat through a uniform material (whenever temperature difference exists) by a conduction mode was given by the French scientist: Joseph Fourier.

## Fourier's Law

Fourier law states that the rate of heat flow by conduction through a uniform material is directly proportional to the area normal to the direction of the heat flow and the temperature gradient in the direction of the heat flow.

### Fourier law of Heat Conduction Derivation

Mathematically, the Fourier law of heat conduction for steady-state heat flow is given by

#### Fourier's law of Conduction Equation

\[Q ∝ A {- dT}/{dn} ---------01\]

\[Q = - kA {dT}/{dn} ---------02\]

\[Q = -kA {∆T}/{∆n}\]

The three dimensional from the Fourier law as given.

\[q = -k ∆T\]

Where

Q is the rate of heat flow/transfer in watts (W).

A is the area normal to the direction of heat flow in m2.

T is the temperature in Kelvin (K°)

n is the distance measured normal to the surface, i.e. the length of the conduction path along the heat flow in m.

dT/dn is the rate of change of temperature with distance measured in the direction of heat flow in K/m.

k is a constant of proportionality and is called thermal conductivity. It is the characteristic property of a material through which heat.

### State Fourier's Law of Conduction

The negative sign is incorporated in equation (2) because the temperature gradient is negative (since with an increase in n there is a decrease in T, i.e. temperature decrease in the direction of heat flow) and it makes the heat flow positive in the direction of temperature decrease.

The Fourier law for a steady state unidirectional (say x-direction) heat conduction then becomes

#### Fourier's law in differential form

\[Q = - kA {dT}/{dx} --------03\]

\[q = Q = - k {dT}/{dx} ---------04\]

Where Q is the rate of heat flow, heat flow per unit time in W, and is the heat flux, i.e. rate of heat flow per unit area in W/m2 in the x-direction. In a further discussion, we will make use of equation (03). The Fourier law is a fundamental differential equation of heat transfer by conduction. It is a simple definition of k.

#### Importance of Fourier law

The importance of Fourier's law is the basic rate of heat flow in the conduction process for the analysis of most conduction problems and indicates propagation of heat through a body is quick or slow.

#### Application of Fourier law

Plastic processing, like plastic and polymer manufacturing, plastic molding, and synthetic fiber manufacturing. Chemical processing, like pharmaceutical, chiral chemical mfg. and chemical mfg.

Take these Notes is, Orginal Sources: Unit Operations-II, KA

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