The law of heat conduction is also known as **Fourier’s law**. Fourier’s law states that

“the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area.”

**Fourier’s equation of heat conduction:**

Q = -kA(dT/dx)

Where,

‘Q’ is the heat flow rate by conduction (W)

‘k’ is the thermal conductivity of body material (W·m

‘A’ is the cross-sectional area normal to direction of heat flow (m

‘dT/dx’ is the temperature gradient (K·m

‘k’ is the thermal conductivity of body material (W·m

^{−1}·K^{−1})‘A’ is the cross-sectional area normal to direction of heat flow (m

^{2}) and‘dT/dx’ is the temperature gradient (K·m

^{−1}).- Negative sign in Fourier’s equation indicates that the heat flow is in the direction of negative gradient temperature and that serves to make heat flow positive.
- Thermal conductivity ‘k’ is one of the transport properties. Other are the viscosity associated with the transport of momentum, diffusion coefficient associated with the transport of mass.
- Thermal conductivity ‘k’ provides an indication of the rate at which heat energy is transferred through a medium by conduction process.

### Assumptions of Fourier equation:

- Steady state heat conduction.
- One directional heat flow.
- Bounding surfaces are isothermal in character that is constant and uniform temperatures are maintained at the two faces.
- Isotropic and homogeneous material and thermal conductivity ‘k’ is constant.
- Constant temperature gradient and linear temperature profile.
- No internal heat generation.

### Features of Fourier equation:

- Fourier equation is valid for all matter solid, liquid or gas.
- The vector expression indicating that heat flow rate is normal to an isotherm and is in the direction of decreasing temperature.
- It cannot be derived from first principle.
- It helps to define the transport property ‘k’.

## All Comments

pls help me show full working of how to determine fourier’s equation of heat conduction in isotropic medium

How to find the critical thickness of a insulation if the convective heat transfer co-efficient between the insulating surface and air is 25 W/m^2K.A wire of 6mm diameter with 2 mm thick insulation is used(K=0.11 W/mK).And also want to find the percentage of change in the heat transfer rate if the critical radius is used.Can you please solve this problem with all the steps i can understand?