see this about Thermal Conduction and Fourier law

Thermal conduction or Heat transfer by conduction is one of the modes of heat transfer. The law which guides heat transmission in engineering is very essential in the design and construction of heat exchange apparatus. Heat can be transferred in three modes which include: conduction, convection and radiation. For the purpose of what we want to achieve in this series, we will limit our discussion on conduction as a mode of heat transfer.

Conduction is the transfer of heat from one isotropic homogeneous substance to another part of that substance and a flow from one substance to another physical substance in contact with it without any displacement of the molecules making up the substance. Conduction can take place in solid, gases and liquid. Heat transfer through conduction can take place in two mechanisms:

• Molecular interaction: the molecules wit high energy moving with high speed (the effect of temperature on kinetic energy of molecules) impact energy on adjacent molecule of lower energy.
• Electron movement: pure metals (good electrical conductors) having free electron constitute an energy that is transferred in the direction of decreasing temperature.

Conduction in gases is similar to the kinetic energy in a molecule as a function of temperature. Recall the Brownian movement and diffusion theory , these molecules are in constant motion and collide together without loss of kinetic energy and momentum.

In liquid, the principle of heat conduction is similar to gases only that the molecules are in close packed than gases (which are far in their average distance) which aids heat transmission effectively in the system. Notice the second law of thermodynamics in general case that energy cannot flow from the region of lower temperature to higher temperature region; this same applies in heat conduction in materials. It is also in this principle that heat apparatus are designed by Engineers especially chemical Engineers.

FOURIER LAW

This states that heat flow rate through a homogeneous solid is directly proportional to the cross sectional area and at right angle to the heat flow direction and to change of temperature with respect to the length of the heat flow.     read more

Mathematically, Q = -Kdt/dx

Where; Q is the heat flow in the body in watt

A is the cross sectional area of the body in m2

dt is the temperature of the body in ˚K

dx is the thickness of the body in the direction of the flow In M

K is the thermal conductivity in W/M˚K

Assumption on Fourier law

• Heat conduction takes place in steady state conduction
• Heat flows in unidirectional path
• The temperature gradient is constant
• No heat is generated internally
• The bonding surface is isothermal in character

The thermal conductivity depends on the following factors: material structure, moisture content, density of material and operating temperature and pressure.

The thermal conductivity (K) of some materials are shown in the table below:

 material K (W/M˚K) material K(W/M˚K) Cast iron 55.65 air 0.024 Glass 0.75 Mild steel 380 water 0.55-0.70 Stainless steel 45 ash 0.20 silver 415 asbestos 0.70

Solid has the highest thermal conductivity followed by liquid and gases. For mono-atomic gases, K varies with about T0.5 but for polyatomic gases, K changes easily with temperature. READ MORE ON THERMAL RADIATION

Heat flow through cylindrical wall

Consider a cylinder with inner and outer diameter of R0 and R1, heat conduction is given by the relation

Q = 2πkl(T1-T0)/In(R0/R1)

Where L is length of the cylinder

T1-T0  is the temperature difference and the denominator of the expression is the logarithmic mean radius.Assuming that a composite wall having two columns and heat is transfer from the higher temperature region passing through the wall to the region of lower temperature. The heat transfer is given by :

The above relation can be presented thus

Q = TEMPERATURE CHANGE/SUM OF TOTAL RESISTANCE