The heat transferscience and its basic concepts have already been discussed in details. This post is about the Conduction phenomena in detail.
What Is Conduction?
Conduction is the phenomena of transfer of energy due to the temperature gradient. On the molecular level, conduction definition can be described as the transfer of kinetic energy between the molecules;due to the elastic and inelastic collisions between the molecules.
The term conduction is basically used for the heat transfer between the solids. In liquids and gases pure conduction can not exist.
Fourier’s Law Of Heat Conduction:
Fourier law is used as the general equation of conduction. Fourier law states that:
“The rate of heat transfer per unit area is directly proportional to the normal temperature gradient.”
\[Q=\ -kA\ \frac{dT}{dx}\]
or
\[q_x=-k\frac{dT}{dx}\]
The negative sign of the equation shows the negative temperature gradient, which ensures that the thermal energy flows in the direction of decreasing temperature.
Where:
Q= rate of heat transfer
A= heat transfer area
k = thermal conductivity of material; W/m.K
q = heat flux ; W/m2
The above two equations are the equations for heat conduction in single direction. As per to the Cartesian coordinates system,the above equations can be simplified as the most general equation of conduction is:
\[q=-k\nabla T\]
Thermal Conductivity Units (k):
Thermal conductivity units in FPS system : Btu/hr·ft⋅F
Where:
1W/(m. oC) = 0.5778 Btu/hr·ft⋅F
One Dimensional Steady State Conduction:
The term steady state conduction describes that the temperatures at any point are independent of the time factor. The one dimensional conduction refers to the fact that the temperature gradients exist along in the single direction only.
- Plane wall:
The heat transfer rate through a plane wall (made up of single material) is :
\[Q_x=-\frac{kA}{\triangle x}(T_2-T_1)\]
Or
\[Q_x=\frac{T_1-T_2}{R_th}\]
Where Rth is the resistance to the heat transfer, which is equal to the Δx/ kA
- Composite wall:
The heat transfer rate through a composite wall made up of more then 1 material is :
\[Q_x=\frac{{\triangle T}_{overall}}{\Sigma R_{th}}\]
- Cylinders:
The heat transfer rate along the cylinder are:
\[Q=\frac{2\pi Lk(T_i-T_o)}{{\rm ln}(\frac{r_0}{r_i})}\]
- Spheres
Spherical systems are also considered as the one dimensional systems. The heat transfer rate along the sphere is:
\[Q=\frac{4\pi k(T_i-T_o)}{\frac{1}{r_i}-\frac{1}{r_0}}\]
