Heat Transfer – Conduction of Heat

Conduction of heat in a solid wall is expressed using Fourier's equation for heat conduction,

T : Temperature at a point in wall
θ : Time
k : Thermal Conductivity of wall material
A : Cross sectional area of the element around the point
x : Distance perpendicular to area element

For steady state heat transfer this equation becomes,

Q : Heat transfer rate

This equation can be further developed to express temperature profiles in various geometries with one dimensional heat transfer.

Heat conduction across flat wall

Assuming that thermal conductivity 'k' is independent of temperature and location and A is independent of location, as is the case for a solid wall with constant cross sectional area.

Radial heat conduction across a hollow cylinder

For radial geometry of a hollow cylinder, following equation expresses the heat transfer rate.

Integral of this equation from inner radius r₁ to outer radius r₂ represents the total heat transfer across the cylindrical wall.

N = length of the hollow cylinder

T₁ and T₂ are inner and outer wall temperature of the hollow cylinder.