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.