Heat Transfer within Members

In the modern physics, the heat transfer by conduction in solids is governed by the Fourier’s equation of heat transfer. Fourier’s equation states that the quantity of heat transferred per unit time across an area A is proportional to the temperature gradient ∂T/∂x as follows:

(1)

where

A	is the area across which heat is transferred [m2];
k	is the thermal conductivity of the material [W/m K];
q	is the heat transfer rate across the area A[W];
T	is the temperature [K];
X	is the distance normal to the area A[m].

In the conduction heat transfer of solids made of construction materials, such as steel, concrete and masonry, the solids are generally assumed to be isotropic. Consequently, for a small rectangular block in a Cartesian coordinate system (x, y and z), the three-dimensional heat conduction equation is given by:

(2)

where

ρ	is the density of the material [kg/m3];
c	is the specific heat of the material [j/kg K];
k	is the thermal conductivity of the material [W/m K];
Q	is the internal energy generated within the element [W/m3];
T	is the temperature [K];
t	is the time [sec].

Eq.(2) can be rewritten in the following form:

(3)

where α = k/ρc is called the thermal diffusivity of the material [m²/sec]. The larger the value of &alpha, the faster heat will diffuse through the material.

To solve the heat conduction equation of a structural member, initial and boundary conditions must be provided, including:

In steady-state condition where the thermal environment of the member is constant, it is possible to establish simple analytical solutions by using finite difference technique for one and two-dimensional heat transfer of conduction, if the material properties are assumed to be independent of temperature. However, in transient-state condition where the thermal environment is not constant, the solutions can only be carried out by using either finite element or finite difference technique.