Page 148 - Industrial Ventilation Design Guidebook

1 10 CHAPTER 4 PHYSICAL FUNDAMENTALS
4.33 Heat Conduction

The heat flow density q of a material depends on the local temperature gradi-
ent, according to Fourier's law:

In simple one-dimensional cases, it is easy to determine the temperature gradi-
ent and calculate the heat flow from Fourier's law.
The general case is that of steady-state flow, and the thermal conductivity
factor is a function of the temperature. In the unsteady state the temperature
of the system changes with time, and energy is stored in the system or released
from the system reduced. The storage capacity is

4.3.3.1 General Heat Conduction Equation
Consider a small control volume V — Sx8y8z (Fig. 4.27), where the
inner heat generation is Q^"(T)(heat production/volume) and the heat
conductivity is A(T). The material is assumed to be homogeneous and iso-
tropic, and the internal heat generation and thermal conductivity are
functions of temperature.
The heat flow to the control volume through area dy8z at x is

The outgoing heat at the point x + 8x is

Similar formulas can be derived for the other directions. The change of inter-
nal energy inside the control volume during time dt is

and the heat generation inside the control volume is

From the first law of thermodynamics,

Substituting Eq. (4.175) and the formulas for other directions into Eq. (4.178)
gives