Page 15 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION
The solution is determined iteratively because of the nonlinearity of the problem.
Equation 1.15 can be simplified by substituting relations for the surface areas. It should be
noted, however, that this is a general equation that can be used for similar systems. 7
It is important to note that the spatial variation of temperature within the plate is
neglected here. However, this variation can be included via Fourier’s law of heat conduc-
tion, that is, Equation 1.1. Such a variation is necessary if the plate is not thin enough to
reach equilibrium instantly.

1.4.2 Incandescent lamp

Figure 1.2 shows an idealized incandescent lamp. The filament is heated to a temperature
of T f by an electric current. Heat is convected to the surrounding gas and is radiated to the
wall, which also receives heat from the gas by convection. The wall in turn convects and
radiates heat to the ambient at T a . A formulation of equations, based on energy balance,
is necessary in order to determine the temperature of the gas and the wall with respect to
time.
Gas:
Rise in internal energy of gas:

dT g
m g c pg (1.17)
dt
Convection from filament to gas:

h f A f (T f − T g ) (1.18)
Convection from gas to wall:

h g A g (T g − T w ) (1.19)

Radiation from filament to gas:
4
4
f A f σ(T − T ) (1.20)
g
f

Glass bulb
Filament

Gas

Figure 1.2 Energy balance in an incandescent light source