Page 11 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION
defined as being one of forced convection. A mixed convection state is one in which both
natural and forced convections are present. Convection heat transfer also occurs in boiling
and condensation processes. 3
All bodies emit thermal radiation at all temperatures. This is the only mode that does
not require a material medium for heat transfer to occur. The nature of thermal radiation
is such that a propagation of energy, carried by electromagnetic waves, is emitted from the
surface of the body. When these electromagnetic waves strike other body surfaces, a part
is reflected, a part is transmitted and the remaining part is absorbed.
All modes of heat transfer are generally present in varying degrees in a real physical
problem. The important aspects in solving heat transfer problems are identifying the sig-
nificant modes and deciding whether the heat transferred by other modes can be neglected.

1.3 The Laws of Heat Transfer

It is important to quantify the amount of energy being transferred per unit time and for that
we require the use of rate equations.
For heat conduction, the rate equation is known as Fourier’s law, which is expressed
for one dimension as
dT
q x =−k (1.1)
dx
2
where q x is the heat flux in the x direction (W/m ); k is the thermal conductivity (W/mK,
a property of material, see Table 1.1)and dT/dx is the temperature gradient (K/m).
For convective heat transfer, the rate equation is given by Newton’s law of cooling as

q = h(T w − T a ) (1.2)
2
where q is the convective heat flux; (W/m ); (T w − T a ) is the temperature difference
2
between the wall and the fluid and h is the convection heat transfer coefficient, (W/m K)
(film coefficient, see Table 1.2).
The convection heat transfer coefficient frequently appears as a boundary condition in
the solution of heat conduction through solids. We assume h to be known in many such
problems. In the analysis of thermal systems, one can again assume an appropriate h if not
available (e.g., heat exchangers, combustion chambers, etc.). However, if required, h can
be determined via suitable experiments, although this is a difficult option.
The maximum flux that can be emitted by radiation from a black surface is given by
the Stefan–Boltzmann Law,thatis,
4
q = σT w (1.3)
2
where q is the radiative heat flux, (W/m ); σ is the Stefan–Boltzmann constant (5.669 ×
4
2
10 −8 ), in W/m K and T w is the surface temperature, (K).
The heat flux emitted by a real surface is less than that of a black surface and is given by
4
q = σT w (1.4)