| |
|
| If u denotes specific internal energy of the medium,
then the left side of Eq. (2.1) yields |
|
where
is the density of a medium. By introducing the specific heat(cp)
of the medium defined by |
|
| where T is the temperature. Equation
(2.2) can be rewritten as |
To obtain an expression for the
rate formula of heat conduction into Vacross S, the Fourier
law of the heat conduction is introduced. This is an empirical
relationship saying that, for a surface with a unit normal vector
 , the
rate at which heat is conducted across the surface, per unit
area, in the direction of ,
is given by |
 |
where k is a property of the medium
termed the thermal conductivity . In this Eq. (2.5),  denotes
differentiation in the direction of and q is termed the heat
flux in this direction. Thus, it follows that |
 |
| where the Divergence theorem has
been applied. If it is assumed that heat generation in the medium
is occurring at a rate Qper unit volume, then |
|
| Combining Eq. (2.4), (2.6) and (2.7)
into (2.1) produces an energy conservation statement as follows
|
|
| and, since the volume V was arbitrarily
chosen initially, it follows that |
 |
| everywhere in the medium. This is
the familiar formula for the heat conduction equation in a non-stationary
system. If the medium is anisotropic, i.e., if the conductivity
depends upon the direction, the formula of the heat conduction
equation is modified, yielding |
|
where  is
the conductivity tensor and, for example, kxy denotes the thermal
conductivity in the xdirection across the surface with a normal
in the y direction. |
If the conductivity k and the specific
heat capacity 
are assumed to be constant and if the heat generation rate Q
is independent of temperature T, Eq. (2.9) will become a linear
function. Recall that the linearized heat conduction equation
is represented by |
 |
where  is
termed a thermal diffusivity of the medium and  denotes
the Laplacian operator defined in Cartesian coordinates as follows
|
 |
| Without heat generation within the
medium, Eq. (2.11) reduces to the standard diffusion equation
, |
 |
| If the temperature does not vary
with time, a steady-state condition is said to exist. Here the
governing equation simplifies the Laplace equation further as
follows |
|
Suppose that the solution of Eq.
(2.9) is required over an arbitrary domain  bounded
by a closed surface  ,
as illustrated in Fig. 2.1. |
|
General domain and boundary for the heat transfer equation. |
 |
|
| If the problem being modeled is independent
of time, the solution will be uniquely defined with appropriate
boundary conditions . For the heat conduction equation in steady
state , one condition should be specified at each point of the
boundary curve . In general, the boundary conditions can be
classified into two types, given by: |
This is in the situation when the
temperature distribution is prescribed at the boundary surface
 , which
is |
|
| This is known as the dirichlet boundary
condition or essential boundary condition . |
| Here the value of the outward normal
heat flux is given by |
|
| This boundary condition is also known
as the neumann boundary condition or natural |
This boundary condition is also known
as the neumann boundary condition or natural boundary condition
. Here f,  ,
and  are
prescribed functions of  and
T, and  ,
. In Eq. (2.16),  denotes
a convective heat flux , defined as follows |
|
where h is a coefficient of surface
heat transfer  and
is a specified ambient temperature of the surrounding medium.
The is
a radiative heat flux that is |
|
where  is
the Stefan-Boltzmann constant and is  an
emissivity of the surface. Here, the emissivity is defined as
the ratio of the heat emitted by a surface to the heat emitted
by a black body at the same temperature. |
| As was shown above, the heat transfer
problem is a branch of engineering applications that deals with
the transfer of thermal energy; the thermal energy moves from
one point to another within the medium or from one medium to
another due to the temperature difference. Therefore, heat transfer
may be take place in one or more of three basic forms: conduction,
convection, and radiation. |
| Conductive heat transfer exists in any material
whether solid, gas, or liquid. This phenomenon is driven
by a temperature gradient within the material and is dependent
on the thermal conductivity of the material. In a typical
electronics problem, this value can range from 400W/mK
for copper to 0.0261W/mK for air - several orders of magnitude
difference. The conductivity may vary locally with parameters
such as local temperature especially in silicon. It may
not be isotropic particularly for circuit boards, where
the copper layers play a critical part in the heat spreading.
|
|
| Convective heat transfer is defined as energy
transport by a moving fluid such as liquid or gas. The
convective heat-transfer efficiency from the solid to
the fluid depends on the fluid velocity over the source
and the convection regime. This regime can be characterized
as laminar (low speed) where the streamlines follow a
kind of parallel pattern, or turbulent (high speed). In
the latter case, a strong mixture occurs between the streamlines
enhancing the heat transfer ultimately. Convection is
the critical part of design problems in electronic systems.
It is usually characterized as either: |
|
 |
Forced convection occurs when
air/fluid movers (fans or pumps) are used to create the
flow. |
|
Natural convection is the flow created by
a variation of the fluid density with temperature (buoyancy
). |
|
Radiative heat transfer is defined
as radiant energy emitted by a medium and is due solely to the
temperature of a medium. Radiant energy exchange between surfaces
or between a region and its surrounding is described by the
Stefan-Boltzmann law . It says that the radiant energy transmitted
is proportional to the difference of the fourth power of the
surface temperatures. Unlike the conduction and convection,
the radiation does not require a conductive medium to take place.
Consequently, heat transfer by the radiation is potentially
present everywhere. The radiation is a major heat transfer mechanism
for space applications but is generally negligible in forced
convection systems. However, many of the applications such as
laptop computers, cellular telephones, PDA's and telecom's exchange
equipment use natural convection cooling. Frequently, these
enclosures are partially, if not completely, sealed and the
radiation is no longer negligible when compared to the convection.
When the heat transfer problem being modeled is time dependent
(transient state) , the solution is uniquely defined with an
initial condition and a boundary condition at each point of
the boundary .
The initial condition should give the distribution of temperature
over the entire domain of at
the initial time, usually taken to be time t=0. Furthermore,
the function f,
,.
|
|
