5 CONCLUSIONES Y RECOMENDACIONES
5.4 Cierre
( ) ( ( ) )
( ) (
T x T
( ))
TTbcosh
cosh (11)
The heat flow through the base of the fin,Qb, is given by:
( )
T m k A
Qb TTb tanh (12)
The tanh(x) function has an asymptotic value of 1 and thus Qb has a and than(3) = 0.995. This means that increasing the fin length L to increase Qb has no effect for mL values beyond §2.5.§
more or less layered structure inside the flow: layers of fluid slide along each other and mixing between layers is low. Turbulent flow is a chaotic movement of fluid particles and there is strong mixing in the flow.
Consequently the boundary layer is smaller for turbulent flow and heat transfer from a surface to the fluid will be greater. The thickness of the boundary layers, velocity and temperature, increases in the down stream direction.
Because of the complexity of convection heat transfer the common approach is to describe the heat flux from surface to fluid by means of a heat transfer coefficient. The equation for the local heat flux is:
f)
s x
x h T T
qx hx(((( s f (13)
with qqqxx the local heat flux (W/m2
m
m ), hx the local heat transfer coefficient (W/mmm K),2 TTTs the surface temperature, and TTTff the bulk temperature of the fluid.
The bulk temperature is the temperature of the fluid outside the thermal boundary layer in the “undisturbed flow”. The local heat flux, and thus the local heat transfer coefficient, varies along the surface. In the downstream direction the thickness of the thermal boundary increases and the heat transfer is reduced. This leads to the definition of the average heat transfer coefficient,h, along the surface A:
=
³
A xdA hx h A1
(14)
The total heat flow from the surface,Q, is then given by:
) (Ts Tf h
A
Q Ah(( (15)
In analogy with the thermal resistance for heat conduction one can define the thermal resistance for convection as:
h Rth A1
= (16)
The flow field can be a result of an external driving force like a pump or ventilator. In that case one speaks of forced convection. The other driving force is the result of density variations in the liquid caused by the temperature variations. This is called natural convection. Obviously the direction of the flow in natural convection is always against the direction of gravity; hot air rises.
Determining the value of the heat transfer coefficient has been the subject of many studies, both experimental as numerical by means of computational fluid dynamics. The current practice is that results of these studies are given in the form of so called Nusselt correlations. The Nusselt number, Nu, is defined as:
k L
Nu = h (17)
withh the average heat transfer coefficient, Lan characteristic length and k the thermal conductivity of the fluid. The Nusselt number is a function of other dimensionless numbers, which are characteristic for the flow and the fluid. For natural convection the characteristic number is the Rayleigh number, while for forced convection the Reynolds and Prandtl numbers are used. In the referenced textbooks correlations are given for various geometries and flow conditions. Some representative values for the heat transfer coefficient are given in Table 1.
Table 1. Typical heat transfer coefficients
Fluid Configuration h (W/m2K)
Air Free flow, vertical plate 5
Air Forced flow along plate 50
Air Flow in tube 100
Water Flow in tube 3000
2.2.1 Heat Transfer Coefficient Correlations
In the previous section, convection is pointed to Nusselt correlations for estimating the convection heat transfer coefficient. For illustration two correlations will be given:
A warning should be given with respect to the use of correlations. In principle the correlations are a fit of measured or calculated data to describe the heat transfer coefficient for a particular situation. This implies a certain error level, it is a best fit, and errors of 5 to 10% are common. Another limitation lies in the geometry and the heat source. Usually the geometry is relatively simple (flat plate, tube) and the heat source is a uniform temperature or heat flux surface. Obviously this not often encountered in a real electronic system. A board with components is a much more complicated situation. Each component influences the flow, produces a thermal boundary layer and interacts with the other components. Examples of two typical correlations are given in the next section.
• Vertical plate with natural convection.
• Flat plate with a forced flow along the surface.
1. Vertical plate – natural convection
For a vertical flat plate with uniform surface temperature the heat transfer coefficient can be calculated using the following equations:
( )
( )
( ) ( )
[ ]
k L Nu h
Nu
[ (
C RaNu Nu
Ra C Nu
L T T g
( ( )
Ra
m V
t
t CC
l l T
f
=
[ ( )
=
=
1 12 3
1 4 1
3
10 1
, 6
, m≈ < RaRa<
ln
((
2.8α ν
(18)
The nomenclature in the equations is:
• Ra, the Rayleigh number (-)
• g, the gravitational acceleration (9.81 m/s2)
• ββ, the thermal expansion coefficient of the fluid (1/K)
• TTTs, the surface temperature (K)
• TTTfff, the fluid temperature (K)
• L, the vertical length of the plate (m)
• ν, the kinematic viscosity (m2/s)
• α, the thermal diffusion coefficient (m2/s)
• k, the thermal conductivity of the fluid (W/mK)
• h, the average heat transfer coefficient (W/m2K)
The constants Cll and CttV depend on the Prandtl number of the fluid. For dry air (0 to 100°C) the values are: Cll= 0.515, CVtt = 0.103.
The fluid properties have to be taken at the average temperature of surface and fluid.
2. Horizontal plate – forced convection
The average convection heat transfer coefficient for forced flow along a flat plate with constant surface temperature can be calculated with the flowing formula:
( ) ( )
k L Nu h
Nu Nu
L V
L
L L
L
L L
L
=
=
=
=
flow turbulent ,
10 5 ReL > × Pr
Re 0369 .
0
( )
flow laminar ,
10 5 ReL < × Pr
Re 664 .
0
( )
6 0 Pr , > . Re
3 5 8 1 0
3 5 1 1
µ ρ
(19)
The nomenclature in the equations is:
• ReeL, the Reynolds number (-)
• ρρ, the fluid density (kg/m3)
• µµ,the fluid dynamic viscosity (Pa·s)
• V, the fluid velocity (m/s) VV
• L, the plate length (m)
• Pr, the Prandtl number (-); for air Pr§0.7
• NuL, the Nusselt number for lengthL (-)
• k, the thermal conductivity of the fluid (W/mK) h, the average heat transfer coefficient (W/m2K)