Download A-Basic on Thermal Management

APPLICATION NOTE # A
Basic on Thermal Management
Abstract
The transfer of heat occurs whenever a temperature gradient exists. There are three modes of heat transfer.
Conduction refers to heat transfer through a conductor across which a temperature difference exists. Convection refers
to heat transfer between a surface and a moving fluid having a difference in temperature. The third modality of heat
transfer is called thermal radiation. Thermal radiation is a modality of transferring energy (heat) throughout
electromagnetic waves that intervene from the hot surfaces to the cold one as electromagnetic waves. It does not
require medium a conductor for its propagation. For these three modes, the heat transfer rates depend on the
transport properties of the substances involved, the geometrical parameters, and the temperatures of the elements,
object of the exchange.
Introduction
Whenever a temperature gradient exists, either within a medium or between media, the transfer
of heat will occur. This may take the form of either conduction, convection or thermal radiation.
Conduction
Heat transfer by conduction involves transfer of energy within a material without any motion of
the material as a whole. The rate of heat transfer depends upon the temperature gradient and the
thermal conductivity of the material.
When neighboring molecules in a fluid collide, energy is transferred from the more energetic to
the less energetic molecules. Because higher temperatures are associated with higher molecular
energy, conduction must occur in the direction of decreasing temperature.
It is possible to quantify heat transfer processes in terms of appropriate rate equations. These
equations can be used to compute the amount of energy being transferred per unit time.
Figure A.1 One-dimensional heat transfer by conduction
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APPLICATION NOTE # A
The rate equation for heat conduction is known as Fourier’s law (A.1). For the one-dimensional
plane having a temperature distribution T(x) (Figure A.1), the rate equation is expressed as:
Fourier’s law
q"x  k
dT
dx
(A.1)
The heat flux q"x (W/m2) is the heat transfer rate in the x direction per unit area, perpendicular to
the direction of transfer, and it is proportional to the temperature gradient, dT/dx, in this
direction. The proportionality constant k is the thermal conductivity (W/m  K), and it is a transport
property characteristic of the wall material.
The minus sign in Fourier’s law is requested by the second law of thermodynamic. Thermal energy
transfer resulting from a thermal gradient must be from a warmer to a colder region.
Where the temperature distribution is linear and under a steady-state conditions (Figure A.1), the
temperature gradient delta and heat flux, respectively, may be expressed as:
dT T2  T1

dx
L
T
T
T T
T
q"x  k 2 1  k 1 2  k
L
dx
L
(A.2)
(A.3)
This equation provides a heat flux, that is the rate of heat transfer per unit area. The heat rate by
conduction, qx (W), through a plane wall of area A, is then the product of the flux and the area:
qx  q"x  A .
Convection
Convection is the transfer of heat energy between a surface and a moving fluid at different
temperatures. The convection is comprosed by two mechanisms. In addition to energy transfer
due to random molecular motion , energy is also transferred by the bulk (or macroscopic) motion
of the fluid.
When neighboring molecules in a fluid collide, energy is transferred from the more energetic to
the less energetic molecules, and for this reason, conduction must occur in the direction of
decreasing temperature.
This phenomenon can be seen both in liquids and gases. However, in liquids the molecular
interactions are stronger and more frequent, as the molecules are closer together. In solids,
conduction is caused by the atomic activity of lattice vibrations.
If a fluid flow over a surface, viscous effects are important in the hydrodynamic (velocity)
boundary layer and, for a Newtonian fluid, the frictional shear stresses are proportional to the
velocity gradient. The thermal boundary layer is the region that expresses a temperature
distribution from that of the freestream T to the surface TS (Figure. A.2).
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APPLICATION NOTE # A
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fbfb
Heat surface
Convection from a surface to a moving fluid
Figure A.2 Hydrodynamic and thermal boundary layer development in convection heat transfer.
Convection heat transfer may be classified according to the nature of the flow. Forced convection
is when the flow is caused by external means, such as a fan, a pump, or atmospheric winds.
Instead Natural convection is when the flow is induced by buoyancy forces, which arise from
density differences caused by temperature variations in the fluid. There is also a distinction
between external flow and internal flow. External flow is associated with immersed bodies for
situations such as flow over plates, cylinders and foils. In internal flow, the flow is constrained by
the tube or duct surface. The corresponding hydrodynamic boundary layer phenomena are
different and for this reason the convection processes for the two types of flow are distinctive.
The appropriate rate equation for heat convection, known as Newton’s law of cooling, is as
follows:
q"  h(Ts  T )
(A.4)
where q " , the convective heat flux (W/m2 ), is proportional to the difference between the surface
and fluid temperatures, Ts and T , respectively, and the proportionality constant h (W/m2  K) is
called convection heat transfer coefficient. The convection heat flux is presumed to be positive if
the heat transfer is from the surface Ts  T  and negative if the heat transfer is to the surface
T  Ts  . However, if T  TS , it is possible to expresses the Newton’s law of cooling as:
q"  h(T  TS )
(A.5)
in this case heat transfer is positive to the surface. The convection coefficient depends on
conditions in the boundary layer, which is influenced by surface geometry, the nature of fluid
motion, and an assortment of fluid thermodynamic and transport properties.
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APPLICATION NOTE # A
Thermal Radiation
The third mode of heat transmission is due to electromagnetic wave propagation, which can occur
in a total vacuum as well as in a medium. Experimental evidence indicates that radiant heat
transfer is proportional to the fourth power of the absolute temperature.
p
Surface of emissivity
 , absorpivity  , and temperature Ts
Surface of emissivity
(a)
 , absorpivity  , and temperature Ts
(b)
Figure A.3 Radiation exchange: (a) at a surface in term of the irradiation G provided by different radiation sources and
the surface emissive power E; and (b) between a small, gray surfaces and its large, isothermal surroundings.
Considering radiation transfer processes for the surface (Figure A.3a), the thermal radiation
emitted by the surface originates from the internal energy of matter bounded by the surface, and
the rate at which this energy is released per unit area (W/m2) is called surface emissive power E.
There is an upper limit to the emissive power, which is prescribed by the Stefan–Boltzmann law:
Stefan–Boltzmann law
Eb    TS4
(A.6)
where Ts is the absolute temperature (K) of the surface and  is the Stefan – Boltzmann constant
(   5,67  108 W/m2 K4). Such a surface is called an ideal radiator or blackbody.
The radiant heat flux emitted by a real surface is less than that of a blackbody’s heat flux at the
same temperature and is given by:
E    TS4
(A.7)
where  is called emissivity and it is a radiative property of the surface.
With values in the range 0≤  ≤1, this property provides a measure of how efficiently a surface
emits energy relative to a blackbody. It depends strongly on the surface material and finish.
Radiation can also be incident on a surface. The radiation can originate from a special source( such
as the sun), or from other surfaces to which the surface of interest is exposed.
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APPLICATION NOTE # A
Irrespective of the source(s), the irradiation G is the rate at which all such radiations are incident
on a unit area (W/m2) of the surface as (Figure. A.3a).
A portion, or all, of the irradiation may be absorbed by the surface, thereby increasing the internal
energy of the material. The rate at which radiant energy is absorbed per unit surface area may be
evaluated from knowledge of a surface radiative property called the absorpitivity α. That is:
Gabs  G
(A.8)
where 0≤  ≤1. If  < 1, a portion of the irradiation is not absorbed and may be reflected or
transmitted. The value of  depends on the nature of the irradiation, as well as on the surface
itself.
Figure A.3b represents a case that occurs frequently: radiation exchange between a small surface
at Ts and a much larger, isothermal surface that completely surrounds the smaller one. The
surroundings could be, for example, a radiant panel whose temperature Tsur differs from that of an
enclosed surface (Tsur ≠ Ts). The irradiation may be approximated by emission from a blackbody at
4
Tsur, in which case G    Tsur
. If the surface is assumed to be the one for which    (called a
diffuse-gray surface), the rate of radiation exchange leaving the surface, expressed per unit area of
the surface, is:
"
qrad

q
4
 Eb (Ts )  G   (Ts4  Tsur
)
A
(A.9)
This expression provides the difference between internal energy that is released due to radiation
emission and that which is gained due to radiation absorption.
It is possible to express the radiation exchange in the form:
qrad  hrad A(Ts  Tsur )
(A.10)
where, hrad is the radiation heat transfer coefficient:
2
hrad   (Ts  Tsur )(Ts2  Tsur
)
(A.11)
hrad depends strongly on temperature, while the temperature dependence of the convection heat
transfer coefficient h is generally weak.
The surfaces of Figure A.3 may also simultaneously expresses convection heat transfer to an
adjoining gas. For the conditions of Figure. A.3b, the total rate of heat transfer leaving the surface
is:
4
q  qconv  qrad  hA(Ts  T )  A (Ts4  Tsur
)
(A.12)
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APPLICATION NOTE # A
In conclusion:
Table 15.2
Heat transfer
mode
d
d
Mechanism
Equation
Conduction
Energy transfer due to molecular
activity
Convection
Energy transfer due to molecular
motion (conduction) in addition
to energy transfer due to bulk
motion
Energy transfer by
electromagnetic waves
Thermal
Radiation
q"x  k
dT
dx
Co-efficient
k (W / m  K )
q"  h(T  TS )
"
qrad

q
4
 Eb (Ts )  G   (Ts4  Tsur
)
A
qrad  hrad A(Ts  Tsur )
h(W / m2  K )

hrad (W / m 2  K )
References:
[1] Michael J. Moran,Howard N. Shapiro,Bruce R. Munson, “David P. DeWitt Introduction to Thermal Systems
Engineering: Thermodynamics, Fluid Mechanics, and Heat Transfer” John Wiley & Sons, Inc., p 342-348.
[2] John H. Lienhard IV and John H. Lienhard IV “A heat transfer textbook”, Phlogiston Press Cambridge
Massachussetts.
[3] Donald R. Pitts, Leighton E. Sissom, “Heat Transfer second edition” Schaum’s outline series McGraw-Hill.
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