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HEAT TRANSFER Prepared by
Mohammad Fazlur Rahman Asst. Professor (AERO)
B. S. Abdur Rahman University
This document contains the basic information regarding the
subject matter
Heat Transfer. The effort is made to help the students getting
exposure to
the subject as well as understand the basic and fundamental
behaviour of
the heat transfer phenomenon. It must be noted that this
document in no way
can avoid the use of text books. For the detailed and deep
understanding of
the subject matter students must refer the text books. While
providing
information the syllabus of the B. S. Abdur Rahman University
has been
targeted.
Forced
Convection over a
Flat Plate
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
1
Contents
Introduction
............................................................................................................................
2
Drag and Heat Transfer in External Flow
.....................................................................
3
Friction Drag and Pressure Drag
.....................................................................................
4
Parallel Flow over Flat Plate
.............................................................................................
6
Friction Coefficient
..........................................................................................................
7
Heat Transfer Coefficient
..............................................................................................
9
Special Cases
.......................................................................................................................
11
Flat Plate with Unheated Starting Length
...............................................................
11
Uniform Heat Flux
..........................................................................................................
11
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
2
Introduction After having studied the forced convection and
having solved the equations involved in it, now
is the time to tackle different cases to discuss and see how the
same finding can be used for the
various cases.
The forced convection differs from the natural one only in the
case of flow as how the flow
takes place and which is the driving force which affects the
flow and its behaviour. First of all
it must be noted that so called free convection is not that in
which there is not force acting. The
force acting in it to cause the bulk motion to occur is the
natural force of buoyancy and hence
it is more appropriate to call it a natural convection rather
than free convection. Now whatever
the flow we have studied in our previous level courses like
Fluid Mechanics and Aerodynamics,
they all contain flow of fluid and thats all are actually forced
flow. So there is a clear analogy
in the governing equations of forced convection and flows in
fluid mechanics and
aerodynamics. The only difference is the energy equation which
involves thermal energy
instead of kinetic energy and flow energy.
The forced flow convection process can be further divided into
two parts 1. External flow and
2. Internal flow. External flows are the cases when fluid is in
contact with one of the solid
surface and other surface of the fluid is free and is in contact
with free air. In the case of internal
flow, the entire fluid surface is surrounded by the solid
surface and this flow is highly affected
by the viscous effect of the fluid. We have a third type also
which is not classified as a separate
class though, in this type of flow top and bottom of the fluid
boundary is in contact with the
solid surface and flow takes place in between them. It is known
as Couette Flow.
Heat Transfer
(Forced Convection over a Flat Plate)
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
3
Drag and Heat Transfer in External Flow The main reason to
discuss all these flows and their behaviour is to get the
appropriate
expressions for the surface friction drag coefficient and the
convection heat transfer coefficient.
There are many natural phenomenon in which the external flow
over a solid surface takes place,
like rain drops fall, flow passing automobile, power lines,
trees and underwater pipelines etc.
Cooling of metal or plastic sheets, steam and hot water pipes
and extruded wires and
rectangular fins etc. these surfaces when exposed to external
fluid flow, they not only cause
the heat transfer, rather there exists a shear drag on the solid
surface owing to the viscous effect
of the fluid. This entire phenomenon is inter related and mutual
interaction of thermal boundary
layer and velocity boundary layer takes place. The knowledge of
heat transfer and momentum
transfer relation helps us understand such phenomenon and make
us solve the problem
normally we encounter.
Since the geometry which we encounter in the normal life is not
always a simple one; the
complicacy of the geometry and their interaction with the fluid
flow forces us to go the
numerical way. Because analytical solution for such problems are
very complicated.
Availability of high speed computers have made our life a bit
easy in this regard, and most of
the data collection and analyses is done through numerical
experimentations quickly by
solving the governing equations numerically. The time consuming
and expensive experimental
testing are done at the final stage of design to collect some
real time data and validate the data
collected numerically.
In this regard we discuss and use two types of velocity
frequently. The velocity far away from
the solid body is called the free stream velocity and this is
the fluid velocity we expect to be
approaching the body encountering the solid body so it is also
the upstream velocity. We call
it approach velocity and denote it by . The subscript denotes
and reminds that this is the
velocity far away from the surface which remains unaffected by
the presence of the solid. The
upstream velocity may vary depending with respect to time but
for the sake of convenience and
ease in analysis we take it to be a steady flow velocity which
remains constant with time.
Another velocity in the near vicinity of the solid is the local
velocity which is affected by the
solid and its value depends upon the geometry of the solid apart
from other influencing
parameters. The local fluid velocity ranges from zero at the
surface to the free stream velocity
far away from the surface.
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
4
Friction Drag and Pressure Drag When the flow over a body takes
place, there exists a drag on the body which tries to pull the
body in the direction of fluid flow. This force is called Drag
force. By the mechanism of its
generation this is of two types. 1. Friction drag and 2.
Pressure drag. They both act in the same
direction (in the direction of flow) but their origination is
different so they have are drag and
anybody in the fluid flow faces a drag which is a combination of
these two drags.
Friction drag is clear by name that it is the result of surface
friction. This generates due to the
realness of the fluid which and this property of the fluid is
called viscosity. This always works
tangentially to the surface. This depends upon the roughness of
the surface.
Pressure drag is also clear by its name that it is coming into
picture due to the pressure
difference between the front portion and back portion of the
body. It is also the integral sum of
the pressure over the surface in the direction of fluid flow. It
exists due to the presence of the
wake in the flow leaving the body. By a proper design we can
minimize the wake but it can
never be removed completely.
If a flat plate is laid along the flow direction, it will
produce very small pressure drag but
comparatively large amount of friction drag due to existence of
very thin wake but large surface
area involved. On the other hand if the plate is laid
perpendicular to the flow direction, a large
wake will be present there and a huge amount of the pressure
drag will be felt and friction drag also called surface
friction
drag will be very small. In the terms of aerodynamics, drag
on any slender body is mostly surface friction drag while
drag on any bluff body is mostly pressure drag. None the
less, at any time or in any case total drag on anybody in
the
fluid motion will be given by the sum of these two drags.
Sometimes pressure drag is also referred to as form drag.
= +
For the friction drag, it is viscosity which causes the drag
force
to come into being, but for the pressure drag too, it is the
viscosity which becomes the reason for its existence. In the
ideal case when the fluid has no viscosity, both types of
drag
are missing no matter what is the shape of the body.
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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We normally deal with the coefficients of the forces. So the
drag force for the pressure and
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
6
Parallel Flow over Flat Plate Let us consider upon a normal flat
plate having a flow over it. The boundary layer is formed
over it due to the realness of the fluid. Initially there is a
laminar boundary layer and then very
soon the flow becomes turbulent following a thin transition
region. This happens due the fact
that it is the distance of the location from the leading edge
which plays the role of a
characteristic length in the Reynolds number expression. The
transition from laminar to
turbulent takes place at a location where Reynolds number
reaches its critical value for
transition.
The transition of the flow from laminar to turbulent flow
depends upon the surface geometry,
surface roughness, upstream velocity, surface temperature, and
the type of fluid among some
more other things. It is best characterised by the Reynolds
number which is expressed as below:
=
=
So the value of Reynolds number rises as the location under
consideration goes away from the
leading edge. The transition of the flow from laminar to
turbulent takes place at about =
1 105 but becomes fully developed turbulent flow typically
around = 3 106. In
engineering a general value accepted for the critical Reynolds
number is 5 105 which gives
turbulent flow in any case. Actually under controlled condition
the flow can be maintained to
be laminar up to a maximum value of Reynolds number equal to 5
105.
=
=
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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Friction Coefficient
We have already derived the expressions for the boundary layer
flow over a flat plate. We can
use them here for finding the friction coefficient.
For the laminar flow:
, =4.91
1 2
and , =0.664
1 2
< 5 105
Same for the turbulent boundary layer will be:
, =0.38
1 5
and , =0.059
1 5
5 105 < < 107
The boundary layer thickness is directly proportional and
friction coefficient is inversely
proportional to 1 2 . Therefore at the leading edge where = 0
boundary layer thickness will
be zero while friction coefficient is supposedly infinite.
Friction coefficient thereafter decreases
by a factor of 1 2 in the flow direction. The local friction
coefficient are higher in turbulent
flow than in laminar flow owing to the intense mixing that
occurs in turbulent boundary layer.
In the transition region the friction coefficient increases till
it becomes highest in the fully
grown turbulent region. Thereafter it starts decreasing by a
factor of 1 5 in the flow direction.
The average friction coefficient over the entire plate is
determined by integrating the local
friction coefficient over the entire length of the plate.
For laminar region:
=1
,
0
=1
0.664
1 2
0
=0.664
(
)
1 2
()1 2
0
=0.664
(
)
1 2 1 2
1
2
|
0
= 1.33 (
)
1 2
=.
For Turbulent region:
=1
,
0
=1
0.059
1 5
0
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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=0.059
(
)
1 5
()1 5
0
=0.059
(
)
1 5 4 5
4
5
|
0
= 0.074 (
)
1 5
=.
So if the flow is entirely laminar or entirely turbulent then we
can find the friction coefficient
as per the expressions obtained above. Sometimes the plate is
very long and in this case though
both the types of flow will exist on the surface, laminar flow
friction coefficient can be ignored
and only turbulent flow can be estimated to find the total
friction coefficient. If the plate is long
enough to have a turbulent region but not long enough to ignore
the laminar region then the
average friction coefficient can be found using the concept
given below:
=1
( ,
0
+ ,
)
The transition region is small enough to be included in the
turbulent region. Now again taking
the critical Reynolds number to be = 5 105 and then substituting
the critical length in
the above expression and integrating it over the entire plate
after the critical length will give
that:
=.
5 105 10
7
The constants in this relation will be different for the
different critical Reynolds numbers. Also,
the surfaces are assumed to be smooth, and the free stream to be
turbulent free. For laminar
flow, the friction coefficient depends on only the Reynolds
number, and the surface roughness
has no effect. For turbulent flow, however, surface roughness
causes the friction coefficient to
increase several fold, to the point that in fully turbulent
region the friction coefficient is a
function of surface roughness alone, and independent of the
Reynolds number. That is the
case of pipe flow.
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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A curve fit of experimental data for the average friction
coefficient in this regime is given by
Schilchting as
Rough surface, turbulent:
(. .
)
.
where is the surface roughness, and L is the length of the plate
in the flow direction. In the
absence of a better relation, the relation above can be used for
turbulent flow on rough surfaces
for > 106, especially when > 104
Heat Transfer Coefficient
We have already determined the Nusselt number for a location for
the laminar flow over a
flat plate:
For Laminar Boundary Layer
=
= 0.332
0.5 1 3 (for > 0.6, 5 10
5)
For turbulent boundary layer
=
= 0.0296
0.8 1 3 (for 0.6 < > 60, 5 10
5 107)
(Note: Remember we have no solution for the flow with Prandtl
number less than 0.6. For the Prandtl number
equal to 1 we get a self-similar flow in which velocity boundary
layer coincides with the thermal boundary layer.)
For the laminar flow heat transfer coefficient is
proportional to 0.5 and thus to 0.5. Therefore it
is infinite at the leading edge i.e. = 0 and
decreases by a factor of 0.5 in the flow direction.
The variation of boundary layer thickness and
the friction and heat transfer coefficients along an
isothermal flat plate are shown in the figure. The
local friction and heat transfer coefficient are
higher in turbulent flow than they are in laminar
flow. Also, reaches its highest values when the flow becomes
fully turbulent, and then
decreases by a factor of 0.2 in the flow direction.
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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The average Nusselt number over the entire plate is determined
by integrating the expression
for the entire length of the plate and then dividing it by the
length itself. By doing so we find
the below equations.
For Laminar Boundary Layer
=
= 0.664
0.5 1 3 (for > 0.6, 5 10
5)
For turbulent boundary layer
=
= 0.037
0.8 1 3 (for 0.6 < > 60, 5 10
5 107)
The equation for the laminar boundary layer gives the heat
transfer coefficient for the entire
plate when the flow is laminar over the entire plate. The second
relation gives the average heat
transfer coefficient for the entire plate only when the flow is
turbulent over the entire plate.
Sometimes when the laminar region is too small relative to the
turbulent flow region, the
turbulent flow equation can be applied to the entire plate and
accepting a little compromise in
the accuracy.
In some other cases when the plate is sufficiently long for the
flow to become turbulent, but
not long enough to disregard the laminar flow region, the
average heat transfer coefficient over
the entire plate must be estimated using both the
relation over the appropriate regions.
=1
( , + ,
0
)
Again taking the critical Reynolds number to
be = 5 105 and performing the integration
the average Nusselt number over the entire plate will
be given by:
=
= (0.037
0.8 871)1/3
for ( 0.6 < > 60,
5 105 107)
In the above case it has been assumed flow over the plate is
partly laminar initially and partly
turbulent afterward. Above relation depends upon the critical
Reynolds number and for
different value of critical Reynolds number it will be
different.
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Notes on Heat Transfer prepared by Asst. Professor Mohammad Page
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Liquid metals such as mercury which have high thermal
conductivities, and are commonly used
in the applications that require high heat transfer rates.
However they have small Prandtl
numbers, and thus the thermal boundary layer develops much
faster than the velocity boundary
layer. Then we can assume the velocity in the thermal boundary
layer to be constant at the free
stream value and solve the energy equation. It gives:
= 0.565()1/2
< 0.05
Above relation has a limitation of the Prandtl number and will
change accordingly as the
Prandtl number changes. It is however desirable to have a single
correlation that applies to all
fluids, including liquid metals. By curve-fitting existing data,
Churchill and Ozoe (1973)
proposed the following relation which is applicable for all
Prandtl numbers and is claimed to
be accurate to 1%.
=
=
. /
/
[ + (. )/]/
These relations have been obtained for the case of isothermal
surfaces, but could also be used
for approximately for the case of non-isothermal surfaces by
assuming the surface temperature
to be constant at some average value. Also, the surfaces are
assumed to be smooth, and the free
stream to be turbulent free. The effect of variable properties
can be accounted for by evaluating
all properties at the film temperature.
Special Cases
Flat Plate with Unheated Starting Length
Uniform Heat Flux