Video summary
This lecture introduces the concept of internal flow, which refers to fluid motion within confined channels such as circular pipes or non-circular cross-sections, marking a shift from the previously discussed external flows like those over flat plates or around cylinders. The fundamental distinction lies in how viscosity affects the fluid; in external flow, the boundary layer grows until it merges far downstream, leaving a large inviscid core, whereas in internal flow, the boundary layers developing along the entire circumference eventually merge at a specific distance from the inlet. Once these layers merge, the entire cross-section becomes dominated by viscous effects, and the region is termed "fully developed," where the velocity profile no longer changes with distance along the pipe, meaning the velocity distribution depends only on the radial direction and remains constant downstream.
In the context of heat transfer within these confined flows, the analysis relies heavily on two primary reference temperatures: the wall temperature and the bulk mean temperature, often called mixing cup temperature. Unlike external flow which utilizes a free-stream temperature at infinity, internal flow defines the fluid temperature as an average value obtained by hypothetically mixing all the fluid in a cross-section. This bulk mean temperature is crucial because heat transfer occurs due to the temperature difference between the solid wall and this averaged fluid temperature. The lecture emphasizes that convective heat transfer cannot be separated from fluid mechanics; therefore, understanding the velocity profile developed in fluid dynamics is essential for analyzing thermal behavior, as the shape of the flow dictates how heat is transported and distributed within the pipe.
The practical application of these concepts extends far beyond simple textbook examples to complex engineering scenarios like nuclear reactor rod bundles. Even though a fuel rod bundle consists of multiple rods generating heat in a complicated geometry, engineers analyze it by simplifying the problem into equivalent circular pipe flows using subchannel analysis. In such real-world applications, boundary conditions are often approximated as either constant wall heat flux or constant wall temperature to make the problem solvable analytically and numerically. For instance, in nuclear reactors, the power distribution along a fuel rod is not perfectly uniform but can be modeled as a series of segments with different constant heat flux values, simulated using electrical heating in experiments. This approach demonstrates that while the conditions might seem idealized, they are highly relevant for solving practical engineering problems where controlling both heat input and fluid temperature simultaneously is impossible.
The lecture concludes by outlining the roadmap for future modules, which will delve deeper into analyzing these two specific boundary conditions: constant wall heat flux and constant wall temperature. The primary goal is to understand the development of thermal profiles within the pipe, analogous to the hydrodynamic fully developed condition already discussed. By examining how the temperature profile evolves from the inlet until it reaches a state where it no longer changes with axial distance, students will gain the tools necessary to predict heat transfer rates in various internal flow systems. This foundational knowledge is critical for designing efficient thermal systems, ensuring that engineers can accurately model and optimize performance in applications ranging from simple piping networks to high-stakes nuclear energy generation.
Read the full video transcript
Yeah, welcome everyone.
Uh
this module and the subsequent three or
four modules, we will uh look at
internal flow. That is flow through
pipes or uh flow through uh non-circular
cross sections. Uh till now we had
looked at uh external flow, flow around
a
cylinder, flow over a flat plate, flow
around a sphere, etc. So, just as in
fluid mechanics, we had uh boundary
layer formation over a flat plate. And
then we also looked at how pipe flows
are going to be characterized. So, the
same logic is being followed in heat
transfer. So, having done external flow,
now we are going to look at internal
flow. So, as you all know already from
fluid mechanics, internal flow is
essentially where
we are talking of flow through a
confined channel. So, by confined
channel, I mean something where you have
a Let us take a circular pipe. It could
be a rectangular channel or any such
channel. You're going to have
the boundary for flow
all around the circumference.
Physical boundary.
So, this imposes a very important
restriction or constraint on the flow
because as we know in external flow, if
if this is my external flat plate flow
situation, this is my coordinate system,
we have what we call as a boundary layer
which is being formed.
And this boundary layer develops as I
move further in the X direction, the
thickness of the boundary layer goes on
increasing. And what this means is the
viscous region goes on becoming more and
more prominent. So, the inviscid part,
which is the external part, which is the
outside of the boundary layer, is there,
but the effect of viscosity gets
propagated to a greater and greater
distance away from the solid surface.
This is what happens in external flow.
Okay, this is because we have uh
no boundary on the top. Okay, suppose we
had a flat plate and flow between two
parallel plates.
Okay, this is one one plate and some
plate is there very far off, whatever.
The boundary layer will develop like
this.
And from the top also a boundary layer
would develop, and somewhere here the
boundary layers may even merge.
So, this is what is going to happen in
case of flow through a circular pipe or
any such closed
uh
uh geometry, where the boundary layer,
which is formed all around the
circumference, would end up merging at
some location
some distance away from the inlet. That
depends on various conditions. Uh we'll
see what they are, but the important
point to notice, if you take a look at
this flow between two parallel plates.
So, this is parallel plate one.
This is one and two.
We will have the viscous region here.
This is also the viscous region.
This is the inviscid part.
And as you proceed along the flow
direction, the inviscid part goes on
decreasing, and eventually, when the
boundary layers merge, the entire flow
region becomes viscous. So, viscous
effect is there throughout the flow.
Okay, unlike having been able to
segregate this into two parts, whereas
this is the
inviscid part, which was decreasing
with respect to distance. So, inviscid
part
decreases with X.
Viscous region
increases.
How much does it go to? It will go to
the entire space that is going to be
there. So, that is what happens. The
flow becomes entirely viscous. So, here
also in flow through a circular pipe, we
do circular pipe as our example because
most real-life situation, engineering
applications would have pipe flow as one
of the simplest geometries of interest.
And I'll tell you something. Even when
you're talking of very, very fancy
uh
height, high level of stuff like flow
through a nuclear reactor rod bundle.
So, a rod bundle essentially consists of
a
set of fuel rods.
Uh you could have 5 by 5, 7 by 7, 9 by 9
such This is a fuel rod bundle. I'm just
drawing a representative here.
Okay, so if I'm looking at this kind of
a
setup, this is a nuclear fuel rod
bundle. This is fuel rod.
And this entire arrangement is called a
bundle. This is
generating heat. These are generating
heat. This is generating heat, etc. So,
if I look at a region which is
here. Region A. If I look at another
region, which is a region B.
If I look at another region, which is a
region C. And region D. These are all
empty spaces through which flow is going
to happen. Flow is happening through
into the plane of the board or outside,
whichever way, and this rod is supplying
heat to it.
Okay, so I'm just shading them to
indicate these are solids and heat
generating surfaces. Um
So, this what happens? The fluid as it
passes through gets hot. Okay, so this
length of this heater rod bundle is
about 3 3.6 m. Diameter of the fuel pin
is about 12 mm, and pitch is probably 15
or 20 mm. So, whatever it is, the area
available is a very small narrow area
through which coolant is flowing, and
the heat generated by the fuel
rods is taken away for steam generation,
etc. The point I'm trying to make is
even such a complicated geometry, okay,
is translated is analyzed as if it is a
pipe flow. So, even this geometry, they
will analyze as if it is a pipe flow
problem.
So, this is going to be analyzed as if
this is a flow through a pipe, and this
heat
The heat that is coming in from these
four rods is as if it is being supplied
through a pipe across the wall. So, this
is the Q double prime from the four
surrounding fuel rods.
So, each region A is surrounded by rod
one, rod two, rod three, rod four. Each
of this is supplying some amount of heat
to this fuel of fluid inside here, and
this heat is taken and said that it is
as if there is a flow through a circular
pipe of some diameter. This diameter DH
I will try to find out what it is, but
same mass flow rate, same heat flux as
is done through the
uh fuel rods channel. This is called a
subchannel analysis. So, point I'm
trying to make is even when you have a
very complicated geometry,
uh very very real-life application,
flow through pipes, circular pipe is
what is used for analysis. Therefore,
whatever we are doing now is not
something which is only confined to
textbooks or is it it's a it's a
approximation, we don't know to do to do
anything better, therefore we are doing
it. It is something which is very very
practical and therefore it's being done.
Okay, so I will just draw the pipe uh
wall a little better. This is the pipe
wall. And as the flow comes in, so we
typically generate
uh show the flow like this. Uh
Now, if you look at external flow, and
if you look at internal flow, velocity
is U infinity.
In internal flow, there is nothing like
U infinity because infinity is where you
have free stream,
where nothing is affected by the
presence of the solid surface. Here,
you please uh think of it like this.
Suppose you are you are walking through,
you know, holding hands, like you know,
your friends are walking and going
through holding hands, and you have to
cross a small gate, and the constraint
is that all of you need to cross at the
same time, pass through the gate at the
same time, you will come closer to each
other and try to squeeze through. That's
what happens to the fluid also. Outside,
when it is coming in here, it is all
free free,
but once it it sees that there is a pipe
wall which I have to take care of, the
the fluid will have to adjust itself to
this particular
contour, to this shape. And therefore,
what happens to the fluid? The The
boundary layer that we have studied in
fluid mechanics will start generating,
grow, grow, grow, grow, and it will
grow around the circumference, and you
will see that the boundary layer will
merge at this point. Whatever is coming
from here
would come like this and
merge at this location. So, imagine this
is like a
plastic sheet which is just being pushed
and it's allowed to merge here. This is
obviously symmetric. It doesn't look
symmetric the way I have drawn because
of the tablet, but this is not a
straight line. This is more of a curved
geometry like this. I don't want to
Yeah, this is better. It's something
like this. And you'll see that the
boundary layer merges at a certain
distance, at a certain location.
Okay? So, this is your viscous part.
This is your viscous part.
And here, this is the inviscid core or
the inviscid central part.
Inviscid central part. And as you see,
the inviscid region goes on decreasing,
and finally, in this After this
location, after this, fully viscous flow
is there.
Okay? Entirely viscous. Okay? So, this
part, of course, we know it's called as
a fully
developed region.
Fully developed region. We know this
from fluid mechanics. And we know in
internal flow, we all we have a quantity
called as average or bulk mean velocity.
Which
takes over the role of what U infinity
would take care of. So, I will draw the
velocity profile, etc. So, here you had
U infinity, here you had bulk mean
velocity. Now, in heat transfer, life is
a little bit different. So, in external
flow, remember, we had this as T wall,
and we had T infinity.
T wall and T infinity, and we had a
local temperature T of X, Y. So, we had
T wall,
T infinity, T of X, Y. We had this.
Now, here, of course, U of X, Y would be
there. Here also U of X, Y would be
there.
Okay.
Now,
from heat transfer point of view, T wall
would remain,
but instead of T infinity, there is no
scope for this T infinity, we have what
we call as
bulk mean temperature.
Mixing cup temperature,
bulk mean temperature, or average
temperature. I'll tell you what this is.
And of course, we have local
temperature.
Bulk mean temperature. When we say water
flows through a pipe at 2 m/s and 30° C,
the 30° C which I'm specifying is the
bulk mean temperature. You take this
water in a
insulated container, put a thermometer,
and measure the temperature. That is
called as the bulk mean. That's why it's
given the name mixing cup temperature.
So, whatever is the fluid, take it into
a cup, mix it well, stir it well, don't
allow any heat to pass out through, put
in a put in a thermocouple and measure
the temperature. It is an average value
which you're going to see. It's called
as the mixing cup temperature.
Okay. So, unlike external flow, where we
had
U infinity, here you have U bulk mean, U
bulk mean, or U bar, whatever, or
whatever you want to call.
I'll just write what whatever you see, U
bar or V bar or U average, anything.
These are all meaning the same.
T infinity doesn't have any value, but
we have T bulk mean.
T wall necessary condition for heat
transfer to occur. So, I have to have a
temperature difference between the
surrounding wall and the T fluid.
This has to be there, otherwise there is
no
uh heat transfer possible. T X Y is in
this one, we would give it as R {comma}
X or R {comma} Z. In internal flow, the
coordinate system will be such that this
is my X or Z direction. This is your
radial direction. Okay, as we had in
pipe flow. I will use Z. Okay, simply
because X can refer to thermodynamic
quality in uh
uh phase change applications. But, we
can use X also because entire course is
on single phase heat transfer.
So, T T wall will also be there. And of
course, T local X {comma} Y will be
there here also T local X {comma} Y
would be there.
Okay. Now,
heat transfer, convective heat transfer,
we all know is coupled with in
coupled with the fluid mechanics part.
You cannot separate the fluid mechanics
from the heat transfer. So, whatever we
did for pipe flow, everything has to be
taken here in the heat transfer part.
That means, whatever I I understood as
velocity profile, velocity distribution,
etc., those things will continue to be
there. I'm just going to revise this
very quickly because this has been part
of your curriculum already in fluid
mechanics.
So, if this is the boundary layer which
is formed,
and let me put in my R and X here.
Okay, this is fluid coming in at
velocity U infinity.
Okay, so if I plot the velocity
distribution at some location here,
I would see
and then I would have a
constant value and then a
varying value like this.
If I plot the velocity distribution at
some other location,
some other location here,
second region,
I would have
constant here again in this part and
doing this. So, this part, which I'm
shading,
is where
velocity is constant.
Because that is a part of the inviscid
region, where the necessary thing to
slow down the velocity, which is the
effect of viscosity, has not propagated.
This is the viscous
region,
entire part, and the shaded part is
where is we call as the inviscid part.
Inviscid.
And this after the boundary layers
merge, we call this part as
fully
developed flow.
And what do we know about fully
developed flow? The velocity
distribution anywhere
is exactly the same at any other
location. So,
U
Z, which is a function of R, but not a
function of Z. So, dU by dZ is equal to
zero. Velocity
is invariant
along the flow direction.
In fully developed
region.
This is extremely important. dU by dZ or
dU by dX, whatever you want to call.
I'm somehow used to using Z but it's
same as X.
Okay?
This condition we all know from fluid
mechanics. That means if I take a
picture or velocity plot at this
location A and do this do the
measurements of velocity at some other
downstream location B and
overlap the profile at A to all the
profile at B, they will be exactly the
same. That means velocity is only a
function of the radial direction and
independent of the actual direction.
This is the actual direction. Actual
direction.
X is the actual direction. Independent
of the actual direction, which means dU
by dX is equal to zero. Please remember
this condition. Okay? So, UZ is only a
function of R. In the part in this part
which we call as the developing region,
that is still here.
Velocity, as you can see, there is the
viscous part velocity is changing, the
inviscid part where the velocity is
constant, and because m. is constant by
continuity equation and at steady state,
the integration of the velocity profile
The integral of the velocity profile
that is whatever you see row UDA
is constant.
That means if I do the velocity
distribution at location A1, A2, A3,
etc.
All the integrals should add up to m.
Okay, I mean should be equal to m. That
means if I have a slow moving portion
here
the center velocity would come out to be
something such that it satisfies the
conservation of mass which is given by
this U infinity business.
Now as I move downstream the viscous
region is increasing between section A1
and section A2 the viscous part has
increased. That is what do I mean by
that? This region is thicker here
compared to here.
Correct? So you have a slower moving
fluid extending to a greater uh distance
away from the wall.
That means mass conservation has to
dictate that the central core which is
inviscid should move faster so that this
constraint is satisfied. That means U
center line
is not constant.
in developing region.
But in fully developed region
U center line is constant because the
profile itself doesn't change. That
means whatever is the U center line or U
max
U center line is same as U max
that is going to increase. So if I talk
of U center line it is going to increase
this way and settle to a value.
Okay, some value. So this is a function
of X.
Okay, I'm not saying it's it will not
start at zero. This need not be a zero
zero, okay? The idea is use centerline
will go on increasing until until the
flow becomes fully developed, and after
that it will become constant. Life is
easy in fluid mechanics, okay? But now
if you go to heat transfer,
the same situation, the same situation
that I have, for heat to flow,
for heat to flow, we need T wall, we
need T fluid.
And this one is proportional.
Correct? Now, what is this? T fluid,
what is the temperature of the fluid
that I'm talking about? So, let us look
at it in a little better way. I'll just
draw a bigger diagram
so that we can draw many things, write
many things here.
So, this is the centerline of the pipe.
Okay? And this is my R, and this is my
Z.
Fluid is
I have In heat transfer, I can have two
conditions. One is
constant wall heat flux.
That is Q double prime is equal to
constant.
Second one
is
constant
wall
temperature.
Which means
T wall is equal to constant.
What is so special about these, and why
is this even being mentioned?
Well,
when I have to do analysis, first of
all, I can control or I can I I
control the wall heat flux
and allow the fluid temperature to vary
or I can control the wall temperature
and allow the heat flux to vary. For me
to control both is impossible. I cannot
control the amount of heat I put in and
also the temperature of the fluid. I can
control only one. If I put in so much
amount of heat, so much is the variation
in
temperature so on and so forth. So, this
is something which I have to be I have
to live with. Otherwise, if I say this
is how my wall temperature is going to
behave, the heat transfer will only
behave in this manner or it is
controlled it is constrained by the fact
that the wall equal to constant. Why not
and why are these two cases important?
Because in real life, in most
engineering application, one or the
other condition
is what is you can approximate a real
life problem to. That means I might have
something which is not a function of I
might have a situation where
uh the wall temperature may not be
constant neither is the wall heat flux
constant. But, you with with certain
approximations,
you can at least have piecewise constant
values for
uh either the wall heat flux or the wall
temperature. And why do we need these?
Because these two cases are the cases
where
both analytical
as well as numerical solutions are
possible.
Experiments can be done.
Experimental data is available.
Because how do you how do you give
constant wall heat flux? So, if I want
to supply heat to a pipe carrying a
fluid, m. Fluid comes in at 30°C, I want
it to go out at 60°C,
I have to supply heat. So, I would have
a
you know, resistance heating, which is
going to be like a wire which is wrapped
around the circumference.
Okay, so Q is equal to V * I.
Okay, so I will give that much amount of
power so that in a given length L, this
will come out to be 60°. So, I have
control over it. So, Q is known. Q
double prime is Q divided by pi d l,
which is known. This is heat flux is
equal to constant case.
Okay, so when I do experiments also, I
will have to incorporate one of these
conditions. So, either this or this
condition will have to be incorporated
or will have to be created in the
experiment. Numerical also, this is what
we'll have to do. And more importantly,
just because it is convenient, I'm not
doing it because it is convenient. I'm
doing it because many real-life
engineering
applications,
either this or this. I'll just give you
one very simple, quick example. Again,
it comes from the reactor nuclear
reactor business only. This is typically
the power shape of a
nuclear fuel rod. So, this is a single
rod, if I'm going to say. This is the
rod, fuel rod.
And this is what I call as a power
profile. That means this is a sign kind
of profile.
This is typically
Z direction. Q double prime is Q double
prime not sin pi Z by L. This is my L.
Okay, length of the rod I will
And this is my Z direction. So,
typically it is this way. Now, uh
when you conduct experiment, when you do
experiments for these kind of things,
you don't have a fission nuclear uranium
dioxide or any such material here. You
do these experiments using electrical
heating. So, you simulate this using
electrical heating.
By that, I mean I have
this this region
split into
Just show this here. This would be the
And then, I mean
Okay? So, this is
for this length L, I will have small
regions where there is electrical
heating of different levels, so that it
mimics this power profile. How is that
possible? So, from zero to say L1, L2,
L3, L4. So, I would I would have a
certain resistance of heating here, so
which will give me so much amount of
heat flux. Next portion, I want to have
higher heat flux, so I would have a
higher resistance. Next small distance,
still higher. How small can this be?
Ideally, it should be like this.
Okay? But, practicality also is there.
So, we cannot have such a
you know,
small step step. So, what is done is
over a small region, this is done. And
this
power profile is essentially mimicking
this.
Midpoints of these, if I take and join,
it's essentially a
what you have seen here.
So, experiments are conducted where in
this region, you have constant wall heat
flux. In this region, you have another
heat flux value, which is different from
Q1 double prime, but it's still a
constant.
Third region, some other constant heat
flux value greater than Q double prime
two, but still it's a constant. So, even
in a nuclear reactor assembly situation,
to conduct experiments of this type, you
have electrical heating, but nothing
changes in in real life abruptly. So,
you you you will have step changes which
will be modeled as or which is which is
seen to mimic this kind of a profile.
And therefore, here, when you do the
analysis for this part, it is constant
wall heat flux situation. In this part,
it's also a different constant wall heat
flux. In this part, another constant
wall heat flux. This is how the analysis
of heat transfer in a nuclear fuel rod
bundle also progresses. So,
uh what I try what I'm trying to say is
even though these seem to be very
textbook-like conditions, these are very
very important, very practical, very
relevant to real-life engineering. In
the next module and further modules, we
will look at
how to analyze both these cases. So,
first we will look at how the
temperature profile is there for a flow
through a pipe situation, try to
understand the concept of thermally
fully developed just as we had velocity
fully developed condition. This one was
for velocity, hydrodynamics. We'll try
to look at what is there for heat
transfer similarly, and then proceed
with the analysis. Thank you.
>> Hey.