Video summary
The lecture begins by distinguishing between hydrodynamically and thermally fully developed flow conditions. In fluid mechanics, a flow is considered hydrodynamically fully developed once the velocity boundary layers merge, resulting in a velocity profile that does not change along the pipe length. However, in heat transfer, maintaining a constant wall temperature or heat flux still involves active heat addition or removal as the fluid moves downstream, meaning the actual temperature cannot remain constant. Consequently, the condition for thermal full development is defined not by a constant local temperature, but by an invariant non-dimensional temperature distribution along the flow direction. This implies that while the absolute temperatures change, the shape of the temperature profile relative to the wall and bulk fluid remains consistent as the fluid progresses from one location to another.
A key derivation in this module demonstrates that in the thermally fully developed region, the local heat transfer coefficient becomes independent of the axial position. By analyzing the energy balance and the definition of the heat transfer coefficient, it is shown that if the wall heat flux is constant and the flow is fully developed, the difference between the wall temperature and the bulk fluid temperature must also remain constant. This leads to a significant physical insight: in this region, the lines representing the variation of wall temperature and bulk mean temperature along the pipe are parallel to each other. Conversely, in the developing region upstream of this point, the heat transfer coefficient decreases as the boundary layer grows, causing the temperature difference between the wall and the fluid to increase progressively until it reaches a constant value where the flow becomes fully developed.
The lecture further explores practical applications and scenarios involving these boundary conditions. A constant wall temperature condition typically arises in phase change processes like boiling or condensation, whereas a constant heat flux condition is common in situations involving solar radiation or electric resistance heating. The instructor emphasizes that while real-world problems often involve varying diameters or non-uniform heat fluxes, the fundamental concept of energy balance remains the governing principle for deriving temperature distributions. For instance, if the pipe diameter varies, the Reynolds number changes, affecting the Nusselt number and complicating the analysis, but the underlying energy conservation equation still applies. Similarly, if the heat flux varies spatially, such as in a reactor with sinusoidal heating, integration is required to find the temperature profile, though the basic methodology remains unchanged.
In summary, the module establishes that for internal flow with constant wall heat flux and constant properties, the bulk mean temperature increases linearly along the pipe once the flow is thermally fully developed. This linear increase results in parallel temperature profiles for both the wall and the bulk fluid, a direct consequence of the constant heat transfer coefficient in this region. The lecture concludes by noting that while the next module will cover the constant wall temperature case and provide specific examples, the current analysis highlights how energy balance principles allow engineers to predict temperature distributions even when geometric or thermal conditions vary, provided the assumptions of steady state, single-phase flow, and negligible kinetic or potential energy changes are maintained.
Read the full video transcript
Yeah, in the last module uh we looked at
the condition for the flow to be
thermally fully developed. In case of
fluid mechanics, we said that du by dx
was equal to zero. And that was coming
from the simple fact that once the
boundary layers merged. So this is where
say the boundary layers have merged. So
if I look at a region location A B C
nothing is changing as far as the flow
is concerned because this entire effect
of viscosity is now all through the core
of the pipe. Okay. So nothing there is
nothing which is changing for the fluid
when it moves from location A to
location B to location C. And therefore
this du by dx necessarily becomes
independent I mean u becomes necessarily
independent of the actual location. But
in heat transfer whatever be the
boundary condition whether it is heat
flux constant or wall temperature
constant there is a process of heat
addition or removal as the fluid moves
from location A to B to C which means
none of these can remain constant. Yeah,
TWW can remain constant in a special
case where you you force the wall
temperature to remain constant but there
is action happening when the flow goes
from A to B to C. So you don't you
necessarily have heat addition
or heat rejection from the fluid as I
move from A to B to C and therefore and
you will never get dt by dx.
This is not possible because dt by dx is
zero means temperature is constant. The
moment temperature is constant. That
means you are not having heat addition
or heat removal which means heat
transfer is not happening. The problem
of heat transfer itself is thrown out of
the window. Therefore this is not right.
We have to look at a nondimensional
temperature distribution
is invariant along the flow direction.
What is this non-dimensional
temperature? The local temperature is in
the numerator. The bulk mean temperature
is in the denominator. Tall minus TM is
like the excess temperature. Okay. T
wall minus T is the
difference in the temperature between
the wall and the local location. Tall
minus TM is the difference between the
wall and the bulk fluid temperature. So
it is in essence like a ratio of two
differences. This is found to remain
constant with actual location and
therefore this is the condition for
thermally fully developed. Now let's
just go and try to understand what this
implies. Okay. So in a thermally fully
developed region the derivative with
respect to x is zero. That means the the
term that I have shown there T- wall or
TS T wall is TWW TS or surface wall and
surface have been used interchangeably.
So T surface minus T local divided by T
surface minus T bulk mean the derivative
with respect to X is zero. That means
that bracketed term is independent of X.
Okay. Then the derivative of this
particular bracketed quantity with
respect to r also must be independent of
x. Right? So if this whole thing is
independent of x, the derivative with
respect to r also must be independent of
x. So if I take the derivative of these
quantities, this is like a constant. So
let me just go through the so derivative
with respect to r must be independent of
x. So d by d r of t wall of x - t of r
comma x divided by t wall of x minus tb
bulk mean of x is going to be
independent of x. This we can do u du by
dx. So this is T wall minus T bulk mean
derivative of wall temperature with
respect to R is zero. This is minus D by
D R
minus T wall minus T of R comma X
into
0 because
V du minus U DV. U is this one. DV is
TWW minus TM. Derivative of two
quantities which are only a function of
X. You are taking derivative with
respect to R. So that is going to be
equal to zero divided by T wall minus T
bulk mean the whole squared is not a
function of X. What does this mean?
minus dt by dr
at r equal to
so this divided by t wall minus tbulk
mean is not a function of x correct so
we will say this gradient let us
evaluate this gradient at the wall
at r equal to r So minus dt by d r at r
= r divided by t wall minus t bulk mean
is not a function of x
correct. Now what is heat flux? QP prime
is H into in external flow what was it
in external flow this was T wall and
this was T infinity
the role of T infinity let's go back I
had written this explicitly many times
yeah the role of T infinity is taken
over by T bulk mean here The role of U
infinity in external flow was taken over
by Ubulk mean. Same way role of T
infinity is taken over by T bulk mean.
And we are going to say H is T wall or T
surface
minus T bulk mean local.
Okay, that means it's just a function of
X. So TS is same as T wall. Okay, sorry
about it.
both are the same. Okay. So this
quantity t ss minus tm. So this is qp
prime over h is t s - tm. And what does
this mean? This is nothing but
so let me just write this in this way.
This is nothing but k dt by d r at r =
r. How do I get this? If this is your
wall, this is my temperature
distribution. This is my uh y direction.
This is my r direction. y is equal to
capital r - r. dy is equal to minus d r
- k dt by dy is equal to k dt by dr.
Okay. So this is nothing but your heat
transfer coefficient. So if I combine
these two k dt by dr r at r = r divided
by t s minus t b b b b b b b b b b b b b
b b b b b b b b b b b b b b b b b b b b
b b b b b b bulk mean. This is nothing
but your heat transfer coefficient
local.
So combine these two, what do I get? The
local heat transfer coefficient is
independent of X in the thermally fully
developed region.
The local heat transfer coefficient is
not a function of X which means it is
in the thermally fully developed region.
Very very very very very important
result. The in thermally fully developed
region the heat transfer coefficient is
not a function of X. Of course it's not
a function of R. It's not a function of
X even. So we get one very useful result
that is going to have farreaching
consequences. Okay, let's quickly go
through what is there in the slides for
this is what is shown. Okay, variation
of friction factor and heat transfer
coefficient are similar. Okay, for
laminina flow uh entrance region is 05
* d into reol's number l / d is
approximately 005.
If you're talking of pantal number
greater than one fluid
l uh thermal entrance length is l / d is
005 * pantal number. So thermal entrance
length is pantal number time hydrodnamic
entrance length roughly. Okay. Similarly
for turbulent flow hydrodnamic entrance
length is as a function of e to the
power.25
turbulent as well as hydrodnamic case or
uh heat transfer case approximated to 10
* d. Okay. So because it is turbulent
lot of these variations etc would get
absorbed because of the turbulence and
you can safely say that beyond 10D the
uh flow can be assumed to be both
hydrodnamically as well as thermally
fully developed. Okay. What you see here
in the picture below are the boundary
layers for this uh thermal boundary
layer which is shown below the
hydrodnamic boundary layer because we
have looked at a case where pantal
number is greater than one where the
velocity boundary layers are merging. We
call that as the beginning of fully
developed region where the thermally
fully developed region is essentially
where the thermal boundary layers are
merging. Ideally, you you would want
both these things to be taken care of
and be in both thermally as well as
hydrodnamically fully developed. So, I
would want my analysis to be in this
part rather than somewhere here or here.
Okay. That's what we are trying to say.
Okay.
Okay. These are some very typical uh
diagrams that you will see in any
textbook. So, these are for the case
where parental number greater than one,
parental number less than one. The blue
lines are the velocity boundary layers.
The red ones are the thermal boundary
layers. As you can see, plant greater
than one. The hydrodnamic boundary layer
is thicker than the thermal boundary
layer. Therefore, they merge earlier,
which means the hydrodnamic entrance
length is smaller than the thermal
entrance length which is coming here.
Whereas for pantal number less than one
case, you have delta less than delta t.
So, delta t merges before this. you have
hydrodnamic entrance length far
extending as compared to the thermal
entrance length. Okay. And these are
basic variation of local nasil number
along a tube uniform surface and uh uh
temperature and uniform surface heat
flux. This is your nassel number plot as
a function of the x / d. x is
essentially the coordinate system from
the stagnation point. This you would
have again seen from your uh the
external flow similar.
Okay.
Okay. Now we come to the very important
part associated with thermal analysis
for internal flow. Okay. So we will do
this analysis for both constant wall
heat flux as well as constant wall
temperature case. Okay. And as I said
earlier,
any engineering problem can be
approximated with reasonable accuracy to
either of these two boundary conditions.
Okay, constant wall temperature happens
in a condensation or boiling process,
phase change process. So if I have a
surface whose temperature needs to be
maintained constant, I should have
either boiling or condensation occurring
below it or above it. so that this
surface temperature remains constant.
Okay. So that is the way in which
constant wall temperature boundary
condition is created in real life. For
constant wall heat flux is uh you can
have solar radiation, you can have
electric resistance heating which is
based on the windings etc. You can have
any such kind of situation which will
create the constant wall heat flux
condition. Okay. Now let's go back to
our paper and pen and do this
derivation.
So we are looking at general energy
balance.
I'm going to do now first for the
constant wall heat flux case.
This is extremely important not just in
your undergraduate heat transfer. This
concept of energy balance you will see
all through in any thermal analysis
you'll see this kind of a uh energy
balance analysis it's being done. First
thing is you take a control volume.
This is a pipe. What we are doing for a
pipe of cross of diameter D can be done
for uh varying cross-sectional geometry
anything. Okay. The idea is E dot in
minus E dot out plus E dot generated
equal to E dot store. So we are going to
write
some assumptions
steady state
no volutric heat generation
I'm just having a fluid
which is entering a pipe at some tbulk
mean I
this is my x and this is my r this is my
x is equal to zero and this is my sum dx
location I have constant wall heat flux
throughout
I'm not showing it on the b on the
bottom part because the figure will get
too crowded the
temperature local is tm this is tm m +
dtm bulk mean temperature
assumptions steady state no volutric
heat generation
negligible changes
in potential and kinetic energy which
means all the heat that I'm adding is
going to increase the temperature okay
no pump
work etc
Constant mass flow rate, constant
diameter,
constant properties.
Okay.
All heat added
goes into fluid.
We are maintaining single phase
that means there is no boiling or phase
change.
Okay. So these are typical assumptions.
Now let's for this control volume let us
write E dot in minus E dot out plus E
dot generated is equal to E dot stored.
This is zero because of steady state.
This is zero because we don't have
energy generation. Now we'll say E dot
in is equal to E dot out. E dot in is M
dot CPT bulk mean plus heat flux into pi
D into DZ Q prime or Q just write QP
prime
perimeter is pi D into DZ would be your
surface area that is equal to M dot into
CP T bulk mean plus dtm.
So if I open the brackets I will get qp
prime pi dz
is equal to m dot cp dtm which means dt
m by dz is equal to qp prime pd by m dot
cp. Let us take a look at this for a
minute.
What did we say? Heat flux is constant,
right?
Diameter is constant, mass flow rate is
constant, properties are constant. What
does this translate to?
Because
QP prime, D, M dot, CP are all
constants.
What does this mean? gradient of
temperature along with X. Sorry, I have
I have I'm accustomed to Zed. We have
used X. So, let us use X only. Sorry
about that. Okay. DT by DX is constant
which means temperature
is a linear function.
Who told me this? Max.
Okay, dy by dx is constant means slope
is constant. Y it's a straight line
equation. So if I this is going to be
varying tbulk mean
that means what if fluid came in at 30°
its variation in case of a constant wall
heat flux case would be linear.
Okay, this is tbulk mean not yet over.
Okay, so tbulk mean of x if I integrate
is qprime pi dx by m dot cp.
Okay,
I I integrate and evaluate at x =0 and x
= some x. So this would be T bulk mean
of X minus T bulk mean at inlet. Right?
This is my left hand side. Right hand
side I'm going to do it.
So T bulk mean of X minus T bulk mean at
inlet is QP prime pi DX by M. CP implies
T M of X is equal to T M I + Q prime PID
by M dot CP into X
Y is equal to
C +
X
right so the slope is like this
it starts Starts at tmi. Not not rocket
science. X is equal to0. Inlet
temperature is tmi. What is the outlet
temperature? It depends on this. The
slope is
qp prime pi d by m do cp which is a
constant.
Okay. So this is your bulk mean
temperature. Now what is this? Q prime
is H into T local wall minus T bulk mean
local general equation. Now if this is
constant
implies
H into T S minus T bulk mean is
constant.
What did we show couple of slides ago?
H
is not a function of X location.
That means in ther uh one other
assumption I forgot to add here
important assumption fully
developed.
I'm I'm I'm putting this at the end for
a simple reason that we are not going to
use it throughout. Okay. So I'll just
tell you where we are going to use it.
So when I'm in fully developed region,
so in fully developed flow,
H is a constant.
What does this mean?
Q prime divided by H is equal to
constant is equal to T wall minus T bulk
mean. In case of fully developed flow
that means this difference is a constant
which means T wall is always greater
than we know T wall is always greater
than T bulk mean correct T- wall or T
surface is always greater than T bulk
mean and this difference is constant
which means the line of bulk mean
temperature
and line of wall temperature are going
to be parallel to each other.
Can I say that?
Right. That means dt s by dx minus dt m
by dx is equal to zero. Which means dt s
by dx is dt m by dx. The lines are
parallel. Same slope
implies parallel lines.
This is when the flow is fully
developed. Now what about the part when
the flow was not fully developed? Well,
we know from analogy of fluid mechanics
and heat transfer. We also saw this that
in the developing region
this is how the
variation of H is going to be.
Okay. So as heat transfer coefficient
decreases
we come to a constant value. So QP prime
divided when in
developing region
ts minus tm is equal to qp prime by h
local these are also local values let me
put give them the respect t ss local
minus t bulk mean local is qp prime by
local which is constant over h of X
H of X is decreasing
which means
TS minus TM
increases
in the developing region.
Correct? This is decreasing. That means
the denominator is decreasing as I move
along the flow direction. which means
this difference has to increase. So that
is why I said I didn't write that
assumption of fully developed flow in
the previous slide. I put it extra
because in the fully developed part
whatever I said before the lines are
parallel is correct. Now in this part
where it is developing we are going to
see a progressively
larger increasing deltat t. Okay, this
is T wall as a function of X or T
surface as a function of X. The line of
T surface as a function of X and T bulk
mean as a function of X are going to be
parallel to each other in the
fully developed region.
In this part which is the developing
region developing region the lines are
not parallel the the lines will diverge
reach a constant value and then remain
parallel. So this is something which you
need to remember you need to be able to
derive this and this is a very simple
case because we have taken constant wall
heat flux is constant diameter was also
constant. Suppose a diameter was
varying. Let us say this is a arbitrary
diverging channel.
Diameter was varying. M dot is constant.
Or we could why why go through
diverging? We we do a converging
channel.
Okay. M dot is constant.
Well, I cannot do anything about
the the boundary layers are merging. All
that is fine. But you would have the
heat transfer coefficient itself
progressively changing because m dot is
constant
diameter
is decreasing along the flow direction.
So reol's number is 4 m dot by pi d mu
is going to change diameter decreases.
So it increases along the flow
direction. Therefore, we also know
nassel number which is a function of re
and pr is not going to remain constant
and therefore things will be complicated
for a non constant diameter. Similarly,
if we had a varying heat flux as we had
in a reactor case, QP prime is Qprime KN
sin pi Z by L pix by L. Sorry, I'll just
I'm so used to zed pix by L case. Then
we will have to
integrate this this part. So you'll
substitute the sine function and then
you will integrate. So dt m by dx would
be
q sin pix by l into
pi d
m dot cp. So this would be qprime
pi d by m dot cp
t of x - t m i integral 0 to x sin pix
by f dx which is x.
Okay. So the concept remains the same.
The mathematics may get a little
complicated depending on whether you
have a varying function varying qp prime
you have d converging or diverging
channels
the idea of energy balance remains the
same. Now whether the lines of
temperature will be parallel all those
things are subject to various other
things but the concept of E dot in minus
E dot out plus E dot generated is equal
to E dot stored is sacred. You take the
fluid element do the energy balance get
the differential equation solve for the
temperature distribution.
Okay, this is how you'll get T bulk mean
of X, T wall of X variation.
Okay, in the next module we'll go ahead
and do a couple of examples and also
look at the same analysis for constant
wall temperature case. Thank you.