Video summary
The lecture begins by refining the derivation of the energy equation, emphasizing the importance of understanding every step rather than skipping details. The instructor explains that enthalpy ($H$) can be expressed as the product of specific heat capacity and temperature ($C_P T$), which transforms the governing equation into a form involving convection, conduction, pressure work, viscous dissipation, and internal heat generation. In this expanded version, the convective terms represent temperature changes due to fluid motion, while the instantaneous change accounts for temporal variations. The instructor uses an analogy of entering an air-conditioned room to illustrate how convection causes a gradual temperature shift along the flow direction, whereas sudden impacts like hitting an ice ball represent instantaneous thermal changes.
The core of the lecture focuses on applying the principle of similarity by non-dimensionalizing both the momentum and energy equations for steady, two-dimensional flows. For the momentum equation, variables such as length, velocity, and pressure are normalized using characteristic scales like plate length or free-stream velocity. After substituting these dimensionless variables and dividing through by appropriate inertial terms, the equation simplifies to reveal that the resulting flow characteristics, including velocity and pressure distributions, are governed primarily by the Reynolds number. This dimensionless group represents the ratio of inertial forces to viscous forces, highlighting its critical role in determining fluid behavior in momentum transfer problems.
When the same non-dimensionalization process is applied to the energy equation, a more complex set of dimensionless parameters emerges. The analysis shows that temperature distribution depends not only on the Reynolds number but also on the Prandtl number and the Eckert number. The Prandtl number is identified as the ratio of momentum diffusivity to thermal diffusivity, linking viscous effects to heat conduction capabilities. Meanwhile, the Eckert number represents the ratio of kinetic energy to enthalpy difference (thermal energy), indicating how much kinetic energy is converted into heat due to fluid motion. Consequently, while friction factors in momentum problems rely solely on the Reynolds number, heat transfer phenomena like the Nusselt number are functions of all three parameters: Reynolds, Prandtl, and Eckert numbers.
The lecture concludes by summarizing these key takeaways and previewing future topics involving scale analysis to better understand boundary layer thicknesses and friction factors. The instructor stresses that non-dimensionalization is a powerful tool that reduces complex physical equations into forms governed by specific dimensionless groups, allowing for broader applicability across different scales and conditions. By identifying which forces or energy mechanisms dominate through these numbers, engineers can simplify problems and predict fluid behavior without solving the full differential equations for every specific scenario. This foundational understanding sets the stage for deeper investigations into heat transfer correlations in subsequent classes.
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So we are we were into the interesting
derivation of the energy equation. It
may sound a little longish but I would
implore upon all the students to derive
the way I have derived. I have tried not
to skip any step so that you are with me
all the time. Okay, please do derive
it's once in a lifetime at least you
should derive once. Okay. So we were
here we were we had derived this
equation. Now we were just about to see
that H in the form of temperature. So to
get that I need to write H as equal to
CP into T. From thermodynamics we know
that. So if I do that, if I do that, H
is equal to CP into T and my equation
row D H by DT equal to K into D ² T by
DX²
+ D ^ 2 T by D Y 2 + DP by DT
+ 5 + Q dot prime
Now if I substitute H is equal to CP
into T and CP is constant. So row cp d
t by dt dt by dt is equal to k into do ²
t by dx²
+ d ² t by d y² +
dp by dt
+ 5. Remember this is the viscous
dissipation term which had all the
velocity gradients. This is this this is
what you can see that viscosity and
velocity gradient. Viscosity velocity
gradient this is stretching this is
stretching. This is this may cause
either rotation or angular deformation.
Okay. So depending on the uh plus or
minus plus q dot triple prime. So if you
divide through this is the whole energy.
This is what we were up to. We can see
that now this is the energy equation.
Okay. And this if you expand this row cp
instead of dt dt let us expand this do t
by d t plus
u do t by dx
plus v do t by dy is equal to
k into d ² t by dx²
+ d ² t by d y²
plus dp by dt this is the total
derivative of the pressure plus viscous
dissipation term plus q dot triple
prime.
So this is the total derivative that is
the temperature is varying with time and
this temperature is varying with x and y
by virtue of convection. If you have to
physically feel this, if I consider this
room, when I'm entering into this AC
room with a velocity U, I experience a
velocity temperature gradient of delt by
d x. That is the convective term. But
someone while entering someone hits me
with an ice ball so that I will
experience instantaneously a temperature
change that is dot t by dt. So these
terms are because of the temperature
variations because the by virtue of
convection that is u and v because they
have the velocity and this is
instantaneous temperature change. So
this is convection term.
This is heat transfer by conduction
and this is pressure work
and this is viscous dissipation. You can
see that it's all because of the
velocity gradients and the viscosity
viscous dissipation term. Okay. So just
recall viscous this is viscous
dissipation term. This is viscous
dissipation term. Okay.
Now
and dp by dt is nothing but dp by d t +
u d p by d x + v dp by d y. And this is
the heat volumetric heat en heat heat
generation either through chemical
reaction or radiation or electrical
heating within my control volume. So now
I have conservation of mass momentum
energy equation. Now the question is we
will apply what is called as principle
of similarity and nondimensionalize this
equations and see what what are the
nondimensional numbers which are going
to emerge while we do this equation. So
in the conservation of momentum if you
just take conservation of momentum
this is your conservation of momentum
equation you can see that this is the x
momentum equation you have the inertia
force this is the inertia force this is
the pressure force these are the viscous
forces and this is the body force and du
by dt if you expand you get du by dt and
these are the convective terms
If you consider, let me consider steady
and twodimensional flows. If I consider
steady and two-dimensional, the moment I
say steady, nothing varies with time.
Then d u by dt is zero. And zed term
that is d u by zed and w term does not
exist because it is a two-dimensional
flow. So this term this term will
vanish. So let me write the steady flow
for steady
and twodimensional
flow.
The x momentum equation I write x
momentum equation. So if I write X
momentum equation, you have row into
[snorts]
u d u by del x + v into del u by del y
is equal to - p by dx
+ mu into d² u by dx²
+ d ² u by d y² 2
+ mu by d by dx of mu by 3 into d u by
dx + dv by dy
plus f of x. So what I do is I
non-dimensionalize x with a
characteristic length L. Characteristic
length. L is characteristic length. For
a flow over a flat plate, it can be
length of the plate. For internal flows
that is pipe, it can be diameter of the
pipe. And yar is y by l. Yar xar are
non-dimensional and u star that is
velocity u non-dimensionalized with u
infinity. U infinity is the free stream
velocity. That means if you take flow
over a flat plate, what is the velocity
with which it is flowing on the flat
plate
and V star equal to V by U infinity
and P star is equal to P by row U
infinity squared. If you do
non-dimensionalization like this, if you
substitute this in this above equation,
if you substitute all of them here, so
what is that you get is row into
u star u becomes u* into u infinity.
Okay. And do of
do xar x is xar into length. So you have
L into U is U infinity into U star. U
infinity is constant. So U infinity into
U infinity U star. Similarly for V you
have Var into U infinity that is 1 U
infinity and do YAR YAR into L you have
L and U do U is do U* into U infinity.
So that's what you get. Now here again
you have P. So you have do p star that
is you have row u infinity squared minus
sign is there divided by do xar divided
by l
plus same way you can do mu into u
infinity by l² into d ² u* divided by
dx* squ + d² u* divided by d Y*²
plus
mu into U infinity
by L²
D by D XAR
into 1 by3
into d u* by dx*
plus do var by d yar
plus fx here I'm neglecting neglecting
body force.
Neglect body force. There are no body
forces. So now divide throughout. Now
divide through out by throughout by U
infinity squared by L. Divide throughout
by U infinity. So left hand side will
become row U infinity squar by L U
divide through. So if you do that you
get u star d u* by dx*
[snorts] plus v*
d u* by d y* that is on the left hand
side and u infinity squar row u infinity
squar by l we'll get so you have minus
row u infinity squar by l into l by row
u infinity squar into dopar by do xar R
+ mu U infinity by L²
into L by sorry L by row U infinity
squared into do ^ 2 U* by D XAR squar +
D ^ 2 U* divided by D Y* squar plus mu U
infinity by L into L by row U infinity²
into d by dx*
into 1x3 do by d of u* xar plus do var
by do yar you get so here in the if this
gets canceled out okay here what is that
I end up getting I end up getting
I end up getting row u mu by row u
infinity l and here also mu by row u
infinity l. So if I write that so the
equation turns out to be u* d u* by dx*
plus v* d u* d yar equal to -1 by row
dopar by dox*
plus
what is that mu by row u infinity
Infinity L that is nothing but 1 by
Reol's number. If you recall Reol's
number is row U infinity L by mu that is
viscous inertia force divided by viscous
force
that is the Reol's number. So you get 1
by R E into do²
u* divided by do xar squar + do ² u*
divided by d y* squar + 1 by r e into d
by d xar
into mu by 3 into d u* divided divided
by dx*
plus d vstar divided by d yar. The point
is the takeaway from this is that when
you do non-dimensionalization
of momentum equation the non-dimensional
number which emerges is reol's number.
So if the anything which comes out of
the momentum equation the velocity
distribution the pressure distribution
all that is decided by reol's number
remember that that is what is emerging
and reol's number is nothing but row u
infinity l by mu so similar exercise let
us do it for energy equation so if you
go to the energy equation
If you go to the energy equation, this
is my energy equation. So my energy
equation is consisting of temperature.
My energy equation is row CP do T by DT
that is this is the convection term.
This is the conduction term. This is the
pressure work. This is the viscous
dissipation. This is the volutric heat
generation. So what I do here is this is
the viscous dissipation term. Okay. And
the pressure term if you put that you
get dp by dt u dp by dx + v dp by dy + 5
q q dotle prime. For steady flows you
don't have d by dt. So let me start off
writing the energy equation for steady
flows and nondimensionalize
that equation. So let me start with
steady flows energy equation
for steady flows.
Okay. So the energy equation would be
row cp
u do t by dx
plus v do t by dy which is the
convection term which is left hand side
is equal to conduction term k into do ²
t by dx²
+ d ² t by d y²
this is the conduction term and the
pressure Work is U del P by del X plus V
del P by del Y plus 5 + Q dot triple
prime. Five is the viscous dissipation
term which is essentially the velocity
gradients that is phi = 2 mu into del u
by del x²
+ 2 mu into delv by del y²
minus of 2x3
mu of d u by d x + del v by del y² ² +
mu into d u by d y + del v by del x²
now to non-dimensionalize I take xar = x
by l y* = y by l and u* = u by u
infinity l u infinity I have already
defined v by u infinity
And P star is P by row U infinity
squared. And T star is what is new for
us. T minus of T S divided by T infinity
minus T S. T S is the surface
temperature which is assumed to be
constant. Surface temperature of the
flat plate. Let us say surface
temperature of the plate or a pipe.
Okay. T infinity is free stream
temperature. Restall we have already
defined. So if I substitute this now
what is that I get? So first let us do
for row CP. Row CP U. If you do you get
U infinity
by L. That is you have U infinity U u*
into U infinity and do YAR this is sorry
this is X
do XAR
into L and now do T star. So what do you
get? do tar is t infinity minus s
and t ss is constant. So do ts by do x
is
not existing because ts is constant. So
that's what you get. Similarly you get
var for v star also you can pull out the
same terms do tar by do y*
is equal to k into t infinity minus t s
you get when you do for do tar
okay divided by for x you get x do xar
squar by l²
plus do ^ 2 tar divided by d y* squar
plus
plus okay plus
you have u for u you have to write u
infinity into u star divided by do xar
that is l for p star p star do p star
into row u infinity squared
Similarly for plus V star do P star by
do Y star.
Now for viscous dissipation term. So let
us take up the viscous dissipation term.
The viscous dissipation term. The first
term let us say you can write I think I
will write directly. You can see that
this this is the viscous dissipation
term. So you have plus mu is common
everywhere. Mu mu mu mu. So I pull out
that mu. And you have velocity squared
that means u infinity squared. And you
have x² that means u infinity squar by
l² that's there velocity squar x²
velocity squared. So you get velocity
squared everywhere. So you can write
this as five star row u infinity squar
by l²
l. So now l². So now what you do is you
divide throughout. You divide through
out with this term.
Okay. Let us take up one by one and
divide. So if I take up this term that
is k into t infinity minus t s divided
by l² into
row cp u infinity into t infinity minus
t s by l. So t row k t t t t t t t t t t
t t t t t t t t t t t t t t t t t t t t
t t t t t t t t t t infinity minus t S T
infinity minus T S. So I have 1 by row
CP U infinity L coming here in this
term. Similarly, what do I get for row U
infinity squared? that is
row u infinity²
into u infinity by l into l by row cp
into t infinity minus t s into u
infinity so what is that l gets canceled
out 1 u infinity 1 u infinity gets
canceled out row and row gets canceled
out so you get u infinity squ divid by
CP into T infinity minus T S. So for
this term you got this and for this term
for this term and this term you got
this. So let us write those term with
those terms this equation. So what I get
here is u*ar
u*
do t star divided by do xar
plus v star do tar by d yar is equal to
k by
row cp l u infinity
into do ^ 2 tar divided by do xar
squared plus do 2 tar divided by do y*
squar
plus u infinity squar divided by cp into
t infinity minus t s into
U do P star by do XAR
plus V do P star by do Y star
plus U infinity squared divided by CP
into T infinity minus T S
okay into mu divided by row U infinity
L.
Okay, that's what you get here.
Yeah, this term I didn't do. So you have
mu u infinity squar by l² if you divide
it by row
row cp u infinity into t infinity minus
t s by l. So
u infinity 1 u infinity gets canceled
out. L and L gets canceled out. So you
get mu by row cp l t infinity by t s
that is what I have written here. U
infinity squared that is you get
mu infinity mu that is mu u infinity
squar divided by l² into l divided by
row cp into u row cp into t infinity
minus t s into u infinity so that I have
written retained that this is what I
have put U infinity squared by CP into T
infinity S I have put this term and the
left term is mu by row U infinity L I
have put here into into five star
now this if you see this K by this is
nothing but
E R so that is E pr equal to row U
infinity L by mu into pantal number is
mu cp by k. So that is you get this as
row cp mu mu gets canceled out row cp l
u infinity by k that is nothing but this
in 1 by r. So now u infinity squared by
cp into t infinity minus t s is called
yakert number.
So what does this represent? It
represents the u infinity squared means
it is kinetic energy. The kinetic energy
with which my fluid particle is moving.
and CP T infinity minus T S is the
thermal energy. So the echert number
represents how much of the kinetic
energy is transformed into thermal
energy when it is moving. So I can now
write this equation as u* deltar by del
xar plus v* del tar by del y* is equal
to 1 by r e p r into del² tar divided by
d xar
squared plus d² tar divide divided by
del y* squared plus yurt number
into this is star
u* del p* by del xar plus var del p* by
del y*
plus this is echert number and this is 1
by e so echert by E 5 star.
So from the energy equation the
non-dimensional numbers which are
emerging are R E P R Y number. So the
takeaway from this class is that
if you nondimensionalize
momentum equation the
non-dimensional number which emerged is
Reol's number. So the velocity profile
the velocity gradients the pressure
gradients the pressure profile is
dependent on Reol's number. But when you
non-dimensionalize
energy equation the non-dimensional
numbers which emerged are R E P R Yakert
number okay where R E is viscous sorry
inertia force divided by viscous force
viscous force that is row U infinity L
by mu. But on the other hand, pantal
number is momentum diffusivity divided
by thermal diffusivity that is muc by K.
And yakert number. Yakert number is U
infinity²
divided by CP into T infinity minus T S
that is kinetic energy divided by
thermal energy that means my temperature
distribution is not only dependent on
reol number it is also dependent on
number and Yakerta number so this is the
takeaway so we are going In the next
class we are going to represent that the
friction factor is a function of reol
number only. Here nasalt number is a
function is a function of not only reol
number but also parental number and a
correct number. So we will study this in
detail. So much for today's lecture. In
the next class, let us derive or do what
is called scale analysis and try to get
much more understanding on boundary
layer thicknesses and the friction
factor and nusled number. Thank you.
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