WEBVTT
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hello everybody so we continue with our discussions
on the flow pattern based modelling approach
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situates started in the last class in the
last class after discussing the reflux model
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sorry after discussing the homogeneous flow
model then we went over to discuss the reflux
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model what the reflux model basically does
it incorporate some particular constraints
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which have been derived considering the distribution
of the two phases in the conduit so therefore
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it incorporates different constraints in the
equations if the flow is bubbly or if the
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flow is slug and based on those particular
constraints it tries to predict the hydrodynamics
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of the different flow patterns now this was
definitely good approach but but more specifically
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if one considers the exact distribution of
the two phases and then tries to modulate
01:15.500 --> 01:22.810
from the basic [vocalized-noise] mass momentum
equations then it it is expected to give much
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better representation or much realistic representation
of the physical phenomena so in this class
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will be discussing one such analysis for the
slug flow pattern keeping in view that the
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slug flow pattern is the most widely occurring
flow pattern in micro channels and as we have
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all ready discussed
01:48.400 --> 01:55.100
now in micro channels also there have been
several attempts to model the slug flow pattern
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and the usual approach of modelling is the
unit cell approach what is the approach let
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us see now in this approach we assume that
the flow is completely or perfectly intermittent
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or periodic where the entire flow passage
can be divided into unit cells and the each
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unit cell comprises of a liquid slug and a
gas plug or a gas dealer bubble so therefore
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how can we imagine this we can imagine the
entire flow passage to be comprising of a
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large number of unit cells and each unit cell
comprising of one gas slug which is immersed
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in an liquid film and a liquid slug in narrow
passages we find that the liquid slugs are
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almost pure or they comprise of single phase
liquid flow so naturally they can be model
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one single phase dynamics and the taylor bubble
region is the two phase region where the [vocalized-noise]
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gas plug and the liquid film which flows in
the annular passage between the taylor bubble
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and the pipe valve it flows counter current
to the taylor bubble
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so therefore this can be modelled by by approximating
the flowing the taylor bubble region to be
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or to resemble the annular flow so therefore
[vocalized-noise] the other assumptions which
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we make here is that the thickness of the
liquid film is constant which is quite a logical
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expression considering the because the thickness
or the liquid film it becomes asymmetric
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it is usually [vocalized-noise] wider in the
lower portion as compared to the upper portion
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and this occurs primary due to the effect
of gravity now since in micro channels there
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is a decreasing effect of gravity and increasing
effect of a surface tension so therefore the
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assumption of absence symmetric gas slugs
and constant thickness of the liquid film
04:05.590 --> 04:12.780
are justified and naturally the other assumption
is the liquid film velocity is much less as
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compared to the velocity of the gas as well
as the liquid slugs and in addition we find
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that in this particular case therefore the
usual step of modelling is we try to find
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out the pressure gradient in the taylor bubble
region we try to find out the pressure gradient
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in the liquid slug region and in addition
we consider another particular pressure gradient
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which occurs because the counter current flowing
liquid film comes and meets [vocalized-noise]
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the upward flowing or the forward flowing
liquid slug as a result of which some mixing
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and some turbulence results at the end of
the taylor bubble which also should be contributing
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to the pressure drop of the entire slug unit
but this usually it is neglected under most
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of the circumstances
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so therefore with this particular assumptions
we start the modelling as as have all ready
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mentioned so as discussed the frictional pressure
gradient i would again like to remind you
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that the pressure gradient it is predominantly
frictional for micro channels and is very
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less effect of orientations since gravity
is [vocalized-noise] is not very important
05:33.840 --> 05:39.521
in this in this particular case so therefore
the frictional pressure gradient this is equal
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to the this should comprise of the pressure
gradient in the taylor bubble region this
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should comprise of the pressure gradient in
the liquid slug region and it should also
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comprise of the pressure gradient due to the
drainage of the liquid film into the liquid
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slug at the taylor bubble and each pressure
gradient it should be multiplied with the
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relative lengths of the individual taylor
plugs and the taylor slugs is in it so this
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is going to give you this this particular
complete expression is going to give you the
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total pressure gradient defectional pressure
gradient which occurs over unit cell rather
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it occurs from here to here which occurs over
the unit cell of the slug flow pattern
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now let us see how to evalue how we go about
evaluating the individual pressure gradient
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terms is naturally start with the simplest
the frictional pressure gradient in the liquid
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slug now since just only liquid is flowing
in this particular case so naturally this
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can be evaluated from near single phase phase
flow hydro dynamics which gives the expression
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of frictional pressure gradient as rho l j
square by two d where naturally f l s it is
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a function of the reynolds number in the liquid
slug which is again a function of the reynolds
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number which can be defined as rho l j t p
d by mu l is in it so therefore in [vocalized-noise]
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sorry i made a mistake this should have been
actually j t p square because remember one
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thing in the liquid slug region the liquid
is flowing over the entire cross section and
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it is flowing at the total mixture velocity
in this particular case so therefore in the
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slide have represented it has j but j in this
case means the two phase volumetric plugs
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more specifically [vocalized-noise] i can
refer to it as j t p so therefore we find
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that the that the frictional pressure gradient
in the liquid slug is it can be obtained from
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single phase hydro dynamics there the unknown
f l the friction factor in the liquid slug
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can be expressed in terms of the reynolds
number which is expressed in this particular
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we and we all ready all ready know from or
single phase knowledge of single phase hydro
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dynamics that f l s is goes to sixty four
be r l s for laminar flow and the blushes
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type equation for the turbulent flow provided
we have to remember the this equation is applicable
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provided we are considering the affection
factor right now again let me tell you that
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for most of the of the cases we neglect the
pressure gradient due to the drainage portion
09:01.050 --> 09:07.350
but in case someone wants to do a very accurate
modelling then this should be consider and
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the person is advised to refer to the paper
of dukler and hubbard who have derived this
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particular expression considering the counter
current motion and relative velocity between
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the film as well as the liquid slug
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well now we come to the modelling of the taylor
bubble portion now what the taylor bubble
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portion what do we find that firstly i would
like to mention that for macro systems usually
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we notice that the [vocalized-noise] frictional
pressure gradient in the taylor bubble region
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it can be neglected because it is much smaller
as compared the frictional pressure gradient
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it is usually much much less as compared to
the frictional pressure gradient in the liquid
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slug region right so therefore very frequently
for macro systems what is the conventional
10:03.009 --> 10:11.149
way or conventional approach is that the frictional
pressure gradient for the entire unit cell
10:11.149 --> 10:18.939
for the entire unit cell for the two phase
slug flow it is often approximated as the
10:18.939 --> 10:24.129
frictional pressure gradient in the liquid
slug its better written as approximated into
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one minus alpha assuming that alpha is the
void fraction and the entire gas flows as
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the taylor bubble and the liquid slugs
are completely unedited so naturally that
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portion and also we neglect the liquid in
the film as [vocalized-noise] it if we assume
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it to be negligible as compared to the liquid
in the slug so therefore we assume that the
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entire one minus alpha portion of the liquid
flows as slug and accordingly the pressure
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gradient of the unit cell is approximated
as one minus alpha times minus d p d z a film
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f l s
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now all the this is fines are macro systems
as well as for mini channels but the pressure
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gradient in the liquid rather in the taylor
bubble region it becomes significant for micro
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channels and also for mini channels for refrigerant
flow right [vocalized-noise] for these two
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cases it becomes almost [vocalized-noise]
it becomes quite important for micro channels
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i also for mini channels for refrigerant flow
because in this case we cannot complete in
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neglect the [vocalized-noise] gas velocity
and density as compared to the liquid velocity
11:58.999 --> 12:06.230
and density so therefore for [vocalized-noise]
micro channels and mini channels during refrigerant
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flow this particular term has to be considered
otherwise we can neglect this now in case
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we have to consider this term then let us
discuss how this term can be evaluated now
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we know minus d p d z frictional taylor bubble
this can be written down just like we are
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written down for the liquid slug region in
the same way we can write down down for the
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taylor bubble region right in the taylor bubble
region then this is sorry this is f at the
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interface between the bubble and the liquid
film rho g square u g minus u i whole square
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divided by four into radius of the taylor
better up to diameter of the taylor bubble
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so we have expressed the frictional pressure
gradient arising due to the friction between
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the taylor bubble and the liquid film so naturally
it is the function of the inter facial friction
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factor and it is the function of the relative
velocity between the bubble and the interface
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now in this particular equation you find that
u i is an unknown so next we try to find out
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or other we next we discussed how to find
out u i but before that i would like to mention
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that just like single phase dynamics as well
as the hydro dynamics of the liquid slug region
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in this case also f i it is nothing but a
function of r e t b where we can write down
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r e t b which is nothing but equal to the
gas phase reynolds number that is equal to
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in the same way r t b into u t b sorry ya
u t b (ReferTime: 14:00) minus u i divided
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by mu g right and remember one thing i think
it should not be using both the symbols it
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will confuse you you further remember one
thing since the entire gas flows as a taylor
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bubble so in this particular case u g equals
to u g b so even if i use both the symbols
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remember one thing its the velocity of the
taylor bubble and the gas phase velocity the
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inset velocity inside the conduit they are
identical right so therefore f i f i can be
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determine from as a function of r e t b again
thus the the same equations or applicable
14:41.199 --> 14:48.139
in this particular case as well that the other
same type of equations are applicable and
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r t b it is nothing but the various of
the taylor bubble portion
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now we come to discussing how to find out
the inter facial velocity once we can find
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out this particular velocity we can very easily
calculate the frictional pressure gradient
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in the taylor bubble region now order to find
out u i what do we do we start from the basic
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momentum balance equation for the liquid film
and the gas core so therefore [vocalized-noise]
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we first start by considering the taylor bubble
to be more or less approximated by the annular
15:26.439 --> 15:38.149
flow pattern ok so we assume that flow in
the taylor bubble region as ideal annular
15:38.149 --> 15:53.839
flow region as ideal annular flow and we neglect
effect of gravity this is definitely justified
15:53.839 --> 16:07.389
for milli and micro channels so naturally
if if if we can neglect they effect of gravity
16:07.389 --> 16:18.069
then the momentum equation for laminar flow
of any particular flow it it can be written
16:18.069 --> 16:27.309
down as minus d p d z plus mu by r d d r in
cylindrical cornets this can definitely be
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written down provided we have neglected the
effect of gravity for any laminar flow we
16:33.160 --> 16:38.699
this is applicable provided we [vocalized-noise]
for the new two this is equally applicable
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to the taylor bubble region as well as in
the liquid film region only the boundary conditions
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where solving this equation are going to be
different for the two cases
16:47.790 --> 16:55.619
now the general solution can be derived for
both phases in laminar flow how to derive
16:55.619 --> 17:06.199
it we apply the no slip boundary condition
at the wall and continuity this is one and
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this is two continuity of velocity and shear
stress at gas liquid interface so based on
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these two assumptions we find that the general
solution to this particular equation which
17:31.370 --> 17:36.850
are [vocalized-noise] also mentioned here
so therefore the general solution is as i
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have given this also would like to write down
so that you can follow it better the general
17:42.650 --> 17:56.950
solution can be given as minus d p d z r square
plus b l n r plus e now this is highly applicable
17:56.950 --> 18:19.789
for both liquid film and taylor bubble provided
the flow in flow in both are laminar one and
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two fully developed then we can you can use
this and and we [vocalized-noise] using this
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we can find out u and and then using the boundary
conditions as i have mentioned the no slip
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boundary condition at the wall and the continuity
of velocity and shear stress at the gas liquid
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interface we can solve it and and we can find
out u i
18:44.309 --> 18:50.480
but here we need to remember one particular
point that is for most of the cases we find
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that the usually the liquid film although
it can be laminar but the gas slug are the
18:58.519 --> 19:04.610
gas flow is usually turbulent so that is that
is the case in what do we do how do we proceed
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in [vocalized-noise] that case for that particular
case as simple semi analytical solution it
19:11.450 --> 19:19.990
can be formulated by first deriving the equation
or the solution for laminar liquid film flow
19:19.990 --> 19:40.419
ok [vocalized-noise] usually liquid flow in
film is laminar that gas flow in core is turbulent
19:40.419 --> 19:53.960
so for such a situation for do we do we find
out a simpler semi analytical solution for
19:53.960 --> 20:09.690
this particular case it can be formulated
how by first deriving for the laminar liquid
20:09.690 --> 20:24.470
film by first deriving for laminar liquid
film ok considering no slip boundary condition
20:24.470 --> 20:36.899
at wall and u l equals to u i at r equals
to r t b can be now do this so therefore if
20:36.899 --> 20:42.389
for the case of laminar liquid flow in the
film and turbulently gas flow in the core
20:42.389 --> 20:49.170
the simpler an semi analytical solution can
be formulated by first deriving the expression
20:49.170 --> 20:55.570
for laminar liquid film and the boundary conditions
for deriving expressions is mostly boundary
20:55.570 --> 21:00.150
condition at the wall and the liquid film
velocity is going to the inter facial velocity
21:00.150 --> 21:02.899
at r equals to r t b
21:02.899 --> 21:10.360
now for such a case we find that the liquid
velocity profile is obtained by the expression
21:10.360 --> 21:18.870
which is given here now one we have derived
this then we go for deriving or further gas
21:18.870 --> 21:27.419
phase in the taylor bubble region so for the
gas in the taylor bubble region what we do
21:27.419 --> 21:33.720
we can write it down the momentum balance
equation as the taylor bubble region this
21:33.720 --> 21:46.110
is again inter facial friction in this particular
case yeah this is the [vocalized-noise] this
21:46.110 --> 21:52.330
is the inter facial friction f i the way i
have told sorry sorry very sorry for the gas
21:52.330 --> 22:01.080
phase in the taylor bubble region i am extremely
sorry the momentum balance equation the momentum
22:01.080 --> 22:16.009
balance equation on the gas core gives mu
l d u d r at r equals to r t b they should
22:16.009 --> 22:25.370
be r t b by two minus d p d z frictional for
the taylor bubble portion right you can write
22:25.370 --> 22:34.240
it down in this particular way and then from
there we can find out the inter facial velocity
22:34.240 --> 22:39.940
from this particular expression and expression
i have shown we can find out the inter facial
22:39.940 --> 22:49.160
velocity as u i is equal to minus one by four
l the pressure gradient in the taylor bubble
22:49.160 --> 22:58.159
region in this particular form so therefore
what do we have now we have the expression
22:58.159 --> 23:04.330
for the frictional pressure gradient in
the taylor bubble region which i [vocalized-noise]
23:04.330 --> 23:09.799
i had mentioned this is that [vocalized-noise]
frictional pressure gradient in the taylor
23:09.799 --> 23:16.620
bubble region here f i can be derived from
this particular expression for u g sorry for
23:16.620 --> 23:22.220
u i we have got this particular expression
right so [vocalized-noise] from these three
23:22.220 --> 23:30.399
equations we we can solve for the three unknown
which we have ok so if we know j g if we know
23:30.399 --> 23:37.649
j l if we know alpha then from here we can
find out minus d p d z frictional for the
23:37.649 --> 23:43.129
entire unit slap the only thing which we do
not know in this particular case which we
23:43.129 --> 23:47.990
need to find out is the relative length of
the liquid slug and the taylor bubble
23:47.990 --> 23:55.720
now this is often derived by considering the
expression of alpha the void fraction usually
23:55.720 --> 24:05.789
we know alpha it is equal to l t b by l t
b plus l l s into r t b square by r square
24:05.789 --> 24:15.149
right and several experimental investigation
have shown that usually r t b by r it is close
24:15.149 --> 24:21.399
to zero point nine ok so therefore this can
easily be substituted in the expression of
24:21.399 --> 24:30.940
alpha and from there one can very well estimate
the relative length of the liquid taylor bubble
24:30.940 --> 24:37.090
and [vocalized-noise] the relative length
of the liquid slug is nothing but one minus
24:37.090 --> 24:43.340
the relative length of the taylor bubble as
is quite evident right so therefore this can
24:43.340 --> 24:49.440
also be derived and once this has been derived
then [vocalized-noise] we know all the terms
24:49.440 --> 24:54.980
here we can substitute the terms and find
out the friction pressure gradient for unit
24:54.980 --> 25:00.230
cell in the slug flow pattern and this gives
us the frictional pressure gradient for the
25:00.230 --> 25:09.379
entire slug flow pattern here i would also
like to mention that even if you do not consider
25:09.379 --> 25:17.720
the value of alpha from experiments we can
find out the value of alpha from several correlation
25:17.720 --> 25:23.779
which are available from the several correlation
[vocalized-noise] also available from which
25:23.779 --> 25:29.120
even without knowing alpha we can find out
the relative lengths and the other things
25:29.120 --> 25:33.440
which i would [vocalized-noise] also like
to mention is that if you if we can either
25:33.440 --> 25:39.429
assume this r t b by r rho equals to point
nine or there are several correlations available
25:39.429 --> 25:46.990
for the liquid film thickness in terms of
the hydraulic diameter as the function of
25:46.990 --> 25:52.700
capillary number such correlations can also
be adopted and they [vocalized-noise] they
25:52.700 --> 25:59.320
can also be used to find out r t b by r square
from which the relative length of the taylor
25:59.320 --> 26:03.510
bubble and the liquid slug region (Refer Time:
26:00 ) can be obtained so therefore this
26:03.510 --> 26:08.710
this is one particular model which we use
for the slug flow pattern and i also like
26:08.710 --> 26:16.090
to mention that since i had i had expressed
that the the [vocalized-noise] when we observed
26:16.090 --> 26:21.929
the flow pattern in a micro channel we find
that several flow patterns exist at the same
26:21.929 --> 26:29.460
time so a better model will be if we consider
we develop the pressure gradient for the taylor
26:29.460 --> 26:34.600
bubble region for the liquid slug region for
single phase liquid flow for single phase
26:34.600 --> 26:41.250
gas flow for the annular flow region and we
also know the relative time of existence of
26:41.250 --> 26:47.169
the different patterns and if you wet each
particular pressure gradient with the time
26:47.169 --> 26:52.799
of existence of that particular distribution
we would be getting much more accurate expression
26:52.799 --> 26:57.910
of pressure gradient but in order to do that
we have to know the time of existence of the
26:57.910 --> 27:01.700
different flow patterns and again we have
to depend on (Refer Time: 27:00 ) experiments
27:01.700 --> 27:03.029
for this particular evaluation
27:03.029 --> 27:11.720
so with this i come to an end [vocalized-noise]
end to on the analytical models for the hydrodynamics
27:11.720 --> 27:19.470
of gas liquid and vapour liquid flows in micro
channels and at the end in the last lecture
27:19.470 --> 27:25.990
i would be dealing with some flow boiling
aspects in micro and milli channels and that
27:25.990 --> 27:31.700
will be the end of my lecture series on adiabatic
two phase flow and flow boiling in micro channels
27:31.700 --> 27:32.659
thank you very much