WEBVTT
Kind: captions
Language: en
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So, last time we were discussing about the
bulk density of this first phase and as we
00:00:31.390 --> 00:00:35.000
said that bulk density is nothing but the
mass of dispersed phase per unit volume of
00:00:35.000 --> 00:00:40.420
the mixture. And based on that we have defined
some formula and that was rho d bar which
00:00:40.420 --> 00:00:45.970
is the bulk density is nothing, but the
del dot M d upon dou V, where V is the volume
00:00:45.970 --> 00:00:50.670
of the mixture. So, complete volume.
And it can also be written as rho d bar
00:00:50.670 --> 00:00:55.469
is equal to n where n is the number density
multiplied by the mass of a single particle.
00:00:55.469 --> 00:01:00.820
So, in that way you can also find the number
density sorry, bulk density of the particle.
00:01:00.820 --> 00:01:07.329
Now, moving to the next the another important
definition which we actually found or we were
00:01:07.329 --> 00:01:12.000
about to discuss and we stopped the last class
is about the inertial effect.
00:01:12.000 --> 00:01:16.159
Now, what we mean by inertial effect? And
the inertial effect is the effect as the name
00:01:16.159 --> 00:01:20.360
suggests that there is an inertia. So, now,
we are discussing about the two phase flow
00:01:20.360 --> 00:01:24.890
or multi phase flow in which more than one
phases are available. And if suppose in the
00:01:24.890 --> 00:01:31.130
pipeline a fluid is flowing and suppose air
is flowing at a particular velocity and there
00:01:31.130 --> 00:01:36.470
is some particle which is suspended ok. So,
suppose there is a one particle which is suspended.
00:01:36.470 --> 00:01:42.930
Now, if I change the velocity of the air by
any means say suppose if I put a divergence
00:01:42.930 --> 00:01:50.150
here ok. So, if I put a divergence what will
happen the velocity of air will change and
00:01:50.150 --> 00:01:56.350
with that ideally the velocity of particles
would also change, but the problem is sometimes
00:01:56.350 --> 00:02:01.009
because of the inertial effect the particle
does not follow the path of the fluid exactly.
00:02:01.009 --> 00:02:05.329
So, suppose what will happen though is suppose
there is a stream line flow and then what
00:02:05.329 --> 00:02:10.590
will happen the fluid will pass something
like this, but it is not needed that the particles
00:02:10.590 --> 00:02:15.879
would also follow exactly the same path this
is say fluid path though is not needed the
00:02:15.879 --> 00:02:20.970
particles should also follow the same path.
It may be possible that because of the inertia
00:02:20.970 --> 00:02:26.200
the particle may go straight it does not move
at all it does not change the path and that
00:02:26.200 --> 00:02:32.780
is called the inertial effect ok. So, inertial
effect is being defined is that basically
00:02:32.780 --> 00:02:37.829
because of the acceleration in the flow field
and the particles are not able to follow the
00:02:37.829 --> 00:02:42.129
fluid motion. And why they are not able to
follow the fluid motion? Because of their
00:02:42.129 --> 00:02:47.379
own mass they have certain mass and because
of that because they are moving with a certain
00:02:47.379 --> 00:02:51.930
velocity they have certain momentum. So, they
actually go with their momentum and they may
00:02:51.930 --> 00:02:58.409
not follow the path of the fluid exactly.
Now, how much it deviates from the path of
00:02:58.409 --> 00:03:05.889
the fluid that is what is called the impaction
or inertial effect. So, we have to find it
00:03:05.889 --> 00:03:10.219
out the two things the first is whether they
will follow the path of the fluid or not and
00:03:10.219 --> 00:03:15.790
if they will not follow how much they will
hm kind of deviate from their path.
00:03:15.790 --> 00:03:21.859
So, impaction effect or inertial effect need
to be found. Now, why this effect is important?
00:03:21.859 --> 00:03:26.589
Because these effect is very critical particularly
for the separation application. Suppose if
00:03:26.589 --> 00:03:32.030
I want to separate there is some dust which
is being suspended in the air or you are doing
00:03:32.030 --> 00:03:38.409
some operation say in fluidized bed and gases
are passed and those gases are actually carrying
00:03:38.409 --> 00:03:43.870
some of the fine particles, or in a boiler
we are passing the gas for the oxidation reaction
00:03:43.870 --> 00:03:49.329
of a combustion reaction and with the gas
or with the flue gas some of the ash particles
00:03:49.329 --> 00:03:53.489
or soot particles are being carried away.
So, what you need? You have to actually separate
00:03:53.489 --> 00:03:58.689
those particles and those particles actually
can be separate by using the mechanism of
00:03:58.689 --> 00:04:02.019
impaction.
Now, how it is possible to separate? What
00:04:02.019 --> 00:04:08.060
I have to do suppose if there is a flow
in which the particles are flowing or the
00:04:08.060 --> 00:04:13.109
fluidant particles both are flowing I need
to just define in such a way I have to give
00:04:13.109 --> 00:04:17.481
the particle such a momentum that it should
have some inertial effect. And then it should
00:04:17.481 --> 00:04:24.509
say if there is a path here and I can if suppose
I put some filter or some kind of a catch
00:04:24.509 --> 00:04:29.099
here where the particle we go and get stuck.
Then what will happen? The fluid actually
00:04:29.099 --> 00:04:34.090
will move this in elbow kind of a shape this
is the fluid, but if suppose there is a particle
00:04:34.090 --> 00:04:40.040
which is moving here because of the impaction
it can go and hit on this wall. So, the there
00:04:40.040 --> 00:04:44.500
can be a separation.
Now, this can be desirable and this can actually
00:04:44.500 --> 00:04:49.150
worsley effect also it can be undesirable
phenomena. So, what can possible? Possibly
00:04:49.150 --> 00:04:54.400
if you are not designing the system properly
and you are having the flow of a gas and solid
00:04:54.400 --> 00:04:59.900
or liquid and solid then in the same bed what
will happen the particle will go all the time
00:04:59.900 --> 00:05:05.610
and hit this bed. Now, if they will hit it
multiple time the erosion will occur and your
00:05:05.610 --> 00:05:10.449
bend make damaged after some time. So, that
is why it is very important to understand
00:05:10.449 --> 00:05:13.840
that how the particle will follow the path
of the fluid.
00:05:13.840 --> 00:05:18.540
Now, again I am telling that it can be used
for the separation mechanism to separate the
00:05:18.540 --> 00:05:25.689
particle clearly. So, that is the way the
impaction has been studied and actually the
00:05:25.689 --> 00:05:32.389
impaction can be studied by the 3 quantities
and that 3 quantities are relaxation time,
00:05:32.389 --> 00:05:38.380
or we say it a response time. We can also
find the stop distance, the stop distance
00:05:38.380 --> 00:05:44.439
means where the particle will go and stop
and third is the stokes number. Now, we will
00:05:44.439 --> 00:05:49.020
define all these 3 separately and we will
see the important of each phases, but the
00:05:49.020 --> 00:05:53.990
impaction generally is being defined with
these 3 numbers ok or these 3 values.
00:05:53.990 --> 00:05:59.590
Now, how these 3 values are defined? So, first
is response time or relaxation time as it
00:05:59.590 --> 00:06:06.430
name suggests that what is the response time
response time is the time which particle will
00:06:06.430 --> 00:06:10.340
take to respond to the change in the fluid
velocity or the continuous phase velocity
00:06:10.340 --> 00:06:14.360
as the name suggests clearly. So, it is the
momentum response because we are talking about
00:06:14.360 --> 00:06:20.289
the momentum we are talking about the velocity
it is a momentum response time that relates
00:06:20.289 --> 00:06:28.080
to the time required for a particle or a droplet
to change of the in the velocity of the continuous
00:06:28.080 --> 00:06:34.550
loop. It means it is the time required a particle
for a particle to respond to change in any
00:06:34.550 --> 00:06:39.229
particle velocity. So, before that it will
not respond it means what it will continue
00:06:39.229 --> 00:06:43.060
its path.
So, suppose if this is a bend again I am taking
00:06:43.060 --> 00:06:48.389
bend is a very simple example. If the pursued
is moving it it in this way and the particle
00:06:48.389 --> 00:06:53.060
response time is say very high then what will
happen particle will not move in this side
00:06:53.060 --> 00:06:57.889
in this direction instead of going in this
direction it will continue its motion in the
00:06:57.889 --> 00:07:02.020
same direction. So, what will happen? It will
go and hit the bends. So, that will depends
00:07:02.020 --> 00:07:07.229
on the particle response time that how fast
it can response to the change in the fluid
00:07:07.229 --> 00:07:13.389
velocity. So, that is called particle response
time. So, that is the one phenomena which
00:07:13.389 --> 00:07:20.460
actually describe that how the relaxation
factor or whether the impaction will play
00:07:20.460 --> 00:07:26.219
a role or not if it will play how much important
role it will play play. So, that is the
00:07:26.219 --> 00:07:30.139
definition of the response time.
Now, how to calculate the response time? So,
00:07:30.139 --> 00:07:35.080
for that we can do a derivation. Now, how
to do the derivation is a very simple thing
00:07:35.080 --> 00:07:40.110
suppose if a particle for the same case is
moving and it is moving horizontally. So,
00:07:40.110 --> 00:07:45.189
I am neglecting the effect of the gravity
and buoyancy will be very small. So, what
00:07:45.189 --> 00:07:50.680
will happen? If suppose there is a particle
which is moving in horizontal direction in
00:07:50.680 --> 00:07:56.889
say direction of x, if suppose this is x this
is y it is moving in x direction along with
00:07:56.889 --> 00:08:01.639
the fluid. So, the fluid is also moving and
particle is also moving. So, what will happen
00:08:01.639 --> 00:08:08.831
say the fluid velocity is v and the particle
velocity is u. So, these are the particle
00:08:08.831 --> 00:08:21.809
velocity u is particle velocity
and this is fluid velocity.
00:08:21.809 --> 00:08:30.529
So, a fluid is moving in which some particle
is suspended and if I am assuming that the
00:08:30.529 --> 00:08:36.039
particle velocity is u meter per second fluid
velocity is v meter per second if they are
00:08:36.039 --> 00:08:39.800
moving then what will happen what are the
forces which are going to act. So, if you
00:08:39.800 --> 00:08:44.600
see the forces which are acting because the
motion is in horizontal there is not gravity
00:08:44.600 --> 00:08:49.360
is not going to play any role if I neglect
the buoyancy and I am just considering the
00:08:49.360 --> 00:08:53.190
motion in the horizontal direction or in the
x direction then the buoyancy will also not
00:08:53.190 --> 00:08:57.430
play a role. So, there will be two forces
which will be acting, one is the particle
00:08:57.430 --> 00:09:02.120
momentum or particle inertia and that will
be opposed with the drag force.
00:09:02.120 --> 00:09:12.410
So, I can write a equation which can say that
m d u by dt. So, rate of change in the momentum
00:09:12.410 --> 00:09:17.710
is nothing but that will be equal to the F
D which is nothing, but a drag force. So,
00:09:17.710 --> 00:09:25.860
you can say this F D is nothing, but drag
ok. Now, how the drag force are defined? Well
00:09:25.860 --> 00:09:30.800
revisit it again, but I hope in your undergraduate
or in your basic chemical engineering courses
00:09:30.800 --> 00:09:35.740
or basic engineering courses this might have
been introduced that drag is nothing, but
00:09:35.740 --> 00:09:46.100
it is half into C D it is rho A u minus v
square. So, because this both are moving we
00:09:46.100 --> 00:09:52.610
are defining it based on the slip velocity.
So, I am saying that it is u minus v and mod
00:09:52.610 --> 00:09:58.490
of u minus v. So, some book says that is u
minus v square I am writing it in the directions
00:09:58.490 --> 00:10:03.590
u minus v into mod of u minus v so that you
can get the direction of the drag force. So,
00:10:03.590 --> 00:10:08.060
finally, these are the forces which is acting
on the particle if it is moving horizontally.
00:10:08.060 --> 00:10:12.780
So now, we can what we can do we can try to
see that how to calculate the response time.
00:10:12.780 --> 00:10:18.449
So, we will try to define or derive the formula
for the response time. Now, to do that we
00:10:18.449 --> 00:10:23.209
did the force balance the only problem is
the C D value as we know that the C D is a
00:10:23.209 --> 00:10:27.070
function of Reynold number and if you keep
on changing the Reynold number the value of
00:10:27.070 --> 00:10:32.240
C D will change and that the value will be
different for the laminar and turbulent flow
00:10:32.240 --> 00:10:36.899
or you can say there is stokes regime and
Taylor regime or turbulent regime the values
00:10:36.899 --> 00:10:40.529
will be different.
So, let us assume that the velocity is less
00:10:40.529 --> 00:10:47.430
and we are still in the stokes regime then
the C D will be actually defined with the
00:10:47.430 --> 00:10:52.410
24 by Re.
Now, this Reynolds number Re is nothing, but
00:10:52.410 --> 00:11:04.280
the Reynolds number ok. And this is Reynold
number is defined based on the slip velocity.
00:11:04.280 --> 00:11:10.540
So, what will be the Reynold number? Is this
nothing, but rho of continuous fluid d of
00:11:10.540 --> 00:11:17.390
particle u of particles because the particle
velocity is v u, I am writing it as a u and
00:11:17.390 --> 00:11:23.040
this will be based on the slip velocity. So,
you will say that u minus v upon mu of continuous.
00:11:23.040 --> 00:11:27.300
So, that is the Reynold number the weight
has been defined, it has been defined based
00:11:27.300 --> 00:11:32.490
on the slip velocity. So, this rho is for
the continuous fluid, these two is for continuous
00:11:32.490 --> 00:11:43.560
fluid rho and mu, and this is for the discrete
phase or discrete phase and the relative motion.
00:11:43.560 --> 00:11:45.959
So, that is the way the Reynold number has
been defined.
00:11:45.959 --> 00:11:51.209
So, we know that if this is following in the
stokes regime or it means the Reynolds number
00:11:51.209 --> 00:11:57.080
is very very less or is less than one then
C D can be written as 24 by Re and the whole
00:11:57.080 --> 00:12:06.680
equation can be further simplified as m d
u by dt is nothing but half C D is 24 by Re
00:12:06.680 --> 00:12:18.769
and Re I am going to this is a Re and rho
of particle ok into area into u minus v to
00:12:18.769 --> 00:12:25.949
mod of u minus v ok. So, that is the way it
is going to be defined. Now, what is this
00:12:25.949 --> 00:12:31.330
area? Area is the area which is on the projected
area. So, it means this area for a spherical
00:12:31.330 --> 00:12:37.810
particle will be nothing, but pi by 4 4 square.
So, I will just open this Reynold number and
00:12:37.810 --> 00:12:45.040
area and this will be equal to 12 upon Reynolds
number is rho of continuous fluid into d of
00:12:45.040 --> 00:12:56.480
particle into u minus v ok and then this is
mu of c continuous because 0 d upon mu it
00:12:56.480 --> 00:13:05.390
will go up it will be rho p it will be pi
by 4 D p square this is u minus v mod of u
00:13:05.390 --> 00:13:10.649
minus v.
So, that is the way one can define this. Now,
00:13:10.649 --> 00:13:15.050
Reynold number is the based on the mod. So,
I will write it here the mod. So, that it
00:13:15.050 --> 00:13:19.811
values would be positive. Now, if you solve
this what will happen? This two will be cancelled
00:13:19.811 --> 00:13:26.089
out ok and we can write it in the most simplified
form and this will be cancelled out 4 and
00:13:26.089 --> 00:13:35.970
this. So, this will be 3. So, you can write
it 3 mu c it will be rho p into pi this D
00:13:35.970 --> 00:13:43.360
p 1 D p will be cancelled out it will be D
p and u minus v and this will be rho p upon
00:13:43.360 --> 00:13:49.740
rho c. So, you will get a value which will
be of this kind.
00:13:49.740 --> 00:13:57.829
Now, if we see that sorry this is not
rho p, this is the rho c actually ok. So,
00:13:57.829 --> 00:14:03.569
in that way this will be actually rho c, and
this rho c rho c will be cancelled out ok.
00:14:03.569 --> 00:14:08.519
So, I think you have written it correctly
here it is rho c. So, this is the rho c. Now,
00:14:08.519 --> 00:14:17.050
if this two will be canceled out what you
will get is thrice mu c into pi D p u minus
00:14:17.050 --> 00:14:26.360
3 or I will write it to a more familiar term
which will be 3 pi into mu c D p into u minus
00:14:26.360 --> 00:14:31.660
v. So, if you see that that is the formula
which you have used for the drag force in
00:14:31.660 --> 00:14:38.269
the stokes regime and in terms of that it
will be 6 pi mu r into v. Now, because based
00:14:38.269 --> 00:14:42.190
on the slip velocity it is two phase flow
we are defining it based on the slip velocity.
00:14:42.190 --> 00:14:51.310
So, you are going to have that is m d u by
dt is nothing, but 3 pi mu c D p into u minus
00:14:51.310 --> 00:14:57.170
c. Now, what I am going to do? I am going
to integrate it, but before that I would like
00:14:57.170 --> 00:15:01.740
to open this mass which mass is nothing, but
the mass of the particle. So, this is mass
00:15:01.740 --> 00:15:10.139
of the particle it can be written as m p can
be written as it will be the rho of particle
00:15:10.139 --> 00:15:15.270
into volume of the particle and volume of
the particle for a spherical particle it
00:15:15.270 --> 00:15:23.009
will be 5 by 6 D p cube. So, I will replace
mp with that and if I do that mp if I replace
00:15:23.009 --> 00:15:32.110
with this I can just write it instead of mp
it will be rho p pi by 6 D p cube.
00:15:32.110 --> 00:15:38.110
Now, I will just try to simplify it if you
will try to simplify I will get the values
00:15:38.110 --> 00:15:47.410
it is like pi pi will be cancelled out you
will get d u by dt will be equal to you just
00:15:47.410 --> 00:15:53.990
bring everything here it will be 18, this
pi will be cancelled out it will be mu c upon
00:15:53.990 --> 00:16:02.960
rho p ok. Now, D p cube will be cancelled
out. So, this will be upon d square D p square
00:16:02.960 --> 00:16:15.069
and u minus v ok. So, this will be what
is being turned out the equation and the value
00:16:15.069 --> 00:16:20.500
if you see this value this is the unit of
this is the reciprocal of the time and that
00:16:20.500 --> 00:16:30.279
is called the response time. So, this is nothing,
but is called response time or relaxation
00:16:30.279 --> 00:16:33.550
time.
So, the response time or relaxation time is
00:16:33.550 --> 00:16:40.170
being defined say if I say that response time
by tau v if I represent then it is nothing,
00:16:40.170 --> 00:16:50.329
but rho p into D p square upon 18 mu of continuous.
So, this is called the response time ok. So,
00:16:50.329 --> 00:16:57.029
that is much time a particle will take if
you are putting a particle in the system which
00:16:57.029 --> 00:17:02.860
is moving a gas which is moving. So, depending
upon how much is the density of the particles
00:17:02.860 --> 00:17:07.990
what is the size of the particle, and what
is the viscosity of the gas it will have certain
00:17:07.990 --> 00:17:13.190
time it will take before it will respond to
any change in the fluid motion velocity of
00:17:13.190 --> 00:17:16.770
fluid velocity.
Now, do you see that it is function of the
00:17:16.770 --> 00:17:22.330
density it is function of the particle diameter
and it is inversely proportional to the part
00:17:22.330 --> 00:17:27.880
fluid viscosity. It means what? In the
fluid viscosity is very high the response
00:17:27.880 --> 00:17:33.300
time is going to be low, if your particle
density is low then the response time is going
00:17:33.300 --> 00:17:37.800
to be the low, if your particle diameter is
low then the response time is going to be
00:17:37.800 --> 00:17:43.680
low. So, it means what if this times will
be low particle will respond to the flow very
00:17:43.680 --> 00:17:48.630
fast if this time is very high particle will
not respond to the fluid or it will respond
00:17:48.630 --> 00:17:54.040
very lately to any change in the fluid velocity
of fluid motion of fluid direction. So, that
00:17:54.040 --> 00:17:59.100
is the way response time has been derived.
Now, it can be further continued and if I
00:17:59.100 --> 00:18:06.330
just want to find it out the stop distance
I can say that d u by dt was equal to if I
00:18:06.330 --> 00:18:17.780
just look here it was u minus v upon tau v.
So, that is the way we can write it. Now,
00:18:17.780 --> 00:18:26.720
if you do it it in this way what you will
get you will get that 1 upon d u upon u minus
00:18:26.720 --> 00:18:37.100
v is equal to dt upon tau v ok. Now, if you
do the integration of the same if I integrate
00:18:37.100 --> 00:18:41.440
it then what I will get I will get the function
which will be in the form of ln. So, this
00:18:41.440 --> 00:18:50.130
will be ln u minus v will be equal to this
will be t upon tau v.
00:18:50.130 --> 00:18:58.690
So, now, if you do that t upon tau v value.
Now, what will happen? I can take the integral
00:18:58.690 --> 00:19:07.700
and it will be u minus v will be equal to
e raised to the power t upon tau v. So, that
00:19:07.700 --> 00:19:13.760
will be what you will get as the formula.
Now, mostly what will happen the u will be
00:19:13.760 --> 00:19:18.470
smaller v will be higher. So, this value is
going to be negative. So, I can take that
00:19:18.470 --> 00:19:22.880
negative sign here to reduce this value
to make this value positive. So, finally,
00:19:22.880 --> 00:19:28.980
you are going to get that u minus v is nothing,
but e raised to the power minus t upon tau
00:19:28.980 --> 00:19:36.130
v. So, that will be your overall time which
it will take before the particle will respond
00:19:36.130 --> 00:19:42.290
to the motion and you can see that how the
tau v is the time and how the u will change
00:19:42.290 --> 00:19:46.480
with the time.
So, you will find it out that u and v correlation
00:19:46.480 --> 00:19:50.560
you can see that will change in the v how
the u is going to be changed. I hope this
00:19:50.560 --> 00:19:55.450
is clear that why we have taken negative,
we have taken it negative because here the
00:19:55.450 --> 00:19:59.770
particle mainly is moving because of the gas
velocity. So, u is going to be smaller than
00:19:59.770 --> 00:20:05.590
the v and because of that this value is going
to be negative and if that is true I have
00:20:05.590 --> 00:20:10.030
to take the minus out and e raise to the power
minus t by tau it will be there. So, once
00:20:10.030 --> 00:20:14.770
we are decaying the velocity we are changing
the velocity the particle motion is also going
00:20:14.770 --> 00:20:19.670
to decay or particle motion is also going
to change and we can calculate that how the
00:20:19.670 --> 00:20:26.930
motion change will take place. So, in that
way we are defining this.
00:20:26.930 --> 00:20:33.310
Now, if we have defined this we can also find
it out suppose if the particle have b started
00:20:33.310 --> 00:20:39.200
with the initial velocity 0, then what will
happen I can include the whole thing from
00:20:39.200 --> 00:20:46.130
0 to say a velocity u if I do that what will
happen. If the time t is equal to 0 u v will
00:20:46.130 --> 00:20:50.510
be equal to 0 I can calculate that what will
be the value after you and then we can say
00:20:50.510 --> 00:20:56.620
that at time t equal to t if the velocity
is u if you do that then what you will get
00:20:56.620 --> 00:21:03.490
you will just do this ln. So, you will get
that u is nothing, but it will be v into 1
00:21:03.490 --> 00:21:11.740
minus e raised to the power t upon tau v.
So, what you need to do? You have to just
00:21:11.740 --> 00:21:18.920
integrate this this integral will be from
0 to time t and at 0 you can say the velocity
00:21:18.920 --> 00:21:25.760
is u is 0 and at time t the velocity is u.
So, if you do that calculation if you solve
00:21:25.760 --> 00:21:31.910
it you will get the value of this kind and
it will show you that how the particle velocity
00:21:31.910 --> 00:21:37.990
is actually going to change with the time.
So, if you change the fluid velocity motion
00:21:37.990 --> 00:21:43.040
how the particle velocity is going to change.
Now, from there you can find it out that whether
00:21:43.040 --> 00:21:49.350
the particle is responding to the change
in the fluid velocity and you can also track
00:21:49.350 --> 00:21:53.070
the velocity of the particle with the time.
So, that is very very important very very
00:21:53.070 --> 00:21:59.840
critical and if you see that it is actually
following if you just try to revise your control
00:21:59.840 --> 00:22:04.770
it is actually following that when the u will
be equal to v once the particle will change
00:22:04.770 --> 00:22:09.770
or it will be nearly equal to b. So, if you
find that the e raise to the power is this
00:22:09.770 --> 00:22:14.950
value whatever we are saying that t minus
v is equal to 1 then what will happen? The
00:22:14.950 --> 00:22:20.160
e raised to the power one value will come
it will be 1 upon 1 then this whole value
00:22:20.160 --> 00:22:28.330
will come 1 upon sorry 2.736. So, this whole
value will come at 63 percent, you 63.2 percent,
00:22:28.330 --> 00:22:35.570
but for the sake of simplicity you can find
that it is 63 percent.
00:22:35.570 --> 00:22:41.720
So, it means what that the momentum response
time can also be defined as a time which is
00:22:41.720 --> 00:22:50.050
required to achieve the 62 percent of the
particle velocity or 63 percent of the velocity
00:22:50.050 --> 00:22:56.940
of the free stream velocity or of the 63 percent
of the particle velocity of the free streamed
00:22:56.940 --> 00:23:02.710
in the particle will as find or will particle
we attain that is called the response time.
00:23:02.710 --> 00:23:07.640
So, that can also be defined it, it in this
way which is coming directly from this. So,
00:23:07.640 --> 00:23:12.320
what will happen u and p will be then how
the response time will be defined. So, you
00:23:12.320 --> 00:23:16.850
can define the response time it it in this
way. So, this is the another way to define
00:23:16.850 --> 00:23:21.110
the response time, but we will go with our
own definition which says that it is nothing,
00:23:21.110 --> 00:23:27.740
but the time required for a particle to follow
or to response to any change in the fluid
00:23:27.740 --> 00:23:31.430
velocity. So, that is the way we have defined
this tau v.
00:23:31.430 --> 00:23:36.110
Now, once the tau v is defined we have also
defined that how the particle velocity will
00:23:36.110 --> 00:23:40.940
change with the time. So, now, what you can
do, you can solve any problem in which I have
00:23:40.940 --> 00:23:45.460
the initial particle velocity is given if
the fluid velocity is given and it is given
00:23:45.460 --> 00:23:50.500
that the flow is very small you can use this
formula to calculate that with the time how
00:23:50.500 --> 00:23:54.010
the particle velocity will change.
Now, once you know that with the time how
00:23:54.010 --> 00:23:58.840
particle velocity is changed you can also
track the position of the particle, that how
00:23:58.840 --> 00:24:04.080
much position how much distance it will travel
before it will stop. So, in that way you can
00:24:04.080 --> 00:24:09.430
find it out that the penetration distance
or stop distance. So, you can also calculate
00:24:09.430 --> 00:24:14.710
the stop distance for these particles that
if they are moving at what distance it is
00:24:14.710 --> 00:24:20.210
going to stop. So, that is called the stop
distance. And it has been defined actually
00:24:20.210 --> 00:24:26.640
the stop distance is nothing, but is that
l this is nothing, but the velocity into the
00:24:26.640 --> 00:24:32.870
tau v. So, if you have know the as stop distance
is nothing, but what is the velocity say is
00:24:32.870 --> 00:24:38.030
you into tau v if you do that we will get
the stop distance that what will be the final
00:24:38.030 --> 00:24:42.720
distance at which it will stop.
But if you want to calculate that how the
00:24:42.720 --> 00:24:48.280
food plot will move or how the particle will
move along the fluid path you can use this
00:24:48.280 --> 00:24:52.950
formula you can convert it to the in terms
of the velocity. The u in terms of the you
00:24:52.950 --> 00:24:57.310
can calculate to the position and you can
track that how the particle positions say
00:24:57.310 --> 00:25:03.730
if there is a pipeline where there is elbow
and the fluid is moving particle is also moving
00:25:03.730 --> 00:25:11.830
we can track that how the particle will start
following the the path of the fluid.
00:25:11.830 --> 00:25:16.330
So, it means if you change the path of the
fluid how fast the particle will able to accommodate
00:25:16.330 --> 00:25:21.570
that change you can calculate that by using
tau b. You can calculate the particle path
00:25:21.570 --> 00:25:27.880
line because calculating the position of the
particle you can also calculate the particle
00:25:27.880 --> 00:25:32.180
velocity how it will decay the velocity and
what will be the final velocity it will get.
00:25:32.180 --> 00:25:36.790
So, all those things can be found with the
stop distance also. So, you can also find
00:25:36.790 --> 00:25:43.650
that when the particle will go and finally,
stop at some place. So, these all things can
00:25:43.650 --> 00:25:48.350
be handled can be completed.
So, with this; now, similar to particle momentum
00:25:48.350 --> 00:25:52.540
response time there is thermal response time
also. So, particle thermal response time is
00:25:52.540 --> 00:26:02.140
nothing, but the. So, what we have discussed
is the momentum response time.
00:26:02.140 --> 00:26:11.070
Similarly, in similar line we can discuss
the thermal response time, and thermal response
00:26:11.070 --> 00:26:18.620
time is nothing but the response time needed
by the particle to respond to any change in
00:26:18.620 --> 00:26:25.150
the temperature of the continuous fluid. So,
if suppose the fluid is moving and I put certain
00:26:25.150 --> 00:26:32.760
a heating zone here at certain location and
the fluid is moving what will happen the fluid
00:26:32.760 --> 00:26:37.100
temperature once it will come here we start
developing and you will see a proper fluid
00:26:37.100 --> 00:26:40.670
velocity.
Now, if there is a particle here then the
00:26:40.670 --> 00:26:46.240
particle will take actually some time before
it will respond to any change in the food
00:26:46.240 --> 00:26:50.910
flow in any change in the temperature of the
continuous fluid or of the fluid which is
00:26:50.910 --> 00:26:56.090
slowing. So, that is called the thermal response
time and it can again calculate the wave we
00:26:56.090 --> 00:27:01.040
have balance the momentum or the force we
can balance the energy and based on the energy
00:27:01.040 --> 00:27:05.920
balance we can calculate that what will be
the response time which will be needed for
00:27:05.920 --> 00:27:08.820
the thermal or thermal what will be the thermal
response time.
00:27:08.820 --> 00:27:13.630
Now, to just for the example that how to do
that how to do the energy balance what will
00:27:13.630 --> 00:27:21.020
be the total energy contained by the particle
it will be nothing, but mc p dou T upon dou
00:27:21.020 --> 00:27:25.760
t you. So, it means this capital T is the
temperature small t is the time. So, that
00:27:25.760 --> 00:27:31.270
is the total energy any particle is having
or the total energy the particle can have.
00:27:31.270 --> 00:27:36.380
Now, this will be equal to nothing, but the
Nusselt number which is the number and kind
00:27:36.380 --> 00:27:42.400
of responsible for the heat transfer it should
be the pi k c, k c will be the thermal conductivity
00:27:42.400 --> 00:27:56.910
of the continuous phase and it will be D p
into T p minus T c. So, once I said T p minus
00:27:56.910 --> 00:28:04.320
T c it is nothing, but the temperature of
the particle this is the particle temperature
00:28:04.320 --> 00:28:10.720
and this is the temperature of the continuous
phase. So, I will say that this is a gas phase
00:28:10.720 --> 00:28:24.290
temperature and this is particle phase temperature.
So, you can calculate similar way that how
00:28:24.290 --> 00:28:28.900
much time it will take for the particle to
respond to any change in the temperature.
00:28:28.900 --> 00:28:33.970
So, you can do all this balance and if you
solve this you can find it out that what will
00:28:33.970 --> 00:28:39.230
be the thermal response time of the particle
and just I am leaving it to you as an assignment
00:28:39.230 --> 00:28:44.040
and maybe we will do this as a assignment.
The tau T is the thermal response time of
00:28:44.040 --> 00:28:50.350
the particle will be nothing, but it will
be rho p into c p d, where the c p d is nothing
00:28:50.350 --> 00:28:57.390
, but the heat capacity value for discrete
phase or the particle phase into d square
00:28:57.390 --> 00:29:06.980
upon 12 k c.
So, this will be the particle thermal response
00:29:06.980 --> 00:29:11.370
time and it is nothing, but you have to just
balance this open the end the way we have
00:29:11.370 --> 00:29:18.100
done in terms of the rho p into vp, the vp
is the volume of the particle. And then ncp
00:29:18.100 --> 00:29:23.270
dp you have to write this and then you just
equate it put the value of Nusselt number
00:29:23.270 --> 00:29:28.880
in terms of that k and then you solve it you
will get that rho t is nothing, but equal
00:29:28.880 --> 00:29:35.460
to rho p c p d D square upon 12 k c.
So, it means what if I try to find it out
00:29:35.460 --> 00:29:44.920
the value of tau v upon tau t and just try
to find it out. So, tau v is nothing, but
00:29:44.920 --> 00:30:00.461
it was rho p d p square upon 18 mu c and tau
t is nothing, but it will be rho p into c
00:30:00.461 --> 00:30:12.860
p d d square upon 12 k c. So, this is the
way this is I c p d it is this if you solve
00:30:12.860 --> 00:30:18.120
it out it will be cancelled out many things
and you will get that 2 by 3 it will be k
00:30:18.120 --> 00:30:25.530
c upon c p d.
So, that you will get the value of tau v upon
00:30:25.530 --> 00:30:35.840
tau t sorry tau v upon tau t that will be
the value which will tell you that whether
00:30:35.840 --> 00:30:41.880
the momentum response time is higher or thermal
response time is higher and if you want to
00:30:41.880 --> 00:30:48.600
correlate it with the this. So, this will
be mu c p d there will be one mu which is
00:30:48.600 --> 00:30:53.570
missing here. So, it will become mu c p d.
So, that is the way it will be found.
00:30:53.570 --> 00:30:58.130
So, if you want to find it out if you want
to correlate it you can also correlate this
00:30:58.130 --> 00:31:05.250
with the Prandtl number and if you want
you have to what you need to do you have to
00:31:05.250 --> 00:31:10.790
just multiply by this cpd up and down
and then you can calculate the Prandtl number
00:31:10.790 --> 00:31:14.900
of continuous flow.
So, you can do that you can write it in
00:31:14.900 --> 00:31:19.150
terms of the Prandtl number also and if you
write in terms of the Prandtl number it will
00:31:19.150 --> 00:31:24.580
be nothing, but 2 by 3 it will be c p c it
means the heat capacity value for continuous
00:31:24.580 --> 00:31:34.040
phase will be c p d into one upon Prandtl
number ok and. So, this is the way you can
00:31:34.040 --> 00:31:39.520
you can define. Now, how this is being
defined? How what I have done to derive that?
00:31:39.520 --> 00:31:47.540
I have just multiplied here in this equation.
So, if I take that tau v upon tau t thermal
00:31:47.540 --> 00:31:56.670
response time is nothing, but it comes 2 upon
3 this will be k c upon mu it will be c p
00:31:56.670 --> 00:32:01.640
d.
Now, if I multiplied with c p c up and down
00:32:01.640 --> 00:32:08.810
it will be c p c then what will happen? We
know that c p mu by k that is nothing, but
00:32:08.810 --> 00:32:19.660
it is the Prandtl number. So, this will be
2 by 3 it will be c p c upon c p d and
00:32:19.660 --> 00:32:31.140
this c p, this c p this mu upon k is going
to be the Prandtl number. So, we will say
00:32:31.140 --> 00:32:38.200
one upon Prandtl number.
So, similarly you can find the response time
00:32:38.200 --> 00:32:42.920
the way we have derived the things remain
same you can find the response time you can
00:32:42.920 --> 00:32:47.420
find that momentum response time, you can
find the thermal response time. The idea is
00:32:47.420 --> 00:32:53.320
that impaction it means if the fluid path
is changing of fluid velocity is changing
00:32:53.320 --> 00:32:58.940
particle may take some time before it will
respond to that change ok and that is called
00:32:58.940 --> 00:33:02.510
the response time.
Now, once we are talking about the momentum
00:33:02.510 --> 00:33:07.240
that is called momentum response time when
we are talking about the temperature or the
00:33:07.240 --> 00:33:12.230
heat we will say that it is the thermal response
time. So, that is the way the response time
00:33:12.230 --> 00:33:18.390
has been derived. And one can use that response
time to calculate that how the particle will
00:33:18.390 --> 00:33:23.130
behave and the same calculation can be used
to calculate the stop distance the way I have
00:33:23.130 --> 00:33:28.190
told you earlier. Same calculation can be
used to track the particle trajectory of particle
00:33:28.190 --> 00:33:33.640
motion, same equation can be used to calculate
the particle velocity. So, if you know this
00:33:33.640 --> 00:33:37.140
you can calculate how the particle will move
with the time.
00:33:37.140 --> 00:33:41.200
Now, moving towards the next. So, we have
defined response time, we have defined this
00:33:41.200 --> 00:33:57.310
stop time. Now, the third one is that stokes
number. Now, what is a stokes number and
00:33:57.310 --> 00:34:01.310
how it has been defined.
So, a stokes number is nothing, but it is
00:34:01.310 --> 00:34:07.490
the ratio of response time to the flow characteristic
time. So, the stokes number is represented
00:34:07.490 --> 00:34:15.260
with St subscript k and it is ratio of tau
v upon tau F, where the tau F is the flow
00:34:15.260 --> 00:34:19.340
characteristic time.
Now, what do you mean by tau F, flow characteristic
00:34:19.340 --> 00:34:26.800
time? So, suppose there is a fluid which is
flowing in a pipe line, it is a fluid which
00:34:26.800 --> 00:34:32.089
is flowing in a pipe line and I have to calculate
that what will be the characteristic flow
00:34:32.089 --> 00:34:39.049
time I will say that what is say a is the
length suppose this is the time line and suppose
00:34:39.049 --> 00:34:44.510
it is changing the dimension, let us assume
that it is changing the dimension and again
00:34:44.510 --> 00:34:49.349
it is going get it in this way.
So, suppose your fluid is flowing here and
00:34:49.349 --> 00:34:53.609
the particle is also suspended. Now, because
the particles will fluid will move particle
00:34:53.609 --> 00:34:57.839
will also move along the fluid. Now, they
will reach to this divergent section they
00:34:57.839 --> 00:35:03.559
will go and diverge to a small section of
throat and then again they will go to again
00:35:03.559 --> 00:35:07.920
separate it and they will kind of instead
of contraction. Now, they will have expansion
00:35:07.920 --> 00:35:11.650
and the fluid will move out and particles
would also move out.
00:35:11.650 --> 00:35:20.569
So, now, if we find it out the tau F, tau
F is will be nothing, but the characteristic
00:35:20.569 --> 00:35:39.410
flow length characteristic length divided
by the velocity
00:35:39.410 --> 00:35:45.900
of the fluid. So, that is the tau F. So, the
characteristic length in this type will be
00:35:45.900 --> 00:35:51.460
what, is the length of the throat or you can
say that. So, that will be the tau F. So,
00:35:51.460 --> 00:35:56.519
how much is the fluid characteristic time
will be there that is the time. So, you can
00:35:56.519 --> 00:36:01.700
calculate that the tau F tau v we already
know we can calculate the tau v and we can
00:36:01.700 --> 00:36:06.790
calculate the stokes number and to find it
out that whether the particle is going to
00:36:06.790 --> 00:36:12.440
follow the path of the fluid or not.
Now, if the stokes number is very very less
00:36:12.440 --> 00:36:18.819
than 1, if suppose St k is very very less
than 1. It means what? It means the particle
00:36:18.819 --> 00:36:26.010
response time is very low compared to the
fluid flow time ok, it means the characteristic
00:36:26.010 --> 00:36:32.009
time of the flow it means what the particle
response time because it is very very low
00:36:32.009 --> 00:36:41.350
it will response is very low it means the
particle has ample time or very long time
00:36:41.350 --> 00:36:46.049
to respond to the change in the fluid motion
because tau v is very very small ok.
00:36:46.049 --> 00:36:52.339
So, it means what tau is very small tau F
is very high, particle length. So, characteristic
00:36:52.339 --> 00:36:57.569
time of the fluid is much higher compared
to the characteristic time of the particle.
00:36:57.569 --> 00:37:02.030
So, it means what? Particle will just follow
the path of the fluid it will have ample time
00:37:02.030 --> 00:37:07.970
to respond. So, in that case if this is less
than one the particle will actually flow and
00:37:07.970 --> 00:37:13.569
it will just change is stay here and then
go out. So, that will be the path of the particle
00:37:13.569 --> 00:37:18.960
exactly same will be the path of the fluid
also. So, it has ample time to respond to
00:37:18.960 --> 00:37:23.119
any change.
Now, if the stokes number is say is very very
00:37:23.119 --> 00:37:30.490
high then 1, then what does it mean that
particle response time is very very high compared
00:37:30.490 --> 00:37:33.970
to the flow characteristic time. It means
the flow characteristic time is very very
00:37:33.970 --> 00:37:38.890
low and particle response time is very high
it means what in that case particle will have
00:37:38.890 --> 00:37:45.300
no time to respond to any change in the
fluid velocity ok. So, it will have its own
00:37:45.300 --> 00:37:51.309
inertia or action is going to work it will
move with its own inertia and it will not
00:37:51.309 --> 00:37:57.990
respond to any change in the fluid velocity
of flow direction. So, what will happen? For
00:37:57.990 --> 00:38:01.670
the similar case now, if suppose this is the
case.
00:38:01.670 --> 00:38:10.849
Now, what will happen? The particle will go
and it will just hit the water. So, that is
00:38:10.849 --> 00:38:15.299
the way it is not going to respond to any
change in the fluid motion. So, the fluid
00:38:15.299 --> 00:38:19.480
motion may change, but particle is not willing
to do anything it is just going and hitting
00:38:19.480 --> 00:38:25.019
the ball. So, that is called once will happen
when the stokes number is very very more than
00:38:25.019 --> 00:38:30.390
1 and its stokes number is less than 1. Then
particle will have ample time and it will
00:38:30.390 --> 00:38:33.960
respond to all the changes you will make with
the fluid velocity.
00:38:33.960 --> 00:38:40.210
So, that is the way we can find the impaction
effect. So, we can calculate the stop distance,
00:38:40.210 --> 00:38:44.779
we can calculate the response time and we
can calculate the stokes number. So, in that
00:38:44.779 --> 00:38:50.670
way you can find the effect of impaction with
the help of stokes number, with the help of
00:38:50.670 --> 00:38:55.849
response time one can also derive that how
the particle will be have how the particle
00:38:55.849 --> 00:39:01.069
location will change the position can be calculated,
one can also calculate that with that time
00:39:01.069 --> 00:39:05.519
how the particle velocity will change. So,
you can see that whether the particle is accelerating
00:39:05.519 --> 00:39:10.980
or it is deaccelerating, if you know how it
is deaccelerating you can calculate that what
00:39:10.980 --> 00:39:15.300
will be the stop distance when the particle
will deaccelerated this much that the velocity
00:39:15.300 --> 00:39:19.299
of the particle will go to 0.
So, all those calculations can be done and
00:39:19.299 --> 00:39:24.269
the similar way as I said the stop distance
has been defined it is nothing, but the initial
00:39:24.269 --> 00:39:31.390
velocity of the particle into the relaxation
time. So, stop distance S is nothing, but
00:39:31.390 --> 00:39:35.299
what is the initial velocity of the particle.
Now, I am saying that particle velocity as
00:39:35.299 --> 00:39:42.109
a u. So, I will denote it as say u naught
into tau v and that is nothing, but the stop
00:39:42.109 --> 00:39:46.970
distance. So, you can also calculate that
if you know the tau v we after how long the
00:39:46.970 --> 00:39:58.049
particle will actually stop. So, this u naught
is the initial velocity of the particle.
00:39:58.049 --> 00:40:10.030
So, these 3 quantities together can actually
help to track the motion of the particle to
00:40:10.030 --> 00:40:14.759
find that if the particle want to stop where
the particle will stop. So, we can suppose
00:40:14.759 --> 00:40:20.849
if I ask a question that there is a vein
high speed vein in which some particles has
00:40:20.849 --> 00:40:27.940
been suspended. And let us assume that initial
velocity of the particle is say 1 meter per
00:40:27.940 --> 00:40:33.109
second and can you tell me after how much
time that particle will stop, if I give you
00:40:33.109 --> 00:40:37.750
the density of the particle, if I give you
the velocity of the particle, initial velocity
00:40:37.750 --> 00:40:42.279
of the particle, if you give the velocity
of the fluid, if I give you the properties
00:40:42.279 --> 00:40:46.890
of the fluid, it means the density and viscosity
you can actually calculate that after this
00:40:46.890 --> 00:40:50.549
much length the particle will actually stop
it will not move.
00:40:50.549 --> 00:40:55.539
So, in that way we can calculate that we can
also see that how the deacceleration will
00:40:55.539 --> 00:41:00.211
take place if the particle velocity is reducing
and we can also track the motion of the particle
00:41:00.211 --> 00:41:05.339
of position of the particle because once you
have the velocity you can just do the dx by
00:41:05.339 --> 00:41:12.819
dt or the dz by dt or dy by dt and integrate
it to get the velocity of position of the
00:41:12.819 --> 00:41:15.470
particle. So, everything can be done by using
this 3 ok.
00:41:15.470 --> 00:41:22.690
Now, the next thing which is critical in multi
phase flow, now we are defining it. So, in
00:41:22.690 --> 00:41:26.869
the definition the one of the most critical
thing is that whether the flow is dilute or
00:41:26.869 --> 00:41:32.220
dense. So, actually speaking of frankly if
you see that the whole multi phase can be
00:41:32.220 --> 00:41:37.329
divided in 2 parts and the treatment will
be entirely different for these 2 parts. Once
00:41:37.329 --> 00:41:41.150
the flow is dilute and another once the flow
is dense.
00:41:41.150 --> 00:41:45.910
Now, there is a critical difference between
these two. Now, what is the difference? Once
00:41:45.910 --> 00:41:50.740
as the name suggests dense means the fraction
of the discrete phase will be very high that
00:41:50.740 --> 00:41:56.520
is called a dense flow and when the fraction
of discrete phase is very low and that is
00:41:56.520 --> 00:42:01.609
called the dilute flow. So, that is the broadly
definition has been defined the way the dense
00:42:01.609 --> 00:42:05.840
and dilute flows are being defined in the
multi phase flow. And the whole treatment
00:42:05.840 --> 00:42:08.750
is different.
We will see the treatment once we will discuss
00:42:08.750 --> 00:42:14.450
the dilute flow and dense flow pertain the
gas solid flows we will see that how those
00:42:14.450 --> 00:42:19.269
differences will occur once the flow is dilute
how you will treat the flow or how you will
00:42:19.269 --> 00:42:24.700
model the flow or how you which technique
can be used to see the flow are kind of diagnosis
00:42:24.700 --> 00:42:29.390
the problem and if the flow is dense how it
will be modeled or how it will be investigated.
00:42:29.390 --> 00:42:33.690
So, we will see that later on.
But what is the definition and how you are
00:42:33.690 --> 00:42:37.790
going to quantify that whether the flow is
dilute or dense it is very very critical and
00:42:37.790 --> 00:42:44.040
rough definition or you can say the layman
definition is if the discrete phase fraction
00:42:44.040 --> 00:42:51.410
is very high the particle is actually dense
the flow is actually dense if the discrete
00:42:51.410 --> 00:42:56.589
phase fraction is very low less than 5 percent
or so, the particle or the flow is called
00:42:56.589 --> 00:42:59.740
as a dilute flow.
That these are the weight definition there
00:42:59.740 --> 00:43:05.869
are certain concrete definition and terms
to find that whether the flow is dilute or
00:43:05.869 --> 00:43:12.009
dense and the concrete definitions whatever
it is is given here and it says that once
00:43:12.009 --> 00:43:17.680
or dilute dilute the dispersed phase is one
in which the particle motion is controlled
00:43:17.680 --> 00:43:23.920
by the fluid forces. It means what? In the
say there is a fluid which is flowing in a
00:43:23.920 --> 00:43:31.359
pipeline and they have only few particles
suspended, so 4 5 particles which are suspended
00:43:31.359 --> 00:43:33.960
here.
Now, what will happen? Because the number
00:43:33.960 --> 00:43:39.980
of particles are very very small the fluid
is going to dominate the motion of the particles
00:43:39.980 --> 00:43:44.450
ok. So, I am not considering the particle
response time here, I am not considering
00:43:44.450 --> 00:43:48.510
that the vein and all these things are there
if that is not there then what will happen
00:43:48.510 --> 00:43:53.470
the particle motion will be primarily depend
on the fluid and it is actually you can say
00:43:53.470 --> 00:43:57.809
that it will depend on the drag as we have
already seen that if one particle is suspended
00:43:57.809 --> 00:44:02.420
how you can calculate, how the particle motion
will take place. So, particle motion will
00:44:02.420 --> 00:44:09.269
finally, depend on the fluid only. So, that
is the cause as a dense or dilute phase. So,
00:44:09.269 --> 00:44:15.349
once the particle motion is mainly governed
or controlled by the fluid forces like drag
00:44:15.349 --> 00:44:21.240
it is called a dilute film.
Now, a dense phase just contrary to each other.
00:44:21.240 --> 00:44:27.569
Suppose if I pack it with lot of particles
say if I just put lot of particles here inside.
00:44:27.569 --> 00:44:32.492
Then what will happen? The particle motion
actually will not only depend on that how
00:44:32.492 --> 00:44:37.930
the fluid motion is taking place, but it also
depends on that how these two particles are
00:44:37.930 --> 00:44:42.380
moving together or how these particles are
moving together, are they having a collision,
00:44:42.380 --> 00:44:47.130
if they are having a collision how they are
responding to that collision. So, whether
00:44:47.130 --> 00:44:51.980
they break whether it is a complete elastic
collision, whether is a inelastic complete
00:44:51.980 --> 00:44:56.730
inelastic collision, whether it is in between
the elastic and inelastic visco elastic kind
00:44:56.730 --> 00:45:00.840
of a collision, whether they are changing
the dimensions after the collision, whether
00:45:00.840 --> 00:45:04.599
they are kind of changing the velocities
after they are having collision.
00:45:04.599 --> 00:45:09.660
So, the particle motion will not only depend
on the fluid forces the particle forces will
00:45:09.660 --> 00:45:14.650
also have a radar meaning role, like particle
collision or particle, particle collision
00:45:14.650 --> 00:45:21.529
we talk about that then the flow is called
as a a dense flow. So, that is the major difference
00:45:21.529 --> 00:45:27.720
between the dilute flow and dense flow.
Now, how to quantify? This all our definition
00:45:27.720 --> 00:45:31.880
once I am saying that the particle forces
are dominating or whether the discrete
00:45:31.880 --> 00:45:36.559
phase fraction resolved our qualitative the
quantitative we do not have the picture till
00:45:36.559 --> 00:45:42.380
now. So, how to find the quantity that whether
it is dilute or dense phase? So, for that
00:45:42.380 --> 00:45:48.839
we actually see the ratio and we see the ratio
of tau v upon tau c, where tau v is nothing
00:45:48.839 --> 00:45:58.400
but the response time
and tau c is nothing but it is the collision
00:45:58.400 --> 00:46:08.920
time collision time.
Now, how to calculate tau c we will see that.
00:46:08.920 --> 00:46:15.930
So, qualitatively we can calculate the
whether the fluid is dense dilute based
00:46:15.930 --> 00:46:21.559
on these two ratios, so tau v by tau c. So,
if the tau v by tau c is less than 1 the flow
00:46:21.559 --> 00:46:26.980
is dilute if it is greater than 1 the flow
is dense. Now, if you see that value it means
00:46:26.980 --> 00:46:33.660
greater than one means what the response time
is very high compared to the your tau c
00:46:33.660 --> 00:46:37.710
collision type which is very very low if the
collisional time is very very low. It means
00:46:37.710 --> 00:46:42.279
what? What you understood about that that
the particles are densely packed with each
00:46:42.279 --> 00:46:46.349
other or they densely packed then well if
the collision time will be low. So, how the
00:46:46.349 --> 00:46:50.700
collision time is being defined? It is the
time between the two successive collision
00:46:50.700 --> 00:46:56.249
of the particles. So, that is called the
collision time ok. So, that is the two successive
00:46:56.249 --> 00:47:02.230
the time between the collisions or two successive
collisions of the particle is called the collision
00:47:02.230 --> 00:47:05.280
time.
Now, if this is greater than 1 collision time
00:47:05.280 --> 00:47:10.559
is this overall tau e upon tau c ratio
is greater than 1, it means what collision
00:47:10.559 --> 00:47:15.519
time is very very small if the collision time
is very small it means the particle will have
00:47:15.519 --> 00:47:20.299
a very frequent collisions and that is possible
only if they are packing fraction of the particle
00:47:20.299 --> 00:47:24.769
is very high. It means the fraction of the
particle is very high, it means it is going
00:47:24.769 --> 00:47:32.289
to be a a kind of dense phase or dense flow.
A project way around if we see that tau v
00:47:32.289 --> 00:47:38.289
upon tau c is very very less less than 1 then
which means what the tau c value is very high
00:47:38.289 --> 00:47:43.930
the tau c value will be high if only if the
frequency of the collision is less and it
00:47:43.930 --> 00:47:46.060
will be less only if the particles are very
far from each other.
00:47:46.060 --> 00:47:51.079
So, what we need? We need to actually find
with this we can classify that whether the
00:47:51.079 --> 00:47:57.720
flow is dilute or dense, but what I need is
the value of tau v and tau c. Now, we already
00:47:57.720 --> 00:48:01.799
know how to calculate the value of tau v.
So, if I know my fluid system if. Now, I know
00:48:01.799 --> 00:48:07.789
my particle if I know the diameter of the
particle then I can easily calculate as rho
00:48:07.789 --> 00:48:20.849
p into D p square upon 18 we can easily calculate
the response time the only problem is how
00:48:20.849 --> 00:48:25.490
to calculate the collision time that is tricky.
Now, once you calculate the collision time
00:48:25.490 --> 00:48:30.019
you can find it out without doing any experiment
or without physically seeing two other things
00:48:30.019 --> 00:48:35.630
you can say that whether the flow is going
to be dilute or the flow is going to be dense.
00:48:35.630 --> 00:48:42.630
So, to calculate the this collision time we
will use the basic collisional approach or
00:48:42.630 --> 00:48:47.500
theory of the collision of frequency and that
theory of collision frequency has been actually
00:48:47.500 --> 00:48:52.650
derived with the granular temperature theory
we discussed that theory later our kinetic
00:48:52.650 --> 00:48:55.630
theory of granular flow which will discuss
later.
00:48:55.630 --> 00:49:02.390
And we will try to see that how the collisions
has been found by using the very simple approach
00:49:02.390 --> 00:49:06.769
which is being used also in the kinetic theory
of granular flow also in the kinetic theory
00:49:06.769 --> 00:49:11.730
of the gases both the things, where this collision
frequency can be calculated exactly in the
00:49:11.730 --> 00:49:18.329
similar manner and how to do that we are just
going to see that. So, to calculate the collision
00:49:18.329 --> 00:49:22.470
frequency what we need to do we have to assume
a system.
00:49:22.470 --> 00:49:35.940
And let us assume that I have a system I am
making a particle which is very big
00:49:35.940 --> 00:49:40.700
and let us assume that this is my one particle
which is going to move. So, I am assuming
00:49:40.700 --> 00:49:45.829
that this only particle is moving ok. And
let us assume that it is moving because we
00:49:45.829 --> 00:49:50.339
are talking about the relative velocity say,
it is moving with a velocity v r they have
00:49:50.339 --> 00:49:54.920
realized the relative velocity and we are
assuming right. Now, that only one particle
00:49:54.920 --> 00:49:59.789
is moving at a time and rest of the particles
are constant ok. So, that is the assumption
00:49:59.789 --> 00:50:05.980
which we use in kind of granular flow also,
we also use in the kinetic theory of the gases.
00:50:05.980 --> 00:50:10.329
So, what I am assuming that suppose there
is a system in which suppose this is a pipeline
00:50:10.329 --> 00:50:16.550
or system you need some particles are suspended
and I am assuming that only one particle is
00:50:16.550 --> 00:50:22.809
moving rest everyone is stationary and I am
giving a velocity of the particle this particle
00:50:22.809 --> 00:50:28.460
to v r which is the relative velocity it means
I am assuming that other velocities other
00:50:28.460 --> 00:50:32.799
particles are also moving physically they
are not changing their position ok. So, in
00:50:32.799 --> 00:50:36.480
that way we are just defining that how the
particle is moving.
00:50:36.480 --> 00:50:40.630
Now, what will happen once the particle will
move? Suppose the diameter of this particle
00:50:40.630 --> 00:50:48.230
is d suppose this particle is d diameter then
what will happen. Once it will move it will
00:50:48.230 --> 00:50:54.039
form a collisional cylinder. Now, while it
will form a cylinder the region is once
00:50:54.039 --> 00:50:59.680
it is moving say it is moving in this direction
in this circular pipe and the particles are
00:50:59.680 --> 00:51:04.170
spherical then what will happen it will travel
certain distance.
00:51:04.170 --> 00:51:11.099
So, let us assume that it is traveling certain
distance. Now, how much distance it will travel
00:51:11.099 --> 00:51:17.609
what will be the distance that is very obvious
that if the particle velocity is v r then
00:51:17.609 --> 00:51:25.460
in a time t or dt it is going to travel a
distance l which is nothing, but v r into
00:51:25.460 --> 00:51:30.509
dt. So, that is the distance it will travel
and that is nothing, but this. Now, during
00:51:30.509 --> 00:51:33.670
the travel it will have some collisions with
the other particles.
00:51:33.670 --> 00:51:37.869
Now, I have already assumed that the particles
other particles are stationary they are not
00:51:37.869 --> 00:51:44.009
changing their position only one particle
changing the position. So, on the hooch which
00:51:44.009 --> 00:51:49.499
particles this particle will start will having
going to have a collision it means what are
00:51:49.499 --> 00:51:54.910
those particles or whose are those particles
the true it is going to have a collision.
00:51:54.910 --> 00:52:00.190
Now, that can be determined by the collisional
cylinder dimensions. So, collisional cylinder
00:52:00.190 --> 00:52:05.089
length I have already calculated that is v
r into dt it means time delta t or dt is very
00:52:05.089 --> 00:52:09.829
small time it will move a distance say dl
and that dl distance is given here.
00:52:09.829 --> 00:52:14.749
Now, with whom it is is going to have the
collision only with the particle if suppose
00:52:14.749 --> 00:52:22.109
this diameter is d it will have a collision
with a particle which is coming in the one
00:52:22.109 --> 00:52:28.700
r space it means the particle is suppose all
the particles are having the same dimensions.
00:52:28.700 --> 00:52:45.820
Means I am assuming that all the particles
are having same dimensions first. Second I
00:52:45.820 --> 00:53:11.759
am assuming that all other particle except
one is not changing their location
00:53:11.759 --> 00:53:27.900
and particle is moving with relative velocity
v r.
00:53:27.900 --> 00:53:35.650
So, these are what we have assumed that the
particles are having the same dimensions
00:53:35.650 --> 00:53:42.089
it means all the particles is of have the
same dimensions t. Then second that the particle
00:53:42.089 --> 00:53:46.740
other than one particle or rest other particles
are stationary and their positions are not
00:53:46.740 --> 00:53:51.529
changing it means the velocity of the other
particles are 0 and I am allotting a velocity
00:53:51.529 --> 00:53:55.799
v r which is the relative velocity of the
particle to the particle which is moving.
00:53:55.799 --> 00:54:01.730
Now, once I allotted that what will happen
it will during its path say for a small time
00:54:01.730 --> 00:54:06.089
delta t it will travel certain distance and
that distance will be nothing, but the dl
00:54:06.089 --> 00:54:10.400
and which we have already discussed that that
is not going to be something, but it is going
00:54:10.400 --> 00:54:16.160
to v r into dt. So, in a very small time it
will travel certain distance that distance
00:54:16.160 --> 00:54:21.210
is d r, but during this distance travel it
will have collision with the other particles.
00:54:21.210 --> 00:54:26.559
Now, which particle it will have the collision
with the particle which is following falling
00:54:26.559 --> 00:54:30.569
between the one diameter of the particle.
It means what?
00:54:30.569 --> 00:54:38.470
Suppose if I am just writing this as I separately
say this is my particle path if any particle
00:54:38.470 --> 00:54:45.500
which is b like that or it is inside of this
is going to have a collision. Now, what is
00:54:45.500 --> 00:54:51.259
this? This will be nothing, but if the particle
say it is not going to because any particle
00:54:51.259 --> 00:55:02.769
center which is actually lying from a distance
at 1 diameter. So, any particle center which
00:55:02.769 --> 00:55:08.859
will be lying on a distance within the 1 diameter
of the particle from the center of the particle
00:55:08.859 --> 00:55:11.819
it is going to have the collision with it
ok.
00:55:11.819 --> 00:55:17.869
So, let me again clarify it. So, suppose these
are the particles ok this is say let me raise
00:55:17.869 --> 00:55:26.809
it and then again kind of draw it to have
more clarity on it. Now, suppose I am just
00:55:26.809 --> 00:55:31.220
assuming this picture here this, this whole
portion I am just doing here. So, one particle
00:55:31.220 --> 00:55:35.920
we have assumed that there is only one particle
which is moving ok and it is moving forwards
00:55:35.920 --> 00:55:41.799
particular distance, so for a small time.
So, say this traveling a distance of dl this
00:55:41.799 --> 00:55:45.400
is the distance it is traveling.
Now, during its path it will have a position
00:55:45.400 --> 00:55:50.349
with the other particle. Now, each particle
it is going to have a collision with the particle
00:55:50.349 --> 00:55:57.519
suppose this is the diameter which is d ok.
So, this will be this distance will be what?
00:55:57.519 --> 00:56:05.940
It will be d by 2, d by 2. So, this will be
the d by 2 distance which is actually nothing,
00:56:05.940 --> 00:56:11.130
but the radius d by 2 ok. So, this is the
centerline radius.
00:56:11.130 --> 00:56:16.880
Now, any particle suppose which is following
their center. So, which is their center is
00:56:16.880 --> 00:56:24.660
it in such a way that the center of this particle
is at a difference at a distance of d from
00:56:24.660 --> 00:56:29.020
the center of this particle. So, say this
is the as you that as the center of the particle.
00:56:29.020 --> 00:56:35.430
So, if this distance is also equal to the
d. If any particle which is following within
00:56:35.430 --> 00:56:42.609
one diameter of the particle which is moving
ok and please remember one diameter from the
00:56:42.609 --> 00:56:47.920
center of the particle not from the wall.
So, from the edge of the particle one radius
00:56:47.920 --> 00:56:52.789
and from the center of the particle 1 diameter,
if any one, any particle which is following
00:56:52.789 --> 00:56:56.099
within that it is going to have a collision
with this particle.
00:56:56.099 --> 00:57:01.930
So, suppose if I draw a line here and I draw
a similar line here. So, any particle which
00:57:01.930 --> 00:57:06.450
will be fall within this is going to have
the collisions with this particle, any particle
00:57:06.450 --> 00:57:14.930
which is falling outside even like that it
is not going to have any collision with the
00:57:14.930 --> 00:57:17.549
particle because this will not touch each
other ok.
00:57:17.549 --> 00:57:23.979
So, in that way we have defined the collisional
dimension or you can say the collisional
00:57:23.979 --> 00:57:28.680
diameter and that collisional diameter will
be nothing, but the 2D and collisional length
00:57:28.680 --> 00:57:33.779
will be nothing, but the v r into dt. So,
suppose now, if I come back to the same place
00:57:33.779 --> 00:57:38.800
I will say that a collision cylinder will
form and any particle which is following that
00:57:38.800 --> 00:57:44.200
in this cylinder is going to have a collision
with the particle. So, I can find it out that
00:57:44.200 --> 00:57:47.089
how many number of collision the particle
will have.
00:57:47.089 --> 00:57:52.410
Now, how to find the number of collision?
If suppose that the packing density of number
00:57:52.410 --> 00:57:57.640
density of the particle which is there present
in this pipeline or in this system is n. So,
00:57:57.640 --> 00:58:08.039
n is the number density and we have already
defined the n as nothing but number of particle
00:58:08.039 --> 00:58:18.789
and divided by v or dou n by dou v, where
n is this is the number of particles per unit
00:58:18.789 --> 00:58:24.849
volume.
So, you can calculate that how much is
00:58:24.849 --> 00:58:29.539
the number density is particle if that is
known then what we can do we can find it out
00:58:29.539 --> 00:58:35.730
that how many number of collision or particle
which is moving in that system is going to
00:58:35.730 --> 00:58:41.499
have. Now, that number of particle which is
going to be printed. So, for that what I am
00:58:41.499 --> 00:58:49.119
going to first calculate that this is my collisional
dimensions ok, this is my collisional cylinder,
00:58:49.119 --> 00:58:53.640
one particle which is moving they are some
other particles is available the some particles
00:58:53.640 --> 00:59:04.299
are actually out of the system and this is
my collision cylinder. So, if I know the number
00:59:04.299 --> 00:59:08.609
density of the system first what I want to
find it out that number of particles which
00:59:08.609 --> 00:59:13.680
are present within this collisional cylinder.
Now, how to calculate that? That number of
00:59:13.680 --> 00:59:19.319
particles within this we can say that dou
n and dou n is nothing, but number density
00:59:19.319 --> 00:59:25.150
into dou v or the volume of the collisional
cylinder and the volume of the collisional
00:59:25.150 --> 00:59:31.520
cylinder will be nothing, but it will be pi
this diameter this whole diameter is actually
00:59:31.520 --> 00:59:38.609
2D for 2D p I will write. So, that you do
not get confused that D p is nothing, but
00:59:38.609 --> 00:59:43.910
the diameter of the particle. So, the radius
will be D p you can say that pi it will be
00:59:43.910 --> 00:59:49.049
D p square. Now, we have to multiply with
the length. Now, length is nothing, but this
00:59:49.049 --> 00:59:53.920
length and that length is nothing, but if
the particle is moving with a speed v r then
00:59:53.920 --> 01:00:00.990
it will be v r into dt. So, it will be this
d square into v r into dt.
01:00:00.990 --> 01:00:07.460
So, this will be the number of particle is
going to present inside the collisional cylinder.
01:00:07.460 --> 01:00:11.940
Now, number of particle present inside the
collisional cylinder means that many number
01:00:11.940 --> 01:00:16.451
of collision it is going to have because whatever
the particle is presented in the collision
01:00:16.451 --> 01:00:18.890
cylinder it is going to have a collision with
it.
01:00:18.890 --> 01:00:23.059
Now, I can write in terms of the frequency
so that I can get the collisional frequency.
01:00:23.059 --> 01:00:32.420
Now, if I have the number density I can write
dou n upon dou t is nothing but n pi D p square
01:00:32.420 --> 01:00:39.400
into v r ok, and this door upon dou t is nothing,
but the number of part collision per unit
01:00:39.400 --> 01:00:46.240
time it means it is going to be the frequency
of the collision I will write it as f c, f
01:00:46.240 --> 01:00:59.150
c is nothing but is the collision frequency
and that is going to be equal to that this
01:00:59.150 --> 01:01:09.380
is and this is going to be equal to n pi
D p square into v r. So, that is going to
01:01:09.380 --> 01:01:17.749
be the collisional frequency of the particle.
So, this many frequency this will be the frequency
01:01:17.749 --> 01:01:21.690
of the collision ok.
Now, what we are interested is we are interested
01:01:21.690 --> 01:01:27.319
in the tau c and tau c is what collision time.
So, if I have that frequency I can find it
01:01:27.319 --> 01:01:31.700
out what is collision time and how we can
find it out because we know that frequency
01:01:31.700 --> 01:01:35.839
is inversely proportional to the time. So,
this will be nothing, but actually will be
01:01:35.839 --> 01:01:44.640
equal to 1 upon tau c. So, this is collision
time
01:01:44.640 --> 01:01:47.880
and this is the collision frequency. So, I
am going to have both.
01:01:47.880 --> 01:01:54.259
Now, I will just modify it. So, tau c is nothing,
but it is going to be one upon f c and that
01:01:54.259 --> 01:02:05.219
is going to be 1 upon n pi D p square into
your v r, that is going to be the your collisional
01:02:05.219 --> 01:02:12.299
time ok. So, this is your collision time and
n please remember is nothing, but this n is
01:02:12.299 --> 01:02:22.019
nothing, but the number density.
Now, whether the flow is dilute or flow is
01:02:22.019 --> 01:02:28.270
dense what we need to find we need to find
the ratio of tau v upon tau c. Now, tau v
01:02:28.270 --> 01:02:35.019
upon tau c if it is less than 1 then the flow
is going to be damaged, now if we do that,
01:02:35.019 --> 01:02:41.229
v by tau c what I am going to do is I am just
going to calculate the value tau b by tau
01:02:41.229 --> 01:02:50.780
c. Now, tau c is nothing, but it is rho p
into D p square upon 18 mu c and tau c is
01:02:50.780 --> 01:03:01.000
nothing, but it is and 1 upon n into pi D
p square into v r. So, this is going to be
01:03:01.000 --> 01:03:05.950
the ratio of tau b upon tau c.
Now, what we can do? We can solve it further
01:03:05.950 --> 01:03:10.559
and we can instead of number density because
that is sometimes very difficult to calculate
01:03:10.559 --> 01:03:16.349
or to find inside the multi phase flow system.
But the calculating the bulk density is relatively
01:03:16.349 --> 01:03:19.780
easier as you already know that if you know
the volume fraction you can calculate in the
01:03:19.780 --> 01:03:25.519
bulk density. I can write this in terms of
the and in terms of the bulk density and how
01:03:25.519 --> 01:03:30.519
to write that we know that the bulk density
rho d bar is nothing, but is equal to number
01:03:30.519 --> 01:03:35.460
density into mass of one particle. So, that
is number density we have already introduced
01:03:35.460 --> 01:03:46.859
that. So, number density into mass of one
particle ok.
01:03:46.859 --> 01:03:54.200
So, if we do that mass of number density and
mass of one particle we can write it. Now,
01:03:54.200 --> 01:03:59.410
mass of one particle we can again calculate
this will be n the mass of one particle is
01:03:59.410 --> 01:04:06.839
nothing, but rho of particle into V of particle,
the V of particle is nothing but the volume
01:04:06.839 --> 01:04:16.219
of particle and volume of particle if the
particle is spherical can be written as n
01:04:16.219 --> 01:04:25.319
rho p it will be pi by 6 D p cube, that will
be equal to rho d bar. So, we can replace
01:04:25.319 --> 01:04:31.440
this place in terms of the rho d bar. So,
if you want to write that then what we have
01:04:31.440 --> 01:04:36.140
to do this n. So, I will just try to simplify
this equation further here.
01:04:36.140 --> 01:04:42.109
So, this tau c upon tau c and I will write
it in such a way that we can separate this
01:04:42.109 --> 01:04:48.329
things together I will write these terms
together actually. So, I will write n, I will
01:04:48.329 --> 01:04:57.849
take rho p, I will take pi, I will take D
p cube, I will write D p cube into D p ok
01:04:57.849 --> 01:05:02.430
and then I will write it as D p v r and D
p.
01:05:02.430 --> 01:05:12.039
So, I have just braked it and I will write
here as 3 this 18, 6 into 3 of mu c. Now,
01:05:12.039 --> 01:05:20.760
if you see that this is n rho t pi D p cube
upon 6 this whole term is nothing, but is
01:05:20.760 --> 01:05:27.249
equal to the rho d bar or you can say is the
bulk density. So, it will be rho d bar q v
01:05:27.249 --> 01:05:35.309
r into D p upon twice mu of c that is nothing,
but tau v upon tau c.
01:05:35.309 --> 01:05:45.660
So, now, if we do this again I will write
it separately tau v upon tau c and that tau
01:05:45.660 --> 01:05:54.140
v upon tau c is nothing, but rho d bar into
D p into V r upon twice mu c.
01:05:54.140 --> 01:06:00.109
Now, if the flow is dilute then what? Then
this value should be less than 1, if this
01:06:00.109 --> 01:06:10.079
value is less than 1 you can say that the
flow is dilute if this value rho d bar D p
01:06:10.079 --> 01:06:17.979
V r upon thrice mu c if this is greater than
1 the slowest dense ok. So, you can. Now,
01:06:17.979 --> 01:06:22.880
classify the flow is dilute of lowest dense
and it can be further seen in terms of the
01:06:22.880 --> 01:06:29.589
particle diameter. So, suppose if I do it
I can say the D p is less than thrice mu c
01:06:29.589 --> 01:06:38.390
upon rho d bar into here. If this is correct
if the particle diameter is less than this
01:06:38.390 --> 01:06:45.809
number or this quantity the flow is going
to be the dilute if it is more than that the
01:06:45.809 --> 01:06:52.190
flow is going to be dense. So, these are the
definition one can use to find that whether
01:06:52.190 --> 01:06:58.200
the flow is dilute or the flow is dense. So,
that is the whole classification one can do
01:06:58.200 --> 01:07:05.099
with finding the dilute phase and dense phase
system. So, this can be done properly.
01:07:05.099 --> 01:07:10.410
So, what we have discussed now, till now,
we have discussed that different terms terminology
01:07:10.410 --> 01:07:15.099
which we are going to use. We have discussed
about the impaction, in impaction we have
01:07:15.099 --> 01:07:22.390
discussed about the stopping time, relaxation
time or stop distance relaxation time and
01:07:22.390 --> 01:07:27.609
then stokes number. Then we have tried to
again divide the flow multi phase in two class
01:07:27.609 --> 01:07:33.640
major classes dilute flow, and dense flow
in the dilute flow we have defined and we
01:07:33.640 --> 01:07:38.349
have tried to see that how to find that whether
the flow is dilute or not and for that we
01:07:38.349 --> 01:07:42.329
have done the collisional cylinder we have
found, we have taken the collision frequency
01:07:42.329 --> 01:07:46.019
fundamental and based on that we have calculated
the collisional time.
01:07:46.019 --> 01:07:52.859
And we found the correlation which will tell
that whether the particle is doing is the
01:07:52.859 --> 01:07:59.140
diameter is dilute whether the flow is dilute
or the fluid states. So, this is the basic
01:07:59.140 --> 01:08:03.489
definition of the multi phase flow or basic
classification of the multi phase flow.
01:08:03.489 --> 01:08:09.080
Now, there are certain other things also
which we would like to cover. I would like
01:08:09.080 --> 01:08:14.619
each other recover as a basic example and
it is very critical to analyze the multi phase
01:08:14.619 --> 01:08:34.540
flow reactors that I will do in the next time.