WEBVTT
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well hello everybody so we have come to almost
the end of this particular course so i
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was continuing my rather i will pick up my
discussions which was continuing on the
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different analytical techniques to be used
to predict hydrodynamics of flow and i was
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telling you that mainly in order to solve
the mass momentum and energy balance equations
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we have to make several assumptions now these
assumptions which we should be making that
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depends that has been decided after
observing the different types of flow patterns
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which we come across during two phase flow
just like for single phase flow we for the
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momentum balance equation deciding on fathered
flow is laminar or turbulent for this is flow
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patterns also we try to develop a similar
criteria and we observed that if we go through
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all the flow patterns more or less that we
have discussed particularly in macro substance
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we find that we can classify the entire set
into a separated flow pattern and dispersed
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flow pattern and the last range of mix flow
patterns fitch can dish or rather rather
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this schematics are shows as follows
so from here it is very evident that for
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as we come down to the micro channels we will
primarily be dealing with the discussed and
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more importantly with the mixed or the transitional
flow patterns accordingly for to we have decided
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is that in this particular lectures series
we shall be discussing the homogeneous flow
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model which assumes the two fluids to be completely
mixed and then after that will be going to
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the flow patterned based model and here one
of this simplest flow patterned based model
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is the drift flux model which is used more
or less frequently in in micro systems so
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will be dealing with it the drift flux model
and then since slug flow is the most predominate
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flow pattern we will be dealing with the simplified
slug flow model to ender end it up
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and while i go through the analysis i will
be taking up cases for adiabatic gas liquid
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flows as well as cases or rather how the equations
they change when we are dealing with your
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flow boiling in micro systems so now with
the homogeneous flow model now in this particular
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model as we know for what do assume we assume
that the two fluids are uniformly mixed and
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they they have flowing as if pseudo fluid
at the uniform mixture now for does this imply
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this implies that u l f must rather u l it
is not u l s it is u l this is equal to u
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g which is equal to u t p isn't it and naturally
if the two phases the velocities they are
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flowing at the same velocity it automatically
implies that the slip between the two which
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is f which is equal to u g by u l that is
naturally equal to one which again implies
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that alpha should be equal to beta under these
particular conditions and along with that
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it also implies that there is homodynamic
equilibrium between the two phases
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with this if you if you developed the model
what do we find we find that in this particular
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case we find so so accordingly as so
where does the the where does the equation
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come down the equation it the the next thing
which we from here the the the model which
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we get from here is that
so the equation which we can develop from
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here is the homogeneous flow equation which
where we get my did pressure gradient minus
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d p d z it is equal to tau w two phase plus
s by a plus g d u d z write two phase density
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at two phase mass flux and du d t plus rho
t p g sin theta right and this particular
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case what is rho t p equal to rho t p is naturally
equal to g sin theta into alpha rho two plus
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one minus alpha rho one where alpha is the
fraction and in this particular case this
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is this is since alpha equal to beta we get
this has beta rho two plus one minus beta
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rho one right
and we can also right we are fine so therefore
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the this comprises of the gravitational pressure
gradient and this is nothing but the acceleration
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pressure gradient and this is the frictional
pressure gradient ok so therefore from here
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can do we get we get from here so therefore
and there is something very important about
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two phase flow which we should be knowing
its its tells you that rho t p and v t p they
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are not exactly reciprocal of one another
or in other words what i mean is that for
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any particular single phase flow we know that
the specific volume is nothing but the inverse
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of at reciprocal of density ok
but if we consider a two phase mixture even
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under homogeneous flow conditions are also
what do we find we find that rho t p it refers
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to the weight of said one meter or one meter
cube of the mixture so naturally this gives
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you rather it comprises of say in that one
meter of a mixture we have alpha litre of
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phase two and we have one minus alpha litre
of phase one right and therefore you are the
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total weight of alpha litres of phase two
is what it is naturally alpha into rho two
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and the total weight of one minus alpha litres
of phase one is one minus alpha into rho one
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right and so therefore rho t p is given by
it is
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on the other hand if we considered v t p what
do we get v find that v t p it is the volume
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of one k g of the mixture so therefore while
rho t p it was the weight of one liter mixture
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or may be one meter cube mixture this is the
volume of one k g mixture so naturally what
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does is imply this means that in this one
k g mixture there is x g phase two and there
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is one minus x g phase one right so the the
weight of this one k g phase two this is going
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to be x into v two and this is going to be
one minus x into v one right
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so therefore we find out that while v t p
is expressed in terms of the quality of the
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mixture rho t p is expressed in terms of the
volume fraction ok and therefore its quite
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evident that rho t p or other v t p it is
not the reciprocal of rho t p in other words
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they expression of rho t p is in terms of
wide fraction where as where as the expression
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of rather one by rho t p this is in terms
of your quality right so therefore this is
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one particular basic thing which you need
to remember in to phase flow this is something
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very important we are quite used to thinking
that the specific volume is the reciprocal
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of density but that does not matter or that
does not happen in this particular case
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so therefore regarding the gravitational pressure
gradient what do we find gravitational pressure
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gradient its can be expressed either in terms
of the volume fraction and if there is
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large amount of change there then if the if
the the wide fraction if changes to a large
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extent a better way of expressing it is g
sin theta by v t p which gives you g sin theta
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by v one plus x v one two right now regarding
the exhilar rather we we go to the frictional
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pressure gradient was have to be find in the
frictional pressure gradient naturally this
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frictional pressure gradient this is equal
to two f t p rho t p i can write it
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down u t p square by d where this is equal
to two f t p g t p square by d in to v t p
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which can other wise be written down as v
one the v t p part can be written down in
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this particular form
now the i am now remember one thing when we
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are dealing with micro systems gravitational
pressure gradient is usually not there so
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it's the frictional pressure gradient which
is there and the acceleration pressure gradient
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which is there what is the acceleration pressure
gradient in this particular case its naturally
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g d t t p t p means two phase of u t p why
does u t p vary can you tell me its can vary
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if the area varies which is true for single
phase flow also which can also vary if the
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densities of the true phases they vary or
if the quality of the two phase mixture it
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varies so accordingly we find that v t p it
can or rather this particular term this can
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be expanded as your g t p into w t p by a
one by rho t p plus g t p w t p by rho t p
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d d z one by a which can again be expanded
as g t p square g t p square into d d z of
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v one plus x v one two right plus sorry minus
g t p square by rho t p d a d z
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so we can write it down in this particular
form where and we have written written it
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down here where we find that there is when
we consider adiabatic and there is no flashing
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so in that case what happens this is the frictional
pressure gradient this is the acceleration
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pressure gradient in this case acceleration
it occurs firstly due to a change in area
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if it is their mostly there is no area change
this term goes away and then there is one
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more term which happens because the specific
volume it changes with pressure so due to
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this also and additional pressure gradient
happens and accordingly we find for adiabatic
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and no flashing case we find that the pressure
gradient can be written down by this particular
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expression
now in this case i like you mentioned that
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we assume that only the gas phase is compressible
it is usually the case we other phase which
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is the solid or the liquid phase is usually
not compressible and we generally do not deal
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with two phase flows but to both the phases
compressible because if you have on gas gas
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mixture that is never the two phase mixture
that's always the miscible mixture and liquids
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can be have as compressible fluid only under
very large pressure gradients if that part
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is there then in that case along with the
denominator we have another additional term
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apart from this particular term we have another
additional term to take into account the compressibility
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of the liquid phase or this was for adiabatic
and no flashing
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now suppose the pressure gradient is so large
that even if you a not giving any heat also
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there is some amount of depressurization and
may be due to that some amount of liquid it
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it that's converted into its vapour right
so therefore in that case what happens at
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the flow proceeds the x the quality of the
mixture if changes and that quality change
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occurs just because there is a change in pressure
so therefore in this particular case when
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i was talking about about finding out the
acceleration pressure gradient in this particular
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case this rho t p which is nothing but v one
plus x v one two until this particular case
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we had assumed at v t is the function of your
pressure and that's why it is changing with
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z x also we comes a function of pressure and
that also starts changing with x sorry with
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z
and so naturally in this case we need to incorporate
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d x d z which is nothing but d x d p into
d p d z so this gives me and additional term
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in the denominator in this case and suppose
we take the most general case where it is
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that both the phases are com compressible
and we have a heat input where flow boiling
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occurs and along with that it is in the micro
channel so good amount of pressure gradient
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also occurs so with all these if we combine
then we find that the general expression its
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is expressed again as the component of frictional
pressure gradient there is from the acceleration
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pressure gradient there there is long particular
component due to the change in quality due
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to the heat and flux there is one particular
component due to area change if at all it
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is there that's the gravitational component
and the in the denominator part it comprises
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of terms due to the compressibility of the
two phases as a result of pressure gradient
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along the flow and also the change in quality
due to the due to the flashing of the mixture
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and as a result of which exchanges
so therefore we are considered all the gen
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all the cases for adiabatic no flashing that
again adiabatic flashing for q equal to or
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d heat input is zero but any how the change
in quality occurs this can occur in micro
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channels when it is flowing under under
significant pressure gradient and then
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we have also considered the general case now
if you look at the equations very minutely
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what do we find we find that we all the equations
almost you know everything g t p so it's a
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mass flux if you know the mass flow rate and
the area you can find it out but change in
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the change in quality along where due to
heat flux and the change in quality due to
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be depressurization can be obtained the compressibility
of the of the two phases once you know
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the phases you can find out the physical properties
from suitable tables and there so this can
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be found out and so you know all the terms
which are available to find out pressure gradient
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for any particular case which you consider
the only unknown which we find is the two
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phase friction pattern
now how to find it out rather how to evaluate
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that two phase friction pattern we know that
from knowledge of single phase flow
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and as well as two phase flow in macro systems
naturally the way to find out f t p will again
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x some q from the general practices in macro
systems in macro systems one conventional
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thing to assume is that f t p it is a function
of reynolds number for the two phase and it
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is a function of sorry epsilon by d where
epsilon is the or other this is the relative
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roughness factor
now how do we defined r e t p for the two
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phase system r e t p for the two phase system
is naturally d g t p into mu t p we know d
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g t p i have already mentioned this is g one
plus g two diameter of the pipe we know only
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thing which we do not know is the mixture
viscosity here i would like to mention that
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if it is rho t p then it can very well be
expressed in terms of a suitable average of
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the individual mixture densities expressed
it expressed as a function of the in sitting
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wide fraction right but remember one thing
mu t p it is lot of mixture property this
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rho t p is the same probably the mixture is
flowing or this and under that condition
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it hardly matters but mu t p is a flow property
therefore it depends upon the shear which
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occurs during the flow and therefore replacing
to mu to p in the way rho t p are be density
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of the two phase mixture is expressed will
not be correct and unless we have we have
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a good knowledge about the type of shear which
occurs it is very difficult to find out and
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analytical expression to find to if estimate
mu t p so what is the conventional thing which
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is done conventional thing is we propose some
averaging loss some suitable averaging loss
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for mu t p in terms of composition and ensured
that mu t p reduces to mu or phase one for
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phase for the quality of phase one to be one
or rather x equals to zero and alpha equals
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to zero and mu t p equals mu two for x equals
to one and alpha equals to one whatever be
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the case
so therefore these are the true constraints
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based on feeds a proper mu t p is expressed
in terms of mu one mu t p and composition
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which can be either the volume fraction or
the mass fraction accordingly in macro systems
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what do we find we find that several mu t
p expressions ha have been proposed and they
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have also been tried in macro channels and
people have observed that for some cases the
21:46.340 --> 21:52.980
duckler expression is good a few cases the
cicchitti expression is not bad but in micro
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channels additional or rather of few other
averaging loss and mu t p has been provided
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and beattie and wha whalley is correlation
is one such important thing
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and also we find that during a few other correlations
have also been proposed this has been proposed
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based on that fact that r twelve is flowing
in capillaries and due to flashing the two
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phase flow was predominantly bubbly so therefore
mu t p has has been expressed in terms of
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a here homogeneous mixture density and the
specific volume of the liquid and then for
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high flux boiling in channel again a settle
second correlation has been proposed now
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once more i would like to emphasis that its
not a mass that you should remember the correlations
22:45.890 --> 22:50.410
the only two correlations which you should
be remember in for this particular class is
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the dukler correlations which is pretty easy
and the beattie and whalley correlation which
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is basically the dukler correlation with one
particular term of one plus plus two point
23:00.460 --> 23:07.620
five beta now different people they have worked
to get different types of correlations and
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then have suggested different things
for example in that oil water flow its
23:13.000 --> 23:19.470
oil water flow paper for under certain
conditions they said that the beattie and
23:19.470 --> 23:24.750
whalley correlation is better and for some
other conditions people said at the dukler
23:24.750 --> 23:31.800
correlation it is better ok now its now for
example then this oil water flows if if you
23:31.800 --> 23:44.650
notice people have said that cichitti correlation
this was better for oil water flow in micro
23:44.650 --> 23:57.100
channels where the dukler correlation this
was go to for oil water flow in glass micro
23:57.100 --> 24:06.870
channels well this was in what micro channels
right and then again some there where other
24:06.870 --> 24:12.150
people who said that beattie and whalley is
correlation it it is suitable for a large
24:12.150 --> 24:18.700
number of cases for example if you if you
observed the experimental results which i
24:18.700 --> 24:27.940
had shown it will find that for five fourty
and for two fifty micro meters circular channels
24:27.940 --> 24:37.250
beattie and whaley correlation was better
where as for fifty and hundred micro meter
24:37.250 --> 24:44.660
channel with duckler correlation was better
now i will definitely not expect you to remember
24:44.660 --> 24:50.990
all this things but it is very important to
know that thrice so many correlations have
24:50.990 --> 24:56.230
been proposed and thrice under difference
circumstances different correlations worked
24:56.230 --> 25:02.230
better just observe the two most common correlations
and due to dukler and beattie and whalley
25:02.230 --> 25:07.370
if you observe to be expressions you can understand
that for this same flow rates beattie and
25:07.370 --> 25:15.400
whalley correlation will give up higher effective
to shows viscosity as compared to the dukler
25:15.400 --> 25:20.810
correlation so therefore for any particular
homogeneous mixture we find that the beattie
25:20.810 --> 25:26.530
and whalley is correlation is going to be
give you a consistently higher mu effective
25:26.530 --> 25:33.450
and therefore you find that in the five forty
and in the two fifty micro meter channel we
25:33.450 --> 25:45.360
find that for these two cases there is a higher
pressure loss due to mixing of the two phases
25:45.360 --> 25:48.760
ok
in these two cases the flow is predominately
25:48.760 --> 25:55.310
bubbly and flux and therefore there is a higher
pressure loss due to mixing of the two phases
25:55.310 --> 26:03.290
and the higher value of mu effective it gives
us a or rather it enable facts to account
26:03.290 --> 26:10.210
for the increase inter facial shear in these
two cases in the bubbly and the slug flow
26:10.210 --> 26:16.000
pattern on the other hand if you if you observe
these two micro channels in this particular
26:16.000 --> 26:23.710
case the flow was predominately laminar here
the flow was predominately laminar and therefore
26:23.710 --> 26:39.480
there was no need to no need for a enhance
mu t p to account for interfacial shear so
26:39.480 --> 26:46.050
therefore it is very important to remember
that under the same condition this gives a
26:46.050 --> 26:54.520
high higher effective this coscity and therefore
send we need to a when there is a extra interfacial
26:54.520 --> 26:58.030
shear due to increase mixing and the inter
phases
26:58.030 --> 27:04.130
naturally this correlation gives us at better
better result where as when the flow is more
27:04.130 --> 27:11.200
or less laminar and and the the interfacial
shear is not very high then in that case the
27:11.200 --> 27:20.320
duklers correlation is better now other than
finding out to suitable f t p a very conventional
27:20.320 --> 27:28.610
technique of finding out the frictional pressure
gradient can be obtained by defining a particular
27:28.610 --> 27:37.970
variable phi square which is known as the
two phase multiplier what does the two phase
27:37.970 --> 27:48.140
multiplier do the two phase multiplier it
finds out the two phase frictional pressure
27:48.140 --> 27:58.090
drop when it is multiplied with a single phase
frictional pressure drop so therefore what
27:58.090 --> 28:03.970
it does it is basically a ratio of the two
phase frictional pressure drop to the ratio
28:03.970 --> 28:11.220
of the single phase pressure drop of either
have the phases flowing alone in the micro
28:11.220 --> 28:14.450
channel
so in this particular case finding out the
28:14.450 --> 28:20.550
single case pressure drop is not a very big
deal so one this can be find out and we can
28:20.550 --> 28:26.100
find out and estimate of two phase multipliers
from known into parameters we can multiply
28:26.100 --> 28:32.700
the two and we can find out the frictional
pressure drop during two phase homogeneous
28:32.700 --> 28:38.720
flow so in this particular case we do not
need the true phase friction factor and on
28:38.720 --> 28:44.080
the contrary we can find out the two phase
friction fact rather two phase frictional
28:44.080 --> 28:52.390
pressure drop by using a multiplier so therefore
we stop for now and we proceed to find out
28:52.390 --> 28:59.440
different relationships to estimate the two
phase multipliers and predict the frictional
28:59.440 --> 29:01.950
pressure drop accordingly
thank you