In this class, we are going to talk about
convective fluid dynamics in microchannels.
Now, we all know that when we have flow through
any normal size channels, the flow maybe laminar,
turbulent or in between. Now, in microchannels
because of the predominance of viscous forces
the flow can remain laminar for quite some
time. And it is also not possible in most
of the cases to have turbulent flow in such
small devices. So, in in in other words, in
order to obtain or in order to attain turbulent
flow in microchannels one has to sacrifice
a huge amount of pressure drop. So, most of
the cases, we will encounter laminar flow
in microchannels, but there will be situations
in which the flow at times or at certain pockets
may become turbulent or the mixing will become
much more.
For example, if there are bends in the pipeline
or if there are internals which are present
in the in in in the path of the in microchannels
or in the path of the fluid or there are entry
and exit, then, it may lead to a near turbulent
situations specially near the corners. Now,
this fact has been used to device mixers at
microscale. Now, we all understand that if
we have two fluids which are flowing side
by side then mixing them in a microchannel
where the flow is predominantly laminar could
be a problem. So, one alternative could be
to use these bends intelligently such that
they they themselves will act as mixers, but
we also have to understand that once this
mixer mixing pattern sets in the flow will
become stabilized, once once again once we
are away from that bend.
So, we need to have some specific studies
in which we can quantify or we can study at
depth, how the flow becomes turbulent or how
the flow becomes irregular, not straight streamline
laminar flow near an near 90 degree bend and
how it stabilizes at a certain length from
the bend.
So, we are going to concentrate today, in
on flow in microchannels which are generally
regarded as straight laminar flow. But in
straight channels, the flow remains laminar
below a Reynolds number of 2300, though we
start to see disturbances with various streamlines
that may appear even lower Reynolds number
and there is obviously, a range a a Reynolds
number beyond 2300 for which we may expect
through the turbulent flow.
Now, as I was talking as I was telling you
before this straight streamline or straight
laminar flow changes when fluid flows through
curves, bends or around obstacles. Now, what
happens at at the bends is that the centrifugal
forces at bends pushes the fluid away from
the bend. So, there will be alternatively
positions of low pressure and high pressure
at at different at at different positions
at the bend and this would this would give
rise to mixing near the bends. So, at the
wall the fluid is forced either upwards or
downwards producing a systematic, symmetric,
double vortex which fills the entire channel.
Now, this vortex will soon dye down depending
on the viscosity of the fluid and the dimension
of the channel and may give rise to straight
streamline flow once again.
So, in order to quantify the flow regime in
curved channel elements, we often bring in
bring in a number, which is known as the Dean
number and the Dean number is Reynolds number
multiplied by D by R c to the power half.
So, this D is the diameter or the diameter
of this channel and R c is the radius of curvature
is the radius of curvature for the bend. Now,
this flow regime in curved elements are often
called dean flow. And it is the viscous flow
viscous wall friction which acts against the
centrifugal forces present in those bends
and they will bring order to the system. So,
depending on the value of the Dean number,
we can classify the the flow at curved bends
under several different distinct regimes and
the flow pattern in each of these systems
have been has been probed.
And depending on the values of Dean number,
we could see rapid mixing between two liquids
which are flowing side by side and suddenly
come across a T junction or an 90 degree bend
and I will give you examples of that in my
subsequent slides.
So, now we will let us say we moved to a situation
in which there is a 90 degree bend and at
this 90 degree bend, what we see is that the
straight streamline of of the flows before
the bend suddenly get mixed in this region
and then, afterwards once we go far from the
bend then the flow becomes stabilized again.
So, here is a the case where viscous forces
are important and in this case the centrifugal
forces would try to push the liquid towards
this side and viscous forces will oppose the
centrifugal force and will try to bring order
over here. So, if we we can see that, over
here there would be if we if we could measure
the pressure at every point, one can see that
the pressure from here would be definitely
have to be higher than the pressure at this
point so that the flow can take place.
And from here to this point, the pressure
may vary in an erratic fashion depending on
whether we are measuring the pressure over
here or we are measuring the pressure at this
point. Beyond a certain length from the bend
the pressure again becomes similar to what
we have before the bend. So, these two pressure
pressure pressure differences will behave
in a similar fashion and the pressure gradient
one can expect from this point to this point
and beyond this region will will will be straight
line will be linear in nature. So, what we
have shown here is the plot of pressure against
channel length and if we think of this as
the 90 degree bend, what we have shown over
here, then the pressure will linearly fall
between the this point and the beginning of
the bend and beyond this line the pressure
will again fall linearly.
But if we if we think conceptually what is
happening at the bend, at this point the pressure
is going to be more and at this point the
pressure is going to be less. So, across this,
there is going to be a pressure jump near
the near the wall where the liquid is; where
the fluid is coming hitting the wall changing
it is direction and moving in moving in another
direction and over here where a low pressure
region is going to form.
So, what we see here is the pressure jump
near the outer wall and a pressure a reduced
pressure near the inner wall. So, this kind
of pressure behavior is quite common even
for laminar flow in a 90 degree bend. So,
this concept that, change in flow direction
is causing mix is causing the straight streamlines
of the flow to deviate from the straight path
can be utilized in a in a number of devices
and one such device is a mixer utilizing a
90 degree bend.
Now, what we can we can summarize? Our observation
is that the investigated flow regimes are
laminar with formation of vortex and we may
not see any onset of turbulence in the bends.
And at the inlet channel we have an uniform
pressure distribution in the cross section
and due to the curvature of the bend the flow
is altered in a new direction and as I say
that the outer side of the bend the pressure
is increased compared to the inner side. And
this difference in pressure at the curved
bend can create mixing due to the presence
of the 90 degree bend and which has further
uses usage in micro flow in microchannels.
It has been shown that approximately 100 micron
behind the bend, uniform pressure gradient
pressure re re reestablishes in the cross
section. So, in the at this point the vortices
are already dampened by the presence of viscous
forces and the straight laminar flow reappears
and will continue till it encounters another
bend in the path. So, the pressure loss in
the bend results in vortex formation and in
is a basis for further calculation in mixing
theory. So, we may get more mixing, but definitely
at the cost at the cost of increased pressure
gradient in the path of the flow. Now, we
we we can also look at the fluid dynamics
in T junctions by we will will come to that
later on.
Let us just think of how 90 degree bend works
and this is some experiment which has been
done with two different fluids. So, I have
fluid 1 which is slowing and in and fluid
2 and the volumetric flow of these two are
the same. So, the fluids are clearly separated
and then it has a 90 degree bend in its path.
So, what would happen is that the pressure
gradient up to this point will be linear and
over here, there is going to be a high pressure
zone and a low pressure zone that would cause
the fluid to move from one side to the other.
And therefore, the green fluid and the red
fluid will mix and they will mix further and
over here at the exit from the 90 degree bend,
we are going to get a mixing bend which will
propagate as we move along this. And very
good mixing can be obtained utilizing a 90
degree bend in a microchannel and that is
one of the principle ways to mix two fluids
at the microscale. Because we we know that
in a in a in a very how to mix two fluids
in the bulk, but how do we mix two fluids
which are in in at microscale and when they
are flowing side by side. So, when they are
flowing side by side in a microchannel, the
only way these two fluids can be mixed is
by diffusion which we know is inherently a
very slow process.
So, in order to speed up the diffusion process,
in order to speed up the the mixing in between
the two purposefully a bend is placed in it
is path so that we we get very good mixing.
Now, this has been used, this has been studied
for the mixing of different protein solutions.
And in T bends this mixing pattern, the study
of this mixing pattern is very interesting
results in term; and the results have been
correlated in terms of mixing parameters and
in terms of the Dean number of the flow. So,
I will show you a simple experimental setup
in which two protein solutions are flowing
side by side and suddenly, when it comes to
a T bend the two protein solutions which were
flowing side by side suddenly starts mixing
and it may so, happen that they will they
will mix it may so happen that, they they
they are going to overlap and it it it has
also been shown that for very large values
of Dean number, the flow may go from one side
and the other flow will go to the other side.
So that the flow reversal that is the fluid
which was flowing from the left it is going
to move to the right and the other fluid is
going to move to the left.
So, there are different kinds of mixing can
which are possible, which can be seen in many
such situations. So, the fluid dynamics in
T junctions with symmetric inlet conditions
they are like this. So, we have a T junction
in which we have liquid 1 which is flowing
in this direction and liquid 2 which is flowing
in the other direction, if the flow is very
small then, we will still have 1 that is moving
from from from right to left and the 2 which
will move in the other direction with very
little mixing in between the two. And this
happens, the classification of the of the
flow patterns, the flow; these are 1 is to
1 mixture of two fluids. And the cfd simulations
and visualization visualizations simply tells
us that for Reynolds number less than 10 and
Dean number less than 10 the the the where
Dean number is defined as Reynolds times D
by R c to the power half.
So, this simply tells us that when when both
are of the same order then D is roughly equal
to R c and if they this condition is satisfied
then we have straight streamline flow even
after even after the after the mixing. So,
the diffusion dominates mass transfer in this
case. So, this is a diffusion only process
and you get very little physical mixing in
between the two.
Now, when we move to a situation, where the
Dean number is; where the Reynolds number
is greater than 10, but it is less than roughly
about 150 or so, but the Dean number is greater
than 10 then we get Symmetric Vortex. Symmetric
Vortex are formed in what what you would see
is that, if this is the boundary 1 and 2 and
then I have the T junction then you are going
to get some sort of mixing in between the
two and part of 1 may go in in in this direction,
a small amount of 2 may come in to the other
other direction. So, the straight streamline
flow which was which was present before, straight
streamline flow gets disturbed and this disturbance
is due to the centrifugal force. As again
as I said Dean number is defined as Reynolds
into D by R c. So, for sharp bends what you
would get is the centrifugal force will will
will start to cause mixing the the this some
sort of mixing at the mixing channel.
Now, the situation becomes even more interesting
when the Reynolds number is greater than 150
and less than about 200, 250 or so, then,
this region this is called the Engulfment
flow. And in the Engulfment flow what you
would you would see is that, the symmetry
totally breaks up and the part of the fluid
is going to move in the opposite direction
and this 1 2 is going to come in the reverse
direction. So, the fluid swaps to the opposite
side and this this leads to short diffusion
length and very high mixing quality. So, the
mixing increases in such systems. For two;
Reynolds number greater than 250, but less
than 400 then the situation becomes even more
chaotic and it the two will not come to this
side on a on a continuous basis or the liquid
one will not move into this direction. Rather
we have periodic flows periodic pulsations
periodic pulsations of liquid of liquid column
from one side to the other.
So, this pulsating flow has very high, very
good quality mixing. And if the Reynolds number
is greater than 500 then, what we have is
a complete mixing of these two liquids. So,
for Reynolds number greater than 500 it becomes
chaotic pulsating flow, the vortex breaks
down and you get the the single flushes of
fluid show ups to the other side. So, the
liquid 2 will move to this side, liquid 1
will to move to the other side and this takes
place at such a high pulsation that the mixing
quality may actually decrease in mixing quality
may actually decrease.
So, depending on the value of Reynolds number
and depending on the value of the Dean number
one would be able to classify this type of
flows into different specific domains. Generally
with increase in Reynolds number the mixing
quality increases, but that increase does
not continue indefinitely, at some point of
time the Engulfment of one flow to the other
is is has a periodic nature in it and as a
result the mixing quality may decrease. But
we have to keep in mind is that these are
cfd simulations, these are highly specific
to the geometry of the system, as well as
nature of the fluid in question.
So, the limits of Reynolds number and Dean
number that I have that I have proposed so
far is system specific. And one has to be
careful in conducting these numerical studies
before one can design an actual mixer based
on these phenomena, where centrifigal centrifugal
forces help in the mixing and the viscous
forces will try to reduce any or to try to
any suppress in a vortex formation. So, these
are interesting areas of research, where one
can think of use this simple concepts to the
design of micromixers.
Next, we we are going to as I as I as I was
mentioning about the the different ranges
of Reynolds number. So, for example, Reynolds
number higher than 1000 are usually avoided
avoided in T shaped micromixers as it it leads
to very high pressure losses and that kind
of pressure loss cannot be sustained in in
most of the microfluidic devices. And mixing
quality as I was referring to is the is the
standardized concentration field and can can
often be used to characterize the state of
mixing.
Now, we are going to come into another subtopic,
which is my heat transfer and micro heat exchangers
so this is a very interesting application
of microscale transport processes. Now, we
all know that the Nusselt number let us say,
for flow through a tube, there are two distinct
conditions, one is a constant heat flux and
the other is a constant wall temperature.
Now, in all cases after the flow becomes fully
developed, thermally fully developed the Nusselt
number defined as h d by K, where d is the
length scale, by K is a constant. The value
of which is a 3.66 and 4 point something and
depending on whether you have a uniform heat
flux condition or uniform wall temperature
condition. Now, as we know that in microchannels,
the value of d is quite small. So, if the
value of d is small then all and if Nusselt
number is constant then the value of H has
to increase. So, this is the reason, that
in microchannels the value of heat transfer
coefficient convective heat transfer coefficient
can be very high. This this fact or this phenomena
of high convective heat transfer coefficient
has have been utilized in devising a number
of number of equipments devices. So, to extract
or to dissipate large quantities of it for
very small area,
The examples that we can think of are for
the cooling of of microchips where a small
amount of heat maybe generated, but the area
where it is generated is very small. So, the
heat flux that needs to be removed can be
very large. So, a small amount of heat generation
can lead to a situation where a large heat
flux has to be removed in order to maintain
the integrity of the device, integrity of
the Ic chip. And for some such situations
microscale heat exchangers based on the or
developed on the principle of high heat transfer
coefficient can be a very effective tool to
dissipate the large heat flux produced in
some in such systems. So, in this in this
part of in this subtopic we are going to look
into the heat transfer in and micro heat exchangers,
how do they perform? What are the governing
equations for such cases and so on.
So, we we we look into, the first the fundamental
heat transfer fundamentals the energy balances
which we are very much familiar with. So,
for any control volume the rate of energy
in minus rate of energy out and plus minus
any kind of loss generation etcetera, all
are rates it is going to be equal to zero.
So, if we if we if we put into the first law
of thermodynamics, then we can add we need
to add a dissipation function phi for close
systems and for open systems this the; these
are the two equations for close systems and
for open systems with phi denoting the dissipation
term. So, the dissipation phi takes into account
the energy conservation from one form to another
form accompanied by natural losses. For example,
if we have flow in a microchannel with a bend,
then the dissipation function over there essentially
tells us it is is related to the pressure
losses in such a system. And the frictional
losses, which which is going to change the
internal energy content of the fluid. So,
these are dissipation functions which could
also be viscous losses need to be added or
need to be included in the energy equation
so as to describe the energy content of the
inflow; of the inlet fluid and the outlet
fluid. Which are essentially all the fundamental
fundamental relations, thermodynamic relations
for energy balance.
So, let us see, we have a process device,
where the mass flow rate is m, the heat flux
q is over the boundary. So, if it is if it
is a tube then, we can think of this q as
the heat flux that is being added through
the walls of the tube and the technical work
which is W t or the mechanical power which
is denoted as the p, and there may be a chemical
reaction and therefore, we can write the energy
equation as given. So, if you see here the
the the Q dot is the amount of heat, the amount
of heat that is being the change in that m;
change in the energy and this change in the
energy has several components. So, if you
see the first term on the right hand side.
So, it is the net energy flowing to the control
volume as a result of flow, as a result of
convective flow and as a could be also as
a result of conduction across the system boundaries.
If you think of the second term, second bracketed
term on the right hand side it refers to pressure.
So, m dot times delta p that that tells you
the contribution of pressure to the overall
overall overall measure of energy of a system,
the third term on the right hand side tells
us about the hydrodynamics hydrodynamics,
tells us about how what is the what is the
head? The hydrodynamic head which is which
is being which is suppose the fluid is moving
from low to high then what is it is going
to be, it is potential energy content. The
fourth term on the right hand side refers
to the kinetic energy, the change in kinetic
energy of the flow in alpha 1 and alpha 2
are the correction factors. So, these are
essentially the kinetic energy terms multiplied
by the mass flow rate and it tells us the
change in kinetic energy between location
one and location two.
The the fourth term E dot q is the energy
produced or consumed, the energy produced
can be a result of a reaction well so a chemical
reaction so it could be an exothermic or an
endothermic reaction and E dot q simply takes
into account the amount of energy produced
or absorbed in such a system. And if you look
at the last term it is a technical work, it
is the work that is being added to the system
by a pump or extracted from the system by
a turbine. So, this equation then gives us
the complete description of the net energy
change, the special energy change for a flowing
fluid, considering the internal energy, the
kinetic energy, the potential energy, the
work done against surface forces such as pressure
the heat generated due to let us say reactions
and the any amount of work that is being done
on the system, for example a pump or by the
system, for example, by a turbine.
So, such energy equation would be equally
valid for microscale processes as well with
certain additional terms which we will see
later on. So, the energy of the fluid which
which is flows in a straight line, without
chemical reactions straight channels without
chemical reactions and technical work is simply
the sum of the internal energy, the kinetic
energy and the gravitational energy.
So, we can add Fourier's law to it, for conductive
heat transfer perpendicular to the channel
axis and one can write the energy equation
in the following form. Which is simply going
to be the time rate of change of internal
and kinetic energy would be equal to the spatial
change of, this W is the mass flow rate internal
plus kinetic energy then there would be a
gravitational force component minus del del
z of pressure times W minus del del z of k
times del t del x. So, this refers to the
conductive heat which is being added to the
system minus del del z of tau times w. So,
tau times W it it it gives us the dissipation
plus W dot g so any amount of work done. So,
this is the gravitational term, this is due
to pressure, this is due to conduction, this
is due to dissipation and this is the the
amount of work that is being done by the system
or on the system.
So, this can also be thought of it is, as
if it is the Bernoulli equation for the energy
balance in channel flow and can be can simplified
to suit each process. Now, if we if we try
to get a more meaningful equation to this,
which is more common to us, we can write it
as rho C p d T by. So, these are substantial
derivatives should be equal to epsilon plus
divergence of k times gradient of T. So, this
is the conduction and in here since, this
is a substantial derivative, we have the we
have the time derivative since, this is the
substantial derivative we have the time derivative
as rho C p del temperature over del time plus
we have v x times del temperature over del
x plus v y times del temperature over del
y and so on. So, this is the time dependent
term and this is the convection so, this refers
to the convection and this refers to conduction
in a microchannel.
So, solving the; this equation essentially
gives us the temperature distribution, if
you could solve this problem then it gives
us the temperature distribution for any any
real process.
Now, heat conduction in small system so, these
are these are common for large systems as
well as for small systems, but in small systems
there would be a complicating factor and the
complicating factor is something that we have
needs, we need to be very careful about.
For example, in Fourier's law we all know
about Fourier's law we simply says that the
heat flux is equal to minus k times delta
T by delta x. So, this Fourier’s law assumes,
that the k in in this case it assumes, that
the k the thermal conductivity is independent
of position it is same as in in all directions.
So, there is no no no question of writing
k x k y k z, because all of them are equal
and equal to one value of thermal conductivity.
But heat, so, this is this is true for a bulk
system, but when we think of a microsystem,
the microsystem is the thermal conductivity
like many other physical property is influenced
by the microstructure of the material.
So, you have grain boundaries and crystal
lattices which form additional resistances
to heat transfer. And the solution of the
three-dimensional heat conduction is often
possible with numerical method. So, what you
need to do then is, the right form of the
right form of Fourier’s law would be the
would be the tensor nature, where the k x
k y k z etcetera are will be allowed to vary
and they are going to be a functions of position.
And the solution of this three-dimensional
heat conduction equation is often possible
only with numerical methods.
So, Fourier’s equation of heat transfer
has to be expanded to the tensor notation
with the heat conductivity tensor. But, there
is one more thing that one would; I would
like to mention is that the miniaturization
will not influence temperature development
and the heat flux of a semi infinite body.
Now, what is what is the common form of semi
infinite body that we know of like the transient
temperature development in a semi infinite
body is in one in one dimensional form is
delta temperature by delta time is the alpha
times del 2 T by del x square.
So, this equation needs to be solved, where
the temperature is a function of x as well
as a function of time. So, let us say we have
a solid object where this is the x direction
and and and then what we need to have is then;
this is the in constant initial temperature
T 0 at time t equal to 0 for all values of
x. So, t 0 is is not a function of x and it
is at time t equal to 0.
Now, when we allow this solid block of; solid
block to come in contact with with another
with another fluid whose temperature is lower
than T 0 then with time what you would see
is that, the temperature will fall, this is
the temperature of the wall and there would
be a thermal boundary layer, a thin thermal
boundary layer and then the temperature will
be equal to T W and this growth of this thermal
growth of this equation the temperature variation
can be can be expressed by this equation.
And there are solutions for solutions for
semi-infinite semi-infinite of semi-infinite
of body and which describes the temperature
development in a in a in a solid body. So,
with constant wall temperature T W becomes
constant and the dimensionless temperature
distribution is govern by an error function,
which is an error function in t and x. So,
this T which is a function of x and time can
be denote can be expressed by an error function.
But, what is important here to know is that
the same equation same equation can be used
for a semi infinite body and for small bodies.
There are certain numbers which we need to;
for small bodies which are small values of
Biot number and Biot number is defined as
h D by k, where this k is the k of this solid,
unlike Nusselt number which is also h D by
k, but this k is for the liquid. So, for a
small value of Biot number let us say, we
are talking about a sphere now in a sphere
the the the diameter is small. Now, if the
diameter of the sphere is small or if the
value of k is large then, what one can think
of is that the temperature inside the sphere
will remain constant, will remain invariant
with time and this is the temperature T infinity,
where as this is the temperature of the solid.
Now, one one can see that for small value
of Biot number that is small d or large case
or small value of h, this T s is going to
be a function of time, but it T s is not going
to be a function of of let us say r in this
case. So, at one instant of time this is going
to be the temperature profile, at the next
instance the temperature will wil will reduce
to another value, but still inside the solid
the temperature will not vary with r and that
is only possible if your Biot number value
is small. And physically it tells us that
if the thermal conductivity of the solid is
large if the size of the system is small or
if you are dealing with a small value of heat
transfer coefficient then the small value
of Biot number indicate that I can assume
space wise Isothermality in any solid.
And once I assume space wise Isothermality
then it it it this modeling that system becomes
quite easy and that is known as the lumped
capacitance model. So, the lumped capacitance
model or LC model in short allows us to get
analytical solution of the temperature profile
inside a solid object. And this is extremely
useful, this is very useful to obtain to obtain
the behavior of of a of a fluid, behavior
of a behavior of a solid when it is quenched.
So, let us say, we have a very small steel
bowl that hot steel bowl which is put into
a liquid and how would the temperature of
the steel bowl change with time that is that
is extremely important from an engineering
perspective.
Similarly, for micro system how does the temperature
inside the micro system vary with temperature?
If you could assume that inside the small
micro systems since the length scale is small
there is no variation of temperature then,
we would be probably, we would be able to
obtain an analytic solution of temperature
variation with time inside such a system.
So, the two numbers of relevance here are
Fourier number, which is alpha t by x square
where alpha is a thermal diffusivity t is
the time, x is the length scale and Biot number
as I have already explained. So, Fourier number
tells us how fast a temperature front will
penetrate into a solid, where as in Biot number
tell us how how the temperature of a solid
is changing. So, these two are important parameters,
the time dependent important parameters in
the second Fourier law.
Now, this these are something, sorry these
are, these I have already described partially.
Is that the total heat transfer in micro devices
can consist of heat conduction through the
walls and convective heat transfer from the
wall into the fluid in the microchannel. So,
for example, if you are if you are if you
are making a micro device and while you are
making a micro device if you consider only
the convective heat losses from let us say
it is it is a tube and it is micro scale in
size. So, the diameter of the tube is 100
micron and it is length is 2 centimeters.
So, now, when you are; it is it is made of
silicon. Now, when you are; when it is let
us say v grouped micro micro v grouped micro
heat pipe. Now, there are situations one has
to consider not only convective heat losses
from the outer wall of the microchannel, but
the heat which is which gets transported through
the wall. So, in micro devices, unlike in
macro devices the conduction heat transfer
through the body of the device itself needs
to be taken into account to calculate the
total amount of heat loss or total amount
of heat transfer from such a system.
Secondly, this we as I said for straight laminar
flow, the dimensionless heat transfer coefficient
is a constant and it is 4.3 or 3.66 for a
wide gap or a narrow slit the Nusselt number
can vary for double sided heat transfer. And
for smaller channels the heat transfer coefficient
increases due to the constant Nusselt number
as I I I as I have already described. And
at the entrance of a channel at channel junctions
expansions etcetera the disturbed flow enhances
the radial transport.
So, so far we have seen that the at the entrance
of the channel or at a 90 degree bend how
the fluid dynamics gets altered resulting
in larger pressure drop. Here, what we are
saying is that, not only the fluid dynamics
get us altered, the the the altered fluid
dynamics helps in transporting more heat due
to the presence of the 90 degree bend. So,
so an obstruction in the path or alteration
of the fluid path not only increases pressure
drop it also increases heat transfer in microchannels
in in microchannels. So, you we get more pressure
loss, but at the same time we have increased
heat or mass transfer and it is just a balance
of which one we would want for our system.
Now, one more thing is that, this is fine.
But, when we talk about Nusselt number, the
constant value of Nusselt number and the smaller
length scale giving rise to high value of
heat transfer coefficient that is that is
fine, but is it really beneficial is the high
heat transfer rate helping us in obtaining
in obtaining large heat removal, the answer
is yes and no. The high heat transfer coefficient
definitely helps us to extract more heat,
but the area in consideration is quite small
since, we are talking about micro devices.
So, we need to somehow increase the area of
contact to utilize the full potential of a
high heat transfer coefficient. Now, how that
can be done. So, there has to be numerous
let us a numerous tubes, micro tubes which
needs to be employed simultaneously to extract
more heat.
Now, whenever that happens it gives it so,
it is a fabrication problem, how do we fabricate
a large number of such channels which will
utilize the high heat transfer coefficient,
high convective heat transfer coefficient
to extract more heat. And it is not only a
fabrication issue it is also an issue of how
do we feed the coolant liquid equally into
all such channels or tubes. So, proper feeding,
uniform feeding of liquid through all the
fabricated channels, at the same time with
minimum pressure drop that is the major challenge
being faced by engineers who are who are who
are trying to build smaller and smaller devices
to take the help of the higher convective
heat transfer coefficient. We also know that
when the flow starts taking place it is not
fully developed, it is not hydro dynamically
fully developed or thermally fully developed.
All the relations or correlations that we
have so far after fully developed flow and
for a micro device it may so happen that before
the flow gets fully developed we reach end
of the channel. Now, if that happens then
what is the equation, what is the relation
to be used for calculating the Nusselt number
which is essentially a design parameter. So,
the convention the normal convention is to
use the use the expression for Nusselt number
for fully developed flow and multiply it with
some factors in order to obtain the Nusselt
number for the for the zone in which the flow
is developing more thermally and Hydrodynamically.
So, for a large tube in a micro system it
is not a problem, because the the entrance
length part the portion where the flow becomes
fully developed is quite small compare to
the overall length of the channel. But in
the microchannel when a micro system the length
is so short that we are not, we may not provide
enough length for the flow to become fully
developed.
So, we have these equations which tells us
the Nusselt number, the Nusselt number which
is which is developing is the Nusselt number
which is which is straight channel flow and
there are certain correlations which are available.
So, this is the Prandtl number and this is
the dimensionless length and the equation
is valid for the entire channel length X star
and for Prandtl number greater than 0.1.
So, since so since turbulent flow is rarely
encountered in micro channels. So, this this
kind of a situation may not arise, but there
are some turbulent conditions at locations
special near the bends and so on. Now, comes
the question of when we start reducing the
size of the devices smaller and smaller and
and and letting it operate at very low pressures,
what happens is that we will probably reach
a situation where we have we have we have
struck the continuum limit.
Now, when we have continuum limit and as we
know that the continuum limit is generally
expressed in terms of Knudsen number which
is the ratio of the mean free path and the
characteristic length. So, if the value of
Knudsen number is less than 10 to the power
minus 2; that means, the mean free path is
small compared to the characteristic length
and for such cases the continuum and the thermodynamic
equilibrium assumptions are appropriate and
the flow situations can be described by conventional
no-slip boundary condition. If the Knudsen
number is in between the two that is 10 to
the power minus 2 to 10 to the power minus
1, then we come to the regime which is known
as the slip flow regime.
So, in the slip flow regime the Navier-Stokes
equation will still remain valid provided
we alter we modify our no slip condition with
a tangential slip-velocity and a temperature
jump incorporated into our boundary condition.
So, you may not be able to use that u and
v the velocity in the x and the y component
are both 0 at y equal to 0, where y equal
to 0 denotes the denotes the wall. But we
will have to provide a slip-velocity and there
are several models which take into account
the possibility of slip. Similar, to slip-velocity
we would also have to have to incorporate
a temperature jump across the interface.
So, previously for total, for continuum continuum
cases, the temperature of the solid and the
temperature of the liquid or the fluid adjacent
to it at the same at the interface, but if
if the flow is verified then we have a temperature
jump across the interface. So, the reduced
flow there since we have a slip velocity.
So, therefore, the flow resistance will be
rather less and the molecular motion etcetera,
there is going to be insufficient momentum
transfer between the wall and the bulk fluid.
So, in order to in order to characterize the
amount of transfer, amount of momentum transfer
and heat transfer there is a concept called
accommodation coefficient beta, which is defined
as the ratio of the average energy which actually
get us transferred between a surface and the
gas molecule, to the average energy which
would theoretically be transferred if the
impinging molecules reached complete equilibrium
with the surface. Now, what happens here is
that the molecule comes and hits the surface
and reflects back into the bulk. Now, if this
is a continuum case then this would come stay
here for some time or stay here for sufficient
amount of time to reach complete thermal equilibrium
and then get will get reflected back, but
the due to the rarefied condition it comes
and then immediately it goes back. So, it
cannot take enough the total amount of momentum,
total quantity of momentum or total quantity
of energy.
So, this beta essentially tells us about how
much, what fraction of energy or what fraction
of it it can take due to the rarefied condition.
So, for rarefied flow the boundary condition,
the the velocity is going to be a function
of zeta this which is called the slip length
and it can be the slip length can be calculated
from the accommodation coefficient and the
mean free path of the molecules.
So, the experimental data on the accommodation
coefficient beta it is defines the efficiency
of the momentum and energy transfer from molecules
to the wall and vice a versa. So, beta can
be calculated from kinetic theory and there
are enough large quantity of data on beta
which can be found in the literature and similar
to the jump the slip there is also temperature
jump across the wall.
So, that the temperature at x equal to 0 is
not t w, but g which is a temperature jump
coefficient like the coefficient zeta in the
in in in the hydrodynamic case times the gradient
at the wall. So, the presence the temperature
jump coefficient can be regarded as an additional
distance to the gap.
So, it may look something like this that at
the at the interface, we have the temperature
profile in the solid and the temperature profile
in the fluid. But, in the case of rarefied
condition, we have the temperature profile
in the solid and a temperature profile in
the liquid with a change in temperature on
the solid side of the interface and on the
liquid side of the interface. So, this is
the temperature jump that we are talking about.
So, we could think of a situation in which
I am adding an additional resistance over
here and this additional resistance is is
the is the is the temperature jump in rarefied
gases, for cases where Knudsen number is greater
than 0.01. So, this temperature jumps coefficient
g it can be regarded as an additional distance
to the curve.
So, for non circular sections, in order to
calculate these half of the hydraulic diameter
can be taken for r A. And since, we have more
mobility of the molecules in rarefied condition
the Nusselt number for constant wall temperature
is 5 percent higher than for the constant
heat flux. And numerical study with Monte-Carlo
method indicated that slip flow model correctly
represents convective heat transfer between
the continuum and the molecular flow. And
the influence of actual heat conduction must
be considered, but the viscous heat dissipation
and evaporative cooling and and expansion
cooling can be neglected for flows in microchannel.
So, in summary what we can say is that, heat
transfer in microchannel is essentially similar
with most of the equations of macroscale that
can be used except for certain cases where
for Fourier's law the thermal conductivity
k needs to be the three dimensional variation
of thermal conductivity has to be incorporated
and the tensor needs to be the three dimensional
variation of thermal conductivity has to be
incorporated and the tensor form of Fourier's
law needs to be used with numerical solution
only possibility is numerical solution. And
Secondly, during a rarefied condition, the
the the accommodation concept of accommodation
coefficient tells us the efficiency of momentum
transfer when a molecule hits a surface. And
the accommodation coefficient thermal accommodation
coefficient tells us that the the the impinging
molecule and the wall has not reached complete
thermal equilibrium and this can be modeled
by having the presence of a velocity at the
wall where it is a slip which is the function
of the slip length and the gradient in velocity.
And similarly, the temperature can be expressed
as a function of a temperature jump at the
interface which is as if we are adding a we
are adding thin layer at the interface. So,
these are some of the factors that need to
be considered for modeling or for using convective
flow both the hydrodynamics and heat transfer
in microchannels. Thank you.
