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So, welcome to the thirtieth lecture of cryogenic engineering under the NPTEL program. In the
earlier lecture, we talked about GM Cryocooler or differed Mc Mahon Cryocooler, we have seen
the schematic and the working of the GM Cryocooler which was invented by Gifford and McMahon,
in 1950. It is high let was it has a valve mechanism to generate the pressure pulse,
the relation between the pressure pulse and the expander displacer motion is vital and
this is what we saw in detail during the last lecture. The basic components of the GM Cryocooler
are compressor flex lines regenerator displacer valve mechanism etcetera. A GM system can
reach much lower temperatures as compared to a Stirling system.
Basically, the difference between the GM and the sterling is GM has a valve and trans at
very low frequency of around 1 to 2 hertz well Stirling operates without a wall and
normally it trans at very high frequency as compared to GM Cryocooler. Now, in GM Cryocooler
multistaging is done to reach very lower temperature and of the order of around of 4.2 k which
is helium boiling point and up to 10 Kelvin. Now, these are temperature which are normally
use for basically various scientific experiments for example, 10 k while 4.2 k is normally
use to Liquify or re-condense helium gas. A typical regenerator material in a single
stage GM Cryocooler which would possibly bring the temperature down to 30 Kelvin is stain
less steel mesh and if you want to come to by using a two stage Cryocooler to reach around
10 Kelvin there in the first stage we will have stainless steel mesh material and in
the second stage we will have around lead balls.
Similarly, if I want to reach using a two stage Cryocooler to 4.2 Kelvin temperature
the first stage may have the stainless steel mesh plus lead ball as originator material.
And in the second stage we may have lead balls plus erebrianethical holmium copper neodymium
etcetera the all spears made out of normally these materials at around 0.2 millimeter diameter.
Now, when I am telling these a very broad reginator configuration SS mesh plus lead
ball it may have 100 percent SS mesh and second stage may have lead balls plus erebrianethical,
but this is what we generally, would at the reginator material for a four point two Kelvin
Cryocooler. Commercially available Cryocoolers have normally
rotary wall to control regulate the flow of the work in fluid the work in fluid normally
will be they helium gas this is what we saw in detail during in the last lecture.
Now, this lecture we will now completed to dedicated to pulse tube Cryocooler which is
very important to Cryocooler right now, amongs all the Cryocooler that are use and you will
understand why it is so. Let us see the working of a pulse tube Cryocooler let us see how
the pulse tube Cryocooler are classified as and normally I will refer pulse to be as now
PT Cryocooler or PT PTC sometimes. So, pulse to classification and let us compare this
Stirling, a Gifford McMahon and pulse tube Cryocooler and let see different applications.
And let us try to understand the pulse to Cryocooler using a freezer analysis, which
you very important concept to understand why it different phase shift mechanism are employed
in pulse tube Cryocooler. And will have the next lecture also, dedicated
to pulse tube Cryocooler where we can see more hardware that has been generated here
in our laboratory.
So, in the earlier lecture we have seen a regenerative Cryocooler and of different types
let us Stirling type and GM type. So, this is Stirling type this is just to recap to
understand what we have done in the last lectures. So, we have got Stirling type machines Stirling
Cryocooler where this now wall in between the compressor and the expander and therefore,
the displacer moves at a same frequency as that are compressor piston. And there are
the fix phase difference between the motion of the compressor piston and the motion of
the displacer while, in a GM Cryocooler you have got a pressures of wall mechanism between
the compressor piston and expander. And there is the phase difference not between the displacer
and the piston, but between the wall mechanism and the displacer which is moving at low frequency
in a GM Cryocooler. These systems have a mechanical expander displacer
to displace the working gas in both the systems we have got one component moving in a expander
we have got a displacer moving here, at higher frequency we are going to displacer moving
here at around low frequency of around 1 to 2 hertz in a GM Cryocooler. The displacers
are either free moving or driven by an external mechanism. So, in order to drive this piston
or displacer we may have to have a different driving mechanism for driving this displacer
up and down or sometimes it could be free displacer, which moves because of the pressure
drop across the displacer. There could be small pressure drop across the displacer the
gas across the displacer, because of which the piston moves up and down the displacer
the normally very light weight moving component.
The cold and displacer pose few problems as given below sometimes, the in GM Cryocooler
especially as you can see there is a seal which works at a low temperature, because
this said is a low temperature. And therefore, will have a rubbing seal at this point and
these seals which have rubbing in this can as can pose some problem from working point
of view. The rubbing seal on displacer is difficult to maintain and therefore, when
I was to really to take care of theses seal and the seal definition is very important
how this seal is composed of. So, at low temperature there could be shrinkages
across and seal, which is perfect and room temperature may not work very well at low
temperature and sometime this is a trouble for lot of displacers moving in this cylinder.
The motion of displacer induces unnecessary vibration at the cold end so, whatever, object
I want to cool the whatever the load comes over here in case of Stirling at this point
in case of GM and at this point. Because the mechanical component moves up and down this
creates lot of vibration for the component to be cooled and sometimes this component
may not function whatever, scientific action you want take over here may get hamper because
the vibrations. That means, this experimental will not tolerant the kind of vibration generated
by this moving component which is displacer in this case.
To overcome this problem pulse tube cooler comes to rescue and therefore, what happens
in pulse tube cooler? Pulse tube cooler overcomes these problems as pulse tube cooler does not
have a mechanical displacer their nothing moving at in the in the cylinder like here
you got a displacer the nothing is moving in case pulse tube. And therefore, one complete
mechanical component absent in case of pulse tube Cryocooler and this is very difference
between Stirling GM and the pulse tube Cryocooler. The moment I do not have something moving
over here, I do not have vibration at this point at the same time I do not have any need
to have drive mechanism. So, as in to move this mechanical component this is a very important
difference between the pulse tube cooler and Stirling and GM coolers.
So, now what happens? Considers schematic of a Stirling system as shown in this figure
this is a normal Stirling type Cryocooler and if I want to convert this to pulse to
Cryocooler what will I do I will just replace as you just saw I will just replace this displacer
by a gas column cylinder or tube filled with gas column. So, in a pulse tube Cryocooler
the mechanical displacer is removed and then oscillating gas flow in the thin walled tube
produces cooling. So, what is going to happen is I will just have a empty tube filled up
with the gas, which is then subjected to oscillating pressurization and depressurization and this
generates cooling and this is simple definition of a pulse to Cryocooler.
So, in effect we have got regenerative Cryocooler, when the pulses oscillating pulses enter a
tube which is subjected to pressurization and depressurization the cold generator, there
is something called PSM which is what I will talk about this gas tube is called pulse tube
and this phenomenon is called as pulse tube action.
The components of a pulse tube system therefore, are compressor heat exchanger regenerator,
pulse tube and phase shift mechanism. So, earlier we had a displacer and as I said every
time the displacer and the compressor have got some relationship in there motion. Now,
we do not have any moving component as such here mechanical moving component here and
therefore, whatever gases moving at this point I have to see that the some kind of a the
gas moves in a cordons with what I want which would generate cooling. So, would ensure that
the oscillation setup in this tube the move in a fix manner the move in a manner, which
generates cooling and this manner in which they should oscillate in this tube is basically
due to the pay shift mechanism PSM that is what is return over here.
We will talk about this PSM in detail in the this lecture and also in the next lecture.
So, details and the requirement of the phase shift mechanism which is very important this
is a very important and this is present only pulse tube cooler not in the Stirling in the
GM type machines. So, therefore, these PSM has to be understood and this will be explained
at the later part of the lecture or in the next lecture also. So, what happens how do
we get cooling? So, the piston comes down and the pressurization happens the piston
goes back and the depressurization happens during pressurization now. The high pressure
gas flows across the regenerator and in to the pulse tube as shown here, by this arrow
the gas comes down the after cooler takes the compression the gas enters the regenerator
and then the gas enters the pulse tube.
The gas in the pulse tube compresses the gas present at the other ends. So, everything
was field up earlier at an average pressure or a charging pressure as soon as the gas
gets charge of as soon as the gas gets compress the depending on the amplitude of the pressurization
the gas enters and these gas will compress the gas present already in the tube which
initially, is present at the charging pressure. So, gas in the pulse tube compresses the gas
present at the other end let us called as the top end let us call as the bottom end
of the pulse tube. Now, these compression this compression of
a earlier gas by the gas which is coming, because of pressurization this compression
result in a rise of temperature at the top end. Infect the temperature rises at every
point, but the temperature rise at the hot end at the top end will be much more as compare
to the gas in this tube below this top end. So, during pressurization the temperature
at this point increases, what happens after that the piston goes back and the depressurization
happens. So, during depressurization the gas expands in the pulse tube resulting in lowering
of temperature across the pulse tube. So, as soon as the piston goes back the depressurization
happens and therefore, the gas in the pulse tube expands, which will result in the lowering
of temperature. So, temperature at the top end also reduces temperature at the bottom
also reduces, but previously the top end was at higher temperature then the temperature
at the bottom end. So, what does happen? First during pressurization we got increase temperature
at top end as compare to the cold end and during depressurization the gas expands and
temperature lows down. As a result of which a temperature gradient will gets setup over
here across the length the temperature start decreasing down. So, gas pressurization happen
temperature at the top end increases depressurization happen temperature decreases as a result a
temperature gradient is setup across the length of the pulse tube.
Now, the cold gas, this temperature will be less than the temperature at the top end temperature
of the bottom end is going to be less than the temperature of the top end and slowly
and steadily the temperature at the bottom end will start reducing. While depressurization
when the gas goes back this gas gives the heat to the regenerator matrix and during
pressurization this gas will get precooled, because of the heat transfer from the regenerator
matrix to the pulse tubes. The cold gas during depressurization transfer cold to the regenerator
matrix and this cold is again given which is use this cold is again use to precool the
incoming gas, during pressurization. The hot end temperature at the top end is
maintained at a ambient temperature now, has I said earlier the temperature increases,
but what I am going to do is the temperature at this top end is going to the maintain at
ambient temperature by running water over here. As the result of which this will be
ambient temperature and the temperature of the bottom end we slowly start coming down
below the ambient and therefore, you will get cooling effect as soon as the temperature
comes below ambient you will get you will generate cooling effect, which is transferred
by the gas we need goes back during depressurization. So, this is how the pulse tube cooler works.
The pulse tube cooler will compress the gas and setup a temperature gradient across the
length pulse tube during the depressurization the gas goes back the hot end of. The pulse
tube is always maintained at ambient temperature, there by the lower end of the pulse tube temperature
starts going down and how much it should go down will depend on what is the capacity of
regenerator matrix material to store the heat, this is true for Stirling cooler this is true
for even GM cooler. So, the basically this are all regenerator Cryocooler, but the major
difference is now, there is no displacer and therefore, the oscillating gas flow sets of
a temperature gradient across the length of the pulse tube.
This is a very simple way of you know understanding the pulse tube action the cooling effect produced
at the bottom end also called as cold end is lifted by using a heat exchanger. So, the
heat exchanger here with basically, exchanges the cold generated because of pulse tube action.
So, normally a pulse tube cooler will be identify the hot end of the pulse tube which is maintained
at the room temperature and cold end the pulse tube were the cooling effect gets generated,
this is very simple terminology the hot end and pulse tube and the cold end of the pulse
tube. Also, the after cooler temperature at the top is maintained at ambient.
So, we have got a after cooler heat exchanger here which maintains the temperature gas to
be ambient temperature. So, when the gas gets compressed the gas enters the reginator at
ambient temperature, because of the after cooling.
The gas movement in the pulse tube does not need any mechanical drive as we understood
that is only oscillating flow and it does not require any mechanical drive. Hence, the
vibration in the pulse tube Cryocooler are less as compared to Stirling and GM Cryocoolers
obvious there is no moving component and therefore, the vibrations are megnititude a megnititude
less then as compared to Stirling and GM Cryocooler. The schematic of the pulse tube Cryocooler
with three heat exchangers namely, after cooler which is called AC, cold end heat exchanger
which can be called as CHX and hot end heat exchanger HHX is as shown in the next slide.
So, now, I will be formatting the entire figure so, as we understand the pulse tube action
in better way vary we call AC, CHX and HHX will be shown on this figure.
So, if I want to show now the temperature variation across the length the pulse tube
Cryocooler it will be like this is a compressor there is the after cooler regenerator cold
and heat exchanger pulse tube and hot and heat exchanger and the PSM. So, this is pulse
tube Cryocooler shown in line and this is called as inline configuration the pulse tube,
were the gas travels in straight line and comes back in a straight line during pressurization
gas goes like this during depressurization gas comes back. Now, if I well to compute
the temperatures across from this point up to this point from after cooler up to the
end of the pulse tube up to the hot end across temperature across position graph would look
like this and this is my ambient temperature. So, during the after cooling the temperature
will be gas is going to be little above the ambient temperature, where Q is given to the
after cooler. In the regenerator at study state we have temperature gradient develop,
because of in coming because of the precooling the gas going back and the gas coming in and
the study state temperature distribution will get generated. In the CHX that is cold and
heat exchanger the temperature normally will be maintained constant because there is load
on a system that is constant load on a system Q c at temperature T c. And across the pulse
to also reliable temperature gradient generated and therefore, temperature gradient would
look like these in the hot and heat exchanger now, again the temperature would remain constant
normally. So, is very important diagram to understand
that the cold is getting generated at this point the lowest temperature is generated
in the cold and heat exchanger or the near the pulse tube end, which is closer to the
regenerator. So, here we generate cooling effect so, whatever, object I want to cool
should be kept attach to the cold and heat exchanger across which the cold is transferred.
Now, let us see the pulse tube Cryocooler in again a different perspective what are
the advantages and disadvantages what are the uses of pulse tube Cryocooler in general?
So, advantage of pulse tube cooler are obvious now, no moving part in the expander hence
less vibrations, which is very important advantage pulse tube cooler as compare to other Cryocoolers,
there is no sealing requirement the moment you do not have a mechanical drive there is
no sealing requirement at low temperature. So, therefore, problem of rubbing seals do
not arise high reliability the moment we do not have in a drive mechanism moment we do
not have any moving component it will high reliability.
So, pulse tube cooler does have high reliability as compared to other expander in which one
of the components is moving the disadvantages; however, are no reliability data due to less
history. So, there is no failure data over period of time and therefore, we do not have
any reliable data as such to come to conclusion that this pulse tube cooler will never failed.
And the same time you can understand the pulse tube cooler being a gas dominant phenomena
smallest angle here and there to the pulse tube cooler, because the gravity driven also.
Because it will some convective current the hot and on the top the cold and the bottom
we can have some gas heat transfer in the gas column itself and therefore, the pulse
tube cooler is subjected to some kind of orientation effect.
So, as soon as the pulse tube cooler is not vertical and is inclined at some angle the
cooling effect generated by the pulse tube cooler will be different so, pulse tube cooler
normally is operated in a vertical mood only. So, will have orientation effect, which is
also a negative characteristic of a pulse tube Cryocooler.
The uses of pulse tube cooler it is used to cool the infinite sensors for space application
re condensing liquid helium and liquid nitrogen which is very important function it can be
use for re-condensation. And therefore, it is normally club to the object to be cooled
where the helium boil of happens where the nitrogen boil of happens and the pulse tube
can be kept coupled to that particular equipment. And sometimes the pulse tube cooler can be
use for nitrogen liquefaction or even helium liquefaction depending on it will be single
stage or two stages for this respective operations. The recent developments one can obtain less
than 4 Kelvin temperature using a two stage pulse tube Cryocooler and miniaturization
is absolutely possible, if you go for Stirling type pulse to Cryocooler and therefore, pulse
tube cooler the very important candidate in Cryocoolers.
So, if I want to see all three together the Stirling the GM and the pulse tube cooler
they would look like these. So, this Stirling Cryocooler the GM type Cryocoolers these the
Stirling type pulse tube cooler. Now, in pulse tube cooler we will have Stirling type pulse
tube cooler where they will not be any wall and therefore, this is a very compact and
miniature unit Stirling type pulse tube cooler. And we can also have a GM type pulse tube
cooler the moment we have a got wall between the compress have a expander this will be
called as GM type pulse tube cooler. So, very important to understand Stirling cooler as
different GM cooler is a different Stirling type pulse tube cooler different and GM type
pulse tube cooler is different. So, one has to really understand whenever
we say particular Cryocooler what Cryocooler we have in talking about this to be fix on
a mind the moment I say GM cooler I got a wall between the compressor and expander,
moment as a pulse tube I do not have a displacer moment as a Stirling I do not have a wall
between the piston and the expander. So, this classification of Stirling type pulse tube
GM type pulse tube cooler has to be completely understood by all of you.
Now, let us come to very important classification under the pulse tube categories. So, pulse
tube Cryocoolers can be classified based on as we know Stirling type or gifford mcmahon
type. So, moment we got a wall Gifford McMahon type moment we do not have a wall is Stirling
type. Under both of this type now, we can have various other classification based on
the geometry of the pulse tube cooler and depending on the phase shift mechanism that
we use in the pulse tube Cryocooler. So, let us come to the geometry first now, based on
the geometry we have inline configuration we have got U type configuration we got a
coaxial configuration and we got a angular configuration will see each of this in detail
this will basically, refer to the position of a regenerator and pulse tube.
If the regenerator and the pulse tube are inline as I shown earlier in the long horizontal
pulse tube Cryocooler this will be called inline configuration if they are parallel
to each other this called U type configuration, if their having a same axis coaxial and angular
will see that. Other classification based on the kind of phase shift being used we have
not at come to understand why this phase shift mechanism is being use in pulse tube Cryocooler,
but what is use for the phase shift mechanism also will decide what kind of pulse tube Cryocooler
it is. So, we can have will understand this in detail later when we understand the phase
shift mechanism and that time I will show this classification to you again.
So, basic pulse tube Cryocooler or orifice pulse tube Cryocooler inertance type pulse
tube Cryocooler and double inlet valve pulse tube Cryocooler. So, will understand about
this later in the lecture and under Stirling Cryocooler now, we know the Stirling cooler
is high frequency as come to the GM cooler, but when I say high frequency how height is?
And therefore, based on the kind of frequency we use in a Stirling type pulse tube cooler
based on the frequency I can again identify the low frequency high frequency and very
high frequency.
Now, let us see this classification in little more detail. So, as you know that depending
on the usage of valve the pulse tube Cryocooler can either be Stirling type pulse tube cooler
or a GM type pulse tube cooler. The moment as a GM type pulse tube cooler I got a wall
over here Stirling systems are high frequency machines whereas, GM systems are low frequency
machine I hope this is now, entirely clear to all of you each of the system as know earlier
is at got own advantages and disadvantages.
Another classification of pulse tube cooler is based on the relative position of the regenerator
and the pulse tube as we had just seen. So, this is my inline configuration where the
pulse tube cooler and the regenerator are inline. Second classification is U type classification,
where the pulse tube cooler and the regenerator are parallel to each other and the U type
connection given for the gas flows the gas comes over here gas goes to these and comes
order. Let us the advantages the merits and demerits each of this type in the next slides
and then a coaxial Cryocooler coaxial pulse tube Cryocooler; that means, the pulse tube
Cryocooler and the regenerator have the same coaxial their basically coaxial they got a
same axis around, which they are basically assembled the gas come in the angular regenerator
from outside and enters in to the pulse tube like this.
So, you could a hot and heat exchanger here, while this are angular regenerator over here
this is what we call as coaxial, when the regenerator is in angular position and we
have got a annular pulse tube when the pulse tube now, is an annular position while the
regenerator is at the center. So, these are four different types inline U type coaxial
and annular now, let us see the advantages and disadvantages of each of this type.
Let us come top first the inline configuration the gas does not change the direction of the
flow hence, the pressure losses are minimum. We can understand that during pressurization
the gas will travel up to entire link and during depressurization the gas will go back.
So, basically during pressurization the gas comes trades there is no change of direction
and therefore, there is no pressure drop losses across this. So, thermodynamically sign, that
this will have minimum pressure drop and therefore, this pulse tube cooler will be most efficient.
So, what is the problem this will take a long length the space occupied this by this pulse
tube cooler in this length dimension will be pretty high as compared to other units.
Also, this arrangement delivers the best performance as compared to others this is what we just
saw thermodynamically, the cold end is at the center of the system this is an important,
if I want to cool something I have to access the middle portion of this pulse tube Cryocooler
which sometimes is not accessible. I have to enter from this side or this side, but
it is not at the sides it is at the center of this thing and therefore, whatever object
I want to cool I will not have a really good accessibility to it and therefore, sometime
this is may not be acceptable for user from uses point of view. The third disadvantage
of the system goes to say the system is less compact since it occupies huge space length
wise. All the thermodynamically it is advantages
is efficient you may have loose on the non-accessibility of the cold and heat exchanger and the huge
length wise dimension required in the in line pulse tube Cryocooler.
The next as you see is a U type Cryocooler now, suddenly you can see that the length
wise dimension has decrease, but what you can see now? The gas flow undergoes a 180
degree change in flow direction so, gas during pressurization will come to regenerator and
then take U turn and then enter the pulse tube cooler. So, it will take a 180 degree
change in the flow direction due to which the system exhibits pressure drop. The cold
end is now; however, exposed I can access the cold and from the bottom so whatever,
I want to cool can easily be attach to the cold and heat exchanger and this is very important
and it is easily accessible. The system becomes more compact now, and it
occupies less space as again is what it was doing earlier the performance is dependent
upon the sharpness of the bend. So, how much pressure drop could happened depend on how
sharp this spend is made basically is fabricated and therefore, if we made a very gradual bend
the pressure drop in the askets will be minimized. So, this is the U type pulse tube Cryocooler
where the regenerator and the pulse the cooler are parallel to each other.
Let us come to now coaxial and annular Cryocoolers and you can see that, the system exhibits
maximum pressure drop due to change in flow direction. So, here the system the gas will
come like that and it will have a sharp bend direction in both this cases and therefore,
the pressure drop in both the coaxial and the pulse tube Cryocooler will be very high.
In this case the gas will come from the center and then go to the pulse tube again having
a it has to turn in a sharp way basically is it not it, because there all assembled
together. So, in the both this cases the pressure drop
process could be tremendous the cold end is exposed and it is easily accept the best part
of about this two configuration is whatever, I want to cool is accessible from this bottom
end. And therefore, from users point of view this is the most important or wanted system
this a very compact system also, because there all you know assembled together is very compact
system the cold and heat exchanger availabilities accessibilities perfect in this case. So,
other than the thermodynamic component that is high pressure drop losses, we have got
the both the advantages system is very compact and system is the cold end is accessible for
the user. The system is more compact, but is a possibility of a heat transfer now the
only disadvantage also other than the pressure drop it as their coupled up together you can
have a heat transfer between the gas in the pulse tube and the gas in the regenerator.
In both this case we can make this system little bit in efficient there are ways to
overcome this; however, normally there can be heat transfer between the gas in the regenerator
and gas in the pulse tube and therefore, we can loose some cooling effect over here in
this particular configuration.
So, this is the four configurations which we just saw. Now, pulse tube Cryocoolers can
also be classified depending on the frequency under the Stirling cooler which I had seen.
So, depending upon the operating frequency the Stirling type pulse tube can be classified
as listed below, I can call this as low frequency if the frequency less than 30 hertz be clear
I am talking about Stirling type pulse tube cooler. Now, the GM type pulse tube cooler
will operate at 1 to 2 hertz only while Stirling type pulse tube cooler will be called a low
frequency a Stirling type Cryocooler, if is frequency less than 30 hertz high frequency
if it operate between 32 hertz 80 hertz and very high frequency when it is frequency is
going to be more than 80 hertz. So, we can call low frequency machine high frequency
or very high frequency depending on the frequency at with Stirling type pulse tube cooler operates.
So, after understanding this pulse tube cooler classification let us see now, the analysis
of the pulse tube Cryocooler. So, just saw that the pulse tube cooler we got a Stirling
type and Gifford McMahon type and we just saw the classification based on the geometry
now, let us understand this phase shift mechanism. We have shown here that based on the phase
shift mechanism we can further classify the pulse tube cooler as Basic type pulse tube
cooler orifice type pulse tube cooler Inertance tube pulse tube cooler and Double inlet valve
type pulse tube cooler. So, basically these are the nothing but the phase shift mechanism,
which are important to bring cooling effect to be get more and more cooling effect in
the pulse tube Cryocooler. So, phase shift mechanism very important part
of the pulse tube Cryocooler and also it is very important to understand what is the requirement
of this phase shift mechanism and how do the induce cooling in case of pulse tube Cryocooler.
So, in order to understand the need of phase shift mechanism it is important to understand
the modeling of a pulse tube Cryocooler. So, modeling can be very you know modeling as
the simulation business can be very difficult in case of pulse tube Cryocooler very simple
to a very a complicated mathematical modeling, that can be done for pulse tube Cryocooler
it is a very important research topic. But in this particular lecture I would to show
a simple modeling in order to understand what is this Basic or orifice Inertance and Double
inlet requirement and how do they create cooling effect in case of pulse tube Cryocooler.
So, with a very simple modeling excise it is simple analysis we can understand that.
Different analyses are published in the literature with varied difficulty and level of accuracy.
And this analysis could be classified we had use this earlier also first Stirling type
Cryocoolers, the following are the methods used to analyze the pulse tube Cryocooler
we have got a first order analysis in this pulse tube cooler phasor analysis. We have
got a second order analysis where we can have isothermal model, thermodynamic non symmetry
model and thing like that and we can have third order analysis where we can numerical
methods computational flow dynamics and etcetera. Depending on the kind of difficulties or computer
time requirement and more and more realistic analysis as you up to top to bottom it becomes
very complicated it requires lot of time to solve these equations. While in order to understand
the phase shift mechanism the first analysis very important and therefore, in this particular
lecture, I am going to take Phasor analysis is going to be explained and I will solve
is small tutorial problem also may be in the next lecture to understand this Phasor analysis.
So, this Phasor analysis make is understand what is the requirement of phase shift mechanism,
while in order to calculate the cooling effect and to get dimensions more complicated models
like second order model third order model could be utilized.
So, let us come to Phasor analysis in the year 1990, Ray Radebaugh from NIST, proposed
Phasor analysis of a pulse tube Cryocooler. The theory was applied to simple orifice pulse
tube Cryocooler with a monatomic gas like helium gas use as a working fluid. So, let
us consider orifice type pulse tube Cryocooler and use helium as a working fluid. The simple
assumption made are, the thermodynamic processes in the pulse tube are adiabatic, in the pulse
tube not in the heat exchanger only in the pulse tube that the processes are adiabatic;
that means, there is no loss of heat energy from the pulse tube cooler which is realistic
assumption. At the same time the pressure is constant
throughout the system; that means, there is no pressure drop in the system both this assumptions
are very realistic. The pressure P and temperature T at any location where is sinusoidal the
variations also sinusoidal. So, because of oscillating pressure flows the temperature
also at every point will vary and let us assume that these variations are sinusoidal.
So, let us see orifice type pulse tube cooler what does it mean? Between the hot end and
the reserve wire this is the reserve wire and this hot end of the pulse tube cooler
as you can see this is a inline kind of pulse tube Cryocooler the compressor after cooler
regenerator cold and heat exchanger pulse tube cooler and hot end. And this hot end
is attach to reserve wire is through is small orifice and therefore, this is called as orifice
pulse tube Cryocooler. Now, what is requirement of this orifice why it is kept and all that
we understand in this analysis? The sinusoidal variation of pressure and temperature are
as given below as we have just given assumption that the pressure and temperature variations
are sinusoidal. So, let us assume pressure P is equal to P
0 plus P 1 cos omega T, while temperature T is equal to T 0 plus T 1 cos omega T this
are the sinusoidal variation of pressure and temperature. In the above equation, P 0 and
T 0 are the average pressure and ambient temperature respectively there average value basically
around which the variations happen where we can cos T 0 as ambient temperature and T 0
as a pressure variation. While we can assume the P 1 and T 1 are the variations basically
there are amplitudes P 1 and T 1 are the amplitudes of the pressure and temperature variation.
Now, let us see the let us come the mass flow rate let m c and m pt, m h and m dot o be
the mass flow rates in the cold end m c as cold end heat exchanger here at this point.
So, as we can see m dot c as cold and heat exchanger mass flow rate m dot pt as the mass
flow rate let say center of the pulse tube m dot h is the mass flow rate in the hot end
heat exchanger while m dot o is the mass flow rate to the orifice at this point. Using the
law of conservation of mass, we have m dot pt is equal to m dot h, if I want to compute
the m dot pt we can assume that by conservation of mass m dot pt is equal to whatever, is
living minus whatever is incoming. So, m dot pt is equal to m dot h minus m dot
c in the OPTC, that means, in the orifice pulse to Cryocooler the following holds true
where we can say that m dot h is equal to m dot o; that means, whatever is the mass
flow rate living the hot and heat exchanger is the mass flow rate the m dot o, which is
very again valid because is orifice is very close to hot and heat exchanger. And also
is perfect assumption that m dot h which is living at this point is the mass flow rate
through the orifice, orifice is the very small opening it opens into reserve wire and the
pressuring the reserve wire is going to be the average pressure is going to be average
pressure in the system, this is not subjected to oscillating pressure rearranging the above
mass equation.
We just computed m dot pt is equal to m dot h minus m dot c and if I want to replace the
same thing in terms of volume we have got the dV pt is equal to dV h minus dV c. That
means, volumetric variations a small variations in volume in the pulse tube is equal to variation
in the hot and heat exchanger minus the variation in the cold and heat exchanger this can be
obtained by multiplying the earlier mass equation by then c t and area that is why we can calculate
this. Upon multiplying this equation by pressure p if you multiply entire thing by pressure
p we get pdV p t is equal to pdV h minus pdV c.
Now, let us come to the ideal gas equation the earlier equation can be use later, we
know pV is equal to m RT or ideal gas differentiating the ideal gas equation we get pdV by d t is
equal to RT and d m by d t RT is constant at any temperature. So, we can get pdV by
dt is equal to RT in to dm by dt where, if I cancel the dt what we get is pdV is equal
to RT in to dm and we have already describe we have already got an expression for pdV
earlier. So, if I want to have expression for pdV pt which is what we are calculated
pdV pt is equal to pdV h minus pdV c this is what we had calculated and I want to replace
these by these now. So, I got a pdV h is equal to RT into dm h now here. So, I can replace
this pdV h by this term I can have pdV c by RT c into dm c.
So, combinem, if I put the values from here into this equation I will get replacing these
by these replacing pdV c by again this I will get pdV pt is equal to RT h into dm h, pdV
h is equal to RT h into dm h minus RT c into dm c. So, basically now I am replacing this
pdV by RT so, and I will have a mass term introduce over here.
Now, let us come to the pressure variation and temperature variations and because there
is sinusoidal they could be return as P is equal to P 0 P 1 cos omega T, T is equal to
T 0 plus T 1 cos omega T. At any cross section in the pulse tube let there be a phase alpha
between the pressure and temperature variation they need not be in face and therefore, we
can introduce the face difference between this pressure and temperature variations in
the pulse tube. So, P is equal to P 0 plus P 1 cos omega T and T is equal to T 0 plus
T 1 cos omega T plus alpha. So, we can just introduce phase difference between the pressure
and temperature variation at any cross section in the pulse tube, at any cross section in
the pulse tube will have this variation and we assume that the entire pulse tube cooler
works in a adiabatic way there is a adiabatic. So, for adiabatic process we have T by T 0
is equal to P by P 0 to the power gamma minus 1 by gamma, this is the temperature and the
pressure relationship connected together by the gamma of particular gas, if I am talking
about helium as work in fluid the gamma for a helium is 1. 67. So, putting that value
of gamma in this equation and putting the expression for P and temperature as shown
over here, we can see that T by T 0 expressed using the keeping the value of T in this expression
so, T by T 0 is equal to putting the value of P by P 0 to the power gamma minus 1 by
gamma is equal to 2 by 5. So, once we put gamma is equal to 1.67 we
will get the value of gamma minus 1 by gamma has to 2.5, expending this further by binominal
theorem and neglecting the second order terms what we get is therefore, T 1 by T 0 is equal
to 2 by 5 P 1 by P 0. So, you can see there if I expand is T 0 by T 0 get canceled T 1
by T 0 P 0 by P 0 get canceled P 1 by P 0 will come and then I expand is further and
neglect the further terms, I will get a relationship between the temperature T 1 by T 0 this is
my amplitude to average value relationship related to the pressure amplitude the average
pressure relationship. This we will use again later connecting temperatures and pressure
variation in the pulse tube cooler.
So, pulse tube cooler an adiabatic law between the pressure and the volume is given below,
we know that pV to the power gamma is equal to constant this is what we say the adiabatic
law. So, let us apply this law for the gas in the pulse tube, because we know that the
gas in the pulse tube behave you know adiabatic manner. So, let us say pdV to the power gamma
is equal to constant differentiating now, differentiating this we have v pt into dp
plus gamma types pdV pt is equal to 0. Just differentiate this and rearrange the terms
what we get here is v pt dp plus gamma type pdV pt is equal to 0 differentiating constant
what you get is 0. Now, what we have derived earlier as mass equation we have already got
it here. So, pdV pt is equal to RT h dm h minus RT
c dm c which we have derived earlier now, what I would like to do is replace this term
pdV pt by this. So, putting this term over here I will replace this by this so, what
I get is v pt dp is equal to and putting this on the right side now, minus gamma taking
gamma common here in to bracket RT h dm h minus RT c dm c. Please understand this steps
we may have to take more time to understand I am just goings step by step, I have try
to give as many steps as possible that you understand this derivations.
So, here v pt dp by gamma if, I do I just put gamma on this side I got this expression
further also, taking the minus side and putting this on this said you got a dm c and you got
a dm h plus term over here. Rearranging the above equation, what I get now minus v pt
dp by gamma is equal to minus RT c dm c plus RT h dm h. So, what I get is dm c basically
now I am finding the relationship between the mass flow rate at the cold end and the
mass flow rate at the hot end it is my entire objective, relating entire thing to the pressure
variation in a system. So, dm c now is equal to v pt upon gamma taking
RT c in the denominator if I put RT c over here R and R will get cancel transferring
on this said I get plus term T h by T c into dm h. Dividing entire theme by dt, now, so
that I will get dm by dt dm h by dt by infinites time step dt we have dm c by dt which nothing
but the mass flow rate at the cold end heat exchanger or the cold end of the pulse tube
is equal to v pt gamma RT c in to dp by dt this is my pressure variation with time T
h by T c into dm h by dt this my mass flow rate variation at the hot end the pulse tube.
So, now, I am relating the mass flow rate at the cold end the pulse tube to, the pressure
variation and the mass flow variation at the hot end to the pulse tube this is a very important
derivation now, basically relating the mass flow rate at the cold end mass flow arte at
the hot end and the pressure variation.
Now, let us understand this in this inline pulse tube, the mass flow rate at the orifice
is directly proportional to the mass flow rate to the pressure drop. Now, I am coming
to m dot h or m dot o we have said that let us find the relationship of m dot h m dot
h is equal to m dot o as far as the orifice type pulse tube cooler is considered. So,
what is m dot h proportional to? The m dot h proportional to the pressure drop across
this orifice correct, because the mass flow rate orifice will depend on what is the pressure
drop at this point and this point. Now, the pressuring the reserve wire is always average
pressure which is p 0 while the pressure on this said of the orifice is always oscillating
flow which is p 0 plus p 1 cos omega t which is what we have earlier seen.
So, m dot h is always proportional to delta p across the orifice. So, what is m dot h
proportional to p 0 plus p 1 cos omega t which is the pressure variation on the left said
of the orifice while, on the right said of the orifice is just the p 0. So, if I want
to calculate to delta p it is p 0 plus p 1 cos omega t that is the pressure on the left
side orifice and minus t 0 which is pressure on the right side the orifice. So, if you
see this p 0 and p 0 will get cancelled and m dot h directly proportional to p 1 cos omega
t. So, p 0 p 0 will get cancelled and I will remove this constant of this proportional
at this signed and introduce a constant of proportionality therefore, I can right m dot
h is equal to C 1 which is constant C 1 p 1 cos omega t this a very important.
Combining the following equations we have p is equal to p 0 p 1 cos omega t we got m
dot h is equal to C 1 p 1 cos omega t now. And now we have got a relationship between
what we have derived earlier which is basically, mass flow rate at the cold end mass flow rate
at hot end and dp by dt, can I get dp by dt from this term I will get dp by dt from this
term I will get m dot h is nothing but d m h by d t. So, I can now put this term over
here I can get dp by dt and now, I can get relationship between of this parameters in
a better way yes. So, get a dp by dt from this get m dot h as a dm h by dt over here
and if I right this things what I get is m dot C is equal to the differential of cos
omega t will be minus sinomical t. So, will get minus term sinomical t and I will put
C 1 p 1 cos omega t as this point over here. So, I got m dot c as a some of two components
one component which is this cos omega t the other component is sin omega t, if I want
to write this sin omega t instead of that if I write in terms of cost than, I will get
minus sine omega t as cos omega t plus pi by 2 this is clear from the trigonometry.
So, m dot c now can be return as one cos term cos omega t and other term is cos omega t
plus pi by 2; that means, it is sure that m dot c is a vectorial addition of one term
which is in one direction. And the other term is going to be at pi by 2 phase difference
with this term is it not it so, m dot c is equal to this plus this.
So, I can right now m dot c as we just wrote is this term and what is this C 1 p 1 cos
omega t is nothing but m dot h and as we know that m dot h is the directly proportional
to the pressure m dot h is always going to be in line with pressure, in other way m dot
h is always in face with the pressure. The mass flow rate at the hot end is always in
face with pressure axis, if I want to plot it will be always pressure axis while the
second term or this term is going to be at 90 degree angle to the pressure axis. So,
pressure axis and m dot h are in the same axis while the second term is going to be
at a phase angle of pi by 2 or 90 degree to this. So, now, in order to calculate m dot
c I got two terms, one term which is always in the direction of phase a pressure which
is m dot h and second term is going to be 90 degree angle to that particular term.
So, from the above equation it is clear that vector, if I want to have a vectorial addition
of this to get to calculate m dot c vector m dot c is a sum of two vectors which are
at 90 degrees two each other is it clear. So, this term is going to be in line with
pressure axis and this term is going to be 90 degree to this term. So, if I want to have
if suppose I pressure is in this action I know that, plotting these two vectors we have
flame figure this is my m dot h which is in line with pressure, but what is amplitude
of this T h by T c in to p 1 this is my axis over here. So, I will get T h by T c in to
m dot h this is my pressure axis on which m dot h matches where in phase, the amplitude
of this vector is T h by T c into m dot h. So, T h being larger than T c this axis will
be T h by T c m dot h larger than the m dot h to which now, I got a vertical axis vertical
Phasor which is having a magnitude of this is having a amplitude of this and which is
going to the 90 degree angle. So, if I want to calculate from here my m dot c is equal
to these vector plus vertical component which is at 90 degree, which has got a magnitude
of omega in to p 1 in to v pt divided by RT c gamma and if I add them together vectorally
I will get the value of m dot c.
What does this mean? This means that from this figure it is clear that there exists
a face angle between the mass flow rate at the cold end which is my m dot c and the pressure
vector. So, we can see in pulse tube cooler the mass flow rate at the cold end and the
pressure axis they are not in face, but they have get a some theta angle between them.
While m dot h has got a same face with the pressure at the hot end the pressure and the
m dot h R is same phase basically they are parallel to each other while m dot c is making
an angle of theta. This is very important and in the next lecture will talk more about
the value of this theta, but what does this theta determine this theta is a phase difference
between the mass flow rate at the cold end and the pressure.
This angle theta depends on what parameter it will depend basically on this is it not
it will depend on this quantity this quantity small we can have, this vertical vector could
be of very small like and therefore, theta in that case will be very small. These parameters
are very large in that case m dot c can be very very large and therefore, the theta also
can be very large quantity. So, this theta angle depends on omega p 1 V pt what is omega
frequency what is p 1 is a amplitude of pressure what is V p t is the volume of the pulse tube.
So, if got a very large pulse tube we can have very large theta, if I got a higher frequency
we can have a very high theta if the amplitude is the pressure variation is very large we
can have very high theta similarly, if we got a low temperatures very very low temperature
this theta could be larger. So, the theta basically the angle theta depends
on the dimensions that mean V pt volume of pulse tube cooler the frequency which is omega
p 1 which is amplitude of pressure variations and other operating parameters like temperature
etcetera. This theta is very important component which will be basically determined by all
this parameters. The importance of the phase angle is explained at a greater detail in
this next lecture, because I cannot explain everything in this today’s lecture. And
all this phase shift mechanism what we have talked about will depend on through this theta
is minimized, we will understand this we want for would pulse tube action theta to be as
minimum as possible and therefore, all this phase shift mechanism would ensure that theta
is getting reduced. So, that the pulse tube cooling action is
better and better and therefore, all this orifice double inlet inheritance to are employed
to basically change this angle of theta for over benefit this will what will be explained
in the next lecture in the Phasor analysis continued during that lecture.
So, I will stop here now, the summary of this lecture therefore, is in a pulse tube Cryocooler
the mechanical displacer is removed and an oscillating gas flow in the thin walled tube
produces cooling this phenomenon is called as pulse tube action. The pulse tube systems
can be classified based on the Stirling type or GM type; that means, presence of wall or
low wall high frequency or low frequency. The geometry and operating frequency geometry
will decide with the inline U type coaxial angular etcetera.
The operating frequency in the Stirling also will be determined whether the low frequency,
high frequency, very high frequency etcetera. Also, they can be classified based on the
kind phase shift mechanism employed in the pulse tube cooler, and they can be call as
basic pulse tube cooler orifice pulse tube cooler double inlet or inertuf pulse tube
cooler. This part we have not seen, and this is what we are basically understanding now,
what we also understand there exist phase angle between the mass flow rate at the cold
end and the pressure vector. And this is what we derived in details to understand why this
theta appears, between m dot c and the pressure vector, because of what parameter this theta
appears; thank you very much.