 So, welcome to the 30th lecture of cryogenic engineering under the NPTEL program. In the earlier lecture, we talked about GM cryocooler or Gifford-McMone cryocooler. We have seen the schematic and the working of GM cryocooler which was invented by Gifford and McMone in 1950. So, the highlight was it has a wall mechanism to generate a pressure pulse. The relation between the pressure pulse and the expanded 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, wall mechanism, etc. A GM system can reach much lower temperature as compared to a sterling system. Basically, the difference between the GM and the sterling is GM has evolved and it runs at a very low frequency of around 1 to 2 hertz while sterling operates without a wall and normally it runs at a very high frequency as compared to GM cryocooler. Now, in GM cryocooler, multistaging is done to reach very low temperature and of the order of around 4.2 K which is a helium boiling point and up to 10 Kelvin. Now, these are temperatures which are normally used for various scientific experiments for example, 10 K while 4.2 K is normally used to liquefy or recondense helium gas. A typical regenerator material in a single stage GM cryocooler which would possibly bring the temperature down to 30 Kelvin is stainless steel mesh and if you want to come to by using a 2 stage cryocooler to reach around 10 Kelvin then 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 2 stage cryocooler to 4.2 Kelvin temperature, the first stage may have stainless steel mesh plus lead ball as a regenerator material and in the second stage we may have lead balls plus Arabium 3 nickel, Holonium copper, Neodymium etcetera. They are all spheres made out of normally these materials at around 0.2 millimeter diameter. Now, when I am telling this is a very broad regenerator configuration SS mesh plus lead ball it may have 100 percent SS mesh and the second stage may have lead balls plus Arabium 3 nickel. But this is what we generally would try the regenerator material for a 4.2 Kelvin cryocooler. Commercially available cryocoolers have normally rotary valve to control regulate the flow of the working fluid. The working fluid normally will be the helium gas. This is what we saw in detail during the last lecture. Now, this lecture we will now completely dedicate to pulse tube cryocooler which is a very important cryocooler right now amongst all the cryocoolers that are used 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 tube as now P T cryocooler or P T P T C sometimes. So, pulse tube classification and let us compare the sterling a Gifford-McMahon and a pulse tube cryocooler and let us see its different applications and let us try to understand the pulse tube cryocooler using a phasor analysis. It is a very important concept to understand why different phase chip mechanisms are employed in a pulse tube cryocooler and we 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 lectures we have seen a regenerative cryocooler and of different types let us say sterling type and GM type. So, this is a sterling type this is just to recap to understand what we had done during the last lectures. So, we have got a sterling type machine sterling cryocooler where there is no wall in between the compressor and expander and therefore, the displacer moves at the same frequency as that of compressor piston and there is a fixed phase difference between the motion of the compressor piston and the motion of the displacer. While in a GM cryocooler you have got a presence of wall mechanism between the compressor piston and expander and there is a phase difference not between the displacer and the piston, but between the wall mechanism and the displacer which is moving at a relatively low frequency in a GM cryocooler. This systems have a mechanical expander displacer to displace the working gas in both these systems we have got one component moving in a expander. We have got a displacer moving here at high frequency we have got a displacer moving here at around low frequency of around 1 to 2 hertz in a GM cryocooler. The displacer 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 is normally a very light weight moving component alright. The cold end displacer pose few problems as given below sometimes. The in a GM cryocooler especially as you can see there is a seal which works at low temperature because this side is at low temperature and therefore, we will have a rubbing seal at this point and these seals which are rubbing in this can pose some problem from working point of view. The rubbing seal on displacer is difficult to maintain and therefore, one has to really take care of the seal and the seal definition is very important how this seal is composed of alright. So, at low temperature there could be shrinkages across and seal which is perfect at room temperature may not work very well at low temperature and sometimes this is a trouble for lot of displacers moving in this cylinder. The motion of the 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 sterling at this point in case of GM at this point because the mechanical component moves up and down. This creates lot of vibrations for the component to be cooled and sometimes this component may not function whatever scientific action you want to take over here may get hampered because the vibrations. That means, this experiment will not tolerate 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? The pulse tube cooler overcomes these problems as pulse tube cooler does not have a mechanical displacer. There is nothing moving at in the in the cylinder like here you got a displacer then nothing is moving in case of pulse tube cooler. And therefore, when complete mechanical component is absent in case of pulse tube cooler and this is a very important difference between sterling GM and the pulse tube cooler. 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 a driving mechanism so as you know to move this mechanical component. This is a very important difference between the pulse tube cooler and sterling and GM coolers. So, now what happens? Consider a schematic of a sterling system as shown in this figure. This is a normal sterling type cryocooler and if I want to convert this to a pulse tube cryocooler what will I do? I will just replace as you just saw I will just replace this displacer by a gas column a cylinder or a tube filled with gas column. So, in a pulse tube cryocooler the mechanical displacer is removed and an oscillating gas flow in the thin wall tube produces cooling. So, what is going to happen is I will just have a empty tube filled up with gas which is then subjected to oscillating pressurization and depressurization and this generates cooling and this is a simple definition of a pulse tube cryocooler. So, in effect we have got a regenerative cryocooler when the pulses oscillating pulses enter a tube which is subjected to pressurization and depressurization the cold is generated. Now, there is something called as PSM which is what I will talk about. This gas tube is called pulse tube and this phenomenon is called 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 their motion. Now, we do not have any moving component as such here a mechanical moving component here and therefore, whatever gas is moving at this point I have to see that there is some kind of a the gas moves in a accordance with what I want which would generate cooling. So, I have to ensure that the oscillation setup in this tube they move in a fixed manner they move in a manner which generates cooling and this manner in which they should oscillate in this tube is basically due to the phase shift mechanism PSM that is what is written 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 very important and this is present only in the pulse tube cooler and not in the sterling and the GM type machines. So, therefore, this PSM has to be understood and this will be explained at the later part of this 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. Now, the high pressure gas flows across the regenerator and into the pulse tube as shown here by this arrow the gas comes down the after cooler takes the heat of compression the gas enter the regenerator and then the gas enters the pulse tube. The gas in the pulse tube compresses the gas present at the other end. So, everything was filled up earlier at an average pressure for a charging pressure as soon as the gas gets charged up as soon as the gas gets compressed the depending on the amplitude of the pressurization the gas enters and this gas will compress the gas present already in the tube which initially is present at a charging pressure. So, gas in the pulse tube compresses the gas present at the other end let us call this as a top end let us call this the bottom end of the pulse tube. Now, this compression this compression of a earlier gas by the gas which is coming because of pressurization this compression results in the rise of temperature at the top end. In fact, the temperature rises at every point, but the temperature rise at the hot end at the top end will be much more as compared to the gas in this tube below this top end alright. So, during pressurization the temperature at this point increases and 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 than the temperature at the bottom end. So, what does happen first during pressurization we got increased temperature at top end as compared to that of the cold end and during depressurization the gas expands and the temperature lows down. As a result of which a temperature gradient will get set up over here across the length the temperature start decreasing down alright. So, gas pressurization happen temperature at the top end increases depressurization happen temperature decreases as a result a temperature gradient is set up across the length of the pulse tube alright. 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 at 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 pre cooled because of the heat transfer from the regenerator matrix to the pulse tube. So, the cold gas during depressurization transfer cold to the regenerator matrix and this cold is again given which is used this cold is again used to pre cool the incoming gas during pressurization. The hot end temperature at the top end is maintained at ambient temperature. Now, as I said earlier the temperature increases, but what I am going to do is the temperature at this top end is going to be maintained at ambient temperature by running water over here. As a result of which this will be ambient temperature and the temperature at the bottom end will 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 when it goes back during depressurization. So, this is how the pulse tube cooler works. The pulse tube cooler will compress the gas and set up a temperature gradient across the length of the pulse tube. During depressurization the gas goes back the hot end of the pulse tube is always maintained at ambient temperature thereby 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 the regenerator matrix material to store the heat. This is true for sterling cooler this is true for even GM cooler. So, the basically these are all regenerative cryocooler, but the major difference is now there is no displacer and therefore, the oscillating gas flow sets up 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 produce at the bottom end also called as cold end is lifted by using a heat exchanger. So, the heat exchanger here which basically exchanges the cold generated because of this pulse tube action. So, normally a pulse tube cooler will be identified by the hot end of the pulse tube which is maintained at room temperature and cold end of the pulse tube where the cooling effect gets generated. This is a very simple terminology the hot end of the pulse tube and the cold end of the pulse tube. Also the after cooler temperature at the top end is maintained at ambient. So, you got a after cooler heat exchanger here which maintains the temperature of the gas to be ambient temperature. So, when the gas gets compressed the gas enters the regenerator 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 sterling and gm cryocoolers obvious. There is no moving component and therefore, the vibrations are magnitude a magnitude less than as compared to sterling and gm cryocooler. The schematic of the pulse tube cryocooler with 3 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 reformatting the entire figure. So, as we understand the pulse tube action in a better way where we will call AC CHX and HHX will be shown on this figure. So, if I want to show now the temperature variation across the length of the pulse tube cryocooler it will be like this is a compressor with the after cooler, regenerator, cold end heat exchanger, pulse tube and hot end heat exchanger and the PSM. So, this is a pulse tube cryocooler shown in line and this is called as in line configuration of the pulse tube where 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 were 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 of the gas is going to be little above the ambient temperature where Q is given to the after cooler. In the regenerator at a steady state we will have a temperature gradient developed because of incoming because of the pre cooling the gas going back and the gas coming in at a steady state temperature distribution will get generated. In the CHX that is cold end heat exchanger the temperature normally will be maintained constant because there is a load on a system with a constant load on a system Qc at temperature Tc and across the pulse tube also we will have a temperature gradient generated and therefore temperature gradient would look like this in the hot end heat exchanger. Now, again the temperature would remain constant normally. So, it is a very important diagram to understand that the cold is getting generated at this point the lowest temperature is generated in the cold end 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 attached to the cold end heat exchanger across which the cold is transferred. Now, let us see 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 a very important advantage of a pulse tube cooler as compared to other cryocoolers there is no sealing requirement the moment you do not have a mechanical drive there is no sealing requirement at low temperatures therefore the problem of rubbing seals do not arise high reliability the moment we do not have any drive mechanism moment we do not have any moving component it will have 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 there is no reliability data due to less history. So, there is no failure data over a 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 fail. At 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 it is a gravity driven also because it will have some convictive current the hot end on the top the cold end at 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 mode only. So, we 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 infrared sensors for space application recondensing liquid helium and liquid which is very important function it can be used for recondensation. And therefore, it is normally clubbed to the object to be cooled where the helium boil off happens where the nitrogen boil off happens and the pulse tube can be kept coupled to that particular equipment. And sometimes the pulse tube cooler can be used for nitrogen liquefaction or even helium liquefaction depending on it will be single stage or two stage for this respective operations. The descent development one can obtain less than 4 Kelvin temperature using a two stage pulse tube cryocooler and miniaturization is absolutely possible if you go for a sterling type pulse tube cryocooler. And therefore, pulse tube cooler is a very important candidate in a cryocoolers. So, if I want to see all three together the sterling the gm and the pulse tube cooler they would look like this. So, this is sterling cryocooler the gm type cryocoolers this is a sterling type pulse tube cooler. Now, in pulse tube cooler we will have a sterling type pulse tube cooler where there will not be any wall and therefore, this is a very compact and miniature unit sterling type pulse tube cooler and we can also have a gm type pulse tube cooler the moment we have got a wall between the compressor and expander this will be called as a gm type pulse tube cooler. So, very important to understand sterling cooler is different gm cooler is different sterling type pulse tube cooler different and gm type pulse tube cooler is different. So, one has to really understand whenever we say a particular cryocooler what cryocooler we have been talking about. This should be fixed on our mind. The moment I say gm cooler I have got a wall between the compressor and expander. Moment I say pulse tube I do not have a displacer. Moment I say sterling I do not have a wall between the piston and the expander. So, this classification of sterling 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 category. So, pulse tube cryocoolers can be classified based on as we know that sterling type or Gifford macmon type. So, moment we have got a wall Gifford macmon type moment we do not have a wall sterling 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 a u type configuration, we have got a coaxial configuration and we have got a annular configuration. We will see each of this in detail. This will basically refer to the position of a regenerator and the pulse tube. If the regenerator and the pulse tube are inline as I had shown earlier in the long horizontal pulse tube cryocooler, this will be called inline configuration. If they are parallel to each other this is called u type configuration, if they are having a same axis coaxial and annular, we will see that thing. Other classification based on the kind of phase shift being used we have not yet come to understand why this phase shift mechanism is being used in pulse tube cryocooler, but what is used for the phase shift mechanism also will decide what kind of pulse tube cryocooler it is. So, we can have we 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 orifice type pulse tube cryocooler, inertons type pulse tube cryocooler and double inlet wall pulse tube cryocooler. So, we will understand about this later in the lecture and under sterling cryocooler now we know that sterling cooler is a high frequency as compared to GM cooler, but when I say high frequency how high it is and therefore, based on the kind of frequency we use in a sterling type pulse tube cooler based on the frequency I can again identify as 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 species of wall the pulse tube cryocooler can either be a sterling type pulse tube cooler or a GM type pulse tube cooler. The moment I say GM type pulse tube cooler I have got a wall over here sterling 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 you know earlier is I got own advantages and disadvantages. Another classification of pulse tube cooler is based on the relative position of regenerator and the pulse tube as we had just seen. So, this is my in line configuration where the pulse tube cooler and a regenerator are in line. The second classification is U type classification where the pulse tube cooler and a regenerator are parallel to each other and the U type connection given for the gas flow. So, the gas comes over here gas goes to these and comes out. Let us see the advantages, the merits and the demerits of each of these types in the next slides and then I say coaxial cryocooler, coaxial pulse tube cryocooler that means the pulse tube cooler and the regenerator have the same coaxial they are basically coaxial they have got the same axis around which they are basically assembled. The gas come in the annular regenerator from outside and enters into the pulse tube like this. So, you got a hot end heat exchange over here while these are annular regenerator over here. This is what we call as coaxial when the regenerator is in annular position and we got a annular pulse tube when the pulse tube now is in annular position while the regenerator is at the center. So, these are four different types in line U type coaxial and annular. Now, let us see the advantages and disadvantages of each of these type. Let us come to first the in line configuration. The gas does not change the direction of the flow hence the pressure losses are minimum. And that during pressurization the gas will travel up to the entire length and during depressurization the gas will go back. So, basically during pressurization the gas comes straight there is no change of direction and therefore there is no pressure drop losses across this. So, thermodynamically saying 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 deliver the best performance as compared to other this is what we just saw thermodynamically. The cold end is at the center of the system this is very important. If I want to cool something I have to access the middle portion of this pulse tube cooler 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 sometimes this is may not be acceptable for users from users point of view. The third disadvantage of this system so to say the system is less compact since it occupies huge space length wise alright. Although thermodynamically it is advantageous it is efficient we may lose on the non-accessibility of the cold end heat exchanger and the huge length wise dimension required in the inline pulse tube curricular. The next as you see is a u type curricular now suddenly you can see that the length wise dimension has decreased but what you can see now the gas flow undergoes a 180 degree change in a flow direction. So, gas during pressurization will come to regenerator and then take a 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 draw. The cold end is now however exposed so I can access the cold end from the bottom so whatever I want to cool can easily be attached to the cold end heat exchanger and this is very important and it is easily accessible. The system becomes more compact now and it occupies less space as against what it was doing earlier. The performance is dependent upon the sharpness of the bed so how much pressure drop would happen depend on how sharp this bend is made basically is fabricated. And therefore if you made a very gradual bend the pressure drop in the skates will be minimized. So, this is a u type pulse tube curricular where the regenerator and the pulse tube corals are parallel to each other. Let us come to now coaxial and annular curricular 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 these cases and therefore the pressure drop in both the coaxial and the pulse tube curricular will be 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 not it because they are all assembled together. So, in the both these cases the pressure drop loss is going to be tremendous. The cold end is exposed and it is easily acceptable. The best part about these two configuration is whatever I want to cool is accessible from this bottom end and therefore from user's point of view this is the most important or wanted system is a very compact system also because they are all you know assembled together is a very compact system. The cold end heat exchanger availability is accessibility is perfect in this case alright. So, other than the thermodynamic component that is high pressure drop losses we have got both the advantages the system is very compact and system is the cold end is accessible for the user. The system is more compact but there is a possibility of heat transfer now the only disadvantage also other than the pressure drop it as they are coupled up together you can have a heat transfer between the gas in the pulse tube and the gas in the regenerator in both these case we can make this system little bit inefficient. 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 lose some cooling effect over here in this particular configurations. So, these are the four configuration which we just saw. Now pulse tube cryocoolers can also be classified depending on the frequency under the sterling cooler banner which I had seen. So, depending upon the operating frequency the sterling type pulse tube cooler can be classified as listed below. I can call this as low frequency if the frequency is less than 30 hertz be clear I am talking about sterling type pulse tube cooler now the GM type pulse tube cooler will operate at 1 to 2 hertz only while the sterling type pulse tube cooler will be called a low frequency sterling type cryocooler if it is frequency is less than 30 hertz high frequency if it operate between 32 hertz 80 hertz and very high frequency when its frequency is going to be more than 80 hertz. So, we can call low frequency machine high frequency or a very high frequency depending on the frequency at which the sterling 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, we just saw that the pulse tube cooler we got a sterling type and Gifford Macbun type and we just saw the classification based on the geometry and 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 or if it is type pulse tube cooler inner term tube type pulse tube cooler and double inlet wall type pulse tube cooler. So, basically these are nothing but the phase shift mechanism which are important to bring cooling effect to get more and more cooling effect in the pulse tube cryocooler alright. So, phase shift mechanism is a 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 they induce cooling in case of pulse tube cryocooler. So, in order to understand the need of a phase shift mechanism it is important to understand the modeling of a pulse tube cryocooler. So, modeling can be very very you know modeling and simulation business can be very difficult in case of pulse tube cryocooler. It can be from very simple to a very 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 like to show a simple modeling in order to understand what is this basic or if it is inner terms and double inlet requirement and how do they create cooling effect in case of pulse tube cryocooler. So, with a very simple modeling exercise with simple analysis we can understand that. Different analysis are published in the literature with varied difficulty and level of accuracy and this analysis could be classified we had used this earlier also for sterling type cryocoolers. The following are the methods used to analyze the pulse tube cryocooler we got a first order analysis in this pulse tube cryocooler we use phasor analysis, we 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 have numerical methods computer health fluid dynamics and etcetera depending on the kind of difficulties or computer time requirement and more and more realistic analysis as you go from top to bottom it becomes very complicated it requires lot of time to solve these equations while in order to understand the phase chip mechanism the first analysis is very important and therefore in this particular lecture I am going to take phasor analysis is going to be explained and we will solve a small tutorial problem also may be in the next lecture to understand this phasor analysis. So, this phasor analysis make us understand what is the requirement of phase chip 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 Radebo from NISD proposed this phasor analysis of a pulse tube cryocooler the theory was applied to a simple orifice pulse tube cryocooler with a monotomic gas like helium gas used as a working fluid. So, let us consider orifice type pulse tube cryocooler and use helium as a working fluid the simple assumptions made are the thermodynamic processes in the pulse tube are adiabatic alright 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 a very realistic assumption at the same time the pressure is constant throughout the system that means there is no pressure drop in the system both these assumptions are very very realistic the pressure P and the temperature T at any location varies sinusoidally the variations are 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 a orifice type pulse tube cooler what does it mean between the hot end and the reservoir this is a reservoir and this hot end of the pulse tube cooler as you can see this is a in line kind of a pulse tube cryocooler a compressor after cooler regenerator cold end heat exchanger pulse tube cooler and hot end and this hot end is attached to reservoir through a small orifice and therefore, this is called as orifice pulse tube cryocooler. Now, what is the requirement of this orifice why it is kept and all that we will understand in this analysis the sinusoidal variations of pressure and temperature are as given below as we have just given the 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 these are the sinusoidal variations of pressure and temperature in the above equation P 0 and T 0 are the average pressures and ambient temperature respectively their average values basically around which the variations happen where we can call T 0 as ambient temperature and P 0 as a pressure variation while we can assume that P 1 and T 1 are the variations basically the amplitudes P 1 and T 1 are the amplitudes of the pressure and temperature variation. Now, let us see the let us come to the mass flow rate let m c m p t m h and m dot o be the mass flow rates in the cold end m c as cold cold end heat exchanger here at this point. So, you can see m dot c as cold end heat exchanger mass flow rate m dot p t as the mass flow rate let us say at the centre 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 p t is equal to m dot h if I want to compute the m dot p t we can assume that by conservation of mass m dot p t is equal to whatever is leaving minus whatever is incoming. So, m dot p t is equal to m dot h minus m dot c in the o p t c that means in the orifice pulse tube curricular 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 leaving the hot end heat exchanger is the mass flow rate to the m dot o which is a very again valid because this orifice is very very close to hot end heat exchanger and also is the perfect assumption that m dot h which is leaving at this point is the mass flow rate through the orifice. Orifice is a very small opening it opens into a reservoir and the pressure in the reservoir is going to be the average pressure alright 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 p t 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 p t is equal to dv h minus dv c that means volumetric variations a small variation in volume in the pulse tube is equal to variation in the hot end heat exchanger minus the variation in the cold end heat exchanger. This can be obtained by dividing earlier mass equation by density and area and that is why we can calculate this. Upon multiplying this equation by pressure p if you multiply entire thing by pressure p we get p dv p t is equal to p dv h minus p dv c. Now let us come to the ideal gas equation the earlier equation can be used later we know p v is equal to m r t for ideal gas differentiating the ideal gas equation we get p dv p dv by d t is equal to r t and d m by d t r t is a constant at any temperature. So, we can get p dv by d t is equal to r t into d m by d t where if I cancel the d t what we get is p dv is equal to r t into d m alright and we have already described we have already got an expression for p dv earlier. So, if I want to have expression for p dv p t which is what we have calculated p dv p t is equal to p dv h minus p dv c this is what we had calculated and I want to replace this by this now. So, I got a p dv h is equal to r t into d m h now here alright. So, I can replace this p dv h by this term I can have p dv c by r t c into d m c. So, combine if I put the values from here into this equation I will get replacing this by this replacing p dv c by again this I will get p dv p t is equal to r t h into d m h right p dv h is equal to r t h into d m h minus r t c into d m c. So, basically now I am replacing this p dv by r t. So, and I will have a mass term introduced over here. Now, let us come to the pressure variation under temperature variations and because they are sinusoidal they could be written as p is equal to p 0 p 1 cos omega t t is equal to t 0 per 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 then it not be in phase and therefore, we can introduce a phase 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 alright. So, we can just introduce a phase difference between the pressure and temperature variation at any cross section in the pulse tube at any cross section in the pulse tube we will have this variation and we assume that the entire pulse tube 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 alright this is the temperature and the pressure relationship connected together by the gamma of particular gas. If I am talking about helium as a working fluid the gamma for helium is 1.67 alright. 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 put in 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 as 2.5 expanding this further by binomial 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 alright. So, you can see there if I expand this t 0 by t 0 get cancelled t 1 by t 0 p 0 by p 0 get cancelled p 1 by p 0 will come and then I expand this 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, for a pulse tube cooler an adiabatic law between the pressure and volume is given below we know that p v 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 in a adiabatic manner. So, let us say p d v to the power gamma is equal to constant differentiating now differentiating this we have v p t into d p plus gamma types p d v p t is equal to 0 just differentiate this and rearrange the terms what we get here is v p t d p plus gamma type p d v p t 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, p d v p t is equal to R T H d m h minus R T C d m c which we have derived earlier. Now, what I would like to do is replace this term p d v p t by this. So, putting this term over here I will replace this by this. So, what I get is v p t d p is equal to and putting this on the right side now minus gamma taking gamma common here into bracket R T H d m h minus R T C d m c all right. Please understand these steps you may have to take more time to understand I am just going step by step I have tried to give as many steps as possible. So, that you understand these derivations. So, here v p t d p 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 side you got a d m c and you got a d m h plus term over here. Rearranging the above equation what I get now minus v p t d p by gamma is equal to minus R T C d m c plus R T H d m h. So, what I get is d m c basically now I am finding a 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 ok. So, d m c now is equal to v p t upon gamma taking R T C in the denominator if I put R T C over here R and R will get cancel transferring on this side I get plus term T H by T C into d m h dividing entire thing by d t now. So, that I will get d m by d t d m h by d t by infinite time step d t we have d m c by d t which is nothing but the mass flow rate at the cold end heat exchanger or the cold end of the pulse tube is equal to v p t gamma R T C divided into d p by d t this is my pressure variation with time T H by T C into d m h by d t this is my mass flow rate variation at the hot end of the pulse tube. So, now I am relating the mass flow rate at the cold end of the pulse tube to the pressure variation and the mass flow variation at the hot end of the pulse tube and this is very important derivation now basically relating the mass flow rate at the cold end mass flow rate at the hot end and the pressure variation. Now, let us understand this in this in line 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 this m dot h proportional to the m dot h is proportional to the pressure drop across this orifice correct, because the mass flow rate to orifice will depend on what is the pressure drop at this point and this point. Now, the pressure in the reservoir is always average pressure which is p 0 while the pressure on this side 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 alright. So, what is m dot h proportional to p 0 plus p 1 cos omega t which is the pressure variation on the left side of the orifice while on the right side of the orifice is just the p 0. See, if I want to calculate the delta p it is p 0 plus p 1 cos omega t that is the pressure on the left side of the orifice and minus p 0 which is the pressure on the right side of the orifice. So, if you see this p 0 and p 0 will get cancelled and m dot h is directly proportional to p 1 cos omega t. So, p 0 p 0 will get cancelled and I will remove this constant of this proportionality sign and introduce a constant of proportionality with therefore, I can write m dot h is equal to c 1 which is a constant c 1 p 1 cos omega t this is very important. Combining the following equations we have p is equal to p 0 plus p 1 cos omega t we got m dot h is equal to c 1 p 1 cos omega t now alright. 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 d p by d t. Now, can I get d p by d t from this term I will get d p by d t 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 d p by d t and now I can get a relationship between all these parameters in a better way yes. So, get a d p by d t from this get m dot h as d m h by d t over here and if I write these things what I get is m dot c is equal to the differential of cos omega t will be minus sin omega t. So, I will get minus term sin omega 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 sum 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 cos then I will get minus sin omega t as cos omega t plus pi by 2 this is clear from the trigonometry. So, m dot c now can be written 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 not it. So, m dot c is equal to this plus this. So, I can write 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 directly proportional to the pressure m dot h is always going to be in line with pressure in other way the m dot h is always in phase with the pressure. The mass flow rate at the hot end is always in phase with pressure axis if I want to plot it it will be always on the 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 have got two terms one term which is always in the direction of phase 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 to each other. Is it clear? So, this term is going to be in line with pressure axis and this term is going to be at 90 degree to this term. So, if I want to have if suppose I say pressure is in this axis I know that plotting these two vectors we have Fulham figure this is my m dot h which is in line with pressure. But what is the amplitude of this T h by T c into P 1 this is my axis over here. So, I will get T h by T c into m dot h this is my pressure axis on which m dot h matches there 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 have got a vertical axis vertical phasor which is having a magnitude of this which is having amplitude of this and which is going to be at 90 degree angle. So, if I want to calculate from here my m dot c is equal to this vector plus vertical component which is at 90 degree which has got a magnitude of omega into P 1 into V p t divided by R T c gamma and if I add them together vectorially 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 phase angle between the mass for it at the cold end which is my m dot c and the pressure vector. So, you can see in pulse tubular the mass for it at the cold end and the pressure axis they are not in phase they have got a some theta angle between them while m dot h has got the same phase with the pressure at the hot end the pressure and the m dot h are in same phase basically they are parallel to each other while m dot c is making angle of theta. Now, this is very important and in the next lecture we will talk more about the value of this theta alright, but what does this theta determine? This theta is a phase difference between the mass for it at the cold end and the pressure. This angle theta depends on what parameter? It will depend basically on this is not it? It will depend on this quantity if this quantity is small we can have this vertical vector could be of very small length and therefore, theta in that case will be very small. This parameter is 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 p t what is omega frequency what is p 1 is the amplitude of pressure what is v p t is the volume of the pulse tube. So, if you got a very large pulse tube we can have a very large theta if you got a very high frequency we can have a very high theta if the amplitude is of pressure variation is very large we can have very high theta. Similarly, if you 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 means v p t volume of pulse tube cooler. The frequency which is omega p 1 which is amplitude of pressure variations and other operating parameters like temperatures etcetera. So, this theta is a very important component which will be basically determined by all these 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 how this theta is minimized. We will understand this we want for a good pulse tube action this theta to be as minimized 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 these orifice double inlet inner terms tube are employed to basically change this angle of theta for our benefit. This is 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 cooler the mechanical displacer is removed and an oscillating gas flow in the thin wall tube produces cooling. This phenomenon is called as pulse tube action. The pulse tube systems can be classified based on the sterling type or gm type that means, presence of wall or no wall high frequency or low frequency. The geometry and operating frequency geometry will decide whether the in line u type coaxial annular etcetera the operating frequency in the sterling also will determine whether the low frequency high frequency very high frequency etcetera. Also they can be classified based on the kind of phase shift mechanism employed in the pulse tube cooler and they can be called as basic pulse tube cooler orifice pulse tube cooler double inlet or inner term pulse tube cooler. This part we have not seen and this is what we are basically understanding now. What we also understand? There exists a phase angle between a mass flow rate at the cold end and the pressure vector and this is what we derived in greater 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.