 So, welcome to the 27th lecture of cryogenic engineering under the NPTEL banner. During the last lecture, we have initiated the study on cryocoolers and the first introductory lecture on cryocooler was given. And if you see the highlights of earlier lecture, I mean only the introductory remarks of cryocooler, we can just go through this following points. A cryocooler is a mechanical device operating in a closed cycle manner which generates low temperature. And I compare this to a domestic refrigerator wherein you get low temperature in a closed cycle manner. What do they do? It basically eliminates cryogenic requirement of a reliable operation and it is cost effective. There are different heat exchangers and these heat exchangers can either be regenerative type or of recuperative type depending on the type of heat exchange we want to have. And depending on these, we had classified the cryocoolers. If we have recuperative heat exchanger, then we can have Joule-Thompson cryocooler, Brighton cryocooler or Claude cryocooler. And if we got regenerative type of heat exchanger, then we got Sterling cryocooler, GM or Gifford McMohen cryocooler and Pulse Tube cryocooler. And this was the broader classification of a closed cycle cryocooler under heat exchange type under the heat exchanger that is used in the cryocoolers. Today in this lecture, we will talk about Sterling cryocoolers. So, today's topic, we will talk about Ideal Sterling Cycle, how is this cycle executed? How does a Sterling cryocooler work? That means, how is this Sterling cycle is executed in Sterling cryocooler? And then we will have a simple Smith's analysis if you want to design a simple Sterling cryocooler broadly. I mean, the first guess of design basically not very accurate, but the first guess. And finally, I will have some conclusions on what we have done during this lecture. So, let us come to first Ideal Sterling Cycle. On that, we will have a history of Sterling cycle and how this Sterling cycle was brought into effect by having different machines. So, a brief history of Sterling cycle, a well-developed and most commonly used cryocooler is the Sterling cycle cryocooler. It is very commonly used and has been used for space application for quite some time and therefore, lot of reliable data is available today and therefore, the efficiency and reliability of Sterling cycle is considered to be very high. This cycle was first considered by Robert Sterling in the year 1815. If you remember Sterling cycle, the Sterling cryocooler works on Sterling cycle and it is named after this inventor called Robert Sterling. When it was invented, it was basically meant for engine cycle and was aimed at producing work. Being an engine cycle, it was producing work and you know the refrigerator cycle is reverse of engine cycle. So, the same cycle what we call as reverse Sterling cycle is used for producing cold. The important events that occurred in the history of Sterling cryocoolers are given in the next slide. So, the chronology of events is if you see 1815 where Robert Sterling first talked about Sterling cycle and talked about a possibility of Sterling engine. In 1834, John Herschel for the first time talked of the concept of using this cycle as cooler. That means, we talked about having a reverse Sterling cycle. In 1861, Alexander Kirk, he got this concept into practice of using Sterling cycle as cooler. So, in 1834, the concept was given by John Herschel while Alexander Kirk realized that concept in practice. Later on in 1873, Davy Postel came with a new idea called free piston system. So, you got a free piston, free displacer kind of Sterling cycle also which we will talk about later and this idea was first proposed by Davy Postel in 1873 and lately after 1950, during 1956 John Kohler first time he showed a commercial machine for air liquefaction. So, air liquefaction at around 78 Kelvin temperature and this Sterling cycle cooler was for the first time used to demonstrate liquefaction of air in 1956 and further in 1965 John Kohler again used the same machine for nitrogen liquefaction where in nitrogen has a liquefaction point or a boiling point of 77 Kelvin and after that this machine become commercial machine and it is available everywhere in the world. So, looking at 1815 to 1965 was a real period during which time Sterling cycle got its birth and then it got evaluated over a period of time and then Sterling cycle based commercial liquefies are now available all over the world. So, let us see how this Sterling cycle works. Working of Sterling cycle has been shown on this PV diagram. So, consider this PV chart as shown in the figure. So, you got a 1 to 2 points here which is isothermal compression at temperature T c. So, this 1 to 2 process occurs at constant temperature and that is why it is called as isothermal compression. So, if I were to write some equations for 1 and 2 process we will have P 1 V 1 is equal to P 2 V 2 temperature remaining constant this is very well known. T 1 is equal to T 2 is equal to T c and the heat transfer is equal to work done heat transfer is nothing but Q c during this time and which is equal to minus R T c log V 2 upon V 1. This is what will happen in isothermal compression at temperature T c. The second action is 2 3 which is constant volume heat rejection as soon as the heat is rejected at constant volume will come down and the pressure will get reduced here. So, 2 to 3 process is constant volume heat rejection here in we have got V 2 is equal to V 3 and the amount of heat rejected during this time is equal to D Q is equal to minus C V T e minus T c. As the heat is rejected we have got a negative sign final temperature minus initial temperature and this is the amount of heat rejected during the process 2 3 which is constant volume heat rejection. So, here the pressure automatically got reduced and then what we do is isothermal expansion. So, 3 to 4 process is again an isothermal expansion wherein P 3 V 3 will be equal to P 4 V 4. So, 0.3, 0.4 and 0.3 will have same temperature and therefore, we have P 3 V 3 is equal to P 4 V 4, T 3 is equal to T 4 is equal to T e and during this time the amount of cooling effect that one gets at during this isothermal expansion process is D Q is equal to R T e log V 4 upon V 3. So, here the heat is rejected Q c while here what you get is a cooling effect or Q e and this is what we get as a refrigerator. And again during the 4 1 process which is a constant volume heat absorption this was constant volume heat rejection 2 3 was constant volume heat rejection and now in 4 1 we have constant volume heat absorption the amount of heat absorbed is going to be at constant volume therefore, V 4 is equal to V 1 D Q is equal to C V into T c minus T e this is the amount of heat which is absorbed during the process 4 1. Now, you know COP is given as Q e upon Q c minus Q e that is the refrigeration effect Q e divided by the work input and this work input is going to be equal to Q c minus Q e this is the COP or the coefficient of performance of any cycle. So, if I were to put to get the value of COP and I put Q e value which is obtained during the process 3 4 and I put respective value of Q c minus Q e my equation comes down to this. If I were to manipulate these values we know V 2 upon V 1 is equal to V 3 upon V 4 alright V 2 upon V 1 is equal to V 3 upon V 4 for isothermal process putting up those values I will replace this V 2 upon V 1 by V 3 upon V 4 and take this minus side also into consideration this will become V 4 upon V 3 and therefore, log of V 4 upon V 3 gets cancelled overall R gets cancelled overall and what you ultimately get is COP is equal to T e upon T c minus T e that means expansion space temperature T e at which cooling effect is obtained divided by compression space temperature T c minus T e. So, T e upon T c minus T e is a COP of ideal sterling cycle and if you remember the same expression exists for COP of Carnot cycle also alright Carnot cycle considered as the ideal cycle operation and therefore, we can conclude from here that COP sterling is equal to COP Carnot cycle right. So, we say the COP of ideal sterling cycle is equal to COP of Carnot cycle and now if I want to show both the diagrams both these cycles on PV chart as well as temperature entropy chart. So, you can see 1 2 3 4 as sterling cycle and the same cycle now is shown on temperature entropy diagram which is normally what we refer in cryogenics 1 2 is the isothermal compression 2 2 3 is a constant volume process heat rejection 3 2 4 is a isothermal cooling effect obtained at this point isothermal expansion and 4 to 1 is constant volume heat addition and the cycle continues. So, this is what a sterling cycle would look like and if I were to plot a Carnot cycle on the same diagram under the same pressure and temperature limits it would look like this. So, we have got now Carnot cycle which is put on the same maximum pressure and minimum pressure maximum temperature and minimum temperature and you can see that COP of Carnot cycle will also be same as COP of sterling cycle ideal Carnot cycle COP will be same as COP of ideal sterling cycle and you have got a different diagrams shown over here as Carnot cycle alright. So, this is what it would look like if I were to compare a sterling cycle with a Carnot cycle. Now, ideal sterling cycle how can it be realized? If I were to have this constant volume processes if I want to have constant isothermal process isothermal compression, constant volume heat rejection, isothermal expansion, constant volume heat addition. If I were to realize this process I have to look for some kind of a process which will be realized in practice. So, if I were to understand how to realize this ideal sterling cycle into practice I have to imagine a process like this in which we have got a compression piston on the right side in a green color and the left side I have got some expander piston or it could be expander displacer connected through a heat exchanger called regenerator and this regenerator is the process through which heat is absorbed for some time which we have seen last time is a regenerative heat exchanger. This heat exchanger stores heat during the heat rejection and gives back the heat during the heat absorption when gas flows back and forth in this regenerative heat exchanger. So, if I want to plot this process on PV chart and to understand exactly what happens we can see over here. So, this is my initial position to begin with. My piston it has the bottom dead center while at this position the expander is at top dead center and the process of compression now begins from 0.1 to 0.2. So, 1 to 2 is a compression process during which Qc is released and a temperature remains constant over here and here you can see that during this time the expander piston remains at the same place where it was initially while the compressor piston has come up during which time the Qc is released over here and this is 1 to 2 process is a isothermal compression process. Now, I have got the next process which is a regenerative cooling or constant volume heat rejection and what will happen during this time? During this time the volume will remain constant the total gas now this is the volume this piston will come forward here and in order to keep the volume constant corresponding to that thing this expander will move back so that the volume of the gas remaining constant alright and that is why that is the way we can achieve constant volume process during ideal sterling cycle. So, 2 to 3 process is a constant volume during this process the gas will give up its heat to this matrix and this matrix will regenerator matrix the matrix will store the heat during which time the pressure will come down. Now, as you can see that this expander piston has come back the volume has been kept constant and this is what we call it regenerative cooling further now the gas is now in the regenerator as well as in the expander this gas will now be expanded and how will it be expanded? The piston has to expander piston has to move back so as soon as the expander piston moves back from this position and comes down over here the gas will get expanded from process 3 to 4 and during which time because the process is isothermal will have QE as a cooling effect that will be realized during this process. So, 3 to 4 process is isothermal expansion process please understand again during this time the piston is at a top dead center ok piston is at the top dead center while the expander piston was top dead center during the process 1 2 but during this process 3 4 the compressor piston was at top dead center while the gas is expanded from 3 to 4 isothermally now during the 4 to 1 process it is a regenerative heating or heat absorption process the expander piston will come back to its original process original place and all this gas which was held over here will be moved back during regenerative heating during this travel the gas will take back the heat from the regenerative matrix during this time the piston the compressor piston will move back to its original position over here 1 in order to accommodate all the gas and the process will repeat. So, we got a 1 to 2 as isothermal compression which have occurs here over here 2 to 3 is constant volume process which happens over here 3 to 4 is isothermal expansion and 4 to 1 is a regenerative heating or constant volume heat addition all right this is the way the piston and the displacer will have to move in order to realize all this isothermal and constant volume processes in practice. Now what you can see from here is how this pistons and how these displacers or compressor piston and expander piston move with respect to each other which is very important thing. So, in order to understand that let us see the next slide and here we can plot the locus of the top portion of the expander piston as well as the compressor piston. So, initially at point 1 we had compressor piston at the bottom dead center while the expander piston was at the top dead center. During the process 1 to 2 during which compression happen this green color line which indicates the locus of the motion of the compressor piston while the red color line gives the locus of the motion of the top portion of the expander piston. So, you can see that during this time 1 to 2 shows that how this piston moved forward up to this point however during this time the expander piston remained at top dead center only. So, this is moving piston is moving but expander piston remained at the same place. Now during this process 2 and 3 which is regenerative cooling or constant volume heat rejection. Now you can see that both the pistons are moving. So, 2 is moving front the compressor piston moved front up to the top dead center while the expander piston has started moving back. So, that this process becomes a process at constant volume. So, that is why you can see that this volume between these 2 is always remaining constant during this process. So, 2 to 3 is the motion of the compressor piston this 2 to 3 is the motion of the expander piston. You can realize from the top I have written here V e is equal to V c is equal to 0 at this central line. So, whenever the piston is at the top at this point the at this point the V c value or the compressor volume is 0 or V e volume which is the volume above the compressor expander piston is also 0. While at the 2 extreme position what we have shown is V c max when the piston is at this point the this is amount this distance amounts to V c max while this distance amounts to V e max on the expander piston side. If we go further from 3 to 4 now the expansion process occurs isothermal expansion process during which time compressor piston remains at top dead center as shown over here while the expander piston goes back up to the bottom dead center. This is what is again shown and during regenerative heating or constant volume process again we can see 4 to 1 is a constant volume heat addition process it comes back to its original position what we had earlier at 0.1 and this way the cycle continues. So, what you can see here that there is a motion for some time then there is no motion for some time again there is a motion both for compressor piston and expander piston or expander displacer. This is what I want to show that the motion is not continuous the motion is for some time there is motion after some time there is no motion. As mentioned in the earlier lecture the characteristic of a sterling cycle are high frequency. We remember that the sterling cycle there are no walls between the compressor and expander and therefore whatever is the frequency of the compressor piston the same is the piston of the expander piston or displacer. So, they move at very high frequency between let us say 30 hertz to 150 hertz or so. There is a regenerative heat exchanger as well as there is a phase difference between the piston and the displacer motion. So, both of them do not go to the top dead center at the same time or both of them do not reach the bottom dead center at the same time which we just saw which we can see from this motion also. This comes to the top dead center much later and expansion piston is already at the top dead center alright. So, this is what basically very important is to understand the importance of this phase difference between the piston and the displacer or expander piston also what is sometimes called as. So, in actual case now if I were to realize such a motion in actual case the discontinuous motion what we just saw cannot be achieved. Can I have a motion which is the motion for some time then it stops abruptly again there is a motion after some time. So, this is not possible. So, what is possible is normally a simple harmonic motion or a sinusoidal motion. So, in order to realize this practice in view of this a sinusoidal motion may be implemented. This is a very important aberration from ideal sterling cycle. So, actual sterling cycle may not have discontinuous motion I actual sterling cycle may have a sinusoidal motion because that is possible to be given in actual practice this motion is realistic. So, whatever motion we just saw that motion was there is no motion then there is a motion and again there is a motion in the reverse direction instead of that can we have a sinusoidal variation like that. So, we have a sinusoidal motion like that which is a simple harmonic kind of a motion which is possible to be to be given in actual practice and therefore, we call this motion is realistic and can be given using the crank or a gas spring mechanism. So, this is something which can be realized in practice actual sterling cycle in reality the actual working cycle may be different from ideal sterling cycle in following ways. So, now what we are doing we are going away from ideal sterling cycle and we are talking about in what ways the actual sterling cycle could be different than ideal sterling cycle what are different possibilities. The first possibility we just pointed out is a discontinuous motion it is difficult to realize in practice. So, in the actual case we may have a sinusoidal motion we cannot possibly have discontinuous motion over there. Also the presence of void volume what we just saw was we got a compressions wave volume we got expansion space volume and we got a regenerator volume. But in order to realize this in practice we may have some piping we may have some tubes through with the gas travels from compressions space to the expander space alright. That means, we got some more volume to what we just saw. So, this volume which is not travelled by piston or displacer is normally called as void volume or dead volume. In fact, the regenerator volume is also called as dead volume alright. So, presence of void volume or dead volume is a very very realistically possible in case of a actual sterling cycle. But having this additional void volume or dead volume is going to kill the COP of the machine. We have to sacrifice COP of the sterling cycle in that case. Also we will have pressure drop because the gas is travelling through regenerator and therefore, gas will realize some resistance to the motion of the gas depending on its viscosity depending on the porosity of the regenerator etcetera alright. So, we will have actual in the actual sterling cycle we will have some pressure drop that also is taking the cycle away from the ideal sterling cycle. Also we talked about having heat exchange between the regenerator matrix and the gas and this heat exchange may not be perfect and therefore, we will have some ineffectiveness associated with this heat exchange. So, this is a very important thing which has to be considered while designing actual sterling cycle. So, we will have ineffectiveness in heat transfer or regeneration. Is the gas transferring all the heat to the regenerator? Is the gas taking all the heat from the regenerator? It will all depend upon how effective this regenerator is and therefore, we will have to consider the effectiveness of heat exchange during this actual sterling cycle. Also the fourth possibility is non-isothermal compression and expansion. Now, in order to realize compression process isothermally it is very difficult as you know this has to be otherwise a very slow process. However, we call sterling cycle process as the speedy process is basically high frequency process and therefore, to realize isothermal compression in actual case is not so simple is rather difficult and therefore, we may not have isothermal compression in actual practice or we may not have actual isothermal expansion in practice and therefore, we will go away from ideality in this case. So, these are different possibilities because of which the actual sterling cycle will go away from ideal sterling cycle and therefore, the COP of actual sterling cycle will be quite less than what you otherwise get from ideal sterling cycle. So, ideal sterling cycle will give a same COP as Carnot cycle, but actual sterling cycle will not be as efficient as the ideal sterling cycle. Now, there are different sterling cryocoolers types and we will just briefly touch upon those types. So, depending on the relative arrangements of piston and displacer or this expander piston we can have a displacer or we can have a piston. Various types of sterling cryocoolers are possible namely alpha type sterling cryocooler, beta type sterling cryocooler and gamma type sterling cryocooler. So, these figures show over here this is alpha type sterling cryocooler, beta type sterling cryocooler and gamma type sterling cryocooler. So, here you can see that we have got a compressor piston, the compressed gas goes through the regenerator and the expander piston again. These are two piston kind of arrangement over here and this is what we call as alpha type. Then here we have got a beta type, here the compressor and the expander displacer or a piston is housed in one unit only, while they are in gamma type there are two different housings here and this is called as gamma all right. So, alpha, beta and gamma or they are also called as by different name which we will see now. So, alpha is also called as two piston arrangements ok. So, here in this two piston arrangement the driving mechanism may be mounted on the same crankshaft. So, we can have a same crankshaft here and it may be having two cranks one is driving the compressor piston one is driving the expander piston. So, here in this case we can have driving mechanism may be mounted on the same crankshaft over here. The other arrangement it is beta type is also called as integral piston and displacer arrangement. That means, the piston and a displacer are housed inside the same cylinder. So, here you can see that the piston is over here and this is compression swept volume, while above the expander displacer here we have got a expansion volume and both of them are now they could be driven by the same crankshaft in this case all right. So, in this arrangement which is the integral arrangement we can have the same crankshaft or the same crank driving the piston and the displacer. Then we got other unit which is called as gamma type or it is also called as split type piston and displacer arrangement. So, the split unit that means, you got a piston over here you got a expander displacer over here. So, this is basically displacer while in this case both are pistons all right. So, the compressor space in this case this is the compressor space and the gas may enter through the displacer over here. So, this is the compressor volume which is connected the compressor volume over here. So, the compressor space is divided with the compressor volume above the piston and below the displacer in this case in gamma type arrangement over here. These systems have variable dead volume in compression space due to the movement of displacer. So, when the displacer starts moving you will have a different dead volume as compared to what we have in other cases. And therefore, gamma type split piston type arrangement also may be used many times and this will have different drive mechanisms because displacer drive will be different while the piston drive will be different in this case. So, these are just the ways how this different mechanisms work and how they are classified as alpha, beta and gamma arrangements. Now, if I were to go for a design of sterling cryocoolers, this is a very important thing to understand. What do I have to do? I have to first understand what are my design parameters. So, you have got a compressor piston and you have got a expander displacer. The gas gets compressed over here it goes to degenerator where the constant volume process happens it comes to the expander space volume and the expansion occurs and gas gets cooled over here and you get the cooling if it at this point. If I were to design what is my compressor piston diameter should be, what is my expander displacer diameter should be or expander piston diameter should be, I should know how my compression space varies. What is the variation in compression space volume? What is the variation in expansion space volume? Corresponding to these volumes, what is the volume of the degenerator? Also at what temperature do I get cooling? At what temperature do I get compression? These are very important design parameters. And therefore, let us see what these design parameters are and these are very important if you as mechanical engineers were to go for designing this sterling cryocooler. So, let us see the various design parameters of a sterling cryocooler are as follows. Evaporator temperature or expansion space temperature which is at T e and this will be at this particular temperature. At this temperature, we got a isothermal expansion and therefore, the cooling effect will be generated at this particular temperature. Then we have got a compressor temperature which is T c. So, here the process of compression happens and we will get the process happening at T c at this point over here which is isothermal compression process and isothermal expansion process as we know what happens in ideal sterling cycle. Then we have got compression volume which is V c. So, what is my maximum compression space volume? What is my maximum expansion space volume also which will come into picture? So, depending on the diameter of this piston, depending on the stroke of this piston, you will have pi by 4 d square into stroke and that is what your compression space volume will be. Similarly, you will have expansion space volume depending on the diameter and stroke of the expander piston or a displacer. Then we have got a regenerator volume which is V r which is coming over here. Then we got out what is my pressure generation because the gas gets compressed, gas gets expanded. So, you got a maximum pressure, minimum pressure and average pressure. These are very important values to be known. Then what is my phase angle between the compression space volume and expansion space volume? We just saw that the compressor piston and expander piston do not reach top dead center at the same time, but they come after a phase lag of alpha and therefore, this is a very critical parameter which we will study in the next slides. So, phase angle is very important alpha and we talk about crank angle. Suppose these drives are given by a crank, then we got a crank angle also. So, all this together will basically form the design parameters which are very important and one has to know that for how much cooling effect is to be obtained at Te which becomes your starting point. If I were to design a sterling cryocoolers, I should know how much amount of cooling is required to be generated at a particular Te and corresponding to that depending on all these parameters, I have to design a sterling cryocooler. So, in order to take care of all these design parameters, Smitz has given his Smitz analysis and this is as I said is a one of the most basic analysis that is used for first guess of different dimensions that could be obtained in order to design a sterling cryocooler. So, in the year 1861 Gustav Smitz, a German scientist presented a sterling cryocooler analysis. This analysis is based on a realistic cycle that is from motion point of view. As we saw that discontinuous motion is not possible and Smitz considered a continuous motion or a sinusoidal motion which is a more realistic kind of a motion and he assumed that this motion to provide a first guess of dimension. The following are different assumptions. He assumed the perfect isothermal compression and expansion process as exists in ideal sterling cycle. He assumed harmonic motion of piston and displacer which is more realistic motion of piston and displacer. He assumed that there is a perfect heat exchange in regeneration. Also, he assumed there is no pressure drop in systems. The non-dimensional parameters in the Smitz analysis which he considered are swept volume ratio which is K vc upon ve. What is vc? The swept volume in compression space by the compression piston and ve is the swept volume in the expansion space by expander displacer. This ratio vc by ve is called as K. Then we got a temperature ratio which is Tc upon Te, compression temperature divided by expansion temperature and this is called as tau and we got dead volume ratio that means we will have some dead volume in the system. This is called as x which is equal to vd upon ve where vd is the dead volume in a system. So, we got a vd by ve, we got a vc by ve and we got Tc by Te. So, Kx and alpha. So, Kx and tau in addition to that we have alpha which is a phase angle between the piston and the displacer which we will see in the next slide. So, expansion space volume variation will be given by ve. The small e shows the variation of expansion space volume with phi which is a crank angle alright. When the crank moves corresponding to that ve will have ve small e as the volume of expansion. When the crank angle is at 180 degrees what you will get here is ve is equal to ve in this case. When cos phi is equal to 1 will have 1 plus 1 as 2 and ve is equal to capital ve or maximum expansion space volume. Corresponding to that we have got a compression space volume variation given by this formulation where we can see that phi is now phi minus alpha, alpha being the phase difference between vc and ve. This is vc and this is ve and this is maximum volume in the compression space. So, this is highlighting the presence of alpha in the compression and the expansion space variations and this will be always there. So, you can see if I were to plot these two variations we have got a ve variation which is a sinusoidal and we have got a vc variation which is also sinusoidal and this is the alpha angle between the two. So, ve is leading the expansion space volume is leading the compression space volume variations by angle alpha which is one of very important parameters. You can see later that if is alpha is made equal to 0 you will not get any cooling effect alright. The cooling effect is obtained basically due to this phase difference we should be optimally designed. So, we got these variations of compression space volume and expansion space volume and here we can write vc as k into ve in that case in that case this formulation will turn out to be this. Now, if I were to do the mass balance for entire cryocooler we have got a mass m t is equal to p v upon r t which is the mass fraction in the expansion space, mass fraction in the compression space and mass fraction in the data volume p d v d upon r t d this is by total mass in a system at any point of time. Let the instantaneous pressure in the system be same throughout the pressure that means there are no pressure drops in a system this is assumption in Smith's analysis that there is no p e p c p d they are all the same as p and in that case also assuming that t e t c are constant temperature which is what we know that their isothermal compression process isothermal expansion process. So, t e t c are assumed to be constant as t e and t c respectively in that case my m t will be given as I can write this entire m t as some constant into ve upon 2 r t c it is just an assumption that there is some constant and I can represent entire this thing as expansion space volume divided by compression space temperature related by k ve upon 2 r t c ok. This k takes care of all other things basically. So, now I can write this m t is equal to all these parameters is equal to k v upon 2 r t c ok. Now, I will manipulate this algebraically. So, if I take p v we know that there is no p e p c and all that is p can be taken common r t c can be taken common and ve also can be taken common my entire expression now will be will look like this. So, 1 plus cos phi t c upon 2 t e t c upon t e nothing but tau then k will come into picture and phi minus alpha term will be come into picture for compression space variation then by dead volume and we know that this is equal to this temperature now this constant into ve upon 2 r t c then putting the value of t c upon t e as tau x is equal to v d upon v e as we have earlier decided to have and assuming that the dead space volume is a mean value between t e and t c. So, t d is equal to t e upon t c by 2 which is what will come over here also we define one more constant as s which is 2 x tau upon tau plus 1. So, if we do some earlier algebraic manipulation this s value also will figure over here. So, I am going to replace all this thing by their non dimensional values over here entire equation now will get reduced to this k by p is equal to tau into 1 plus cos phi plus k into 1 plus cos phi minus alpha plus 2 s is very simplified now and all the constants are defined over here. If I define further constants as a b and delta as a by b and putting these values over here in this equation on the mass equation I will further go as tan theta I have defined one more angle as tan theta which is k sin alpha upon tau plus k cos alpha which is given in this a substituting a b theta and delta in the mass equation and rearranging them that is algebraic manipulation can be done what we get is a pressure expression this is a very important expression for pressure which is k upon b into this. In this case I will get pressure as minimum value that is my p minimum when my denominator is maximum and the denominator is maximum when cos theta minus phi is equal to 1 and therefore I will get 1 plus delta at this will happen when phi equal to theta and I get p max p is equal to p max when my denominator is minimum this will happen when my phi is equal to theta minus phi or when this particular parameter is minus 1 in this case alright. So, I get maximum pressure as a function of constant divided by p b into 1 minus delta and I get minimum pressure as k upon b 1 plus delta. So, if I got to pressure ratio that is p max upon p minimum is nothing but 1 plus delta upon 1 minus delta and what is delta? Delta was equal to a by b where a by b has been defined earlier and this is a typical expression for Smith's analysis for pressure and if I integrate that over complete cycle 0 to 2 pi what I get is a mean pressure which is defined and this mean pressure can be expressed as p max into 1 minus delta divided by 1 plus delta under root and if I were to now calculate cooling effect which is nothing but integral p d V e which is what you know I put the value of pressure in this and I put the value of d V e in this case and I will get now what is cooling effect over here and this term gives me cooling effect similarly if I were to find out what is the work done during compression which is integral which is q c which is integral p d V c if I do the similar integration keeping the value of p here I get this expression alright. Now, I am to calculate what is the c o p of the machine I know c o p of the machine is equal to cooling effect divided by work input which is equal to cooling effect q e obtained as this over here divided by q c minus q e and if I were to put all this value at the respective position I will get c o p ultimately equal to T e upon T c minus T e which proves that in this case also by Smith's analysis the c o p of the sterling cycle is same as Carnot cycle. So, T e upon T c minus T e by doing all these things now I am relating pressure generated in the system pressure ratio generated in the cryocooler and it is all related to what is my compression space volume what is my expansion space volume and thing like that. This is a very important analysis which relates all these parameters together and based on all these parameters we can find out what is the cooling effect and also what is the power input to the system which is q c minus q e and what is the c o p of the system again this is based on the assumption what Smith's analysis has assumed. Now, there are different losses in the system which Smith's analysis has not taken into account. So, in the earlier slide we saw the cooling effect based on Smith's analysis, but in actual case there are many losses as given below what are these losses ineffectiveness of heat exchanger or ineffectiveness of a regenerator which has not been taken into account pressure drop in the system we had assumed all the pressures to be the same we got solid conduction because we got a high temperature and low temperature across the solid members in the system we got a shuttle conduction because the motion of the displacer up and down and we got losses in power input because of mechanical efficiency. So, all these losses have to be taken into account in order to get net cooling effect that is available from the system and also net power that is required to be given to the system and therefore, we will have a net c o p of the system also. So, normally considering the above mentioned losses the net cooling effect and gross power input to the system is given as following correlations q net is equal to whatever we have got from the Smith's analysis q e minus sigma losses this is my net cooling effect my net power input w total is equal to whatever w t I have calculated based on Smith's analysis plus sigma losses. Now, in Smith's analysis we do not calculate all these losses over here while they are taken as some factor of q e all right. So, as I said that this is the first guess of analysis which we use to have first case for the dimensions of the sterling cryocoolers we assume that out of q e calculated from the Smith's analysis about 60 to 70 percent is considered as loss in cooling. So, if I calculate 100 watts as q e I will understand that 65 percent is loss as losses and what is available q net is only 35 now. So, first calculate q e based on various dimensions assume that 60 to 70 percent is lost and what is net available is q net after that. Similarly, I have to take mechanical losses into account to get to calculate what is my net power input to the sterling cryocooler and this is what will give me a first case of different dimensions for sterling cryocoolers. So, in summary I will to write the summary of this lecture a sterling cycle was first conceived by Robert sterling in the year 1815. We know that COP sterling is equal to COP Carnot in reality the actual working cycle has discontinuous motion pressure drop ineffectiveness and non isothermal process. Depending on the relative arrangement of piston and displacer piston alpha beta gamma are different types of sterling cryocoolers Smith's presented a sterling cycle analysis in the year 1861 it is assumed to provide a first guess of dimensions the net cooling effect and gross power input is given by following correlations q net is equal to q e minus sigma losses w total is equal to w t plus sigma losses a self assessment is given based on this lecture kindly assess yourself for this lecture and these are the different questions please try to answer those thank you very much.