So, welcome to the 17th lecture of cryogenic engineering under the NPTEL program. In the last lecture, I will just give a brief outline just to see what we did in the last lecture. In the earlier lecture, we have seen that Capitza and Helant systems, which are nothing but modifications of Claude system. In the Claude system, what we had was 2 to 3 heat exchangers and 1 expander. While in Capitza and Helant, we had only 2 heat exchangers and 1 expander and it has a kind of modification from the Claude system. Then we went to Collins-Seiskal or Collins system, which is nothing but an extension of Claude system to reach lower and lower temperature. For example, liquid helium temperatures of around 4.2 Kelvin, where in now 2 to 6 expansion devices are used. Collins system basically use more than one expansion device. The expansion device produce cold and in order to reach lower and lower temperature and depending on the mass for rates or the amount of cooling effect what one wants to have, we will decide the number of expansion devices that one wants to use in a Collins cycle, which is nothing but a modified Claude system. For a given pressure condition, that means if P 1 and P 2 or the inlet pressure to the compressor and outlet pressure to the compressor are known, the yield y and the work done per unit mass of gas which is liquefied, this depend on the fraction of the gas and diverted to expander 1 and 2 which are nothing but x 1 and x 2 as we saw last time and the temperature at the inlet to the temperature that is T 3 and T 5. So, if you see a Collins cycle, the yield from the Collins cycle and the work that is done per unit mass of gas liquefied, it depends on what are the values of x 1 and x 2 that is what are the fractions of the gas which are diverted through expansion devices 1 and 2. Assuming that there are 2 expansion devices and also at what temperatures this gas is diverted from expansion number 1 and expansion number device number 2. Now, this is basically going to decide what is the value of y that is the yield and the work done per unit mass of gas liquefied for Collins system. Now, in today's lecture, I am again going to talk about the gas liquefaction and this is going to be last lecture now regarding gas liquefaction and refrigeration. What I am going to discuss now in brief are different components in a gas liquefaction cycle and also in a refrigeration cycle. The different components are heat exchangers, the compressors and expanders. In addition to that, having seen all these devices, having understood all these systems, having understood all these cycles, I would like to show you now how a particular cycle works. How does one get liquid nitrogen or how does one get liquid helium? That means, how does one reach 77 Kelvin to liquefy nitrogen and also how does one reach 4.2 Kelvin to get liquid helium. So, I have got some videos to show it to you because we cannot take you to the plants but I have got some videos from these plants and let us try to understand how this cycle will happen. But in cryogenesis, you cannot see anything closer because everything is enclosed, everything is under vacuum condition. So, one cannot really see what is happening inside and therefore, what you see is only from outside, everything is under vacuum, everything is at very low temperature and not accessible from room temperature. So, you will have to see what I will tell you as to what exactly they are inside this vacuum's device. Also I will like to have comparison of different systems to so that one gets feel of different liquefaction cycles. There are different liquefaction systems where I will just show a small comparison assuming that the pressure ratios are known to us and finally, I will summarize what all we did during the last 10 or 12 lectures in which we covered gas liquefaction and refrigeration system. In the earlier lectures, we have seen that various gas liquefaction and refrigeration system and we have seen that the various components like compressors, expanders, heat exchangers are absolutely critical to the performance of the system. We had seen that how effectiveness of heat exchangers matter, how does it affect the yield, also different efficiencies related to compressor and expanders, how do they affect the yield and the work done per unit mass of gas which is liquefied. These processes of compression, expansion, heat exchange that occur in these components are irreversible and deteriorate the performance of the system. Hence, why are we studying all these things? Because we know that these are very significant, they have very significant effect on the liquefaction and therefore, there is a need to study about these components that are used in this system. So, we should know what kind of heat exchangers are used, what kind of expanders are used, what kind of compressors are used, what are the problems with these units. Although we cannot go in the details of all these system components, at least what we can do is we can touch upon briefly to see how these components work. So, first we come to the heat exchangers which is a very very important component of the liquefier. The heat exchangers are the most critical components of any liquefaction system. They are used to conserve cold by heat exchange between the high pressure hot gas and the low pressure cold gas. We know that the low pressure gas after liquefaction, the remaining gas goes back and this is cold gas and therefore, whatever cold it is carrying, we use this cold to pre cool the incoming hot pressure gas, high pressure gas. Now, this graph tells you the variation in the value of y that is a yield with the epsilon value. So, you can see that if the epsilon value or the effectiveness of heat exchanger is 100 percent, we got a very high yield or the highest yield around let us say 0.09 or something like that. But you can see as the epsilon value goes down from 100 percent to 95 percent to 90 percent, the yield starts coming down. So, this curve tells you how important it is and what you can also see from this, if your epsilon value or the effectiveness value is less than 85 percent, the yield is absolutely 0. That means, it is essential that your yield or the your heat exchanger effectiveness is somewhere close to 95 percent. So, that you get a good value of y. So, that you get a reasonable value of y. Definitely, when one cannot reach 97 percent, 98 percent because there are fabrication limitation, there are assembly limitations, there is pressure drop in the system. All these things have to be considered while design a heat exchanger, but at least one could strive to attain at least 95 percent effectiveness for heat exchangers and therefore, we know that the effectiveness of heat exchanger has to be more than 85 and if one wants to have a good y value or a reasonably good y value, the effectiveness has to be let us say more than 90 or 95 percent. What are the requirements of heat exchangers? Now, in order to qualify to be a good cryogenic heat exchanger for gas liquefaction, it has to satisfy certain criteria, it has to satisfy certain requirements and what are these requirements? The requirements are, it has to have high effectiveness with minimum pressure drop. It has to be a compact heat exchanger and this is very important because in cryogenic, if you got a very big heat exchanger, for example, if you got a shale and tube heat exchanger, lot of cryogenic will be utilized to cool the heat exchanger itself. That means, it will have a lot of mass and therefore, in cryogenic, the cool down time will be very high. When you start the liquefier, it will take a lot of time when the liquefaction starts. So, therefore, one has to have a very compact heat exchanger and what does it mean? It should have a high heat transfer area per unit volume. If we say that, if the heat transfer area per unit volume is more than 700, then it is normally called as compact heat exchanger. So, one should strive to go in a compact heat exchanger area zone as far as cryogenic heat exchangers are concerned. We should have, therefore, minimum mass with multi-channel capabilities. If you got multi-channel capabilities, you can have more flow rates, more heat transfer area also. One should have definitely high reliability with minimum maintenance and these are very general criteria. What is most important is high effectiveness and the compact heat exchanger. Now, there are different configurations that are possible for heat exchangers to be used in the cryogenic heat exchangers and they are tubes in tube, bundle tubes, fin tubes and plate fin heat exchangers. So, different kind of heat exchangers could be used and in order to maximize the heat transfer area per unit volume, we can have various structure. We can have many tubes in tube. We can see that in the next slide. We can have fin tubes. We can have plate fin heat exchangers also. And as you know from different cycles what we have seen, heat exchanger can be of 2 fluid or 3 fluid types depending on the heat exchanger and cycle we are talking about. So, normally the heat exchangers are called linde tube heat exchangers which are commonly used in liquefaction systems because they were invented by linde and used by linde. So, you can see here a tube in tube heat exchanger with a high pressure gas goes inside the tube and the low pressure gas comes over the tube. So, it is a tube in tube kind of heat exchanger or linde concentric tube heat exchangers. Then instead of 1 tube, we can have 3 tubes in 1 tube. So, linde multiple tubes heat exchangers. They are pretty simple to understand from here. So, here we can have a coiled structure of the heat exchangers and the high pressure gas can go in the inner tubes while the low pressure gas can come above this tube. Now, one can have a linde concentric tube heat exchanger with a wire spacer. So, the low pressure gas which is coming over can see this turbulence which is created by this wire spacer. And therefore, the heat transfer area on the low pressure side could be increased in that case. We can have bundled heat exchangers like this. So, so many solar structure is available of so many tubes over here and you can have heat transfer between different tubes. The tubes in tube type heat exchangers as you can see over here are the simplest of all types in terms of construction. So, all these heat exchangers are actually called tube in tube and they are simple heat exchanger or manufacturing point of they are simple, but the amount of heat transfer area that is available here could be limited. They may not be classified under what you call as a compact heat exchangers, but they are still used in cryogenic heat exchangers. These have low cost and are available suited for high pressure applications. All these heat exchangers for large flow rates one goes for 3 tubes are used in a bigger tube or a 3 channel heat exchanger. So, we can have different flows through different tubes also. The use of a wire spacer or a tribulator like what you can see over here on low pressure side acts as an extended surface and enhances heat transfer on the low pressure side. I think this is mostly clear because many of these things possibly you have understood or you have covered in a heat transfer or heat exchanger courses. Now, very typical kind of heat exchanger which was used by Collins and therefore, it is normally called as Collins heat exchangers and what you can see here is a fin tube and shell kind of heat exchanger. Basically what you can see are different fin tubes wound around a mandrel. So, you can see a mandrel over here around which concentrically arranged fin tube are there which you can see in top view here there are all fin tubes. The high pressure gas goes to the inner tube over here while the low pressure gas the high pressure gas goes in a coiled form it will come down the end it will get expanded and the low pressure gas will go over these fins and then it will come out of the top and you can have various layers of this fin tubes. So, that you get higher heat transfer area available in a given volume. So, the Collins type heat exchanger consists of several concentric copper tubes with an age wound copper helix wrapped in annular surface. So, what I am calling as fins could be a age wound copper helix which acts like a fin alright and therefore, what is most important is what is the contact of this copper helix with the main tubes over here. Therefore, the fin efficiencies or the contact resistance should be 0 and the fin efficiencies should be higher in this case. This helix acts as a fin and enhances the heat transfer area. The heat transfer area on the low pressure side the low pressure side gas sees this fins while the high pressure gas goes inside the tube and as you can see we can have two layers three layers depending on the kind of flow rates which we are talking about and corresponding to that whatever heat transfer area required one has to calculate and accordingly design this heat exchanger. In this heat exchanger the high and low pressure streams flow in the inner and outer passages respectively. So, the high pressure gas goes through this inner tube while the low pressure gas come over the fins in a coiled manner the tubes are stacked like this and the gas will come over this fin tubes. You can see the same thing over here where you can see that the coiling is done over here. The high pressure gas will come through this tube while the low pressure gas will go over this tube and this is basically kind of a shell what you can call it and therefore, sometimes there are also called as not only coiled fin tube heat exchanger, but fin tube and shell also can be this can be referred as the most important or most prominently used heat exchanger is normally which is called as aluminum brazed plate fin heat exchanger. This is very costly item and very difficult to manufacture. However, this is a very compact heat exchanger that means it has got a heat transfer area more than 700 meter square per unit volume. So, aluminum brazed plate fin heat exchangers are most compact heat exchanger with high heat transfer area to volume. This can either be single or multi stream these are widely used in air separation plants and helium plants and how does it work? What you can see here is you have got a fins over here that could be normal fins could be u type fins could be of different characteristic fins over here and this fins has a top plate and a bottom place and a side plate also. So, this makes one layer of the fin and here you can see that these are made in layers and there could be 10 layers or 20 layers depend on the flow rate and all these are basically arranged alternately. So, that high pressure gas goes through or the hot stream can go through one layer and if it is a cross fluid exchanger above it comes the other gas. So, there are two fluids one fluid this way one fluid this way and therefore, you can find you can understand that is a cross flow kind of a aluminum brazed plate fin heat exchanger. I got a small unit to show to you here and this unit can be seen over here. So, this is basically what you can see on if I show like this you can see as I have just shown in the figure that you have got a one layer over here and this is a one layer. So, you got a top plate bottom plate in which what you have got is a kind of this kind of a model basically not for this for experiments and you can see different fins which are kept over here and the whole thing is aluminum material and it is a brazed structure. So, gas can go in this direction all this will be having one header and high pressure gas can go through these layers while alternately you can see that there is a other gas which can go through here. So, hot gas can go through in direction the other gas cold gas can go in this direction which is alternate to this. So, this minimum thickness is basically through which the other gas is going this is what you can see between this two layer is the other gas which is going from this direction. So, I hope you are understanding that one gas is going this direction below and above of these two layers and they are alternately stacked. So, one gas goes this way the other gas can also come down from this if it is a counter flow in this particular what you can see model it is a cross flow arrangement. So, one gas is going this way and then alternate layer one gas is going like this. So, all these things can have one gas in a coupled through a header the gas can go in this direction one header can here and other gas is going through direction. Now, there are different fluids that that could go in these directions also. So, we can have one fluid going through this top two layers the other fluid is going through bottom three layers also similar thing can happen over here. So, you can have a multiple fluid facilities heat exchange over here in a plate finite exchanger and one can see how compact it is and you can imagine such units made in big length of 1 meter or something like that with 10 layers or something like that then you will have a tremendous heat transfer area that is available in a given unit volume and this is what we call as a aluminum brazed plate finite exchanger. So, coming back the critical requirement in this is the thermal design and fabrication because aluminum brazing is not a very simple process and therefore, because aluminum gas oxidize immediately and therefore, the aluminum brazing requires lot of pre processing and post processing. So, thermal design is very important depending on the fins efficiency types of fin etcetera. At the same time the fabrication process also is a very demanding process and aluminum brazing as it is done in a vacuum environment or a controlled environment and therefore, this is a very special class of manufacturing or brazing. But this is a very important device because plate finite exchangers are very compact heat exchanger they are definitely are used for all the liquefaction high flow liquefaction system they are commonly used. So, the same thing I have just shown your photographic manner so that this is always with you. So, you can see one gas is going over here the other gas is coming through in this direction. Having seen heat exchangers let us come to compressors and in compressors we can see now a compressor is a source of high pressure gas for any liquefaction or a refrigerating system. So, it is clear to us that compressor has to be there so that we get high pressure gas. It is also the biggest source of heat generation due to the motor inefficiency and a gas compressor. Again this is known to us that the isothermal efficiency of a compressor is a very important thing overall compressor efficiency also the very important parameter. The two broad classes of compressors are reciprocating compressors and a rotary type of compressors any of this could be used depending on the flow rates and pressure ratio which we are talking about. The reciprocating types are used for high pressures applications with low flow rates whereas, the rotary types are used for high flow rates and at moderate pressures. So, we can have reciprocating as well as rotary depending on the flow rates and the pressure ratios we are talking about. The losses associated with the compressors are given by isothermal efficiency, adiabatic efficiency or mechanical efficiency or overall efficiency. All these efficiency have to be taken into account in order to calculate the work input to the compressor. Normally screw compressors and scroll compressors are used for helium gas or nitrogen gas. So, screw and scroll compressors higher isothermal efficiency, low initial cost and more reliability and they offer a vibration pre-performance. So, most of the places you will find either screw or scroll compressor they are actually rotary kind of a compressors and they are being used for most of the cryogenic applications. The compressors being oil lubricated both these compressors are oil lubricated the oil content in the compressed gas is reduced by using oil filters. As you know cryogenics cryocoolers or cryogenic heat exchangers or liquefies do not like oil because oil will get frozen and therefore, all the gas which is coming out of this compressor should be free of oil and therefore, oil filters are a must. This is a very important requirement that this gas has to be stripped of the oil and only oil free gas should go to the low temperature area. It is further purified in a gas purifier system consist of activated charcoal bed. So, that all the moisture etcetera is removed from this gas and therefore, only dry gas without moisture without oil should go to the heat exchangers and expanders. Apart from these centrifugal compressors have better reliability and are used in liquefaction and separation of gases and air separation plant. Depending on the again for a very high flow rates centrifugal compressors could be used they are also reliable and are used for liquefaction. Screw compressors are oil lubricated and are generally used for high pressure ratios. As I said screw compressors and scroll compressors generally used for liquefaction of a moderate quantity of gases. So, having seen heat exchangers and compressors now, let us come to the expanders in short and I will have 3, 4 slides on the expanders. So, expanders are used to produce cold as you know that expanders are very important because it is the one because of which the gas undergoes isentropic expansion and which produces cold in the system. This system must be well insulated to avoid heat in leak from the ambient. In fact, all the heat exchangers all the expanders should be well insulated here because the heat should not come from the ambient. On the similar lines of a compressor reciprocating type of expanders are used for low flow rates and high pressure ratio. If you want to expand the gas from for a high pressure ratio, then one should go for a reciprocating expander, but then it can take a limited flow rate in this case. On the other hand, a turbo expander is used or a rotary expander is used for high flow rates and low pressure ratios. The design involves high technology and almost zero maintenance. Mostly, the new technology is using turbo expanders which run at a very high speed. They are really very good for high flow rates. However, the technology demand is too much. The design is very sophisticated. Sometimes have almost zero maintenance requirements. The rough schematic of a turbo expander I am showing over here. So, we have got a shaft which is moving at a very high RPM. One end is a turbine wheel which could be a very smaller wheel. The other one is a compressor. On this side, basically this compressor is kept to basically control the speed of this turbine over here. This turbine speed is very high. The RPM could be of the order of 1 lakh or 2 lakhs or around 6000 to very high RPM in this case. It is an expander turbine wheel and a compressor mounted on the carbon shaft. The work produced in expansion across the turbine wheel is used for the compressor. So, basically compressor is the amount of work which is produced by turbine is not substantial. Therefore, normally this work actually dumped in an alternator or sometimes this is and the compressor is basically used to control the speed of this turbine. To ensure high speed, high efficiency for high mass flow rates, turbo expander in small diameters are operated at very high speeds. So, as I said 3000 to 4000 rps which means around 2 lakhs above kind of a RPM which we are talking about and such a small turbo expander is going to be running at very high speed and this is the most important design requirement for. As soon as we talk about very high speed, the shaft has to have correct bearings and therefore, very high technology requirements will be there in order to basically take loads of this kind. However, efficiency degrades due to various non ideal conditions like leakage around turbine wheel, windage loss, finite number of flow passages etcetera. So, therefore, as a result of all these losses, you will have some expander efficiency which could be of the order of 75 percent to 70 percent like that or sometimes it could be as as 80 percent also and this has to be taken into account while designing the expander. Turbine bearings, balancing and manufacturing are still matter of research. So, lot of work is still being carried out in order to design turbine bearings, balancing of turbine on a shaft, manufacturing and thing like that. So, lot of research work is being carried out on the turbo expander. With this background having done compressors, heat exchangers, expander. Let us now see the working of liquid helium plant and liquid nitrogen plant. You can see here liquid helium plant. I have got a schematic of the liquid helium plant and this is the cold box which basically houses, expander, heat exchanger wall, piping, sensors and insulations. So, it has got everything as I said different heat exchangers and expander. So, high pressure gas from compressor gets into the cold box and the liquid is produced in the cold box which goes to the liquid receiver. So, the mist helium that is the vapor plus liquid come over here, the liquid gets received over here, it is stored over here and the remaining gas goes back at low pressure and cycle continues. Now, this is a basically cycle of which we have talked till now. This is a closed cycle, compressor, expander and heat exchangers, the liquid is stored here and the gas goes back and cycle continues. So, from here the users take liquid, use the liquid or the liquid is transferred from here to even diva vessels which are taken by different laboratories for end usage. So, liquid helium is directly used by the users right here in the site or they could be transferred to different divas and different divas are taken to different labs. Now, this helium being a very costly gas wherever it is used, this liquid will get converted to gas, this gas cannot be left to atmosphere and therefore, this gas is recovered and this is therefore, there are lot of recovery lines may attached to different utility points. So, all the gas or all the liquid which gets evaporated which become gas is stored in a gas bag with this recovery lines. So, here a gas bag is normally a kind of a rubber bag in which the gas is stored and once it has recovered most of the gas, the gas goes to a recovery compressor and it is then a pressurized by using a recovery compressor which is different from the main compressor and this actually this gas is normally called as impure gas because it is collected from various end users and therefore, they are stored in impure cylinder over here. Now, this impure gas stored in impure cylinder has to be first purified alright and the pure cylinder gas also sometimes needs some purification alright. The purification is a very important phenomena because you have to remove all the possible contaminants like air, moisture, carbon dioxide etcetera from this gas. So, the purifier will purify the gas and then it will go to some intermediate pressure buffer and it will stored at this point and from here the gas will be supplied to main compressor and the cycle continues. So, what is important to see here the liquid helium whatever liquid helium is used the helium is not let allowed to go outside to the atmosphere all this liquid. In fact, one says that almost 98 percent gas which is given in the liquid form should be collected. So, that the recovery is almost 98 to 99 percent alright. So, whatever liquid is given to the users the helium liquid will get evaporated all this evaporated gas. Now, should be stored it should be stored in a gas bag it should be purified using a recovery compressor stored in the cylinder the gas should be purified and again the cycle continues and this is what a liquid helium plant will look like. Now, let us see a liquid helium plant and some specs associated with this at IIT Bombay we have got a liquid helium plant and the specifications of these plants are this is basically a linde 14 tin machine model number belonging to linde company the output is 15 liter per hour that means, in one hour it produces 15 liters of liquid helium. The liquefier inlet pressure is 17 bar 17 bar is basically inlet pressure to the heat exchanger that means, the gas gets compressed from almost 1 bar to 17 bar and the expanders in this case are reciprocating time to understand that they are not basically the rotary type there is no turbo expander because, we are talking only about 15 liter per hour the moment I talk about having around 100 liter per hour or 200 liter per hour it will be justified to go for a high flow rate system that is turbo expander nowadays even for 15 liter per hour turbo expander are being used. The RPM of this expander is around 230 and sometimes they could be liquid nitrogen cooled also. So, you can have pre cooling done on this gas the liquid helium plant has a main compressor which is a screw type compressor it has got to have a chill water cooling because of the compressor needs cooling oil lubricated the suction is 1.33 bar and deliveries 18 bar and the power input is 80 kilowatt. Now, we can see that 80 kilowatt power input electrical power input is given to get 15 liter per hour liquid helium from this plant it is a very it demands lot of power basically. In addition to this main compressor we have got other compressor which is a recovery compressor it is a 4 stage reciprocating type air cool and oil lubricated the suction is around ambient and delivery as 17 bar and this also requires electrical power input of 11 kilowatt. So, once you are operating both the plants you got a both the compressors you got to have 80 kilowatts you are here and 11 kilowatts at the recovery for the recovery compressors. So, the power input is of the order of 90 kilowatt in this case then these are got a buffer volume of 1 meter cube capacity the cylinder pressures are 133.3 bars chiller for the main compressor is blue star the temperatures of the water which come out is 11 to 15 degree centigrade over here liquid helium which we are producing use for various end users one of which is p p m s or physical property measurement system the consumption is 15 litre per day while there are other system like NMR apparatus which requires continuous replenishment of the liquid helium. With this background I would like to show you a video of the liquid helium plant right. So, with this background with this specification which you just saw of the liquid helium plant let us have a look at actual liquid helium plant that is we have with IIT. Now, this is a linde plant giving you around 15 litres per hour and what you can see now on the screen is basically the all the containers or the divorce in which the liquid helium which is produced is kept. So, what you can see here are different containers and this is around let us say 200 litre capacity around 100 litre capacity and this is around less than 100 around 80 litre capacity for liquid helium. So, these are different divorce and when you add liquid helium to this some liquid helium will get evaporated which is then taken care of in the recovery bags this is stored in the recovery bag and this is what you will see now. So, this is a video and this is around 200 litre and you can see this is a kind of experimental divorce where some experiments could be carried out. From all these things what you see is a there are recovery lines.  So, on every divar wherever you see a helium divar you will see some recovery line and the helium which is getting boiled off from this is getting stored in a recovery bag which is stored above this and we will see that later at the end of this video. So, there is all the recovery lines which are where boiled off liquid helium gas will be you know stored it is not let go to the atmosphere as what you see otherwise in nitrogen and these are the piping and now let us come to the actual helium liquefier machine. So, this is the helium liquefier machine and here you can see that in this it houses heat exchangers as well as the expanders. Let us see the video ahead and this is the most important place where all the action takes place. So, this is our cold box this cold box houses heat exchangers the entire liquefier works on colline cycles. So, know that you know that there are 5 or 6 heat exchangers and there are 2 reciprocating expansion engines alright. So, this is a cold box which houses all the heat exchangers and the expansion engines and there will be inside divar outside divar and there is a vacuum in between with insulation and the vacuum is always you know whenever you want to start the plant the vacuum will be made first on. So, that the vacuum is always maintained over there is a rotary pump and whenever you want to start the plant first you start the vacuum rotary vacuum pump. So, that you get a good vacuum out there and then you will start the compressor later. So, this is normally called as cold box which houses the heat exchangers and the expansion engines and there are various pressures which are recorded the vacuum which is recorded all those dials are shown over there many times nowadays these are all digital dials also. So, just have a close up of the machine and what you see now here is some display. So, you can see the display where you can always see what are the temperatures at various points in the machine. For example, the inlet temperature the J T temperature or the gas temperature before the J T expansion also what you can see is what is the RPM of the expansion engine number 1 and expansion number engine number 2 and things like that. So, all the data which what is essential for an operator to know just to know that if the machine is working properly or not what are the temperatures at various locations what is the RPM of the expansion engines all this data can be seen over there and accordingly the action may be automatically taken or in some cases if the plant is of older generation the action has to be taken manually. So, this is the display which will be there and this is now the liquid helium which is produced in the cold box it will be transferred using this transfer line through the main divar. Now, this main divar is of around 500 litre capacity alright. So, this is a transfer line which transfers liquid helium from your cold box through this and enters the main divar and from this divar if suppose somebody want to take liquid helium from our laboratory it will be transferred from this to the user this is never touched this will never go this always will be associated using this transfer line to the cold box. So, this part is always untouched from this main divar or sometimes called as mother divar it will transfer the liquid helium to other divars alright. So, let us see the video ahead and here you will have some liquid level indicator which tells you how much is the liquid helium content over there in the divar. Now, what you see here are the moving components and what you see now here in this direction you can see now these are the two expansion engines and the engines are basically kept open in the cold box below which we just saw and these are the two reciprocating expansion engine one on this side one on the other side. As I said there are always two expansion engines there could be four expansion engine depending on the flow rates and the fly wheel here. Like any IC engine you have got an inlet and outlet walls. So, there are puppet type walls you can see here on either side with the inlet wall to the gas the outlet wall to the gas and this expansion engine may be moving at 250 rpm 300 rpm and as the gas gets cooler and cooler the gas becomes denser and denser and in that case the rpm of this will be as low as around 60 rpm. So, that feedback control loop will be there and depending on the temperatures inside the rpm of this machine will change alright. So, you can see these two expansion engines which are moving right now here. So, essentially what you have is a heat exchanger and the expansion engines. So, now let us come to the compressor that was what the cold box was all about and this is now the compressor which is the most important thing and this compressor now is a screwed up compressor and it compresses the gas from around 1.2 bar to around 17 or 18 bar alright with some flow rate of 12 to 18 gram per second of helium. So, you can see this compressor this compress is going to be water cooled because the temperature after compression going to be very high alright. Now, what is most important is we just saw the other time in from the from the diva from different user divas that the gas goes to the recovery bag and the recovery bag will get inflated as the gas comes over there and this inflation is major in some terms of height of the recovery bag. So, as soon as this inflation of the bag exceeds a particular value the recovery gas now is going to will be compressed in a recovery compressor and it will then be stored in impure cylinder and this cylinder gas also will be used for liquefaction over a period of time. So, what you see is this is a recovery line and this is the recovery line connection which indicates gets connected to the recovery bag meant out of a rubber and this gets because it is flexible because of being in the rubber material of rubber material it gets inflated. So, you can see now this is a recovery bag and as the recovery bag becomes bigger and bigger then it will be the gas will be getting transferred to recovery compressor and this is the recovery compressor alright this is around 11 kilowatt recovery compressor and what you see now is a user alright some property measurement is being done over here in this container here where liquid helium is transferred and again the recovery line of that will be connected to the recovery bag. So, you can see normally wherever helium production is done the user also has to normally preferably sit near the liquid helium diva so that you will not have losses in transferring helium liquid helium from place A to place B. The users are normally kept nearer to the liquid helium production and whatever liquid helium boiled off will be there it will be collected in the recovery bags and the recovery will be faster and efficient. Whatever helium is delivered all that helium gas has to be recovered we should have a recovery ratio of above 98 percent that means we should not lose any of the helium that has been delivered in the liquid form ok. So, this is a user's community who is now nearer to the liquid helium production and you can see again that the helium gas will go to the recovery connection later ok. So, the piping plumbing has to be taken care of properly to ensure that gas alright and then they need to have a chiller and cylinders the infrastructure required what we saw earlier was water chiller we supplies chilled water for the compressor and these are the different cylinder helium cylinder these are actually manifolds of pure cylinder and impure cylinder. The gas from the recovery compressor is actually stored in the impure cylinders while there are pure cylinders with which we start the plant. So, we will have a pure cylinder manifold below the impure cylinder manifold alright and they both are put to action while operating the helium plant and this is what in short the video tells you how the helium plant operates. Having seen the liquid helium plant now let us see liquid nitrogen plant at IIT Bombay following other details now this liquid nitrogen plant is not a open system over here it uses a sterling cryocooler with helium as working fluid used to liquefy nitrogen. So, this works on a different principle it is not a open loop system this cryocooler generates cooling effect at 77 Kelvin and it has got a condenser the nitrogen gas comes on this condenser and it gets liquefied. So, this is a closed cycle cryocooler which is running all the time and the nitrogen gas will come from the top it will get condense and what you get is a liquid nitrogen. So, it is a completely different system as compared to what you saw for liquid helium or whatever systems we have studied earlier. So, it basically is a closed cycle while all are the earlier systems were open cycle that means the working fluid itself used to get liquefied here the working fluid just produces cold while other fluid nitrogen comes over there gets liquefied and therefore, you get liquid nitrogen from here. Now, what are the specifications of this unit the specification of this plant are the model is sterling cryogenics the output is 50 liter per hour. So, you can see the production rates are higher over here now it has to best first produce nitrogen from air. So, air compressor is used and then from air oxygen and nitrogen are separated and nitrogen is then sent to the cryocooler to get liquefied or to get condense. So, one has to have a air compressors the power input to air compressors 25 kilowatt the pressure is 15 bar the nitrogen plant has a motor power of 45 kilowatt if the gas input is of the nitrogen it will take 45 kilowatt to condense this nitrogen in order to get 50 liter per hour production. The speed is 1480 rpm the operating temperature could be 67 to 200 Kelvin what we have in this case will be 77 Kelvin to liquefied nitrogen cooling effect is around 4.4 kilowatt at 66 Kelvin, but what we want at 77 Kelvin and the capacity could be around 5 kilowatt in this range. The working fluid inside the cryocooler is helium and it is a very high purity helium of 69 99.999 purity. So, very high purity gas is used to run this cryocooler with a mean pressure of 22 bar. So, pressure of helium inside the cryocooler is 22 bar and therefore, the high pressure could be around 30 bar and the low pressure could be around 12 to 15 bar. The shielder is used to cool the cryo generator and the cooling capacity requirement around 48 kilowatt the condenser is water cooled and here is what a schematic looks like. This is the air compressor which takes air from atmosphere the air gets compressed to around 8 bar and stored in air vessel. This air then is dried so that all the moisture is taken off from here and then it goes to a something called as a PSS system or pressure swing adsorption system and here only nitrogen is allowed to go ahead while oxygen is not allowed to go ahead assuming that air is a mixture of nitrogen and oxygen. Because of different wall arrangement over here only nitrogen comes out of this system the nitrogen gets stored in the buffer vessel at this point and this nitrogen gas then comes to the cryo generator where continuous 77 Kelvin temperature is produced and the cooling effect of around 5 kilowatt is produced. So, that when this nitrogen gas comes from this side it will get converted to liquid and this liquid is stored in the storage vessel and from this storage vessel the liquid is given to all the users. So, this is a schematic of the entire liquid nitrogen plant and now I will show the video of the liquid nitrogen plant. So, let us see the video of our liquid nitrogen plant which we have recently bought at IIT Bombay and what you see here is a 4 cylinder cryo generator which works on sterling cycle and this is the common condenser to all the 4 cylinder machine this is a sterling cryo generator and the liquid nitrogen is produced from this side and it is stored in this 2000 liter mother divar and from here it will be transferred to anybody who wants to take liquid nitrogen from our laboratory. Now, this crank case you can see this crank case houses the crankshaft which drives which compresses the gas and does the compression of the helium gas inside this cryo generator and this is driven by one motor and the specification of this motor is as you know is around 45 kilowatts. The liquid nitrogen generated from this is around 50 liters per hour while the gas is charged here at 22 bar alright. So, that means around 10 to 12 liters per hour from each cylinder. Now, what you see from other side is the outlets of the air which are going out the hot gas the hot air because this air which is taken from atmosphere to cool the compressor at this point and to cool the chiller at other point alright. This is water chiller requirement because the gas when gets compressed over here the temperature increases and there is a chilled water requirement which is supplied by the chiller which is sitting over here and the air which is getting compressed basically get nitrogen from is also having air compressor which also needs to be cooled. So, cooling is done by air only and the hot air which leaves this place is you know left outside at some higher point. So, that people around will not be getting affected. So, this is the overall structure of the plant the air gets compressed over here then air gets in the PSA plant here which is pressure swing adsorption the nitrogen is allowed to flow ahead while oxygen is not allowed to go over here. Here the nitrogen gas comes and nitrogen gas enters the cryo generator will get condensed and condensed nitrogen or the liquid nitrogen will be stored in this divaar. This is the in general the plant structure. Now, let us see each component from close distance ok. So, these are all the outlet for the air the hot air which leaves our laboratory importantly you can see that it is a very well ventilated laboratory alright. This is the air outlet when it cools the compressor this is the air compressor around 25 kilowatt is the electrical power requirement for the compressor and 45 kilowatt for the cryo generator here and then we need. So, this is the compressor. So, you can see this is the air compressor which takes air from outside here and the compressor also uses the outside air to cool this compressor and the hot air will leave from here ok and we just saw earlier. So, once the air gets compressed to around 8 bar this air will be stored in this air vessel ok. You can see it here also in this small schematic over there. So, when air is getting stored over there the air then will be dried that means whatever water component is there moisture is there it will be taken care of. So, we want only dry air to go after that from the air buffer and from that dry air this air will go to pressure spring adsorption devices PSA system and as you know that PSA will not allow assuming that air to be mixture of nitrogen and oxygen. The PSA will not allow oxygen to go ahead and only nitrogen will be allowed to go and this is done by using molecular sieves it is like a filter mechanism wherein oxygen cannot go through the sieves only nitrogen can go through and once whatever gas leaves this PSA it will be 100 percent nitrogen alright. So, now what is coming out is nitrogen and this nitrogen is now stored in the buffer vessel. So, here at the end of PSA nitrogen will be stored and it will be nitrogen pure gas. This nitrogen gas then will go to the cryogenerator for condensation. So, this nitrogen gas will come over here and it will get condensed due to the cold which is produced by this sterling cryocooler and it will be stored in this and then we have got a chiller also to get chilled water from this machine alright and again chilled water plant is cooled by the air and hot air leaves the laboratory at some higher temperature. So, this is chiller which is a very important device in this machine and this is our liquid nitrogen plant functions and you can see liquid nitrogen being delivered from the main diva and put in some 5 liter cryo can cryo containers. Having seen both the videos of liquid helium and liquid nitrogen now I will actually summarize various systems which you have studied right now till now during all the last almost 10 to 11 lectures. The following parameters are kept constant to compare various liquefaction systems studied so far. Working fluid is nitrogen and initial condition is 1 atmosphere 300 Kelvin. The final condition is around 200 atmosphere for these cycles. 40 atmosphere for Claude, Capitan, Helian cycle while 15 atmosphere around Collins and for Helium for Collins cycle using Helium as a working fluid. All the equipments are assumed to be perfect. So, if I see this table now here. So, first we had a ideal thermodynamic cycle where we get y as 100 percent and these are the FOM value and W by MF value. Then I went from ideal to simple Lindy-Hamson system and y got reduced my work of liquefaction also got increased and FOM decreased. Then I went to a pre cooled Lindy-Hamson cycle at T3 is equal to 243 Kelvin. What you can see the y increased over here at the same time W by MF decreased as compared to simple Lindy-Hamson system, but what I had to do is a pre cooling. Then I went for Lindy-Duell pressure system where the intermediate pressure was taken at 50 atmosphere and here y decreases, but what is important is W by MF decreases that the work per unit mass of gas liquefied decreases and this was basically the advantage of Lindy-Duell pressure system. If I go for now 1 atmosphere to 40 atmosphere nitrogen is a working fluid the Claude cycle I get y of almost 27 percent which is very high W by MF is 810 well figure of merit also is very high. Then I had a Capitza system and Helian system where you can again see that y value is quite high W by MF is quite low and FOM is also quite good basically in this case, but they are all giving me temperature of around 80 Kelvin. They are all using single expansion device. If I go for now 1 atmosphere to 15 atmosphere using Helium as a working fluid which is now I am using Quallian cycle and expanders are having inlet temperature of 60 Kelvin and 15 Kelvin with a ratio of 0.4 and 0.2 you get y at 4.2 Kelvin around 6 percent W by MF is of course very high because you got a very small value of y and the figure of merit in this case is 0.271. All these calculations I would ask you basically to calculate yourself and compare with this table. The table is shown basically to compare different systems for different end pressure conditions and just to give you a feel of what could be the y W by MF and figure of merit for these cycles. In summary a system which produces cold or maintain such low temperature is called refrigerating system. This process is called as refrigeration. This ratio of delta T by delta P at constant enthalpy is called as JT coefficient. The ratio delta T by delta P at constant entropy is called isentropic expansion coefficient and ideal gas does exhibit a cooling effect when it undergoes isentropic expansion unlike JT expansion. JT expansion has to worry about the inversion temperature of the gas. Isentropic expansion of gases such as Hydrogen, Helium does not produce cold when expanded from room temperature because their inversion temperature are less than room temperature whereas gases like Oxygen, Nitrogen result in cooling when expanded isentropically. The isentropic expansion always results in cooling irrespective of T inversion.  So, we have studied all the systems with their T S diagram with the schematic of those things and we have solved various problems or we have taken various tutorials to understand how is cycle got evolved, how is cycle become better and better or how one reaches lower and lower temperatures using various expansion devices etcetera. Also we studied the effect of compressors, heat exchangers and expanders and effect of those efficiencies of the performance of these systems. So, what is important is to understand how this system function and one has to be able to do minimum calculations like to compute Y W by M F figure of merit for all these systems. Thank you very much.


Hey there friends, how's it going? My name is Kevin and today we are playing Tamadachi Life.  I just don't mit the Tamadachi part because you know, it's just nice to be able to say I have a life for a change. Alright, let's go into my fictional village with many of my split personalities. Alright, donations please. Give me, I can't remember how it said. You know when you give money to a church? I'm gonna say teeth. Give me your teeth. Aw little Kev. I give all my pocket money. 70 quid, not bad people. You have appeased your dear leader. Alright, let's check in how everyone's doing. Turret looks kind of pissed off for some reason. God, you don't feed someone for a week and they immediately turn on you. I want to be friends with default. We all do, trust me. She's like a breakout star of this series. Alright, what would default like? Keep it brief I think. She's the perfect match for Turret, you know why? Because he'll show up looking like that, like a fecking monster and she'll be like, I don't care. I have no strong feelings one way or the other. What do you think they're talking about? Like she has no personality whatsoever and he has too much personality but no way of like getting it across. He can't articulate himself. Usually just a series of grunts and turgs. Alright, well whatever they talked about it worked their friends now. What's he giving me? He gave me some cold medicine as a gift. Well did you say it's the thought that counts? Well shouldn't he give me a cold medicine? Sucks to be you.  Well to be fair I have an allergy so I assume little Kev does.  It can't be good for his allergies. What do you think about me in default?  Little Kev wants to be with default. Aw she's so out of your league. Yeah he's definitely having some reaction here. He's scratching non-stop and he's sneezing. Aw well you're staying in the cage little man. Let's do a gorg once. He's rolling around on his floor. It's been a while. Oh my god he sounds pissed. He's like where have you been? You like a hat would you? Alright I'll get you a feckin hat. So what's on the shopping list for today? We're getting him a hat. Grogneck what do you want? I want a sneeze. I've got this cold medicine if that's any help. Tickle the me's nose to get them to sneeze. Okay sneeze on me please. I'm really into it. Come on sneeze. Oh yeah come on. This is so disturbing. Did I get it? No? There you go. There's your sneeze.  But I'm not gonna question that Grogneck. Cause I'm a good friend. What would you like to give? Give me a song. I think personally that this is an alternate reality. So like popular music here hasn't been written in their universe yet. So let's just plagiarize a popular song. What's the most popular song right now? I don't know like any of these songs. Am I so out of touch? No. No it's the kids who are wrong. Alright we're just gonna do a famous song that I already know of. Okay here we go. A famous iconic song that really is probably the number one of all time. If we really want to go into ghost territory. Oh this is awful. Very good Grogneck. I don't know if you're into the deep lore of Smash Mouth. But there's a nice little clip of them where they were playing a little outdoor concert. And someone threw bread at the main singer and he got really upset. So that was kind of an homage to that really. We were trying to do something outside the box. No I don't think I'll save my changes. Thank you. That was probably our worst song ever. Anything for the shopping list Mr. Jim Peckens?   This is his version of the Last Supper. This is feckin demented. Jesus Christ they all just keep asking each other how are things going in your country? Your country's economy is doing well isn't it? Not bad darling and then they all laugh. What do I do here? Is this it? Like is this a dream or a nightmare? Cause I know it's a nightmare for me at least. I can't listen to that anymore. How are things going in your country by the way? Yeah he woke up with a bit of shock. I think it was a nightmare for him too. You're starving. Okay I'll get you some food from the shop and I'll get a gorg. Oh wait I've got some food here actually. You take this. So I just need to get gorg a hash. That's my shopping run for the day.  I'll give you a song but I'll come back to write it later. I think an opera would be nice. Actually I don't think we've done techno. You have a techno song. Sorry. I put techno on the floor where I belong. Alright let me go get gorg a hat. That's the main thing. There's so much to digest on this map. There's a lot going on. And we gotta make a new person and everything. Yeah Fez. He'll love that. Or maybe a cat ears headband. Oh god it's tough. It's tough. Now give me the Fez. Give me the Fez. That'll suit him. I think I want to add two more people to the island today. And one of them is a very forgotten character from the Call Me Kevin cinematic universe. Mikasso. Who would fit in perfectly in this village. Who honestly I'm thinking might be somewhere related to Terg. Just because of the nose placement alone. There we go Mikasso. It's been a while. I must admit. I'm sorry we forgot about you. Looks gentle. Relaxed kind. And honest. Sensitive and innocent. A dreamer. Living at their own pace. And never rushes into anything. Alright. And one more addition. Will be Ortigame. Otherwise known as Dan the Villain. Nickname is Rumble Tumble. Aw. Can't be. It's just Rumble Tumble. Sounds like an MMA name. And in the blue corner. Rumble Tumble. I made them a child. Okay. I don't know what I'm doing. I'm somehow making the sloth from Ice Age. Okay. Now I'm making a ghoul from Fallow. Hey smooth skin. I'm just making him as bad as I possibly can. To bring revenge on the way he treated me in the Sims. Okay. That's his mouth. That's almost falling off his chin. This is honestly the most horrifying thing we've made yet. Looks nothing like Dan. But it is scary. That's all I know. Oh my God. What kind of a voice is that? Oh my God. His face isn't even on his face. It's floating off it. He looks like a heroin addict. Perfect. Extremely shy. Doesn't show much emotion. But there is a lot going on deep down. I mean I can tell. Oh my God. Even his room is complete darkness. That is so scary. I've made a lot of paralysis demons. But I think this is a good one. I'm orty game. I'm orty game. That's it. I'd like some peanuts. And you would like some peanuts. I don't think I have any for you. I'll try and get you some. I'd like some peanuts. I know you want peanuts. I get it. I'll go get peanuts. Yes. You want peanuts. Okay. I'm going to get you peanuts. These YouTube stars dude. They're so demanding. A default. Everyone's favorite. Everybody clap for default. Let's play default. I just clicked don't play. All right. Sorry default. I'm sure they don't mind. They don't have strong feelings one way or the other. Despite them freaking out like that. I'm going to say they're doing a workout. Please play with me. Oh, they still want to play. Okay. See, they didn't even mind that I neglected them. They just ask again. That's default for you. Yes. They do. They do. They're both leaders. That is false. Unless something has changed in the meantime. Yeah. Wait. Yeah. You just met earlier. Actually. So no, you're not. Ah, Grogne can run at the same birthday. Could be possible. Because I've just been leaving them default a lot of the time. Sure. Fools. F**ker. F**kers. She gave me a roll of toilet paper. Probably to wipe away my tears from losing. I'm pissed. Let's play together. Welcome, Mikasso. God, you're terrifying. But somehow Rumble Tumble Games over there is more terrifying. You get to keep the item if you catch it before it hits the floor. You lose if you don't. Okay. Too bad. What? What was that about? Too bad. F**ker.   You got a box of tissues. I'm just going to be crying all day. Okay. I don't know why you were making mouse noises, Grogne, but I got you a hand. I like how he brings up the change curtain just with the fez on. It looks like a bucket got stuck up there. A news flash. Oh my God. It's Lil Kev behind the desk. Compatibility tester.  Compatibility tester. Let's take a look at this. All right. I want to see how Lil Kev and Dan the Villain are going to get along. They get along better on some days than more than others. I don't understand. Let's see. Turg and Default, which is just a match made in heaven. Oh my God. Look at this. Love. A lack of rain leads to high expectations. Okay. I don't know what that means. I clearly don't understand relationships. Oh my God. Turg and Default's love rating. Don't give them bodies like that. It's really scary. What? It's hopeless. Why are you playing such cheery music? Default is no opinion one way or the other. As usual, she really didn't care. Her heartbeat has remained a solid 79. Her entire life. All right. What about Jim then? Do you like Jim? 25%. I mean, how are you supposed to make a match made in heaven when they literally don't care about anything? They're indifferent no matter what happens. Where's Lil Kev? Even with luck, this will never work. Someone's incompatible here and I don't know which one it is. Let me try Jim and Turg. A balance match. Time together is bound to be relaxing. Look at them shaking each other's hands. Yes, fellow businessman. I too like your company at the water cooler. Who's at the beach? Someone's there anyway. Does that run for a keeper distance? We can make an excuse at social distancing. But honestly, I just don't want to talk to him. All right. I really got to pick up those nuts. I need some peanuts, please. Oh my God. They don't even sell peanuts. He's going to be pissed. In the meantime, give me 10 stewed beefs, please. I just spent 150 euros and stewed beef. Look at Kevin.  He's just, well, I was going to say happy to be there. No, he's actually sad to be there, but he's making the most of it. All right. RTK, what's up with you? Let's take a look. Oh my God.  No, nothing in your tummy. All right. Let me feed you. You're going to have this stewed beef. I always arrive anywhere. God, he sounds like nails on a chalkboard. I need to get him a song to cheer him up. I have some jelly. Oh no. He didn't like the jelly. Here's a highlander outfit. If that'll make you feel any better. Here you go. It was actually quite expensive, so I hope you like it. You look great. And what's he giving me in return? The hell is that? A swing. Why are you carrying around a swing? Come back in his normal clothes. He's not appreciating the gift. I'll give you something else as well. Here, you can finally have a fan. I know it's what you've been craving for years, so it's not happened yet. He's just sitting there staring into the fan. Oh my God, his voice. Yes. Okay. That leveled him up. Let's make you a song. All right. Little Kev and Rumble Tumble Games. We're going to play a little song for us. Our band is called Dire Land. All right. Famous band, Dire Land. Come on, you two. Knock it out of the pack. Subscribe to me. It's all I have. Little Kev is beautiful. Offer gets a pocket. How can we sleep in our destiny? I don't know about Rumble T's voice. Please watch us and subscribe to the Red Cross. I hope we're smitten with a song. Let's go, let's go, let's skip it.  It doesn't help that RT's voice is literally... Turk, what's wrong? You got a little rain cloud over you there. I can't go online. Oh my God. What's wrong with him? He's usually so happy. He's got a cold. Well, lucky for you. I'm pretty sure you gave me cold medicine earlier. Maybe you shouldn't give it away next time. Okay. Maybe you should have read the label first. You should probably go to the hospital in your stomach bump. He gave me a hypnotism set as a reward. I'm happiest when I'm asleep. God, I think Turk is smiling at me outside by crying on the inside. Let's play together. Oh, Mikasa wants to play again. Okay, fine. What do you want to play this time?  That could be fecking anything. Offer feck's sake. It's the equivalent of that thing in Pokemon that's like, can you guess this Pokemon? It's like, it's a jigglypuff from above.  I got set up. Here, I'll try and give him a little gift, too. Here, have some jelly. I like jelly, don't you? I just make them all happy with jelly. I just want them to level up and get as many as I can doing techno for the final song. Oh, yeah, I forgot. He fecking loves jelly. It's so scary when he eats jelly. Here's the swing. I hope you like it. Okay, am I pushing them? Okay, this is so fecking weird. I'm fine. This game is so bizarre. I wonder if can I hypnotize someone with that set or is it just a gift? I'm not sure if I'm giving the villain here the real threat in the village. A weapon of mass destruction or if I'm going to be hypnotizing him. Oh, no, I'm going to be hypnotizing him. Great. You are feeling very murdery. Okay, he's asleep. Now what? Oh, don't dream. Please don't dream. I don't need another dream here. Wonderful. Oh, my God, what is this? Like, why is he even going under it for that half coconut? Okay, he got captured. Is that a bit traumatic? That seemed like a nightmare. All right, Turg is leveled up. You're going to take part in the techno musical as well. Wait, what did I get? I got a swimming certificate. So that's why it was on your mind. That was one of the prizes. Why? Why does that matter? Oh, God, someone's dreaming. Okay, let's see what default dream is about. Surely it's got to be neutral. What?  Did we just flash forward? I got a walking stick. I have no idea what's happening. Oh, no, no, no. We don't want to watch you bathe. It's like a siren's call. Okay, default is almost leveled up. Let's feed you. That might help. You're not really used to eating. I don't really feed you often, but you're growing on me a lot. I got to admit. So I don't mind feeding you now. It's like the fecking blues, brothers. I'm going around recruiting everyone for my techno band, but I got to do them favors and feed them and stuff to actually get them in the band. It's impossible to dislike. It's impossible to dislike Turg. That is the truest thing you've ever said. Grognak is out working her part time job. Where does Grognak work? Oh, my God. Of course. She's selling dates? Grognak. Ron, we have literally not visited you once today. I would like to eat something. You'd like to eat something sour. I don't have anything sour. Here's a bean. I didn't please him enough to get him into the band, though. All right. How about some jelly? It's the ultimate bribe. Will you join my band? No. How about if I gave you some jelly?  I'm in. All right. Here are your lines. The song isn't even fecking written yet. Just assembling the greatest band of all time. Gorg's going nuts. I think he wants to be in the band. Me fact. Okay. Good. I have a good chance at this one. We're born on the same day. Why doesn't it have to be birthdays? It's probably true. True. Yes. Yes. Look at that. Default is default. That's the main reason I went with true on that one. And Mikasa, I just don't care enough about to put in a birthday. They like the same color. No. I think he likes orange and he likes brown. Yes. Oh my God. I know all my split personalities so well. Kevin and Grognak. Oh, sure. They're the neighbors. Yeah. I think they are. I think they are actually. Yeah. Grognak and then Kevin, right? Yes. Yes. I am correct. Congratulations. You don't look too happy about it, to be honest. I got to confess that to you. Do you get it? It looks even more angry. Okay. I'll take the middle box, please.  Some kind of sex toy? You know what? Never mind. I'll take the magnifying glass. Yes. Leveled up. He's in the band. Here's some hair color spray. Join my band. What color do you think? What color? I think a slightly different shade of gray would just look lovely on you. Oh yeah. Silver fox. You're looking great. Oh my God.  You even dyed your eyebrows and mustache. Here. Now that you've dyed your hair, here's a wig. I'm just trying to bribe them into joining this band. Yeah. You look so nice. Oh, they don't like it. All right. Make edit group. Make a group. Everyone who was willing to do techno get on in here. It's pretty much everyone except for Dan, the villain and Grognak. What is this group's name? Well, beef brought us together. So we're going to be the beef boys. It's like, I love the beach boys. This is the beef boys. Totally different band. Okay. Let's see what they sound with the default song first. I'm feeling it so far. God, Ron is good at techno. Oh, I don't like the chorus. Two other keys is a blessing though. All right. Your sound is great, but we have to make these lyrics more relatable because honestly, I don't know what you're singing about. All right. Now a fresh new sound. All right. Same beat, but relatable words. Yeah. Gorg swear. Yes. Yes. He's showing them. Is that it? Okay. It ends kind of abruptly there. It feels like you could have written more. I feel like you paid someone to write this song on Fiverr, but you didn't want to get the premium package. So you had to keep it within like a word limit there. But you know what? It was still good. I'll join your cult. That is legally not a cult. You know, for legal reasons it's not a cult, but like join our cult, like subscribe and stuff. All right. You know what? This was a fantastic day of adding to my village. Dan the Villain is welcome edition as well as Mikaso being an addition. Not as welcome. I just feel like they're the new reject them and Ron default has grown on me so much. I love default now. But I hope you guys enjoy the video. I appreciate you watching. If you want to see more of Tamadachi life, do let me know because it seems like it's going down well. It's one of the few series that I've done that is actually growing in popularity. It's like that in Sims. Usually I post the first one.  But yeah, I hope you guys enjoyed. Let me know if you did appreciate you watching. And Dan has been itching to sing a song. So we'll leave him do the outro. But for now, I'll just say thanks again. And I hope to see you next time. Bye for now.


We are in March, we don't play Gunterson against Alliance We've got Forest, Sheffey Knight, Burnley and Brentford, and Liverpool so take Liverpool off for a minute Day differ games we have to win but we've just had Palace and West Ham which looked like two winnable games anyway So for me that's what I will be doing and I know after Mind York, they've got three weeks with our game …eith y ddweud y wneud I'm getting someone else in a fresh voice. I'm really doing intense work only trying to play and see what we can come up with. Right now, it's the area where. Defensively, even with 10 3 goals you look at the bottom 6 or 7. We're comfortably the best goal difference with only minus 8. You know, I still say bull. We haven't considered tons of goals but offensively we are dreadful. Mae'r bwysig oherwydd mae'n cael ei wneud. Mae'r bwysig oherwydd, mae'r bwysig oherwydd mae'r bwysig oherwydd.


In March 2016 we conducted a total count survey to assess the current population status of the heart-south and parsnip caribou herds. Our primary objective was to count the total number of caribou and identify the proportion of calves in each herd. We compared the data collected in 2016 with past estimates to assess population size trends over time. The black line shows the boundaries of the management units, the blue lines show the census blocks, and the brown line shows the flight path of the survey. Ministry biologists have been monitoring the heart-south and parsnip caribou herds since the late 1990s. Long-term trend data suggest both the parsnip and the heart-south herds have declined by 45 to 50 percent since 2006. Calf survival appears to be low in the heart-south and the herd may continue to decline. Caribou and the heart ranges are on the provincial red list in British Columbia. A number of management actions have been implemented to support mountain caribou recovery in the parsnip and the heart-south herds. Ongoing management actions promoting recovery are essential to ensure the persistence of these herds in the long term. Continued monitoring of the heart-south and parsnip caribou herds is essential to determine population status and assess recovery actions.


This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory, and will be about a sketch of the proof of the prime number theorem. So we recall the prime number theorem says that the number of primes less than x is approximately x over log of x. In the previous lecture, I gave some background to this and gave a rough overview of the proof. And what we're going to do is to go through the various steps mentioned in the previous lecture in a bit more detail. So as I said, the first main step is to show that zeta of s has no zeros if the real part of s is greater than or equal to one. So when the real part of s is greater than one, this follows very easily from Euler's infinite product, that this is the product overall primes of one over one minus p to the minus s, as I discussed in the previous lecture. So the problem is to show that it's got no zeros on the boundary of this region where the product converges. And the proof of this is quite short but involves a rather clever idea which is not at all obvious. And the clever idea is to look at this function.  And you may wonder where this is funny looking function come from. Well, the exponents come from binomial coefficients. So you recall that if we've got a number z then z plus c to the minus one to the four is equal to z to the four plus four z squared plus six plus four z to the minus two plus z to the minus four. So we have these exponents, these coefficients one four six four one which are appearing in the exponents one four six four one. So how do you use this? Well, we recall the infinite product for zeta of s. And if you take the logarithm of that we find that the logarithm of zeta of s is sum over all n and all primes of p to the minus ns divided by n. We're just using the usual formula that the logarithm of one minus x is x over x minus x over sorry plus x over squared over two plus x cubed over three and so on. So if we apply that to the product formula for zeta we get this. And this means if we take the logarithm of this funny product of zeta functions we get a sum over all p and n of p to the minus ns times p to the i n t over two plus p to the minus i n t over two all to the power of four divided by n. Here we're using the fact that this thing here is p to the two i t plus four p to the i t plus six plus four p to the minus i t plus p to the minus two i t. And so all we're doing is substituting in the sum for zeta of s for each term of this product. And this expression looks like a bit of a mess but it does one really nice property. You notice this is always greater than or equal to naught zero for t real because this expression is a number plus it's complex conjugate. So it's real and the fourth power of that is real and so on. So this expression here is always greater than or equal to zero. Well, if the logarithm of something is always greater than or equal to zero then that something is always greater than or equal to one. So we've got this key inequality that this expression here is always at least one. Now that's why we write it down. Well, now let's look at it in a bit more detail.  This is greater than or equal to one and here we need the real part of s to be greater than zero. So it's going to be greater than one so that everything converges. And now let's fix t and assume that zeta of one plus i t is equal to zero and this implies zeta of one minus i t is equal to zero because it's just the complex conjugate of zeta of one plus i t. And now let's take a look at what happens at the point s equals one. So all of these become meromorphic functions. So at the point s equals one, this thing has a pole of order six and zeta of s plus one plus i t has a zero order four and this also has a zero of order four. Well, if we've got all together, so this has a zero of order at least eight and a pole of order at most six. So altogether this function has a zero of order at least two at s equals one. Well, this contradicts the fact that it's greater than or equal to one or s greater than one. I mean, if it's at least one for s greater than one, it can't suddenly become zero at s equals one. So this is a contradiction. So zeta of one plus i t cannot be equal to zero. So that's the first step of the proof of the prime number theorem. The next step is to prove Newman's Tauberian theorem which I'm not actually going to do that because it involves a slightly tricky bit of complex analysis. So we're going on to step three, which is to show the integral from one to infinity of psi of x minus x over x squared dx converges. Here, we just recall that psi of x is the sum over n less than or equal to x of lambda of n where lambda has the property that lambda of a power of a prime is equal to the logarithm of the prime and is zero otherwise, as I discussed in the previous lecture. And what we do to prove this is to be observed that the derivative of zeta of s over zeta of s which we recall is equal to sum over n of lambda n over n to the s can also be written as sum over n of lambda n times the integral from n to infinity of s over x to the s dx by elementary calculus which turns out to be s times the integral from one to infinity of psi of x over x to the s plus one dx. And you're using that formula for psi. So we have zeta prime of s over zeta of s minus one over s minus one is equal to s times the integral from one to infinity of psi of x. Minus x over x to the s plus one dx, okay. And now we're going to apply Neumann's Tauberian theorem and Neumann's Tauberian theorem says that if the integral from one to infinity of f of x times x to the minus s dx converges for the real part of s greater than one and if you can extend this to a holomorphic function with no poles for real part of s equal to one then it converges for s equals one. Well, so if we take f of s to be psi of s minus x over x then we see from this that this function here has no poles for the real part of s equal to one because we showed that zeta of s is no zeros for real part of s equals to one. So Neumann's Tauberian theorem works for this and implies that this function converges for s equals one which shows that this interval converges which is what we needed to show. So now step four we want to show that psi of x is asymptotic to x and this follows easily because if psi is a increasing function so if psi of x is any increasing function and the improper integral integral from zero to infinity of psi of u minus u over u squared du is less than infinity then this automatically implies that psi is asymptotic to x and the proof of this suppose that psi of x is greater than lambda x for some x with lambda greater than one then the integral from x to lambda x of psi of t minus t over t squared dt is going to be greater than or equal to the integral from x to lambda x of x lambda minus t over t squared dt which is greater than or equal to some constant k so if the integral converges this can't happen for arbitrarily many values of x because then the integral will be bigger than k plus k plus k plus k plus k and so on so this condition here can't happen for arbitrarily large values of x so psi of x so the limb's soup of psi of x over x is less than or equal to one a similar argument shows that the limb inf of psi of x over x is greater than or equal to one which shows that the limit of psi of x over x as x tends to infinity is actually equal to one which is what we're trying to show that psi of x is asymptotic to x so this is a slightly tricky piece of analysis but really has nothing to do with number theory it's just a theorem about arbitrary increasing functions so now we've got psi of x is asymptotic to x and from this step five deduces the prime number theorem which says that x over log of x is asymptotic to the number of primes less than x and this follows easily from the following key step which is that log of x is almost constant well you probably don't think log of x is almost constant because if you've seen graphs of log of x it kind of looks a bit like that and that is clearly not constant no we're near being constant well that's because you haven't looked at the graph of log of x for really large values of x so suppose I look at it at a really large scale so here's nought and here's ten to the ten here's nought and here's ten to the ten so what does the graph of log of x look like well it looks like this it's indistinguishable from the x-axis then it's indistinguishable from the negative y-axis there's it's got a very sharp bend in it well that said that log of x is approximately zero so I want to focus in a little bit more on it so let's expand the scale of the y-axis a bit so what I'm going to do is I'm going to look at nought here's ten to the ten and here's nought and here's let's just go up to a hundred and then the graph of log of x looks something like this it's very nearly log of ten to the ten for all values until you get very very close to zero that's because whenever you go down by a factor of e log of x goes down by one well if we go down by two or three factors of e we're already down here but that means that log of this number has only gone down very slightly to two or three or something so we hardly notice it and the graph of log of x looks almost like something with the right angle in it if you look at something on a very big scale and you can see that what this is saying is that log of x so if x is less than ten to ten then log of x is approximately log of ten to ten unless x is very very small so the logarithm of x is very close to being constant when x is very large so now we've got psi x is asymptotic to x and what's psi of x well that's lambda of two plus lambda of three plus lambda of four and so on and what's lambda of n well lambda of p to the k is x equal to log of p so we've got a sum over prime so we've also got a sum over prime powers and now we notice that we can ignore prime powers and that's because prime powers are pretty rare so the number of what's the number of prime squares p you know two squared three squared five squared up to p squared less than x well it's going to be at most the square root of x because the number of squares less than x whether or not their prime squares is going to be at most the square root of x and similarly the number of prime cubes is going to be at most the cube root of x so the number of prime powers less than x is going to be at most the square root of x plus cube root of x plus fourth root of x and so on up to plus some kth root of x where two to the k is equal to x because there's no need to go further than that so k is going to be about log of x times some constant so the number of prime powers less than x that aren't primes is going to be at most you know about log of x times x and this will be very so log of x times root of x and this will be very much less than x divided by log of x so we can ignore prime powers and if we ignore prime powers we see that psi of x is going to be about log of 2 plus log of 3 plus log of 5 and so on and since log of x is approximately so log of p is going to be approximately log of x when x is large as long as p isn't too small so this is going to be approximately log of x plus log of x and so on where the number of terms here is just the number of primes less than x so we see that psi of x if we ignore prime powers and pretend log of x is constant is going to be about log of x times pi of x so since psi of x is asymptotic to x this gives us that pi of x is going to be asymptotic to x over log of x so that ends the sketch of the proof of the prime number theorem if you want to see the details I'm adding a link to a paper by Don Zagie which in particular gives a proof of Newman's Tauberian theorem and all together his proof of the prime number theorem is only about four pages long ok next lecture will be about the Dirichlet's theorem about prime numbers in arithmetic progression


Good morning everybody on there that's watching Y254 remember that we're so glad for you guys to be watching us remember you can follow us we want you to follow us and I want you to subscribe to us because we've got wonderful content especially today I know you've seen the interviews that Barry Moses and Hilda Wadili did the interviews with the other models the beautiful ladies so today this stage just filled with pretty pretty mamas and I'm about to interview one but before I do that I just wanted to let you know remember we are on Y254 channel on YouTube and on Facebook we're on Y254 channel as well on Instagram we're on Y254 excuse me underscore channel so remember to subscribe remember to follow us remember tell us what you thought and I just want to introduce our beautiful beautiful beautiful model who's here today Karibusana and her name is actually Teresa and Jogu Teresa and Jogu you're so welcome to this set thank you so much yes I just wanted to ask you before people get to learn about you where can they get you uh well I'm on Instagram okay uh Teresa underscore and Junge okay yeah Junge yes Junge okay yeah Teresa underscore and Junge just Instagram yeah just Instagram okay so you guys you've heard that nice and clear and let's just dive straight in I wanted to ask how did you become a model how did it all begin this wonderful journey well um I think as any other normal young girl you start out from um wearing your mom's shoes uh trying on her makeup and passes so I'll say I basically began there but um after high school that's when I grew up I interested in modeling I'll actually wear heels to Nairobi and I'm going for a class really yeah heels to class yes so I was always on heels always on heels so I meet people and they're like hey do your model are your model I'm like not really so um from there I just started actually actively participating in modeling yes okay well just to let you know there's nothing wrong with wearing heels to class whatever you like do whatever you do right exactly all right and I also wanted to say um I heard that you actually have a few titles one of them being Miss J quad yes what year was that um that was 2016 2016 oh that's not long ago yes okay yeah how was that experience for you both the training and then the competition yeah also getting that title okay well um first of all I um I wasn't actually going to do the Miss J quads I think actually did it from pressure from my friends I like just try it you never know you know so I auditioned and then I went through training and then we got to the finals which are very intense by the way but it was a very very good experience and I I was I got the crown and I didn't feel oh god it's like the best feeling ever you know especially when you're not expecting yes exactly because I obviously never saw it coming yeah okay yeah and I wanted us to show the clip of um you getting the training done at J quad oh okay yes if someone can please run that as and I wanted to also know what what exactly they teach you are training oh in that video yes uh well that was actually um in 2017 uh when they were going through the training for the Miss uh Mr and Miss for 2017 I did pop up for a few tips uh huh yes all right yeah well I think that they've lost the video but I'm sorry about that I'm sure they'll find it somehow let's just move right along um tell me about your favorite photographer if you have one that one when you hear I'm working with so and so today yes good vibes it's just going to be amazing I know my photos will be a bomb well to be honest this will come as a surprise because um this this photographer calls don paul he don paul yes okay don paul he's uh on facebook at uh don paul photography he has amazing images uh and funny thing I've never worked with him but we have been in contact before so I think this might be the year when I'm actually going to work with him okay his work is just mind-blowing oh so that's someone you're aspiring to work with oh okay so and you said he's a Kenyan yes okay and speaking of Kenya your work is here in Kenya okay have you had a chance to move out of uh on modeling yes on modeling business no not really not really but it will come I'm sure it will come you're such a talented lady it will come at the right time uh so what is it like in Kenya I know that um modeling is something that okay right now it's generally accepted yeah and it's always been accepted my opinion but then there are those families where you have a little bit of opposition yeah or you have some people who are not comfortable with it because they have that let's say old school mindset yeah and they're like modeling is modeling really anything is it really a career and why do you want to go out there and do this you know is this really something you want to partner with our family name have you had any of those no no they were all for you yes wow that's wonderful I meet my aunties okay and my uncles when we're having the family get together they're like you're so pretty why are you not an air hostess or why are you not like an international model and like I'm comfortable with who I am but with the exception of growth you know yes so I've never heard um anyone telling me or um saying that modeling is not something that our family would normally do yeah so I'll basically say I've had my family in support you've had your family support all through yes that's great yeah that's great and how do you okay friends out there yeah now that we've talked about family how would they describe you as a person okay the one we don't see modeling oh the one who is a friend okay how can I behind the camera yeah how would they describe you I think they're the sweetest selfless person they know oh for real yes okay yeah all right that's wonderful so then and I have to ask this because I'm a lady okay and I know there are a lot of young ladies who are watching us right now please what are some of the ways you keep fit because your body is banging thank you do you do a particular exercise is there a particular routine that you do to keep yourself fit uh no no you're one of those lucky ones you can just stuff your face and nothing happens um actually I never had a routine like I wake up I'll eat this or I'll do this but I take a lot of water and I um I eat enough not too much just enough and I at least go for evening runs at least twice a week okay that's basically what I do okay yeah so all right and um I wanted us to move into the game because I want to see Teresa who is not the model but before I do that what is some of the advice you have for young girls who are watching you and they're thinking to themselves oh man I wish I could do this I really want this from the bottom of my heart what would you tell them how would you advise them maybe the first step that they should do and then how do they go about it okay so first of all um I mean they have to um be sure that they want to do it then the moment you actually identify that I want to go out and do modeling you start by you start small um go to modeling agencies if you can sign in with them sign up with them they do have um ready jobs um maybe promotions or um activations or whatever or even start um with photographers um I think when we're fast venturing into modeling it's usually advisable that you have um presentable um photos so you start with a headshot and a full body shot that will actually help you yes because mostly when there's modeling um gigs but say they first of all ask for your photos you see so your photos sell you fast and then when they do a short listing they will ask you now in person okay yes so start small start with the agencies you know if these um modeling auditions runway auditions go for the auditions if there's it's a beauty pageantry go for the auditions and plus keep trying I know a lot of people who do pageantry you know they go for auditions they feel but they still go because it's something that they really want to do like three years and on end they're still pursuing it you see because it's in them they just want that crown but then they can you know if they the time it's just uh good timing you know okay yeah so just go then just do it all right yeah so you guys have heard that thank you so much for that advice let's play a game with our wonderful wonderful Teresa who's up here with us it's a very easy game don't worry I can see the expression on your face you're freaking out a bit you're like hey what's this about don't worry I won't make you do like um something physical yes it's just some questions it's more like a trivia game a very very easy trivia yeah so this game is called body cherry I'm not sure why but that's the name of it and what I wonder okay how this goes is I'm going to give you some clues okay and the first clue I'll give you is the name of a celebrity the first letter of the first name and the first letter of the last name okay that's all I'll tell you about their name no no and then I'll tell you a few more clues about them okay we'll build up and if you're not getting there I'll tell you a very obvious one so don't worry there's no failing this game there's no failing you will do well either way okay so then the first one is her first name starts with L okay yeah and the last name starts with N uh-huh that's the first clue okay and this lady is a model uh-huh well not well she used to be at least I think she still is and right now she's actually very big in acting yes she's an actress well she started here then she became international she's one of ours oh my god she has won an Oscar crickets yeah wow okay I'll tell you this one I'll tell you this one but the next one is Guambi okay Lupita young oh Lupita young oh yes yes you see that was very simple and the next one is also one of ours okay and then became big out there as well and he's so big his name the first one starts with B the second one starts with O his wife was a first lady which makes him oh Barack Obama yes yeah perfect uh-huh you people are watching give applause even though we can't hear their applause and uh this one she's right here in Kenya with us okay and she's a tv presenter news slash news anchor and she has her own show and uh recently she was caught being serenaded by another celebrity her first name starts with B and the second name starts with K she used to have a salon and she doesn't have it anymore for some reason oh I know that one yes good yeah so you see you're getting the hang of it yeah getting the hang of it so you see I didn't lie yeah we have the last one this okay sorry I'm not like it's easy easy yes it is very easy and our very last one um he's a man he's here in Kenya with us he's he's a rapper oh the first name starts with K the last name with J and he's known for having a big big african-american accent calligraph jones that perfect good job you've done really well that's a guy that was our final one yeah I've had such a great time with you and please I wanted you to know that what the advice you've given the young ladies is wonderful thank you for caring about them thank you for being open about them and thank you once again it's great to meet you Teresa Joge very nice to meet you real quick yes so actually do a bit of humanitarian jobs I have an initiative for 10 months and we've had it running for one and a half years now and we started out with three girls um but um you know it's it's just like any normal race not everyone makes it to the finish line so at the end of it all we um finished up with one girl who actually we took to a technical school she did the hairdressing and right now she's actually doing quite well she's supporting herself she's actually starting her own salon pretty soon so if anybody will be interested in joining my group um they can what's the name of the initiative it's teen mums initiative just teen mums we're actually also on instagram okay and they can also what's the handle on instagram for nine of the teen mums initiative yes okay yeah this is and you say just started with three yes we started with three how many do you have now one just her she was the only one left standing tall and it's so funny but any things happen yes but at least from us working with her she has shown the people around her that it's possible you know just because you had a a baby when you're so young and you quit school and all that stuff that's not the end of the road yeah so she has she has been calling us and she's like most of our friends actually they kind of they want to join with us you know to if to facilitate changing the lives which is a really good thing so if anybody would want to work with us as well they can just hit me up yeah yeah so you guys you heard that you can hit her up if you want to work with teen mums initiative yeah it's a wonderful venture from what i'm hearing and is there any other venture that you'd like people to know besides modeling um being a humanitarian no i think that's it but that's wonderful it's a great other side of you it's a great other side of the coin yeah a Santa so much so i saw you guys we're going to sign out our next interview is going to be coming from hilda wadidhi so please stay tuned in and remember my name is joy much higher