 So, let us come to the 37th lecture on cryogenic engineering under the NPTEL program. The earlier lecture we were talking about vacuum and during the earlier lecture we talked about heat in leak is minimized by having vacuum between two surfaces of different temperatures. Then we talked about lambda which is mean free path of the molecule and lambda is defined as the average distance travelled by the molecules between the subsequent collisions. And we had seen that based on this lambda we had classified the different vacuum levels. So, based on the Knudsen number again we have got a number analogous to Reynolds number in the continuum flow. We had Knudsen number which is equal to lambda by D where D is the characteristic dimension. And based on Knudsen number NkN we had continuum flow when the Knudsen number is less than 0.01. We have a mixed flow when the Knudsen number between 0.01 and 0.3 and we got a free molecular flow when Knudsen number is more than 0.3. So, normally in very good vacuum we will always encounter a situation when we got a free molecular flow. And we also studied what is this conductance and when you found that conductance will depend on the length of the pipe diameter of the pipe, the gas, the temperature, the pressure etcetera. We have got conductances in series that means, we got a pipes in series the conductances would get added like this. So, 1 upon C 0 is a overall conductance of the entire pipe structure which will depend on as sigma 1 by C i where i will go from 1 to 2 or 1 to 3 depending on how many pipes you have. So, 1 upon C 1 plus 1 upon C 2 plus 1 upon C 3 is equal to 1 upon C 0. And in that way we can calculate overall conductance. Similarly, if you have got pipes in parallel the conductances will be added C 0 is equal to sigma C i they will be directly added to each other and they will with that we will get a overall conductance for the parallel pipes. And this will be utilized ultimately in order to get the vacuum speed the pumping speed and that is what we will see in this particular lecture. So, in this lecture now we will continue with what this conductance business is all about. Then we will come to very important parameters quite regularly used in vacuum technology which is pumping speed and pump down time alright. So, we will come across what is this terminologies which are normally used in vacuum technology. And then we will have some tutorial based on this pumping speed conductances pump down time etcetera. And therefore, all these things which we are understanding in this particular lecture we will have some tutorial based on it and we can have some values associated with all these parameters alright. In the earlier lecture we have seen the importance of vacuum in cryogenics we have seen the mean free path lambda and the degree of vacuum will decide the fluid flow regime alright. And then we had Knudsen number also in order to classify these different flow regimes. The conductance for circular pipe for continuum mixed and free molecular flow regimes respectively where derived as shown below. If you remember the derivation which we did last time we found that conductance for continuum flow is proportional to d to the power 4. And this is what the expression we had got at Poiseuille equations basically. Then for a mixed flow we had a conductance which is a typically a little mixed behavior and you got a very not direct relationship p to the diameter over here. But it is inversely proportional to the length one can see and diameter d to the power 4 is here in the numerator while d is there in the denominator also. And when we come for a free free molecular regime the conductance is now proportional to d cube and inversely proportional to l. So, there are there are the conductances which we found that they vary with different diameters, they vary with lengths, they vary with pressures, they vary with temperatures and of course, the gas property like viscosity etcetera. So, using this formula depending on the flow regimes you are in will have to calculate for a given pipe of a given diameter and length what is the conductance of that pipe. And this will determine ultimately help us determining what is my pumping speed or what is my system pumping speed etcetera and this is what we will see in this particular lecture. So, conductance is a vacuum as mentioned in the earlier lecture vacuum systems involve complex piping arrangements. These piping arrangements may involve circular straight tubes rectangular straight tubes 90 percent elbow joints etcetera and actually lot of other structures which has not been mentioned over here. We can have some couplings also involved over here, we can have some bends over there, they may not be 90 percent again alright. So, in order to calculate the conductances for such mechanism because I have got a vacuum pump I have got a system to be vacuum and in between I will have some piping arrangements alright. So, vacuum pump cannot directly go and sit on the system, the vacuum pump is going to be away from the system to be vacuum and therefore, these piping arrangements effectively will determine what is the conductance which is responsible to get particular system pumping speed. So, if I got some structures which are circular straight tubes, rectangular straight tubes or 90 percent elbow joints and many other possibilities, but I am just showing you three possibilities then there are some standard equations to be used in order to calculate the conductances for these particular pipes joints etcetera. The table on the next slide gives the conductance equations for some commonly used pipes and pipe joints and we have to use such correlations that are available to calculate the conductance in order to come to calculate overall conductance of the system. For example, we got a long tube and long tube is defined as when L by D is more than 30. So, L is length D is diameter and for a continuum flow we have got a relationship as we have derived earlier as pi D to the power 4 P 125. So, you use this equation to calculate conductance if we are in continuum region or if we could prove that we are in free molecular region then use this formula which we have ultimately derived. So, use such formula first find out what is the Knudsen number, ensure that you are in a free molecular region, mixed region or a continuum region and use corresponding formula to calculate the conductance for that pipe for a given dimensions T, D and L. If you got a short tube where L by D is less than 30 then again we have got different formula and if you got a free molecular region we have got a different formula again. Where D 1 and D 2 we can have two joints two diameters also having D 1 as large and D 2 as smaller diameter. So, use all these things in order to calculate the conductances for such given pipes. If you got a 90 percent elbow which may look like this you can see that the gas flow in this direction and you can see there is a you know radius of curvature and this is the diameter of the pipe and again we have got expressions for calculating the conductance for a continuum flow for a free molecular region and the value of K which is showing up here over here depends on particular R by D this is basically get is shown up only in the this only alright. So, find out that calculate the value of conductances accordingly and put those values here and for different structures therefore, for different pipes for different joints for different curved elbows etcetera you calculate the conductance in this way and then effectively you will calculate if there are there are pipes in parallel if there are pipe in series one can calculate the overall conductance of this combination of pipes. In addition to this we will have several couplings we can have several non characterized kind of a items which could be placed in place or pipe of different structures also could be placed and then correspondingly you will have a very big table that is available in literature which you have you have to look for and you can get the conductances for those particular structures which are employed to connect vacuum pump to the system to be vacuumed. Now, let us come to know what is this pumping speed. So, what you can see here is a cavity or a cylinder which is shown to be vacuumed and then the vacuum pump kept on the right the vacuum pump is a is a known device to me and therefore, vacuum pump will have is known characteristics. However, this vacuum pump is directly not mounted on the system to be vacuum, but it is connected to some piping arrangement and therefore, the conductances of this pipe also becomes very important. So, consider the closed cavity and a vacuum pump system. So, closed cavity vacuum pump and connections given by pipes over here this pipe can have different diameters different elbows different joints and corresponding to that will have a different overall conductance for this pipe. So, you can see a closed cavity vacuum pump system as shown in this figure for the above system. Let us assume the following parameters P i is the pressure at the inlet to the vacuum system which is P i at this point P is the pressure in the cavity. So, this P and this P i will be different the vacuum at this level will be very very high because I have got a vacuum pump here. So, the pressure at this point will be less well the pressure at this point will be little bit more and how much more will depend on the conductance of this all right. So, we know that we can suck the gas molecule from this only when that the pressure at this point is going to be less than the pressure at this point and therefore, this vacuum will have a different pressure P i while this system to be vacuum or the cavity to be vacuum will have pressure little bit higher than what you get at the vacuum pump. Q is the throughput of the pump. So, so many bar later per minute or you know pressure into later per minute what is the unit of throughput of the pumps which will be known to you and the CO is the overall conductance of the piping. So, depending on the pipes that have been used the couplings that have been used the elbows that have been used will have some overall conductance of this pipe. In order to analyze the above system the following quantities are defined and what are those quantities the capacity of a vacuum pump is generated in terms of pumps speed. So, we got a pumping speed sp and this is known to me because the capacity of vacuum pump when I buy a particular pump I should know what is the what its capacity. So, how many meter cube per minute meter cube per hour liter per minute liter per hour that this particular vacuum pump can suck all right and again this this capacity normally is given in for a particular gas let us say nitrogen normally it is defined as. So, every vacuum pump when you buy it will have its capacities defined all right. So, a pump speed is normally defined for a given vacuum pump. So, what do we define this pump speed as it is the ratio of the throughput Q to the pressure at the inlet to the vacuum pump all right. So, Q upon P i and therefore, we can see mathematically sp is equal to Q upon P i. So, whatever throughput it is basically getting right pressure into volumetric flow rate divided by P i at this particular point we will get that as throughput divided by P i is going to be my pumping speed and the unit of this will be like volumetric flow rate meter cube per second it could be liter per minute liter per hour depending on the kind of pump we have been talking about. So, pumping speed as we said is a standard for a given pump depending on the pumping speed my cost of the vacuum pump also will increase or decrease. So, sp is equal to Q upon P i P i is the pressure at the vacuum pump on the similar line there is a system pumping speed. So, we got a vacuum pumping speed which is a pumping speed normally referred to as and then we have a system pumping speed which is specified for the system here all right this is my system to be vacuum this is my vacuum pump I have been utilized and this is the way I have connected this vacuum pump to the system all right. So, on the similar line system pumping speed SS is defined as a ratio of throughput Q to pressure in the cavity or the vacuum now I am talking vacuum space that is I am talking about the pressure of the system. So, this pressure as I said is going to be more than pressure what you see at a vacuum pump. So, system ultimately what is important for me is what is my system pumping speed because I am interested in vacuuming in this particular cavity to get this vacuum over here I am using this pump. But if I use a different connections as given over here my system pumping speed is going to be much different than what my vacuum pumping speed will be all right and let us see how they are related basically. So, mathematically we see that SS which is my system pumping speed is equal to Q upon P Q is the throughput, but the P is now the pressure of the system over here all right. So, this pressure is going to be little bit more as compared to what you get P i at this point. So, definitely what you understand from here SS is going to be less than SP. So, system pumping speed is going to be less than the vacuum pump speed or the pump speed again the units are meter cube per second liter per minute or whatever units you want to use. The conductance now everything depends what is the value of SS depending on the value of SP, but SP the pump is connected through the system to be vacuum through a conductance all right. So, I can have a very large vacuum pump, but I can connect it through a very small tube and therefore, the conductance in this case is going to be very very small and therefore, that will affect the system pumping speed. So, the conductance is given by Q upon P minus P i which is what we have seen earlier. So, what is the difference of pressure across this piping P minus P i this will determine what is my conductance as you know P is more than P i and this is again unit will be conductance unit will be same as pumping speed. So, we know now SP is equal to Q upon P i SS is equal to Q upon P and C 0 is equal to Q upon P minus P i. So, now, what a relationship between P P i P minus P i and how are these related how are this pressures related P i is equal to now Q upon SP P is equal to Q upon SS and P minus P i is equal to Q upon C 0 Q upon overall conductance. See, if I put these values of P and P i over here I will get the relation which is connecting now C 0 SS and SP I get connection between pumping speed system pumping speeds and overall conductance. So, if I did that eliminating the pressures P i and P from above equations what we get therefore, is Q upon SS is equal to Q upon SP and transpose it little bit I will get Q upon SS is equal to Q upon SP plus Q upon C 0 and I can eliminate Q Q Q from these equations and what you get ultimately is 1 upon SS is equal to 1 upon SP plus 1 upon C 0. Now, this is a very important relationship which connects system pumping speed to the pump speed to the conductance of the pipe which is going to be connecting my pump to the system. So, it depends on what is my pumping speed and what is my conductance these two will determine together what is my system pumping speed and therefore, this is a very important relationship to decide if I want to design a particular system vacuum system to get a particular SS I will have to worry about both this parameter SP as well as C 0 and let us see how do how do they show up. So, this is my relationship 1 upon SS is equal to 1 upon SP plus 1 upon C 0 from the above equation it is clear that SS is lower than the minimum of SP and C 0 this is a simple mathematical thing you got a reciprocal rule SS which is a resultant of SP and CO is going to be less than the lowest of these two. So, if SP out of SP or C 0 whoever is lower lower than that value will be value of SS normally I would like to be SS to be very very high, but you have to understand that SS cannot be more than the low I mean it will basically be depending on what is the lowest of these two all right. So, SP depends on vacuum pump let us understand now what is the parameter which that is that dominates the value of SS. So, SP is a vacuum pump which I have bought and therefore, I know everything about SP I know that so many liters per minute or so many meter cube per second is what my pumping speeds of a available pump is all about. So, SP depends on a vacuum pump and therefore, in order to maximize SS if I want to maximize SS I have to maximize my CO. So, if I increase my CO I can have SS coming closer to the minimum of SP and CO. In principle what could be the maximum value of SS the maximum volume of SS could be SP provided CO is infinite if CO is infinite 1 by infinity becomes equal to 0 and we can say that 1 upon SS is equal to 1 upon SP in that case we can say that the maximum value of system speed is going to be can be equal to the pumping speed. So, in principle SS can maximum be equal to SP and SS can maximum when CO is infinite. So, when CO is infinite what we get is this if CO is infinite then we have SS is equal to SP. Can we have CO as infinite is it possible we will see that later. For a given connecting pipe now conductances increase with decrease in length and increase in diameter. So, can I make my diameter infinite or length to be equal to 0 then only my CO is going to be infinite. So, in principle I cannot do. So, what I can do I want to increase my diameter as much as possible and I want to decrease my length connecting vacuum speed vacuum pump to the system to be vacuum to a minimum distance. So, keep minimum length of a pump away from the system to be vacuum and keep maximum diameter of the connecting pipe. So, that the conductance is going to be higher in that case and therefore, SS can go as high as possible, but as you understood SS can maximum be equal to SP and not more than that. So, let us try to go as close as much to the value of SP, but everything then depends on what is my conductance. Similarly, if CO is equal to SP let us say in a case just an extreme case if CO is equal to SP then my SS is going to be half of SP. So, you can see that if CO is equal to SP then we have SS is equal to SP by 2. So, increase go on increase in conductance as much as you can. So, that this parameter become very small in that case my SS will approach to SP value and that is what we want to ultimately achieve. So, my system pumping speed should approach the vacuum pump speed. Now, having understood what is system pumping speed, what is pump speed or what is conductance all these things having understood let us now come to what is pump down time. So, consider a close system as shown in the figure this is and this is my high pressure to begin with let the initial pressure in the system be P i before I start vacuuming my pressure is going to be initial pressure which is P i and after vacuuming I reach a final pressure which is P f. So, I can see that less number of molecules over here and this is my completely low pressure which I have got after vacuuming from P i to P f. So, when I connect my vacuum pump to this you need ultimately I reach the value of P f and this time it will require lot of time basically it will start sucking the molecules and how much time it takes to reach a value P f will depend on the conductance I have used the vacuum pump I have used and all those parameter will come into picture. So, the amount of time taken by a vacuum pump to reduce the pressure from P i to P f is called the pump down time. So, it will depend all the parameters it will depend on the vacuum pump it will depend on the conductance I have used to connect this vacuum pump to this to the system to be vacuum. So, all these parameters will determine what is my pump down time and this pump down time is a very important criteria because if I want to do an experiment which will last for only 2 hours and in one day I would like to do this experiment 2 or 3 times my pump down time should not be more than 2 or 3 hours I should get a vacuum within half an hour or 10 to 15 minutes and therefore, I have to design my vacuum system in such a way that I should get my vacuum P f what is my pressure P f within 10 to 15 minutes or half an hour or something like that alright. So, this is very important and it depends on what is my P f I am talking about what is my conductance how do I connect that conductance to the vacuum pump and all those parameters will determine what is my pump down time. So, application decides the degree of vacuum if I want to decide a particular experiment that experiment will decide what degree of vacuum do I require what is my P f requirement what I want minus 5 tor minus 6 tor minus 3 tor vacuum levels and this is what we call as degree of vacuum. So, decide first what is that lowest pressure I want what is vacuum I want what is degree of vacuum I want depending on the application the require a pump down time is determined. So, according to that I will decide what is my pump down time should I have 15 minutes 20 minutes half an hour 2 hour 5 hour complete day alright it depends on application it depends on how big my system is also because bigger the system bigger time it is going to take to pump down all the molecules alright. So, pump down time helps in selection of vacuum pump. So, I have to decide what is my volume to be sucked what is my volume to be vacuumed according to that I will decide what my conductance should be and what my pump should be. So, ultimately depending on my application depending on my pump down time I will decide the vacuum pump to be employed for this purpose. So, the vacuum pump is the pump down time is a very important criteria to decide a vacuum system. Hence, there is a need to study the pump down time of vacuum system. So, can I calculate this pump down time and that is why we are coming over here. So, if I know pump down time can I select a particular vacuum system or a vacuum pump for a given operation and therefore, this parameter pump down time is a very important parameter. Now, this parameter will also depends on what are the leakages paths there the gas is going to get leaked over a period of time and the gas is going to get leak leakage paths could be definitely there as long as the fabrication happens you could always joints you got some seal matters you got a material which out gases and therefore, gas leaks is always going to be there. So, apart from heat in leak gas in leak and out gassing are the major problems posed by a cryogenic system all right. So, there is during fabrication some gas has gone into the material and this will out gas over a period of time this will come out over a period of time and therefore, there will be constant gas in leak in the system and this has to be taken care by the vacuum pump. These leakage paths have to be considered for calculation of pump down time and selection of a pump. So, I should know what are these leakage paths and what is the leakage quantity also what are the flow rates also I should know. The leakage paths increase the pump down more the leakage path it will take more time to reach a required vacuum for a particular application. So, I should know all these leakage paths and I should know all these quantities that are leaking and what are the gases also I should have an estimate for those things also all right. So, gas enter the system due to actual leak through the vessel wall joints. So, this should be minimized actually you know. So, you got a weld it will have some leak however and that whatever that leak is it has to be considered. So, actual leak through the vessel walls and joints suppose you have got a joint which is a mechanical joint or which has got a seal across oaring across it then it will start leaking over a period of time it will have some its leak rate associated with that thing that also has to be considered very calculating the pump down time. Then the trapped gas release from the pockets within the system also called as virtual leaks. So, there are material surface is there and there are some trapped gases inside this material it will get released once you start vacuuming that also has to be estimated. See this this this parameter become very important if you want to go down to minus 8, minus 9 kind of vacuum all right. So, these are very important to be considered while calculating pump down time. Outgassing of the metal walls or seals and this is what you saw just talked about all the non metals all the metals in the system will start outgassing and we got some this outgassing quantities in available table also for vacuum technology. So, one should know what are these outgassing rates for a given material for a given seal etcetera. All these have to be considered in order to consider gas leak and in return it will be basically taken into consideration to calculate the pump down time. Outgassing is the release of adsorbed gases either from surface or interior or both when exposed to vacuum all right. This is a very slow process outgassing happens over a period of time over 6 months time over a 1 year time and it all depends on what fabrication the material has gone through. And therefore, all these things have to be understood when we start designing for a particular application of vacuuming and especially where vacuuming is very plays a very important and crucial role. The major contribution to the mass in leak is the outgassing. So, out of all this parameter the outgassing is a very important parameter that has to be considered. The contribution from outgassing is going to be a major contribution to mass in leak and that is why it has to be considered correctly while designing a particular vacuum system. So, now let us understand how do I derive an expression for calculating pump down time. So, let us see this system. So, consider the closed cavity as shown over here with a vacuum pump system. So, you got a entire thing will be vacuum pump system, a closed cavity connected to a vacuum pump through a given conductance. All right. So, consider a closed cavity vacuum pump system as shown above and we have got two parameters. Let the mass flow rate leaving the system is m dot out which is coming from over here and this is my system to be vacuumed. This is my piping which has got its conductance overall conducting as C0. The throughput could be called as Q from here and this is my P i which is the pressure at the vacuum pump or the vacuum level basically at the vacuum pump. So, mathematically I will say m dot out is equal to rho into ss. What is my ss? s is the system pumping speed. So, depending on ss I got some meter cube per second multiplied it by kg per meter cube and therefore, what you get is a mass flow rate which is leaving the system which is m dot out in let us say kg per second kg per hour whatever. So, m dot out is equal to density of the gas into ss. All right, where rho and ss are the density and the system pumping speed respectively. Please understand this clearly. Now, similarly I will have some m dot i and let the total inflow due to gas leak and out gassing be m dot i. It can also be written as m dot i is equal to Q i divided by R T. So, depending on heat in leak now, depending on the gas in leak in a system. So, you got several joints, you got several out gassing possibilities and depending on all these seal materials used etcetera will have some Q i and therefore, as a result of which will have some m dot i. All right and this could be again a constant or it could be a question of question of time. It could be dependent on time. All right. So, it can change with time also. So, let the total inflow due to gas in leak and out gassing be m dot i. It can also be written therefore, as m dot i is equal to Q i upon R T. Now, applying the mass conservation to this system. So, I am talking about system only the cavity to be vacuumed right now, where m dot i is mass in leak and m dot out is mass which is going out of the system. So, we have m dot i minus m dot i was equal to d m by d t and this is a very standard expression for log conservation of mass basically. From the definition of density, we have m is equal to rho into v. This is my volume and therefore, m is equal to rho into v and therefore, m dot i minus m dot out is equal to v into d rho by d t. So, if I write d m by d t basically, I can write d m by d t in terms of d rho by d t multiplied by volume. All right. So, now, I have got expression which is talking about d rho by d t and therefore, can I relate it to d p by d t now? Yes, I can. So, m dot i minus m dot out is equal to v into d rho by d t, which is a change of density with time. Using the ideal gas law now, what we have is equal to p v is equal to R T. So, we could have p is equal to rho R T and therefore, now, I have got an expression for rho in terms of p and R T, where R T could be considered to be the constants. And therefore, now expression changes and I get an expression which is talking about not d p by d t, which is a pressure change with time and this is what we are aiming at basically when we are trying to vacuum this system out basically. So, m dot i minus m dot out is equal to v upon R T, R T being constant is coming out of this and what we get is the d p by d t. We get expression for d p by d t now. So, combining the following equations now, we have we know that m dot i is equal to q i upon R T, we know m dot out is equal to rho into S S. This is what we have seen earlier, put those values over here what we get therefore, is equal to q i upon R T minus replacing this rho as p upon R T what we have got here, we get p upon R T into S S that is p s upon R T is equal to v upon R T d p by d t. And therefore, all the R T's would get cancelled and what you get ultimately is d p by d t is equal to q q i upon v minus S S into p upon v. So, you got expression now which depends on the system pumping speeds, the pressure, the volume to be vacuumed and q i which is talking about basically the mass in leak throughput. So, I got expression like this now and if I want to now calculate the time to get a pressure from p 1 to p 2, I can basically integrate this and this is what I will do some mathematical treatment on this. This equation is valid for any vacuum system in general, for any vacuum system now which has got mass leak as given by q i, we got a system what we want, we got a system pumping speed is what we desire to have, given volume of the cavity to be vacuum and the pressure we are going to talk about what is the vacuum level that we are talking for this volume. If I know this, this is my general expression for d p by d t and then one can get a pumping system pumping speed and then I have to decide what my pump speed should be for in order to get my S S for a given pre requirement. So, it is a very general expression and whatever you want to do with this now, you can you know do all the mathematical treatment to this. It is important to note that both q i and S S are time dependent functions. So, my mass in leak throughput can change with time, it can change because gas in leak because outgassing being dependent on time can change with time. Also, I can say my system pumping speed also could change with time. So, I can write q i as a some function of time and S S also is some function of time. So, f q i is a f 1 of t, S S is function f 2 of t, basically this to show the time dependence of these two parameters. The equation can be integrated analytically or numerically if the transient variation of q i and S S are known. If I know the function f 1 or it is polynomial function or any other function if I know the dependence of q i on time, then I can put that function in this value and then integrate it to get. So, I should know basically some time dependent function of q i, similarly I should know some time dependent function of S S and I could in then put these values put these expressions analytically in this particular expression and then integrate it and I can get the time to get some pressure from P 1 to P 2 or P i to P f. So, in this expression we say at steady state. So, ultimately when I run this vacuum system for a long time, let us say many hours many days depending on what is the volume we are talking about, what is the q i we are talking about and what is the pressure in our site is, what is my vacuum I am talking about. So, at steady state I will find that after particular time there is no change in the value of pressure, I have reached some steady state value of the vacuum. In that case I will say d P by d t equal to 0 at that steady state. So, at steady state or after a long time the changes in pressure with time are negligibly small and therefore, I can say mathematically my d P by d t equal to 0. And therefore, if I say d P by d t is equal to 0, I can say now I can have a relationship between q i and at that time I got the pressure as ultimate pressure, the pressure at the steady state is called as ultimate pressure P u. So, this P value has become my P u, it is the minimum possible pressure that can be achieved using a certain pump. So, whatever vacuum system I have used, whatever conductances I have used after connecting those, after running it for a long time what pressure I have got is something like P u, at that particular time I have got d P by d t equal to 0. And therefore, I have got a pressure P as P u and this is the minimum possible pressure that I could get at the minimum vacuum I could achieve choosing a certain pump. And therefore, I will get P is equal to P u and putting that value of P is equal to P u in this I will get a relationship now as P u is equal to q i upon s s. Canceling V out here transposing on this side I get ultimate pressure that can be obtained depends on q i and system pumping speed alright. So, q i upon s s is what minimum pressure I can achieve. If I know the value of q i I can put this and I can get my P u value if I know the system pumping speed I can put it over here I can know that my ultimate pressure ultimate vacuum that is possible with such a you know mass flow in leak throughput and such a system pumping speed I can get a value of P u to be equal to this. It is important to note that for most of the pumps S P is constant in its operating pressure range. This is the fair assumption to say that if I got a particular pump it has got its own pumping speed. And therefore, I will say that this pumping speed is constant right from when you keep the machine on and you fairly constant S P over a period of time alright. So, just as a fair assumption I will say that for a given pump pumping speed remains constant. So, that means, S P does not depend it is not a function of time S P is constant throughout. Also in a free molecular region conductance C is also independent of pressure. So, if you see our expression if my pressure is constant or if I am in free molecular region C is because C depends largely on the gas properties and C largely depends on the diameter and the lengths geometric parameters of that pipe which is going to be connecting the vacuum pump to the system. So, I can say again that conductance is also not a function of time. So, S P is not a function of time conductance is not a function of time. And therefore, in this relationship I can say 1 upon S S is equal to 1 upon S P plus 1 upon C O we can say from here if S P and C O are a constant from the avoid expression it is clear that S S is also a constant or it is independent of pressure. I can see S S can be a function of time depending on whether these two are a function of time. If I know the time variance time dependence I can put that in earlier expression, but because S P and C O is fairly constant with time they do not depend on time I can say that S S also can be considered as independent of time that means S S is constant or it is independent of the pressure we are talking about all right. So, we can have this assumption of having system pumping speed as constant. Therefore, for a constant S S the equation can be integrated with falling limits now. So, if you go to earlier expression now I can assume S S to be constant and therefore, this is my expression and we can now if I want to find a time dependence on pressure the only parameter which is depending on time will be now q i. And therefore, my time will be at t is equal to 0 p is equal to p 1 let us say to begin with and at t is equal to some time any time I am talking about t p my p is equal to p 2 all right. So, this is my time limit I am talking about where the pressure changes from p 1 to p 2 where the pressure decreases from p 1 to p 2 and if I have integration therefore, I will have integration p 1 to p 2 V d p upon q i minus S p p you can see from here expression and ultimately I will have on this side only d t parameter basically. So, 0 to t p d t all right transposing d t on one side and getting all other expressions on the other side we can we can get this particular expression. And then I can you know solving using a simple numerical simple technique of solving an integration I can put q i minus S S p is equal to x and you know put a value of d p in terms of that and then integrating and puts some integration rules in place what I get is minus V upon S S log q i minus S p p in the limits of p 1 to p 2 is equal to t in the limits of 0 to t p. So, you just integrate based on available integration rules what you can get is this expression. So, ultimately now put the value of p is equal to p 2 and p 1 in this expression and I will get parameter and we know that p u is equal to q i upon S S putting the value of p u also over there replacing q i as p u into S S in place I get an expression like this. So, I get t p is equal to now V upon S S log p 1 minus p u upon p 2 minus p u. So, p 1 in a given time t p pressure decreases from p 1 to p 2 while my ultimate pressure value is p u which is equal to q i upon S S. I have got an expression now which talks about pumping time pump down time for a given volume to be vacuumed which S S as a pumping speed which is constant it does not change with time while the pressure is during this t p time comes down from p 1 to p 2 while the ultimate pressure that could be achieved is going to be q i upon S S. So, here is an expression I can now compute for a given S S given volume what is the pump down time I can calculate for this expression and this will be more clear when I do the tutorials ahead alright. So, this is an expression for calculating of the pump down time. So, what are more important expression is 1 upon S S is equal to 1 upon SP plus 1 upon CO that is the most important expression and second is the expression for pump down time. With this background now let us have a look at the tutorials and wherein you can understand all the basics more clearly. So, tutorial 1 we got 2 pipes of 2 different lengths and 2 different diameters in series I think the lengths are same 400 millimeter the diameters are 40 millimeters of 1 and 30 millimeter of 1 and they are connected in series and first problem is to calculate the overall conductance of the pipe assembly shown above. The pressure on the right end of the 40 millimeter tube is 150 MPa, 1 30 millipascal on this side while the pressure on the left of the 30 millimeter pipe is 10 MPa. So, you can see that you got a 10 millipascal and 150 millipascal as to pressure the ambient temperature is 300 Kelvin alright. So, we got one pressure on this side we got a one pressure on this side the molecular weight and the viscosity of the error given alright. So, basically it is a calculation of the overall conductance of 2 different diameters pipes of 2 different lengths or same lengths. So, what you have apparatus is basically series combination of pipe we know the working fluid we know the temperature we know dimensions 40 millimeter 400 millimeter length pipe to 40 millimeter 400 millimeter length. So, calculate the overall conductance CO calculation of flow regime first in order to ensure what is my in width flow regime I am because depending on the flow regime I have to use the expressions for conductances. So, first is Knudsen number for pipe 1 we know D is equal to 0.4 0.04 meter L is equal to 0.4 meter T is equal to 300 Kelvin R mu and P is known to me alright. So, the Knudsen number calculation is lambda by D this is expression put those values all over here and I get Knudsen number as equal to 1.132 which means I am in free molecule region because Knudsen number is more than 0.3. Similarly, I can do it for pipe 2 also similarly calculating Knudsen number for pipe 2 what we have is Knudsen number of 0.1 is pipe 1 is 1.132 and Knudsen number for pipe 2 is 22.65 which is very very high and we are not shown the calculations again and therefore, we can see that Knudsen number has been directly calculated for the pipe 2. The Knudsen number for both the pipes are greater than 0.3. Therefore, the flow is free molecular throughout the series combination that is first thing to be established because that will give you now what expression for conductance to be used from the available table. The L by D ratios of each of this pipe being less than 30 these are classified as short pipes. So, we got expression for short pipe free molecular flow and the expression for the conductance could be used for this case. So, conductance for pipe which is D 1 0.04, D 2 is 0.04, L is 0.4, T is 300 Kelvin and all these things are shown over here. We got a C 1 for the conductance of the first pipe put those values and get C 1 is equal to 0.0173 meter cube per seconds. I am not going through all these parameters just go back. The expressions for D 1 D 2 in the short formula could be used as 0.04 straight away. Similarly, calculate conductance for pipe 2 now we got again pipe 1 as C 1 as 0.013 and pipe C 2 we get 0.007 and the conductance for point the pipe C 2 is quite less as compared to what because the diameters are different. The overall conductance now for a series combination is given as 1 upon C 0 is equal to 1 upon C 1 plus 1 upon C 2 and therefore, I can calculate the overall conductance by this formula is going to be 0.00542. That means, I get overall conductance now which is less than any of these two. So, it is less than 0.017 which is less than 0.007 also. So, as a result of this C 1 and C 2 I get overall conductance which is going to be less than A 1 C 1 and C 2 and this is my final conductance. Now, which could be utilized to calculate the system pumping speed if I know what is the vacuum pumping speed is. So, this is my first tutorial to calculate the overall conductance for a series combination of a given pipe. Now, tutorial is if this arrangement what we have just seen is connected to vacuum system of 1 meter cube volume and I got a vacuum pump at the other end over here. So, now I can see that I have got an entire system with a known conductance which we have just calculated with a vacuum pump on one side and system to be vacuum on the other side. So, my problem statement now consider a vacuum system of 1 meter cube with an initial pressure of 1 atmosphere at 300 Kelvin. So, it is all at 1 atmosphere to begin with it is connected to a vacuum pump via connecting pipe as shown above. So, it is using the same system which we have just calculated the CO 4. So, it is connected to a vacuum pump. The ultimate pressure of the system is 0.1 millipascal as shown over here determine the system pumping speed if the required vacuum in the cavity is 1 kilo Pascal in 1 hour. So, I know what is my P u which is ultimate pressure I know why I am what my P 1 is which is 1 atmosphere I know what my P 2 is which is 1 kilo Pascal and I know my what T P is which is 1 hour. So, what is important to be known is what is my system pumping speed how much what is the system pumping speed for a given pump and for given values of pressures. So, what is given to me vacuum pump system has been given to me working fluid is air at 1 atmosphere vacuum is 1 kPa vacuum requirement is 1 kPa then temperature 300 Kelvin I have got a connecting pipe 440 millimetre 400 millimetre 40 millimetre 400 millimetre then time is 1 hour volume is 1 meter cube ultimate pressure is 0.1 millipascal system pumping speed is SP. So, calculate the value of SS. So, what I know is all these parameters are known to me V P 1 is 1.013 1 atmosphere to begin with 10 to power 5 Pascal P 2 is 1000 Pascal alright then P u is ultimately what I have got is a 0.1 millipascal which is 0.1 into 10 to the power of minus 3 Pascal and T P is given as 1 hour which is 3600 second. So, I know P 1 P 2 I know P u. So, all the things in the formula is known to me. So, I have got a formula which is SS is equal to V upon T P log P 1 minus P u divided by P 2 minus P u I know almost all the values and I calculate what my SS is and therefore, I can put the values over here and I can get my SS to be equal to 0.0012 meter cube per second this is my system pumping speed is. So, if I know system pumping speed if I know the conductances over here I can now calculate my pumping speed or vacuum pump speed requirement for this particular operation so that I can have T P to be equal to 3600. So, from earlier tutorial we have understood that Knudsen number for pipe 1 is 1.132 Knudsen number for 0.2 is 22.65 Knudsen number is more than 0.3 and the flow is free molecular region this is what I know also I know the conductance of this short pipes I know conductance for pipe 1 is this conductance for pipe 2 is this as a result of which we have got overall conductance of CO as 0.00542 meter cube per second. So, if I were to connect all this parameter together I got the relationship of calculating my SP what is required for that is to know what is my overall conductance and is to know what my system pumping speed I know both this parameters and therefore, I got expression now 1 upon SS is equal to 1 upon SP plus 1 upon C 0 and therefore, from there I get 1 upon SP is equal to 1 upon SS minus 1 upon C 0 alright. So, put the value of system pumping speed put the value of conductance and I will get now what my SP should be what is my pumping speed should be. So, 1 upon SP is equal to 1 upon 0.0012 minus 1 upon 0.005 per 2 putting this value I get my pump speed to be equal to 0.00154 meter cube per second alright. So, SP is going to be 0.00154. So, this is my pumping speed and therefore, converting that into liter per minute is SP is 92.4 liter liter per minute. So, I should choose such a pump now which is having a pumping speed around this approximately ideally theoretically we are talking about. So, actually I should collect all the outgassing in a connections here also here and therefore, I should have a pump which has got a pumping speed more than this at least 50 percent more than this and therefore, this will give me first calculation to understand for a given pump. If I want to do vacuuming from a given P 1 to P 2 in 1 hour with this such a connections over here I should have a pump of capacity 92.4 liter per minute and this is the way we do this initial calculations and then put all the reality factors in this in order to decide which pump I should buy. With this we got a summary of the lecture. Correlations for conductance for some commonly used pipes and pipe joints are given. So, you got a short pipe, long pipe, you got a elbow and we know the conductances expressions for these. If you want to know more such expressions you will have to refer to some vacuum books where you can get some intricately shaped T points and thing like that where you can calculate the conductance for such a joints such joint and such pipes also. We know that the pumping speed is given by SP which is equal to Q upon PI and we know that the system P is nothing but SS which is equal to Q upon P where P is my pressure in the system while PI is the pressure near the vacuum pump. We also know the relationship between SS and SP and 1 upon SS is equal to 1 upon SP plus 1 upon C 0 from where we understood that maximum SS can be equal to SP if C 0 is infinite. If conductance is infinite SS can maximum be equal to SP but SS can never be more than SP alright. So, this is the maximum relationship that SS can be equal to pumping speed only. SP depends on a vacuum pump and therefore, in order to maximize SS CO should be maximum. So, my conductance or the connection from the pump to the vacuum system has to be of a bigger diameter and minimum length. If required this pump should be directly mounted on the system to be vacuumed there is no point in having a low conductance in this. So, if I got a very high pumping speed there is no pointing in decreasing pumping speed by connecting it to some capillary tube or something like that. I should ensure that this pump directly systems directly sits on a system to be vacuum. So, that my CO is actually close to infinite value and that is the way in practice the decisions will be done. For constant SS we have an expression as T P is equal to V upon SS log P 1 minus P U divided by P 2 minus P U and this is the expression which we use to calculate the pump down time for a given volume V to be vacuumed from P 1 to P 2. Thank you very much.