 Welcome to the 36th lecture of cryogenic engineering under the NPTEL program. We have covered various topics till now and they are as follows. And the last we finish was cryogenic insulations under cryogenic engineering. So, now the current topic which we are going to talk about is vacuum technology. A brief has already been mentioned about vacuum technology in the earlier lecture regarding cryogenic insulation. However, vacuum is a very important technology and it will be used everywhere in cryogenic engineering. And therefore, it is very important to understand what is this vacuum about and if I want to get good vacuum, what is a good vacuum practice, what are different vacuum pumps that could be used and things like that. What is most important to understand is what are vacuum fundamentals. So, under this topic we will understand what is the need of vacuum in cryogenics although it is little bit clear in the earlier topic when we covered cryogenic insulation. We will know what are vacuum fundamentals, very basic knowledge to understand how can good vacuum be obtained, what are different technologies, what are different nomenclatures that are used to get good vacuum. Then what is conductance and electrical analogy applied to vacuum technology. Pumping speed, we use various pumps, how do we get different pumping speeds and what are pumped down times for respective pumps and then we will talk about in a brief about some vacuum pumps that are normally used. This topic will be covered in around 3 lectures and we will have some tutorial and assignments at the end of in various topics at various points in time. So, in today's lecture we will understand what is the need of vacuum in cryogenics, some basics of vacuum that under vacuum fundamentals and conductance and electrical analogy to understand what is this conductance business that is primarily used in vacuum technology used in cryogenics. We know that the net heat in leak into a cryogenic vessel is due to various modes of mechanism of heat transfer. So, I can write q net, the net heat in leak in a cryogenic vessel will be because of q gas conduction, q gas convection, q solid conduction and q radiation. This is also we have seen in cryogenic insulation and if I produce good vacuum, the gas component can be completely ruled out, it will not be present over there. It will not be equal to 0 but we can say that the amount of gas available in a particular space would be negligible or we can neglect q gas conduction and therefore we can also neglect q convection which is going to be caused presence of gas in that particular space. So, if I would have good vacuum, the first two q I can say q gas conduction and q convection can be completely taken care of or they will get minimized alright. So, gas conduction and convection are minimized, the losses occurring because of these or heat in leak occurring because of gas convection and conduction are minimized by having vacuum between two surfaces of different temperatures. So, if I have good vacuum, the first two losses or heat in leaks can be completely taken care of if I have got good vacuum. Second thing we also know that use of evacuated or opacified powder which basically uses vacuum over there, it will decrease k a apparent thermal conductivity and we also know that multilayer insulation which takes care of solid conduction as well as q radiation, solid conduction is taken care by different aspects having you know very small size per light powder or having spacers. We know that optimum number of MLIs could be used so that q solid conduction also could be minimized. So, MLI can definitely reduce q radiations also it will minimize q solid conduction and we know that all these powders and also multilayer insulation work only when good vacuum is there. So, that means that all these modes of heat transfer which bring about heat in leak into the system can be taken care of if we are having very good vacuum. So, vacuum technology forms a very important aspect in cryogenics. We have just seen that all these modes of heat transfer gets minimized when we have a very good vacuum and therefore, vacuum becomes an integral part of cryogenic engineering or a very important technology to be used always with cryogenics. So, what do we mean by vacuum? The word vacuum comes from Latin roots it means empty or the void so actually nothing. A perfect vacuum can be defined as a space with no particles of any states so no presence of solid liquid gas particles in a space if that is so then we will say it is perfect vacuum. It is important to note that the upper definition is theoretical understanding we cannot have a perfect vacuum and therefore, practically it is impossible to achieve perfect vacuum you can have a very low pressure very very low pressure, but you cannot get perfect vacuum that means nothing is present over there no particles are present over there this is something which is very idealistic situation and therefore, practically it is impossible to achieve perfect vacuum. The pressures in vacuum are lower than atmospheric pressures that is why we call it vacuum. The degree of vacuum is decided by the mean free path we have touched upon lambda mean free path in the earlier lecture also and depending on the value of this mean free path we have got different degrees of vacuum and before that let us understand what is this mean free path although it has already been touched upon in cryogenic insulation. So, what is mean free path? Mean free path is defined as the average distance travelled by the molecules between the subsequent collisions that means from one collision to second collision whatever path it travels whatever distance it travels it is called as mean free path and as you know that as you go on reducing the pressure the number of molecules will be getting reduced and reduced and therefore, the path will increase the path between two collisions will start increasing and as the pressure gets lower down to a very low level there are molecular collisions may stop and the molecule may hit the walls directly in that case that is the maximum mean free path possible in that case. So, lambda is given by a mean free path is given by this particular expression lambda is equal to mu by P into priority by 0.2 to the power 0.5 where mu is a viscosity of the gas P is the pressure of the gas and T is the temperature of the gas. So, there are different parameters and one can calculate the value of lambda or mean free path accordingly and R is a specific gas constant. So, for every gas now we can calculate the value of lambda also we understand from these that lambda increases with decrease in pressure. So, as the pressure gets reduced the value of lambda increases that means, the distance between two collisions or the path starts increasing between the two collisions which is understandable because as the pressure gets reduced down the number of particle will get reduced. So, the number of collisions also will get reduced and molecule to molecule collision will take more path to be achieved by one molecule. So, as the pressure gets reduced the value of lambda increases. It also increases with increase in temperature that can be understood from this because temperature happens to be in the numerator. The value of mean free path plays an important role in deciding the flow regimes in vacuum. So, this lambda is a very important characteristic or very important parameter which decides what is this different flow regimes in vacuum. Similar to the fact that we have got different flow regimes in continuum region we have got different flow regimes in a vacuum also. So, let us see what is this flow regime. Consider a closed system as shown in the figure and there are different molecules and it is a at high pressure that means in comparative atmospheric pressures we got different molecules crammed together in a given space or given volume. But as you go down loading the pressure the number of molecules will get reduced and you may land up in a situation like this where the number of molecules are reduced as the pressure is reduced and the residual molecules are now pulled apart that means the distance between the two molecules will start increasing and therefore, we say that here the molecules will have collisions at will I mean the time they move the time that start having movement they will have a collision over here while here is a probabilistic model. The depending on what direction the molecules move it will have a collision and therefore, we say here the length between the two collision or the lambda value the mean free path in this case will be more as compared to what it is over here and that is obvious from here. As a result mean free path of a residual molecules becomes larger than the dimension of the system. So, sometimes this mean free path will become more than the dimension of the if it travels over here it will possibly never collide with any other molecule in that case the mean free path may be more than the diameter which is the characteristic dimensions of this particular enclosure and mean free path may be more than this or it will be comparable to that. In such system molecules collide only when only with the walls of the container and they will never collide with the molecules themselves. So, molecule to molecule collision will be rare as the pressure goes down or as we go down to lower pressures such a flow of fluid is called free molecular flow. So, now we are talking about having a free molecular zone over here and this is where vacuum would come into picture basically because normally we will deal with such flows in vacuum. If lambda is much smaller than the characteristic length such flows are called continuum flow if lambda is very very small that means the length between the collisions or the time between the collisions is very very small or as soon as the molecule starts moving it collides with other molecules. So, in this case lambda is going to be very small and this will not normally happen in continuum flow which is what normally we will deal about atmospheric pressures. In fluid mechanics Reynolds number is used to categorize the pipe flow regime as shown in the figure. So, we got a normally we deal with Reynolds number in continuum flow and we know that the Reynolds number is between 0 to 2300 2300 2300 what we call the flow is laminar while if the Reynolds number is between 2300 and 4000 we say that the flow is transition flow while above 4000 we say the flow is turbulent flow and this is very much known to all of you. In these flows molecules collides with each other as well as with physical boundaries if any that means the molecules are in constant collision conditions they are constantly colliding with each other and they also may collide with the walls or the physical boundaries of this pipe. Now such thing may not exist at low pressures alright. So, in this case pressures are in the range of atmospheric values while if you go to the vacuum now we have got a different flow regime now and we do not have Reynolds number over in vacuum now. What we have now here is called Knudsen number normally written as NkN is used to categorize the flow regimes in vacuum. So, here we always deal with a different number and not Reynolds number and the moment we specify Knudsen number we know that we are talking about flow regimes in vacuum now and what is this Knudsen number now. This concept is analogous to Reynolds number in fluid mechanics and the Knudsen number NkN is given as NkN is equal to lambda by D where lambda is mean free path and D is the characteristic diameter for dimension. So, if you know that as we have lower and lower pressure the value of lambda will go on increasing that means the Knudsen number will go on increasing as we have got better and better and better vacuum and according to the value of this Knudsen number now we will have different flow regimes in vacuum. So, I will have different Knudsen number now. So, based on the Knudsen number NkN the above figure characteristics the flow regimes in vacuum. So, when I say my Knudsen number is less than 0.01 I will say I am in continuum region that lambda is much smaller than D in this case, but as lambda starts increasing Knudsen number starts increasing. So, if I am between 0.01 to 0.3 I have something called as mixed flow over here. So, my lambda is now increasing because the pressure is decreasing and if my Knudsen number is between 0.01 to 0.3 I will have something called as mixed flow and if my Knudsen number is more than 0.3 that means lambda is 0.3 times dimension of the characteristic dimension I am in something called as free molecular region. So, my molecules are in a free molecular flow region here they are in the mixed flow regime in this case when the lambda when the Knudsen number is between 0.01 and 0.3 and if the Knudsen number is less than 0.01 I am actually calling I am still in continuum region here the lambda value is going to be very very small as compared to the dimensions or the characteristic dimension of the container. Summarizingly we say that we have got a continuum flow for Knudsen number less than 0.01 mixed flow between Knudsen number having between 0.01 and 0.3 and free molecular flow for Knudsen number greater than 0.3 the lambda value will go on increasing Knudsen number will go on increasing and I will reach to a free molecular regime when the Knudsen number is more than 0.3. So, let us call now because as we go on lowering the pressures we normally would call the pressures by different unit in this case. So, let us understand what are normal units of pressure and then we will define how do we refer to different vacuum levels. So, in SI unit pressure is measured in Pascal or Newton per meter square very often bar is also used for pressure measurement you know this. For example, the standard atmospheric pressure can be expressed as 1.013 into 10 to the power 5 Pascal 1.013 into 10 to the power 5 Newton per meter square this is what normally we would use in SI or in thermodynamics also we got a 1 bar or we got a 760 millimeter of mercury column at standard C level. This is what our standard units of pressure this is what we use normally in thermodynamics and heat transfer and fluid mechanics and thing like that. Now in vacuum normally unit of pressure is Tor T o double r Tor or sometimes also milli bar this unit is named after Evangelista Torricelli you must have heard about Torricelli and Italian physicist in the year 1644. So, to honor this scientist Torricelli we have got a unit of pressure specifically used in vacuum. The vacuum therefore will be called as so many tors or sometimes in milli bar also and this unit is named after this scientist Torricelli in the year 1644. So, one tor what is this one tor one tor is defined as 1 millimeter of mercury column at standard C level. So, imagine that 760 millimeter of mercury column makes 1 atmosphere 1 bar while 1 millimeter makes 1 tor here or we can say 1 by 760 bar is nothing but 1 tor or 1 by 760 atmosphere is nothing but equal to 1 tor. So, naturally the pressures in this range are much below atmosphere much below 1 bar. Therefore, 1 tor is equal to 133.28 Pascal or 1 Pascal is nothing but 1 Newton per meter square. So, 133.28 Newton per meter square. So, there is a conversion table for your reference which is going over here. So, we can have different units we can have vacuum could be presented sometime in Pascal bar atmosphere or tor normally it will be referred in tor and here we got different conversions from Pascal to atmosphere Pascal to tor. So, I can say that 1 Pascal is equal to 10 to the power minus 5 bar 1 Pascal it is equal to 7.5 into 10 to the power of minus 3 tor. Similarly, 1 bar is equal to 10 to the power of 5 Pascal's 1 bar is equal to 750.06 tor or 1 tor which is what we vacuum is referred to as is equal to 133.3 Pascal or 1.3 1 into 10 to the power of minus 3 atmosphere. So, one can understand from this if I were to convert different units of vacuum or pressures into other units this is the kind of a conversion table for your usage. Similarly, we know that 1 milli is nothing but 10 to the power minus 3 and 1 kilo is 10 to the power 3. So, sometimes kilo Pascal could be written over here. So, I can say 1.33 kilo Pascal is equal to 1 tor also alright because sometimes you can understand like this also. Now, there are different degrees of vacuum depending on the levels of different pressures. So, as mentioned earlier pressures are lower than atmospheric pressure in vacuum spaces and depending on how low they are depending on their values we got something called as degrees of vacuum. So, depending upon the pressure in the system the degree of the vacuum is categorized the table on the next slide correlates the pressure and the degree of vacuum. So, normally we call I should have low vacuum I should have very low vacuum I should have ultra low vacuum and thing like that they are all classified based on what is the level of pressure over there and therefore, let us have a look at this degree of vacuum. So, degree of vacuum normally called as rough vacuum in most of the industrial applications we got rough vacuum wherein the pressure is between 760 tor which is nothing but 1 atmosphere approximately and 25 tor. So, when your pressure is between 25 tor and 760 tor we have got rough vacuum or in kPa we got a 3 kPa and 1 0 1 0 3 kPa. So, if my pressure is between 1 0 3 kPa and 3 kPa we got a rough vacuum then you got a medium vacuum between 25 tor to 0.001 tor 10 to the power minus 3 tor are also given in Pascal's over here then we got a high vacuum. So, we got a rough vacuum medium vacuum high vacuum the high vacuum is between 10 to the power minus 3 to 10 to the power minus 6 tor. So, again you can understand that I am come down to 10 to the power minus 6 tor which is between high vacuum when the pressure is between minus 3 and minus 6 tor sometimes it is referred as when one would not say 10 to the power minus 3 to 10 to the power minus 6 one can always say that vacuum is of the order of minus 3 to minus 6 or it is between minus 3 and minus 6 normally it is understood that this is 10 to the power minus 3 and 10 to the power minus 6 there are different levels of vacuum and then we got a very high vacuum between minus 6 to minus 9 levels alright when we got a pressure of the order of 10 to the power minus 8 minus 9 we say we are in very high vacuum region and then we got a ultra high vacuum which is used for very special purposes and this is now less than pressure less than 10 to the power minus 9 tor. So, you can understand normally most of our operation happens between medium vacuum to high vacuum in this region at least in cryogenics unless very high levels of vacuum are expected then you know very special applications would expect ultra high vacuum to be there in that case what kind of material do you use what processes do you use all these aspects become very important because you need a very high level of vacuum in this case and therefore, kind of processes you use here are very very important. So, slide number 16 correction so, we can say that from the here 1 tor is equal to 0.133 kilo Pascal. So, we have understood what are various degrees of vacuum now and therefore, now we will go to little practical applications or how to get good vacuum, but before that we will have to understand the flow rates the pressure drop because if I want to vacuum particular space I will connect that space to a vacuum pump and I will connect that space to vacuum pump through a tube or a pipe and to begin with now we will have a pressure drop across this pipe and therefore, we will have molecular flow region we will be now talking about not a continuum region, but maybe we will be talking about molecular flow regimes and therefore, we will have to understand what is the pressure drop that happens what are the mass flow rate that happens when we are talking about molecular flow and therefore, let us understand the pressure drop the mass flow relationships as we go down from continuum flow to mix flow and mix flow to molecular flow region and this is very important to understand to calculate the mass flow rate and the pressure drop delta P across a tube of let us say length L and therefore, let us understand what is this pressure drop and mass flow rate relationships for continuum region mix flow region and molecular flow region. So, we know that in a pipe if the fluid is travelling across the pipe of constant cross section area as shown in the figure will have pressure drop across this region from 1 to 2 will have a pressure drop with passage of time for simplicity let the flow region be continuum let us assume it is continuum region now as the fluid flows from 0.1 to 0.2 there is the pressure drop due to viscosity and therefore, what will have pressure at this point 2 will be less than pressure at this point 1. So, P 2 is going to be less than P 1 because of the friction because of the surface conditions because of the viscosity of the fluid the difference between the inlet and exit pressures which is P 1 minus P 2 is what we call as pressure drop. Let this be this be denoted as delta P and therefore, we can say delta P is equal to P 1 minus P 2 and it is a very standard fluid mechanics practice to calculate delta P across the region when the flow rate is around let us say m dot or whatever it is. So, pressure drop for a laminar continuum flow and when I say laminar flow we are talking about Reynolds number less than around 2300 less than 2300 and this is a very standard equation delta P is equal to 128 mu l m dot upon pi into d to the power 4 into rho and this is called as Poiseuille's equation in fluid mechanics which correlates pressure drop delta P to mass flow rate m dot. So, if the length is l and the mass flow rate is m dot and the diameter is d then during this travel I will have a delta P amounting to this and one can see from here that delta P is directly proposed to the length it travels delta P is directly proposed to the m dot mass flow rate. If the mass flow rate is high delta P is going to be high the length is high delta P is going to be high while if the d is small delta P is going to be very very large delta P is dependent on d to the power 4 which is a very important parameter. So, in continuum flow we can see that delta P is inversely proportional to the fourth power of d alright this is a very important parameter please note that. So, here mu and rho are viscosity and density here you can say mu and rho are viscosity and density of the fluid while l and d are the length and the diameter of the tube and the delta P is directly proposed to the length and delta P is inversely proportional to the fourth power of d as far as we are talking about laminar continuum flow. For rough vacuum now when the Knudsen number is less than 0.1 as mentioned earlier the operating parameter pressure is now between 25 to 760 torr which we know now I am talking about Knudsen number less than 0.01 and ideal gas behavior assumed here and hence the correlation between the average pressure and density is rho is equal to P m by R t. Gas law here alright. So, here rho and t are density and temperature of gas and we are talking about now a rough vacuum Knudsen number less than 0.01 or operating pressure between 25 to 760 torr we are having that ideal gas relationship is valid m is a molecular weight of the gas R is a universal gas constant while P is the average pressure. So, we have found that for this continuum region Knudsen number less than 0.01 we would delta P is 128 mu l m dot upon pi into d to the power of 4 into rho. We know also rho is equal to P m by R t and if I put the value of rho over here combining the above two equations the pressure dot delta P for a continuum laminar flow is this. And from here I get relationship it for m dot having m dot over here I will get m dot and delta P related. So, m dot is equal to pi into d to the power 4 P m into delta P divided by 128 mu l R t. So, basically delta P and m dot are directly proportional and you got a relationship between delta P and l and d therefore, you got a relationship between m dot and d and P d and delta P. From the above equation it is clear that the mass flow rate m is directly proportional to the pressure drop. So, if the pressure drop is higher mass flow rate going to be higher and it is directly proportional to the 4th power of diameter. Here delta P we are talking about when delta P had inverse relation with the 4th power of d while mass flow rate has now direct relationship between the to the 4th power of diameter. Now, that was with the continuum flow and let us talk about now mixed flow or with the lowering of pressure in the tube when the Knudsen number is between 0.01 and 0.3 and intermediate flow regime between the continuum and the free molecular flow exists now as we have seen different flow regimes in vacuum now. This regime is called as mixed flow or slip flow. So, what is happening now? We are somewhere in between molecular flow region and continuum regime and that is why it is called as mixed flow regime and in such condition the gas molecules close to the wall appear to slip past the wall with the finite velocity parallel to the axis of the tube and hence it is called as slip flow. In the earlier case in the continuum regime we know that the velocity of the particles at the wall is equal to 0. We know that the pressure is the flow rate is maximum in the center and is 0 at the walls of this pi. But as the number of molecules go on lessening as the Knudsen number is getting increased or as the lambda starts getting increased, this molecule at the wall will also have some finite velocity and therefore, it will not be velocity equal to 0 over in this case. It will have some finite velocity and therefore, we say that the molecules appear to slip past the wall. It is going it is not stopping over there it is going it is slipping past the wall with a finite velocity and therefore, this velocity will also be showing up in the calculation of delta P or mass flow rate. And therefore, in slip flow we will have something what was happening in continuum flow plus because of the change in velocity or because of the velocity existing at the finite velocity at the walls of this pipe will have one more parameter appearing in the pressure or in the mass flow rate calculations. So, from the kinetic theory of gases mass flow rate and pressure drop for slip flow in a circular tube is given by this formula. So, you can see now m dot is equal to something into 1 plus something. So, this is the same as what it was in a continuum flow and this is what we had in a continuum flow. This is the expression for m dot in continuum flow and this is nothing but same. On comparison of above equation with mass flow rate for continuum laminar flow, we know that the first term accounts for the internal laminar flow away from the walls. That means at the centre near the centre of the tube this will account for that, but the second term is now accounting for the slip or the finite velocity the gas molecules will have near the walls. The second term accounts for the finite velocity correction near the tube walls. This will make the matter little complicated because in earlier case mass flow rate was directly proportional to d to the power 4. In this case however, we got a d to the power 4 here and we got a d in the denominator over here and therefore, the relationship between the mass flow rate and diameter is not so much clear. So, we can understand however m dot is directly proportional to still delta p. So, from the above equation the mass flow rate m dot is directly proportional to the pressure dot which is delta p. However, the dependence of diameter d of the tube is more complex as compared to the fourth power relationship in laminar continuum flow. So, earlier we had m directly proportional to d to the power 4 while in this case we got a m which is a complex relationship with diameter because diameter appears in the finite velocity here in the denominator while in the continuum region it was appearing in the numerator only. So, you can see the change that has occurred because we shifted from a continuum region to a mixed flow region now of the Knudsen number between 0.01 and 0.3. Let us if you go down the pressure earlier Knudsen number will be more than 0.3 with further lowering of pressure when Knudsen number is more than 0.3 the number of molecules are reduced as well as the residual gas molecules are pulled apart. So, we have seen earlier the lambda value has increased now and therefore, the collision path the path between having two collision has become comparable with the characteristic dimension the Knudsen number is more than 0.3 the pressures are less the distance between the molecules will increase this flow regime is called as free molecular flow. In such conditions mean free path lambda of the molecules is larger or comparable with the diameter of the tube the flow is limited due to collisions of molecules with the walls this is what we had talked about earlier. In this case now the mass for it m and the pressure drop in a free molecular flow are related by this formula. So, we got a m dot now directly problem to d cube over here while m dot is still directly problem to delta p. However, its dependence on diameter has changed. So, if you recollect in the continuum region it was dependent directly on d to the power 4 in the slip flow it was a complex relationship while here in the molecular region m dot is now directly dependent not power 3 of the diameter d cube basically. So, things are changing as we shift from continuum region to free molecular region and this is very important because we will have to calculate or we will have to understand what is diameter should I use what is the length should I use if I were to connect my vacuum pump with the space to be vacuum alright or if I know that region to be vacuum has so much diameter and length I will have to understand the dynamics of molecular flow in that particular space. From the above equation it is clear that the mass flow rate m is directly proportional to the pressure drop delta p and directly proportional to the third power of diameter this is what we have understood. Now, let us come to the next nomenclature which is prevalently used in vacuum technology which is nothing but throughput and normally designated as q. So, apart from mass flow rate m the rate of fluid flow is often measured by quantity called as throughput q. So, we can call something as mass flow rate which is kg per second or something like that but then we have got one more parameter called q and it is very important because it talks about both pressure and volume flow rates also. So, what is this q? The throughput is defined as a product of volumetric flow rate v and pressure measured at the point where v is measured wherever we are measuring mass flow rate or volumetric flow rates at the same place will measure pressure also and pressure into multiplied by that volumetric flow rate will give me what is called as throughput and therefore, it will have units of pressure as well as volumetric flow rates. So, mathematically we have q is equal to p into v dot. So, volumetric flow rates let us say liter per second, liter per minute, liter per hour or any other dimensions we can have and pressures we can have pressure dimensions as torque or bar or milli bar kilo Pascal's or whatever. So, the assignments of throughput therefore, are Pascal meter cube per second Pascal for pressure meter cube per second for volumetric flow rate very often at low pressure it is also expressed in torque liter per second. Mostly I have seen that various places we could torque liter per second or milli bar liter per second. So, torque liter per second torque is for pressure liter per second or meter cube per second will be for the volumetric flow rate and this is the way normally the throughput is mentioned. Assuming an ideal gas behavior the volumetric flow rate v is expressed using ideal gas law which is nothing but v dot is equal to m dot r t divided by p and r by m is nothing but specific gas constant. So, therefore, we have got this m as molecular weight of the respective gases from the above definition and this has come from basically p v is equal to m r t from the definition of throughput. Now, can we put the value of v dot in that equation and therefore, what we have is q is equal to p v dot and if you put the value of v dot in this equation this pressure and this pressure will get cancelled out and therefore, I will get q is equal to m dot r t upon capital M. So, I got a relationship between now mass flow rate and specific gas constant and temperature to value of q and here m is a mass flow rate m capital M is molecular weight of the gas, t is a temperature or is a universal gas constant. Now, it is important now to note that vacuum system involves complex piping arrangements. If I want to vacuum a particular space I will connect that space by different by having different piping arrangement and these pipes could be in parallel to each other this pipes could be in series with each other and to analyze if I connect pipe of a particular dimensions d and l to another pipe of d and l it will give me different throughputs, it will give me different something called as resistance to the flow or delta p and therefore, it is very important to understand what happens if I connect these pipes of a given dimension in series or in parallel and therefore, this series connection and parallel connection could be understood could be well understood by having electrical analogy, we know Ohm's law basically we know the resistances in series resistances in parallel and therefore, understanding this piping in connection in series and parallel based on the electrical analogy is always very simple. So, let us try to understand that parameter. So, in order to analyze these systems a mathematical theory is developed based on an analogy between electrical circuits and piping systems linear transport laws like Ohm's law and Fourier laws are used in formulating the problem. So, let us understand electrical analogy, we know that when a small current of I is passed through a resistance which has got a potential difference of delta v because of certain resistance which is automatically connected by Ohm's law. So, consider a small electrical conductor as shown when a current I flows across this conductor there is a voltage drop delta v due to the resistance R offered by the conductor and this is very simple and they are related to each other by Ohm's law. These quantities are mathematically related by Ohm's law as given below. So, Ohm's law says delta v is equal to I into R. Now, when we compare this electrical circuits of having voltage difference across a given resistance of length L through which a current I flows. So, now let me compare this with a pipe and this I say that a pipe flow occurs Q which is comparable with I and this Q occurs only when the delta p occurs across the length of the pipe and the length of the pipe is L and delta p is the pressure drop across the length of the pipe L. So, similarly consider a fluid flowing across a small pipe as shown above. So, I can now compare electrical circuit with a pipe. So, I can see that I gets compared with Q and delta v can get compared with delta p while the resistance is nothing but maybe some kind of conductance I can get you know have a different parameters which will be analogous to which will be related to the resistance in the electrical circuit. So, for a throughput Q there is the pressure drop delta p due to conductance C offered by this pipe. So, in pipe normally we do not call resistance, but we refer to this as conductance and therefore, what is conductance area reciprocal of R is nothing but conductance. So, if I have got a parameter R in electrical analogy I will say 1 by R is nothing but conductance. So, from electrical circuit I can compare this with the conductance of the pipe and conducts of the pipe C is related to 1 by R of electrical circuit alright. So, I have got three different comparisons I with Q delta v with delta p and C in conductance of pipe to the 1 by R value in the electrical circuits. So, comparing the above figures we have delta v analogous to delta p I analogous to Q and R is analogous to 1 by C. Now, therefore, we can write delta v is equal to I into R and therefore, I can write delta p is equal to Q into 1 by C Q is nothing but similar to R i and 1 by C is nothing but similar to R and therefore, I will get delta p is equal to Q by C in our equation for fluid flow in a small pipe and this is what basically a comparison of fluid mechanics or a flow regime in vacuum will be with the electrical energy when the current flows through a conductor. So, let us we have just floated a term called C, but we are not yet defined and therefore, we will try to understand what this conductance in a vacuum defined as. So, conductance is related to delta p by this equation delta p is equal to Q by C and therefore, I can say Q is equal to C into delta p. It is clear that for a given pressure drop delta p across a pipe throughput Q is directly proportional to conductance C alright. So, for if I have got a some value of Q given over here then I will have throughput Q directly connected or directly related to delta p or directly connected to conductance C. So, if C increases Q increases if my delta p is given to me. So, Q will vary in accordance to the variation in C directly they are directly related to each other. So, if I want to have more and more Q I should increase the conductance of that particular pipe. So, increase the C increase the value of Q and this will be good for creating vacuum in a particular space. For an ideal gas following equation is holding true Q is equal to m dot r t upon m. This is what we have talked about earlier and we know from here delta p is equal to delta rho into r t by m. This is true with the gas which follows p v is equal to m r t and from where we could conclude that Q is equal to m dot r t upon m dot and delta p is equal to Q upon C or delta p directly related to the gas law by this equation delta rho r t by m. Now substituting these values over in this equation Q and this Q we can put this Q is equal to this Q and we can have delta p as this over here putting these values we will get now m dot r t upon m is equal to C into delta rho r t upon m delta rho is the density difference. So, now from here we can cancel out r t and r t we can cancel out m and m and we get therefore relationship between m dot and conductance and delta rho. From here we can see that therefore this will get cancelled and I get C is equal to m dot upon delta rho. So, now my conductance is nothing but mass flow rate divided by the delta rho the density difference which is occurring because of the pressure difference p 1 and p 2 or the pressure drop occurring across the length l. So, conductance for a pipe for different flow regimes can be now derived by this formula can be derived by rearranging the pressure drop mass flow rate equations derived earlier. So, let us see all the derivations earlier for continuum flow now we will get conductance C is equal to pi into d to the power of 4 p because now we know that from the earlier calculation we know that C is equal to m dot upon delta rho. I will put this into all other equations which we have calculated for m dot for various regimes just divided by delta rho it will cancel out delta rho in that particular equation and therefore what we will get is for continuum flow we will get C is equal to this conductance is directly propelled to the d to the power 4. For a mixed flow for the Knudsen number between 0.01 and 0.3 we will get C is equal to this formula similar to what we had done earlier and similarly for free molecular region now we will get conductance is equal to this. What does it mean? It means that if I know the length of a pipe if I know the diameter of the pipe if I know the average pressure in the system if I know the temperature if I know the gas to be vacuum I can calculate conductance for that pipe for that gas for a given delta p for a given pressure for a given length and given diameter of the pipe I can calculate conductance of different pipes. Many times now we can have standard equations to calculate conductance for a bend for a conductance for a straight length conductance for diameter changes in the pipe lengths and these are the formulae which will help us to calculate conductance for different pipes of different length and diameters for different gases and therefore it is very important to know such relationships which are normally standardized to understand what happens to the conductance when there is a bend in a pipe when the pipe is straight when the pipe has got some you know curvature for example all these equations help us to calculate conductance of pipe which ultimately would help us to calculate the what will the pressure drop that will happen if I have this vacuum pump over there or how much time it will take for me to reach down to the lower and lower vacuum levels for a given pump of when I connect the vacuum pump to a given space through such pipes alright. So, very important to understand what is this conductance and how it is related to the operating parameters and the dimensions of the pipe. So, conductance is vacuum now we can have conductance when the two pipes are connected in series and the pipes could be connected in parallel the way we have in resistances connected in series and parallel. So, consider a series combination of two pipes with C 1 and C 2 as individual conductances respectively with delta P 1 and delta P 2 as the pressure drop happening let Q be the throughput for this pipe it is clear that a series combination of Q is the same for each pipe Q is going to be the same because whatever Q happens to this pipe the same Q is going through this pipe is similar to I I is the same when it is entering at this point and it is going through this particular resistances. The pressure drop in each pipe are going to be delta P 1 and delta P 2 respectively and therefore, we can say delta P 1 is equal to Q by C 1 C 1 and C 2 are conductances of this pipe of different dimensions and length. Accordingly we will have delta P 1 is equal to Q by C 1 delta P 2 is equal to Q by C 2 and what we know let us have a overall conductance conductance as C 0 and let us try to relate that to this and we also know that the overall pressure drop delta P is equal to delta P 1 plus delta P 2. So, let the overall conductance and the total pressure drop in the system be C 0 and delta P respectively and what we know therefore, is total for the equation for this is delta P is equal to Q upon C 0 alright and we know also delta P 1 is equal to Q by C 1 delta P 2 is equal to Q by C 2. Now, I know delta P is equal to delta P 1 plus delta P 2 and therefore, I will get an equation as 1 by C 0 is equal to 1 by C 1 plus 1 by C 2 where Q and Q will get cancelled if they are put in this particular equation. What does it mean? It means that when the two pipes are connected in series the effective or overall conductance of this pipes when they got a C 1 and C 2 of respective conductances will get 1 by C 0 is equal to 1 by C 1 plus 1 by C 2 and this is what happens in the resistance in series 1 by C 2 nothing, but overall resistance R 0 is equal to R 1 plus R 2 same thing happen over here when the conductances are getting connected to each other in series. So, if I got this we did only for two pipes I can have now n number of pipes. So, extended to n pipes in series I will get 1 upon C 0 is equal to sigma 1 by C i i ranging from 1 to n in that case. So, if I know different pipes connected in series what will I do? I will calculate their respective conductances and get overall conductance calculated by this formula. So, 1 upon C 0 is equal to 1 upon C 1 plus 1 upon C 2 plus 1 upon C 3 etcetera would give me the overall conductance of a given series pipe connection. Now, let us see what happens if the pipes are connected in parallel. So, similarly consider a parallel combination of two pipes let us say C 1 and C 2 are the conductances of these two pipes while as you can say that some Q 1 will go through this pipe some Q 2 will go through this pipe depending on the diameters of this pipe. Let us see that they are having same length and therefore, will have the delta P happening across them is going to be the same because the same gas will get split up or the flow will get split up here and the flow will come back here as a result of which the pressure at the entrance and the pressure at the exit will be same and therefore, both in both these pipes which are connected in parallel the pressure drop is going to be the same. This was not the case when they are connected in series. So, let C 0 and Q be connect be given and then we have delta P be the same we know that Q is equal to C 0 delta P which is overall Q and we know that Q is equal to Q 1 plus Q 2 also because Q is coming from here Q is coming out from here when they are travelling in this parallel pipes they getting split up into Q 1 and Q 2 and Q 1 is nothing but C 1 into delta P delta P is same all over Q 2 is equal to C 2 into delta P and therefore, what we know is Q is equal to Q 1 plus Q 2 and if I write them together delta P delta P we will get this is common basically therefore, I will get C 0 is equal to C 1 plus C 2. So, when the pipes of C 1 and C 2 conduct instances are connected in parallel to each other the overall conductance will get added as C 1 plus C 2 directly and therefore, this is the overall conductance for pipes which are connected in parallel with each other. So, extending this to n pipes we will have one of C 0 is equal to sigma C i when i is extending from 1 to n pipes. So, we saw what is conductance when they are connected in series what is overall conductance when they are connected in parallel to each other. Summarizing this lecture heat in leak is minimized by having vacuum between two surfaces of different temperatures lambda which is nothing but mean free path is defined as the average distance travelled by the molecules between the subsequent collisions based on Knudsen number which is lambda by D we have continuum flow when Knudsen number is less than 0.01, we have got a mixed flow when Knudsen number is between 0.01 and 0.3 and we got a free molecular flow when the Knudsen number is more than point through when the lambda is increasing from continuum flow to free molecular flow and we know that conductances of pipe which are connected in series is 1 by C 0 is equal to sigma 1 by C i and when they are connected in parallel we know that C 0 is equal to sigma C i they will just get added C 1 plus C 2 plus C 3 etcetera. Thank you very much.