 Hello and welcome. In this lecture, we will establish some of the basic concepts of what multistaging in rocket is all about and also understand the kind of benefits that we can derive by creating a multistage rocket for the mission that we have in mind. So, let us begin. So, let us start the discussion by looking at the basic concept itself. So, multistaging in rockets is the design strategy that aims to make ascent missions more efficient by dividing the total rocket into a number of sub-rockets called stages. Typically, each sub-rocket is a complete rocket by itself and at the end of the burnout of the sub-rocket shell and other inert mass are separated from the remaining rocket. This is the fundamental idea of the staging that you divide the total rocket into smaller segments and each segment after it finishes the operation is removed from the main body of the rocket and we immediately realize that this would reduce the overall mass of the rocket which will be required for operating the next stage. We see that the starting mass now for the next stage is significantly smaller as the inert mass which we were carrying earlier we are no longer carrying and we are throwing it away. And this obviously means that the remaining propellant which is in the other stages which have not yet operated will need to accelerate a smaller mass and hence they will become more efficient from the overall loss point of view. There is a side benefit that we automatically get by dividing the rocket into segments and that is in terms of using different technologies for different segments because each rocket is a unit by itself that unit can be treated separately and it need not have any commonality with the other units which means it can use a different propellant like solid or liquid it can use a different structure like metal, composite, frame structure, honeycomb and in the process make the operation of each stage also more efficient and optimal. So not only the overall rocket becomes more efficient it is possible for us to also individually make each stage more efficient. Of course, there is always a flip side that the multi staging makes the mission design its implementation in terms of fabrication assembly and its operation in terms of the processes that help in getting rid of the inert mass a lot more complex configuration. Let us before going through the actual formulation for this kind of a configuration make a quick assessment of what are we likely to get out of this so that we can decide whether it is worth making the effort or not. So let us take the same problem of a rocket with 80 ton lift of mass carrying 60 ton of propellant having ISP of 240 seconds and let us now try and see whether the philosophy that we have just now discussed can be worked on this to understand what kind of performance are we likely to get. So here let us make certain assumptions let us assume that I am going to divide rocket equally which means if I divide the rocket into two segments each segment will contain 30 tons of propellant and 9 tons of inert mass and if I divide this into three equal stages each stage will contain 20 tons of propellant and 6 tons of inert mass and let us now use the simplified ideal burnout velocity expression just to understand how it gets impacted by performing this particular split. So let us first look at the first stage or one stage operation in one stage operation I am burning all the 60 tons in single shot so that at the end all the 18 tons of inert mass is attached to the 2 tons of propellant very payload mass and we have already seen that we get 3264 meters per second velocity. Let us now convert this into a two stage operation so when we say a two stage operation its first stage will contain 30 tons of propellant and 9 tons of inert mass which will burn first so that at the end of the first stage operation you will have 50 tons of mass left that is the first term in our velocity expression but now you will find that you will not start from 50 tons the next stage you will get rid of the 9 tons of inert mass of the first stage before you start the second stage so the second stage will start with only 41 tons burn the 30 ton propellant and end up with only 11 tons of final mass of which 2 ton will be the payload and 9 ton will be the remaining shell of the second stage which anyway cannot go anywhere and you realize that in this process the velocity increment is almost of the order of 1 kilometer per second fairly significant increase in your velocity capability by just converting it to two stage operation let us now go to three stage operation in three stage operation first stage has only 20 tons of propellant and 6 tons of inert mass so the first term is 80 by 60 and now using the same strategy that we have done in the two stage we remove 6 tons from 60 tons so that we start only with 54 tons but another 20 tons of propellant and reach 34 tons at the end of second stage and then remove another 6 tons from this 34 and start with 28 tons for the third stage operation which ends up at the end of the mission as 8 tons of which 2 ton is the payload and 6 ton is the shell you can clearly see that if I were to now do this let us say for fourth stage or five stage incrementally the inert mass which is connected to the payload will start reducing and in the limited if you use infinite stages the final stages will have only the payload and nothing else so that will be the most efficient which means that if you perform this with infinite stages that is the best possible scenario and you can get a best possible velocity the question is is this really workable so let us look at a picture so this picture shows a generic scenario that we are likely to get when we start increasing the number of stages the scenario is created in terms of the two important parameters of the rocket performance that is the payload mass fraction and the speed ratio that is amount of speed which is increased as a ratio of the exhaust velocity this capital C is the exhaust velocity which is typically of the order of about three kilometers per second for most common fuel samploid and let us now look at this little bit closely so let us first say that we are going to fix the velocity at this point which is about 2.4 times the exhaust velocity which is approximately equal to the velocity required for establishing a circular orbit at around 200 to 250 kilometer altitude that is about 7500 to 7600 meters per second so let us say that we are interested in such a mission now in single stage operation you immediately realize that you are going to get on the 1 kg of payload per 1000 kg of the lift of mass that is the efficiency of your mass fraction the moment you make it is a same mission a two-stage mission I simply move the cursor vertically and come to this curve for n equal to 2 and then I move it to the left horizontal and I find that with the same lift of mass same propellant isp everything same I can achieve a payload of practically 7 percent fraction which means for every 1000 kg of lift of mass I can get 70 kg payload 70 times increase just by going from 1 stage to 2 stage of course I also see that if I go from 2 stage to 3 stage the benefit is not very significant even though we are using a logarithmic scale on the vertical side let us now argue the other way let us say that we are going to fix our payload requirement as 7 percent so let us come to this point now if I want to have a payload fraction of 7 percent so that for a 1000 kg rocket I would like to have a 70 kg payload I come to n equal to 1 and I find that the velocity I am going to get will be roughly of the order of about 6000 to 6200 meters per second not sufficient to even establish an orbit which means that if I want to establish an orbit for this payload not possible with a single stage operation so let us now move over to a 2 stage operation now immediately I see that for 2 stage operation the velocity will be of the order of 7600 meters per second and we can establish so which means that if I want to launch a 70 kg of payload with a 1000 kg rocket I must have at least 2 stage operation of course if I go to a 3 stage operation there is only a marginal benefit in being velocity now I see another interesting pattern that if I were to lower my payload requirement to let us say 1 kg per 1000 kg with a 3 stage operation I can get a velocity which is almost 5 times the exhaust velocity which is almost like say 12 to 15 kilometers per second it is way beyond escape velocity for interplanetary missions so now you can see if that from this picture by bringing in the multi-stage configuration either you can take a small payload for interplanetary missions or a large payload for near earth missions and that is what you will find most of the launch vehicles do that same launch vehicle can also serve a space station mission or a mission to moon or Mars it is just a question of appropriately deciding on what payload fraction you are going to use of course we also see from this picture that the infinite stages is not really a workable option for two reasons one you have law of dimension return so that beyond three stages the actual benefits in terms of percentage are significantly smaller in relation to the complexities that will increase because every time you add a stage you need to create a mechanism that will handle the separation of the inert mass of that stage and will also hand over the control to the next higher stage so obviously it is not a very useful concept beyond a certain point and that is why you will also find if you look at the data sheet of many launch vehicles that we have currently with us that beyond four or maybe in some various rare cases five stages you do not have any rocket configuration most of the rocket configurations lie within two three and four stages and now we have a basis for using these number of stages so let us now understand the broad framework and philosophy within which the staging operation is going to be carried out so let us recall that each stage is a rocket by itself with its own propellant and structure including systems and mass which is altogether termed inert mass while the final payload mass is separately prescribed as m star as we have seen in the example so this is the terminology that is going to get used from now onwards the m star will always be used as the final useful mass that is going to be released in the orbit and now we have in addition to that other masses for each stage the inert mass for each stage and the propellant mass of each stage or what we would call the propellant loading of the each stage of course in the example that we have seen now we have not really changed the ISP of the propellant and you will immediately note that if we do that the performance that we can get is significantly different for example if you use a higher ISP fuel in the higher stages the amount of velocity increment that you can get even for two stages is very high and that also is one of the reason why in most launch vehicles which are used for launching spacecraft for inter-penetration missions the last stage is typically the cryogenic which has the highest ISP now you see the reason why the cryogenic stage is the last stage because that is the most efficient stage it is required to accelerate the smallest possible mass so the largest amount of delta v we can generate by using even a smaller amount of propellant in the final stage so to summarize the multi-stage rocket design provides significant benefits in terms of energy or velocity and also very high efficiency of conversion of the propellant energy to the mechanical energy however these benefits rapidly decrease with increase in number of stages while the operational complexities increase and results in a cap on the number of stages that are commonly employed for most practical launch vehicle to between three and four stages so we have seen the benefit in a broad sense about what multi-staging can do to rocket performance at the terminal point of course we have used idealized equations but as you can see as a kernel or as an idea this definitely has a lot of potential to give you a much better performance and what we will do is in the next lecture we will look at some basic formulation steps that using the framework and the guideline you will be able to set up the equations that will give us the solution for a configuration of the rocket that in the multi-stage context will give us the desired performance so bye see you in the next lecture thank you