 Good morning. We start from where we left last time. Last time we, in fact there were two lectures in which we looked at the basic methods and formulae for estimating the static lift generated by any LTA system. First we derived the basic formulae and then we tried to simplify them and also apply them to specific LTA systems like rigid airship and then a balloon and also a super pressure balloon. Now what is more important for us is to know what happens to this static lift when there are variations. So today we are going to look at several operational parameters and their effect in the changing values of static lift or to be more precise what is the methodology to calculate the variation in static lift as these parameters change. The first parameter that we will look for is infraction fraction, this inflation fraction. The first parameter that we will look is inflation fraction. This I hope you remember is the ratio of the lifting gas occupied in the volume upon the total volume of the envelope. So if there is no balloon, this fraction is 1. Then we will look at the change in the atmospheric pressure. I do not call it as PA but PS because PS is the standard pressure. We will look at the effect of change in super pressure PS which if you recall is the pressure above the atmosphere which is or if the envelope is filled with gas at a higher pressure than ambient that pressure is the additional pressure is the super pressure. We will look at what happens when there is a slow increase in the ambient temperature. When I say slow increase I mean we give time for the system to come into equilibrium and we assume that the heat transfer takes place through the envelope. We will look at the effect of super heat which is exposure to the envelope to temperatures higher than ambient. Then we will look at what happens when you suddenly bring an LTA system into high temperature without giving time for the system to equalize that is called as rapid increase in the ambient temperature. We will look at the effect of humidity. We will look at the effect of lifting gas purity and finally we will look at the effect of changing the lifting gas volume. So our mandate for today is for all these 9 parameters 1 by 1 what is the variation in static lift as 1 by 1 these parameters change that is our mandate for today. Remember that when these parameters are changed except for the you know the second last and the third last that is the humidity and the purity of the gas when these parameters change the amount of air inside the envelope will also change either because the amount of air in the balloon A increases and hence or for some other reason so the infraction for the inflection infraction I will also change when these parameters change okay and when that happens the weight of the air in the balloon A will also change. So for every case we might have to calculate the effect of the change in the balloon A air also. Let us look at the first of these factors which is the inflection fraction I. Now apply a simple gas law which says that P V by T equal to constant for any lifting gas inside a closed envelope we can always say that P lifting gas into I into V upon T lifting gas is a constant why do we say this because I into V is the multiplication of inflation fraction with the volume. Now in my previous slides I have used V E and V for the envelope volume because there is a mix up with velocity also. But then we are in aerostatics so therefore whenever you see capital V from now on you understand that that means envelope volume so I have dropped the superscript E and V. One reason is it makes the formula very large unnecessarily as you will see the formula with which we will be deriving are anyway very large. So if I remove E and V there should be no confusion there is no velocity here there is only volume. So P lifting gas into inflation fraction into V divided by T of lifting gas that will be constant for the lifting gas and if there is any operating condition 1 and from there you go to condition number 2 then P L G 1 that is the pressure of the lifting gas in condition 1 into I 1 into V upon T L G will be equal to the same things for the second condition. But keep in mind there is a very very big assumption we are making when we equate these two quantities. So can you now think and tell me what are we assuming when we are saying that P L G 1 I 1 V upon T L G 1 is equal to the values for the second operating condition or if I ask you a specific question under what condition will this equality be invalid? Density variation will take place no density variation is implicit. So you cannot say that this will be true only at the same altitude or at the same so density variation is permitted envelope volume will not change because there is V on both sides. So yes you can say that assuming envelope volume remains constant in all our analysis we are assuming envelope volume remains constant externally the volume remains constant internally the volume available for the lifting gas might change correct. So mass of the lifting gas is the same. So even though the ballonet is insulating or deflating the volume available for the lifting gas may change but its mass is not changing under what condition does the mass remain constant. But which operating condition will ensure that mass does not remain constant one is leakages so with time there will be leakages. So we are looking at the quasi-study situation so we cannot take care of time dependent changes in the mass. So that is another assumption that we are looking at a particular instance we are looking at constant outside envelope volume okay. So can you recall is there any condition operating condition which makes you throw out the lifting gas if you want to exceed the pressure altitude. Do you remember the concept of pressure altitude when you are at ground you have air in the envelope and a ballonet the ballonet is full on the ground as you go up the ballonet air is being pushed out you reach an altitude at which the ballonet is flush if you still want to go higher up then the only option you have is to throw out some gas. So we are assuming that we are below the pressure altitude so that is a key assumption that we do not exceed the pressure height during this analysis. So as far as operating from any operating condition to pressure altitude this equation is valid for a constant outer envelope gas shape envelope shape hence volume okay. Now let us continue so the same formula I have reproduced here for continuity. Now what you can do is you can take the ratio of I2 by I1 by rearrangement in the formula. So I2 by I1 so I1 will come this I1 will come here down this TLG will go there so you get PLG1 TLG2 upon PLG2 TLG1 this is the ratio of the infraction inflation fractions okay. Now PLG is the pressure of the lifting gas that will be equal to the pressure at which you are operating plus the super pressure that you will provide. Now why do you provide super pressure to maintain the shape? Yes to give the pressure slightly more than atmospheric so that in case of any dynamic pressure acting on the balloon the shape so if you want to keep V constant you need to ensure that there is some pressure inside. So that is why P of the envelope will be ambient pressure or the standard pressure at operating altitude plus delta PSP similarly T of the lifting gas that is the temperature of the lifting gas will be equal to the ambient temperature plus any super heat that is present. So reproduce it so do not look at the screen just do it yourself so in this particular expression you replace PLG1 and PLG2 with PS1 plus delta PSP1 and PS2 plus delta PSP2 because at two operating conditions the super pressure may not remain same similarly the ambient pressure also will not be the same it will be P1 and P2. So if you do that the expression becomes as shown on the screen and further if I plot the variation from C level to higher altitudes. Now this is a double scale figure so the figures on the right side are in feet pressure altitude in feet and those are shown by the dotted lines or the scale dotted lines. The figure the number on the left pressure altitude in meters so from 0 to 5000 meters or 0 to 16000 feet approximately is the altitude variation here and on the x axis you have the inflation fraction varying from 0.6 to 1. So will someone tell me if the inflation fraction is 0.7 what will be the percentage volume occupied by the ballonet at C level if the inflation fraction is 0.7 look at this point if it is 0.7 or let us say 0.6 at C level the inflation fraction is 0.6 you are at C level 0 altitude and inflation fraction is 0.6 what will be the volume occupied by the ballonet percentage wise 40% inflation fraction 0.6 means 60% of the envelope is full of the LTA gas therefore remaining 40% is full of the ballonet which is the air and as you go higher up if you move along this line we see that for an airship with inflation fraction 0.6 at C level as the altitude is increased or you can also say as the ambient pressure level is decreased you reach a stage here at around 5000 feet where the inflation fraction becomes 1 which means what has happened Sunira what has happened volume okay which volume is 0 the ballonet correct the entire envelope is full of the same amount of LTA gas we have not increased the LTA gas LTA gas mass is the same but the volume available to it has been changed we did that because we pushed out some ballonet air we pushed it out because we wanted the envelope not to get over stressed okay. So what can you say about the pressure height of this particular airship 5000 meters is the pressure height of this airship because the inflation fraction becomes equal to 1 at 5000 feet and therefore in normal circumstances we should not go beyond above 5000 meters if you have to go you have to throw some LTA gas out okay only then you will be able to go up okay. So keep this in mind inflation fraction 0.6 C level condition it becomes 1 almost 1 at 5000 meters okay. So that much is what we have to learn about the effect of inflation fraction. What we have learnt is that as the ambient pressure decreases because of change in the altitude or increase in the altitude the inflation fraction increases till it hits equal to 1 and the line which it follows depends on from where it started these are almost parallel lines almost okay. So it follows a non-linear but as I should say a slightly non-linear variation.