 Today I am going to cover a portion that we had not covered before the midsim because it was getting too theoretical and I thought let us give a break. But this is again part of the previous capsule. So let us revisit pressure height and first of all let us see if we can define pressure height and from now on we will call it as HPH. So this is the definition of pressure height about which you are all familiar. It is a height as the airship keeps climbing, we release the air from the ballonet and you reach a height at which the ballonet become flush and there is no air left in the ballonet. So that is the pressure height and the analogous definition will be the ceiling for an aircraft. So let us remember or recall the inflation fraction relationship applicable. So if we ignore super pressure, we have seen earlier that the inflation fraction at any operating condition 2 is equal to the ratio of the pressures at any condition 1 PS1 upon PS2 into the ratio of the temperature at altitude 2 upon altitude 1 times the inflation fraction at the condition 1 and in the temperature I have also included super heat terms. So I have ignored super pressure but I have retained the super heat terms. So this is something that we had already derived earlier and just reproducing here for continuity. So if you consider ISA conditions, so what will happen in the ISA condition is that the ambient temperature Ta will become the standard temperature under ISA at that altitude called as Ts and the delta Ts or the super heat contribution will be 0. So we will get a very simple expression I2 is equal to PS1 by PS2 into Ts2 by Ts1 times I1. So this simple relationship directly connects inflation fraction at any altitude with inflation fraction at some other altitude with the pressures and the temperatures. Once again we are ignoring super heat, we are ignoring super pressure, we are considering ISA conditions and we are also ignoring in this case any lifting gas purity or the effect of humidity etc. This is a very simplistic explanation. Now under ISA conditions the ratio of pressures PS1 to PS2 is called as delta and the ratio of temperatures is called as theta. So you can replace PS1 by PS2 as delta1 by delta2 because delta1 is PS1 by P0 and delta2 is PS2 by P0, P0 being the sea level standard pressure and similarly Ts2 by P0 is theta S2 and Ts1 by T0 is theta S1. So you can replace this by the standard relationships and once again we also know that the value of delta by theta is sigma of the density ratio. So therefore inflation fraction at any operating condition under standard ISA will be equal to the density altitude sigma S2 upon sigma S1 times I1. So this simple relationship connects any two operating altitudes with their densities. Now let us look at what happens when the airship for some reason has to fly above the pressure altitude and also let us look at the relationships. So I am replacing the same expression from the previous slide here. I am calling it as IG with it stands for the inflation fraction at ground level equivalent to the subscript 1 in the previous slide. And I am putting pH as the subscript for 2 considering pH to be the pressure altitude or the pressure height. So once again we get IG inflation fraction at ground level is equal to PSPH by PSG times temperature ratio times IPH. Is this clear? So we have just taken the simple relationship between any two points and we have considered those 2 points to be the ground and the pressure altitude and we are assuming that 1 to 1 relationship exists. Please note here I have not assumed anything like ISA although I have ignored super pressure I am retaining superheat okay. Now what is the value of IPH inflation fraction at pressure altitude 100 percent or 1 ratio is 1 correct. So therefore you can replace it by 1 and hence you can say that inflation at ground would be equal to just the ratio of pressures and the ratio of temperatures including the superheat if present. Because the inflation fraction at the pressure altitude is 1 there is no air left the ballonet is flush okay the same formula we copy ahead. Now let us look at these expressions once again. So if we assume ISA conditions and zero superheat then the TA will become TS and the delta T SHPH will vanish okay. So therefore the equation will become PSPH by PSG into TAG plus delta T SHG upon TSPH instead of A it becomes S and what you can also do is you can basically leave the pressure ratios on one side and take the other terms on that side. So PSPH by PSG as shown here will become IG into TSPH okay IG into PSG divided by TAG plus delta T SHG so I am just inverting the sides and if we now continue look at the ISA conditions or if we look at standard conditions we have delta and theta as the ratios. So when you put those values you can get an expression like this what is the advantage of doing this? The advantage of doing this is that you bring in T0 and P0 which are constants T0 being 288.16 degrees Kelvin and P0 being 101325 Newton per meter square. So these two numbers are constants. So therefore the density ratio at the pressure altitude okay that is sigma SPH will be equal to two standard quantities their ratio times pressure at the ground under standard conditions temperature at the ground and if any super heat is present times IG. So it simplifies now the same expression I will just copy and paste here. So it is the same thing which I am transferring to the next slide. Now let us look at these numbers. So I would like you to substitute for T0 and P0 T0 101325 sorry P0 101325 and T0 288.16 okay. I expect people to bring calculators in the class. So can you please do this find this ratio. So all you can do in this is replace for T0 and P0 and what would the value be for TAG if you assume G as the ground level or as the sea level ambient temperature at ground or at operating at ground 0 altitude 288.16 same as T0 for sea level. So what do you get? Now the expression which says sigma SPH which is the density ratio for a given pressure altitude in terms of the Indian pressure on the ground, temperature on the ground, ground level inflation fraction and delta T s h G. How much will it be? Yeah 2.84 minus 3 0.0 0.284 correct. So if you now replace TAG in centigrade then it will be 273 degrees plus TAG. So while this simple expression you can get the value of the sigma. So how does it help you? How does this expression help you? Why are we doing this? Everything is in ground level. So therefore what does it get you? What does it help you obtain? So that is why you are operating your airship from some place. So you know the pressure at that place PSG, you also know the TAG temperature at that place and you also know the super heat, temperature increase because of super heat. So I assume that I have a temperature sensor inside the envelope, I keep this airship on the ground for a long time and I find that because of the heating of the sun, there is some temperature increase of the gas. So when I know all these things what do I get? Sigma Sph. That you will get sigma Sph if you know the value of I G. Now think reverse. If you want to have a particular pressure altitude that means let us say you want to fly at least up to 5000 feet. From sigma Sph you can get from the tables. So with that you can get now the value of I G that is what should be the inflation fraction on the ground so that this airship from these operating conditions can go to a particular pressure altitude. So let us say you are planning a profile or a journey for an airship from Pune to Mumbai or Mumbai to Pune. So you need to know what will be the altitude which I should cover. So you can back calculate. So you can come to know how much should be the volume of air in the balune as a percentage of the total gas for me to create the possibility of flying up to a pressure height. Because in normal circumstances we do not wish to exceed the pressure height. Today we will see what happens when you exceed the pressure height also but you would not plan for it. So now we know that I 2 is equal to sigma 1 by sigma 2 times I 1 for any operating condition if we ignore the heat and super pressure. So therefore the inflation fraction under ISA conditions will be equal to the ratio of density altitude at pressure height upon the sigma at the ground into I ph. But I ph will be 1. So therefore you can easily get the value of sigma sph. So the place where you are operating you also know the density of the air at that particular condition P by RT. So therefore you know the density ratio sigma. So this will tell you, this will easily tell you, so if you fix the inflation fraction at ground level you will know how much high you can go by getting the sigma value. Please understand that sigma sph is the number which from the atmospheric tables can give you the h or we can do the rollers. If I want to go to that height I should have the inflation fraction up to a particular minimum value.