@Johnkurveen @Altaïr
Remember back when I said something about finding out a way to compensate for delta velocity change due to free stream pressure loss? Well I looked into it and did some math about it on paper.
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On the first image, since free stream pressure only affects thrust directly, I converted the Isp in the delta velocity equation to thrust divided by mass flow rate multiplied with the gravity constant, then I converted the thrust value to its full thrust equation to reveal the pressure factor. So far out of all the values in the equation, Po or Free Stream pressure is so far the only factor that changes, the rest in the thrust equation are constant.
On the second image, I am trying to figure out how to compensate for a changing delta velocity value due to pressure loss over altitude, I tried logarithmic and exponential functions to deduce the nature of the pressure ratio graph. I then came up with the final function of f(x)=0.2*2^x which was rather faulty, but later I noticed that delta velocity is not an addition system, so using integration to combine or compensate specific impulses of two different thrust values is completely wrong.
As you can see in the finalized (circled) formula which states delta thrust and delta mass, for delta thrust we know that it is affected by pressure, but pressure is affected by altitude and altitude gain is dependent on mass, so for delta mass? For this I thought about applying the time factor of the flight which can then be used to deduce the altitude of rocket, which then can allow us to find out the pressure, but all this depends on the payload of the rocket.
Look at it this way, depending on the payload of the rocket and Newton's 2nd law of F=ma, a rocket with greater payload may reach a lower altitude in for example 50 seconds, than a rocket with less payload, this means that the local pressure of the heavier rocket will be greater than the lighter rocket.
These writings still didn't help me in visualizing what factors I need nor the changes involved, I drew an entire graph which compares altitude, pressure ratio, thrust, seconds of specific impulse and time which can be seen below.
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So we move on the final image, you can see that I have compared Isp, thrust and pressure ratio on the y-axis against altitude on the x-axis, the time factor is also drawn into the x-axis but on top for better visualization. As you can see, I am using the RD-107 engine as an example, both thrust and Isp increases with decreasing rate against the pressure ratio that decreases with decreasing rate. Fuel consumption is stated as constant as mass flow rate never changes.
After I have drawn out all of the factors, I drew a line down to intersect all the lines of thrust, Isp, mass, pressure and altitude, and labelled the time line as time constant b or Tb. I then deduced these info to the two equations below, and noticed that with decreasing free stream pressure which leads to increasing thrust, the dry-wet mass ratio reduces as well, this means that each rocket has its own unique delta velocity gain, and in theory hydrogen rockets like the SLS will receive the least gain due to its smaller dry-wet mass ratio.
I am still yet to experiment on using the "area under the exponential curve" method to find the compensated dV value, and with so many factors I have to involve, I don't think it will be as easy as putting in an integration and x-function and be done with it. I think using a computer aided software to simulate this will make it much easier.
So what do you guys think?
P.S. if all of these mumbo jumbo sound like the ramblings of a drunk asshole, its because I am writing this at 3 in the morning.