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  1. Thanks. I think the equations we discussed will be able to give the ideally optimized engine nozzle Isp’s. What needs to be worked on now are the trajectory equations including the gravity turn and the air drag equations. Any suggestions? Bob Clark
  2. Thanks for doing the calculation. I assume you take into account that with fixed nozzles there is also a backpressure term given at the end of the equation here: F = q × Ve + (Pe - Pa) × Ae Also, as you know sea level engines are somewhat overexpanded at sea level because they also want to get good performance at vacuum. So the exit pressure for the fixed nozzle engine isn't really 1 atm at sea level. For the SSME's I think it's at around 1/3rd atm. For the Vulcain engines versions 1 and 2 the overexpansion is even worse. I think around at 1/4th atm for the Vulcain 1 and 1/5th atm for the Vulcain 2. You can get an idea about the degree of overexpansion for the Vulcain 1 at sea level from this graphic: At sea level the ideal Isp is significantly better than what the Vulcain 1 gives. I did a rough estimate and found for the currently-used Vulcain 2 because of its even higher level of overexpansion, its sea level thrust could be increased ca. 30%(!) by given it ideal expansion at sea level. This is important because higher lightoff thrust can reduce gravity drag. The increase for most engines wouldn't be this great though because most aren't this greatly overexpanded. For some other engines I tried I estimated the increase was less than 10% better sea level thrust by giving it ideal sea level expansion. For the parameters that go into the equation for the exhaust velocity, combustion temperature, molecular weight of combustion products, specific heat, etc., I've used the shareware program Rocket Propulsion Analysis, http://propulsion-analysis.com/index.htm. Bob Clark
  3. Assuming we can use the equation for the exhaust velocity ve dependent on ambient pressure, we still need to calculate the flight path to orbit. For simplicity sake we can use the “gravity-turn” trajectory: Gravity turn A gravity turn or zero-lift turn is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory controlled solely through the vehicle's own thrust. First, the thrust is not used to change the spacecraft's direction, so more of it is used to accelerate the vehicle into orbit. Second, and more importantly, during the initial ascent phase the vehicle can maintain low or even zero angle of attack. This minimizes transverse aerodynamic stress on the launch vehicle, allowing for a lighter launch vehicle.[1][2] https://en.m.wikipedia.org/wiki/Gravity_turn#Launch Doing a google search turned up several references on calculating gravity turn trajectories. For this first level analysis I’m looking for some easily implemented ones if anyone knows of any. Also, some kerbal RealSolarSystem mod simulations have been done using aerospike nozzles for SSTO’s. Aerospike nozzles are the closest we have to ideal adaptive nozzles. Anyone want to give it a try with aerospike engines replacing the engines on the F9, Delta IV, Atlas V or other first stage boosters? Bob Clark
  4. Dragon01 mentioned the equation for Isp was linear. It’s not really, but this reminded me of something I’m puzzled about. Fixed nozzles on a sea level engine are a compromise. They are overexpanded for sea level operation so they can get good Isp in vacuum.This should mean they get optimal Isp at some intermediate altitude, not at sea level and not in vacuum. But actually in graphics of engines they show the Isp either constant or increasing towards vacuum conditions. The graphic would be expected instead to look like this: Taken from this page: http://www.braeunig.us/space/sup1.htm But using the first image above we might be able to model approximately the Isp according to altitude by two straight lines, both for the fixed nozzle case and for the adaptive nozzle case. For the fixed nozzle case it would be an inclined straight-line to the altitude of 15,000m, then switching to a constant, i.e., flat-line thereafter at the vacuum Isp value. For the adaptive nozzle case, it would be two inclined straight lines. The first would be steeper than the second with the transition at around 15,000m to 20,000m. Bob Clark
  5. One of the few papers that calculated the flight averaged Isp for the standard bell nozzle version of an engine and one fitted with alt.comp was this paper by Dana Andrews et.al.: Rocket-powered single-stage-to-orbit vehicles for safe economical access to low Earth orbit. July 1992Acta Astronautica 26(8-10):633-642 DOI: 10.1016/0094-5765(92)90153-A Dana G. Andrews E.E. Davis E.L. Bangsund https://www.researchgate.net/publication/245138678_Rocket-powered_single-stage-to-orbit_vehicles_for_safe_economical_access_to_low_Earth_orbit (This is behind a paywall but you can get a free copy through interlibrary loan from any university or public library.) I was surprised it showed the flight averaged Isp was 447s for the standard engine. The alt.comp version was a little higher at 460s. The flight averaged Isp is important since it allows you to make a rocket equation estimate of the payload using a single number for the Isp. Then you can get quite significant payload as an SSTO either for the standard version or the alt.comp version. However, it is known an SSTO is better realized using dense propellants. This is because their lower Isp is more than made up for by their higher density. Bob Clark
  6. Quite correct. But these were quite expensive engines. The point of this exercise is to match the best vacuum Isp for vacuum optimized upper stage engines, using the less expensive lower chamber pressure engines, while still being able to launch from sea level. The F9 first stage engine’s vacuum Isp would be extended from 312s to ca. 365s and the Delta IV’s from 412s to ca. 465s. Because of the exponential nature of the rocket equation this would result in significant increase in payload. Robert Clark Not necessarily. Just as a TSTO doesn’t necessarily have to have return capability. IF it is found with alt.comp it can offer significant payload then it can be determined if there is sufficient payload to add reusability systems. Bob Clark
  7. You are quite right; there are a lot of variables. For simplicity sake, you can imagine the nozzle attachment is one that can be extended so the the exhaust gas pressure matches the ambient pressure. This was the idea behind an extensible nozzle investigated for the Apollo Saturn V rocket though not implemented: https://www.alternatewars.com/BBOW/Space_Engines/Rocketdyne_Engines.htm For the Falcon 9 and Delta IV first stages this would result in a quite high increase in the vacuum Isp’s. From 312s to ca. 365s for the F9 first stage and from 412s to ca. 480s for the Delta IV first stage. Bob Clark
  8. I think I know which equation you mean. It’s the first one on this page: http://www.braeunig.us/space/sup1.htm Here it is on that page:: F = q × Ve + (Pe - Pa) × Ae where F = Thrust q = Propellant mass flow rate Ve = Velocity of exhaust gases Pe = Pressure at nozzle exit Pa = Ambient pressure Ae = Area of nozzle exit The problem is the Ve is quite complicated depending on ambient pressure. Bob Clark
  9. Perhaps you can point me to the equation you mean. The equation that gives the exhaust velocity, which equals g*Isp, is quite complicated in its dependence on ambient pressure therefore altitude: from, https://en.m.wikipedia.org/wiki/Rocket_engine_nozzle#One-dimensional_analysis_of_gas_flow_in_rocket_engine_nozzles Bob Clark
  10. I’m actively seeking collaborators to calculate the payload possible by adding altitude compensating attachments to existing rockets: https://www.researchgate.net/project/Single-stage-to-orbit-SSTO Elon Musk said the Falcon 9 booster could be SSTO, but with small payload. Altitude compensation can increase the payload, but by how much? Bob Clark
  11. I like your calculation. I noticed you gave a dry mass for the BFR Starship, which is the version of the BFR upper stage with the passenger quarters for 100 colonists on a Mars flight.. But you did not give a dry mass for the tanker, which instead has just a big empty fairing in that space What do you estimate the dry mass of the tanker version of the upper stage to be? What do you estimate the delta-v of the empty tanker upper stage to be? Bob Clark
  12. Expendable SSTO would also be useful to test. Remember all rockets including the F9 were tested in an expendable mode. Bob Clark
  13. Not the recent version from the 90’s. But the 70’s version can be downloaded from the NASA technical reports server: Design of Liquid Propellant Rocket Engines Second Edition. https://ntrs.nasa.gov/search.jsp?print=yes&R=19710019929 Bob Clark
  14. The delta-v to orbit is commonly taken to be 9.1 km/s for equatorial orbits because you get a 400 m/s boost by the Earth’s rotation for free. Also the dry mass for the BFS upper stage is given as 85 tons by wiki. But this is the version with the passenger quarters for 100 Mars colonists. The tanker version would weigh much less without the passenger quarters, perhaps only in the 50 ton range. Bob Clark
  15. You get 400 m/s for free by launching near the equator, such as from Cape Canaveral. Taking this into account, the delta-v to LEO is often taken to be about 9,100 m/s or 30,000 ft/s: From Modern Engineering for Design of Liquid-Propellant Rocket Engines, p. 12.https://books.google.com/books?id=TKdIbLX51NQC&pg=PA12&source=gbs_toc_r&cad=4#v=onepage&q&f=false Because of the exponential nature of the rocket equation that 900 m/s difference between 10 km/s and 9.1 km/s accounts for a significant amount of payload. Bob Clark