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If you additionally send \$100 to the Skylon team, they'll do this in 2019.

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Posted (edited)

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

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]

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

Edited by Exoscientist

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Posted (edited)

Hey, @Exoscientist. I ran the numbers for the simplest assumptions and got this for exhaust velocities with upgraded nozzles:

Where:
v3 is the exhaust velocity for our engine with the bigger nozzle
v2 is the exhaust velocity for our engine as it is now (the 1atm version)
P3 is the exhaust exit pressure for our engine with the bigger nozzle
P2 is the exhaust exit pressure for our engine as it is now (~1atm)
P1 is the chamber pressure
R is the universal gas constant (8.314 J/molK)
Cp is the exhaust gas' effective heat capacity (averaged on a molar basis between the constituents, and using specifically the molar version, denoted with a big Cp rather than a little cp)

This assumes constant Cp, which is a really terrible assumption. It also assumes frozen equilibrium and all sorts of things! These assumptions will by-and-large decrease the calculated value of v3 relative to what it would actually be. Still, this is the first order equation. If it's enough let me know! Otherwise I'll probably keep chugging until I make a better version. The next step up would need a spreadsheet to work. What's the thoughts?

You can get Cp values from JANAF, but I'm sure there's easier sources out there.

Edited by Cunjo Carl
Put in a better simplification.

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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

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6 hours ago, Exoscientist said:

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

No, the equation is just for the Ve term (v3 is the exhaust velocity). The Pe-Pa is something you'll need to calculate within your model/sim.

I went ahead and typed in the subscripts so it's a bit easier to read:

Cheers!

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9 hours ago, Exoscientist said:

I've used the shareware program Rocket Propulsion Analysis, http://propulsion-analysis.com/index.htm.

Bob Clark

I took a quick peek at the web page for the program you found, and it looks like it does everything my spreadsheet would and then some! My plan of attack was:

1. Find an engine with known: Propellants, mix ratio, expansion ratio, Isp (ASL), chamber pressure, and exhaust exit pressure (guesstimate this last one by looking at exhaust expansion as it exits the nozzle)

2. Use the Isp (ASL) to back-calculate the temperature in the combustion chamber. Just use guess and check iteration until the right temp is found- the adiabatic flame temp can be used as a starting guess. I was planning on using JANAF thermodynamics values and an assumption of fast kinetics and adiabatic expansion (so all points are at equilibrium, and all of the losses are modeled as a simple reduction in combustion chamber temp). The one thing I wasn't sure about is how to deal with the non-ideality of the gas. I was planning to use Redlich-Kwong, but there's quite a few exhaust components I'm not confident I could find critical temps/pressures for. For starters, the ideal gas assumption may be 'close enough' though.

3. Using this combustion chamber temperature, calculate the Isp of the engine for different expansion ratios.

Of course real life doesn't really work like this, but the problems caused by the assumptions made here will tend to cancel eachother out. It should come out pretty close! Since you've got the program, maybe give this a go?

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Posted (edited)

(4. Expansion ratio may vary for spikes, adjusting to the flight profile.)

Edited by kerbiloid

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2 hours ago, Cunjo Carl said:

2. Use the Isp (ASL) to back-calculate the temperature in the combustion chamber. Just use guess and check iteration until the right temp is found- the adiabatic flame temp can be used as a starting guess. I was planning on using JANAF thermodynamics values and an assumption of fast kinetics and adiabatic expansion (so all points are at equilibrium, and all of the losses are modeled as a simple reduction in combustion chamber temp). The one thing I wasn't sure about is how to deal with the non-ideality of the gas. I was planning to use Redlich-Kwong, but there's quite a few exhaust components I'm not confident I could find critical temps/pressures for. For starters, the ideal gas assumption may be 'close enough' though.

This takes me back about 30 years or so. I worked in the combustor group of a turbine engine manufacturing company. We used a little program called TSSST that was originally written to run on something like an HP-45 calculator. Anyway, all it did was make a few assumptions about luminosity and convection and it did a simple 1-d "thin-shell steady state (heat) transfer" calculation to come up with combustor temperatures. Interestingly enough, it worked really, really well considering the (lack of) computing power available back then.

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On 7/22/2019 at 10:04 PM, mikegarrison said:

This takes me back about 30 years or so. I worked in the combustor group of a turbine engine manufacturing company. We used a little program called TSSST that was originally written to run on something like an HP-45 calculator. Anyway, all it did was make a few assumptions about luminosity and convection and it did a simple 1-d "thin-shell steady state (heat) transfer" calculation to come up with combustor temperatures. Interestingly enough, it worked really, really well considering the (lack of) computing power available back then.

Glory days of the RPN calculator right there! Also I totally get what you mean. If you know what assumptions to make it can feel rather cheeky what you're able to get away with!

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Posted (edited)

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

Edited by Exoscientist

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Altitude compensation only affects the first 2 km/s of the orbital trajectory. Single stage to orbit is not the right approach.

Actually, altitude compensation would be most effective on the first stage of a THREE stage to orbit rocket. Get out of the atmosphere, start a vac optimized second stage, land the first stage.

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