amount of gas needed to keep up mass flow rate

[jedidia]

So here's a question that's hopelessly above my math paygrade but that would be cool to know more exactly than multiplying my thumb with Pi...

Let's assume a fictional spacecraft, built on the moon. I'm saying this because it's a hybrid Aluminium/LOX rocket, and those wouldn't make much sense otherwise. According to some sources, these could theoretically produce 290 kN at a mass flow rate of 110 kg/s. That's adequate thrust for my purposes, so I'm going with that.

Aluminium burns with oxygen to Al2O3, which is 53% aluminium and 47% oxygen in mass. My current design has a top propellant capacity of 30 tons, of which 14.1 tons are LOX. It requires a volume of 12.357 m^3 in total to store.
BUT!
I need an oxygen flow rate into the combustion chamber of 51.7 kg/s. That means I need to be able to give some pressure on those tanks. The cheapest gas on the moon seems to be oxygen, so one would expect that they'd use oxygen also as the pressurizer.
So now comes the prize question: If I want to keep a flow rate of 51.7 kg/s to burn the entire 14.1 tons of oxygen, how much of it do I need additionally to get the rest there?

 

[Thunder Chicken]

I need an oxygen flow rate into the combustion chamber of 51.7 kg/s. That means I need to be able to give some pressure on those tanks. The cheapest gas on the moon seems to be oxygen, so one would expect that they'd use oxygen also as the pressurizer.
So now comes the prize question: If I want to keep a flow rate of 51.7 kg/s to burn the entire 14.1 tons of oxygen, how much of it do I need additionally to get the rest there?

It depends on the pressure needed to push the mass flow required, which depends on injector/nozzle geometry, whether you are injecting liquid oxygen or gaseous oxygen, etc..

As an upper limit you could assume the tank at the end of the burn is holding saturated vapor oxygen at the highest pressure the tank would hold.

 

[Urwumpe]

Also, how much you need extra can be calculated from your tank volume at the point the tank should be empty. If you operate with gas, this means it reached the minimum pressure into the injector for your mass flow. With liquid, either all of it turned into gas or the liquid is expended. The ideal gas law should help you to calculate how much the gas at the desired pressure and temperature in the volume of the tank should weight.

Generally, the higher the pressure, the more it might pay off to switch to a pump fed cycle, especially since higher pressure also increases the tank dry mass.

 

[jedidia]

Generally, the higher the pressure, the more it might pay off to switch to a pump fed cycle

I'm thinking about that, the question is how much power would the pump require. The problem with a hybrid is, it's not very feasible to power it by burning some of your propellant, and it might not be feasible to feed it from batteries. Also, pressure is a lot more failsafe than an electrical pump, especially if divided over several tanks...

 

[Urwumpe]

I'm thinking about that, the question is how much power would the pump require. The problem with a hybrid is, it's not very feasible to power it by burning some of your propellant, and it might not be feasible to feed it from batteries. Also, pressure is a lot more failsafe than an electrical pump, especially if divided over several tanks...

Does not mean that you can't have both. Think for example about a tap-off cycle, where a part of the gases in the combustion chamber are routed into the turbine and then dumped overboard. In that case, tank head pressure is used to start the engine and thus the turbine, but you can quickly ramp up mass flow to reach a higher thrust. This is also very robust, but means in case of most hybrid engines, that you make a low ISP engine a bit worse.

If you like oxidizer rich combustion, you can of course also use a preburner for the turbine and direct the exhaust gas into the hybrid combustion chamber for more thrust.

 

[Thunder Chicken]

I'm thinking about that, the question is how much power would the pump require.

This can be calculated by calculating the volume flow rate (mass flow rate / density) and multiplying that result by the pressure rise across the pump, then dividing by the pump efficiency. Pump efficiency is somewhat difficult to estimate without data, but would probably be closer to 50% than 100% as by their nature pumps have fluid dragging over impeller and volute surfaces which is all frictional loss. This effect gets worse as pumps get smaller due to the surface area/volume ratio.

The problem with a hybrid is, it's not very feasible to power it by burning some of your propellant, and it might not be feasible to feed it from batteries. Also, pressure is a lot more failsafe than an electrical pump, especially if divided over several tanks...

Sometimes you will still need a pump even if you have a pressurized tank, because even with that pressure difference you may not be able to get the mass flow needed through the resistance of the injector nozzles on that alone.

 

[francisdrake]

The problem with pressure-fed rocket engines is, that the combustion chamber pressure is limited by the tank pressure minus some dP required to inject the fuel into the combustion chamber. This may be feasible for smaller tanks like those used for reaction control thrusters, but not for big rockets.

The chamber pressure of state-of-the art rocket engines is:

• RD-180 (RP1/LOX) 256 bar
• SSME (LH2/LOX) 206 bar

Typical tank pressures for rockets are 6 - 8 bar (gauge). The remaining pressure is created by fuel pumps.

So, the first question is: What is my tank pressure?

Then it is like Thunder Chicken said:

an upper limit you could assume the tank at the end of the burn is holding saturated vapor oxygen at the highest pressure the tank would hold

Assuming you start with no gas at startup, and a tank full of gas at the end of the burn, this is your total volume.
Divided by the burn time, this gives the gas flow in m³/s.
Remember we are talking working volume (at tank pressure). So for 1 m³ gas at 6 bar (gauge) = 7 bar (abs) you need 7 m³ to be generated, or supplied from a high-pressure reservoir.

 

[jedidia]

The chamber pressure of state-of-the art rocket engines is:

• RD-180 (RP1/LOX) 256 bar
• SSME (LH2/LOX) 206 bar

Right... The propellant has to overcome that too. Completely forgot about that...

 

[jedidia]

• RD-180 (RP1/LOX) 256 bar
• SSME (LH2/LOX) 206 bar

I see though that these are massively more powerful motors than what I have in mind (even the SSME outthrusts my little contraption here almost 10fold, much less something like a first stage main engine like the RD-180). The 290 kN I mentioned above should really be about all that I need. Enough to push some 40 tons around in a vacuum with manageable acceleration. So I guess the chamber pressure would be quite a bit lower? No idea, really, not an engineer, and I hadn't thought I'd need to go into this much detail...

 

[Urwumpe]

I see though that these are massively more powerful motors than what I have in mind (even the SSME outthrusts my little contraption here almost 10fold, much less something like a first stage main engine like the RD-180). The 290 kN I mentioned above should really be about all that I need. Enough to push some 40 tons around in a vacuum with manageable acceleration. So I guess the chamber pressure would be quite a bit lower? No idea, really, not an engineer, and I hadn't thought I'd need to go into this much detail...

Well, for producing a certain amount of thrust, you need to reach a certain chamber pressure and mass flow. Most parameters are actually fixed by the propellants that you use there. I can only find the properties from using powdered aluminum and LOX, but it should still approach your engine pretty well.

https://ntrs.nasa.gov/api/citations/19940017287/downloads/19940017287.pdf

As expected, the characteristic velocity (c* or "c star" ) will be terrible (944 m/s calculated from experiments) and thus specific impulse can only become terrible as well. Raising the chamber pressure can improve this, but it can not provide miracles (Lets start calling it a miracle that your hybrid engine has a specific impulse of 5609 m/s in vacuum)

Using a very good thrust coefficient of 2.2, your engine would need to reach a characteristic velocity of ~2500 m/s, a pretty strong improvement over the 944 m/s measured in low-pressure experiments, thus you need to reach pretty huge (for a hybrid propellant engine) chamber pressure and combustion temperature compared to the small NASA experiment, about 2.5 times the chamber pressure and six times the combustion temperature.

 

[jedidia]

https://ntrs.nasa.gov/api/citations/19940017287/downloads/19940017287.pdf

Huh, the PDF times out for some reason, but I know it. One of the few things one can find on throttlable aluminium thrusters.

Lets start calling it a miracle that your hybrid engine has a specific impulse of 5609 m/s in vacuum

That would be very miraculous. I thought the best we can hope for out of Al-LOX is somewhere around 2600 to 2700 m/s, which is what I'm currently designing with.
Ah, the PDF is working again. They're talking about a chamber pressure of 480 kPa on page 5, which is roughly 5 bar, so even if I'd have 2.5 times or even 5 times as much, that pressure would still seem very manageable compared to the 200 bar of an SSME...

 

[K_Jameson]

Assuming you start with no gas at startup, and a tank full of gas at the end of the burn, this is your total volume.
Divided by the burn time, this gives the gas flow in m³/s.
Remember we are talking working volume (at tank pressure). So for 1 m³ gas at 6 bar (gauge) = 7 bar (abs) you need 7 m³ to be generated, or supplied from a high-pressure reservoir.

This seems to clarify a lot. Very interesting topic!

 

[Urwumpe]

Huh, the PDF times out for some reason, but I know it. One of the few things one can find on throttlable aluminium thrusters.


That would be very miraculous. I thought the best we can hope for out of Al-LOX is somewhere around 2600 to 2700 m/s, which is what I'm currently designing with.
Ah, the PDF is working again. They're talking about a chamber pressure of 480 kPa on page 5, which is roughly 5 bar, so even if I'd have 2.5 times or even 5 times as much, that pressure would still seem very manageable compared to the 200 bar of an SSME...

Yes, but could you also get a material that withstands a combustion temperature of about 6 * 2800 K? Again, these are just rules of thumb derived from the idealized equations for characteristic velocity and assuming a constant ratio of specific heats. Of course, in reality, many more parameters would change at higher pressure.

For comparison, here is a more accurate calculation done with H2 +O2:

https://ntrs.nasa.gov/api/citations/19980227353/downloads/19980227353.pdf

Note how increasing the chamber pressure here doesn't actually change c* much once the injectors operated in their ideal range. At this point, you can simply create a more powerful engine just by scaling things up, but increasing the specific impulse by increasing pressure works a bit harder than in my simplified calculations since the specific heat in a constant pressure process (determined at nozzle exist) also changes with increased pressure, temperature and different composition of the exhaust until leaving the chamber.

If you want a certain headache, just look at this page:

Specific Heats - Calorically Imperfect Gas

 

[Thunder Chicken]

Yes, but could you also get a material that withstands a combustion temperature of about 6 * 2800 K? Again, these are just rules of thumb derived from the idealized equations for characteristic velocity and assuming a constant ratio of specific heats. Of course, in reality, many more parameters would change at higher pressure.

For comparison, here is a more accurate calculation done with H2 +O2:

https://ntrs.nasa.gov/api/citations/19980227353/downloads/19980227353.pdf

Note how increasing the chamber pressure here doesn't actually change c* much once the injectors operated in their ideal range. At this point, you can simply create a more powerful engine just by scaling things up, but increasing the specific impulse by increasing pressure works a bit harder than in my simplified calculations since the specific heat in a constant pressure process (determined at nozzle exist) also changes with increased pressure, temperature and different composition of the exhaust until leaving the chamber.

If you want a certain headache, just look at this page:

Specific Heats - Calorically Imperfect Gas

C* is the speed of sound at the nozzle throat where the Mach number = 1. The speed of sound is purely a function of temperature. The subsonic gas in the combustion chamber must accelerate to Ma = 1 at the throat, then the gas expands and accelerates to Ma > 1 in the nozzle.

 

[Urwumpe]

C* is the speed of sound at the nozzle throat where the Mach number = 1. The speed of sound is purely a function of temperature. The subsonic gas in the combustion chamber must accelerate to Ma = 1 at the throat, then the gas expands and accelerates to Ma > 1 in the nozzle.

I think I remembered something like that from lectures 20 years ago. Good to have the confirmation. But the definition of measured characteristic velocity just contains throat area, chamber pressure and mass flow.

Also in theory, combustion should be already done when the exhaust reaches the throat. In reality, a part of the combustion reaction does still happen afterwards and can not fully contribute to the engine thrust.

 

[jedidia]

Yes, but could you also get a material that withstands a combustion temperature of about 6 * 2800 K?

Uhm, no... Haven't even thought about temparature yet. But 16000 Kelvin seems somewhat excessive for a chemical thruster, as it seems a bit excessive for... well, pretty much anything apart from maybe a fusion drive

 

[jedidia]

This can be calculated by calculating the volume flow rate (mass flow rate / density) and multiplying that result by the pressure rise across the pump

What units do these need to be in? From what I got together now, the volume flow rate would be 58.1 litres/s, and the chamber pressure would be 710 kPa.

Ok, from what I gather, I'd have to multiply m^3/s * Pa, so that would be equivalent to litres/s * kPa. Which leaves me with 41 kW, before factoring in efficiency. That's a small car! No idea how I'd power that from batteries... I'll be over there calculating how much pressurizer I'd need, probably just to be shocked again.

 

[Urwumpe]

What units do these need to be in? From what I got together now, the volume flow rate would be 51.8 litres/s, and the chamber pressure would be 710 kPa. I hope they are not multiplied like this to result in Watts, because 37 kilowatts (without factoring in efficiency yet) would mean I'd need a small car battery in there to drive the pump. Which weighs a ton. Probably literally.

The density of LOX is not 1 kg/l (but close to it) and you would need cubic meters anyway in SI units....

Also your pump must achieve the pressure difference between your tank head pressure (tank pressure + hydrostatic pressure) and your injector pressure (which is higher than the chamber pressure, for the mass flow). And the chamber pressure affects your pump as additional back pressure!

 

[jedidia]

Ok, had some things messed up. The density was a typo, I tend to mix up digits in numbers. It was still wrong though because of some unit conversion kerfuffle. Unfortunately I kerfuffled multiple units and they evened each other out, so after having the (I really think) correct numbers of 0.045m^3/s * 710000 Pa, I still end up with 32.2 kW. It just seems way too much to feed from a battery.

 

[Urwumpe]

Ok, had some things messed up. The density was a typo, I tend to mix up digits in numbers. It was still wrong though because of some unit conversion kerfuffle. Unfortunately I kerfuffled multiple units and they evened each other out, so after having the (I really think) correct numbers of 0.045m^3/s * 710000 Pa, I still end up with 32.2 kW. It just seems way too much to feed from a battery.

Depends on your burn time.... for Rocket Lab and the Electron launcher, that is no problem.

Each engine has two small motors that generate 37 kW (50 hp) while spinning at 40 000 rpm. The first-stage battery, which has to power the pumps of nine engines simultaneously, can provide over 1 MW (1,300 hp) of electric power.

But these engines have only a mass flow of less than 8 kg/s.