Propellant volume vs ratio equation ?

[N_Molson]

Hello,

I'm trying to design a rocket (addon) and I'm searching for an equation that would give me the volume of the propellant tanks given the following variables :

- Oxydizer : LOX
- Fuel : LH2
- The sum of the height of the two tanks (cylinders) must be 24 meters (variable "th").
- The diameter of the tanks is 3.95 meters.
- Combustion ratio : 6.03 (SSME value, but that could be named variable "cr" to make the equation more universal).
- LOX density (comes from pressure and temperature : 1141 kg/m3, say variable "od"). And, does that value makes sense for a cryogenic rocket ?
- LH2 density (idem : 70,973 kg/m3, say variable "fd"). Same question as above.

:hmm:

 

[Thorsten]

You need the total mass of propellant (obviously...), otherwise the volume can't be known - or I misunderstood the question...

Anyway, if that is M,

* use the combustion ratio to get M1 and M2 (oxidizer and fuel mass) using M1 + M2 = M and M1 = const. * M2
* use M1 = rho1 * V1 and M2 = rho2 * V2 to get the volumes of each
* make height of the tank proportional to volumes
* use V = R^2 * pi * h to solve for R to get the radius of your rocket

Voila.

 

[jedidia]

You need the total mass of propellant (obviously...), otherwise the volume can't be known - or I misunderstood the question...

By looking at his variables, it seems what he has is the total of available volume for propellant, although he neglected to state the radius of the tanks. He's kind of engineering a rocket backwards.

 

[N_Molson]

Yes, added the diameter of the tanks. But no, I don't even have the total volume. But that can be a variable in the equation, obviously. And yes, it should be somehow be linked to the propellants density (I hope my values are consistent) to get the mass.

 

[Loru]

You can try this spreadsheet: (be aware that mixture ratio is given here as oxidiser to fuel)

http://loru.bluespiderhosting.net/random/mark3-LV.xls

Edit: Hmm. I think that won't help you. How are you designing rocket without fuel mass?

---------- Post added at 04:09 PM ---------- Previous post was at 03:54 PM ----------

Gimme few moments

 

[Urwumpe]

It should work... But it is not an easy question, since you also need assumptions on what you are looking for.

Is the question to put as much LOX and LH2 at the necessary mixture ratio, including different boil-off behaviors into the given volume?

How bad is sloshing for you? What tank pressure is required and how should it be achieved?

 

[Loru]

up till now I'm getting Height of LOX tank = 3.07 * Height of LH2 tank

 

[Thorsten]

Yes, added the diameter of the tanks. But no, I don't even have the total volume.

If you have the diameter of the tanks, you have the the total volume as

V = pi R^2 h

since h=24 m = h1 + h2.

Then you have

V1 + V2 = V
rho1 * V1 + rho2 * V2 = M
M1 = const. * M2 or rho1 * V1 = const * rho2 * V2

That's three equations and three unknowns (V1, V2 and M) - you can solve that

(Then you can add all the extras for boiling and sloshing).

 

[N_Molson]

including different boil-off behaviors into the given volume?

Of course. We're talking about the rocket first stage for now, so it won't be exposed to boil-off as much as a space tug. Still, indeed there is boil-off on the pad and even during ascent (which will probably last around 4 minutes for that stage).

The thermal environnement at the aft of the stage will be a bit nasty, as I plan to use a belt of GEM-60 booster stages, pretty much like it is done with GEM-46 on the Delta 2.0

How bad is sloshing for you? What tank pressure is required and how should it be achieved?

The idea is that the rocket diameter is 4 meters. From there I take 2.5 centimeters as a margin and I get 3.95 meters tanks. The height of the propellant tanks section of the stage should be no more than 25 meters. I don't want the stage to be too heavy or too "thin". So it gives a frame.

I've no idea if I can go beyond the Shuttle ET tank pressure (140–150 kPa for LOX, 220–230 kPa for LH2). Maybe it is possible given the tanks are much smaller (and then probably structurally stronger) ? I was planning to use extra Helium tank(s) to keep the pressure to acceptable level, though again I have little idea about how much gas is required to do that.

I'll also have a couple of rotation control thrusters, given that the rocket has only 1 engine, which requires a small tank of hypergolics. Should be placed in the interstage (Ariane 5 has something like that). I guess they'll have to be linked to the He tank too.

 

[Urwumpe]

I'll also have a couple of rotation control thrusters, given that the rocket has only 1 engine, which requires a small tank of hypergolics. Should be placed in the interstage (Ariane 5 has something like that). I guess they'll have to be linked to the He tank too.

Depends on where the thrusters are. The tank(s) should be close to it. The Ariane 5 has the thrusters and the tanks in the VIB ring.

A common bulkhead between the tanks would put the most fuel into the same height, but has some challenges (different temperatures of LOX and LH2, where to place equipment for filling and draining the tanks, sensors, etc).

The tank flight pressure depends on the demands of the rocket engine, BTW. It has to be high enough so that the pumps don't cavitate. A more complex rocket engine can operate with lower tank pressures by having special low-pressure boost pumps in advance (like the SSME). If the chamber pressure in the engine is fairly low (like in gas generator cycle engines), a low tank pressure is good enough.

The Raptor engines of SpaceX for example expect a pretty high tank pressure and seems to have only high pressure pumps.

 

[N_Molson]

Depends on where the thrusters are. The tank(s) should be close to it.

All that stuff is going to be in the interstage, but again the volume of the He tank(s) will dictate the layout.

A common bulkhead between the tanks would put the most fuel into the same height, but has some challenges

I had a "classic" 2 in-line separated tanks architecture in mind.

The tank flight pressure depends on the demands of the rocket engine, BTW. It has to be high enough so that the pumps don't cavitate.

The engine will actually be a RS-25E or F.

 

[Loru]

Was bit wrong in previous post (decimal point error)

Disclaimer: many calculations are rounded.

Diameter: 3.95
Radius: 1.975
Height=24
Base area of the tank: (12,247)
Total volume (V): 293

LH2 density: 0,07
LOX density: 1,14

LH2_per_ton_volume: 14,285
LOX_per_tone_volume: 0,877

Mass mixture ratio (ox/fuel): 5
Volume mixture ratio per ton of propelant (ox/fuel): 0,3070175439

and that gives me:

v_LOX = v_LH2 * 0,307 and v_LH2 = v_LOX / 0.307 which sounds about right

so

293 = v_LH2 + (v_LH2 * 0,307)

1.307 * v_Lh2 = 293

v_LH2 = 224,17 cubic meters and from that v_LOX = 68 cubic meters


lets divide both by base area and we have

H_LOX = 5,55 meters and
H_LH2 at 18,45 meters

Which sounds about right here.

 

[Urwumpe]

flat cylindrical tanks or domed tanks?

 

[Loru]

I assumed cylindrical. Done mostly by hand. In orbiter dev work I just use 3d studio max as it can measure volume of any shape, and TBH I usually start with fuel mass and thrust not other way around.

 

[Urwumpe]

I assumed cylindrical. Done mostly by hand. In orbiter dev work I just use 3d studio max as it can measure volume of any shape, and TBH I usually start with fuel mass and thrust not other way around.

Same here, but then, I also had to fit domed tanks into an aerodynamic shape once and remember the problem of how to find the right configuration of the tanks to either fit into the shape with only minimal scaling of the vehicle or to scale the vehicle for a given tank system.

 

[N_Molson]

Domed is better of course but I guess that calculating the volume of a dome is going to be a bit complicated (how to define the curvature, also what kind of shape is going to be the best compromise between saving room and resisting to internal pressure ?)...

And I don't have 3ds Max, needless to say.

---------- Post added at 06:53 PM ---------- Previous post was at 06:46 PM ----------

Mass mixture ratio (ox/fuel): 5

Just curious, they give 6 as the Oxidizer to Fuel Ratio on Astronautix (http://www.astronautix.com/s/ssme.html), and even 6.03 on Wikipedia ?

 

[Loru]

I think 5 is a remnant of fuel rich LH2/LOX engine in HCLV. My bad again

You should easilly recalculate it for 6.

 

[Urwumpe]

Another project that could really make good use of a "multistage rocket design tool" ... :hmm:

 

[C3PO]

If you ignore domed tanks, the heights of the tanks can be solved by relative fractions.

Mixture ratio: 6.03
Density ratio: 70.973/1141
"Volume ratio": (6.,03x70.973)/1141 = 0.3751

Since the cross section isn't changing, the height ratio is the same as the volume ratio.

LOX tank: 24x[0.3751/(1+0.3751)] = 6.55m
LH2 tank: 24x[1/(1+0.3751)] = 17,45m

You can calculate the "wasted" space of domed tanks once you've decided on the shape. Then use the volume ratio to calculate each tank.

Or did I miss something? :hmm:

 

[Urwumpe]

Or did I miss something? :hmm:

Well, the tank domes are usually constants, so no deal. Of course, you miss Ullage space.