I did some rough calculations for a Nuclear Thermal Rocket Engine of the Phoebis-2 class, which was a 5,000 MWt class engine capable of generating nearly 250,000 lbf of thrust with greater than 860 sec of specific impulse. Had Project NERVA continued, this engine could have flown as Saturn upper stage as part of an Expanded Advanced Apollo Program...but alas it was not to be. NERVA was cancelled and soon so was Apollo...

Anyways the question asked was: if a nuclear thermal rocket engine of 250,000 lbf were augmented by injecting liquid oxygen downstream of the nozzle, could such a hybrid engine be used for a practical SSTO?

Without doing the detailed nozzle calculations (which are needed for a responsible and thorough analysis!) some tentative indications are that: yes, a 250 Klbf NTR, augmented with LO2 to bring its thrust up to nearly 1 million lbf, could infact be used to create a single stage to orbit vehicle given quite a few conditions.

I can't seem to attach a file (I was going to upload a PDF) so I guess I'll just dump the whole thing in here. My apologies for any formatting errors, and I apologize for the length, but here goes:


Estimation of some important parameters for a Reusable SSTO utilizing a 250 Klbf-class NTR in a
LANTR boost-mode

LEO Orbital Velocity: 7700 m/s
Total mission delta-V requirements accounting for Gravity and Aerodynamic Losses: 10,000 m/s

Base engine performance:
Isp, vacuum operation, not augmented: 960 sec
equivalent exhaust velocity: 9,414 m/s
Isp, vacuum operation, augmented: 880 sec
equivalent exhaust velocity: 8,630 m/s
Isp, boost phase, augmented, sea level: 700 sec
equivalent exhaust velocity: 6,864 m/s

Thrust, vacuum, non-augmented: 250,000 lbf (1112 KN)
liquid hydrogen consumption rate: 118.1 kg/s

kinetic power of exhaust jet: 5,235 MW
assume 98% nozzle efficiency to estimate total reactor power: 5,340 MWt


Liquid Oxygen Augmentation of Thrust:

O/F = 4.0:1
Liquid oxygen consumption rate: 472.5 kg/s
Lower (non condensing) heat of combustion of hydrogen: -286 KJ/mole (141.9 MJ/kg)
Pcombustion = 118.1 kg/s * 141.9 MJ/kg
= 16,758 MWt (total combustion power)

Ptotal jet power = 5,235 MW + 16,758 MW
= 21,993 MW
total propellant consumption rate: 118.1 kg/s + 472.5 kg/s
: 590.6 kg/s

from K=1/2*m*v^2, differentiating with respect to time, keeping v constant:
P=1/2*mdot*v^2
solving for v gives:

v=sqrt(2*P/mdot)

Let P=21,993 MW, mdot=590.6 kg/s

v=8630 m/s (Isp = 880 sec)


Without doing detailed nozzle calculations (required for a thorough analysis), experience suggests that a sea level performance reduction of 20% from vacuum Isp can be expected:

Isp, sea level, augmented = 880 sec * 0.8
= 700 sec
v sea level = 6,865 m/s

This suggests we can expect a sea level thrust to be near:

F = g*Isp*mdot

= (9.80665 m/s^2) * (700 s) * (590.6 kg/s)
= 4,054 kN (911,440 lbf)

This suggests a GLOW (gross lift off weight) of about F/1.2 = 750,000 lb (340,000 kg)

Assume a boost phase, augmented thrust, burn time: t = 139 sec

"1st Stage" boost phase LO2 consumed, 100% throttle: 65,680 kg
assume +5% for ullage: 69,000 kg.

boost phase LH2 consumed, 100% throttle: 16,416 kg
total boost phase propellants consumed: 82,093 kg.

Average Isp, boost phase, augmented thrust:
60 % sea level to 40% vacuum level: 0.6*700 s + 0.4 * 880 s = 772 s
average equivalent exhaust velocity: 7570.6 m/s

mission delta-V achieved to t=139 s:

deltaV=c*ln(mi/mf)

let c=7,570.6 m/s; mi=340,000 kg; mf = mi - 82,093 kg
deltaV=2,092 m/s (note: this is not the actual velocity achieved at t=139 s, since there are gravity and drag losses...)

mission deltaV remaining: 10,000 m/s - 2,092 m/s =7,908 m/s

"2nd Stage"
assume principly vacuum operation, no augmentation:
Isp, vac, no augmentation: 960 s

total vehicle mass at "2nd Stage": 257,907 kg

V=c*ln(mi/mf) solve for mf:

mf = mi*EXP(-V/c)

Let mi = 257,907 kg; c = 960 s * 9.80665 m/s^2, V=7,908 m/s:
mf = 111,342 kg.

This is the total mass delivered to orbit: this includes the mass of the structure, the engine, remaining propellants (residuals,) and payload.

Total mass of propellants consumed during this phase: mi-mf = 146,565 kg of which this is entirely liquid hydrogen.

Total Hydrogen consumed in flight: 146,565 kg + 16,416 kg
: 162,981 kg
density of liquid hydrogen at 20K boil point: 70.7 kg/m^3

Volume of Liquid Hydrogen tank: 2,305 m^3.
assume 5% ullage space: 2,420 m^3 (actual tank volume)

total liquid oxygen consumed in flight: 65,680 kg
assume +5% ullage mass: 68,960 kg
density of liquid oxygen at 90 K boil point: 1,140 kg/m^3
volume of liquid oxygen: 60.5 m^3
assume +5% volume: 63.5 m^3 actual tank volume

Total mass of Propellants consumed: 231,941 kg

Back GLOW = 231,941 kg + 111,342 kg = 343,283 kg, very close to our original estimate of 340,000 kg.

At this point I would point out that by shifting the boost-burn time back, burning more oxygen, we would end up with a more compact vehicle. The optimal result occurs when the total tankage mass is minimized while simultaneously maximizing total delta-V. This is more complex than it sounds, because you pay a slight performance penalty in Isp by burning longer with oxygen, but you save tankage volume (and hence tankage weight) by burning less hydrogen. Also a more compact vehicle will naturally have less atmospheric drag, and hence drag losses will also be less. Now since the hydrogen tank is so big, making it smaller will save you More mass than by making the oxygen tank bigger (because oxygen is denser than hydrogen by a factor 16 to 1, and since tankage mass scales approximately with volume.) The complexity of these interelated factors makes a detailed analysis exceedingly difficult--which is why such an optimization is usually done iteratively on a computer (requiring detailed mathematical models of the various configurations.)

If one wants to get an idea of how much liquid hydrogen 163,000 kg represents look at the Sace Shuttle ET. The LH2 section carries 226,000 lb of liquid hydrogen (102,500 kg.) This RSSTO carries 60% more hydrogen than that...that starts to give us an idea of how big this machine would be...

Estimating the mass of the tankage, which if the vehicle is like a large right frustrum cone, will also be very difficult because of the non-standard geometry and the fact the the tankage must be intregral with the vehicle structure (to save weight.)

I tend to use something I am familiar with for comparison: The Space Shuttle ET.

The Super-Light Weight ET has a mass of about 26,560 kg and contains about 735,550 kg of propellants for a containment ratio of: 27.7 : 1, that is for every 27.7 kg of propellant, you only have 1 kg of tank.

Assuming we can do no better than 15:1 (which incidently is very close to the Apollo Saturn 5 1st stage) because of the complexity of the vehicle tankage geometry, we may expect to have a tankage mass in the range of: 231,941 kg/15 = 15,462 kg. Actually this sounds a bit too light to me...

This may, or may not be accurate. Only a detailed analysis of an actual tankage structural design will give a really meaningful answer.