GEO to Lunar transfer. 50,000 kg payload. Hydrogen and oxygen, lit twice.
Size a liquid hydrogen and liquid oxygen propulsion system to move a 50,000 kg payload from GEO to a circular lunar orbit using a two-burn Hohmann transfer. Vacuum Isp of 450 seconds. Oxidizer to fuel ratio of 5.5. Ten percent Delta-v margin. One-way.
A Hohmann transfer connects two coplanar circular orbits with a single ellipse tangent to both. The first burn raises apogee from GEO to lunar distance. The second, 5 days later, circularizes at the Moon. It is the minimum-energy two-impulse transfer between circles.
Speed anywhere on any conic orbit, given the gravitational parameter, radius, and semi-major axis. Everything else falls out of this.
The transfer ellipse touches both circular orbits. Its semi-major axis is the average of their radii.
At perigee, speed up from GEO circular to transfer perigee. At apogee, speed up from transfer apogee to lunar circular. Both burns prograde.
The tyrant. Exhaust velocity times the log of the mass ratio equals the Delta-v you can achieve. Everything hard about rocketry lives in that logarithm.
Invert the rocket equation around dry mass. Given the target Delta-v and the fixed 50,000 kg payload, this is exactly how much propellant you must carry.
At an O/F of 5.5, roughly 85 percent of the propellant by mass is liquid oxygen. By volume the hydrogen tank is still the monster.
LOX is 16 times denser than LH2. The mass split is five parts oxidizer to one part fuel, but the volume split almost flips. The hydrogen tank dominates the stage.
Why the hydrogen tank dwarfs the LOX tank. Density is the whole story. At 71 kg/m3, liquid hydrogen is one of the least dense cryogens you can carry. LOX at 1,141 kg/m3 is packed tight. So even though you burn 5.5 kg of oxidizer per kg of fuel, the hydrogen needs roughly 2.9 times the tank volume.