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After reviewing this amazing graphic posted in another question:
I noticed that the LH/LOX ratio is slightly different in the Second and Third stages. The mass stoichiometry of the reaction:
2 H2 + O2 -> 2 H2O
Is 1:8 H to O , while these are close to 3:1 H to O , so obviously very fuel rich. Is there a reason that the upper stage is 3.28:1 while the second stage is only 3.06:1 ?
The J-2 engine used on the second and third stages of the Saturn V has a "PU valve" (propellant utilization) on the oxidizer turbopump. Adjusting the mixture ratio with this valve primarily provides a mechanism to ensure that the hydrogen and oxygen propellants are depleted at the same time. Secondarily, it allows a tradeoff between specific impulse (fuel mass efficiency) and total thrust.
The PU valve has three settings: 5.5:1, 5:1, and 4.5:1. These differ from your calculated numbers because rocket combustion ratios are normally given in oxidizer mass to fuel mass form, rather than by fuel-to-oxidizer volume as you used. (Hydrogen is extremely low-density, about 1/14 the density of water or LOX). The higher the oxidizer flow rate, the higher the thrust. The 5.5:1 ratio yields about 35% more thrust than 4.5:1; the specific impulse goes from about 422 at 5.5:1 to 427 seconds at 4.5:1 — the efficiency gain is much smaller than the thrust penalty.
On the S-II second stage, the ratio starts off at 5.5:1 for maximum thrust; even so, the rocket's thrust-to-weight ratio is only about 0.8:1 at second stage ignition! The stage switches from 5.5:1 to 4.5:1 near the end of the burn, shortly after the center engine cutoff. At this point, the stage is much lighter than at ignition, so it's not necessary to maximize thrust. The timing of this "EMR shift" (engine mix ratio) was dynamically chosen in flight on the early Saturn V flights to ensure simultaneous depletion of the hydrogen and oxygen tanks, should the actual consumption rates not match the expected rates — if you ran out of one or the other first, you'd wind up with excessive unburned propellant at the end of the burn, which is dead weight. On later flights they set the shift to occur when the second stage reached a particular velocity instead.
You can see the thrust drop associated with the EMR shift here; it's the second dogtooth step in the second-stage middle portion of the plot, labeled point 3:
(Hey, @uhoh, "CECO" is "center engine cutoff" and "OECO" is "outboard engine cutoff". On Saturn I/IB documentation you'll also see "IECO" for "inboard engine cutoff", as those first stages have 4 inboards, 4 outboards, and no single center engine.)
The S-IVB third stage burns at 5.0:1 for the duration of the first (Earth-orbital insertion) burn, using only a small portion of the stage's fuel. The Apollo mission plan then provides two chances to do the trans-lunar injection burn, sending the spacecraft on its way to the moon. If the TLI burn occurs as planned at the first opportunity, that burn will start at 4.5:1 then change to 5:1 about two minutes in. If the TLI burn is delayed to the second opportunity, the entire burn will occur at 5:1 — I believe that this is because some hydrogen will be "boiled off" and lost during the delay, and once again it's more efficient to burn all the propellant, regardless of the ratio, than to potentially leave unburned propellant at cutoff.
