The key assumptions were 5% gravity loss, and 5% drag loss for the first stage, and 5% gravity loss only for the second
stage, on a fast ascent trajectory, plus 4.5% stage inert weights, stage payload included in that
accounting. I assumed stage 1 burnout
outside the “sensible atmosphere”, at
3.05 km/s achieved velocity. LEO was
assumed to be 7.79 km/s achieved velocity.
I completely ignored the “boost” effect of the Earth’s eastward
rotation.

I did some crude engine ballistics based on characteristic
velocity c* at 1000 psia chamber pressures (c* = 5900 ft/s for LOX-RP1, from the vintage 1970 version of the old
Pratt & Whitney “Vest-Pocket Handbook”),
and a bell divergence-corrected thrust coefficient CF chart computed for
specific heat ratio 1.20. The divergence
thrust correction factor is 0.983,
pretty much an average 15-degree half angle, and equivalent to most modern curved
expansion bells.

The first stage is expanded “perfectly” to 14.7 psia (101.3
Kpa, 1013 mbar, 760 mm Hg) backpressure. The chart says CF = 1.57 at expansion ratio
9.00 when read at pressure ratio 68.
There is a simple relationship among CF,
c*, and Isp, leading to Isp = 287.9 s for the first
stage. A very slightly-oversimplified
model then estimates exhaust velocity as 2.835 km/s.

For the second stage,
the ambient backpressure is zero (out in the vacuum). I looked at the chart for pressure ratio set
to an arbitrary 1000:1, and got CF =
1.826 at expansion ratio 65:1. For that
same 1000 psia LOX-RP1, c* = 5900
ft/s, I got Isp = 334.8 s, and exhaust velocity 3.284 km/s.

5% gravity loss plus 5% drag loss is a total 10% loss for
the first stage, making the effective
required velocity not 3.05 but 3.355 km/s.
That corresponds to a mass ratio of 3.2820, and a propellant fraction of 0.6953. The corresponding stage payload fraction is
0.2597. For a nominal liftoff weight of
500 metric tons, the stage 1 payload
(stage 2 plus the “real payload”) is 129.85 tons. That ignores any interstage weights, which might be around a ton, for a second stage ignition weight of 128.85
tons.

5% gravity loss on the second stage velocity increment of
4.74 km/s results in an effective velocity requirement on the second stage of
about 4.977 km/s. At the higher second
stage Isp, the mass ratio is 4.55183, and the propellant fraction for that stage is
0.7803. For the same stage 4.5% inert
fraction, that leaves 0.1747 for the
stage 2 “real payload” fraction. That’s
near 22.51 metric tons for a stage 2 ignition weight of 128.85 tons.

That payload has to ride inside some sort of
protective, aerodynamic shroud. For the sake of argument, assume that shroud also weighs about 1
ton. Therefore, the real delivered payload is nearer 21
metric tons for a vehicle massing 500 tons at launch.

The real Falcon-9 is a little over 500 tons at
ignition, and is rated to deliver 13
metric tons to LEO. My 21 tons is in the
ballpark, clearly, but still quite a ways “off” for purposes of
“exact” estimates. All in all, I’d say these ballpark estimating techniques
are actually quite good, especially for
relative-comparison calculations. But it
takes a better model than this, to
really “pin things down”.

For example, use the
5%-5% loss factors on the entire trajectory to LEO, with 4.5% inerts, and the lower-performing first stage engine
performance, but as a single-stage to
orbit vehicle. It really does calculate
as technically feasible, just at only
1.48 ton payload, and with propellant
tanks “stretched” by 37% (volume) to maintain a 500 ton ignition weight.

That 1.48 tons gets compared to the 21 tons for the 500 ton
two-stage bird. For one stage versus two, the launch cost could only be factor 2 lower
for the one-stage bird, at the very
most. On a per-unit-delivered payload
basis, the one stage version will always
deliver less mass for a higher cost.
That basic effect is why staging was invented, over 6 decades ago.

Actually the first F9 weigh 333t, the 1.1 version is 505t. Even some other estimates put its payload to LEO at about 16t. Possibly they made a reserve for landing the first stage back. Well see on 16th or 19th...

ReplyDeleteAlso, the fairing detaches just after second stage ignition, therefore it should not be counted into second stage mass ratio.