Thursday, June 30, 2016

Colonize Mars - part 1, a really big rocket (updated 12 July with corrections)

 I had a burning vision of a spacecraft in my head that simply demanded to be written down. In a rush I threw together a bulky, dense post that isn't terribly useful for me so it must be opaque and meandering for anyone else reading. This is not the direction I want to go.

 The solution is to break the whole project into smaller pieces, go deeper into the details, show my work and hopefully try to put some graphics together. We'll see how well it goes.

{{I made a pretty serious mistake in the initial version of this post, one that led to inflated payload capacity. This has since been corrected throughout; I've also removed some commentary that is no longer supported by the numbers.}}

 The first thing you need for a giant transit habitat is a giant rocket. I want a single module to be big enough for Mars gravity at no more than 4 rpm, which works out to 43 meters in diameter. That means a massively large rocket is needed to loft this module in one piece.

 Working on rumor, innuendo, published interviews and Wikipedia it is clear that Elon Musk plans to build such a giant rocket. Nobody knows what the exact size will be, but 12 and 15 meters have been mentioned.

Let's make a bunch of wild assumptions after the jump.

{{Warringer pointed out a web tool for evaluating launcher payloads in the comments; worth a look.}}

If you don't care about the process, the results of those assumptions are LEO payloads of 620 tons for the 12-meter rocket and 1,200 tons for the 15-meter rocket.

First off, let's look at the SpaceX state of the art, the Falcon 9 full thrust (Spaceflight101, Space Launch Report). See also the official page (SpaceX)
There are some details of the craft that are proprietary but thanks to physics, fanboys and recorded flights many of those details have been calculated.
{{In retrospect and using some additional tools it seems a safer value for payload is 20 tons. It also seems that the published mass on the pad doesn't include payload.}}

Published values:
Diameter:  3.66 meters
Height (all inclusive):  70 meters, with 13.9 meter payload fairing
Launch mass:  549 tons
Payload to LEO:  22.8 tons (that's 74% better than the v1.0, way to go SpaceX)
Estimated values:
First-stage engine height:  2.9 meters (M1C, not sure about D)
Second-stage engine height:  5.6 meters (M1Cvac, not sure about D)
Interstage:  6.75 meters
Second-stage length without payload:  14.3 meters
Stage 1 fuel:  123.1 tons
Stage 1 oxidizer:  286.4 tons
Stage 1 tank length:  37.75 meters
Stage 1 dry mass:  22 tons
Stage 1 tank length to diameter:  10.314 to 1
Stage 2 fuel:  32.3 tons
Stage 2 oxidizer:  75.2 tons
Stage 2 dry mass:  4 tons
Stage 2 tank length:  10.83 meters (my guess)
Stage 2 tank length to diameter:  2.959 to 1

With that data I've modeled each stage's tanks so I can get a better idea of how they scale with diameter. This is using my tank estimator spreadsheet, so it's good enough for a first cut but with some uncertainty.

 - The overall length to diameter ratio remains the same for the larger versions
 - The design limits are 5g acceleration and 140% safety margins
 - Tanks are modeled as monocoque; reinforced tanks will be slightly heavier
 - Tanks are formed of lithium-aluminum alloy, 2641kg/m³ and 455 MPa yield
 - All propellants are subcooled (aka densified, LOX at 1230kg/m³)
 - Ullage is by pressurized helium gas in Kevlar composite overwrap spheres
 - F9 uses RP-1 fuel (854kg/m³) while BFR uses methane fuel (452kg/m³)

Tank model results F9 FT BFR-12 BFR-15 (units)
Diameter:          3.66            12.00            15.00 m
Stage 1 tank length:        37.75          123.77          154.71 m
Stage 1 fuel:     123.80      3,271.90      6,378.20 ton
Stage 1 oxidizer:     288.20      7,614.90    14,845.50 ton
Stage 1 dry mass:          4.10          291.20          698.20 ton
Stage 1 mass fraction: 2.08% 3.71% 4.29%
Stage 2 tank length:        10.83            35.51            44.39 m
Stage 2 fuel:        32.70          865.30      1,689.10 ton
Stage 2 oxidizer:        76.00      2,013.80      3,931.50 ton
Stage 2 dry mass:          0.50            28.00            64.90 ton
Stage 2 mass fraction: 1.53% 2.09% 2.27%
Total propellant:     520.70    13,765.90    26,844.30 ton

 For comparison, other people's estimates of the total propellant load for the F9 FT are 517 tons; my model returns 520.7 tons, within 1%. I don't include things like the internal LOX line and other plumbing components, plus I assume hemispherical endcaps. I've modeled a variety of other spacecraft with the same sheet, but none are as large as these proposed rockets. There is some risk that the model loses accuracy outside the range of values that can be compared to flight hardware, but I am comfortable with these numbers.
 The increasing mass fraction for the first stage tanks is due to high dynamic pressure under thrust; shorter tanks would require less material as can be seen by the better second stage numbers. For reference, I came up with 4.4 mm average wall thickness for the first stage oxidizer section of the F9, vs. 32.7mm for the BFR-12 and 50.5mm for the BFR-15. Given centimeters-thick walls I'd be tempted to do a composite overwrap to take some of the hoop stress, but that could add a lot of complexity to the manufacturing and QA process.

 As for engines, I am simply assuming that SpaceX can achieve the same thrust to weight ratio in their large Raptor methane-burners as they have in their current Merlin kerosene-burners. That means the engine mass will scale linearly with fuel mass to a first approximation, which can be revisited after other dry mass is constrained. I assumed that the Raptor engines will achieve the same efficiency as the Merlin engines relative to theoretical maximums for their propellants, which for the upper stage is 96% of vacuum Isp and for the lower stage is 94% of sea level Isp + 87% of vacuum Isp.

 Other components like thrust structures, avionics, landing hardware, etc. are less straightforward; assuming a linear scale would result in pitifully tiny payloads so we will have to apply some intelligence. I started with the estimated dry mass of each stage and subtracted the engine and tank masses. The remainder is thrust structure, avionics,etc., which I sectioned off using some mass estimation relationships. I applied the same relationships to the larger versions based on an initial guess at stage mass (which was iteratively updated later). That still left a fair amount of mass with no allocated purpose, so I assume that is the mass of the interstage, fairing and payload adapters and I scaled those values by surface area of the relevant section. This may be a low estimate, but it shouldn't be outrageously low.

 Once a good estimate of the dry mass and propellant mass of each stage is settled, I did a trial and error fit of payload masses until the total dV came out high enough. I̶n̶t̶e̶r̶e̶s̶t̶i̶n̶g̶l̶y̶,̶ ̶t̶h̶e̶ ̶e̶s̶t̶i̶m̶a̶t̶e̶d̶ ̶t̶o̶t̶a̶l̶ ̶d̶V̶ ̶o̶f̶ ̶t̶h̶e̶ ̶F̶9̶ ̶F̶T̶ ̶w̶i̶t̶h̶ ̶m̶a̶x̶i̶m̶u̶m̶ ̶p̶a̶y̶l̶o̶a̶d̶ ̶i̶s̶ ̶9̶,̶4̶0̶3̶ ̶m̶/̶s̶;̶ ̶t̶h̶e̶r̶e̶ ̶i̶s̶ ̶n̶o̶ ̶p̶u̶b̶l̶i̶s̶h̶e̶d̶ ̶v̶a̶l̶u̶e̶ ̶f̶o̶r̶ ̶t̶h̶e̶ ̶v̶e̶h̶i̶c̶l̶e̶'̶s̶ ̶d̶V̶ ̶a̶t̶ ̶m̶a̶x̶ ̶p̶a̶y̶l̶o̶a̶d̶ ̶b̶u̶t̶ ̶t̶h̶a̶t̶ ̶i̶s̶ ̶a̶l̶m̶o̶s̶t̶ ̶e̶x̶a̶c̶t̶l̶y̶ ̶t̶h̶e̶ ̶'̶g̶e̶n̶e̶r̶a̶l̶l̶y̶ ̶a̶c̶c̶e̶p̶t̶e̶d̶'̶ ̶v̶a̶l̶u̶e̶ ̶t̶o̶ ̶a̶c̶h̶i̶e̶v̶e̶ ̶l̶o̶w̶ ̶o̶r̶b̶i̶t̶ ̶a̶f̶t̶e̶r̶ ̶f̶a̶c̶t̶o̶r̶i̶n̶g̶ ̶i̶n̶ ̶g̶r̶a̶v̶i̶t̶y̶ ̶a̶n̶d̶ ̶a̶e̶r̶o̶d̶y̶n̶a̶m̶i̶c̶ ̶d̶r̶a̶g̶.̶  {{With revised payload and launch masses, the F9 FT comes to 9.5km/s almost exactly.}} I decided to target 9.5km/s rather than 9.4km/s to leave a little room for errors in my assumptions, particularly the mass relationships.

Here's the big mess: {{extraneous data removed, rows clarified and explicit Minitial (as launchpad mass and stage 2 ignition mass) and Mfinal (as burnout mass) values for each stage are presented for ease of verification.}}

Performance estimate F9 FT BFR-12 BFR-15 (units)
loaded tank mass      525.30        14,085.10        27,607.40 tons
launchpad mass      562.20        14,988.12        29,323.15 tons
entered payload         20.00              620.00           1,200.00 tons
calculated payload         20.00              620.00           1,200.00
Stage 1 fuel ratio         0.733                 0.726                 0.724
Stage 1 fuel      412.00        10,886.80        21,223.70
Stage 1 burnout mass      150.20           4,101.32           8,099.45
Stage 1 Isp (average) 291.67 300.67 300.67 s
Stage 1 dV         3,775                 3,821                 3,794 m/s
Stage 1 thrust         6.804              186.50              361.86 MN
Stage 1 total dry mass               17                    526                 1,131 tons
Stage 2 fuel ratio           0.82                   0.82                   0.81
Stage 2 ignition mass      133.20           3,575.20           6,968.56
Stage 2 fuel      108.70           2,879.10           5,620.60
Stage 2 burnout mass         24.50              696.10           1,347.96
Stage 2 Isp 345 355.3 355.3 s
Stage 2 dV         5,728                 5,701                 5,724 m/s
Stage 2 thrust         0.934              24.128              47.050 MN
Stage 2 total dry mass             4.5                   76.1                 148.0 tons
Total dV         9,504                 9,522                 9,518 m/s
Some points of interest:
 ̶-̶ ̶T̶h̶e̶ ̶s̶w̶i̶t̶c̶h̶ ̶t̶o̶ ̶m̶e̶t̶h̶a̶n̶e̶ ̶c̶o̶s̶t̶s̶ ̶s̶o̶m̶e̶ ̶t̶a̶n̶k̶ ̶m̶a̶s̶s̶ ̶w̶h̶i̶c̶h̶ ̶i̶n̶ ̶t̶u̶r̶n̶ ̶c̶o̶s̶t̶s̶ ̶f̶i̶r̶s̶t̶-̶s̶t̶a̶g̶e̶ ̶d̶V̶.̶ ̶S̶e̶p̶a̶r̶a̶t̶i̶o̶n̶ ̶w̶o̶u̶l̶d̶ ̶o̶c̶c̶u̶r̶ ̶a̶t̶ ̶a̶ ̶l̶o̶w̶e̶r̶ ̶a̶l̶t̶i̶t̶u̶d̶e̶ ̶a̶n̶d̶ ̶v̶e̶l̶o̶c̶i̶t̶y̶ ̶t̶h̶a̶n̶ ̶t̶h̶e̶ ̶b̶a̶s̶e̶l̶i̶n̶e̶ ̶F̶9̶ ̶f̶l̶i̶g̶h̶t̶ ̶p̶a̶t̶h̶.̶ ̶T̶h̶e̶ ̶s̶a̶m̶e̶ ̶s̶w̶i̶t̶c̶h̶ ̶p̶r̶o̶v̶i̶d̶e̶s̶ ̶a̶ ̶v̶a̶c̶u̶u̶m̶ ̶I̶s̶p̶ ̶b̶e̶n̶e̶f̶i̶t̶ ̶l̶a̶r̶g̶e̶ ̶e̶n̶o̶u̶g̶h̶ ̶t̶o̶ ̶s̶w̶i̶n̶g̶ ̶t̶h̶e̶ ̶n̶e̶e̶d̶l̶e̶ ̶t̶h̶e̶ ̶o̶t̶h̶e̶r̶ ̶w̶a̶y̶ ̶f̶o̶r̶ ̶t̶h̶e̶ ̶s̶e̶c̶o̶n̶d̶ ̶s̶t̶a̶g̶e̶.̶ ̶A̶s̶ ̶l̶o̶n̶g̶ ̶a̶s̶ ̶t̶h̶i̶s̶ ̶d̶o̶e̶s̶n̶'̶t̶ ̶i̶n̶t̶e̶r̶f̶e̶r̶e̶ ̶w̶i̶t̶h̶ ̶a̶e̶r̶o̶d̶y̶n̶a̶m̶i̶c̶s̶ ̶i̶t̶ ̶m̶a̶y̶ ̶b̶e̶ ̶a̶ ̶n̶e̶t̶ ̶p̶o̶s̶i̶t̶i̶v̶e̶,̶ ̶a̶l̶l̶o̶w̶i̶n̶g̶ ̶r̶e̶c̶o̶v̶e̶r̶y̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶i̶r̶s̶t̶ ̶s̶t̶a̶g̶e̶ ̶w̶i̶t̶h̶ ̶a̶ ̶l̶o̶w̶e̶r̶ ̶r̶e̶s̶e̶r̶v̶e̶ ̶p̶r̶o̶p̶e̶l̶l̶a̶n̶t̶ ̶l̶e̶v̶e̶l̶ ̶r̶e̶q̶u̶i̶r̶e̶d̶.̶ {{This was an artifact of my calculation errors.}}
 ̶-̶ ̶P̶a̶y̶l̶o̶a̶d̶ ̶f̶r̶a̶c̶t̶i̶o̶n̶ ̶i̶m̶p̶r̶o̶v̶e̶s̶ ̶f̶r̶o̶m̶ ̶4̶.̶1̶5̶%̶ ̶t̶o̶ ̶5̶.̶9̶1̶-̶6̶.̶7̶2̶%̶ ̶o̶f̶ ̶g̶r̶o̶s̶s̶ ̶m̶a̶s̶s̶.̶ ̶T̶h̶i̶s̶ ̶i̶s̶n̶'̶t̶ ̶t̶e̶r̶r̶i̶b̶l̶y̶ ̶m̶e̶a̶n̶i̶n̶g̶f̶u̶l̶ ̶e̶x̶c̶e̶p̶t̶ ̶a̶s̶ ̶a̶ ̶c̶o̶s̶t̶ ̶f̶a̶c̶t̶o̶r̶;̶ ̶h̶i̶g̶h̶ ̶p̶a̶y̶l̶o̶a̶d̶ ̶f̶r̶a̶c̶t̶i̶o̶n̶s̶ ̶m̶e̶a̶n̶ ̶a̶ ̶c̶h̶e̶a̶p̶e̶r̶ ̶p̶r̶i̶c̶e̶ ̶p̶e̶r̶ ̶k̶g̶ ̶d̶e̶l̶i̶v̶e̶r̶e̶d̶,̶ ̶a̶l̶l̶ ̶e̶l̶s̶e̶ ̶b̶e̶i̶n̶g̶ ̶e̶q̶u̶a̶l̶.̶{{This was an artifact of my calculation errors. Actual payload fractions are very similar, 4.09 to 4.15%.}}
 - The spent second stage provides enormous capacity as a fuel depot, tug or EDS. T̶h̶r̶e̶e̶ ̶ Fiveflights of a given vehicle could provide a full refill to a previously-launched stack, enough dV to push the payload to Mars and return empty. Lots of opportunity there if you're in the market for a 150 ton dry mass tug with ridiculous thrust.
 - These vehicles would be pushing either 20 or 35 *tons* of helium as a pressurant. That's simply not sustainable. The first stage at the least should convert to warm gas pressurization, using O2 in the LOX tank and methane gas in the fuel tank. The mass of helium and COPV tanks saved should hopefully offset the mass of plumbing and heat exchangers necessary, but no guarantees.
 - The Falcon series capitalizes on the use of kerosene as a fuel; it nicely doubles as hydraulic fluid and is already available at very high pressure from the turbopump. The BFR series will need to introduce a separate hydraulic system or convert to either pneumatics or electrical actuators.
 - I had made a first cut estimate of payload based solely on the relative propellant masses of the three vehicles using my tank estimator for the other post. This turned out to be a̶ ̶p̶r̶e̶t̶t̶y̶ ̶s̶i̶g̶n̶i̶f̶i̶c̶a̶n̶t̶ ̶u̶n̶d̶e̶r̶e̶s̶t̶i̶m̶a̶t̶e̶ ̶(̶~̶2̶0̶%̶)̶.̶ ̶T̶h̶e̶ ̶I̶s̶p̶ ̶b̶o̶o̶s̶t̶ ̶f̶r̶o̶m̶ ̶m̶e̶t̶h̶a̶n̶e̶ ̶m̶a̶k̶e̶s̶ ̶a̶ ̶m̶u̶c̶h̶ ̶b̶i̶g̶g̶e̶r̶ ̶i̶m̶p̶a̶c̶t̶ ̶t̶h̶a̶n̶ ̶I̶ ̶h̶a̶d̶ ̶i̶m̶a̶g̶i̶n̶e̶d̶.̶ {{Rough estimate was actually 33% over the detailed estimate.}}
 - The 15-meter version if fitted with nine engines would require *each engine* to produce about 41 MN of thrust. That's more than the total first stage thrust of the Saturn V at 34 MN.
 ̶-̶ ̶I̶f̶ ̶t̶h̶e̶ ̶f̶i̶r̶s̶t̶ ̶s̶t̶a̶g̶e̶ ̶w̶a̶s̶ ̶k̶e̶r̶o̶s̶e̶n̶e̶-̶p̶o̶w̶e̶r̶e̶d̶ ̶i̶t̶ ̶c̶o̶u̶l̶d̶ ̶b̶e̶ ̶m̶a̶d̶e̶ ̶s̶h̶o̶r̶t̶e̶r̶ ̶t̶h̶a̶n̶ ̶t̶h̶e̶ ̶m̶e̶t̶h̶a̶n̶e̶-̶p̶o̶w̶e̶r̶e̶d̶ ̶f̶i̶r̶s̶t̶ ̶s̶t̶a̶g̶e̶ ̶(̶d̶u̶e̶ ̶t̶o̶ ̶k̶e̶r̶o̶s̶e̶n̶e̶'̶s̶ ̶s̶u̶p̶e̶r̶i̶o̶r̶ ̶d̶e̶n̶s̶i̶t̶y̶)̶,̶ ̶s̶i̶g̶n̶i̶f̶i̶c̶a̶n̶t̶l̶y̶ ̶i̶m̶p̶r̶o̶v̶i̶n̶g̶ ̶t̶h̶e̶ ̶t̶a̶n̶k̶ ̶m̶a̶s̶s̶ ̶r̶a̶t̶i̶o̶.̶ ̶I̶ ̶h̶a̶v̶e̶n̶'̶t̶ ̶d̶o̶n̶e̶ ̶t̶h̶e̶ ̶m̶a̶t̶h̶ ̶y̶e̶t̶ ̶b̶u̶t̶ ̶t̶h̶a̶t̶ ̶s̶h̶o̶u̶l̶d̶ ̶r̶e̶s̶t̶o̶r̶e̶ ̶t̶h̶e̶ ̶f̶i̶r̶s̶t̶ ̶s̶t̶a̶g̶e̶ ̶d̶V̶ ̶p̶e̶r̶f̶o̶r̶m̶a̶n̶c̶e̶ ̶r̶e̶l̶a̶t̶i̶v̶e̶ ̶t̶o̶ ̶F̶9̶ ̶w̶h̶i̶l̶e̶ ̶k̶e̶e̶p̶i̶n̶g̶ ̶t̶h̶e̶ ̶s̶u̶p̶e̶r̶i̶o̶r̶ ̶s̶e̶c̶o̶n̶d̶ ̶s̶t̶a̶g̶e̶ ̶p̶e̶r̶f̶o̶r̶m̶a̶n̶c̶e̶ ̶w̶i̶t̶h̶ ̶m̶e̶t̶h̶a̶n̶e̶.̶ {{No longer relevant from a dV balance perspective, but there are program advantages to a kerolox first stage.}}


  1. I would like to point towards the Launch Vehicle Performance Calculator for any calculations for existing and not-existing launch vehicles.

    It's a good web-program to help with getting a payload out of a rocket.

    1. It seems to work nicely for existing rockets, thanks for the tip. My numbers make it angry. I'm not sure if that is due to an error on my part or if it is because I am entering values far above the tool's valid range. No error is returned, so I can't tell where the problem lies. I tried scaling things up and down without success. It does return 18.9 tons for the F9 (my numbers) / Cape Canaveral / 185x185x28.5° which sounds reasonable.

      As a quick gut check, I can use the rocket equation to get the expected dV for each stage.
      For stage 1 of BFR-15 I get:
      dV = Ve ln(Minitial / Mfinal)
      dV = 2948.5m/s * ln(29,897,400kg / 8,673,700kg)
      dV = 2948.5m/s * ln(3.4469)
      dV = 2948.5m/s * 1.2375
      dV = 3,648.7m/s
      For stage 2 I get:
      dV = 3484.3m/s * ln(7,535,400kg / 1,914,800kg)
      dV = 3484.3m/s * ln(3.9353)
      dV = 3484.3m/s * 1.3700
      dV = 4773.5m/s

      The stage 2 number doesn't agree with my table, so I've made a mistake in my spreadsheet somewhere.

    2. It was a pretty big mistake; I was using a temporary table to explore masses and get a few dependent variables to converge and ended up using the wrong manually-entered payload for the dV calculations. Fixed, and the post has been updated.
      I was able to get good results from the calculator you referenced, but only after scaling everything down. I'm guessing the absurdly high values are causing an overflow somewhere. One interesting result is that increased first stage thrust (more than the 1.258 to 1 shown here) would generate a higher net payload; the methodology paper explains why this is so.

      Thanks again for the reference; this is a very useful tool for estimating losses during ascent.

    3. Always happy to help.

      And I have noticed the same for a few rockets I've developed for my SciFi universe myself.

    4. I also think that the your table of dry mass vs full mass is a little misleading.

      And your Mass Ratio calculation is off. Its M/Me for the Mass Ratio, for a single stage without any additional stages.

    5. The value for stage 1 dry mass is the mass remaining after stage 1 engine cutout which is then detached from the second stage. (seen as the difference between stage 1 burnout mass and stage 2 ignition mass.) That includes engines, tanks, the interstage fairing and other hardware as well as unburned fuel. This would be useful information if you wanted to calculate the fuel required for first stage return.
      Likewise the value for stage 2 dry mass is the mass remaining after stage 2 engine cutout which is then detached from the payload. (Stage 2 burnout mass minus payload.) That includes engines, tanks, unburned fuel, an allowance for fairing (even though it would have been discarded earlier) and payload adapter plus other components. This would be useful information if you were considering repurposing the second stage in orbit since you know its mass, thrust and propellant capacity from the values in the table.

      The fuel ratio given is the fraction of the vehicle at ignition which is fuel. For the BFR-15 first stage that's 72.4% fuel and 27.6% everything else (stage 1 dry mass, stage 2 dry mass, stage 2 propellant and payload). For a given fuel ratio Rf the final mass Mf equals the initial mass M0 * (1 - Rf).
      There seem to be varying conventions on what fraction to specify and what name to use for it; I've seen mass fraction, fuel fraction, fuel ratio and payload fraction (in that case taken to mean both vehicle mass and payload). Each of those has a specific technical meaning but I've found people rarely use them as precisely as their definitions warrant.

      That said, it's certainly not the clearest table I've put together and it went through a few iterations. Can you suggest a better way to show this information?

  2. I believe that this is a good way of presenting the data. Much clearer, to me at least.

    And the dry mass of a stage is just that stage itself, without any of the next stages. That way its easier to work with if you change the next stages out. For example when you stretch the second stage for more fuel.

    The burnout mass for a stage is unnecessary, as staging will get rid of the stage dry mass before the ignition of the next stage. The payload of a stage is only of interest if you want to turn that stage into an SSTO.

    As for the 'Fuel Ratio', usually its the Mass Ratio that gets noted down, meaning that the fuel weights x times the dry mass.

    From Atomic Rockets:

    R = M / Me // M = Total (wet) Mass, Me = Empty (dry) Mass

    R = (Mpt / Me) + 1 // Mpt = Total Propellant Mass (propellant in stage)