Monday, September 24, 2018

Relevant Data for 2018 BFS Second Stage

I did the very best I could,  reverse-engineering what the 2018 version of the BFS second stage might be able to do.  This is based on the Musk presentation of a paying moon passenger,  at Spacex,  and posted on their website in Sept. 2018.   I had to use the 2017 rocket engine data at 250 bar chamber pressures,  as the 300 bar performance figures for 2018 are not yet available.  I did not revisit the first stage BFR (see ref. 1).  Things begin with a best-cut guess at the 2018 weight statement: 

All metric ton                     2017                      2018
Payload                                150                         100
Inerts                                    85                           135
Burnout                               235                         235
Propellants                         1100                       1100
Ignition                               1335                       1335

Note:  payload reduced by 50 tons while inerts are increased by that same 50 tons,  due to fins and the longer payload section.  The 3 fins are heavier than the old 4 landing legs,  but inherently span wider.

                                             2017                       2018
Mass ratio                           5.6809                   5.6809
Propellant fraction           0.82397                 0.82397
Payload fraction                0.11236                 0.07491
Inert fraction                      0.06367                 0.10112
Sum fraction                      1.00                        1.00

About the Engines:

                                             2017                       2018
Engines                                6                              7
SL                                        2                              7(nom.) to 0
Vac                                       4                              0(nom.) to 7

                                             2017                       2018
Pc, bar                                  250                         300
Throttle, %                          20-100                   xxx
SL Fth @ SL, KN                 1700                       xxx
SL Isp @ SL, s                      330                         xxx
SL Isp @vac, s                      356                         xxx
SL De, m                               1.3                          xxx
Vac Fth @ vac, KN              1900                       xxx
Vac Isp, s                              375                         xxx (380?)
Vac De, m                            2.4                          xxx

“Xxx” means actual data hinted at,  but not available yet

BFS weights more-or-less determine SL-vac mix of engines:

Scenario               M, ton  1-g KN   .384-g KN
Landing                235         2305       885
Takeoff                1335       13,092    5027

To land on Earth with SL engine design (250 bar):  use 3 engines at 45+%,  if 1 lost,  remaining 2 at 68+%

To land on Mars with vac engine design (250 bar):  use 2 engines at 23+%,  if 1 lost,  remaining 1 at 47+%

To land on Mars with SL engine design (250 bar):  use 2 engines at 26+%,  if 1 lost, remaining 1 at 52+%

To take off fully loaded on Earth with SL design (250 bar):  use all 7 engines at 110% (fully loaded takeoff not feasible)

To take off fully loaded on Mars with vac design (250 bar):  Use 4 engines at 66+%,  if 1 lost,  remaining 3 at 88+%

To take off fully loaded on Mars with SL design (250 bar):  Use 4 engines at 74+%,  if 1 lost,  remaining 3 at 99+%

Engine-Mix Conclusions (for 250 bar designs):

#1. BFS used only at Earth could use all SL engines,  or use 3 SL engines to land,  and 4 vac engines for better Isp to LEO.  Requires BFR first stage with 31 SL engines.

#2. BFS used at Mars and returning to Earth must use 3 sea level engines for Earth landing,  and 4 vac engines for best takeoff from Mars,  as well as powering to LEO.  Requires BFR first stage with 31 SL engines.

For Figuring Performance:

3 SL engines for Earth landing:  Isp = 330 s,  Vex ~ 3.236 km/s

4 Vac engines for Mars landing,  or for powering to LEO:  Isp = 375 s,  Vex ~ 3.677 km/s

Test flight BFS-only takeoff 7 SL engines:  Isp = 330 s,  Vex ~ 3.236 km/s,  max TO mass (to hover only) 1213 metric tons vs 1335 tons fully loaded

Estimating BFS Performance:

These are jigger-factored rocket equation estimates,  per the methods of ref. 2.  The orbital mechanics delta vee requirements come from ref. 3.  For landing at Mars,  retro-burn starts near end of hypersonics at very low altitude,  near 0.7 km/s flight velocity (see landing estimates below).  For landing on Earth,  the “skydiver” descent rate at low altitude appears from Musk’s presentation to be ~ 0.2 km/s.  For getting to LEO from the stage point from BFR,  a slightly-factored delta-vee is orbit velocity minus stage velocity.  Staging velocity is presumed to be ~ 3 km/s. 

Powering to LEO on 4 Vac engines (250 bar design):

Stage velocity 3 km/s,  orbit velocity 7.9 km/s,  theo. dV = 4.9 km/s.  Apply 5% grav-drag loss:  dV = 5.1 km/s.  Req’d MR = exp(5.1/3.677) = 4.00;  Wp/Wig = 1 – 1/MR = 0.75,  vs 0.82 available (8.5% margin).  Margin is 3% at 2.5 km/s staging velocity.  Therefore,  the presumption of 3 km/s staging velocity,  or perhaps slightly lower,  is thus verified. 

Departing LEO and landing upon Mars,  using 4 vac engines (250 bar design):

Depart LEO dV = 3.9 km/s,  land on Mars dV = 1.0 km/s (factored from 0.7 km/s by 1.4),  total = 4.9 km/s.

Req’d MR = exp(4.9/3.677) = 3.791;  Wp/Wig = 1 – 1/MR = 0.736,  add 10% for boiloff to 0.810,  with only 0.823 available (1.5% margin implies,  at full payload,  Hohmann min energy transfer only!!!!).

Departing Mars on 4 vac engines,  and landing upon Earth on 3 SL engines (250 bar design):

Earth free fall = theo. min dV to land = 0.2 km/s,  factor by 1.5 to 0.3 km/s;  req’d MR = exp(0.3/3.236) = 1.0971 (figured from SL perf.);  dWp/Wig = 1 – 1/MR = 0.089;  add 10% for boiloff:  dWp/Wig = 0.098.

Loaded Mars takeoff on 4 vac engines direct to min energy Hohmann interplanetary trajectory:  min theo. dV = 5.35 km/s,  factor up 2% for gravity and drag,  dV = 5.46 km/s;  req’d MR = exp(5.35/3.677) = 4.284 (figured for vac perf.);  dWp/Wig = 1 – 1/MR = 0.767;  total Wp/Wig = 0.865,  with only 0.823 available at full rated payload!  Therefore,  payload must reduce!

Estimate takeoff Wig = 1100 tons propellants/.865 = 1272 tons.  The difference 1335-1272 = 63 tons is the required payload reduction for the return trip,  with no propellant margin at all.  Max return payload = 100 – 63 = 37 tons,  and that is for a min-energy Hohmann transfer trip!!! 

Miscellaneous Information:

There is not much change,  if any,  to the 31-engine first stage (BFR).  The real changes are a lengthened payload section and 3 large fins,  for the second stage (BFS).  The vertical fin is fixed (and termed more of a landing leg than a fin by Musk),  while the other two articulate about hinge lines for aerodynamic control during entry and landing.  These 3 fins replace the four folding landing legs previously shown

The articulation varies from roughly 45 degrees away from the vertical fin during entry and descent,  to a 120 degree separation at landing,  and during initial boost at launch.  Per Musk,  actuation forces for the articulated fins are “in the mega-Newton class”.  See Figure 1 and Figure 2.

The best-estimated landing sequences are shown in Figure 3.  Musk’s September 2018 presentation included a landing computer simulation video that he showed twice.  It was clearly an Earth entry and landing,  as effective deceleration to subsonic in the vertical-descent “skydiver” broadside-to-the-wind mode,  would be impossible to achieve in the thin air on Mars. 

For the Mars landing,  the 2017 presentation’s computer simulation video is still the best guide,  leading to a very low-altitude transonic pitch-up into a sort of tail-slide maneuver,  to position the vehicle tail-first for its final touchdown.  However,  it is likely that thrust must be used to effect the pitch-up into the tail-slide,  because lift equal to weight requires Mach 2-to-3 speed in such thin air.  

That means landing thrust must start at end-of-hypersonics at about Mach 3 (about 0.7 km/s).

For those worried about the fin tips digging into the soil on Mars,  here are some allowable soil bearing pressure data for selected Earth materials,  which might be similar to some soils on Mars.  Design practice requires static exerted pressures be less than these allowables.  For dynamic events,  design practice says stay under half these allowables.  The ton in the data is the 2000 lb US ton.

Ton/sq.ft             MPa                       type

1-2                          0.1-0.2                  fine loose sand

4-6                          0.38-0.58              compact sand and gravel,  requiring picking

8-10                       0.76-0.96              hardpan,  cemented sand and gravel,  difficult to pick

10-15                     0.96-1.43              sound shale or other medium rock,  requiring blasting to remove

25-100                   2.4-9.56                solid ledge of hard rock,  such as granite,  trap,  etc.

Eyeballed Fin Dimensions, Etc.:

Looking at the BFS images in Figures 1 and 2,  we might estimate fin dimension root-to-tip as about equal to basic body diameter,  which is said to still be 9 m.  That puts the fin tips about 13.5 m off of vehicle centerline.  With articulation to 120 degree spacing,  these tips form an equilateral triangle as the “footprint”. 

That puts the shortest distance from the vehicle centerline to the adjacent footprint edge (halfway between two tips) at about 6.75 m.  The “span” from there to the opposite fin tip is 6.75+13.5 = 20.25 m.  The vehicle itself is over 50 m long,  so the height to effective span ratio is about 2.5 to 3.  For the 2017 design with 4 landing legs,  this fell in the 3-4 range.  Some slight improvement in rough-field landing stability may have been obtained,  by going to the fin-as-landing-leg approach. 

The rounded tips on the rear tips of the fins cannot be more than 1 m diameter,  as eyeballed from the images.  That puts the total supporting bearing area for 3 fins at about 2.35 sq.m.  Exerted static bearing pressure at landing weight on Earth is 0.98 Mpa,  and on Mars is 0.38 MPa.  Exerted static bearing pressure at BFS-only takeoff weight on Earth is 5.6 MPa,  and 2.1 MPa on Mars. 

Mars regolith in many places looks like sand and gravel requiring picking,  in other places like loose fine sand.  It would appear the BFS could land on the sand and gravel requiring picking,  but not the loose sand.  However,  it cannot take off from that sand and gravel,  because the weight after refilling with propellant requires a medium rock to support it without sinking-in,  and getting stuck,  or possibly toppling over and exploding.  Prepared hard-paved pads appear to be fundamentally necessary for this design,  unless the fin tip landing pad area can be at least tripled.

Issues Not Fully Explored Here,  But Still Quite Critical:

#1. Rough field landings:  both soil bearing pressures and overturn stability on rough ground or because of obstacles under a landing pad. This requires serious attention!!!

#2. How to seal organic-binder carbon composite propellant tank structures against propellant leakage,  and also have this sealing (and the basic structures) survive at cryogenic temperatures.  None of this has been made public yet. 

#3. How to keep hot slipstream gases from scrubbing the leeside windows and composite structure.  These hot scrubbing flows result from the flow fields at high angle-of-attack,  that are induced by vortices shed from the strong body crossflow component,  and from the nose-mounted canard tips.  See sketch in Figure 4!  This can be a very serious issue for window failure.  It was for the Space Shuttle.

#4. How much internal pressurization is required to resist broadside airloads during entry and descent?

#5. No designs have yet been presented for cargo and tanker versions.  In particular,  the tanker design affects how many tanker refilling flights are necessary for BFS to depart from LEO. 

#6. Estimated costs per launch from Spacex are unavailable.  Some things seen recently on the internet suggest ~ $300 million per launch.  For 100-ton payloads,  that is ~ $3 million per ton,  for the one flight.  Such figures are entirely unreliable as yet,  and likely will remain so,  until several flights into LEO have been made. 

References:

#1. Article dated 4-17-2018 and titled “Reverse-Engineering the 2017 Version of the Spacex BFR” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.

#2. Article dated 8-23-2018 and titled “Back-of-the-Envelope Rocket Propulsion Analysis” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.

#3. Article dated 9-11-2018 and titled “Velocity Requirements for Mars” located on this site at http://exrocketman.blogspot.com,  authored by G. W. Johnson.




 Figure 1 – BFS/BFR at Launch,  2018 Version

 Figure 2 – BFS/BFR at Staging,  2018 Version

 Figure 3 – Best-Estimate Analysis of BFS Earth and Mars Entry and Landing,  2018 Version


Figure 4 – How Crossflow Vortices Greatly Enhance Lee-Side Heating Rates

UPDATE 9-28-18:  The shortage of fin tip bearing area can be addressed fairly-easily by a relatively minor shape change as indicated in Figure 5.  Instead of a tip pod with a round landing pad,  make the tip installation a larger part of the fin tip,  with an elongate pad.  Figure 5 shows the bearing area comparison between three 1-m dia round pads,  and three elongate pads 3.6 m x 1 m overall. 

This reduces the fully-loaded takeoff bearing pressure on Mars from 2.1 MPa to 0.49 MPa.  That reduction falls within the safe range for desert hardpan,  and might even be allowable for some simple compacted sand and gravels (requiring picking).  Landing (lighter vehicle weight) at 0.087 MPa becomes no problem for these types of soils on Mars,  even simple loose sand.  Although,  that loose sand is still quite unacceptable for supporting refilled takeoff weight.

Being able to land and take off from loose Mars sand is governed by takeoff weight (5027 KN),  and requires a total bearing area of about 50.3 sq.m to stay under 0.1 MPa bearing pressure.  That is probably far outside what is geometrically feasible. 


Therefore,  the unimproved landing sites are restricted to compacted sand and gravel requiring picking,  or better,  even with the elongate pads shown here.  




Figure 5 – How to Increase Landing Pad Area In the Simplest Way


UPDATE 10-1-18:  A few astute individuals have expressed a concern about BFS landing pads exposed to hypersonic heating,  if built as a streamlined item a the rear of the fin tips,  or as a part of fin trailing edges,  as I proposed just above. 

The best shape for a landing pad is not known to me,  but it is unlikely to be anything like a streamlined shape.  Better to design it to support the weight of a BFS fully-fueled on Earth (some 13,092 KN),  for purposes of short-hop flights.  It seems likely this is a relatively flat-surfaced shape,  whether round in footprint,  or elongated,  as advocated just above. 

This pad is also very likely to be of substantial weight,  bearing as it does the full Earth weight of a fully-fueled BFS,  with due allowance for impact effects during the landing transient.  It is also very likely to a surface that is hydraulically extended,  with shock-absorbing partial retraction,  much as any shock absorber.  And it is very likely to need the bending strength to endure hogging and sagging over obstructions,  instead of uniform pressure. 

If you retract the heavy pad itself just inside the otherwise wide-open fin trailing edge,  it sees no hypersonic scrubbing action,  only simple subsonic wake turbulence,  albeit at a high temperature.  Given the short duration of the entry event,  and the weight of a substantial structure,  the pad needs no heat protection to heat-sink its way through the entry event. 

This situation is sketched in Figure 7.  Bear in mind that the original Mercury and Gemini capsules had metal surfaces in contact with the hypersonic wake.  These were thin but structurally-unloaded corrugated skin panel structures capable of considerable radiative cooling,  surviving quite well at 8 km/s entry speeds from Earth orbit.  Using the old rule-of-thumb,  that’s around 8000 K gas temperatures,  with considerable ionization into plasma.  Free-entry/above-escape entry interface speeds are in the 6-7 km/s range at Mars (about 6000-7000 K),  and 11-17 km/s at Earth (about 11,000-17,000 K,  with very considerable radiation heating from the plasma sheath at all speeds above 10 km/s). 

The landing pad structure should probably be cellular,  in order to have lots of bending strength,  while not allowing any significant debris accumulation.  This is also shown in Figure 7.  If one pad can support the entire weight of the ship,  for obstructions touching anywhere on the pad undersurface,  then we have factor-3 redundancy to cover transient impact loads during landing. 

Looking at load,  shear,  and moment diagrams for the rock-under-the-middle and rock-under-the-end cases,  we find the same max moment magnitude to resist,  just opposite signs.  That value under these assumptions is 5.9 MN-m = 52 E6 inch-lb.  For the lateral dimension of 0.5 m,  the moment arm and moment magnitude are less than for the rock-under-the-end case.  Base the cellular spacing on 5.9 MN-m,  and the pad will be strong enough. 

For a typical high-alloy hardened steel,  yield strength might fall in the 50,000 to 100,000 psi range.  Use 75,000 psi,  and find the necessary pad section modulus (for the long direction) S ~ 700 in3.  Ignoring the section modulus effects of the top and bottom surfaces,  and just considering rectangular-section verticals 15 inches tall and half an inch thick,  the section modulus per vertical is 141 in3.  We only need about 5 such verticals spanning a meter-wide enclosure,  so the spacing is about 7.8 inches = 3 cm.  That’s 85% open volume in a square grid. 

Make the bottom out of the same kind of half-inch alloy steel plate,  and the top out of sheet metal.  That’s about 4000 lb = 1800 kg each for a 3.6 x 1 m pad footprint.  Very small for the 3 pads (5.4 metric tons) compared to the mass of the vehicle (~ 1300 metric tons). 

My conclusion is that there is really no reason why this cannot be made to work safely and reliably.  It will take very careful detailed structural-thermal design,  more than what I did here.  The max soak-out temperatures need to fall below the annealing temperature of the selected alloy,  so that properties do not change with age and number of flights. 


Looking at the reactions in the figure,  each hydraulic cylinder (of a pair per elongated pad) should be capable of providing the Earth weight of the BFS vehicle as an extension force.  That impacts the design of the landing pad hydraulics,  something beyond scope here.  


Figure 7 – Landing Pad Rough-Cut Design Data


Update 10-3-18:  For those who want to see how I calculated performance numbers,  see Figure 8.  This is simple rocket equation work,  with the required kinematic delta-vees jigger-factored upwards to account for gravity and drag losses,  or for severe uncertainties landing.  I used factor 1 for in-space departure from LEO,  factor 1.02 for the gravity and drag-affected departure from Mars,  and factor 1.5 upon the touchdown burn delta-vees. 

The “kicker” that throws off the simplest calculation is the 10% evaporation or boiloff loss for cryogenic propellants during the 9 month transit to Mars.  Propellant remaining after the departure burn is knocked-down 10% in the weight statement to start the arrival sequence. 

Another “kicker” is the change in specific impulse for the Earth landing with sea level engines.   The vacuum bell design cannot be used for that.

What I get doing it this more realistic way is a propellant-remaining safety margin upon landing that is a single-digit percentage of the original propellant load at departure.  It corresponds to approximately 1 km/s extra speed from the departure burn,  without really affecting the landing.  This does eliminate all the safety margin for obstacle avoidance or correcting trajectory errors during the arrival.  It is something I would not recommend!

I was surprised and pleased to find that these performances were not so very sensitive to the actual payload carried.  Raising the payload to Mars from 100 tons to 150 tons cut the 9% margin to 5%. Raising the payload back to Earth from 37 tons to 50 tons cut the 7% margin to 6%.   


Reducing propellant load from 1100 tons to 900 tons cut the to-Mars margin from 9% to 5%,  and the Earth-return margin from 7% to 4%.  These margins are thus demonstrably more sensitive to the initial propellant load carried.  The lesson is:  always top off the tanks completely,  before you fly.  


Figure 8 – Some Details for BFS Performance Estimation

2 comments:

  1. Really, really good work and I have nothing to pick on regarding the figures!

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    1. Thanks again, sir! I hope Spacex has noticed these reverse-engineering articles. It's easy to overlook things early in a design's development. Extra pairs of eyes do a better job spotting them. -- GW

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