The future presence of bases and settlements upon Mars
brings the need for transportation of freight and people about the planet. A little thought reveals the two
categories, freight and people, are fundamentally different in their
requirements. Most freight is not
time-sensitive, while people are.
Freight not time-sensitive needs to go by slow surface
transportation, without emplacing
expensive and effort-intensive infrastructure to make that possible. Such infrastructure is often very expensive
here, and the costs there are likely to
be quite catastrophic. Freight needs
something similar to rail transport here,
but without the tracks. A robot
“truck train” on a graded dirt road is the answer.
People are time-sensitive,
and need to fly in order to cover large distances rapidly. But physics in an atmosphere so thin, that it is first cousin to the vacuum of
space, rules out air transport by
airplanes or helicopters as we know them on Earth. It also rules out any form of
“lighter-than-air” transport, since
buoyant lift forces are proportional to differences in densities, and those are so vanishingly-low on
Mars. That pretty much leaves suborbital
rocket travel (the “rocket hopper”), the
topic of this document. The same could
be used for that tiny portion of freight that really is time-sensitive.
Suborbital Trajectories
I used my “orbit basics” spreadsheet, “R V q orbits” worksheet, to model suborbital trajectories about the planet, using ellipses that lie mostly within the planet. It was necessary to sharply limit the trajectory angle below local horizontal at entry interface altitude (135 km for Mars), in order to limit peak entry deceleration gees and peak heating, and to achieve altitudes from which rocket-braked landings were feasible in terms of timelines and deceleration gees. The results I was able to obtain are shown in Figure 1. The longest-range of these trajectories is very nearly antipodal (10,637 km).
Figure 1 – Suborbital Trajectories
Entry,
Descent, and Landing
I used the “entry spreadsheet” spreadsheet file, worksheet “Mars variations”, to compute estimates for the worst case
(highest entry speed) entry trajectory,
using the old 1953-vintage analysis used for warheads by H. Julian Allen
and his colleagues. This stuff was
declassified and taught in engineering schools in the late 1960’s. It is not the most accurate thing to use, but it gets you “well into the ballpark” with
very simple, essentially-by-hand, calculations.
These are easily put into a spreadsheet.
The highest speed at entry is the most challenging, since all the entry angles are about 15
degrees, so that analysis is depicted
in Figure 2. It was crucial to get
the hypersonic ballistic coefficient of the hopper vehicle down under 200
kg/sq.m, and preferably under 180
kg/sq.m. It proved feasible with the
final design to get well under that requirement with the value shown in the figure, although most design approaches will fail in
that respect! Getting the heating down
to something easily handled required the largest-possible “nose radius”. The value shown was dictated by the design
approach taken here, and is quite easily
handled, although it is too much for
metal exposed at the stagnation zone.
Figure 2 – Estimates for Entry Conditions
The hypersonic aerobraking is largely over once the vehicle
has decelerated to about local Mach 3,
which is roughly 0.7 km/s speed in the Martian atmosphere. Slower than Mach 3, a blunt object is no longer hypersonic, and the assumption of a constant ballistic
coefficient fails. It would be folly to
continue the hypersonic estimate past that point, for that very reason.
Not included in the approximate analysis is the effect of
trajectory “droop” to steeper angles due to gravity. That mostly happens after the peak
deceleration point, which is actually
rather close to the end-of-hypersonics (Mach 3) point. Peak heating occurred slightly earlier. For purposes of “reasonable
approximation”, I just used the
hypersonic endpoint altitude as obtained,
but I presumed the trajectory was headed about 45 degrees downward at
the Mach 3 point.
Ignoring the effect of the potential energy associated with
the Mach 3 point altitude, it is the
same Mach 3 point speed of 0.7 km/s that we have to “kill” with last-second rocket
braking, regardless of that
altitude. Appropriately factored for
losses and maneuvering, that is the
rocket braking delta-vee (dV) that we must have. I used factor 2, and would never use less than 1.5 under any
circumstances, in order to account for
losses, plus hover and divert
requirements. The Mach 3 point altitude essentially
determines how much time we have left before surface impact, which sets the required average gees, and thus the thrust required for any
particular vehicle mass.
I used simple high school-level geometry and physics/kinematics
to establish the average deceleration gees during the rocket-braking
landing. This is a very simple hand
calculation, illustrated in Figure 3. The 6 km altitude translates to an 8.5 km
path length down a straight line at 45 degrees.
At an undecelerated constant 700 m/s,
we are about 12.1 sec from impact,
as shown. Thus there is far too
little time available to deploy a chute,
much less expect any deceleration from it.
If we rocket-brake decelerate to zero, the average velocity down the path is only
350 m/s, and we have 24.3 sec to
touchdown, as shown. The change in speed is the 700 m/s. The change in time is the 24.3 sec. Their ratio is the average deceleration
required, which is 28.8 m/sec2, or some 2.94 standard gees.
Figure 3 – Hand Calculations for the Rocket-Braked Landing
Therefore, we are
looking at roughly a 3-gee rocket-braked landing, with the gees felt over an interval 24-25 sec
long. That is easily handled by persons
not trained in resisting gees, if seated, even more so if reclined. Roller coaster riders endure worse all the
time.
The hypersonic aerobrake peak gees fall in the 6.6 to 6.7
range, but again for a short interval above
5 gees that is only around 20-25 sec long,
as well. That is more
difficult, but it is endurable, even by untrained persons as long as they are
fully physically fit, as also
experienced by some roller coaster riders.
It is anticipated that passengers will be riding while wearing some sort
of pressure suits. It would help if
these pressure suits had “gee-suit” features as well. Otherwise,
some passengers might temporarily faint,
if sitting up. Any crew must
be trained to endure such gees, and
they must be wearing suits with “gee-suit” features.
“Rocket Hopper” Vehicle Design Concept and Estimates
I looked at 3 classes of possible design configurations
trying to meet the requirements of low hypersonic ballistic coefficient and
large “nose radius” simultaneously. Only
one approach satisfied those needs, and
still offered ways to mount landing legs for rough-field operations, plus a simple unobstructed engine bay. That was the cylindrical stack depicted in
Figure 4, but flown dead-broadside
to the oncoming stream during entry!
If the cylinder L/D ratio falls in the 4-to-6 range, that is enough larger blockage area to
greatly-reduce the hypersonic ballistic coefficient, despite the low hypersonic drag coefficient
of the cylinder shape. That shape was
required to keep tank construction lightweight.
As for the tanks,
these are main tanks that are integral components of the vehicle
airframe, but they also contain header
tanks. As the numbers worked out, about 15% of the tank volume is in the
headers, for course correction and
rocket-braking, with the other 85% in
the main tank volume for launch.
Figure 4 – Sketch Layout and Characteristics of the “Rocket
Hopper” Design Concept
As shown, the
smallest item was the engine bay, and
the largest item the cabin in which people ride. The tanks are stacked in the middle, so that center-of-gravity travel is not very
large as the propellants burn off. The
figure shows the layout, a weight
statement, seating arrangements, the basic trajectory-related notions, and some entry heat protection numbers.
Most but not quite all of the surface of this craft could be
exposed metal construction, if something
like a 316L stainless steel or an Inconel X-750 is used. Only near the stagnation line on the windward
side is something more heat-resistant required.
That could be a strip of ceramic tiles of some kind. Even low-density alumino-silicate ceramics
could be used, if blackened for high
emissivity, as peak entry pressures are actually
quite low.
Peak entry pressures are easily rough-estimated by simple
hand calculations at the peak deceleration gee point in the hypersonic entry
trajectory. If you know the mass at
entry, the peak gees acting upon that
mass give you the peak force decelerating the vehicle. Dividing that force by the frontal blockage
area gives you the average pressure applied to that area. The peak is at the stagnation zone, crudely twice the average value.
To run some of these numbers, I did create a custom “rocket hopper”
spreadsheet. It has multiple
worksheets, of which two are relevant here. Worksheet “veh” is set up to make the
calculations illustrated in Figure 5.
Worksheet “sections” is set up to make the calculations shown in
Figure 6.
In worksheet “veh”,
user inputs are yellow,
significant outputs are blue, and
things requiring iteration or verification are green. (The same color code applies to all my
spreadsheets.) It works in terms of mass
ratio-effective dV, which is the
end-of-burn V multiplied by an appropriate factor to account for gravity and
drag losses (about 1.02 on Mars).
Figure 5 – Numbers Run for the Design Concept, Part 1
The 3.6 km/s orbit speed at launch multiplied by that factor
is the 3.672 km/s dV shown. The course
correction budget is 8% of the orbit apoapsis speed, which is where corrections should be
made. For the highest-speed case, that is the 0.215 km/s shown. And at factor 2, the 1.4 km/s dV is what the rocket-braking
burn must be capable of, the same for
all cases.
I used a launch liftoff 1.5 gees as required for initiating
good ascent kinematics, same as here on
Earth. 0.1 gee for course corrections is
probably just a lower limit. I used 3.5
gees at landing to get some margin over the average 3 gees determined above.
The specific impulse input of 352 sec shown is justified by
the analysis given below. The sum of the
dV values is the total dV to be delivered for the mission, which sets vehicle mass ratio and propellant
mass fraction. 1 – propellant fraction –
inert fraction is the available payload fraction.
There is an “inert mass fraction build-out” block
shown. We can argue about the component
inputs, but their sum is likely “in the
ballpark” no matter exactly what inputs one uses. The same is true of the size payload
block. You are looking at the weights
for a person, his pressure suit, a few hours worth of oxygen, water,
and food, plus some luggage. It’ll be around 0.2-0.25 metric tons per
person, almost no matter what, when you sum it up. That and how many people
are aboard, sets the payload mass, which ultimately sets the weight statement.
The “heat shield re-radiation” block presumes convective
heating is balanced by re-radiation from hot exposed surface materials. It uses the peak heating from the entry
analysis, plus good guesses for surface
emissivity and the effective temperature of the surroundings that must receive
that re-radiated heat. A highly-emissive
surface is typically 0.8, while low is
0.2. 400 R for the surroundings is -60
F.
The “run weight statement and size thrusts” block does
exactly that. Payload mass divided by
payload fraction is ignition mass.
Ignition mass time inert fraction is inert mass, and times propellant fraction is propellant
mass. Payload plus inert is
burnout, and burnout plus propellant is
ignition. Each burn has a dV that sets
its mass ratio. That in turn sets start
and end-of-burn masses, the difference
being propellant used for that burn. The
sum of those propellant masses used must equal the total propellant mass
already figured.
There is a “check pressure on heat shield” block that uses
the hypersonic ballistic coefficient and the hypersonic drag coefficient as
inputs, plus the initial mass at
entry, and the peak entry gees. It computes the mass per unit blockage area
from the ballistic coefficient and drag coefficient, then the peak entry force from that and the
entry mass and peak entry gees. It then
divides force by blockage area for average pressure, and doubles that for the stagnation pressure
estimate, reported in a variety of units
of measure.
The ”heating other locations” block gets you equilibrium
temperatures in degrees F for stagnation,
“typical” lateral, and separated
wake zone locations. The stagnation peak
heating is reduced by a factor of 3 for “typical” lateral surfaces, and by a factor of 10 for separated wake zone
surfaces.
The only other block shows launch dV available as a function
of percent max propellant loaded on board.
The course correction and landing dV values are not changed. This would be useful trying to relate range
to propellant load required.
Figure 6 – Numbers Run for the Design Concept, Part 2
The “sections” worksheet works out the proportions of the
engine bay, tankage, and cabin sections of the ship, plus some other pertinent results. It needs the r-ratio (oxidizer/fuel fuel mass
ratio) for the engine, and the specific
gravities of the fuel and oxidizer materials.
It also needs as inputs the mass at start of entry (after course
corrections from the weight statement),
and a hypersonic drag coefficient in crossflow for the presumed vehicle
shape, which in this case is a circular
cylinder.
It also needs as inputs the propellant masses for each
burn. It works out from these the masses
of oxidizer and fuel for each burn, and
their volumes. This is done in an
untitled block top center of the page.
There is an “engine resize” block that takes the engine
characteristics modeled elsewhere, and
rescales them to the correct thrust size.
Those inputs are the modeled vacuum thrust, the required vacuum thrust, the throat and exit diameter sizes, and the effective average half-angle of the
supersonic expansion bell.
There is a “tanks figured on totals, with headers inside TBD” block. It has vehicle diameter and a sort of
interstage length between the tanks as inputs.
It works out all the lengths and L/D ratios, and requires the “right” vehicle diameter to
keep a full sphere as the smaller tank,
as well as to be consistent with the seating arrangements in the “cabin”
block. The seating is an input number per
level, and number of levels, consistent with the total number of
people. There has to be room for seating
at the input diameter in the “tanks” block,
and the input seat pitch is the spacing between levels. Note how the empty main tank shells protect
the propellants in the header tanks from the effects of entry heating.
The “engine bay” block takes the resized dimensions of the
engine (overall rough estimate of length,
and the exit diameter, and uses
these with inputs for number of engines and the spacing requirements for
gimballing, to estimate min dimensions
of the engine bay. Its diameter should
never exceed the input diameter in the “tanks” block.
From there, the
“overall vehicle” block puts together these results into estimates for the
length and L/D ratio (which ought to be in the 4 to 6 range) of the entire
vehicle stack, and with inputs for entry
mass and hypersonic drag coefficient,
estimates the broadside-entry ballistic coefficient. Too low an L/D will
get you too low a blockage area and too high a ballistic coefficient. Too high an L/D is a topple-over risk, or at least a risk of bigger, heavier landing legs.
Creating the Re-Sizable Engine Data
I used the “rocket nozzle” spreadsheet file, worksheet “rocket noz”, to rough-size a suitable engine and calculate
a reliable performance estimate for it.
This was actually one of the first things I did. This engine burns liquid methane fuel with
liquid oxygen oxidizer, similar to
SpaceX’s Raptor and Blue Origin’s BE-4.
These propellants are thought to be manufacturable in situ on Mars. This entire design study assumes that to be
true.
I chose not to push the state of the art, given the troubles SpaceX had getting to
current Raptor-2 performance levels.
This engine is only sized for a chamber total pressure delivered to the
nozzle entrance of 2000 psia, and only a
3:1 pressure turndown ratio, although
this analysis does presume a full-flow cycle with no dumped bleed gas. The nozzle is assumed to be an 18-8 degree
curved bell, with a throat discharge
coefficient of 0.995. Both c* and r are
presumed functions Pc, using the data shown
in Figure 7 as the inputs. Similar
data for other propellant combinations are in the “prop comb” worksheet.
Figure 7 – The Resizable Engine For This Study
The nozzle expansion was designed at full Pc = 2000
psia, expanded to 5.97 psia at its exit
plane. This gave us a nozzle right at
incipient separation when operated at 2/3 Pc, at sea level on Earth, allowing easy open-air testing on Earth. It is unseparated for that power setting (or
higher), but cannot be operated at lower
settings at sea level, because the
nozzle will separate.
Thrust was sized at an arbitrary 10,000 lb in vacuum at full
Pc. Performance in the near-vacuum of
Mars’s atmosphere is indistinguishable from true vacuum performance, so the vacuum data were used for this
study. There is very little specific
impulse variation from 1/3 to full thrust,
reflecting pretty much only the variation of c* with Pc. I used the 2/3
power value of 352 sec as “typical” of operation at any throttle setting. Thrust and flow rates scale with throat
area, dimensions scale with throat
diameter.
Final Note
Most of the spreadsheets used here are part of the course
materials that I created for orbits and vehicle sizing.
Only the custom spreadsheet used for vehicle characteristics as calculated here, is not part
of those course materials. Those course materials are available on the New Mars forums, in the "interplanetary transportation" topic, "orbit mechanics class traditional" thread.
No comments:
Post a Comment