A lot of people with whom I correspond are interested in
making their own calculations of what rocket vehicles might do. A part of that is the velocity requirements
coming from orbital mechanics, and that
is not covered here, except for some typical values. Another part is estimating the performance
available from a given rocket vehicle or stage design concept. That is really the topic addressed here.
What I will try to do is introduce the tools and techniques
for “ballpark” back-of-the-envelope estimates that are actually quite
realistic. This is where you start when
you want to properly pursue a real design.
You must work out the configuration characteristics that bound (and provide
budgets for) the real system and subsystem design work, to be done later by real experts with better
tools.
I will make this presentation algebra-based. You need not be a trained scientist or
engineer to do this. Just follow the
recipe. The only assumption made here is
that your velocity requirement coming from orbital mechanics is appropriate and
realistic, for whatever it is that you
want to investigate.
This is
about making estimates using the rocket equation, with appropriate “jigger factors” for
overcoming losses and other non-ideal phenomena, and some organized way to guess realistic
inert mass fractions. It is also about
selecting realistic values for specific impulse. Appropriate “jigger”
factors, good inert fractions, and realistic specific impulse are the real
keys to making this work.
Where Thrust Comes
From
In a word, it comes
from Newton’s Laws when you expel mass.
These Laws not only have over 3 centuries’ worth of verification in a
variety of practice venues, but also over
2 centuries’ successful application specifically in rocketry, beginning with the design calculations for
Mr. Congreve’s rockets of the early 1800’s.
The concepts shown in Figure 1 can be found in any high
school physics book, specifically the
chapter on impulse and momentum. This
does require establishing the coordinate system against which measurements are
made. The preferred choice is on the
rocket vehicle, as shown.
The rocket does not “push against” anything behind it in
order to fly, except its own expelled
plume of mass. In order to accelerate
that expelled mass to any given speed with respect to the rocket, requires an effective force exerted upon that
expelled mass. That is the “action”
here.
Newton says that for every action, there is an equal and oppositely-directed
reaction. The reaction to the force
expelling the mass is the “thrust” that moves the rocket, shown very explicitly in Figure 1.
This process works as well in the vacuum of space as it does down here in the atmosphere. Only the expelled plume behavior is different between those two regimes, a topic covered later. The rocket does not push against the atmosphere or the ground, it only pushes against its own expelled mass plume.
How that plume might interact with the atmosphere, or any physical objects in its path, does not enter into the calculation of thrust! Figure 1 shows the conceptual simplified
algebra of formulating thrust force as equal to massflow rate times velocity, as well as the calculus-based version.
This formulation is exactly true as long as the nozzle
expansion is perfect, meaning the
expanded gas pressure at the exit plane is equal to the ambient pressure
surrounding the nozzle. But it is not
far off, for any realistic nozzle
design, even if the expanded pressure
does not match ambient. That is covered
further just below, along with notions
of nozzle efficiency.
Real Nozzle Plume
Behavior
This comes from compressible fluid mechanics analysis, the exposition of which is quite simply out
of scope here. Suffice it to say that
the pressure mismatch at the nozzle exit plane influences downstream plume
behavior in predictable ways. These are
shown as part of Figure 2. Imperfect
expansion also affects rocket thrust with a pressure-difference term in
addition to the momentum term.
For perfect expansion,
the expanded pressure right at the exit plane Pe exactly equals the
ambient back pressure Pb, and the plume
travels downstream essentially unchanged in diameter, meaning it does not spread or contract. Its speed downstream is essentially fixed at
the expanded exit plane speed Vex.
If Pb > Pe, we
call the situation “over-expanded”,
because Pe is expanded too low relative to Pb. The plume will contract a little to a smaller
diameter before proceeding downstream,
still essentially at Vex.
If Pb < Pe, we
call the situation “under-expanded”,
because Pe has not been reduced by enough expansion to reach Pb. The great bulk of the plume will spread a
little, at roughly the nozzle half-angle,
to a larger size before proceeding
downstream at essentially Vex. There is
a small minority of the plume mass that spreads a little more sharply, but it has little effect in any practical
sense.
The extreme under-expanded case happens when operating in
vacuum, for which Pb = 0. Pe can never be made 0 with “enough
expansion”, because compressible fluid
mechanics would require an exit area of infinite size, something quite impossible to physically
build. In most cases, the fundamental exit area limit is driven by
the size of the rocket (or stage) cross section area.
The picture of plume behavior in vacuum shows the bulk of
the plume spreading at the nozzle half-angle and Vex speed, with the small minority of plume mass turning
very sharply outward at a bit more than 90 degrees.
These pictures obtain “up close”, right in the vicinity of the nozzle exit
plane. What happens farther away depends
upon the presence or absence of a surrounding atmosphere. That is shown on the right side of Figure 2. The illustration for rocket plume behavior in
an atmosphere looks just like the jet blast predictions for airports and
military bases, because it really is the
same fundamental physics.
In an atmosphere, the
expelled plume experiences both friction drag effects, and mixing effects. The illustration shows an unmixed core
essentially moving downstream at Vex,
but with a finite contour shape.
There is a region outside that contour where both frictional dissipation
of plume momentum, and dilution of its
momentum over mixed-in extra air mass,
take place.
This mixing region contains contours of constant reduced
velocity not actually shown in Figure 2,
but similar in shape to those illustrated. Ultimately,
an imperceptibly-low velocity contour is the outer boundary of the
region affected by the plume. (Theoretically, it is never quite zero velocity.)
Out in the vacuum of space,
the phenomena are quite different.
There is no atmosphere for the rocket plume to interact with. It merely proceeds downstream at essentially
Vex “to infinity” (given enough time),
while spreading at roughly the nozzle half angle. There are no contours, no limits on plume extent downstream. The interplanetary/interstellar medium
actually does have a sort of in-the-atmosphere effect, but the dimensions of its contours are better
measured in astronomical units.
The sharp lateral spread of that minority of the hot plume massflow
could easily impact nearby structures on the rocket vehicle and damage
them. That is why the rocket nozzles
protrude aft out of the tail of the vehicle or stage, in most designs that could fly in
vacuum, or near-vacuum, conditions.
Real Nozzle
Efficiency and Engine Thrust Behavior
Nozzle performance (and overall engine performance) depends
upon exit/throat area ratio Ae/At, exit
cone effective half-angle “a”, the
effective backpressure Pb, and the
chamber pressure feeding the nozzle assembly Pc, plus the gas specific heat ratio γ. These are not independently-selected
variables!
We are interested in figuring realistic thrust and flow
rate, because their ratio is the
specific impulse Isp. That factor goes
into figuring vehicle performance potential,
as described below. It needs realism,
too.
If throttleable, we
have to make the nozzle work at min Pc with the design point Pb, without incurring too much pressure-term
thrust reduction, or worse yet, internal flow separation. This will set the expansion ratio Ae/At. Then performance is limited to that expansion
ratio at the high value of Pc, where the
pressure difference term increases thrust,
but not as effectively as more expansion ratio.
This is most conveniently book-kept as thrust coefficient
CF, for an easy thrust equation F = CF
Pc At. The flow rate depends upon a
“characteristic velocity” c* that is a function of chamber temperature and chamber
gas properties.
There is little or no measurable friction loss in a good
nozzle design, so the only “efficiency”
impact is streamlines not in perfect axial alignment because of the conical
bell shape (the half-angle effect).
For a conical bell, “a” is the geometric cone half-angle. For a curved bell profile, there are local half-angles near the throat and at the exit lip. Arithmetically average them for use as “a”. The streamline divergence factor (kinetic energy efficiency) is nKE = (1 + cos a)/2, as shown in Figure 3.
A good general default value for “a” = 15 degrees. The corresponding efficiency is about
98%. While this is a high
efficiency, performance done with the
rocket equation will exponentially magnify the effect of any such errors. See performance, below.
All of this is shown in Figure 3. Not shown is the effect of engine cycle, which deals with how the propellant
turbopumps are driven. Staged chambers
and injector plate pressure drops make the ultimate feed pressure to the engine
quite a bit higher than the nozzle-feed Pc shown here.
The effect of cycle shows up in a mass loss fraction, which is turbopump drive massflow dumped
overboard, divided by the nozzle
massflow. You add that fraction to unity
for a scaling factor: if 2% of massflow
is dumped, the factor is 1.02. For the oversimplified ballistic calculations
recommended here, propellant feed rates
need to be scaled up by that factor, and
specific impulse gets scaled down by that same factor. Some cycles dump a lot, there are a few that dump nothing. Pressure-fed dumps nothing.
Figure 3 – Real Nozzle Efficiency and Thrust Behavior
There’s a lot of published reference data for various
propellant combinations that includes Isp data,
which people then just use in the rocket equation uncorrected. That is an error! Such data are usually reported for two
conditions, one representing a perfect-nozzle
efficiency sea level design, the other
representing a typical (but also perfect nozzle efficiency) vacuum design at
arbitrary expansion ratio.
Sea level is usually reported for Pc = 1000 psia and perfect
expansion (whatever that area ratio turns out to be) for Pe = Pb = 14.696 psia
= 101.325 KPa. This is done inside a
thermochemical code at 100% (perfect) nozzle efficiency nKE. It does not take not account any
cycle-related mass loss factor, either.
It is usually done for all-shifting (or all-frozen)
composition in that thermochemical code,
when shifting-to-the-throat and frozen-to-the-exit is actually the
better model. Of those 3 error sources, it is nozzle efficiency and dumped-mass cycle-related
factor that are more significant, by
far.
The reference vacuum estimate is usually done for Pc = 100
psia and Pb = 0 psia, with a fixed
expansion ratio Ae/At, usually about
40. Again, nozzle efficiency is assumed 100%, and no account is taken of the cycle-related
dumped mass factor. Thermochemical
shifting versus frozen composition methods are the same as in the sea level
case.
Shortcut
to Specific Impulse With Reference Data
If you shortcut actually doing the engine and nozzle
ballistics for your initial estimates,
then just pick the appropriate sea level or vacuum Isp out of the
reference data. But, before you use it, multiply it by a default “typical” 0.98
nozzle efficiency factor, and then
divide it by your mass-dump cycle factor (which for 2% dumped overboard is
factor = 1.02). Every cycle is
different, make the best dumped-mass factor
guess that you can. Such data are very
hard to find as published items.
You do need to come back and check your engine Isp values
once your configuration “gels”, and
before you believe your vehicle performance estimates. To do that,
you need to run a simplified liquid rocket ballistic design procedure at
a suitable design point with a known Pb.
Simplified
Ballistics for Better Estimates of Specific Impulse
Size the nozzle expansion at your design point expanding
from your min (throttled-down) Pc to Pe = Pb at design (not necessarily sea
level). You can use a default specific
heat ratio γ
= 1.20 for this.
Me =
{[(Pc/Pe)^c1 – 1]/c2}^0.5 where c1 = γ/(γ –
1) and c2 = (γ – 1)/2
Ae/At =
{[(1 + c2*Me^2)/c3]^c4}/Me where c3 = (γ +
1)/2 and c4 = 0.5*(γ + 1)/(γ – 1)
Once Me is known,
Pc/Pe = (1 + c2*Me^2)^c1 is a constant for unseparated operation, which can be used to find Pe for any given value
of Pc. The CF equation itself is given
in Figure 3, including the effects of
nozzle kinetic energy efficiency. Thrust
is just F = CF Pc At. Expanding to an
Ae/At is iterative.
Massflow wdot = Pc CD At gc / c*, where c* is a weak power function of Pc, as shown in Figure 3. (CD is just the nozzle mass discharge
efficiency, reflecting the boundary
layer displacement effect in its throat.
It is usually 99% or better in a good design, to a max of essentially 100%.) For early configuration work, CD and the rather weak variable-c* effect can
be ignored, as the empirical exponent is
usually in the vicinity of 0.01. Values
of c* are often reported in the same reference tables as Isp. Unlike Isp,
these c* values are actually pretty good just as they are, since they are figured from chamber
conditions only.
Once you have a ballistically-calculated thrust and the
corresponding massflow, then Isp = F/wdot = CF
c* / gc, which then already includes the
nozzle efficiency effects because of their inclusion in the CF equation. Don’t forget to ratio up your reported propellant
massflow by the cycle-related factor,
and to ratio down your reported Isp by the same cycle-related factor. That is not in the CF calculation!
Estimates obtained this way (with simplified ballistics) are
more reliable than just picking Isp numbers out of a reference table, and correcting them for typical nozzle (multiplier
near 0.98) and mass dump (divisor on the order of 1.02) effects. In turn,
corrected reference table Isp numbers are better than just using the
reference Isp values uncorrected.
Separated Nozzle Flow
This estimate is entirely empirical: mine is Psep/Pc = (1.5*Pe/Pc)^0.8333. Find Psep from Pc and this ratio, and make sure that your Pb is always less
than Psep in all the ways that you operate your engine.
Typical Reference Data for a Few Selected
Propellant Combinations (1969 P&W Handbook)
................................Sea level to 14.696
psia
Oxidizer.....fuel.......r (ox/f)..Pc, psia..Tc, R....c*,fps.....Isp, sec
O2..............H2........4.0.........1000.......5330....7950.......388
O2............CH4.......3.15........1000.......6350....6120.......310
O2............RP-1.......2.55.......1000........6590....5900.......299 NTO...hydrazine.....1.33........1000.......5870.....5860......292
NTO.......MMH.......2.17........1000.......6110.....5730......288
IRFNA.....RP-1.......5.0..........1000.......5670.....5180......263
.................................Vacuum at Ae/At = 40
Oxidizer.....fuel.......r (ox/f)..Pc, psia..Tc, R....c*,fps.....Isp, sec
O2..............H2........4.5.........100........5610.....7840.......454
O2............CH4.......3.25.......100........5840......5960.......365
O2............RP-1.......2.6.........100........6030.....5730........351
NTO...hydrazine.....1.36.......100........5500......5740.......342
NTO.......MMH.......2.26.......100........5660......5570.......338
IRFNA.....RP-1.......5.1.........100........5300......5070.......309
Vehicle Performance
Potential Via the Rocket Equation
There is a preferred order to how one approaches this. It is assumed that for a given propellant
combination, one has a realistic and
appropriate value of Isp, sec, per above discussions. As shown in Figure 4, this is converted to an effective exhaust velocity
in the desired units, usually km/sec to
match with the typical reporting of dV values from orbital mechanics. The gc value is 32.174 if US customary units
of ft/sec are desired, and 9.807 if
metric m/sec units are desired. Convert
ft/sec to km/sec with 3280.83 ft/km as a divisor, and m/sec to km/sec with 1000 m/km as a
divisor.
There are 3 mass classifications in this process: payload,
inert, and propellant. They combine by addition to an overall
“weight statement” as shown in the figure,
which gives burnout and ignition masses.
Inert is the tankage,
engine, and any other vehicle
structures.
The overall mass ratio MR achievable with this weight
statement is also shown in the figure:
MR = Wig/Wbo. The overall
theoretical delta-vee (dV) obtainable from this weight statement is obtained
from the rocket equation as shown, using
the effective exhaust velocity in km/s.
That is dV = Vex LN(MR), where
“LN” means the natural, or base e, logarithm.
This theoretical rocket stage or vehicle dV performance must
equal or exceed the sum of all the orbital mechanics-derived dV requirements
you wish to satisfy, plus all the
gravity and drag losses. If it does not
satisfy the factored dV requirements,
you need a higher Isp, or a
different weight statement, or you need
to stage your rocket, or some
combination of these.
In this dV-based sizing process, one uses the rocket equation-in-reverse to
find the MR from dV and Vex, as MR =
exp(dV/Vex), and then from it, the propellant mass fraction Wp/Wig = 1 –
1/MR. “Exp” means base e
exponentiation. The payload fraction is
then unity minus propellant fraction and minus inert fraction. If payload fraction is zero or negative, the design concept is infeasible. Period.
For a given payload mass Wpay, the payload fraction gets you a value for ignition
mass Wig. From that, the other mass fractions size Win and
Wp, allowing the weight statement to be
constructed all the way to Wbo by the additions in the table, and then checked to make sure everything adds
up.
Estimating Realistic
Inert Mass Fractions
Clearly, having a
good guess for the inert fraction Win/Wig is critical! This is where most people attempting this process
will go wrong. The more you ask your
structure to do, the heavier it is going
to be! Period. Here is an organized way to guess realistic
inert mass fractions. It starts from a
nominal 5% inert for a simple throwaway tankage and engine set. That is pretty much the current state of the
art.
Item.........factor...........descriptions
Basic.......1...................basic
minimal one-shot tankage and engine
Cryo........0 or 1...........double-wall
Dewar tankage with cryocoolers
Reusable.0 or 1............added
structural beef for many, many flights
Lander.....0 or 1...........add
heat shield/aeroshell, load ramps, landing legs
Volume 0 or 0.5 or 1 add
small (0.5) or large (1.0) pressurized cargo bay
Total........sum factors..add
up all the factors
Inert
faction: multiply sum by 0.05 for Win/Wig never less than 0.05
If you stage your rocket,
remember that the ignition mass of an upper stage is the payload mass of
the next lower stage. Start sizing with
your uppermost stage, and work your way
down to the first stage. Each stage will
shoulder its portion of the mission dV. A
good startpoint is equal portions among the stages, but you will adjust that later, as lower stages with lower Isp in the
atmosphere, and more gravity and drag
losses, will need to shoulder a little
less than an equal dV portion, for best
results.
Gravity and Drag
Losses
For a two stage launcher sending payload to low Earth
orbit, the staging velocity is usually
in the vicinity of 10,000 ft/sec or 3 km/s,
at an altitude outside the sensible atmosphere, and at a trajectory path angle that is almost
horizontal. The first stage sees all the
drag losses, and most (if not
effectively all) of the gravity losses.
The second stage sees no drag losses,
and very little if anything in the way of gravity losses, because the gravity vector is very nearly
perpendicular to its trajectory for its entire burn.
A good rule-of-thumb guess for Earthly launches is adding 5%
to the theoretical delta vee demanded of the first stage for drag losses, and another 5% for gravity losses, such that the total increase is 10% over
theoretical. This does assume an
aerodynamically clean vehicle launched vertically onto a gravity-turn
trajectory! The second stage needs no
such increases over theoretical. If the
staging velocity is Vstage, and the
orbital velocity Vorbit, then the
adjusted dV to be demanded of the first stage is dV = Vstage*(1 + .05drag
+ .05gravity), and the dV
demanded of the second stage is simply Vorbit-Vstage.
This is not exactly right,
but it is very close, close
enough for adequate realism. One could
run a good trade study by starting at Vstage = Vorbit/2, and running some decreasing Vstage
cases, looking for max delivered payload
divided by overall ignition mass.
To rescale these corrections for a similar flight to low
orbit on another world, multiply .05drag
by the ratio of surface density to sea level Earth standard density, and .05gravity by the ratio of
surface gravity acceleration to the Earth surface value. You may not need two stages. If so,
apply the correction to the full orbital velocity, and demand it as the delivered dV of the
single stage.
Generally speaking,
you don’t need gravity (or drag) corrections for the dV requirements to
escape from orbit. The exception would
be electric propulsion, with its very
long “burn” times accumulating a really large gravity loss. That is out-of-scope here. (So far,
I have used factor = 2, but that
is just a bad guess.)
Theoretical Delta-Vee
Requirements from Orbital Mechanics
The surface escape velocity (a theoretical dV value) has
been long published for Earth and many other bodies in the solar system. The surface circular orbit velocity (also a
theoretical value) is surface escape divided by the square root of two, if not also listed in the publication you
consult.
For typical low orbit velocity, I use the surface circular value for eastward
launches (with the aid of Earth’s rotation).
This is a slightly higher speed than at the real orbital altitude, but the excess covers the potential energy of
being at orbital altitude, all in a
quickly and easily available number. Use
this surface circular orbit velocity as the unadjusted (theoretical) dV
required to reach low eastward Earth orbit.
The same procedure can be used for any other body, based on its surface escape speed.
It is easy enough to compute the surface velocity due to
Earth’s rotation from its radius,
rotation rate, and latitude. For a polar launch, add one of these to your surface circular
orbit velocity-as-dV. If launching
westward against the rotation, add two
of these to the surface circular orbit velocity. The same can be done for any body in the
solar system for which rotation rate and radius are reliably known.
Here are some selected data from an old CRC Handbook (53rd
edition, 1972):
Body.....Vesc
km/s....Ravg., km...Rot., day...mass, kg....surf gees...dens/std
Earth.....11.179..........6371.3........1.00...........5.98E24....1.000........1.00
Moon.....2.3735.........1738.3.........27.3..........7.35E22....0.1652.......0.00
Mars......5.0282..........3380...........1.0257......6.42E23.....0.3814......0.007
Calculating reliable dV data for interplanetary trajectories
is beyond scope here. Suffice it to say
the numbers given here are worst-case Hohmann min-energy transfer to and from
Mars, with the smallest semi-major axis
and largest perihelion and aphelion velocities.
The Earth orbit departure/arrival dV could be a little smaller, and the Mars arrival/departure dV could be a
little larger. The one-way transfer time
could be about a month longer.
Mars
Mission dV Requirements Data (worst case)
Earth
dep/arr dV from LEO..3.937
km/s..(orbital assembly presumed)
Mars arr/dep
from LMO.......1.594 km/s..(docking in orbit with assets presumed)
One-way transfer time:..........234
days......(shortest case, longest
exceeds 270)
Missions to Mars that use direct entry from the
interplanetary trajectory need only let the planet run over them “from behind”, but on a nearly tangential-to-the-planet
trajectory, so that entry angle is
sufficiently shallow. The only burn in
this scenario is the final retro-propulsive touchdown burn.
Missions to Mars that orbit the planet in a low circular
orbit require the arrival burn figure (which for the other extreme case might
be as high as 2 km/s). From there, the deorbit burn is trivial (on the order of
dV = 50 m/s), but there is the final touchdown
retro-propulsive burn (see below).
Missions returning from Mars will be rapidly overtaking the
Earth from behind. Those that posit a
free aerobraking entry will hit the atmosphere at higher-than-escape
speed, and must be more-or-less
tangential to the planet to maintain a shallow-enough entry angle. But not too shallow, or else the craft will bounce off the
atmosphere above escape speed, and never
return. Heat protection requirements are
very stressful.
Downlift may be required early in the entry to prevent
bounce-off, and uplift later in the
entry to prevent over-steepening. From
there landing requirements vary with the vehicle design approach. End-of-hypersonics will be ~0.7 km/s at 40-50
km altitudes. Very subsonic terminal
chute velocities may be obtained, up to
some size limit beyond scope here. See
landing requirements below.
Missions intended to recover in low Earth orbit must make
the arrival burn listed above. From
there, the deorbit burn is fairly
trivial, and the final landing may take
many different forms, depending upon the
vehicle design. See landing requirements
below. Heat protection requirements are
far less stressful than the direct entry case.
Missions to the moon need not quite exceed Earth escape
velocity, but the theoretical necessary
speed is very close to escape (10.84 km/sec vs 11.18 km/sec). The moon will be overtaking the craft from
behind, at its transfer orbit
apogee. The least-costly entry into low
lunar orbit is a retrograde orbit about the moon, which is the least favorable for
landing. However, the moon’s slow rotation rate makes this
effect negligible. Departure for the
moon can be from low Earth orbit, or
direct from the surface. The variations with
orbital eccentricity are so small, that
only departure and arrival data for average orbital conditions are shown (it is
a second or third decimal variation):
Moon
Mission Earth departure dV, km/sec (return
into orbit same as departure)
From low
orbit.....3.286 (unfactored in space)
From surface........11.595
(factored to orbit speed, unfactored
from there)
Moon
Mission Arrival-at/Departure-from the Moon dV, Km/sec
Into/from
low lunar orbit.....0.759
(unfactored in space)
Direct to/from surface..........2.376
(slight gravity loss factored)
Multiple Burns from a
Single Stage
This requires a multiple-burn weight statement. It presumes the same payload and inerts as an
overall one-burn weight statement. The
usual case is splitting the on-board Wp into two allowances for two burns, based on the individual adjusted dV’s
required for the burns, whose sum is the
total required adjusted dV that set the overall stage design.
This is quite often the case for a restartable second stage for
an Earth orbit launch vehicle. Such a
stage will burn most but not all its propellant putting itself into a transfer
ellipse orbit, followed by a short burn
at apogee to circularize into the final desired circular orbit.
For a two-burn case,
there is an intermediate burnout Wbo1,
and two propellant weights Wp1 and Wp2 that sum to the total Wp allowed
in the design (Wp = Wp1 + Wp2). This is
shown in the two-burn weight statement format just below.
For a three-burn case,
there would be three propellants expenditures Wp1, Wp2,
and Wp3, that sum to Wp, and two intermediate burnout masses Wbo2 =
Wbo + Wp3, and Wbo1 = Wbo2 + Wp2, such that Wbo1 + Wp1 = Wig. The pattern is otherwise the same.
For the two-burn case illustrated, the mass ratio for the second burn is MR2 =
Wbo1/Wbo, and for the first burn MR1 =
Wig/Wbo1. Similar results obtain for the
3-burn case not shown. Delivered dV’s
for each burn come from the rocket equation.
These should sum to the overall delivered dV (for invariant payload and
inerts). Each should equal or exceed the
corresponding demanded dV from orbital mechanics, as factored for gravity and drag losses.
Item.............description
Wpay...........payload,
presumed invariant
Win inert structural weights invariant
Wbo.............final burnout mass, all propellants expended
Wp2 propellant expended in second burn
Wbo1...........intermediate burnout mass before burn 2 and after
burn 1
Wp1 propellant expended in first burn
Wig..............initial ignition
mass, before either burn
Odd-Ball
Requirements: Retro-Propulsive Needs for
Landing
On an airless place like the moon, landing must be all-retro-propulsive, and is essentially launch-in-reverse. There is no drag loss, but there is a small gravity loss for
launch. Use the launch propellant figure
as the min figure for landing. Then adjust
it with an allowance for hovering and maneuvering around, to avoid hazards at touchdown. This is a margin factor applied to a min
Wp, not the theoretical dV, because only the final seconds are affected
with hazard avoidance. As a guess, use something like factor 1.20 to 1.30
increase to min Wp. From low lunar
orbit, the min theoretical Wp is
determined by a dV that is lunar orbit speed,
adjusted by a small gravitational loss.
On Mars, landing is
quite different from launch, because of
an atmosphere that, while quite
thin, is substantial enough for
hypersonic aerobraking. One comes out of
hypersonics at about local Mach 3 (~0.7 km/s),
at a very low altitude compared to Earthly entries: something nearer 5 km or less, for large multi-ton vehicles. Altitude depends sharply on ballistic
coefficient: beyond about half a ton to
a ton of entry mass of ordinary density and size, end-of-hypersonics altitude is just too low
for any effective use of parachutes. You
are but seconds from impact.
If small enough to use a chute for additional
deceleration, the final velocity
downward with the chute is high subsonic on Mars, roughly ~0.2 km/sec. That is the theoretical dV for the touchdown
burn. It needs to be factored-up for a
hover/maneuver allowance to avoid hazards.
An unmanned probe might only need a factor of 1.2 or so. A manned item probably ought to use a factor
in the 1.4 to 1.5 range.
If too large for chutes,
that means retro-propulsion must start as the hypersonics end, or even sooner. The theoretical dV to “kill” is that ~0.7
km/s end-of-hypersonics speed, but there are altitude effects and a big hover
allowance needed to hit the target location and avoid obstacles. My rough guess is a factor of about 1.4 or
1.5 applied to the min theoretical 0.7 km/s dV for landing.
In either case, all
the rest of the speed-at-entry is “killed” by the hypersonic aerobraking, which is true for both entry-from-orbit, and for direct entry from deep space. Only the heat protection requirements
differ, direct entry being considerably
more stressful.
On Earth, one comes
out of hypersonic aerobraking at about the same Mach 3 speed (~0.7 km/s), but at much higher altitudes: perhaps 40 to 50 km. For smaller objects like space capsules, parachutes are quite practical, and have long been used. Depending on the size, and whether a water or dry-surface
landing, there may (or may not) be final
small touchdown burn requirement. That
is beyond scope.
There is a limit to the size of the vehicle that can use a
chute on Earth. Above it, your choices are (1) a winged vehicle making
horizontal landings like the Space Shuttle,
or (2) pitching-up hard (more than 90 degrees) during the descent, to a tail-first retro-propulsive landing, like nothing we have seen before, except in science fiction. The vehicle must be able to withstand
dead-broadside air loads to do that!
For the winged case,
there is no landing burn. For the
retro-propulsive landing option, a wild
guess would apply a factor of 1.5 to 2 times the 0.7 km/s end-of-hypersonics
speed, as the “adjusted landing dV”
requirement.
Calculating Jet Blast
Effects
This may come up during engine testing on the ground: blast screens for safety purposes. Basically,
whatever the nozzle thrust is, of
whatever device is producing that thrust,
that is an accurate and convenient number for designing the strength of
any blast screens around the test.
As illustrated in Figure 5,
here on Earth, the plume is
finite, because the atmosphere gradually
decelerates it with fluid friction and mixing.
If such a test were run on an airless world (such as the moon), there would be no plume deceleration, and the picture would be the vacuum case
illustrated in the figure.
Figure 5 – Calculating Jet Blast Effects
There are multiple correlations available for calculating
the extent of jet blast effects here on the Earth’s surface. Those details are beyond scope here.
A Note on Solid
Rockets
The ballistics of thrust coefficient, nozzle design, and massflow used for the liquids here, also applies directly to solid rockets, but there is much more “interior ballistics”
to deal with, in the solids. That is beyond scope here. Further,
in solids, the inerts figure
differently, because the typical application
is quite different (usually a strap-on booster). I may at some time post an article about
solids, but nothing is in the works
right now.
Related Articles
I have been doing this sort of configuration-sizing design
feasibility analysis, in one or another
form, for a very long time. I’ve been doing it since I first went to work
after finishing graduate school, back in
1975. Even after changing careers to
mostly teaching in 1995, I still do
it. I’ve been fully retired since 2015, but I still do this sort of analysis for my
own projects. I’ve had this blogspot
site since 2009.
There are many related articles posted here on this
site, most of which are listed
below. Use the navigation tool on the
left: click first on the year, then the month, then the title. The “fundamentals” list has things like
atmosphere data, a ballistic entry
model, and orbital velocity requirements
covered.
The “studies” list has articles where I conducted vehicle
configuration studies using some or most of these techniques. There is also a related “costs” list, which has little to do with estimating
performance, but everything to do with
making decisions about what is affordable and what is not.
I did not include those studies where I looked at ramjet assist. There are a lot of them. But that is a whole other topic area, quite different from rocket propulsion. There is little that can be done with ramjet without a cycle analysis computer code. Not a CFD code with real-gas effects built-in, just compressible fluid flow with ideal gas models. And not just a simple pressure-ratio model like those in the textbooks that work so very well for gas turbines. The cycle code is something in between those extremes, and I do in fact write my own.
If you want to see the ramjet stuff, find one,
then click on search keyword “ramjet”.
The site will show you only those articles sharing that keyword. The
latest one is 12-10-16 “Primer on Ramjets”.
I also did not include any of the articles addressing important
stuff like spacesuit technology,
radiation hazards, artificial
gravity, or construction techniques on
Mars, etc. While crucial, those generally have nothing fundamental to
do with vehicle configuration sizing and design feasibility analysis. You can isolate them pretty easily by
locating any one listed here dealing with launch, Mars,
or the space program, and then
clicking on one of the search keywords “space program”, “Mars”,
“launch”, or “spacesuit”. It will show only those articles with the
keyword you clicked.
These lists have the date and title. That’s enough to use the navigation tool quickly and easily.
“fundamentals”
8-2-12
"Velocity Requirements for Mars Orbit-Orbit Missions"
7-14-12
"Gravity Data on All the Interesting Worlds"
7-14-12
"“Back of the Envelope” Entry Model"
6-30-12
"Atmosphere Models for Earth, Mars, and Titan" (this is the Justus & Braun
stuff)
6-24-12 "Mars Atmosphere Model (Glenn RC)" superseded by 6-30-12 posting
“studies”
8-6-18
"Exploring Mars Lander Configurations" (most recent stuff)
4-17-18
"Reverse-Engineering the 2017 Version of the Spacex BFR" (best version)
10-23-17
"Reverse-Engineering the ITS/Second Stage of the Spacex BFR/ITS
System"
3-18-17
"Bounding Analysis for Lunar Lander Designs"
3-6-17
"Reverse-Engineered "Dragon" Data (about as good as anything publicly available)
5-28-16
"Mars Mission Outline 2016" (most recent version,
and the best so far)
11-26-15
"Bounding Analysis: Single Stage to
Orbit Spaceplane, Vertical Launch"
12-13-13
"Mars Mission Study 2013"
10-2-13
"Budget Moon Missions"
9-24-13
"Single Stage Launch Trade Studies"
8-31-13
"Reusable Chemical Mars Landing Boats Are Feasible"
12-13-12
"On the 12-12-12 North Korean Satellite Launch"
9-3-12
"Using the Chemical Mars Lander Design at Mercury"
8-28-12
"Manned Chemical Lander Revisit"
8-12-12
"Chemical Mars Lander Designs “Rough-Out""
7-19-12
"Rough-Out Mars Mission with Artificial Gravity"
12-14-11
"Reusability in Launch Rockets"
7-25-11
"Going to Mars (or anywhere else nearby) the posting version" (Mars Society paper)
1-8-11
"Update to Manned Mars Mission Concept"
12-20-10
"Feasibility of a Manned Mars Exploration Mission Concept"
11-29-10
"Fast Transit To and From Mars"
11-26-10 "Mars in 39 Days One-Way"
“costs”
2-9-18 "Launch
Costs Comparison 2018" (latest
and best version so far)
8-7-15
"Access to Space: Commercial vs
Government Rockets"
9-13-12
"Revised Launch Cost Update"
5-26-12
"Revised, Expanded Launch Cost Data"
1-9-12 "Launch Cost Data"
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