## Thursday, August 23, 2018

### Back-of-the Envelope Rocket Propulsion Analysis

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.

Figure 1 – Where Thrust Comes From

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.

Figure 2 – Real-World Nozzle Plume Behavior

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.

At this point ,  one must take notice of the mass fraction constraint shown in the figure,  because quite often,  one starts this process with a loss-adjusted overall dV figure,  and uses it to set the weight statement by means of the mass fractions.  That mass fraction constraint is very hard:  the sum of the payload,  inert,  and propellant mass fractions absolutely must be unity.

Figure 4 – Performance Potential Via the Rocket Equation

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
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
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)
9-13-12 "Revised Launch Cost Update"
5-26-12 "Revised, Expanded Launch Cost Data"

1-9-12 "Launch Cost Data"

## Monday, August 6, 2018

### Exploring Mars Lander Configurations

I have corresponded on-line with many people who want to explore what might be feasible as a Mars lander vehicle.  I worked out a configuration sizing and performance analysis,  and coded it into a spreadsheet.   This uses nothing more than algebra and the rocket equation.  I used it to get these results in this article.  If you want a copy of this spreadsheet,  email me.  I would be happy to forward it to you.  This article,  among other things,  is a user's manual for that spreadsheet.  It addresses one-stage,  two-way,  reusable vehicles;  and two-stage,  two-way,  non-reusable vehicles.

There are 4 worksheets in the spreadsheet.  One ("misc"),  is where among other things,  you can guess realistic estimates of vehicle inert mass fraction,  by stage if multi-stage.  Two other worksheets address one-stage and two-stage vehicles.  If two-stage,  all of the descent is handled by one stage,  and all of the ascent by the other.  In all cases,  the same max payload is used for both ascent and descent calculations.

The fourth worksheet is an exploration of re-engining a vehicle with a different propellant combination,  at the same propellant total volume.  That proved not very practical,  because the oxidizer and fuel tank volumes must also change.  That is a core rebuild,  not just a re-engine effort.

If instead,  you want to know what the now-cancelled Red Dragon might have done at Mars,  I worked out a reverse-engineering analysis with minimal but realistic assumptions some time ago,  based on posted Spacex data.  It covers cargo Dragon,  crewed Dragon,  and Red Dragon.  That article is posted at http://exrocketman.blogspot.com as the article dated 3-6-17 and titled "Reverse-Engineered Dragon Data".

"Mars Landers My Way"

Here is a crude but well-in-the-ballpark way to estimate the size and performance of aerobrake/retropropulsion-type Mars landers from the payload masses they must carry.  These would be large vehicles operating between low Mars orbit and the surface.  End-of-aerobraking would be too low for using chutes.

Flight requirement conditions are 3.55 km/s orbital velocity,  0.70 km/s remaining velocity about 45 degrees downward at around 3-5 km altitude for heavy vehicles coming out of aerobraking entry,  about a 2% empirical "kitty" for gravity and drag losses applied to ideal delta-vee,  and a factor 1.5-to-2 scaleup on the min landing delta-vee,  for direct retropropulsive landing without chutes. That's about 1.05-to-1.4 km/s min to land.  An on-orbit rendezvous and maneuver "kitty" is modeled as a user input factor Frm (greater than 1) applied to the ascent delta-vee. This is only slightly greater than one,  if no orbital plane change is required.

Vehicle design requirements depend upon whether the vehicle is one stage or two.  If one stage,  for design purposes, the ascent and descent payloads are assumed the same at the user-input design maximum value.  If two stage,  the same payload assumptions are true,  plus stage 1 is sized for the descent delta-vee,  while stage 2 is designed for the ascent delta-vee.

The inert mass fractions for each stage are user-input values.  Caution should be applied to select reasonable values for designs that must include landing legs,  heat shields,  enough structural robustness to handle rough landings and whatever degree of reusability that the user intends,  and excess vehicle volume to enclose bulky,  low-density payload during aerobraking.  This volume may require pressurization,  and may even need to be compartmentalized under pressure.  It is recommended that these inert fractions be 10% or greater,  even for a simple 1-shot design.

The propulsion model is very simple:  a user-input value for the average delivered Isp out of the engines.  This should be a realistic value factored for real efficiencies and any off-angle mounting effects.  The effective exhaust velocity model is Isp x gc, for use in the rocket equation. Metric gc = 9.807 m/s^2.

Descent engine thrust sizing is based upon 0.7 km/s to be "killed",  along a slant path about 7 km long (assuming a 5 km altitude at 45 degrees),  at the vehicle descent ignition mass,  and Mars gravity (0.384 gee). This is for the total of the descent engines at near full thrust.  It is effectively a 3.6 standard-gee initial deceleration requirement applied to the max descent mass.  (This velocity change is factored-up for losses and maneuvering effects.)

Ascent engine thrust sizing is based upon a force ratio factor (3) applied to the local weight of the ascent ignition mass at Mars gravity (0.384 gee) to achieve about 2 Mars gravity's net upward acceleration.  It works out to about a 3 standard-gee requirement. Rapid accelerations ease control problems by taking advantage of pitch and yaw inertia.  If accelerated rapidly,  the vehicle does not have time to change attitude significantly.

For single stage designs,  the greater thrust requirement of the two,  is actually used.  For two-stage designs,  the thrust requirement (and user-input Isp) can be different for each stage.  Throttle-down capabilities and number of engines making up the thrust are not specified in this sizing procedure.

One-Stage Vehicle Sizing:

User inputs are max payload mass Wpay (kg),  specific impulse Isp (sec),  inert mass fraction Win/Wig,  landing velocity scale-up factor RV (min 1.5,  max 2),  and ascent rendezvous and manuever "kitty" Frm (recommend a number between 1.05 and 1.10 as long as no plane changes are required). This Frm factor applies to delta-vee,  not sized propellant mass.  A summary of results takes the form of an overall weight statement with overall mass ratio and total delta-vee shown.

Exhaust velocity Vex (km/s) = Isp (sec) x 9.807 / 1000
Required delta-vee capability dV (km/s) = 0.7 * RV + 3.55 * 1.02 * Frm (first term descent delta-vee,  second term ascent delta-vee)
Mass Ratio MR = exp(DV/Vex)
Propellant fraction Wp/Wig = 1 - 1/MR
Payload fraction Wpay/Wig = 1 - Win/Wig - Wp/Wig (infeasible if negative or zero)
Ignition Mass Wig (kg) = Wpay/(Wpay/Wig)
Propellant Mass Wp (kg) = Wig*(Wp/Wig)
Burnout mass Wbo (kg) = Wig - Wp
Thrust requirement Fth (KN) = larger of {Wig * 3 * 9.807 / 1000 or 3.6 * 9.807 * Wig / 1000} pending number of engines and which are used

Two-Stage Vehicle Sizing (stage 1 descent,  stage 2 ascent):

User input max payload mass Wpay (kg),  specific impulses Isp1 and Isp2 (sec) for stages 1 and 2,  inert mass fractions Win1/Wig and Win2/Wig for stages 1 and 2,  landing velocity scale-up factor RV (min 1.5,  max 2),  and ascent rendezvous and maneuver "kitty" Frm. A summary of results takes the form of descent and ascent weight statements,  with stage mass ratios and delta-vees shown.

Exhaust velocities Vex1 and Vex2 (km/s) = Isp (sec) x 9.807 / 1000 for stages 1 and 2

For the ascent stage required delta-vee capability dV2 (km/s) = 3.55 * 1.02 * Frm
Ascent stage mass ratio MR2 = exp(dV2/Vex2)
Ascent propellant fraction Wp2/Wig2 = 1 - 1/MR2
Ascent payload fraction Wpay/Wig2 = 1 - Win2/Wig2 - Wp2/Wig2
Ascent ignition mass Wig2 (kg) = descent payload mass = Wpay/(Wpay/Wig2)
Ascent propellant mass Wp2 (kg) = Wig2*(Wp2/Wig2)
Ascent burnout mass Wbo2 (kg) = Wig2 - Wp2
Ascent thust requirement Fth2 (KN) = Wig2 * 3 * 9.807 / 1000

For the descent stage required delta-vee capability dV1 (km/s) = 0.7 * RV
Descent stage mass ratio MR1 = exp(dV1/Vex1)
Descent propellant fraction Wp1/Wig1 = 1 - 1/MR1
Descent payload fraction Wpay/Wig1 = 1 - Win1/Wig1 - Wp1/Wig1
Descent ignition mass Wig1 (kg) = Wig2/(Wpay/Wig1)  recall that the ascent stage ignition mass is the payload for the descent stage
Descent propellant mass Wp1 (kg) = Wig1*(Wp1/Wig1)
Descent burnout mass Wbo1 (kg) = Wig1 - Wp1
Descent thrust requirement Fth1 (KN) = Wig1 * 3.6 * 9.807 / 1000

User inputs include the average bulk density of the payload materials,  and an effective packing fraction for how tightly-together the individual items are spaced.  A minimum value of the payload bay volume is then based on the sized max payload mass:  vol (cu.m) = Wpay (kg) / (1000 * specific gravity * packing fraction).  This is true for any design.  The vehicle dimensions depend upon what specific configurations are to be analyzed. That is beyond scope here.

One-Stage Vehicle Performance Estimates

The nominal design performance estimates presume the same max payload in ascent as in descent.  The vehicle is presumed to be fueled on-orbit about Mars,  and must return to orbit for any refueling.  The descent and ascent delta-vees are used separately to determinine mass ratios,  to define the individual descent and ascent propellant masses,  and thus the ascent ignition mass.  Results take the form of a two-burn weight statement,  with two mass ratios and two delta-vees shown,  and an overall payload fraction.  Also shown are the mass percentages of propellant capacity used in each burn.  Because thrust is defined from initial descent ignition mass,  this ignition mass value is taken to be a design limit.  Because the propellant tankage is of fixed size,  the design total propellant mass is taken to be another design limit.

If used for a one-way descent,  there is no need to carry the ascent propellant,  and its mass equivalence may be added directly to max payload.  This stays within both the ignition mass and propellant mass design limits.  The results are reported as a one-burn mass ratio and delta-vee,  with percent of propellant capacity that is actually loaded also shown,  plus an overall payload fraction.  This payload is very much larger than the nominal design value for two-way operation.  To recover and re-use the vehicle,  there must be propellant refueling capability on the Martian surface. Using user inputs for payload mean density and packing fraction,  a min payload containment volume is also defined and shown.  This payload is the largest for the one-way descent scenario.

If propellant refueling capability is available on the surface,  then a one-way ascent also becomes possible at somewhat-increased payload.  One will hit the max ignition weight limit before hitting the design propellant capacity limit,  for any properly-sized design.  Using the ascent delta-vee and max ignition mass,  the burnout mass can be determined,  and with it,  plus the inert mass of the design,  can also be determined the ascent propellant mass,  and the total payload that can be carried,  which will exceed the two-way design value.  These results are presented as a one-burn weight statement,  with mass ratio,  delta-vee,  and percent of propellant capacity actually loaded.  Payload mass fraction is also shown. Such a vehicle could be refuelled on-orbit,  and re-used.

Overall,  the nominal one-stage two-way vehicle configuration design result is a very large vehicle that carries a rather small payload on the design two-way mission.  This very same design can carry enormous descent cargo if used one-way,  then refueled on the surface.  It also gets a significantly improved ascent cargo,  if operated one-way from the surface,  having been fueled there.

Typical One-Stage,  Two-Way,  Reusable Results

Figure 1 shows where some of the numbers used in the analysis came from.  The 3 ton payload presumes a man with suit and spares,  and a month's open-cycle food, water,  and oxygen,  all masses half a ton.  A crew of 3,  plus half a ton of instruments and equipment,  and a one-ton rover car,  totals to 3 tons of payload.

Figure 1 -- Design Analysis Assumptions for One-Stage Two-Way Vehicle

This figure includes a result from a semi-organized way to guess a ballpark inert mass fraction for the vehicle structure.  That figure also shows the basic assumptions made about the lander configuration.  There is a center cylindrical core containing the propellant tanks,  a sealed engine compartment,  and a crew control cabin that could also be an abort-to-surface escape capsule,  somewhat similar to the Red Dragon concept from Spacex.

Around this core is a conical volume containing the ascent or descent cargo.  There are a heat shield and extendible landing legs attached to the cargo deck,  that in turn is the frame tying the vehicle together.  The conical cargo volume is pressurizable in a compartmented sense,  and can serve as considerable crew habitation volume.  It is deliberately sized to contain a large amount of low density cargo at low packing fraction.  The overall shape resembles the old Gemini capsule.

Because the engine compartment is otherwise sealed,  the engines can fire through openings in the heat shield without any closures during aerobraking,  since there is no through-flow into a dead-end passage.  A static gas column is the best insulator of all.

Figure 2 shows the sized results at 330 sec specific impulse (a typical figure for a fairly large expansion bell,  using storables like MMH-NTO).  The most notable result item is the low payload mass fraction,  because of the high inert fraction more-or-less inherent with this kind of design.  A higher-specific impulse propellant combination (such as liquid methane - LOX) would offset the inert mass fraction effect some,  and push the vehicle to a higher payload fraction. Liquid methane - LOX is thought to be producible on Mars,  given an adequate source of water ice. At 330 sec (storables) and 0.2 inert fraction,  payload is just over 2.1% of ignition weight.  At 360 sec Isp (more like liquid methane-LOX),  this rises to just over 8.0%.

Figure 2 -- Sizing and Nominal Performance Results for One-Stage Two-Way Vehicle

Figure 3 shows the descent and ascent performance possibilities obtained with surface refueling of this very same vehicle,  using the sized ignition mass and sized propellant quantity as hard limits.  Ignition mass sized the engine thrust,  which requires a thrust increase if it grows.  The propellant tanks are of fixed volume,  which renders max loaded propellant mass a constant. Note the remarkably-large payload mass fraction available in this vehicle if operated one-way in descent,  assuming surface refueling for re-use.  The effect during ascent is much smaller,  but still considerable,  for a surface-fueled,  one-way ascent. For 330 sec and 0.2 inert,  the descent payload fraction rises from 2.1% to over 52%,  and the ascent payload fraction rises from that 2.1% to about 9.2%.

Figure 3 -- Performance of One-Stage Two-Way Vehicle Operated with Surface Refuelling

Overall, the conclusion here is that,  given the "right" propellants compatible with surface refueling, this rather limited two-way payload capability dramatically grows into a very versatile one-way capability,  made reusable in that mode with that surface refueling. This kind of design approach offers great promise of long and versatile service life without any need to develop new vehicles.

One final observation:  it might be wise to upsize the design thrust per engine in a multi-engine cluster,  in order to cover an engine-out situation.  This requires increasing further the engine turndown ratio,  plus shutting down even more engines later in the trajectory, in order to stay within a nominal 3-4 gee ride limitation.

Two-Stage Vehicle Performance Estimates

These estimates are easier and less extensive.  The vehicle is fundamentally non-reusable.  The ascent stage and mission payload are together the payload of the first stage (descent stage).  Each stage is already a one-way,  one-use item.  There is only nominal design performance at max payload to evaluate for each stage.  You do the ascent stage first at the ascent delta-vee,  then use its ignition weight as the effective payload for a descent stage evaluated at its delta-vee.  These numbers are computed as part of the sizing process.  They get reported as a two-stage combined weight statement,  with mass ratios and delta-vees for the two flight segments.  Overall payload fraction is also shown.  Propellant tanks are always filled to capacity in a vehicle delivered to low Mars orbit.

The only variation would be to replace the ascent stage with a simple payload pod of equivalent total mass to the ascent stage.  This is the only way available to increase the payload mass deliverable by a descent stage to the surface.  The disadvantage is that this delivery of increased cargo delivers no ascent stage at all.

Typical Two-Stage,  Two-Way,  One-Shot Results

The assumptions and configuration approaches for the two-stage,  two-way,  one-shot design are given in Figure 4.  Note that there is no backshell for the hypersonic entry.  Instead,  exposed structures need a thin coat of an ablative,  perhaps Avcoat.  The plasma is quite hot,  but the flow velocity and its heat transfer scrubbing action are much reduced,  compared to the windward side of the heat shield.  The descent stage propellant tanks are tapered,  so as to be out of direct windblast,  even at fairly high pitch or yaw angles during hypersonic entry.

Figure 4 -- Assumptions and Source Data for Two-Stage One-Shot Vehicle Design

What obtained was a very much smaller lander-and-ascent vehicle,  as described in Figure 5,  which gives sized data and nominal performance,  plus a max cargo delivery variant without an ascent stage. Replacing the ascent stage with a cargo pod significantly increases the deliverable payload mass.  I used a 10% inert fraction for this cargo pod,  on the assumption that it be designed as pressurizable and compartmentalizable.  That way,  additional habitable space could be brought down by this variant.  The "standard" form with the ascent vehicle has a pressurizable cargo bay,  but it is quite small,  too small for an extended visit without some augmentation sent down by other means.

Figure 5 -- Nominal and Max-Cargo Variant Performance for the Two-Stage One-Shot Lander

There is one design possibility here that could possibly reduce inert weight further,  something that directly increases payload fraction in any scenario.  That would be to use the ascent engines as part of the descent engine count.  That would reduce the number of engines to be incorporated in the design,  but it would require a switchable propellant interconnection between the two stages,  one that must be disconnected entirely,  before the ascent stage can lift off.  Whether the engine count reduction reduces inert weight more than the propellant interconnection hardware increases it,  is an unknown that remains to be seen. That level of detail requires real detail design,  which this configuration study is not.

One-Shot Versus Reusable Effects

The one-shot,  two-stage vehicle carries the same payload as the one-stage reusable vehicle,  but is very,  very much smaller overall.  In part,  this is the staging effect,  which for this application is going to be inherently non-reusable.  When required delta-vee is high,  the propellant mass fraction is also very high,  with an exponential dependence.  Since propellant mass fraction,  inert mass fraction,  and payload mass fraction must add linearly to unity,  then for a demanding delta-vee,  often the payload fraction is quite tiny,  or even infeasible as a negative number. Staging is a way to reduce the required delta-vee on each portion of the vehicle,  so that feasible payload fractions result.

With staging and inherent non-reusability,  there is no need to build the structures capable of withstanding the rigors more than once.  That leads to lower inert fractions,  reflecting the more fragile structure.  That is why the stage inert fractions in the two-stage non-reusable vehicle are lower than the inert fractions in the one-stage vehicle that is intended to be reusable.  Lower inert fractions also lead to larger payload fractions,  for any given delta-vee and its corresponding propellant fraction.  The two effects together produce the great disparity in ignition masses for the two designs.

Propellant Selection Effects

The propellant combination assumed for both configuration designs was the well-known storable combination MMH-NTO.  Storables require simple,  lightweight tankage,  and are good for long times between firings (days,  months,  even years).  Cryogenics can only use simple,  lightweight tankage if the time between loading propellant and its use is relatively short (hours-to-days only).  Otherwise,  they need insulated tanks or Dewar flasks,  and perhaps powered cryocooler rigs.  This increases inert mass fractions,  a choice which has since been added to the spreadsheet's "guess-the-inert-fraction" feature.

For nozzles firing into vacuum, or near-vacuum as is the Martian atmosphere,  expansion bells can be large,  and the specific impulse higher than at sea level on Earth.  That is where the 330 sec value used in the design study came from.  This value of specific impulse is easily converted to a good approximation of the exhaust velocity,  for the purpose of doing configuration studies with the rocket equation.  For a real detailed design,  you need to do real engine-nozzle ballistics with a real engine design. That is out-of-scope here.

Higher specific impulse is higher exhaust velocity,  leading to smaller propellant fraction for a given delta-vee demand. The one-stage reusable vehicle configuration is right at the "hairy edge" of feasibility with 330 sec of specifc impulse,  with the result of a very low payload fraction.  For a given payload requirement,  that makes the vehicle ignition weight very large.

MMH-NTO is not a combination currently contemplated as a possible thing to manufacture on Mars from local resources.  The mild cryogen combination liquid methane-LOX is a good candidate for local manufacture on Mars.  Its vacuum-bell specific impulse will be nearer 360 sec.  That is a significant increase in specific impulse and effective exhaust velocity over the storables,  leading to a significant decrease in required propellant mass fractions.  For the same inerts otherwise,  this could be a significant increase in payload mass fractions.

The first inclination is to try this higher specific impulse in both designs.  However,  mission practicalities say otherwise.  The two-stage one-shot lander is sent from Earth to Mars orbit,  or to a direct entry.  The journey there is months long.  That is fine for the storables,  but not for the mild cryogens.  Inert mass fraction must increase for the insulated Dewar tankage and cryocoolers required for the months-long voyage to Mars.  Plus,  the design is inherently one-shot.  It will never be refueled for any reuse,  precisely because it is two-stage.  Thus the storable design is simply the better choice for that application.

The one-stage reusable design is similar,  in that at least initially,  the right choice is storables,  because of the months-long journey to Mars.  The design as it is,  simply cannot afford the weight of insulated Dewars and cryocoolers,  even with the higher specific impulse.  When I plug in 25% inert and 360 sec specific impulse,  payload fraction drops to zero.  Such a vehicle would have to be shipped empty to Mars,  and its propellant supply shipped separately,  until propellant production is established on the surface of Mars.  That is just not very practical.  An empty vehicle is no good for direct entry,  and cannot maneuver itself,  even if separately braked into Mars orbit.

However,  once propellant production is actually established on Mars,  vehicles with simple lightweight tankage,  previously operating on storables based from Mars orbit,  could be landed and re-engined with liquid methane-LOX engines,  and operated for relatively short flights from the surface of Mars reusably,  and locally refuelled.  In that case,  the specific impulse is now 360 sec,  with the inert fraction still only about 20%.   Unfortunately,  the fuel to oxidixer volume ratios are wrong for this,  being about 2-to-2.7 by mass for the storables,  and in the vicinity of 3.25 for the mild cryogens.  What that really says is that you design the thing to use liquid methane-LOX from the outset,  or else you design it to use MMH-NTO storables from the outset.  Re-engining is not really an option.

Not only the engines,  but also the tankage volume ratio in the center core,  must be substantially altered to allow the use of mild cryogen propellants made on Mars.  That is a major structural design change.  This is not a very practical thing to attempt.

Overall Conclusions

Either design approach will work.

If one-shot,  use a two-stage vehicle using storable propellants.  It can land about 3 tons for a 22.2 ton vehicle,  using those storable propellants,  from either low Mars orbit or direct entry.  Replacing the ascent stage with a cargo pod,  it can land 12.4 tons in that same 22.2 ton vehicle.  The ascent stage can take that same 3 tons back to Mars orbit.  Available cargo volume convertable to habitation space is quite limited,  being around 20 cubic meters nomimal,  and only 82.9 cubic meters if the ascent stage is replaced by a cargo pod.

If one-stage and reusable,  and refuelled in Mars orbit from a storable propellant supply kept there,  the vehicle can land about 3 tons in each flight,  in a vehicle massing 269 tons at ignition.  It can carry the same 3 ton payload back to Mars orbit.  These are the same storable propellants as the two-stage non-reusable configuration.

Replacing the ascent propellant with cargo as a one-way descent trip,  means that 140.7 tons can be landed without the possibility of reuse. Similarly,  if refuelled with storables on the surface somehow,  it can bring 24.7 tons of payload back to Mars orbit.  The same 269 ton ignition mass limit applies to the two-way and one-way cases.  Cargo volume potentially habitable is over 469 cubic meters,  based on the cargo descent with no ascent.

There is no scenario where re-engining the two-stage one-shot vehicle to use liquid methane-LOX produced on Mars makes any sense.  That is because nothing about this design is reusable in any way.  That eliminates the point of any surface refuelling.

Re-engining the one-stage reusable vehicle to use liquid methane-LOX produced on Mars also makes no sense,  because the oxidizer and fuel tank volumes are all wrong in the very core of the vehicle.  This is a major redesign and rebuild problem,  not just an engine replacement problem.

Finally,  it makes no sense to design the vehicles from the outset to use liquid methane-LOX propellants,  because of the increases in inert fractions to counter boil-off losses during the months-long transit to Mars,  plus the fact that,  initially,  such propellants are simply not available at Mars.  Those increases in inert fraction negate the gains in propellant fraction at the higher specific impulse.  In point of fact,  payload fraction gets zeroed.

Whichever approach you choose,  go with storable propellants such as MMH-NTO.  Personally,  I like the one-stage reusable design,  because of what it can do if used as a one-way descent vehicle.  The tankage is just twice as lightweight.  I also like two-way reusability based from Mars orbit,  if sufficient propellant can be shipped there from Earth.