## 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)