Frontal Thrust Density in Rockets

The design liftoff thrust performance of rocket vehicles is something I,  and many on-line correspondents,  are interested in,  and have discussed extensively.  I have long maintained that solid rockets offered the highest possible liftoff frontal thrust density,  but I never had the analysis and numbers to back that assertion up.  Now I do.

I started with a rather simple pair of point analyses,  using perfect expansion from a chamber pressure to sea level to size the nozzle expansion ratio Ae/At and sea level thrust coefficient C_F.  It turns out that the variable of real interest is the ratio of the sum of nozzle exit areas to the vehicle base area Ae/Abase.  I did this for a solid design similar to a Shuttle SRB at 1000 psia,  and for a liquid operating at up to about 4000 psia,  as in SpaceX’s Raptor methane-oxygen engines.  For either,  a gas specific heat ratio γ = 1.20 is pretty close to the best available model.

The first thing that “jumped off the page at me” was the single-engine vs multiple-engine configuration trade that one must deal with,  in liquid stages.  This not only determines how far the engine bell or bells extend aft of the stage,  it also determines the effective Ae/Abase ratio,  as Figure 1 indicates.  For this initial calculation,  I just guessed Ae/Abase = 0.75 for a 3-engine configuration.  Of course Ae/Abase is 1 for the solid,  but at a Shuttle SRB chamber pressure of only 1000 psia in a 10 ft diameter size.

Be aware that extra spacing between the bells of a multi-engine installation is required of liquid engine designs that use thrust vector control (TVC).  The imposition of a TVC requirement thus lowers Ae/Abase further than the geometry of simple nested circles would suggest. 

Figure 1 – Initial Point Analysis

There’s enough variable interplay going on here to confuse these results.  So I ran a little study of these results,  versus chamber pressure Pc variations all the way from 500 to 4000 psia,  expanding perfectly to a fixed sea level backpressure of P_SL = 14.696 psia. Perfect expansion is when expanded pressure equals backpressure.

For these calculations,  I used γ = 1.20,  and η_KE = 0.99,  with the perfectly-expanded pressure ratio PR = Pc/Pe = Pc/P_SL.  In compressible flow,  the expanded temperature ratio TR = Tc/Te = PR^{(γ – 1)/γ}.  Expanded Mach number Me = [(TR – 1)*2/(γ – 1)]^0.5.  The expansion area ratio Ae/At is then (1/Me)*[TR*2/(γ + 1)]^{0.5*(γ + 1)/(γ – 1)}.  The resulting vacuum thrust coefficient C_Fvac = [(Ae/At)/PR]*(1 + γ*η_KE*Me²).  The backpressure correction term is (Ae/At)*P_SL/Pc,  which subtracts from C_Fvac to create the sea level thrust coefficient C_F.  The expanded pressure term is already in C_Fvac.

Once you know the sea level C_F,  you can easily estimate the thrust per unit exit area as F_th/Ae = C_F*Pc*At/Ae.  Multiplying that by the Ae/Abase ratio gives you F_th/Abase = C_F Pc (At/Ae) (Ae/Abase),  which is the frontal thrust density that the design can achieve.  Plots of this for the solid and the liquid are given in Figure 2.

Figure 2 – More Detailed Trends From Plots vs Pc

First,  notice that the frontal thrust density trend vs Pc of the solid is exactly the same as the max frontal thrust density of a 1-engine liquid,  since both have Ae/Abase = 1.  This is independent of the rocket specific impulse performance!  It only depends upon the expansion you get from Pc into a fixed sea level backpressure.  (In the real world,  the odds are against an existing engine design having an exit area exactly equal to the base area of the stage into which you want to install it.)

What I added here to the solid plot is the resulting trend of stage diameter vs Pc under the assumption that the same case material at tensile strength and thickness is used at all Pc’s.  That makes stage diameter inversely proportional to Pc,  because:  Pc ID = 2 σ t,  and ID ~ stage diameter.  The reference point is the approximately 1000 psia Pc,  at 10 ft stage diameter,  used in the Shuttle SRB’s.  2 σ t is a constant representing the best that can be done,  under these realistic assumptions.

I ran these trends for Ae/Abase = 1.0,  0.5,  and 0.25 for the liquids,  just to see the effects of Ae/Abase.  There is also a constraint here,  as to what the designer can really do.  The single-engine configuration that maximizes at Ae/Abase = 1,  leads to a very long expansion bell (of a very large engine),  protruding from the back of the stage.  Almost no designer does this in a first stage booster anymore!  Nearly all such boosters are multi-engine now,  driven to be so,  in order to get a much shorter bell protrusion length,  plus some level of engine-out capability.

One should be aware that TVC requires significant extra space around the bell exit area,  for the bell to move without hitting other engines that may vector differently,  or not at all.  That simply means that geometrically-nested circles that fit the stage base diameter are a large over-estimate of Ae/Abase.  Thus,  the value of Ae/Abase = 0.5 (versus the 0.75 I assumed initially) is actually rather realistic for a multi-engine liquid stage employing TVC.

Under those circumstances of the various design constraints driving where you are on these curves,  a 5-foot diameter SRB offers around 32 or 33 Klb/ft² of frontal thrust density,  while a multi-engine liquid core with TVC,  at the more typical 3000 psia Pc,  only offers in the vicinity of 18 Klb/ft² frontal thrust density at Ae/Abase = 0.5.  The solid is some factor 1.78 larger in its individual frontal thrust density!  Add only two such SRB’s to a core,  and your liftoff thrust is 2*1.78 + 1-for-the-core,  or some factor 4.56 larger than the core alone.  (That is figured for equal strap-on and core diameters.) 

Add instead four of them,  for a factor 8.12 thrust increase over the liquid core alone.  That is one whopping increase in liftoff thrust!

If the solid were instead 10 feet in diameter and only 1000 psia,  its frontal thrust density is about 25 Klb/ft².  That is still some factor 1.39 larger than the multi-engine 3000 psi liquid with TVC.  Again,  adding only two such SRB’s to the liquid core produces a liftoff thrust some factor 3.78 larger than the liquid core alone.  Adding four would still be a whopping increase at factor 6.56!

The worst case scenario for the solid under these constraints is a 10 ft diameter solid at 1000 psia,  with the 25 Klb/ft² density,  versus a multi-engine liquid with TVC,  at the full 4000 psia Pc,  with 20 Klb/ft² density at Ae/Abase = 0.5.  That solid is still factor 1.25 “thrustier” than the liquid.  Adding only two of them produces a total lift thrust some factor 3.5 larger than the core alone.  Add four for factor 6.

Consider also that most launch vehicles designed to use solids as SRB’s,  have SRB’s with a smaller diameter than the core liquid stage.  That means the SRB can have a larger length/diameter ratio than the liquid core stage,  up to the same length as the liquid core (but also limited to being within the ballistics limitations of the solid propellant’s burn rate).  The extra length can partially compensate for the specific impulse deficit that solids have,  with respect to the liquids.

So,  under realistic design constraints,  yes,  the solids offer significantly more frontal thrust density potential than the liquids.  Which explains their popularity for added liftoff thrust among the many launch vehicles currently flying,  reusable cores or not.  Liquid strap-ons simply cannot add as much thrust as the solids can,  because liquid frontal thrust density potential is simply lower,  in any sort of practical design.  (Strap-ons do not need TVC,  but liquid strap-ons are still likely multi-engine). 

Final Comments

The main effect here has nothing to do with the type of propellant (liquids or solids).  It has mostly to do with the size of (the sum of) the nozzle exit areas relative to the size of the base area of the vehicle.

As it turns out,  with solids,  that area ratio is usually near 1,  because there is only 1 nozzle,  and it quite commonly has an exit area very near the base area.  That simply cannot happen with a cluster of liquid engines,  and it could only happen with a single liquid engine,  if one specifically designs it to be so.  That would actually be quite rare.

Chamber pressure driving the nozzle expansion also plays a large role,  and here the constraints upon that variable are again quite different for the solids versus the liquids.  This shows up in the thrust coefficients achievable,  which vary quite strongly with chamber pressure level.  For these purposes,  “chamber pressure” is that at the entrance to the nozzle,  not earlier in the cycle with liquids.

The solids have a trade between case diameter and design chamber pressure level,  because the steels from which the cases are made,  have a tensile strength that does not scale with size.  That is exactly why the larger sizes are associated with lower chamber pressure levels.  Liquids are more limited by the ability of the pumps to deliver adequate flow at extreme outlet pressures,  compounded by the “staged pressures” of the pump drive cycles.

As it turns out,  you get the highest frontal thrust density,  regardless of type,  when the exit diameter of the engine bell is just about equal to the base diameter of the vehicle.  That is common with solids,  and rare with liquids,  which is why the historical experience has been that solids have a higher frontal thrust density than liquids.

That frontal thrust density potential is higher at higher chamber pressures,  also quite the strong variable in the ranges where the technology is,  and has been.  Earlier in history,  when stage diameters were smaller and liquid chamber pressures were lower,  the solid typically achieved more of its potential frontal thrust density advantage over the liquid.  In recent times with larger diameters and higher liquid chamber pressures,  the disparity has lessened.

There is absolutely nothing innate about this.  It’s all in how the design constraints combine!

As for the effect of gas properties,  the specific heat ratio γ is pretty close to the same,  at about 1.20,  for all the solids and all the liquids.  Only γ,  and not molecular weight MW,  appears in the thrust coefficient C_F,  and the engine thrust requirement defines throat area via Fth = C_F Pc At,  without any gas properties. 

It is the chamber c* velocity that contains MW and chamber temperature,  as well as γ.  That reflects the propellant type,  and determines propellant massflow through the nozzle.  In solids,  that massflow determines the required burning surface and burn rate.  In liquids,  it determines how much pump you must supply.

Gas properties are also in the speed of sound equation,  similar to,  but not the same as,  chamber c*.  One would need that,  to find the exit velocity from the expanded Mach number,  but that velocity does not appear in the C_F equation;  only Mach number and Mach number-related ratios appear there.

We can safely conclude that gas properties (propellant type) do not determine the thrust capability achievable with a propellant,  they only determine the associated massflow (and thus specific impulse).