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kbd512

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Registered: 2015-01-02

Posts: 7,853

SSTO Minimum Isp to Achieve Orbit

One of the most interesting aspects to note about what total impulse, thrust-to-weight, and materials strength math tells us about what will or won't work, with respect to attaining orbit using a single stage, is the following:

1. Any viable propellant candidates for providing 9.5km/s+ of total impulse will greatly resembles densified LOX/RP1, as far as total propellant volume is concerned.

Explanation:
Markedly higher-Isp propellants such as LOX/LH2 have exceptionally low bulk propellant densities that make them infeasible to enclose with sufficiently strong and stiff materials when the volume of propellants provides equal total impulse to densified LOX/RP1.  LH2 works so well for the upper stages of TSTOs specifically because the total volume of propellant is quite small relative to its booster stage, a lighter upper stage contributes to a higher booster stage burnout velocity or altitude or both, and the thrust-tow-eight ratio demanded of a viable upper stage is far lower than that of a booster stage.  In point of fact, thrust-to-weight ratios of less than 1 are perfectly viable for LH2-fueled upper stages.  For a SSTO that executes a single burn from sea level all the way to orbit, that is not a workable proposition, because gravity losses eat up all available payload.

If none of that makes any sense, then think about it this way:
If your propellant's bulk density (oxidizer plus fuel) is not either nearly as dense or (ideally) denser than water, then your tank volume grows at a faster rate than the strength of the materials used to enclose it and support its mass and the aero loads acting upon it.  SSTO viability is entirely dependent upon dry / structural mass fraction and engine performance per unit weight.  If the low density of the selected propellant combination dramatically increase the mass of the propellant tanks, then it's probably not viable or at the far edge of feasibility.  Even if you can make LOX/LH2 work, it will never be as performant as a denser propellant combination that allows for smaller / stronger / stiffer structures, because your vehicle suffers from a doubled performance penalty on dry mass- much larger propellant tanks and much lower engine thrust-to-weight.  The combination of those two factors is catastrophic to payload performance.

2. Sea level thrust-to-weight ratio (TWR) of viable engine candidates must meet or exceed 150:1, and preferably 200:1.  Higher is always better, so long as the engine remains intact over the duration of the burn.

Explanation:
Apart from not exceeding 1/3rd of the vehicle's dry mass fraction, high-TWR engines also provide the thrust required to accelerate quickly to minimize gravity losses by reducing the time spent clearing the lower atmosphere to complete the gravity turn and begin accelerating perpendicular to the surface of the Earth.  The longer your vertical thrusting period, the higher your gravity losses.  There is no way around that problem.  This applies equally to both TSTOs and SSTOs, but is of even greater importance to a SSTO because it's acting upon a greater vehicle mass for the duration of the ascent to orbit.

3. Tensile strength of viable candidate materials for constructing the propellant tanks, wings (if any), and any other structures with significant volume must be 500ksi or greater.

Explanation:
The very essence of SSTO is a highly efficient rocket wherein almost all of the mass of the rocket is propellant mass.  The greater the tensile strength and stiffness of your selected materials, the less material is required to enclose the propellants.  This strength-to-weight ratio effectively rules out the use of any metals as major structural materials.  High strength steel pins and shear bolts will be required.  Metals will be required for certain engine components.  Apart from that, propellant tanks, wings, and other structures with significant volume must be constructed from high strength and stiffness fiber composites.  Lower strength-to-weight ratios renders SSTO infeasible for large payloads.

If all 3 of those preconditions are met, then you have all the fundamentals in place for a viable SSTO vehicle suitable for carrying a large number of passengers to orbit.  You will never compete with TSTO on payload mass fraction, regardless of how strong and light your materials are, although at a certain point the propellant mass differential becomes negligible.  The trade space for a viable SSTO is, therefore, tightly constrained.  Selecting a propellant combination on the basis of Isp alone simply does not work, because it ignores the volume of propellant tank material required to enclose the propellant.  Propellant tank and engine mass dominates the dry mass fraction of the vehicle.  Even if you have TPS included for a reusable SSTO, since all modern TPS solutions are so light, your propellant tank and engine mass determines your vehicle's feasibility.

Some of this may seem very counter-intuitive, especially reduced propellant Isp leading to better performance, but it's all fundamental to building an efficient rocket.  Total vehicle dry mass greatly affects the payload performance of all rockets.  All attainable payload mass fractions will be less than that achievable using TSTO, especially if the same advanced composites technologies and integrated vehicle design are applied to TSTO, yet large payloads are indeed attainable using SSTO.

For an expendable SSTO, this is an interesting thought experiment:
1,927,600kg / 1,065m^3 of HTPB-1912 (69% AP; 19% Al; 12% HTPB; Isp 280s; Tc 3550K; ρ=1,810kg/m^3)

At 278.5s Isp, this provides 10,044m/s of ΔV Capability with a 50t vehicle dry mass fraction, same as 1,000m^3 of LOX/RP1.  It's 57.5% heavier than 1,000m^3 of densified LOX/RP1, but generates more thrust to compensate.

That amount of propellant is approximately equal to 4 Space Shuttle SRBs.  If the fiber was strong enough, then you could have an expendable SSTO using solids.  At 912psi maximum pressure in the SRB, a vehicle dry mass fraction of 2.59% would be challenging, to say the least.  However, if the fiber plus aerogel insulation allows the stage to complete its burn without rupturing, then it looks an awful lot like LOX/RP1 from a volume perspective.  Despite HTPB's markedly lower Isp, total propellant volume to achieve 10,044m/s is very near to being precisely the same as 1,000m^3 of densified LOX/RP1 with the same 50t vehicle mass fraction.  Apart from casing strength, the cost of the propellant and the speed with which it can be mixed and poured, as well as the difficulty of transporting something so heavy, becomes the limiting operational factor.  It would have to be a monolithic motor without multiple mixes and pours, meaning not segmented like a Space Shuttle SRB.

Only 65m^3 of propellant volume increase over the top of 1,000m^3 of densified LOX/RP1 propellants, a 6.1% volume penalty for solid propellant with an Isp 46 seconds lower than densified LOX/RP1 run through staged combustion engines, because HTPB-1912 has 161% of the bulk density of densified LOX/RP1.

Densified LOX/LCH4 with avg Isp of 341.5s, providing the same 10,044m/s of ΔV Capability provided by 1,000m^3 of densiified LOX/RP1, weighs 953,500kg and occupies 1,071m^3.

Densified LOX/LH2, with avg Isp of 409s, providing the same 10,044m/s of ΔV Capability provided by 1,000m^3 of densiified LOX/RP1, weighs 611,700kg and occupies 1,549m^3.

The mass of the LH2 propellant is not the problem with LOX/LH2.  LOX/LH2's mass figure is highly desirable.  It's almost exactly 50% of densified LOX/RP1, which is fantastic.  It's 54.9% greater volume is, unfortunately, a serious issue because somehow you must enclose 54.9% greater propellant volume using the same dry mass fraction, that material must survive LH2 exposure with acceptable permeation rates, and LH2-fueled engines have less than 1/3rd of LOX/RP1 or LOX/LCH4 sea level thrust-to-weight performance per unit of engine weight.  LH2 also requires higher internal tank pressurization to force-feed Hydrogen into the turbopump inlet.  The combination of those factors is catastrophic to overall vehicle performance.  The tank volume increase could be dealt with, but the low engine performance in concert with increased tank volume is not a solvable problem.  The payload mass gets cannibalized by engine and propellant tank mass.  That is why all SSTO studies show greatly reduced payload performance for LH2 when compared to RP1 or LCH4.  Something has to give, and that something will be vehicle dry mass fraction.

As you consider different propellant combinations, it becomes increasingly apparent just how "boxed-in" the viable tradespace is for candidate SSTO propellant combinations.  You're operating within a highly constrained environment.

tahanson43206

Moderator

Registered: 2018-04-27

Posts: 19,383

Re: SSTO Minimum Isp to Achieve Orbit

This post is reserved for an index to posts that may be contributed by NewMars members over time.

(th)

SpaceNut

Administrator

From: New Hampshire

Registered: 2004-07-22

Posts: 29,431

Re: SSTO Minimum Isp to Achieve Orbit

gw post on fuels

For all ... this post is about rocket sizing ... GW Johnson provides a spreadsheet and user manual

I'll put the text after the links...

Here is the spreadsheet:
https://www.dropbox.com/scl/fi/600qk03p … yqmag&dl=0

Here is the user manual:
https://www.dropbox.com/scl/fi/cuvcuddj … iijai&dl=0

Here is an addendum to the user manual:
https://www.dropbox.com/scl/fi/btl8weqm … 0xh7t&dl=0

And here is a pdf comparing SSTO and TSTO designs ...
https://www.dropbox.com/scl/fi/rjv90o1o … pg5ak&dl=0

And here is the text about the links:

(for NewMars members):

As you know,  I did a little simplified vehicle sizing spreadsheet “launch sizing.xlsx”,  responding to the latest round of SSTO talk on the forums.  These were simplified to one weight statement per stage,  no linked burn calculations.  That rules out complicated missions and reusability. But it works for simple all-expendables on a simple mission:  LEO.

I wrote up a user’s manual for it.  At that time,  there were only worksheets for SSTO and TSTO expendables.  The default examples in those worksheets were just approximate guesses.  I went back and adjusted those data by running the “r noz alt.xlsx” spreadsheet to get better Isp data,  and I revised the TSTO second stage acceleration requirement at its ignition.

Those data got written up in a document titled “Effects of Better SSTO and TSTO Modeling Data”.  It covers both LOX-LH2 and LOX-LCH4 SSTO’s,  and a LOX-LH2/LOX-RP! TSTO.  I was surprised to find the hydrogen SSTO had about the same payload fraction as the TSTO,  close enough to be a serious competitor in the expendable launcher market.  I was unsurprised that the methane SSTO turned out as poorly as it did.  We have known since before WW2 that low Isp requires more stages,  and that is exactly the dilemma where the methane SSTO falls.

I then added a third worksheet “tank volumes” to “launch sizing.xlsx” that estimates the tank volumes and stage dimensions for designs created in the “SSTO exp” and “TSTO exp” worksheets.  This is designed to work with engine data from “r noz alt.xlsx”,  beyond just Isp.  The engine configuration is thrust-rescaled to fit the required thrusts coming from the “SSTO exp” and “TSTO exp” worksheets.  And the r-value is used to split propellant into oxidizer and fuel masses.

This third worksheet is documented in an addendum document for the user’s manual for the “launch sizing.xlsx” spreadsheet.   Attached hereto are 4 of the 5 listed items (there is no need to transmit the original spreadsheet without the tank sizing worksheet):

1.        Excel spreadsheet “launch sizing.xlsx”  (original:  only worksheets “SSTO exp” and “TSTO exp”,  not attached hereto)

2.        User’s manual:  “User’s Manual Launch Sizing.pdf”  (cruder examples for “how-to”)

3.        Refined models by “r noz alt.xlsx”:  “Effects of Better SSTO and TSTO Modeling Data.pdf” (these are still only weight statement results,  no tank volumes or stage dimensions)

4.        Added tank sizing to “launch sizing.xlsx”:  “Addendum to User’s Manual.pdf”,  refined examples complete with tank volumes and stage dimensions (this version attached)

5.        Revised spreadsheet file “launch sizing.xlsx”:  added worksheet “tank sizing”

I suspect you will want to put these things in that dropbox thingie,  with links in a posting somewhere on the forums.  The text of this letter could go in that posting with those links.

What I found with these simple tools:

When I tried tank sizing for the methane SSTO,  I found I needed an engine cluster that simply would not fit behind a stage that met the “aerodynamically-clean” criterion of stage L/D = 6.  The much larger takeoff mass (and required thrust) prevents that.  It’s not just unattractive from a payload fraction standpoint,  it’s actually geometrically infeasible!   It would have to be shorter and fatter for the engines to fit,  and that is more drag,  more drag loss,  and a higher required mass ratio-effective dV,  which drives payload fraction even closer to zero.

With the advent of TSTO launchers that have partial reusability (SpaceX’s Falcons),  which demonstrably impacts cost even more than payload fraction,  I see no point to offering SSTO expendable launchers anymore.  They are feasible,  and competitive in terms of payload fraction with all-expendable TSTO’s,  but they cannot offer any reusability,  while the TSTO can (flyback first stages),  at the “cost” of somewhat-reduced payload fraction.  The expendable market will vanish!

And,  by the way,  when I do the engine ballistics thing versus altitude,  I can get a higher ascent-averaged Isp out of an “ascent compromise” fixed bell,  than from any conceivable free-expansion nozzle.  Meanwhile,  bell extensions,  which do work,  reduce engine thrust/weight considerably.  That adds extra inert mass,  costing payload fraction out of otherwise the same design.  Plus,  they add failure modes the fixed bell simply does not have.

GW

(th)

SpaceNut

Administrator

From: New Hampshire

Registered: 2004-07-22

Posts: 29,431

Re: SSTO Minimum Isp to Achieve Orbit

Hadn't heard anything I recognized as a failure.  Did hear one very short test about 3 days ago,  though.  That was probably it. 

Meanwhile,  I have comments relevant to SSTO concepts:

Many correspondents on these forums may make different assumptions than I did,  trying to bound what an SSTO can do.  It almost doesn't matter,  as long as your assumptions are in the ballpark,  your result will be,  too. 

Mission:  just assume something like 9.3 km/s of delivered dV covers getting to an orbital destination.  That would be for a low-drag configuration with 0.5+ net upward gee at liftoff. 

Engine:  you probably won't like my modest-technology engines,  with only a 2000 psia chamber pressure at the nozzle entrance,  only a 2:1 turndown ratio,  and about 5% dumped turbopump bleed gas. I've been using an ascent-compromise fixed bell with that,  one that is just barely non-separated at sea level and about 80% max chamber pressure.  That sacrifices some sea level thrust and Isp for better vacuum thrust and Isp,  and pretty much produces a good ascent-average Isp:  near 428 s with LOX-LH2.  That's a Vex = 4.197 km/s.

You may want to use a bit higher.  Fine.  Do so.  How about 440 sec?  Corresponds to Vex =  4.315 km/s.  We'll use that.

The mass ratio required to reach destination is exp(dV/Vex) = 8.630.  The corresponding propellant mass fraction Wp/Wig = 1 - 1/MR = 0.884,  or some 88.4% of the liftoff mass is propellant.  THERE IS NO WAY AROUND THAT.  My other numbers are right in that same ballpark.  Go look for yourself.  Go do the calculation for yourself.  I just gave you the equations.

That leaves 0.116 (or 11.6%) of the liftoff mass to be both inert mass plus payload mass.  THERE IS NO WAY AROUND THAT,  EITHER!  And my other numbers from the other studies are also right in that same ballpark. 

Just remember,  the sum of payload plus inert is your dry-tanks burnout mass.  Add the propellant to it for your filled-tanks ignition mass.  MASS MUST BE CONSERVED!  Cannot play games here!

Now if you have 11.6% allowance,  and you believe you can build an all-expendable stage with 4.5% inert,  which has been done,  then that leaves 7.1% max payload fraction.  My other studies fell in the 6-8% range,  too!  Surprise,  surprise.  LOX-LCH4 did NOT.  The Isp is too low,  pretty much regardless of what you assume for engine technology.  And LOX-RP1 is even a bit lower.  Those are SIMPLY NOT candidates for an expendable SSTO application!  But LOX-LH2 is,  and I got just about the same payload fraction out of a LOX-LH2/LOX-RP1 all-expendable TSTO at 7-8%. 

But,  based on what SpaceX has already done with its Falcons,  the all-expendable launcher market is going to disappear in the next few years.  Partial reusability is a bigger contributor to lower cost than payload fraction.  We've already seen that in practice.

You have ONLY 11.6% inert+payload allowance using LOX-LH2 in an SSTO,  give or take a single % point,  depending upon what you assume for your engine and nozzle technologies.  SpaceX's "Starship" inert has been running about 120 m.tons in configurations still not fully equipped like a production vehicle.  Up to now,  the max propellant load has been 1200 tons,  and no payload has been carried.  That's a stage ignition mass of about 1320 m.tons.  The inert is 9.1% at payload 0% and propellant 90.9%.  (As I already said,  mass must be conserved.)

However,  that design has yet to survive entry,  and I'm betting its inert mass is going to grow before it is successful.  And then they still have to land the thing.  So also would a reusable SSTO have to land in some way.  That’s going to increase not only the inert but maybe also the dV requirement.  Which means your non-propellant allowance is going to go down!

My best guess is that a truly reusable SSTO vehicle that can land by any conceivable means will have an inert mass fraction nearer 15% than any 10% figure.   And if that is truly the case,  your payload fraction is 2-to-4% NEGATIVE,  or worse,  which says IN NO UNCERTAIN TERMS that a reusable SSTO is infeasible,  even with LOX-LH2. 

Even if you could succeed in entering and landing at ~10% inert,  your payload fraction will be down under 2%. Partly or fully reusable TSTO's will outcompete you by carrying larger payload fractions than that.

GW

SpaceNut

Administrator

From: New Hampshire

Registered: 2004-07-22

Posts: 29,431

Re: SSTO Minimum Isp to Achieve Orbit

"Densified" (Oxidizer and Fuel Melting Points + 10C) Bulk Propellant Densities with Proper Mixture Ratios
LOX: 1,262kg/m^3; RP1: 867kg/m^3; LOX/RP1: 1,124kg/m^3
LOX: 1,262kg/m^3; LCH4: 438kg/m^3; LOX/LCH4: 890kg/m^3
LOX: 1,262kg/m^3; LH2: 77kg/m^3; LOX/LH2: 395kg/m^3

Staged Combustion Sea Level Isp, Vacuum Isp, Straight Average Isp for Fuel of Choice
LOX/RP1 (RD-180): 311s; 338s; 324.5s
LOX/LCH4 (Raptor 3): 327s; 356s; 341.5s
LOX/LH2 (RS-25E): 366s; 452s; 409s

Propellant Mass and ΔV Results with 50,000kg Vehicle Mass and 1,000m^3 of Propellants of Choice, using Straight Avg Isp
LOX/RP1 Propellant Mass - 1,124t; Vehicle Mass - 50t; Total Mass - 1,174t; ΔV Capability - 10,044m/s
LOX/LCH4 Propellant Mass - 890t; Vehicle Mass - 50t; Total Mass - 940t; ΔV Capability - 9,825m/s
LOX/LH2 Propellant Mass - 395t; Vehicle Mass - 50t; Total Mass - 445t; ΔV Capability - 8,768m/s

Demonstrated Engine Sea Level Thrust-to-Weight Ratios
RP1 (Merlin-1D): 184.5:1
LCH4 (Raptor 2): 143.8:1 (Raptor 3 has achieved 169.4:1 in testing)
LH2 (RS-25E): 53.4:1

Engine Percentage of Vehicle Dry Mass Fraction to Achieve 1.5:1 Thrust-to-Weight at Sea Level
RP1: Thrust Required - 1,761t; Engine Mass (184.5:1) - 9.545t; Percentage of Vehicle Mass: 19.09%
LCH4: Thrust Required - 1,410t; Engine Mass (169.4:1) - 8.323t; Percentage of Vehicle Mass: 16.646%
LH2: Thrust Required - 667.5t; Engine Mass (53.4:1) - 12.5t; Percentage of Vehicle Mass: 25%

RS-68A, RS-83, J-2X, Vulcain 2.1, and RD-0120 all have inferior thrust-to-weight performance when compared to the RS-25.  There may be a few experimental LH2 engines I don't have numbers for, but I doubt they were any more performant than our RS-25, else they'd have been touted for their performance, far and wide.

Why does thrust-to-weight matter so much for SSTO?

LH2 is 1/2 the density of the lightest hydrocarbon fuel, and all engines burning LH2 are 3X the weight of any hydrocarbon burning engine, because the pumps required to feed the engines are massive relative to pumps feeding much denser fuels.  LH2's 114s Isp advantage over RP1 simply cannot make up for the heavy engines and heavy fuel tanks required to provide equivalent payload performance.  Greater Isp is an enabler for SSTO, but heavy engines and excessively voluminous fuel tanks are not, because both of those problems eat so heavily into the available vehicle dry mass fraction, and therefore payload fraction.  It's counter-intuitive, or at least it was for me.

Using LH2 is stacking the deck heavily against SSTO, as well as TSTO if LH2 is used in the first stage.  LH2 with 70% CH4 gellant by weight, only gives up 50s of Isp to neat LH2, meaning 410s vs 460s, yet it's density (177kg/m^3 vs 77kg/m^3) and O/F ratio (4:1) is such that it out-performs neat LH2.  Your tank volume ends up between LH2 and LCH4 for equal ΔV, but you have much more propellant mass to work with.  I would imagine that thrust improves.  Presumably, you'd have much lighter turbopumps and therefore engines because they're sucking down a more viscous gel instead of a rarefied liquid with almost no mass per unit volume.  That means you can have lower tank pressures, too, which helps reduce tank mass, since tank pressurization is one of the major loads to contend with.

O = Oxidizer
F = Fuel
MR = Mixture Ratio
ρ avg = (ρO * ρF * (1 + MR)) / ((ρF * MR) + ρO)

I think that's the correct formula to use to determine propellant bulk density.  Gelled LH2's bulk density is about 567kg/m^3 if I did my math right.  I get the right numbers for LOX/LH2 at 6:1, so I presume it works.  My straight average Isp is 363s, and my ΔV capability is therefore 8,945m/s with a 50t vehicle mass and 1,000m^3 of propellant.

Relative to RP1, using LCH4, I need 7% more propellant tank volume for equal ΔV, but only 78% of RP1's propellant mass.  There might be a worthwhile trade here, presuming near-identical engine thrust-to-weight, which is presently 91.8% that of RP1, at sea level.  Careful evaluation of what engine dry mass does to total vehicle mass is still required.  It's probably not possible to make the engine as performant as a RP1 engine, but if it's close enough then that might be good enough.
Relative to RP1, using 70% gelled LH2, I need 39% more propellant tank volume for equal ΔV.
Relative to RP1, using neat LH2, I need 42% more propellant tank volume for equal ΔV.

Needless to say, increasing tank volume by 40% is disastrous to the vehicle's dry mass, because stiffness becomes the limiting factor, rather than strength.  Modern composites provide strength galore.  Their ability to resist deformation, an intrinsic property of the fiber and resin matrix, is only on-par with high strength steel (not by unit weight, obviously).  The low thrust-to-weight ratio of LH2 engines only adds insult to injury.

My propellant load is only 561.7t for LH2 vs 1,224t for RP1 for equal ΔV, but increasing volume by 40% is far more problematic than the strength of CFRP.  Isp is clearly much better with LH2, so that's good, but even if I use 90% of vacuum Isp for LH2, I can't do anything about the tank volume.  At best, my tanks are 38% larger than they are with RP1, using 90% of vacuum Isp for LH2.  I know that cube-square works in my favor with LH2, but is that enough to overcome a 40% volume increase?  That seems unlikely.  My thrust is also 1/4 to 1/3 per unit of engine weight, as compared to RP1, which would be fine if both the tanks and propellant were lighter to compensate, but LH2 propellant weight is only 1/2 of RP1 for equal ΔV, ignoring all other factors.  Basically, LH2 is no help to me at all.  I can't overcome the volume and thrust-to-weight problems with higher Isp.  Using RP1, I can devote more volume of material to enclose a sufficient volume of propellant capable of producing the required ΔV, because the propellant is 3.1X denser and the mass of my engines 3.46X lighter for sufficient thrust.

All those compounding factors make LH2 very unattractive for SSTO.  On top of that, there's a long term issue with LH2 permeation of composites.  RP1 doesn't have that problem.  LOX isn't even much of a problem.  LCH4 has free Hydrogen molecules within it, but not that much.  For a TSTO upper stage, LH2 is much more attractive, specifically because the booster has to accelerate its own weight plus the upper stage weight.  That's where LH2 really shines, because raw thrust is not what produces spectacular upper stage payload performance.  About 80% of a TSTO's liftoff mass is the booster stage, so throwing a lightweight upper stage on top, with stellar Isp, really makes it perform quite well for high orbits and escape trajectories.  It's a different class of problem, though.  You can't fix the booster thrust and propellant density problem with Isp.  Incidentally, nuclear engines suffer from the same problems in booster applications- their thrust is too low to make up for their weight.  Gravity losses dominate.  LH2 nuclear SSTO would be even worse than a conventional LOX/LH2 SSTO.

SpaceNut

Administrator

From: New Hampshire

Registered: 2004-07-22

Posts: 29,431

Re: SSTO Minimum Isp to Achieve Orbit

Seems that BE-4 engine and the Raptor are coming up quite strong.
Blue Origin's BE-4 Vs. SpaceX's Raptor Engine: What's The Difference Between Them?

In a while we may not how much more stable it will be.

kbd512

Administrator

Registered: 2015-01-02

Posts: 7,853

Re: SSTO Minimum Isp to Achieve Orbit

SpaceX Merlin-1D (LOX/RP1 staged combustion; fictive SC vs GG cycle variant)
Thrust: 845kN (sl); 981kN (vac)
Weight: 470kg;
Isp(sl): 311s; Isp(vac): 365s; Isp(avg): 338s
Mass Flow: 277.0755kg/s (ṁ sl); 274.07kg/s (ṁ vac); 275.57275kg/s (ṁ avg)
TSFC: 327.9g/s⋅kN (sl);  279.4g/s⋅kN (vac); 303.65g/s⋅kN
Densified Propellant Bulk Density: 1,124kg/m^3

Densified LOX/RP1
1,124,000g/m^3 / 327.9g/s⋅kN
Total Impulse (sl TSFC) provided by 1m^3 of propellant: 3,427.874kNs
1,124,000g/m^3 / 303.65g/s⋅kN
Total Impulse (avg TSFC) provided by 1m^3 of propellant: 3,701.630kNs
1,124,000g/m^3 / 279.4g/s⋅kN
Total Impulse (vac TSFC) provided by 1m^3 of propellant: 4,022.906kNs

RS-25 (LOX/LH2 staged combustion)
Thrust: 1,859kN (sl); 2,279kN (vac)
Weight: 3,177kg;
Isp: 366s (sl); 452.3 (vac); 409.15 (avg)
Mass Flow: 517.9kg/s (ṁ sl); 513.8kg/s (ṁ vac); 515.85 (ṁ avg)
TSFC: 278.6g/s⋅kN(sl); 225.45g/s⋅kN (vac); 252.025g/s⋅kN (avg)
Densified Propellant Bulk Density: 395kg/m^3

Densified LOX/LH2
395,000g/m^3 / 278.6g/s⋅kN
Total Impulse (avg TSFC) provided by 1m^3 of propellant: 1,417.803kNs
395,000g/m^3 / 252.025g/s⋅kN
Total Impulse (avg TSFC) provided by 1m^3 of propellant: 1,567.305kNs
395,000g/m^3 / 225.45g/s⋅kN
Total Impulse (vac TSFC) provided by 1m^3 of propellant: 1,752.051kNs

Tank Volume Ratio (LOX/LH2:LOX/RP1) for equal Total Impulse
SL: 2.418:1
Avg: 2.362:1
Vac: 2.296:1

Let's think about the problem this way.  For any LOX/LH2 rocket, regardless of the number of stages, you need more than double the equivalent tank volume for equivalent total impulse.

Q: What is the principle problem that all SSTOs and TSTOs must overcome?

A: Providing sufficient thrust and total impulse to attain orbital velocity.

Q: Can you more than double tank volume for less dry mass fraction, despite the fact that LOX/RP1 is a "heavier" / "denser" propellant combination?

A: No, you cannot.

Q: Can you more than double tank volume for equivalent aerodynamic drag in a practical aerodynamic vehicle design?

A: No.

Q: Can you more than double tank volume while maintaining equivalent tank weight and airframe stiffness (resistance to deformation under load)?

A: No.

Q: Can you switch to a fuel with 1/11th the density of RP1 while feeding it to the propellant feedline inlet to the engines using the same internal tank pressurization?

A: No.

Some material, regardless of its weight, must enclose the total propellant volume while maintaining sufficient strength and stiffness to survive the pressurization loads, aero loads, and thrust loads it will be subjected to.  Assuming acceleration is at or under 5g, then the greatest load on the tank will be the internal pressurization load.  LH2 for a large orbital class vehicle has to be pressurized to more than 2X (45psi) what LOX and RP1 (20psi to 22psi) are pressurized to.  A simple hoop stress calculation will adequately illustrate why the LH2 tank is under equivalent or greater stress from internal pressurization than it is from a heavier / denser propellant, thus why it will not be significantly lighter, despite LH2's very light weight, relative to RP1.  Remember that both fuels also require LOX, which is far heavier / denser than LH2 or RP1.

45psi = 31,638.1kgf/m^2
22.5psi = 15,819.05kgf/m^2
867kg/m^3, spread over 1m^2, is still only 867kgf/m^2

We can easily see why the difference between densified LH2 (77kg^3) and densified RP1 (867kg/m^3), is an utterly meaningless contribution to the overall mass of the propellant tank to provide adequate strength and stiffness, relative to internal pressurization, which is a factor of 18X less at 22.5psi, or 36X less at 45psi.  The real force or load on said propellant tank comes from internal pressurization, NOT the weight of the propellant itself, regardless of what kind it is.  Subject the tank to a 5g acceleration, and RP1 still represents 27.4% of the total load on the tank.  LH2 represents a lot less, but that still doesn't matter when the LH2 fueled engines require 45psi of propellant feed line pressure to prevent cavitation in the turbopumps.  Beyond that, by the time you can accelerate to 5g, your rocket is already much lighter, given a 1.5:1 thrust-to-weight ratio at liftoff.

Apart from sufficient thrust per unit of engine weight, Mass(initial), Mass(final), and Total Impulse are what make rockets "go into orbit", or not.  It's not Isp alone, nor does thrust alone get that job done.  You need both, at the lowest dry mass readily achievable.  Switching to LH2 from RP1 doesn't actually help you achieve that.  LH2 is a SSTO disabler.  Roughly speaking, it cuts your useful payload mass fraction in half.  If you could get 20,000kg to LEO using RP1, then for equivalent vehicle weight you could get 8,000kg to LEO using LH2.

You cannot have a lower final mass when your propellant tank is 2X+ larger for LOX/LH2 and the very large LH2 tank(s) operates at 45psi, as compared to LOX/RP1 operating at 22psi, while your LH2 fueled engine provides 585kN of thrust per kilo of engine weight vs 1,798kN of thrust per kilo of RP1 fueled engine weight.  The LOX/RP1 engine delivers 3X more thrust per kilo of engine weight.

The net-net of this situation is that you end up with engine weight as a greater total percentage of your dry mass fraction by using LOX/LH2, when that mass really needs to be sunk into propellant tank mass to resist LH2 pressurization while enclosing 2X+ greater total propellant volume, since your LH2 tank requires 2X the pressurization of LOX or RP1.  LOX/LH2 simply will not allow propellant tank and engine mass to ever be lighter than LOX/RP1 for eqivalent total impulse.  The Isp advantage of LOX/LH2 cannot overcome the inherent propellant tank and engine mass disadvantages, unless the material the tank is made from becomes 2X stronger and stiffer for a given weight and the engines produce more thrust for a given weight.

On top of that, for reusability you must also increase the surface area of TPS materials to protect your vehicle during reentry.  Ye olde cube-square law means this problem is not as bad as it would superficially seem, but you do in fact need more TPS mass allocated for LOX/LH2 powered vehicles.  That trifecta of SSTO disablers works against a practical LOX/LH2 SSTO.  Whatever you're gaining in terms of Isp improvement with LH2, you're immediately giving up in terms of Total Impulse per unit of vehicle and engine dry mass.

For any SSTO, dry vehicle mass is something you drag along with you, from the surface of the Earth, all the way to orbit.  That is why you either require a much larger vehicle made from much stronger materials for a given weight, or else you need to use a propellant combo with a significantly higher Density Impulse (with a still-reasonable Specific Impulse).  This is why I stated that all practical SSTO propellant combinations look remarkably similar to LOX/RP1 in terms of Density Impulse, Specific Impulse, and thrust provided.

If you need to put a payload of a given mass into orbit and you require a given thrust to get off the pad, then your LOX/LH2 solution ends up meaningfully heavier than LOX/RP1, due to the extra tank and engine mass required.  This is anathema to a practical fully reusable SSTO vehicle, which must remain as small as possible, in order to remain as light / strong / stiff as possible.