Starhopper+Starship for heavy. Triple-cored Starship for super heavy.

kbd512 · 2020-08-09 11:49:31

Bob,

Use the link below:

Plug in the dry mass, wet mass, thrust, and Isp to see what you get. If the vehicle gets 360s of overall Isp, ground-to-orbit, then you get what you get. The mass fraction of propellant only increases as Isp drops. It's not hard to understand. With LOX/RP1 and 360s Isp, the propellant mass is more than 90% of the total mass of the vehicle, period.

900,000 kg of propellant
50,000 kg of dry mass (I presume you want this thing to come back and to be reusable; Falcon 9 has 31.7t of mass at burnout for an Aluminum gas tank with no thermal protection)
12,000kN of thrust (same as a single STS SRB's worth of thrust)

The tool estimates 13,578kg of payload, with a 95% performance confidence interval falling between 2,626kg and 26,634kg.

Falcon 9 FT weighs 549,054kg (total vehicle mass) and the tool estimates 16,627kg of payload with downrange recovery of the booster on the drone ship.

The SSTO is burning about twice as much gas for worse performance when compared to Falcon 9 and the dry mass figure I used is grossly unrealistic for a reusable vehicle without resorting to advanced composites.

Yes, it really is that bad. What you want to happen won't happen because it can't happen.

RGClark · 2020-08-09 13:23:01

kbd512 wrote:

Bob,
Use the link below:
Launch Vehicle Performance Calculator
Plug in the dry mass, wet mass, thrust, and Isp to see what you get. If the vehicle gets 360s of overall Isp, ground-to-orbit, then you get what you get. The mass fraction of propellant only increases as Isp drops. It's not hard to understand. With LOX/RP1 and 360s Isp, the propellant mass is more than 90% of the total mass of the vehicle, period.
900,000 kg of propellant
50,000 kg of dry mass (I presume you want this thing to come back and to be reusable; Falcon 9 has 31.7t of mass at burnout for an Aluminum gas tank with no thermal protection)
12,000kN of thrust (same as a single STS SRB's worth of thrust)
The tool estimates 13,578kg of payload, with a 95% performance confidence interval falling between 2,626kg and 26,634kg.
Falcon 9 FT weighs 549,054kg (total vehicle mass) and the tool estimates 16,627kg of payload with downrange recovery of the booster on the drone ship.
The SSTO is burning about twice as much gas for worse performance when compared to Falcon 9 and the dry mass figure I used is grossly unrealistic for a reusable vehicle without resorting to advanced composites.
Yes, it really is that bad. What you want to happen won't happen because it can't happen.

For the hypothetical SSTO you discuss at 900,000 kg propellant and 50,000 kg dry mass, it’s about twice the gross mass of the Falcon 9 FT, so give it a vacuum thrust twice that of the F9 FT, so at 16,000kN. So plug this into the Launch calculator you linked.

There are a few quirks of this launch calculator you need to keep in mind also. First, the calculator always takes the vacuum values of the Isp and thrust even for the first stage. This is because it already takes into account the diminution at sea level. Secondly, the “Restartable Upper Stage” option should be clicked “No”. This always reduces the payload when clicked “Yes”, eventhough in this case there’s no upper stage.
Thirdly, taking the launch site as Cape Canaveral you should set the launch inclination as 28.5 degrees to match the launch site latitude. This is a fact of orbital mechanics that altitude, or payload, is maximized by launching in a direction to match the latitude of the launch site.

Doing this, I get a payload of 26 tons for the expendable mass. But as I said this calculation has limited accuracy. The reason is this calculator was designed for fixed nozzles. What really needs to be done is a true trajectory simulation that takes into account the actual variation of Isp with altitude.

Bob Clark

Last edited by RGClark (2020-08-09 13:31:41)

GW Johnson · 2020-08-09 14:37:32

Bob:

Separation backpressure is not a fixed ratio of atmospheric at sea level. It depends upon the nozzle expansion ratio, in turn controlling the ratio of expanded exit plane pressure to chamber (stagnation) pressure.

The correlation I use is: Psep/Pc = (1.5 Pexit/Pc)^0.8333

Depending upon your chamber pressure Pc and your nozzle expansion ratio, results for the critical backpressure vary rather wildly.

GW

SpaceNut · 2020-08-09 15:14:31

The main thing with SSTO ships is the hauling of dead weight in structure an fuel tanks once they are nearly empty.

kbd512 · 2020-08-09 16:34:42

Bob,

How unrealistic do you expect me to get?

1. I'm already using an Isp 12 seconds higher than what Merlin 1D with the Vacuum nozzle actually achieves in a vacuum.

2. The SSTO design has twice as much LOX/RP1 propellant mass as the Falcon 9 and I presume we're including some heat shielding mass to reuse the rocket. If I merely presumed we were using Aluminum-Lithium alloy, then the mass figure I provided is grossly unrealistic for a throwaway SSTO.

3. If you can't restart the "upper stage" / "only stage" (in our case), then how are you circularizing your orbit?

4. If this vehicle has 16 vs 9 Merlin engines to provide 16,000kN of thrust, then 10,080kg of mass is devoted to the engines alone. That means 41,450kg of weight for everything else. That's a dry mass fraction, exclusive of engines, of 4.2%, whereas Falcon 9 has a dry mass fraction of 5.8%. Merlin already has the highest TWR we've managed thus far. A pair of RD-171's that produce 15,000kN would weigh 19,000kg for comparison's sake. The stainless steel Centaur upper stage has the highest propellant mass fraction of any upper stage, period, and it's 95% propellant by weight. If they made it from Aluminum-Lithium alloy, then it would be heavier. The only way they could "go lighter", was to use advanced composites. ULA already did both the experiment and the math, and those are the results, so you're using composites that can minimally withstand aerodynamic heating on ascent, which requires adding insulation mass, or it's never making it to orbit.

5. This is America, so I presume our rocket will launch from an American launch pad, thus I selected KSC. However, I also want to know what kind of payload it can deliver to an off-nominal orbit. I did like-for-like comparison.

What you want is unrealistic. I already pulled out all the stops on inert mass fraction. There are no free lunches to be had here. You're going to pay dearly for an ultra-expensive advanced composite throwaway SSTO, merely to burn a lot more gas and get the same or worse performance as Falcon 9.

RGClark · 2020-08-10 07:38:09

SpaceNut wrote:

The main thing with SSTO ships is the hauling of dead weight in structure an fuel tanks once they are nearly empty.

However, for the reusable case, the ideal scenario for a SSTO, the tanks will be needed on succeeding flights just like on aircraft, so in that sense are not dead weight.

Bob Clark

RGClark · 2020-08-10 11:50:28

GW Johnson wrote:

Bob:
Separation backpressure is not a fixed ratio of atmospheric at sea level. It depends upon the nozzle expansion ratio, in turn controlling the ratio of expanded exit plane pressure to chamber (stagnation) pressure.
The correlation I use is: Psep/Pc = (1.5 Pexit/Pc)^0.8333
Depending upon your chamber pressure Pc and your nozzle expansion ratio, results for the critical backpressure vary rather wildly.
GW

Thanks for that equation. I think I can find the answer to the question following that.

By the way, here’s an article I’m reading about various methods to improve rocket nozzle performance:

JOURNAL OF PROPULSION AND POWER Vol. 14, No. 5, September– October 1998
Advanced Rocket Nozzles.
Off-design operations with either overexpanded or under- expanded exhaust flow induce performance losses. Figure 3 shows calculated performance data for the Vulcain 1 nozzle as function of ambient pressure, together with performance data for an ideally adapted nozzle. Flow phenomena at different pressure ratios Pc/Pamb are included in Fig. 4. [The sketch with flow phenomena for the lower pressure ratio Pc/Pamb shows a normal shock (Mach disk). Depending on the pressure ratio, this normal shock might not appear, see, e.g., Fig. 2.] The Vulcain 1 nozzle is designed in such a manner that no uncon- trolled flow separation should occur during steady-state oper- ation at low altitude, resulting in a wall exit pressure of Pw,e ~ 0.4 bar, which is in accordance with the Summerfeld criterion. The nozzle flow is adapted at an ambient pressure of Pamb ~0.18 bar, which corresponds to a ? flight altitude of h ~ 15,000 m, and performance losses observed at this ambient pressure are caused by internal loss effects (friction, diver- gence, mixing), as shown in Fig. 1 and Table 1. Losses in performance during off-design operations with over- or un- derexpansion of the exhaust flow rise up to 15%. In principle, the nozzle could be designed for a much higher area ratio to achieve better vacuum performance, but the ?flow would then separate inside the nozzle during low-altitude operation with an undesired generation of side-loads.
https://www.researchgate.net/profile/Ge … ozzles.pdf

I’m trying to understand that passage I quoted. It seems to be saying the exit nozzle pressure is both 0.4 and 0.18.

Bob Clark

Last edited by RGClark (2020-08-10 11:53:18)

RGClark · 2020-08-10 12:05:11

kbd512 wrote:

Bob,
How unrealistic do you expect me to get?
1. I'm already using an Isp 12 seconds higher than what Merlin 1D with the Vacuum nozzle actually achieves in a vacuum.
2. The SSTO design has twice as much LOX/RP1 propellant mass as the Falcon 9 and I presume we're including some heat shielding mass to reuse the rocket. If I merely presumed we were using Aluminum-Lithium alloy, then the mass figure I provided is grossly unrealistic for a throwaway SSTO.
...

Since we already know the Launch calculator has limited accuracy in being based on fixed nozzle engines, it’s not going to be useful to argue the payload numbers it gives anyway. What needs to be done is a true trajectory simulation with a varying expansion ratio nozzle.

Bob Clark

GW Johnson · 2020-08-10 12:07:25

Bob:

I may be wrong, but I believe that article was a French look at the uneven-expansion technology built into the shuttle engines. In the core, the expanded pressure is lower than at the over-recurved wall. This fends off backpressure-induced separation a little, relative to the correlation estimate I gave you. That correlation is for an evenly-expanded (method of characteristics design) with a constant static pressure all across the section at each flow station. It's not that large of an effect, but the shuttle engine showed it to be a real thing you can do.

The best thing to do (best in the sense of lightest inert masses and simplest designs) is design for perfect expansion at about 20 to 30 kft, which will be on the verge of separation at sea level, but has a bigger expansion ratio than a sea level design, for better vacuum performance. You pay for that with not as large a sea level thrust, which hurts your at-launch thrust/weight ratio. You sidestep that problem by adding solid strap-ons, which have the greatest frontal thrust density of anything you could do.

Spacex's Raptor has a 40:1 expansion for the sea level design, and a 200:1 expansion for the vacuum design, otherwise the same gas generator chamber. They cannot ground test the vacuum engine, but they do have a ground test version with about 120:1 expansion, if memory of what I read serves.

That will just barely avoid separation tested at sea level, at whatever passes these days for "near full thrust" with the Raptor. But its thrust is lower than the real sea level design, at sea level, all else being equal, because of the backpressure term on thrust, and despite the higher exit velocity.

GW

Last edited by GW Johnson (2020-08-10 12:08:55)

kbd512 · 2020-08-10 17:56:53

Bob,

As near as I can tell, an altitude-compensating nozzle would make the most sense in the booster stage of a two-stage vehicle. Even if you could spare the structural mass necessary to take the entire launch vehicle to orbit, reenter, and land it, a two-stage vehicle would still be cheaper and easier to produce using common metal alloys and fabrication techniques. Whatever performance boost you would get from the altitude compensating nozzle design would translate into more payload performance than a SSTO using the same engine because you get no performance benefit from such a design in the upper atmosphere or in space and incur whatever mass penalty you must pay for the altitude compensating nozzle and the inert mass of the now-dry propellant tank.

At the end of the booster's burn on Falcon 9, you're effectively in space- nominally, about 80km (from NASA's CRS-6 mission press kit). The atmospheric pressure at that altitude is effectively zero, so all meaningful performance enhancement provided by the altitude compensating nozzle has ended by the time you achieve that altitude.

Someone did an analysis on Falcon 9 to determine which parameters had the greatest effect on vehicle performance:

Reddit - https://www.reddit.com/r/spacex/comments/34fd80/i_created_a_ascent_profile_analysis_report_using/

Rocket Propulsion Project - Analysis of Rocket Ascent Profiles by Raul Maldonado

A little bit on ascent trajectory optimization for two stage vehicles without so much math:

A Convex Approach to Rocket Ascent Trajectory Optimization

I think I've posted this before in another thread, but will repost it here:

Optimal trajectory designs and systems engineering analyses of reusable launch vehicles

SpaceNut · 2020-08-10 18:17:42

Lets play the what if a Falcon 9 was converted to SSTO by moving a second stage merlin to the base with minimal alterations.
Problem 1 for reuse is a heat shielding for return of the rocket and less payload to orbit.

kbd512 · 2020-08-10 19:07:13

SpaceNut,

STS had ~5,250ft^2 worth of tiles. The surface area of a "straight cylinder" Falcon 9 rocket is at least 835.17ft^2 / 8,990ft^2, excluding any surface area increases associated with a larger / more useful payload fairing surface area. So, we're closing in on double the heat shield area. It could be made from PICA-X to reduce costs, but even PICA-X has mass associated with it. Let's say PICA-X is just 0.2g/cm^3 and the insulation and mounting solution is 0.1m thick, then 83.566m^3 * 200kg/m^3 = 16,713.2kg seems like a reasonable heat shield mass.

There is no way that this fictional SSTO Falcon 9 made from CNT composite and PICA-X would beat a TSTO Falcon 9 made from conventional metals in terms of payload performance if all stages of both vehicles are designed to be reusable. It's physically impossible.

For maximum payload performance, Falcon 9 should be an unpainted stainless steel booster with altitude compensating nozzles and Falcon's upper stage should be a disposable filament-wound carbon composite tank with a single J-2X, equipped with ULA's IVF technology instead of batteries and He2 bottles, and a conventional vacuum nozzle. The J-2X engine core, less nozzle extension, should be the only component supplementally shielded for reentry using a disposable Nextel fabric HIAD, much like ULA's Vulcan booster engines. That's the greatest payload performance improvement and maximum recovery of expensive vehicle hardware components that Falcon 9 would realistically see.

SpaceNut · 2020-08-10 19:33:43

The interstage is a composite structure that connects the first and second stages, so why not use it for the complete shell.

The stage seperation for expendable Falcon 9 rocket, is around Mach 10.
For a reusable Falcon 9, it is around Mach 6, depending on the mission.

kbd512 · 2020-08-10 20:09:38

SpaceNut,

Do you mean use the interstage as a heat shield / reentry vehicle to save the J-2X?

Both the Oxygen and the Hydrogen will trash the composite propellant tanks, so I say we build it just well enough to survive a single mission.

If we have robust stainless steel boosters, recover upper stage engines, recover payload fairings, then I think we've struck the best compromise between performance and reusability. Cheap composite propellant tanks are an acceptable loss. A new tank could feasibly be fabricated in a matter of minutes using Lamborghini's new "carbon forging" process that uses heat and extreme pressure to rapidly pop out parts made from cheap chopped carbon fiber, with equivalent strength to all but the most expensive woven high modulus carbon fiber fabrics.

Ultimately, everything's a compromise here. The dV increment is simply too high for a practical SSTO using chemical propellants.

RGClark · 2020-08-12 08:18:58

kbd512 wrote:

SpaceNut,
STS had ~5,250ft^2 worth of tiles. The surface area of a "straight cylinder" Falcon 9 rocket is at least 835.17ft^2 / 8,990ft^2, excluding any surface area increases associated with a larger / more useful payload fairing surface area. So, we're closing in on double the heat shield area. It could be made from PICA-X to reduce costs, but even PICA-X has mass associated with it. Let's say PICA-X is just 0.2g/cm^3 and the insulation and mounting solution is 0.1m thick, then 83.566m^3 * 200kg/m^3 = 16,713.2kg seems like a reasonable heat shield mass.
There is no way that this fictional SSTO Falcon 9 made from CNT composite and PICA-X would beat a TSTO Falcon 9 made from conventional metals in terms of payload performance if all stages of both vehicles are designed to be reusable. It's physically impossible.
...

Note it doesn't have to equal the two stage F9 in payload, but only be able to lift enough payload to be profitable. The partially reusable, two-stage F9 is able to lift in the range of 16 tons to LEO. Suppose a fully reusable SSTO F9 is able to lift, say, 8 tons to LEO. If a customer only needs 8 tons to LEO, why should he pay for the higher capacity? Keep in mind most payloads to LEO do not need the full 16 tons to LEO of F9. SpaceX wanted to give the F9 FT the higher capacity so it could cover those customers that needed it. But in reality it's a small proportion of the customers who do.

The heat shield mass probably can be less than your estimate. According to the Spaceflight101 page on the F9 FT, the diameter is 3.66 m and the height of the first stage is 42 m. The full area of the cylindrical first stage is Pi*(3.66)*(42) = 490 m^2. But we'll only need to cover the bottom half, so only 245 m^2. Then the mass of the PICA-X heat shield will be in the range of 5,000 kg.

Also, nobody knows how much the F9 with altitude compensation can get to LEO because nobody ever calculated it for dense propellants.

Bob Clark

Last edited by RGClark (2020-08-12 12:50:21)

kbd512 · 2020-08-12 10:42:14

Bob,

Apologies, but I still don't understand your point. If a customer only needs 8t, then they're going to choose an existing smaller / lighter TSTO. Alternatively, they go half-in with another customer to completely fill the Falcon 9 payload bay. Ride sharing is "a thing" when it comes to rockets. If a rocket is SSTO and reenters the atmosphere, then you have to pay for the refurbishment of a heat shield in addition to the engines. The Delta IV essentially has the vacuum-Isp of Merlin or Raptor vacuum models, but it has that Isp at sea level. The performance of a LOX/RP1 vehicle will never be equal to that, much less better that. Delta IV puts 8.5t in LEO with no solids. Delta IV weighs 226t for the booster core and 24t to 30t for the RL-10 powered upper stage. The Delta IV CBC dry mass was 26.76t (engine, Aluminum, insulation). Falcon 9 Block V is 27.2t at burnout.

The Delta IV common booster core has the same dry mass as the reusable Falcon 9 Block V booster core. Make Delta IV the same tonnage as Falcon 9 by adding solids and it'll lift nearly identical payload.

There's no there there. This is a basic rocket performance problem. If Delta IV used stainless steel balloon tanks for the CBC and composite landing gear like Falcon 9, then dry mass and payload performance would be no different and no better. The mass differential in the 4m vs 3.66m tank is practically nothing and if you used stainless steel balloon tanks, then it would be nothing. This might be why nobody has "done the calculations". There's no possibility of better performance with identical wet mass and TWR, much less more mass added for reusability and lower payload mass fraction inherent in SSTO on account of carrying all that dry mass to orbit with you.

If you want more payload performance per pound or kilo of vehicle mass, then you use LOX/LH2 with solids. You can't get more from LOX/RP1. You can burn more gas and achieve equivalent performance, though. If gas is cheap and easy to come by, no big deal. If not, then marginal cost per flight must go up.

I'm totally onboard with using altitude compensating nozzles in boosters for TSTO to improve performance, but payload performance per unit mass of propellant expended simply doesn't work in favor of SSTO on account of vehicle dry mass dragged all the way uphill to orbit, with or without altitude compensating nozzles.

RGClark · 2020-08-13 09:08:22

The advantage is SpaceX would be fully reusable, so the cost would be less. The other launch companies only show middling interest in reusability.
SpaceX making a whole new launcher to cover this smaller launch market would be very expensive. The SSTO approach though would only use an extra alt. comp. attachment to the nozzle and heat shield for the reusable SSTO F9 booster.

Bob Clark

kbd512 · 2020-08-13 21:34:31

Bob,

Here are my points:

1. I don't believe there's any such thing as a "fully reusable" vehicle after reentry. I've never seen any such animal in the entire history of rocketry. If it's possible, then it's never been demonstrated by anybody, including SpaceX. We have seen "fully refurbish-able" vehicles, but those were and are incredibly expensive to actually operate. I want to see someone launch a crewed or un-crewed vehicle into orbit, recover it after reentry, and re-fly it without refurbishing the heat shield and numerous other components (apart from refilling propellants or consumables and running electronics-based diagnostics checks of critical components). If we can't do that, then building another gigantic STS-like vehicle is probably a waste of time and money.

If SpaceX intends to try something fundamentally different with TPS by using transpiration cooling with LCH4, then let's see how that works out. The PICA-X TPS doesn't change anything at all, with respect to heat shield technology. It's a lightweight ablative, but once you reenter, in actual practice it's trash. That's why SpaceX replaces it on Dragon. If you built the TPS so thick that you could simply scrape off the outer char layer and re-fly, then you're necessarily reducing your payload performance by carrying extra TPS mass. Worth it? Depends on the mass differential. If you carry an extra 100kg of TPS, but you payload was 10,000kg, I'd say so. If it's 1,000kg out of 10,000kg, then probably not.

2. The TSTO advantage is that the largest part of the vehicle (the booster) requires no thermal protection. RP1 and LCH4 engines are smaller and lighter for the thrust performance provided, but payload performance suffers from their lower specific impulse in comparison to LH2, with or without altitude compensating nozzles. That's fine because the booster needs to lots of thrust to clear the atmosphere quickly, but not lots of specific impulse to pour on the speed. Either way, nearly 90% of the vehicle mass has to be propellant and vehicle wet mass will always be worse with RP1 or CH4 compared to LH2.

Put another way, the loss of thrust per unit mass of propellant expended at sea level is not something that an altitude compensating nozzle can actually compensate for because there will always be 14.7 pounds of pressure per square inch restricting the expansion of the exhaust product. It can compensate for loss of thrust from under-expansion as the vehicle ascends through the atmosphere, but that's it. Vacuum Isp is only obtainable in a vacuum. You can increase chamber pressure and lose less thrust at sea level, but you're never going to achieve Vacuum Isp at sea level, period and end of story.

3. The booster is the only part of the vehicle that would see any improved performance from an altitude-compensating nozzle design. The upper stage never will because the vehicle is effectively operating in a vacuum at the time the engine starts. Therefore, you will be carrying all of that vehicle dry mass and extra nozzle mass all the way to orbit with a SSTO.

As such, a TSTO with stainless steel tanks that's also equipped with altitude compensating nozzles for the booster stage is the most payload performance you can possibly get in a practical design. Ultimately, it's payload mass vs dollars spent. If it's my dollars, then I want to refine the booster performance to the extent possible, recover the payload fairing and upper stage engine if feasible, and pay a slight cost penalty for dumping a minor quantity of carbon fiber in the form of propellant tankage in the ocean at the end of the flight. If the upper stage is integrated with the vehicle, then all of that has to reenter and you pay a payload performance penalty for the increased TPS mass as a result. It might be worth it or it might not be, but if it requires TPS then the TPS will be inspected and repaired or replaced after every flight- something that's never proven to be cheap or fast to do, thus far.

A robotically-fabricated carbon fiber composite tank (automated tape laying; takes about a week, but a single operator stands there and watches the machine work) might cost in the range $100K/t and the upper stage might have a mass of 2t, so $200K. If we start using Lamborghini's "carbon forging" process that uses scrap or chopped carbon fiber, then the cost is not much more than the cost of Aluminum alloy because the entire fabrication process is automated and over with in mere minutes, meaning you could crank out a year's supply far faster than the RP1 or CH4 or LH2 to fill it with can be made. At that point, propellant cost starts to dwarf the marginal cost of fabricating a new tank, which takes us right back to the original payload performance problem- and TSTO will always provide more of that using less gas because it's not pushing the dry mass of the entire vehicle all the way to orbit.

If SpaceX ever manages to make their rocket landing system so reliable and their vehicle refurbishment processes so fast that they fly several times per week, then the fuel bill will quickly become the biggest cost and only significantly better Isp will help. SSTO will never improve that performance metric.

tahanson43206 · 2020-08-14 04:45:35

Re #43 by kbd512

SearchTerm:CompareSSTOvsTSTO

http://newmars.com/forums/viewtopic.php … 81#p171081

(th)

GW Johnson · 2020-08-16 12:42:36

All right, here is a bounding calculation, based on technologies-in-hand now.

Required: reach 7.9 km/s LEO speed eastward from a modest-latitude launch site, allowing for potential energy as well as kinetic energy. Use surface circular orbit speed 8.0 km/s to model both energies. If the orbit were polar, I would add the rotational velocity of the Earth at launch site latitude to that. If it were westward, I would add two of those velocities. But we are launching eastward.

Propulsion: LOX-LH2 has the best Isp potential of anything we have, but suffers severe volume requirements for the LH2 that increase drag and stage inert mass. Use instead LOX-LCH4 as the second-highest Isp potential that is currently flying, and with far more reasonable volume requirements for the LCH4.

Use a fixed-geometry conventional bell nozzle sized at sea level for the first stage of a TSTO, or the only stage of an SSTO. Use a vacuum engine for the second stage of an SSTO. Spacex says its Raptor at full throttle with 40:1 expansion gets Isp = 330 s at sea level, and 356 s in vacuum. Average those for the trip up to vacuum as Isp = 343 s. Spacex says its vacuum Raptor gets 380 s out in vacuum at 200:1 expansion, but cannot be used deep in the atmosphere. Use those Spacex LOX-LCH4 Raptor engine data.

Accounting for Losses: factor-up the required velocity for estimated gravity and drag losses, but not for altitude Isp performance; let the values of Isp used in the calculation account for that variation. That puts the greater realism into the estimates. Typical for Earth vertical-launch gravity-turn trajectories with aerodynamically-clean vehicles is 5% drag loss and 5% gravity loss, for a factor of 1.1. These apply to the first stage burn of a TSTO, or to the staging point speed reached during an SSTO gravity turn. Treat the second stage burn of a TSTO trajectory (or the remainder of an SSTO trajectory) as loss-less. 5% is an arbitrary assumption, but use it as "realistic".

Trajectory Assumptions: one-shot TSTO vehicle designs typically have staging speed near 3 km/s achieved velocity, essentially exoatmospheric at around 50-60 km altitude, and very nearly horizontal. Reusable vehicles compromise that by reducing the staging speed to nearer 2 km/s, near 50 km altitude, and very nearly horizontal. This is to save propellant for the powered entry and landing of the first stage. That puts a higher required delta-vee on the second stage, but that cannot be avoided. Since we are looking at reusable SSTO vs TSTO, we will use 2 km/s, and apply the factor 1.1 to that speed and 1.0 to the remainder (6 km/s), for both.

Analysis technique: factoring-up the required kinematic delta-vee data converts it to a mass ratio-effective delta-vee, suitable for use in the rocket equation to size mass ratios. 1 - 1/MR is the propellant mass fraction required of the stage. We assume the inert mass fraction, using "typical" data. For our purposes here, use 5% inert for a booster stage, and 8% for a recoverable SSTO or second stage vehicle of a TSTO. Those are arbitrary, but "realistic". Note that for any stage, inert + propellant + payload mass = mass at stage ignition. For a first stage of a TSTO, "payload is the ignition mass of the second stage. For the second stage, payload is what mass is to be unloaded and delivered to orbit. For an SSTO, payload is just that, there is only one stage.

SSTO Results: MR-effective dV = 1.1*2 km/s + 6 km/s = 8.2 km/s; Vex = avg Isp * gc = 343 s * 9.80667 / 1000 = 3.364 km/s; dV/Vex = 8.2/3.364 = 2.4376; MR = exp(dV/Vex) = exp(2.4376) = 11.445; propellant mass fraction = 0.9126, payload fraction = 1 - propellant fraction - inert fraction = 1 - .9126 -.08 = 0.0074. To deliver 10 metric tons to orbit, the SSTO ignition mass would have to be some 1352 metric tons. Delivered payload mass/launch mass = 0.74%!!!

TSTO results: second stage MR-effective dV = 6 km/s *1.0 = 6.0 km/s; Vex = 380 s * 9.80667 / 1000 = 3.727 km/s; dV/Vex = 1.6099; 2nd stage MR = 5.0022; propellant fraction = 0.8001; payload fraction = 1-.8001-.08 = 0.1199. To deliver 10 tons to orbit, the second stage ignition mass must be 83.4 metric tons. First stage MR-effective dV = (2 km/s to stage point plus the 3rd km/s to cover recovery) * 1.1 = 3.3 km/s; Vex = 343 s * 9.80667 / 1000 = 3.364 km/s; dV/Vex = 0.98098; MR = 2.6671; propellant fraction = 0.6251; payload fraction = 1 - propellant - inert = 1 - .6251 - .05 = 0.3249. The first stage ignition mass uses that payload fraction and the second stage ignition mass: stage 1 ig = 83.4 m.tons/.3249 = 256.7 metric tons. Delivered payload mass/launch mass = 10/256.7 = 0.0390 = 3.90%.

Comments: That 3-4% size payload/launch figure is rather similar to what is being done currently with TSTO, actually, so my estimates and my estimating techniques are just not that far wrong.

In my opinion, SSTO suffers from two problems. (1) yes, there is a low average Isp problem induced by being forced to use a sea level nozzle out into vacuum. (2) there is also a mass ratio-"overload" problem induced by ratioing too many km/s of dV to too few km/s of Vex. Because of the exponential nature of the rocket equation, the second problem is worse than the first. The resulting propellant mass fractions are almost-infeasibly high with LOX-LCH4.

Chemical propulsion Isp is just too low for SSTO to be feasible at currently-acceptable payload fractions, unless you use hydrogen fuel! To get "reasonable" payload fractions (3.9%) with SSTO at 8% inert, you cannot exceed 88.1% propellant fraction, which is MR = 8.40. For the same 8.2 km/s MR-effective dV, your Vex must equal or exceed 3.852 km/s, which corresponds to an average Isp during ascent of some 393 s. And that does not reflect any changes in drag loss or inert fraction for the far larger tank size.

That means only LOX-LH2 could be used, and in turn, you then have to solve the volume problem LH2 induces: drag and inert fraction. The 5% drag and 5% inert fraction assumptions used here are just wrong for that propellant choice.

On the other hand, the TSTO could indeed go to LOX-LH2 for the second stage, without increasing the diameter of the first stage, only the length of the second stage. Greater than 4% payload/launch becomes feasible.

Revised/Modified Design Sizings: The RS-25 engine is listed as 366 s Isp at sea level, and 452.3 s in vacuum, using LOX-LH2. Those average to 409.2 s for the ascent into vacuum. I added 1% each to drag loss factor and inert fraction, for using LH2, for the SSTO, because its overall diameter must increase, as well as its length. For the TSTO, I stay with LOX-LCH4 in the first stage so that its diameter may stay the same. Only the second stage uses LOX-LH2, at the first stage diameter, and an increased length. So I add 1% to the second stage inert fraction, and did not change the drag loss factor. All else is the same.

Mod SSTO: MR-effective dV = 1.11*2 km/s + 6 km/s = 8.22 km/s; Vex = avg Isp * gc = 409.2 s * 9.80667 / 1000 = 4.013 km/s; dV/Vex = 8.22/4.013 = 2.0483; MR = exp(dV/Vex) = exp(2.0483) = 7.755; propellant mass fraction = 0.8711, payload fraction = 1 - propellant fraction - inert fraction = 1 - .8711 -.09 = 0.03895. To deliver 10 metric tons to orbit, the SSTO ignition mass would be 256.7 metric tons. Delivered payload mass/launch mass = 3.90%.

Mod TSTO: second stage MR-effective dV = 6 km/s *1.0 = 6.0 km/s; Vex = 452.3 s * 9.80667 / 1000 = 4.4356 km/s; dV/Vex = 1.3527; 2nd stage MR = 3.8679; propellant fraction = 0.7415; payload fraction = 1-.7415-.09 = 0.1685. To deliver 10 tons to orbit, the second stage ignition mass must be 59.35 metric tons. First stage MR-effective dV = (2 km/s to stage point plus the 3rd km/s to cover recovery) * 1.1 = 3.3 km/s; Vex = 343 s * 9.80667 / 1000 = 3.364 km/s; dV/Vex = 0.98098; MR = 2.6671; propellant fraction = 0.6251; payload fraction = 1 - propellant - inert = 1 - .6251 - .05 = 0.3249. The first stage ignition mass uses that payload fraction and the second stage ignition mass: stage 1 ig = 59.35 m.tons/.3249 = 182.7 metric tons. Delivered payload mass/launch mass = 10/182.7 = 0.0547 = 5.47%.

Summary of results (you decide which is the better way to go):

item.........................SSTO..TSTO....SSTO-mod..TSTO-mod
payload, m.ton..........10......10........10..............10
launch mass, m.ton....1352..257......257.............183
payload/launch, %.....0.74...3.90.....3.90............5.47
propellants...............LOX-LCH4----...LOX-LH2.....LOX-LCH4(1)/LH2(2)
engines....................SL......SL+vac..SL..............SL+vac
inert fraction.............8%.....5/8%....9%.............5/9%
drag loss too low?......no-5%.no-5%..no-6%........no-5%

An Observation: going with two stages allows you to use different propellants in the two stages, and to take advantage of the fact that the first stage is very much bigger than the second stage. Not covered here is the added fact that the thrust required of the first stage is very much larger than the thrust required of the second stage. That is very difficult to accommodate with an SSTO without having very high vehicle accelerations near burnout, unless you use a large number of throttleable engines.

Final remark: if you know what you are doing so that you make good assumptions and use good "jigger factors", such calculations really are that easy, and that accurate. I did not use anybody's sizing software for this. Just a hand calculator. I could have used my old slide rule, and gotten exactly the same results at 2-3 significant figures instead of 3-4 significant figures. So what? It shows the very same trend. This staging stuff has been known since Von Braun's time at Peenemunde. The "trick" has been figuring out how to reuse the stages. That has taken nearly 80 years, but is now an accomplished fact.

GW

Last edited by GW Johnson (2020-08-16 12:48:56)

kbd512 · 2020-08-16 15:10:02

GW,

For Falcon 9, according to the user's guide from the SpaceX website (s = seconds):

MECO: +145s for a LEO mission
SES-1: +156s
Payload Fairing Release: +195s
SECO-1: 514s
SES-2: + 3086s
SECO-2: +3090s

9 Sea Level Merlin: 145s total burn time, excluding boost-back and landing burns
1 Vacuum Optimized Merlin: 362s total burn time

Using the TSTO method, ~2.5X as much total burn duration is spent running 1 vacuum-optimized engine as compared to 9 sea level-optimized engines that would have to be deeply throttled, probably losing Isp as a result.

I think the best we can hope for are cheap reusable boosters constructed from durable stainless steel, with disposable upper stages fabricated from carbon fiber. In order to recover the entire upper stage, a TPS has to be applied, which necessarily eats into our payload performance. If we desperately want the entire upper stage back, then we have to burn more gas to achieve that and pay for TPS refurbishment, which will necessarily make the upper stage more expensive to construct and to operate. If the fuel cost is reasonable, then the juice might be worth the squeeze, but probably not.

GW Johnson · 2020-08-16 15:28:07

PS/Update to post 45: I got to thinking about the details of how I modeled the TSTO and SSTO designs. I am satisfied with the TSTO, both all-LOX-LCH4 and LOX-LCH4 first stage with LOX-LH2 second stage. I am not so satisfied with how I modeled the SSTO with either propellant combination, because the bulk of the burn is in vacuum, not as well modeled by an average of sea level and vacuum performance of a sea level-capable engine design.

Now with an SSTO, booster recovery is not an issue, so the nominal 3 km/s "stagepoint" at 50-60 km altitude is a representative endpoint for the "average Isp" portion of the burn. From about there, the burn can use full vacuum Isp performance. In effect, you should really model it as two sequential burns, one at average Isp to 3 km/s, and the other at vacuum Isp for the remainder of the delta-vee (some 5 km/s).

Here is how that plays into the LOX-LCH4 SSTO case: first burn MR-eff dV = 3.0 km/s * factor 1.10 = 3.30 km/s; avg Isp = 343 s for Vex = 3.364 km/s; dV/Vex = 0.9810; MR = 2.6671. Second burn MR-eff dV = 5.0 km/s * factor 1.00 = 5.0 km/s; avg Isp = vac Isp = 356 s for Vex = 3.491 km/s; dV/Vex = 1.4323; MR = 4.1881. Compound (overall) MR for the whole burn is the product of the two mass ratios at the two effective Isp values: MR = 11.1701. The overall propellant fraction is thus 1 - 1/MR = 0.9105. For inert= 8%, the payload fraction is 0.0095 = 0.95%. For delivering 10 metric tons to orbit, the ignition mass is 1052.6 metric tons.

Here is how that plays into the LOX-LH2 case with 1% added to both drag loss and inert fraction: first burn MR-eff dV = 3.0 km/s * factor 1.11 = 3.33 km; avg Isp = 409.2 s for Vex = 4.013 km/s; dV/Vex = 0.8298; MR = 2.2929. Second burn MR-eff dV = 5.0 km/s * factor 1.00 = 5.00 km/s; avg Isp = 452.3 s for Vex = 4.4356 km/s; dV/Vex = 1.1273; MR = 3.0872. The compound overall MR required for the total burn is MR = 7.0786. Overall propellant fraction is 0.8587. For inert = 9% with LH2 tankage, the payload fraction is 0.05127 = 5.13%. To deliver 10 metric tons to orbit, the required launch mass is 195.0 metric tons.

That is a slight improvement over what I had calculated before, in both cases. Bear in mind that the "original" designs used all LOX-LCH4 propellants, while the "mod" designs used either LOX-LH2 all the way in the SSTO or in only the second stage of the TSTO. There are only changes to launch mass and payload percentage of launch mass. Launch mass changes from 1052 to 1053 metric tons and 0.74 to 0.95% for LCH4. It changes from 257 to 195 metric tons and 3.90 to 5.13% for LH2 fuel. I'm not convinced the old 0.74% figure was right, I may have miss-keyed something. The disparity between the launch masses for the two SSTO's traces directly to the tiny payload fraction left over after subtracting propellant and inert. It is far worse with methane than it is hydrogen.

Isp affects the velocity ratio, which gets exponentiated for the mass ratio, producing a propellant fraction. The interplay of propellant, inert, and payload fractions is only linear, yet that controls how big (and expensive) a vehicle you must build. Require a more demanding orbit (such as geosynchronous transfer), and even hydrogen quickly fades into infeasibility for SSTO application. Here are the revised tables:

item...........................SSTO....... TSTO.... SSTO-mod......... TSTO-mod
payload, m.ton...........10........... 10........ 10..................... 10
launch mass, m.ton.......1053..... 257.......195.................. 183
payload/launch, %......... 0.95......3.90..... 5.13.................. 5.47
propellants.................... LOX-LCH4----..... LOX-LH2............ LOX-LCH4(1)/LH2(2)
engines......................... SL......... SL+vac..SL..................... SL+vac
inert fraction................. 8%........ 5/8%.....9%................... 5/9%
drag loss too low?......... no-5%...no-5%.. no-6%............... no-5%

Remarks: For methane, the changes are negligible. For hydrogen, they matter somewhat (a 195 ton vs a 257 ton vehicle). This confirms the importance of stage dV-loading upon mass ratio, and the basic infeasibility of everything but LOX-LH2 for consideration in chemical SSTO designs for low Earth orbit.

SSTO really looks good only with nuclear propulsion far better than NERVA, with high engine T/W > 20-ish, and Isp well above 600-700 s. That's really gas core stuff. Use my methods and run those numbers for yourself. I have spent enough effort on this bounding calculation.

The key to getting this right is use of the correct average Isp during the ascent gravity turn, to around 2-3 km/s at 40-60 km altitude, where the path becomes nearly horizontal and essentially exoatmospheric. That same portion of the trajectory needs to be factored for drag and gravity losses at around 5% each. After that, conditions are essentially vacuum and essentially horizontal, and thus essentially unfactored for drag and gravity losses, and using vacuum Isp performance for the engine design.

If you are doing 2 or more stages, use a near-sea-level engine design in the first stage, and vacuum engine designs in stage 2 (and beyond). If you are doing a single-stage-to-orbit design, use a sea level engine design, but use its average performance during only the initial portion, and its vacuum performance once the vehicle is essentially exoatmospheric and horizontal in the second portion. This vacuum performance is NOT the performance of a vacuum engine design!

For the first part of the gravity turn trajectory, I recommend 5% gravity and 5% drag losses for factor = 1 + .05 + .05 = 1.10 on kinematic delta-vee, for all propellants except hydrogen fuel. Add 1% to the drag loss (and the inert mass fraction), with hydrogen, because the tank diameter and length are far higher. That would be factor 1.11 on dV, and "nominal inert" + 0.01 on stage inert mass fraction.

For the second part of the gravity turn, zero the gravity and drag losses due to exoatmospheric operation very near horizontally, which is factor = 1.00 on the kinematic dV. Add 1% to the nominal inert fraction if using hydrogen, due to the larger diameter and length of tankage required.

Based on the most recent vehicles, a good stage "nominal inert" fraction for booster stages is 5%. For upper stages capable of entry and landing, 8% is likely a better figure, because of the need for landing features, and a heat protection scheme that serves for entry speeds near low Earth orbit speeds. That would be for all propellant combinations that do not incorporate hydrogen fuel. Because of its lower density and larger required tank size, if you use hydrogen, add another 1% to those inert fractions that I recommended.

The nominal breakpoint between the "first" and "second" portions of the gravity turn trajectory is near 3 km/s for non-reusable designs, based on a lot of prior TSTO performance from a variety of manufacturers. It is nearer 2 km/s for reusable TSTO designs, based on what Spacex does with its Falcon-9 and Falcon-Heavy vehicles.

Because there is no booster to recover with an SSTO design, you can use near-3 km/s as the "breakpoint" velocity, for your two sequential burn analyses. That's about as good a breakpoint as anything you can assume.

And that is the best I have to offer, for now.

GW

RGClark · 2020-08-17 02:30:02

GW Johnson wrote:

...
Use a fixed-geometry conventional bell nozzle sized at sea level for the first stage of a TSTO, or the only stage of an SSTO. Use a vacuum engine for the second stage of an SSTO. Spacex says its Raptor at full throttle with 40:1 expansion gets Isp = 330 s at sea level, and 356 s in vacuum. Average those for the trip up to vacuum as Isp = 343 s. Spacex says its vacuum Raptor gets 380 s out in vacuum at 200:1 expansion, but cannot be used deep in the atmosphere. Use those Spacex LOX-LCH4 Raptor engine data.
...
GW

Thanks for the calculation, but quite key is even for the first stage a significant portion of the flight is in near vacuum conditions. That is why first stage engines are overexpanded, with a nozzle size larger than optimal for sea level operation to be able to get better performance in vacuum.

For instance, the F9 first stage engine cuts off after about 2 and 1/2 minutes at about 60 km altitude. At that altitude and even before a vacuum optimized engine would be getting close to full vacuum Isp. So most of the 9 minute or so flight to orbit is actually at near vacuum conditions, so would be getting close to full vacuum Isp of a vacuum optimized engine.

Then the average Isp of an alt. comp. kerolox engine would be close to the 360 s max Isp of a vacuum optimized engine. Actually, the max Isp could probably even be in the 370s to 390 s range. But 360 s is in the range achieved for upper stage kerolox engines:

http://www.friends-partners.org/mwade/p … xosene.htm

Likewise, for a methanolox alt. comp. engine, the average Isp would be close to the 380 s max Isp of the vacuum optimized engine.

Here’s an image illuminating most of the flight to orbit place where the Isp would be near the max Isp level:

This is for hydrolox, though. I’ve not seen any analysis of a dense fuel engine using alt. comp.

Bob Clark

Last edited by RGClark (2020-08-17 02:35:54)

RGClark · 2020-08-17 06:33:00

GW Johnson wrote:

...
Here is how that plays into the LOX-LCH4 SSTO case: first burn MR-eff dV = 3.0 km/s * factor 1.10 = 3.30 km/s; avg Isp = 343 s for Vex = 3.364 km/s; dV/Vex = 0.9810; MR = 2.6671. Second burn MR-eff dV = 5.0 km/s * factor 1.00 = 5.0 km/s; avg Isp = vac Isp = 356 s for Vex = 3.491 km/s; dV/Vex = 1.4323; MR = 4.1881. Compound (overall) MR for the whole burn is the product of the two mass ratios at the two effective Isp values: MR = 11.1701. The overall propellant fraction is thus 1 - 1/MR = 0.9105. For inert= 8%, the payload fraction is 0.0095 = 0.95%. For delivering 10 metric tons to orbit, the ignition mass is 1052.6 metric tons.
...
GW

For the second, vacuum part of the flight you should have used the Isp of the vacuum optimized engine, ca. 380s.

By the way we can probably get a good estimate of an actual trajectory simulation by using the curve in my last post to represent the Isp according to altitude, at least for hydrolox engines.

Bob Clark

kbd512 · 2020-08-17 10:09:21

Bob,

Increasing the first stage / burn Isp to 360s and the second stage (TSTO) or second burn (SSTO) Isp to 380s won't produce a significantly different result than what GW already provided. The payload performance of the TSTO is at least 4X better for the same 2nd lowest performance propellant combination. Increasing first burn Isp by 4.7% and the second burn Isp by 5.5% is NOT going to make a vehicle that's 4 times heavier a cost-competitive solution. The inert mass fraction of the LCH4 SSTO is 84,240kg. The inert mass fraction of the LH2 SSTO is 17,550kg. You're NEVER going to overcome that with a 5% thrust performance improvement to the Isp of RP1 or LCH4. The documents I previously linked to that do parametric analysis of Falcon 9 performance proves that (actual performance variation 1% or less). For the same payload, the LH2-fueled SSTO requires half an order of magnitude less propellant and structural mass when compared to the LCH4-fueled SSTO. More interestingly, the LH2-fueled SSTO is just 12,000kg heavier than the lightest possible solution using current technology, the all-LH2-fueled TSTO. If altitude compensating nozzles were readily available, the TSTO would still beat the SSTO with any propellant combination and the only LH2-fueled SSTOs would be remotely comparable in terms of propellant burn and structural / vehicle mass.

We have absolutely mastered staging and we implemented it on every single orbital launch vehicle actually flying because it provides more payload performance than any other practical solution using chemical-fueled rocket engines. If there was a practical way to compensate for the pressure decrease with increasing altitude, then we would absolutely use it to improve booster performance. However, we're never ever going to get a better payload mass fraction with a lower performance propellant combination or a solution that requires dragging the entire launch vehicle into orbit and protecting it from reentry if reusability is at all important.

We've come full circle on this altitude compensating nozzle performance improvement issue:

1. TSTO beats SSTO on payload performance in a major way if using the 2nd less performant but easier to store fuel, LCH4.

2. SSTO using the most performant LH2 fuel is the only propellant that comes close to TSTO payload performance, but it will always be heavier than TSTO if reusability after reentry is a consideration.

3. A 5% increase in thrust performance per unit mass of propellant expended by using an altitude compensating nozzle does very little to reduce vehicle structural mass or propellant consumption. The density impulse delta between LH2 and LCH4, per unit mass of propellant, ensures that the structural mass of the LCH4 fueled vehicle could never be better than half as much as the LH2 fueled vehicle, yet in practice we see that the 15% Isp improvement of LH2 vs LCH4, with or without altitude compensating nozzles, ensures that that will never happen.

4. The solution to maximize payload performance and vehicle durability, while minimizing LH2 headaches, is still a TSTO with a booster fueled with RP1 or LCH4 and an upper stage fueled with LH2.

5. The only quantum leap in booster performance comes from doubling the booster Isp using nuclear thermal engines or microwave radiation measured in multiple gigawatts or an electromagnetic launch system that negates the requirement for a booster stage entirely. Between those 3 options, the electromagnetic launch system is the most technologically advanced solution and the only one with a working prototype busily flinging 30t aircraft off the deck of an aircraft carrier.

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