Using Payload Fraction Expressions in an Example

Now that I’ve gotten the math and derivations out of the way, let’s us the payload fraction expressions in a real-world example.

Let’s say you work for the chief technologist of NASA, and he’s thinking about sending humans to Mars. He’s considering whether or not to invest in a seemingly-promising new technology: nuclear thermal propulsion. He’s intrigued by the higher levels of specific impulse that you can achieve with nuclear thermal propulsion, nearly twice that of chemical, but he knows that it will cost billions to develop and test. He wants to know if the technology improves the payload enough to make it worth developing, so he asks you to do a study. He says:

“Assume that you have a nuclear thermal rocket engine with an Isp of 850 seconds and a chemical engine with an Isp of 460 sec. You have a heavy-lift launch vehicle that will put 80 metric tonnes into LEO. How much more payload will the nuclear thermal rocket get over the chemical rocket?”

You being a diligent engineer point out that you need a bit more data to do the analysis, so your boss tells you that you can make some more assumptions.

The NTR has a vacuum thrust of 15000 lbf and a weight of 5000 lbm with an Isp of 850 sec. It uses hydrogen with a density of 71 kg/m3 at a mixture ratio of zero. The chemical engine is based on an RL10 burning LH2 and LOX at a mixture ratio of 5.5 with an Isp of 460 sec. The RL10 has a vacuum thrust of 22,000 lbf and weighs 370 lbm. The LOX has a density of 1142 kg/m3.

For both vehicles, he tells you that you can assume that the thrust structure weighs 0.3% of the total vacuum thrust, and that the LH2 tank has a factor of 10 kg/m3 and the LOX tank is 14 kg/m3. The ullage in both tanks is 3%. You can “rubberize” the engines so that any particular thrust you need to get the stage thrust-to-weight can be calculated. Otherwise you’ll get weird effects from integer numbers of engines.

Since the delta-V to do a trans-Mars injection (TMI) burn from LEO varies from opportunity to opportunity, he wants you to run a sweep of DVs from 3800 m/s to 4400 m/s, incremented by 200 m/s. He’s also unsure of the initial thrust-to-weight that the injection stage should have in LEO, so he tells you to run a sweep from 0.2 to 1.0, incremented by 0.2. With four values of delta-V, five values of thrust-to-weight, and two different engine technologies for each case, he figures that you’re going to be pretty busy for the next few weeks designing forty different trans-Mars injection stages.

Little does he know that you’re a Selenian Boondocks reader, and that you think this would be a good chance to use the payload fraction derivation to simplify your workload substantially. So you reluctantly agree to take on this “huge” analysis effort, and with your head down trudge out of his office.

Meanwhile, you get to your office and call your wife and tell her not to worry, your beach vacation is still a go, and that you’ll be able to finish the analysis he wants by the afternoon.

     \begin{displaymath} {MR} = \exp\left(\frac{\Delta v}{v_e}\right) = \exp\left(\frac{\Delta v}{g\:I_{sp}}\right) \end{displaymath}

     \begin{displaymath} {PMF} = \left(\frac{MR - 1}{MR}\right) = 1 - \left(\frac{1}{MR}\right) \end{displaymath}

     \begin{displaymath} \lambda = \dfrac{(MXR) \left(\dfrac{f_{OT}}{\rho_{ox}}\right) + \left(\dfrac{f_{FT}}{\rho_{fuel}}\right)}{(1 + MXR) (1 - f_{ullage})} \end{displaymath}

     \begin{displaymath} \phi = \frac{T/W_{vehicle-initial} (1 + f_{TSW} (T/W_{engine-vacuum}))}{T/W_{engine-initial}} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{gross}} = 1 - {PMF} - \lambda ({PMF}) - \phi \end{displaymath}

First you’ll need the propellant-sensitive term for both the chemical (LH2/LOX) and the NTR cases.

Let’s assume RL10-class technology for the LH2/LOX rocket engine. That would give us a mixture ratio of 5.5 (5.5 kg of LOX for every kg of LH2) and LOX has a density of 1142 kg/m3 while LH2 has a density of 71 kg/m3. Let’s also assume an ullage factor of 3% and that the tank factor for the hydrogen tank is 9 kg/m3 while that of the oxygen tank is 12 kg/m3. The nuclear thermal rocket will use the same LH2 density value and the same tank technology, but the NTR has a mixture ratio of 0 (0 kg LOX for each kg LH2).

So for the LH2/LOX rocket:

     \begin{displaymath} \lambda = \frac{(5.5) \left(\dfrac{12 \text{kg/m}^3}{1142 \text{kg/m}^3}\right) + \left(\dfrac{9 \text{kg/m}^3}{71 \text{kg/m}^3}\right)}{(1 + 5.5) (1 - 0.03)} \end{displaymath}

     \begin{displaymath} \lambda = \frac{(0.0578) + (0.1268)}{6.305} = 0.0293 \end{displaymath}

and for the NTR:

     \begin{displaymath} \lambda = \frac{(0) \left(\dfrac{12 \text{kg/m}^3}{1142 \text{kg/m}^3}\right) + \left(\dfrac{9 \text{kg/m}^3}{71 \text{kg/m}^3}\right)}{(1 + 0) (1 - 0.03)} \end{displaymath}

     \begin{displaymath} \lambda = \frac{0.127}{0.97} = 0.131 \end{displaymath}

The factor is far, far worse in the case of a nuclear thermal rocket, at over 13% compared to less than 3% for the LH2/LOX case. This is because the only propellant is hydrogen and it has a very low density. So this is a major factor is performance difference between an NTR stage and a chemically-fueled stage–the tankage for the chemically-fueled stage will be far less.

Next calculate the gross-mass sensitive term for the chemical engine and the NTR:

     \begin{displaymath} \phi = \frac{T/W_{vehicle-initial} (1 + f_{TSW} (T/W_{engine-vacuum}))}{T/W_{engine-initial}} \end{displaymath}

In this case, the engine is operating in vacuum the entire time, so the initial engine T/W and the vacuum T/W are the same value.

     \begin{displaymath} \phi = T/W_{vehicle-initial}\left(\frac{1}{T/W_{engine-vacuum}} + f_{TSW}\right) \end{displaymath}

What should the initial T/W of the entire vehicle be? It depends on the circumstance, but a good value for an Earth departure stage would be about 0.5, meaning that when the engine starts the crew will feel about 0.5g of acceleration, which will increase as the stage consumes its propellant but maintains its thrust. Let’s also assume a thrust-structure-factor of 0.003.

The RL10 family of engines have various vacuum T/W ratios. The A4 variant has a vacuum T/W of 60.3, the B2 has 37.3, and the RL-60 engine variant has a projected vacuum T/W of 59.1. Let’s assume a vacuum T/W of 50 for the LH2/LOX engine:

     \begin{displaymath} \phi = 0.5\left(\frac{1}{50.0} + 0.003\right) = 0.5(0.02 + 0.003) = 0.0115 \end{displaymath}

For the NTR case, the engine has a far worse vacuum T/W. Information I’ve read of Stan Borowski’s NTR design (15,000 lbf thrust, 5,000 lbm) yields a vacuum T/W of 3:

     \begin{displaymath} \phi = 0.5\left(\frac{1}{3.0} + 0.003\right) = 0.5(0.333 + 0.003) = 0.1682 \end{displaymath}

Some people favor NERVA technology, which projected a vacuum T/W ratio of roughly 4.0.

     \begin{displaymath} \phi = 0.5\left(\frac{1}{4.0} + 0.003\right) = 0.5(0.25 + 0.003) = 0.1265 \end{displaymath}

Within about ten minutes, you’ve finished your analysis and have the results in front of you:


You are somewhat surprised. Although at higher levels of delta-V the NTR stage has more payload than the chemical stage, it is not nearly the improvement over chemical that you would have first expected by looking at the much higher value of Isp. And the NTR only has more payload fraction than the chemical stage at low values of initial thrust-to-weight. You know that the initial T/W can’t be too low, or else the stage will incur large gravity loss penalties in the form of higher DV. Using two separate perigee burns to do TMI might reduce this somewhat, but that will subject the crew to two extra passes through the Van Allen radiation belts, as well as bring a “hot” nuclear thermal core back within close proximity of the Earth’s atmosphere for the next perigee burn, and that might cause somebody some heartache.

You note that at initial T/Ws of around 0.6 and greater, the chemical stage actually has BETTER payload fraction than the NTR stage. There are two reasons for this–one is hydrogen and the other is engine thrust-to-weight. The hydrogen propellant of the NTR has a really low density, so you get very large tanks and a significantly greater tank penalty than the chemical case. But even worse is the wretched thrust-to-weight ratio of the NTR engine (3.0) versus the chemical engine (~60). With such a low thrust-to-weight, getting the required initial stage thrust-to-weight is very penalizing. So if the initial T/W is not kept low, then the NTR stage can’t beat the chemical stage for payload performance, and there’s really no reason to spend billions to develop it.

But all of this you keep to yourself for just now. Your boss expects you to be cranking the numbers for a few weeks, so you keep the door shut and let him keep thinking that.

(in case you’re curious, here’s the spreadsheet–took me about ten minutes to write)

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MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999

About Kirk Sorensen

MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999
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47 Responses to Using Payload Fraction Expressions in an Example

  1. A_M_Swallow says:

    The other payload fractions worth calculating are:
    solar thermal, similar Isp to nuclear thermal, permitted in LEO;
    nuclear electric;
    solar electric (solar cells and/or solar Stirling).

  2. Solar thermal wouldn’t have enough thrust to be treated by the impulsive delta-V assumption. It would have to be modeled using a low-thrust trajectory tool. Nuclear and solar electric have the same problem.

  3. Nice work Kirk.. I’ll be saving that image with the combined equations, thanks.

    The last 12 hours I’ve been giggling at my handiwork here:

    .. Bolden’s obviously a believer in NEP/SEP, his magic rocket is clearly VASIMR hype.

  4. Good video Trent. Remember STS 61A, January 1986. On board were commander Charlie Bolden, mission specialist Franklin Chang-Diaz, and future senator Bill Nelson. I’m really glad you made this video.

  5. Ann Onymous says:

    Excellent ! So that’s the reason why Stan Borowski desesperately tries to push NTR specific impulse toward 925 – 1000 seconds.
    First, using a “magical” russian NTR engine; later, with the Triton paper engine.
    Does 925 or 1000 seconds change anything ?

  6. I ran the spreadsheet at 925 and 1000 seconds and it bumped the payload performance up a bit, but no where near enough to justify the enormous expense of developing NTR.

  7. Ann Onymous says:

    Or resurrecting NERVA (try to say space cadets that NTR is no better than chemical propulsion !)

  8. Trent,
    Having watched the videos, it’s pretty clear that Charlie didn’t mean under a week by “days”. He was definitely being a bit sloppy with language, but in a way that many people do, especially when under pressure and in front of a camera. But even by your videos, it shows that he immediately clarified in two of the three mentions “or under half the time it takes now”. And in the same comments mentioned it currently takes several months. And in other venues he’s mentioned weeks. I think you’re making the man an offender for the word.

    Is VASIMR overhyped? Sure. Is the 39 days one-way trip that Chang-Diaz mentions ridiculously overoptimistic? Probably. It requires much, much higher power densities than we’ve ever done before in space power systems. But is something close to that physically impossible? No. It might not be engineering possible anytime soon, but that doesn’t mean it’s magic.

    Do I wish he’d focus more on other game changers? Sure. But really you guys need to cut the guy some slack, and quit trying to make him an offender for the word.


  9. Also, regarding the NTR stuff, I’m personally not a huge fan, but aren’t you ignoring some of the other potentially higher T/W concepts like DUMBO and its derivatives? They have issues, but it looks like people are still working on them. Theoretically they should be capable of getting thrust to weights much higher than NERVA, due to the much higher surface area to volume ratio, and much shorter characteristic heating lengths, which should help immensely.

    All that said, I’m fine with just having a depot in L2, using chemical propulsion to get there, and either chemical from there to Mars, or a lower thrust solar electric. You can’t cut the travel times down too far with either of those approaches, but they’re probably good enough for starting things.


  10. A “fast” transit to Mars is about 6 months. A minimum-energy transfer is about 9 months. A 3-month transfer requires an excessive amount of delta-V on both ends and hits you hard in payload fraction.

  11. Martijn Meijering says:

    A 3-month transfer requires an excessive amount of delta-V on both ends and hits you hard in payload fraction.

    On the other hand, if you start from a Lagrange point you lop off 3.2km/s in delta-v. You could use that to buy some reduction in trip time. It’s not free and it would affect IMLEO, but it would be good for mass fractions.

  12. Mmm, kinda. Something had to get you out to that Lagrange point in the first place. It’s not something for nothing.

  13. Martijn Meijering says:

    Sure, it’s just something you’d do to get around the limitations of your transfer stage, not something that would reduce IMLEO. You could even use it both ways, using a Mars Lagrange point too. Huntress et al proposed something similar in their Mars orbit architecture. Their aim was to avoid the need for NTR, although that would still be a very useful addition. And by using SEP to preposition propellant they were able to keep IMLEO to reasonable levels. With NTR you might also be able to transport heavy payloads with a higher Isp than with chemical propulsion. SEP would be more difficult for heavy payloads since it would require very large solar arrays. Interestingly this approach also obviates the need for HLV. That still may or may not be useful for EDL of course.

  14. The characteristics of NTR apparently help you avoid the need for NTR. “Heavy” payloads don’t benefit from NTR anymore than “light” ones. These payload fraction expressions are independent of gross mass.

  15. Martijn Meijering says:

    The characteristics of NTR apparently help you avoid the need for NTR.

    Heheh, nice phrase.

    But the we can widen the trade space a bit. Suppose you used ammonia NTR instead of LH2, which gives you slightly worse Isp, but much greater density. Also consider using a very high delta-v for speed (as opposed to a lot of mass for shielding). Then there’s the option of starting from a Lagrange point without an Earth swingby. That costs you a bit more delta-v, but now you need much less T/W, as the Lagrange points take a month for a whole revolution.

    There might still be a useful application for NTR in there. Of course, my main interest in this is that it avoids the need for HLV, NTR and some other things, but if it’s a nice addition, then that’s good to know.

  16. Throw the ammonia NTR in the spreadsheet and see if it makes a difference.

  17. Leaving from the Lagrange point is a lot more DV, not a little bit more.

  18. Martijn Meijering says:

    Leaving from the Lagrange point is a lot more DV, not a little bit more.

    You mean compared to a swingby, because you now don’t use the Oberth effect? That’s true, but you do have the NTR to compensate for that. It also relaxes your T/W requirements enormously.

    Another interesting application for an NTR transfer stage would be as an uncrasher stage for fully propulsive Mars EDL. If you use the NTR stage to come to a dead stop above the atmosphere, you can do a vertical powered descent with a chemical lander with relatively small delta-v. I don’t like putting NTR on the critical path, but if this eliminates the need for a large heat shield and thus an HLV I’m all for it.

    But your general point is of course correct, you don’t need NTR to go to Mars orbit. With L1/L2 as a staging point, you don’t even need very high chemical Isp.

    I wouldn’t want NTR on the critical path, but I think it would be a very useful addition.

  19. Chris (Robotbeat) says:

    If you’re on NSF’s forums, you know I’m a proponent of solar-electric propulsion (mostly because not enough space-nuts are, not because I think other solutions are incredibly stupid). Are you ever going to tackle low-thrust propulsion, like SEP, NEP, or others?

    BTW, the reason I am pro-SEP instead of NEP is simply because the performance of solar panels are so much greater than nuclear electric, even out to Mars. Thin-film solar panels made like solar sails (i.e. deposited on a gossamer structure) could actually produce 17000W/kg of electricity (although 4000W/kg and 2000W/kg deployed are more near-term). Granted, thin-film solar panels are less efficient than concentrating triple-junction (20% efficient on sunlight received versus 40% for conc 3-junction, both state-of-the-art but improving steadily) so must be larger.

    What is nuclear rated at? 100W/kg, unshielded? Once you put the reactor far away from the crew and add a shield and the support for a large radiator, the weight and structure advantages of nuclear electric power evaporate. We already can do 175W/kg for solar power (it was going to be demonstrated on the ST-8 New Millenium tech demo mission and was already built before it was canceled in 2008/9), and with state-of-the-art concentrating solar power, 10MW at 1AU from the Sun requires only 2 solar arrays each 170 meters square (the ISS is longer than 110m) which isn’t unreasonable if you don’t have any high-thrust orbit boosts. 350W/kg is feasible in the near-term for high-efficiency solar power, with 500W/kg feasible in the near-mid-term with a little extra development. 1kW/kg for high-efficiency concentrating solar is possible mid-term.
    As far as SEP Mars reference missions, this one from 1991 assumes about 350W/kg and the SEP has the lowest IMLEO of any of the other options (cryogenic chemical rockets with aerobraking of the whole MTV, NEP, or NTR) for an opposition-class mission while also being easiest to reuse:

    Specific power determines what your maximum possible acceleration can be at whatever Isp you choose. (There are other electric propulsion systems out there to choose from besides VASIMR, BTW.)

    10MW would probably be enough for a four-crew Mars transfer vehicle if the ascent vehicle was already checked out on Mars’ surface before you left (and again before you separated from the MTV for Martian EDL), especially if you’re doing a Conjunction-style mission. 10MW is huge for a SEP tug.

    From what the Obama budget says, SEP is probably going to be what would be used for getting people to Mars, either for pre-placing equipment (prop depots, etc) or for a Mars transfer vehicle. Of course, make sure that the crew travels through the Van Allen belts with chemical propulsion to avoid unnecessary radiation exposure.

  20. I really wish we could talk about the equations and their implications rather than get off on technology that wasn’t discussed in this thread.

  21. Yes, Chris, I’ve actually got a lot to say about low-thrust stuff. I’ve done a lot of work in that area, which has led me to form some opinions about how it should be done.

  22. Jon,

    Bah!! It’s dishonest to say “days” to mean weeks. So either Bolden means under 2 weeks or he’s being dishonest. It’s not “sloppy” it’s, as we say here, bullshitting. And, frankly, I think saying “days” to mean anything over a week is dishonest too.

    In any case, let’s not be sloppy with language like Bolden. Let’s use numbers. How much delta-v do you need to go to Mars in 39 days? How much delta-v do you need to go to Mars in 7 days? Either way it’s more than anything we have or are likely to have before some new Administrator is in the driver’s seat.

  23. Kirk, sorry for hijacking your thread. The equations are awesome.

  24. A_M_Swallow says:

    39 days is one and a bit months or 5.57143 weeks. Of the 3 alternative times I prefer the integer number.

  25. You’re allowed Trent because you’re straight up right on with what you said.

  26. Robert Clark says:

    Thanks for the calculations. I especially like the fact of your mentioning that propellant tank weight scales by tank *volume*, not the mass of the contents. So the same size and same mass tanks can hold about 3 times the mass of kerosene/LOX than of LH2/LOX simply by kerosene being denser.
    This has extreme importance for creating a SSTO vehicle because the the higher fuel load for a kerosene vehicle due to the lower Isp is significantly less than 3 times more. When you take into account the T/W ratio of the kerosene engines are also about twice that of the hydrogen ones, you conclude the kerosene vehicle using the same size tanks and similar dry mass, would be able to carry *significantly* greater payload.
    See for instance this analysis of a SSTO, though expendable, hydrogen/LOX vehicle from a systems design course:

    Mass Estimating Relations.
    ENAE 483/788D – Principles of Space Systems Design.

    It gives a vehicle capable of a 5,000 kg payload to LEO. But when you switch to kerosene/LOX keeping the same size tanks, you can get about *4 to 5 times higher payload*. To calculate this use the fact that kerosene engines have about twice the T/W of hydrogen ones and a good kerosene engine such as the NK-33 has an Isp in the 330s to 340s range.

    Bob Clark

  27. john hare says:

    The calculations are awesome and very useful to the industry. Even for first order discussions it is good to have a place to point opponents (and ourselves) toward for fact checking. I have been in too many discussions that became a “yes it is, no it isn’t” partly because there were no acceptable methods of calculating the true differences. I’m refering to the honest people, not the closed ear fanatics that won’t listen to or calculate anything.

    Good series, we are enjoying the ride.

  28. Bob,
    Unless you’ve got some volume constraint on your stage, you really need to look at both sides of the rocket equation. For the SSTO case, even with the most optimistic assumptions on the LOX/Kero side (only 8900m/s of dV required due to lower gravity and drag losses, 120 T/W ratio on the engines, 330s Mission Averaged Isp) it still had a lower payload fraction than LOX/LH2 (with 9500m/s required dV, 45 T/W ratio on the engines and 430s Mission Averaged Isp). Now this doesn’t include reusability or other terms. Maybe once you cram those in, they come closer to a wash, or even a win for LOX/Kero, but at this point, based on Kirk’s approach it doesn’t look like LOX/Kero actually wins hands down unless you’re for some reason volume constrained. And all of that’s before adding any cleverness like TAN which tends to help LOX/LH2 more than LOX/Kero.

    BTW Kirk, are you planning on showing how this extrapolates to TSTO approaches.


  29. Yup, Jon, that comes next. I took this Mars stage diversion to show how you could use the equations in a simpler framework before we got to something more complex.

  30. Robert Clark says:

    Jon, are you looking at the numbers for Kirk’s Mars vehicle or the numbers for the SSTO vehicle in that online lecture I linked to? The dense propellant advantage for the Mars vehicle might not be as great since it is travelling in zero-G so the T/W advantage for the dense propellant engines is not as important, i.e., the engines would not have to have higher thrust than the weight of the vehicle, which forces a heavy engine weight for the Earth lift-off case.
    However, the most important advantage is in the propellant tank weight. Most people get the idea of the advantage in dense propellant engines for Earth lift off. It’s the reason why first stages by and large use dense propellant engines or solids for example.
    But not as familiar is the weight advantage of dense propellants for the propellant tanks. The reason as I said is that tank weight scales by volume, *NOT* by the mass of the material contained. It is easy to see why this is so, though. See the formulas for the mass of pressure vessels here:
    Pressure vessel.

    Note that the mass of the tank depends only on the dimensions of the tank, the internal pressure, and the density and strength of the tank material, *NOT* on the mass of the fluid inside. Then since the internal pressure in propellant tanks for dense fuels and hydrogen is about the same, for proper operation of the turbopumps, the same size tank will hold *much* more of the dense propellant mass *without an increase in tank mass*. This is important since the tank mass is the single biggest component of the rocket dry mass. See the masses of the LH2 and LOX tanks in that online lecture on a SSTO. Note their total mass is in the range of 30% to 40% of the vehicle dry mass.
    Then simply filling the same size tanks with dense propellants such as kerosene and LOX, you get about 3 times the propellant mass, *while the mass of the tanks stays the same*.
    The engine weight also will not have to be 3 times as heavy. Since kerosene engines are about twice as good in T/W ratio, they will only have to be about 1.5 times as heavy. Now look at the weights of the LH2/LOX engines in that SSTO study. You see this results in a relatively small increase in the dry weight when switching to kerosene. However, what’s important is that the payload can now be *4 to 5 times higher* using the Isp that kerosene engines such as the NK-33 can achieve.

    Bob Clark

  31. Bob,
    You’re still missing my point. Even after you factor in the larger tank volumes due to the lower density, it only makes so much of a difference. Tankage usually isn’t the heaviest component in a stage, even for low density combinations like LOX/LH2. And tanks aren’t only pressure vessels. For dense propellants, you have higher hydrostatic head than for lower density propellants, so you end up losing some of the advantage, especially for pump-fed tanks which are run at pretty low pressures. Also, lower temp propellants tend to be colder, which for metal tanks makes the tanks stronger…there’s lots of factors.

    My only point was that just using the stuff Kirk has shown in the last several posts, including historical data on tank factors and such, it doesn’t make as much of a difference as you think, unless the vehicle is volume constrained instead of mass constrained. Sure, the tanks for LOX/LH2 will end up weighing more than a similar mass of LOX/Kero due to the lower bulk density, but tanks usually aren’t the heaviest component on a stage, and LOX/LH2 has a higher dry fraction to play with to start with. I can send you the spreadsheet based on Kirk’s approach from his past four posts that shows this.


  32. Brad says:

    I think the presumptions used are unfairly pessimistic towards NTR.

    For example, the biggest drawback of NTR in the results is due to the crappy 3.0 T/W assumed for NTR. But why assume that low a value when a real world example exists that shows a higher 4.0 ratio?

    So how does the stage comparison turn out when the NTR T/W is increased to 4?

    In addition, the NERVA’s low 4.0 T/W was due in part to radiation shielding used to protect the turbopumb. But what if a different design was used instead of NERVA, such as a pressure fed ammonia NTR? Getting rid of shielding for the turbopump should greatly increase the T/W, and I believe loss in ISP from ammonia would be directly proportional to an increase in T/W.

  33. Brad says:

    RE: VASIMR, the magic rocket

    Awesome video Trent! I’m glad I’m not the only one to have noticed the wild claims made recently, and not just by Bolden.

    I’m amazed at the claims being made. VASIMR is nothing but an electric rocket, and one that doesn’t seem to me to have much if any advantage over more mature technology like a Hall effect plasma thruster.

    The only way to achieve super short flight times to Mars is by combining high ISP AND high thrust, and all electric rockets are low thrust. No self-powered pure electric rocket will ever spiral out from Earth and spiral down to Mars in less than a years flight time.

    Only something like the Orion nuclear pulse drive combines high ISP and high thrust, and fat chance of that ever coming to be.

  34. Brad says:

    I wish I could edit post 32, as I posted too hastily. I apologize for the imprecision and sloppy referencing. Oh well this update will have to do.

    In fairness the small NTR engine selected for the stage study does indeed have the low T/W indicated. But I still believe the NTR stage was not fairly evaluated. Because for the thrust level required not only was the small NTR engine the wrong choice, since so many engines were necessary, but also the small NERVA Alpha engine with a 850 ISP was selected rather than the small NERVA Gamma with a 975 ISP. And even the NERVA Alpha was supposed to be capable of 875 ISP in single use high power mode.

    However, the most suitable NTR for the 80 tonne stage study would have been this one…

    Engine Model: Nerva NTR. Propellants: Nuclear/LH2. Thrust(vac): 333.400 kN (74,951 lbf). Isp: 925 sec. Mass Engine: 8,500 kg (18,700 lb). Diameter: 5.00 m (16.40 ft). Chambers: 1. Area Ratio: 500.00. Thrust to Weight Ratio: 4.00. Country: USA. Status: Study 1991.
    Late 1980’s update of 1960’s Nerva design.

    Those numbers reflect most closely the potential capability of the actual hardware that was tested during the 1960’s.

  35. Brad, one of the most fundamental problems of NTR is ground testing. Borowski and his crew made a big push for the 15 klbf engine because it was considered “ground-testable”. The testing techniques used back in the 1960s would never be allowed today, and so this represents a basic constraint on NTR development.

    I think 850 sec Isp was actually being kind because the higher Isps have very speculative performance features. Notice that 500 to 1 expansion ratio on the engine you mentioned? Good luck launching it, much less ever trying to test it.

  36. Brad says:


    I highly recommend the book, “To the ends of the solar system: the story of the nuclear rocket”.

    You might be as amazed as I was to discover just how far the program got in actual development and proof of performance and testing precautions before the program was finally shut down. Not speculative at all.

  37. Read it a long time ago.

  38. Brad says:


    Ah. Then you know they successfully ran a +1,100 MW power engine core for longer than 60 minutes? Of the stable running, and easy to predict nature of NTR? That newer core materials were successfully tested at temperatures that would provide performance exceeding 850 ISP? That an enclosed testing facility was actually built towards the end? In fact I believe the 1,100 MW engine itself was successfully run at levels that would have exceeded 850 ISP.

    Of course, I’m recalling this from memory. Please correct me if I am wrong.

  39. You’ve got all my assumptions and my spreadsheet. Go prove to me that it’s worth the mega-billions to develop and will make even a smattering of difference to a Mars mission that will likely never fly anyway.

  40. 1100 MW sounds impressive until you realize that the thrust power of a large LH2/LOX rocket is typically measured in the gigawatts. Then it’s not so impressive.

  41. Brad says:

    For an in space engine, actually 1,100 MW is pretty impressive. Of course that makes sense when one realized that the hydrogen turbopump for the American nuclear rocket program, which absorbed much of the budget of the program, was reused as the hydrogen turbopump for the mighty J-2 rocket engine used in the Saturn V upper stages. How ironic that the nuclear rocket program was instrumental in winning the moon race.

  42. Robert Clark says:

    I invite anyone interested in this question of the best fuel to use for a SSTO to read this paper by Dr. John C. Whitehead of Lawrence Livermore Labs:

    Single Stage To Orbit Mass Budgets Derived From Propellant Density and Specific Impulse.
    John C. Whitehead
    32nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference.
    Lake Buena Vista, FL July 1-3, 1996

    I believe this will come to be regarded as a seminal paper in the field of SSTO vehicles. Whitehead has his “props” among the amateur rocketry and NewSpace communities because he showed how to make, and he built, reciprocating pumps for small-scale pump fed rocket engines, turbopumps not scaling well to small sizes. I believe his design is what XCOR’s reciprocating pumps are based on for example.
    Whitehead’s paper was also referred to in the book “The Rocket Company”:

    The Rocket Company
    by Patrick J. G. Stiennon & David M. Hoerr
    Illustrated by Doug Birkholz
    ——————————————————————————— Chapter 4: Build Big, Build Many, or Use it Again.
    “Back in 1996, a man named John Whitehead wrote a paper (AIAA96-3108) in which he showed that the weights of typical launch vehicle stages had not changed substantially over time. If tank weights have not improved in fifty years, it’s best not to assume you can dramatically beat history. Whitehead also noted that tank weight as a function of propellant volume was essentially constant over a wide range of tank sizes, from a few tens of thousands of pounds to well over a million pounds. He highlighted the fact that the weight of a tank capable of holding a given quantity of propellant was directly proportional to the propellant density, because the tank weight was almost completely dependent on the volume of the propellants.”

    This is a fictionalized account of the building of a non-governmentally funded orbital rocket written by rocket engineers. But it’s nice to see Whitehead get his props here. The point they are referring to here is that the mass of rocket tanks large and small is still going to be determined by the volume, *NOT* the mass of the contents. So a dense propellant tank will contain *much* more propellant at the *same size and same mass* as one filled with hydrogen.
    In his paper, Whitehead shows why dense propellants such as kerosene are better suited to a SSTO despite their lowered Isp because of the two factors of tankage weight and engine weight on which they are *markedly* better than hydrogen. This is important because the tankage weight and engine weight are the first and second largest components of a rockets dry weight, at least for chemical propulsion.
    Note that even in Kirk’s example the total tankage weight is the biggest component of the dry weight, bigger also than the engine weight. See the numbers in his “Payload Fraction Example Proof” post. There is a slight typo on that page. He wrote “oxidizer tank volume” twice. The second listing must mean “oxidizer tank mass” because the units are given in kg.
    Note that the total of the fuel tank mass and oxidizer tank mass is 1,554 kg, while the total weight of the 4 engines is 1480 lbs., 672 kg. The difference is this big because the engines don’t have to be too heavy since they don’t have to have enough thrust to lift off from Earth. But this is the usual state of affairs even for rockets to LEO, the tank weight being higher than the engine weight.
    Note also that you can calculate the dry weight of the vehicle as 2,347 kg by subtracting the propellant weight and payload weight from the gross weight. So the tankage weight is well over 60% of the vehicle dry weight. This is high again because the proportion of the engine weight is low. But even for usual rockets that have to lift off from Earth, 30% to 40% is common. So the tankage is weight is always a major contributor to vehicle dry weight.

    On the question of SSTO propellant, rocket engineers have been too focused just on the Isp. If they would just run the numbers it would become apparent that dense propellants are the way to go. If this had been realized we would have already had a fully reusable SSTO ten years ago and with launch costs in the $100 to $200 per kilo range. We wouldn’t have this problem of worrying what are we going to do about manned flight with the retirement of the shuttle.

    Bob Clark

  43. Jonathan Goff Jonathan Goff says:

    You’re missing my point still. I know that tanks scale mostly with volume. That’s why Kirk came up with tank scaling factors based on historical masses of tanks made for that specific propellant two posts back. It is true, and I’ve never disagreed that LOX/LH2 takes a hit compared to LOX/Kero. My point has been that you’re not doing the overall numbers, you’re just handwaiving. I’ve run the numbers and the benefit in higher Isp isn’t entirely outweighed by the heavier tanks.

    Taking one realworld example, the Centaur tanks are less than 40% of the dry mass of the Centaur upper stage. They are not 80% or 90% but 40%. Most of the rest of the weight is engines, RCS engines and tanks, electronics, payload support structures, etc. Propellant tank weights do matter, but only so much. If you want to actually run a spreadsheet to prove your point, that might be more convincing than just repeating “propellant tanks only scale with volume” over and over again. Unless your vehicle is volume constrained, the fact that the tanks weigh a bit less than 3x as much for LOX/LH2 isn’t the only important fact, and for SSTO levels of performance is still counterbalanced by the fact that you have over twice the dry fraction to start working with.

    And all of this is before you look at options like Afterburning Rocket Engines (aka Thrust Augmented Nozzles), which can allow you to shift your effective bulk density and T/W on a LOX/LH2 vehicle to something much more competitive with LOX/Kero for at least the boost phase.


  44. Robert Clark says:

    Jon, I did some calculations in the “Payload Fraction Example Proof” thread.
    For the Centaur, it is an upper stage vehicle where high Isp is most important and T/W not nearly so since you don’t even need to have it be greater than 1. Also, quite key for the upper stage case is that making it have a 3 times greater, say, fuel load would have an extreme effect on your lower stage capabilities. This is the main reason why hydrogen is most suited for upper stages.
    I agree with you that tripropellant engines would be even more advantageous for a SSTO because they would allow you to use the dense propellant at launch and low altitude where they are best suited and hydrogen at high altitude. I do want those to be developed as quickly as possible.
    However, I am also arguing that by using dense propellants such as kerosene we can have a SSTO and reusable vehicle *now*.

    Bob Clark

  45. Pete says:

    LH2/LOX has a much greater margin for mass reduction than RP-1/LOX. But historically, I am under the impression that LH2 systems are typically around an order of magnitude more expensive, although the DC-X was comparatively inexpensive.

    A wonder what the cost difference would be between LH2 and RP-1 (or alternate hydrocarbon) in the modern New Space context. No one in New Space seems to be using LH2 just yet, is this likely to change in the not to distant future? Will LH2 systems become much cheaper to develop and use?

  46. Daniel Cooper says:

    Hi Kirk
    I tried to download your spreadsheet but received a “server error” notice.
    If its not too much trouble, can you email me the spreadsheet at
    many thanks – in advance

  47. Sorry about that Daniel. I’ve restored the spreadsheet and the image that was previously missing from the post.

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