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.
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:
and for the NTR:
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:
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.
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:
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:
Some people favor NERVA technology, which projected a vacuum T/W ratio of roughly 4.0.
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)