Quick notes on MAV / Mars Direct surface refueling... · 11:07pm Sep 21st, 2020
So, in The Martian (and The Maretian) the Mars Ascent Vehicle manufactures its own fuel for taking off from the Martian surface very slowly, using a radioisotope thermal generator and stored hydrogen plus compressed Martian atmosphere to cook up methane and oxygen.
How much sense does this actually make?
Going by the round numbers, a fully fueled MAV ascent vehicle carries about twenty tons of fuel at launch. Since The Martian doesn't say word one about oxidizer, when I was writing Maretian I treated that number as being inclusive of fuel and oxidizer alike. But I'll also make a second calculation that assumes (a) that the number ONLY refers to methane fuel, and (b) that the fuel plant also produces oxygen and stores it, so oxidizer doesn't have to be hauled from Earth.
So, first let's suppose that twenty tons is the fuel-and-oxidizer load all put together. How much is which?
If memory serves, the equation for methane combustion is:
CH4 + 2 O2 = 2 H2O + CO2
(check: yep, got it right)
One mole of methane is 16 grams. Two moles of molecular oxygen is 64 grams. So, by that logic, four-fifths of the weight of the rocket fuel is oxygen. That means, if 20 tons includes fuel and oxidizer alike, all that's necessary is enough hydrogen to make four tons of methane. On an atomic level, one-quarter of the mass of methane is hydrogen, so one ton of hydrogen would be enough to make four tons of methane. And since the carbon for the methane and the oxygen all come from the carbon dioxide of the Martian atmosphere, one ton is all that needs to be shipped- saving nineteen tons of mostly oxidizer.
(Note: in The Martian at one point one of the characters claims that "one kilogram of hydrogen is enough to make thirteen kilograms of fuel". That doesn't work out to me. A single carbon atom (C) has an atomic weight of 12 CH would be atomic weight 13, but CH is an impossible molecule; it would work as an ion with an enormous charge (+3 if I remember high school chemistry right), and it would react with something IMMEDIATELY. And since Andy Weir used the Mars Direct proposal as the foundation for Ares in his story (with substantial changes), I'm going to presume the fuel being made is methane and not some other fuel I know nothing about which only requires one mole of hydrogen per twelve moles of Everything Else. Instead I'm choosing to believe that the other nine kilograms of stuff Bruce Ng talks about (pg. 327 hcvr ed) is the oxygen left over from splitting CO2 for its carbon atom- which justifies my initial thought that the numbers given were fuel and oxidizer combined.)
Now let's do the math if we're talking about making twenty tons of methane and enough oxygen to go with it.
First, as I mentioned, it takes two moles of oxygen, or 64 grams, to burn 16 grams of methane. So if we're burning twenty tons of methane, we're using up eighty tons of oxygen. And, again, one-fourth of methane's mass is hydrogen, so to make twenty tons of methane you need five tons of hydrogen. In this scenario, making your fuel and oxidizer on-site saves a whopping ninety-five tons of Earth launch weight- or a bit more than a fully loaded, legal limit container truck.
Sounds good, right?
There's one huge caveat, though.
Depending on scenario, you're saving either nineteen tons or ninety-five... less the weight of the fuel plant.
Now, the RTG is man-portable in Mars gravity and is never described as particularly heavy or bulky in the story. That's the least of our worries. And the equipment to first split carbon dioxide apart and then cook methane, though probably not man-portable for this purpose, is fairly lightweight on the scales we're talking about, especially considering the whole has to be able to run on one hundred watts or less.
But then there's the problem of compressing these products down to liquified gases for storage, pumping and use. Methane has to be chilled down to 112 Kelvin for compression and storage. Oxygen? Even colder- 90 Kelvin. (The mean surface temperature of Mars, by the way, is about 210 Kelvin.) Now, we know this can be done feasibly using Ares-era equipment- after all, the Hab atmospheric regulator uses fraction-separation to manage oxygen and nitrogen levels, and Watney was able to move the regulator by himself from the Hab into the "trailer" for the drive to Schiaparelli. But we also know the regulator is bulky- big enough that it's taller than the rover cabin. So the weight here is, put simply, non-trivial, and Watney probably had one hell of a job making it happen. We can also expect its equivalent in the MAV fuel plant to be on the bulky side.
Still, managable, so probably under a ton of weight. Still plenty of savings, right?
Nope. Because here comes the biggest problem of all: keeping the hydrogen you sent up from Earth stored until you need it.
Liquid hydrogen is COLD- 33 Kelvin at the warmest. And it has this annoying habit of warming ITSELF up, because hydrogen. And gaseous hydrogen not only seeks out seals and pipe threads and the like and flits through them, it has been known to even leak THROUGH SOLID TANK WALLS without a breach- the hydrogen molecules, so far as anyone knows, just find gaps in the molecular lattice of the metal. Long-term storage of liquid hydrogen is a problem humanity hasn't solved yet except by brute force, i. e. bringing along a lot more hydrogen than the minimum you need.
So there's the real killer: systems to keep the hydrogen chilled and compressed for up to three years, while slowly drawing off hydrogen all the while. That will be HEAVY, especially since, with Mars's very thin atmosphere and the fact that you have red-hot electrolyzing action (reducing carbon dioxide) RIGHT NEXT DOOR in the fuel plant, refrigeration will have to be INDUSTRIALLY strong to maintain safe operating temperatures for the hydrogen.
But, you say, isn't there a lightweight compound with hydrogen that could be used instead, to keep it stable in containment? And the answer is: yes, and it's called METHANE. Substituting lighter elements for carbon- boron, beryllium, or lithium- gets you less hydrogen and, thus, less energy density for more weight. (Also, beryllium hydride only decomposes under fairly high temperatures, and lithium hydride has a melting point only slightly less than that of lead, so getting hydrogen atoms off of either is, as engineers put it, a non-trivial difficulty.)
So, how much weight would the Ares MAV fuel plant actually save? I don't know, because I don't know how much the systems required to contain the hydrogen would weigh. But it sure as hell doesn't save nineteen or ninety-five tons, depending on scenario.
Anyway, enough note-taking for my maybe-never companion volume about The Science of The Maretian. I have a stream beginning in less than an hour, and I need to get ready for that.
https://twitch.tv/redneckgaijin - Tonight and Wednesday, 7 PM Central, it's more Stellaris.
Turns out there are already some alloys that are known not to be susceptible to hydrogen embrittlement, like austenitic stainless steel — namely the 300-series alloys that SpaceX is presently working with for Starship and Super Heavy. There’d likely still be diffusion through the tank walls, but at least you wouldn’t have to worry about the hydrogen tank cracking upon landing.
Instead of shipping liquid hydrogen, though, how about shipping solid (frozen) hydrogen instead? In solid form I’d think there’d be fewer problems with diffusion. Yes, it would have to be kept even colder than liquid hydrogen, but it’s already been used as a cryostat to actively cool several space telescopes via sublimation cooling, so there’s already experience with handling it. And given that the James Webb Space Telescope will be relying on radiative cooling rather than solid hydrogen, I’d be surprised if some way for long-term storage of solid hydrogen couldn’t be developed in the next couple of decades. Heck, we might even get lucky and find a metastable form of metallic hydrogen and a way to produce large quantities of it — though I wouldn’t hold my breath.
Not gonna lie, FiMFiction is not the place I ever expected to get a crash course on space travel and high technology, but, here we are.
Whats the hydrogen, mass ratio for Ammonia? Hmm. lose one, gain 2, ah.. I subtracted instead of adding, dont mind me.
Advanced material, graphene? weirdly, its claimed hydrogen cant pass through it at all, but water vapour can, at least through graphene oxide sheet? I claim molecular quantum chaos.
If the hydrogen is going to leak out of the tank anyway, why not use that as the long term vent rating, where some is scavanged off to drive fuel cells to power sytems for pumping the stuff back before it gets too warm? the External Tank was supposed to be good for a month on Earth at 6 inch foam? Maybe with cold nitrogen gas buffer or one of those Saturns F1 engine or SABREs helium cooling loops?
I thought it was possible, between Water, CO2, Methane, Nitrogen, Oxygen, to create a cascade refrigeration system, which if theres 20 ton of water locally, you dont even need to haul ALL the hydrogen over? Maybe they can do that experiment with the next skycrane? Have it drop to ground level and run as long as it can to try and blast the surface away to ice or at least permafrost?
At least they aint so desperate yet to take flax seeds with them to grow the fibrous material needed to build a Pycrete MAV onsite.
Well, RTG's tiny energy output is kinda show-stopper: 1kg of stoichiometric fuel-oxidizer pair burns with ~3KWh of energy output, which someone needs to put in first. Hydrogen stash helps a bit, but even with optimal reaction:
2H2 + 3CO2 -> CH4 + 2O2 + 2CO
enthalpy change is only tiny bit better than what burning yields. Although, rocket engines are usually run somewhat fuel-rich.
There is no actual way you can store liquid hydrogen for 3 whole years without losing a sizeable chunk of it through vapourization and leakage, even with our best available tech today. That's why hydrogen-fueled rockets keep refueling until the last second before launch. Also why leaks are factored into any efficiency calculation for hydrogen fuel cells.
The most realistic thing I think is for that hydrogen to be kept in a form stable at room temperature until it is needed, i.e. water. I don't remember every detail of the book but I do remember Wattney urinating to fill up the H20 tank on the MAV's fuel plant and joking about being so epic he pisses rocket fuel. That kind of implies not all the H20 the MAV fuel plant needed was available when he arrived in Schiaparelli, so they probably didn't bring everything from earth. They probably harvest Martian permafrost to top up that H20 reserve.
5360990
Remember that Watney needed the MAV to not only reach Mars orbital velocity (as it was designed to, for rendezvous with a Hermes parked in Mars orbit) but Mars escape velocity in order to roughly match Hermes’ velocity as it flew by on the Rich Purnell trajectory. He needed to squeeze every bit of delta v out of the MAV as possible — hence the MAV stripped-down soft top conversion with full-to-bursting propellant tanks.
5360874
Solid Hydrogen, kept at a balmy 13K, has a density ~825 times worse than liquid hydrogen. To get minuscule, momentary amounts of what we think is metallic hydrogen, you're looking at >400GPa shock wave pressures, which just isn't doable.
5360990
I tried looking up what the actual rate of hydrogen leakage from a normal gas cylinder, and all I could really find is that it's close to that of methane, see here. Another article claims ~3x methane leak rates, which, as methane has 4 hydrogen atoms, means that methane loses more hydrogen initially?
I think it was quoted to me once that liquid h2 leaks between 5-15%/mass per year. Probably lower on larger tanks, but higher with higher pressures? I forget if it's only for mixed gasses, but part of me remembers that when going from liquid to gas phase, you can get a spike in pressures that can rupture a tank.
The real concern, I think, would be long-term hydrogen embrittlement, paired with cosmic ray exposure on the tank walls on the journey over. I feel that while methane might have a static rate of leakage, that the hydrogen storage solution would just accelerate over time.
Since both carbon dioxide and water are available locally, I think the plan has to be to create and use the hydrogen in situ, not to send a vast quantity of hydrogen and hope it won't evaporate.
5360990 Your suggestion is essentially to reverse the combustion equation using a lot of time, electricity, and Martian CO2.
Take the combustion formula again:
CH4 + 2 O2 = 2 H2O + CO2
Now let's render that in atomic / mole weights:
16 + 64 = 36 + 44
Reactants and products balance out at a total of 80 units each side.
The original plan is to send up hydrogen - 4 units- and get the other 76 units on-site.
Sending up enough water to make the fuel and oxidizer means sending up 36 units and getting the other 44 on-site (basically only the CO2).
Now, this would definitely solve the hydrogen storage problem... short term. But you now have the problem of capturing, compressing, and storing hydrogen, which is VASTLY more difficult than merely keeping hydrogen in storage once you've got it. And you've got a lot less weight savings in exchange for solving that problem.
And if you're going to do that, you're slightly better off just sending the methane (which weighs two grams per mole less than water does). The only difficulty then becomes how your oxygen plant disposes of all the carbon it's prying out of the CO2 while it makes your oxidizer.
5360990 The MAV had used up all the hydrogen it had by the time Mark got to Schiaparelli. But apparently it hadn't used up its fuel/oxidizer storage, which makes sense because you'd want some fudge factor in case hydrogen leakage is greater or less than anticipated. That said, the MAV fuel plant wasn't equipped to electrolyze water. Mark had to do it himself in a rigamarole that involved the air pump in the rover's airlock.
5361076 Water is there, yes, but for the most part it's not in a form that an unmanned probe can access. You'd pretty much have to plan for human intervention. Otherwise you're going to have a situation much like, well, the Insight probe's mole.
I mean, I can't really speak to the tech that was available at the time of Andy Weir's writing, but iirc there's already been substantial work done on a sort of metal-alloy sponge a-la aerogel that can store hydrogen with very little leakage.
From what I can find, modern spacecraft hydrogen tanks generally have a tank-fuel mass ratio of around .12 or so (the Space Shuttle lightweight tank, the second of three variants used and the only one for which I can find separate masses for the fuel, oxidizer, and intertank sections, was .124, which is absolutely abysmal by the standards of other rocket propellants- its oxygen tank was about 40% of the hydrogen tank's mass and contained about six times as much propellant for a ratio of .009. First stage tanks are probably going to be heavier than payload tanks because they have to be insulated against Earth's fairly warm atmosphere at sea level and the enormous amount of heat generated by rocket engines and they have to be structurally strong enough to handle the aerodynamic forces of launch since they're not safely enclosed in a fairing, but since the MAV's tanks would also have to be able to handle the heat from the fuel plant and presumably bear at least some of the landing stage's structural loads, they'd probably be similar).
Assuming ChrisCornflake's stated most pessimistic estimate of 15% lost right through the tank wall per year over a three-year mission (in engineering, especially aerospace engineering, you have to always design things around the assumption that your worst prediction is what will wind up happening), you'd lose about 45% of your tank's contents- let's call it 50% here for both a safety margin and a nice round number to work with. If you need five tons of hydrogen for your methane, then you need to send 10 tons, plus 1.2 tons of tank to store it. Ultimately, you save 7.8 tons of mass, plus 80 tons of oxidizer. The actual numbers would be different if 20 tons is the complete mass of propellant including oxygen, but the proportions would be the same- you'd need one ton of hydrogen to make four tons of methane, so take two tons and a 240-kg tank, save about 1.76 tons of fuel and 16 tons of oxygen. You do add some additional weight from the fuel plant itself, but you save more than that again by being able to use smaller engines and less fuel for the descent stage's landing system, since it has much less weight to land.
I think the 100 tons of propellant total is probably more accurate- googling "delta v mars surface to orbit" just brings up results for getting to LMO from Earth rather than from the surface of Mars, so it's difficult to figure out the actual requirements, but from what information we have about the MAV capsule (seats six with minimal elbow room plus 500 kg of samples and experiments and a week's worth of food, has no abort or landing systems), I'd estimate a fully loaded MAV capsule probably weighs about the same as an empty Dragon 2 or Starliner (note that the figures I could find for those include their service modules), putting it around 9-10 tons. Considering that a rocket that can put about that much mass into low Earth orbit generally uses around 300 tons of fuel to do it (for example, the Atlas V 401 carries about 305 tons of fuel and can put about 9.7 tons in LEO, and its combination of kerosene and hydrogen fuels make it probably the closest equivalent to an all-methane rocket currently operating and I can't find planned fuel capacities for Vulcan, New Glenn, and Starship), I think 100 tons sounds reasonable for Mars.
In summary, you're saving something like 85-90 TONS off your Earth launch mass by sending hydrogen even if you send twice as much as you actually need. That's an insane amount of mass savings- only four rockets capable of launching just that much mass into LEO, never mind to Mars, have ever been built, and none of them could get all that fuel and the spacecraft using it to Mars. And even that comes with some substantial caveats- the N1 and Energia could have ONLY carried the fuel (and the N1 not quite even that, as its LEO capacity was 95 tons) even if they'd worked and not been cancelled, the Space Shuttle's ~125 tons to LEO includes the mass of the orbiter, which includes all sorts of necessary stuff for the mission after the launch like, you know, the crew cabin and its occupants, but also means that 95-99 (depending on the orbiter) tons of that capacity isn't going to be going any further than LEO, and over half the Saturn V's 140 ton LEO payload was the third stage and most of its fuel, which was used for trans-lunar injection. A Starship fully refueled in orbit that itself makes no attempt to land on Mars or return to Earth might just about be able to pull that mission off, but it'd be close and also pointless because Starship is designed to be able to do that sort of mission itself without using a separate lander.
5360893
Have you read CSP?
5361033
Well, how about hydrogen slush, then? You’d still need a lower storage temperature than liquid hydrogen requires, plus added equipment to maintain the slush state over the life of storage (vacuum evaporation to cool, mechanical agitation to break up the solid hydrogen that forms in the liquid, rinse, repeat as necessary). It’d be 16% - 20% denser than liquid hydrogen, according to Wikipedia, so the reduction in tankage mass might just offset that of the more complex cooling equipment sustaining long term storage of the slush. And NASA did a good bit of research back in the mid ’90’s into using hydrogen slush as a propellant for the late National Aerospace Plane project, so there’s already a starting point.
5361101 Some numbers from The Martian:
Low Mars orbital velocity is 3.8 km/s, as opposed to just over 7 km/s for Earth. (Not mentioned in the book: Martian surface gravity is 3.8 m/s, as opposed to Earth's 9.8 m/s. That by itself means a significant fuel savings compared to an LEO launch.)
The two ascent stages plus capsule of the MAV, put together and fully loaded, are just under thirteen tons plus fuel/oxidizer. (I'm guessing the capsule alone has a tare weight of three tons or so, justified because it's a one-use short-term craft that only has to withstand Mars air resistance after initial launch, so it doesn't have to be as robust as Dragon.)
And finally, the spacecraft hydrogen tanks you mention are for craft with less than three weeks of deployment time. I still believe a really long-term hydrogen storage system would have to be a lot more robust.
5361033 I think the bigger problem is that as the tank becomes emptier, the rate of hydrogen escape will increase, as more gaseous hydrogen builds up in the tank and produces more spontaneous warming which, you guessed it, causes more hydrogen to evaporate. And embrittling will eventually lead not to faster leakage but a tank explosion.
Remember talk in Apollo about the "helium burst disc"? The lunar module descent engine used helium to backfill its fuel tanks to maintain pressure and thus make it possible to re-ignite the engine several times as needed (which was lucky for Apollo 13). The problem was, once you started using helium you couldn't keep the rest of it contained, so pressure in the helium storage tank would build rapidly as liquid helium turned to gas. In order to control the point of failure and make sure a rupture didn't damage anything, a weak point was built into the system- the "burst disc"- to direct the rupture in a safe direction.
They did this because they couldn't contain helium in a less-than-full tank for longer than three days. Hydrogen is worse.
5361355
Couldn't you counter the partially empty helium tank issue by having semi-compartmentalized tanks? I'm imagining a bunch of bubbles with small tubes connected them, each tube with a cutoff.
I was under the impression the embrittlement was caused by hydrogen making physical paths through material, nudging other atoms around a bit. Combined with cosmic rays which tend to punch holes in everything, yeah, not good times. But this is known ahead of time, so one can over-spec the tank that I would expect it to have a MTBF well past mission spec. I don't know how one could prevent the leakage though, as multitudes of holes measured in angstroms are not easy to deal with.
Actually, I was under the impression that helium is worse than hydrogen due to it being completely neutral. Guess I was wrong, it's only around 2/3's as permeable as hydrogen.
5361139
Yeah, that might work. I didn't know about that before.
5361362 Compartmentalized tanks or multiple tanks are heavier. The LM was built so an incautious fat-finger moment in the right place would have breached the hull, it was that lightweight. And it only had to work for a few days. My point, though, was about the containment problem getting worse as the tanks empty.
As for helium, yes, it's unreactive. But it's also four times the mass- four times as large, basically- as hydrogen. That makes it harder to squeeze between atoms of other stuff.
5360990
Modern MRI machines don't require helium refill for their whole lifespan, which may be decades (well, in principle --- if you're repairing something you'll probably need to refill). Weight, lack of maintenance and cost seem like more relevant constraints.
5361407
Kinda counterintuitively, helium nucleus has twice the charge and as so keeps electrons closer and it's atom is actually smaller. Although, for the same reason it likes to polarize more, so it has marginally greater van der Waals radius (which is more relevant for diffusion). Last detail here is that hydrogen is molecular and in the end H2 is bigger than He but twice lighter and as so faster (under same temperature).
Thanks for the dive into this. :)
5361355
Ah. I forgot the most obvious source- the actual book being discussed. Well, I feel stupid. So, according to that, it takes 4.1 km/s to get the MAV to Hermes. Unfortunately, it still doesn't tell us the mass of the stages individually or the specific impulse of the engines (or even how many of them it has- just that the first stage has more than one), but if we take some rough numbers and assume the first stage represents roughly 60% of the total mass and the combined second stage and capsule account for the other 40%, and the engines are comparable to vacuum-optimized Raptors (the only methane engine that's gotten as far as airborne tests and is actually designed to work on Mars) in terms of efficiency, we have some numbers to work with. If we assume a three-ton empty capsule plus 600 kg of people, spacesuits, and personal effects, 500 kg of samples and experiments, and, say, 250 kg of miscellaneous mass- RCS fuel, cabin air, food and water in case of a missed orbit scenario, etc.- then we can call the capsule 4,350 kg and the booster is 8,250 kg of tanks and engines.
Based on all of this, I still think it's 20 tons of fuel and 100 tons of total propellant- it looks like most real rockets have a roughly 1:10 ratio of tanks/engines to fuel (generally not including payload), which lines up reasonably well with 8 tons of booster to 100 tons of fuel here, and I can't imagine they'd ever downgrade to such an appalling ratio as 1:2.5 for use on Mars.
As for hydrogen tanks, I did a bit more digging. Most discussion of long-term hydrogen storage is for use as a form of energy storage on Earth, generally as a way to store power generated by renewable sources during periods of sun and wind so that things keep working in dark and calm, so they don't need to worry about their weight. However, I did find this paper summarizing the various cryo tanks built by Ball Aerospace (formerly Beech Aircraft) over its history, including the fuel cell reactant tanks for Gemini, Apollo, and the Space Shuttle. While these hydrogen tanks still operated on much shorter scales than a Mars mission would, it's at least longer than the mere hours a launch stage tank would need. They also describe a couple experimental tanks built for NASA during the 70s that, while they never flew themselves, were the basis for the Shuttle tanks and designed as technology demonstrators for aerospace tanks. The Hydrogen Thermal Test Article had a tank-hydrogen mass ratio of 1.3 and the paper indicates they leaked at a rate of about 8% per year. It also mentions that their tanks are actually heated during operation to keep their pressure high enough for their contents to remain a supercritical fluid, so being near the fuel plant might not actually be such a problem. The best indication of where technology is at at the time of The Martian would probably be the hydrogen tanks on Blue Origin's Artemis lander, which will need to have a mission endurance of a couple months, but pretty much no information on these tanks other than that they contain hydrogen. is publicly available right now, and with it being privately developed it might never be.
5361563 The LM ascent module had a fuel:dry weight mass ratio of less than 1.
If NASA could make a 5:2 fuel:dry weight ratio rocket work from the Martian surface to low orbit work, you bet they would- and they'd keep looking for ways to reduce that even more, because that savings means a smaller, lighter rocket that can carry more payload.
The 10:1 ratio you mention for rockets is purely a function of the fact that we live in the fifth deepest gravity well in the solar system excepting the sun's itself.
5361795
Upon further consideration, you were right. I've decided to break out the actual rocket equation here. Unfortunately, the numbers will be VERY fuzzy because we know next to nothing about how the rocket's staging is set up aside from that there's two of them. Assuming the ratio of tanks/engines to fuel is the same for both stages (which it probably isn't, but since all we know about the engines is that they run on methane and oxygen, are optimized for near-vacuum pressures, and weigh less than four tons, there just isn't enough information to go on here) and they get the same specific impulse and use the same fuel mix, the mass of each stage based on my earlier estimate of 60-40 first stage:second stage and capsule can be determined by getting the ratio of the entire stack and subtracting the capsule's estimated mass from the 40% to determine what percentage of the fuel and booster goes to the second stage. If we assume the book's stated 19,397 kg (which from this point will be rounded to 19,400 for convenience- it's three kilograms, I think I can get away with that considering how rough all the other numbers here are anyway) of propellant includes both fuel and oxidizer, then the total mass of a fully loaded MAV stack is 32 tons. 40% of that is 12.8 tons, so that's the entire second stage and capsule, which I estimated at 4,350 kg fully loaded. The total mass of booster is therefore 27,650 kg. 19,400 kg of that is fuel, which accounts for 70.2% of the booster's total mass and the vehicle itself 29.8%. Applying this ratio, the second stage carries 5,123 kg of propellant (all masses rounded to the nearest kg for these purposes). The first stage masses 19,200 kg, of which 13,748 kg is fuel.
Since the only methane-fueled vacuum-optimized engine in a reasonable state of development we really have the numbers for is the SpaceX Raptor, I'll use its specific impulse of 380 seconds (obviously the MAV wouldn't actually use Raptors both because it's indicated that Lockheed-Martin built it and I doubt they'd use SpaceX engines and because a single Raptor produces about 484 tons of thrust, which would be pretty much instantly fatal to anyone riding a 32-ton rocket with that sort of ridiculous amount of power- and the first stage has more than one engine, too). This provides all the numbers necessary for the rocket equation: Δv=veln(m0/mf).
ve is effective exhaust velocity, or specific impulse in units of time multiplied by standard gravity in meters per second squared (standard gravity is Earth's 9.8m/s2 regardless of what celestial body or lack thereof the maneuver is taking place on or around). The effective exhaust velocity here is 3,724. m0 and mf respectively are the vehicle's initial total mass and final mass after performing the maneuver, both in kilograms. These need to be calculated separately for each stage- we already got m0 for each stage as 32,000 kg for the whole stack and 12,800 kg for the second stage and capsule, and the final mass is a simple matter of subtracting the propellant masses from their respective stages, giving mf as 18,252 for the first stage and 7,677 for the second stage. With these numbers, the first stage has 2,091 m/s and the second has 3,410 m/s, for a total of 5,501, well over the ~4,100 the book says the MAV has. This suggests my numbers for the mass ratios were well off the mark and/or Weir was imagining substantially less efficient engines than the Raptors, but it still definitely shows that 19,400 kg is enough total propellant for the job even if you used less efficient engines. So, yeah, my bad on that. I should have broken out the actual rocket equation from the start rather than comparing it to existing rockets that operate in a very different environment.
So, what does all this mean for the original question of whether they'd really benefit from shipping up liquid hydrogen (or more accurately, supercritical fluid hydrogen, since that generally stores more easily) and having to send large tanks and expect to lose about half of it? If you need one ton of hydrogen, assume you'll lose half of what you launch over the mission (considering that the Beech Aircraft HTTA that's been around since the 70s only had a loss rate of around 8% per year, probably a ridiculously pessimistic estimate, even before considering that Watney's ability to have the MAV fully fueled and then some by the time he leaves long before Ares IV was planned to use it indicates that it doesn't actually need the entire time it's on the surface to make fuel anyway, but you can't be too careful here), and your tank(s) are 1.3 times the mass of their contents, you'd need 4.6 tons of hydrogen and tanks, compared to the 19.4 tons of propellant you would have had to launch- and some of that 2.6-ton tank mass can also be a load-bearing part of the descent stage's structure.
If you launched that hydrogen as water, you wouldn't need as much of it, but you'd also be carrying a bunch of oxygen with you, and the oxygen is the heaviest part of the equation. An oxygen atom weighs 16 times as much as a hydrogen atom, and therefore for one ton of hydrogen in water you'd need eight tons of oxygen, plus however much the tank weighs- not nearly as much as the hydrogen tank, undoubtedly, but since you're sending up nine tons of material to the hydrogen-only setup's 4.6 tons even before you consider the tank, you're not saving anything unless you manage to make a tank with negative mass- and if you've got negative mass, all talk of making methane for rocket fuel is irrelevant because negative mass lets you travel faster than light. However, water does have one significant advantage: no extra carbon.
One molecule of methane needs two molecules of oxygen to burn. Each oxygen molecule has two atoms of oxygen, meaning that the carbon dioxide that gets split to provide the one carbon atom in this equation only provides half the oxygen needed. If you send water, then that covers the rest of the oxygen, meaning you only need to extract enough atmospheric carbon dioxide to produce the methane and nothing more, since that'll give you a total of sixteen tons of oxygen between the carbon dioxide on Mars and the water you brought with you. However, if you just send hydrogen, then you have to process twice as much Martian atmosphere to get the oxygen and just discard half that carbon. And that carbon has to go somewhere. Carbon's not super reactive, so these free carbon atoms are unlikely to eat through the fuel plant like free oxygen might, but carbon loves to form polymers and massive crystalline structures like graphite and diamond. This stuff will wreak havoc on a machine it forms in- the polymers that manage to form in the extreme conditions found in jet and rocket engines burning kerosene are bad enough. Imagine if you had solid graphite forming in the fuel plant's mechanism. I have no idea how the MAV gets rid of this extra carbon or how much that system weighs, and certainly not whether it would be enough to justify the extra 5-ish ton mass of sending hydrogen as water rather than just hydrogen. And the carbon issue would go double if you launched just the methane itself and extracted all your oxygen from the atmosphere- you'd just be launching four tons of fuel rather than 4.6 tons of hydrogen and its tanks, but you'd have to get rid of all the carbon you pulled out of the atmosphere.
5362453 Re: the excess delta-V in your calculations, remember that efficiency is lost on launch between:
(a) cancelling out Mars's gravity (though I think you cover that in the equation... maybe...);
(b) air resistance (Mars doesn't have much, but it has some, and it goes up a lot higher than Earth's); and
(c) trajectory adjustments (because you're not starting up four hundred miles above the surface pointed the exact correct way; you're on the surface pointing at right angles to the trajectory you want to achieve).
As for the engine(s), it's safe to guess that some combination of Lockheed and NASA developed lighter-weight engines that provide sufficient thrust while not being as heavy as the Raptor. Looking it up, a single Raptor engine has a dry weight of 1.5 tons (or that's their design goal, anyway). For the first stage three or four smaller engines can get almost the thrust of one big one, and for the second stage you really, really want a smaller engine.
And yeah, carbon disposal is a problem the boffins will have to solve to make this work. The whole rigamarole really only saves weight if hydrogen is the only thing shipped up. Shipping up water requires an electrolysis / hydrogen capture system be added to the stack, which means more weight. (Andy Weir made it clear such a rig was NOT in the MAV, hence the "pissing rocket fuel" scene.)
To do the math in round numbers, out of the twenty tons (20,000 kg) of propellant in a maximum load MAV, water shipped from Earth would make up exactly nine tons of that. This saves eleven tons of weight at Earth launch... less the water tanks, less the electrolysis plant, less the fuel and oxidizer plant, less the tankage for the water, less the RTG, and less the structure that has to be built to hold all of it.
At what point do the fuel savings not justify the danger of system failure forcing a mission abort?
No, it's really got to be hydrogen-only for this to work on a single-use basis. A permanent Mars base could have a spaceport with a fuel plant that could refuel multiple ships, which really would save launch mass from Earth, but not a single-use ship system.
Fun fact, the mass of a molecule is almost always less than the sum of the mass of it's constituent atoms. But the difference is not very significant and for your purposes it's negligible.