# Gasping for Fumes

Finally we take a look at which propellant we should use for our rocket motor here. As mentioned in earlier posts, the two prime candidates for near future warship propulsion are the combustion rocket, and solid core Nuclear Thermal Rocket (NTR).

Some reminders. Combustion rockets tend to achieve up to 5 km/s of exhaust velocity, and NTRs achieve almost twice that at their best. At the same time, NTRs suffer from lower thrust generally. NTRs in Children of a Dead Earth generally achieve around 3000 K temperatures in their reactor, limited by the materials that make up the core. Combustion rockets can achieve greater temperatures, but it’s wholly dependent upon the reaction used and the stoichiometric mixture ratio of the propellants (if using a multi-propellant reaction).

A popular combustion rocket is the LOX/LH2 engine, which uses liquid oxygen and liquid hydrogen to achieve almost 3000 K as well (assuming a 1:1 mixture ratio). This is the main engine of the space shuttle orbiter, and I’ll refer it to primarily when discussing combustion rockets, though later we’ll explore its limitations, and switch to different reactions.

Looking at the energy densities of nuclear energy versus chemical energy, one comes to the realization that nuclear power is roughly 600,000 more energy dense than hydrogen. So how on earth is a hydrogen combustion rocket even remotely comparable to a nuclear rocket?

After all, remember the rocket power equation:

$P=\frac{1}{2} T v_e$

Where $T$ is the Thrust, $v_e$ is the exhaust velocity, and $P$ is the power. If you increase the power by 600,000, either the thrust or the exhaust velocity must also increase by 600,000.

The trouble comes with releasing that power all at once. We have the ability to do so: it’s called a nuclear bomb. However, releasing it in a way that we can control is difficult, and must be done with a nuclear reactor. (This is one reason why the theoretical Nuclear Salt Water Rocket is so powerful: it tries to unlock that 600,000x power factor and still control it.)

A nuclear reactor’s rate of energy release can be seen through how high the temperature of the core can get. This means that between a NTR with a chamber temperature of 3000 K and a combustion rocket with a chamber temperature of 3000K, if they have identical mass flow rates, they must have identical rocket power.

Mass flow rate is how fast you can feed propellant into the rocket, which is governed by the turbopump injector you use to feed the rocket. The flow rate increases with pump size and with pump speed, and in general, is the same between an NTR and a combustion rocket.

This means, given an NTR and a combustion rocket of similar sizes and similar temperatures, the total power is roughly the same. And if we assume the exhaust velocity of the NTR is roughly twice that of the combustion rocket, the thrust of the NTR must be roughly half that of the combustion rocket. By extension, if the NTR has the same exhaust velocity as the combustion rocket, then the thrust must be the same.

A more direct way to see this is to look at the rocket thrust equation:

$T=\dot{m} v_e$

Where $T$ is the thrust, $\dot{m}$ is the mass flow rate, and $v_e$ is the exhaust velocity. It’s obvious from this that given a constant mass flow rate, exhaust velocity and thrust are inversely proportional. On the other hand, in order to increase your rocket’s thrust, you simply need to increase the mass flow rate by using a bigger turbopump.

Essentially, this means the biggest advantage of NTRs, their high exhaust velocity, is the root cause of their lower thrust. Additionally, NTRs which do not have this advantage, the high exhaust velocities, have comparable thrust as combustion rockets!

In Children of a Dead Earth, Methane is the primary propellant used, because it achieves slightly better exhaust velocities than the best combustion rockets, which means in terms of thrust, it’s only slightly worse than combustion rockets. Decane and Water are also other NTR propellants that see heavy use.

But at that point, is there a purpose to using NTRs at all? If we only use NTRs that yield roughly similar stats to combustion rockets, why not just go with combustion rockets altogether? After all, combustion rockets are cheaper, don’t spew neutron radiation, and are somewhat less massive.

The trouble with combustion rockets, particularly the LOX/LH2 rocket, is the propellants. As mentioned in the previous post (Slosh Baffles), each propellant tank has an ultimate mass ratio ceiling. Roughly speaking, higher density propellants have higher allowed mass ratios. Given standard tank materials, water has an excellent mass ratio ceiling (in the hundreds), while hydrogen has an awful mass ratio limit (< 10 generally).

When using a bipropellant (like LOX/LH2), this mass ratio limit is primarily governed by the worst propellant. So in the case of LOX/LH2, the mass ratio limit is extremely low, because hydrogen’s mass ratio limit is low. Compare that to a Water NTR. A Water NTR will achieve comparable exhaust velocities and thrusts, but water is very high density compared to hydrogen, allowing much higher mass ratios.

On top of this, high density propellants allow your ships to be much smaller, making them much harder to hit in combat, and as indicted in earlier posts, minimizing your targetable surface area is critical.

Much more dense chemical propellants are needed to compete with the mass ratio limits. At this point, we have to discard the assumption that we are using the LOX/LH2 reaction. However, when looking into different chemical reactions, one finds that more dense chemical propellants tend to yield much higher exhaust molar mass.

In thermal rockets, the exhaust velocity is based primarily upon the temperature and the molar mass. Chemical reactions that have competitive or better mass ratio limits tend to yield somewhat lower exhaust velocities.

On the flip side, combustion rockets with certain reactions (particularly those involving fluorine) can achieve greater chamber temperatures than NTRs using clever cooling techniques not viable for NTRs. This means the total power of these rockets exceeds that of solid core NTRs. However, they tend to have low exhaust velocities once again, which means the power manifests as much higher thrusts.

Finally, what about the costs of propellants? Unlike just about every other equation in Children of a Dead Earth, determining the cost of something has no hard and fast rules. As a result, propellant costs are estimated primarily based on solar abundance, and on ease of extraction from common celestial bodies. In this way, common NTR propellants tend to be quite cheap. Combustion rockets with high density propellants end up being much more expensive comparatively.

So where does this leave us?

If you want thrust, thrust, thrust, you should go with combustion rockets with high density propellants. Find a reaction with a high chamber temperature and a low exhaust velocity. The high density propellants might comparatively pricey against NTR propellants, though. This sort of drive is generally what most drones and smaller capital ships in Children of a Dead Earth use.

If you want high thrust but still want a reasonable amount of delta-v, NTRs tend to win out with certain propellants like Methane or Decane. This is what ended up going on most large capital ships. These drives tend to be the good-at-everything, excel-at-nothing choice.

And if you want middling thrust and an even higher delta-v, go all the way and grab a Hydrogen Deuteride NTR. Very few ships ended up falling into this use case, though.

Finally, if you want cheap, go for a monopropellant combustion rocket. Good thrust, awful exhaust velocity, but cheaper than dirt. This is what most small, disposable missiles use in game.

And of course, if thrust is totally irrelevant to you, maybe go for an ion thruster. Only a real option if you’re making a non-combat ship and don’t ever plan to dodge. And if you are okay with taking years to get anywhere.

One final note: It may surprise some readers to find that Hydrogen Deuteride ($HD$) NTRs performs better (9.1 km/s) than pure Hydrogen NTRs (9.0 km/s), especially considering that Hydrogen ($H_2$) has a lower molar mass than Hydrogen Deuteride. This surprised me when I saw it as well.

As it turns out, the Gibbs free energy of formation of monatomic Deuterium is lower than that of monatomic Hydrogen, which yields a much lower dissociation temperature. At 3000 K, $H_2$ dissociation is less than 1%, while $HD$ dissociation is nearly 100%, yielding higher exhaust velocities. As a result, $HD$ is both denser (has a higher mass ratio limit) than $H_2$ and has a higher exhaust velocity, making it better in nearly every way. The only real advantage is that $H_2$ is slightly cheaper than $HD$.

And there you have an analysis of the major near future rocket engines that would see use in space warfare.

In the end, however, I am eager to see what sort of rockets that the players of Children of a Dead Earth can come up. Everything from the propellants to the stoichiometric mixture ratio, to the dimensions and shape of the rocket nozzle, to the turbopump injector attributes are editable in game. There are likely plenty of unexplored designs here that may beat out the designs I’ve made.

# Slosh Baffles

So much time is spent discussing rocket engines that one of the most important parts of a rocket is glossed over: the propellant tanks. A significant amount of engineering goes into them, despite their simple appearance.

A rocket’s liquid propellant tank involves a number of considerations, such as propellant boil off, corrosion of the tank material, cryogenic insulation, slosh compensation, and pressurization. In space, without one g of gravity constantly pushing down, simply getting the propellant to the engine is a problem, since there is no force pushing the propellant into the rocket’s thrust chamber.

All of those issues must be dealt with using the least amount of mass, because of the rocket equation.

Indeed, propellant tanks tend to be one of the largest contributors of mass to a spacecraft. An oft made comparison is that rocket propellant tank walls tend to be proportionally thinner than aluminum cans in order to skimp on mass. In Children of a Dead Earth, armor tends to beat out propellant tanks in terms of mass, which immediately begs the question: why don’t we use a Monocoque design?

A Monocoque design builds the propellant tank into the outer skin of the spacecraft, rather than having separate propellant tanks at all, and it is used by a number of rockets in modern times, such as SpaceX’s Falcon 1. Thus, armor doubles as propellant tank as well.

However, when a propellant tank suffers an external force such as that from a projectile, a water hammer, or a fluid shockwave forms within the tank, which can and usually will destroy the propellant tank. Thus, a monocoque propellant tank need not even be penetrated to be disabled. Enemies can simply hit the armor of the spacecraft hard enough to trigger a water hammer, and never even have to get close to penetrating the armor to render the spacecraft unable to move.

As a result, propellant tanks are kept separate from the armor skin. It costs more mass, but not significantly more, because it means the propellant tanks can be made much thinner.

Since Children of a Dead Earth takes place entirely in space without gravity, getting the propellant from the tanks into the rocket engine doesn’t happen automatically. There are a number of solutions to this issue, from ullage rockets (small solid fuel rockets designed to push the spacecraft forward, forcing the liquid propellant to the rear of the craft) to pressure diaphragms to piston expulsion devices.

In Children of a Dead Earth, surface tension devices, or systems which use the surface tension of the propellant to pull it towards the engine, in tandem with a turbopump injector are used. They do not need the additional pressurized gas that pressure diaphragms require, and the turbopump feed needs only slight pressurization of the tanks, yielding thinner walls.

The lower pressure of the propellant tank also is valuable for avoiding water hammers for when the spacecraft undergoes rapid acceleration. Low pressure tanks also makes it easier to compensate for slosh effects with anti-slosh baffles within the tank.

On top of those considerations, cryogenic propellants may need to be insulated, although this is much less of a problem in space, as spacecrafts are not built in atmosphere, and so there is no convection of room temperature air always around the craft. Some propellants are corrosive to a lot of materials, and can only be stored in propellant tanks of certain materials. Finally, cryogenic propellants boil away at a slow but steady rate, and this too needs to be taken into account.

A last note on tank mass. Spherical tanks once again save the most mass, however, cylinders fit much better in cylindrical spacecrafts (discussed in blog post Why Does it Look Like That? (Part 2)), so capsule shaped propellant tanks are used.

All in all, there are a lot of details to worry about, and they all add mass, which is very troublesome for any spacecraft designer. After all, recall the rocket equation:

$\Delta v = v_e \ln \frac{m_0}{m_f}$

Where $\Delta v$ is the final delta-v of your spacecraft, $v_e$ is the rocket engine’s exhaust velocity, and $\frac{m_0}{m_f}$ is the mass ratio, or the wet mass divided by the dry mass. Recall that the wet mass is the total mass of the spacecraft including propellant, and the dry mass is the total mass except for the propellant.

In a previous blog post, Burn Rockets Burn, I went over the fact that the only drives with reasonable thrust for space combat are chemical propulsion and nuclear propulsion. In our case, combustion rockets or solid core nuclear thermal rockets. This limits your exhaust velocity to single digits of km/s. Also recall that while thrust scales up with the number of engines, exhaust velocity remains constant.

This means that the only way to squeeze out any more delta-v for your craft is by increasing your mass ratio. In particular, reducing your dry mass, or adding more and more propellant mass. Assuming you’ve reduced your dry mass as much as possible, getting more and more delta-v is simply a matter of adding enormous amounts of propellant tanks.

Since the delta-v scales with the logarithm of the mass ratio, consider this example. Suppose you have a spacecraft with an exhaust velocity of 5 km/s, either an excellent chemical rocket, or a middling nuclear thermal rocket. A mass ratio of $e$ (Euler’s number, ~2.7) will give you 5 km/s of delta-v. To get twice that, 10 km/s, you need a mass ratio of $e^2$ (~7.4), and to get three times that, 15 km/s, you need a mass ratio of $e^3$ (~20).

To make things less abstract, remember that a mass ratio of 20 means your spacecraft is 19 parts propellant, 1 part actual spacecraft. At that point, your actual spacecraft will be ballooning in size, because even the densest propellants tend to be lower density than the actual alloys and ceramics of spacecraft components.

And how much delta-v do you need? Maneuvering in combat around a planet or high gravity moon, from what I’ve seen, requires around 5-10 km/s to remain effective in combat (for evading missiles, drones, and even the enemy fleet). Somewhat more is needed when planning interlunar transfers, and tens of km/s are needed for interplanetary traveling.

For anyone who has studied the rocket equation, this is all pretty elementary. However, there is one complication that appeared from the rocket equation which was not immediately apparent to me. Once you have a drive chosen, the exhaust velocity is absolute. Thus, to get more delta-v, you get a higher mass ratio, by adding more propellant.

But there is an ultimate limit to your mass ratio, and by extension, an ultimate limit to the delta-v of your spacecraft. Not only that, you can bump into that limit very quickly.

Most of my capital ships run Nuclear Thermal Rockets using Methane as a propellant (for reasons I’ll outline in later posts), which yields an exhaust velocity of about 6 km/s. I started making a tanker with the same engine to refuel my capital ship fleets, and I wanted them to have an enormously high delta-v in order to get just about anywhere. Yet, almost immediately, I started hitting a low delta-v ceiling, no matter how many propellant tanks I added.

This is because every propellant tank added has dry mass in addition to its propellant. Thus, each tank has it’s own separate mass ratio, and a spacecraft can never have a mass ratio that exceeds the mass ratio of its propellant tanks. This propellant tank mass ratio approaches very quickly, and it depends heavily on a number of different factors, primarily the propellant type, the tank material, and the aspect ratio of the tank.

Some numbers here. In game, I optimized Methane propellant tanks to yield a mass ratio of about 30, around 50 for Decane tanks, and several hundred for Water. Most propellants cap out at around 50. Given very exotic and expensive materials, this can be doubled or tripled, though this often runs into corrosion or insulation issues.

So, given a Methane rocket with an exhaust velocity of 5 km/s, and an ultimate mass ratio of 25 from the propellant tanks, the most delta-v your spacecraft can ever conceivably have is about 16 km/s. Yikes! That spacecraft will never make any significant interplanetary journey unless there are copious fuel depots along the way.

This emphasizes just how critical propellant depots are in space travel, and especially in space warfare.

One final way to push these limits are through rocket staging, which involves discarding your propellant tanks after use. However, if your spacecraft is armored, this involves likely dumping off a lot of expensive armor as well. A better way to do this is to take a number of propellant tankers with you, and then scuttling them after you’ve drained them out.

In Children of a Dead Earth, this is the primary way to stage an interplanetary invasion. When there are no allied propellant depots along the way, one has to take a huge number of tankers along for the ride, and then simply scuttle them at various points along the way as they are depleted.

Whew! That’s all for propellant tanks! We didn’t even get to a comparison of propellants in this post. I’ll have to get to them in a later post, but for now, you have all the challenges involved with managing your propellant tanks in space warfare.