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The question Is there a maximum $\text{I}_{sp}$? reminded me that I once read somewhere that the maximum possible $\text{I}_{sp}$ for a rocket engine based on expansion driven by exothermic, chemically reacting propellant(s) is about 450 seconds.

(Actually I read 4500 m/s and I'm just dropping a zero instead of dividing by 9.8.)

Is this about right? if so, how was that actually determined? Was there just limits on kCal/mole and kg/mole and some thermodynamic wisdom, or something more?

note: if there is a more accepted, concise term than "exothermic chemical reaction rockets" or "rocket engine based on expansion driven by exothermic chemically reacting propellant(s)" please tell me! I want to say 'ya know - normal rockets'.

edit: the search site:space.stackexchange.com 4500 m/s turns up a number of locations here. For example this question says:

$\Delta v$ from surface to LEO is 9000 m/s,

typical exhaust velocity $v_e$ = 4500 m/s

But is this a practical limit? I don't mean what's the highest $\text{I}_{sp}$ chemical engine demonstrated, I really would like to learn about a thermodynamic limit based on chemistry and thermodynamics.

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  • $\begingroup$ (also, I asked a helper question on Chemistry.SE) $\endgroup$
    – SF.
    Jul 11 '16 at 6:48
  • $\begingroup$ @SF. OK then that really brackets reality. That can be taken as an absolute upper limit with the constraints of a chemical reaction driving thrust composed of the reaction products - 'a normal rocket' for lack of a better term. It also explains the quote in the answer below. Thanks! $\endgroup$
    – uhoh
    Jul 11 '16 at 7:58
  • $\begingroup$ IIRC from Thermodynamics of Propulsion, the limiting factor in an exothermic chemical rocket ends up being the nozzle--which makes sense, because that's the part that changes the high pressures and temperatures produced by combustion into exhaust momentum. Most of the relevant equations are at web.mit.edu/16.unified/www/SPRING/propulsion/UnifiedPropulsion6/… $\endgroup$
    – Erin Anne
    Oct 12 '16 at 4:55
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450-455s Isp is typical of H2/O2; according to the Huzel and Huang data, a hydrogen-beryllium mix combusted with oxygen can hit ~540s. The numbers in that table are for moderate chamber pressure and expansion ratio; higher values are possible.

According to Wikipedia:

The highest specific impulse for a chemical propellant ever test-fired in a rocket engine was 542 seconds (5,320 m/s) with a tripropellant of lithium, fluorine, and hydrogen.

I don't know what the theoretical limits are. I know that with complex molecules in the exhaust, substantial kinetic energy is held in the form of vibration within the molecular bonds, not contributing to thrust -- this is one of the reasons H2/O2 engines are run hydrogen-rich; H2 doesn't flex the way H2O does -- so looking for high-energy reactions in complex compounds comes with diminishing returns.

A Bruce Dunn post on yarchive claims without citation:

Isps in the mid 700s are not even theoretically possible, let alone a practical proposition.

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    $\begingroup$ OK! I think this summarizes the chemical $\text{I}_{sp}$ landscape nicely. The general neighborhood of 300 has many possibilities, including solids, hybrids, and LOX/Kerosene; low-to-mid 400's is the home of LOX/LH2, the highest $\text{I}_{sp}$ propellant system that is widely used; exotics up to the low 500's; and an absolute ceiling (for chemical reaction product thrust, for lack of a better term) below 700. Thank you! $\endgroup$
    – uhoh
    Jul 11 '16 at 8:21
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    $\begingroup$ Fluorine-containing oxidizers are horribly impractical as discussed here: space.stackexchange.com/questions/1415/… -- so in practice when you want high Isp you look to hydrolox, and if that's not sufficient you go to electric thrusters and get an order of magnitude Isp advantage at the price of extremely low thrust. $\endgroup$ Jul 11 '16 at 17:58
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The theoretical limit is set by the specific energy of the reaction of combustion of the propellant.

Knowing specific energy $e$ of given substance, we can put a cap on obtainable specific impulse $I_{sp}$ by assuming 100% of conversion of chemical energy to kinetic energy.

$$ I_{sp} = {v_e \over g_0} $$

$$ E_{chem} = e m \geqslant E_k = {1 \over 2 }{ m v_e^2} $$

$$ v_e \leqslant \sqrt{2e } $$

$$ I_{sp} \leqslant {\sqrt{2e } \over g_0} $$

How close we can approach to this theoretical limit is the matter of engineering and construction of the engine. For example, for the common LH2+LO2 cryofuel, the specific energy is 13.43MJ/kg.

$$I_{sp} \leqslant {\sqrt{2 \cdot 13430000 {J\over kg}} \over 9.8 {m \over s^2}} = 528.8s$$

The practically obtainable 455 seconds of Specific Impulse for the Space Shuttle mean the SSME achieved 86% of that theoretical maximum (the rest obviously dissipated as heat in the exhaust gasses).

The most energetic reaction seems to be (although the claim is unsourced) oxidation of beryllium At 23.9MJ/kg it would purely theoretically allow 705 seconds of specific impulse. Purely theoretically, because beryllium oxide is a powder, so there's no adiabatic expansion of gas which creates propulsion.

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  • $\begingroup$ I wish we had a better word than "theoretically" here. I can't think of one, but it's really just an absolute upper limit. Well know, unavoidable thermodynamic realities will probably lead to a lower number without question, so there's no actual "theory" that says it can be 705. But ya people frequently use "theoretically" like this. I never would have thought to use kinetic energy to establish the upper limit but it makes sense. Thank you for putting this into such nice, quantitative terms! $\endgroup$
    – uhoh
    Jul 11 '16 at 8:35
  • $\begingroup$ @uhoh: The upper cap might be reached if the exhaust gas temperature drops to substrates temperature before it stops accelerating within the engine (providing propulsive force). You might exceed the cap even, if you start with a very hot propellant and considerably cooler exhaust gas. (technically, that's how a steam rocket would work.) But since the exhaust gasses are very hot, and building a nozzle long enough would be totally counter-productive (surface flow friction!) this is not going to happen. $\endgroup$
    – SF.
    Jul 11 '16 at 8:42
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For most rocket fuels calculating the specific energy (the energy released per unit mass), assuming a 100% conversion to kinetic energy (as this is a theoretical limit) and calculating velocity from that will give you a good estimate of specific impulse.

If you want a better estimate you can adjust for energy lost from the enthalpy change of vaporisation and the initial thermal energy of the fuel, however both of these are pretty negligible and will only change your number by a few seconds.

As for ways that engines could be made more efficient I would suggest checking out rolling detonation engines

(link to Scott Manley's excellent video on the subject) these are a type of engine which uses detonation rather than deflagration (burning). This means that the expansion of the fuel would take place at constant volume rather than constant pressure. This process is massively - around 25% - more efficient - note this is a 25% increase in efficiency, not specific impulse (25% more efficiency means 25% more kinetic energy and therefore 11% higher velocity) although this would still be a gigantic leap forward, giving us Hydrolox engines with around 500s Isp.

Current engines also tend to run mixture ratio's which are fuel rich. This leaves some fuel un-combusted/partially combusted, reducing the specific energy (energy density) of the O/F (oxidiser/fuel) mixture. However, it is worth it because it results in the exhaust species having a lower molecular mass.

Thermal energy is stored in three forms (rotation, vibration and translation), of these, only translation (movement) can be converted into kinetic energy of the rocket. Therefore it is worth it (to a point) to sacrifice energy density (and therefore temperature) for a lower molecular mass and the resulting increased efficiency in the conversion of chemical to kinetic energy. This is shown by the the equation for thermal velocity (below)

v = sqrt(3kT/m)

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    $\begingroup$ "...of these, only translation (movement) can be converted into kinetic energy of the rocket." That's sort-of true, but these degrees of freedom are not necessarily decoupled. During expansion, as the temperature associated with translational motion drops, energy stored as rotation and vibration will tend to re-equilibrate with that of translation at each collision. A "hot, vibrating" molecule can give a kinetic-energy, translational "kick" to another molecule when they collide. $\endgroup$
    – uhoh
    Sep 15 at 12:23
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    $\begingroup$ Strictly speaking the equipartition theorem applies to systems in equilibrium, but it also tells us that even when temperature (and pressure) is changing, the different partitions will try to equilibrate amongst themselves if given an opportunity. $\endgroup$
    – uhoh
    Sep 15 at 12:25
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Cosgrove and Snyder (1952) found the heat of formation of BeO from the combustion of beryllium foil in oxygen gas to be 143.1kcal/mol, corresponding well with the "Std enthalpy of formation" -599kJ/mol stated in the "Thermochemistry" section for Beryllium oxide at Wiki. Each is equivalent to 23.9MJ/kg with three sig-figs.

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  • $\begingroup$ Thanks for your answer. This value is already reported in this answer along with a statement that the reaction product BeO is a solid (even at very high temperatures). This means that the reaction would not produce a high velocity exhaust, especially after expansion. And without a high Isp, this isn't really an answer to the question. $\endgroup$
    – uhoh
    Feb 20 '18 at 23:50
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    $\begingroup$ @uhoh: your link states "the claim is unsourced". I provided a source, along with additional verification. $\endgroup$ Feb 21 '18 at 1:16
  • $\begingroup$ Ah! OK I understand. This would be a good and helpful comment, but the Stack Exchange interface requires a reputation (points) of 50 to start leaving comments on posts other than your own. This can be a little frustrating I know. In general it's not allowed to post comments as answers in SE. Someone may come along and move this answer to a comment there, or you can try (though I'm not sure how) to expand this to a proper SE answer. A question receives +5 per up vote, an answer +10 so it doesn't take long to get to 50. $\endgroup$
    – uhoh
    Feb 21 '18 at 1:24
  • $\begingroup$ Welcome to Stack Exchange! If you have a minute you can take the tour or visit the help center for more info. $\endgroup$
    – uhoh
    Feb 21 '18 at 1:24
  • $\begingroup$ It's probably worth mentioning that beryllium oxide is carcinogenic and a lung irritant. $\endgroup$
    – PM 2Ring
    Sep 15 at 23:03

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