Is there a maximum Isp for "exothermic chemical reaction rockets"?

Question

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.

Answer 1

450-455s I_sp 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.

Answer 2

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.

Answer 3

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)

Answer 4

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.