Rocket Engine Fuel ratio

[ciscosdad]

I have just been reading the Jenkins book on the space shuttle.
There was a brief mention in the section on engine development, that the F/O ratio differs with altitude (along with bell nozzle size) for the same engine design when used at Sea level vs use at altitude. I assume that more H at high altitude and more O at launch from density considerations.
Is anyone out there familiar with the fundamentals in this case? I could find no direct reference in Sutton.
I am hoping for some reference that can clarify the application in my mind, especially from an engineering perspective.
eg
Could the ratio be varied continuously or have to be fixed at engine start?
Does any engine use continuously variable O/F ratios during operation?
Would there be enough benefit to justify the complexity?
I must admit the only engine I'm aware of that operates from launch to orbit is the SSME.
I'm assuming that the driving issue is the overall fuel density vs Specific Impulse. First stage benefits aerodynamically from  higher density while upperstages benefit more from the higher impulse.
Any rocket scientists out there willing to take pity on a techno-peasant?

[butters]

I am not a rocket scientist, but it's my understanding that the optimum mixture ratio (for engine performance) becomes more oxidizer-rich from sea level to vacuum.  But for LV mass ratio and aero drag, many hydrolox stages run oxidizer-rich throughout. 

Wikipedia says that hydrolox mixture ratios are optimally 4.13 at sea level and 4.83 at vacuum.  SSME (and RS-68) run at 6.  However, Vulcain (Ariane 5), the other major launch-to-orbit hydrolox engine, runs at 4.7, close to the ideal stoichiometric ratio.

SSME uses two separate fuel-rich preburners to drive the fuel and oxidizer turbopumps.  The mixture ratio is controlled via the two valves that throttle oxidizer flow into the fuel and oxidizer preburners. 

These valves control the thrust level additively and the mixture ratio differentially.  The Main Engine Controller uses the fuel flowmeter and the combustion chamber pressure transducer to calculate both thrust and mixture ratio, actuating the preburner valves to maintain the commanded thrust levels and the fixed mixture ratio.

Theoretically, the MEC *could* receive mixture ratio commands and actuate the preburner valves accordingly.  But I assume that designing injector plates and combustion chambers that can operate at different mixture ratios is non-trivial, much like designing for deep throttle ratios is difficult.

And as you've probably deduced by this point, the engine performance and mass ratio  considerations diverge through ascent, with the engines wanting more oxidizer at high altitude and the vehicle structure wanting to discharge more oxidizer at low altitude.

In the end (with hydrolox), the structure wins out, and the vehicle always runs oxidizer-rich at sea level if not also at vacuum.

[sdsds]

I am neither a rocket scientist nor a chemical engineer but it is my understanding that most hydrolox engines are run hydrogen rich because H2 is a (much) smaller molecule than H2O, and smaller molecules in the nozzle provide better performance than large ones.

Am I deceived, butters?

[Proponent]

The stoichiometric mixture ratio for hydrolox is O/F = 8.  Hydrolox engines always run fuel-rich.

My recollection is--and this could be wrong--that the mixture ratio in the J-2 was varied continuously to ensure that both fuel and oxidizer were depleted at the same time.  At the very least, the ratio changed from one burn to the next:  see the entries for "Launch Vehicle Propellant Usage" in Apollo by the Numbers (and following pages).

EDIT:  Note that second-stage burns were less fuel-rich than third-stage burns, which agrees with the OP's suggestion that higher density and lower Isp are optimal at earlier phases of flight.

[Proponent]

A chemical rocket engine does two things.  Firstly it burns propellants to produce heat.  This process is most efficient, in terms of heat produced per mass of propellant, at the stoichiometric ratio.

In the second step, the engine converts heat into kinetic energy, i.e., into the flow of exhaust gases out the nozzle.  For a given amount of energy, the momentum is maximized by making the gas molecules as light as possible.  Hence, in this phase a hydrolox engine ideally has a large excess of H₂ molecules over H₂O and other combustion products.

Optimum performance is necessarily a balance between the requirements of the combustion and acceleration phases.  In the case of hydrolox, broadening the scope to consider not just the engine but the entire vehicle pushes one to leaner mixtures because of hydrogen's low density.

EDIT:  There's a nice discussion of mixture ratios in Chapter 7 of Clark's book Ignition!.  It addresses other issues, like the need to keep chamber temperature low enough.

[93143]

Actually, to first order Isp is independent of molecular weight, for a given mass flow rate and combustion power.  You can show this with the equations of momentum and energy:  F = mdot*v_exh, eta*P = 0.5*mdot*v_exh^2.  Notice how molecular weight doesn't show up?  Given mdot and P, and choosing a fixed value of the efficiency eta, you can find F and v_exh without ever knowing M.

The big reason hydrogen is so great for chemical rocketry is that it has a ridiculously high specific heat of combustion, so you don't need very much of it mass-wise; the ratio P/mdot is large.

The advantage of running fuel-rich comes from the vagaries of the chemical behaviour of the combustion products during expansion through the nozzle (resulting in efficiency becoming dependent on the mixture ratio), and from the lower chamber temperature (since lower molecular weight means lower temperature for the same energy per unit mass).

If you've got an engine that's solidly temperature-limited, like a nuclear thermal rocket, molecular weight is the dominant parameter, which is why no one considers propellants other than hydrogen and maybe ammonia for NTRs.

[Proponent]

Maximal embarrassment.   

[gospacex]

Actually, to first order Isp is independent of molecular weight, for a given mass flow rate and combustion power.  You can show this with the equations of momentum and energy:  F = mdot*v_exh, eta*P = 0.5*mdot*v_exh^2.  Notice how molecular weight doesn't show up?

Molecular weight should improve v_exh, no?

[93143]

No.  Work the equations.  With fixed efficiency eta, combustion power P, and mass flow rate mdot, they are solvable as given.

The trick comes in with the efficiency.  Mixture ratio affects it.

[butters]

The stoichiometric mixture ratio for hydrolox is O/F = 8.  Hydrolox engines always run fuel-rich.

So I'm misunderstanding the numbers here?:

http://en.wikipedia.org/wiki/Liquid_rocket_propellants#Bipropellants

Now that I think about it, the atomic masses for H2O suggest O/F = 8...

[ciscosdad]

Interesting.
Its my understanding that the exhaust velocity is inversely proportional to the average molecular wt of the gases. Perhaps that is where the MW enters your equations (hidden in the Ve term).
In any case, low mol wt is desirable. My query was regarding variation in the mixture ratio. Is there any benefit in continuous variation?
I expect its complex to implement, but would there be any theoretical basis for such a practice?

[93143]

Its my understanding that the exhaust velocity is inversely proportional to the average molecular wt of the gases. Perhaps that is where the MW enters your equations (hidden in the Ve term).

Nope.  Take a close look at the energy equation:

eta*P = 0.5*mdot*v_exh^2

You can solve for v_exh without knowing M.  All you need are mdot (a controllable design parameter), P (which is mdot times specific heat of combustion of the mixture),  and eta (which is generally fairly close to 1, at least for a wide-nozzle engine in vacuum).

Given these parameters, what the molecular weight of the combustion products tells you is what the chamber temperature is (temperature is proportional to energy per particle, and when you have more particles sharing the same total energy, the temperature is lower even if the total mass is the same).  Chamber temperature can be important, of course, but it doesn't directly affect the performance.

A chemical rocket will always have the highest ratio of P/mdot at the stoichiometric mixture ratio.  However, the finite expansion ratio, combined with second-order chemical effects during the expansion, mean that the highest value of eta*P/mdot, at least for a hydrolox engine, is achieved with a fuel-rich mixture.

In other words, molecular weight is hidden in my equations, but in the efficiency eta, not the exhaust velocity.

On the other hand, a nuclear rocket works not by dumping a given amount of energy per unit mass into the propellant, but by heating it to a certain temperature.  At a given chamber temperature, P (and thus v_exh) is strongly dependent on molecular weight.  Thus low M is absolutely paramount for a nuclear thermal rocket.

In any case, low mol wt is desirable. My query was regarding variation in the mixture ratio. Is there any benefit in continuous variation?
I expect its complex to implement, but would there be any theoretical basis for such a practice?

Yes.  The efficiency depends on the exit pressure.  I'm not entirely sure how it works with a fixed nozzle, but with an altitude-compensating one the optimum definitely moves with altitude.

[ciscosdad]

This is fascinating.
And rather different to my previous understanding. Fuel rich but not for the reasons I expected
Just for clarification:
Is the 6:1 ratio for the SSME by Wt,Volume or molecular ratio? I would have expected Wt.

[93143]

Yes, 6:1 is the LOX/LH2 mass flow ratio.

[ciscosdad]

OK.

What actual effect does changing the ratio make?
If I were to change from (say) 5:1 to 6:1 to 7:1, what effect would this have on the operation of the motor?
How would the Isp change? I assume temp would rise as the stoichometric ratio approaches.

BTW: Thank you for your patience and your responses