Rocket fuel efficiency and payload capacity rule of thumb

[jonesclai]

"A rule of thumb is that for every ten-percent increase in efficiency for rocket fuel, the payload of the rocket can double.”

Can someone assist me with a simple formula to model this rule-of-thumb?
What happens to the payload if the rocket fuel efficiency increases twenty to thirty percent?

 

[K^2]

I'm not sure what you mean by "efficiency" of the rocket fuel.

At any rate, what you are going to be looking for is rocket formula.

$$v_f = v_i + v_p ln\left(\frac{m_i}{m_f}\right)$$

This applies to a single stage. m_i is the total mass of the rocket when the stage engages, and m_f is the total mass right before the stage separates, or the final weight if this is the final stage or single stage rocket. Both of these include payload.

v_p is the effective exhaust velocity. It is also the specific impulse (I_SP) per mass of propellant. If specific impulse is given per weight of propellant (s^-1 units), you have to divide it by 9.8m/s² to get the effective exhaust velocity.

10% increase in efficiency might mean a 10% increase in v_p. Or it can mean a 10% increase in fraction of kinetic energy that goes into exhaust, in which case, you are only looking at 5% increase in v_p. Or it could be something entirely different.

The effect on the payload will depend on the number of stages, the mass of the rocket itself, and on the specific impulse you started out with.

For insertion at LEO, the rocket needs to achieve approximately 9km/s, including losses to gravity and drag. A good fuel, like cryogenic H2 with O2 oxidizer, with good engine will have I_SP of a little over 3km/s. So the m_i/m_f is roughly exp(3), or around 20. Because of the engines, tanks, and all the other heavy junk you need for rocket to fly, payload fraction ends up closer to 1:100. In other words, only about 20% of the mass that's being lifted is in the payload. This is neglecting all the complexity of the rocket actually being multi-stage, but it is good for an estimate.

Suppose I increased v_p by 10%. The ratio goes down from 20 to 15. Assuming that the weight of the rocket itself did not have to be increased, all this goes into payload. So if before for every unit of payload, you needed 4 units of rocket mass and 95 units of fuel, now the same 95 units of fuel will lift 5.8 units, of which 4 are still the rocket, and you end up with 1.8 units of payload.

This is very close to doubling the payload that the rule of thumb suggests, so I suspect that this is what it is all about. Hopefully, you can follow all this logic to see what would happen with 20% and 30% increase.