Rocket equation revisited

Rocket equation has been the primary operating principle and the most limiting factor to space travel. Space is expensive due to the energy and mass of the propellant required to perform any of the orbit launches or maneuvers. This is a well know equation:

$$$\Delta v = u \ln \frac {m_0} {m_e}$$$

Common knowledge derived from rocket equation is that better $$$I_\text{sp}$$$ means better performance. So the holy grail of rocketry in general have been the pursue of higher exhaust velocities regardless of costs. Liquid hydrogen/liquid oxygen was frequently observed and recommended as the best performing fuel/oxidizer combination that outperforms all other combinations. It is just not powerful enough for initial lift from the ground.

The goal of this blog is to investigate behavior of rocket equation not in relation of masses, but volumes and densities. That is, how to compare different propellants based on the same volume, and not mass. The reason for this fallacy is when comparing masses, propellant with better specific impulse always wins. But the cost of the rocket components do not scale with mass of the propellant, they scale with the mass of the structure needed to be manufactured. So we can rewrite the rocket equation as follows:

$$$\Delta v = u \ln \frac {{m_p} + {m_s} + {m_f}} {{m_p} + {m_s}}$$$

where $$${m_s}$$$ is the mass of the structure (tanks, engines, pressurants, valves, avionics etc.), $$${m_p}$$$ is the mass of the payload and $$${m_f}$$$ is the mass of the propellant in the tanks.

The goal here is to see how does the payload mass $$${m_p}$$$ depends on tank volume and propellant density. We can replace propellant mass with $$${m_f} = {\rho _f} {V_f}$$$, where $$${\rho _f}$$$ is average propellant density and $$${V_f}$$$ is the total propellant volume of the stage.

Now comes the key assumption: the mass of the structure is proportional to propellant volume, not to the mass of the full stage. Typical stage has to have structural strength to carry itself and the payload through all acceleration stages, but typically much larger contributor to the mass is tank pressure, and is the main cause why tank mass is proportional to its volume and pressure. So is the mass of the pressurization gas (typically helium) and its tanks. For engines, their mass is mainly proportional to the volume flow of the propellant (turbo pumps, pipes, combustion chamber and nozzle), not to mass flow. Since main components of the stage dry weight are proportional to its volume, I assume that the same holds for the whole stage as the approximation. In this case, structure mass can be written as follows:

$$${m_s} = {\rho _s} {V_f}$$$

Finally we shall also introduce function $$$E(\frac {u} {\Delta v} )$$$ in order to simplify equations. This function is interesting because needed velocity change can be viewed as a constant needed by the target mission profile, while specific impulse depends on the choice of engine and its propellant

$$$E(\frac {u} {\Delta v} ) = \frac 1 {e^{\frac {\Delta v} {u}} - 1}$$$

This function is nearly linear in the range between 0.5 and 1, which is important to note in later discussion why propellant density is more important than specific impulse in the given range.

Now, the rocket equation can be rewritten as:

$$${m_p} = {m_f} E - {m_s} = {\rho _f} {V_f} E - {\rho _s} {V_f} = {V_f} ( {\rho _f} E - {\rho _s})$$$

We can define new rocket parameter, named "specific payload density", which shows payload mass per unit of propellant volume.

$$${\rho _p} = \frac {m_p} {V_f} = {\rho _f} E - {\rho _s} = \frac {\rho _f} {e^{\frac {\Delta v} {u}} - 1} - {\rho _s} $$$

The final equation expresses payload mass as a function of propellant volume, propellant density, structure density, required velocity change and specific impulse. This function will be very useful when evaluating different propellant combinations.

The interesting part is to investigate how our revised rocket equation behaves for different values of input parameters. Although rocket equation seem to have exponential dependency between specific impulse and payload mass, it is important to note that in the typical region of current stages and propellant combinations this dependency can be approximated with a linear function. Therefore we can conclude that for a typical launch vehicle, payload mass is linearly proportional to propellant volume, propellant density and specific impulse for any of the target $$${\Delta v}$$$. Therefore, ideal propellant is not the one with the highest specific impulse.