- #1
Leo Liu
- 353
- 156
Sutton writes that the mass ratio of a multistage rocket is the product of the individual vehicle stage mass ratios, does it mean the expression below?
$$MR=\prod_{i} \frac{m_{f,i}}{m_{0,i}}$$
I don't think that would be correct. The final mass of a stage contains plenty of mass which you drop before the next stage ignited. That's the very point of having multiple stages.Filip Larsen said:the final mass of a stage is the initial mass of next stage...
For the payload ratio, the structural mass of the i-th stage is included in ##m_{0,i}##. Its both described on Multistage rocket (Wikipedia) and e.g. by Wiesel in "Spaceflight Dynamics" (using the symbols from this thread):Rive said:I don't think that would be correct.
W.E. Wiesel said:... the initial mass ##m_{0,k}## of the ##k##th stage is the mass of everything above the separation plane for that stage. The final mass of the ##k##th stage, ##m_{f,k}##, is the structural mass of that stage, plus the mass of stages still remaining. The ##(n+1)##th stage in an ##n## stage rocket is the payload, mass ##m_*##.
Sorry, nothing to add to the thread, just wanted to say that I love the phrase "Rocket Stack". It's both very high-tech and very low-tech sounding at the same time.Filip Larsen said:but to the rocket stack after each individual staging
You can plug that ratio into the (single-stage) rocket equation as effective mass ratio. The mass ratio passed by each stage combined. Between stages you have to remove the inert mass that's dropped - otherwise this would be a telescopic product and you would simply get the initial to final mass ratio, which is incorrect.Filip Larsen said:I can see that Sutton indeed mentions the overall mass ratio as the product shown in first post, but I do not recall any particular use for it. Anyone?
The equivalent single-stage equation would then be $$\Delta v = -v_e \log MR,$$ with ##v_e## being the effective total ejection speed. If this has to match the multi-stage rocket equation $$\Delta v = - \sum_i v_{e,i} MR_i$$ with ##v_{e,i}## being the effective ejection speed for stage ##i##, then $$v_e = \frac{\sum_i v_{e,i}\log MR_i}{\sum_i \log MR_i}$$ which only reduce to something simple if ##v_{e,i}## is constant for all stages, i.e. if ##v_e = v_{e,i}##.mfb said:You can plug that ratio into the (single-stage) rocket equation as effective mass ratio.
The mass ratio of a multi-stage rocket is the ratio of the initial mass of the rocket (including propellant) to the final mass of the rocket after all stages have been separated and the propellant has been expended.
The mass ratio can be calculated by dividing the initial mass of the rocket by the final mass of the rocket. This can be expressed as a decimal or a percentage.
The mass ratio is important in rocket design because it directly affects the performance and efficiency of the rocket. A higher mass ratio means that the rocket can carry more payload and travel farther, while a lower mass ratio results in a shorter range and less payload capacity.
The mass ratio of a multi-stage rocket is influenced by several factors, including the efficiency of the rocket engines, the amount of propellant carried, the structural weight of the rocket, and the desired velocity and altitude of the rocket.
The mass ratio of a multi-stage rocket can be optimized by carefully designing and selecting rocket engines with high efficiency, minimizing the structural weight of the rocket, and determining the optimal amount of propellant needed for the desired velocity and altitude. Additionally, staging the rocket at the right time can also improve the mass ratio.