# Problem in finding Propellant Mass using Rocket equation where total mass includes components dependant on Propellant Mass

Formulation - $$e^{\Delta v/v_{exhaust}} - 1=mass_{fuel}/mass_{structure}$$ $\text{let } k=e^{\Delta v/v_{exhaust}} - 1$ $$\implies mass_{fuel}= k \times mass_{structure}$$ According to SMAD (Space Mission Analysis and Design book), overall tank weight is $1.25 (0.1\times mass_{fuel})$ (10% of propellant and 25% extra of the weight for PMD's and hardware) According to this assumption, $$mass_{fuel}= k \times (mass_{structure-tank}+mass_{tank})$$ $$mass_{fuel}= k \times (mass_{structure-tank}+0.125 \times mass_{fuel})$$ $$mass_{fuel}\times (1-0.125k)=k \times mass_{structure-tank}$$ As k can reach 50, $mass_{fuel }$ would become -ve which should not be so. I would be grateful if you could tell me what exactly I have missed. Thanks!

• If I'm understanding correctly, the 1.25(0.1 x mf) formulation determines what k is; you can't vary k while keeping the 1.25 and 0.1 constant. (Not sure why tank weight isn't just 0.125mf; is there something else being let out?) – Russell Borogove Mar 14 '18 at 18:17
• @RussellBorogove $k=e^{\Delta v/v_{exhaust}}-1$, so k is a constant. Tank weight is 0.125 mf. – Infi Mar 15 '18 at 2:52

K value is too large. In most cases major $\Delta V$ is provided by gravity assists. So if you consider $\Delta V$ performed just by the propulsion systems, $k$ value would decrease drastically. An excerpt from SMAD $-$