Question bank — Pressure-fed cycle — simplest, used in upper stages

Flow sirf high pressure se low pressure ki taraf jaata hai, isliye tank ko har downstream loss "pehle se pay" karni padti hai. Hydrostatic/acceleration head include karte hue: $p_{\text{tank}}=p_c+\Delta p_{\text{inj}}+\Delta p_{\text{feed}}+\Delta p_{\text{cool}}-\rho gh$ Sign convention: $h>0$ ka matlab hai liquid surface injector ke upar hai, toh uska weight (ya thrust-acceleration column) liquid ko neeche push karne mein madad karta hai — isliye minus sign, yeh woh pressure kam karta hai jo gas ko supply karna padta. Coasting ($g\to 0$) ya bahut chhota column is term ko negligible bana deta hai.

$p_{\text{tank}}\approx 1.5p_c$ kyun (loss breakdown) Extra half-chamber-pressure ek typical losses ka sum hai, koi magic number nahi:

• Injector drop $\Delta p_{\text{inj}}\approx 0.25p_c$ — sabse bada chunk, atomization ke liye zaroori.
• Feed-line friction $\Delta p_{\text{feed}}\approx 0.1p_c$.
• Cooling-jacket friction $\Delta p_{\text{cool}}\approx 0.15p_c$. Jod kar: $p_{\text{tank}}\approx p_c(1+0.25+0.1+0.15)=1.5p_c$. Alag-alag designs 1.3–1.5 range mein kahin bhi land karti hain, depending on how these split.

Hoop stress se thin sphere par, $\sigma =\frac{pr}{2t}\Rightarrow t=\frac{pr}{2\sigma }$. Wall mass = (density)(surface area)(thickness): $m_{\text{tank}}={\rho }_m(4\pi r^2)\left(\frac{pr}{2\sigma }\right)=\frac{{\rho }_{m}p}{2\sigma }(4\pi r^3)=\frac{3{\rho }_{m}pV}{2\sigma }$ $V=\frac{4}{3}\pi r^3\Rightarrow 4\pi r^3=3V$ use karte hue. Toh $\boxed{m_{\text{tank}}=\frac{3{\rho }_{m}pV}{2\sigma }}$ — mass seedha pressure ke saath step mein badhta hai.