• Fuel-rich staged combustion (FRSC): hydrogen-rich turbine gas — SSME / RS-25.
• Oxidizer-rich staged combustion (ORSC): oxygen-rich turbine gas — RD-180 / RD-170.
• Full-flow staged combustion (FFSC): do preburners, ek har ek ka — SpaceX Raptor.

Blank: main combustion chamber mein route karke dobara jalaya jata hai. Kyun matter karta hai: koi bhi propellant waste nahi hota, isliye chamber pressure aur specific impulse dono ko zyada push kiya ja sakta hai. Yahi "do baar jalao" trick hai — staged combustion ke efficient hone ki poori wajah. Dekho Specific impulse and chamber pressure.

Stoichiometric LOX/LH₂ ~3300 K par jalta hai, jo turbine blades ko turant pighla deta hai. Rich chalane ka matlab hai ek reactant excess mein hai; woh excess mass bina zyada energy add kiye heat absorb karta hai, isliye flame temperature drop ho jata hai ek range tak (~900–1000 K) jo turbine metal survive kar sake. Combustion deliberately incomplete hai — yeh main chamber mein finish hota hai. Dekho Combustion stoichiometry and flame temperature.

Kya karein: $P_{\text{pump}}=\frac{\dot{m}\Delta p}{\rho {\eta }_p}$ mein plug in karo. Kyun: pressure energy per volume hai; volume flow $\dot{m}/\rho$ se multiply karne par power milti hai, aur ${\eta }_p<1$ se divide karne par waste heat account hoti hai. $P=\frac{90\times 3.5\times 10^7}{423\times 0.7}=\frac{3.15\times 10^9}{296.1}\approx 1.064\times 10^7\text{W}\approx 10.6\text{MW}$

Kya karein: pehle exponent evaluate karo, phir power, phir 1 se subtract karo. Kyun: bracket woh fraction hai jo ek ideal expansion gas enthalpy se shaft work mein convert karta hai — 0 aur 1 ke beech ek pure number. Exponent: $\frac{\gamma -1}{\gamma }=\frac{0.3}{1.3}=0.2308$. ${\pi }_t^{-0.2308}=1.8^{-0.2308}$. Kyunki $\ln 1.8=0.5878$, toh milega $e^{-0.2308\times 0.5878}=e^{-0.1357}=0.8731$. $\left[1-0.8731\right]=0.1269\approx 0.127$

Unit flow per power: $c_p{T}_{in}{\eta }_t\times \text{bracket}$. $=4500\times 800\times 0.75\times 0.127=342,900\text{J/kg}\approx 3.43\times 10^5\text{J/kg}$ Aage kya karein: required power ko power-per-kg se divide karo. Kyun: required power ÷ (har kg/s deta hai jitna power) = kitne kg/s chahiye. ${\dot{m}}_{pb}=\frac{10.6\times 10^6}{3.43\times 10^5}\approx 30.9\text{kg/s}$ Staged combustion mein yeh ~31 kg/s waste nahi hote — yeh chamber mein flow karte hain aur dobara jalte hain.

Balance: turbine ko saare pumps ki power supply karni hoti hai, isliye right side har pump ka sum hai (fuel pump plus oxidizer pump, har ek apna $\dot{m}\Delta p/(\rho {\eta }_p)$ contribute karta hai): ${\dot{m}}_{pb}{c}_p{T}_{in}{\eta }_t[\cdot ]=\underset{\text{har pump ki demand add karo}}{\underbrace{\sum _{\text{pumps}}\frac{\dot{m}\Delta p}{\rho {\eta }_p}}}$ $\sum$ (Greek letter sigma, matlab "yeh sab add karo") simply kehta hai: fuel pump aur oxidizer pump ki power demand total karo. $p_c$ badhane se har pump ka $\Delta p$ badhta hai (har pump ko higher chamber pressure tak pahunchne ke liye zyada push karna hota hai), isliye right side badhta hai. Equality maintain karne ke liye, left side bhi badhni chahiye — matlab zyada preburner flow ${\dot{m}}_{pb}$ ya hotter $T_{in}$.

• Gas-generator: woh extra ${\dot{m}}_{pb}$ bahar dump ho jaata hai aur almost koi thrust produce nahi karta, isliye jaise jaise $p_c$ badhta hai efficiency girti hai.
• Staged combustion: extra ${\dot{m}}_{pb}$ chamber mein wapas milta hai aur dobara jalta hai, full thrust contribute karta hai — isliye $p_c$ "free mein" badh sakta hai, 250–300 bar tak. Compare karo Gas-generator cycle se.

Curve padhna: ${\pi }_t=1$ par (koi pressure drop nahi, left edge jahan arrow point karta hai) bracket exactly $0$ hai — koi expansion nahi, koi work nahi. Jaise jaise ${\pi }_t$ badhta hai, zyada enthalpy extractable hoti hai, isliye bracket badhta hai. Lekin exponent $-(\gamma -1)/\gamma$ ek chhota negative number hai, isliye ${\pi }_t^{-(\gamma -1)/\gamma }$ dheere dheere girta hai — bracket saturate hota hai (right-hand arrow yeh flattening mark karta hai), diminishing returns deta hai. Design matlab: butter-coloured band dikhata hai jahan real staged engines hote hain — ${\pi }_t$ deliberately chhota rakhte hain (~1.5–2). Bada ${\pi }_t$ us gas se pressure cheen leta jo high-pressure chamber mein enter karni hoti hai. Flat curve batata hai ki chhota pressure ratio already available work ka zyaatar hissa capture kar leta hai — isliye aap woh pressure chamber par spend karo, turbine par nahi. Turbopump design se relevant.

Toolbox se yaad karo: $\Delta T=Q/(m{c}_p)$, jahan $Q$ combustion se release hua heat hai aur $m$ woh mass hai jise warm karna hai. Fixed heat zyada mass par spread karne se chhota temperature rise hota hai. Kya cool karta hai: aur zyada excess oxidizer add karna (stoichiometric se aur door jaana, zyada oxidizer-rich) thanda mass add karta hai — yeh $m$ ko badhata hai bina zyada extra $Q$ ke — isliye $\Delta T=Q/(m{c}_p)$ girta hai: thanda flame. Stoichiometric ki taraf jaana (b) zyada completely jalta hai, zyada $Q$ release karta hai kam excess mass par, use hotter banata hai — jo aap chahte hain uska ulta. Danger: aur zyada oxygen-rich chalana gas ko aur zyada aggressively oxidising banata hai, isliye alloy oxidation problem aur bigadti hai. Yahi fundamental ORSC trade hai — aap temperature se ladte ho excess oxygen se lekin corrosion mein pay karte ho. Dekho Combustion stoichiometry and flame temperature aur Regenerative cooling.

(a) Pump power: $P_{\text{pump}}=\frac{120\times 3\times 10^7}{1140\times 0.72}=\frac{3.6\times 10^9}{820.8}\approx 4.386\times 10^6\text{W}\approx 4.39\text{MW}$ (b) Bracket: exponent $=\frac{0.3}{1.3}=0.2308$; $1.6^{-0.2308}=e^{-0.2308\ln 1.6}=e^{-0.2308\times 0.4700}=e^{-0.1085}=0.8972$. Bracket $=1-0.8972=0.1028$. Power per kg/s $=1200\times 750\times 0.72\times 0.1028=66,614\text{J/kg}\approx 6.66\times 10^4$ J/kg. ${\dot{m}}_{pb}=\frac{4.386\times 10^6}{6.66\times 10^4}\approx 65.9\text{kg/s}$ Note karo ki heavy oxygen-rich gas ka low $c_p$ (1200 vs H₂ ka ~4500) ek bada turbine flow force karta hai — yahi ek real reason hai ki ORSC turbines bade hote hain.

Power per kg/s $=5000\times 750\times 0.72\times 0.1028=277,560\text{J/kg}\approx 2.78\times 10^5$ J/kg. ${\dot{m}}_{pb}=\frac{4.386\times 10^6}{2.78\times 10^5}\approx 15.8\text{kg/s}$ Hydrogen-rich gas apne huge $c_p$ ki wajah se roughly ek-chauthai flow se same power deliver karta hai. FFSC mein fuel-rich (H₂/CH₄-rich) turbine light aur efficient hota hai, jabki ox-rich turbine ko kahin zyada mass move karna padta hai — yahi exactly wajah hai ki FFSC dono ko alag alag optimised turbopumps mein split karta hai, ek turbine ko dono kaam karne ke liye force karne ki bajaye.

(a) $P_{\text{pump}}=\frac{140\times 3.6\times 10^7}{423\times 0.7}=\frac{5.04\times 10^9}{296.1}\approx 1.702\times 10^7$ W $\approx 17.0$ MW. (b) Exponent $\frac{0.35}{1.35}=0.2593$; $1.7^{-0.2593}=e^{-0.2593\ln 1.7}=e^{-0.2593\times 0.5306}=e^{-0.1376}=0.8714$; bracket $=1-0.8714=0.1286$. (c) Power per kg/s $=3800\times 850\times 0.78\times 0.1286=323,983\text{J/kg}\approx 3.24\times 10^5$ J/kg. ${\dot{m}}_{pb}=\frac{17.02\times 10^6}{3.24\times 10^5}\approx 52.5\text{kg/s}$ (d) Fraction $=52.5/140\approx 0.375$, yani lagbhag 37.5% fuel preburner se guzarta hai. FFSC mein woh gas vent nahi hoti — yeh turbine se exit karti hai abhi bhi unburnt methane se bhari aur chamber mein dump hoti hai baaki jalne ke liye ("dobara jalti hai"). Isliye 37.5% abhi bhi full thrust contribute karta hai; sirf cost plumbing ki hai jo hot aur high-pressure chalti hai. Rocket thrust equation aur Specific impulse and chamber pressure se ties.

Proportions ki chain: pump power linearly scale karta hai $\Delta p$ ke saath ($P_{\text{pump}}=\dot{m}\Delta p/(\rho {\eta }_p)$ se), aur ${\dot{m}}_{pb}$ linearly scale karta hai us power ke saath (bracket, $c_p$, $T_{in}$, ${\eta }_t$ sab unchanged hain). Isliye factor simply do pressures ka ratio hai: $\frac{\Delta p_{\text{new}}}{\Delta p_{\text{old}}}=\frac{4.5\times 10^7}{3.6\times 10^7}=1.25$ Required preburner flow isliye 25% badhta hai. L5.1 ki value ${\dot{m}}_{pb}\approx 52.5$ kg/s lete hue: ${\dot{m}}_{pb,\text{new}}=1.25\times 52.5\approx 65.6\text{kg/s}$ Ek line: staged combustion mein woh extra ~13 kg/s chamber mein wapas milte hain aur thrust produce karte hain, isliye higher chamber pressure pay off karta hai; gas-generator cycle mein woh bahar dump ho jaate, isliye efficiency loss gain ko cancel kar deta.

Hydrogen-rich turbine gas ko same power ke liye oxygen-rich se kam flow kyun chahiye? ::: Kyunki ${\dot{m}}_{pb}\propto 1/c_p$, aur hydrogen-rich gas ka $c_p$ kahin zyada bada hota hai (~4500–5000 vs ~1200 J/kg·K). Chamber pressure badhana power balance mein pehle konsa term badhata hai? ::: Pump side, ek bade $\Delta p$ ke through. High ${\pi }_t$ par turbine bracket kyun flat hota hai? ::: Chhota negative exponent $-(\gamma -1)/\gamma$ ${\pi }_t^{-(\gamma -1)/\gamma }$ ko dheere dheere girata hai, isliye returns diminish hote hain. FFSC mein propellant ka kitna fraction chamber se pehle gasify hota hai? ::: Sab kuch — yahi "full flow" ki definition hai.