Expander cycle — hydrogen-cooled nozzle drives turbine

3.3.24 · Physics › Rocket Propulsion

WHY karta hai yeh cycle exist?

WHAT problem solve kar rahe hain? Ek rocket engine ko propellant ko enormous chamber pressure ke against combustion chamber mein force karna hota hai. Iske liye pumps chahiye. Pumps ko drive karne ke liye ek turbine chahiye. Turbine ko ek hot, high-pressure working gas chahiye.

Woh gas kahan se laoge? Do bure options aur ek clever option:

Gas generator cycle: thoda propellant ek side chamber mein jalao turbine gas banane ke liye, phir usse overboard dump karo → propellant waste, lower efficiency.
Staged combustion: fuel-rich preburner jalao → bahut high performance lekin brutal turbine temperatures aur pressures → complex, expensive.
Expander cycle (yeh note): kuch extra mat jalao. Nozzle walls pehle se blazing hot hain aur unhe cool karna hi padega. Toh fuel ko cooling channels se route karo; woh heat jo usne free mein li hai woh usse boil karti hai aur itna pressurize karti hai ki turbine chal sake.

HOW kaam karta hai loop (hydrogen ko trace karo)

Liquid H₂ tank se nikalta hai → fuel pump mein jaata hai (pressure badhta hai).
Yeh nozzle/chamber ke around lage regenerative cooling channels mein flow karta hai. Yahaan yeh combustion se heat $Q$ absorb karta hai, aur hot high-pressure gas ban jaata hai. Isse nozzle cool hota hai taaki woh melt na ho.
Hot H₂ gas turbine se expand hoti hai, pressure drop hota hai aur work $W_{t u r b}$ milta hai.
Woh work shaft ko spin karti hai → fuel pump aur oxidizer pump drive hote hain.
Ab lower-pressure H₂ (abhi bhi gas hai) combustion chamber mein inject hoti hai aur oxygen ke saath burn hoti hai.

Figure — Expander cycle — hydrogen-cooled nozzle drives turbine

Power balance first principles se derive karna

Step 1 — Cooling jacket mein pick up hui heat

Hydrogen channels mein $T_{in}$ temperature par enter karti hai aur $T_{o u t}$ par bahar nikaalti hai. Specific heat ki definition se (temperature raise karne ki energy), $Q = \overset{m}{˙}_{H_{2}} c_{p} (T_{o u t} - T_{in})$

Yeh step kyun? $c_{p}$ define hota hai constant pressure par heat per unit mass per kelvin ke roop mein. Mass flow $\overset{m}{˙}$ aur temperature rise se multiply karo → heat rate absorbed milti hai. Yeh $Q$ exactly wahi heat hai jo otherwise wall ko melt karti — hum ise recycle kar rahe hain.

Step 2 — Available enthalpy = turbine ka fuel

Woh absorbed heat gas ki enthalpy badhati hai. Turbine enthalpy drop ko shaft work mein convert karta hai. Ek ideal (isentropic) turbine ke liye jo pressure $p_{1}$ se $p_{2}$ tak expand karta hai: $W_{t u r b} = \overset{m}{˙}_{H_{2}} c_{p} T_{1} [1 - (\frac{p _{2}}{p _{1}})^{\frac{γ - 1}{γ}}] η_{t}$

Yeh step kyun? Ideal gas ke liye, enthalpy $h = c_{p} T$ . Isentropic expansion temperature ratio ko pressure ratio se link karta hai $T_{2} / T_{1} = (p_{2} / p_{1})^{(γ - 1) / γ}$ ke through. Work enthalpy drop $\overset{m}{˙} c_{p} (T_{1} - T_{2})$ hai; $T_{1}$ factor out karo aur turbine efficiency $η_{t}$ se multiply karo.

Step 3 — Required pump power

Ek pump $Δ p$ se pressure badhata hai $\overset{m}{˙} / ρ$ volume flow par: $W_{p u m p} = \frac{m ˙ Δ p}{ρ η _{p}}$

Yeh step kyun? Fluid ko pressure step par push karne ke liye per unit volume work $Δ p$ hota hai ( $W = \int p d V$ se). Volume flow $\overset{m}{˙} / ρ$ hai. Pump efficiency $η_{p}$ se divide karo kyunki real pumps kuch input waste karte hain.

Step 4 — Self-sustaining condition

$W_{t u r b} \geq W_{p u m p, f u e l} + W_{p u m p, o x}$

Square-cube limit (WHY expanders bade nahi ho sakte)

Worked Examples

Common Mistakes (steel-manned)

Flashcards

Hydrogen expander cycle mein turbine ko kaun sa working fluid drive karta hai, aur uski energy kahan se aati hai?

Gaseous hydrogen; energy nozzle/chamber walls se absorb ki gayi heat se aati hai (regenerative cooling), kisi bhi combustion se nahi.

Closed aur open (bleed) expander cycle mein kya difference hai?

Closed saari heated H₂ ko turbine se guzaar ke chamber mein bhejta hai (kuch dump nahi); open turbine flow ka ek fraction overboard dump karta hai zyada power ke liye lekin kuch loss hota hai.

Expander cycles bahut large thrust tak scale kyun nahi ho sakte?

Available heat wall area (

\propto L^{2}

) ke saath scale hoti hai lekin pump power demand propellant flow (

\propto L^{3}

) ke saath scale hoti hai; demand supply se aage nikal jaati hai — square-cube law.

Cooling jacket mein absorbed heat ka formula?

Q = \overset{m}{˙} c_{p} (T_{o u t} - T_{in})

Hydrogen kyun aur kerosene kyun nahi coolant/turbine gas ke roop mein?

Hydrogen ka

c_{p}

bahut high hai (~14 kJ/kg·K), cleanly boil hoti hai, aur coke nahi hoti; kerosene channels mein carbonize hoti hai aur bahut kam heat store karti hai.

Ideal isentropic turbine work formula?

W = \overset{m}{˙} c_{p} T_{1} [1 - (p_{2} / p_{1})^{(γ - 1) / γ}] η_{t}

Cycle ke liye self-sustaining condition?

W_{t u r b} \geq W_{p u m p, f u e l} + W_{p u m p, o x}

Turbine hydrogen ko near-zero pressure tak max work ke liye expand kyun nahi kar sakta?

Exhaust ko abhi bhi chamber mein

\geq

chamber pressure par inject karna hota hai, toh downstream pressure

p_{2}

neeche se bounded hai.

Pump power formula?

W_{p u m p} = \overset{m}{˙} Δ p / (ρ η_{p})

Ek real closed expander engine ka naam batao.

RL10 (Centaur/DCSS upper stage).

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek rocket jisme engine ka bottom part (bell) super hot ho jaata hai. Ise melt hone se bachane ki bajaye, tum thanda liquid hydrogen — fuel — ko tiny pipes se guzaarte ho jo uske around wrapped hain, jaise fridge ki cooling coils mein paani. Hydrogen saari woh heat soak kar leti hai aur hot, fast gas ban jaati hai. Woh gas phir ek chhote pinwheel (turbine) se blows karti hai aur use spin karti hai. Spinning pinwheel un pumps ko chalata hai jo aur fuel andar push karte hain. Toh engine khud ko cool karta hai AUR apne pumps ko usi heat se power karta hai — woh heat jo otherwise waste ho jaati. Pinwheel spin karne ke baad, hydrogen fire mein jaati hai aur sach mein jali jaati hai. Clever! Lekin yeh sirf chhote-ish engines ke liye kaam karta hai, kyunki ek giant engine ko bahut zyada pumping chahiye hoti hai jitna uski hot walls pay nahi kar sakti.

Connections

Regenerative cooling — woh heat-transfer mechanism jo is cycle ko feed karta hai.
Gas generator cycle — contrast: extra propellant jalata hai, turbine exhaust dump karta hai.
Staged combustion cycle — contrast: fuel-rich preburner, highest performance.
Turbopump design — woh load jo turbine ko drive karna hota hai.
Specific impulse — closed expander Isp gain karta hai kyunki koi propellant waste nahi hota.
Isentropic flow relations — $T$ – $p$ turbine equation ka source.
Square-cube law — woh geometric scaling jo expander thrust ko cap karti hai.
Liquid hydrogen properties — kyun $c_{p}$ aur boiling point matter karte hain.