Expander cycle — hydrogen-cooled nozzle drives turbine: Foundations

3.3.24 · D1 · Physics › Rocket Propulsion › Expander cycle — hydrogen-cooled nozzle drives turbine

Turbines aur cooling channels ki baat karne se pehle, humein ek shared vocabulary chahiye. Har ek symbol neeche ek measurement hai moving fluid ki. Hum unhe ek ek karke banate hain, har ek pichle ke upar.

1. Flow: "moving fluid" ka matlab asal mein kya hai

Picture. Ek pipe imagine karo. Uske across ek imaginary flat window draw karo. Ek second mein us window se kitna mass guzarta hai — woh count hi $\overset{m}{˙}$ hai. Upar chhota dot standard notation hai "per second" ke liye (ek rate).

Figure — Expander cycle — hydrogen-cooled nozzle drives turbine

Topic ko yeh kyun chahiye. Parent note mein har energy quantity ek rate hai — heat per second, work per second (yaani power). Energy ka rate paane ke liye hamesha ek energy-per-kilogram ko $\overset{m}{˙}$ se multiply karte hain. Toh $\overset{m}{˙}$ "per kilogram" facts aur "per second" facts ke beech bridge hai.

Recall

$\overset{m}{˙}$ mein dot ka kya matlab hai? ::: "per second" — yeh ek rate banata hai.

2. Pressure $p$ — fluid kitni zor se push karta hai

Picture. Ek balloon squeeze karo: andar ki hawa skin ke har hisse par equally push karti hai. Woh push-per-area pressure hai. Ek rocket chamber mein burning gas baahir ki taraf ~ $10 MPa$ par push karta hai — atmospheric se lagbhag 100 guna zyada.

Topic ko yeh kyun chahiye. Pump ka poora kaam hai pressure badhana taaki fuel ko high-pressure chamber mein force kiya ja sake. Hum us rise ko $Δ p$ likhte hain (Greek capital delta = "change in"). Turbine ulta karta hai: woh pressure giraata hai aur released energy harvest karta hai. Toh pressure — aur uska rise/fall — poore cycle ki currency hai.

3. Density $ρ$ — kitna tight pack hua hai

Picture. Ek cubic metre liquid oxygen ka wazan hai $1140 kg$ ; wahi box liquid hydrogen ka sirf $71 kg$ hai. Oxygen dense hai, hydrogen fluffy hai.

Topic ko yeh kyun chahiye. Pump volume handle karta hai, lekin hum mass track karte hain. Density dono ke beech convert karti hai: volume flow $= \overset{m}{˙} / ρ$ . Yeh ek fact hai jis wajah se parent note kehta hai oxygen pump "sasta" hai — dense oxygen ka matlab hai ek kilogram per chhota volume, toh use push karne mein kam kaam. Halka hydrogen ka matlab hai ek kilogram per zyada volume, toh ek zyada bhookha pump. Dekho Liquid hydrogen properties.

Recall

$\overset{m}{˙}$ aur $ρ$ ke terms mein volume flow? ::: $\overset{m}{˙} / ρ$ (kilograms-per-second divided by kilograms-per-cubic-metre gives cubic-metres-per-second).

4. Temperature $T$ aur uska change $Δ T$

Picture. Thanda H₂ wall channels mein enter karta hai; bahar ki flame heat pump karti hai usmein; woh zyada garam hokar nikalta hai. Rise hai $Δ T = T_{out} - T_{in}$ .

Topic ko yeh kyun chahiye. Cycle ki poori trick yeh hai ki heat hydrogen ka temperature badhati hai. Har kelvin per kitni heat chahiye woh depend karta hai agle symbol par.

5. Specific heat $c_{p}$ (aur uska partner $c_{v}$ )

Yeh tool kyun, aur kuch simpler kyun nahi? Humein ek bookkeeping number chahiye jo "kitne degree zyada garam" ko "kitne joules absorb hue" mein convert kare. Yahi exactly hai $c_{p}$ ka definition. Toh heat rate directly aata hai:

$Q = \overset{m}{˙} c_{p} Δ T$

Left se right padho: (kilograms per second) × (joules per kg per kelvin) × (kelvin) = joules per second = watts. Har unit perfectly cancel hokar power banata hai.

Picture. Har kilogram ko ek bucket samjho. $c_{p}$ bucket ki size hai: hydrogen ka bucket bahut bada hai (14 000 J per degree), toh thodi si mass bahut zyada heat soak kar leti hai. Isliye sirf hydrogen is cycle ko kaam kara sakta hai — dekho Regenerative cooling jacket jahan yeh saari heat grab hoti hai.

6. Enthalpy $h$ — gas ki usable energy content

Picture. Hot gas ko ek loaded spring plus uski heat samjho. Jaise woh turbine se flow karta hai aur uska temperature $T_{1}$ se $T_{2}$ tak drop karta hai, enthalpy bhi drop karta hai, aur released energy $c_{p} (T_{1} - T_{2})$ blades spin karti hai.

Topic ko yeh kyun chahiye. Turbine work per kilogram enthalpy drop hai, $c_{p} (T_{1} - T_{2})$ . Power paane ke liye $\overset{m}{˙}$ se multiply karo. Yeh direct link hai "humne jo heat churaayi" (Step 1) se "hum jo power banate hain" (Step 2) tak. Deeper flow relations ke liye dekho Isentropic flow relations.

7. Heat-capacity ratio $γ$ aur pressure–temperature link

Yeh tool kyun? Jab ek gas smoothly aur bina heat leak kiye expand karta hai (ek "isentropic" expansion — clean, ideal; yahan §6 ki ideal-gas assumption bhi in force hai), uska temperature aur pressure ek doosre se locked hain:

$\frac{T _{2}}{T _{1}} = (\frac{p _{2}}{p _{1}})^{\frac{γ - 1}{γ}}$

Humein yeh isliye chahiye kyunki ek turbine pressure drop se drive hota hai, lekin jo energy woh release karta hai woh ek temperature drop hai. $γ$ dono ke beech exact conversion knob hai. Iske bina hum " $12$ se $8 MPa$ tak pressure gira" ko "yeh kitne degree thanda hua, isliye kitni power aayi" mein nahi badal sakte.

Picture. Zyada steep pressure fall → zyada steep temperature fall → zyada kaam. Exponent $\frac{γ - 1}{γ}$ (lagbhag $0.286$ for $γ = 1.4$ ) control karta hai kitna steep.

Recall

Clean expansion mein temperature ratio ko pressure ratio se kya link karta hai? ::: $T_{2} / T_{1} = (p_{2} / p_{1})^{(γ - 1) / γ}$

8. Efficiency $η$ — "real machines kuch waste karti hain" factor

Yahan yeh kyun matter karta hai. Turbines $η_{t}$ se multiply karte hain (tumhe ideal se kam milta hai). Pumps $η_{p}$ se divide karte hain (tumhe ideal se zyada supply karna padta hai). Dono self-sustaining balance ko failure ke paas le jaate hain — cycle tabhi "close" hota hai jab turbine, apne losses ke baad, bhi pumps ko beat kare, unke losses ke baad. Dekho Turbopump design.

9. Symbols ko do work formulas mein daalna

Ab jab har symbol define ho gaya hai, hum woh do expressions likh sakte hain jo parent note bookkeep karta hai.

Yeh sab cycle ko kaise feed karta hai

Har node us symbol ke roop mein padho jo woh represent karta hai: m-dot hai $\overset{m}{˙}$ , cp hai $c_{p}$ , delta-T hai $Δ T$ , Q hai heat rate, h hai enthalpy, gamma hai $γ$ , p hai pressure, rho hai $ρ$ , delta-p hai $Δ p$ , eta-t aur eta-p hain do efficiencies, aur do W nodes hain turbine aur pump power.

Ise top-down padho: raw measurements (mass flow, specific heat, temperature) heat rate aur enthalpy build karte hain; $c_{v}$ aur $c_{p}$ $γ$ build karte hain, jo pressure ke saath turbine ka expansion build karta hai; density aur $Δ p$ pump demand build karte hain; efficiencies dono ko trim karti hain; aur sab kuch us single inequality par milta hai jo decide karta hai ki engine khud ko power kar sakta hai ya nahi.

Equipment checklist

Right side cover karo aur khud test karo — tum parent note ke liye ready ho tabhi jab har cheez instantly aaye.

\overset{m}{˙}

means

mass flow rate, kilograms passing a point per second.

p

and

Δ p

mean

pressure (force per area, in Pa) aur uska change; pump ke liye

Δ p = p_{out} - p_{in} > 0

ρ

means

density, mass per cubic metre; mass flow ko volume flow mein convert karti hai

\overset{m}{˙} / ρ

ke zariye.

T

and

Δ T

mean

temperature in kelvin aur cooling jacket ke across uska rise.

c_{p}

and

c_{v}

mean

specific heats at constant pressure aur constant volume; hamesha

c_{p} > c_{v}

Q = \overset{m}{˙} c_{p} Δ T

gives

heat rate (watts) jo walls mein hydrogen absorb karta hai.

h = c_{p} T

enthalpy of an ideal gas, turbine-extractable energy per kilogram.

R

and "ideal gas" mean

gas constant in

p = ρR T

, with

c_{p} - c_{v} = R

; real H₂ ke liye ek approximation.

γ

c_{p} / c_{v}

(~1.4), woh knob jo pressure ratio ko temperature ratio se link karta hai.

The isentropic link is

T_{2} / T_{1} = (p_{2} / p_{1})^{(γ - 1) / γ}

W_{pump}

\overset{m}{˙} Δ p / (ρ η_{p})

— pump power demanded.

W_{turb}

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

— turbine power supplied.

The self-sustaining condition is

W_{turb} \geq W_{pump,fuel} + W_{pump,ox}