Under-expanded nozzle — Prandtl-Meyer expansion, efficiency loss: Foundations

3.3.15 · D1 · Physics › Rocket Propulsion › Under-expanded nozzle — Prandtl-Meyer expansion, efficiency

Parent note ki ek bhi line padhne se pehle, humein har letter ko earn karna hoga jo woh use karta hai. Neeche kuch bhi assume nahi kiya ki aapne yeh notation pehle dekha hai. Hum order mein build karte hain: har idea sirf uske upar wale par lean karta hai.

1. Pressure — "push per patch" $p$

Socho balloon phoolane ke baare mein. Andar ki hawa rubber ke har bit par outward push karti hai equally. Woh outward push, area ki per unit measure ki gayi, pressure hai.

Hum teen alag alag pressures se milenge, aur poora topic actually ek story hai ki kaun sa bada hai:

Symbol	Plain meaning	Picture
$p_{e}$	exhaust ka pressure bilkul nozzle ke munh par	gas ekdum lip par, abhi bhi push kar rahi hai
$p_{\infty}$	bahar ki hawa ka pressure door se (ambient)	woh sky jisme rocket fly karta hai
$p_{0}$	pressure andar deep mein jahan gas ne shuru kiya tha (chamber / "total" pressure)	pressurised furnace gas ke hone se pehle

Subscript $\infty$ (infinity) ka matlab sirf "door, undisturbed" hai — surroundings ka ordinary air pressure.

Figure — Under-expanded nozzle — Prandtl-Meyer expansion, efficiency loss

Figure dekho: same gas ke teen pressures hain depending on kahan aap measure karte ho. Topic ka central question ek simple comparison hai — kya $p_{e}$ , $p_{\infty}$ se bada hai, equal hai, ya chota hai?

Humein $p$ chahiye kyunki "under-expanded" define hota hai ek pressure comparison se. Opposite case ke liye dekho 3.3.14-Over-expanded-nozzle-shock-diamonds।

2. Area $A$ aur nozzle ka shape

Ek rocket nozzle ek tube hai jo pehle tight throat tak narrow hoti hai, phir wide ho jaati hai — classic bell shape. Jaise gas pass karti hai aur area change hota hai, gas speed up hoti hai. Woh link area aur speed ke beech mein hai 3.2.7-Isentropic-flow-area-Mach-relation.

Humein $A_{e}$ chahiye kyunki thrust mein ek pressure-times-area term hota hai: $A_{e} (p_{e} - p_{\infty})$ . Ek push per area, uss area se multiply karo jis par woh act karta hai, ek total force milti hai.

3. Gas ki speed $v$ aur mass flow $\overset{m}{˙}$

Dot notation par ruka jaana zaruri hai: $\overset{m}{˙}$ "m times kuch" nahi hai — dot ka matlab time mein rate of change hai. Toh $\overset{m}{˙}$ = "mass, per second."

4. Force aur Thrust $F$

Newton ke third law ke through, gas ko backward uchhaalna rocket ko forward push karta hai. Thrust ke do contributions hain, aur topic ka key thrust equation unhe bas add karta hai:

Notice karo: jab $p_{e} > p_{\infty}$ (under-expanded) toh doosra term positive hai — woh thrust add karta hai. Phir bhi topic kehta hai hum efficiency lose karte hain. Uss puzzle ka resolution hi poore parent note ke baare mein hai, aur humein 2.5.12-Thrust-coefficient-definition chahiye use measure karne ke liye.

5. Angles — hum "turning" ko kaise describe karte hain

Figure mein red arrow dekho. Woh axial (straight back) se shuru hua tha. Gas ke nozzle ke bahar expand hone ke baad, woh angle $θ_{turn}$ se outward point karta hai. Sirf woh part jo straight back point karta hai rocket ko push karne mein help karta hai — aur woh part poore arrow se chota hota hai.

$cos θ$ kya hai, zero se?

Cosine kyun aur kuch nahi? Kyunki "ek tilted arrow kitna forward point karta hai" exactly cosine ki definition hai — woh shadow jo arrow forward axis par daalta hai. Jab $θ = 0$ (koi turning nahi), $cos 0 = 1$ : poora arrow useful hai. Jab $θ$ badhta hai, $cos θ$ 1 se neeche shrink hota hai, aur missing fraction $(1 - cos θ)$ woh thrust hai jo sideways jaa ke waste ho jaata hai.

6. Mach number $M$ — speed "sound-lengths" mein measure ki gayi

Speed is tarah kyun measure karo instead of metres per second mein? Kyunki gas ka behaviour completely flip ho jaata hai $M = 1$ par:

$M < 1$ (subsonic): disturbances (pressure ki news) upstream travel kar sakti hain — flow "jaanta hai aage kya hai."
$M > 1$ (supersonic): kuch bhi flow ko outrun nahi kar sakta — pressure ki news upstream travel nahi kar sakti.

Yeh single fact hi wajah hai ki expansion smooth aur uniform nahi ho sakta: kyunki exhaust supersonic hai ( $M_{e} > 1$ ), exit lip par pressure drop, gas ko advance mein warn nahi kar sakta. Adjustment waves ke through honi chahiye jo lip se fan out hoti hain. Wahi Prandtl-Meyer fan hai.

7. Mach waves aur Mach angle $μ$

Ek speedboat ki picture socho: jitna faster woh jaaye, utna flatter aur zyada swept-back uski wake lines. Same yahan — faster gas (bada $M$ ) smaller, zyada swept angle $μ$ deta hai.

$arcsin$ (aur $arctan$ ) kya hai?

Parent note $arctan$ aur $arcsin$ se bhara hua hai. Yeh "un-doing" functions hain — yeh reverse question ka jawaab dete hain.

Humein inki zaroorat hai kyunki Prandtl-Meyer formula ek speed ratio se ek angle compute karta hai — yeh precisely "un-do" direction hai.

8. Isentropic flow aur $γ$

Yeh key assumption hai jo humein gas track karne deti hai: kyunki $p_{0}$ preserved hai, $p_{e} / p_{\infty}$ jaanna humein new Mach number batata hai.

$γ$ derive nahi karte; apni gas ke liye lookup karte ho. Yeh poore Prandtl-Meyer function aur pressure relations mein appear karta hai.

9. Prandtl-Meyer function $ν (M)$ — "turning budget"

Ab har piece earn ho chuki hai. Parent ka central tool:

Prerequisite map

Upar har box ek symbol hai jo humne is page par build kiya; har arrow hai "yeh pehle chahiye woh se pehle." Arrows follow karo topic node mein aur tum parent note line by line padh sakte ho.

Equipment checklist

Right side cover karo aur khud test karo. Agar tum sab answer kar sako, tum parent note ke liye ready ho.

Nozzle ke baare mein

p_{e} > p_{\infty}

kya batata hai?

Yeh under-expanded hai — exit pressure ambient se zyada hai, toh gas nozzle ke bahar expand karti rehti hai.

\overset{m}{˙}

mein dot ka matlab kya hai?

Ek rate per second — yahan, har second nikalne wali gas ke kilograms.

F = \overset{m}{˙} v_{e} + A_{e} (p_{e} - p_{\infty})

mein do terms kya hain?

Ek momentum term (mass/sec × speed) plus exit area par ek pressure-imbalance term.

Flow ko

θ_{turn}

turn karne se thrust kyun lose hota hai?

Sirf axial part

v cos θ_{turn}

rocket ko push karta hai; fraction

1 - cos θ_{turn}

sideways waste ho jaata hai.

cos θ

geometrically kya mean karta hai?

Ek tilted arrow ka woh fraction jo forward point karta hai — axis par uska shadow.

Mach number

M

kya hai?

Gas speed local speed of sound se divide ki gayi;

M > 1

supersonic hai.

Exit expansion waves ke through kyun honi chahiye, smoothly nahi?

Kyunki flow supersonic hai, pressure ki news upstream travel nahi kar sakti, toh adjustment Mach waves ke fan ke through hoti hai.

arctan (x)

kya poochta hai?

"Kaun sa angle hai jiski tangent

x

ke equal hai?" — yeh tangent ko un-do karta hai.

"Isentropic" kya guarantee karta hai constant rehne ki?

Total pressure

p_{0}

(smooth, reversible, koi heat loss nahi).

γ

kya hai?

Gas ka heat-capacity ratio, roughly

1.2

–

1.4

, tag karta hai ki woh energy kaise store karta hai.

Shabdon mein,

ν (M)

kya measure karta hai?

Woh total angle jitna flow turn hua hai Mach

M

tak pahunchne ke liye,

M = 1

se count karte hue.

Recall Self-check: poori story ek breath mein

$p_{e} > p_{\infty}$ par exit gas abhi bhi supersonic hai, toh woh lip ke bahar ek Prandtl-Meyer wave fan ke through expand hoti hai, $θ_{turn}$ se turn hoti hai; us angle ka cosine axial thrust churaata hai — woh churaya hua fraction efficiency loss hai.

Yeh bhi dekho 3.3.13-Optimal-expansion-ratio, 3.3.16-Altitude-compensation-nozzles, aur oblique-shock counterpart 4.1.3-Oblique-shock-theory।