Over - under expanded nozzle flows: Foundations — Hinglish deep dive

3.1.18 · D1 · Physics › Compressible Flow & Aerodynamics › Over - under expanded nozzle flows

Parent note ko bina atke padhne se pehle, tumhe har letter apna banana hoga. Ye page har ek ko bilkul zero se build karta hai, us order mein jo har idea ko pehle wale pe lean karne deta hai.

1. Pressure $p$ — area ke per unit push

Ek deewar se tukkron ki tarah bounce karte tiny gas molecules ki bheed imagine karo. Har bounce ek tiny shove hai. Ek square metre pe saare shoves ko add karo aur tumhe pressure milti hai.

Figure — Over - under expanded nozzle flows

Topic ko ye kyun chahiye? POORA subject do pressures ka comparison hai — nozzle se nikalne wali gas ki pressure versus uske aas-paas ki hawa ki pressure. Agar pressure ko tum "molecular pushing" ki tarah feel nahi karte, to "too high" aur "too low" phrases ka koi matlab nahi.

2. Topic jin pressures ko compare karta hai

Parent note mein char pressure symbols hain. Ye rahe, har ek jahan rehta hai wahan pin kiya gaya.

$p_{e}^{d es i g n}$ ko alag kyun introduce karein? Kyunki "over-" aur "under-expanded" ka naam design point ke relative rakha gaya hai, current $p_{b}$ ke relative nahi. Jab parent kehta hai ek nozzle "sea level ke liye designed" altitude pe under-expanded ho jaata hai, to $p_{e}$ khud kabhi nahi badalti — ye hamesha $p_{e}^{d es i g n}$ ke barabar hoti hai. Jo badalti hai wo hai $p_{b}$ . To humein "ye hardware hamesha jo pressure produce karta hai" uska ek fixed naam chahiye jo moving target $p_{b}$ se alag ho; wo fixed naam hai $p_{e}^{d es i g n}$ , aur ek fully-supersonic nozzle ke liye $p_{e} \equiv p_{e}^{d es i g n}$ .

Topic ko inki zarurat kyun: poora classification ek comparison hai — kya $p_{e}$ , $p_{b}$ se upar hai, neeche hai, ya barabar? Baaki sab $p_{e}$ calculate karne ki machinery hai.

Read line

p_{e} = p_{b}

matlab "perfectly expanded",

p_{e} < p_{b}

matlab "over-expanded",

p_{e} > p_{b}

matlab "under-expanded".

3. Nozzle ka shape aur uske areas $A$ , $A_{e}$ , $A^{*}$

Ek converging–diverging (de Laval) nozzle ek aisi tube hai jo pehle narrow hoti hai, phir wide. Sabse narrow slice ko throat kehte hain.

$A^{*}$ pe star kyun? Star ek convention hai jiska matlab hai "sonic point pe value" — wo jagah jahan flow exactly sound ki speed se move kar rahi hoti hai. Is nozzle mein throat wahi jagah hai (§6 mein dekhenge kyun). To $A^{*}$ aur "throat area" yahan ek hi cheez hain.

Topic ko areas kyun chahiye: parent ka punchline ye hai ki $p_{e}$ sirf area ratio $A_{e} / A^{*}$ se fixed hoti hai. Geometry, bahar ki hawa nahi, exit pressure set karti hai. Wo sentence samajhna possible nahi agar pata na ho ye dono areas kya hain.

4. Speed of sound $a$ aur Mach number $M$

$M$ kyun aur $V$ kyun nahi: same speed pe do flows bilkul alag behave karti hain agar ek sound speed ke upar ho aur ek neeche. Sirf ratio $M$ batata hai tum sonic line ke kis side pe ho — ye wahi ratio hai jis par physics actually dhyan deti hai.

5. Perfect gas, $γ$ , aur "isentropic"

Pictures ko formulas mein convert karne ke liye humein ek gas model chahiye.

Topic ko $γ$ aur isentropic kyun chahiye: parent $p_{e}$ derive karta hai ye assume karke ki nozzle ke andar flow isentropic hai. Yahi assumption exactly humein pressure ko Mach number ka clean function likhne deti hai. Shocks (over-expanded case mein) wahi ek jagah hain jahan flow not isentropic hoti — isliye inhe special events ki tarah treat kiya jaata hai.

"Isentropic" shock pe kyun fail ho jaata hai?

Shock ek sudden, thin jump hai — ye smooth nahi hai, isliye entropy badhti hai aur

p / ρ^{γ}

uske across constant nahi rehta.

6. Choking: throat $M = 1$ kyun hold karta hai

Mass conservation ( $ρ A V$ tube mein constant hai) ko smooth isentropic rules ke saath combine karo, aur kuch remarkable samne aata hai: kisi aisi gas ko speed up karne ke liye jo already supersonic hai, tube ko widen karna hoga, jabki subsonic gas ko speed up karne ke liye tube ko narrow karna hoga. Crossover — $M = 1$ — sirf wahan ho sakta hai jahan area shrink karna band kare aur grow karna shuru kare: throat.

Isliye $A^{*}$ (sonic area) throat area ke barabar hoti hai, aur isliye hum kehte hain nozzle choked hai: ek baar jab $M = 1$ throat pe baith jaata hai, mass flow max out ho jaati hai aur peg ho jaati hai.

Intuition Ye formulas is shape ke kyun hain (koi algebra nahi chahiye)

Pehla relation ( $p_{0} / p$ ). Energy conserved hoti hai: total energy $=$ (heat energy) $+$ (motion energy). Jab gas speed up hoti hai (bada $M$ ), motion energy badhti hai, to heat energy — aur isliye pressure — girati chahiye. Bracket $(1 + \frac{γ - 1}{2} M^{2})$ exactly "kitni motion energy tumne build up ki hai" hai; ise power $\frac{γ}{γ - 1}$ pe raise karna sirf isentropic rule $p / ρ^{γ} =$ const ka wo energy statement ko pressure statement mein translate karna hai. To: bada $M$ ⇒ bada bracket ⇒ choti $p$ . Yahi poori content hai.
Doosra relation ( $A / A^{*}$ ). Mass flow $ρ A V$ tube mein sab jagah fixed hai. Throat pe $M = 1$ $ρ V$ ko as large as possible banata hai, to wahan area $A^{*}$ as small as possible hoti hai. Throat se door $ρ V$ choti hai, to same mass pass karne ke liye $A$ badi honi chahiye — isliye $A / A^{*} \geq 1$ har jagah hoti hai, sirf $M = 1$ pe $1$ hit karti hai. Messy exponent sirf bookkeeping hai jo isentropic $ρ$ -vs- $M$ dependence carry karta hai.

Pehla ek Mach number ko pressure ratio mein turn karta hai. Doosra ek area ratio ko Mach number mein. Inhe chain karo aur akele geometry se $p_{e}$ milti hai — full builds ke liye Isentropic Flow Relations, Area-Mach Number Relation aur Choked Flow & the Throat (M=1) dekho.

7. Mass flow $\overset{m}{˙}$ aur thrust symbols

Parent ka thrust equation $F = \overset{m}{˙} V_{e} + (p_{e} - p_{b}) A_{e}$ inhe sab chahiye: ek momentum term ( $\overset{m}{˙} V_{e}$ ) plus ek pressure term $(p_{e} - p_{b}) A_{e}$ . Notice karo pressure term exactly hamara do-pressure comparison hai, ab exit area se multiplied — Rocket Nozzle Design & Thrust Optimization dekho.

8. Fans aur shocks — mismatch settle karne ke do tarike

Do words pehle hi sentence se appear hote hain, to inhe abhi pictures ke saath pin karo.

Dono kyun chahiye: ek fan sirf pressure gira sakta hai, ek shock sirf pressure badha sakta hai. Mismatch jis direction mein chale wo tool pick karta hai. Ye single choice — badhao ya ghataao — topic ka poora branching logic hai.

Foundations topic ko kaise feed karte hain

Recall Node legend for the map below

Har box is page se ek foundation hai. "pe" $= p_{e}$ , "pb" $= p_{b}$ , "p0" $= p_{0}$ , "Astar" $= A^{*}$ , "Me" $= M_{e}$ , "M" $=$ Mach number.

Equipment checklist

Self-test: right side cover karo aur reveal karne se pehle har ek answer do.

Pressure

p

physically kya measure karta hai?

Force per unit area jo molecules surface se bounce karke exert karte hain.

p_{0}

p_{e}

p_{b}

kya hain?

Chamber (stagnation) pressure, exit-plane static pressure, aur outside (back/ambient) pressure.

p_{e}^{d es i g n}

kya hai aur use

p_{b}

se alag kyun rakha jaata hai?

Wo fixed exit pressure jo hardware hamesha banata hai; regimes us relative named hain, jabki

p_{b}

moving ambient value hai.

A^{*}

kya hai aur kahan hoti hai?

Sonic area jahan

M = 1

; de Laval nozzle mein ye throat (narrowest slice) hoti hai.

Mach number

M

define karo.

Flow speed divided by local sound speed,

M = V / a

Ek supersonic exit

p_{b}

kyun nahi "sun" sakta?

Pressure messages speed

a

pe travel karti hain; jab

M > 1

flow unhe outrun kar leta hai to downstream info exit tak nahi pahunch sakti.

Subsonic exit (

M_{e} < 1

) pe kya hota hai?

Messages upstream travel karti hain, isliye

p_{e}

simply

p_{b}

ke barabar adjust ho jaati hai — koi shocks ya fans nahi; over/under-expansion sirf supersonic mein hoti hai.

γ

kya describe karta hai?

Ratio of specific heats — dabane par gas kitni strongly garm hoti hai (air

\approx 1.4

"Isentropic" ka matlab kya hai aur ye kahan break hota hai?

Smooth, lossless flow jismein

p / ρ^{γ}

constant ho; ye shock ke across break ho jaata hai.

Area ratio

A_{e} / A^{*}

kis kaam aata hai?

Supersonic exit Mach number

M_{e}

find karne ke liye, jo phir

p_{e}

fix karta hai.

Expansion fan vs shock kya hota hai?

Fan dheere se pressure giraata hai (Prandtl–Meyer, tab aata hai jab

p_{e} > p_{b}

); shock abruptly badhata hai (tab aata hai jab

p_{e} < p_{b}

Thrust equation likho aur har term ka naam batao.

F = \overset{m}{˙} V_{e} + (p_{e} - p_{b}) A_{e}

— momentum thrust plus pressure thrust.

Recall Aage le jaane ke liye ek-line summary

Har symbol ek contest tak reduce hota hai: geometry $p_{e}$ set karti hai; sky $p_{b}$ set karta hai; jo jeete wo decide karta hai shocks vs fans.