Foundations — Converging-diverging (de Laval) nozzle — subsonic, supersonic flow

3.1.10 · D1 · Physics › Compressible Flow & Aerodynamics › Converging-diverging (de Laval) nozzle — subsonic, supersoni

Parent note de Laval nozzle padhne se pehle, tumhe us mein aane wala har letter khatam karna hoga. Yeh page har ek ko zero se build karta hai — pehle simple words mein, phir ek picture, phir reason ki topic ko iska zaroorat kyun hai.

1. Flow, aur "steady 1-D" ka matlab

Topic ko iska zaroorat kyun hai: yeh hume poore flow ko un quantities se describe karne deta hai jo sirf tube ke saath-saath position par depend karti hain, isliye ek single number jaise "yahan area" ya "yahan speed" meaningful hai. s01 figure dekho — har vertical slice gas ki ek "state" hai.

2. Area $A$ — ek slice ka size

• Converging = $A$ downstream jaate waqt shrink hota hai (tube pinch in hoti hai).
• Throat = woh jagah jahan $A$ sabse chhota hota hai.
• Diverging = $A$ phir se badh raha hai (tube flare out hoti hai).

Kyun zaroorat hai: poora topic tube ke saath-saath $A$ ko shape karne ke baare mein hai. Master equation literally ek statement hai ki $A$ mein change hone se speed mein change force hoti hai.

3. Speed $V$ — gas kitni fast move karti hai

Kyun zaroorat hai: nozzle ka poora point $V$ ko bada banana hai. "Accelerate" ka matlab sirf yeh hai ki $V$ downstream jaate waqt badhti hai.

4. Density $\rho$ — gas kitni tightly packed hai

Kyun zaroorat hai: gases compressible hoti hain — unhe squeeze karo aur $\rho$ badh jaata hai; expand hone do aur $\rho$ gir jaata hai. Liquids mein $\rho$ bahut kam badlta hai, yahi reason hai ki nozzle physics water pipes se alag hoti hai. Counter-intuitive supersonic behaviour poori tarah $\rho$ ke $A$ badhne se bhi tez girne se aata hai.

5. Mass flow rate $\dot{m}$ aur continuity

Upar ke teen quantities combine karo: ek second mein ek slice par gas $V$ distance aage badhti hai, $A\times V$ volume ka ek tube-chunk fill karti hai; mass paane ke liye density se multiply karo: $\dot{m}=\rho AV$

Kyun zaroorat hai: yeh single conserved product $A$, $V$, $\rho$ ko saath baandhta hai — yeh parent ki derivation ka Step 1 hai.

6. Pressure $p$ aur Euler momentum idea

Yahan ek quantity ke saamne letter $d$ ka matlab hai "usme ek tiny change" — $dp$ tube ke neeche ek small step par ek small pressure change hai. Hum tiny changes use karte hain kyunki area, speed aur pressure sab nozzle ke saath smoothly aur continuously vary karte hain; small steps hume unke slopes relate karne dete hain.

7. Temperature $T$ aur speed of sound $a$

Kyun zaroorat hai: $a$ referee hai. Yeh woh magic speed set karta hai jis par nozzle ka rule flip hota hai. Yeh pressure aur density changes ko bhi connect karta hai: ek gentle isentropic disturbance ke liye, $a^2=dp/d\rho$ (parent ka Step 3).

8. Constants $\gamma$, $R$, aur specific heats $c_p,c_v$

Kyun zaroorat hai: $\gamma$ aur $R$ temperature ko speed of sound mein convert karte hain, aur $\gamma$ har isentropic ratio mein exponents set karta hai.

9. Mach number $M$ — speed "sounds" mein measure hoti hai

• $M<1$ subsonic — sound se slower; pressure news upstream travel kar sakti hai.
• $M=1$ sonic — exactly sound barrier.
• $M>1$ supersonic — sound se faster; gas apni khud ki pressure news se aage nikal jaati hai.

Kyun zaroorat hai: $M$ single knob hai jo $(M^2-1)$ ka sign decide karta hai, aur isliye decide karta hai ki narrow karna ya wide karna gas ko accelerate karta hai. Yeh master equation ka star hai.

10. Isentropic flow — "clean" assumption

Kyun zaroorat hai: yeh assumption hi hai jo pressure, density aur temperature ko tidy formulas mein lock hone deti hai ($T_0/T$, $p_0/p$, $A/A^{\ast }$ ratios). Ise hatao aur algebra explode ho jaati hai.

11. Stagnation state (subscript $0$) aur starred sonic state (${}^{\ast }$)

Kyun zaroorat hai: har isentropic ratio ek local quantity ko stagnation ya starred se compare karta hai. Choking, aur poori Choked Flow & Mass Flow Limit ki kahani, inhi symbols mein batayi jaati hai.

12. Master equation ke symbols padhna

Parent ka boxed result hai $\frac{dA}{A}=(M^2-1)\frac{dV}{V}$

• $\frac{dA}{A}$ = area mein fractional change (maan lo 2% widening). Tiny change $dA$ ko $A$ se hi divide karne par yeh ek percentage ban jaata hai, isliye kisi bhi size ki tubes fairly compare hoti hain.
• $\frac{dV}{V}$ = speed mein fractional change.
• $(M^2-1)$ = sign-flipping gatekeeper: $M<1$ par negative, $M=1$ par zero, $M>1$ par positive.

Kyun zaroorat hai: jab tum har symbol padh lete ho, equation words mein kehti hai — "gas ko speed up karne ke liye, area ko us direction mein change karo jo gatekeeper allow karta hai; exactly Mach 1 par gatekeeper zero hai, isliye speed sirf wahan sonic ho sakti hai jahan area change karna band kar de (throat)."

Do aur devices jo parent passing mein mention karta hai:

• Normal shock — ek paper-thin jump jahan supersonic flow subsonic mein snap back ho jaata hai, $p$ aur $T$ abruptly badh jaate hain (Normal Shock Waves).
• Yeh foundations baad mein Rocket Propulsion & Thrust aur Steam Turbine Nozzles power karte hain.

Prerequisite map

Equipment checklist

Parent note par jaane se pehle kya tum har ek ka jawab de sakte ho?