3.1.10
Ek nozzle ek tube hai jo pressure/thermal energy ko kinetic energy (speed) mein convert karti hai . Ek slow (subsonic) gas ke liye, usse speed up karte ho squeeze karke (converging) tez jaane ke liye tumhe ise expand karna hoga (diverging) . Toh ek gas ko rest se supersonic tak le jaane ke liye tumhe dono chahiye: ek converging section, phir ek throat (jahan woh Mach 1 hit karta hai), phir ek diverging section.
Definition de Laval nozzleEk duct jiska cross-sectional area A pehle decrease hota hai (converging), throat par minimum tak pahunchta hai, phir increase hota hai (diverging). Yeh ek standard device hai jo ek compressible gas ko subsonic se supersonic speed tak accelerate karne ke liye use hoti hai, rockets aur steam turbines mein.
Flow ki state ko Mach number se track kiya jaata hai:
M = a V , a = γ R T V a γ = c p / c v R T
M < 1 subsonic M = 1 sonic (yeh sirf throat par hota hai — ise choked flow kehte hain)M > 1 supersonic
Yeh is topic ka core hai. Hum ise steady, 1-D, isentropic (no friction, no heat transfer) flow ke liye first principles se DERIVE karte hain.
Intuition Master equation padhoA d A = ( M 2 − 1 ) V d V
Subsonic (M < 1 ( M 2 − 1 ) < 0 d V > 0 d A < 0 area shrink honi chahiye (converging). ✅ everyday garden hose se match karta hai.Supersonic (M > 1 ( M 2 − 1 ) > 0 d A > 0 area grow honi chahiye (diverging). 😮 yeh counter-intuitive part hai.Sonic (M = 1 0 d A = 0 jahan area minimum hai , yaani throat . Isliye M = 1
Intuition Supersonic flow expand hone par speed kyun badhti hai?Jab M > 1 area grow hone se zyada tez girta hai. ρ A V V
Energy conservation h 0 = h + 2 1 V 2 h = c p T a 2 = γ R T stagnation-to-static ratios milte hain (subscript 0 V = 0
Intuition Do-solution factKisi bhi diverging area ke liye do possible flows hain: ek subsonic decelerating wala aur ek supersonic accelerating wala. Nature kaunsa pick karta hai yeh back pressure p b
Intuition Operating regimes (lowerp b
High p b flow har jagah subsonic rehta hai; converging speed up karta hai, diverging slow down karta hai (venturi ki tarah kaam karta hai). Throat choked nahi hai.p b M = 1 choked ho jaata hai — m ˙ p b Thoda aur low: throat ke baad flow supersonic ho jaata hai, phir diverging part mein ek normal shock khada hota hai, ise wapas subsonic mein jump karta hai.Design pressure: exit tak smooth supersonic flow, perfectly matched — koi shocks nahi.p b
Jab throat par M = 1 maximum aur fixed hoti hai: m ˙ ma x = ρ ∗ A ∗ a ∗ p b
Worked example Example 1 — Area ratio se Exit MachEk nozzle ka exit area A e = 4 A ∗ γ = 1.4 M e A / A ∗ = 4 Solve karo → numerically M e ≈ 2.94 Yeh step kyun? A / A ∗ > 1 supersonic root pick karte hain kyunki ek choked throat ke baad diverging section gas ko accelerate karta hai.
Phir exit pressure: p 0 / p e = ( 1 + 0.2 ⋅ 2.9 4 2 ) 3.5 ≈ 33 p e ≈ p 0 /33
Worked example Example 2 — Kya yeh choked hai?Reservoir p 0 = 10 γ = 1.4 p 0 p ∗ = ( γ + 1 2 ) γ − 1 γ = ( 2.4 2 ) 3.5 ≈ 0.528 p ∗ ≈ 5.28 Kyun? M = 1 ≤ 5.28 m ˙ Throat temperature: T ∗ / T 0 = 1/ ( 1 + 0.2 ) = 0.833
Worked example Example 3 — Sirf converging nozzle supersonic kyun nahi ja saktiAkela converging nozzle: sabse chhota area exit hai, jahan zyada se zyada M = 1 M > 1 d A > 0 p b Kyun? Area–Velocity law M = 1 d A = 0
Common mistake "Squeeze karna hamesha gas ko speed up karta hai."Kyun sahi lagta hai: paani/subsonic air ke liye (garden hose, everyday experience) narrow karna karta hai flow speed up. Fix: yeh sirf M < 1 ( M 2 − 1 ) expanding speed up karta hai. Master equation yaad karo, hose nahi.
Common mistake "Back pressure aur neeche karne se mass flow aur badhti hai."Kyun sahi lagta hai: zyada suction matlab zyada flow sunne mein aata hai. Fix: ek baar choked hone par (M = 1 m ˙ m ˙ ma x
Common mistake "Maximum speed throat par reach hoti hai."Kyun sahi lagta hai: throat sabse narrow hai, aur subsonic intuition ke liye narrow = fast. Fix: throat par sirf M = 1 highest speed exit par hoti hai, diverging section mein.
M = 1 Kyun sahi lagta hai: pressure speed control karta hai. Fix: A d A = ( M 2 − 1 ) V d V d A = 0 M = 1
Recall Feynman: 12-saal ke bachche ko samjhaoSocho ek straw se phoonk maar ke ek toy car push kar rahe ho. Agar straw ko pinch karo, hawa tez nikalti hai — yeh "squeeze to speed up" trick hai, aur yeh kaam karta hai jab tak hawa slow hai. Lekin ek magic speed hai: speed of sound. Hawa ko sound se tez banane ke liye, iska ulta karna padta hai — tube ko wider hone do, narrow nahi! Toh ek rocket nozzle ek hourglass ki tarah shaped hota hai: beech mein pinch hota hai (throat, jahan hawa exactly speed of sound hit karti hai), phir flare out hota hai taaki hawa super fast blast ho sake. Pinched throat ek one-way gate ki tarah bhi kaam karta hai: jab wahan hawa sound barrier hit kar le, chahe bahar se kitna bhi suck karo, same amount of hawa flow hoti hai — yeh "choked" hai.
"Sub Squeezes, Super Spreads."
M < 1 Squeeze . M > 1 Spread (expand). Handover throat par hoti hai (M = 1 d A = 0
Compressible flow ke liye Area–Velocity relation kya hai? A d A = ( M 2 − 1 ) V d V
Subsonic flow mein gas ko accelerate kaise karte hain? Area decrease karo (converging section), kyunki M 2 − 1 < 0
Supersonic flow mein gas ko accelerate kaise karte hain? Area increase karo (diverging section), kyunki M 2 − 1 > 0
Nozzle mein exactly Mach 1 flow kahan reach kar sakti hai? Sirf throat par, jahan d A = 0
"Choked flow" ka matlab kya hai? Throat par M = 1
γ = 1.4 p ∗ / p 0 ( 2/ ( γ + 1 ) ) γ / ( γ − 1 ) ≈ 0.528
Stagnation temperature ratio T 0 / T 1 + 2 γ − 1 M 2
Stagnation pressure ratio p 0 / p ( 1 + 2 γ − 1 M 2 ) γ / ( γ − 1 )
Converging-only nozzle M = 1 Uska minimum area exit hai; diverging part ke bina, M > 1 d A > 0
Properly running de Laval nozzle mein highest speed kahan hoti hai? Exit par (diverging section ke end mein), throat par nahi.
Agar back pressure choke aur design ke beech ho toh diverging section mein kya khada hota hai? Ek normal shock jo flow ko supersonic se wapas subsonic mein jump karta hai.
Mach number aur speed of sound ki definition? Accelerate gas to supersonic
Subsonic sonic supersonic
Speed of sound a^2=dp/drho
Sign of M^2-1 sets area change
