Foundations — De Laval nozzle geometry — conical, bell (Rao contour), 80% bell

3.3.17 · D1 · Physics › Rocket Propulsion › De Laval nozzle geometry — conical, bell (Rao contour), 80%

Parent note mein ek bhi formula padhne se pehle, tumhe vocabulary chahiye. Yeh page har symbol ko zero se build karta hai — ek plain sentence, ek picture, aur woh reason ki topic uske bina kaam kyun nahi kar sakta. Upar se neeche padho; har idea usse pehle wale par rely karta hai.

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1. Nozzle ki shape khud: throat, exit, axis

Ek wasp-waist imagine karo: gas left se andar aati hai, beech mein narrow throat se squeeze hoti hai, aur right par baahar flare karti hai.

• Throat woh jagah hai jahan gas exactly sound ki speed tak pahunchti hai. Iske downstream sab kuch supersonic hai.
• Exit woh rim hai jahan gas finally nikalta hai aur baahri hawa se milta hai.
• Axis woh imaginary straight line hai jis par hum chahte hain ki saara exhaust travel kare, kyunki thrust ek force hai axis ke along.

Topic ko yeh kyun chahiye: "conical vs bell" ki poori discussion throat aur exit ke beech ki wall shape ke baare mein hai — isliye tumhe exactly pata hona chahiye ki hum kaunse do ends ki baat kar rahe hain.

Recall

Nozzle ka kaunsa point woh hai jahan flow pehli baar sound ki speed tak pahunchti hai? ::: Throat (sabse narrow neck).

Yeh deep story ki narrowing phir widening gas ko speed up kyun karta hai, iske liye dekho Converging-Diverging Nozzle Basics.

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2. Radius $R$ — pipe kitni wide hai

Do special radii baar baar aate hain:

Upar wali picture mein, $R_t$ waist par chota red arrow hai, $R_e$ mouth par lamba red arrow hai. Subscript bas ek chota sa label hai jo ek letter ke saath lagta hai yeh batane ke liye ki hum kaunsa radius mean kar rahe hain — isse zyada mysterious kuch nahi.

Recall

$R_e$ mein subscript tumhe kya batata hai? ::: Yeh label karta hai ki kaunsa radius — yahan woh jo exit par hai.

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3. Area $A$ aur expansion ratio $ϵ$

Wall har cross-section par ek circle hai, isliye uska area hai $A=\pi R^2$ (circle ka area: $\pi$ times radius squared).

Kyunki $A=\pi R^2$ hai, radii ka ratio areas ke ratio ka square root hota hai: $\frac{R_e}{R_t}=\sqrt{\frac{A_e}{A_t}}=\sqrt{ϵ}\Rightarrow R_e=R_t\sqrt{ϵ}$

Woh single line parent note ke har worked example mein use hoti hai (jaise $R_e=10\sqrt{16}=40$ cm). Ab tum jaante ho square root kahan se aata hai: yeh circle ke area mein "squared" ko undo kar raha hai.

Recall

Agar $ϵ=25$ aur $R_t=8$ cm hai, to $R_e$ kya hai? ::: $R_e=8\sqrt{25}=8\times 5=40$ cm.

$ϵ$ aur final gas speed ke beech ka link Expansion Ratio and Area-Mach Relation ka subject hai.

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4. Angles: half-angle $\alpha$ aur wall angles $\theta$

Yeh poore topic ka core hai, isliye hum slowly chalte hain.

Figure mein, blue axis aur wall ke beech ka yellow wedge hi $\alpha$ hai. Bada $\alpha$ matlab mota, zyada sharply flaring cone.

Recall

Kaunsa exit angle ${\theta }_e$ sabse kam thrust waste karta hai, aur kyun? ::: ${\theta }_e=0°$ — flow axis ke parallel nikalta hai, isliye uski saari momentum seedha peeche point karti hai.

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5. Tools $\cos$, $\sin$, $\tan$ — ek slanted line padhna

Parent note $\cos \alpha$, $\sin \theta$, aur $\tan \alpha$ use karta hai. Yahan hai ki har ek kya poochta hai, sab ek hi right triangle par.

Exhaust velocity ko ek arrow ki tarah draw karo jo axis se $\alpha$ angle par tilted ho. Ise ek right triangle par drop karo:

• axis ke along wali side (useful part) adjacent side hai,
• axis ke across wali side (wasted part) opposite side hai,
• arrow khud hypotenuse hai (sabse lambi side, length = full speed $v_e$).

Parent ka number confirm karte hain. $R_t=10$, $ϵ=16$, $\alpha =15°$ ke saath: $L=\frac{10(\sqrt{16}-1)}{\tan 15°}=\frac{10\times 3}{0.2679}\approx 112\text{ cm}✓$

Recall

Kaunsa ratio exhaust speed ka woh fraction deta hai jo thrust produce karta hai? ::: $\cos \alpha$ (adjacent over hypotenuse = along-axis fraction).

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6. Efficiency score $\lambda$

Parent derive karta hai $\lambda =\frac{1+\cos \alpha }{2}$ poori conical sheet of exhaust par $\cos \theta$ average karke. Tumhe abhi calculus ki zaroorat nahi — bas $\lambda$ ko padhho as "jo thrust bach gaya, ideal ka fraction ke roop mein."

Har case mein sanity checks:

• $\alpha =0°$: $\lambda =\frac{1+1}{2}=1$ — ek straight pipe kuch bhi waste nahi karta. ✓
• $\alpha =15°$: $\lambda =\frac{1+0.9659}{2}=0.983$ — parent se match karta hai. ✓
• $\alpha =90°$ (wall flat-out sideways, degenerate): $\lambda =\frac{1+0}{2}=0.5$ — aadha momentum lost; extreme limit. ✓

Recall

$\lambda =1$ physically kya mean karta hai? ::: Koi divergence loss nahi — exhaust perfectly axis ke along nikalta hai.

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7. Flow-rate symbols: $\dot{m}$ aur $v_e$

Recall

$\dot{m}{v}_e$ ke units force kyun hain? ::: Mass-per-second times velocity = momentum-per-second, aur force hi momentum ke change ki rate hai.

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Foundations topic ko kaise feed karte hain

Har box ek aisa symbol hai jo tum ab padh sakte ho. Milaakar yeh tumhe parent note (De Laval nozzle geometry) mein har formula, table, aur worked example follow karne dete hain.

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Equipment checklist

Khud test karo — right side cover karo aur zor se answer bolo.

• Throat vs exit vs axis ::: Throat = narrow neck (sonic point); exit = wide mouth; axis = centre line jis par hum thrust chahte hain.
• $R_t$ aur $R_e$ ka matlab ::: Throat aur exit par radii (axis-to-wall distance).
• Expansion ratio $ϵ$ ::: $A_e/A_t$ — exit area throat area se kitni baar bada hai.
• $ϵ$ se $R_e$ nikalna ::: $R_e=R_t\sqrt{ϵ}$ (square root $A=\pi R^2$ mein "squared" ko undo karta hai).
• Half-angle $\alpha$ ::: Conical wall ki axis se tilt; full opening $2\alpha$ hai.
• ${\theta }_n$ vs ${\theta }_e$ ::: Bell ka steep throat-start angle vs gentle exit angle; ideal ${\theta }_e=0°$.
• $\cos \alpha$ ka matlab ::: Exhaust speed ka fraction jo forward point karta hai (adjacent/hypotenuse).
• $\tan \alpha$ ka matlab ::: Wall ki steepness, sideways/along-axis — length ke liye use hota hai.
• Efficiency $\lambda$ ::: Thrust-survived score, $0$ se $1$ tak; cone ke liye $\lambda =\frac{1+\cos \alpha }{2}$.
• $\dot{m}$ aur $v_e$ ::: Mass per second nikalta hai, aur exit gas speed; inका product ideal thrust force hai.