Foundations — Injector design — impinging, coaxial, swirl injectors

3.3.34 · D1 · Physics › Rocket Propulsion › Injector design — impinging, coaxial, swirl injectors

Is page mein assume kiya gaya hai ki aap kuch nahi jaante. Hum har letter build karte hain jo parent note use karta hai, usse pehle ki aap use kisi equation mein dekhen. Upar se neeche padhen — har idea agli ke liye ek building block hai.

1. Pressure $p$ — ek push jo area pe spread hoti hai

Ek crowd ko ek door pe press karte huye imagine karen. Das log ek wide door pe lean kar rahe hain toh shayad kuch nahi hoga; wahi das log ek chhote door panel pe squeeze ho jayein toh usse snap kar sakte hain. Total push same, chhota area → zyada pressure.

$p=\frac{\text{force}}{\text{area}}$

Parent note $p_1$ use karta hai (hole ke andar feed manifold ka pressure, pehle), $p_2=p_c$ (chamber pressure, hole ke baad), aur $\Delta p=p_1-p_2$.

Figure 1 — Ek injector wall ka cross-section. Baayein taraf manifold high pressure $p_1$ par hai jahan fluid almost rest mein hai; daayein taraf chamber lower pressure $p_2=p_c$ par hai. Bachha hua push $\Delta p=p_1-p_2$ (black arrow) fluid ko hole ke through dhakelta hai aur woh speed $v$ ki fast red jet ke roop mein bahar aata hai.

WHY topic ko yeh chahiye: woh bachha hua push $\Delta p$ poore injector ka engine hai — yahi woh cheez hai jo stored high-pressure liquid ko ek fast jet mein convert karti hai.

2. Density $\rho$ — kitna mass packed hai

Ek shoebox feathers se bhara hua versus wahi box lead se bhara hua imagine karen. Same box (same volume), wildly different mass. Lead ki density zyada hoti hai.

$\rho =\frac{\text{mass}}{\text{volume}}$

WHY topic ko yeh chahiye: bhaari fluid ko accelerate karna mushkil hota hai. Jab same $\Delta p$ kerosene (dense) versus hydrogen gas (light) ko push karta hai, toh light wali cheez bahut tez move karti hai. Yahi ek fact hai jis ki wajah se coaxial injectors bahar ki taraf fast light gas rakhte hain.

3. Velocity $v$ — ek direction mein speed

Ek dhara mein behta hua patta imagine karen — uski velocity ek arrow hai: length = speed, arrowhead = direction. Injector hole se bahar aane wali jet mein, $v$ woh arrow hai jo hole se bahar point karta hai.

WHY topic ko yeh chahiye: combustion ek race hai clock ke against. Fluid jo tez move karta hai aur mist mein shred hota hai woh chamber ki door wall tak pahunchne se pehle jal jaata hai. Velocity pressure (jo speed create karta hai) aur mass flow (jo speed carry karta hai) ke beech middle-man hai.

4. Mass flow rate $\dot{m}$ — hole ke through kilograms per second

Ek doorway par khade hokar log per second count karte huye sochen. Agar woh tez chalte hain, ya doorway wider hai, ya woh zyada densely packed hain, toh zyada log per second cross karte hain. Fluid bhi aisa hi hai:

$\dot{m}=\rho Av$

• $\rho$ = kitna densely packed hai (mass per volume),
• $A$ = opening kitna wide hai,
• $v$ = kitni tez cross karte hain.

Figure 2 — Log area $A$ ke doorway se speed $v$ (red arrow) pe stream kar rahe hain; baayein taraf black dots dikhate hain ki woh kitne densely packed hain (density $\rho$). Zyada density, wider opening, ya higher speed mein se koi bhi count per second badha deta hai, jo exactly $\dot{m}=\rho Av$ hai.

WHY topic ko yeh chahiye: "metering" ka poora point ek target $\dot{m}$ hit karna hai. Engine ka thrust is baat se set hota hai ki har second kitna propellant jalta hai, isliye $\dot{m}$ woh number hai jo injector ko exactly deliver karna hai.

5. $\Delta p$ se jet speed tak — factor of 2 kahan se aata hai

Ab hum parent ka key result $v=\sqrt{2\Delta p/\rho }$ step by step build kar sakte hain, taaki mysterious factor of $2$ ab mysterious na rahe.

Derivation (parent ka Bernoulli step, spell out karke):

HUM KYA KARTE HAIN — ek hi blob ke path par do points ke liye "energy per cubic metre streamline ke sath conserved hai" likhte hain: point 1 manifold ke andar (pressure $p_1$, essentially rest mein toh $v\approx 0$) aur point 2 hole exit par (pressure $p_2=p_c$, speed $v$). KYU — teen assumptions ke under unke beech koi energy add ya lost nahi hoti, toh total (pressure energy + motion energy) match karni chahiye: $\underset{\text{point 1}}{\underbrace{p_1+\frac{1}{2}\rho (0{)}^2}}=\underset{\text{point 2}}{\underbrace{p_2+\frac{1}{2}\rho v^2}}$

AAGE KYA — dono sides se $p_2$ subtract karo taaki sara pressure left side par aa jaye. KYU — hum motion-energy term ko isolate karna chahte hain: $p_1-p_2=\frac{1}{2}\rho v^2⟹\Delta p=\frac{1}{2}\rho v^2$

Wahan hai: pressure drop one-half density times speed-squared ke barabar hai. AAKHIR MEIN KYA — $\frac{1}{2}$ aur square ko undo karo taaki $v$ free ho jaye. KYU — dono sides ko $2$ se multiply karo, $\rho$ se divide karo, phir square root lo (square ko undo karne ka yahi ek tool hai, dekhen §6): $v^2=\frac{2\Delta p}{\rho }⟹v=\sqrt{\frac{2\Delta p}{\rho }}$

Poore statement ke liye Bernoulli Equation dekhen jisme height terms bhi hain (jo hum yahan drop karte hain kyunki injector hole chhota aur horizontal hai).

6. Square root x​ — ek square ko undo karna

WHY topic ko yeh chahiye — aur kyun yahi tool na koi aur: humne abhi find kiya $\Delta p=\frac{1}{2}\rho v^2$, yaani $\Delta p$ speed squared ke sath badhta hai. $v$ ko akele pull karne ke liye, ek square ko undo karne wala ek hi operation hai aur woh hai square root. Isliye flow law mein ​ hai: $v=\sqrt{\frac{2\Delta p}{\rho }}$ Isliye pressure drop double karne se speed double nahi hoti — yeh speed ko sirf $\sqrt{2}\approx 1.41$ se multiply karti hai. Square root har incompressible injector flow law mein baked in hai.

7. Orifice area $A$ aur diameter $d_{\text{h}}$ — hole ka size

Ek coin imagine karen: use across measure karo taaki $d_{\text{h}}$ mile; flat face ka area $A$ hai. $\pi /4$ isliye aata hai kyunki ek circle side $d_{\text{h}}$ ke square ke andar "boxed" hota hai — circle us square ka lagbhag $78.5\%$ fill karta hai.

WHY topic ko yeh chahiye: designers $A$ (aur $d_{\text{h}}$) choose karte hain flow tune karne ke liye. Parent mein Example 1 exactly yahi solve karta hai: given jo flow aap chahte ho, hole kitna bada hona chahiye?

8. Discharge coefficient $C_d$ — "real hole" penalty

Paani ko ek doorway se squeeze hote huye imagine karen: woh pura doorway fill nahi karta — stream andar ki taraf neck karti hai ("vena contracta") aur edges se rub karti hai (friction). Toh effective opening drilled wale se chhoti hai. $C_d=0.75$ matlab aapko actually ideal flow ka $75\%$ milta hai. Yahan woh friction hai jo humne §5 ke inviscid Bernoulli step mein ignore kiya tha, use quietly wapas daala ja raha hai.

WHY topic ko yeh chahiye: $C_d$ ke bina aapka predicted $\dot{m}$ bahut zyada hoga aur engine lean ya rich run karega. Yeh clean theory se machined part tak ka bridge hai. §4, §5 aur $C_d$ combine karke parent ka metering law milta hai $\dot{m}=C_{d}A\sqrt{2\rho \Delta p}$.

9. Jab fluid ek gas ho — compressible aur choked flow

Upar sab kuch assume kiya tha ki fluid ek liquid hai (roughly constant density $\rho$ — §5 ki teesri assumption). Lekin coaxial injectors gas (jaise gaseous hydrogen) ko unke outer holes se push karte hain, aur gases springy hoti hain — unhe dabaya ja sakta hai. High pressure ratios par yeh rules change kar deta hai, aur topic ko aise pretend nahi karna chahiye.

Figure 4 — Ek hole par pressure ratio ke versus gas mass flow $\dot{m}$. Jab aap downstream pressure lower karte ho (right move karte ho), flow badhti hai — phir flatten ho jaati hai. Red critical point ke baad throat speed of sound tak pahunch gayi hai aur flow choked hai: $\dot{m}$ fix rehta hai chahe chamber pressure kitna bhi neeche jaye.

WHY topic ko yeh chahiye: ise chhod dene par ek novice liquid $\sqrt{\Delta p}$ law ko ek choked gas stream par apply karta aur galat $\dot{m}$ paata — real rocket coaxial injectors ke liye ek critical regime.

10. Angles $\theta$, $\alpha$, $\varphi$ aur tangent — right triangle se direction

Parent note teen spray directions angles se build karta hai. Yeh sab ek right triangle par ruke hain, aur teeno angles chamber axis se measure kiye jaate hain — chamber ki straight-down centre-line.

Figure 3 — Ek velocity arrow (red) ek right triangle ki slanted side ki tarah draw kiya gaya. Uska along-axis piece $v_x$ horizontal (adjacent) side hai; uska across piece $v_t$ vertical (opposite) side hai. Arrow aur axis ke beech ka angle $\tan (\text{angle})=v_t/v_x$ = across ÷ along ki tarah padha jaata hai. $\theta$, $\alpha$, $\varphi$ mein se har ek yahi angle hai, bas alag arrow ke liye.

WHY yeh tool: injector ko care hai ki spray kis taraf jaati hai. Tangent do velocities (sideways $v_t$ aur forward $v_x$) ko ek angle mein turn karta hai. Yahi exactly swirl cone law hai $\tan \varphi =\frac{v_t}{v_x},$ aur wahi triangle logic impinging spray direction $\alpha$ deta hai (wahan "sides" sideways aur forward momenta hain velocities ki jagah).

Ab teen specific angles, har ek neeche Figure 3b mein define aur picture kiya gaya:

Figure 3b — Teeno angles side by side, har ek chamber axis (dashed) se measure kiya gaya. Baayein: do impinging jets har ek aim angle $\theta$ par; unka collision ek sheet ko resultant angle $\alpha$ (red) par throw karta hai — yahan balanced, toh $\alpha =0$ seedha neeche. Daayein: ek swirl injector ka hollow cone half-angle $\varphi$ (red) par khulta hua.

11. Momentum $\dot{m}v$ aur momentum flux ratio $J$

Ek fire hose ko aapko peechhe knock karte huye imagine karen: paani ka mass times speed woh dhakka hai jo aap feel karte ho. Parent ka impinging law do jets ka sideways oomph add karta hai aur demand karta hai ki jab woh collide hote hain toh total conserved rahe — sheet bachhe hue oomph ke along fly off hoti hai (yahan se §10 ka $\alpha$ aata hai).

WHY topic ko yeh chahiye: coaxial atomization do streams ke beech ek ladaai hai. $J$ scoreboard hai.

12. Droplet diameter (droplet $d$) aur $d^2$ label

Ek ice cube versus crushed ice imagine karen: crushed ice bahut tez pighalta hai kyunki tiny pieces ki surface-to-volume huge hoti hai. Droplet size half karne se burn time quarter ho jaati hai. Poori derivation ke liye Atomization and the d-squared Law dekhen.

WHY topic ko yeh chahiye: yahi pura reason hai ki atomization matter karta hai — chhote droplets chamber ke andar jal jaate hain; bade unburned nikal jaate hain aur performance waste karte hain (Characteristic Velocity c-star se measure kiya gaya).

13. Chamber pressure $p_c$ aur stability rule of thumb

WHY topic ko yeh chahiye: injector ka apna drop $p_c$ ka ek healthy fraction rehna chahiye (rule of thumb $\Delta p\gtrsim 0.15p_c$) taaki roaring chamber oscillations ko holes se wapas push na kar sake. Agar aisa hota hai, feed aur chamber ek feedback loop mein lock ho jaate hain — dekhen Combustion Instability. (§9 ka ek choked gas orifice yeh decoupling automatically karta hai.)

Foundations topic ko kaise feed karte hain

Neeche ka map top to bottom padha jaata hai. Har box upar se ek foundation hai; cryptic-looking node names bas short tags hain, yahan expand kiye gaye hain taaki aap unhe trace back kar sako: p = pressure, pc = chamber pressure, dp = pressure drop $\Delta p$, rho = density, A = orifice area, v = velocity, mdot = mass flow $\dot{m}$, energy = Bernoulli energy balance, sqrt = square root, Cd = discharge coefficient, meter = metering law, gas = compressible/choked flow, mom = momentum, J = momentum flux ratio, tri = velocity right triangle & tan, imp/coax/swirl = teen injector spray laws, drop = droplet diameter, dsq = $d^2$ burn law.

Equipment checklist

Daayein side cover karo aur apne aap ko test karo. Agar koi jawab aapko surprise kare, toh woh section dobara padhen.