Foundations — Characteristic velocity c - and its relation to flame temperature, MW

3.3.21 · D1 · Physics › Rocket Propulsion › Characteristic velocity c - and its relation to flame tempe

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Koi bhi algebra se pehle, woh machine dekho jise hum describe kar rahe hain.

Rocket chamber ek mota pipe hai jo ek side se band hai. Propellant andar jalti hai aur bahut saari garam gas ka badal banti hai. Gas sirf ek tang jagah se bahar ja sakti hai — throat — jiske baad pipe bell ke shape mein khul jaata hai yaani nozzle. $c^{\ast }$ sirf mote hisse aur tang jagah ko grade karta hai. Phailne wala bell alag kahaani hai (woh $C_F$ hai, dekho Thrust Coefficient CF).

Yeh drawing apne dimaag mein rakho. Neeche har symbol us par kisi hisse ka label hai.

Symbols ko ek ek karke build karna

1. Pressure $p$ — gas kitni zor se deewaaron ko dhakka deti hai

Picture: laakhon tiny gas molecules baar baar chamber ki deewar se bounce karti hain. Har bounce ek chota sa dhakka hai. Ek square metre par saare dhakke jodo — woh total dhakka hi pressure hai.

Yeh topic kyun zaroori hai: rocket chamber ka poora maqsad pressure build karna hai taaki gas bahar bhaage. $c^{\ast }$ ke numerator mein literally pressure shuru hoti hai.

2. Stagnation (chamber) pressure $p_0$ — pressure jahan gas ruki hui hai

Picture: figure s01 mein, baaya mota hissa almost still gas hai. Woh shant, high-pressure zone wahan hai jahan hum $p_0$ padhte hain. Jaise jaise gas throat ki taraf speed badhati hai, woh kuch pressure speed ke liye exchange karti hai, isliye throat par pressure $p_0$ se kam hoti hai — yeh ek important idea hai jo hum baad mein use karte hain.

Yeh topic kyun zaroori hai: $p_0$ woh "quality" hai jo combustion produce karta hai. Zyada hot, better burn zyada $p_0$ build karta hai. Yeh teen chezon mein se pehli hai jo aap test stand par measure kar sakte hain.

3. Throat aur throat area $A_t$ — sabse tang hole

Picture: pipe ki kamar ko seedha kaat do — ek chota sa circle milega. Uska area $A_t$ hai.

Yeh topic kyun zaroori hai: throat woh bottleneck hai jo control karta hai ki kitni gas escape karti hai. Uska area $c^{\ast }=p_0{A}_t/\dot{m}$ mein doosra measurable quantity hai.

4. Mass flow rate $\dot{m}$ — har second kilograms mein gas nikalna

Picture: throat par stopwatch aur scale lekar khade ho jao. Ek second mein, saari gas ka wazan karo jo se guzri. Woh wazan (kg mein) $\dot{m}$ hai.

Yeh topic kyun zaroori hai: $\dot{m}$ teesra measurable quantity hai. Milke $p_0$, $A_t$, $\dot{m}$ aapko $c^{\ast }$ dete hain bina nozzle ki koi jaankari ke — yahi isolation ki wajah se $c^{\ast }$ exist karta hai.

5. Characteristic velocity $c^{\ast }$ khud — woh number jiske liye yeh sab build kiya

Picture: figure s01 mein, $c^{\ast }$ ek scorecard hai jo mote chamber + waist ke upar hover kar raha hai. High score matlab burn ne bahut pressure build kiya har kilogram escape hone wali gas ke liye.

Yeh topic kyun zaroori hai: yahi poore topic ka star hai. Isse pehle sab kuch (symbols 1–4) iske ingredients hain; iske baad sab kuch (symbols 6–15) explain karta hai ki kyun score temperature aur molecular weight par depend karta hai.

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

Picture: gas ka ek dabba. Andar ke kilograms gino, dabbe ke cubic metres mein volume se divide karo. Dense gas = zyada molecules per box.

Yeh topic kyun zaroori hai: nikalne wali gas ki matra depend karti hai ki woh kitni packed hai. Mass-flow derivation $\dot{m}={\rho }_t{A}_t{v}_t$ se shuru hoti hai — density × hole × speed.

7. Continuity — "kuch disappear nahi hota" ka bookkeeping

Picture: socho log darwaaze se file kar rahe hain. Har second wale = woh kitne closely packed hain × darwaaze ki chaudai × woh kitni tez chal rahe hain. Same logic, logon ki jagah gas.

Yeh topic kyun zaroori hai: parent ki derivation ka yeh Step 1 hai. Har rocket flow calculation yahan se shuru hoti hai.

8. Temperature $T$ aur flame temperature $T_0$ — molecules kitne energetic hain

Picture: zyada hot gas = molecules zyada tezi se zoom aur bounce kar rahe hain. Figure s01 mein, mota chamber temperature $T_0$ par glowing hot hai.

Yeh topic kyun zaroori hai: hot gas fast gas hoti hai. $c^{\ast }$ $\sqrt{T_0}$ ke saath badhta hai — "hot and light wins" mein "hot." Dekho Adiabatic Flame Temperature ki $T_0$ ka number kahan se aata hai.

9. Molecular weight $\mathcal{M}$ — har molecule kitna bhaari hai

Picture: ek "mole" (ek fixed bada count, $6.022\times 10^{23}$) molecules ko scale par rakho. Bhaare molecules (bada $\mathcal{M}$) zyada wazan karte hain; halke molecules (chota $\mathcal{M}$) kam wazan karte hain.

Yeh topic kyun zaroori hai: same energy ke liye, halke molecules tezi se move karte hain (socho ping-pong ball vs bowling ball ko same dhakke se maaro). $c^{\ast }$ $\sqrt{1/\mathcal{M}}$ ke saath badhta hai — "hot and light wins" mein "light." Dekho Propellant Selection and Molecular Weight.

10. Gas constants $R_u$ aur $R_s$ — energy ko speed mein convert karna

Picture: figure s02 mein same energy budget share hote dikhti hai. Per mole yeh fixed hai ($R_u$). Per kilogram depend karta hai ki aapke molecules kitne bhaare hain — halki gas ko har kilogram mein zyada speed milti hai.

Yeh topic kyun zaroori hai: final boxed form $c^{\ast }=\frac{1}{\Gamma }\sqrt{R_u{T}_0/\mathcal{M}}$ mein ek $R_s$ chupi hai — yeh $R_u$ sirf isliye rakhta hai kyunki $\mathcal{M}$ waheen divide kar raha hai.

11. Speed of sound $a$ — gas mein push kitni tezi se travel karta hai

Picture: haath bajao; compressed air ka "bump" speed $a$ par bahar ki taraf daudta hai. Hot light gas mein woh bump tezi se travel karta hai kyunki molecules use jaldi relay karte hain.

Yeh topic kyun zaroori hai: throat par gas exactly apni khud ki speed of sound par move karti hai. Yeh fact throat speed fix karta hai aur poori derivation ka dil hai.

12. Mach number $M$ aur choked flow — "M = 1" gate

Picture: figure s03 — gas chamber mein almost still ($M\approx 0$) se accelerate karti hai, tang kamar par exactly $M=1$ hit karti hai, phir flaring bell mein supersonic ($M>1$) ho jaati hai.

Yeh topic kyun zaroori hai: parent ki derivation ka yeh Step 2 hai, $v_t=a_t=\sqrt{\gamma R_s{T}_t}$. Dekho Choked Flow and the Throat.

13. Heat-capacity ratio $\gamma$ — gas ki "springiness"

Picture: gas ko ek spring ki tarah socho. $\gamma$ measure karta hai woh spring kitna "bouncy" hai — gas compress ya expand hone par pressure kitni tezi se change hota hai.

Yeh topic kyun zaroori hai: $\gamma$ speed of sound aur neeche isentropic ratios mein appear hota hai. Yeh $c^{\ast }$ mein sirf weakly enter karta hai, $\Gamma$ ke andar bundled hokar.

14. Isentropic flow — smooth, no-loss flow

Picture: jaise gas throat ki taraf rush karti hai woh speed up hoti hai, aur us speed ki keemat pay karne ke liye woh thandi aur patli ho jaati hai — pressure aur density exactly upar wale factors se drop karte hain.

Yeh topic kyun zaroori hai: yeh ratios (Step 3) throat conditions jo hum easily measure nahi kar sakte unhe chamber conditions $p_0,T_0$ mein convert karte hain jo hum kar sakte hain. Dekho Isentropic Flow Relations.

15. Vandenkerckhove function $\Gamma$ — $\gamma$ ka seedha bundle

Picture: $\Gamma$ ko ek single dial ki tarah socho jo summarise karta hai "gas ki springiness flow ko kaise affect karti hai." Yeh barely hilta hai, isliye $\gamma$ ka $c^{\ast }$ par sirf weak effect hota hai.

Yeh topic kyun zaroori hai: sirf abhi — $\Gamma$ define hone ke baad — hum clean final form $c^{\ast }=\frac{1}{\Gamma }\sqrt{R_u{T}_0/\mathcal{M}}$ likh sakte hain, springy-$\gamma$ part ko hot-and-light part se alag karte hue. Dekho Vandenkerckhove Function Γ.

16. Exhaust velocity $c_e$, $C_F$ aur $I_{sp}$ — $c^{\ast }$ kya NAHI hai

Picture: phir se figure s01 — $c^{\ast }$ = mota chamber + waist, $C_F$ = flaring bell, aur unhe multiply karne par real exhaust speed $c_e$ milti hai.

Yeh topic kyun zaroori hai: taaki aap kabhi $c^{\ast }$ (ek chamber figure of merit, real speed nahi) ko exhaust speed se confuse na karo. Dekho Specific Impulse Isp aur Thrust Coefficient CF.

Foundations topic ko kaise feed karte hain

Equipment checklist

Test yourself — you should be able to answer each before reading the parent note.

Connections

• 3.3.21 Characteristic velocity c - and its relation to flame temperature, MW (index 3.3.21) — woh parent jiske liye yeh page aapko prepare karta hai.
• Choked Flow and the Throat — symbol 12 mein built $M=1$ gate.
• Isentropic Flow Relations — symbol 14 ke smooth-flow ratios.
• Adiabatic Flame Temperature — $T_0$ kahan se aata hai (symbol 8).
• Propellant Selection and Molecular Weight — kyun chota $\mathcal{M}$ jeetta hai (symbol 9).
• Vandenkerckhove Function Γ — symbol 15 ka $\gamma$-bundle.
• Thrust Coefficient CF — nozzle half, $c_e=c^{\ast }{C}_F$.
• Specific Impulse Isp — efficiency score $I_{sp}{g}_0=c_e$.