Visual walkthrough — Characteristic velocity c - = P_c A - ṁ — derivation, combustion efficiency measure

3.3.10 · D2 · Physics › Rocket Propulsion › Characteristic velocity c - = P_c A - ṁ — derivation, combu

Algebra-heavy version ke liye yahan wapas aao: the parent note.

Characters ki cast (use se pehle define karo)

Ek bhi equation se pehle, chalo har symbol ko picture mein naam dete hain taaki kuch bhi anjaana na rahe.

• Chamber left side ki moti cavity hai jahan propellant jalta hai. Yeh (almost) still gas hai: pressure ==$P_c$, temperature $T_c$, density ${\rho }_c$==. Chhota $c$ matlab "chamber". "Almost still" important hai — iska matlab yeh hai ki yeh stagnation values hain (woh values jo gas ke nearly rest mein hone par hoti hain).
• Throat woh pinch hai — sabse narrow slice. Star ${}^{\ast }$ wali har cheez yahan rehti hai: area ==$A^{\ast }$==, temperature $T^{\ast }$, density ${\rho }^{\ast }$, speed $v^{\ast }$.
• ==$\dot{m}$== (kaho "m-dot") mass flow rate hai: har second kisi bhi slice se guzarne wale gas ke kilograms. Upar ka dot engineering shorthand hai "per second" ke liye.
• ==$\gamma$== (gamma) gas ko describe karne wala ek single number hai (hot rocket exhaust ke liye lagbhag $1.2$). Ise "gas kitna springy hai" socho — yeh set karta hai ki gas speed badhne par kitna thanda hota hai. Hum ise Step 3 mein theek se milenge.
• ==$R$== is particular gas ke liye gas constant hai: $R=R_u/\mathcal{M}$, bada universal constant $R_u=8314$ divided by molecular weight $\mathcal{M}$. Halke molecules → bada $R$. Yeh yaad rakho; yahi punchline hai.

Step 1 — Throat se guzarne wala mass count karo

KYA. Likho ki har second throat se kitna mass nikalti hai.

$\dot{m}=\underset{\begin{array}{c}\text{how packed}\\ \text{the gas is}\end{array}}{\underbrace{{\rho }^{\ast }}}\cdot \underset{\begin{array}{c}\text{how wide}\\ \text{the hole is}\end{array}}{\underbrace{A^{\ast }}}\cdot \underset{\begin{array}{c}\text{how fast}\\ \text{it moves}\end{array}}{\underbrace{v^{\ast }}}$

KYUN. Mass create ya destroy nahi ho sakta. Area $A^{\ast }$ ki ek thin disc of gas imagine karo jo ek second mein $v^{\ast }$ distance aage slide karti hai. Joh chhota cylinder yeh sweep karta hai uska volume $A^{\ast }{v}^{\ast }$ hai; density ${\rho }^{\ast }$ (mass per volume) se multiply karo aur mass per second milta hai. Yeh bas "swept volume × har cube kitna heavy hai" hai.

PICTURE. Neeche wala teal cylinder exactly woh swept volume hai. Notice karo ki hum abhi ${\rho }^{\ast }$ ya $v^{\ast }$ nahi jaante — yeh throat values hain, aur throat ek special jagah hai. Agle steps unhe dhundenge.

Step 2 — Throat exactly Mach 1 par run karta hai

KYA. Unknown throat speed ko local speed of sound se replace karo:

$v^{\ast }=a^{\ast }=\sqrt{\gamma R{T}^{\ast }}$

Yahan $a^{\ast }$ throat par gas mein speed of sound hai, aur $\sqrt{\gamma R{T}^{\ast }}$ woh formula hai ki ideal gas mein sound kitni tez travel karti hai — hotter gas (bada $T^{\ast }$) sound ko tez carry karta hai.

KYUN. Enough pressure wale converging–diverging nozzle mein, gas tab tak speed up hoti hai jab tak yeh exactly speed of sound tak pahunch jaaye sabse narrow point par, wahan aur nahi. Woh condition — flow speed equals sound speed — choking kehlati hai, aur yeh poore page ka load-bearing idea hai. ("Sonic aur faster kyun nahi" ki poori kahani ke liye dekho Choked Flow and the de Laval Nozzle.) Kyunki throat lock hai Mach 1 par, uski speed ab free unknown nahi hai — yeh $\sqrt{\gamma R{T}^{\ast }}$ par pin ho gayi hai.

PICTURE. Neeche wala gauge nozzle ke saath Mach number dikhata hai: yeh chamber mein nearly $0$ se climb karta hai aur pinch par exactly $1$ touch karta hai. Woh single dot $M=1$ par hi $\dot{m}$ ko predictable banata hai.

Step 3 — Throat conditions ko chamber conditions mein translate karo

KYA. Humare paas abhi bhi $T^{\ast }$ aur ${\rho }^{\ast }$ (throat) hain lekin hum sab kuch chamber terms $(T_c,{\rho }_c)$ mein chahte hain. Gas smoothly aur bina heat leak ke accelerate hui (isentropic process), aur uske liye fixed ratios hain jab hum jaante hain ki Mach number $1$ hai:

$\frac{T^{\ast }}{T_c}=\underset{\text{throat is cooler}}{\underbrace{\frac{2}{\gamma +1}}},\frac{{\rho }^{\ast }}{{\rho }_c}=\underset{\text{throat is thinner}}{\underbrace{{\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}}}$

Inhe literally padho: kyunki $\gamma >1$, fraction $\frac{2}{\gamma +1}$ 1 se kam hai, isliye throat gas chamber gas se thanda aur less dense dono hai. Yeh physically sense karta hai — gas ne apni internal heat speed kharidne mein kharch kar di.

Yeh exact numbers kyun? Jab gas accelerate hoti hai toh thermal energy ko kinetic energy mein trade karti hai. "Isentropic" (koi heat lost nahi, koi turbulence waste nahi) temperature, density aur speed ke beech exact bookkeeping force karta hai. Mach $=1$ un bookkeeping rules mein plug karne par $\frac{2}{\gamma +1}$ factors milte hain. $\gamma$ appear hota hai kyunki yeh govern karta hai ki gas speed badhne par kitni tezi se thandi hoti hai. In ratios ka full derivation Isentropic Flow Relations mein hai.

PICTURE. Do thermometers aur do density-dot patches: chamber vs throat. Dekho throat visibly thanda hai aur uske dots thinner spread hain.

Step 4 — Ideal gas law se chamber density pin karo

KYA. Humein abhi bhi ${\rho }_c$ chahiye. Ideal gas law use karo:

${\rho }_c=\frac{P_c}{R{T}_c}$

Har symbol: $P_c$ pressure molecules ko ek saath push karta hai (more density), $T_c$ temperature unhe apart jostle karta hai (less density), aur $R$ batata hai ki specific gas behave kaise karta hai.

KYUN. Yeh woh ek equation hai jo gauge par measure kiya ja sakne wale pressure ko Step 1 mein zarurati density se connect karti hai. Iske bina, $\dot{m}$ mein abhi bhi unmeasurable ${\rho }_c$ hota. Notice karo sneaky win: $R=R_u/\mathcal{M}$, isliye halke molecules matlab bada $R$ aur lower density — ek thread jo hum final step mein khinchte hain.

PICTURE. Ek pressure gauge molecules ke box mein feeding karta hua: zyada squeeze karo (raise $P_c$) → box mein zyada dots; heat karo (raise $T_c$) → dots fly apart, box mein kam.

Step 5 — Sab kuch ek $\dot{m}$ mein assemble karo

KYA. Steps 1–4 stack karo. $\dot{m}={\rho }^{\ast }{A}^{\ast }{v}^{\ast }$ se shuru karo, substitute karo ${\rho }^{\ast }={\rho }_c{\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}$, substitute karo $v^{\ast }=\sqrt{\gamma R{T}^{\ast }}=\sqrt{\gamma R{T}_c\cdot \frac{2}{\gamma +1}}$, aur ${\rho }_c=P_c/(R{T}_c)$. $T_c$ aur $R$ jo top aur bottom mein appear hote hain unhe cancel karne par milta hai:

$\dot{m}=\frac{P_c{A}^{\ast }}{\sqrt{R{T}_c}}\underset{\text{call this }\Gamma }{\underbrace{\sqrt{\gamma }{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}}}$

Exponent $\frac{\gamma +1}{2(\gamma -1)}$ kahan se aaya? Yeh bas $\frac{1}{\gamma -1}$ (density ratio se) plus $\frac{1}{2}$ (speed mein square root se) hai. Woh do fractions add karo aur exactly $\frac{\gamma +1}{2(\gamma -1)}$ milta hai — kuch bhi mysterious nahi, bas powers multiply karne par exponents add karna.

KYUN. Sab unknown (throat values) ko chamber values mein rewrite kar diya gaya hai. $\gamma$'s ka poora clump ek single symbol ==$\Gamma$== mein bundle ho gaya hai (Vandenkerckhove function) taaki formula clearly padhe.

PICTURE. Substitutions ka ek flow-chart: teen arrows unknowns ${\rho }^{\ast },v^{\ast },{\rho }_c$ ko master equation mein feed karte hain aur bahar aata hai $\dot{m}$.

Step 6 — Flip karo: $c^{\ast }$ ki definition khud nikal aati hai

KYA. Hum $\frac{P_c{A}^{\ast }}{\dot{m}}$ chahte the. Master equation ko across divide karo:

$c^{\ast }\equiv \frac{P_c{A}^{\ast }}{\dot{m}}=\frac{\sqrt{R{T}_c}}{\Gamma }=\frac{\sqrt{\gamma R{T}_c}}{\gamma }{\left(\frac{\gamma +1}{2}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$

Middle form $\frac{\sqrt{R{T}_c}}{\Gamma }$ sabse clean hai: right side par koi $P_c$ nahi, koi $A^{\ast }$ nahi, koi $\dot{m}$ nahi — sab cancel ho gaye. Sirf chamber-gas properties bachi hain.

KYUN yeh poora point hai. Left side woh hai jo test stand measure karta hai (pressure, throat width, flow). Right side pure chemistry hai $(T_c,R=R_u/\mathcal{M},\gamma )$. Inhe equal set karna kehta hai: tumhara measured pressure-per-flow reveal karta hai ki tumhari combustion kitni achi thi, aur kuch nahi. Isliye $c^{\ast }$ ek combustion report card hai aur Combustion Chamber Thermochemistry mein rehta hai, nozzle theory mein nahi.

PICTURE. Equation do mein split: left half tinted "stand par measured," right half tinted "chemistry se computed," unke beech ek bridge labelled yeh equality $c^{\ast }$ hai.

Step 7 — Edge cases (koi bhi scenario unshown mat chhorho)

Har honest derivation ko apne extremes se survive karna chahiye. Teen yahan matter karte hain.

Case A — Throat NOT choked ($M<1$). Agar chamber pressure bahut weak hai, flow throat par kabhi Mach 1 nahi pahuanchti. Tab Step 2 collapse ho jaata hai ($v^{\ast }\ne a^{\ast }$) aur tidy constant $c^{\ast }$ exist nahi karta — $\dot{m}$ ab downstream pressure par depend karta hai. Lesson: $c^{\ast }$ sirf tab meaningful hai jab tum choked ho.

Case B — $\gamma \to 1$ (bahut complex, "soft" molecules). Exponent $\frac{\gamma +1}{2(\gamma -1)}$ blow up hota hai jab $\gamma \to 1$, lekin $\Gamma$ finite rehta hai aur $\sqrt{e^{-1}\cdot 2}\approx$ ek well-behaved limit tak pahunchta hai; $c^{\ast }$ finite rehta hai. Toh formula explode nahi hota — yeh bas hotter, heavier-atom gases ke chote-ish $\gamma$ par lean karta hai.

Case C — Molecular weight extremes. Kyunki $c^{\ast }\propto \sqrt{T_c/\mathcal{M}}$, $\mathcal{M}\to$ tiny bhejne par (pure hydrogen exhaust) $c^{\ast }$ upar jaata hai; $\mathcal{M}$ huge bhejne par (heavy metal oxides) neeche jaata hai. Koi sign flip ya blow-up nahi hai — $c^{\ast }$ smoothly move karta hai aur sab physical inputs ke liye positive rehta hai.

PICTURE. Teen mini-panels: (A) ek Mach curve jo 1 touch karne mein fail hoti hai, greyed out; (B) $\Gamma$ vs $\gamma$ tame rehta hai; (C) $c^{\ast }$ vs $\mathcal{M}$ smoothly neeche slide karta hai.

Ek-picture summary

Upar sab kuch, compressed: mass conservation → choke at Mach 1 → isentropic ratios → ideal gas → assemble → flip. Left column physical chamber-to-throat story hai; right column algebra hai jo $c^{\ast }=\sqrt{R{T}_c}/\Gamma$ par collapse ho raha hai.

Cousins jo engine story complete karte hain: nozzle half hai Thrust Coefficient C_F, aur mile kar woh poore engine ka effective exhaust velocity dete hain.

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