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

3.3.10 · Physics › Rocket Propulsion

$c^{\ast }$ KYA hai?

Rocketry mein teen cousins:

• $c^{\ast }$ (characteristic velocity) → sirf combustion chamber + throat measure karta hai.
• $C_F$ (thrust coefficient) → sirf nozzle expansion measure karta hai.
• $c$ (effective exhaust velocity) → poora engine: $c=c^{\ast }{C}_F$.

Yeh clean split hi poori wajah hai ki $c^{\ast }$ exist karta hai: yeh chemistry ko geometry se alag karta hai.

Aisa number exist kyun karta hai? (choking argument)

Isse first principles se kaise derive karte hain

Hum choked mass-flow formula derive karenge, phir rearrange karenge.

Step 1 — Throat par mass flow. $\dot{m}={\rho }^{\ast }{A}^{\ast }{v}^{\ast }$ Yeh step kyun? Mass conservation: mass flow = density × area × speed, throat par evaluate kiya jahan sab kuch sonic hai.

Step 2 — Throat par gas sonic hai, $v^{\ast }=a^{\ast }=\sqrt{\gamma R{T}^{\ast }}$. Kyun? Definition ke hisaab se throat wahan hai jahan Mach $=1$ hota hai, aur ideal gas ke liye local sound speed $\sqrt{\gamma RT}$ hoti hai.

Step 3 — Throat conditions ko chamber (stagnation) conditions se relate karo. Chamber values $T_c,P_c$ aur Mach $M=1$ ke saath isentropic relations use karte hue: $\frac{T^{\ast }}{T_c}=\frac{2}{\gamma +1},\frac{P^{\ast }}{P_c}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}},\frac{{\rho }^{\ast }}{{\rho }_c}={\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}$ Kyun? Flow chamber mein ~rest se throat par Mach 1 tak isentropically accelerate hoti hai; yeh $M=1$ par standard isentropic ratios hain.

Step 4 — Chamber mein ideal gas: ${\rho }_c=\frac{P_c}{R{T}_c}$.

Step 5 — $\dot{m}$ assemble karo. Density aur speed substitute karo: $\dot{m}=\underset{{\rho }^{\ast }}{\underbrace{{\rho }_c{\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}}}\cdot A^{\ast }\cdot \underset{a^{\ast }}{\underbrace{\sqrt{\gamma R{T}_c\frac{2}{\gamma +1}}}}$ ${\rho }_c=P_c/(R{T}_c)$ daalo aur $T_c$ aur $R$ factors simplify karo: $\dot{m}=\frac{P_c{A}^{\ast }}{\sqrt{R{T}_c}}\sqrt{\gamma }{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$

Exponent $\frac{\gamma +1}{2(\gamma -1)}$ kyun? Kyunki density ratio se $\frac{1}{\gamma -1}$ aur speed mein $\sqrt{2/(\gamma +1)}$ se $\frac{1}{2}$ milke us ek single power mein combine ho jaate hain.

Step 6 — $P_c{A}^{\ast }/\dot{m}$ solve karo, jo ki $c^{\ast }$ hi hai: $\boxed{c^{\ast }=\frac{P_c{A}^{\ast }}{\dot{m}}=\frac{\sqrt{\gamma R{T}_c}}{\gamma }{\left(\frac{\gamma +1}{2}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}}$

Combustion efficiency

Yeh sahi efficiency metric kyun hai? Kam ${\eta }_{c^{\ast }}$ matlab hai ki real chamber pressure theory se neeche hai us propellant ke liye jo tumne feed kiya — yaani incomplete combustion, walls ko heat loss, ya mixing losses. Iske liye kharab nozzle ko blame nahi kiya ja sakta (woh $C_F$ ka kaam hai).

Worked examples

Common mistakes

Connections

• Choked Flow and the de Laval Nozzle — kyun throat par $M=1$ $\dot{m}$ lock karta hai.
• Thrust Coefficient C_F — nozzle ka score; $c=c^{\ast }{C}_F$.
• Effective Exhaust Velocity and Specific Impulse — $I_{sp}=c/g_0$.
• Isentropic Flow Relations — $2/(\gamma +1)$ ratios ka source.
• Combustion Chamber Thermochemistry — jahan se $T_c$, $\mathcal{M}$, $\gamma$ aate hain.
• Vandenkerckhove Function Γ — $\gamma$-dependence ka packaging.

Flashcards

Concept Map