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

3.3.21 · Physics › Rocket Propulsion

WHAT is $c^{\ast }$?

Aise kyun define karte hain? Kyunki $p_0$, $A_t$, $\dot{m}$ woh teen cheezein hain jo tum test stand pe measure kar sakte ho bina nozzle exit ke baare mein kuch jaane. Yeh chamber performance ko isolate karta hai.

Thrust coefficient $C_F$ nozzle handle karta hai; specific impulse saaf split ho jaata hai: $I_{sp}{g}_0=c_e=c^{\ast }{C}_F$ Toh $c^{\ast }$ = chemistry & chamber, $C_F$ = nozzle. Yahi separation reason hai ki $c^{\ast }$ exist karta hai.

HOW to derive $c^{\ast }$ from first principles

Hum 1-D isentropic flow use karte hain ek perfect gas ki jo throat pe choked (Mach 1) hai.

Step 1 — Mass flow through a choked throat. $\dot{m}={\rho }_t{A}_t{v}_t$ Yeh step kyun? Continuity: throat area se mass in = mass out.

Step 2 — Throat pe $M=1$ hai, toh $v_t=a_t=\sqrt{\gamma R_s{T}_t}$ jahan $R_s=R_u/\mathcal{M}$ specific gas constant hai. Kyun? Choking ka matlab hai flow minimum area pe local speed of sound tak pahunch jaati hai.

Step 3 — Throat conditions ko chamber (stagnation) conditions se relate karo. Isentropic flow mein $M=1$ pe: $\frac{T_t}{T_0}=\frac{2}{\gamma +1},\frac{p_t}{p_0}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}},\frac{{\rho }_t}{{\rho }_0}={\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}$ Kyun? Yeh isentropic energy equation $T_0/T=1+\frac{\gamma -1}{2}{M}^2$ se aate hain, $M=1$ pe evaluate karne pe.

Step 4 — $\dot{m}$ assemble karo. ${\rho }_0=p_0/(R_s{T}_0)$ use karke: $\dot{m}=\underset{{\rho }_t}{\underbrace{{\rho }_0{\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}}}{A}_t\underset{v_t}{\underbrace{\sqrt{\gamma R_s{T}_t}}}$ $T_t=T_0\frac{2}{\gamma +1}$ aur ${\rho }_0=p_0/(R_s{T}_0)$ substitute karo aur simplify karo: $\dot{m}=\frac{p_0{A}_t\gamma }{\sqrt{\gamma R_s{T}_0}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$

Step 5 — $c^{\ast }=p_0{A}_t/\dot{m}$ mein invert karo:

Yeh final form kyun matter karta hai: yeh dependence seedha tumhare saamne rakh deta hai.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Thrust Coefficient CF — nozzle ka aadha hissa; $c_e=c^{\ast }{C}_F$.
• Specific Impulse Isp — $I_{sp}{g}_0=c^{\ast }{C}_F$.
• Choked Flow and the Throat — Step 2 mein use ki gayi $M=1$ condition.
• Isentropic Flow Relations — $\frac{2}{\gamma +1}$ ratios ka source.
• Adiabatic Flame Temperature — $T_0$ kahan se aata hai.
• Propellant Selection and Molecular Weight — kyun hum low $\mathcal{M}$ ke peeche bhaagte hain.
• Vandenkerckhove Function Γ — sirf-$\gamma$ wala lumped factor.

Concept Map