Thrust coefficient C_F = F - (P_c A - ) — derivation

3.3.9 · Physics › Rocket Propulsion

$C_F$ KYA hai?

Thrust ko is tarah kyun split karte hain? Thrust depend karta hai dono par — propellant chemistry (hot, high-molecular-energy gas) aur nozzle shape. Rocket engineers in dono ko alag karte hain: $F=\underset{\text{nozzle}}{\underbrace{C_F}}\times \underset{\text{chamber+throat}}{\underbrace{P_c{A}^{\ast }}}\text{aur}{c}^{\ast }=\frac{P_c{A}^{\ast }}{\dot{m}}(\text{characteristic velocity, propellant quality})$ To $F=C_F\cdot c^{\ast }\cdot \dot{m}$. $C_F$ = nozzle ka report card; $c^{\ast }$ = combustion ka report card.

Isse first principles se KAISE derive karte hain

Hum shuru karte hain fundamental thrust equation aur isentropic nozzle relations se, phir $P_c{A}^{\ast }$ se divide karte hain.

Step 1 — Thrust equation

Thrust = momentum thrust + pressure thrust: $F=\dot{m}{u}_e+(P_e-P_a)A_e$ Yeh step kyun? Newton ka 2nd/3rd law: rocket mass $\dot{m}$ ko exit speed $u_e$ par baahir dhakelta hai (momentum term), aur agar exit pressure $P_e$ ambient $P_a$ se alag hai, to exit area $A_e$ par yeh unbalanced pressure force add (ya subtract) karta hai.

Step 2 — Choked throat se mass flow

Choked (Mach 1) throat ke liye, one-dimensional isentropic flow deta hai: $\dot{m}=\frac{P_c{A}^{\ast }\gamma }{\sqrt{\gamma R{T}_c}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$ Yeh step kyun? Mass flow bilkul throat par hi set ho jaata hai jab flow choked hoti hai — isliye $A^{\ast }$ aata hai. Yahan $\gamma$ specific heat ratio hai, $R$ specific gas constant hai, $T_c$ chamber temperature hai. Ek compact group define karo: $\Gamma \equiv \sqrt{\gamma }{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}\Rightarrow \dot{m}=\frac{\Gamma P_c{A}^{\ast }}{\sqrt{R{T}_c}}$

Step 3 — Energy conservation se exit velocity

Isentropic expansion enthalpy ko kinetic energy mein convert karta hai. $h_c=h_e+\frac{1}{2}{u}_e^2$ se shuru karo (chamber gas at rest, $h=c_{p}T$): $u_e=\sqrt{2c_p(T_c-T_e)}=\sqrt{2c_p{T}_c\left(1-\frac{T_e}{T_c}\right)}$ $c_p=\frac{\gamma R}{\gamma -1}$ aur isentropic relation $\frac{T_e}{T_c}={\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}$ use karke: $\boxed{u_e=\sqrt{\frac{2\gamma R{T}_c}{\gamma -1}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}}$ Yeh step kyun? Energy conserve hoti hai: hot chamber gas ki "heat energy" directed kinetic energy ban jaati hai. Pressure ratio $P_e/P_c$ control karta hai ki nozzle kitna expansion (aur isliye kitni speed) achieve karta hai.

Step 4 — Momentum term $\dot{m}{u}_e/(P_c{A}^{\ast })$ assemble karo

$\frac{\dot{m}{u}_e}{P_c{A}^{\ast }}=\frac{\Gamma }{\sqrt{R{T}_c}}\sqrt{\frac{2\gamma R{T}_c}{\gamma -1}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}$ $\sqrt{R{T}_c}$ cancel ho jaata hai (isliye $C_F$ $T_c$ se independent hai!). $\Gamma$ substitute karke aur constants simplify karke: $\frac{\dot{m}{u}_e}{P_c{A}^{\ast }}=\sqrt{\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}$

Step 5 — Pressure term add karo

Pressure thrust ko $P_c{A}^{\ast }$ se divide karo: $\frac{(P_e-P_a)A_e}{P_c{A}^{\ast }}=\frac{P_e-P_a}{P_c}\cdot \frac{A_e}{A^{\ast }}$

Special cases (80/20 wali baat)

Worked examples

Common mistakes

Flashcards

Connections

• Thrust Equation — $F=\dot{m}{u}_e+(P_e-P_a)A_e$ starting point.
• Characteristic Velocity c-star — $C_F$ ka combustion-quality partner.
• Specific Impulse Isp — $I_{sp}{g}_0=c^{\ast }{C}_F$ ke through related.
• Choked Flow and the Throat — $\dot{m}$ relation ka source.
• Isentropic Nozzle Flow — Step 3 energy/pressure relations.
• Nozzle Expansion Ratio Ae-over-Astar — $P_e/P_c$ aur pressure term control karta hai.
• Over- and Under-Expanded Nozzles — pressure term ka sign.

Concept Map