Nozzle area ratio ε = A_e - A - — choosing for optimal performance

3.3.18 · Physics › Rocket Propulsion

Area ratio matter kyun karta hai?

KYA chahiye: maximum thrust. Thrust ke do parts hain: $F=\dot{m}{v}_e+(p_e-p_a)A_e$

• $\dot{m}{v}_e$ = momentum thrust (mass flow × exit velocity)
• $(p_e-p_a)A_e$ = pressure thrust (exit pressure minus ambient, exit area ke upar)

KYUN ε yeh control karta hai: Supersonic flow mein, ek bada diverging area gas ko zyada expand karne deta hai, jo $v_e$ badhata hai lekin $p_e$ ghatata hai. Toh ε exit velocity aur exit pressure ke beech trade karta hai. Ek sweet spot hota hai.

$p_e=p_a$ optimal kyun hai — derivation (scratch se)

KAISE: Chamber conditions aur $\dot{m}$ ko fixed rakho. Jab hum ε change karte hain, $v_e$ aur $A_e$ dono change hote hain lekin woh mass conservation $\dot{m}={\rho }_e{v}_e{A}_e$ se linked hain. Guess karne ki jagah, thrust ko $A_e$ ke respect mein differentiate karo.

Total exit force = momentum flux $+p_e{A}_e$. Diverging section ke along, steady momentum balance se $(\dot{m}v)$ ka incremental change ek slice mein net pressure force ke barabar hota hai. Added exit area $d{A}_e$ ke liye wall pressure $\approx p_e$ act karta hai. Dhyan se:

$dF=\dot{m}d{v}_e+d(p_e{A}_e)-p_{a}d{A}_e$

Flow ke liye steady 1-D momentum equation deti hai $\dot{m}d{v}_e=-Adp$, aur nozzle ke along internal pressure force balance karta hai taaki $\dot{m}d{v}_e+d(p_e{A}_e)=p_{e}d{A}_e$ ho (internal terms combine hokar local wall pressure times area change ban jaate hain). Substitute karne par:

$dF=p_{e}d{A}_e-p_{a}d{A}_e=(p_e-p_a)d{A}_e$

$dF/d{A}_e=0$ set karo:

$\boxed{p_e=p_a}\text{(optimum expansion)}$

$p_e/p_0$ se area ratio ε tak

KYUN yeh link chahiye: Hum ek design $p_e$ choose karte hain ($p_a$ se matched), phir use geometry ε mein convert karna hota hai taaki nozzle actually build kar sakein.

Isentropic 1-D flow with ratio of specific heats $\gamma$. Do ingredients:

(1) Mach–pressure relation (energy + isentropic $p{\rho }^{-\gamma }=$const se): $\frac{p_0}{p_e}={\left(1+\frac{\gamma -1}{2}{M}_e^2\right)}^{\frac{\gamma }{\gamma -1}}$

(2) Area–Mach relation (mass conservation se, throat par $M=1$, area $A^{\ast }$): $\frac{A_e}{A^{\ast }}=\frac{1}{M_e}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}_e^2\right)\right]}^{\frac{\gamma +1}{2(\gamma -1)}}$

Derivation sketch of (2): Mass flow constant hai: $\rho Av={\rho }^{\ast }{A}^{\ast }{v}^{\ast }$. $\rho$, $v$ ko $M$ ke terms mein likho isentropic + sound-speed $a=\sqrt{\gamma RT}$ use karke, throat par $M=1$ set karo, aur simplify karo. $1/M$ front factor ki wajah se same ε do Machs ke corresponding ho sakta hai (ek subsonic, ek supersonic) — hum hamesha throat ke downstream supersonic branch lete hain.

Worked Example 1 — Design altitude ke liye ε nikalo

Chamber $p_0=70$ bar, exhaust $\gamma =1.20$, design $p_e=p_a=0.5$ bar.

Step 1 — pressure ratio. $p_0/p_e=70/0.5=140$. Kyun: Yeh woh hai jo (1) ko chahiye; bada ratio → zyada expansion → bada ε.

Step 2 — $M_e$ solve karo. $140={\left(1+0.1M_e^2\right)}^6\Rightarrow 1+0.1M_e^2=140^{1/6}=2.276$ $M_e^2=12.76,M_e=3.57$ Exponent 6 kyun: $\gamma /(\gamma -1)=1.2/0.2=6$.

Step 3 — area ratio. $M_e=3.57$ ke saath, $(\gamma +1)/[2(\gamma -1)]=2.2/0.4=5.5$: $\epsilon =\frac{1}{3.57}[\frac{2}{2.2}(2.276){]}^{5.5}=\frac{1}{3.57}(2.069{)}^{5.5}\approx \frac{1}{3.57}(54.6)\approx 15.3$ Yeh magnitude kyun: Vacuum-ish engines ko tens ya hundreds mein ε chahiye; sea-level engines chhote hote hain (ε ≈ 5–15).

Worked Example 2 — Off-design penalty (over-expansion)

Wahi nozzle (ε≈15.3, $p_e=0.5$ bar) sea level par fire kiya ($p_a=1.0$ bar). Exit area $A_e=0.3{\text{m}}^2$.

Pressure thrust term: $(p_e-p_a)A_e=(0.5-1.0)\times 10^5\times 0.3=-1.5\times 10^4\text{N}$ Negative kyun: $p_e<p_a$ → over-expanded → yeh term 15 kN thrust subtract karta hai aur flow separation trigger kar sakta hai. Practice mein fix: sea-level stage ke liye chhota ε use karo, upper stages ke liye bada ε.

Flashcards

Connections

• Thrust Equation and Effective Exhaust Velocity
• De Laval Converging-Diverging Nozzle
• Isentropic Flow Relations
• Choked Flow and the Throat Condition
• Flow Separation in Over-expanded Nozzles
• Altitude Compensation — Aerospike Nozzles
• Specific Impulse Isp

Concept Map