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

3.3.10 · D1 · Physics › Rocket Propulsion › Characteristic velocity c - = P_c A - ṁ — derivation, combu

Koi formula aane se pehle: rocket engine ki picture

Is topic mein sab kuch ek hi shape ke andar hota hai. Ise ek baar draw karte hain aur baar baar isi ki taraf point karte rahenge.

Figure 1 — Rocket engine ki anatomy.

Figure 1 dekho. Left se right padhte hue:

• Ek mota combustion chamber jahan fuel aur oxidiser jalkar hot gas bante hain jo almost still khade hain (bahut dheere move kar rahe hain).
• Walls andar ki taraf squeeze hokar ek narrow throat banate hain — yeh pinch hai. Yeh sabse tight cross-section hai.
• Throat ke baad walls baahir ki taraf flare hokar nozzle banate hain, jahan gas speed up hoti hai aur exit se shoot karti hai.

Characteristic velocity sirf left half ka report card hai (chamber + throat, Figure 1 mein blue shaded). Nozzle (orange) ko alag number se grade kiya jaata hai. Yeh split apne dimaag mein rakho — yahi reason hai is poore topic ke exist karne ka.

Symbol 1 — Area, aur hum $A^{\ast }$ kyun likhte hain

Chhota star ${}^{\ast }$ ek flag hai jiska matlab hai "throat par measure kiya gaya jahan gas sound speed hit karti hai". Hum exit area $A_e$ yahaan use nahi karte — yeh ek bahut common trap hai. Physics jo characteristic velocity ko special banati hai woh throat par hoti hai, isliye $A^{\ast }$ woh area hai jo humein chahiye.

Symbol 2 — Mass flow rate $\dot{m}$

Throat par ek checkpoint imagine karo aur ek counter jo har kilogram ko count karta hai jo cross karta hai. Agar har second $25$ kg cross karte hain, toh $\dot{m}=25\text{kg/s}$. Yeh "kitna maine dhakel diya" wali quantity hai — ek-line idea se.

Symbol 3 — Density $\rho$ aur velocity $v$: $\dot{m}$ build karna

Gas kaise mass flow banata hai? Throat par teen ingredients milte hain.

Figure 2 — Mass flow bas ek door se kilograms count karna hai.

Ab Figure 2 master relation explain karta hai. Ek second mein, woh gas jo throat cross karti hai ek tube bhaarta hai jiska length $v$ hai (woh $v$ metres chali) aur cross-section $A^{\ast }$ hai. Us tube ka volume hai $A^{\ast }\times v$. Mass paane ke liye density $\rho$ se multiply karo: $\dot{m}=\rho A^{\ast }v$

Symbol 4 — Pressure $P$ aur chamber value $P_c$

Hot burning ne sealed chamber ko fast-jiggling molecules se bhar diya; woh walls par hammer karte hain, aur wahi hammering hai $P_c$. Yeh "kitna pressure maine hold kiya" wali quantity hai — jo payoff hum $\dot{m}$ ke against grade karte hain.

Symbol 5 — Temperature $T$ aur chamber flame temperature $T_c$

Hotter gas zyada tezi se jiggle karta hai, toh woh zyada tezi se push karta hai aur tezi se baahir nikalti hai. Isliye $T_c$ tab dikhta hai jab hum chemistry se characteristic velocity compute karte hain: zyada $T_c$ matlab zyada high score.

Symbol 6 — Speed of sound $a$, aur "sonic / Mach 1 / choked"

Yeh heart hai why throat is special, isliye hum ise ek picture dete hain.

Figure 3 — Sound speed temperature ke saath badh jaati hai; throat Mach 1 par lock ho jaata hai.

Symbol 7 — Gas constant $R$, $R_u$ aur molecular weight $\mathcal{M}$

Halke molecules (chhota $\mathcal{M}$) ek bada $R$ dete hain, isliye fast sound speed aur high $c^{\ast }$ — isliye hydrogen exhaust ko itna value diya jaata hai. Dekho Combustion Chamber Thermochemistry.

Symbol 8 — Heat-capacity ratio $\gamma$

Tumhe yahaan $\gamma$ derive karne ki zaroorat nahi — bas jaano ki yeh tumhare gas mixture ki ek fixed property hai, Combustion Chamber Thermochemistry se deliver hoti hai, aur yeh har isentropic ratio mein appear hoti hai.

Symbol 9 — Isentropic ratios ("star over c" factors)

Jab gas still chamber se sonic throat tak smoothly aur bina heat loss ke flow hoti hai (isliye isentropic kehte hain), fixed fractions throat values ko chamber values se relate karte hain. $M=1$ par: $\frac{T^{\ast }}{T_c}=\frac{2}{\gamma +1},\frac{P^{\ast }}{P_c}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}},\frac{{\rho }^{\ast }}{{\rho }_c}={\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}$

Symbol 10 — Vandenkerckhove function $\Gamma$

Jab bhi $\Gamma$ dekho, mentally padho "saara $\gamma$-wala stuff, bottled". Parent note ki choked mass flow ki derivation mein end hota hai $\dot{m}=\frac{P_c{A}^{\ast }}{\sqrt{R{T}_c}}\Gamma ,$ kyunki messy $\sqrt{\gamma }(2/(\gamma +1){)}^{\dots }$ ka pile exactly $\Gamma$ hai. Us ratio $P_c{A}^{\ast }/\dot{m}$ ke liye rearrange karne par clean theoretical form milta hai: $c_{\text{ideal}}^{\ast }=\frac{P_c{A}^{\ast }}{\dot{m}}=\frac{\sqrt{R{T}_c}}{\Gamma }.$

Symbol 11 — Characteristic velocity $c^{\ast }$ khud

Symbol 12 — Teen cousin velocities

Parent note teen "velocities" naam karta hai. Inhe seedha rakho:

Foundations topic ko kaise feed karte hain

Equipment checklist

Right side cover karo aur khud ko test karo — tum derivation ke liye ready ho sirf tab jab har line asaani se aaye.