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

3.3.9 · D2 · Physics › Rocket Propulsion › Thrust coefficient C_F = F - (P_c A - ) — derivation

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Step 0 — Characters ki cast (kisi bhi formula se pehle)

• $P_c$ — chamber ke andar ka pressure. Pressure = gas kitni zor se har wall pe push karta hai, pascals (Pa) mein measure hota hai. Socho chhote gas molecules walls pe hammer maar rahe hain.
• $T_c$ — us chamber gas ka temperature (molecules kitni tezi se hilte hain).
• $A^{\ast }$ — throat area: nozzle ka sabse sankra cross-section, pinch-point.
• $A_e$ — exit area: woh chauda munh jahan gas finally nikalti hai.
• $P_e$ — exit pe gas ka pressure (usually $P_c$ se bahut kam — yeh expand ho chuki hai).
• $P_a$ — bahar ka ambient pressure (sea-level air ≈ $0.1$ MPa; space = $0$).
• $u_e$ — gas ki exit speed (metres per second).
• $\dot{m}$ — mass flow rate: kilograms of gas leaving per second. Dot ka matlab "per second" hai.

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Step 1 — Thrust equation: push kahan se aata hai?

KYA. Hum total push ko do pieces ka sum likhte hain: $F=\underset{\text{momentum thrust}}{\underbrace{\dot{m}{u}_e}}+\underset{\text{pressure thrust}}{\underbrace{(P_e-P_a)A_e}}$

Symbols ko seedha wahin padhna jahan woh baithe hain:

• $\dot{m}{u}_e$ — (kg/s) × (m/s) = kg·m/s² = newtons. Har second tum $\dot{m}$ kilograms peeche $u_e$ speed se phenkte ho; Newton ke third law se woh tumhe utna hi aage push karte hain. Yeh rocket-throwing-mass term hai.
• $(P_e-P_a)$ — munh pe bacha hua pressure difference. Agar exit gas bahar ki hawa se zyada push kar rahi hai, toh woh surplus engine ko push karta hai.
• $A_e$ — woh area jis par surplus push karta hai. Pressure × area = force.

YEH FORM KYUN. Ek rocket zameen ya hawa ke against push nahi karta — woh us mass ke against push karta hai jo woh phenkta hai. Pressure term ek correction hai: nozzle rarely gas ko exactly bahar ke pressure pe dump karta hai, aur koi bhi mismatch exit area par act karta hai. (Poori baat Thrust Equation mein hai.)

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Step 2 — Mass flow throat pe set hota hai

KYA. Ek choked throat ke liye, one-dimensional isentropic flow (dekho Isentropic Nozzle Flow) se mass pour hota hai: $\dot{m}=\frac{P_c{A}^{\ast }\gamma }{\sqrt{\gamma R{T}_c}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}$

Naye symbols, yahan define hain:

• $\gamma$ (gamma) — gas ka specific-heat ratio, $1.2$ jaisi ek number jo batati hai gas compress hone par kitna "springy" hai. Pure gas property hai.
• $R$ — specific gas constant (yeh particular gas ka ek kilogram kitna expand hota hai garam hone par).

YEH STEP KYUN. Figure dekho: kyunki flow throat par exactly Mach 1 hit karta hai, downstream se koi signal upstream nahi ja sakta ise change karne ke liye. Throat ek valve fixed wide open hai — mass flow sirf chamber conditions aur throat size $A^{\ast }$ par depend karta hai. Isliye precisely $A^{\ast }$, $A_e$ nahi, engine ka reference area hai.

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Step 3 — Energy conservation se exit speed

KYA. Gas chamber mein almost still shuru hoti hai (saari energy "heat" hai) aur tez nikalti hai (energy ab "motion" hai). Energy in = energy out: $h_c=h_e+\frac{1}{2}{u}_e^2$

• $h$ — enthalpy, per kilogram heat-energy content; $h=c_{p}T$.
• $c_p$ — specific heat at constant pressure: ek kilogram ko ek degree warm karne ke liye joules chahiye.

Exit speed ke liye solve karte hue aur $c_p=\frac{\gamma R}{\gamma -1}$ plus isentropic link $\frac{T_e}{T_c}={\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}$ daalte hue: $\boxed{u_e=\sqrt{\frac{2\gamma R{T}_c}{\gamma -1}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}}$

Term by term:

• $\frac{2\gamma R{T}_c}{\gamma -1}$ — total heat energy available (bada $T_c$ → zyada garam → zyada tez).
• ${\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}$ — exit pe heat ke roop mein abhi bhi trapped energy ka fraction. Chhota $P_e/P_c$ (zyada expansion) → yeh term shrink karta hai → bracket barta hai → tez gas.
• Bracket $[1-\cdots ]$ — heat ka woh fraction jo actually speed mein convert hua.

YEH STEP KYUN. Figure ek pahaadi se valley mein girna dikhata hai: enthalpy (height) kinetic energy (speed) mein turn hoti hai. Pressure ratio $P_e/P_c$ literally hai kitna neecha gas pahaad se slide kar pata hai. (Isliye Nozzle Expansion Ratio Ae-over-Astar itna matter karta hai.)

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Step 4 — Momentum term ko reference force se divide karo

KYA. Ab hum $C_F$ piece by piece banate hain. Pehle momentum thrust $\dot{m}{u}_e$ ko reference force $P_c{A}^{\ast }$ se divide karo: $\frac{\dot{m}{u}_e}{P_c{A}^{\ast }}=\frac{1}{P_c{A}^{\ast }}\cdot \frac{\Gamma P_c{A}^{\ast }}{\sqrt{R{T}_c}}\cdot u_e=\frac{\Gamma }{\sqrt{R{T}_c}}{u}_e$

Step 3 se $u_e$ substitute karo: $\frac{\dot{m}{u}_e}{P_c{A}^{\ast }}=\frac{\Gamma }{\sqrt{R{T}_c}}\sqrt{\frac{2\gamma R{T}_c}{\gamma -1}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}$

Cancellation kyun matter karta hai. Figure mein red boxes dekho: neeche $\sqrt{R{T}_c}$ aur $u_e$ ke andar chupi $\sqrt{R{T}_c}$ exactly cancel ho jaati hain. Saari propellant-specific, temperature-specific cheezein gayab ho jaati hain. Jo bachta hai woh $\gamma$ aur pressure ratio ka pure function hai: $\frac{\dot{m}{u}_e}{P_c{A}^{\ast }}=\underset{C_{F,\text{mom}}}{\underbrace{\sqrt{\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}\left[1-{\left(\frac{P_e}{P_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}}}$

• $\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}$ — ek pure-$\gamma$ constant (purana $\Gamma$ squared, tidy kiya hua).
• Bracket — Step 3 se expansion factor, unchanged.

Yeh $C_{F,\text{mom}}$ hai, momentum thrust coefficient. Ise $T_c$ ya $R$ ke baare mein kuch nahi pata. Yahi poore page ka deep point hai.

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Step 5 — Pressure term add karo, full $C_F$ pao

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

• $\frac{P_e-P_a}{P_c}$ — normalised pressure mismatch (ek chhota dimensionless number).
• $\frac{A_e}{A^{\ast }}$ — expansion ratio, munh throat se kitna zyada chauda hai.

Dono pieces add karo:

KYUN. Figure dono contributions ko coloured bars ki tarah stack karta hai: ek bada blue momentum bar plus ek chhota orange (ya red, agar negative ho) pressure bar. Total height = $C_F$. Jo kuch bhi reader ne dekha hai — throat, exit, chamber pressure, expansion ratio — ab ek formula mein visible hai.

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Step 6 — Teen limiting cases (kabhi surprise mat hona)

KYA AUR KYUN. Har real engine in teen pictures ke beech mein rehta hai. Figure dikhata hai pressure bar shrink karta, sign flip karta, aur gayab hota hai.

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Ek picture ka summary

Recall Feynman: poora walkthrough plain words mein

Ek rocket do reasons se aage push karta hai: woh hot gas peeche phenk ta hai, aur munh pe bacha hua pressure bhi push karta hai. Humne unhe momentum thrust aur pressure thrust likha. Phir humne pucha: gas nikal bhi kitni sakti hai? Kyunki sankra throat Mach 1 par chalta hai, woh akela flow set karta hai — toh throat area hamara natural yardstick ban gaya. Phir humne puch a gas kitni tez nikalti hai, aur energy se jawab diya: chamber ki heat pahaadi se neeche slide karti hai aur speed mein turn hoti hai, control hoti hai is baat se ki pressure kitna girta hai. Finally humne poora thrust "reference force" (chamber pressure times throat area) se divide kiya. Kuch wonderful hua: saara temperature aur gas-chemistry stuff cancel ho gaya, ek aisa number bachha jo sirf gas springiness $\gamma$, pressure ratio, aur nozzle ki shape par depend karta hai. Woh number, $C_F$, nozzle ke liye ek pure report card hai — aur is baat par depend karte hue ki bahar ki hawa sea level par hai ya vacuum mein, chhota pressure term ise neeche ya upar nudge karta hai.

Recall Quick self-test

$A^{\ast }$ (throat) kyun reference area banta hai aur $A_e$ (exit) kyun nahi? ::: Kyunki choked throat par flow Mach 1 par hota hai aur downstream se kuch bhi mass flow change nahi kar sakta — throat akela $\dot{m}$ fix karta hai, isliye woh engine ka natural yardstick hai. Kaunsa term $P_e=P_a$ hone par gayab hota hai, aur us condition ka physically kya matlab hai? ::: Pressure-correction term; iska matlab hai nozzle perfectly expanded hai (ambient se matched), us pressure ratio ke liye maximum thrust deta hai. $C_F$ chamber temperature $T_c$ se independent kyun hai? ::: Mass flow mein $\sqrt{R{T}_c}$ aur $u_e$ ke andar $\sqrt{R{T}_c}$ $P_c{A}^{\ast }$ se divide karne par exactly cancel ho jaate hain; temperature effects $\dot{m}$ aur $c^{\ast }$ mein rehte hain. Sea level par over-expanded nozzle mein pressure term positive hai ya negative? ::: Negative — $P_e<P_a$, toh ambient air push back karta hai aur $C_F$ ko momentum value se neeche le jaata hai.