Characteristic velocity c - = P_c A - ṁ — derivation, combustion efficiency measure: Question bank

3.3.10 · D5 · Physics › Rocket Propulsion › Characteristic velocity c - = P_c A - ṁ — derivation, combu

In do faces ko yaad rakho jo poore topic mein kaam aate hain: $c^{*} = \frac{P _{c} A ^{*}}{m ˙} (measured), c_{ideal}^{*} = \frac{γ R T _{c}}{γ} (\frac{γ + 1}{2})^{\frac{γ + 1}{2 ( γ - 1 )}} (theory)$ jahan $P_{c}$ = chamber pressure, $A^{*}$ = throat area, $\overset{m}{˙}$ = mass flow, $T_{c}$ = flame temperature, $R = R_{u} / M$ (universal gas constant divided by exhaust molecular weight), $γ$ = ratio of specific heats.

True or false — justify

TF1. "

c^{*}

woh speed hai jis par exhaust gas chamber se nikalta hai."

False.

c^{*}

ek bookkeeping velocity hai (

P_{c} A^{*} / \overset{m}{˙}

); koi bhi gas particle physically

c^{*}

par move nahi karta. Asli throat gas speed local sound speed

a^{*}

hoti hai, jo smaller hoti hai.

TF2. "

c^{*}

sirf chamber aur throat conditions par depend karta hai, nozzle ke downstream par kabhi nahi."

True. Jab throat par flow choke ho jaaye,

\overset{m}{˙}

sirf

(P_{c}, T_{c}, A^{*})

se lock ho jaata hai; nozzle exit shape upstream aa kar inhe change nahi kar sakti.

TF3. "Throat area

A^{*}

ko double karne se

c^{*}

double ho jaata hai."

False.

A^{*}

double karne se fixed

P_{c}

par choked

\overset{m}{˙}

bhi roughly double ho jaata hai, isliye ratio

P_{c} A^{*} / \overset{m}{˙}

wahin rehta hai —

c^{*}

essentially area-independent hai.

TF4. "Ek fixed propellant chemistry ke liye,

c_{ideal}^{*}

chamber size se independent ek constant hai."

True. Theoretical

c^{*}

sirf

γ

T_{c}

, aur

M

par depend karta hai — pure chemistry — kisi bhi geometric dimension par nahi.

TF5. "Zyada hot flame temperature, sab kuch equal rehe toh, hamesha

c^{*}

badhata hai."

True.

c^{*} \propto T_{c} / M

, isliye

T_{c}

badhane par (

M, γ

fixed rahe)

c^{*}

increase hota hai — yeh design mantra ka "burn hot" wala half hai.

TF6. "Heavier exhaust molecule (bada

M

) switch karne se

c^{*}

badhta hai."

False. Kyunki

R = R_{u} / M

, bada

M

R

ko ghataata hai aur isliye

c^{*} \propto 1/ M

ko bhi ghataata hai — yahi reason hai ki light

H_{2}

-rich exhaust jeet jaata hai.

TF7. "

c = c^{*} C_{F}

, isliye poor combustion wale rocket ko ek great nozzle bacha sakta hai."

Aadha false.

C_{F}

overall

c

ko boost kar sakta hai, lekin ek low

c^{*}

phir bhi wasted chemistry signal karta hai; nozzle woh energy recover nahi kar sakta jo propellant ne kabhi release hi nahi ki.

TF8. "Well-tuned engine ke liye combustion efficiency

η_{c^{*}} = c_{measured}^{*} / c_{ideal}^{*}

1 se zyada ho sakti hai."

False. Ideal value thermodynamic ceiling hai (complete burn, no losses); real engines

\approx 0.92

–

0.99

par hote hain, hamesha unity se neeche.

TF9. "Agar throat choked nahi hai, toh formula

c^{*} = P_{c} A^{*} / \overset{m}{˙}

phir bhi sahi

c^{*}

deta hai."

False. Poori derivation throat par sonic (

M = 1

) flow assume karti hai; choking ke bina fixed

P_{c}

–

\overset{m}{˙}

relationship toot jaata hai aur number meaningless ho jaata hai.

Spot the error

SE1. "

c^{*} = P_{c} A_{e} / \overset{m}{˙}

, nozzle exit area use kar ke."

Error

A_{e}

hai — yeh zaroor throat area

A^{*}

honi chahiye, kyunki choking (physical basis) throat par hoti hai, exit par nahi.

SE2. "Ek student low

c^{*}

measure karta hai aur conclude karta hai ki nozzle under-expanded hai."

Galat culprit. Nozzle expansion mismatch

C_{F}

mein dikhta hai,

c^{*}

mein nahi; low

c^{*}

combustion problems ki taraf point karta hai (incomplete burn, heat loss, poor mixing).

SE3. "

c_{ideal}^{*} = γ R T_{c}

Γ

factor drop kar ke."

γ

-dependent bracket drop nahi kar sakte. Poora expression

γ R T_{c} / γ \cdot (\frac{γ + 1}{2})^{\frac{γ + 1}{2 ( γ - 1 )}}

hai; equivalently

c^{*} = R T_{c} / γ /Γ

jahan

Γ = γ (\frac{2}{γ + 1})^{\frac{γ + 1}{2 ( γ - 1 )}}

SE4. "Kyunki

c^{*}

ki units m/s hain, thrust equation mein exhaust velocity mein seedha add kar do."

Units match karti hain lekin meaning nahi karta.

c^{*}

ek physical velocity nahi hai jo add kar sako; yeh multiplicatively enter karta hai

c = c^{*} C_{F}

ke roop mein.

SE5. "

η_{c^{*}}

raise karne ke liye, longer nozzle lagao."

Longer nozzle expansion change karta hai (yeh

C_{F}

effect hai).

η_{c^{*}}

sirf zyada completely burning se improve hoti hai — better injectors, mixing, ya longer chamber residence time se.

SE6. "Formula mein

\overset{m}{˙}

exit mass flow hai, jo throat mass flow se alag hota hai."

Mass conservation se steady flow mein mass flow har jagah same hoti hai; ek hi

\overset{m}{˙}

hota hai, aur yeh choked throat se fix hota hai.

SE7. "

c^{*}

ambient (baahri) pressure par depend karta hai, kyunki rockets vacuum mein alag perform karte hain."

Vacuum-vs-sea-level difference ek nozzle/back-pressure effect hai jo

C_{F}

mein rehta hai.

c^{*}

sirf internal chamber aur throat conditions use karta hai aur ambient se blind hai.

Why questions

WHY1. "

c^{*}

ek alag quantity ke roop mein kyun exist karta hai sirf

c

use karne ki bajaye?"

Chemistry ko geometry se cleanly split karne ke liye:

c^{*}

chamber+throat (combustion) score karta hai,

C_{F}

nozzle (expansion) score karta hai, taaki ek problem ko localize kiya ja sake.

WHY2. "Throat kyun controlling station hai, chamber ya exit nahi?"

Kyunki throat woh jagah hai jahan gas Mach 1 hit karti hai aur choke hoti hai; wahan sonic flow

\overset{m}{˙}

ko cap karta hai aur use downstream conditions se independent banata hai.

WHY3. "

c_{ideal}^{*}

mein

Γ

(Vandenkerckhove) function kyun appear karta hai?"

Γ = γ (\frac{2}{γ + 1})^{\frac{γ + 1}{2 ( γ - 1 )}}

choked-flow algebra ki saari

γ

-dependence ko ek clean factor mein pack karta hai, taaki baaki sirf thermochemistry

R T_{c}

reh jaaye. Dekho Vandenkerckhove Function Γ.

WHY4. "

c^{*}

combustion-efficiency metric ke roop mein

I_{s p}

se zyada fair kyun hai?"

I_{s p}

chamber aur nozzle performance mix karta hai, isliye kharab nozzle use drag karta hai;

c^{*}

nozzle ko poori tarah ignore karta hai, isolate karta hai ki propellant actually kitna accha jala.

WHY5. "Derivation chamber aur throat ke beech isentropic relations kyun invoke karti hai?"

Kyunki gas chamber mein near-rest se throat par Mach 1 tak negligible entropy change ke saath accelerate karti hai, jo hume

T^{*}, P^{*}, ρ^{*}

ko stagnation

T_{c}, P_{c}, ρ_{c}

se relate karne deta hai. Dekho Isentropic Flow Relations.

WHY6. "

c^{*} \propto T_{c} / M

hydrogen ko kyun favor karta hai?"

H_{2}

-rich exhaust ki molecular weight

M

bahut low hoti hai, aur chhota

M

moderate flame temperature par bhi

T_{c} / M

boost karta hai — halka, fast-thermal-speed gas.

WHY7. "Hum

\overset{m}{˙}

ko 'locked' kyun treat kar sakte hain jis moment throat choke ho jaaye?"

Downstream disturbances local sound speed par travel karti hain; jab throat par flow sonic ho jaaye toh woh upstream propagate nahi kar sakti, isliye downstream kuch bhi throat mass flow alter nahi kar sakta. Dekho Choked Flow and the de Laval Nozzle.

Edge cases

EC1. "Agar chamber pressure ambient pressure ke equal ho (no expansion), toh

c^{*}

ka kya hota hai?"

c^{*}

unchanged rehta hai — usne kabhi ambient pressure reference nahi ki. Sirf

C_{F}

aur thrust collapse hote hain; chamber apna choked

P_{c} / \overset{m}{˙}

ratio phir bhi build karta hai.

EC2. "

γ \to 1

ki limit mein (ek gas jo internally bahut energy store karti hai),

c^{*}

kya karta hai?"

Bracket

(\frac{γ + 1}{2})^{\frac{γ + 1}{2 ( γ - 1 )}}

exponent ko blow up karta hai, isliye

Γ

apni limiting value tak jaata hai aur

c^{*}

ek finite, relatively large value tend karta hai — soft, high-heat-capacity gases efficient thermal-to-flow conversion deti hain.

EC3. "Agar

\overset{m}{˙} \to 0

jabke

P_{c}, A^{*}

finite rahe, formula

c^{*} \to \infty

deta hai — kya yeh real hai?"

Nahi — yeh unphysical limit hai. Steady choked flow

\overset{m}{˙}

P_{c}

aur

A^{*}

se tie karta hai; tum zero flow ke saath

P_{c}

hold nahi kar sakte, isliye yeh scenario choking assumption violate karta hai.

EC4. "Do engines same propellant same

T_{c}

par burn karte hain lekin ek das guna bada hai. Same

c^{*}

Haan.

c_{ideal}^{*}

sirf

γ, T_{c}, M

par depend karta hai; scale (throat area, mass flow)

P_{c} A^{*} / \overset{m}{˙}

mein cancel ho jaata hai, isliye dono identical

c^{*}

share karte hain.

EC5. "Ek cold-gas thruster (no combustion) — kya

c^{*}

ka koi matlab bhi hai?"

Haan, formally: choked cold gas phir bhi

c^{*} = P_{c} A^{*} / \overset{m}{˙}

follow karta hai stored-gas

T_{c}, M, γ

use kar ke. Bas koi "combustion efficiency" ki baat nahi, kyunki kuch jalta hi nahi.

EC6. "Ek perfect, loss-free engine ke liye

η_{c^{*}}

kya hai?"

Exactly 1 — measured

c^{*}

thermochemical ceiling se milta hai. Real engines short fall karte hain kyunki walls ko heat loss hoti hai, finite mixing hoti hai, aur incomplete reaction hoti hai.

EC7. "Agar mid-burn mein measured

c^{*}

suddenly drop ho jaaye, physically kya change hua?"

Kuch combustion completeness ko hurt kiya — injector fouling, off-nominal mixture ratio, ya extra wall heat loss — kyunki

P_{c}

feed ho rahe

\overset{m}{˙}

ke relative fall gaya. Nozzle implicated nahi hai.

Recall Ek-sentence summary

$c^{*}$ chamber+throat (chemistry) score karta hai, deliberately nozzle se blind hai, sirf tab kaam karta hai jab throat choked ho, aur hot, light exhaust ke saath badhta hai — upar ke har trap mein se ek bhi in chaar facts ko bhool jaane ka variation hai.