Chamber-to-exit relation - all quantities as f(M_e, γ): Question bank

3.3.12 · D5 · Physics › Rocket Propulsion › Chamber-to-exit relation - all quantities as f(M_e, γ)

Shuru karne se pehle, char symbols jo tumhe bilkul seedha yaad hone chahiye (sab parent note mein banaye gaye hain):

$M_{e}$ = exit Mach number, exit flow speed aur local speed of sound ka ratio. Ye master control variable hai.
$γ$ = ratio of specific heats ( $c_{p} / c_{v}$ ), gas mixture ki ek property hai (hot rocket products ke liye lagbhag $1.2$ , air ke liye $1.4$ ).
$R$ = specific gas constant, per-kilogram gas constant jo $P V = m R T$ aur speed of sound $a = γ R T$ mein aata hai. Iske units hote hain, isliye koi bhi dimensional speed ke liye isko chahiye.
$A^{*}$ = sonic throat area, wo cross-section jahan $M = 1$ hota hai; saare area ratios isi ke against measure hote hain.
Subscript $0$ = stagnation (total) state — wo reference state jo tumhein milta agar flow ko isentropically rest par la diya jaata. Near-still chamber mein, static $\approx$ stagnation.

True or false — justify karo

Saare exit ratios sirf

M_{e}

aur

γ

par depend karte hain, actual chamber temperature

T_{0}

par kabhi nahi.

True — ratios ke liye (

T_{e} / T_{0}

P_{e} / P_{0}

ρ_{e} / ρ_{0}

A_{e} / A^{*}

) — ye

M_{e}, γ

ke pure functions hain. Lekin dimensional value

V_{e}

ko phir bhi

T_{0}

(aur

R

) chahiye kyunki velocity ke units hote hain.

M_{e}

badhane se exit temperature

T_{e}

badhti hai.

False —

T_{e} / T_{0} = (1 + \frac{γ - 1}{2} M_{e}^{2})^{- 1}

mein

M_{e}

denominator mein hai, isliye zyada

M_{e}

matlab thanda exit; thermal energy ordered kinetic energy mein convert ho rahi hai.

γ = 1.4

ke liye pressure-ratio exponent

- 35

hota hai.

False — ye

- γ / (γ - 1) = - 1.4/0.4 = - 3.5

hai.

- 3.5

- 35

samajhna ek factor-of-ten error hai jiske baare mein parent note explicitly warn karta hai.

Throat par flow hamesha exactly sonic hoti hai,

M = 1

M_{e}

ki parwah kiye bina.

True sirf tab jab nozzle choked ho — yaani back-pressure ratio itna kam ho ki throat

M = 1

tak pahunch jaaye. Ek baar choked hone par, throat

M = 1

hold karta hai (

A^{*}

define karta hai) aur saare exit ratios us sonic point ko reference karte hain; high back pressure par flow poori tarah subsonic rehti hai aur koi true

A^{*}

exist nahi karta.

Kyunki flow isentropic hai, entropy conserve hoti hai lekin stagnation pressure

P_{0}

nahi.

False — isentropic ka matlab hai koi entropy rise nahi, aur isentropic flow ke liye stagnation pressure

P_{0}

nozzle ke saath exactly constant rehta hai.

P_{0}

sirf tab girta hai jab irreversibilities (shocks, friction) aate hain.

Zyada exit Mach number hamesha zyada area ratio

A_{e} / A^{*}

require karta hai.

True sirf supersonic branch mein.

M_{e} > 1

ke liye,

A_{e} / A^{*}

M_{e}

ke saath badhta hai (diverging section chauda hota hai).

M_{e} < 1

ke liye area ratio bhi 1 se zyada hota hai lekin throat ki taraf ghatta hai — ek hi

A / A^{*}

do Mach numbers ko serve karta hai.

Gas expand hone par density pressure se zyada slowly girti hai.

True — density exponent

- 1/ (γ - 1)

hai jabki pressure exponent

- γ / (γ - 1)

hai, aur

γ > 1

pressure exponent ko steeper banata hai. Temperature ye difference carry karta hai (pressure

=

density

\times

temperature).

Exit velocity

V_{e}

bina bound ke badhti rehti hai jab

M_{e} \to \infty

False —

V_{e} = M_{e} a_{0} (1 + \frac{γ - 1}{2} M_{e}^{2})^{- 1/2}

saturate ho jaati hai. Jab

M_{e} \to \infty

tab ye finite limiting velocity

a_{0} 2/ (γ - 1)

par pahunchti hai; thermal energy convert karne ke liye khatam ho jaati hai.

Error pakdo

P_{0}

chamber ke andar ki static pressure hai."

P_{0}

stagnation (reference) pressure hai. Ye chamber static pressure ke lagbhag equal isliye hoti hai kyunki chamber flow almost at rest hoti hai; conceptually

P_{0}

ek state ko name karta hai, kisi location ko nahi.

"Kyunki

h_{0} = h_{e} + V_{e}^{2} /2

, kinetic energy add karne se total enthalpy badhti hai."

Stagnation enthalpy

h_{0}

adiabatic nozzle mein conserved hai; kinetic energy sirf static enthalpy

h_{e}

ko draw down karke badhti hai (isliye

T_{e}

girta hai). Total fixed rehta hai.

"Density ratio ke liye main sirf

P_{e} / P_{0}

use karunga kyunki pressure aur density saath track karte hain."

Ye proportionally track nahi karte — tumhe temperature ratio se divide karna hoga:

ρ_{e} / ρ_{0} = (P_{e} / P_{0}) / (T_{e} / T_{0})

. Temperature ignore karna galat exponent deta hai.

"Speed of sound nozzle mein har jagah same hai, isliye exit par

a_{0}

use kar sakta hoon."

Speed of sound

a = γ R T

local temperature par depend karti hai, jo exit ki taraf girta hai.

a_{0}

(chamber value) use karna

a_{e}

ko overestimate karta hai; isliye

V_{e}

mein

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

correction factor hota hai.

A_{e} / A^{*}

nikalne ke liye main exit density-velocity ko chamber density-velocity se compare karunga."

Tum exit ko throat (

M = 1

, starred quantities) se compare karte ho, chamber se nahi — mass flux

ρ V

conserve hoti hai aur throat choking reference hai, jahan chamber velocity

\approx 0

infinite area degi.

"Lower

γ

matlab gas zyada tezi se expand aur cool hoti hai."

Ulta — lower

γ

same

M_{e}

ke liye chhota temperature drop deta hai (

T_{e} / T_{0} = 1/1.9

γ = 1.2

1/2.8

γ = 1.4

). Wo warmer, more energetic exhaust isliye hai jab low-

γ

propellants zyada specific impulse dete hain.

"Area–Mach relation se main ek unique

M_{e}

nikaal sakta hoon diye gaye

A_{e} / A^{*}

se."

Ek single supersonic area ratio do Mach numbers ko map karta hai — ek subsonic, ek supersonic — kyunki

A / A^{*}

ka minimum 1 hai

M = 1

par. Tumhe pata hona chahiye ki tum kaunsi branch (converging vs diverging) par ho.

Why questions

M_{e}

ko master variable kyun choose kiya jaata hai na ki

P_{e} / P_{0}

ko?

Chaaron thermodynamic ratios plus geometric area ratio

M_{e}

ke clean explicit functions hain, jabki

P_{e} / P_{0}

ke through express karne mein fractional power invert karni padti hai.

M_{e}

har quantity ko one-line formula banata hai.

Derivation temperature ke liye energy conservation se kyun shuru hoti hai lekin area ke liye mass conservation se?

Temperature set hoti hai is baat se ki kinetic energy enthalpy ke against kaise trade karti hai (energy balance), jabki nozzle ka cross-section set hota hai is baat se ki fixed mass flux

ρ A V

ko kitna area chahiye (mass balance). Alag physics, alag conservation law.

Static pressure itni steeply kyun girta hai (exponent

- 6

γ = 1.2

) jabki temperature mild hoti hai?

Pressure dono density drop aur temperature drop ko isentropic law

P \propto T^{γ / (γ - 1)}

ke through compound karta hai, isliye iska exponent magnitude mein sabse bada hota hai — ye steep drop hi flow ko accelerate karta hai aur thrust create karta hai.

In relations ke liye flow ko isentropic assume karna kyun zaroori hai?

Pressure–temperature link

P / P_{0} = (T / T_{0})^{γ / (γ - 1)}

P V^{γ} = const

se derive hota hai, jo sirf reversible adiabatic flow ke liye valid hai. Ek shock ya heat loss us relation ko tod deta hai, isliye

P_{e} / P_{0}

formula follow nahi karta.

Temperature-ratio factor har doosre relation ke andar kyun aata hai?

Saari exit quantities usi energy balance se anchored hain, jo group

1 + \frac{γ - 1}{2} M_{e}^{2}

produce karta hai; pressure, density, velocity aur area sab isko inherit karte hain, sirf alag-alag powers ke saath jo isentropic aur mass-flow physics se set hoti hain.

Edge cases

M_{e} = 0

par har ratio ka kya hota hai?

Har ratio exactly 1 ho jaata hai (

T_{e} = T_{0}

P_{e} = P_{0}

ρ_{e} = ρ_{0}

), exit velocity 0 hoti hai, aur area ratio

A_{e} / A^{*} \to \infty

— ek stationary gas infinitely bade "exit" ko fill karti hai sonic throat ke relative mein.

Exactly

M_{e} = 1

par,

A_{e} / A^{*}

kya hota hai?

Ye 1 ke equal hota hai — exit hi throat condition hai, area function ka geometric minimum. Koi bhi

M_{e} \neq = 1

(super- ya subsonic) zyada area maangta hai.

M_{e} \to \infty

par limiting exit velocity kya hai?

V_{e} \to a_{0} 2/ (γ - 1)

, ek finite maximum —

γ = 1.2

ke liye ye

a_{0} 10 \approx 3.16 a_{0}

hai. Saari available thermal energy convert ho chuki hai, isliye

T_{e} \to 0

aur aur koi speed extract nahi ki ja sakti.

Purely subsonic nozzle (

M_{e} < 1

) ke liye, kya wahi formulas apply hote hain?

Haan — isentropic ratios kisi bhi

M_{e}

ke liye valid hain; sirf dA/dM ka sign alag hota hai.

M = 1

se neeche area ko accelerate karne ke liye shrink karna padta hai, isliye sirf converging nozzle apne throat par zyada se zyada

M_{e} = 1

tak pahunch sakta hai.

Bade

M_{e}

ke saath formula limit

T_{e} \to 0

physically kya signal karta hai?

Denominator

1 + \frac{γ - 1}{2} M_{e}^{2}

hamesha

> 1

hota hai, isliye

T_{e}

positive rehta hai lekin

M_{e} \to \infty

ke saath

0

ki taraf jaata hai — ye ek ideal-gas idealization hai. Real gases liquefy ho jaate hain ya continuum assumption fail ho jaati hai usse bahut pehle, isliye model breakdown ho jaata hai.

Agar

γ \to 1

, pressure exponent ka kya hota hai aur ye unphysical kyun hai?

Exponent

- γ / (γ - 1) \to - \infty

, infinitely steep pressure collapse predict karta hai. Physically

γ > 1

hamesha real gas ke liye hota hai (specific heats differ karte hain), isliye

γ = 1

sirf ek degenerate limit hai, achievable propellant nahi.

Recall Traps ka ek-line summary

Ek hi fixed group $1 + \frac{γ - 1}{2} M_{e}^{2}$ sab kuch drive karta hai; zyada $M_{e}$ gas ko cool karta hai aur velocity ko saturate karta hai, ratios ko sirf $M_{e}, γ$ chahiye jabki $V_{e}$ ko $T_{0}$ chahiye, " $0$ " ka matlab stagnation hai koi jagah nahi, aur har quantity sonic throat ko reference karti hai.