Nozzle thermodynamics — isentropic expansion from chamber to exit: Question bank

3.3.11 · D5 · Physics › Rocket Propulsion › Nozzle thermodynamics — isentropic expansion from chamber to

Yeh ek reasoning gym hai parent topic ke liye. Koi bhaari arithmetic nahi — har item neeche ek specific misconception ko pakadne ke liye bana hai ki gas thermal chaos ko directed jet mein kaise convert karti hai. Prompt padho, apna jawab zor se commit karo, phir reveal karo.

Shuru karne se pehle, ek shared vocabulary reminder taaki neeche kuch bhi mystery na rahe:

Flow speed $V$ : ek gas parcel nozzle axis ke saath kitni tez chalti hai (metres per second). Yeh moving stream ki ordinary bulk velocity hai — woh cheez jo eventually exhaust thrust banti hai.
Density $ρ$ (Greek "rho"): har cubic metre mein packed gas ka mass, $kg/m^{3}$ . Hot chamber gas dense hoti hai; jaise jaise yeh nozzle mein expand aur speed up karti hai, $ρ$ girta hai.
Cross-sectional area $A$ : axis ke saath ek given point par nozzle ke circular slice ka area, $m^{2}$ — chamber ke neck mein chhota, throat par sabse chhota, exit ki taraf phir se barta hua. Yeh "geometry knob" hai jo designer control karta hai.
Stagnation quantity (subscript $0$ , jaise $T_{0}$ , $P_{0}$ ): woh value jo gas hogi agar tum usse smoothly rest ( $V = 0$ ) par le aao. Socho "poora energy tank pehle koi bhi motion banne se."
Static quantity (koi subscript nahi, jaise $T$ , $P$ ): jo ek thermometer/gauge moving gas ke saath chalte hue read kare.
Throat (star, jaise $A^{*}$ ): nozzle ka sabse sankra cross-section.
$γ$ (gamma): heat capacity ratio $c_{p} / c_{v}$ , hot rocket exhaust ke liye roughly $1.2$ .
$R$ : exhaust ka specific gas constant — universal gas constant ko combustion products ke mean molar mass se divide kiya, $J/(kg\cdotK)$ mein (typical rocket gas ke liye approximately $320$ ). Yahi hai jo $P = ρR T$ ko per kilogram kaam karta hai, aur yeh dono $c_{p} = γ R / (γ - 1)$ aur sound speed $a = γ R T$ mein aata hai. Dekho Ideal Gas Law.
$M$ : Mach number, flow speed ko local speed of sound se divide kiya, $M = V / a$ — dekho Mach Number and Sound Speed.

Woh ek geometric picture jo neeche har item pe lean karti hai:

Figure — Nozzle thermodynamics — isentropic expansion from chamber to exit

Figure s01 se kya extract karein: wall shape ko left se right trace karo — passage dashed coral throat line tak sankri hoti hai, phir exit ki taraf chaudi hoti hai. Teen labelled zones note karo: green flow arrows right move karne par lambe hote jaate hain, jo batata hai $V$ poore waqt badhta rehta hai, chaude hote hisse mein bhi. Apne aap confirm karo ki throat (min area) exactly wahan hai jahan label kehta hai $M = 1$ — yeh pivot hai jis par poora chapter ghoomta hai.

Figure s02 se kya extract karein: yeh factor $\frac{1}{M ^{2} - 1}$ plot karta hai jo decide karta hai ki chauda karna gas ko speed up karta hai ya slow down. Dekho coral (subsonic) curve kahan baith raha hai — zero line ke neeche, toh factor negative hai; phir mint (supersonic) curve uske upar baith raha hai, positive. Dekho ki dono curves infinity ki taraf jaati hain jaise woh lavender dashed $M = 1$ line ke paas aati hain: wahan finite acceleration tabhi possible hai jab $d A = 0$ , isliye throat exactly $M = 1$ par hona chahiye.

Formula Differential area–velocity relation (kai items ka engine)

Continuity ( $ρ V A = const$ ), isentropic momentum balance, aur $a^{2} = γ R T$ ko combine karne par milta hai $\frac{d V}{V} = \frac{1}{M ^{2} - 1} \frac{d A}{A}$ YEH SIGN KYU? $\frac{d A}{A} + \frac{d V}{V} + \frac{d ρ}{ρ} = 0$ se shuru karo (log-differentiated continuity — mass in = mass out). Aage humein pressure changes ko velocity changes se link karna hai: yeh steady frictionless flow ke liye Euler momentum equation hai, $d P = - ρ V d V$ (yeh sirf ek gas parcel ke liye Newton ka doosra law hai — ek net pressure push $- d P$ mass ko accelerate karta hai, aur isse integrate karne par Bernoulli's equation milti hai). Sound-speed fact $d P = a^{2} d ρ$ ke saath combine karne par, yeh $d ρ / ρ$ ko $- M^{2} d V / V$ se replace karne dete hain. Substitute karne par milta hai $\frac{d A}{A} = (M^{2} - 1) \frac{d V}{V}$ , yaani boxed form. YEH KAISA DIKHTA HAI: Figure s02 upar factor $\frac{1}{M ^{2} - 1}$ plot karta hai — $M < 1$ ke liye negative, $M > 1$ ke liye positive, $M = 1$ par explode karta hua.

Subsonic ( $M < 1$ ): $M^{2} - 1 < 0$ , toh $d A > 0$ deta hai $d V < 0$ — chauda karo aur yeh slow ho jaata hai (garden-hose intuition).
Supersonic ( $M > 1$ ): $M^{2} - 1 > 0$ , toh $d A > 0$ deta hai $d V > 0$ — chauda karo aur yeh speed up karta hai.
$M = 1$ par: factor blow up karta hai, $d A = 0$ force karta hai — throat.

True or false — justify

True or false: Ek isentropic nozzle mein gas speed up hone par thandi hoti jaati hai.

True. Total enthalpy

c_{p} T + V^{2} /2

(jahan

V

flow speed hai) fixed hai, toh

V

badhane par

T

zaroor girta hai — thermal energy literally kinetic energy khareedne mein kharch ho rahi hai.

True or false: "Isentropic" ka matlab hai "walls ke saath koi heat exchange nahi."

False (incomplete). Isentropic = adiabatic aur reversible. No-heat-exchange sirf adiabatic half hai; friction wala nozzle adiabatic hai lekin isentropic nahi kyunki friction entropy generate karta hai.

True or false: Stagnation temperature

T_{0}

nozzle mein poore waqt constant rehta hai, chahe static

T

girta rahe.

True. Adiabatic flow ke liye bina kisi shaft work ke, total enthalpy — isliye

T_{0}

— conserved rehta hai chahe static temperature velocity ke liye trade ho raha ho.

True or false: Real nozzle mein stagnation pressure

P_{0}

bhi constant rehta hai.

False. Ideal isentropic case mein

P_{0}

conserved hota hai, lekin real friction aur shocks

P_{0}

giraa dete hain — woh lost stagnation pressure exactly performance loss hai.

True or false: Nozzle ke diverging (chauda hota) hisse mein flow accelerate hoti rehti hai.

True — lekin sirf isliye kyunki yeh already supersonic hai.

M > 1

ke liye,

d V / V = + \frac{1}{M ^{2} - 1} d A / A

mein

M^{2} - 1 > 0

hai, toh area badhne par speed badhti hai. Ek subsonic stream ko chauda karo toh yeh slow ho jaata hai.

True or false: Throat woh jagah hai jahan gas sabse tez chalti hai.

False. Throat wahan hai jahan

M = 1

; sabse tez velocity exit par hai, supersonic diverging section mein gehrai mein. Highest Mach ≠ velocity ke maximum ke liye wahi point jahan sabse sankra area hai.

True or false: Agar tum chamber pressure

P_{0}

double karo, toh exit Mach number roughly double ho jaata hai.

False. Exit Mach area ratio

A_{e} / A^{*}

(geometry) se set hota hai, chamber pressure se nahi.

P_{0}

double karne par saare pressures scale up hote hain lekin

M_{e}

essentially unchanged rehta hai.

True or false: Ek converging-only nozzle supersonic exhaust produce kar sakta hai agar tum zyada push karo.

False. Ek converging duct flow ko sirf

M = 1

tak hi accelerate kar sakta hai apne exit par. Mach 1 exceed karne ke liye tumhare paas physically throat-then-diverge geometry hona chahiye — dekho Nozzle Flow Regimes.

True or false:

M = 1

par local static pressure rocket ke bahar ambient pressure ke barabar hoti hai.

False. Throat pressure

P^{*} = P_{0} (2/ (γ + 1))^{γ / (γ - 1)}

chamber se fix hoti hai, sky se nahi. Ambient pressure exit plane par matter karta hai, throat par nahi.

Spot the error

Ek student likhta hai "

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

". Error dhundho.

Relation reciprocal hai:

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

. Jaise likha hai yeh claim karta hai ki gas speed ke saath garam hoti hai, energy conservation ko contradict karta hua.

Ek student kehta hai "kyunki exhaust supersonic hai, sound (aur pressure info) bahar se upstream travel karke chamber change kar sakti hai." Flaw kahan hai?

Ulta hai. Kyunki exit supersonic hai, downstream pressure signals throat ke paas se upstream travel nahi kar sakti — yahi precisely reason hai kyunki chamber aur throat conditions ambient pressure ke liye insensitive hain.

Ek derivation likhti hai "

c_{p} = \frac{γ R}{γ + 1}

." Isse correct karo.

Yeh hona chahiye

c_{p} = \frac{γ R}{γ - 1}

(jahan

R

specific gas constant hai). Denominator

γ - 1

hai;

γ + 1

use karne par neeche ke har temperature relation kharab ho jaayenge.

Ek student exit sound speed ke liye

a = γ R T_{0}

(stagnation temperature) use karke exit conditions compute karta hai. Yeh galat kyun hai?

Sound speed local static temperature par depend karta hai:

a = γ R T

jahan

T = T_{e}

T_{0}

nahi.

T_{0}

use karne par

a

overestimate hota hai aur isliye

V_{e} = M_{e} a

galat compute hota hai.

Koi kehta hai "

P_{e} = P_{a}

exit par hamesha hold karta hai kyunki pressures match karni chahiye." Error pakdo.

Sirf ek perfectly expanded nozzle mein

P_{e} = P_{a}

hota hai. Real nozzles aksar over- ya under-expanded hoti hain, toh

P_{e} \neq = P_{a}

; mismatch exactly Thrust Equation Derivation mein pressure-thrust term hai.

Ek student critical ratio

T^{*} / T_{0} = 2/ (γ - 1)

likhta hai. Fix karo.

Yeh

T^{*} / T_{0} = 2/ (γ + 1)

hai.

γ = 1.2

ke saath correct value

\approx 0.91

hai; galat formula

10

deta hai, ek impossible ratio jo 1 se upar hai.

Ek derivation area–velocity relation ko

\frac{d V}{V} = - \frac{1}{M ^{2} - 1} \frac{d A}{A}

likhti hai. Sign error pakdo.

Correct relation

\frac{d V}{V} = + \frac{1}{M ^{2} - 1} \frac{d A}{A}

hai. Extra minus sign ke saath, widening duct (

d A > 0

) mein supersonic flow (

M > 1

) decelerate hoti dikhai degi — yeh contradict karta hai ki diverging nozzle actually kaise kaam karti hai.

Why questions

Increasing area supersonic flow ko accelerate kyun karta hai, jab everyday intuition (garden hose) kehti hai chaura matlab slower?

M > 1

ke liye, density velocity se tez girta hai, toh mass flow

ρ V A

constant rakhne ke liye area grow karna chahiye jabki

V

phir bhi chadh raha ho — yeh

d V / V = + \frac{1}{M ^{2} - 1} d A / A

se capture hota hai jiska

M > 1

hone par positive factor hota hai.

Sonic point (

M = 1

) exactly minimum area par kyun hona chahiye, pehle ya baad mein nahi?

d V / V = \frac{1}{M ^{2} - 1} d A / A

mein factor

\frac{1}{M ^{2} - 1}

M = 1

par blow up karta hai; wahan finite acceleration tabhi possible hai jab

d A = 0

. Toh

M = 1

throat par pin hota hai.

Jab real nozzles mein friction hai toh hum nozzle ko isentropic kyun model karte hain?

Isentropic case theoretical ceiling hai: yeh maximum possible exhaust velocity deta hai, toh real performance (95–98%) iska fraction quoted hoti hai. Dekho Entropy and Reversibility.

Rocket exhaust ke liye pressure exponent

γ / (γ - 1)

itna bada (approximately 6) kyun hai?

Kyunki

γ \approx 1.2

banata hai

γ / (γ - 1) = 6

; chhota

γ

(kai internal molecular modes energy absorb karte hue) matlab pressure Mach number ke saath steeply plummet karti hai, huge expansion ratios deti hai.

Area–Mach relation ek hi area ratio ke liye do alag Mach numbers kyun deta hai?

Kyunki

A / A^{*}

one-to-one nahi hai: ek subsonic aur ek supersonic solution har area ratio share karte hain (throat ko chhod ke). Geometry akele decide nahi karti konsi branch — pressure/flow regime decide karta hai. Dekho Nozzle Flow Regimes.

Hum exhaust ko ideal gas kyun treat kar sakte hain jabki yeh ek hot reacting mixture hai?

High temperature aur moderate density par combustion products near-ideally behave karte hain, humein

P = ρR T

effective specific gas constant

R

aur

γ

ke saath use karne dete hain — expansion ke liye ek bahut achha engineering approximation.

Edge cases

M \to 0

limit mein

T / T_{0}

P / P_{0}

ρ / ρ_{0}

ka kya hota hai?

Teeno ratios

\to 1

: koi motion nahi hone par gas stagnation par hai, toh static aur stagnation values coincide karte hain — formulas par ek sanity check.

Edge case:

M \to \infty

(infinite expansion). Ratios kya approach karte hain?

T / T_{0}

P / P_{0}

ρ / ρ_{0}

sabhi

\to 0

— gas theoretically saari thermal energy ko motion mein convert kar leti hai. Real nozzles bahut pehle ruk jaati hain kyunki area ratio aur pressure impractical ho jaate hain.

Edge case: nozzle choked hai aur tum ambient pressure aur neeche girate ho. Throat ke upstream kya badalta hai?

Kuch nahi. Ek baar choked hone par (

M^{*} = 1

), mass flow aur saare throat/chamber conditions lock ho jaate hain; ambient girna sirf downstream expansion/shock pattern ko bahar affect karta hai.

Edge case: exit-plane static pressure exactly ambient ke barabar hai. Yeh kaunsi special condition hai?

Perfect (ideal) expansion —

P_{e} = P_{a}

. Yahan pressure-thrust term vanish ho jaata hai aur, ek given chamber ke liye, thrust aur specific impulse maximize hota hai.

Edge case: exit pressure ambient se kaafi neeche (extreme over-expanded). Kaun sa physical event isse limit karta hai?

Flow wall se separate ho sakti hai aur/ya diverging section ke andar shocks form kar sakti hain, toh ideal isentropic relations shock ke baad ab hold nahi karti — ek real-loss regime.

Degenerate case:

γ \to 1

(enormous internal energy storage wali gas). Kya toot jaata hai?

γ / (γ - 1)

jaise exponents infinity tak blow up karte hain, toh pressure infinitely fast giregi — signalling karta hai ki truly

γ = 1

gas ko in relations se treat nahi kiya ja sakta; yeh ek limiting idealization hai, real propellant nahi.

Boundary check: throat par, kya flow speed chamber sound speed ke barabar hai ya local throat sound speed ke?

Local throat sound speed

a^{*} = γ R T^{*}

, cooler throat temperature

T^{*}

par evaluate ki gayi, hotter chamber value par nahi.

V^{*} = a^{*}

kyunki wahan

M^{*} = 1

hai.

Recall Self-test: teen assumptions ke naam bolo jo isentropic model ko tod dein

Friction (viscous losses), shock waves, aur significant wall heat transfer — koi ek bhi entropy add karta hai aur $Δ S = 0$ violate karta hai.

Shock ke through kaun si quantity conserved hai, aur kaun si NAHI?

Stagnation enthalpy (

T_{0}

) shock ke across conserved hai, lekin stagnation pressure (

P_{0}

) girta hai — shock entropy-generating culprit hai.