Characteristic velocity c - and its relation to flame temperature, MW: Question bank

3.3.21 · D5 · Physics › Rocket Propulsion › Characteristic velocity c - and its relation to flame tempe

Shuru karne se pehle, is bank mein use hone wala har symbol define karte hain — plain words mein aur units ke saath. Koi bhi symbol trap mein tabhi aayega jab usse pehle yahan earn kiya gaya ho.

Do pictures jo poore bank ko anchor karti hain

Pehla figure dikhata hai kyun $A_{t}$ cancel hota hai — throat geometry aur mass-flow formula kaise mil kar kaam karte hain — aur doosra dikhata hai sound speed kahaan se aati hai jo $c^{*}$ ko tick karti hai.

Figure — Characteristic velocity c - and its relation to flame temperature, MW

True or false — justify karo

Bada throat area

A_{t}

c^{*}

badhaata hai

False. Mass-flow law

\overset{m}{˙} \propto p_{0} A_{t}

ka matlab hai ki bada

A_{t}

proportionally zyada mass bahar jaane deta hai, isliye

\overset{m}{˙}

saath-saath badhta hai; jaise figure dikhata hai,

c^{*} = p_{0} A_{t} / \overset{m}{˙}

mein

A_{t}

\overset{m}{˙}

ko cancel kar deta hai jo usne produce kiya, sirf

T_{0} / M

bachta hai.

c^{*}

m/s mein hai, isliye yeh engine mein kisi jagah ki real gas speed hai

False. Yeh velocity units ke saath ek figure of merit hai; koi bhi molecule actually

c^{*}

pe travel nahi karta. Throat par real gas speed sonic speed

a_{t} = γ R_{s} T_{t}

hai, jo ek alag (chhoti) number hai.

Do chambers jo same

c^{*}

deliver karte hain, woh same exhaust velocity bhi denge

False. Exhaust velocity hai

c_{e} = c^{*} C_{F}

; nozzle coefficient

C_{F}

phir bhi differ karega. Same

c^{*}

ka sirf yahi matlab hai ki same chamber quality — dekho Thrust Coefficient CF.

Flame temperature

T_{0}

double karne se

c^{*}

roughly double ho jaata hai

False. Kyunki

c^{*} \propto T_{0}

T_{0}

double karne se

c^{*}

sirf

2 \approx 1.41

se multiply hota hai — square root payoff ko cap kar deta hai.

Exhaust molecular weight

M

half karna temperature

T_{0}

double karne se behtar hai

True.

M

half karna

c^{*}

2

se multiply karta hai, same as

T_{0}

double karna — lekin

M

half karna chemically kaafi sasta hai compared to flame temperature double karne ke, isliye

M

ghataana bada lever hai.

Perfectly efficient chamber mein

η_{c^{*}} > 1

ho sakta hai

False.

η_{c^{*}} = c_{m e a s}^{*} / c_{t h}^{*}

real ko ideal se compare karta hai, isliye yeh

1

pe cap hota hai.

1

se upar ki values measurement error ya galat theoretical baseline indicate karti hain, super-engine nahi.

Vandenkerckhove function

Γ

propellant ki temperature par depend karta hai

False.

Γ

sirf

γ

(specific-heat ratio) par depend karta hai. Temperature aur

M

poori tarah

R_{u} T_{0} / M

factor mein rehte hain — dekho Vandenkerckhove Function Γ.

Throat par choked flow ek assumption hai jo humne choose ki; derivation iske bina bhi kaam karti

False. Choking (

M = 1

, Mach one) hi hai jo throat velocity ko sonic speed se fix karti hai, jisse hum

v_{t} = a_{t}

likh sakte hain Choked Flow and the Throat mein. Iske bina,

v_{t}

unknown hai aur clean

c^{*}

formula collapse ho jaata hai.

Error dhundho

γ R_{u} T_{0}

mein sound speed nikalne ke liye seedha

R_{u} = 8.314

use karo."

Galat constant. Sound speed ko specific gas constant

R_{s} = R_{u} / M

(per kg) chahiye, universal

R_{u}

(per mole) nahi.

R_{u}

use karne se crucial

1/ M

dependence kho jaati hai.

c^{*} = p_{0} A_{t} / \overset{m}{˙}

, isliye

c^{*}

boost karne ke liye bas chamber pressure

p_{0}

badha do."

Galat.

p_{0}

badhana (zyada propellant feed karke)

\overset{m}{˙}

proportionally badhaata hai, kyunki

\overset{m}{˙} \propto p_{0} A_{t}

; ratio

p_{0} A_{t} / \overset{m}{˙}

chemistry se set hota hai (

T_{0} / M

), is baat se nahi ki tum kitna push karo.

"Stoichiometric

H_{2} / O_{2}

sabse high

c^{*}

deta hai kyunki yeh sabse hot jalata hai."

Galat. Fuel-rich mixtures jeette hain: excess

H_{2}

mean

M

ko kaafi zyada drop karta hai compared to

T_{0}

drop ke, aur

c^{*} \propto T_{0} / M

M

drop ko reward karta hai — dekho Propellant Selection and Molecular Weight.

"Kyunki

c^{*}

nozzle ko ignore karta hai, isliye yeh throat area ko bhi ignore karta hai."

Galat.

c^{*}

throat ke downstream diverging nozzle ko ignore karta hai, lekin throat hi woh jagah hai jahan choking

\overset{m}{˙}

set karti hai;

A_{t}

dono definition aur derivation mein central hai.

"Isentropic ratio

T_{t} / T_{0} = 2/ (γ + 1)

exit plane ke liye derive kiya gaya tha."

Galat. Yeh general isentropic relation

T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}

hai (

M

Mach number ke saath) jo

M = 1

par evaluate ki gayi hai — yaani throat par, exit par nahi. Dekho Isentropic Flow Relations.

"Kyunki

Γ \approx 0.64

almost constant hai,

γ

c^{*}

par koi effect nahi."

Overstated.

γ

Γ

ke through enter karta hai, bas weakly —

Γ

roughly

0.63

–

0.66

range mein hai. "No effect" bolna ek real few-percent shift chhupata hai.

"Kam measured

c^{*}

matlab nozzle badly shaped hai."

Galat.

c^{*}

construction se nozzle-independent hai; kam value poor combustion, heat loss, ya chamber ke andar heavy products ki taraf point karta hai. Nozzle problems

C_{F}

mein dikhte hain,

c^{*}

mein nahi.

Why questions

c^{*}

T_{0}

par directly kyun nahi balki

T_{0}

par depend karta hai?

Kyunki

c^{*}

throat sound speed

a_{t} = γ R_{s} T_{t}

se bana hai, aur sound speed

T

ki tarah scale karti hai; square root seedha sound ki physics se inherit hota hai.

Poori cheez ko

Γ

se divide kyun karte hain instead of ise temperature term mein absorb karne ke?

Γ

choked isentropic flow ki sabhi

γ

-only geometry ko ek clean number mein collect karta hai,

R_{u} T_{0} / M

ko pure "hot & light" energy factor ke roop mein chhodta hai. Concerns ka separation physics ko readable banata hai.

c^{*}

raw thrust se better chamber diagnostic kyun hai?

Thrust chamber quality aur nozzle performance ko mix karta hai (

F \sim c^{*} C_{F} \overset{m}{˙}

c^{*}

nozzle strip out kar deta hai, isliye kharab number sirf chamber problem matlab ho sakta hai — cleaner signal.

c^{*}

T_{0}

aur

M

se measure karne ki jagah

p_{0}, A_{t}, \overset{m}{˙}

se kyun measure karte hain?

Test stand par tum pressure, geometry, aur flow directly pad sakte ho, lekin

T_{0}

(flame temperature) aur true exhaust

M

ek fireball ke andar measure karna mushkil hai. Definition sirf wahi use karta hai jo ek gauge dekh sake.

Wahi

η_{c^{*}} < 1

chahe heat leakage ho ya incomplete burning — dono mein kyun dikhta hai?

Dono actual chamber energy ghataate hain, isliye given

\overset{m}{˙}

ke liye actual

p_{0}

, isliye measured

c^{*}

η_{c^{*}}

ek lumped chamber-health score hai; yeh problem flag karta hai lekin batata nahi kaun si hai.

Ek longer, fancier nozzle low

c^{*}

ko rescue kyun nahi kar sakta?

Ek nozzle sirf chamber output ko

C_{F}

se multiply karta hai:

c_{e} = c^{*} C_{F}

. Agar

c^{*}

chhota hai kyunki chamber weak hai, toh best nozzle sirf ek chhote number ko scale karta hai — dekho Specific Impulse Isp.

Edge cases

Kya hoga

c^{*}

ka jab

γ \to 1

(bahut complex, many-atom molecules)?

Exponent

\frac{γ + 1}{2 ( γ - 1 )}

blow up karta hai kyunki iska denominator

γ - 1 \to 0

, lekin saath mein base

\frac{2}{γ + 1} \to 1

bhi hota hai, aur

1

se thoda neeche ki number huge power par raise hone ke baad bhi finite rehti hai — dono effects draw par ladte hain. Isliye

Γ = γ (\frac{2}{γ + 1})^{\dots}

0.65

ke paas settle karta hai aur

c^{*}

T_{0} / M

se governed rehta hai.

Agar combustion fail ho aur gas thandi rehe (

T_{0}

ambient ke paas), toh

c^{*}

kya karega?

c^{*} \propto T_{0}

ek chhoti value ki taraf collapse ho jaata hai — chamber har unit flow ke liye almost koi pressure nahi banata. Yahi woh "poor combustion" signature hai jo low measured

c^{*}

report karta hai.

Maano flow choked nahi hai (throat Mach 1 se neeche) — kya

c^{*}

tab bhi meaningful hai?

Derived formula fail ho jaata hai, kyunki usne

v_{t} = a_{t}

assume kiya tha. Un-choked operation (bahut low chamber-to-ambient pressure ratio) matlab throat velocity unknown hai aur

c^{*}

apna clean interpretation kho deta hai.

Kya hoga agar

\overset{m}{˙} \to 0

jabki

p_{0}

finite rahe?

Definition

c^{*} = p_{0} A_{t} / \overset{m}{˙}

diverge karti hai. Physically yeh persist nahi kar sakta: choked flow

\overset{m}{˙}

p_{0}

se tie karta hai (

\overset{m}{˙} \propto p_{0} A_{t}

), isliye finite

p_{0}

ek matching

\overset{m}{˙}

force karta hai — tum

\overset{m}{˙}

ko independently zero nahi kar sakte.

Ek engine heavy metallic propellant jalaata hai isliye

M

bada hai — yeh kya predict karta hai?

c^{*} \propto 1/ M

c^{*}

ko neeche push karta hai, isliye heavy exhaust inherently chamber merit ko cap kar deta hai chahe flame hot hi kyun na ho. Metallized propellants

c^{*}

ko density ke liye trade karte hain — ek deliberate compromise.

Perfect mixing aur no heat loss ki limit par,

η_{c^{*}}

kya hai, aur kya yeh reachable hai?

Yeh

1

ki taraf approach karta hai lekin real hardware mein kabhi poora nahi pahunchta; chhoti boundary-layer heat loss aur finite-rate chemistry hamesha ek-do percent kaat leti hai.

0.93

–

0.98

ki values practical "excellent" range hain.

Recall Yahan har trap ki one-line summary

$c^{*}$ chamber-only hai, $T_{0} / M$ ki tarah scale karta hai (square root, isliye gentle), specific gas constant $R_{s} = R_{u} / M$ use karta hai, choked flow ( $M = 1$ ) chahiye, real speed nahi hai, aur $C_{F}$ se sirf tabhi multiply hota hai jab tum nozzle add karo.

Connections

Thrust Coefficient CF — nozzle half jo $c^{*}$ deliberately exclude karta hai.
Specific Impulse Isp — jahan $c^{*}$ aur $C_{F}$ phir se combine hote hain.
Choked Flow and the Throat — $M = 1$ assumption jis par kai traps hinge karte hain.
Isentropic Flow Relations — throat/chamber ratios ka source (aur ideal-gas fine print).
Vandenkerckhove Function Γ — kyun $γ$ sirf weakly enter karta hai.
Adiabatic Flame Temperature — $T_{0}$ set karta hai, do levers mein se ek.
Propellant Selection and Molecular Weight — fuel-rich $H_{2} / O_{2}$ trap.