Burn rate r = a·P^n — Vieille's law: Question bank

3.3.36 · D5 · Physics › Rocket Propulsion › Burn rate r = a·P^n — Vieille's law

Shuru karne se pehle, poora law ek saanch mein yaad karo:

$r$ = speed jis par solid surface peeche hatati hai (mm/s), gas speed nahi.
$P$ = chamber pressure jo us surface par push karta hai.
$a$ = burn-rate coefficient (chemistry aur starting temperature dono).
$n$ = pressure exponent, combustion sensitivity, usually $0.2$ – $0.5$ .

True ya false — justify karo

Vieille's law kehta hai ki pressure badhne par exhaust gas ki speed badh jaati hai.

False.

r

solid surface ki recession speed hai (mm/s scale), exhaust velocity nahi (km/s scale jo nozzle se aati hai) — woh Thrust Equation and Specific Impulse mein hoti hai, yahan nahi.

Agar

n = 0

ho toh propellant ki burn rate chamber pressure se bilkul independent hoti hai.

True.

r = a P^{0} = a

, ek constant — yeh idealised "plateau" propellant hai jiska burn rate kabhi pressure ke saath respond nahi karta, jo sabse safe case hai.

Pressure double karne par burn rate hamesha double ho jaati hai.

False. Isse

r

2^{n}

se multiply hoti hai; sirf

n = 1

par hi

P

double karne se

r

double hoti hai. Ek real

n = 0.4

propellant ke liye,

P

double karne par

r

sirf

2^{0.4} \approx 1.3 \times

badhti hai.

n = 1.2

wala propellant zyada powerful hai isliye better hai.

False.

n > 1

ka matlab hai gas generation (

\propto P^{1.2}

) nozzle exhaust (

\propto P

) se tez badhti hai, toh koi bhi pressure bump runaway hokar explosion mein badal jaata hai — high

n

dangerous hai, desirable nahi.

r

versus

P

ke ordinary (linear) axes wale plot par ek seedhi line milti hai jiska slope

n

hota hai.

False.

r = a P^{n}

linear axes par ek curve hai; yeh seedha sirf log–log axes par hota hai, jahan slope

= n

aur intercept

= lo g a

hota hai.

Coefficient

a

sirf chemical formula ki fixed property hai.

False.

a

mein initial grain temperature bhi shamil hoti hai, isliye same motor ka

a

(aur burn rate) thandi winter launch aur garmi summer launch par alag hoga — yeh Temperature Sensitivity of Propellants hai.

n < 1

condition motor ko thermally stable guarantee karta hai.

True (is feedback loop ke liye). Generation

\propto P^{n}

, exhaust

\propto P^{1}

se dheemare badhti hai, toh pressure rise gas ko add karne se zyada tez remove karta hai — disturbance khud hi correct ho jaata hai.

Agar

n < 1

ho, toh pressure badhne par burn rate kam ho jaati hai.

False.

r

phir bhi kisi bhi

n > 0

ke liye

P

ke saath badhti hai;

n < 1

ka matlab sirf yeh hai ki yeh linearly se dheemare badhti hai, jo motor ko stable banata hai.

Error dhundho

r = a P^{n}

, aur kyunki power, force times something hai,

r

ke units watts hone chahiye."

Error yeh hai ki

r

ko power samjha gaya.

r

ek speed hai (mm/s) — poora law is tarah calibrated hai ki

a P^{n}

speed units mein aaye;

a

dimensionless nahi hai balki jo bhi units chahiye woh absorb karta hai.

"Kyunki reaction rate

\propto P^{2}

hai aur flame high

P

par closer baithti hai, exponent

n

zaroor

2

hona chahiye."

Error yeh hai ki reaction-rate exponent ko burn-rate exponent se confuse kiya gaya. Flame standoff

δ \propto P^{- k}

ek heat balance ke through enter karta hai, aur

r \propto 1/ δ \propto P^{n}

; resulting

n

chhota hota hai (

0.2

–

0.5

2

nahi.

"Ek cold-soaked motor aur ek warm motor ka

a

same hota hai, toh woh identically jalaate hain."

Error yeh hai ki temperature

a

ke andar rehti hai — yeh bhool gaya. Thanda solid ignition temperature tak pahunchne ke liye zyada heat maangta hai, toh woh dheemare peeche hatata hai — same

P

, different

r

"Do data points se

n

nikaalane ke liye, burn rates subtract karo aur pressures se divide karo."

Error yeh hai ki linear slope use kiya gaya. Kyunki law ek power law hai, log lena zaroori hai:

n = \frac{ln ( r _{2} / r _{1} )}{ln ( P _{2} / P _{1} )}

, log-slope lene se pehle

a

cancel karne ke liye divide karo.

"Zyada pressure ka matlab zyada gas molecules, toh flame surface se door chali jaati hai."

Ulta hai. Dense gas reactions ko chhoti distance mein khatam karta hai, toh flame karib baithti hai; karib, hotter flame steep temperature gradient drive karti hai aur tezi se jaalaati hai.

"Steady burning ka matlab hai ki supply heat aur demand heat aapas mein unrelated hain."

Error us balance ko negate karta hai jo law produce karta hai. Steady recession ke liye flame ki surface par supplied heat bilkul us heat ke barabar honi chahiye jo aane wale solid ko warm karne ke liye demanded hai — supply = demand set karna hi

r

deta hai.

Why questions

Physically, burn rate pressure ke saath badhti kyun hai?

Higher

P

par dense gas flame ko karib aur hotter baithne deta hai, jo thandi solid mein temperature gradient steep karta hai; Fourier's Law of Heat Conduction ke zariye zyada heat wapas conduct hoti hai, toh agli layer jaldi ignition tak pahunchti hai aur surface tezi se peeche hatati hai.

Engineers deliberately chhote

n

(

0.2

–

0.5

) wale propellants kyun choose karte hain?

Chhota

n

burn rate ko pressure ke baare mein weakly sensitive banata hai, toh accidental pressure bumps gas generation mein bahut kam change laate hain aur motor self-stabilise hota hai, runaway nahi hota.

r = a P^{n}

ke logs lene se engineers ko kyun faayda hota hai?

Yeh curve ko straight line

lo g r = lo g a + n lo g P

mein badal deta hai, toh slope aur intercept directly

n

aur

lo g a

ke roop mein strand-burner data se read ho jaate hain.

n = 1

ko knife-edge kyun kehte hain?

n = 1

par generation (

\propto P

) aur exhaust (

\propto P

) bilkul same rate par badhte hain, toh pressure disturbance na badhti hai na decay hoti hai — motor stable aur explosive ke beech boundary par baitha hota hai.

r

aur nozzle se gas ka mass flow rate ek hi cheez kyun nahi hai?

r

measure karta hai ki burning surface andar kitni tezi se move karti hai; mass flow

r

aur total burning surface area aur propellant density par depend karta hai — geometry Solid Rocket Motor Grain Geometry se set hoti hai.

Flame-standoff distance

δ

pressure ka power kyun hoti hai?

δ

diffusion aur reaction ke beech ek race se set hoti hai; reaction rate concentration ka power scale karti hai (isliye

P

ka), toh

δ \propto P^{- k}

, aur kyunki

r \propto 1/ δ

hai, hume power law

r \propto P^{n}

milta hai.

Edge cases

Agar

n

exactly

1

ho toh stability argument ka kya hota hai?

Generation aur exhaust ek doosre ko perfectly track karte hain, toh pressure neutrally stable hoti hai — disturbance na self-correct hoti hai na explode karti hai, jo design ko practice mein unacceptably risky banata hai.

n > 0

ke liye

P = 0

par

r = a P^{n}

kya predict karta hai?

r = a \cdot 0^{n} = 0

— zero chamber pressure ke saath koi flame heat wapas push nahi karti, toh surface peeche nahi hatati; burning effectively band ho jaati hai.

Kisi bhi

n

ke liye

P = 1

(chosen pressure unit mein) par

r = a P^{n}

kya predict karta hai?

r = a \cdot 1^{n} = a

, toh

a

literally unit pressure par burn rate hai — graph se

a

read karne ke liye ek useful anchor.

Agar koi propellant

n < 0

dikhaye, toh physically kya matlab hoga aur kya yeh usable hoga?

Yeh higher pressure par dheemare jaalaega, jo thermodynamically unusual hai; trivially "stable" hone ke bawajood, aisa behaviour ek ill-behaved combustion regime signal karta hai aur normal design target nahi hai.

Bahut high pressures par kuch propellants mein

n

badhkar

1

ke pass ya us se aage jaata hai. Yeh dangerous kyun hai?

n \geq 1

regime mein enter karna ek pehle se stable motor ko runaway mein badal deta hai, isliye safe operating band poore pressure range mein check hona chahiye, sirf ek point par nahi — dekho Combustion Instability aur Chamber Pressure and Nozzle Throat.

Do identical grains, ek

- 2 0^{\circ} C

par launch hua, ek

+ 4 0^{\circ} C

par — same chamber pressure. Kya woh same rate par jaalaate hain?

Nahi. Warm grain ko ignition temperature tak pahunchne ke liye kam heat chahiye, toh uska effective

a

bada hai aur woh tezi se peeche hatata hai; cold grain dheemare jaalta hai — yeh purely temperature ka effect hai, pressure ka nahi.

Recall Khud ko test karne ke liye ek-sentence summary

Trap-free mental model: $r$ ek surface speed hai, yeh $P$ ke saath hamesha badhti hai (kisi bhi $n > 0$ ke liye) lekin sirf $P^{n}$ ki tarah, aur akela number $n$ sab kuch decide karta hai — chhota $n$ = weak response = stable, $n \geq 1$ = runaway.