Chapman-Jouguet detonation; deflagration vs detonation

5.3.4 · Chemistry › Combustion Chemistry (Propulsion Bridge)

Intuition Bada picture (YE KYU EXIST KARTA HAI)

Jab ek fuel–oxidizer mixture jalti hai, tab reaction front do fundamentally alag tareekon se unburnt gas mein travel kar sakta hai:

Deflagration — ek subsonic flame jo heat conduction + radicals ke diffusion se aage badhti hai. Pressure almost nahi badlta (thoda sa girta bhi hai).
Detonation — ek supersonic shock wave jo gas ko mechanically compress karke ignite karti hai. Pressure ~15–40 ke factor se jump karta hai.

Dono same chemical energy release karte hain, lekin propagation ka mechanism (diffusion vs shock) bilkul opposite hai. Chapman–Jouguet (CJ) condition woh rule hai jo mathematically allowed infinitely many solutions mein se unique stable detonation speed select karti hai. Yahi propulsion ka bridge hai: rotating detonation engines (RDEs) detonation chahte hain kyunki ye constant-volume (zyada efficient) heat release ke kareeb hoti hai.

1. Front ke across conservation laws (SETUP KAISE KARTE HAIN)

Front ko ek patli discontinuity maano. Uss frame mein baitho jo front ke saath move kar raha hai, to gas usme $u_{1}$ speed se andar aati hai aur $u_{2}$ se bahar jaati hai. Subscript 1 = unburnt (reactants), 2 = burnt (products). Per unit area:

Ye teen kyun? Reaction front sirf ek control volume hai. Kuch magic nahi — wahi bookkeeping jaise shock tube mein, bas energy equation mein chemistry ka source term $q$ bhi add ho jaata hai.

Rayleigh line derive karna (momentum se pressure aur volume ka link KAISE milta hai)

Maano $v = 1/ ρ$ specific volume hai. Mass se: $u_{1} = ρ_{2} u_{2} / ρ_{1}$ , aur mass flux $\overset{m}{˙} = ρ_{1} u_{1} = ρ_{2} u_{2}$ . To $u_{1} = \overset{m}{˙} v_{1}$ , $u_{2} = \overset{m}{˙} v_{2}$ .

Momentum $p_{1} + \overset{m}{˙}^{2} v_{1} = p_{2} + \overset{m}{˙}^{2} v_{2}$ mein substitute karo:

$p_{2} - p_{1} = - \overset{m}{˙}^{2} (v_{2} - v_{1}) (Rayleigh line)$

Hugoniot curve derive karna (energy se final state KAISE pin hoti hai)

Energy equation lo aur velocities eliminate karo. Use karo $\frac{1}{2} u_{1}^{2} - \frac{1}{2} u_{2}^{2} = \frac{1}{2} \overset{m}{˙}^{2} (v_{1}^{2} - v_{2}^{2})$ . Momentum relation $\overset{m}{˙}^{2} = - (p_{2} - p_{1}) / (v_{2} - v_{1})$ ke saath combine karo:

$\frac{1}{2} u_{1}^{2} - \frac{1}{2} u_{2}^{2} = \frac{1}{2} (p_{2} - p_{1}) (v_{1} + v_{2})$

Energy mein plug karo ( $h_{2} - h_{1} = q + \frac{1}{2} u_{1}^{2} - \frac{1}{2} u_{2}^{2}$ ):

Actual final state DONO lines satisfy karti hai: ye Rayleigh line aur Hugoniot curve ka intersection hai.

Figure — Chapman-Jouguet detonation; deflagration vs detonation

2. Chapman–Jouguet condition (topic ka core)

Ek given $\overset{m}{˙}$ ke liye Rayleigh line Hugoniot ko do points par kaat sakti hai (strong & weak), ek point par (tangent), ya bilkul nahi. CJ point special hai:

Typical numbers: stoichiometric H₂–O₂ ke liye $D_{C J} \approx 2840$ m/s, $p_{2} / p_{1} \approx 18$ .

3. Deflagration vs Detonation — contrast table

Definition Side-by-side

Property	Deflagration	Detonation
Propagation mechanism	heat conduction + radical diffusion	shock compression
Speed	subsonic ( $\sim$ 0.1–10 m/s lab flames)	supersonic ( $\sim$ 1500–3000 m/s)
$p_{2}$ vs $p_{1}$	thoda lower ( $p_{2} < p_{1}$ )	strongly higher ( $p_{2} ≫ p_{1}$ )
$v_{2}$ vs $v_{1}$	gas expands ( $v_{2} > v_{1}$ )	gas compressed ( $v_{2} < v_{1}$ )
Front se nikalna wali flow	accelerate hoti hai, $M_{2} < 1$	CJ par $M_{2} = 1$
Hugoniot branch	lower-right	upper-left
Propulsion relevance	normal flames, ramjets	RDE, PDE — near constant-volume, higher efficiency

4. Worked examples

5. Common mistakes (Steel-man + fix)

Recall Feynman: 12-year-old ko samjhao

Socho ek corridor mein log ek line mein khade hain aur "burn" pass kar rahe hain.

Deflagration: har person agale ko shoulder tap karta hai burn start karne ke liye. Slow, gentle, corridor shant rehta hai.
Detonation: koi corridor mein itni tez cannonball chodta hai ki har person smash hoke jal jaata hai. Ye sound se faster hai, aur peeche high pressure ki wall chhodta hai. Chapman–Jouguet woh rule hai jo kehta hai: cannonball ek exact speed par settle ho jaata hai — woh speed jis par peeche ke log aage instructions shout nahi kar sakte (unki awaaz pakad nahi sakti). Woh self-running speed detonation speed hai.

Flashcards

Deflagration aur detonation mein mechanism ke hisaab se kya farq hai?

Deflagration heat conduction + radical diffusion se propagate hoti hai (subsonic); detonation shock compression se propagate hoti hai (supersonic).

Deflagration vs detonation mein pressure ke saath kya hota hai?

Deflagration:

p_{2}

thoda lower than

p_{1}

(gas expands). Detonation:

p_{2} ≫ p_{1}

(gas compressed).

Teen Rankine–Hugoniot jump conditions batao.

Mass

ρ_{1} u_{1} = ρ_{2} u_{2}

; momentum

p_{1} + ρ_{1} u_{1}^{2} = p_{2} + ρ_{2} u_{2}^{2}

; energy

h_{1} + \frac{1}{2} u_{1}^{2} + q = h_{2} + \frac{1}{2} u_{2}^{2}

Rayleigh line kya hai aur uska slope kya hai?

p_{2} - p_{1} = - \overset{m}{˙}^{2} (v_{2} - v_{1})

;

(v_{1}, p_{1})

se guzarti straight line jiska slope

- \overset{m}{˙}^{2}

hai (hamesha negative).

Chapman–Jouguet condition define karo.

Rayleigh line Hugoniot curve ko tangent hoti hai; equivalently burnt gas front se exactly local sound speed par bahar nikalta hai,

u_{2} = a_{2}

(

M_{2} = 1

Detonation CJ speed par kyun settle hoti hai?

M_{2} = 1

par flow choked hoti hai, to rearward rarefaction waves front ko overtake karke kamzor nahi kar sakti — ye minimum detonation speed par self-sustaining ban jaati hai.

CJ velocity ka ideal-gas estimate do.

D_{C J} \approx 2 (γ^{2} - 1) q

(strong-detonation limit); zyada heat release

q

→ faster detonation.

Detonation kaun sa Hugoniot branch hai? Deflagration kaun sa?

Detonation = upper-left (

p_{2} > p_{1}

v_{2} < v_{1}

); deflagration = lower-right (

p_{2} < p_{1}

v_{2} > v_{1}

Detonation propulsion (RDE/PDE) ke liye attractive kyun hai?

Ye constant-volume heat release approximate karta hai, jo constant-pressure deflagration cycles se zyada thermodynamic efficiency deta hai.

Steeper Rayleigh line physically kya mean karti hai?

Bada

\overset{m}{˙}

, yani faster front — deflagration ki bajaye detonation ki characteristic.

Connections

Rankine-Hugoniot jump conditions
Shock waves and Mach number
Adiabatic flame temperature
Speed of sound in gases
Rotating Detonation Engine (RDE)
Pulse Detonation Engine (PDE)
Constant-volume vs constant-pressure combustion cycles
Laminar flame speed (deflagration)