Combustion thermodynamics — stoichiometry, adiabatic flame temperature

3.3.19 · Physics › Rocket Propulsion

1. Stoichiometry — atoms ki bookkeeping

HUMEIN KYA CHAHIYE: coefficients jo atom conservation satisfy karein. KAISE MILENGE: skeletal reaction likho, har element conserve karo, solve karo.

Worked balance — hydrogen/oxygen (ek real rocket propellant, jaise RS-25): $2 H_{2} + O_{2} \to 2 H_{2} O$

Yeh step kyun? Left mein 4 H, 2 O hain. H balance karne ke liye 2 water molecules chahiye (→ 4 H, 2 O). Ab O bhi balance ho gaya. Ho gaya.

2. Enthalpy of reaction — heat kahan se aati hai

3. Adiabatic flame temperature $T_{a d}$ — first principles se derive karna

Derivation (constant pressure, $Q = 0$ ):

Constant $P$ par energy balance: total enthalpy conserve hoti hai, $H_{r e a c t an t s} (T_{0}) = H_{p r o d u c t s} (T_{a d}) .$

Kyun? Adiabatic + constant-P ⇒ poore process ke liye $Δ H = Q = 0$ .

Har side ko "formation at $T_{0}$ " + " $T_{0}$ se $T$ tak heating" mein split karo. Kyunki reactants $T_{0}$ par enter karte hain: $H_{r e a c t} (T_{0}) r e a c t \sum n_{j} Δ H_{f, j}^{\circ} = products T_{0} par banao p r o d \sum n_{i} Δ H_{f, i}^{\circ} + products ko heat karo p r o d \sum n_{i} \int_{T_{0}}^{T_{a d}} c_{p, i} d T$

Rearrange karo (§2 se $Δ H_{r x n}$ use karke):

Figure — Combustion thermodynamics — stoichiometry, adiabatic flame temperature

4. $T_{a d}$ aur $M$ performance kyun decide karte hain

Active Recall

Recall Khud test karo (answers chhupao)

Combustion ko chamber mein adiabatic kyun maana jaata hai?
Stoichiometric balance mein kya conserve hota hai?
Rockets fuel-rich kyun chalte hain?
Naive $T_{a d}$ reality se zyada kyun hoti hai?
Exhaust speed ke liye asal mein kaun si quantity matter karti hai, aur sirf $T$ kyun nahi?

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek super-fast campfire ek sealed metal box mein. Tum exactly utna "jalane wala stuff" (fuel) aur "jalane mein help karne wala stuff" (oxygen) daalo ki kuch bhi waste na ho — yahi stoichiometry hai, jaise har bachche ke liye exactly ek kursi match karna. Aag itni fast jalti hai ki heat box se bahar nahi ja sakti, toh SAARI heat andar reh jaati hai aur dhuaan ko paaglon jitna garam kar deti hai — yahi flame temperature hai. Lekin agar woh bahut zyada garam ho jaata hai, toh dhuaan ke molecules literally khud ko tod dete hain, aur todne mein energy lagti hai, toh woh thoda thanda ho jaata hai. Isliye real aag tumhari pehli guess jitni garam nahi hoti. Phir rockets is garam dhuaan ko ek nozzle se bahar phenkte hain — garam aur halka dhuaan = tez phenkna = zyada push.

Flashcards

"Stoichiometric mixture" ka kya matlab hai?

Woh exact fuel:oxidizer ratio jo dono ko poori tarah consume kare, koi excess reactant na bacha.

H₂ + O₂ → water balance karo.

2 H_{2} + O_{2} \to 2 H_{2} O

H₂/O₂ ke liye stoichiometric O/F mass ratio?

\frac{1 \cdot 32}{2 \cdot 2} = 8

(8 kg O₂ per kg H₂).

Equivalence ratio

ϕ

define karo.

ϕ = (F / O)_{a c t u a l} / (F / O)_{s t}

;

ϕ > 1

fuel-rich,

ϕ < 1

fuel-lean,

ϕ = 1

stoichiometric.

Rockets aksar fuel-rich kyun chalte hain?

Excess light fuel (jaise H₂) product molar mass

M

kam karta hai,

v_{e} \propto T / M

badhata hai, aur

T

ko material limits se neeche rakhta hai.

Heat of reaction ke liye Hess's law?

Δ H_{r x n} = \sum n_{i} Δ H_{f, p r o d} - \sum n_{j} Δ H_{f, r e a c t}

Standard state mein element ki standard enthalpy of formation?

Definition se zero.

Adiabatic flame temperature define karne wali physical condition?

Constant pressure,

Q = 0

⇒ saari reaction enthalpy products ko heat karti hai;

H_{r e a c t} (T_{0}) = H_{p r o d} (T_{a d})

T_{a d}

ke liye governing equation?

- Δ H_{r x n} = \sum_{i} n_{i} \int_{T_{0}}^{T_{a d}} c_{p, i} d T

Constant-

c_{p}

estimate of

T_{a d}

T_{a d} \approx T_{0} + \frac{- Δ H _{r x n}}{\sum _{i} n _{i} c _{p, i}}

Real

T_{a d}

naive estimate se kam kyun hoti hai?

Products ka endothermic dissociation aur high temperature par badhta hua

c_{p} (T)

energy absorb kar lete hain.

Maximum

T_{a d}

max fuel par kyun nahi hoti?

Excess reactant unreacted mass hai jo heat absorb karta hai; peak

T

ϕ = 1

ke paas hoti hai.

Exhaust velocity kaunse combustion outputs set karte hain?

Chamber temperature

T_{c} \approx T_{a d}

aur product molar mass

M

v_{e} \propto T_{c} / M

ke zariye.

Connections

Specific Impulse and Exhaust Velocity — $v_{e} \propto T_{c} / M$ directly $T_{a d}$ use karta hai.
Nozzle Theory and Isentropic Expansion — garam high- $P$ gas ko kinetic energy mein convert karta hai.
Chemical Equilibrium and Dissociation — Gibbs minimization se $T_{a d}$ correct karta hai.
First Law of Thermodynamics — $T_{a d}$ ke neeche enthalpy balance.
Hess's Law and Enthalpy of Formation — $Δ H_{r x n}$ ka source.
Propellant Selection — kyun H₂/O₂, RP-1/LOX $T$ aur $M$ mein alag hain.