Combustion thermodynamics — stoichiometry, adiabatic flame temperature

1. Stoichiometry — the bookkeeping of atoms

WHAT we want: coefficients that satisfy atom conservation. HOW we get them: write skeletal reaction, conserve each element, solve.

Worked balance — hydrogen/oxygen (a real rocket propellant, e.g. RS-25): $2\,\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\,\mathrm{H_2O}$

Why this step? Left has 4 H, 2 O. To balance H we need 2 water molecules (→ 4 H, 2 O). Now O balances too. Done.

2. Enthalpy of reaction — where the heat comes from

3. Adiabatic flame temperature $T_{ad}$ — deriving from first principles

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

Energy balance at constant $P$ : total enthalpy is conserved, $H_{reactants}(T_0) = H_{products}(T_{ad}).$

Why? Adiabatic + constant-P ⇒ $\Delta H = Q = 0$ for the whole process.

Split each side into "formation at $T_0$ " + "heating from $T_0$ to $T$ ". Since reactants enter at $T_0$ : $\underbrace{\sum_{react} n_j \Delta H_{f,j}^\circ}_{H_{react}(T_0)} = \underbrace{\sum_{prod} n_i \Delta H_{f,i}^\circ}_{\text{form products at }T_0} + \underbrace{\sum_{prod} n_i \int_{T_0}^{T_{ad}} c_{p,i}\,dT}_{\text{heat products up}}$

Rearrange (using $\Delta H_{rxn}$ from §2):

Figure — Combustion thermodynamics — stoichiometry, adiabatic flame temperature

4. Why $T_{ad}$ and $M$ decide performance

Active Recall

Recall Test yourself (hide answers)

Why is combustion treated as adiabatic in a chamber?
What conserves in a stoichiometric balance?
Why do rockets run fuel-rich?
Why is the naïve $T_{ad}$ higher than reality?
What quantity actually matters for exhaust speed, and why isn't it just $T$ ?

Recall Feynman: explain to a 12-year-old

Imagine a super-fast campfire in a sealed metal box. You throw in exactly the right amount of "burny stuff" (fuel) and "air-that-helps-burning" (oxygen) so nothing is wasted — that's stoichiometry, like matching every kid with exactly one chair. The fire burns so fast the heat can't escape the box, so ALL the heat stays inside and makes the smoke insanely hot — that's the flame temperature. But if it gets too hot, the smoke molecules literally rip themselves apart, and ripping takes energy, so it cools back down a bit. That's why the real fire isn't as hot as your first guess. Rockets then squirt this hot smoke out a nozzle — hotter and lighter smoke = faster squirt = more push.

Flashcards

What does "stoichiometric mixture" mean?

The exact fuel:oxidizer ratio that fully consumes both, leaving no excess reactant.

Balance H₂ + O₂ → water.

2H_2 + O_2 \to 2H_2O

Stoichiometric O/F mass ratio for H₂/O₂?

\frac{1\cdot32}{2\cdot2}=8

(8 kg O₂ per kg H₂).

Define equivalence ratio

\phi

\phi=(F/O)_{actual}/(F/O)_{st}

;

\phi>1

fuel-rich,

\phi<1

fuel-lean,

\phi=1

stoichiometric.

Why do rockets often run fuel-rich?

Excess light fuel (e.g. H₂) lowers product molar mass

M

, raising

v_e\propto\sqrt{T/M}

, and keeps

T

off material limits.

Hess's law for heat of reaction?

\Delta H_{rxn}=\sum n_i\Delta H_{f,prod}-\sum n_j\Delta H_{f,react}

Standard enthalpy of formation of an element in standard state?

Zero by definition.

Physical condition defining adiabatic flame temperature?

Constant pressure,

Q=0

⇒ all reaction enthalpy heats the products;

H_{react}(T_0)=H_{prod}(T_{ad})

Governing equation for

T_{ad}

-\Delta H_{rxn}=\sum_i n_i\int_{T_0}^{T_{ad}}c_{p,i}\,dT

Constant-

c_p

estimate of

T_{ad}

T_{ad}\approx T_0+\dfrac{-\Delta H_{rxn}}{\sum_i n_i c_{p,i}}

Why is real

T_{ad}

lower than the naïve estimate?

Endothermic dissociation of products and rising

c_p(T)

at high temperature absorb energy.

Why isn't maximum

T_{ad}

at max fuel?

Excess reactant is unreacted mass that absorbs heat; peak

T

is near

\phi=1

What combustion outputs set exhaust velocity?

Chamber temperature

T_c\approx T_{ad}

and product molar mass

M

, via

v_e\propto\sqrt{T_c/M}

Connections

Specific Impulse and Exhaust Velocity — $v_e\propto\sqrt{T_c/M}$ uses $T_{ad}$ directly.
Nozzle Theory and Isentropic Expansion — converts hot high- $P$ gas into kinetic energy.
Chemical Equilibrium and Dissociation — corrects $T_{ad}$ via Gibbs minimization.
First Law of Thermodynamics — enthalpy balance underlying $T_{ad}$ .
Hess's Law and Enthalpy of Formation — source of $\Delta H_{rxn}$ .
Propellant Selection — why H₂/O₂, RP-1/LOX differ in $T$ and $M$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, rocket ka combustion chamber basically ek super-fast, sealed campfire hai. Sabse pehle stoichiometry — iska matlab hai atoms ka hisaab-kitaab. Reaction ko balance karo taaki fuel aur oxidizer dono poori tarah khatam ho jaayein, kuch bacha na rahe. Jaise H₂ aur O₂ ke liye: $2H_2 + O_2 \to 2H_2O$ . Isse nikalta hai ki 1 kg hydrogen ke liye theek 8 kg oxygen chahiye — yeh stoichiometric O/F ratio hai. Equivalence ratio $\phi$ batata hai ki hum stoichiometric se rich (zyada fuel) chala rahe hain ya lean (zyada oxidizer).

Ab adiabatic flame temperature. "Adiabatic" ka matlab garmi bahar nahi ja sakti ( $Q=0$ ), kyunki combustion itni fast hai ki heat leak hone ka time hi nahi. Toh reaction se jitni chemical energy release hoti hai ( $-\Delta H_{rxn}$ ), woh poori ki poori product gases ko garam karne mein lag jaati hai. Energy balance: heat released = heat absorbed by products, yani $-\Delta H_{rxn}=\sum n_i \int c_p\,dT$ . Simple estimate mein $T_{ad}=T_0 + \frac{-\Delta H_{rxn}}{\sum n_i c_{p,i}}$ .

Ek important trap: naïve calculation H₂/O₂ ke liye ~5600 K deta hai, lekin asli $T_{ad}$ sirf ~3300 K hai. Kyun? Kyunki itni high temperature par molecules dissociate ho jaate hain ( $H_2O \to OH + H$ ) — yeh energy khaa jaata hai — aur $c_p$ bhi temperature ke saath badhta hai. Isliye naïve number ek upper bound hai, real value hamesha kam.

Aur last cheez — yeh sab matter kyun karta hai? Rocket ki exhaust speed $v_e \propto \sqrt{T_c/M}$ . Yani sirf temperature nahi, product ka molar mass $M$ bhi important hai. Isiliye rockets thoda fuel-rich chalate hain: bacha hua halka hydrogen $M$ ko kam kar deta hai, aur $\sqrt{T/M}$ badh jaata hai — matlab tez exhaust, zyada thrust. Yaad rakho: chase "Hot divided by Heavy", sirf hot nahi.

Combustion thermodynamics — stoichiometry, adiabatic flame temperature

1. Stoichiometry — the bookkeeping of atoms

2. Enthalpy of reaction — where the heat comes from

3. Adiabatic flame temperature $T_{ad}$ — deriving from first principles

4. Why $T_{ad}$ and $M$ decide performance

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Rocket Propulsion

Connections