CEA (Chemical Equilibrium with Applications) — using NASA-CEA tool to compute Isp, Tc, products

5.3.10 · Chemistry › Combustion Chemistry (Propulsion Bridge)

CEA actually kya compute karta hai

Teen headline outputs:

Symbol	Name	Meaning
$T_{c}$	Chamber (flame) temperature	Combustion ka adiabatic equilibrium temp
$X_{i}$	Product mole fractions	Species ka equilibrium "soup"
$I_{s p}$	Specific impulse	Propellant ke unit weight-flow per thrust (efficiency)

CEA products kaise dhundhta hai: Gibbs free energy minimization

First principles se derivation

Step 1 — Total Gibbs energy likho. $N$ species ke mix ke liye jisme mole numbers $n_{i}$ hain: $G = \sum_{i = 1}^{N} n_{i} μ_{i}$ Yeh step kyun? $G$ extensive hai; har species apna chemical potential $μ_{i}$ contribute karta hai times kitna usme hai.

Step 2 — Ek gas ka chemical potential. Partial pressure $p_{i}$ par ek ideal gas ke liye: $μ_{i} = μ_{i}^{\circ} (T) + R T ln \frac{p _{i}}{p ^{\circ}}, p_{i} = \frac{n _{i}}{n _{tot}} p$ Yeh step kyun? $μ_{i}^{\circ}$ standard chemical potential hai (JANNAF/CEA thermo tables se, $T$ mein polynomials ke roop mein fit kiya gaya). Log term encode karta hai ki dilute hona kisi species ki "escaping tendency" ko kaise kam karta hai — yahi mixing aur dissociation ko drive karta hai.

Step 3 — Constraints: atoms conserve karo. Har element $j$ ke liye (C, H, O, N…), jahan $a_{ij}$ = species $i$ mein element $j$ ke atoms ki sankhya, aur $b_{j}$ = propellant dwara supply kiye gaye element $j$ ke total moles: $\sum_{i} a_{ij} n_{i} = b_{j} for every element j$ Yeh step kyun? Reactions atoms ko rearrange karte hain; woh unhe kabhi create ya destroy nahi karte. Yahi cheezein fixed rakhi jaati hain.

Step 4 — Lagrange multipliers. $G$ ko constraints ke subject minimize karo. Form karo: $L = \sum_{i} n_{i} μ_{i} - \sum_{j} λ_{j} (\sum_{i} a_{ij} n_{i} - b_{j})$ $\partial L / \partial n_{i} = 0$ set karo: $μ_{i} = j \sum λ_{j} a_{ij}$ Yeh step kyun? Yeh master equilibrium condition hai: equilibrium par har species ka chemical potential "elemental potentials" $λ_{j}$ ke weighted sum ke barabar hota hai. Equivalently, kisi bhi reaction $\sum ν_{i} (species) = 0$ ke liye, yeh force karta hai $\sum_{i} ν_{i} μ_{i} = 0$ — exactly har reaction ke liye ek saath law of mass action ( $K_{p}$ relations). CEA in coupled equations ko numerically solve karta hai ( $λ_{j}$ aur $n_{i}$ par Newton iteration).

$I_{s p}$ kaise aata hai (nozzle expansion)

Step 1 — Energy → velocity. Steady adiabatic flow stagnation enthalpy conserve karta hai: $h_{c} = h_{e} + \frac{1}{2} v_{e}^{2} \Rightarrow v_{e} = 2 (h_{c} - h_{e})$ Yeh step kyun? Hot chamber gas mein "stored" enthalpy exhaust kinetic energy ban jaati hai.

Step 2 — Specific impulse define karo. Mass flow $\overset{m}{˙}$ aur (matched-pressure) thrust $F = \overset{m}{˙} v_{e}$ ke saath: $I_{s p} = \frac{v _{e}}{g _{0}} (seconds), g_{0} = 9.81 m/s^{2}$ Yeh step kyun? $I_{s p}$ thrust per unit weight flow rate hai. Seconds ki units ise engine size se independent banati hain — ek clean efficiency figure.

Figure — CEA (Chemical Equilibrium with Applications) — using NASA-CEA tool to compute Isp, Tc, products

Worked examples

Common mistakes (Steel-man → fix)

Active recall

Recall CEA kaunsi quantity minimize karta hai, aur kaunse constraint ke subject?

Total Gibbs free energy $G = \sum n_{i} μ_{i}$ minimize karta hai, har chemical element ke conservation ke subject ( $\sum_{i} a_{ij} n_{i} = b_{j}$ ).

Recall Peak

$I_{s p}$ stoichiometric par nahi, fuel-rich kyun hota hai? Kyunki $I_{s p} \propto T_{c} / M$ . Fuel-rich exhaust molar mass $M$ (leftover $H_{2}$ ) ko $T_{c}$ se zyada kam karta hai, toh ratio rich par peak hota hai.

Recall CEA "equilibrium" aur "frozen" nozzle modes mein kya difference hai?

Equilibrium: gas thandi hone par composition re-solve hoti hai, recombination heat release karta hai → higher $I_{s p}$ . Frozen: composition fixed → lower $I_{s p}$ . Real engines beech mein aate hain.

Recall Dissociation include karna predicted

$T_{c}$ ko kyun kam karta hai? Dissociation reactions endothermic hain; woh combustion heat absorb karte hain, toh equilibrium flame temperature "complete combustion" estimate se kam hoti hai.

Recall

$I_{s p}$ ke liye use ki gayi exit velocity derive karo. Adiabatic flow $h_{c} = h_{e} + \frac{1}{2} v_{e}^{2}$ se, toh $v_{e} = 2 (h_{c} - h_{e})$ aur $I_{s p} = v_{e} / g_{0}$ .

Recall (Feynman, ek 12-saal ke bacche ko explain karo)

Socho ki ek super-hot blender (rocket chamber) mein bahut saare LEGO bricks (atoms) phenk rahe ho. Bricks alag-alag chhote toys (molecules) mein snap hote hain, aur us crazy heat mein woh snap apart hote rehte hain aur wapas aate rehte hain jab tak mix "sab se aaram-pasand," lowest-energy arrangement mein settle na ho jaaye. CEA ek smart calculator hai jo figure out karta hai ki aakhir mein tumhare paas exactly kaun se toys hain, blender kitna hot hota hai, aur hot gas peechhe se kitni tezi se shoot karta hai — jo tumhare rocket ki quality batata hai. Trick: tum kabhi LEGO bricks lose ya gain nahi kar sakte, sirf rearrange kar sakte ho.

Flashcards

NASA-CEA equilibrium products dhundhne ke liye kaunsa thermodynamic potential minimize karta hai?

Total Gibbs free energy

G = \sum_{i} n_{i} μ_{i}

fixed

T, p

par.

Minimization ke dauran kaunse constraints fixed rakhe jaate hain?

Har chemical element ke moles ka conservation:

\sum_{i} a_{ij} n_{i} = b_{j}

Lagrange multipliers se master equilibrium condition?

μ_{i} = \sum_{j} λ_{j} a_{ij}

har species

i

ke liye (sabhi reactions ke liye mass-action laws ke equivalent).

Kaunsa CEA problem type chamber

T_{c}

compute karta hai?

Constant enthalpy & pressure (HP) — fixed

p

par adiabatic combustion.

Exhaust velocity ke terms mein specific impulse ka formula?

I_{s p} = v_{e} / g_{0}

, jahan

g_{0} = 9.81

m/s².

Enthalpy balance se ideal exit velocity?

v_{e} = 2 (h_{c} - h_{e})

I_{s p}

T_{c}

aur

M

ke kaunse combination ke saath scale karta hai?

I_{s p} \propto T_{c} / M

(hot aur low molar mass accha hai).

H2/O2 ke liye peak

I_{s p}

fuel-rich kyun hai?

Fuel-rich exhaust

M

(extra H₂) ko

T_{c}

se zyada tezi se kam karta hai.

Dissociation predicted flame temperature kyun kam karta hai?

Dissociation endothermic hai; yeh combustion heat absorb karta hai.

Equilibrium vs frozen nozzle flow — kaunsa higher

I_{s p}

deta hai aur kyun?

Equilibrium; gas thandi hone par recombination heat release karta hai, flow mein energy add karta hai.

H2/O2 ke liye stoichiometric O/F (mass basis)?

8 (

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

se).

CEA output mein

c^{*}

(characteristic velocity) kya measure karta hai?

Nozzle se independent chamber efficiency:

c^{*} = p_{c} A_{t} / \overset{m}{˙}

Connections

Adiabatic Flame Temperature — $T_{c}$ jo CEA compute karta hai woh equilibrium adiabatic flame temperature hai.
Gibbs Free Energy and Equilibrium — CEA ko underpin karne wala minimization principle.
Chemical Equilibrium and Kp — $μ_{i} = \sum λ_{j} a_{ij}$ sabhi reactions ke liye ek saath mass action hai.
Hess's Law and Enthalpy of Formation — enthalpy balance mein use kiya gaya $Δ_{f} h^{\circ}$ supply karta hai.
Rocket Nozzle and Thrust Equation — jahan $v_{e}$ aur $I_{s p}$ physically realize hote hain.
Dissociation at High Temperature — kyun products ek "soup" hain, sirf CO₂+H₂O nahi.
Tsiolkovsky Rocket Equation — $Δ v$ paane ke liye $I_{s p}$ use karta hai.