Real gas effects — dissociation, recombination

3.3.20 · Physics › Rocket Propulsion

KYA ho raha hai?

KYUN matter karta hai rocket ke liye?

Rocket performance exhaust velocity se govern hoti hai: $v_e=\sqrt{\frac{2\gamma }{\gamma -1}\frac{R_u{T}_c}{M}\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}$

Dissociation isko do tareekon se attack karta hai:

1. Yeh $T_c$ ko giraa deta hai — jo energy gas ko heat karni chahiye thi woh bonds todne mein chali jaati hai, toh flame temperature ek naive "saara fuel burn hokar final products banega" estimate se zyada thandi hoti hai.
2. Yeh molar mass $M$ ko giraa deta hai — zyada bade molecules nahi, zyada halke atoms/radicals. Yeh $v_e$ ko help karta hai (kyunki $v_e\propto 1/\sqrt{M}$).

Toh ek sachchi tug-of-war hai. $v_e$ par net effect is baat par depend karta hai ki nozzle mein recombination kaafi tez hoti hai ya nahi.

KAISE: dissociation energy balance ko first principles se derive karna

Ek single dissociating reaction lo, jaise ${\text{A}}_2\rightleftharpoons 2\text{A}$, jisme ${\text{A}}_2$ ke ek mole ke toote par bond dissociation enthalpy $\Delta H_d$ ho.

Step 1 — Chamber mein energy conservation. Combustion se release hone wali heat, $Q_{comb}$, ko temperature raise bhi karni hai aur dissociation ka "bill" bhi bharna hai: $Q_{comb}=\underset{\text{heats the gas}}{\underbrace{n{\bar{c}}_p(T_c-T_0)}}+\underset{\text{spent breaking bonds}}{\underbrace{\alpha n\Delta H_d}}$ Yeh step kyun? Energy gayab nahi ho sakti; ek bond todne mein jaane wala har joule $T_c$ nahi badhata.

Step 2 — Degree of dissociation define karo. $\alpha =\frac{\text{moles dissociated}}{\text{moles initially present}},0\le \alpha \le 1$ Kyun? Hume ek number chahiye jo bataye "kitna toot chuka hai."

Step 3 — Equilibrium $\alpha$ fix karta hai. Total pressure $p$ par ${\text{A}}_2\rightleftharpoons 2\text{A}$ ke chemical equilibrium se: $K_p(T)=\frac{p_A^2}{p_{A_2}}=\frac{(2\alpha {)}^2}{(1-\alpha )(1+\alpha )}\frac{p}{p^{\circ }}$ Kyun? Thermodynamics ka $K_p$ ($\ln K_p=-\Delta G^{\circ }/R_{u}T$) humein balance point batata hai; isko solve karne par $\alpha (T,p)$ milta hai.

Key consequence — pressure dissociation ko suppress karta hai. Kyunki dissociation moles ki sankhya badhata hai, Le Chatelier kehta hai ki $p$ badhane se equilibrium wapas molecules ki taraf shift ho jaata hai ($\alpha \downarrow$). Isliye high chamber pressure achha hota hai: kam dissociation → zyada hot, zyada complete combustion.

Frozen vs. Equilibrium flow — asli baat

Jaise gas nozzle mein accelerate karti hai woh thandi hoti hai. Do idealized limits hain:

Reality in dono ke beech mein hoti hai. Controlling number hai Damköhler number: $Da=\frac{{\tau }_{flow}}{{\tau }_{chem}}=\frac{\text{residence time in nozzle}}{\text{chemical reaction time}}$

• $Da\gg 1$: kaafi time hai → equilibrium flow.
• $Da\ll 1$: time nahi → frozen flow.

Kyun matter karta hai: Real nozzle performance equilibrium se kuch % neeche aur frozen se upar hoti hai. Dissociation ko ignore karna $T_c$ aur $v_e$ ko overpredict karta hai; frozen flow assume karna recovered energy ko underpredict karta hai.

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Chemical Equilibrium and Kp
• Adiabatic Flame Temperature
• Nozzle Flow and Exhaust Velocity
• Specific Impulse
• Le Chatelier's Principle
• Damköhler Number

Concept Map