Real gas effects — dissociation, recombination: Question bank

3.3.20 · D5 · Physics › Rocket Propulsion › Real gas effects — dissociation, recombination

Shuru karne se pehle, teen symbols jo is page par baar baar aayenge, plain words mein:

$T_{c}$ — chamber temperature, burnt gas kitna hot hai nozzle mein jaane se pehle.
$v_{e}$ — exhaust velocity, gas peeche se kitni tez nikalti hai; jitni zyada utna zyada thrust.
$M$ — mean molar mass, gas mixture ke ek mole ka average "weight". Bade molecules ko tode chhote pieces mein toh $M$ kam ho jaata hai.
$α$ — degree of dissociation, $0$ se $1$ tak ka ek number (kuch nahi toota se sab toot gaya).

True or false — justify

Recall T/F: Dissociation ek endothermic process hai.

True ::: Chemical bond todne mein energy lagti hai, isliye reaction gas se heat absorb karta hai — yahi wajah hai ki $T_{c}$ "perfect combustion" ke estimate se kam nikalta hai.

Recall T/F: Nozzle mein recombination woh energy release karta hai jo dissociation ne pehle churayi thi.

True ::: Recombination ulti (exothermic) reaction hai; jaise jaise gas thandi hoti hai, molecules dobara ban jaate hain aur bond energy flow mein wapas daal dete hain — agar uske liye waqt ho.

Recall T/F: Ek calorically perfect gas phir bhi dissociate ho sakti hai.

False ::: "Calorically perfect" ka matlab hai fixed composition aur constant $c_{p}, γ$ by definition; dissociation composition badal deta hai, isliye yahi woh cheez hai jo perfect-gas assumption ko todo deti hai.

Recall T/F: Chamber pressure badhane se degree of dissociation hamesha badhti hai.

False ::: Dissociation mole count badhata hai (jaise $A_{2} \to 2 A$ ), toh Le Chatelier's Principle ke according zyada pressure balance ko waapas molecules ki taraf dhakelta hai — $α$ kam hota hai, zyada nahi.

Recall T/F: Equilibrium flow mein exhaust frozen flow se tez ho sakti hai jab same propellant aur nozzle ho.

True ::: Equilibrium flow recombination energy recover karta hai aur $M$ ko kam rakhta hai jaise pieces dobara jodti hain; frozen flow woh energy lock karke rakhta hai, isliye equilibrium $v_{e}$ ka zyada wala ideal bound hai.

Recall T/F: Dissociation hamesha rocket performance ke liye buri hoti hai.

False ::: Yeh kam $T_{c}$ se hurt karta hai lekin kam $M$ se help karta hai (kyunki $v_{e} \propto 1/ M$ ); equilibrium flow mein churayi gayi energy wapas bhi aati hai, toh net effect kabhi kabhi favorable bhi ho sakta hai.

Recall T/F: Damköhler number compare karta hai ki gas nozzle mein kitna waqt bitaati hai aur chemistry kitna waqt leti hai.

True ::: $D a = τ_{f l o w} / τ_{c h e m}$ ; bada $D a$ matlab chemistry saath chalti hai (equilibrium), chhota $D a$ matlab flow reactions se aage nikal jaati hai (frozen).

Recall T/F:

$α = 0$ par reacting-gas model aur calorically-perfect model same answer dete hain. True ::: Kuch bhi dissociate nahi hua toh composition fixed hai aur koi bond energy divert nahi hoti, isliye dono models same $T_{c}$ , $M$ , aur $v_{e}$ dete hain — yeh degenerate limiting case hai.

Spot the error

Recall "Kyunki

$v_{e} \propto T_{c}$ , maximum thrust ke liye hume $T_{c}$ ko jitna ho sake utna badhana chahiye." Error ::: Ek point ke baad, zyada heat temperature badhane ki jagah sirf zyada bonds todne mein jaati hai, aur $v_{e}$ actually $T_{c} / M$ par depend karta hai, sirf $T_{c}$ par nahi — isliye blindly $T_{c}$ maximize karne par diminishing returns milte hain.

Recall "Frozen flow mein

$v_{e}$ zyada hoti hai kyunki composition fixed aur simple hota hai." Error ::: Frozen flow woh recombination energy khota hai jo equilibrium flow recover karta hai; yeh $v_{e}$ ka lower bound hai, upper bound nahi.

Recall "Kyunki dissociation

$M$ ghata deta hai, dissociation hamesha $v_{e}$ badhata hi hoga." Error ::: Yeh doosra lever ignore karta hai — dissociation $T_{c}$ bhi ghata deta hai. Yeh $T_{c} ↓$ aur $M ↓$ ke beech ek tug-of-war hai; winner is baat par depend karta hai ki recombination downstream kitni energy wapas karta hai.

Recall "

$K_{p}$ batata hai ki reaction kitni tez jati hai, toh bada $K_{p}$ matlab gas jaldi equilibrium pe pahunch jaati hai." Error ::: $K_{p}$ ek thermodynamic balance point hai ( $ln K_{p} = - Δ G^{\circ} / R_{u} T$ se), rate nahi; equilibrium kitni tez reach hoti hai yeh kinetics ka sawaal hai jo Damköhler Number se capture hota hai, $K_{p}$ se nahi.

Recall "Real nozzle performance equilibrium-flow prediction ke barabar hoti hai."

Error ::: Real flow frozen aur equilibrium ke beech hoti hai, usually equilibrium se kuch percent neeche, kyunki recombination tez hoti hai lekin infinitely tez nahi — finite $D a$ .

Recall "Dissociation ignore karna

$T_{c}$ estimate ke liye ek safe, conservative simplification hai." Error ::: Ise ignore karna $T_{c}$ ko overpredict karta hai (tum bond-breaking ka cost kabhi nahi pay karte), isliye yeh optimistic hai, conservative nahi.

Why questions

Recall Bonds todne se chamber temperature kyun kam hoti hai jabki same fuel jalaya jaata hai?

Because ::: har ek joule jo bond todne mein lagti hai woh ek joule hai jo gas temperature badhane mein nahi lagti — energy conserved hai, isliye split hota hai temperature-heating aur bond-breaking ke beech.

Recall Zyada chamber pressure combustion completeness kyun improve karta hai?

Because ::: zyada pressure dissociation ko suppress karta hai (Le Chatelier, kyunki dissociation moles badhata hai), zyada energy heat ke roop mein rakhta hai aur hotter, zyada complete burn deta hai.

Recall

$M$ kam karna helpful kyun hai jab $T_{c}$ same rahe? Because ::: $v_{e} \propto T_{c} / M$ , toh same thermal energy ke liye halki species nozzle se tez nikalti hain — chhote pieces bade pieces se zyada tez "daudti" hain.

Recall Recombination mainly nozzle mein kyun hoti hai, chamber mein kyun nahi?

Because ::: recombination cooling se favored hoti hai; chamber sabse zyada hot hota hai (dissociation drive karta hai), jabki nozzle gas ko cool karta hai jaise woh accelerate hoti hai, balance ko molecules dobara banana ki taraf shift karta hai.

Recall Chhota nozzle frozen flow ki taraf kyun tend karta hai?

Because ::: kam residence time $τ_{f l o w}$ se $D a = τ_{f l o w} / τ_{c h e m}$ chhota ho jaata hai, toh gas recombination chemistry ke catch up karne se pehle hi nikal jaati hai.

Edge cases

Recall

$α \to 0$ limit mein tug-of-war ka kya hota hai? Nothing to trade off ::: koi bonds nahi tode matlab $T_{c}$ nahi ghata aur $M$ nahi ghata — reacting gas bilkul calorically perfect gas jaisi behave karti hai.

Recall

$α \to 1$ limit mein kya hota hai (fully dissociated)? Maximum energy divert aur minimum $M$ ::: $T_{c}$ sabse zyada depress hota hai aur mixture sab halke fragments ka hota hai; yeh net gain hai ya nahi yeh poori tarah is baat par depend karta hai ki nozzle kitna recombination recover kar sakta hai.

Recall

$D a ≫ 1$ limit mein kaun si idealization apply hoti hai aur kyun? Equilibrium flow ::: chemistry effectively flow ke relative infinitely tez hai, toh gas har point par chemical equilibrium mein rehti hai aur poori recombination energy recover karti hai.

Recall

$D a ≪ 1$ limit mein kaun si idealization apply hoti hai aur kyun? Frozen flow ::: flow chemistry se kahin zyada tez hai, toh composition apni chamber value par lock ho jaati hai aur bond energy nozzle se unrecovered nikal jaati hai.

Recall Agar

$Δ H_{d}$ (bond energy) zero hoti, toh dissociation $T_{c}$ ka kya karta? Nothing ::: "bond" todne mein koi energy cost nahi hogi, molecules split karna na heat churaata na wapas karta — $T_{c}$ untouched rehta aur sirf $M$ ka effect bachta.

Connections

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