Question bank — Film cooling — effectiveness, coverage fraction

False. $\eta$ protection measure karta hai, heat nahi: $\eta =1$ wall ko coolant temperature par rakhta hai (sabse thanda possible), $\eta =0$ use poori hot-gas temperature par chhod deta hai.

True. Wahan film bilkul unmixed coolant hai, isliye $T_{aw}=T_{cool}$ aur $\eta =\exp (0)=1$ — exponential decay abhi shuru nahi hui.

False. $\eta =\frac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}}$ best case mein sirf 1 tak pahunch sakta hai jab $T_{aw}=T_{cool}$; wall coolant se thandi nahi ho sakti jo use garam kar raha hai, isliye ratio 1 par cap hai.

True — ek fixed threshold ke liye. Kyunki $x_{prot}=\frac{{\dot{m}}_c{c}_p}{h_g}\ln \frac{1}{{\eta }_{min}}$, length ${\dot{m}}_c$ mein linear hai; nonlinearity $\ln$ mein rehti hai jo tabhi kaatti hai jab ${\eta }_{min}$ change karo.

True as written ($f=\frac{{\dot{m}}_c{c}_p}{h_{g}L}\ln \frac{1}{{\eta }_{min}}$ ${\dot{m}}_c$ mein linear hai) — lekin sirf tab tak jab $f$ 1 tak pahunche; poori wall se zyada protect nahi kar sakte, isliye $f=1$ ke baad extra coolant waste hai.

False. Yeh $T_{aw}$ use karta hai, adiabatic wall temperature (wall mein zero conduction). Yeh gas-side film physics ko wall ki khud ki conduction aur regenerative cooling se alag karta hai.

False. Zyada $h_g$ ka matlab hai hot gas film ko zyada tezi se heat karta hai, isliye $\eta$ tezi se decay karta hai aur $x_{prot}$ chhota ho jaata hai — $h_g$ coverage length ke denominator mein hai.

False. Zyada ${\eta }_{min}$ ek sakht requirement hai, isliye kam stations qualify karte hain; $\ln \frac{1}{{\eta }_{min}}$ chhota hota jaata hai jaise ${\eta }_{min}\to 1$, coverage kaat deta hai.

Yahan sahi hai, lekin reason check karo: $T_{hot}-T_{aw}=\eta (T_{hot}-T_{cool})$ $\eta$ mein linear hi hai, isliye $\eta =0.8$ par drop exactly $0.8\times 2500=2000$ K hai. Trap yeh hai ki is linear $T_{aw}$ relation ko exponential $\eta (x)$ relation se confuse kar lete hain — alag variables hain.

Half-error. Length ${\dot{m}}_c$ mein linear hai isliye fixed ${\eta }_{min}$ ke liye yeh kaam karta hai, lekin real slot ki maximum injection rate aur momentum-ratio limits hoti hain (bahut zyada coolant film ko wall se utha deta hai). Engineers ek aur slot add karte hain — re-establishing the film.

Error: $e^0=1$ hota hai, 0 nahi. $x=0$ par exponent zero hai aur $\eta =1$ (perfect protection); reader ne $e^{-0}$ ko $e^{-\infty }$ se confuse kiya.

Error: yeh dimensionless hai — ek Stanton-number-times-length parameter. $h_{g}x/({\dot{m}}_c{c}_p)$ ki units hain $\frac{({\text{W/m}}^2\text{K})(\text{m})}{(\text{kg/m⋅s})(\text{J/kg⋅K})}$, aur W = J/s sab kuch cancel kar deta hai.

Error: $\eta$ dimensionless hai (0–1) lekin $T_{aw}$ kelvin mein ek real temperature hai, hamesha $T_{cool}$ aur $T_{hot}$ ke beech rehti hai — ek blend, fraction nahi.

Error: formula $T_{aw}$ chahta hai, conducting-wall $T_w$ nahi. Ek real wall conduction/regen cooling se heat lose karti hai, isliye $T_w\ne T_{aw}$; $T_w$ use karne se do alag physics blend ho jaate hain aur $\eta$ corrupt ho jaata hai.

Error: $f=x_{prot}/L$ actual wall length ka ek fraction hai; formula 1 se zyada value compute kar sakta hai, jiska matlab sirf yeh hai ki ek slot over-protect kar raha hai — physically $f$ 1 par saturate hota hai aur surplus coolant redundant hai.

Kyunki $\eta$ distance ke saath exponentially decay karta hai; exponential ko invert karna ki "kahan $\eta$ ${\eta }_{min}$ tak pahunchta hai" ek $\ln$ produce karta hai. Diminishing returns built-in hain — deeper protect karna disproportionately cost karta hai.

Concerns alag karne ke liye: $T_{aw}$ sirf gas-side film behaviour capture karta hai zero wall conduction ke saath, taaki film performance ko jo bhi wall material ya regenerative loop peeche ho usse alag characterize kiya ja sake.

${\dot{m}}_c{c}_p$ film ka heat capacity hai — uska "cold budget." Zyada budget ka matlab hai ki same entrained hot gas film temperature ko per metre kam badhata hai, isliye $\eta$ zyada dheere girta hai.

Film–core interface par turbulent mixing hot gas ko film mein entrain karta hai (entrainment); har hot mass piece add hone se film dono garam hoti hai aur uski coolness dilute hoti hai, $\eta$ ko 0 ki taraf drive karta hai.

Kyunki wall ek mixture of hot core aur cool film dekhti hai, aur $\eta$ precisely mixing fraction hai — us station par surviving "coolness" ki amount weights set karti hai.

$h_g$ gas-side heat transfer coefficient hai (Bartz-type correlation se); yeh set karta hai ki hot gas temperature difference ke per degree film mein kitni tezi se heat deliver karta hai — decay ODE ka driver.

${\eta }_{min}$ material limit se set hota hai: usse neeche wall apni safe temperature exceed kar leti hai. ${\eta }_{min}$ artificially kam karna kaagaz par coverage "extend" karta hai jabki actual wall pighilti hai — yeh ek design lie hai, fix nahi.

$\eta =e^{-\infty }=0$: film bilkul khatam ho gayi, $T_{aw}\to T_{hot}$, aur wall poori hot gas dekhti hai — yahi reason hai multiple slots ka.

Exponent $\frac{h_{g}x}{{\dot{m}}_c{c}_p}\to \infty$ kisi bhi $x>0$ ke liye, isliye $\eta \to 0$ instantly — bina coolant ke koi film nahi aur koi bhi jagah protection nahi.

Exponent $\to 0$, isliye $\eta \to 1$ har jagah — wall poore chamber mein $T_{cool}$ par rehti hai. Physically unreachable (finite propellant), lekin yeh sahi limit dikhata hai.

Definition $\frac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}}$ zero se divide ho jaati hai — $\eta$ undefined hai kyunki protect karne ke liye koi temperature gap hi nahi; "coolant" koi benefit nahi deta.

$\ln \frac{1}{1}=0$, isliye $f=0$: perfect protection sirf exactly slot par ($x=0$) milti hai, measure zero ka ek set — ek slot se kisi finite length par $\eta =1$ demand nahi kar sakte.

Exactly 1 — film threshold ko far end par barely meet karti hai, isliye poori chamber length protected qualify hoti hai aur coverage complete hai.

$T_{aw}=(1-1)T_{hot}+1\cdot T_{cool}=T_{cool}$ — wall exactly coolant temperature par hai, kisi bhi station ka best possible result.