Film cooling — effectiveness, coverage fraction

3.3.29 · Physics › Rocket Propulsion

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WHAT is film cooling effectiveness?

WHY this definition? Hume ek aisa number chahiye jo:

• $\eta =1$ ho jab wall fully protected ho ($T_{aw}=T_{cool}$, perfect film),
• $\eta =0$ ho jab film khatam ho jaaye ($T_{aw}=T_{hot}$, koi protection nahi).

Yeh definition sirf ek linear interpolation flipped inside-out hai. Isse wall temperature ke liye solve karo:

$\boxed{T_{aw}=T_{hot}-\eta (T_{hot}-T_{cool})=(1-\eta )T_{hot}+\eta T_{cool}}$

Toh $T_{aw}$ ek weighted average hai: weight $\eta$ coolant par, weight $(1-\eta )$ hot gas par. Yeh bilkul waise hi hai jaise do streams ka "blend" hona chahiye — isse confirm hota hai ki definition sahi hai.

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HOW does $\eta$ decay along the wall? (Derivation from scratch)

Film ek slot par inject hoti hai aur phir downstream flow karte waqt hot gas ko entrain karti hai, dhire dhire garam aur patli hoti jaati hai. Chalo decay derive karte hain.

Step 1 — Film ki "coolness budget" define karo. Unit wall area per, coolant hot gas ke relative ek enthalpy deficit carry karta hai. Maano ${\dot{m}}_c$ coolant mass flow per unit width hai, $c_p$ uski specific heat hai. Total "cold capacity" ${\dot{m}}_c{c}_p(T_{hot}-T_{cool})$ ke proportional hai.

Why this step? Effectiveness depend karti hai iss baat par ki tune kitna thanda "material" inject kiya aur hot gas usse kitni jaldi consume kar rahi hai.

Step 2 — Entrainment model karo. Jaise film distance $x$ travel karti hai, turbulent mixing hot mass ko local temperature difference ke proportional rate par entrain karta hai (Newton-style). Film ki coolness ka differential loss hai:

$\frac{d(\Delta T)}{dx}=-\frac{h_g}{{\dot{m}}_c{c}_p}\Delta T,\Delta T\equiv T_{hot}-T_{aw}$

Why this step? Hot gas film ko convective coefficient $h_g$ ke through garam karta hai; zyada film heat-capacity (${\dot{m}}_c{c}_p$) heating ko slow karta hai. Yeh ek first-order linear ODE hai — "decay" problems ka workhorse.

Step 3 — Integrate karo.

\;\Rightarrow\; \Delta T = \Delta T_0\,e^{-\dfrac{h_g\,x}{\dot m_c c_p}}$$ Injection par $x=0$: film pure coolant hai, toh $\Delta T_0 = T_{hot}-T_{cool}$, jisse $\eta_0=1$ milta hai. **Step 4 — $\Delta T_0$ se divide karo.** Kyunki $\eta = \dfrac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}} = \dfrac{\Delta T}{\Delta T_0}$: > [!formula] Exponential film effectiveness decay > $$\boxed{\eta(x) = \exp\!\left(-\frac{h_g\,x}{\dot m_c\,c_p}\right)}$$ > Effectiveness injection slot par 1 se start hoti hai aur distance ke saath exponentially decay karti hai. Zyada coolant flow $\dot m_c$ → slower decay → longer protection. Group $\dfrac{h_g x}{\dot m_c c_p}$ dimensionless hai (ek Stanton-number-times-length parameter) — ek accha sanity check. --- ## WHAT is coverage fraction? > [!definition] Coverage fraction $f$ > Wall area (ya chamber length) ka woh fraction jo film se adequately protected rehta hai, yaani jahan effectiveness ek design threshold $\eta_{min}$ se zyada ho: > $$f \equiv \frac{x_{protected}}{L}, \qquad x_{protected} = \text{length where } \eta(x)\ge \eta_{min}$$ **HOW to compute it** decay law se. $\eta(x_{protected})=\eta_{min}$ set karo: $$\exp\!\left(-\frac{h_g x_{protected}}{\dot m_c c_p}\right)=\eta_{min} \;\Rightarrow\; x_{protected} = \frac{\dot m_c c_p}{h_g}\ln\!\frac{1}{\eta_{min}}$$ > [!formula] Coverage fraction > $$\boxed{f = \frac{\dot m_c\,c_p}{h_g\,L}\,\ln\!\frac{1}{\eta_{min}}}$$ *Log kyun hai?* Kyunki effectiveness exponentially decay karti hai, protected length double karne ke liye double coolant **nahi** chahiye — sirf $\ln$ increase chahiye. Diminishing returns built-in hain. ![[3.3.29-Film-cooling-—-effectiveness,-coverage-fraction.png]] --- ## Worked examples > [!example] Example 1 — Effectiveness se wall temperature > Diya hai $T_{hot}=3400$ K, $T_{cool}=900$ K, aur ek station par $\eta=0.4$. $T_{aw}$ nikalo. > $$T_{aw}=(1-\eta)T_{hot}+\eta T_{cool}=(0.6)(3400)+(0.4)(900)=2040+360=2400\text{ K}$$ > **Why this step?** Humne inverted definition use ki — $T_{aw}$ $\eta$-weighted blend hai. Modest $\eta=0.4$ bhi wall se 1000 K shave kar deta hai. Yahi film cooling ka point hai. > [!example] Example 2 — Downstream station par effectiveness > $h_g=1500\ \text{W/m}^2\text{K}$, $\dot m_c=0.05\ \text{kg/(m·s)}$ (per unit width), $c_p=2000\ \text{J/kgK}$, station $x=0.30$ m. > Exponent $= \dfrac{h_g x}{\dot m_c c_p}=\dfrac{1500\times0.30}{0.05\times2000}=\dfrac{450}{100}=4.5$. > $$\eta = e^{-4.5}=0.011$$ > **Why this step?** Dimensionless group 4.5 tak blow up kar gaya, toh film essentially khatam ho gayi — real design mein yahan se pehle ek aur injection slot add karna padega. > [!example] Example 3 — Coverage fraction > Ex.2 jaisa hi $h_g,\dot m_c,c_p$, chamber length $L=0.50$ m, require $\eta_{min}=0.2$. > $$x_{prot}=\frac{\dot m_c c_p}{h_g}\ln\frac1{0.2}=\frac{100}{1500}\times\ln 5 = 0.0667\times1.609 = 0.1073\text{ m}$$ > $$f=\frac{0.1073}{0.50}=0.215 \;\;(\approx 21\%)$$ > **Why this step?** Ek slot sirf ~21% wall ko $\eta\ge0.2$ par protect karta hai. Poore chamber ko cover karne ke liye ~5 slots distribute karne padenge — *engineers isi tarah slots count karte hain*. --- ## Common mistakes > [!mistake] "Zyada $\eta$ matlab garam wall." > **Kyun sahi lagta hai:** $\eta$ ek temperature-jaisa quantity lagta hai, aur log "high number = high heat" associate karte hain. > **Fix:** $\eta$ *protection* measure karta hai, temperature nahi. $\eta=1$ → wall coolant temperature par (sabse thanda possible). Definition dobara padho: yeh hot se cool ki taraf jaane ka *fraction* hai. > [!mistake] Effectiveness ko *adiabatic* wall temperature se reference karna bhool jaana. > **Kyun sahi lagta hai:** Experiments mein tum ek real (heat-conducting) wall measure karte ho, toh us measured $T_w$ ko plug in karna tempting lagta hai. > **Fix:** $\eta$ $T_{aw}$ ke saath define hota hai — woh temperature jo wall *tabhi* reach karti jab **wall mein zero conduction** hoti. Yeh wall ki apni conduction/regenerative cooling se *gas-side film* physics ko alag karta hai. > [!mistake] Coverage ko coolant flow mein linear treat karna. > **Kyun sahi lagta hai:** "Double coolant, double protected length" sahi lagta hai. > **Fix:** Coverage $\ln(1/\eta_{min})$ ke saath scale karta hai threshold ke through, aur protected *length* $\dot m_c$ ke saath linearly scale karti hai **fixed threshold ke liye** — phir bhi effectiveness exponentially decay karti hai, toh $\eta_{min}$ badhane par coolant super-linearly cost karta hai. Dono knobs ko mix mat karo. --- ## Active recall > [!recall]- Test yourself (jawab dene ke baad reveal karo) > - $\eta$ define karo aur uski do limiting values batao. > - Entrainment ODE se $\eta(x)=e^{-h_g x/(\dot m_c c_p)}$ derive karo. > - Coverage fraction mein logarithm kyun hai? > - Definition mein kaun sa temperature ($T_w$ ya $T_{aw}$) jaata hai, aur kyun? > [!recall]- Feynman: ek 12-saal ke bacche ko explain karo > Socho tum ek bonfire ke paas khade ho. Agar tum apne aur flames ke beech ek geela towel pakad lo, tumhe bahut thanda feel hoga — towel tumhari skin ki jagah heat leta hai. **Film cooling rocket ki andar ki wall ke saath thande gas ka ek "geela-towel" spray karta hai.** Lekin towel dhire dhire sukh jaata hai (thanda gas fire ke saath mix ho jaata hai), toh door downstream mein woh help karna band kar deta hai. **Effectiveness** kisi bhi jagah par towel kitna accha hai (1 = perfect, 0 = gone). **Coverage** woh hai ki towel sukhne se pehle kitni wall ko manage karta hai. Zyada wall cover karne ke liye ya tum zyada gas spray karo, ya aage ek naya towel rakh do. > [!mnemonic] > **"E-CoLd"** — **E**ffectiveness = wall kitni **Co**ol rehti hai, **L**ike an exponential decay karti hai (**e** to the minus something). Aur coverage = protection ka **L**og → **"cover needs a log."** --- ## Connections - [[Regenerative cooling]] — film cooling ka complement; real wall temp conduction ke through set karta hai. - [[Convective heat transfer coefficient]] — woh $h_g$ jo film decay drive karta hai (Bartz equation). - [[Adiabatic wall temperature & recovery factor]] — yahan use hone wala $T_{hot}$ define karta hai. - [[Stanton number]] — $h_g/(\dot m c_p)$ directly exponent mein appear karta hai. - [[Combustion chamber thermal design]] — jahan coverage $f$ se slot count choose ki jaati hai. - [[Boundary layer & entrainment]] — mixing model ka physical basis. #flashcards/physics Film cooling effectiveness $\eta$ ki definition ::: $\eta=(T_{hot}-T_{aw})/(T_{hot}-T_{cool})$; 1 = wall coolant temp par, 0 = wall hot-gas temp par. $\eta$ ke terms mein wall temperature ::: $T_{aw}=(1-\eta)T_{hot}+\eta T_{cool}$ (hot aur coolant ka η-weighted blend). Distance ke saath effectiveness ka decay law ::: $\eta(x)=\exp(-h_g x/(\dot m_c c_p))$, slot par 1 se start hota hai. Exponential decay ki physical origin ::: First-order entrainment ODE: film $h_g\,\Delta T$ rate par coolness khoti hai, heat capacity $\dot m_c c_p$ se damped. Coverage fraction formula ::: $f=(\dot m_c c_p)/(h_g L)\cdot\ln(1/\eta_{min})$. Coverage mein logarithm kyun hai ::: Kyunki effectiveness exponentially decay karti hai, protected length sirf required threshold ke $\ln$ ke roop mein badhti hai — diminishing returns. $\eta$ definition mein kaun sa temperature jaata hai ::: Adiabatic wall temperature $T_{aw}$ (zero wall conduction), gas-side film physics ko isolate karta hai. $\dot m_c$ badhane ka effect ::: Exponential decay slow karta hai → film zyada der tak effective rehti hai → zyada coverage. ## 🖼️ Concept Map ```mermaid flowchart TD HOT[Hot core gas ~3500 K] WALL[Chamber wall would melt] FILM[Film cooling: inject cool gas layer] ETA[Effectiveness eta 0 to 1] TAW[Adiabatic wall temp T_aw] ENT[Entrainment of hot gas] ODE[First-order decay ODE] DECAY[Exponential decay of eta along x] COV[Coverage fraction f] HOT -->|would melt| WALL FILM -->|protects| WALL FILM -->|quantified by| ETA ETA -->|blends into| TAW TAW -->|weighted avg of hot and cool| HOT ENT -->|heats film via h_g| ODE ODE -->|integrated gives| DECAY DECAY -->|reduces| ETA ENT -->|erodes film so limits| COV COV -->|fraction of wall protected| WALL ```