Film cooling — effectiveness, coverage fraction

---

WHAT is film cooling effectiveness?

WHY this definition? We want a number that is:

• $\eta = 1$ when the wall is fully protected ($T_{aw}=T_{cool}$, perfect film),
• $\eta = 0$ when the film is gone ($T_{aw}=T_{hot}$, no protection).

The definition is just a linear interpolation flipped inside-out. Solve it for the wall temperature:

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

So $T_{aw}$ is a weighted average: weight $\eta$ on the coolant, weight $(1-\eta)$ on the hot gas. That is exactly what a "blend" of two streams should look like — reassuring the definition is sane.

---

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

The film is injected at one slot and then entrains hot gas as it flows downstream, getting warmer and thinner. Let's derive the decay.

Step 1 — Define the film's "coolness budget." Per unit wall area, the coolant carries an enthalpy deficit relative to the hot gas. Let $\dot m_c$ be coolant mass flow per unit width, $c_p$ its specific heat. The total "cold capacity" is proportional to $\dot m_c c_p (T_{hot}-T_{cool})$.

Why this step? Effectiveness must be governed by how much cold "stuff" you injected versus how fast hot gas eats it.

Step 2 — Model entrainment. As the film travels distance $x$, turbulent mixing entrains hot mass at a rate proportional to the local temperature difference (Newton-style). The differential loss of the film's coolness is:

$\frac{d(\Delta T)}{dx} = -\frac{h_g}{\dot m_c c_p}\,\Delta T, \qquad \Delta T \equiv T_{hot}-T_{aw}$

Why this step? Hot gas heats the film through convective coefficient $h_g$; more film heat-capacity ($\dot m_c c_p$) slows the heating. This is a first-order linear ODE — the workhorse of "decay" problems.

Step 3 — Integrate.

\;\Rightarrow\; \Delta T = \Delta T_0\,e^{-\dfrac{h_g\,x}{\dot m_c c_p}}$$ At injection $x=0$: film is pure coolant, so $\Delta T_0 = T_{hot}-T_{cool}$, giving $\eta_0=1$. **Step 4 — Divide by $\Delta T_0$.** Since $\eta = \dfrac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}} = \dfrac{\Delta T}{\Delta T_0}$: > [!formula] Exponential film effectiveness decay > $$\boxed{\eta(x) = \exp\!\left(-\frac{h_g\,x}{\dot m_c\,c_p}\right)}$$ > Effectiveness starts at 1 at the injection slot and decays exponentially with distance. Larger coolant flow $\dot m_c$ → slower decay → longer protection. The group $\dfrac{h_g x}{\dot m_c c_p}$ is dimensionless (a Stanton-number-times-length parameter) — a good sanity check. --- ## WHAT is coverage fraction? > [!definition] Coverage fraction $f$ > The fraction of the wall area (or chamber length) that stays adequately protected by the film, i.e. where effectiveness exceeds a design threshold $\eta_{min}$: > $$f \equiv \frac{x_{protected}}{L}, \qquad x_{protected} = \text{length where } \eta(x)\ge \eta_{min}$$ **HOW to compute it** from the decay law. Set $\eta(x_{protected})=\eta_{min}$: $$\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}}}$$ *Why the log?* Because effectiveness decays exponentially, doubling the protected length does **not** need double the coolant — only a $\ln$ increase. Diminishing returns are built in. ![[3.3.29-Film-cooling-—-effectiveness,-coverage-fraction.png]] --- ## Worked examples > [!example] Example 1 — Wall temperature from effectiveness > Given $T_{hot}=3400$ K, $T_{cool}=900$ K, and at a station $\eta=0.4$. Find $T_{aw}$. > $$T_{aw}=(1-\eta)T_{hot}+\eta T_{cool}=(0.6)(3400)+(0.4)(900)=2040+360=2400\text{ K}$$ > **Why this step?** We used the inverted definition — $T_{aw}$ is the $\eta$-weighted blend. Even modest $\eta=0.4$ shaved 1000 K off the wall. That's the point of film cooling. > [!example] Example 2 — Effectiveness at a downstream station > $h_g=1500\ \text{W/m}^2\text{K}$, $\dot m_c=0.05\ \text{kg/(m·s)}$ (per unit width), $c_p=2000\ \text{J/kgK}$, station $x=0.30$ m. > Exponent $= \dfrac{h_g x}{\dot m_c c_p}=\dfrac{1500\times0.30}{0.05\times2000}=\dfrac{450}{100}=4.5$. > $$\eta = e^{-4.5}=0.011$$ > **Why this step?** The dimensionless group blew up to 4.5, so the film is essentially spent — a real design would add another injection slot before here. > [!example] Example 3 — Coverage fraction > Same $h_g,\dot m_c,c_p$ as Ex.2, chamber length $L=0.50$ m, require $\eta_{min}=0.2$. > $$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?** One slot protects only ~21% of the wall at $\eta\ge0.2$. To cover the whole chamber you'd distribute ~5 slots — this is *how engineers count slots*. --- ## Common mistakes > [!mistake] "Higher $\eta$ means hotter wall." > **Why it feels right:** $\eta$ looks like a temperature-ish quantity, and people associate "high number = high heat." > **The fix:** $\eta$ measures *protection*, not temperature. $\eta=1$ → wall at coolant temperature (coldest possible). Re-read the definition: it's the *fraction of the way from hot toward cool*. > [!mistake] Forgetting effectiveness must be referenced to the *adiabatic* wall temperature. > **Why it feels right:** In experiments you measure a real (heat-conducting) wall, so it's tempting to plug in that measured $T_w$. > **The fix:** $\eta$ is defined with $T_{aw}$ — the temperature the wall *would* reach with **zero conduction into the wall**. It isolates the *gas-side film* physics from the wall's own conduction/regenerative cooling. > [!mistake] Treating coverage as linear in coolant flow. > **Why it feels right:** "Double the coolant, double the protected length" seems fair. > **The fix:** Coverage scales with $\ln(1/\eta_{min})$ only through the threshold, but protected *length* scales linearly with $\dot m_c$ **for a fixed threshold** — yet effectiveness itself decays exponentially, so pushing $\eta_{min}$ higher costs coolant super-linearly. Don't mix the two knobs. --- ## Active recall > [!recall]- Test yourself (reveal after answering) > - Define $\eta$ and state its two limiting values. > - Derive $\eta(x)=e^{-h_g x/(\dot m_c c_p)}$ from an entrainment ODE. > - Why does coverage fraction contain a logarithm? > - What temperature ($T_w$ or $T_{aw}$) belongs in the definition, and why? > [!recall]- Feynman: explain to a 12-year-old > Imagine standing near a bonfire. If you hold a wet towel between you and the flames, you feel much cooler — the towel takes the heat instead of your skin. **Film cooling sprays a "wet-towel" of cool gas along the rocket's inside wall.** But the towel slowly dries out (the cool gas mixes with the fire), so far downstream it stops helping. **Effectiveness** is how good the towel is at any spot (1 = perfect, 0 = gone). **Coverage** is how much of the wall the towel manages to cover before it dries out. To cover more wall, you either spray more gas, or put down a fresh towel further along. > [!mnemonic] > **"E-CoLd"** — **E**ffectiveness = how **Co**ol the wall stays, decays **L**ike an exponential (**e** to the minus something). And coverage = **L**og of protection → **"cover needs a log."** --- ## Connections - [[Regenerative cooling]] — complements film cooling; sets the real wall temp via conduction. - [[Convective heat transfer coefficient]] — the $h_g$ driving film decay (Bartz equation). - [[Adiabatic wall temperature & recovery factor]] — defines $T_{hot}$ used here. - [[Stanton number]] — $h_g/(\dot m c_p)$ appears directly in the exponent. - [[Combustion chamber thermal design]] — where slot count is chosen from coverage $f$. - [[Boundary layer & entrainment]] — physical basis for the mixing model. #flashcards/physics Film cooling effectiveness $\eta$ definition ::: $\eta=(T_{hot}-T_{aw})/(T_{hot}-T_{cool})$; 1 = wall at coolant temp, 0 = wall at hot-gas temp. Wall temperature in terms of $\eta$ ::: $T_{aw}=(1-\eta)T_{hot}+\eta T_{cool}$ (η-weighted blend of hot and coolant). Decay law for effectiveness with distance ::: $\eta(x)=\exp(-h_g x/(\dot m_c c_p))$, starting at 1 at the slot. Physical origin of the exponential decay ::: First-order entrainment ODE: film loses coolness at rate $\propto h_g\,\Delta T$, damped by heat capacity $\dot m_c c_p$. Coverage fraction formula ::: $f=(\dot m_c c_p)/(h_g L)\cdot\ln(1/\eta_{min})$. Why coverage has a logarithm ::: Because effectiveness decays exponentially, protected length grows only as $\ln$ of the required threshold — diminishing returns. Which temperature goes in the $\eta$ definition ::: The adiabatic wall temperature $T_{aw}$ (zero wall conduction), isolating gas-side film physics. Effect of increasing $\dot m_c$ ::: Slows the exponential decay → film stays effective longer → larger 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 ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, rocket chamber ke andar flame ka temperature $\sim3500$ K hota hai — koi bhi metal wall seedha is heat ko sehen nahi kar sakti, pighal jayegi. Iska smart solution hai **film cooling**: wall ke bilkul paas ek patli si thandi gas ki layer inject kar dete hain (aksar unburnt fuel). Yeh layer ek "geeli towel" ki tarah kaam karti hai — heat wall ke bajaye is film pe padti hai. **Effectiveness $\eta$** batata hai ki kisi point pe film kitni acchi protection de rahi hai: $\eta=1$ matlab wall bilkul coolant jitni thandi, $\eta=0$ matlab film khatam, wall ko full hot gas jhel rahi hai. > > Ab yeh film downstream jaate jaate hot gas ke saath mix hoti jaati hai (entrainment), toh dheere-dheere garam aur patli hoti jaati hai. Isi liye effectiveness distance ke saath **exponential** girta hai: $\eta(x)=e^{-h_g x/(\dot m_c c_p)}$. Yaad rakho — jitna zyada coolant flow $\dot m_c$, utni slow decay, utni lambi protection. Exponent ek dimensionless number hai (Stanton-type), toh units ka bhi cross-check ho jaata hai. > > **Coverage fraction $f$** ka matlab hai wall ka kitna hissa "kaafi protected" rehta hai, yaani jahan $\eta$ apne design threshold $\eta_{min}$ se upar hai. Formula nikalta hai $f=(\dot m_c c_p)/(h_g L)\cdot\ln(1/\eta_{min})$. Yahan $\ln$ isliye aata hai kyunki decay exponential hai — thoda aur protection chahiye toh coolant thoda hi zyada, par threshold bahut ऊँcha rakhoge toh coolant tezi se badhta hai. Practical baat: ek slot pura chamber cover nahi karta, isliye engineers thodi-thodi doori pe naye injection slots lagate hain — bilkul geeli towel ko refresh karne jaisa. > > Exam me steps yaad rakho: (1) definition se $T_{aw}$ nikalna, (2) exponent compute karna, (3) coverage me $\ln$ lagana. Ek common galti — $\eta$ ko temperature samajh lena; nahi, yeh protection ka fraction hai, aur hamesha **adiabatic** wall temperature $T_{aw}$ use hota hai, real conducting wall ka nahi. ![[audio/3.3.29-Film-cooling-—-effectiveness,-coverage-fraction.mp3]]