Question bank — Film cooling — effectiveness, coverage fraction

False. $\eta$ measures protection, not heat: $\eta =1$ puts the wall at the coolant temperature (coldest possible), $\eta =0$ leaves it at the full hot-gas temperature.

True. There the film is pure unmixed coolant, so $T_{aw}=T_{cool}$ and $\eta =\exp (0)=1$ — the exponential decay hasn't started yet.

False. $\eta =\frac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}}$ can at best reach 1 when $T_{aw}=T_{cool}$; the wall cannot get colder than the coolant that is warming it, so the ratio is capped at 1.

True — for a fixed threshold. Since $x_{prot}=\frac{{\dot{m}}_c{c}_p}{h_g}\ln \frac{1}{{\eta }_{min}}$, length is linear in ${\dot{m}}_c$; the nonlinearity lives in the $\ln$ that only bites when you change ${\eta }_{min}$.

True as written ($f=\frac{{\dot{m}}_c{c}_p}{h_{g}L}\ln \frac{1}{{\eta }_{min}}$ is linear in ${\dot{m}}_c$) — but only until $f$ hits 1; you can't protect more than the whole wall, so past $f=1$ the extra coolant is wasted.

False. It uses $T_{aw}$, the adiabatic wall temperature (zero conduction into the wall). This isolates the gas-side film physics from the wall's own conduction and regenerative cooling.

False. Larger $h_g$ means the hot gas heats the film faster, so $\eta$ decays faster and $x_{prot}$ shrinks — $h_g$ sits in the denominator of the coverage length.

False. A higher ${\eta }_{min}$ is a stricter requirement, so fewer stations qualify; $\ln \frac{1}{{\eta }_{min}}$ shrinks as ${\eta }_{min}\to 1$, cutting coverage.

Correct here, but check why: $T_{hot}-T_{aw}=\eta (T_{hot}-T_{cool})$ is linear in $\eta$, so at $\eta =0.8$ the drop is exactly $0.8\times 2500=2000$ K. The trap is confusing this linear $T_{aw}$ relation with the exponential $\eta (x)$ relation — different variables.

Half-error. Length is linear in ${\dot{m}}_c$ so this works for a fixed ${\eta }_{min}$, but a real slot has a maximum injection rate and momentum-ratio limits (too much coolant lifts the film off the wall). Engineers add another slot instead — re-establishing the film.

Error: $e^0=1$, not 0. At $x=0$ the exponent is zero and $\eta =1$ (perfect protection); the reader confused $e^{-0}$ with $e^{-\infty }$.

Error: it is dimensionless — a Stanton-number-times-length parameter. $h_{g}x/({\dot{m}}_c{c}_p)$ has units $\frac{({\text{W/m}}^2\text{K})(\text{m})}{(\text{kg/m⋅s})(\text{J/kg⋅K})}$, and W = J/s makes everything cancel.

Error: $\eta$ is dimensionless (0–1) but $T_{aw}$ is a real temperature in kelvin, always lying between $T_{cool}$ and $T_{hot}$ — a blend, not a fraction.

Error: the formula wants $T_{aw}$, not the conducting-wall $T_w$. A real wall loses heat by conduction/regen cooling, so $T_w\ne T_{aw}$; using $T_w$ blends two separate physics and corrupts $\eta$.

Error: $f=x_{prot}/L$ is a fraction of the actual wall length; the formula can compute a value above 1, which simply means one slot over-protects — physically $f$ saturates at 1 and the surplus coolant is redundant.

Because $\eta$ decays exponentially with distance; inverting an exponential to find "where does $\eta$ hit ${\eta }_{min}$" produces a $\ln$. Diminishing returns are built in — protecting deeper costs disproportionately.

To separate concerns: $T_{aw}$ captures only the gas-side film behaviour with zero wall conduction, so film performance can be characterized independently of whatever wall material or regenerative loop is behind it.

${\dot{m}}_c{c}_p$ is the film's heat capacity — its "cold budget." A bigger budget means the same entrained hot gas raises the film temperature by less per metre, so $\eta$ falls off more slowly.

Turbulent mixing at the film–core interface entrains hot gas into the film (entrainment); each bit of hot mass added both heats the film and dilutes its coolness, driving $\eta$ toward 0.

Because the wall sees a mixture of hot core and cool film, and $\eta$ is precisely the mixing fraction — the amount of "coolness" surviving at that station sets the weights.

$h_g$ is the gas-side heat transfer coefficient (from the Bartz-type correlation); it sets how fast the hot gas delivers heat into the film per degree of temperature difference — the driver of the decay ODE.

${\eta }_{min}$ is set by the material limit: below it the wall exceeds its safe temperature. Lowering ${\eta }_{min}$ fictitiously "extends" coverage on paper while the actual wall melts — a design lie, not a fix.

$\eta =e^{-\infty }=0$: the film is completely spent, $T_{aw}\to T_{hot}$, and the wall sees the full hot gas — the reason multiple slots exist.

The exponent $\frac{h_{g}x}{{\dot{m}}_c{c}_p}\to \infty$ for any $x>0$, so $\eta \to 0$ instantly — with no coolant there is no film and no protection anywhere.

The exponent $\to 0$, so $\eta \to 1$ everywhere — the wall stays at $T_{cool}$ along the whole chamber. Physically unreachable (finite propellant), but it shows the correct limit.

The definition $\frac{T_{hot}-T_{aw}}{T_{hot}-T_{cool}}$ divides by zero — $\eta$ is undefined because there is no temperature gap to protect against; the "coolant" offers no benefit.

$\ln \frac{1}{1}=0$, so $f=0$: perfect protection is only met exactly at the slot ($x=0$), a set of measure zero — you cannot demand $\eta =1$ over any finite length from one slot.

Exactly 1 — the film just barely meets the threshold at the far end, so the whole chamber length qualifies as protected and coverage is complete.

$T_{aw}=(1-1)T_{hot}+1\cdot T_{cool}=T_{cool}$ — the wall is exactly at coolant temperature, the best any station can do.