Foundations — Ablative cooling — charring, blowing

This page assumes you know nothing. We will not use a single letter from the parent note until it is defined in plain words and pinned to a picture. Read top to bottom; each block earns the next.

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0. The stage: hot gas, a wall, and a flux

Before any symbol, picture the situation.

On the left is very hot gas (thousands of kelvin). On the right is the cold structure we must protect. In between sits our sacrificial material. Heat flows left to right, from hot to cold — that is the only direction heat ever moves on its own.

The little dot on top, $\dot{q}$, is worth its own line.

Why do we need a rate and not a total? Because a heat shield can absorb a huge total, yet fail if the heat arrives too fast. Rate is the fair way to compare arrival against removal.

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1. Temperature $T$ and the two subscripts

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2. The convection law: ${\dot{q}}_{conv}=h(T_{gas}-T_{wall})$

Before we can read the parent's first equation we must own the letter $h$ that sits inside it — so we define $h$ first, then assemble the law.

Now every symbol in the law is defined, so we may write it.

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3. Mass, density, and a moving front

The wall pays for cooling in mass. We need three linked symbols.

Now the front. As the surface burns inward, an imaginary line — the pyrolysis front — creeps deeper into the block.

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4. Energy per kilogram: $Q_p$, $c_p$, $Q^{\ast }$

Ablation is a trade: kilograms out buy joules absorbed. So we need "joules per kilogram" symbols.

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5. Enthalpy $H$ — temperature's grown-up cousin

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6. The blowing parameter $B^{\prime }$ and the logarithm

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7. Sensible heating — where the solid's $c_p$ lives

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Prerequisite map

Every arrow says "you need the left box to make sense of the right box." Follow any path and you reach the parent topic: Ablative cooling — charring, blowing (index 3.3.30).

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Where these lead next

• The convection law and $h$ deepen in Convective Heat Transfer (Stanton number) and Boundary Layer Theory.
• $Q_p$ and pyrolysis chemistry are unpacked in Heat of Reaction and Pyrolysis.
• The rival cooling schemes: Regenerative Cooling and Radiative Cooling.
• The extreme heating environment: Re-entry Aerothermodynamics.

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Equipment checklist

Recall Self-test — can you say each in one plain sentence?

• What does a dot over a symbol mean? ::: a rate — that quantity per second.
• What does the double-prime ${}^{\prime \prime }$ add to a symbol? ::: "per unit area" (per square metre).
• $\dot{q}$ in words and units? ::: heat crossing one m² each second; W/m².
• Why does heat flow at all in this problem? ::: because $T_{gas}>T_{wall}$; a temperature gap is the push.
• What is $h$ and what happens to it during blowing? ::: the conductance from gas to wall; blowing shrinks it.
• How do you turn ${\dot{m}}^{\prime \prime }$ into a recession speed $\dot{s}$? ::: divide by density, $\dot{s}={\dot{m}}^{\prime \prime }/{\rho }_v$.
• What does ${\dot{q}}_{net}$ include? ::: convective heat in, minus radiation and blowing losses — the heat the wall must actually handle.
• What does ${\dot{m}}_{total}^{\prime \prime }$ comprise? ::: all mass lost per area per second — pyrolysis gas plus char removed by reaction/erosion.
• Difference between $Q_p$ and $Q^{\ast }$? ::: $Q_p$ is only the chemistry sink; $Q^{\ast }$ sums all cooling channels per kg lost.
• Whose $c_p$ appears in $B^{\prime }$ versus in the sensible term? ::: in $B^{\prime }$ the injected gas's $c_p$; in the sensible term the solid material's $c_p$.
• Why use enthalpy $H$ not temperature $T$ at high heat? ::: $H$ counts bond energy from dissociation that a thermometer misses.
• What does $B^{\prime }$ compare? ::: injected gas rate vs the gas's heat-delivery ability; it is dimensionless.
• Why does $\ln (1+B^{\prime })/B^{\prime }$ mean "diminishing returns"? ::: the logarithm flattens, so extra blowing buys ever less cooling.