Foundations — Burn rate r = a·P^n — Vieille's law

This page assumes you know nothing. Before you can read $r=a{P}^n$ you need to understand what each letter is, what picture it points to, and why the topic can't do without it. We build them in dependency order — every symbol earned before it is used.

0. The scene: what are we even looking at?

Picture a tube (the rocket motor) packed with a rubbery solid fuel called the grain. There's a hole down the middle. We light the surface of that hole. The flame doesn't blow the whole block up — it slowly chews the solid away, surface receding outward, while hot gas builds up inside and rushes out the back.

Everything in Vieille's law is a number attached to some part of this picture. Let's name each part.

1. Pressure $P$ — the push of the gas

Why the topic needs it: $P$ is the cause in Vieille's law. Change nothing but $P$ and the burn rate changes. It is the one knob nature turns.

The chamber pressure links directly to how gas escapes — see Chamber Pressure and Nozzle Throat.

2. Burn rate $r$ — the speed of the eating

Why the topic needs it: $r$ is the effect — the quantity the whole law predicts. Multiply $r$ by the burning surface area and you get how much gas is being made per second, which drives thrust (Thrust Equation and Specific Impulse).

3. Exponents and powers — reading $P^n$

Before we meet $n$, make sure the notation $P^n$ is not scary.

Why the topic needs it: real burn rate does not double when pressure doubles. It grows more gently. A fractional power like $P^{0.35}$ is exactly the shape that says "responds, but weakly." No other simple function captures "grows, but dampened" so cleanly.

4. Pressure exponent $n$ — how strongly $r$ listens to $P$

Why the topic needs it: $n$ is the single most important number for safety. It decides whether a random pressure bump self-corrects or runs away into an explosion (that stability story is built in the parent note). The whole reason engineers care about $n$ is: keep it below 1.

5. Burn-rate coefficient $a$ — the chemistry-and-temperature dial

Why the topic needs it: $a$ sets the scale of the burn (how fast in absolute terms), while $n$ sets the shape (how it responds to pressure). You need both to get a number out.

6. Logarithm $\log$ — the tool that straightens a power law

Vieille's law is a curve. Curves are hard to read constants off. The logarithm is the tool that turns the curve into a straight line.

Why the topic needs it: this is how the constants are measured. Burn a strand at several pressures, plot $\log r$ against $\log P$, draw the best straight line — slope hands you $n$, intercept hands you $\log a$. No other tool reads off a power-law's exponent so directly.

7. The heat-balance cast (why the law even has this form)

The parent note derives $r=a{P}^n$ from a heat balance. Here are the supporting characters, each a physical thing in the picture.

Why the topic needs these: they explain why pressure controls burn rate. Higher $P$ shrinks $\delta$ (flame moves closer), the temperature gradient $\frac{T_f-T_s}{\delta }$ gets steeper, heat conducts back faster, and the surface recedes faster. When $\delta$ shrinks as a power of pressure, $r$ rises as a power of pressure — and that power is $n$.

The prerequisite map

Related deeper reading: Solid Rocket Motor Grain Geometry, Combustion Instability, and the parent Hinglish version.

A tiny sanity calculation with all symbols in place

Equipment checklist

Cover the right side and answer before revealing.