Foundations — Solid propellants — AP - HTPB - Al composition; burn rate dependence on pressure (Vieille's law)

This page assumes you know nothing. Before you can read the parent note (the Hinglish version is here), you need to earn every letter it throws at you. We build them one at a time, each on top of the last.

0. The one picture the whole topic lives in

Before any symbol, look at the burning block.

The solid propellant is a block. Its top face is on fire — but the flame floats a hair above the surface, not touching it. Heat from the flame reaches down, cooks the solid into gas, that gas burns, and the fire feeds itself. The surface moves down as it eats the solid. Everything below is a name for a piece of this picture.

1. Rate of change — the dot and the idea of "per second"

Picture: water leaving a tap. If 2 litres come out every second, the flow rate is 2 L/s. The dot is just shorthand for "flow rate of".

Why the topic needs it: a rocket makes hot gas continuously. We care about how much gas per second the burning surface produces, because that gas is what pushes the rocket. That quantity is $\dot{m}$.

2. Density $\rho$ — how much stuff is packed in

Picture: a brick versus a sponge of the same size. The brick has more mass squeezed into the same volume — higher $\rho$.

Why: to know the mass of gas made, we need to know how much mass was sitting in the solid before it burned. Denser propellant = more mass per chunk consumed.

3. Burning surface area $A_b$ — the size of the fire

Picture: a candle flame across a fat candle burns more wax per second than across a thin candle, simply because more surface is exposed. Same fire speed, bigger area, more gas.

Why: more burning surface makes more gas at once. Total gas depends on area, not just on how fast each spot burns.

4. Burn rate $r$ — the speed the surface retreats

This is the star of the whole topic.

Picture: mark the surface at time zero. One second later the flame has eaten $r$ millimetres deeper. That distance-per-second is $r$.

Why this letter and not a chemical rate? Chemists usually mean "moles reacting per second" by "rate". Here we want something geometric — how far the surface travelled — because that is what tells us how the grain shape changes over time. So $r$ is a velocity, measured in mm/s.

Why this multiplies: area × speed = volume swept per second (m² × m/s = m³/s). Multiply by density (kg/m³) and you get kg/s. The units force the formula to be right.

5. Pressure $P$ — how hard the gas pushes

Picture: a crowded room. The more people crammed in (denser, hotter gas), the harder they shove against the walls. That shove per unit wall is pressure.

Why: pressure is the cause in this whole story. Squeeze the gas harder and the flame gets pushed closer to the solid surface, heat arrives faster, and the surface burns faster. So $r$ will turn out to depend on $P$.

6. Powers and the exponent $n$ — "to the power of"

Picture: three curves of $r$ against $P$.

• $n=0$: flat line — burn rate ignores pressure.
• $n=1$: straight line through origin — double the pressure, exactly double the rate.
• $0<n<1$: a curve that rises but bends over — the higher $P$ gets, the less each extra bit of pressure helps. This gentle bending is what keeps a motor stable.

Why we need a fractional exponent: real propellants sit between "ignores pressure" and "proportional to pressure". Only a fractional power like $P^{0.35}$ can describe that middle behaviour. That is exactly why the law is written $r=a{P}^n$ and not $r=aP$.

7. The coefficient $a$ — the overall scale

Picture: $n$ is the shape of the curve; $a$ is how high the whole curve is stretched up the page. Two curves with the same $n$ but different $a$ are parallel-ish, one just higher than the other.

Why: the pressure shape alone gives you a ratio, not an actual mm/s number. $a$ supplies the real units and the real magnitude.

8. The Greek letters of the heat balance

The parent note derives Vieille's law from heat flow. Those steps use four more symbols. Meet them now so no line surprises you.

Picture: three thermometers stacked from deep-cold below to blazing-hot flame above. The gap $T_f-T_s$ is the temperature drop the heat crosses on its way down.

Why these four: the whole derivation is a heat budget. Heat produced at the flame ($T_f$) crosses the gap $\delta$ (conductivity $\lambda$) and must supply enough energy ($c$, over the temperature rise $T_s-T_0$) to cook the surface. Pressure squeezes $\delta$ smaller, heat arrives faster, surface burns faster — and that is why $r$ grows with $P$. The tool used to describe heat crossing the gap is Heat Conduction (Fourier's Law).

9. Nozzle and outflow symbols

The parent uses two more when it discusses the motor staying at a steady pressure.

Picture: a balloon with a pinhole. Gas is made inside by the burning (${\dot{m}}_{gen}$); gas leaves through the pinhole $A_t$. When make-rate equals leave-rate, the pressure holds steady.

Why: stability of the whole motor is a race between "gas made" and "gas escaping". $A_t$ and $c^{\ast }$ describe the escaping side; this is where Characteristic Velocity c-star and the balance with Specific Impulse enter.

10. Prerequisite map

Read it upward: the basic quantities on top feed the gas-rate and the heat balance; those feed Vieille's law; Vieille's law plus the nozzle side gives the equilibrium chamber pressure that decides if the motor is stable.

Equipment checklist

Cover the right side and check you can say each one before opening the parent note.