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

This bank refines your grasp of the parent topic and its dependence on Vieille's Law. Cross-links: Specific Impulse, Characteristic Velocity c-star, Combustion Thermodynamics, Heat Conduction (Fourier's Law), Ammonium Perchlorate Decomposition, Liquid Propellants.

Symbol table — read this first

Before the traps, here is every symbol used on this page, in plain words. Nothing below uses a letter you haven't met here.

Definition The cast of symbols

$r$ = burn rate — the linear speed (mm/s or m/s) at which the burning surface eats inward. Picture the face of a candle retreating.
$P$ = chamber pressure — how hard the hot gas presses on everything inside the steel can (units MPa).
$a$ = pre-exponential coefficient in $r = a P^{n}$ — a "gain dial" that packages all the thermal/chemical constants; it shifts with initial grain temperature $T_{0}$ .
$n$ = pressure exponent — a pure number (~0.2–0.5) saying how sensitively burn rate responds to pressure.
$A_{b}$ = burning surface area (m²) — total area of grain currently on fire.
$A_{t}$ = nozzle throat area (m²) — the narrowest cross-section the gas squeezes through on the way out.
$ρ_{p}$ = propellant density (kg/m³) — mass packed per unit volume of solid.
$c^{*}$ ("c-star") = characteristic velocity (m/s) — a chamber-efficiency number that sets how much gas the throat can vent per unit pressure. See Characteristic Velocity c-star.
$δ$ = flame stand-off distance (mm) — the tiny gap between the solid surface and where the flame actually sits.
$q^{''}$ = heat flux (W/m²) — power per unit area conducted from flame back to surface. (Double-prime is the engineering tag for "per area".)
$λ$ = thermal conductivity of the gas gap (W/m·K) — how readily heat travels across $δ$ . See Heat Conduction (Fourier's Law).
$T_{f}$ = flame temperature (K) — hottest point, up in the gas.
$T_{s}$ = surface temperature (K) — the (cooler) temperature the solid must reach to gasify.
$T_{0}$ = initial grain temperature (K) — how warm the propellant was before ignition.
$c$ = specific heat of the solid (J/kg·K); $\overset{m}{˙}$ = mass flow rate (kg/s).

The one picture: flame stand-off and pressure

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

Why $P_{e q} = (ρ_{p} a c^{*} A_{b} / A_{t})^{1/ (1 - n)}$ — derived in one breath

True or false — justify

A composite propellant needs air from outside to burn.

False. The whole point of the oxidiser (AP) is that it carries its own oxygen inside the solid; a rocket in vacuum has no air to draw on.

In Vieille's law

r = a P^{n}

, the coefficient

a

is a universal constant like

g

c

False.

a

bundles thermal and chemical constants and depends on the initial grain temperature $T_{0}$ — a cold-soaked motor and a sun-warmed one have different

a

, hence different burn rates at the same pressure.

Raising the pressure exponent

n

toward 1 makes a more powerful, better motor.

False. Near

n = 1

the equilibrium pressure

P_{e q} \propto (A_{b} / A_{t})^{1/ (1 - n)}

becomes hypersensitive; a tiny pressure rise runs away and the motor can explode. Designers deliberately keep

n < 0.5

Aluminium is added mainly to make the exhaust smoke more visible.

False. Al is a high-energy fuel:

A l \to A l_{2} O_{3}

is strongly exothermic, raising flame temperature (~3500 K) and specific impulse. The white

A l_{2} O_{3}

smoke is a side effect, not the purpose.

You can throttle a solid motor mid-flight by adjusting a valve, like a liquid engine.

False. Fuel and oxidiser are pre-mixed in the solid grain; once lit it burns until spent. Thrust is shaped only by grain geometry, not real-time control — unlike a liquid engine.

Higher chamber pressure pushes the flame farther from the burning surface.

False. Higher

P

means faster chemistry and denser gas, so the flame stand-off

δ

shrinks (

δ \propto P^{- k}

). The flame sits closer, dumps more heat back, and burn rate rises.

Doubling the burning surface area

A_{b}

doubles the burn rate

r

False.

r = a P^{n}

depends on pressure, not directly on area. Doubling

A_{b}

doubles gas generation (

\overset{m}{˙} = ρ_{p} A_{b} r

), which raises equilibrium pressure, which then raises

r

— an indirect effect, not a doubling.

HTPB is only glue; the actual fuel is the aluminium.

False. HTPB is both binder and fuel — its carbon and hydrogen burn. Aluminium is an additional high-energy fuel, not the sole one.

Spot the error

"Burn rate has units of kg/s because it measures how much propellant is consumed."

Error:

r

is a linear regression speed (mm/s or m/s) — how fast the surface moves inward. Mass consumption is

\overset{m}{˙} = ρ_{p} A_{b} r

in kg/s; don't confuse the two.

"Since

q^{''} = λ (T_{f} - T_{s}) / δ

and

δ

is fixed, burn rate is independent of pressure."

Error: This is Fourier's law across the gas gap — heat flux

q^{''}

equals conductivity

λ

times the temperature drop

(T_{f} - T_{s})

from flame to surface, divided by the gap

δ

. But

δ

is not fixed: it shrinks with pressure (

δ \propto P^{- k}

), so

q^{''}

rises with

P

— exactly what produces

r \propto P^{n}

"To find

n

from two data points, subtract the burn rates and divide by the pressure difference."

Error: The law is a power relation, not linear. Take the ratio (which cancels

a

) and use logs:

n = ln (r_{2} / r_{1}) / ln (P_{2} / P_{1})

. A slope on a linear plot gives the wrong number.

P_{e q} = (ρ_{p} a c^{*} A_{b} / A_{t})^{1/ (1 - n)}

works fine for

n = 1

; you just get exponent 1."

Error: For

n = 1

the exponent

1/ (1 - n)

divides by zero — undefined. Physically the generation (

\propto P^{n}

) and outflow (

\propto P^{1}

) both scale linearly with

P

, so the lines never cross to give a restoring balance; pressure is indeterminate/unstable.

"In the schematic reaction

10 N H_{4} C l O_{4} + 8 A l \to \dots

, AP acts as the fuel and Al as the oxidiser."

Error: Reversed. AP supplies the oxygen (oxidiser); Al grabs that oxygen to form

A l_{2} O_{3}

and is the fuel.

c^{*}

(characteristic velocity) is the speed of the exhaust gas leaving the nozzle."

Error:

c^{*}

(Characteristic Velocity c-star) measures combustion/chamber efficiency and sets the mass-flow–pressure relation

\overset{m}{˙}_{o u t} = P A_{t} / c^{*}

; it is not the exhaust exit velocity (that governs Specific Impulse).

Why questions

Why does the burn rate depend on pressure at all, mechanistically?

Higher

P

speeds gas-phase chemistry and increases gas density, shrinking the flame stand-off distance

δ

; more heat flux

q^{''}

(Fourier conduction) reaches the surface, so it regresses faster.

Why is the equilibrium pressure self-correcting only when

n < 1

Generation

\propto P^{n}

grows slower than outflow

\propto P^{1}

when

n < 1

; if pressure drifts up, outflow outpaces generation and pulls it back down. If

n \geq 1

generation keeps up or overtakes → no restoring force → runaway.

Why do engineers plot

ln r

ln P

instead of

r

P

Taking logs of

r = a P^{n}

gives

ln r = ln a + n ln P

— a straight line whose slope is $n$ and intercept is $ln a$ . Linearising makes both constants readable directly.

Why must the propellant carry chlorine and nitrogen, not just carbon fuel?

They come packaged in the oxidiser AP (

N H_{4} C l O_{4}

); on decomposition they leave as gaseous

H C l

N_{2}

H_{2} O

— the high-velocity, low-molar-mass exhaust that produces thrust (Ammonium Perchlorate Decomposition).

Why does adding aluminium raise specific impulse despite

A l_{2} O_{3}

being a heavy solid?

The reaction's huge

Δ H

raises flame temperature so much that the extra thermal energy more than compensates; hotter chamber gas gives higher exit velocity. (Two-phase

A l_{2} O_{3}

does impose a small efficiency loss, but net

I_{s p}

rises.)

Why is a cold-soaked solid motor a safety concern before launch?

Temperature sets

a

; a colder grain has lower

a

and burns slower, but grain brittleness and pressure-mismatch on ignition can crack the grain, exposing extra surface and spiking pressure.

Edge cases

What happens to

P_{e q}

n \to 1^{-}

The exponent

1/ (1 - n) \to + \infty

, so

P_{e q}

becomes catastrophically sensitive to any change in

A_{b} / A_{t}

— the mathematical warning sign of an unstable, explosion-prone motor.

What if a crack forms in the grain during burning?

The crack exposes fresh burning surface, suddenly increasing

A_{b}

; gas generation jumps, pressure rises,

r

rises further — a positive-feedback "chuff" or overpressure event.

At zero chamber pressure (

P \to 0

), what does

r = a P^{n}

predict, and is it physical?

It predicts

r \to 0

(with

n > 0

): no pressure, effectively no burning. Physically the flame cannot sustain itself below a minimum pressure $P_{min}$ , so the law only applies for

P > P_{min}

— extrapolating to

P = 0

is meaningless.

Roughly what is the minimum sustaining (extinction) pressure, and why does it exist?

For typical AP/HTPB motors

P_{min}

is around 1–2 MPa (some formulations extinguish near ~0.5 MPa). Below it the flame stand-off

δ

grows so large that heat flux

q^{''}

back to the surface can't keep it above

T_{s}

, and the flame goes out — this is the basis of "depressurisation extinguishment."

n = 0

exactly, how does the motor behave?

r = a P^{0} = a

— burn rate becomes independent of pressure (a "plateau" propellant). This is maximally stable: pressure changes don't feed back into burn rate at all.

What if

A_{b}

stays constant but the nozzle throat

A_{t}

erodes larger during burn?

P_{e q} \propto (A_{b} / A_{t})^{1/ (1 - n)}

falls as

A_{t}

grows, so chamber pressure and thrust drop off through the burn — a common real-world thrust-tailing effect.

Can burn rate ever exceed the flame's ability to supply heat, stalling the burn?

The steady state is set by the heat-flux balance

ρ_{p} r c (T_{s} - T_{0}) = q^{''}

;

r

self-adjusts so supply equals demand. A mismatch (e.g. sudden pressure drop) can lower

q^{''}

below what sustains ignition, extinguishing the surface.

Recall One-line summary of the traps

Burn rate is a linear speed driven by pressure via flame proximity; $a$ depends on temperature; $n$ must stay below 1 for stability; AP is oxidiser, HTPB and Al are fuels; below $P_{min}$ (~1–2 MPa) the flame dies.