Chapman-Jouguet detonation; deflagration vs detonation: Foundations

This page assumes you know nothing about the notation the parent note uses. We build every letter and picture from the ground up, in the order the topic needs them.

0. The one picture the whole topic lives in

Before any symbol: picture a tube full of unburnt gas. Somewhere inside it there is a razor-thin sheet — the reaction front — with cold fresh gas on one side and hot burnt gas on the other. That sheet sweeps down the tube.

Figure — Chapman-Jouguet detonation; deflagration vs detonation

Look at the figure. We label everything on the cold, unburnt side with subscript 1 and everything on the hot, burnt side with subscript 2. That single convention — ==1 = before, 2 = after== — is used by every equation in the parent note. Memorise it now and the rest is easy.

1. Sitting still on the front: velocities $u_{1}$ , $u_{2}$

The front is moving through the tube, which is awkward. The classic trick: ride on the front. Imagine you shrink down and stand on the thin sheet. Now you are still, and the gas seems to rush into you at speed $u_{1}$ and out the other side at speed $u_{2}$ .

The detonation velocity $D$ that the parent note wants is simply how fast the front eats into the still lab gas — which, in this riding frame, equals $u_{1}$ . That is why the parent writes $D_{C J} = u_{1}$ .

2. How crowded is the gas: density $ρ$ and specific volume $v$

The parent note constantly flips to the upside-down of density:

The whole " $(v, p)$ plane" the parent draws on has roominess $v$ across the bottom and pressure $p$ up the side. The figure shows why compression moves you left (smaller $v$ ) and expansion moves you right (bigger $v$ ).

3. Pressure $p$ and why it drives momentum

Pressure matters twice. First, it is one axis of the graph. Second, an imbalance of pressure across the front is a net push that changes the gas's momentum — that is the whole content of the momentum jump condition.

4. The flow of stuff each second: mass flux $\overset{m}{˙}$

The two equalities are literally the conservation of mass: whatever pours in ( $ρ_{1} u_{1}$ ) must pour out ( $ρ_{2} u_{2}$ ) — gas cannot pile up inside a razor-thin sheet.

Because $\overset{m}{˙} = ρ u = u / v$ , we can rewrite $u_{1} = \overset{m}{˙} v_{1}$ and $u_{2} = \overset{m}{˙} v_{2}$ . This one substitution is what turns the messy velocity equations into the clean straight line (Rayleigh line) the parent draws — a bigger $\overset{m}{˙}$ means a steeper, faster front.

5. Energy words: enthalpy $h$ , heat capacity $c_{p}$ , and heat release $q$

Combustion is about energy, so we need words for energy stored in hot gas and energy released by burning.

6. Springiness of the gas: the ratio $γ$

Why does the topic need $γ$ ? Two reasons:

It lets us write enthalpy purely from $p$ and $v$ : $h = \frac{γ}{γ - 1} p v$ . That is the substitution that turns the energy law into the Hugoniot curve.
It sets the speed of sound (next section), which is the whole point of the CJ condition.

7. The speed limit of small pushes: sound speed $a$ and Mach number $M$

8. Putting the symbols on the map

Read the map top to bottom: the basic labels feed the mass, momentum, energy books, which draw the Rayleigh line and Hugoniot curve; where those meet tangentially at $M_{2} = 1$ is the Chapman–Jouguet point that picks the one true detonation speed.

9. Quick self-check of meanings

Recall What does each subscript mean, and which is the burnt gas?

1 = unburnt gas ahead of the front; 2 = burnt gas behind it (the products).

Recall Why do we ride on the front instead of watching from the lab?

To make the picture steady — same gas in and out every second — so the tidy conservation laws apply.

Recall If gas is compressed, does specific volume

$v$ go up or down? Down: $v = 1/ ρ$ , and compression raises density $ρ$ .

Recall In words, what is mass flux

$\overset{m}{˙}$ ? Kilograms of gas crossing one square metre of the front per second; equal on both sides because mass is conserved.

Recall What single number decides subsonic vs supersonic, and what value marks the CJ point?

The Mach number $M = u / a$ ; the CJ point is $M_{2} = 1$ .

Equipment checklist

Cover the right side and see if you can state each before revealing.

Subscripts 1 and 2 ::: 1 = unburnt (ahead), 2 = burnt (behind) the front.
Flow speeds $u_{1}, u_{2}$ ::: gas speed into / out of the front in the frame riding with it (m/s).
Detonation velocity $D$ ::: how fast the front eats lab gas; equals $u_{1}$ in the riding frame.
Density $ρ$ ::: mass per cubic metre (kg/m³) — how crowded the gas is.
Specific volume $v = 1/ ρ$ ::: roominess per kilogram (m³/kg); the graph's horizontal axis.
Pressure $p$ ::: push per unit area (Pa or atm); the graph's vertical axis.
Mass flux $\overset{m}{˙} = ρ_{1} u_{1} = ρ_{2} u_{2}$ ::: kg crossing 1 m² of front per second; equal both sides (mass conserved).
Enthalpy $h = c_{p} T$ ::: thermal energy per kg of flowing gas (J/kg).
Heat of reaction $q$ ::: chemical energy released per kg of fuel (J/kg); the term that makes a shock self-powered.
Heat-capacity ratio $γ = c_{p} / c_{v}$ ::: gas springiness; ~1.4 for air, ~1.2–1.3 for burnt gas.
Speed of sound $a = γ p v$ ::: speed of a tiny pressure ripple (m/s).
Mach number $M = u / a$ ::: flow speed in units of sound; $M < 1$ deflagration, $M > 1$ detonation, $M = 1$ = CJ.

Once every reveal comes easily, you are ready for the parent topic and its Rayleigh line, Hugoniot curve, and Chapman–Jouguet condition.