## Richard Nakka's *Experimental Rocketry* Web Site

__Derivation of equation describing chamber pressure due to igniter charge__

From the ideal gas law, pressure within the chamber may be expressed as

Introducing the term *G* relating the fraction of igniter charge burned at any time* t*, and considering that the initial chamber pressure is ambient

letting ("effective force") gives

which may be rewritten as

letting (loading fraction) gives

multiply numerator and denominator by r (charge density)

since , this gives

where:

P = pressure in combustion chamber at time, t lb_{f}/in^{2}

r= density of charge material lb_{m}/in^{3}

D = loading density = C / V lb_{m}/in^{3}

C = original mass of charge lb_{m}

V = free volume of combustion chamber at time, t in^{3}

l = R T/ M "effective force" (energy), in-lb_{f} /lb_{m}

R = universal gas constant in-lb_{f} /°
R-lb_{m}

M = effective molecular weight of combustion products (system mass divided by number of moles of gas) lb_{m} /
mole

T = adiabatic flame temperature °
R

G = fraction of original charge mass consumed by time, t dimensionless

P_{a} = atmospheric pressure lb_{f}/in^{2}

Note that the assumption is made that combustion of the igniter charge occurs rapidly with no pressure loss due to gas seepage through the nozzle opening.
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