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IN WORK...
On this page, the dot symbol
"●" will be used for multiplication as a
standard algebraic symbol. This is to avoid confusion with the
asterisk used in C* (pronounced see star), a common term in rocket
propulsion equations. Also, two letters or a number and a letter
adjacent to each other will also indicate multiplication.
A very simple formula is at the heart of solid propellant rocket motors.
It is the Burn Rate Formula. The formula is:
r = a●Pn
where r = burn rate in in./sec
a = burn rate coefficient in in./sec.
n = burn rate exponent (dimensionless)
P = chamber pressure in psi.
Each propellant has different exponents and coefficients and so behave
differently. As pressure increases, burn rate increases. It
would be nice if propellants burned at the same rate at all pressures.
That would mean "n" was zero. Pn with n=0 is 1.
Any number raised to the zero power is 1. That would mean regardless
of the pressure, the burn rate would be the same. So as n gets larger,
the rate of burn rate increases. As "a" gets larger, the burn rate
increases at all pressures. If the burn rates at different pressures
are plotted on a log-log chart, the plot will be a straight line.
Because it is a straight line, if you have two plot points, you can draw the
line and calculate "a" and "n" for the propellant. This is called
characterizing the propellant because you are finding the character of the
propellant or how it behaves so you can predict its performance.
Actually, because propellants seldom can be measured so accurately, a person
needs as many points as possible. Three is a lot better than two and 5
to 7 would be even better, then a best fit straight line can be drawn.
Not all propellants follow the standard burn rate
performance but most follow it at least for a range of pressures.
Sometimes different coefficients and exponents can be found for two or three
ranges that together will define or characterize the propellant over the
range of pressures expected.
Kn
I am going to present Kn actually a little early on this page
because it is so widely used and heard talked about concerning experimental
rocketry and motor development.
Kn = Ap / At
Kn is the ratio of the propellant burn area to the nozzle throat
area. Ap is the Area of the propellant that is burning, At
is the cross sectional area of nozzle throat, the smallest diameter in
a nozzle between the convergent cone and the divergent cone (see the
rocketry overview page). For a
given propellant, the pressure and so thrust is totally dependant on Kn.
As Kn goes up, the pressure goes up. After designing rocket
motors for a while, you will get to have a feel for what Kn a
specific motor and fuel should have to work. If you are designing
motors using PVC pipe for cases, you will have a lower Kn and if
you are designing reloads for commercial aluminum motors, you will use a
higher Kn. Kn tells you that for a given
pressure, if you add grains to a motor and so surface area, you will have to
increase the nozzle diameter.
When I say the pressure is directly related to Kn, that doesn't
mean it is a 1:1 ratio, that the pressure goes up exactly as much as the
ratio goes up. It is actually controlled by some other constant
factors and an exponent with the "n" exponent in it from the burn rate
equation.
P = [Kn ● a ● ρ ● (C*/g)]1/(1-n)
where
ρ = density of the propellant
C* is the characteristic velocity
g = the gravitational constant, 32.2 ft/sec2
P, a, Kn and n have already been defined.
The only term left is C*.
C* = [(R ● T) /
γ ] ● [ (γ + 1) / 2](γ+1)/(γ-1)
γ = ratio of specific heats = Cp / Cv.
R = the universal gas constant / (molecular weight of the propellant gas).
T = propellant gas temperature in the chamber.
Actually, C* can be looked up or can be calculated using free chem software.
Thrust Formula
Thrust = P ● At ● Cf
where P = chamber pressure in psi
At = area of nozzle throat in in2
Cf = Nozzle or Thrust Coefficient
Cf = A●(1-BC)●DE+F●G
A = 2γ2/(γ-1)
B = Pe/Pc
C = (γ-1)/γ
D = 2/(γ+1)
E = (γ+1)/(γ-1)
F = (Pc-Pa)/Pc
G = Ac/At
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