Why are my empirical and theoretical value of mass flow rate in solid propellant rockets not similar?

Question

For a rocket of:

• fuel mass $m_f$ = 6 kg,
• Thrust = 3.1 kN, (vs 4k using reducing catalyst to delay burn time)
• total powered burn time, $t_b$ = 3.5 s, (using reducing catalyst vs 1.8 sec)

This gives me the burn rate mass flow rate, 6/3.5 = 1.71 kg/s. But theoretically, it should be around 4.8 kg/s. Using

$$ \dot{m} = \rho_p A_b r $$ where

• $\rho_p$ = Density of propellant
• $A_b$ = Burning surface area of the cylinder propellant (circular bore grain shape)
• $r$ = rate of propellant burn

The rate is caluculated using: $$ r = a P_c^n $$ where $a$, $n$ are empirical coefficients and $P_c$ is the chamber pressure

So, if reverse the steps to find pressure, I get 25 psi or 0.177 MPa.

But, theoretically this should be 4.8 kg/s for my propellant KNSU with $A_b = 0.19 m^2$ ($h=0.8m$, $r_{in} = 0.076m$, using $A_b = \pi r_{in} h $) and for chamber press of 4.8 MPa or 700 psi with $\rho_p$=1.889 g/$cm^3$ a=8.26 and n=0.319 using the two relation above. Could anyone share what could cause such discrepancy?

Answer 1

There is work to be done straightening out your analysis as commented on above.above. But, your goal seems to be to determine the effectiveness of a catalyst that has been added to a propellant (unspecified) formulation. The parameters of the burn rate expression "a", and the pressure exponent "n" are properties of your propellant formulation. After decades of solid rocket development these have defied theoretical determination and are still determined experimentally in "strand" or other small scale test apparatus. Tests are run for a propellant formulation at various chamber pressures (and also grain temperatures). Plotting experimental burning rate (determined from a pressure or thrust vs time curve) at various pressures gives values for "a" and "n".

Likely a known well characterized standard propellant was doped with a catalyst. Likely there are values of these parameters to be found in the standard literature. Unlikely that you could find values for the propellant formation with your catalyst added.

You can use your motor as a test apparatus if you run a few more tests (varying chamber press - throat diameter), you could develop parameters for your propellant mixture for your future use. As a baseline also run a test without the catalyst and see if your data matches predicted. Once you have firm values for "a" and "n" for you mixture, you can then proceed to design motors to accomplish what you seek. This is the same procedure as employed by the solid rocket manufacturers.