I have been trying to calculate the performance of a solid rocket engine I am going to build for months on and off now. I still have not been able to. All I ask is if there is an equation I could use to find the nozzle throat dimensions because every equation I have found relies on mass flow rate, which is found by the throat dimensions. So it seems as though it is not able to be done, at least from what I have seen. Is there any equation I can use for that?

  • $\begingroup$ Navier Stokes equation might help $\endgroup$ – InquisitiveMind Dec 25 '16 at 4:50

You really need to know the physical and burn properties of the solid rocket propellant before you can go sizing the nozzle. Things you will need to know before sizing the nozzle:
1.) Solid propellant type (ammonium perchlorate, ammonium nitrate, KNO$_3$/Sucrose, etc.)

2.) Solid propellant density ($\rho_b$) prior to start.

3.) Solid propellant burn rate ($r$), which increases with higher chamber pressures.

4.) The characteristic velocity ($c*$), which is purely a function of the propellant characteristics. For instance, $$ c^* = \sqrt[]{\frac{R T_1}{\gamma\left(\frac{2}{\gamma+1}\right)^{(\gamma +1)/(\gamma-1)}}}$$ Hence, you will need to know the specific heat ratio of the burned products ($\gamma$), the chamber temperature of the burned products ($T_1$), and the specific gas constant of the burned products ($R$). All of these can be calculated using chemical equilibrium analysis.

5.) Lastly, you will need to know what is the desired steady state chamber pressure of your solid rocket ($p_1$). This is usually a constraint on the motor casing allowable hoop stress and scaled by a safety factor of your choosing. You will definitely want to avoid yielding stress in the motor casing. Hence, you can usually pick the desired chamber pressure.

Now the equation of interest for you is,

$$ \frac{A_b}{A_t} = \frac{p_1}{\rho_b r c^*} $$

This is the ratio of the burn area to nozzle throat area, which can only be determined provided you know the properties of your propellant and desired chamber pressure. Now you will usually know the burn area once you pick a grain configuration. Common types are below with the associated neutral, progressive, or regressive burn profiles.

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You will probably go with the central bore hole geometry or tubular (very common for amateur rockets), then you can just take the average burn area ($A_b$) because the burn area will be increasing as the propellant burns, which gives the progressive thrust profile. Anyways, with this now known, you have the required dimension for your nozzle throat ($A_t$). The mass flow rate through the engine/nozzle is now given by, $$\dot{m} = A_b r \rho_b $$

Now you need to size the nozzle exit. Most likely you will desire to expand the burned gas through the nozzle to atmospheric pressure at sea level. Thus, the exit Mach number for your nozzle can be obtained from the isentropic relation, $$\frac{p_1}{p_e} = \left(1 + \frac{\gamma-1}{2}M_e^2\right)^{\gamma/(\gamma-1)} $$ which can be solved for the exit Mach number ($M_e$) with the selected chamber pressure $p_1$, burned gas ratio of specific heats, and exit pressure $p_e$ (ambient 1 atm). Now the nozzle expansion area ratio required to expand the gas to this Mach number is given by, $$\frac{A_e}{A_t} = \frac{1}{M_e} \left(\frac{2 + (\gamma-1)M_e^2}{\gamma+1}\right)^{(\gamma+1)/(2(\gamma-1))}$$ Now you have sized your nozzle geometry required to expand the gas from the chamber pressure to the ambient condition. The final step now is to calculate the performance of the rocket engine. The thrust is given by, $$F = \dot{m} v_e + (p_e - p_a)A_e $$ At sea level we will not have the pressure area force (since we sized the nozzle to expand to atmospheric pressure), and since you won't be launching the rocket to very high altitudes, we can assume $(p_e - p_a) \approx 0$. Therefore, the thrust equation simply becomes, $$F = \dot{m} v_e $$ where we already know the mass flow rate through the engine, $\dot{m} = A_b r \rho_b$. So all we have left to find out is the nozzle exit velocity. This simple expression is given by, $$ v_e = \sqrt[]{\frac{2\gamma}{\gamma-1} R T_1 \left[1-\left(\frac{p_e}{p_1}\right)^{(\gamma-1)/\gamma}\right]} $$ Therefore, your thrust equation becomes, $$ F = A_b r \rho_b \sqrt[]{\frac{2\gamma}{\gamma-1} R T_1 \left[1-\left(\frac{p_e}{p_1}\right)^{(\gamma-1)/\gamma}\right]} $$ and the specific impulse is simply, $$ I_{sp} = \frac{F}{g\dot{m}} $$

Granted all of the above analysis is for the ideal scenario, which will usually get you within 10-15% of the actual non-ideal performance. Hope this helps. Also, disclaimer, I am not responsible for any mishaps or whatever your intentions might be. Just passing information from one rocketeer to another.

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Mostly I think people build a rocket for a specific purpose. They require a delta-v out of it, and work in the opposite direction to determine required thrust and mass flow and thus throat size.

In your case choose an arbitrary height and see if you can complete the design in that direction. Bear in mind the legal restrictions for launch. If you go to high you need a flight plan or something, never done it myself. There are rocket building clubs around, perhaps you should get in touch.

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