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LS2b: Combustion & Propellants
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Remember when we told you that this was the equation for the delta-v of a rocket?
We kinda lied. This equation makes a few assumptions that aren’t always realistic. First, it assumes that all the propellants go through the combustion chamber. This is often not the case in rocket engines, as some of the propellants are combusted separately to power the pumps, and then dumped overboard. Second, it assumes a matched nozzle (aka exit pressure = atmospheric pressure).
In reality, this is the more accurate rocket equation:
Where Isp is specific impulse (and g0 is 9.8 m/s^2). Specific impulse is a very important quantity that measures rocket engine efficiency. It is defined as the impulse produced per unit weight of propellant consumed. Put another way, it is proportional to the thrust of the engine divided by the mass flow rate of the propellants:
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Recall the isentropic relation introduced in Topic 2a:
As a reminder, this equation relates the stagnation & static pressures to the mach number. The greater the mach number of a flow, the lower the static temperature (as Tc is considered constant throughout the isentropic flow). We know how to take our Mach number and get velocity from the following equation:
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2. Exhaust Velocity
Assume that the nozzle is matched, and that all propellants flow through the combustion chamber. What is the exhaust velocity of this nozzle?
What is the required mass flow rate to achieve the thrust target?
What is the required throat area?
What is the Isp of this engine?
Bonus: An engineer recommends that you decrease the exit pressure of the flow to 0.5 atm to increase the exit velocity of the flow. (This requires increasing the exit area of the nozzle. This nozzle will no longer be matched.). Keep the same throat area as before. Show that this will have a net negative impact on the total thrust of the engine, and on Isp. (hint: the final equation on the equation sheet (added 10/10/2020) should help with this problem. It gives the relationship between area ratio and mach number).
3. The engine is modified for vacuum operation (nozzle designed for a Pe of 0.1 psi). The team is looking to increase performance. Someone recommends increasing the chamber pressure to 6 MPa (while keeping nozzle geometry the same). What effect will this change have on the thrust of the engine? What effects will it have on the exhaust velocity? What effect will it have on Isp?
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