MEAM.Design - MEAM 247 - P1: Launch - Fundamental Rocket Relationships


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Air Expansion

We will begin with the rocket motor, filled with a certain volume of water, V_w(0), and air, V_a(0), at an initial pressure, P_a(0). Upon launch, we will assume adiabatic expansion of the air as water is quickly ejected through the nozzle. The pressure and volume of the air are thus related by:

\frac{P_a(t)}{P_a(0)} = \left(\frac{V_a(t)}{V_a(0)}\right)^{-\gamma}

where \gamma =1.4 is the adiabatic index for air.


Water Nozzle Exit Velocity

Recognizing that the water is essentially incompressible, we can use Bernoulli's equation to find the velocity of the water through the nozzle. Bernoulli's equation states that at any point in a streamline

\frac{v^2}{2} + \frac{p}{\rho} + g h = \text{constant}

where v is the velocity, p is the pressure, \rho is the density, g is gravity, and h is the height above a reference plane. Given that the motor is relatively short, we can ignore gravitational effects. We can therefore apply Bernoulli's equation to compare the nozzle exit to the air/water interface:

\frac{v_e(t)^2}{2} + \frac{P_{atm}}{\rho} = \frac{v_a(t)^2}{2} + \frac{P_a(t)}{\rho}

where v_e is the nozzle exit velocity, P_{atm} is the atmospheric pressure, and v_a is the velocity of the air/water boundary.. Now, solving for v_e:

v_e(t) = \sqrt{v_a(t)^2 + \frac{2(P_a(t) - P_{atm})}{\rho}}

Thrust

The linear momentum of a water particle of mass \Delta m expelled from the nozzle at velocity of v_e(t) can be expressed as

\Delta L = \Delta m \; v_e(t)

dividing by \Delta t and taking the limit as \Delta t goes to zero results in

\dot{L} = \dot{m} \; v_e(t)

which is equal to the sum of external forces acting on the water particle, and therefore equal to the thrust acting on the rocket

F(t) = \dot{m} \; v_e(t)

Drag

Courtesy of Dr. Kothmann, here are some notes on aerodynamic drag that you might find useful:

The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematicians
Lyle Long and Howard Weiss
The American Mathematical Weekly, Vol. 106, No. 2. (Feb 1999)
pp 127-135