Theory:
A bottle rocket consists of a pressurized chamber and a nozzle. The nozzle can be
modeled as an isentropic device meaning, the change in entropy of the nozzle is zero. For
simplicity we will be considering the nozzle to be adiabatic and analyzing only the initial
state of the nozzle just after it is launched. The enthalpy of the liquid inside the chamber
depends on its temperature and pressure. As pressure increases, the enthalpy of the
flowing liquid increases.
Analysis:
A mass balance reveals that the mass of the control volume is not changing and thus there
is only one mass flow rate across the nozzle.
(1)
We are assuming that the inlet kinetic energy is negligible. An energy balance of the
process as shown in equation (2) and (3). The terms in red cancel due to the nozzle
properties discussed in the theory section revealing that the enthalpy of the fluid at the
inlet is equal to the enthalpy of the fluid at the outlet plus its kinetic energy.
(2)
(3)
After solving for initial exit velocity and accounting for units we arrive at equation (4).
(4)
Using the enthalpy of water at different pressures we calculated corresponding initial
water velocities. This pressure dependence is graphed in Figure 1.
Results & Discussion:
We assumed an outlet state set by a pressure of 101 kPa and 300K. This demands an outlet
specific enthalpy of 92.3810066 kJ/kg. We assumed an inlet velocity of ~0m/s, a temperature of
300K and a variable input pressure. These inlet pressures varied from 110 kPa to 200 kPa. As the
pressure increased, the inlet enthalpy increased. Figure 1 was obtained by using equation (4) at
each inlet enthalpy and graphing the outlet velocity versus inlet pressure. The initial outlet
velocity of the water increases as the initial chamber pressure increases. However, the shape of
the curve in Figure 1 is a positive slope and concave downward. From this we can conclude that
the inlet pressure’s influence on the initial water velocity is smaller at higher pressures.