ASSUMPTIONS
Although a pressurized water rocket is a rather straight forward design, the complexities on an infinitesimal scale are vast. To simplify the theoretical modeling process, a variety of assumptions are made. A brief description of each follow, along with justification and an explanation for how the assumption simplifies the problem.
INCOMPRESSIBLE FLUID
The assumption that liquid water is incompressible allows for a simplified Bernoulli’s Equation by declaration that the density remains unchanged along the streamline. The incompressible fluid assumption is justified by realizing the bulk modulus of water is 2.5x109 N/m2, or 2500000000 Pa! This value is significantly greater than the pressures within the rocket, and beyond the containment of the PET bottle.
CONSTANT BOTTLE VOLUME
Here we have assumed that as the air within the bottle is pressurized, that the bottle does not expand and increase in volume. While the bottle will expand slightly, the amount is negligible and very difficult to verify and measure. For this reason, we have chosen to declare the control volumes once pressurized and neglect expansion until launch.
INITIAL PRESSURE HEAD
While the rocket is on the launch pad and filled with water, the pressure of the water’s height is considered the pressure head. We have assumed that the pressure head is negligible while compared to the magnitude of air pressure inside of the rocket.
FREE SURFACE VELOCITY
As the rocket accelerates upwards, the free surface of water is accelerated in the opposite direction of travel until fuel is exhausted. The assumption that the acceleration in the negative direction of travel is small enough to be ignored while compared to the acceleration in the positive direction of the rocket.
ADIABATIC EXPANSION
During launch, the water is expelled from the rocket, and the air expands to fill the vacated volume. Our calculations show that the water is fully exhausted in 0.2 s. The assumption that expansion of air is quick enough that a negligible amount of thermal energy is exchanged outside of the rocket system and with the environment. This assumption allows us to use the relationship of pressure change given under the pressure tab.
STEADY FLOW
The assumption that the flow of the fluids from inside the rocket through the nozzle are both inviscid and steady. That is that the fluid has no viscosity and that the flow field remains unchanged along the streamline. These assumptions allow is to use Bernoulli’s Equations concerning fluid flow.
TAIL GEOMETRY
Simplification of rocket drag characteristics were most notable done by assuming that the body of the rocket maintains a constant, smooth slope from full body diameter to end of the nozzle. This design is often referred to a boat tail design, for which equations of drag are given. For the true geometry of a 2-ℓ bottle, the shape is much more complex and difficult to solve for theoretically.