PENGHITUNGAN ROKET AIR

Water Rocket Analysis (Page 1)

We searched the web for a suitable analysis to use and decided to adopt the approach presented by Dr. Peter Nielsen of the University of Queensland in Australia. Dr Nielsen has presented a summary of the equations as well as the results of a simulation (Nielsen_Rocket.pdf [164k file]), which we will use as the default values in our exercises. In the following we have developed the rather complex analysis leading to the differential equation for the compressed air volume variation, essentially following the presentation by Dr. Nielsen. In subsequent analyses we continue to develop the differential equations for upwards acceleration, velocity and height attained as well as the numerical integration technique for evaluating these variables as functions of time.
Consider a typical water rocket such as the "Rolling Rock" Rocket, proudly demonstrated by students of a previous ME100 class. (rumour has it that beer was used instead of water in this rocket).

A cross section of this rocket illustrating the principle of operation is shown in the following diagram:

Thus the compressed air in the bottle forces the water through a nozzle (bottle neck) which produces the thrust required to accelerate the bottle (hopefully) vertically upwards. We determine the time derivative of its vertical velocity by Newton's second law of motion:

where:
m is the instantaneous total mass of the rocket [kg]
u is the upwards velocity [m/s]
Fthrust is the thrust forsce (due to the expelled water) [N]
Fdrag is the drag force from the surrounding air [N]
g is the acceleration due to gravity [9.81 m/s2]

Thrust Force Fthrust

The thrust force is proportional to the exhaust mass flow through the nozzle times the velocity of the exhaust relative to the rocket.

where:
is the rate of mass flow of the expelled water [kg/s]
uex is the exhaust velocity of the expelled water through the nozzle [m/s]
is the density of water [1000 kg/m3]
AN is the area of the nozzle [m2]

Bernoulli's Equation

Bernoulli's equation can be derived from the energy equation applied to the water flowing through the nozzle. It relates the kinetic energy of the exhaust water to the compressed air pressure applied at the water surface. Neglecting potential energy terms, we have: where P is absolute pressure inside the bottle and Pa is the outside (atmospheric) pressure [Pa] However usurface << uex and can be neglected, thus:

Combining equations (2) and (3) above we obtain:

We now continue with Page 2 of the water rocket analysis, leading to the compressed air volume variation differential equation. Solving this equation will allow us to evaluate the rocket performance, leading ultimately to the altitude attained by the rocket.

The basic rocket Force equation:

The thrust force in terms of the water expelled through the nozzle:

Bernoulli's equation, relating the pressure applied by the compressd air to the velocity of the exhausting water:

Note that the resulting thrust force is twice the pressure difference times the nozzle area:

Unfortunately the compressed air pressure P is not a constant during the thrust phase, but varies in a nonlinear manner with the expanding volume of the compressed air. This is the main reason for the extremely complex relations resulting from this analysis.

Adiabatic Expansion

As the water escapes, the air volume increases, causing a decrease in pressure and a corresponding decrease in thrust. We consider this process to be adiabatic (no transfer of heat during the split-second expansion process), which allows us to relate the time variation of the pressure to that of the volume.

The adiabatic expansion process is derived from the energy equation applied to an ideal gas, and is developed in the section on Adiabatic Expansion Analysis leading to the following equation:

where:
P0 is the initial absolute pressure at liftoff [Pa]
V0 is the initial volume of the compressed air [m3]
k is the ratio of specific heat capacities [k = 1.4 for air]
P, V are the respective time varying pressure and volume of the compressed air during the thrust phase.

Compressed Air Volume Variation

The volume variation of the compressed air due to the water escaping through the nozzle is given by:

Substituting equations 3 and 5 into equation 6 and simplifying, we obtain:

Equation 7 is the differential equation for the volume variation of the compressed air as a function of time t. It cannot be solved explicity since the volume V is deeply embedded in a nonlinear manner in the equation, hence we resort to a numerical solution.
The numerical solution of ordinary differential equations (ODEs) is an important generic problem in engineering, and you will learn various methods of solving them (such as the Runge-Kutta methods) when you study Math 344. The approach adopted by Dr. Nielsen uses an approximate numerical integration method by replacing the derivative by a first order difference method, as follows:

where:
t is the elapsed time [s], thus V(t) refers to the volume at elapsed time t
is the time step increment
and P is obtained from equation 5 as:

This leads to the following solution for V(t):

In the forthcoming exercises we will use this method as well as a similar technique called the Trapezoidal method for numerical integration in order to evaluate the upward velocity and height of the rocket. This entire project will be developed over six programming exercises, and in this first exercise we wish to set up the basic class structure which will allow us to define specific rocket objects.

Ms - the mass of the rocket solid parts (shell + payload) [kg]
Vtotal - the total volume in the bottle (water + air) [m3]
AN - the area of the nozzle normal to the flow of expelled water [m2]
P0 - the initial compressed air absolute pressure in the bottle (at launch) [Pa]
V0 - the initial compressed air volume in the bottle (at launch) [m3]
Cd - the coefficient of drag for the rocket (reflecting streamlining, surface roughness etc.)
Abottle - the projected area of the rocket normal to the direction of flight [m2]

The five public variables are as follows:

g - the acceleration due to gravity [9.81 m/s2]
- the density of water [1000 kg/m3]
- the density of the outside air [assume = 1.2 kg/m3]
k - the ratio of specific heat capacities of air [k = 1.4]
Patm - the absolute pressure of the atmosphere [assume = 100,000 Pa]

Source : ohio edu