Water Rocket Version 0.2 06.06.2017

Harald von Fellenberg

<hvf@hvf.ch>

Table of Contents

1. The Challenge

2. The Model

2.1. Numerical Solution

3. Results

1. The Challenge

Construct a simulation model to calculate the height our (or your own) bottle will achieve.

For instance, the bottle has a volume 0f 1.5 liters and weighs 40 grams, has a 10.5 cm diameter and a 2.5 cm diameter of the opening. Through that opening, 1 liter of water is expelled in 0.3 seconds. The drag coefficient is thought to be 0.6. Finally, the 5 pumps of Prof. Hoekstra each addeed 0.25 liter of air to the bottle, giving a net pressure of about 250000 Pa (350000 inside against 100000 ambient pressure).

Try to compute the height that the bottle will reach. Also try to investigate the effect of the mass of the water in the bottle: what is the optimum amount of water?

2. The Model

Main Parameters

dt = 0.01 # sec time step

p0 = 250000 # Pa initial pressure in air chamber (empty volume)

Vtot = 1.5e-3 # 1.5 liter total volume of water rocket

V0 = 0.5e-3 # empty volume, 0.5 liter; rest is filled with water

A = 0.105**2*math.pi/4 # m2, diam 10.5 cm; cross section of water rocket (cylinder of 10.5 cm diameter)

Q = 0.025**2*math.pi/4 # m2, diam 2.5 cm; cross section of bottle opening, 2.5 cm diameter

Q = Q / 1.7 # throttling, arbitrary throttling factor; we aim at a propulsion time of about 0.3 sec for 1 liter of water

M = 0.040 # kg, 40 gr empty weight of rocket (bottle)

dwater = 1000. # kg/m3 water density

dair = 1.225 # kg/m3 air density at sea level, ISA

Cdrag = 0.6 # drag coefficient of bottle

The differential equations for velocity v and altitude h are solved numerically. A "small" time step of 0.01 sec was arbitrarily chosen, and a simple forward integration scheme was implemented

2.1. Numerical Solution

v1 = v0 + a * dt

h1 = h0 + (v0 + v1) / 2 * dt

3. Results

The simulation is written in Python, both version 2 and 3 work on my Linux box. You can find the program here:

download rocket-launch.py here

Some results are shown in the table and the plots.

Table 1. Results for different parameters

Pressure [Pa]	Air volume [ml]	Altitude [m]	Velocity [m/s]	Jet time [s]	Jet speed [m/s]	Mass at takeoff [kg]	Mass at landing [kg]	Plot file
350000	400	13.2	14.1	0.480	22.4	1.140	0.140	pressure 350 kPa, 400 ml air volume
350000	500	18.4	32.7	0.302	22.4	1.040	0.040	pressure 350 kPa, 500 ml air volume
350000	600	20.4	42.0	0.223	22.4	0.940	0.040	pressure 350 kPa, 600 ml air volume
350000	750	21.9	51.4	0.158	22.4	0.790	0.040	pressure 350 kPa, 750 ml air volume
350000	825	22.1	54.0	0.134	22.4	0.715	0.040	pressure 350 kPa, 825 ml air volume
350000	900	22.3	56.2	0.114	22.4	0.640	0.040	pressure 350 kPa, 900 ml air volume
350000	1000	22.7	60.3	0.090	22.4	0.540	0.040	pressure 350 kPa, 1000 ml air volume
300000	400	5.2	8.0	0.400	20.0	1.140	0.340	pressure 300 kPa, 400 ml air volume
300000	500	15.3	20.2	0.496	20.0	1.040	0.040	pressure 300 kPa, 500 ml air volume
300000	600	17.5	32.0	0.273	20.0	0.940	0.040	pressure 300 kPa, 600 ml air volume
300000	750	19.6	41.9	0.184	20.0	0.790	0.040	pressure 300 kPa, 750 ml air volume
300000	825	20.0	45.1	0.155	20.0	0.715	0.040	pressure 300 kPa, 825 ml air volume
300000	900	20.6	48.8	0.130	20.0	0.640	0.040	pressure 300 kPa, 900 ml air volume
300000	1000	20.5	49.7	0.102	20.0	0.540	0.040	pressure 300 kPa, 1000 ml air volume

It can be seen that the maximum altitude is not very sensitive to the air volume - the main reason for this is that the air drag reduces the attainable altitude drastically.

For small air volumes (400 ml), there is not enough pressure to expell all of the water,

Comparison to the actual experiment: looking at the video and measuring the altitude with a ruler on the screen, I arrive at somewhat above 10 meters, but the rocket did not fly exactly vertically. The agreement with my calculation is therefore nor very good.