## Sample Calculations

### Initial Values

time step | 0.001 s |

Volume of air | 0.0012 m^{2} |

Volume of water | 0.0008 m^{2} |

Atmospheric pressure | 101325 Pa |

Cross-sectional area of nozzle | 4.08E-4 m^{2} |

Average cross-sectional area of bottle | 8.81E-3 m^{2} |

Initial air pressure | 273300 Pa |

Density of water | 1000 kg/m^{3} |

Density of air | 1.225 kg/m^{3} |

Gravitational acceleration | 9.81 kg*m/s^{2} |

Drag coefficient | 0.82 |

Specific heat capacity of air | 1005 J/kgK |

Initial Temperature of air in bottle | 298 K |

### Calculations & Formulas

Excel Column Letter — Value description(Bottle Mechanics)

Equation

Excel formula taken at t = 0.001

**H — Air volume (2.4)**

=H2+$B$2*$B$6*SQRT((2*($B$8*($B$3/H2)^(1.4)-$B$5))/($B$9))

**I — Thrust (2.4)**

=2*($B$8*(($B$3/H3)^(1.4))-$B$5)*$B$6

**J — Drag (3.0)**

=$B$12*0.5*$B$10*$B$7*K3^2

**K — Velocity (4.0)**

=K2+$B$2*L2

**L — Acceleration (4.0)**

=(I3-(($B$9*(0.002-H3)+0.7)*$B$11)-J3)/(0.7+$B$9*(0.002-H3))

**M — Weight**

=$B$9*R3*$B$11

**N — Mass of water**

=($B$9*(0.002-H3))

**O — Air pressure (2.4)**

=O2*(H2/H3)^(1.4)

**P — Temperature of air (5.0)**

=P2-((O3*10^-3)*(H3-H2))/($B$13*$B$15)

**Q — Height of bottle**

=K3*$B$2+Q2

**R — Volume of water**

=0.002-H3

**S — Exit velocity**

=SQRT((2*(O3-$B$5))/$B$9)

**T — Gauge pressure**

=(O3-101300)/1000

### Summary of Outputs

Time | Vol. of air | Height | Velocity | Acceleration | Thrust Force | Air Pressure (Gauge) |

s | L | m | m/s | m/s^{2} | N | kPa |

0.001 | 1.215 | 8.297E-5 | 8.298E-2 | 82.15 | 136.56 | 167.3 |

0.002 | 1.223 | 2.481E-4 | 0.1651 | 81.35 | 134.68 | 164.9 |

0.003 | 1.230 | 4.946E-4 | 0.2688 | 80.56 | 132.85 | 162.7 |

0.01 | 1.281 | 0.004433 | 0.7947 | 75.45 | 121.04 | 148.3 |

0.1 | 1.777 | 0.3239 | 5.732 | 39.91 | 46.46 | 56.93 |

0.15 | 1.971 | 0.6566 | 7.454 | 29.19 | 28.68 | 35.15 |

## Validation

Our model is able to give results that resemble reality closely, except the temperature drop inside the bottle. This may due to the assumption that the bottle is isolated and no heat transfer happens between the bottle and environment while in reality the effect of heat transferring is not negligible.

When the initial volume of water is 1.2 L and initial air gauge pressure is 172 kPa, the prediction from *Hydrodynamics and thrust characteristics of a water-propelled rocket* [1] gives:

The outputs of our model:

When the initial volume of water is 1.2 L and initial air gauge pressure is 275 kPa, the prediction from *Hydrodynamics and thrust characteristics of a water-propelled rocket* [1] gives:

The outputs of our model:

When the initial volume of water is 0.83 L and initial air gauge pressure is 172 kPa, the prediction from *Hydrodynamics and thrust characteristics of a water-propelled rocket* [1] gives:

The outputs of our model:

When the initial volume of water is 0.83 L and initial air gauge pressure is 172 kPa, the data from *A more thorough analysis of water rockets: Moist adiabats, transient flows, and inertial forces in a soda bottle *[5]*: *

The output of our model: