Modeled Data



Sample Calculations

Initial Values

time step0.001 s
Volume of air0.0012 m2
Volume of water0.0008 m2
Atmospheric pressure101325 Pa
Cross-sectional area of nozzle4.08E-4 m2
Average cross-sectional area of bottle8.81E-3 m2
Initial air pressure273300 Pa
Density of water1000 kg/m3
Density of air1.225 kg/m3
Gravitational acceleration9.81 kg*m/s2
Drag coefficient0.82
Specific heat capacity of air1005 J/kgK
Initial Temperature of air in bottle298 K

Calculations & Formulas

Excel Column Letter — Value description (Bottle Mechanics)
Equation
Excel formula taken at t = 0.001

H — Air volume (2.4)

V_{a}(t + \mathrm{\Delta}t) = V_{a}\left( t \right) + \mathrm{\Delta}t\ \left\lbrack A_{n}\sqrt{2\ \frac{p_{0}\left( \frac{V_{0}}{V\left( t \right)} \right)^{1.4} - p_{\text{atm\ }}}{\rho_{w}}} \right\rbrack
=H2+$B$2*$B$6*SQRT((2*($B$8*($B$3/H2)^(1.4)-$B$5))/($B$9))

I — Thrust (2.4)

F_{t}(t) = 2A_{n}\left\lbrack p_{0}\left( \frac{V_{0}}{V\left( t \right)} \right)^{1.4} - p_{\text{atm\ }} \right\rbrack
=2*($B$8*(($B$3/H3)^(1.4))-$B$5)*$B$6

J — Drag (3.0)

D\left( t \right) = C_{d}\left( \frac{1}{2}\rho_{\text{air}}A_{c}{v(t)}^{2} \right)
=$B$12*0.5*$B$10*$B$7*K3^2

K — Velocity (4.0)

v\left( t + \mathrm{\Delta}t \right) = v\left( t \right) + a(t)\mathrm{\Delta}t
=K2+$B$2*L2

L — Acceleration (4.0)

a\left( t \right) = \ \frac{2A_{n}\left\lbrack p_{0}\left( \frac{V_{0}}{V(t)} \right)^{1.4}{- p}_{\text{atm\ }} \right\rbrack}{\left( V_{\text{total}} - V_{\text{air}}(t) \right)\rho_{w} + m_{b}} - g - \frac{C_{d}\left( \frac{1}{2}\rho_{\text{air}}A_{c}{v(t)}^{2} \right)\ }{\left( V_{\text{total}} - V_{\text{air}}(t) \right)\rho_{w} + m_{b}}
=(I3-(($B$9*(0.002-H3)+0.7)*$B$11)-J3)/(0.7+$B$9*(0.002-H3))

M — Weight

W\left( t \right) = \rho_{w}V_{w}\left( t \right)g
=$B$9*R3*$B$11

N — Mass of water

m_{w}(t) = \rho_{w}(0.002 - V_{w}\left( t \right))
=($B$9*(0.002-H3))

O — Air pressure (2.4)

p(t) = p_{0}\left( \frac{V_{0}}{V(t)} \right)^{1.4}
=O2*(H2/H3)^(1.4)

P — Temperature of air (5.0)

T(t + \mathrm{\Delta}t) = T(t) - \frac{p\lbrack V\left( t + \mathrm{\Delta}t \right) - V\left( t \right)\rbrack}{c_{p}m_{\text{air}}}
=P2-((O3*10^-3)*(H3-H2))/($B$13*$B$15)

Q — Height of bottle

h\left( t + \mathrm{\Delta}t \right) = v\left( t + \mathrm{\Delta}t \right)\mathrm{\Delta}t + h(t)
=K3*$B$2+Q2

R — Volume of water

V_{w}(t) = 0.002 - V_{a}(t)
=0.002-H3

S — Exit velocity

v_{e}(t) = \sqrt{2\ \frac{p_{0}\left( \frac{V_{0}}{V(t)} \right)^{1.4} - p_{\text{atm\ }}}{\rho_{w}}}
=SQRT((2*(O3-$B$5))/$B$9)

T — Gauge pressure

p_{\text{in}}(t) = \frac{p_{\text{air}}(t) - p_{\text{atm}}}{1000}
=(O3-101300)/1000

Summary of Outputs

TimeVol. of airHeightVelocityAccelerationThrust ForceAir Pressure (Gauge)
sLmm/s m/s2 NkPa
0.0011.2158.297E-58.298E-282.15136.56167.3
0.0021.2232.481E-40.165181.35134.68164.9
0.0031.2304.946E-40.268880.56132.85162.7
0.011.2810.0044330.794775.45121.04148.3
0.11.7770.32395.73239.9146.4656.93
0.151.9710.65667.45429.1928.6835.15

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:

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