Ethanol Rocket

Theory

Ethanol 3d model

• 
  The Ethanol
• 
  C2H6O
• 
  The Ethanol

Atom positions

-1.1712	0.2997	0.0000	O
-0.0463	-0.5665	0.0000	C
1.2175	0.2668	0.0000	C
-0.0958	-1.2120	0.8819	H
-0.0952	-1.1938	-0.8946	H
2.1050	-0.3720	-0.0177	H
1.2426	0.9307	-0.8704	H
1.2616	0.9052	0.8886	H
-1.1291	0.8364	0.8099	H

HCH bonds are assumed to be undistorded tetraherdal angle 109.5 degrees. Actually the electrons repeal each other. The HOC bond is 104.5 deg because. . .

Valence shell electron-pair repulsion theory (VSEPR theory). . .

Van der Waals radius

Atom	Radius	Relative radius
C	170 pm	1.42
H	120 pm	1.00
O	152 pm	1.27

Use CPK coloring convention, white (hydrogen), black (carbon) and red (oxygen).

Ethanol reaction with Oxygen and Air

Ethanol reaction with oxygen

${\text{C}}_2{\text{H}}_6\text{O}+3{\text{O}}_2\to 2{\text{CO}}_2+3{\text{H}}_2\text{O}$

The molecular weight of ethanol is $46.069$ g/mol, and the molar weight of oxygen is 32 g/mol. The oxygen--ethanol fuel ratio is $3\times 32/46=2.1$. We need $2.1$ kg of oxygen to $1$ kg of ethanol. The air consists of 23.2 mass-% of oxygen, thus the air--ethanol ratio is $2.1/0.232=9$.

The volume-% of oxygen in air is 20.9%. The volume of the bottle is $V=500$ ml which gives the amount of oxygen to be $0.209V=0.209\times 500=104.5$ ml $=0.1045\times 1.314=0.137$ g of oxygen, which gives the amount of ethanol $0.137/2.1=0.0659$ g $=0.0659/0.789=0.0834$ ml. OR directly using air--ethanol ratio we have $\rho V/9=1.2041\times 0.5/9=0.066$ gram. That amount equals to $V/{\rho }_{\text{Ethanol}}=0.066g/(789g/l)=0.084$ ml $=84$mm³. Almost the same result using the methods. See the attached spreadsheet for detailed calculations.

The energy released by burning ethanol is 17.9 kJ/ml. Thus, $0.084$ ml of ethanol releases $1,50$ kJ of energy. This energy is converted into heat, sound and projectile motion (plus others).

Densities

	Density $\rho$	At
Oxygen (g)	1.429 g/l	STP
Oxygen (g)	1.314 g/l	20 °C
Ethanol (l)	789.45 g/l	20 °C
Air (l)	1.2041 g/l	20 °C

File:Ethanol oxygen combustion.ods

The inner diameter of the rocket bottle is 25 mm. The height of the ethanol in the cap need to be $V=\pi r^{2}h⟺h=\frac{V}{\pi r^2}=\frac{84m{m}^3}{\pi 12.5^{2}m{m}^2}=0.17$ mm.

Exothermic Reaction and Energy release

Specific Heat

The energy is transferred into pressure, sound, etc. The isochoric specific heat $C_v$ of air is $C_v=0.7171$ kJ/(kgK) at 18 centigrade. At 180 degrees Celsius $C_v=0.7352$ kJ/(kgK). Thus, the energy released heats

$\begin{array}{cc}\Delta E& =C_v\Delta Tm\\ \Delta T& =\frac{\Delta E}{C_{v}m}\\ & =\frac{1.5\text{kJ}}{0.7\text{kJ/(kgK)}\times 0.5\times 10^{-3}}\text{kg}\\ & =2100K\end{array}$

Ideal gas law

The simplest idea is to use ideal gas law for isochoric process $p/T=$ constant gives $p_2=\frac{T_2}{T_1}{p}_1$ which gives force $F=pA$ where $A$ is the diameter of the nozzle.

Thus, the propulsive force is $\begin{array}{cc}F& =pA\\ & =\frac{T_2}{T_1}{p}_{1}A\\ & =\frac{T_1+\Delta T}{T_1}{p}_{1}A\\ & =(1+\frac{\Delta T}{T_1})p_{1}A\\ & =(1+\frac{\Delta E/(C_{v}m)}{T_1})p_{1}A\\ \end{array}$

Detonation velocity

Burn rate, detonation velocity. https://en.wikipedia.org/wiki/Table_of_explosive_detonation_velocities

http://www.explosionsolutions.co.uk/110411016.pdf

https://link.springer.com/article/10.1007/s00193-015-0554-7 $v=1500$ m/s???

Horizontal(?) accelaration due to rapidly expanding air.

Benjamin Robins:

$F(x)=\frac{RPAc}{x}$

where $x$ distance in barrel, $R$ is the initial ratio of hot gas pressure to atmospheric pressure, $P$ is the atmospheric pressure, $A$ is the cross-sectional area of the ball or bore, $c$ is the length of the barrel occupied by the powder charge before ignition.

$E_k={\int }_0^{L}F(x)dx⟺v_0^2=\frac{2RP}{m}\frac{\pi d^{2}c}{4}\ln (L/c)$

where $d$ is the barrel diameter (the bore)

The powder change $p$ is given by $p=\frac{\pi d^{2}c}{4}\eta$ where $\eta$ is the density of gunpowder.

--

https://www.arc.id.au/RobinsOnBallistics.html

The pressure falls as $1/x$.

--

Bernoulli?

--

$\sum F=0$

$F=ma=pA⟺F=m\frac{d^{2}x}{d{t}^2}$

$F=m\frac{dv}{dt}=mv\frac{dv}{dx}=pA⟺mvdv=pAdx$

$\int mvdv={\int }_0^{L}pAdx$

where

$\frac{1}{L}{\int }_0^{L}Pdx=averagepressure=\overrightarrow{p}$

If $A$ is constant

Failed to parse (syntax error): {\displaystyle \frac12 mv^2 = A \int_0^L p dx = A \vec p L }

then

$v=\sqrt{\frac{2\overrightarrow{p}AL}{m{c}_f}}$ and $\overrightarrow{p}=\frac{c_{f}m{v}^2}{2AL}$

Projectile friction, rotational energy, heat transfer: correction factor $c_f$.

The average pressure is 25% of peak pressure

--

http://closefocusresearch.com/calculating-barrel-pressure-and-projectile-velocity-gun-systems

https://www.arc.id.au/CannonBallistics.html

Maximum Flying Distance

By the conservation of energy, all explosive energy is transferred into the kinetic energy.

$E_{\text{explosion}}=E_{\text{kinetic}}$

The distance covered by a projectile with initial velocity $v_0$ is

$s=\frac{v_0^2}{g}\sin (2\alpha )=\frac{}{}$

The drag coefficient $c_d$ needs to be found. For the circular disc (a coin) the drag coefficient is almost constant for all velocities (Reynold numbers). The coefficient of drag for a cylinder in this orientation is about 0.81 so long as the length to diameter ratio is greater than 2, see http://www.aerospaceweb.org/question/aerodynamics/q0231.shtml. The cone in either end gives some complications.

The drag equation for the drag force is $F_D=\frac{1}{2}\rho v^2{c}_{d}A$.

Piezo Crystal

To ignite the air--ethanol mixture, we use piezo crystal. When tjhe piezo crystal is compressed, it will generate an electric charge which creates a spark.

References

https://nptel.ac.in/content/storage2/courses/123106002/MODULE%20-%20I/Lecture%201.pdf

https://www.peacesoftware.de/einigewerte/o2_e.html

http://www.users.miamioh.edu/sommerad/NSF%20Files/drag_coefficient_calculation.pdf

DRAG COEFFICIENTS FOR FLAT PLATES , SPHERES, AND CYLINDERS MOVING AT LOW REYNOLDS NUMBERS IN A VISCOUS FLUID by ALVA MERLE JONES

http://www.aerospaceweb.org/question/aerodynamics/q0231.shtml

Ethanol rockets

https://www.youtube.com/watch?v=zTwz6FGobCA Ethanol Rocket - Cool Science Experiment

https://www.youtube.com/watch?v=4s-SZypWxeg Ethanol Explosion - Cool Science Experiment