Can someone help me figure out how to solve the following thought problem which involves magnetic induction, electrolysis of water, and buoyancy?
A system containing a spool of wire is dropped into a well of seawater of unknown depth (to be determined in the problem). The well is in a magnetic field, and a current is induced in the spool of wire as it falls. The induced current in turn drives an electrolytic reaction which generates hydrogen and oxygen gases which are collected in a balloon attached to the spool.
As more gases are generated, they displace more and more water increasing the buoyancy of the combined spool + balloons system.
Assume that the system is at Standard Temperature (25 C) and Pressure (1 Atmosphere) at the surface of the well and temperature of the sea water in the well is also 25 C.
Using these equations:
Buoyancy acceleration:
g(ρV-m) where: a = acceleration, V= volume of displaced water,
a= -----------, m= mass of system, ρ = density of H20 (0.999505 at 20° C)
m+ ρV g is the gravitation acceleration constant (9.8m/sec2)
Faraday's Law:
-N d\(\Phi\) ε= emf in volts,
ε= ------ N = number of turns of wire,
dt \(\Phi\) = electromotive force in volts,
ideal gas law
pV =nRT p = pressure in pascals, V = volume of gas in cubic meters,
n = amount of gas in moles,
R = gas constant (8.314472 J·K−1·mol−1),
T = absolute temperature in degrees Kelvin
Nernst Equation (for electrolysis):
E(cell) = E(0) - RT ln Q where E(cell) is the cell potential (emf)
--- E(0) is the standard cell potential at the temperature
zF z is the number of moles of electrons transferred
F = Faraday constant (= 9.648 533 99(24)×104 C mol−1)
Also Assume:
Density of sea water = 1025 kg/m³
pressure increases by 1 atm with each 10 m of depth in sea water.
1 atm = 01.325 kPa
Electrolysis of 1 mole of H20 requires 282.1 kJ of energy yielding 1 mole H2 and 1/2 moles of O2 gas.
mass of 1 mole of H2 = 1.0794g.
1 mole of O2 gas weighs = 16g.
The system:
1 cylindrical spool of 16 gauge magnet wire (with 1000 windings). The inner diameter of the spool is 2.5 cm and the height of the spool is 5 cm. The spool is weighted at the bottom and a deflated rubber balloon is stretched over the entire spool. The 2 ends of the wire are bare and turned up inside the balloon and will act as electrodes. The whole assembly of spool, wire and balloon occupies 100 cubic cm of space and weighs 1 kg.
The spool is released at water level and begins falling down the middle of a well that is 10 cm in diameter, with height to be determined according to the problem defined below.
On opposite sides of the well are 2 bar magnets. Each magnet is 1.25 cm wide, and .3125 cm thick, and stretches the height and depth of the well. Each is magnetized through its thickness, with one magnet having its N pole on the inside of the well and the other magnet having its S pole on the inside of the well. The surface field strength of each magnet separately is 1000 gauss.
As the spool falls through the magnetic field, it generates an electrolysis current that generates H2 and O2 gases that inflate the balloon.
Since the spool-balllon system occupies currently displaces 0.1L of water, 1.015 L of gas will have to be produced to displace an additional 1L of sea water (at a total weight of 1.025kg) and reach zero buoyancy.
The speed of descent is determined by buoyancy. The rate of acceleration at each point is g *(V - m) / (m + V) where V = the total volume of the system (including the .1 L for the spool component), m is total mass of the system ( including the 0.999505 kg spool) and g is the gravitational constant of 9.8 m/sec2.
Note that volume, V, occupied by the gas produced varies with Pressure of the surrounding water at its current depth.
Obviously, the rate of descent is going to be decelerating as more gas is produced, and so is the number of field lines the spool falls through each second, so the rate of production of gas will also decline. Also since pressure is increasing with depth, the volume of a fixed amount of gas will be less at greater depths with increased pressure.
The questions to be answered are:
1) How deep does the well have to be for the system to fall all the way to its point of neutral buoyancy without hitting the bottom of the well first?
2) How long will it take from release until spool-balloon system reaches neutral buoyancy?
3) What is the shape of the curve of distance traveled (depth) vs. time plot, and what is the equation governing it?
4) What is the shape of the volume of gas produced vs. time plot, and what is the equation governing it?
5) What is the impact of doubling the surface strength of the magnets to 2000 gauss, or halving it to 500 gauss? Of doubling the number of turns of wire to 2000 turns, of halving it to 500 turns?
A system containing a spool of wire is dropped into a well of seawater of unknown depth (to be determined in the problem). The well is in a magnetic field, and a current is induced in the spool of wire as it falls. The induced current in turn drives an electrolytic reaction which generates hydrogen and oxygen gases which are collected in a balloon attached to the spool.
As more gases are generated, they displace more and more water increasing the buoyancy of the combined spool + balloons system.
Assume that the system is at Standard Temperature (25 C) and Pressure (1 Atmosphere) at the surface of the well and temperature of the sea water in the well is also 25 C.
Using these equations:
Buoyancy acceleration:
g(ρV-m) where: a = acceleration, V= volume of displaced water,
a= -----------, m= mass of system, ρ = density of H20 (0.999505 at 20° C)
m+ ρV g is the gravitation acceleration constant (9.8m/sec2)
Faraday's Law:
-N d\(\Phi\) ε= emf in volts,
ε= ------ N = number of turns of wire,
dt \(\Phi\) = electromotive force in volts,
ideal gas law
pV =nRT p = pressure in pascals, V = volume of gas in cubic meters,
n = amount of gas in moles,
R = gas constant (8.314472 J·K−1·mol−1),
T = absolute temperature in degrees Kelvin
Nernst Equation (for electrolysis):
E(cell) = E(0) - RT ln Q where E(cell) is the cell potential (emf)
--- E(0) is the standard cell potential at the temperature
zF z is the number of moles of electrons transferred
F = Faraday constant (= 9.648 533 99(24)×104 C mol−1)
Also Assume:
Density of sea water = 1025 kg/m³
pressure increases by 1 atm with each 10 m of depth in sea water.
1 atm = 01.325 kPa
Electrolysis of 1 mole of H20 requires 282.1 kJ of energy yielding 1 mole H2 and 1/2 moles of O2 gas.
mass of 1 mole of H2 = 1.0794g.
1 mole of O2 gas weighs = 16g.
The system:
1 cylindrical spool of 16 gauge magnet wire (with 1000 windings). The inner diameter of the spool is 2.5 cm and the height of the spool is 5 cm. The spool is weighted at the bottom and a deflated rubber balloon is stretched over the entire spool. The 2 ends of the wire are bare and turned up inside the balloon and will act as electrodes. The whole assembly of spool, wire and balloon occupies 100 cubic cm of space and weighs 1 kg.
The spool is released at water level and begins falling down the middle of a well that is 10 cm in diameter, with height to be determined according to the problem defined below.
On opposite sides of the well are 2 bar magnets. Each magnet is 1.25 cm wide, and .3125 cm thick, and stretches the height and depth of the well. Each is magnetized through its thickness, with one magnet having its N pole on the inside of the well and the other magnet having its S pole on the inside of the well. The surface field strength of each magnet separately is 1000 gauss.
As the spool falls through the magnetic field, it generates an electrolysis current that generates H2 and O2 gases that inflate the balloon.
Since the spool-balllon system occupies currently displaces 0.1L of water, 1.015 L of gas will have to be produced to displace an additional 1L of sea water (at a total weight of 1.025kg) and reach zero buoyancy.
The speed of descent is determined by buoyancy. The rate of acceleration at each point is g *(V - m) / (m + V) where V = the total volume of the system (including the .1 L for the spool component), m is total mass of the system ( including the 0.999505 kg spool) and g is the gravitational constant of 9.8 m/sec2.
Note that volume, V, occupied by the gas produced varies with Pressure of the surrounding water at its current depth.
Obviously, the rate of descent is going to be decelerating as more gas is produced, and so is the number of field lines the spool falls through each second, so the rate of production of gas will also decline. Also since pressure is increasing with depth, the volume of a fixed amount of gas will be less at greater depths with increased pressure.
The questions to be answered are:
1) How deep does the well have to be for the system to fall all the way to its point of neutral buoyancy without hitting the bottom of the well first?
2) How long will it take from release until spool-balloon system reaches neutral buoyancy?
3) What is the shape of the curve of distance traveled (depth) vs. time plot, and what is the equation governing it?
4) What is the shape of the volume of gas produced vs. time plot, and what is the equation governing it?
5) What is the impact of doubling the surface strength of the magnets to 2000 gauss, or halving it to 500 gauss? Of doubling the number of turns of wire to 2000 turns, of halving it to 500 turns?
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