Active Structures: Difference between revisions

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Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.
Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.


The advantage of active structures is that they can be much more massive than passive structures <ref group="Footnote">Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.</ref>, enabling structures many kilometers tall without requiring significant tapering. Some proposals for non-rocket launch infrastructure rely on active support, with the advantage of the option for being built by modern, existing materials.
The advantage of active structures is that they can be much more massive than passive structures <ref group="Footnote">Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.</ref>, enabling structures many kilometers tall without requiring significant tapering. Some proposals for [[non-rocket launch infrastructure]] rely on active support, with the advantage of the option for being built by modern, existing materials.


Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.
Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.
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Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.
Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.


The force required to accelerate the active structure is given by Newton’s second law of motion<ref> https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion <br/>Wikipedia article about Newton’s second law of motion</ref>.
Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:
:


<math alt=>F = MA</math>
<math alt=> F=MA_g</math>
* Where <math alt=>F</math> is the force exerted on the deflector.
* Where <math alt=>M</math> is the mass of the structure.
* Where <math alt=>A</math> is the antigravity acceleration.


This also applies to the mass stream, the cumulative force of the stream must be equal to this force. If the total mass of the mass stream is lower than the mass of the structure, the acceleration for the stream must be higher.
* Where F is the weight of the active structure.
* Where M is the mass of the active structure.


The acceleration required for each particle or pellet of the stream is calculated with:
With that information in mind, the kinetic energy of the stream is given by:


<math alt=>A_s = F/N_m/M_m</math>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
* Where <math alt=>A_s</math> is the acceleration of the particle.
<div style="font-weight:bold;line-height:1.6;">Long form</div>
* Where <math alt=>N_m</math> is the amount of particles or pellets in the stream. The amount for particles is given by:
<div class="mw-collapsible-content">
: <math alt=>M_s / N_s</math>
<math alt=> KE_s = (v-\sqrt{v^2-2A_gh} )/A_g \cdot F/(2 \sqrt{v^2-2A_gh}) v^2 </math>
<div style="text-indent:50px;>
</div></div>
• Where <math alt=>N_s</math> is the molar mass of the stream material.
 
</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="text-indent:50px;>
<div style="font-weight:bold;line-height:1.6;">Simple form</div>
• Where <math alt=>M_s</math> is the mass of the stream.  
<div class="mw-collapsible-content">
</div>
<math alt=> KE_s = hF(b(\frac{1}{\sqrt{1-1/b }}-1)) </math>
* Where <math alt=>M_m</math> is the mass of the particle or the pellet.
<br/><math alt=> b = v^2/2A_gh</math>
</div></div>
 
* Where <math alt=>KE_s</math> is the kinetic energy of the stream.
* Where <math alt=>v</math> is the velocity of the stream.
* Where <math alt=>h</math> is the altitude of the deflector.
 
Notes:
# As a rule of thumb, whenever <math alt=>v>\sqrt{10A_gh}</math> energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with <math alt=> KE_s \approx 0.6hF</math> instead. The choice of pellet characteristics can then fall under engineering constraints.
# The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.
 
Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:
 
<math alt=>KE_p=KE_s/N</math>
 
* Where <math alt=>KE_p</math> is the kinetic energy of the particle/pellet in the stream.
* Where <math alt=>N</math> is the number of pellets/particles in the stream.
 
Additionally, the mass of the pellet/particle can be calculated with:
 
<math alt=>m=2KE_p/v^2</math>
 
* Where <math alt=>m</math> is mass of the particle/pellet.
 
A calculated example:
 
: Given a 1,000 ton structure on Earth (implying a <math alt=>A_g</math> of 9.8066 m/s<sup>2</sup>), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.


=Active Structures=
=Active Structures=
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* [https://www.youtube.com/watch?v=i4kTF0xw3y8 Small-scale demonstration of incomplete mass stream technology using water].
* [https://www.youtube.com/watch?v=i4kTF0xw3y8 Small-scale demonstration of incomplete mass stream technology using water].


* The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.
* The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.</code>


==Proposed==
==Proposed==
* The [[Lofstrom Launch Loop]] is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream.
* The [[Lofstrom Launch Loop]] is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream; the mass stream is a solid continuous iron rotor. The loop suffers from some unaddressed instability concerns.


* The [[Bolonkin Kinetic Space Tower / Anti-Gravitator]], using cables and rollers instead of using mass streams to support various things.
* The [[Space Cable]] is a similar concept to the launch loop. It differs from the launch loop in that it uses magnetically interacting bolts instead of a continuous rotor, is smaller in length, and has addressed instability concerns.


* The [[Space Fountain / Space Tower]], which aims to make a space elevator using active support. Its technology can also extend to shorter buildings.
* The [[Bolonkin Kinetic Space Tower / Anti-Gravitator]], using cables and rollers instead of using rotors or bolts or pellets to support various things. It has completely unaddressed instability concerns.


* The [[Orbital Ring]], which uses a mass stream travelling faster than the orbital velocity in order to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.
* The [[Space Fountain / Space Tower]], which aims to make a [[Space Elevator|space elevator]] using active support. Its technology can also extend to shorter buildings.


* The [[Pneumatic Freestanding Tower]], which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks.
* The [[Orbital Ring]], which uses a mass stream travelling faster than the [[Orbits|orbital velocity]] to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.
 
* The [[Pneumatic Freestanding Tower]], which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks. The main concerns are buckling due to the height of the tower, though it has mechanisms in place to prevent this.


=Control Systems=
=Control Systems=
[[Control Systems]]
[[Control Systems]]
<br/>Active structures can suffer from stability issues, such as, for example in the launch loop, unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. the Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration<ref>https://en.wikipedia.org/wiki/Tuned_mass_damper <br/>https://en.wikipedia.org/wiki/Active_structure <br/> https://www.youtube.com/watch?v=f1U4SAgy60c<br/>Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.</ref>
<br/>Active structures can suffer from stability issues as mentioned before, such as for example in the launch loop unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. The unpredictable winds in the atmosphere are also a concern. Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration<ref>https://en.wikipedia.org/wiki/Tuned_mass_damper <br/>https://en.wikipedia.org/wiki/Active_structure <br/> https://www.youtube.com/watch?v=f1U4SAgy60c<br/>Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.</ref>.
.
 
=Safety Engineering=
Active structures are subject to the problem of how to ensure that they don't fail, or a bit worse, only fail gracefully, when something in their active systems breaks down. This is not a question of if; entropy breaks everything. All electrical and mechanical systems have a mean time between failure. If an active structure is only supported by a single active support "string", the failure of that "string" will cause a catastrophic failure.
 
By adding redundancy to our structure, we can ensure it can tolerate the failure of some of its components, and possible "fail gracefully" with a time period allowing for evacuation and response measures to be taken, and/or a "controlled failure" of the structure in which terminal velocity of the falling structure and the production of energetic debris is reduced.
 
Accepting an increase in mass, we split the support power required between some <math>N</math> "strings" operating in parallel. Each string only operates at a partial power, with an oversize factor of <math>1/N</math> added on top. If one or a few of the strings in the parallel system fail, the other strings are ramped to full power, generating sufficient support power to ensure the active structure remains standing despite the failure of some of its "strings". We can also use this to shut down strings intentionally for inspections, maintenance, overhauls, or other work, overall allowing us also to keep the active structure alive over time by incrementally replacing and upgrading its parts.
 
Usually safety redundancies have at least three systems operating in parallel. You may consider having a larger number of systems.
 
Note: in terms of safety engineering, no degree of redundancy reduces the chance of a total, catastrophic failure to zero. There is some chance that even a very redundant system may experience the failure of all its critical components at once. But this is given for any system, and you could consider pushing the safety factor of an active structure to the same point (or beyond) any passive structure.


=Additional Reading=
=Additional Reading=
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<references group="GAE"/>
<references group="GAE"/>


==Derivation of the Particle/Pellet Acceleration Equation==
==Derivation of the Stream Kinetic Energy Equations==
# If only the total force is known, we must operate on it with something in order to get the force per particle/pellet. We divide the total force by the amount of particles/pellets in the stream.
===Long Form===
# <math alt=>A = F/m</math> gives the acceleration needed for the particle to exert that particular amount of force on the deflector.
# Given a classical particle accelerating upwards, it will achieve a final velocity of <math alt=>v_f=\sqrt{v^2-2A_gh}</math> where v is the initial speed.<ref group="SKE">https://www.school-for-champions.com/science/energy_potential.htm <br/> The expression is derived from the expression for the velocity of falling objects: <math alt=>V=\sqrt{2A_gh}</math>. This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.</ref>.
# The necessary force for deflecting the particle down is given by <math alt=>F=2\dot{m}v_f</math><ref group="SKE"> <math alt=>\dot{m}</math> is the mass of the particle <math alt=>\bullet</math> the repetition rate, the repetition rate being analogous to <math alt=>N</math>, and <math alt=>\dot{m}</math> analogous to <math alt=>KE_s/N</math>.</ref>. The total energy content is then <math alt=>mv^2/2</math>.
# The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time <math alt=>T</math> where <math alt=>v_f = v - A_gT \Rightarrow \color{green}{T = (v-v_f)/A_g}</math>, for a full circulation it takes <math alt=>2T</math>.
# Therefore <math alt=>m = 2T\dot{m}</math>, so <math alt=> KE_s = T\dot{m}v^2</math>. To get rid of <math alt=>\dot{m}</math> we divide <math alt=>F</math> by <math alt=>2v_f</math>.
# This gives our stream kinetic energy equation, taking the limit as <math alt=>v</math> monotonically approaches infinity <math alt=>hF/2</math> from above.
 
===Simple Form===
# The simple form is just a factorized version of the long form, where <math alt=>k</math> is defined as <math alt=>2A_gh/v^2</math> or <math alt=>1/b</math>.
 
===Regarding the notes===
# The simple form shows that whenever <math alt=>v^2/2A_gh > 5</math>, the energy content in the stream is less than <math alt=>0.6hF</math>. Rearranging the equation for <math alt=>v</math> and accounting for 5 gives the rule of thumb.
# The upper bound nature of the equations means that:
:: For <math alt=>v > \sqrt{10A_gh}</math>, <math alt=>0.5hF < E < 0.6hF</math> for any <math alt=>h</math> and any gravitational field that is decreasing.
 
----
'''Reference for the Derivation of the Stream Kinetic Energy Equations'''
<references group="SKE"/>


=Credit=
=Credit=
Line 93: Line 144:
* To SOPHONT SIMP and pMXoTJFu for sweeping the article.
* To SOPHONT SIMP and pMXoTJFu for sweeping the article.
* To AdAstraGames for contributing useful information and sweeping the article.
* To AdAstraGames for contributing useful information and sweeping the article.
* To Sevoris for writing the safety engineering section.
* To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.
[[Category:Transportation & Infrastructure]][[Category:Infrastructure]][[Category:Structures]][[Category:Physics & Math & Engineering‏‎]]

Latest revision as of 13:25, 23 April 2024

Active structures rely on constant power input, in addition to the material and mechanical properties of their construction materials (active support). This is in contrast to passive structures, which solely rely on the aforementioned properties (passive support). An example of an active structure is the force of a jet of water holding up a tethered lid of a trashcan in the air, versus the passive structure of a concrete pillar.

Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.

The advantage of active structures is that they can be much more massive than passive structures [Footnote 1], enabling structures many kilometers tall without requiring significant tapering. Some proposals for non-rocket launch infrastructure rely on active support, with the advantage of the option for being built by modern, existing materials.

Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.

  1. ↑ Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.

Active Support Principles

As gravity[1]is what pulls down objects, active support must counteract gravity. Since it is the acceleration that causes objects to be pulled down, it follows that active support should accelerate in the opposite direction; the acceleration must be equal to gravity to support the structure.

The gravitational acceleration of the planet is given by:

  • Where is the gravitational acceleration.
  • Where is the universal gravitational constant, defined to be 6.6743e11 m3/kg/s2.
  • Where is the mass of the planet.
  • Where is the radius of the planet.

On Earth, equals 9.80665 m/s2, a constant known as .

Mass Streams

Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.

Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:

  • Where F is the weight of the active structure.
  • Where M is the mass of the active structure.

With that information in mind, the kinetic energy of the stream is given by:

Long form

Simple form


  • Where is the kinetic energy of the stream.
  • Where is the velocity of the stream.
  • Where is the altitude of the deflector.

Notes:

  1. As a rule of thumb, whenever energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with instead. The choice of pellet characteristics can then fall under engineering constraints.
  2. The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.

Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:

  • Where is the kinetic energy of the particle/pellet in the stream.
  • Where is the number of pellets/particles in the stream.

Additionally, the mass of the pellet/particle can be calculated with:

  • Where is mass of the particle/pellet.

A calculated example:

Given a 1,000 ton structure on Earth (implying a of 9.8066 m/s2), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.

Active Structures

Existing

  • The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.

Proposed

  • The Lofstrom Launch Loop is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream; the mass stream is a solid continuous iron rotor. The loop suffers from some unaddressed instability concerns.
  • The Space Cable is a similar concept to the launch loop. It differs from the launch loop in that it uses magnetically interacting bolts instead of a continuous rotor, is smaller in length, and has addressed instability concerns.
  • The Orbital Ring, which uses a mass stream travelling faster than the orbital velocity to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.
  • The Pneumatic Freestanding Tower, which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks. The main concerns are buckling due to the height of the tower, though it has mechanisms in place to prevent this.

Control Systems

Control Systems
Active structures can suffer from stability issues as mentioned before, such as for example in the launch loop unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. The unpredictable winds in the atmosphere are also a concern. Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration[2].

Safety Engineering

Active structures are subject to the problem of how to ensure that they don't fail, or a bit worse, only fail gracefully, when something in their active systems breaks down. This is not a question of if; entropy breaks everything. All electrical and mechanical systems have a mean time between failure. If an active structure is only supported by a single active support "string", the failure of that "string" will cause a catastrophic failure.

By adding redundancy to our structure, we can ensure it can tolerate the failure of some of its components, and possible "fail gracefully" with a time period allowing for evacuation and response measures to be taken, and/or a "controlled failure" of the structure in which terminal velocity of the falling structure and the production of energetic debris is reduced.

Accepting an increase in mass, we split the support power required between some "strings" operating in parallel. Each string only operates at a partial power, with an oversize factor of added on top. If one or a few of the strings in the parallel system fail, the other strings are ramped to full power, generating sufficient support power to ensure the active structure remains standing despite the failure of some of its "strings". We can also use this to shut down strings intentionally for inspections, maintenance, overhauls, or other work, overall allowing us also to keep the active structure alive over time by incrementally replacing and upgrading its parts.

Usually safety redundancies have at least three systems operating in parallel. You may consider having a larger number of systems.

Note: in terms of safety engineering, no degree of redundancy reduces the chance of a total, catastrophic failure to zero. There is some chance that even a very redundant system may experience the failure of all its critical components at once. But this is given for any system, and you could consider pushing the safety factor of an active structure to the same point (or beyond) any passive structure.

Additional Reading

Additional References

  1. ↑ https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
    Wikipedia article about gravity in classical mechanics.
  2. ↑ https://en.wikipedia.org/wiki/Tuned_mass_damper
    https://en.wikipedia.org/wiki/Active_structure
    https://www.youtube.com/watch?v=f1U4SAgy60c
    Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.

Derivation of the Gravitational Acceleration Equation

  1. The gravitational force equation[GAE 1] is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = G(m_1m_1/r^2)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is the force, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is the universal gravitational constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} is the mass of the first object, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} is the mass of the second object and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the distance between their centers of mass.
  2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} becomes the radius of the planet from the frame of reference of a planet.
  3. The equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = MA} , which gives the force needed to accelerate an object is rearranged to give acceleration, thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=F/M}
  4. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A =F/M} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the acceleration, divide by the mass of the second object and therefore cancel out its inclusion in the expression, giving you the gravitational acceleration equation.

Reference for the Derivation of the Gravitational Acceleration Equation

  1. ↑ https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
    Reference for the gravitational force equation.

Derivation of the Stream Kinetic Energy Equations

Long Form

  1. Given a classical particle accelerating upwards, it will achieve a final velocity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_f=\sqrt{v^2-2A_gh}} where v is the initial speed.[SKE 1].
  2. The necessary force for deflecting the particle down is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=2\dot{m}v_f} [SKE 2]. The total energy content is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle mv^2/2} .
  3. The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_f = v - A_gT \Rightarrow \color{green}{T = (v-v_f)/A_g}} , for a full circulation it takes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T} .
  4. Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 2T\dot{m}} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KE_s = T\dot{m}v^2} . To get rid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} we divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2v_f} .
  5. This gives our stream kinetic energy equation, taking the limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} monotonically approaches infinity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle hF/2} from above.

Simple Form

  1. The simple form is just a factorized version of the long form, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2A_gh/v^2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/b} .

Regarding the notes

  1. The simple form shows that whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^2/2A_gh > 5} , the energy content in the stream is less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.6hF} . Rearranging the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} and accounting for 5 gives the rule of thumb.
  2. The upper bound nature of the equations means that:
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v > \sqrt{10A_gh}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.5hF < E < 0.6hF} for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} and any gravitational field that is decreasing.

Reference for the Derivation of the Stream Kinetic Energy Equations

  1. ↑ https://www.school-for-champions.com/science/energy_potential.htm
    The expression is derived from the expression for the velocity of falling objects: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\sqrt{2A_gh}} . This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.
  2. ↑ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} is the mass of the particle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet} the repetition rate, the repetition rate being analogous to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} analogous to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KE_s/N} .

Credit

To Tshhmon for writing the article

  • To SOPHONT SIMP and pMXoTJFu for sweeping the article.
  • To AdAstraGames for contributing useful information and sweeping the article.
  • To Sevoris for writing the safety engineering section.
  • To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.
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