HOW BIG IS OUR UNIVERSE

As technology has evolved, astronomers are able to look back in time to the moments just after the Big Bang. This might seem to imply that the entire universe lies within our view. But the size of the universe depends on a number of things, including its shape and expansion. Just how big is the universe? The truth is, scientists can't put a number on it.
The observable universe
Astronomers have measured the age of the universe to be approximately 13.8 billion years old. Because of the connection between distance and the speed of light, this means they can look at a region of space that lies 13.8 billion light-years away. Like a ship in the empty ocean, astronomers on Earth can turn their telescopes to peer 13.8 billion light-years in every direction, which puts Earth inside of an observable sphere with a radius of 13.8 billion light-years. The word "observable" is key; the sphere limits what scientists can see but not what is there.
But though the sphere appears almost 28 billion light-years in diameter, it is far larger. Scientists know that the universe is expanding. Thus, while scientists might see a spot that lay 13.8 billion light-years from Earth at the time of the Big Bang, the universe has continued to expand over its lifetime. Today, that same spot is 46 billion light-years away, making the diameter of the observable universe a sphere around 92 billion light-years.

Ways to improve eye sight.

During the second world war, British propagandists circulated rumours that RAF pilots were such good night fliers because all the carrots they ate helped them to see in the dark. In reality the British were trying to keep their use of radar secret.
Yet it turns out that there is some truth to the idea that diet can affect our eyes. Retinal cells contain three yellow pigments – lutein, zeaxanthin and meso-zeaxanthin – which absorb near-ultraviolet light, protecting the eye from its damaging effects and reducing glare. These pigments are concentrated in the centre of the retina that produces the sharp central area of our vision, the macula. "It's like wearing internal sunglasses," says Billy Hammond of the University of Georgia in Athens. "It reduces the light intensity and absorbs scatter."


PHONE FOCOSSING FOR YOUR EYES
The lenses in our eyes get stiffer as we age, making it harder to focus on things that are close up. This is why people start to need reading glasses from their 40s onwards. Eventually, nearly everyone will own a pair. But it can be a pain putting on your glasses every time you look at your phone, for instance, assuming you can remember where you put them.
Glasses work by partly focusing light before it hits the eye – so if you are looking at a screen, why not make it do the focusing for you? A team at the Massachusetts Institute of Technology has shown that plastic screen covers can correct for all kinds of vision problems – effectively, the screen wears the glasses. But rather than making plastic covers tailored to individuals' eyes, the team want to exploit the ability of

CONJUGATE BEAM METHOD(SLOPES ANDDEFLECTIONS)

Properties of conjugate beam1.The length of a conjugate beam is always equal to the length of the actual beam.
2.The load on the conjugate beam is the M/EI diagram of the loads on the actual beam.
3.A simple support for the real beam remains simple support for the conjugate beam.
4.A fixed end for the real beam becomes free end for the conjugate beam.
5.The point of zero shear for the conjugate beam corresponds to a point of zero slope for the real beam.
6.The point of maximum moment for the conjugate beam corresponds to a point of maximum deflection for the real beam.

Supports of Conjugate Beam
Knowing that the slope on the real beam is equal to the shear on conjugate beam and the deflection on real beam is equal to the moment on conjugate beam, the shear and bending moment at any point on the conjugate beam must be consistent with the slope and deflection at that point of the real beam. Take for example a real beam with fixed support; at the point of fixed support there is neither slope nor deflection, thus, the shear and moment of the corresponding conjugate beam at that point must be zero. Therefore, the conjugate of fixed support is free end.
 
Real beam support and its corresponding conjugate beam support

Examples of Beam and its Conjugate
The following are some examples of beams and its conjugate. Loadings are omitted.
 

TURBINES.

Spillways

Generating station construction and refurbishment



  

Turbines


Turbine type according to available head
Propeller up to 15 metres
Kaplan up to 30 metres
Francis 10 to 300 metres
Pelton 300 metres and over


Francis turbine

The Kaplan turbine is similar to the propeller turbine except that its blades are adjustable.

Pelton turbine
Turbines convert the energy of rushing water, steam or wind into mechanical energy to drive a generator. The generator then converts the mechanical energy into electrical energy. In hydroelectric facilities, this combination is called a generating unit.
Francis turbine
The most commonly used turbine in Hydro-Québec's power system. Water strikes the edge of the runner, pushes the blades and then flows toward the axis of the turbine. It escapes through the draft tube located under the turbine. It was named after James Bicheno Francis (1815-1892), the American engineer who invented the apparatus in 1849.
Kaplan turbine
Austrian engineer Viktor Kaplan (1876-1934) invented this turbine. It's similar to the propeller turbine, except that its blades are adjustable; their position can be set according to the available flow. This turbine is therefore suitable for certain run-of-river generating stations where the river flow varies considerably.
Each Kaplan turbine at Brisay generating station weighs 300 tonnes... That's the weight of 50 African elephants.
Propeller turbine
Since they can reach very high rotation speeds, propeller turbines are effective for low heads. Consequently, this type of turbine is suitable for run-of-river power stations.
Pelton turbine
Named after its American inventor, Lester Pelton (1829-1908), this turbine uses spoon-shaped buckets to harness the energy of falling

STRING THEORY

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.[1] String theory aims to explain all types of observed elementary particles using quantum states of these strings. In addition to the particles postulated by the standard model of particle physics, string theory naturally incorporates gravity and so is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. Besides this potential role, string theory is now widely used as a theoretical tool and has shed light on many aspects of quantum field theory and quantum gravity.[2]
The earliest version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It was then developed into superstring theory, which posits that a connection – a "supersymmetry" – exists between bosons and the class of particles called fermions. String theory requires the existence of extra spatial dimensions for its mathematical consistency. In realistic physical models constructed from string theory, these extra dimensions are typically compactified to extremely small scales.
String theory was first studied in the late 1960s[3] as a theory of the strong nuclear force before being abandoned in favor of the theory of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. Five consistent versions of string theory were developed until it was realized in the mid-1990s that they were different limits of a conjectured single 11-dimensional theory now known as M-theory.[4]
Many theoretical physicists, including Stephen Hawking, Edward Witten and Juan Maldacena, believe that string theory is a step towards the correct fundamental description of nature: it accommodates a consistent combination of quantum field theory and general relativity, agrees with insights in quantum gravity (such as the holographic principle and black hole thermodynamics) and has passed many non-trivial checks of its internal consistency.[citation needed] According to Hawking, "M-theory is the only candidate for a complete theory of the universe."[5] Other physicists, such as Richard Feynman,[6][7] Roger Penrose[8] and Sheldon Lee Glashow,[9] have criticized string theory for not providing novel experimental predictions at accessible energy scales.


Overview
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ALBERT EINSTEIN-QUOTE ON SCIENCE AND RELIGION

Context of Albert Einstein's quote “Science without religion is lame; religion without science is blind”
This much-circulated Einstein quote has some interesting sidelights. It was written in the paper 'Science, Philosophy and Religion', that Einstein prepared for initial meeting of the Conference on Science, Philosophy and Religion in Their Relation to the Democratic Way of Life, at the Jewish Theological Seminary of America, New York City (9-11 Sep 1940).
In Ralph Keyes, The Quote Verifier (2006), 51, Keyes compares Einstein's own subsequent quote:
“Epistomology without contact with science becomes an empty scheme. Science without epistomology is—insofar as it is thinkable at all—primitive and muddled.”
Alice Calaprice, in The Quotable Einstein (1996), 153, compares the earlier quote by Immanuel Kant:
“Notion without intuition is empty; intuition without notion is blind.”
Calaprice states Einstein made this quote in a written contribution to the Symposium, and gives its date as 1941, the date of publication (?) of the Symposium proceedings. Calaprice cites Einstein Archive 28-523; and Einstein's Ideas and Opinions, 41-49.
Einstein laid out his agnostic views on religion late in his life, when on 3 Jan 1954, he wrote a letter of thanks, in German, to Jewish philosopher Eric Gutkind, who had sent him a copy of his book, Choose Life: The Biblical Call to Revolt. In this correspondance, written in the year before his death, Einstein explained his view of religions as “childish superstition:”
“The word god is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honorable, but still primitive legends which are nevertheless pretty childish. No interpretation no matter how subtle can (for me) change this.”
“For me the Jewish religion like all other religions is an incarnation of the most childish superstitions,” the letter continues. “And the Jewish people to whom I gladly belong and with whose mentality I have a deep affinity have no different quality for me than all other people. As far as my experience goes, they are also no better than other human groups, although they are protected from the worst cancers by a lack of power. Otherwise I cannot see anything ‘chosen’ about them.”
The letter has changed hands at ever-increasing prices, and in 2012 was sold on eBay, with bids starting at $3 million, just four years after it fetched $404,000 at auction. The lot included the letter and envelope, with its stamp and postmark. With two bids, the final sale price to an anonymous buyer when the auction closed on 18 Oct 2012 for was reported as $3,000,100.

BLACK HOLE-A MYTH EXPLAINED BY SOME PHYSICISTS.

No Such Thing as a Black Hole?
New theories question just about everything we thought we knew about nature’s bottomless pits.
By Corey S. Powell|Thursday, February 26, 2015
RELATED TAGS: BLACK HOLES, PHYSICS






blackhole2
blackhole2
The black hole from Interstellar is based on current theory, but that theory may not match reality.
Double Negative/Paramount/Warner Brothers
It takes a special kind of personality to argue passionately about the nature of objects that are hundreds of trillions of miles away, impossible to see and impossible even to describe using the known laws of physics. And yet, in the woolly world of modern astrophysics, such personalities are not in short supply. Lately they’ve engaged in a full-on debate about the nature of black holes, with some of the most fundamental notions about these strange objects suddenly under attack.
Despite what you saw in the movie Interstellar, black holes may not be black, and they may not be holes, either. Some theorists argue that the event horizon of a black hole — the boundary where light, matter and Matthew McConaughey vanish from our universe — is actually a brilliant, blistering inferno. Others propose that black holes are more properly described as “gray holes” with fuzzy, leaky outer boundaries. And a few agitators argue that the whole debate is off track because nature makes it impossible for black holes to form in the first place.
All of which might seem like so much theoretical navel-gazing, except that the debate over black holes is a proxy in a much grander battle. Right now, physics is split in two: Quantum mechanics describes small, fast phenomena while general relativity describes large, slow ones.
But in the extreme conditions around a black hole, time and space get so stretched that the two theories are forced to overlap. Making sense of what happens at that intersection is crucial for developing a “theory of everything”— a unified set of physical laws that describe the entire cosmos, from the Big Bang to a Big Mac.




radiation-illustration
radiation-illustration
As material (dust in this illustration) falls toward a black hole, it heats up and emits radiation. No one knows what happens next.
Richard Kail/Science Source
Fire in the Hole!
The trouble began, like so many confounding concepts in modern physics, in the brain of Stephen Hawking. Four decades ago, he realized that a black hole’s event horizon is inherently leaky; quantum processes allow a slow but steady flow of particles away from the black hole, a process now known as Hawking radiation. Given enough time, a black hole can evaporate completely. The idea of matter escaping the alleged point-of-no-return was surprising (it’s a central plot point in that other recent movie about black holes, the biographical The Theory of Everything), but the fate of information that falls into the black hole was what really troubled Hawking’s colleagues.
In the current view, which Hawking helped formulate, every event in the universe contains quantum information. When objects fall into a black hole, they take their information with them. That’s fine so long as the information stays in there, but if the black hole evaporates, things go all wonky. Coherent information goes in, but what comes out is just noise — Hawking radiation is 100 percent content-free.
Falling into a black hole seems to destroy a slice of reality, which just makes no sense. Hawking and other theorists began searching desperately for ways to prevent information from getting into the black hole in the first place, for ways to let it back out, or even for ways to make peace with the possibility that some information really can be lost forever.
In 2012, Joseph Polchinski of the University of California, Santa Barbara, offered a novel solution: As soon as an object gets pulled across the event horizon, it hits a firewall — an inferno so intense that it erases all the quantum information that object contained. No information gets lost because no information actually makes it through to the black hole’s interior. Polchinski pictures the event horizon as a kind of quantum eraser that neatly makes the information paradox go away.
If you think that sounds like a dodgy answer, rest assured that many of Polchinski’s colleagues (and even Polchinski himself) thought so, too — though perhaps not for the reasons you’d expect. The problem is that the rules of general relativity mandate physical continuity everywhere in the universe, even around a black hole. There should be no gap in the experience of, say, a doomed astronaut falling across the event horizon.
Or think of it this way: In relativity, the laws of physics look the same from all perspectives. The astronaut may get squished and stretched on the way in, but should still observe physics operating normally. The firewall, on the other hand, is about as abnormal and discontinuous as it gets. Now we’ve got a new paradox to deal with.
At this point, the story circles back to Hawking, who decided there must be a better way out. In a short paper presented last year, he suggests that the event horizon is not a defined boundary at all, but rather a zone of chaos where space and time are completely scrambled. No specific physical event occurs there — no Polchinski firewall — but any information passing through is rendered completely meaningless. Although Hawking dispenses with the traditional idea of the event horizon, he doesn’t dispute that black holes exist (as some breathless news stories claimed). Rather, he proposes that black holes are more like gray holes, defined by fuzzy edges that shed energy and garbled information.
My brain was starting to hurt, so I called on Juan Maldacena of the Institute for Advanced Study, a leading black hole theorist and a neutral third party. He doesn’t think much of the firewall concept because it fails miserably at the key task of reconciling quantum mechanics and relativity. “It’s telling us how not to do it,” he says. Does he like the Hawking approach, then? “No. He did not propose anything concrete,” Maldacena replies tersely.
When you are trying to solve the riddle of the black hole, nobody gets special treatment, not even Stephen Hawking.
Event Horizon Denialists
Here is where the deniers come in: If black holes keep sprouting paradoxes, the thinking goes, maybe the problem lies with our understanding of the black hole itself. I first encountered black hole denialism during a cosmology conversation with Laura Mersini-Houghton of the University of North Carolina, who mentioned, as a casual aside, “Oh, I also proved black holes don’t exist.” Over the past year, she has written two papers (one published, one in press) laying down the gauntlet.




stephen-hawking
stephen-hawking
Even Stephen Hawking can’t quite explain how black holes actually work in our universe.
Karwai Tang/Getty Images

Mersini-Houghton is flipping around Hawking’s old ideas. He considered the radiation released by black holes after they form. She looks instead at the radiation generated by a massive, collapsing object before it ever crosses the event horizon. In Mersini-Houghton’s analysis, the resulting energy and opposing pressure become so intense that it stops the infall, reverses it, and flings everything outward. The result is “fireworks, not firewalls,” as she puts it.
Event horizons, and the paradoxes that go with them, do not exist because the laws of physics guarantee that imploding stars self-destruct before they can become black holes. Cenalo Vaz, a physicist at the University of Cincinnati, recently came to similar conclusions by looking at outward pressure exerted by the structure of space around a collapsing object.
If black holes don’t exist, I ask Mersini-Houghton, what are those things at the hearts of most galaxies? Astronomers have found clear evidence of tiny but supermassive objects there, pulling on stars and stirring up hot disks of gas. Her answer cuts to the heart of one of the weirdest aspects of a black hole. As matter moves closer to the event horizon, time slows down from the perspective of an outside observer — that is, from the perspective of us and everyone else on the outside. At the horizon itself, time appears to us to come to a dead halt.
Even if matter never quite reaches the horizon, as in Mersini-Houghton’s theory, time could be stretched so drastically that a sudden rebound or explosion (from the collapsing object’s perspective) might look like a single, motionless moment (from ours).




mersini-houghton
mersini-houghton
Laura Mersini-Houghton thinks the laws of physics prevent black holes' forming.
Dan Sears/the University of North Carolina at Chapel Hill
Put plainly: To a human observer, black holes do not exist because we can never see them form. In the traditional view, a star keeps collapsing past the event horizon down to a dot known as a singularity. In the dissenting view, the star collapses to the edge of the event horizon and then hovers there, or rebounds and explodes. But from our outside perspective, there is essentially no difference. All we see is a frozen frame when the star is infinitesimally close to the event horizon. An astronaut falling toward the star would know the answer instantly, but any message that he or she sent would take a near-infinite amount of time to reach us.
Nevertheless, because such an experience is possible, there must be a theory to describe it.
Maldacena, like most mainstream physicists, dismisses black hole denialism. The real way to make sense of all this black hole madness, he insists, is to address the underlying clashes between relativity and quantum physics. “A full theory of quantum gravity must resolve them,” he emphasizes. And that is, perhaps, the ultimate paradox of black holes: They embody our very deepest scientific understanding of how the universe works, and yet in many ways we do not understand them at all.

QUANTUM GRAVITY

Quantum gravity (QG) is a field of theoretical physics that seeks to describe the force of gravity according to the principles of quantum mechanics.
The current understanding of gravity is based on Albert Einstein's general theory of relativity, which is formulated within the framework of classical physics. On the other hand, the nongravitational forces are described within the framework of quantum mechanics, a radically different formalism for describing physical phenomena based on probability.[1] The necessity of a quantum mechanical description of gravity follows from the fact that one cannot consistently couple a classical system to a quantum one.[2]
Although a quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, difficulties arise when one attempts to apply the usual prescriptions of quantum field theory to the force of gravity.[3] From a technical point of view, the problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.[4] A recent development is the theory of causal fermion systems which gives quantum mechanics, general relativity and quantum field theory as limiting cases.[5][6][7][8][9][10]
Strictly speaking, the aim of quantum gravity is only to describe the quantum behavior of the gravitational field and should not be confused with the objective of unifying all fundamental interactions into a single mathematical framework. Although some quantum gravity theories such as string theory try to unify gravity with the other fundamental forces, others such as loop quantum gravity make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces. A theory of quantum gravity that is also a grand unification of all known interactions is sometimes referred to as a theory of everything (TOE).
One of the difficulties of quantum gravity is that quantum gravitational effects are only expected to become apparent near the Planck scale, a scale far smaller in distance (equivalently, far larger in energy) than what is currently accessible at high energy particle accelerators. As a result, quantum gravity is a mainly theoretical enterprise, although there are speculations about how quantum gravity effects might be observed in existing experiments.

Warmholes

A wormhole, also known as "Einstein-Rosen Bridge", is a hypothetical topological feature that would fundamentally be a shortcut through spacetime. A wormhole is much like a tunnel with two ends, each in separate points in spacetime.
For a simplified notion of a wormhole, visualize space as a two-dimensional (2D) surface. In this case, a wormhole can be pictured as a hole in that surface that leads into a 3D tube (the inside surface of a cylinder). This tube then re-emerges at another location on the 2D surface with a similar hole as the entrance. An actual wormhole would be analogous to this, but with the spatial dimensions raised by one. For example, instead of circular holes on a 2D plane, the entry and exit points could be visualized as spheres in 3D space.
Researchers have no observational evidence for wormholes, but the equations of the theory of general relativity have valid solutions that contain wormholes. Because of its robust theoretical strength, a wormhole is one of the great physics metaphors for teaching general relativity. The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them.
The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at a given point.[1] Many physicists, such as Stephen Hawking,[2] Kip Thorne[3] and others,[4][5][6] therefore argue that such effects might make it possible to stabilize a traversable wormhole. Physicists have not found any natural process that would be predicted to form a wormhole naturally in the context of general relativity, although the quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,[7][8] and stable versions of such wormholes have been suggested as dark matter candidates.[9][10] It has also been proposed that, if a tiny wormhole held open by a negative-mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.[11]
The American theoretical physicist John Archibald Wheeler coined the term wormhole in 1957; the German mathematician Hermann Weyl, however, had proposed the wormhole theory in 1921, in connection with mass analysis of electromagnetic field energy.

HYDROELECTRICITY

Hydroelectricity is the term referring to electricity generated by hydropower; the production of electrical power through the use of the gravitational force of falling or flowing water. It is the most widely used form of renewable energy, accounting for 16 percent of global electricity generation – 3,427 terawatt-hours of electricity production in 2010,^[1] and is expected to increase about 3.1% each year for the next 25 years.

Hydropower is produced in 150 countries, with the Asia-Pacific region generating 32 percent of global hydropower in 2010. China is the largest hydroelectricity producer, with 721 terawatt-hours of production in 2010, representing around 17 percent of domestic electricity use.

The cost of hydroelectricity is relatively low, making it a competitive source of renewable electricity. The average cost of electricity from a hydro station larger than 10 megawatts is 3 to 5 U.S. cents per kilowatt-hour.^[1] It is also a flexible source of electricity since the amount produced by the station can be changed up or down very quickly to adapt to changing energy demands. However, damming interrupts the flow of rivers and can harm local ecosystems, and building large dams and reservoirs often involves displacing people and wildlife.^[1] Once a hydroelectric complex is constructed, the project produces no direct waste, and has a considerably lower output level of the greenhouse gas carbon dioxide (CO₂) than fossil fuel powered energy plants.^[2]

HISTORY

Hydropower has been used since ancient times to grind flour and perform other tasks. In the mid-1770s, French engineer Bernard Forest de Bélidor published Architecture Hydraulique which described vertical- and horizontal-axis hydraulic machines. By the late 19th century, the electrical generator was developed and could now be coupled with hydraulics.^[5] The growing demand for theIndustrial Revolution would drive development as well.^[6] In 1878 the world's first hydroelectric power scheme was developed atCragside in Northumberland, England by William George Armstrong. It was used to power a single arc lamp in his art gallery.^[7]The old Schoelkopf Power Station No. 1 near Niagara Falls in the U.S. side began to produce electricity in 1881. The first Edisonhydroelectric power station, the Vulcan Street Plant, began operating September 30, 1882, in Appleton, Wisconsin, with an output of about 12.5 kilowatts.^[8] By 1886 there were 45 hydroelectric power stations in the U.S. and Canada. By 1889 there were 200 in the U.S. alone.^[5]

At the beginning of the 20th century, many small hydroelectric power stations were being constructed by commercial companies in mountains near metropolitan areas. Grenoble, France held the International Exhibition of Hydropower and Tourism with over one million visitors. By 1920 as 40% of the power produced in the United States was hydroelectric, the Federal Power Act was enacted into law. The Act created the Federal Power Commission to regulate hydroelectric power stations on federal land and water. As the power stations became larger, their associated dams developed additional purposes to include flood control, irrigation and navigation. Federal funding became necessary for large-scale development and federally owned corporations, such as the Tennessee Valley Authority (1933) and the Bonneville Power Administration (1937) were created.^[6]Additionally, the Bureau of Reclamation which had begun a series of western U.S. irrigation projects in the early 20th century was now constructing large hydroelectric projects such as the 1928 Hoover Dam.^[9] The U.S. Army Corps of Engineers was also involved in hydroelectric development, completing the Bonneville Dam in 1937 and being recognized by the Flood Control Act of 1936 as the premier federal flood control agency.^[10]

Hydroelectric power stations continued to become larger throughout the 20th century. Hydropower was referred to as white coal for its power and plenty.^[11] Hoover Dam's initial 1,345 MW power station was the world's largest hydroelectric power station in 1936; it was eclipsed by the 6809 MW Grand Coulee Dam in 1942.^[12] The Itaipu Dam opened in 1984 in South America as the largest, producing 14,000 MW but was surpassed in 2008 by the Three Gorges Dam in China at 22,500 MW. Hydroelectricity would eventually supply some countries, including Norway, Democratic Republic of the Congo, Paraguay and Brazil, with over 85% of their electricity. The United States currently has over 2,000 hydroelectric power stations that supply 6.4% of its total electrical production output, which is 49% of its renewable electricity.^[6]

Generating methods

Turbine row at Los Nihuiles Power Station inMendoza, Argentina

Cross section of a conventional hydroelectric dam.

A typical turbine and generator

Conventional (dams)

See also: List of conventional hydroelectric power stations

Most hydroelectric power comes from the potential energy of dammed water driving a water turbine and generator. The power extracted from the water depends on the volume and on the difference in height between the source and the water's outflow. This height difference is called the head. The amount of potential energy in water is proportional to the head. A large pipe (the "penstock") delivers water to the turbine.^[13]

Pumped-storage

Main article: Pumped-storage hydroelectricity

See also: List of pumped-storage hydroelectric power stations

This method produces electricity to supply high peak demands by moving water between reservoirs at different elevations. At times of low electrical demand, the excess generation capacity is used to pump water into the higher reservoir. When the demand becomes greater, water is released back into the lower reservoir through a turbine. Pumped-storage schemes currently provide the most commercially important means of large-scale grid energy storage and improve the daily capacity factor of the generation system. Pumped storage is not an energy source, and appears as a negative number in listings.^[14]

Run of the river

Main article: Run of the river hydroelectricity

See also: List of run-of-the-river hydroelectric power stations

Run of the river hydroelectric stations are those with small or no reservoir capacity, so that the water coming from upstream must be used for generation at that moment, or must be allowed to bypass the dam. In the United States, run of the river hydropower could potentially provide 60,000 MW (about 13.7% of total use in 2011 if continuously available).^[15]

Tide

Main article: Tide power

See also: List of tidal power stations

A tidal power station makes use of the daily rise and fall of ocean water due to tides; such sources are highly predictable, and if conditions permit construction of reservoirs, can also be dispatchable to generate power during high demand periods. Less common types of hydro schemes use water's kinetic energy or undammed sources such as undershot waterwheels. Tidal power is viable in a relatively small number of locations around the world. In Great Britain, there are eight sites that could be developed, which have the potential to generate 20% of the electricity used in 2012.^[16]

CASTIGLIANO THEOREM

• Castigliano's first theorem – for forces in an elastic structure

Castigliano's method for calculating forces is an application of his first theorem, which states:

If the strain energy of an elastic structure can be expressed as a function of generalised displacement q_i; then the partial derivative of the strain energy with respect to generalised displacement gives the generalised force Q_i.

In equation form,

where U is the strain energy.

• Castigliano's second theorem – for displacements in a linearly elastic structure.

Castigliano's method for calculating displacements is an application of his second theorem, which states:

If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q_i; then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q_i in the direction of Q_i.

As above this can also be expressed as: