formulasearchengine - User contributions [en]

Semilattice

2014-01-26T19:20:52Z

185.13.228.87: Sorry, wrong correction.

{{Use dmy dates|date=May 2012}}
{{Use British English|date=May 2012}}
{{Infobox scientist
|name = James Hopwood Jeans
|image = James Hopwood Jeans.jpg
|birth_date = {{Birth date|1877|9|11|df=y}}
|birth_place = [[Ormskirk]], [[Lancashire]], [[England]]
|death_date = {{death date and age|1946|9|16|1877|9|11|df=y}}
|death_place = [[Dorking]], [[Surrey]], [[England]]
|residence =
|citizenship =
|nationality = [[United Kingdom|British]]
|ethnicity =
|field = [[astronomy]], [[mathematics]], [[physics]]
|work_institutions = [[Trinity College, Cambridge]]; [[Princeton University]]
|alma_mater = [[Merchant Taylors' School, Northwood|Merchant Taylors' School]]; [[Cambridge University]]
|doctoral_advisor =
|notable_students = [[Ronald Fisher]]
|known_for = [[Rayleigh–Jeans law]] [[Jeans mass]] [[Jeans length]]
|influences =
|influenced =
|prizes =
|religion =
|signature =
}}
'''Sir James Hopwood Jeans''' [[Order of Merit|OM]] [[Fellow of the Royal Society|FRS]]<ref name="frs">{{cite doi|10.1098/rsbm.1947.0019}}</ref> MA DSc ScD LLD<ref name="jeans1938">Sir James Jeans 1938 (reprint of 1931's edition of 1930 book): ''[[The Mysterious Universe]]''.</ref> (11 September 1877{{spaced ndash}}16 September 1946<ref>GRO Register of Deaths: SEP 1946 5g 607 SURREY SE – James H. Jeans, aged 69</ref>) was an [[England|English]] [[physicist]], [[astronomer]] and [[mathematician]].

==Background==
Born in [[Ormskirk]], [[Lancashire]], Jeans was educated at [[Merchant Taylors' School, Northwood]], [[Wilson's School|Wilson's Grammar School]],<ref>Allport, D.H. & Friskney, N.J. "A Short History of Wilson's School", Wilson's School Charitable Trust, 1987, pg 234</ref> [[Camberwell]] and [[Trinity College, Cambridge]].<ref>{{acad|id=JNS896JH|name=Jeans, James Hopwood}}</ref>
His coach for [[Cambridge Mathematical Tripos]] was [[Robert Alfred Herman]], and in 1898 he came out [[Second Wrangler]].
Jeans was elected [[Fellow]] of Trinity College in October 1901,<ref>{{Cite newspaper The Times |articlename=University intelligence - Cambridge|day_of_week=Friday |date=11 October 1901 |page_number=4 |issue=36583| }}</ref> and taught at Cambridge, but went to [[Princeton University]] in 1904 as a professor of applied mathematics. He returned to Cambridge in 1910.

He made important contributions in many areas of physics, including [[Quantum mechanics|quantum theory]], the theory of [[radiation]] and [[stellar evolution]]. His analysis of rotating bodies led him to conclude that [[Pierre-Simon Laplace|Laplace]]'s theory that the solar system formed from a single cloud of gas was incorrect, proposing instead that the planets condensed from material drawn out of the sun by a hypothetical catastrophic near-collision with a passing star. This theory is not accepted today.

Jeans, along with [[Arthur Eddington]], is a founder of British [[cosmology]]. In 1928 Jeans was the first to conjecture a [[steady state cosmology]] based on a hypothesized continuous creation of matter in the universe.<ref>Astronomy and Cosmogony, Cambridge U Press, p 360</ref> This theory was ruled out when the 1965 discovery of the [[cosmic microwave background]] was widely interpreted as the tell-tale signature of the [[Big Bang]].

His scientific reputation is grounded in the monographs ''The Dynamical Theory of Gases'' (1904), ''Theoretical Mechanics'' (1906), and ''Mathematical Theory of Electricity and Magnetism'' (1908). After retiring in 1929, he wrote a number of books for the lay public, including ''The Stars in Their Courses'' (1931), ''The Universe Around Us,'' ''Through Space and Time'' (1934), ''The New Background of Science'' (1933), and ''[[The Mysterious Universe]].'' These books made Jeans fairly well known as an expositor of the revolutionary scientific discoveries of his day, especially in [[theory of relativity|relativity]] and [[physical cosmology]].

In 1939, the [[Journal of the British Astronomical Association]] reported that Jeans was going to stand as a candidate for parliament for the [[Cambridge University (UK Parliament constituency)|Cambridge University constituency]]. The election, expected to take place in 1939 or 1940 did not take place until 1945, and without his involvement.

He also wrote the book "Physics and Philosophy" (1943) where he explores the different views on reality from two different perspectives: [[science]] and [[philosophy]].

On his religious views, Jeans was an agnostic.<ref>{{cite book|title=The Future of Man|year=2004|publisher=Random House LLC|isbn=9780385510721|page=212|author=Pierre Teilhard De Chardin|accessdate=18 July 2013|quote=We can hardly wonder, in the circumstances, that agnostics such as Sir James Jeans and Marcel Boll, and even convinced believers like Guardini, have uttered expressions ol amazement (tinged with heroic pessimism or triumphant detachment) at the apparent insignificance of the phenomenon of Life in terms of the cosmos— a little mold on a grain of dust...}}</ref>

Jeans married twice, first to the American [[poet]] [[Charlotte Tiffany Mitchell]] in 1907,<ref>{{cite web | url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Jeans.html | title=Sir James Hopwood Jeans | publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland | date=October 2003 | accessdate=28 November 2011 | author=J J O'Connor and E F Robertson}}</ref> then the [[Austria]]n [[organ (music)|organ]]ist and [[harpsichord]]ist [[Susi Jeans|Suzanne Hock]] (better known as [[Susi Jeans]]) in 1935. He died in [[Dorking]], [[Surrey]].

At Merchant Taylors' School there is a James Jeans Academic Scholarship for the candidate in the entrance exams who displays outstanding results across the spectrum of subjects but notably in Mathematics and Sciences.

==Major accomplishments==
One of Jeans' major discoveries, named [[Jeans length]], is a critical radius of an [[interstellar cloud]] in space. It depends on the temperature, and density of the cloud, and the mass of the particles composing the cloud. A cloud that is smaller than its Jeans length will not have sufficient gravity to overcome the repulsive gas pressure forces and condense to form a star, whereas a cloud that is larger than its Jeans length will collapse.

:<math>\lambda_J=\sqrt{\frac{15k_{B}T}{4\pi Gm\rho}}</math>

Jeans came up with another version of this equation, called Jeans mass or [[Jeans instability]], that solves for the critical mass a cloud must attain before being able to collapse.

Jeans also helped to discover the [[Rayleigh–Jeans law]], which relates the energy density of blackbody radiation to the temperature of the emission source.

:<math> f(\lambda) = 8\pi c \frac{k_{B}T}{\lambda^4}</math>

==Awards and honours==
* [[Fellow of the Royal Society]] in May, 1906
* [[Bakerian Lecture]] to [[Royal Society]] in 1917.
* [[Royal Medal]] of the [[Royal Society]] in 1919.
* [[Hopkins Prize]] of the [[Cambridge Philosophical Society]] 1921–1924.
* [[Gold Medal of the Royal Astronomical Society]] in 1922.
* He was [[knight]]ed in 1928.
* [[Franklin Medal]] of the [[Franklin Institute]] in 1931.
* In 1933 Hopwood-Jeans was invited to deliver the [[Royal Institution Christmas Lectures|Royal Institution Christmas Lecture]] on ''Through Space and Time''.
* Mukerjee Medal of the [[Indian Association for the Cultivation of Science]] in 1937.
* President of the 25th session of the [[Indian Science Congress Association#Indian Science Congress|Indian Science Congress]] in 1938.
* Calcutta Medal of the [[Indian Science Congress Association]] in 1938.
* Member of the [[Order of Merit]] in 1939.
* The crater [[Jeans (lunar crater)|Jeans]] on the [[Moon]] is named after him, as is the crater [[Jeans (Martian crater)|Jeans]] on [[Mars]].
* The String Quartet No.7 by [[Robert Simpson (composer)|Robert Simpson]] was written in tribute to him on the centenary of his birth, 1977.

==Bibliography==
* Jeans, James Hopwood. (1947). ''The Growth of Physical Science''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00565-4)
* Jeans, James Hopwood. (1942). ''Physics and Philosophy''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00567-8)
* Jeans, James Hopwood. (1940). ''An Introduction to the Kinetic Theory of Gases''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00560-9)
* Jeans, James Hopwood. (1937). ''Science and Music''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00569-2)
* Jeans, James Hopwood. (1934). ''Through Space and Time''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00571-5)
* Jeans, James Hopwood. (1933). ''The New Background of Science''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00572-2)
* Jeans, James Hopwood. (1931). ''Stars in Their Courses''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00570-8)
* Jeans, James Hopwood. (1930). ''The Mysterious Universe''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00566-1)
* Jeans, James Hopwood. (1928). ''Astronomy and Cosmogony''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00562-3)
* Jeans, James Hopwood. (1925). ''Mathematical Theory of Electricity and Magnetism''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00561-6)
* Jeans, James Hopwood. (1926). ''Atomicity and Quanta''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00563-0)
* Jeans, James Hopwood. (1919). ''Problems of Cosmology and Stellar Dynamics''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00568-5)
* Jeans, James Hopwood. (1904). ''The Dynamical Theory of Gases''. Cambridge University Press (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00564-7)

;Available online from the [[Internet Archive]]
*1904. ''[http://www.archive.org/details/dynamicaltheoryo00jeanrich The Dynamical Theory of Gases]''
*1906. ''[http://www.archive.org/details/elementarytreati00jeanuoft Theoretical Mechanics]''
*1908. ''[http://www.archive.org/details/mathemattheoelec00jeanrich Mathematical Theory of Electricity and Magnetism]''
*1947. ''[http://archive.org/details/growthofphysical029068mbp The Growth of Physical Science]''
Other:
*1929. ''[[The Universe Around Us]]''
*1930. ''[[The Mysterious Universe]]''
*1931. ''The Stars in Their Courses''
*1933. ''The New Background of Science''
*1937. ''Science and Music''
*1942. ''Physics and Philosophy''

==References==
{{reflist}}

==External links==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=James Jeans | s=Author:James Hopwood Jeans | v=no | voy=no | species=no | d=q315545}}

* MacTutor (St. Andrews Univ.): [http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Jeans.html More biographical information.], including photos
* [http://www.britannica.com/eb/article-9043471/Sir-James-Jeans Britannica article] includes photo

{{Authority control|VIAF=36999529|GND=105857807|LCCN=n/82/490|BNF=12383081}}

{{Persondata 
| NAME = Jeans, James Hopwood
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = British mathematician and astronomer
| DATE OF BIRTH = 11 September 1877
| PLACE OF BIRTH = [[Southport]], [[Lancashire]], [[England]]
| DATE OF DEATH = 16 September 1946
| PLACE OF DEATH = [[Dorking]], [[Surrey]], [[England]]
}}
{{DEFAULTSORT:Jeans, James Hopwood}}
[[Category:1877 births]]
[[Category:1946 deaths]]
[[Category:English agnostics]]
[[Category:English astronomers]]
[[Category:English mathematicians]]
[[Category:Idealists]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:Academics of the University of Cambridge]]
[[Category:Princeton University faculty]]
[[Category:People from Ormskirk]]
[[Category:People educated at Merchant Taylors' School, Northwood]]
[[Category:Royal Medal winners]]
[[Category:Second Wranglers]]
[[Category:Recipients of the Gold Medal of the Royal Astronomical Society]]
[[Category:People educated at Wilson's School, Wallington]]
[[Category:Fellows of the Royal Society]]
[[Category:Presidents of the British Science Association]]
[[Category:British physicists]]

Absorption law

2014-01-26T17:07:18Z

185.13.228.87: lattice meet or join should be commutative, associative and also idempotent.

{{About||the human activity|settler|the audio drama|The Settling}}
[[File:Kalová laguna Sojovice.jpg|thumb|Settling pond for iron particles at water works]]
'''Settling''' is the process by which particulates settle to the bottom of a liquid and form a [[sediment]]. Particles that experience a force, either due to gravity or due to [[Centrifuge|centrifugal motion]] will tend to move in a uniform manner in the direction exerted by that force. For gravity settling, this means that the particles will tend to fall to the bottom of the vessel, forming a [[slurry]] at the vessel base.

Settling is an important operation in many applications, such as [[mining]], [[wastewater treatment]], biological science, [[outer space|space]] [[rocket propellant|propellant]] reignition,<ref name=aiaa20100902>
{{cite web|last=Zegler|first=Frank |title=Evolving to a Depot-Based Space Transportation Architecture |url=http://www.ulalaunch.com/site/docs/publications/DepotBasedTransportationArchitecture2010.pdf |work=AIAA SPACE 2010 Conference & Exposition |publisher=AIAA |accessdate=2011-01-25 |coauthors=Bernard Kutter |date=2010-09-02 |quote=It consumes waste hydrogen and oxygen to produce power, generate settling and attitude control thrust.}}</ref>
and [[particle mechanics]].

== Physics ==

[[Image:Stokes sphere.svg|thumb|right|180px|Creeping flow past a sphere: [[Streamlines, streaklines, and pathlines|streamline]]s, drag force ''F''d and force by gravity ''F''g.]]
For settling particles that are considered individually, i.e. dilute particle solutions, there are two main forces enacting upon any particle. The primary force is an applied force, such as gravity, and a [[Drag (physics)|drag]] force that is due to the motion of the particle through the [[fluid]]. The applied force is usually not affected by the particle's velocity, whereas the drag force is a function of the particle velocity.

For a particle at rest no drag force will exhibited, which causes the particle to accelerate due to the applied force. When the particle accelerates, the drag force acts in the direction opposite to the particle's motion, retarding further acceleration, in the absence of other forces drag directly opposes the applied force. As the particle increases in velocity eventually the drag force and the applied force will [[limit (mathematics)|approximately equate]], causing no further change in the particle's velocity. This velocity is known as the [[terminal velocity]], ''settling velocity'' or ''fall velocity'' of the particle. This is readily measurable by examining the rate of fall of individual particles.

The terminal velocity of the particle is affected by many parameters, i.e. anything that will alter the particle's drag. Hence the terminal velocity is most notably dependent upon [[Particle size|grain size]], the shape (roundness and sphericity) and density of the grains, as well as to the [[viscosity]] and [[density]] of the fluid.

=== Single particle drag ===
==== Stokes' drag ====
{{main|Stokes' law}}
[[File:Reynolds-drag.svg|thumb|200px|right|Dimensionless force versus Reynolds number for spherical particles]]
For dilute suspensions, [[Stokes Law|Stokes' law]] predicts the settling velocity of small spheres in [[fluid]], either air or water. This originates due to the strength of viscous forces at the surface of the particle providing the majority of the retarding force. Stokes' law finds many applications in the natural sciences, and is given by:

:<math>w=\frac{2(\rho_p-\rho_f)gr^2}{9\mu}</math>

where ''w'' is the settling velocity, ''ρ'' is density (the subscripts ''p'' and ''f'' indicate particle and fluid respectively), ''g'' is the acceleration due to gravity, ''r'' is the radius of the particle and ''μ'' is the dynamic viscosity of the fluid.

Stokes' law applies when the [[Reynolds number]], Re, of the particle is less than 0.1. Experimentally Stokes' law is found to hold within 1% for <math> Re \leq 0.1</math>, within 3% for <math>Re \leq 0.5</math> and within 9% <math> Re \leq 1.0</math>.<ref name="Rhodes">{{cite book| title=Introduction to Particle Technology | author=Martin Rhodes }}</ref> With increasing Reynolds numbers, Stokes law begins to break down due to the increasing importance of fluid inertia, requiring the use of empirical solutions to calculate drag forces.

==== Newtonian drag ====
Defining a [[drag coefficient]], <math>C_d</math>, as the ratio of the force experienced by the particle divided by the [[impact pressure]] of the fluid, a coefficient that can be considered as the transfer of available fluid force into drag is established. In this region the inertia of the impacting fluid is responsible for the majority of force transfer to the particle.

:<math> C_d = \frac{F_d}{\frac{1}{2}\rho_f U^2 A} </math>

For a spherical particle in the Stokes regime this value is not constant, however in the Newtonian drag regime the drag on a sphere can be approximated by a constant, 0.44. This constant value implies that the efficiency of transfer of energy from the fluid to the particle is not a function of fluid velocity.

As such the [[terminal velocity]] of a particle in a Newtonian regime can again be obtained by equating the drag force to the applied force, resulting in the following expression

:<math> w = 2.46\left( \frac{(\rho_p-\rho_f)gr}{\rho_f}\right)^{\frac{1}{2}} .</math>

==== Transitional drag ====
In the intermediate region between Stokes drag and Newtonian drag, there exists a transitional regime, where the analytical solution to the problem of a falling sphere becomes problematic. To solve this, empirical expressions are used to calculate drag in this region. One such empirical equation is that of Schiller and Naumann, and may be valid for <math> 0.2 \leq Re \leq 1000</math>:<ref name="CoulsonAndRichardsonV2">{{cite book|title=Chemical Engineering, Volume 2.|year=1955|publisher=Pergamon press}}</ref>

:<math> \frac{F}{\rho_f U^2 A} = \frac{12}{ \mathrm{Re}} \left( 1 + 0.15\mathrm{Re}^{0.687} \right) . </math>

=== Hindered settling ===
Stokes, transitional and Newtonian settling describe the behaviour of a single spherical particle in an infinite fluid, known as free settling. However this model has limitations in practical application. Alternate considerations, such as the interaction of particles in the fluid, or the interaction of the particles with the container walls can modify the settling behaviour. Settling that has these forces in appreciable magnitude is known as hindered settling. Subsequently semi-analytic or empirical solutions may be used to perform meaningful hindered settling calculations.

== Applications ==
The solid-gas flow systems are present in many industrial applications, as dry, catalytic reactors, settling tanks, pneumatic conveying of solids, among others. Obviously, in industrial operations the drag rule isn’t simple as a single sphere settling in a stationary fluid. However, this knowledge indicates how drag behaves in more complex systems, which are designed and studied by engineers applying empirical and more sophisticated tools.

For example, ''Settling [[Storage tank|tanks]]'' are used for separating solids and/or oil from another liquid. In [[food processing]], the vegetable is crushed and placed inside of a settling tank with water. The oil floats the top of the water then is collected. In water and waste water treatment a [[flocculant]] is often added prior to settling to form larger particles that settle out quickly in a settling tank leaving the water with a lower [[turbidity]].

In [[winemaking]], the [[French (language)|French]] term for this process is ''[[:fr:Débourbage|débourbage]]''. This step usually occurs in white wine production before the start of [[fermentation (wine)|fermentation]].<ref name="Oxford pg 223">Robinson, J. (ed) (2006) ''"The Oxford Companion to Wine"'' Third Edition p. 223 Oxford University Press, ISBN 0-19-860990-6</ref>

===Settleable solids analysis===
'''Settleable solids''' are the particulates that settle out of a still fluid. Settleable solids can be quantified for a [[suspension (chemistry)|suspension]] using an Imhoff cone. The standard Imhoff cone of transparent glass or plastic holds one liter of liquid and has calibrated markings to measure the volume of solids accumulated in the bottom of the conical container after settling for one hour. A standardized Imhoff cone procedure is commonly used to measure suspended solids in [[wastewater]] or [[stormwater runoff]]. The simplicity of the method makes it popular for estimating [[water quality]].

The water sample to be measured should be representative of the total stream. Samples are best collected from the discharge falling from a pipe or over a weir, because samples skimmed from the top of a flowing channel may fail to capture larger, high-density solids moving along the bottom of the channel. The sampling bucket is vigorously stirred to uniformly re-suspend all collected solids immediately before pouring the volume required to fill the cone. The filled cone is immediately placed in a stationary holding rack to allow quiescent settling. The rack should be located away from heating sources, including direct sunlight, which might cause currents within the cone from thermal density changes of the liquid contents. After 45 minutes of settling, the cone is partially rotated about its axis of symmetry just enough to dislodge any settled material adhering to the side of the cone. Accumulated sediment is observed and measured fifteen minutes later, after one hour of total settling time.<ref>Franson, Mary Ann (1975) ''Standard Methods for the Examination of Water and Wastewater'' 14th edition, APHA, AWWA & WPCF ISBN 0-87553-078-8 pp. 89–91, 95–96</ref>

==See also==
*[[Drag equation]]
*[[Sedimentation]]
*[[Settling basin]]
*[[Suspension (chemistry)]]
*[[Total suspended solids]]

==References==
{{Reflist}}

==External links==
{{Commons category|Settling}}
* [http://www.ne-wea.org/LabManual/settleable_solids.htm Settleable solids methodology]
* [http://www.ajdesigner.com/phpstokeslaw/stokes_law_terminal_velocity.php Stokes Law terminal velocity calculator]

[[Category:Analytical chemistry]]
[[Category:Earth sciences]]
[[Category:Unit operations]]
[[Category:Chemical engineering]]
[[Category:Colloidal chemistry]]