formulasearchengine - User contributions [en]

Group ring

2014-07-18T08:11:48Z

134.99.156.36: /* Group rings over an infinite group */

Valery is what people call me and that i feel comfortable when people use complete name. New Mexico could be the only place he's been residing in and his parents live nearby. The job he's been occupying for years is a people manager and he'll be promoted soon. What he loves doing is collecting kites and he's been doing it for a real while. I'm not effective in webdesign we might to help check my website: [http://valdezy9.jigsy.com/entries/general/beneficios-de-los-componentes-enriquecidos-con-fibra-de-vidrio-reutilizado http://valdezy9.jigsy.com/entries/general/beneficios-de-los-componentes-enriquecidos-con-fibra-de-vidrio-reutilizado]

Flux balance analysis

2013-11-26T08:49:49Z

134.99.147.23: Added new citation for R Software package sybil.

The '''Lippmann-Schwinger equation''' (named after [[Bernard A. Lippmann]] and [[Julian Schwinger]]<ref>{{cite journal | last=Lippmann|first=Bernard A.|coauthors=Schwinger, Julian|journal=Physical Review Letters|year=1950|volume=79|pages=469|doi = 10.1103/PhysRev.79.469}}</ref>) is one of the most used equations to describe particle colisions - or, more precisely, [[scattering]] - in [[quantum mechanics]]. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in [[atomic, molecular, and optical physics]], [[nuclear physics]] and [[particle physics]]. It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and allows to calculate the relevant experimental parameters ([[scattering amplitude]] and [[cross section (physics)|cross section]]s).

The most fundamental equation to describe any quantum phenomenon, including scattering, is the [[Schrödinger equation]]. In physical problems, this [[differential equation]] must be solved with the input of an additional set of initial and/or [[boundary condition]]s for the specific physical system studied. The Lippmann–Schwinger equation is equivalent to the Schrödinger equation plus the typical boundary conditions for scattering problems. In order to embed the boundary conditions, the Lippmann–Schwinger equation must be written as an [[integral equation]].<ref>[[#Joachain|Joachain, Charles J., 1983]] page 112</ref> For scattering problems, the Lippmann–Schwinger equation has been proved to be mathematically and intuitively more convenient than the original Schrödinger equation.

The Lippmann-Schwinger equation general shape is (in reality, two equations are shown below, one for the <math> + \,</math> sign and other for the <math> - \,</math> sign):
:<math> | \psi^{(\pm)} \rangle = | \phi \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)} \rangle. \,</math>

In the equations above, <math> | \psi^{(+)} \rangle \,</math> is the wave function of the whole system (the two colliding systems considered as a whole) at an infinite time before the interaction; and <math> | \psi^{(-)} \rangle \,</math>, at an infinite time after the interaction (the "scattered wave fuction"). The potential energy <math> V \,</math> describes the interaction between the two colliding systems. The [[hamiltonian]] <math> H_0 \,</math> describes the situation in which the two systems are infinitely far apart and do not interact. Its [[eigenfunction]]s are <math> | \phi \rangle \,</math> and its [[eigenvalue]]s are the energies <math> E \,</math>. Finally, <math> i \epsilon \,</math> is a mathematical technicality necessary for the calculation of the integrals needed to solve the equation and has no physical meaning.

==Usage==
The Lippmann-Schwinger equation is useful in a very large number of situations involving two-body scattering. For three or more colliding bodies it does not work well because of mathematical limitations; Faddeev equations may be used instead.<ref>[[#Joachain|Joachain, Charles J., 1983]] page 517</ref> However, there are approximations that can reduce a [[many-body problem]] to a set of [[two-body problem]]s in a variety of cases. For example, in a collision between electrons and molecules, there may be tens or hundreds of particles involved. But the phenomenum may be reduced to a two-body problem by describing all the molecule constituent particle potentials together with a [[pseudopotential]].<ref>[[#Joachain|Joachain, Charles J., 1983]] page 576</ref> In these cases, the Lippmann-Schwinger equations may be used. Of course, the main motivations of these approaches are also the possibility of doing the calculations with much lower computational efforts.

==Derivation==
We will assume that the [[Hamiltonian (quantum mechanics)|Hamiltonian]] may be written as
:<math>H = H_0 + V \,</math>
where ''H'' and ''H''0 have the same [[eigenvalues]] and ''H''0 is a free Hamiltonian. For example in nonrelativistic quantum mechanics ''H''0 may be
:<math>H_0 = \frac{p^2}{2m}. \,</math>

Intuitively <math>V \,</math> is the interaction energy of the system. This analogy is somewhat misleading, as interactions generically change the energy levels ''E'' of steady states of the system, but ''H'' and ''H''0 have identical spectra ''E''α. This means that, for example, a bound state that is an eigenstate of the interacting Hamiltonian will also be an eigenstate of the free Hamiltonian. This is in contrast with the Hamiltonian obtained by turning off all interactions, in which case there would be no bound states. Thus one may think of ''H''0 as the free Hamiltonian for the boundstates with effective parameters that are determined by the interactions.

Let there be an [[eigenstate]] of <math>H_0 \,</math>:
:<math>H_0 | \phi \rangle = E | \phi \rangle. \,</math>

Now if we add the interaction <math> V \,</math> into the mix, we need to solve
:<math>\left( H_0 + V \right) | \psi \rangle = E | \psi \rangle. \,</math>

Because of the continuity of the energy eigenvalues, we wish that <math> | \psi \rangle \to | \phi \rangle \,</math> as <math> V \to 0 \,</math>.

A potential solution to this situation is
:<math> |\psi \rangle = | \phi \rangle + \frac{1}{E - H_0} V | \psi \rangle. \,</math>

However <math>(E-H_0)</math> is [[Mathematical singularity|singular]] since <math> E </math> is an eigenvalue of <math> H_0 </math>.

As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:
:<math> | \psi^{(\pm)} \rangle = | \phi \rangle + \frac{1}{E - H_0 \pm i \epsilon} V |\psi^{(\pm)} \rangle. \,</math>

==Interpretation as in and out states==

===The S-matrix paradigm===

In the [[S-matrix]] formulation of [[particle physics]], which was pioneered by [[John Archibald Wheeler]] among others, all physical processes are modeled according to the following paradigm.

One begins with a non-interacting multiparticle state in the distant past. Non-interacting does not mean that all of the forces have been turned off, in which case for example [[proton]]s would fall apart, but rather that there exists an interaction-free [[Hamiltonian (quantum mechanics)|Hamiltonian]] ''H''0, for which the bound states have the same energy level spectrum as the actual Hamiltonian ''H''. This initial state is referred to as the '''in''' state. Intuitively, it consists of bound states that are sufficiently well separated that their interactions with each other are ignored.

The idea is that whatever physical process one is trying to study may be modeled as a [[scattering]] process of these well separated bound states. This process is described by the full Hamiltonian ''H'', but once it's over, all of the new bound states separate again and one finds a new noninteracting state called the '''out''' state. The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.

This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy. But many interesting physical phenomena do not obviously fit into this paradigm. For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into. In other words, one may be interested in measurements that are not in the asymptotic future. Sometimes an asymptotic past or future is not even available. For example, it is very possible that there is no past before the [[big bang]].

In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In [[S-matrix theory]], it was stated that any quantity that one could measure should be found in the S-matrix for some process. This idea was inspired by the physical interpretation that S-matrix techniques could give to [[Feynman diagrams]] restricted to the [[mass-shell]], and led to the construction of [[dual resonance model]]s. But it was very controversial, because it denied the validity of [[quantum field theory]] based on local fields and Hamiltonians.

===The connection to Lippmann–Schwinger===

Intuitively, the slightly deformed eigenfunctions <math> \psi^{(\pm)}</math> of the full Hamiltonian ''H'' are the in and out states. The <math>\phi</math> are noninteracting states that resemble the '''in''' and '''out''' states in the infinite past and infinite future.

===Creating wavepackets===

This intuitive picture is not quite right, because <math> \psi^{(\pm)}</math> is an eigenfunction of the Hamiltonian and so at different times only differs by a phase. Thus, in particular, the physical state does not evolve and so it cannot become noninteracting. This problem is easily circumvented by assembling <math> \psi^{(\pm)}</math> and <math> \phi</math> into wavepackets with some distribution <math>g(E)</math> of energies <math>E</math> over a characteristic scale <math>\Delta E</math>. The [[uncertainty principle]] now allows the interactions of the asymptotic states to occur over a timescale <math>\hbar/\Delta E</math> and in particular it is no longer inconceivable that the interactions may turn off outside of this interval. The following argument suggests that this is indeed the case.

Plugging the Lippmann–Schwinger equations into the definitions

::<math> \psi^{(\pm)}_g(t)=\int dE\, e^{-iEt} g(E)\psi^{(\pm)}</math>

and

::<math> \phi_g(t)=\int dE\, e^{-iEt} g(E)\phi</math>

of the wavepackets we see that, at a given time, the difference between the <math>\psi_g(t)</math> and <math>\phi_g(t)</math> wavepackets is given by an integral over the energy ''E''.

===A contour integral===

This integral may be evaluated by defining the wave function over the complex ''E'' plane and closing the ''E'' contour using a semicircle on which the wavefunctions vanish. The integral over the closed contour may then be evaluated, using the [[Cauchy integral theorem]], as a sum of the residues at the various poles. We will now argue that the residues of <math> \psi^{(\pm)}</math> approach those of <math> \phi</math> at time <math>t\rightarrow\mp\infty</math> and so the corresponding wavepackets are equal at temporal infinity.

In fact, for very positive times ''t'' the <math>e^{-iEt}</math> factor in a [[Schrödinger picture]] state forces one to close the contour on the lower half-plane. The pole in the ''V'' from the Lippmann–Schwinger equation reflects the time-uncertainty of the interaction, while that in the wavepackets weight function reflects the duration of the interaction. Both of these varieties of poles occur at finite imaginary energies and so are suppressed at very large times. The pole in the energy difference in the denominator is on the upper half-plane in the case of <math> \psi^{-}</math>, and so does not lie inside the integral contour and does not contribute to the <math> \psi^{-}</math> integral. The remainder is equal to the <math>\phi</math> wavepacket. Thus, at very late times <math> \psi^{-}=\phi</math>, identifying <math> \psi^{-}</math> as the asymptotic noninteracting '''out''' state.

Similarly one may integrate the wavepacket corresponding to <math> \psi^{+}</math> at very negative times. In this case the contour needs to be closed over the upper half-plane, which therefore misses the energy pole of <math> \psi^{+}</math>, which is in the lower half-plane. One then finds that the <math> \psi^{+}</math>
and <math> \phi</math> wavepackets are equal in the asymptotic past, identifying <math> \psi^{+}</math> as the asymptotic noninteracting '''in''' state.

===The complex denominator of Lippmann–Schwinger===

This identification of the <math>\psi</math>'s as asymptotic states is the justification for the <math>\pm\epsilon</math> in the denominator of the Lippmann–Schwinger equations.

==A formula for the S-matrix==

The [[S-matrix]] ''S'' is defined to be the inner product

::<math> S_{ab}=(\psi^-_a,\psi^+_b)</math>

of the ''a''th and ''b''th [[Heisenberg picture]] asymptotic states. One may obtain a formula relating the ''S''-matrix to the potential ''V'' using the above contour integral strategy, but this time switching the roles of <math> \psi^+</math> and <math> \psi^-</math>. As a result, the contour now does pick up the energy pole. This can be related to the <math>\phi</math>'s if one uses the S-matrix to swap the two <math>\psi</math>'s. Identifying the coefficients of the <math>\phi</math>'s on both sides of the equation one finds the desired formula relating ''S'' to the potential

:<math> S_{ab}=\delta(a-b)-2i\pi\delta(E_a-E_b)(\phi_a,V\psi^+_b).</math>

In the [[Born approximation]], corresponding to first order [[perturbation theory]], one replaces this last <math> \psi^+</math> with the corresponding eigenfunction <math> \phi</math> of the free Hamiltonian ''H''0, yielding

:<math> S_{ab}=\delta(a-b)-2i\pi\delta(E_a-E_b)(\phi_a,V\phi_b)\,</math>
which expresses the S-matrix entirely in terms of ''V'' and free Hamiltonian eigenfunctions.

These formulas may in turn be used to calculate the reaction rate of the process <math>b\rightarrow a</math>, which is equal to <math>|S_{ab}-\delta_{ab}|^2.\,</math>

==Homogenization==
With the use of Green's function, the Lippmann–Schwinger equation has counterparts in homogenization theory (e.g. mechanics, conductivity, permittivity).

== See also ==
*[[Bethe-Salpeter equation]]

==References==

{{reflist|30em}}

==Bibliography==

*{{cite book | author=Joachain, Charles J. | title=Quantum collision theory | publisher=North Holland | year=1983 | url=http://pt.scribd.com/doc/76844441/Charles-J-Joachain-Quantum-Collision-Theory | isbn=0-7204-0294-8 | ref=Joachain}}

*{{cite book | author=Sakurai, J. J. | title=Modern Quantum Mechanics | publisher=Addison Wesley | year=1994 | isbn=0-201-53929-2}}

*{{cite book|authorlink=Steven Weinberg|author=Weinberg, S.|title=The Quantum Theory of Fields|publisher=Cambridge University Press|year=1995|isbn=0-521-67053-5}}

{{DEFAULTSORT:Lippmann-Schwinger equation}}
[[Category:Scattering]]

Entropy of entanglement

2013-11-26T08:11:26Z

134.99.64.58:

{{multiple issues|
{{Underlinked|date=December 2012}}
{{Orphan|date=December 2012}}
}}

'''CFD''' stands for [[computational fluid dynamics]] (and heat transfer). As per this technique, the governing differential equations of a flow system or thermal system are known in the form of [[Navier–Stokes equations]], thermal energy equation and species equation with an appropriate equation of state.<ref>{{citation | journal=ASHRAE Journal, Proquest education journal, Pages: 44-48 | first=Ladeinde | last=Foluso | year=1997 | title=CFD application in the HVAC & R }}</ref>
In the past few years, CFD has been playing an increasingly important role in building design, following its continuing development for over a quarter of a century. The information provided by CFD can be used to analyse the impact of building exhausts to the environment, to predict smoke and fire risks in buildings, to quantify indoor environment quality, and to design natural ventilation systems, etc.

==Applications==
Recently CFD finds very wide application in different areas of science and engineering; some examples are:<ref>{{cite book|last=Versteeg|first=H.|authorlink=H.Versteeg|title=An Introduction to Computational Fluid Dynamics|year=2009|publisher=Pearson Publications|isbn=978-81-317-2048-6}}</ref>

* [[Aerodynamics]] of aircraft and vehicles : lift and drag
* Hydrodynamics of ships
* Power plant : combustion in internal combustion engines and [[gas turbines]]
* [[Turbo machinery]]: Flows inside rotating passages, diffusers etc.
* [[Electrical and electronics engineering]]: cooling of equipment including microcircuits.
* Chemical process engineering: mixing and separation and polymer moulding.
* [[Marine engineering]]: loads on off-shore structure.
* [[Environmental engineering]]: distribution of pollutant and effluents.
* [[Hydrology]] and [[oceanography]]: flows in rivers, estuaries and oceans.
* [[Meteorology]]: weather prediction.
* [[Biomedical engineering]]: blood flows through arteries and veins.
* External and internal environment of buildings: wind loading, [[ventilation (architecture)|ventilation]] analysis and heating/cooling load calculations.

In early age of construction, the most of [[building]] related issues such as ventilation analysis, wind loading, wind environment etc. were conducted by the [[wind tunnel]] tests, but today all these test can be done effectively with CFD technique. CFD technique can resolve all above mentioned issues in very short time period and it is very economical as well as strong approach than the older one (experimental):.<ref>{{cite book|last=|first=Tom|authorlink=Tom Lawson|title=Building Aerodynamics|year=2010|publisher=Imperial College Press|isbn=978-81-7596-757-1}}</ref> Recently Computational fluid dynamics is used as a sophisticated airflow modelling method and can be used to predict airflow, heat transfer and contaminant transportation in and around buildings. CFD plays an important role in building design, designing a thermally-conformable, healthy and energy-efficient building. CFD can examine the effectiveness and efficiency of various Heating ventilation and air conditioning (HVAC) systems by easily changing the different types and location of different components of diffuser types and locations, supply air conditions and system control schedules. Furthermore, CFD helps in developing passive heating/cooling/ventilation strategies (e.g. natural ventilation) by modelling and optimizing building site-plans and indoor layouts.<ref>{{citation | journal=SAGE, Indoor and Built Environment, Pages: 305-313 | first=Zhiqiang | last=Zhai | year=2005 | title=Application of Computational Fluid Dynamics in Building Design: Aspects and Trends }}</ref>
Globally building sector shares approximately 40% of total energy consumption
.<ref>
{{citation
| journal=The Appraisal journal, Pages: 115
| first=Leopolds
| last=Berger
| year=2011
| title=Energising property valuation: putting a value on energy-efficient buildings
}}</ref>
In present era, there is a huge gap in [[energy consumption]] and energy production. As building sector share a huge amount of the total consumption, hence it becomes essential to investigate the optimum configuration for building to reduce the building's share of energy. In order to achieve this, CFD can play an important role. Energy simulation and CFD programs are important [[building design]] tools which are used for evaluation of building performance, including thermal comfort, indoor air quality mechanical system efficiency and energy consumption
.<ref>
{{citation
| journal=building energy and cfd simulation, Pages: 333-344
| first=johan zhai
| last=Zhiqiang
| year=2005
| title=Energising property valuation: putting a value on energy-efficient buildings
}}</ref>
. CFD in buildings mainly used for one or more followings purposes:
# [[Thermal analysis]]: through walls, roof and floor of buildings
# [[Ventilation (architecture)|Ventilation]] analysis.
# Orientation, site and location selection of buildings based on local geographical and environmental conditions.

===Thermal analysis===

In buildings, [[heat transfer]] takes place in its all modes i.e. [[Conduction (heat)|conduction]], [[convection]] and [[radiation]]. In order to reduce heat losses from buildings, CFD analysis can be done for the optimum configuration of [[composite walls]], roof and floor. The differential form of the general transport equation is as follows
.<ref>{{cite book|last=Versteeg|first=H.|authorlink=H.Versteeg|title=An Introduction to Computational Fluid Dynamics|year=2009|publisher=Pearson Publications|isbn=978-81-317-2048-6}}</ref>
:<math> { \frac{\partial{(\rho \phi)}}{\partial t}} + { div\, (\rho u \phi )} ={div\, (k\, grad\, \phi )} + {S_{\phi}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,(1) </math>
 
The numerical solution of above equation can be obtained by finite difference method (FDM), finite volume method (FVM) and finite element method (FEM). In buildings, for heat transfer analysis, the scalar function ф in equation (1) is replaced by Temperature (T), diffusion coefficient Γ is replaced by thermal conductivity k and the source term <math>S_{\phi} </math> is replaced by heat generation term e or by any heat radiation source <math>Q_i </math> or by both (depending upon the nature of source available) and we have different forms of equation for different cases. For simplicity and easy understanding, only 1-Dimensional cases have been discussed. 
In buildings the heat transfer analysis can be done for all parts of buildings (walls, roof and floor) in following two ways
# Steady State Thermal Analysis
# Transient Thermal Analysis

====Steady state thermal analysis ====
The steady state thermal analysis consist the following type of governing differential equations.
 
'''Case-1''': General steady state heat conduction equation.
 
For this case the governing differential equation (GDE) (1) becomes as follows:
:<math> { div\, (\rho u T )} ={div\, (k\, grad\, T )}+ {S_{T}} \, </math>
 
'''Case-2''': Steady state heat conduction equation (no heat generation)
 
For this case the governing differential equation (GDE) (1) becomes as follows:
:<math> { div\, (\rho u T )} ={div\, (k\, grad\, T )} \, </math>
 
'''Case-3''': Steady state heat conduction equation (no heat generation and no convection)
 
For this case the governing differential equation (GDE) (1) becomes as follows:
:<math> {div\, (k\, grad\, T )} = 0 \, </math>

====Transient thermal analysis ====
The transient thermal analysis consist the following type of governing differential equations.
 
'''Case-1''': Transient heat conduction
 
For this case the governing differential equation (GDE) (1) becomes as follows:
:<math> { \frac{\partial{(\rho T)}}{\partial t}} + { div\, (\rho u T )} ={div\, (k\, grad\, T )}+ {S_{T}} \, </math>
 
'''Case- 2''': Transient heat conduction (no heat generation)
For this case the governing differential equation (GDE) (1) becomes as follows:
 
:<math> { \frac{\partial{(\rho T)}}{\partial t}} + { div\, (\rho u T )} ={div\, (k\, grad\, T )} </math>
 
'''Case-3''': Transient heat conduction (no heat generation and no convection)
 
For this case the governing differential equation (GDE) (1) becomes as follows:
:<math> { \frac{\partial{(\rho T)}}{\partial t}} = {div\, (k\, grad\, T )} \, </math>
 
We can solve these above mentioned governing differential equation (GDE) equations using CFD technique.

===Ventilation analysis===
The ventilation study in buildings is done to find the thermally comfortable environment with acceptable indoor air quality by regulating indoor air parameters (air temperature, relative humidity, air speed, and chemical species concentrations in the air). CFD finds an important role in regulating the indoor air parameters to predict the ventilation performance in buildings. The ventilation performance prediction provides the information regarding indoor air parameters in a room or a building even before the construction of buildings
.<ref>
{{citation
| journal=Building and Environment, Pages: 848-858
| first=Q.
| last=Chen
| year=2009
| title=Ventilation performance prediction for buildings:A method overview and recent applications
}}</ref>
 
These air parameters are crucial for designing a comfortable indoor as well as outdoor environment. This is because the design of appropriate ventilation systems and the development of control strategies need detailed information regarding the following parameters;

*Airflow
*Contaminant dispersion
*Temperature distribution
 
The aforesaid information’s are also useful for an architect to design the building configuration. From the last three decade, the CFD technique is widely used with considerable success in building.<ref>
{{citation
| journal=International Journal on Architectural Science, Pages: 14-29
| first=Q.
| last=Chen
| first=J..
| last=Srebric
| year= 2000
| title=Application of CFD Tools for Indoor and Outdoor Environment Design
}}</ref>
 
Recently ventilation and its related fields has becomes a great part of wind engineering. A ventilation study can be done using wind tunnel investigation (experimentally) or by CFD modeling (theoretically). Natural ventilation system is always preferred over the forced ventilation system, as it causes to save the burning of fuel, which is economical as well as nature friendly. In present era, due to development of a lot of CFD software and other building's energy simulation software, it becomes quite easy to assess the possibility of natural/forced ventilation system in a building. CFD analysis is quite useful than the experimental approach because here we can find other related relations among the variables in post-processing. The data obtained either experimental or numerically is useful in two ways
.<ref>{{cite book|last=|first=Tom|authorlink=Tom Lawson|title=Building Aerodynamics|year=2010|publisher=Imperial College Press|isbn=978-81-7596-757-1}}</ref>
 
1) better comfort of user.
 
2) It provides the data which is used as input to the heat balance calculation of the buildings .

===Orientation, site and location selection===
[[File:Figuree onea.jpg|thumb|right|Figure-1 (a): Flow around a building (collection of air at height and delivering it at ground level)]]
[[File:Figuree oneb.jpg|thumb|right|Figure-1 (b): Flow around a building (center of the front face )]]
Earlier, the choice of dwelling location was made mindful of the need for water, so most of earlier development started in valley area. In present era, due to advancement in science and technology, it becomes easy to select a proper orientation, site and location of buildings based on local geographical and environmental conditions. In selection of building site and location, wind loading plays and important role. In case of two buildings at a location exists side by side having some gap, when volume of [[wind]] blows round the ends of building through the gap is, in first instant the sum of flow around each building separately, then its velocity must increase above that around the end of a single building at the expense of pressure loss.
So, there will be a built of pressure, entering the gap, which will lead to higher wind loads on the sides of buildings. When wind blows over the face of a high rise building, a vortex is created by the downward flow on the front face (as shown in figure-1). The wind speed in the reverse direction near the ground level may have 140 percent of the reference wind speed. So, if any building exist in such region, then that may be subjected to damage (especially the roof of building may get severe damage). Such damage to buildings can be prohibited successfully, if the effects of wind loading are considered in the early stage of construction of a building. In early age of construction all these wind loading effects were determined by the wind tunnel test, but today all these test can be successfully through CFD analysis. The importance of providing pleasant environment to buildings is increasing and architect and wind engineer are often asked to look over the design (orientation, site, location and gaps between the surrounding buildings) in the formative stage of buildings & planning stage of construction.<ref>{{cite book|last=|first=Tom|authorlink=Tom Lawson|title=Building Aerodynamics|year=2010|publisher=Imperial College Press|isbn=978-81-7596-757-1}}</ref> So by using CFD analysis, we can find the suitable information (local wind velocity, convective coefficients, and solar radiation intensity) for orientation, site and location selection of buildings.

==CFD approach for heat transfer analysis in buildings==
 
CFD technique can be used for the analysis of heat transfer in each part of a building.CFD technique finds the solution by following ways:
# Discretization of the governing differential equation using numerical methods (Finite difference method has been discussed).
# Solve the discretized version of equation with high performance computers.

===Discretization of the governing differential equations for the steady state heat transfer analysis===
Consider a building having a plane wall with thickness L, heat generation e and constant thermal conductivity k. The wall is subdivided into M equal regions of thickness <math>\Delta x</math> = X/T in x-direction, and the divisions between the regions are selected as nodes as shown in figure-2.
 
[[File:Fig one.jpg|thumb|right|Figure-2:the nodal points and volume elements for the finite difference formulation of 1-D conduction in a plane wall]]
 
The whole domain of wall in x-direction is divided in elements as shown in figure and the size of all interior elements is same while for exterior elements it is half.

Now to obtain the FDM solution for the interior nodes, consider the element represented by the node m which is surrounded by neighboring nodes m-1 and m+1. The FDM technique presumes that temperature varies linearly in walls (shown in figure-3).
FDM solution is:( for all interior nodes except to 0 and last node)

:<math> \frac{(T_{m-1}^i - 2T_{m}^n +T_{m}^i )}{\Delta {x}^2} + \frac {e}{k} = 0 </math>
 
[[File:Fig two.jpg|thumb|right|Figure-3:Linear temperature variation in finite difference formulation]]

====Boundary conditions====
Above equation is valid only to interior nodes only. To obtain the solution for exterior nodes we have to apply the boundary conditions (as applicable), which are as follows.<ref>{{cite book|last=A. Cengel|first=Yunus|authorlink=Yunus A. Cengel|title=Heat and mass transfer|year=2008|publisher=Tata McGraw-Hills|isbn=978-0-07-063453-4}}</ref>
'''1.Specified heat flux boundary condition'''
:<math> q_{0} A+ k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
When boundary is insulated (q=0)
:<math> k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
 
'''2. Convective boundary condition'''
:<math> h A {(T_{\infty} - T_{0} )}+k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
 
'''3. Radiation boundary condition'''
:<math> \epsilon \sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
 
'''4. Combined convective and radiation boundary condition''' (shown in figure-4).
:<math> h A {(T_{\infty} - T_{0} )}+\epsilon \sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
 
or
when radiation and convection heat transfer coefficient are combined, above equation becomes as follows;
:<math> h A_{combined} {(T_{\infty} - T_{0} )}+k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 \, </math>
 
[[File:Fig three.jpg|thumb|right|Figure-4: Schematic for the FDM formulation of combined convective and radiation on the left boundary of a plane wall]]
 
'''5. Combined convective, radiation and heat flux boundary condition'''
:<math> q_{0}A+h A {(T_{\infty} - T_{0} )}+\epsilon \sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\frac{(T_{1} - T_{0} )}{\Delta {x}} + \frac {e_{0}}{2}A \Delta {x} = 0 </math>
 
'''6.Interface boundary condition : when there is an interface''' (in composite walls) of different walls having different thermo-physical properties, the two different solid media A and B are assumed to be perfect contact and thus have same temperature at interface at node m (as shown in figure-5).
:<math> k_{A} A\frac{(T_{m-1} - T_{m} )}{\Delta {x}} +k_{B} A\frac{(T_{m+1} - T_{m} )}{\Delta {x}}+ \frac {e_{A,m}}{2}A \Delta {x}+\frac {e_{B,m}}{2}A \Delta {x} = 0 </math>
 
[[File:Fig four.jpg|thumb|right|Figure-5: Schematic for the FDM of the interface boundary condition for two mediums A and B having perfect thermal contact]]
 
In above equations q_0 = denotes specified heat flux is in <math>(W/m^2)</math>, h =convective coefficient, <math> h_{combined}</math> = combined convective and radiation heat transfer coefficient,<math>T_sur</math> = Temperature of surrounding surface,<math>T_(\infty)</math> =Ambient Temperature, <math>T_0</math> = Temperature of at initial node.
Note: For interior side of wall we can apply the suitable boundary condition from above (as applicable), in that case <math>T_(\infty)</math> will be replaced by <math>T_r</math> (Room Temperature), <math>T_0</math>= will be replaced by <math>T_l</math> (Temperature of last node).

===Discretization of the governing differential equations for the transient heat transfer analysis===
Transient thermal analysis is more important than the steady thermal analysis, as this analysis include the variable ambient condition with time. In transient heat conduction, the temperature changes with time as well as position. The finite difference solution of transient heat conduction requires discretization in time in addition to space, as shown in figure-6.
 
[[File:Fig five.jpg|thumb|right|Figure-6: FDM fotirmulation of time dependent problem involves discrete points in time as well as in space]]
 
The nodal points and volume elements for the transient FDM formulation of 1-D conduction in a plane wall exist as shown in the figure-7.
 
[[File:Fig six.jpg|thumb|right|Figure-7:The nodal points and volume elements for the transient FDM formulation of 1-D conduction in a plane wall]]
 
For this case the FDM explicit solution for equation (1) will be as follows,
 
:<math> k A\frac{(T_{m-1}^i - T_{m}^i )}{\Delta {x}} +k A\frac{(T_{m+1}^i - T_{m}^i )}{\Delta {x}}+ {e_{m}}A \Delta {x}= (\rho c_{p} \Delta x A) \frac{(T_{m}^{i+1} - T_{m}^i )}{\Delta x} </math>
The above equation can be solved explicitly for the temperature <math>(T_{m}^{i+1})</math> to give
:<math> {T_{m}^{i+1}}= \tau {(T_{m+1}^i - T_{m}^i )}+{(1-2\tau)}T_{m}^i +\tau \frac {(e_{m} \Delta {x}^2)}{k} </math>
 
where,
:<math> \tau = \frac{(\alpha \Delta t )}{\Delta x^2} \, </math>
and
:<math> \alpha = \frac{k}{\rho c_p} \, </math>
here,<math> \tau </math> represents the cell Fourier no,<math> \alpha </math> represents thermal diffusivity, <math> c_p </math> represents specific heat at constant pressure, <math> \Delta t </math> represents time step,<math> \Delta x </math> represents space step.

Above equation is valid for all interior nodes and to find the relation for first and last node, apply boundary conditions (as applicable) as discussed in steady state heat transfer. For a convective & radiation boundary if solar radiation data <math> q_{solar} </math>\, in (<math> (W/m^2) </math>) is available and absorptivity-transmissivity constant K is known, the relation for temperature is obtained as follows;
:<math> h A {(T_{\infty}^i - T_{0}^i )}+ \kappa A q_{sol} = (\rho c_{p} \Delta x A) \frac{(T_{1}^i - T_{0}^i )}{\Delta x} </math>
Note: the thermal analysis for the roof and floor of a building can be done in same way, as discussed for walls.\\

== See also ==
*[[Computational fluid dynamics]]
*[[CFD]]{{dn|date=July 2013}}
*[[Natural ventilation]]
*[[JPMorgan Chase Tower (Houston)]]
*[[Environmental Systems Design, Inc.]]
*[[Dynamic insulation]]
*[[Thermal management of high-power LEDs]]
*[[Vented balance safety enclosure]]
*[[Different Types of Boundary Conditions in Fluid Dynamics]]
*[[Wind tunnel]]
*[[Greenhouse]]

== External links ==
* [https://www.cibse.org/content/Groups/Building_Simulation_Group/CMLETBE10/Richard%20Chitty%20%28BRE%29%20-%20Analysis%20of%20Building%20Performance%20using%20Computational%20Fluid%20Dynamics%20%28CFD%29%20-%20121010.pdf]-Analysis of Building Performance using Computational Fluid Dynamics
* [https://www.google.co.in/search?q=cfd+analysis+in+buildings&hl=en&tbm=isch&tbo=u&source=univ&sa=X&ei=ynyIUZvCA4WyrAe85AE&ved=0CD8QsAQ&biw=1920&bih=1028] - Images for cfd analysis in buildings
* [http://www.designbuildersoftware.com/docs/designbuilder/DesignBuilder_CFD_DraftManual.pdf]-DesignBuilder CFD
* [http://www.inive.org/members_area/medias/pdf/Inive/IAQVEC2007/Tominaga.pdf]-CFD ANALYSIS OF FLOW AND CONCENTRATION FIELDS AROUND A BUILDING WITH A ROOF STACK
* [http://www.bse.polyu.edu.hk/researchCentre/Fire_Engineering/summary_of_output/journal/IJAS/V2/p.67-82.pdf]-CFD AS A BUILDING SERVICES ENGINEERING TOOL
* [https://engineering.purdue.edu/~yanchen/paper/2005-1.pdf]-Performance of Coupled Building Energy and CFD Simulations
* [http://www.ibpsa.org/proceedings/BS2009/BS09_0489_496.pdf]-APPLICATION OF CFD IN BUILDING PERFORMANCE SIMULATION FOR THE OUTDOOR ENVIRONMENT
* [http://www.sciencedirect.com/science/article/pii/S0360132302000458]-Integrating CFD and building simulation
* [http://www.mechartes.com/casestudy/Building_Design_Services.html]-Building Design and Analysis
*[http://www.flowanalysis.co.uk/buildings.html]-Building Simulation
*[http://www.halcrow.com/Documents/fire_safety/cfd_soc_building_serv.pdf]-Capability in Computational Fluid Dynamics (CFD) for Building Services
*[http://www.cfd-online.com/Forums/main/72725-cfd-hvac-green-building.html]- CFD in HVAC and green building
*[http://www.engr.colostate.edu/~meroney/PapersPDF/CEP09-10-1.pdf]-CFD Prediction of Airflow in Buildings for Natural Ventilation
*[http://www.glumac.com/greenresources/gr_cfd_advantages.html]-CFD Advantages and Practical Applications
*[https://www.modelica.org/events/modelica2011/Proceedings/pages/papers/12_3_ID_179_a_fv.pdf]- Numerical coupling of Modelica and CFD for building energy supply systems

==References==
{{Reflist}}

[[Category:Computational fluid dynamics]]

Ernst angle

2013-04-16T13:16:07Z

134.99.165.175:

[[File:TransmissionLineDefinitions.svg|thumb|310px|A [[transmission line]] is drawn as two black wires. At a distance ''x'' into the line, there is current [[phasor]] ''I(x)'' traveling through each wire, and there is a [[voltage difference]] phasor ''V(x)'' between the wires (bottom voltage minus top voltage). If <math>Y_0</math> is the '''characteristic admittance''' of the line, then <math>I(x) / V(x) = Y_0</math> for a wave moving rightward, or <math>I(x)/V(x) = -Y_0</math> for a wave moving leftward.]]
'''Characteristic admittance''' is the mathematical inverse of the [[characteristic impedance]].
The general expression for the characteristic admittance of a transmission line is:

:<math>Y_0=\sqrt{\frac{G+j\omega C}{R+j\omega L}}</math>
where
:<math>R</math> is the [[Electrical resistance|resistance]] per unit length,
:<math>L</math> is the [[inductance]] per unit length,
:<math>G</math> is the [[Electrical conductance|conductance]] of the dielectric per unit length,
:<math>C</math> is the [[capacitance]] per unit length,
:<math>j</math> is the [[imaginary unit]], and
:<math>\omega</math> is the [[angular frequency]].

The current and voltage [[Phasor (electronics)|phasor]]s on the line are related by the characteristic admittance as:
:<math>\frac{I^+}{V^+} = Y_0 = -\frac{I^-}{V^-}</math>
where the superscripts <math>+</math> and <math>-</math> represent forward- and backward-traveling waves, respectively.
==See also==
* [[Characteristic impedance]]

==References==
*{{cite book
| last = Guile
| first = A. E.
| title = Electrical Power Systems
| year = 1977
| isbn = 0-08-021729-X }}
*{{cite book
| last = Pozar
| first = D. M.
| title = Microwave Engineering
| edition = 3rd edition
|date=February 2004
| isbn = 0-471-44878-8 }}
*{{cite book
| last = Ulaby
| first = F. T.
| title = Fundamentals Of Applied Electromagnetics
| edition = media edition
| year = 2004
| publisher = Prentice Hall
| isbn = 0-13-185089-X }}

{{DEFAULTSORT:Characteristic Admittance}}
[[Category:Electricity]]
[[Category:Physical quantities]]
[[Category:Distributed element circuits]]

Transition dipole moment

2012-04-26T11:22:48Z

134.99.254.46: /* Definition */

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