Chemical Engineering 412
|
|
- Ambrose Eaton
- 5 years ago
- Views:
Transcription
1 Chemical Engineeing 41 Intoductoy Nuclea Engineeing Lectue 16 Nuclea eacto Theoy III Neuton Tanspot 1
2 One-goup eacto Equation Mono-enegetic neutons (Neuton Balance) DD φφ aa φφ + ss 1 vv vv is neuton speed Fo eacto, ss νν ff φφ νν is neutons/fission In eigenfunction fom and at steady state DD φφ aa φφ + νν kk ffφφ 0 φφ aa νν kk f DD φφ φφ + BB φφ 0
3 Mateial Buckling φφ BB φφ φφ + BB φφ 0 νν BB kk ff aa DD kk νν ff φφ DDBB φφ + aa φφ eigenfunction fom of eacto equation one-goup, steady-state, eacto equation mateial buckling (neuton geneation absoption)/diffusion νν ff DDBB + aa eacto multiplication facto multiplication facto neuton geneation ate/(leakage + absoption)
4 Fuel utilization and k ss ηη aaaa φφ ηη aaaa aa aa φφ ηηηη aa φφ ff aaaa aa kk ηηηη aaφφ aa φφ ff fuel utilization facto neutons absobed by fuel / (those absobed by fuel + by othe means coolant, modeato, etc.) ηηηη kk k-value fo infinite (no oveall leakage) eacto a mateial popety η fission neutons geneated pe absobed neuton
5 Opeating Citical eacto Equation kk 1 DDBB φφ aa φφ + kk kk aaφφ 1 vv DDBB φφ + kk 1 aa φφ 0 eacto opeating at steady state BB φφ + kk 1 aa DD φφ BB φφ + kk 1 LL φφ 0 LL DD aa One-goup diffusion aea BB kk 1 LL One-goup buckling
6 Pespective Pevious equations show how to solve fo neuton flux pofile φφ as a function of space How to detemine citical eacto dimensions BB kk 11 LL kk 11 aa, fo a bae eacto (B DD gb mat) Fist find solutions to the eacto equations 1D, D, o 3D Then find dimensions fo a citical eacto Assumptions: Bae, homogeneous eactos Constant (special and tempoal) popeties None ae valid but, but help to develop insight into eacto opeations Because souce tems ae popotional to the flux, the geneally inhomogeneous diffeential equations ae now homogeneous equations.
7 Bae Slab eacto Solution dd φφ ddxx BB φφ eacto equation φφ aa φφ aa φφ 0 0 bounday conditions φφ xx AA cos BBBB + CC sin BBBB CC 0 φφ xx AA cos BBBB φφ aa/ 0 BB nn nnnn aa dd φφ geneal solution fom symmety o by substitution Eigenvalues fom bounday conditions all n impotant in tansient solution, only n1 impotant fo steady solution BB 1 is buckling (pop. to flux cuvatue) BB 1 1 φφ ddxx The constant A is as yet undetemined and elates to the powe. Thee ae diffeent solutions to this poblem fo evey powe level. aa aa Infinite plane indicates no net flux fom sides
8 PP EE ff Bae Slab eacto Powe aa/ aa/ PP AA aaee ff ππ φφ xx φφ xx dddd sin ππππ aa ππππ aaee ff sin ππππ aa cos ππππ aa EE is the ecoveable enegy pe fission Powe Scales with flux! aa aa Infinite plane indicates no net flux fom sides
9 Absobe/emitte vs eacto 1 LL φφ dd φφ ddxx ss DD dd φφ ddxx BB φφ tanspot in an absobe/emitte tanspot in a eacto souce popotional to flux φφ xx ss 1 cosh xx LL aa + dd aa cosh LL ππππ φφ xx aaee ff sin ππππ cos ππππ aa aa flux in an absobe/emitte flux in a eacto
10 1 dd dddd dddd dddd Spheical eacto BB φφ φφ φφ 0 0 φφ AA sin(bbbb) + C cos(bbbb) CC 0 sin BBBB φφ AA BB nn nnnn eacto tanspot equation bounday conditions fom symmety o by substitution Eigen values specific solution geneal solution BB 1 ππ buckling φφ AA sin ππ
11 Spheical eacto Powe Integate ove symmetic dimensions tansfom volume integal to adial integal PP EE ff φφ dddd 4ππEE ff PP 4ππEE ff AA ππ ππ sin ππππ 0 cos ππππ φφ dd again, powe is popotional to flux and highest at cente φφ PP sin ππππ 4 EE ff
12 Infinite Cylindical eacto 1 dd dddd dddd dddd BB φφ dd φφ dd + 1 dddd dddd eacto tanspot equation φφ φφ 0 0; φφ < bounday conditions dd φφ dd + 1 dddd dddd + BB mm φφ 0 φφ AAJJ 0 BBBB + CCYY 0 BBBB φφ AAJJ 0 BBBB BB nn xx nn BB 1 xx 1 φφ AAJJ zeo-ode (m0) Bessel equation geneal solution involves Bessel functions of fist and second kind flux is finite oots of Bessel functions - φφ is zeo at bounday fist oot solution (powe poduction detemines A)
13 Bessel Functions J0 J1 J Y0 Y1 Y Bessel Function x
14 Infinite Cylindical eacto Powe PP EE ff φφ dddd ππee ff.405 PP ππee ff JJ 0 0 xxxjj 0 xx ddxxx xxjj 1 xx 0 PP ππee ff AAJJ φφ 0.738PP EE ff JJ dddd 0 dddd tansfom volume integal to adial integal becomes powe pe unit length again, powe is popotional to powe and highest at cente
15 1 Finite Cylindical eacto + φφ zz BB φφ eacto tanspot equation φφ, zz φφ 0, zz φφ, HH φφ, HH 0 φφ, zz ZZ(zz) φφ zz ZZ + 1 ZZ ZZ BB zz sepaation of vaiables bounday conditions + ZZ zz BB ZZ(zz) since and ZZ vay independently, both potions of the equation must equal (geneally diffeent) constants, designated as BB and BB ZZ, espectively HH HH
16 1 BB.405 AA JJ 0 ZZ zz BB ZZ ZZ Finite Cylinde Solution a poblem we aleady solved, w/ same bcs again a poblem we aleady solved, w/ same bcs ZZ zz AA cos ππππ HH φφ, zz AA JJ BB BB + BB HH cos ππππ HH solution is the poduct of the infinite cylinde and infinite slab solutions Buckling is highe than fo eithe the infinite plane o the infinite cylinde. Buckling geneally inceases with inceasing leakage, and thee ae moe sufaces to leak hee than eithe of the infinite cases. HH HH
17 Neuton Flux Contous Neuton flux in finite cylindical eacto 3D contous of neuton flux at high powe 3D contous w colo scaled to magnitude intemediate powe 3D contous of neuton flux at low powe
18 Neuton Flux Contous Neuton flux in finite paallelepiped (cubical) eacto 3D contous of neuton flux at high powe 3D contous w colo scaled to magnitude intemediate powe 3D contous of neuton flux at low powe
19 Citical Buckling kk νν ff aa + DDBB value of k fo citical eacto BB νν ff aa DD BB cc νν ff aa DD BB ll kk 1 LL value of B when k 1 citical mateial buckling geometic buckling νν ff aa DD kk 1 LL geometic and mateial buckling must be equal fo a citical eacto
20 Citical Equation (One goup) kk 1 + BB LL 1
21 Physical Intepetation k 1+ B k φdv a a L a + DB a φdv + DB φdv k k P L The pobability of non-leakage is invesely popotional to k k P k L a k a φdv φdv P L ηfp L geometic and mateial buckling must be equal fo a citical eacto
22 a f Themal eactos Fou Facto Fomula af a af + am af af + total coss section sum of fuel and modeato am themal fuel utilization facto η T η ( E) σ ( E) φ( E) σ af af ( E) φ( E) de de neutons emitted/themal neuton absobed in the fuel k pεηt f φ a T a φ T pεη f T infinite multiplication facto popotional to pobability of escaping esonance absoption, fast fission contibution,
23 Citicality Calculations T T ( 1 ) ( + B L 1+ B τ ) T T 1/ pob themal neuton1/ pob fast neuton doesn' t leak doesn' t leak while slowing L τ k T D D P P 1 1 F a k 1 ( )( 1 1 ) 1 ( + B L + B τ + B L + τ ) T k k T k T T Two-goup equation fo a bae (themal) eacto Modified one-goup citical equation k 1+ B M T 1
24 Bae eacto Summay / cos 3.64 / 3.63 cos.405 J / J / 3.85 cos cos cos / 1.57 cos f f f f f E P A sphee VE P H z A H z D cylinde E P A D cylinde VE P c z b y a x A c b a D plate ae P a x A a D plate π π π π π π π π π π π geomety Buckling (B ) Flux A φ av Ω φ max
25 eflected eactos L A L C L A B A B C B A L L k B B c c c c c exp ' exp ' exp ' sin cos sin φ φ φ φ φ φ φ φ coe tanspot equation coe mateials popeties eflecto tanspot equation geneal solution fo coe flux must be finite at the cente geneal solution fo the eflecto flux must be finite as inceases
26 φc J c D A D ( ) φ ( ) ( ) n J ( ) n φ c ( ) Dφ ( ) sinb exp( / L) c AD c c A B cosb B cot B 1 D B cot B 1 D 1 B cot B L sinb D c eflected eactos L A' D 1 L fluxes equal at coe-eflecto inteface cuent densities also equal equate fluxes and cuent densities 1 L 1 + exp L divide cuent density by flux equation citical equation fo eflected eacto (tanscendental equation) citical equation when D D c (not tanscendental in )
27 eflected eactos < Bae eactos B cot B D Dc L
28 Detemine emaining Unknown ( ) ( ) B B B E PB A B B B B A E d B A E P dv E P B L A A f f f c f cos sin 4 cos sin 4 sin 4 sin exp ' 0 0 π π π φ
29 Some details eflected eactos lend themselves less easily to analytical solution commonly eactos ae consideed as sphee equivalents athe than tying to solve the equations. easonable epesentation fo fast neutons not fo themal eactos eflecto savings in size is typically about the thickness of the extapolated distance.
30 Flux Compaisons
31 Themal Flux Vaiations
32 Two-enegy, detailed model
33 Position 1 (cente of od) fast themal
34 Position (outside od at :30/7:30) fast themal
35 In a Vacuum fast themal
36 Azimuthal Dependence fast themal
Chemical Engineering 693R
Chemical Engineering 693 eactor Design and Analysis Lecture 8 Neutron ransport Spiritual hought Moroni 7:48 Wherefore, my beloved brethren, pray unto the Father with all the energy of heart, that ye may
More informationFlux Shape in Various Reactor Geometries in One Energy Group
Flux Shape in Vaious eacto Geometies in One Enegy Goup. ouben McMaste Univesity Couse EP 4D03/6D03 Nuclea eacto Analysis (eacto Physics) 015 Sept.-Dec. 015 Septembe 1 Contents We deive the 1-goup lux shape
More information1D2G - Numerical solution of the neutron diffusion equation
DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both
More informationOne-Dimensional, Steady-State. State Conduction with Thermal Energy Generation
One-Dimensional, Steady-State State Conduction with Themal Enegy Geneation Implications of Enegy Geneation Involves a local (volumetic) souce of themal enegy due to convesion fom anothe fom of enegy in
More informationI( x) t e. is the total mean free path in the medium, [cm] tis the total cross section in the medium, [cm ] A M
t I ( x) I e x x t Ie (1) whee: 1 t is the total mean fee path in the medium, [cm] N t t -1 tis the total coss section in the medium, [cm ] A M 3 is the density of the medium [gm/cm ] v 3 N= is the nuclea
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More information8 Separation of Variables in Other Coordinate Systems
8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies
More informationHeat transfer has direction as well as magnitude. The rate of heat conduction
cen58933_ch2.qd 9/1/22 8:46 AM Page 61 HEAT CONDUCTION EQUATION CHAPTER 2 Heat tansfe has diection as well as magnitude. The ate of heat conduction in a specified diection is popotional to the tempeatue
More informationPhysics 2212 GH Quiz #2 Solutions Spring 2016
Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationBlack Body Radiation and Radiometric Parameters:
Black Body Radiation and Radiometic Paametes: All mateials absob and emit adiation to some extent. A blackbody is an idealization of how mateials emit and absob adiation. It can be used as a efeence fo
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More information, and the curve BC is symmetrical. Find also the horizontal force in x-direction on one side of the body. h C
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Dynamics (Stömningsläa), 2013-05-31, kl 9.00-15.00 jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics.
More information2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0
Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee
More informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More information$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer
Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =
More informationAstronomy 111, Fall October 2011
Astonomy 111, Fall 011 4 Octobe 011 Today in Astonomy 111: moe details on enegy tanspot and the tempeatues of the planets Moe about albedo and emissivity Moe about the tempeatue of sunlit, adiation-cooled
More information( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is
Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More informationFlux. Area Vector. Flux of Electric Field. Gauss s Law
Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is
More informationMONTE CARLO SIMULATION OF FLUID FLOW
MONTE CARLO SIMULATION OF FLUID FLOW M. Ragheb 3/7/3 INTRODUCTION We conside the situation of Fee Molecula Collisionless and Reflective Flow. Collisionless flows occu in the field of aefied gas dynamics.
More informationLECTURER: PM DR MAZLAN ABDUL WAHID PM Dr Mazlan Abdul Wahid
M 445 LU: M D MZL BDUL WID http://www.fkm.utm.my/~mazlan hapte teady-tate tate One Dimensional eat onduction M bdul Wahid UM aculty of Mechanical ngineeing Univesiti eknologi Malaysia www.fkm.utm.my/~mazlan
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationWelcome to Physics 272
Welcome to Physics 7 Bob Mose mose@phys.hawaii.edu http://www.phys.hawaii.edu/~mose/physics7.html To do: Sign into Masteing Physics phys-7 webpage Registe i-clickes (you i-clicke ID to you name on class-list)
More informationIn the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
More informationPHYS 301 HOMEWORK #10 (Optional HW)
PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2
More informationPhys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations
Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces
More informationLiquid gas interface under hydrostatic pressure
Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More information1) Consider an object of a parabolic shape with rotational symmetry z
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), 01-06-01, kl 9.00-15.00 jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics.
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More informationPOISSON S EQUATION 2 V 0
POISSON S EQUATION We have seen how to solve the equation but geneally we have V V4k We now look at a vey geneal way of attacking this poblem though Geen s Functions. It tuns out that this poblem has applications
More informationx x2 2 B A ) v(0, t) = 0 and v(l, t) = 0. L 2. This is a familiar heat equation initial/boundary-value problem and has solution
Hints to homewok 7 8.2.d. The poblem is u t ku xx + k ux fx u t A u t B. It has a souce tem and inhomogeneous bounday conditions but none of them depend on t. So as in example 3 of the notes we should
More informationTHE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2
THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace
More informationMathematical Model of Magnetometric Resistivity. Sounding for a Conductive Host. with a Bulge Overburden
Applied Mathematical Sciences, Vol. 7, 13, no. 7, 335-348 Mathematical Model of Magnetometic Resistivity Sounding fo a Conductive Host with a Bulge Ovebuden Teeasak Chaladgan Depatment of Mathematics Faculty
More information2.5 The Quarter-Wave Transformer
/3/5 _5 The Quate Wave Tansfome /.5 The Quate-Wave Tansfome Reading Assignment: pp. 73-76 By now you ve noticed that a quate-wave length of tansmission line ( λ 4, β π ) appeas often in micowave engineeing
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationf(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4
Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p 2 + 2 (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y
More informationAbsorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere
Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More information11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.
Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings
More information2. Plane Elasticity Problems
S0 Solid Mechanics Fall 009. Plane lasticity Poblems Main Refeence: Theoy of lasticity by S.P. Timoshenko and J.N. Goodie McGaw-Hill New Yok. Chaptes 3..1 The plane-stess poblem A thin sheet of an isotopic
More informationb Ψ Ψ Principles of Organic Chemistry lecture 22, page 1
Pinciples of Oganic Chemisty lectue, page. Basis fo LCAO and Hückel MO Theoy.. Souces... Hypephysics online. http://hypephysics.phy-ast.gsu.edu/hbase/quantum/qm.html#c... Zimmeman, H. E., Quantum Mechanics
More informationEM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)
EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq
More informationUniversity Physics (PHY 2326)
Chapte Univesity Physics (PHY 6) Lectue lectostatics lectic field (cont.) Conductos in electostatic euilibium The oscilloscope lectic flux and Gauss s law /6/5 Discuss a techniue intoduced by Kal F. Gauss
More information13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the charge N C m 8.
CHAPTR : Gauss s Law Solutions to Assigned Poblems Use -b fo the electic flux of a unifom field Note that the suface aea vecto points adially outwad, and the electic field vecto points adially inwad Thus
More informationHomework # 3 Solution Key
PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More information-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.
The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the
More information1 Spherical multipole moments
Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationToday s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call
Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationΦ E = E A E A = p212c22: 1
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flux. lectic Flux Φ though an aea A ~ Numbe of Field
More informationChapter 2 ONE DIMENSIONAL STEADY STATE CONDUCTION. Chapter 2 Chee 318 1
hapte ONE DIMENSIONAL SEADY SAE ONDUION hapte hee 38 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS empeatue pofile A B X X 3 X 3 4 X 4 Χ A Χ B Χ hapte hee 38 hemal conductivity Fouie s law ( is constant) A A
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationPROBLEM SET #3A. A = Ω 2r 2 2 Ω 1r 2 1 r2 2 r2 1
PROBLEM SET #3A AST242 Figue 1. Two concentic co-axial cylindes each otating at a diffeent angula otation ate. A viscous fluid lies between the two cylindes. 1. Couette Flow A viscous fluid lies in the
More informationProblem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by
Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,
More informationNumerical solution of diffusion mass transfer model in adsorption systems. Prof. Nina Paula Gonçalves Salau, D.Sc.
Numeical solution of diffusion mass tansfe model in adsoption systems Pof., D.Sc. Agenda Mass Tansfe Mechanisms Diffusion Mass Tansfe Models Solving Diffusion Mass Tansfe Models Paamete Estimation 2 Mass
More informationc n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0.
Poblem {a} Fo t : Ψ(, t ψ(e iet/ h ( whee E mc α (α /7 ψ( e /a πa Hee we have used the gound state wavefunction fo Z. Fo t, Ψ(, t can be witten as a supeposition of Z hydogenic wavefunctions ψ n (: Ψ(,
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. D = εe. For a linear, homogeneous, isotropic medium µ and ε are contant.
ANTNNAS Vecto and Scala Potentials Maxwell's quations jωb J + jωd D ρ B (M) (M) (M3) (M4) D ε B Fo a linea, homogeneous, isotopic medium and ε ae contant. Since B, thee exists a vecto A such that B A and
More informationA method for solving dynamic problems for cylindrical domains
Tansactions of NAS of Azebaijan, Issue Mechanics, 35 (7), 68-75 (016). Seies of Physical-Technical and Mathematical Sciences. A method fo solving dynamic poblems fo cylindical domains N.B. Rassoulova G.R.
More informationAnyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1
Anyone who can contemplate quantum mechanics without getting dizzy hasn t undestood it. --Niels Boh Lectue 17, p 1 Special (Optional) Lectue Quantum Infomation One of the most moden applications of QM
More informationV7: Diffusional association of proteins and Brownian dynamics simulations
V7: Diffusional association of poteins and Bownian dynamics simulations Bownian motion The paticle movement was discoveed by Robet Bown in 1827 and was intepeted coectly fist by W. Ramsay in 1876. Exact
More informationModule 05: Gauss s s Law a
Module 05: Gauss s s Law a 1 Gauss s Law The fist Maxwell Equation! And a vey useful computational technique to find the electic field E when the souce has enough symmety. 2 Gauss s Law The Idea The total
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chapte The lectic Field II: Continuous Chage Distibutions A ing of adius a has a chage distibution on it that vaies as l(q) l sin q, as shown in Figue -9. (a) What is the diection of the electic field
More informationElectric field generated by an electric dipole
Electic field geneated by an electic dipole ( x) 2 (22-7) We will detemine the electic field E geneated by the electic dipole shown in the figue using the pinciple of supeposition. The positive chage geneates
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More information6.641 Electromagnetic Fields, Forces, and Motion Spring 2005
MIT OpenouseWae http://ocw.mit.edu 6.641 Electomagnetic Fields, Foces, and Motion Sping 2005 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems. 6.641 Electomagnetic
More informationIntroduction to Arrays
Intoduction to Aays Page 1 Intoduction to Aays The antennas we have studied so fa have vey low diectivity / gain. While this is good fo boadcast applications (whee we want unifom coveage), thee ae cases
More informationLecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
More informationQuantum theory of angular momentum
Quantum theoy of angula momentum Igo Mazets igo.mazets+e141@tuwien.ac.at (Atominstitut TU Wien, Stadionallee 2, 1020 Wien Time: Fiday, 13:00 14:30 Place: Feihaus, Sem.R. DA gün 06B (exception date 18 Nov.:
More informationDOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS
DOING PHYIC WITH MTLB COMPUTTIONL OPTIC FOUNDTION OF CLR DIFFRCTION THEORY Ian Coope chool of Physics, Univesity of ydney ian.coope@sydney.edu.au DOWNLOD DIRECTORY FOR MTLB CRIPT View document: Numeical
More informationVector d is a linear vector function of vector d when the following relationships hold:
Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd
More informationElectromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Electomagnetic scatteing Gaduate Couse Electical Engineeing (Communications) 1 st Semeste, 1390-1391 Shaif Univesity of Technology Geneal infomation Infomation about the instucto: Instucto: Behzad Rejaei
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 10-1 DESCRIBING FIELDS Essential Idea: Electic chages and masses each influence the space aound them and that influence can be epesented
More informationQuantum Mechanics II
Quantum Mechanics II Pof. Bois Altshule Apil 25, 2 Lectue 25 We have been dicussing the analytic popeties of the S-matix element. Remembe the adial wave function was u kl () = R kl () e ik iπl/2 S l (k)e
More informationEKT 356 MICROWAVE COMMUNICATIONS CHAPTER 2: PLANAR TRANSMISSION LINES
EKT 356 MICROWAVE COMMUNICATIONS CHAPTER : PLANAR TRANSMISSION LINES 1 Tansmission Lines A device used to tansfe enegy fom one point to anothe point efficiently Efficiently minimum loss, eflection and
More informationSupplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in
Supplementay Figue 1. Cicula paallel lamellae gain size as a function of annealing time at 50 C. Eo bas epesent the σ uncetainty in the measued adii based on image pixilation and analysis uncetainty contibutions
More information1.2 Differential cross section
.2. DIFFERENTIAL CROSS SECTION Febuay 9, 205 Lectue VIII.2 Diffeential coss section We found that the solution to the Schodinge equation has the fom e ik x ψ 2π 3/2 fk, k + e ik x and that fk, k = 2 m
More informationPHYS 1444 Lecture #5
Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1 Coulom s Law The Fomula QQ Q Q F 1 2 1 2 Fomula 2 2 F k A vecto quantity. Newtons Diection of electic
More informationEKT 345 MICROWAVE ENGINEERING CHAPTER 2: PLANAR TRANSMISSION LINES
EKT 345 MICROWAVE ENGINEERING CHAPTER : PLANAR TRANSMISSION LINES 1 Tansmission Lines A device used to tansfe enegy fom one point to anothe point efficiently Efficiently minimum loss, eflection and close
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationStress Intensity Factor
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo Stess Intensity Facto We have modeled a body by using the linea elastic theoy We have modeled a cack in the body by a flat plane, and the
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.
More information[Griffiths Ch.1-3] 2008/11/18, 10:10am 12:00am, 1. (6%, 7%, 7%) Suppose the potential at the surface of a hollow hemisphere is specified, as shown
[Giffiths Ch.-] 8//8, :am :am, Useful fomulas V ˆ ˆ V V V = + θ+ φ ˆ and v = ( v ) + (sin θvθ ) + v θ sinθ φ sinθ θ sinθ φ φ. (6%, 7%, 7%) Suppose the potential at the suface of a hollow hemisphee is specified,
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More information