Coupled Mass Transport and Reaction in LPCVD Reactors

Size: px
Start display at page:

Download "Coupled Mass Transport and Reaction in LPCVD Reactors"

Transcription

1 ople Ma Tanpo an eaion in LPV eao ile A in B e.g., SiH 4 in H Sepaae eao ino o egion, inaafe & annla b -

2 oniniy Eqn: : onveion-iffion iffion-eaion Eqn Ampion! ile peie i in majo aie ga e.g., H isih 4! Iohemal! onan i an eniy No xn in he ga i i ~ Ame agnan in he inaafe egion i i i Seay ae

3 Eqaion an Bonay oniion fo he Eqaion an Bonay oniion fo he inaafe inaafe egion egion b fae eaion ae ne lo ae of SiH 4 ; elae o he epoiion ae! If i linea in hen hee i an analyi olion! If i nonlinea e have o ee nmeial olion

4 Appoximae olion Appoximae olion Fin Appoximaion Fin Appoximaion! Tae & aveage ove ieion! Aveaging ove he mall imenion i alle he fin appoximaion : an appoximaion hih i vey goo fo - egion ih high ape aio,, le volme aea fae π π noe

5 Nonimenionaliaion & Solion ξ θ b θ ξ θ ξ ξ θ θ ξ ξ ξ ξ Beel eqaion Miley Sheoo & ei, Applie Mah in he p. 74 i he Thiele Mol eaion ae iffion ae iffion eaion ime ime ale ale θ AI ξ o BK o ξ Zeo oe Beel fnion of he n in

6 onenaion pofile in he inaafe egion K o ha logaihmi inglaiy a ξ B Uing θ a ξ AI o θ Io ξ I o! a nifomiy ge bee an ineae; eaion limie; iffion fae han xn! a >> nifomiy egae an eeae; iffion limie; xn fae han iffion eaion ae iffion ae

7 Qanifying Unifomiy Qanifying Unifomiy η Unifomiy Inex hogho ae if epoiion epoiion ae aal b b π π η I I I I o b o b b ξ ξ θ ξ ξ I I o b b π π η 3 f I I o η lim ie iffion epoiion nonnifom lim ie xn epoiion nifom >> << η η

8 Goh ae iibion a a fnion of afe paing

9 Goh ae iibion a a fnion of A << η nifom epoiion xn lim ie B >> η nonnifom epoiion iffion lim ie

10 Annla egion Annla egion i i i i i No homogeneo Ga phae xn Seay ae < < < < L L Bonay oniion: L L o η π π η o o { ane Bonay oniion:

11 -aveage Eqn.. in he Annla egion! We ae moly ineee in afe-o-afe hange vaiaion an o no ae abo pofile in he annl.! We ill aveage ove he imenion an obain a ingle -aveage eqaion fo hee π le π θ, θ, π π θ θ

12 Loo ho bonay oniion in en p in he iffeenial eq Loo ho bonay oniion in en p in he iffeenial eqaion aion fo a if hey ae in he ga phae i.e, he omain of if fo a if hey ae in he ga phae i.e, he omain of iff eq eq.; fae.; fae eaion a aial all appea a if hey ae ga phae eai eaion a aial all appea a if hey ae ga phae eaion on η η ga Loo lie

13 imenional analyi imenional analyi L b η L L L iffive anpo onveiveanpo L ae onveion eaion ae epoiion on all L a ae onveion eaion ae epoiion on afe L a [ ] η a a

14 Bonay oniion an Solion [ β exp { M β M β } β exp{ M β M β } β exp{ M β } β exp{ M β } 4 a ηa hee β an M ane onino flo yem hemial Engineeing Siene, 953.

15 Limiing ae: PF limi! i.e., o mall iffion ae ompae o onveion Plg flo eao PF limi 4 a ηa β an M a* a a η M β 4 a a η a* 4a* a* lim e a*

16 Limiing ae: PF limi! Anohe ay of eeing hi i o loo a of he iffeenial eqaion an bonay oniion a beome [ a ηa ] a* a*~ a*>> e a*

17 Limiing ae: ST limi Limiing ae: ST limi! i.e., o mall flo ae onveion ompae o iffion onino ie an eao ST limi lim onan a a* a

18 Limiing ae: ST limi Limiing ae: ST limi! To fin he onenaion e aveage ove -ieion oo [ ] a* a* a* a* ST limi a*~ a*>> a* Ineaing a*

19 Wafe-o o-afe nifomiy Unifomiy inex ofen efine a U max avg min max avg min max avg min! U : goo nifomiy! U % i vaiane ih epe o aveage! A U ineae nifomiy egae! Fo fixe a* a U alo! mean! Sine ~ P loe P bee nifomiy L onveiveanpo iffive anpo

20 epoiion ae! Ofen e ae ineee in epoiion ae, in hineime, e.g., Å, Åmin, nm, nmmin, µmmin, e. epoiion flx # m film aea m film eniy # m V 3 & m 3 A N f A N f N f N h A f N f ρ M f f N Avogao

Control Volume Derivation

Control Volume Derivation School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass

More information

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence

More information

ESCI 343 Atmospheric Dynamics II Lesson 8 Sound Waves

ESCI 343 Atmospheric Dynamics II Lesson 8 Sound Waves ESCI 343 Amoheri Dynami II Leon 8 Sond Wae Referene: An Inrodion o Dynami Meeorology (3 rd ediion), JR Holon Wae in Flid, J Lighhill SOUND WAVES We will limi or analyi o ond wae raeling only along he -ai,

More information

Deterministic and stochastic internet-style networks with a single link, and one or two user under information delay

Deterministic and stochastic internet-style networks with a single link, and one or two user under information delay Deeminii ohai inene-yle neok ih a inle link one o o e ne infomaion elay Gabiela Miea Mihaela Neam Mailen Piea Dmi Opi ba In hi pape e ineiae he ynami of he Inene-Syle Neok ih elay in a inle link one o

More information

Consider a Binary antipodal system which produces data of δ (t)

Consider a Binary antipodal system which produces data of δ (t) Modulaion Polem PSK: (inay Phae-hi keying) Conide a inay anipodal yem whih podue daa o δ ( o + δ ( o inay and epeively. Thi daa i paed o pule haping ile and he oupu o he pule haping ile i muliplied y o(

More information

Practice Problems: Improper Integrals

Practice Problems: Improper Integrals Pracice Problem: Improper Inegral Wrien by Vicoria Kala vkala@mah.cb.ed December 6, Solion o he pracice problem poed on November 3. For each of he folloing problem: a Eplain hy he inegral are improper.

More information

Integration of the constitutive equation

Integration of the constitutive equation Inegaion of he consiive eqaion REMAINDER ON NUMERICAL INTEGRATION Analyical inegaion f ( x( ), x ( )) x x x f () Exac/close-fom solion (no always possible) Nmeical inegaion. i. N T i N [, T ] [ i, i ]

More information

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes] ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,

More information

Lecture 17: Kinetics of Phase Growth in a Two-component System:

Lecture 17: Kinetics of Phase Growth in a Two-component System: Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien

More information

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN

More information

6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson

6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson 6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen

More information

Support Vector Machines

Support Vector Machines Suppo Veco Machine CSL 3 ARIFICIAL INELLIGENCE SPRING 4 Suppo Veco Machine O, Kenel Machine Diciminan-baed mehod olean cla boundaie Suppo veco coni of eample cloe o bounday Kenel compue imilaiy beeen eample

More information

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion

More information

( )( )( ) ( ) + ( ) ( ) ( )

( )( )( ) ( ) + ( ) ( ) ( ) 3.7. Moel: The magnetic fiel is that of a moving chage paticle. Please efe to Figue Ex3.7. Solve: Using the iot-savat law, 7 19 7 ( ) + ( ) qvsinθ 1 T m/a 1.6 1 C. 1 m/s sin135 1. 1 m 1. 1 m 15 = = = 1.13

More information

Computer Aided Geometric Design

Computer Aided Geometric Design Copue Aided Geoei Design Geshon Ele, Tehnion sed on ook Cohen, Riesenfeld, & Ele Geshon Ele, Tehnion Definiion 3. The Cile Given poin C in plne nd nue R 0, he ile ih ene C nd dius R is defined s he se

More information

Global Solutions of the SKT Model in Population Dynamics

Global Solutions of the SKT Model in Population Dynamics Volm 7 No 7 499-5 ISSN: 3-88 rin rion; ISSN: 34-3395 on-lin rion rl: h://ijm ijm Glol Solion of h SK Mol in Polion Dnmi Rizg Hor n Mo Soilh USH El li Ezzor lgir lgri rizg@gmilom USH El li Ezzor lgir lgri

More information

8.5 Circles and Lengths of Segments

8.5 Circles and Lengths of Segments LenghofSegmen20052006.nb 1 8.5 Cicle and Lengh of Segmen In hi ecion we will how (and in ome cae pove) ha lengh of chod, ecan, and angen ae elaed in ome nal way. We will look a hee heoem ha ae hee elaionhip

More information

CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE

CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE Fameal Joal of Mahemaic a Mahemaical Sciece Vol. 7 Ie 07 Page 5- Thi pape i aailable olie a hp://.fi.com/ Pblihe olie Jaa 0 07 CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE Caolo

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual

More information

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission

More information

Computer Propagation Analysis Tools

Computer Propagation Analysis Tools Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion

More information

Molecular Evolution and Phylogeny. Based on: Durbin et al Chapter 8

Molecular Evolution and Phylogeny. Based on: Durbin et al Chapter 8 Molecula Evoluion and hylogeny Baed on: Dubin e al Chape 8. hylogeneic Tee umpion banch inenal node leaf Topology T : bifucaing Leave - N Inenal node N+ N- Lengh { i } fo each banch hylogeneic ee Topology

More information

C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f

C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f C H A P T E R I G E N E S I S A N D GROWTH OF G U IL D S C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f i n a v a r i e t y o f f o r m s - s o c i a l, r e l i g i

More information

Fall 2004/05 Solutions to Assignment 5: The Stationary Phase Method Provided by Mustafa Sabri Kilic. I(x) = e ixt e it5 /5 dt (1) Z J(λ) =

Fall 2004/05 Solutions to Assignment 5: The Stationary Phase Method Provided by Mustafa Sabri Kilic. I(x) = e ixt e it5 /5 dt (1) Z J(λ) = 8.35 Fall 24/5 Solution to Aignment 5: The Stationay Phae Method Povided by Mutafa Sabi Kilic. Find the leading tem fo each of the integal below fo λ >>. (a) R eiλt3 dt (b) R e iλt2 dt (c) R eiλ co t dt

More information

International Journal of Mathematical Archive-5(6), 2014, Available online through ISSN

International Journal of Mathematical Archive-5(6), 2014, Available online through   ISSN Inenaional Jonal o Mahemaical Achive-6, 0, 09-8 Availale online hogh www.ijma.ino ISSN 9 06 EXISENCE OF NONOSCILLAORY SOLUIONS OF A CLASS OF NONLINEAR NEURAL DELAY DIFFERENIAL EQUAIONS OF HIRD ORDER K

More information

3D Coordinate Systems. 3D Geometric Transformation Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker. Right-handed coordinate system:

3D Coordinate Systems. 3D Geometric Transformation Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker. Right-handed coordinate system: 3D Geomeric ransformaion Chap. 5 in FVD, Chap. in Hearn & Baker 3D Coordinae Ssems Righ-handed coordinae ssem: Lef-handed coordinae ssem: 2 Reminder: Vecor rodc U V UV VU sin ˆ V nu V U V U ˆ ˆ ˆ 3D oin

More information

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow. CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

PREDICTION OF THE TRANSIENT FORCE SUBSEQUENT TO A LIQUID MASS IMPACT ON AN ELBOW OF AN INITIALLY VOIDED LINE

PREDICTION OF THE TRANSIENT FORCE SUBSEQUENT TO A LIQUID MASS IMPACT ON AN ELBOW OF AN INITIALLY VOIDED LINE PEDICTION OF THE TNSIENT FOCE SUBSEQUENT TO IQUID MSS IMPCT ON N EBOW OF N INITIY OIDED INE THESIS SUBMITTED TO THE GDUTE SCHOO OF NTU ND PPIED SCIENCES OF MIDDE EST TECHNIC UNIESITY BY BÜENT BBS KYHN

More information

Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Sharif University of Technology - CEDRA By: Professor Ali Meghdari Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion

More information

2. Fluid-Flow Equations

2. Fluid-Flow Equations . Fli-Flo Eqation Governing Eqation Conervation eqation for: ma momentm energy (other contitent) Alternative form: integral (control-volme) eqation ifferential eqation Integral (Control-olme) Approach

More information

Equilibria of a cylindrical plasma

Equilibria of a cylindrical plasma // Miscellaneous Execises Cylinical equilibia Equilibia of a cylinical plasma Consie a infinitely long cyline of plasma with a stong axial magnetic fiel (a geat fusion evice) Plasma pessue will cause the

More information

AQA Maths M2. Topic Questions from Papers. Differential Equations. Answers

AQA Maths M2. Topic Questions from Papers. Differential Equations. Answers AQA Mahs M Topic Quesions from Papers Differenial Equaions Answers PhysicsAndMahsTuor.com Q Soluion Marks Toal Commens M 600 0 = A Applying Newonís second law wih 0 and. Correc equaion = 0 dm Separaing

More information

6.8 Laplace Transform: General Formulas

6.8 Laplace Transform: General Formulas 48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy

More information

Reinforcement learning

Reinforcement learning Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback

More information

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay) Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens

More information

Chapter 5. Long Waves

Chapter 5. Long Waves ape 5. Lo Waes Wae e s o compaed ae dep: < < L π Fom ea ae eo o s s ; amos ozoa moo z p s ; dosac pesse Dep-aeaed coseao o mass

More information

The Ind ian Mynah b ird is no t fro m Vanuat u. It w as b ro ug ht here fro m overseas and is now causing lo t s o f p ro b lem s.

The Ind ian Mynah b ird is no t fro m Vanuat u. It w as b ro ug ht here fro m overseas and is now causing lo t s o f p ro b lem s. The Ind ian Mynah b ird is no t fro m Vanuat u. It w as b ro ug ht here fro m overseas and is now causing lo t s o f p ro b lem s. Mynah b ird s p ush out nat ive b ird s, com p et ing for food and p laces

More information

Algorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov)

Algorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov) Algorihm and Daa Srucure 2011/ Week Soluion (Tue 15h - Fri 18h No) 1. Queion: e are gien 11/16 / 15/20 8/13 0/ 1/ / 11/1 / / To queion: (a) Find a pair of ube X, Y V uch ha f(x, Y) = f(v X, Y). (b) Find

More information

Classification of Equations Characteristics

Classification of Equations Characteristics Clssiiion o Eqions Cheisis Consie n elemen o li moing in wo imensionl spe enoe s poin P elow. The ph o P is inie he line. The posiion ile is s so h n inemenl isne long is s. Le he goening eqions e epesene

More information

Single Phase Line Frequency Uncontrolled Rectifiers

Single Phase Line Frequency Uncontrolled Rectifiers Single Phae Line Frequency Unconrolle Recifier Kevin Gaughan 24-Nov-03 Single Phae Unconrolle Recifier 1 Topic Baic operaion an Waveform (nucive Loa) Power Facor Calculaion Supply curren Harmonic an Th

More information

Time-Space Model of Business Fluctuations

Time-Space Model of Business Fluctuations Time-Sace Moel of Business Flucuaions Aleei Kouglov*, Mahemaical Cene 9 Cown Hill Place, Suie 3, Eobicoke, Onaio M8Y 4C5, Canaa Email: Aleei.Kouglov@SiconVieo.com * This aicle eesens he esonal view of

More information

TRAVELING WAVES. Chapter Simple Wave Motion. Waves in which the disturbance is parallel to the direction of propagation are called the

TRAVELING WAVES. Chapter Simple Wave Motion. Waves in which the disturbance is parallel to the direction of propagation are called the Chapte 15 RAVELING WAVES 15.1 Simple Wave Motion Wave in which the ditubance i pependicula to the diection of popagation ae called the tanvee wave. Wave in which the ditubance i paallel to the diection

More information

Chapter 9 - The Laplace Transform

Chapter 9 - The Laplace Transform Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC

More information

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion

More information

Exponential Sawtooth

Exponential Sawtooth ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able

More information

ME 141. Engineering Mechanics

ME 141. Engineering Mechanics ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics

More information

Actuarial Models 1: solutions example sheet 4

Actuarial Models 1: solutions example sheet 4 Actuarial Moel 1: olution example heet 4 (a) Anwer to Exercie 4.1 Q ( e u ) e σ σ u η η. (b) The forwar equation correponing to the backwar tate e are t p ee(t) σp ee (t) + ηp eu (t) t p eu(t) σp ee (t)

More information

A L A BA M A L A W R E V IE W

A L A BA M A L A W R E V IE W A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N

More information

More on ODEs by Laplace Transforms October 30, 2017

More on ODEs by Laplace Transforms October 30, 2017 More on OE b Laplace Tranfor Ocober, 7 More on Ordinar ifferenial Equaion wih Laplace Tranfor Larr areo Mechanical Engineering 5 Seinar in Engineering nali Ocober, 7 Ouline Review la cla efiniion of Laplace

More information

CSE 5365 Computer Graphics. Take Home Test #1

CSE 5365 Computer Graphics. Take Home Test #1 CSE 5365 Comper Graphics Take Home Tes #1 Fall/1996 Tae-Hoon Kim roblem #1) A bi-cbic parameric srface is defined by Hermie geomery in he direcion of parameer. In he direcion, he geomery ecor is defined

More information

Scalar Conservation Laws

Scalar Conservation Laws MATH-459 Nmerical Mehods for Conservaion Laws by Prof. Jan S. Heshaven Solion se : Scalar Conservaion Laws Eercise. The inegral form of he scalar conservaion law + f ) = is given in Eq. below. ˆ 2, 2 )

More information

Graphs III - Network Flow

Graphs III - Network Flow Graph III - Nework Flow Flow nework eup graph G=(V,E) edge capaciy w(u,v) 0 - if edge doe no exi, hen w(u,v)=0 pecial verice: ource verex ; ink verex - no edge ino and no edge ou of Aume every verex v

More information

Maximum Flow 5/6/17 21:08. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015

Maximum Flow 5/6/17 21:08. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Maximm Flo 5/6/17 21:08 Preenaion for e ih he exbook, Algorihm Deign and Applicaion, by M. T. Goodrich and R. Tamaia, Wiley, 2015 Maximm Flo χ 4/6 4/7 1/9 2015 Goodrich and Tamaia Maximm Flo 1 Flo Neork

More information

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as

More information

(V 1. (T i. )- FrC p. ))= 0 = FrC p (T 1. (T 1s. )+ UA(T os. (T is

(V 1. (T i. )- FrC p. ))= 0 = FrC p (T 1. (T 1s. )+ UA(T os. (T is . Yu are repnible fr a reacr in which an exhermic liqui-phae reacin ccur. The fee mu be preheae he hrehl acivain emperaure f he caaly, bu he pruc ream mu be cle. T reuce uiliy c, yu are cniering inalling

More information

Design of Controller for Robot Position Control

Design of Controller for Robot Position Control eign of Conroller for Robo oiion Conrol Two imporan goal of conrol: 1. Reference inpu racking: The oupu mu follow he reference inpu rajecory a quickly a poible. Se-poin racking: Tracking when he reference

More information

T h e C S E T I P r o j e c t

T h e C S E T I P r o j e c t T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T

More information

Electric Potential and Gauss s Law, Configuration Energy Challenge Problem Solutions

Electric Potential and Gauss s Law, Configuration Energy Challenge Problem Solutions Poblem 1: Electic Potential an Gauss s Law, Configuation Enegy Challenge Poblem Solutions Consie a vey long o, aius an chage to a unifom linea chage ensity λ a) Calculate the electic fiel eveywhee outsie

More information

Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example

Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional

More information

Chapter 8 Objectives

Chapter 8 Objectives haper 8 Engr8 ircui Analyi Dr uri Nelon haper 8 Objecive Be able o eermine he naural an he ep repone of parallel circui; Be able o eermine he naural an he ep repone of erie circui. Engr8 haper 8, Nilon

More information

A 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions

A 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions EN10: Continuum Mechanics Homewok 7: Fluid Mechanics Solutions School of Engineeing Bown Univesity 1. An ideal fluid with mass density ρ flows with velocity v 0 though a cylindical tube with cosssectional

More information

c- : r - C ' ',. A a \ V

c- : r - C ' ',. A a \ V HS PAGE DECLASSFED AW EO 2958 c C \ V A A a HS PAGE DECLASSFED AW EO 2958 HS PAGE DECLASSFED AW EO 2958 = N! [! D!! * J!! [ c 9 c 6 j C v C! ( «! Y y Y ^ L! J ( ) J! J ~ n + ~ L a Y C + J " J 7 = [ " S!

More information

Example

Example Chapte.4 iffusion with Chemical eaction Example.4- ------------------------------------------------------------------------------ fluiize coal eacto opeates at 45 K an atm. The pocess will be limite by

More information

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,

More information

P a g e 3 6 of R e p o r t P B 4 / 0 9

P a g e 3 6 of R e p o r t P B 4 / 0 9 P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J

More information

Derivatives of Inverse Trig Functions

Derivatives of Inverse Trig Functions Derivaives of Inverse Trig Fncions Ne we will look a he erivaives of he inverse rig fncions. The formlas may look complicae, b I hink yo will fin ha hey are no oo har o se. Yo will js have o be carefl

More information

8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1

8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1 8. a For ep repone, inpu i u, U Y a U α α Y a α α Taking invere Laplae ranform a α e e / α / α A α 0 a δ 0 e / α a δ deal repone, α d Y i Gi U i δ Hene a α 0 a i For ramp repone, inpu i u, U Soluion anual

More information

Vorticity equation 2. Why did Charney call it PV?

Vorticity equation 2. Why did Charney call it PV? Vorici eqaion Wh i Charne call i PV? The Vorici Eqaion Wan o nersan he rocesses ha roce changes in orici. So erie an eression ha incles he ime eriaie o orici: Sm o orces in irecion Recall ha he momenm

More information

THE SINE INTEGRAL. x dt t

THE SINE INTEGRAL. x dt t THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his-

More information

P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

More information

UT Austin, ECE Department VLSI Design 5. CMOS Gate Characteristics

UT Austin, ECE Department VLSI Design 5. CMOS Gate Characteristics La moule: CMOS Tranior heory Thi moule: DC epone Logic Level an Noie Margin Tranien epone Delay Eimaion Tranior ehavior 1) If he wih of a ranior increae, he curren will ) If he lengh of a ranior increae,

More information

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9 OH BOY! O h Boy!, was or igin a lly cr eat ed in F r en ch an d was a m a jor s u cc ess on t h e Fr en ch st a ge f or young au di enc es. It h a s b een s een by ap pr ox i ma t ely 175,000 sp ect at

More information

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security 1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,

More information

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen

More information

Buckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or

Buckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or Buckling Buckling of a rucure mean failure due o exceive diplacemen (lo of rucural iffne), and/or lo of abiliy of an equilibrium configuraion of he rucure The rule of humb i ha buckling i conidered a mode

More information

MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING

MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens

More information

Lecture 4 Notes (Little s Theorem)

Lecture 4 Notes (Little s Theorem) Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,

More information

EGN 3353C Fluid Mechanics

EGN 3353C Fluid Mechanics eture 5 Bukingham PI Theorem Reall dynami imilarity beteen a model and a rototye require that all dimenionle variable mut math. Ho do e determine the '? Ue the method of reeating variable 6 te Ste : Parameter

More information

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship

To become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,

More information

EN221 - Fall HW # 7 Solutions

EN221 - Fall HW # 7 Solutions EN221 - Fall2008 - HW # 7 Soluions Pof. Vivek Shenoy 1.) Show ha he fomulae φ v ( φ + φ L)v (1) u v ( u + u L)v (2) can be pu ino he alenaive foms φ φ v v + φv na (3) u u v v + u(v n)a (4) (a) Using v

More information

EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?

EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal? EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of

More information

Algorithmic Discrete Mathematics 6. Exercise Sheet

Algorithmic Discrete Mathematics 6. Exercise Sheet Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap

More information

Lecture Notes 2. The Hilbert Space Approach to Time Series

Lecture Notes 2. The Hilbert Space Approach to Time Series Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship

More information

Application of Combined Fourier Series Transform (Sampling Theorem)

Application of Combined Fourier Series Transform (Sampling Theorem) Applicaion o Combined Fourier Serie ranorm Sampling heorem x X[] m m Sampling Frequency Deparmen o Elecrical and Compuer Engineering Deparmen o Elecrical and Compuer Engineering X[] x We need Fourier Serie

More information

IrrItrol Products 2016 catalog

IrrItrol Products 2016 catalog l Ps Valves 205, 200 an 2500 eies Valves M Pa Nmbe -205F 1" n-line E Valve w/ FC - se 2500 eies 3* -200 1" E n-line Valve w/ FC F x F 3-200F 1" n-line Valve w/ FC F x F -2500 1" E Valve w/ FC F x F -2500F

More information

Power of Random Processes 1/40

Power of Random Processes 1/40 Power of Random Processes 40 Power of a Random Process Recall : For deerminisic signals insananeous power is For a random signal, is a random variable for each ime. hus here is no single # o associae wih

More information

A note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics

A note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics PobSa Foum, Volume 6, July 213, Pages 35 41 ISSN 974-3235 PobSa Foum is an e-jounal. Fo eails please visi www.pobsa.og.in A noe on chaaceizaion elae o isibuional popeies of anom anslaion, conacion an ilaion

More information

u(t) Figure 1. Open loop control system

u(t) Figure 1. Open loop control system Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference

More information

June Further Pure Mathematics FP2 Mark Scheme

June Further Pure Mathematics FP2 Mark Scheme Jne 75 Frher Pre Mheis FP Mrk Shee. e e e e 5 e e 7 M: Siplify o for qri in e ( e )(e 7) e, e 7 M: Solve er qri. ln or ln ln 7 B M A M A A () Mrks. () Using ( e ) or eqiv. o fin e or e: ( = n = ) M A e

More information

The Production of Polarization

The Production of Polarization Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview

More information

EE202 Circuit Theory II

EE202 Circuit Theory II EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C

More information

2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V

2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V ME 352 VETS 2. VETS Vecor algebra form he mahemaical foundaion for kinemaic and dnamic. Geomer of moion i a he hear of boh he kinemaic and dnamic of mechanical em. Vecor anali i he imehonored ool for decribing

More information

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts.

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts. Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to

More information

Suppose we have observed values t 1, t 2, t n of a random variable T.

Suppose we have observed values t 1, t 2, t n of a random variable T. Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).

More information

MIXED BOUNDARY VALUE PROBLEM FOR A QUARTER-PLANE WITH A ROBIN CONDITION

MIXED BOUNDARY VALUE PROBLEM FOR A QUARTER-PLANE WITH A ROBIN CONDITION Jornal o Science, Ilamic Repblic o Iran 3: 65-69 Naional Cener For Scieniic Reearch, ISSN 6-4 MIXED OUNDRY VLUE PROLEM FOR QURTER-PLNE WITH ROIN CONDITION ghili * Deparmen o Mahemaic, Facl o Science, Gilan

More information

Micro-bunching: Longitudinal Bunch Profile Measurements at TTF

Micro-bunching: Longitudinal Bunch Profile Measurements at TTF Shot Pulses in Rings Mico-bunching: Longitudinal Bunch Pofile Measuements at TTF ) The time vaying fields in a tansvese mode cavity kick the font of a bunch up, and the back of the bunch don. ) A betaton

More information

Particle dynamics class, SMS 618, (Emmanuel Boss 11/19/2003) Van Rijn s TRANSPOR lab: computation of sediment transport in current and wave direction.

Particle dynamics class, SMS 618, (Emmanuel Boss 11/19/2003) Van Rijn s TRANSPOR lab: computation of sediment transport in current and wave direction. Patile dynami la SMS 618 (Emmanuel Bo 11/19/003 Van Rijn TRANSPOR lab: omputation of ediment tanpot in uent and ave dietion Handout: Appendix A in van Rijn 1993 Piniple of ediment tanpot in ive etuaie

More information

Millennium Theory Equations Original Copyright 2002 Joseph A. Rybczyk Updated Copyright 2003 Joseph A. Rybczyk Updated March 16, 2006

Millennium Theory Equations Original Copyright 2002 Joseph A. Rybczyk Updated Copyright 2003 Joseph A. Rybczyk Updated March 16, 2006 Millennim heoy Eqaions Oiginal Copyigh 00 Joseph A. Rybzyk Updaed Copyigh 003 Joseph A. Rybzyk Updaed Mah 6, 006 Following is a omplee lis o he Millennim heoy o Relaiviy eqaions: Fo easy eeene, all eqaions

More information

Sample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems

Sample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI

More information