Some Approximate Analytical Steady-State Solutions for Cylindrical Fin
|
|
- Georgia Chase
- 5 years ago
- Views:
Transcription
1 Some Appoxmate Analytcal Steady-State Solutons fo Cylndcal Fn ANITA BRUVERE ANDRIS BUIIS Insttute of Mathematcs and Compute Scence Unvesty of Latva Rana ulv 9 Rga LV459 LATVIA Astact: - In ths pape we constuct some appoxmate analytcal thee-dmensonal solutons fo one element of cylndcal wall and fn We assume that the heat tansfe pocess n the wall and the fn s statonay These solutons ae otaned y the ognal method of consevatve aveagng and they ae compaed to some one-dmensonal solutons whch ae well known n lteatue We gve some cteons when t s possle to eplace thee-dmensonal fomulaton of polem wth two- o one-dmensonal statement ey-wods: - steady-state thee-dmensonal heat exchange cylndcal fn analytcal solutons consevatve aveagng Intoducton Otanng effcent coolng fo the components of devces s a dffcult challenge n moden ndusty It s elated to efgeatos adatos engnes and moden electoncs etc Usually ts mathematcal modelng s ealed y one dmensonal steady-state assumptons [][5] In ou pevous papes we have constucted two dmensonal analytcal appoxmate []-[4] and exact [3] solutons In ths pape we otan few new appoxmate analytcal thee dmensonal solutons y the ognal method of consevatve aveagng and some ts smplfcatons (specal cases In [] the so-called Muay Gadne assumptons ae fomulated They ae: The heat flow n the fn and the tempeatue at any pont on the fn eman constant wth tme; The fn mateal s homogeneous; ts themal conductvty s the same n all dectons and emans constant; 3 The heat tansfe coeffcent etween the fn and the suoundng medum s unfom and constant ove the ente suface of the fn; 4 The tempeatue of the medum suoundng the fn s unfom; 5 The fn wdth s so small compaed wth ts heght that tempeatue gadents acoss the fn wdth may e neglected; 6 The tempeatue at the ase of the fn s unfom; 7 Thee ae no heat souces wthn the fn tself; 8 Heat tansfe to o fom the fn s popotonal to the tempeatue excess etween the fn and the suoundng medum; 9 Thee s no contact esstance etween fns n the confguaton o etween the fn at the ase of the confguaton and the pme suface; The heat tansfeed though the outmost edge of the fn (the fn tp s neglgle compaed to that though the lateal sufaces (faces of the fn Mathematcal Fomulaton of 3D Polem and educton to D We wll stat wth accuate thee-dmensonal fomulaton of steady-state polem fo one element of peodcal system fo cylndcal wall and fn The one element of the wall (ase s placed n the doman { [ R ] R [ Z] ϕ [ φ ]} and we desce tempeatue feld V ( ϕ n the wall wth the equaton: V ( The cylndcal fn of length L occupes the doman { % [ R R L] % [ Z] ϕ [ φ ]} and the tempeatue feld V ( ϕ fulflls the equaton:
2 V ( And we have followng ounday condtons n accodance wth M-G pont 5 n ϕ decton % % ; (3 ϕ ϕ φ ϕ ϕ φ We can educe polem ( and ( fom 3D to D usng followng aveage ntegal fo agument ϕ φ U ( V ( ϕ dϕ φ and U% ( % % φ V% ( ϕ dϕ φ % % (4 Descpton of Tempeatue Feld n the Wall We wll use followng dmensonless aguments paametes to tansfom ou polem to dmensonless polem: R R L R l Z Z Z Z Z Z hz hz Z h Z k k k And tempeatues: U% ( Ta U% ( Ta U ( U( T Ta T Ta Hee k( k - heat conductvty coeffcent fo the fn (wall h( h - heat exchange coeffcent fo the fn (wall Z - wdth (thckness of the fn L - length of the fn Z - thckness of the wall T - the suoundng tempeatue on the left (hot sde (the heat souce sde of the wall T a - the suoundng tempeatue on the ght (cold - the heat snk sde sde of the wall and the fn and U % ( ( U % ( One element of the wall (ase placed n the doman now s { [ ] [ ] } and we can desce the dmensonless tempeatue feld U ( n the wall wth the equaton: U U (5 We add needed ounday condtons as follow: ( U [] (6 U [] (7 And homogeneous ounday condtons [ ] (8 We assume the conjugatons condtons on the suface etween the wall and the fn as deal themal contact - thee s no contact esstance: U U (9 Descpton of Tempeatue Feld n the Fn The cylndcal fn of length l occupes the doman { [ l] [ ] } and the tempeatue feld U ( fulflls the equaton U U ( We have followng ounday condtons fo the fn: U [ l ] ( U l [ ] ( And homogeneous ounday condtons [ l] (3 3 Appoxmate Soluton of D Polem fo Peodcal System We wll use the ognal method of consevatve aveagng We wll stat wth the case of peodcal system wth cylndcal fns 3 Reducton of the D Polem fo the Fn Smlaly as n papes [][4] we wll use ognal method of consevatve aveagng and appoxmate the D tempeatue feld U ( fo the fn n followng fom: ρ U( f( ( e f( (4 ρ - ( e f ( ρ wth thee unknown functons f ( Fo ths pupose we ntoduce the ntegal aveage value of functon U ( n the - decton: u( ρ U( d (5
3 Ths equalty togethe wth two ounday condtons (at and allow us to exclude all the unknown functons f ( fom the epesentaton (4 The ounday condton (3 fo the functon U ( at gves mmedately the equalty f( f ( The susttuton of epesentaton (4 n (5 gves expesson: u( f( f ( (6 ( snh( and epesentaton (4 takes fom: cosh( ρ U( u( snh( (7 snh( cosh( ρ ( snh f ( Fnally y the use of the ounday condton ( we can exclude f ( fom last expesson and epesent the D soluton U ( fo the fn n followng fom: U ( u ( Φ ( (8 It s easy to check that the functon Φ( looks lke ( ρ snh( cosh( cosh( Φ ( (9 snh( (cosh( snh( The second stage fo the method of consevatve aveagng s to tansfom the patal dffeental equaton (5 fo the functon U ( to the dffeental equaton fo one aguments functon u ( To eale ths goal we ntegate the man dffeental equaton ( n the - decton and usng (5 get: d du ( d d By usng the ounday condton (3 at fo the functon U ( and expessng fom the ounday condton ( at the fst devatve tough the functon U ( we otan followng dffeental equaton: d du U ( d d It emans to expess n dffeental equaton ( the functon U ( though the functon u ( wth the help of the equalty (8 and we eceve the new dffeental equaton whch desces the D dmensonal tempeatue feld u ( n the fn: d du d d μ u( ( Hee μ ( Φ (3 Solvng dffeental equaton ( we gan soluton though Bessel s modfed functons I : u( C( μ I ( μ ( μ (4 whee μ( μ ( μ μ l (5 μi( μ I ( μ l And fom (8 and (4 we get soluton fo fn whch ncludes only one unknown constant C U C μ I ( μ ( μ Φ( (6 ( ( 3 Reducton of the D Polem fo the Wall We wll use same method of consevatve aveagng and appoxmate the D tempeatue feld U ( fo the fn n followng fom: U ( g( g( (7 g( wth thee unknown functons ( g Fo ths pupose we ntoduce the ntegal aveage value of functon U ( n the - decton: u ( U ( d (8 Ths equalty togethe wth equalty (7: u g ( ϕ g ( ϕ g ( (9 ( whee ϕ ϕ ( 3( Fndng devaton of (7 and usng ounday condton (6 we get ( ( g g( (3 ( g( Expess functon g ( fom equaton (9 and put to expesson (3 gves g( A g( Bu ( D (3 whee
4 ( ϕ(( ϕ ( ( A ϕ ( ( B D ϕ ( Expess functon g ( fom equaton (9 and put to expesson (3 gves g ( A g ( B u ( D (3 whee ( ϕ (( ϕ A B ( ϕ ( ϕ D Equatons (3 and (3 gve g( ag ( u ( d (33 g( ag( u ( d A B D whee a d Puttng (33 to (7 we gan a a U( g( ( ( u( (34 d d ( Now we stll have two unknown functons g ( and u ( Theefoe we wll use dffeent ounday and conjugatons condtons on the wall to exclude these functons 33 Soluton fo uppe Wall Puttng (34 to ounday condton (7 we get g ( u ( d (35 B D whee d B a a Puttng (35 n (33 we get U ( Φ ( u ( ψ ( (36 a a whee Φ ( ( ( da d d da and ψ ( ( d Now ntegatng patal dffeental equaton (5 and takng account equaton (8 get d u (37 d Usng ounday condtons ( (3 and (33 d u k u Q (38 d whee ( Φ ( Φ( k Q ( ( ψ( ψ( Soluton of dffeental equaton (38 usng ounday condton ( looks as follows Q u ( C cosh( k( k (39 As we can see we have solved functon u ( and now polem educes to the polem of fndng constant C 33 Soluton fo lowe Wall Usng equaton (34 and (6 and puttng them n to conjugaton condton (9 at value g( C( μ I ( μ ( μ Φ( (4 Equaton (9 we can ewte as Φ ( F F cosh( ρ (4 whee F and snh( cosh( snh( snh( F cosh( F We can contnue wth equaton (4 and (4 and get g( C( F F cosh( ρ whee (4 F ( μ I ( μ ( μ F Usng (6 and (4 we get followng devaton at value C ( ( μ μi μ ( μ Φ( (43
5 But fom (9 (4 and (43 we get C ( μ μi μ ( μ Φ( ( ( F F cosh( ρ C whee (44 F μ μ I ( μ ( μ F ( Usng equaton (4 and devaton of equaton (34 at value we get a a C ( F% F% cosh( ρ d d u( (45 Now sumttng equatons (4 and (4 n equaton (37 we get dffeental equaton du ug FC H ECcosh( ρ (46 d whee G C a a E F % F % a a F F % F% d d H Soluton of dffeental equaton looks followng G G u( C3e C4e EC FC H (47 cosh( ρ ρ G G Fom condton (8 and (47 follows that constants C theefoe u ( can ewte n the 3 C 4 followng way u( C3cosh( G EC FC H (48 cosh( ρ ρ G G puttng togethe (34 (4 and (48 we get a a U( Cu ( Φ( ( ( (49 o EC FC H C G ρ G G 3 cosh( cosh( ρ d d ( Now soluton fo wall contans only two unknown constants C and C 3 34 Soluton We have two addtonal condtons on functon u ( espectvely ( u( u (5 and u u Also n pont ( (5 values of functons U ( and U ( must e equvalent U ( U( (5 To satsfy equaton (5 we need to use (39 and (48 Q H C cosh( k( k G (53 E F C3cosh( G C cosh( ρ G G ut to satsfy equaton (5 - devatve of (39 and (48 Eρ GC3snh( G C snh( ρ G (54 Ck snh( k ( And to satsfy equaton (5 we wll use equatons (6 and (36 Cu ( Φ ( d ECcosh( FC H (55 C3cosh( G ρ G G And now polem educes to soluton of thee lnea equatons (53 (54 and (55 fo thee unknown constants C 3
6 Cu ( Φ ( d ECcosh( FC H C3cosh( G ρ G G Eρ GC3snh( G C snh( ρ G (56 Ck snh( k ( Q H C cosh( k( k G E F C3cosh( G C cosh( ρ G G Afte solvng system (56 we can put constants C 3 to equaton (7 o (49 and calculate value of tempeatue at any pont of ou D doman 4 D Soluton as the Smple Case of the 3D Soluton Usng followng ntegal values fo equatons (5 and (8 v ( U ( d (57 and fo equaton ( v( U ( d (58 we get new D polem d dv ( d d (59 d dv v ( l d d (6 wth followng ounday condtons dv ( v d (6 dv v l d (6 v v (63 dv d dv d (64 dv dv v ( (65 d d The soluton of polem (59-(65 can e wtten n followng fom: v( Cln C (66 v ( CI 3 C 4 whee I s Bessels modfed functons Hee the fou unknown constants can e easy detemned fom the fou ounday and conjugatons condtons (6-(64: ( C C ln ( l ( l ( l C3 C 4 ( l I ( l I ( l C ( ( ln C( C3 I C4 Cln C C3I C4 (67 5 Concluson We have constucted some appoxmate thee dmensonal analytcal solutons fo a peodcal system wth cylndcal fn when the wall and the fn consst of mateals whch have dffeent themal popetes Acknowledgements: Reseach was suppoted y Euopean Socal Fund and Councl of Scences of Latva (gant 555 Refeences: [] aus AD Analyss and evaluaton of extended suface themal systems Hemsphee pulshng Copoaton 98 [] Buks A Buke M Closed two-dmensonal soluton fo heat tansfe n a peodcal system wth a fn Poceedngs of the Latvan Academy of Scences Secton B Vol5 N5 998 pp8- [3] Buks A Buke M Gusenov S Analytcal two-dmensonal solutons fo heat tansfe n a system wth ectangula fn Advanced Computatonal Methods n Heat Tansfe VIII WIT pess 4 pp [4] Buks A Buke M Some analytcal 3-D steady-state solutons fo systems wth ectangula fn IASME Tansactons Issue 7 Vol Septeme 5 pp -9 [5] Wood AS Tupholme GE Bhatt MIH Heggs PJ Pefomance ndcatos fo steady-state heat tansfe though fn assemles Tans ASME Jounal of Heat Tansfe pp 3-36
ϕ direction (others needed boundary conditions And we have following boundary conditions in
Exact Analytical 3-D Steady-State Solution fo Cylindical Fin ANITA PILIKSERE, ANDRIS BUIKIS Institute of Mathematics and Compute Science Univesity of Latvia Raina ulv 9, Riga, LV459 LATVIA saulite@latnetlv,
More informationIntegral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
More informationPHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More informationSet of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
More informationCOMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS
ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant
More informationChapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
More informationTest 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What
More informationChapter 23: Electric Potential
Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done
More informationV. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum
More informationA. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
More informationDistinct 8-QAM+ Perfect Arrays Fanxin Zeng 1, a, Zhenyu Zhang 2,1, b, Linjie Qian 1, c
nd Intenatonal Confeence on Electcal Compute Engneeng and Electoncs (ICECEE 15) Dstnct 8-QAM+ Pefect Aays Fanxn Zeng 1 a Zhenyu Zhang 1 b Lnje Qan 1 c 1 Chongqng Key Laboatoy of Emegency Communcaton Chongqng
More informationUNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
More informationCSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by
More informationContact, information, consultations
ontact, nfomaton, consultatons hemsty A Bldg; oom 07 phone: 058-347-769 cellula: 664 66 97 E-mal: wojtek_c@pg.gda.pl Offce hous: Fday, 9-0 a.m. A quote of the week (o camel of the week): hee s no expedence
More information4.4 Continuum Thermomechanics
4.4 Contnuum Themomechancs The classcal themodynamcs s now extended to the themomechancs of a contnuum. The state aables ae allowed to ay thoughout a mateal and pocesses ae allowed to be eesble and moe
More informationEnergy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
More informationOptimal System for Warm Standby Components in the Presence of Standby Switching Failures, Two Types of Failures and General Repair Time
Intenatonal Jounal of ompute Applcatons (5 ) Volume 44 No, Apl Optmal System fo Wam Standby omponents n the esence of Standby Swtchng Falues, Two Types of Falues and Geneal Repa Tme Mohamed Salah EL-Shebeny
More information19 The Born-Oppenheimer Approximation
9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A
More informationMHD Oscillatory Flow in a Porous Plate
Global Jounal of Mathematcal Scences: Theoy and Pactcal. ISSN 97-3 Volume, Numbe 3 (), pp. 3-39 Intenatonal Reseach Publcaton House http://www.phouse.com MHD Oscllatoy Flow n a Poous Plate Monka Kala and
More informationON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION
IJMMS 3:37, 37 333 PII. S16117131151 http://jmms.hndaw.com Hndaw Publshng Cop. ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION ADEM KILIÇMAN Receved 19 Novembe and n evsed fom 7 Mach 3 The Fesnel sne
More information8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
More informationMultistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 58-64 007 ISSN 549-3644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm
More informationStellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:
Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: 1 1 1 = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y + 4 1+ z A Z We ntoduced the pessue
More informationIf there are k binding constraints at x then re-label these constraints so that they are the first k constraints.
Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
More information3. A Review of Some Existing AW (BT, CT) Algorithms
3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms
More informationCorrespondence Analysis & Related Methods
Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense
More informationOptimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis
Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each
More informationGenerating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences
Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu
More informationThe Greatest Deviation Correlation Coefficient and its Geometrical Interpretation
By Rudy A. Gdeon The Unvesty of Montana The Geatest Devaton Coelaton Coeffcent and ts Geometcal Intepetaton The Geatest Devaton Coelaton Coeffcent (GDCC) was ntoduced by Gdeon and Hollste (987). The GDCC
More informationConservative Averaging Method and its Application for One Heat Conduction Problem
Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) Consevative Aveaging Method and its Application fo One Heat Conduction Poblem
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationPhysics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
More informationAsymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics
Jounal of Appled Mathematcs and Physcs 6 4 687-697 Publshed Onlne August 6 n ScRes http://wwwscpog/jounal/jamp http://dxdoog/436/jamp64877 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent
More informationGroupoid and Topological Quotient Group
lobal Jounal of Pue and Appled Mathematcs SSN 0973-768 Volume 3 Numbe 7 07 pp 373-39 Reseach nda Publcatons http://wwwpublcatoncom oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs
More informationMechanics Physics 151
Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc
More informationChapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23,
hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A n n
More informationMultipole Radiation. March 17, 2014
Multpole Radaton Mach 7, 04 Zones We wll see that the poblem of hamonc adaton dvdes nto thee appoxmate egons, dependng on the elatve magntudes of the dstance of the obsevaton pont,, and the wavelength,
More informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationEE 5337 Computational Electromagnetics (CEM)
7//28 Instucto D. Raymond Rumpf (95) 747 6958 cumpf@utep.edu EE 5337 Computatonal Electomagnetcs (CEM) Lectue #6 TMM Extas Lectue 6These notes may contan copyghted mateal obtaned unde fa use ules. Dstbuton
More informationThe Unique Solution of Stochastic Differential Equations With. Independent Coefficients. Dietrich Ryter.
The Unque Soluton of Stochastc Dffeental Equatons Wth Independent Coeffcents Detch Ryte RyteDM@gawnet.ch Mdatweg 3 CH-4500 Solothun Swtzeland Phone +4132 621 13 07 SDE s must be solved n the ant-itô sense
More informationImplementation in the ANSYS Finite Element Code of the Electric Vector Potential T-Ω,Ω Formulation
Implementaton n the ANSYS Fnte Element Code of the Electc Vecto Potental T-Ω,Ω Fomulaton Peto Teston Dpatmento d Ingegnea Elettca ed Elettonca, Unvestà d Cagla Pazza d Am, 0923 Cagla Pegogo Sonato Dpatmento
More information(8) Gain Stage and Simple Output Stage
EEEB23 Electoncs Analyss & Desgn (8) Gan Stage and Smple Output Stage Leanng Outcome Able to: Analyze an example of a gan stage and output stage of a multstage amplfe. efeence: Neamen, Chapte 11 8.0) ntoducton
More information9/12/2013. Microelectronics Circuit Analysis and Design. Modes of Operation. Cross Section of Integrated Circuit npn Transistor
Mcoelectoncs Ccut Analyss and Desgn Donald A. Neamen Chapte 5 The pola Juncton Tanssto In ths chapte, we wll: Dscuss the physcal stuctue and opeaton of the bpola juncton tanssto. Undestand the dc analyss
More informationANALYSIS OF AXIAL LOADED PILE IN MULTILAYERED SOIL USING NODAL EXACT FINITE ELEMENT MODEL
Intenatonal Jounal of GEOMATE, Apl, 8 Vol. 4, Issue 44, pp. -7 Geotec., Const. Mat. & Env., DOI: https://do.og/.66/8.44.785 ISS: 86-98 (Pnt), 86-99 (Onlne), Japan AAYSIS OF AXIA OADED PIE I MUTIAYERED
More informationAn Approach to Inverse Fuzzy Arithmetic
An Appoach to Invese Fuzzy Athmetc Mchael Hanss Insttute A of Mechancs, Unvesty of Stuttgat Stuttgat, Gemany mhanss@mechaun-stuttgatde Abstact A novel appoach of nvese fuzzy athmetc s ntoduced to successfully
More informationCOMPUTATIONAL METHODS AND ALGORITHMS Vol. I - Methods of Potential Theory - V.I. Agoshkov, P.B. Dubovski
METHODS OF POTENTIAL THEORY.I. Agoshkov and P.B. Dubovsk Insttute of Numecal Mathematcs, Russan Academy of Scences, Moscow, Russa Keywods: Potental, volume potental, Newton s potental, smple laye potental,
More informationGENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS
#A39 INTEGERS 9 (009), 497-513 GENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS Mohaad Faokh D. G. Depatent of Matheatcs, Fedows Unvesty of Mashhad, Mashhad,
More informationThermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
More informationA Study about One-Dimensional Steady State. Heat Transfer in Cylindrical and. Spherical Coordinates
Appled Mathematcal Scences, Vol. 7, 03, no. 5, 67-633 HIKARI Ltd, www.m-hka.com http://dx.do.og/0.988/ams.03.38448 A Study about One-Dmensonal Steady State Heat ansfe n ylndcal and Sphecal oodnates Lesson
More informationChapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29,
hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A
More informationPhysics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
More informationThermoelastic Problem of a Long Annular Multilayered Cylinder
Wold Jounal of Mechancs, 3, 3, 6- http://dx.do.og/.436/w.3.35a Publshed Onlne August 3 (http://www.scp.og/ounal/w) Theoelastc Poble of a Long Annula Multlayeed Cylnde Y Hsen Wu *, Kuo-Chang Jane Depatent
More information24-2: Electric Potential Energy. 24-1: What is physics
D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a
More informationTransport Coefficients For A GaAs Hydro dynamic Model Extracted From Inhomogeneous Monte Carlo Calculations
Tanspot Coeffcents Fo A GaAs Hydo dynamc Model Extacted Fom Inhomogeneous Monte Calo Calculatons MeKe Ieong and Tngwe Tang Depatment of Electcal and Compute Engneeng Unvesty of Massachusetts, Amhest MA
More informationAsymptotic Waves for a Non Linear System
Int Jounal of Math Analyss, Vol 3, 9, no 8, 359-367 Asymptotc Waves fo a Non Lnea System Hamlaou Abdelhamd Dépatement de Mathématques, Faculté des Scences Unvesté Bad Mokhta BP,Annaba, Algea hamdhamlaou@yahoof
More informationPO with Modified Surface-normal Vectors for RCS calculation of Scatterers with Edges and Wedges
wth Modfed Suface-nomal Vectos fo RCS calculaton of Scattees wth Edges and Wedges N. Omak N. Omak, T.Shjo, and M. Ando Dep. of Electcal and Electonc Engneeng, Tokyo Insttute of Technology, Japan 1 Outlne.
More informationLie Subalgebras and Invariant Solutions to the Equation of Fluid Flows in Toroidal Field. Lang Xia
Le Subalgebas and Invaant Solutons to the Equaton of Flud Flows n Toodal Feld Lang a Emal: langxaog@gmalcom Abstact: Patal dffeental equatons (PDEs), patculaly coupled PDE systems, ae dffcult to solve
More informationChapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.
Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,
More informationRotating Disk Electrode -a hydrodynamic method
Rotatng Dsk Electode -a hdodnamc method Fe Lu Ma 3, 0 ente fo Electochemcal Engneeng Reseach Depatment of hemcal and Bomolecula Engneeng Rotatng Dsk Electode A otatng dsk electode RDE s a hdodnamc wokng
More informationKhintchine-Type Inequalities and Their Applications in Optimization
Khntchne-Type Inequaltes and The Applcatons n Optmzaton Anthony Man-Cho So Depatment of Systems Engneeng & Engneeng Management The Chnese Unvesty of Hong Kong ISDS-Kolloquum Unvestaet Wen 29 June 2009
More informationLarge scale magnetic field generation by accelerated particles in galactic medium
Lage scale magnetc feld geneaton by acceleated patcles n galactc medum I.N.Toptygn Sant Petesbug State Polytechncal Unvesty, depatment of Theoetcal Physcs, Sant Petesbug, Russa 2.Reason explonatons The
More informationReview of Vector Algebra and Vector Calculus Operations
Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost
More informationHamiltonian multivector fields and Poisson forms in multisymplectic field theory
JOURNAL OF MATHEMATICAL PHYSICS 46, 12005 Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles
/4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla
More informationA New Approach for Deriving the Instability Potential for Plates Based on Rigid Body and Force Equilibrium Considerations
Avalable onlne at www.scencedect.com Poceda Engneeng 4 (20) 4 22 The Twelfth East Asa-Pacfc Confeence on Stuctual Engneeng and Constucton A New Appoach fo Devng the Instablty Potental fo Plates Based on
More informationThe Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution
oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,
More informationApplied Statistical Mechanics Lecture Note - 13 Molecular Dynamics Simulation
Appled Statstcal Mechancs Lectue Note - 3 Molecula Dynamcs Smulaton 고려대학교화공생명공학과강정원 Contents I. Basc Molecula Dynamcs Smulaton Method II. Popetes Calculatons n MD III. MD n Othe Ensembles I. Basc MD Smulaton
More informationExact Simplification of Support Vector Solutions
Jounal of Machne Leanng Reseach 2 (200) 293-297 Submtted 3/0; Publshed 2/0 Exact Smplfcaton of Suppot Vecto Solutons Tom Downs TD@ITEE.UQ.EDU.AU School of Infomaton Technology and Electcal Engneeng Unvesty
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationCOMPARISON OF METHODS FOR SOLVING THE HEAT TRANSFER IN ELECTRICAL MACHINES
POZNAN UNIVE RSITY OF TE CHNOLOGY ACADE MIC JOURNALS No 75 Electical Engineeing 2013 Zbynek MAKKI* Macel JANDA* Ramia DEEB* COMPARISON OF METHODS FOR SOLVING THE HEAT TRANSFER IN ELECTRICAL MACHINES This
More informationEngineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems
Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationAnalytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness
Appled Mathematcs 00 43-438 do:0.436/am.00.5057 Publshed Onlne Novembe 00 (http://www.scrp.og/jounal/am) Analytcal and Numecal Solutons fo a Rotatng Annula Ds of Vaable Thcness Abstact Ashaf M. Zenou Daoud
More informationONE DIMENSIONAL TRIANGULAR FIN EXPERIMENT. Technical Advisor: Dr. D.C. Look, Jr. Version: 11/03/00
ONE IMENSIONAL TRIANGULAR FIN EXPERIMENT Techncal Advsor: r..c. Look, Jr. Verson: /3/ 7. GENERAL OJECTIVES a) To understand a one-dmensonal epermental appromaton. b) To understand the art of epermental
More informationApplication of Complex Vectors and Complex Transformations in Solving Maxwell s Equations
Applcaton of Complex Vectos and Complex Tansfomatons n Solvng Maxwell s Equatons by Payam Saleh-Anaa A thess pesented to the Unvesty of Wateloo n fulfllment of the thess equement fo the degee of Maste
More informationTHE TIME-DEPENDENT CLOSE-COUPLING METHOD FOR ELECTRON-IMPACT DIFFERENTIAL IONIZATION CROSS SECTIONS FOR ATOMS AND MOLECULES
Intenatonal The Tme-Dependent cence Pess Close-Couplng IN: 9-59 Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... REVIEW ARTICE THE TIME-DEPENDENT COE-COUPING METHOD FOR EECTRON-IMPACT
More informationPhysics Exam II Chapters 25-29
Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do
More informationRotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1
Rotatonal Knematcs Rgd Object about a Fxed Axs Westen HS AP Physcs 1 Leanng Objectes What we know Unfom Ccula Moton q s Centpetal Acceleaton : Centpetal Foce: Non-unfom a F c c m F F F t m ma t What we
More informationScaling Growth in Heat Transfer Surfaces and Its Thermohydraulic Effect Upon the Performance of Cooling Systems
799 A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 61, 017 Guest Edtos: Peta S Vabanov, Rongxn Su, Hon Loong Lam, Xa Lu, Jří J Klemeš Copyght 017, AIDIC Sevz S..l. ISBN 978-88-95608-51-8; ISSN 83-916
More information1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume
EN10: Contnuum Mechancs Homewok 5: Alcaton of contnuum mechancs to fluds Due 1:00 noon Fda Febua 4th chool of Engneeng Bown Unvest 1. tatng wth the local veson of the fst law of themodnamcs q jdj q t and
More informationDynamics of Rigid Bodies
Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea
More informationA NOTE ON ELASTICITY ESTIMATION OF CENSORED DEMAND
Octobe 003 B 003-09 A NOT ON ASTICITY STIATION OF CNSOD DAND Dansheng Dong an Hay. Kase Conell nvesty Depatment of Apple conomcs an anagement College of Agcultue an fe Scences Conell nvesty Ithaca New
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 informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationOn Accurate Stress Determination in Laminated Finite Length Cylinders Subjected to Thermo Elastic Load
Intenatonal Jounal of Mechancs and Solds ISSN 0973-1881 Volume 6, Numbe 1 (2011), pp. 7-26 Reseach Inda Publcatons http://www.publcaton.com/jms.htm On Accuate Stess Detemnaton n Lamnated Fnte Length Cylndes
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationVEKTORANALYS FLUX INTEGRAL LINE INTEGRAL. and. Kursvecka 2. Kapitel 4 5. Sidor 29 50
VEKTORANAYS Ksecka INE INTEGRA and UX INTEGRA Kaptel 4 5 Sdo 9 5 A wnd TARGET PROBEM We want to psh a mne cat along a path fom A to B. Bt the wnd s blowng. How mch enegy s needed? (.e. how mch s the wok?
More informationConsequences of Long Term Transients in Large Area High Density Plasma Processing: A 3-Dimensional Computational Investigation*
ISPC 2003 June 22-27, 2003 Consequences of Long Tem Tansents n Lage Aea Hgh Densty Plasma Pocessng: A 3-Dmensonal Computatonal Investgaton* Pamod Subamonum** and Mak J Kushne*** **Dept of Chemcal and Bomolecula
More informationOn Maneuvering Target Tracking with Online Observed Colored Glint Noise Parameter Estimation
Wold Academy of Scence, Engneeng and Technology 6 7 On Maneuveng Taget Tacng wth Onlne Obseved Coloed Glnt Nose Paamete Estmaton M. A. Masnad-Sha, and S. A. Banan Abstact In ths pape a compehensve algothm
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More information5-99C The Taylor series expansion of the temperature at a specified nodal point m about time t i is
Chapte Nuecal Methods n Heat Conducton Specal opc: Contollng the Nuecal Eo -9C he esults obtaned usng a nuecal ethod dffe fo the eact esults obtaned analytcally because the esults obtaned by a nuecal ethod
More informationJournal of Naval Science and Engineering 2015, Vol.11, No.1, pp FINITE DIFFERENCE MODEL OF A CIRCULAR FIN WITH RECTANGULAR PROFILE.
Jounal o Naval Scence and Engneeng 05 Vol. No. pp.53-67 FINIE DIFFERENCE MODEL OF A CIRCULAR FIN WIH RECANGULAR PROFILE İbahm GİRGİN Cüneyt EZGİ uksh Naval Academy uzla Istanbul ukye ggn@dho.edu.t cezg@dho.edu.t
More informationA Method of Reliability Target Setting for Electric Power Distribution Systems Using Data Envelopment Analysis
27 กก ก 9 2-3 2554 ก ก ก A Method of Relablty aget Settng fo Electc Powe Dstbuton Systems Usng Data Envelopment Analyss ก 2 ก ก ก ก ก 0900 2 ก ก ก ก ก 0900 E-mal: penjan262@hotmal.com Penjan Sng-o Psut
More informationChapter 3 Vector Integral Calculus
hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the
More informationUser-friendly model of heat transfer in. preheating, cool down and casting
ANNUAL REPORT 2010 UIUC, August 12, 2010 Use-fendly model of heat tansfe n submeged enty nozzles dung peheatng, cool down and castng Vaun Kuma Sngh, B.G. Thomas Depatment of Mechancal Scence and Engneeng
More informationChapter 13 - Universal Gravitation
Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen
More informationExact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section *
ans. Japan Soc. Aeo. Space Sc. Vol., No., pp., Exact hee-dmensonal Elastcty Soluton fo Bucklng of Beam of a Ccula Coss-Secton * oshm AKI ) and Kyohe KONDO ) ) Aeospace Company, Kawasak Heavy Industes,
More information