CHAPTER X PHASE-CHANGE PROBLEMS
|
|
- Vivian Kelley
- 5 years ago
- Views:
Transcription
1 Chapter X Phae-Change Problem December 3, CHAPER X PHASE-CHANGE PROBLEMS X.1 Introducton Clacal Stefan Problem Geometry of Phae Change Problem Interface Condton X. Analytcal Soluton for Soldfcaton n Half Space (Neumann Soluton) X.3 Analytcal Soluton for Meltng n Half Space (Eerce)
2 918 Chapter X Phae-Change Problem December 3, 18 X.1 Inroducton ranent heat tranfer problem nvolvng evaporaton, meltng, condenaton or oldfcaton wth movng nterface between phae. Clacal Stefan Problem heat tranfer problem wth lqud-old phae-change. Problem non-lnear due to nterface condton., ubcrpt referred to old and lqud phae, repectvely. m meltng or freezng temperature (temperature at whch lqud oldfe or old melt) or equlbrum phae change temperature (temperature at whch both old and lqud phae can tay together n thermodynamc equlbrum). temperature n the old phae. ( ) temperature n the lqud phae. S( t ) S( t) equaton of the phae-change boundary. ρ denty (t aumed n ome problem, that ρ ρ ρ ) Phae-change problem n the em-nfnte regon: Soldfcaton old lqud heat flu m Meltng lqud S( t ) old heat flu m S( t)
3 Chapter X Phae-Change Problem December 3, Interface Condton: conder the control urface contanng the nterface boundary S( t ) Soldfcaton: ds t A m ρ q h dt h A A freezng ρh ds t dt rate of heat releaed durng oldfcaton per unt area of nterface kg J m 3 m kg A S V A k k S v t ds t dt v ( t) S( t ) Energy balance: ds t dt k k ρh ds dt k k ρh Meltng: ρh ds t dt rate of heat aborbed durng meltng per unt area of nterface k k S( t) Energy balance: ds t + dt k k ρh ds Same condton both for oldfcaton and meltng. dt k k ρh
4 9 Chapter X Phae-Change Problem December 3, 18 X. Analytcal oluton for oldfcaton n half pace (Neumann Soluton): old lqud m Mathematcal Model: S( t ) Heat Equaton: 1 t 1 t < < S( t) t > (8.1) S( t) < < t > (8.) Boundary Condton:,t t > (8.3) Interface Condton:,t t > (8.4) S t,t S t,t m contnuty t ds ( t ) ρ S( t) dt k k h Intal Condton: > (8.5) t > (8.6), > (8.7) Soluton: Recallng the oluton of the Heat Equaton n the doman > (Secton IX.1, p., and Eerce), note that the functon erf t, erfc, any contant C cont t (where, 1 1 a ) erf ( ) and any ther lnear combnaton are oluton of the homogeneou Heat Equaton (8.1-). hen look for the oluton of the equaton (8.1) n the form 1 erfc ( ) (,t ) + A erf A cont (8.8) t whch atfe the Heat Equaton (8.1) and the boundary condton (8.3). And look for the oluton of the equaton (8.) n the form + B erfc B cont (8.9) t whch atfe the Heat Equaton (8.), condton (8.4) and the ntal condton (8.7). hen ue the nterface condton (8.5-6) to determne the coeffcent A, B and S( t ).
5 Chapter X Phae-Change Problem December 3, Subttute the tral oluton (8.8-9) nto the contnuty condton (8.5): S( t) S t + A erf + B erfc m t t S( t) S t + A erf + B erfc m t t + A erf ( ) + B erfc m (8.1) S( t) where S( t) t (8.11) t Equaton (8.1) mple (becaue both erf and erfc are monotonc functon) that S t cont t Solve Equaton (8.1) for coeffcent A and B : m A erf m B erfc and ubttute them nto equaton (8.8-9): (,t ) m + erf erf ( ) t m + erfc t erfc (8.1) (8.13) he remanng contant ha to be determned from the nterface condton (8.6): 1 ep t π t m erf ( ) 1 π erf t ( ) t m ep 1 m π ep t t erfc Dfferentate equaton (8.11): ds ( t) dt d t dt 1 t t 1 m ep π t t erfc
6 9 Chapter X Phae-Change Problem December 3, 18 Apply the nterface condton (8.6): k 1 m S ( t) S ep ( t 1 m ) k ep π erf t ( ) t π t t erfc ρh ds t dt k S( t 1 1 m ) ep π erf t ( ) t S m ( t 1 1 ) k ep π t t erfc ρh t k 1 S m ( t) ep π erf ( ) t 1 S m ( t) k ep π t erfc ρh k 1 m π erf ( ) 1 m ep k ep π erfc ρh e e k ( m ) ( m ) π ρ erf ( ) ρc p k k h erfc Equaton for contant (ha to be olved numercally): e k e π h ( m ) ( m ) erf ( ) k c p erfc (8.14) After found, the oluton of the phae-change problem for oldfcaton gven by the equaton ( ): S( t ) t (8.11) erf t (,t ) + ( m ) S( t) erf ( ) (8.1) + ( ) m erfc t erfc S t (8.13) where a potve root of Equaton (8.14)
7 Chapter X Phae-Change Problem December 3, Eample: Conder oldfcaton of the materal under the followng condton: 1 1 m k 1. ρ 1 cp 5 ρcp k. h 3 k 5. ρ 1 p c 1 k.5 ρ c p Solve equaton (8.14) for contant :.547 Phae-change boundary: S( t) ranent temperature profle: t 5 t t 5 m
8 94 Chapter X Phae-Change Problem December 3, 18 Maple Soluton: 1 PHASE-CHANGE SOLIDIFICAION > retart; > wth(plot): > :-1;:1;m:; > k:1.;kl:5.; > r:1;rl:1; > cp:5;cpl:1; > a:k/r/cp;al:kl/rl/cpl; > h:3; : -1 : 1 m : k : 1. kl : 5. r : 1 rl : 1 cp : 5 cpl : 1 a :. al :.5 h : 3 > W(v):(m-)*ep(-v^)/erf(v)-kl/k*qrt(a/al)*(m-)*ep(- v^*a/al)/erfc(v*qrt(a/al))-qrt(p)*h/cp*v; 1 e ( v ) erf( v) > plot(w(v),v.1..1,y-5..1); e (.4v) erfc( v) W( v ) : + 6 π v > v:folve(w(v),v.1..1); (correpond to contant ) Phae-Change Boundary: v : > S(t):*v*qrt(a*t); > plot(s(t),t..5); S( t ) : t
9 Chapter X Phae-Change Problem December 3, emperature Profle: > S(,t):+(m-)*erf(//qrt(a*t))/erf(v); S (, t ) : erf t > L(,t):+(m-)*erfc(//qrt(al*t))/erfc(v*qrt(a/al)); L (, t ) : erfc t > SH:S(,t)*(Heavde()-Heavde(-S(t))); > LH:L(,t)*Heavde(-S(t)); SH : erf t ( Heavde( ) Heavde ( t )) LH : erfc Heavde ( t ) t > anmate({sh,lh,,m,},..15,t..1,frame,aeboed); > S1:ub(t5,SH):L1:ub(t5,LH): > S:ub(t,SH):L:ub(t,LH): > S3:ub(t5,SH):L3:ub(t5,LH): > plot({s1,l1,s,l,s3,l3,m,,},..15,colorblack,aeboed);
10 96 Chapter X Phae-Change Problem December 3, 18 X.3 Eerce: Analytcal Soluton for Meltng n Half Space lqud old m S( t) Mathematcal Model: Heat equaton: 1 t 1 t < < S( t) t > (8.1) S( t) < < t > (8.) Boundary Condton:,t t > (8.3) Interface Condton: ( ) ( ),t t > (8.4) S t,t S t,t Intal Condton: m contnuty t ds ( t ) ρ S( t) k k h dt > (8.5) t > (8.6), > (8.7) Soluton: Look for the oluton of the equaton (8.1) n the form + A erf A cont (8.8) t whch atfe the Heat Equaton (8.1) and the boundary condton (8.3). And look for the oluton of the equaton (8.) n the form (,t ) + B erfc B cont (8.9) t whch atfe the Heat Equaton (8.), condton (8.4) and condton (8.7). hen ue the nterface condton to determne the coeffcent A, B and S( t ).
11 Chapter X Phae-Change Problem December 3, 18 97
12 98 Chapter X Phae-Change Problem December 3, 18
Introduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015
Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015 Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal.
More informationProblem #1. Known: All required parameters. Schematic: Find: Depth of freezing as function of time. Strategy:
BEE 3500 013 Prelm Soluton Problem #1 Known: All requred parameter. Schematc: Fnd: Depth of freezng a functon of tme. Strategy: In thee mplfed analy for freezng tme, a wa done n cla for a lab geometry,
More informationSmall signal analysis
Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea
More informationStatistical Properties of the OLS Coefficient Estimators. 1. Introduction
ECOOMICS 35* -- OTE 4 ECO 35* -- OTE 4 Stattcal Properte of the OLS Coeffcent Etmator Introducton We derved n ote the OLS (Ordnary Leat Square etmator ˆβ j (j, of the regreon coeffcent βj (j, n the mple
More informationSpecification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationChapter 5: Root Locus
Chater 5: Root Locu ey condton for Plottng Root Locu g n G Gven oen-loo tranfer functon G Charactertc equaton n g,,.., n Magntude Condton and Arguent Condton 5-3 Rule for Plottng Root Locu 5.3. Rule Rule
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationScattering of two identical particles in the center-of. of-mass frame. (b)
Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and
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 informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationRoot Locus Techniques
Root Locu Technque ELEC 32 Cloed-Loop Control The control nput u t ynthezed baed on the a pror knowledge of the ytem plant, the reference nput r t, and the error gnal, e t The control ytem meaure the output,
More informationOn the SO 2 Problem in Thermal Power Plants. 2.Two-steps chemical absorption modeling
Internatonal Journal of Engneerng Reearch ISSN:39-689)(onlne),347-53(prnt) Volume No4, Iue No, pp : 557-56 Oct 5 On the SO Problem n Thermal Power Plant Two-tep chemcal aborpton modelng hr Boyadjev, P
More informationEstimation of the composition of the liquid and vapor streams exiting a flash unit with a supercritical component
Department of Energ oltecnco d Mlano Va Lambruschn - 05 MILANO Eercses of Fundamentals of Chemcal rocesses rof. Ganpero Gropp Eercse 8 Estmaton of the composton of the lqud and vapor streams etng a unt
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationCaractérisation dans le plan de matériaux isolants par thermographie infrarouge et méthode convolutive
Journée SFT du Jeud 3 Jun 3 Caractérsaton Température / Température Caractérsaton dans le plan de matérau solants par termogrape nfrarouge et métode convolutve B. REMY, A. DEGIOVAI & D. MAIET B.Rémy, A.Degovann
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationIce crytal nucleaton, growth and breakage modellng n a craped urface heat exchanger H. Benkhelfa (a, M. Arellano (a,b G. Alvarez (b, D. Flck (a (a UMR n 45 AgroParTech/INRA, Ingénere-Procédé-Alment, 6
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationrisk and uncertainty assessment
Optmal forecastng of atmospherc qualty n ndustral regons: rsk and uncertanty assessment Vladmr Penenko Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SD RAS Goal Development of theoretcal
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
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 informationSMOOTHED PARTICLE HYDRODYNAMICS METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM
The 5th Internatonal Symposum on Computatonal Scences (ISCS) 28 May 2012, Yogyakarta, Indonesa SMOOTHED PARTICLE HYDRODYNAMICS METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM Dede Tarwd 1,2 1 Graduate School
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationMethod Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems
Internatonal Workhop on MehFree Method 003 1 Method Of Fundamental Soluton For Modelng lectromagnetc Wave Scatterng Problem Der-Lang Young (1) and Jhh-We Ruan (1) Abtract: In th paper we attempt to contruct
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationNewton s Method for One - Dimensional Optimization - Theory
Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton You are free to Share to copy,
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationSolving Fractional Nonlinear Fredholm Integro-differential Equations via Hybrid of Rationalized Haar Functions
ISSN 746-7659 England UK Journal of Informaton and Computng Scence Vol. 9 No. 3 4 pp. 69-8 Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalzed Haar Functons Yadollah Ordokhan
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 informationCHEMICAL ENGINEERING
Postal Correspondence GATE & PSUs -MT To Buy Postal Correspondence Packages call at 0-9990657855 1 TABLE OF CONTENT S. No. Ttle Page no. 1. Introducton 3 2. Dffuson 10 3. Dryng and Humdfcaton 24 4. Absorpton
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer
More informationThe Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites
7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT
More informationInterconnect Modeling
Interconnect Modelng Modelng of Interconnects Interconnect R, C and computaton Interconnect models umped RC model Dstrbuted crcut models Hgher-order waveform n dstrbuted RC trees Accuracy and fdelty Prepared
More informationS-Domain Analysis. s-domain Circuit Analysis. EE695K VLSI Interconnect. Time domain (t domain) Complex frequency domain (s domain) Laplace Transform L
EE695K S nterconnect S-Doman naly -Doman rcut naly Tme doman t doman near rcut aplace Tranform omplex frequency doman doman Tranformed rcut Dfferental equaton lacal technque epone waveform aplace Tranform
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More informationOutline. Unit Eight Calculations with Entropy. The Second Law. Second Law Notes. Uses of Entropy. Entropy is a Property.
Unt Eght Calculatons wth Entropy Mechancal Engneerng 370 Thermodynamcs Larry Caretto October 6, 010 Outlne Quz Seven Solutons Second law revew Goals for unt eght Usng entropy to calculate the maxmum work
More informationThermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)
hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIX, 013, f.1 DOI: 10.478/v10157-01-00-y ALGORITHM FOR THE CALCULATION OF THE TWO VARIABLES CUBIC SPLINE FUNCTION BY ION
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationElectrochemical Equilibrium Electromotive Force
CHM465/865, 24-3, Lecture 5-7, 2 th Sep., 24 lectrochemcal qulbrum lectromotve Force Relaton between chemcal and electrc drvng forces lectrochemcal system at constant T and p: consder Gbbs free energy
More informationmodeling of equilibrium and dynamic multi-component adsorption in a two-layered fixed bed for purification of hydrogen from methane reforming products
modelng of equlbrum and dynamc mult-component adsorpton n a two-layered fxed bed for purfcaton of hydrogen from methane reformng products Mohammad A. Ebrahm, Mahmood R. G. Arsalan, Shohreh Fatem * Laboratory
More information% & 5.3 PRACTICAL APPLICATIONS. Given system, (49) , determine the Boolean Function, , in such a way that we always have expression: " Y1 = Y2
5.3 PRACTICAL APPLICATIONS st EXAMPLE: Gven system, (49) & K K Y XvX 3 ( 2 & X ), determne the Boolean Functon, Y2 X2 & X 3 v X " X3 (X2,X)", n such a way that we always have expresson: " Y Y2 " (50).
More informationMass Transfer Processes
Mass Transfer Processes S. Majd Hassanzadeh Department of Earth Scences Faculty of Geoscences Utrecht Unversty Outlne: 1. Measures of Concentraton 2. Volatlzaton and Dssoluton 3. Adsorpton Processes 4.
More informationTotal solidification time of a liquid phase change material enclosed in cylindrical/spherical containers
Appled Thermal Engneerng 25 (2005) 488 502 www.elsever.com/locate/apthermeng Total soldfcaton tme of a lqud phase change materal enclosed n cylndrcal/sphercal contaners Levent Blr *, Zafer _ Ilken Department
More informationElectrical Circuits II (ECE233b)
Electrcal Crcut II (ECE33b) Applcaton of Laplace Tranform to Crcut Analy Anet Dounav The Unverty of Wetern Ontaro Faculty of Engneerng Scence Crcut Element Retance Tme Doman (t) v(t) R v(t) = R(t) Frequency
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More information(b) i(t) for t 0. (c) υ 1 (t) and υ 2 (t) for t 0. Solution: υ 2 (0 ) = I 0 R 1 = = 10 V. υ 1 (0 ) = 0. (Given).
Problem 5.37 Pror to t =, capactor C 1 n the crcut of Fg. P5.37 was uncharged. For I = 5 ma, R 1 = 2 kω, = 5 kω, C 1 = 3 µf, and C 2 = 6 µf, determne: (a) The equvalent crcut nvolvng the capactors for
More informationLECTURER: PM DR MAZLAN ABDUL WAHID PM Dr Mazlan Abdul Wahid
H E A R A N S F E R HEA RANSFER SME 4463 LECURER: PM DR MAZLAN ABDUL WAHID http://www.fkm.utm.my/~mazlan C H A P E R 3 Dr Mazlan - SME 4463 H E A R A N S F E R Chapter 5 ranent Conducton PM Dr Mazlan Abdul
More informationCinChE Problem-Solving Strategy Chapter 4 Development of a Mathematical Model. formulation. procedure
nhe roblem-solvng Strategy hapter 4 Transformaton rocess onceptual Model formulaton procedure Mathematcal Model The mathematcal model s an abstracton that represents the engneerng phenomena occurrng n
More informationMean Field / Variational Approximations
Mean Feld / Varatonal Appromatons resented by Jose Nuñez 0/24/05 Outlne Introducton Mean Feld Appromaton Structured Mean Feld Weghted Mean Feld Varatonal Methods Introducton roblem: We have dstrbuton but
More informationExercises of Fundamentals of Chemical Processes
Department of Energ Poltecnco d Mlano a Lambruschn 4 2056 MILANO Exercses of undamentals of Chemcal Processes Prof. Ganpero Gropp Exercse 7 ) Estmaton of the composton of the streams at the ext of an sothermal
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationDiffusion Mass Transfer
Dffuson Mass Transfer General onsderatons Mass transfer refers to mass n transt due to a speces concentraton gradent n a mture. Must have a mture of two or more speces for mass transfer to occur. The speces
More informationOutlet temperature of a WGS reactor (Stage I) for the conversion of CO, applied for the abatement of CO to a fixed value.
Department of Energy Poltecnco d Mlano Va Lambruschn 56 MILAN Exercses of Fundamentals of Chemcal Processes Prof. Ganpero Gropp Exercse utlet temperature of a WGS reactor (Stage I for the converson of
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationLecture 2 Thermodynamics and stability October 30, 2009
The tablty of thn (oft) flm Lecture 2 Thermodynamc and tablty October 30, 2009 Ian Morron 2009 Revew of Lecture 1 The djonng preure a jump n preure at the boundary. It doe not vary between the plate. 1
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationNumerical Transient Heat Conduction Experiment
Numercal ransent Heat Conducton Experment OBJECIVE 1. o demonstrate the basc prncples of conducton heat transfer.. o show how the thermal conductvty of a sold can be measured. 3. o demonstrate the use
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationVariable Structure Control ~ Basics
Varable Structure Control ~ Bac Harry G. Kwatny Department of Mechancal Engneerng & Mechanc Drexel Unverty Outlne A prelmnary example VS ytem, ldng mode, reachng Bac of dcontnuou ytem Example: underea
More information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
More informationTEST 5 (phy 240) 2. Show that the volume coefficient of thermal expansion for an ideal gas at constant pressure is temperature dependent and given by
ES 5 (phy 40). a) Wrte the zeroth law o thermodynamcs. b) What s thermal conductvty? c) Identyng all es, draw schematcally a P dagram o the arnot cycle. d) What s the ecency o an engne and what s the coecent
More informationGasometric Determination of NaHCO 3 in a Mixture
60 50 40 0 0 5 15 25 35 40 Temperature ( o C) 9/28/16 Gasometrc Determnaton of NaHCO 3 n a Mxture apor Pressure (mm Hg) apor Pressure of Water 1 NaHCO 3 (s) + H + (aq) Na + (aq) + H 2 O (l) + CO 2 (g)
More information1 cos. where v v sin. Range Equations: for an object that lands at the same height at which it starts. v sin 2 i. t g. and. sin g
SPH3UW Unt.5 Projectle Moton Pae 1 of 10 Note Phc Inventor Parabolc Moton curved oton n the hape of a parabola. In the drecton, the equaton of oton ha a t ter Projectle Moton the parabolc oton of an object,
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More informationPETE 310 Lectures # 24 & 25 Chapter 12 Gas Liquid Equilibrium
ETE 30 Lectures # 24 & 25 Chapter 2 Gas Lqud Equlbrum Thermal Equlbrum Object A hgh T, Object B low T Intal contact tme Intermedate tme. Later tme Mechancal Equlbrum ressure essels Vale Closed Vale Open
More informationModeling of Wave Behavior of Substrate Noise Coupling for Mixed-Signal IC Design
Modelng of Wave Behavor of Subtrate Noe Couplng for Mxed-Sgnal IC Degn Georgo Veron, Y-Chang Lu, and Robert W. Dutton Center for Integrated Sytem, Stanford Unverty, Stanford, CA 9435 yorgo@gloworm.tanford.edu
More informationPrinciples of Food and Bioprocess Engineering (FS 231) Solutions to Example Problems on Heat Transfer
Prncples of Food and Boprocess Engneerng (FS 31) Solutons to Example Problems on Heat Transfer 1. We start wth Fourer s law of heat conducton: Q = k A ( T/ x) Rearrangng, we get: Q/A = k ( T/ x) Here,
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 05-03 A comparson of numercal models for one-dmensonal Stefan problems E. Javerre 1, C. Vuk, F.J. Vermolen, S. van der Zwaag ISSN 1389-6520 Reports of the Delft Insttute
More information3. Be able to derive the chemical equilibrium constants from statistical mechanics.
Lecture #17 1 Lecture 17 Objectves: 1. Notaton of chemcal reactons 2. General equlbrum 3. Be able to derve the chemcal equlbrum constants from statstcal mechancs. 4. Identfy how nondeal behavor can be
More informationCHAPTER 9 LINEAR MOMENTUM, IMPULSE AND COLLISIONS
CHAPTER 9 LINEAR MOMENTUM, IMPULSE AND COLLISIONS 103 Phy 1 9.1 Lnear Momentum The prncple o energy conervaton can be ued to olve problem that are harder to olve jut ung Newton law. It ued to decrbe moton
More informationAssignment 4. Adsorption Isotherms
Insttute of Process Engneerng Assgnment 4. Adsorpton Isotherms Part A: Compettve adsorpton of methane and ethane In large scale adsorpton processes, more than one compound from a mxture of gases get adsorbed,
More information), it produces a response (output function g (x)
Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans
More informationUniversiti Malaysia Perlis, Pauh Putra Campus, Arau, Perlis, Malaysia. Jalan Permatang Pauh, Pauh, Pulau Pinang, Malaysia
Mathematcal modelng on thermal energy storage systems N. A. M. Amn 1,a, M. A. Sad 2,b, A. Mohamad 3,c, M. S. Abdul Majd 4,d, M. Afend 5,e,. Daud 6,f, F. Bruno 7,g and M. Belusko 8,h 1,3,4,5,6 Mechancal
More informationNot at Steady State! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Yes! Class 15. Is the following possible?
Chapter 5-6 (where we are gong) Ideal gae and lqud (today) Dente Partal preure Non-deal gae (next tme) Eqn. of tate Reduced preure and temperature Compreblty chart (z) Vapor-lqud ytem (Ch. 6) Vapor preure
More informationTopic 5: Non-Linear Regression
Topc 5: Non-Lnear Regresson The models we ve worked wth so far have been lnear n the parameters. They ve been of the form: y = Xβ + ε Many models based on economc theory are actually non-lnear n the parameters.
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More information6. Stochastic processes (2)
Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space
More informationHomogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface
Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n
More informationElectrostatic Potential from Transmembrane Currents
Electrostatc Potental from Transmembrane Currents Let s assume that the current densty j(r, t) s ohmc;.e., lnearly proportonal to the electrc feld E(r, t): j = σ c (r)e (1) wth conductvty σ c = σ c (r).
More information16 Reflection and transmission, TE mode
16 Reflecton transmsson TE mode Last lecture we learned how to represent plane-tem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such
More information6. Stochastic processes (2)
6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More information: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:
764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationNo! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Survey Results. Class 15. Is the following possible?
Survey Reult Chapter 5-6 (where we are gong) % of Student 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Hour Spent on ChE 273 1-2 3-4 5-6 7-8 9-10 11+ Hour/Week 2008 2009 2010 2011 2012 2013 2014 2015 2017 F17
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT - A mathematcal model for the dssoluton of stochometrc partcles n mult-component alloys F.J. Vermolen, C. Vuk and S. van der Zwaag ISSN 389-652 Reports of the Department
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationScatter Plot x
Construct a scatter plot usng excel for the gven data. Determne whether there s a postve lnear correlaton, negatve lnear correlaton, or no lnear correlaton. Complete the table and fnd the correlaton coeffcent
More informationand Statistical Mechanics Material Properties
Statstcal Mechancs and Materal Propertes By Kuno TAKAHASHI Tokyo Insttute of Technology, Tokyo 15-855, JAPA Phone/Fax +81-3-5734-3915 takahak@de.ttech.ac.jp http://www.de.ttech.ac.jp/~kt-lab/ Only for
More information