Determination of one dimensional temperature distribution in metallic bar using green s function method

Transcription

1 American Journal o Applied Mahemaics ; (4: 55-7 Published online Ocober, (hp:// doi:.648/.aam.4.4 Deerminaion o one dimensional emperaure disribuion in meallic bar using green s uncion mehod Virginia Mwelu Kieu, Thomas Onyango, Jacson Kioo Kwanza, icholas Muhama Muua Deparmen o Mahemaics and Compuer Science, Faculy o Science, he Caholic niversiy o Easern Arica, airobi, Kenya Deparmen o Pure & Applied Mahemaics, Jomo Kenyaa niversiy o Agriculure & Technology, airobi, Kenya Deparmen o Mahemaics and Inormaics, School o Science and Inormaics, Taia Tavea niversiy College, Voi, Kenya address: virginiamwelu@yahoo.com(v. M. Kieu, onyangom@yahoo.com(t. Onyango, wanzaioo@yahoo.com(j. K. Kwanza, nicholasmuhama@uc.ac.e(. Muhama To cie his aricle: Virginia Mwelu Kieu, Thomas Onyango, Jacson Kioo Kwanza, icholas Muhama Muua. Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green S Funcion Mehod. American Journal o Applied Mahemaics. Vol., o. 4,, pp doi:.648/.aam.4.4 Absrac: The presen sudy ocuses on deerminaion o emperaure disribuion in one dimensional bar using Green s uncion mehod. The Green s Funcion is obained using separaion o variables, variaion ormulaion principles and Heaviside uncions. The Boundary Inegral Equaion is obained using he undamenal soluion, Divergence heorem, Green Ideniies, Dirac dela properies and inegraion by pars. The soluion o hea equaion given by he Green s Funcion and he boundary inegral equaion has saisied he uniqueness, regulariy and sabiliy o hea equaion. The uniqueness, regulariy and sabiliy have been proved using smooh properies o class uncion, Lyapunov uncion and L orm. The BEM implemenaion is perormed using FORTRA 95 soware. Soluions o he es problems are presened and ime dependence resuls are in agreemen wih compued analyical soluions. Keywords: Green s Funcion, Heaviside Funcion, Separaion o Variables, Inegraion by Pars, Lyapunov Funcion. General Inroducion.. General Descripion and Terminologies... Temperaure Temperaure is he measure o coldness or honess o a body. In a qualiaive manner, we can describe he emperaure o an obec as ha which deermines he sensaion o warmh or coldness el rom conac wih i. A convenien operaional deiniion o emperaure is ha i is a measure o he average ranslaional ineic energy associaed wih he disordered microscopic moion o aoms and molecules. The Inernaional Sysem o nis (SI or emperaure is Kelvin (K. Oher unis are degree Celsius, Fahrenhei, Ranine, Delisle, ewon, Reaumur and Romer.... Hea Flux Temperaure and hea low are wo imporan quaniies in he problems o hea conducion. Temperaure a any poin in he solid is compleely deined by is numerical value because i is a scalar quaniy, whereas hea low is deined by is value and direcion. The basic law which gives he relaionship beween he hea low and emperaure gradien is by French mahemaician Jean Bapise Joseph Fourier. He ocused on a saionary, homogeneous, isoropic solid (maerials in which hermal conduciviy is independen o direcion.the law is o he orm q T (... Modes o Hea Transer Hea is energy ranserred rom a high emperaure sysem o lower emperaure sysem. Is Inernaional Sysem o unis (SI is Joule (J. Hea ranser is a discipline o hermal engineering concerned wih generaion, use, conversion and exchange o hermal energy and hea beween physical sysems. There are hree modes o hea ranser namely conducion, convecion and radiaion. Conducion: reers o he hea ranser ha occurs across he medium (solid or luid. Hea can be conduced beween wo bodies which are in conac wih each oher; hea lows rom one body o he oher. Maerials ha allow hea o ravel hrough hem are called conducors. Meals are good

2 56 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod conducors o hea. on-conducors (insulaors include plasic, clay, wood and paper. The rae o hea lux (rae o hea ranser per uni area in a solid obec is proporional o he emperaure gradien. The Fourier law is dt q x A dx ( Convecion: reers o he hea ranser ha occurs beween a surace and luid when hey are a dieren emperaures. Convecion hea ranser mode is comprised o wo mechanisms such energy ranser due o random molecular moion (diusion and energy ranserred by he bul or macroscopic moion o he luid (advecion. Convecion leads o he ac ha good insulaors (lie air can ranser hea eicienly -as long as he air is allowed o move reely. Trapped air, as beween panes o a double window, canno ranser hea well because i canno mix wih air o a dieren emperaure. Convecion hea ranser may be classiied according o he naure o he low. Forced convecion aes place when he low is caused by an exernal agen such as an, pump or amospheric winds. aural convecion is caused by buoyancy orces due o densiy dierences caused by emperaure variaions in he luid. Free convecion low ield is sel-susained low driven by he presence o a emperaure gradien. Mixed convecion is experienced when naural and orced convecion occurs ogeher. ewon s law o cooling is ( T q' ' h T Radiaion is he ranser o hea in orm o waves hrough space (vacuum. Hea is carried by elecromoive waves rom one medium o anoher. Radiaion aes place beween wo suraces by emission and laer absorpion. Sean Bolzmann s equaion is given by.. Lieraure Review s 4 4 ( q ASe T T Ozisi (968 sudied he physical signiicance o Green s uncion in hea conducion. He presened he general expressions or he soluion o inhomogeneous ransien hea conducion problems wih energy generaion, inhomogeneous boundary condiion, and a given iniial condiion, in erms o Green s uncion or one, wo and hree dimensional problems o inie and semi-ininie regions. Chang and Tsou (977 in heir boo eniled Hea conducion in Anisoropic Medium Homogeneous in cylindrical Regions-nseady Sae discussed abou he analyical soluion o hea conducion in an anisoropic medium ha is homogeneous in circular cylindrical co-ordinaes. They considered boundary condiions o Dirichle, eumann and mixed ypes o solid cylinder and hollow cylinder o inie and ininie lenghs. Cannon and John (984 in heir boo eniled The one ( (4 dimensional Hea Equaion researched on some Green's uncion soluions in D.A variey o elemenary Green's uncion soluions in one-dimension are recorded here. In some o hese, he spaial domain is he enire real line (-,. In ohers, i is he semi-ininie inerval (, wih eiher eumann or Dirichle boundary condiions. Bec (984 derived he Greens uncion soluion or he linear, ransien hea conducion equaion in he orm ha included ive inds o boundary condiions and also demonsraed he condiions under which i was permissible o use he produc propery o one dimensional Green s uncions. Greenberg (986 in his boo Applicaion o Greens Funcion in Science and Engineering analyzed some applicaion o Green s uncion o solve conducion hea ranser, acousic graviaional poenial and vibraion problems. James and Jerey (987 applied Green s uncion o solve hea conducion equaion in hree dimensions obaining an inegral equaion or emperaure in erms o he iniial and boundary values o he emperaure and lux. Copper and Jeery (998 sudied he low o hea in one dimension hrough a small hin rod. They used he derivaion o he hea equaion, and Ma lab s o model he moion and showed graphical soluions. They used wo mehods o solve he rae o hea low hrough an obec. The irs mehod was derived rom he properies o he obec. The second mehod was derived by measuring he rae o hea low hrough he boundaries o he obec. Eduardo ( ormulaed a able o Green s Funcion ha enable us o derive ransien conducion soluions or recangular co-ordinaes sysem and also provided a numbering sysem ha enables eicien caaloging and locaing o Green s Funcion. The Green s Funcion was obained using Eigen uncions expressions. Praproni e al. ( discussed he numerical soluion o he wo dimensional Hea Equaion. An approximaion o he soluion uncion is calculaed a discree spaial mesh poins, proceeding in discree ime seps. The saring values are given by an iniial value condiion. They explained how o ransorm he dierenial equaion ino a inie dierence equaion which can be used o compue he approximae soluion. Viun (4 analyzed piezoelecric parallel Bimorph using BEM. He concluded ha BEM solves hese problems aser han he oher mehods. He also prediced emperaure disribuion using combined BEM and FEM or hea ranser in uel cell. Onyango e al. (8 in heir sudy on resoraion o boundary condiions in one-dimensional ransien inverse hea conducion problems used BEM o represen boundary condiions wih linear relaions beween emperaure and he hea lux, ogeher wih he space or ime-dependen ambien emperaure o environmen surrounding he hea conducor. Venaaramam e al. ( used he physical approach o he mehod o images o obain Green s uncion or cylinders and spheres. They ound emperaure disribuion in ininie cylinders and spheres wih dieren ypes o discree hea generaion sources such as ring and spiral

3 American Journal o Applied Mahemaics ; (4: sources and showed ha or discree sources Green s uncion deermined by mehod o images yielded analyical soluions. Misawo F. ( researched on a soluion o one dimensional ransien Hea ranser problem using Boundary Elemen Mehod (BEM.She modeled he low o hea rom a ho cylindrical meal bille o a liquid o uniorm emperaure. The cylinder was considered o have many hin slabs across he volume which when heaed summed up o orm he whole volume. Duhamel s mehod relaes he soluion o boundary-value problems o hea conducion wih ime-dependen boundary condiions and hea sources o he soluion o similar problem and hea sources by means o simple relaion.duhamel s mehod is a useul ool or obaining he soluion o a problem wih ime-dependen boundary condiions and hea sources whenever he soluion o a similar problem wih ime- independen boundary condiions and hea source is available. A proo o Duhamel s mehod is given by Barels and Churchil (94 or a building condiion o irs ind. This mehod is used o solve hea conducion in semi-ininie solid, semi-ininie recangular srip and long solid cylinder... Research Obecives The purpose o his wor is o deermine one dimensional emperaure disribuion on a meallic bar using he Green s Inegral Mehod. The speciic obecives are:. To deermine a general Green s uncion soluion equaion applicable o he soluion o hea ranser by conducion.. To apply he Green s uncion soluion equaion considered o deermine boundary inegral equaion or emperaure disribuion.. To deermine emperaure and lux proile using BEM..4. Signiicance o he Sudy In recen years, applied mahemaics has developed a srong ineres in he science o hea and mass ranser. This is mainly due o is many applicaions in engineering, asrophysics, and geophysics and power generaion, among ohers Hea ranser is o grea signiicance o branches o science and engineering. Hea ranser is very imporan o engineers who have o undersand and conrol he low o hea hrough he use o hermal insulaion, hea exchangers such as boilers, heaers, radiaors, rerigeraors and oher devices. Thermal insulaors are maerials speciically designed o reduce he low o hea by limiing conducion, convecion or boh. Radian barriers are maerials which relec radiaion and hereore reduce he low o hea rom radiaion sources. A hea exchanger is a device buil or eicien hea ranser rom one luid o anoher, wheher he luid is separaed by a solid wall so ha hey never mix, or luids are direcly conaced. Hea exchangers are widely used in rerigeraors, air condiioning, space heaing, power producion, and chemical processing, he radiaor in car, in which he ho radiaor luid is cooled by he low o air over he radiaor surace. The modern Elecric and elecronic plans require eicien dissipaion o hermal losses, hence a horough hea ranser analysis is imporan or proper seizing or uel elemens in he nuclear reacor cores o minimize burn ou. The perormance o an aircra also depends upon he ease wih which he srucure and he engine can be cooled. The uilizaion o solar energy which is widely used in rerigeraors, air condiioning space heaing, power producion and chemical producion readily available requires a horough nowledge o hea ranser or proper design o he solar collecors and associaed equipmens.. Governing Equaions Hea ranser problems can be described by se o equaions namely: he hea equaion, Heaviside uncion, Dirac-Dela Funcion, he Divergence heorem, Green Ideniies and Laplace equaion.. The Hea equaion Consider a heaed solid body wih consan hermal conduciviy,, wih soluion domain < x <, he conservaion law or hea ranser in he body in one dimension is given he hea equaion (5 or >. This equaion which is also nown as he diusion equaion can be solved by imposing boundary condiions. The mos common our boundary condiions applicable o i are: Dirichle boundary condiion, which speciy he dependen variable u(x, a each poin o he boundary, I (, l o he region wihin which soluion is required, mahemaically we can represen Dirichle Boundary condiions as:,,, eumann boundary condiion, which speciy he lux o he dependen variable u(x, a each poin o he boundary, I(,l o he region wihin which soluion is required, mahemaically can be represened as:,,, Robin boundary condiion or mixed boundary condiion is a combinaion o Dirichle and eumann boundary condiions. I speciies he lux and he uncion u(x, a each poin o he boundary I (, l o he region wihin which he soluion is required, mahemaically is represened as:,,, Periodic boundary condiion, which speciy he dependen

4 58 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod variable u(x, in periodic shape. Mahemaically is represened as:.. Heaviside Funcion,, I is uni sep uncion, named aer he elecrical engineer Oliver Heaviside, i is deined as: ( x a i i x < H (6 x > Heaviside uncion and Dirac-Dela uncion is relaed by he ollowing propery dh dx.. Dirac-Dela Funcion ( x δ ( x (7 Someimes is called he ni Impulse Funcion. I models phenomena o an impulse naure such as acion o hea low over a very shor ime inerval or over a very small region. Siuaion lie his occurs in mechanics especially when a orce concenraed a a poin caused by deormaion on solid surace, impulse orces in rigid body dynamics, poin mass in graviaional ield heory, poin charges and mulipoles in elecrosaics. And more imporan or his sudy is as poin hea sources and pulses in heory o hea conducion. The Green s Funcion is he response o a dierenial equaion, and he Dirac Dela uncion describes he impulse. The Dirac Dela Funcion δ ( x is zero when x, and ininie a x in such a way ha he area under he uncion is uniy i.e.: (8a (8b (8c... Physical Inerpreaion o Dirac Dela and Fundamenal Soluion The Dirac dela uncion is used in physics o represen a poin source. A coninuous emperaure disribuion in -dimensional space is described by a emperaure densiy, ypically denoed ρ ( x. The oal emperaure o he disribuion is given by inegraing he emperaure densiy all over space: (9 ow suppose ha I have a single poin charge, q, a posiion x '. The charge densiy o a poin charge should be zero everywhere excep a x x', since here is no charge anywhere excep a his poin. On he oher hand, a x ', we have a inie charge in an ininiely small volume, so he densiy should be ininie here. Finally, i mus saisy, Q d xρ( x since q is he oal charge. These requiremens are uniquely saisied by ρ ( x qδ ( x x' (Temperaure densiy o a poin charge q a x ' One would have similar expressions or he (mass densiy o a poin mass m. δ x is essenially a booeeping device; The uncion ( i is a singular uncion which is zero everywhere excep a he posiion x (he origin o he coordinae sysem where i is ininie. Some basic properies o Dirac dela are: Q d xρ( x ρ ( x qδ ( x x' d x δ x ( ( a δ ( ax δ ( x dx ( x δ ( x a ( a ( x ( x x' dx δ δ ( x x' δ ( x' x.4. The Divergence Theorem This heorem is also nown as Gauss Theorem. I saes ha he lux o a smooh vecor ield F hrough a closed µ boundary equal o inegral o is divergence. Mahemaical saemen is Fndx ˆ divfd µ ( where nˆ is he ouward normal o surace µ and is enclosed region. Gauss heorem is very imporan in vecorial calculus. I F is coninuous vecor ield and is componens have coninuous parial derivaives in, hen Fd Fndµ ˆ µ.5. Green s Ideniies F V ( Tae vecor uncion where and V are arbirary scalar ields, deined in. Then F V + V V F nˆ V nˆ F nˆ V nˆ V ( (a (b Subsiuing equaions ( and (b ino equaion (

5 American Journal o Applied Mahemaics ; (4: yields Green s irs ideniy: I ( V + V d V dµ F V ( resuls ino:, µ (4 subsiuing his in divergence equaion V ( V V + V d dµ µ (5 Subracing equaion (5 rom (4 gives Green s Theorem (Green s second ideniy i.e. ( V V d V V µ n.6. Fundamenal Soluion o he Problem (6 A undamenal soluion or a linear parial dierenial operaor L is a ormulaion in he language o disribuion heory o a Green's uncion. In erms o he Dirac dela uncion δ(x, a undamenal soluion G is he soluion o he inhomogeneous equaion. LG δ ( x (7 The useulness o he Green uncion is eviden once you mae he ollowing realizaion. Any disribuion o source (i.e. charge densiy or insance can be wrien as a sum, or inegral in he coninuous case, o poin sources. Thereore, i we now how he sysem reacs o a poin source, hen we should be able o deermine how i reacs o any disribuion o source, since we can sum up all he conribuions. I is absoluely criical here ha he dierenial operaor is linear. The undamenal soluion is an approximaion which is used in boundary inegral equaion. Consider hea ranser equaion (5, a modeled or hea ranser hrough a meallic bar, is obained using mehod o separaion o variables see equaion 9, under prescribed iniial condiions u(x,(x, Boundary condiions are as x ± (8 ( x F( G( x, (9 Subsiuing equaion (9 ino hea equaion (5 resuls ino, F G FG, dividing his equaion by FG gives T xx F Gxx Conan, where he consan is D F G F Solving D i.e. + D F F F resuls ino: c where is consan. Solving, G G xx G D i.e. G + D G xx Dx id ( x c ei + c e x, yields: ( Where c and c are consans. Subsiuing equaions ( and ino (9 gives: D idx idx c e ( c e c e _ ( Subsiuing he boundary condiions (,(-, ino ( resuls ino: The physical signiicance o boundary condiion is ha using Dirac dela uncion, δ ( x i x, he inegraion on inerval is zero. Ae( idx D ( Inegraing ( wih respec o D on (-, resuls ino: ( ( ( idx D x, A D e ( ( ( idxd x, A D e dd (4 Subsiuing he iniial condiion, (x,(x ino (4 gives ix ( x A( D e DdD (5 Then using Fourier ransorm heorem, equaion (5 becomes: A ' ( D ( x' e ix D dx' (6 π Insering (6 ino equaion (4 yields ( idxd ix' D e ( x' e dx' dd π (7 Rearranging equaion (7, we obain: π i( xx' DD e dd ( x' dx' And equaion (8 reduces ino, (8a F D ( c e (

6 6 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod or G G x', ( x' dx' (8b π i( xx' x', e DD dd G x x x x D i x x D e dd π (9a D (, ', { cos ( ' + sin ( ' } (9b The second inegrand in equaion (9b is zero since sine is an odd uncion. G π D { De }dd x', cos( x x' (9c Inegraing equaion (9c using inegraion ax π 4a rule, e cosbxdx e a Where oura, b ( x x', hen equaion (9c becomes he Green s uncion or hea equaion x', b ( xx' 4 G e 4π ( Subsiuing equaion ( ino equaion (8b resuls ino: ( x e 4π ( xx' 4, When ( x ( x x' ( x a dx ( x' dx' (a δ and using Dirac dela uncion, δ, see equaion (8c Equaion (a reduces o, undamenal soluion o one dimension, hea equaion given by G x', ( x x' e (b 4π 4π sing Heaviside uncion, H, which is inroduced o emphasize he ac ha he undamenal soluion is zero i o equaion (b reduces ino:, x', o H 4π ( o ( o e 4 ( xx' ( o (c Where o i he ime period..7. The Boundary Inegral Equaion We are using Greens heorem equaion (6, aing VG(x,, when we irs rewrie he hea equaion using he Del operaor, Where x. ( x, ( x x' and < o. Given ha he Green s uncion G(x,, saisies properies o hea and Green s uncions, namely: G G (a ( x on G, (b ( x x' G o δ (c Muliplying he hea equaion ( by Green s uncion, G x', and inegraing over he volume and over ime inegral we ge: o o ( G dxd ( G dxd (4 Inegraing he LHS o (4 by pars by leing ug and! Thereore, " and v, hen he LHS o (4 becomes o o ( G dxd ( G dx ( G dxd (5a sing equaion (a, o simpliy equaion (5, we obain o o ( G dxd ( G dx + ( G dxd (5b Inegraing RHS o (5b wice by pars by aing ug and dv hen simpliy we have, n o o o ( ( + ( o G dxd Gdxd G ds x d ds ( x d Subsiuing equaion (b, on resuls ino: o o (6a ino above equaion

7 American Journal o Applied Mahemaics ; (4: o o o G dxd Gdxd ds x d (6b ( ( ( unnown emperaures are obained numerically using BEM echnique. Equaing (5b and (6b gives, (. ( ( o G dx + G dxd Gdxd ds ( x d (7 Equaion (7 reduces ino: ( G. ( o dx ds x d (8 Wriing he Le Hand Side o (8 as ( G. o dx ( xx' dx G x';, dx (9a Which by propery o dela uncion we ge, ( G. o dx ( x', G x';, dx (9b Insering (9b ino (9a we ge, x G x x dx ds x d ( ', (, ';, ( (4a Maing ( x', he subec o he ormula gives hea uncion, (BIE in he soluion domain which will hen be discriised using BEM. x G x x dx ds x d ( ', (, ';, ( (4b.8. Saemen o he Problem People who wor under high emperaure condiions need o now he emperaure disribuion in hose areas aer a cerain period o ime. They should be equipped wih nowledge on emperaure disribuion rom high emperaure o low emperaure and insulaion mehods o minimize hea lose. Hea disribuion in a meallic rod has been deermined using non linear mehods and ound no accurae. We are using numerical mehod, Green uncion mehod o improve he accuracy hrough he use o boundary inegral mehod assuming he meallic bar has he ollowing dimensions. Consider hea low rom boundary x o x, wih speciied Dirichle and Robin boundary condiions, Figure.. Geomerical Coniguraion o he research problem.9. Assumpion o he Sudy i. The meallic bar is o uni lengh. ii. Temperaure is discreely disribued. iii. Thermal conduciviy o he meallic bar is one.. Mehodology The Green s uncion inegral mehod is seleced because he soluion is always in he orm o an inegral and can be seen as a recasing o boundary value problem ino inegral orm. The Green s uncion mehod is useul i he Green s uncion is nown and i he inegral expressions can be evaluaed. Overcoming hese limiaions oers Green s uncion several advanages or soluion o linear hea conducion problems. Even i he inegral has o be evaluaed numerically his is generally more accurae han numerical soluion such as inie dierences especially or discree sources. The advanages o he Green s uncion mehod s are he ollowing;. I is lexible and powerul. For a given geomery i can be used as building bloc o he emperaure resuling rom: space-variable iniial condiions, ime and space variable boundary condiions and ime and space variable generaion.. I has a sysemaic soluion procedure. For a given Geomery, he Green s uncion or a paricular ype o boundary condiions can be deermined once.this can be used or any ype o source, and he soluion or Temperaure can be wrien immediaely in he orm o inegrals. The sysemaic procedure saves ime and reduces he possibiliy o error, which is paricularly imporan or wo and hree dimensional geomerics.. I has minimal compuaion labor when compared o purely numerical inie dierence or inie elemen mehods. I has much beer accuracy han inie dierence echniques. 4. Alernaive orm o he soluion can improve series convergences. For hea conducion in inie bodies,

8 6 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod ininie series soluion or hea conducion problems driven by non-homogeneous boundary condiions problem exhibi slow convergences, requiring a very large number o erms o obain accurae numerical values. For some o hese problems, an alernaive ormulaion o he Green s uncion soluion reduces he number o required series erms... Mehod o Soluion In his sudy or simpliciy we reduce he hea equaion o one dimension, unseady case which is hen wrien using Greens uncion mehod which uses variaion ormulaion principles. The undamenal soluion o hea equaion G(x,x`, is obained using mehod analyical approach. The hea equaion is hen discriised using BEM when in cooperaing he Greens uncion. The boundary condiions are hen prescribed by describing he emperaure disribuion on he speciied boundaries o he soluion domain... umerical Discreisaion o he BIE This echnique is necessary because i ransorms he Boundary Inegral Equaion ino a linear sysem o equaions ha can be solved by a numerical approach. In choosing he inerval poins i is imporan ha any corner o he boundary µ and also any poin where a prescribed boundary condiion changes are included. This ensures each o he inervals o he boundary is smooh, so ha he normal is well deined a each nodal poin, and a single boundary condiion applies wihin each inerval. The inerval o he boundary is symmerically disribued abou any o he axes o symmery, and such axes inersec wih he boundary a inerval poins raher han a nodal poins. By he classical BEM mehodology, Brebbia e al (984, and using he undamenal soluion or he ime dependen hea equaion in one dimension given by equaion (c he hea equaion can be ransormed ino he ollowing boundary inegral equaion, G x',, x', ds s + s (,, ', G x x ds (4c The inegral over S vanishes due o he Heaviside uncion in expression (c. The discreisaion o he BIE (4c is perormed using he ollowing seps: The boundaries SandS are discreised ino a series o small boundary elemens. { } { } S (, (, (4a lux { } { } S (, (, (4b T he boundary S is discreised in a series o small cells S [,] { } { x, x ] { } (4c Over each boundary elemen he emperaure and he he midpoin, are assumed o be consan and ae heir values a + ~ (4 i.e. ( ( or (,,, o ( (,, (4 or (, (, (, (, (, ' ' (44 or (, (45 or (, (46 Also, over each cell he emperaure is assumed o be consan and aes is value a he midpoin, x + x ~ x ( (47 ( xɶ, or x x, x ] (48 Wih he approximaions (4 o (48 and, he boundary inegral equaion (4b is discreised as: ' ' (, (,,, + (,,, (,,, (,,, x G x d G x d For + x ' x (,,, ', x d x d G x x d (, [,] (,] x (49 Where # $ and # represen he ouward normal a he boundaries x and x respecively. Equaion (5 can be rewrien in he equivalen

9 American Journal o Applied Mahemaics ; (4: discreised boundary inegral orm o he ime-dependen hea equaion in one dimension as: ' ' ' ' (, C + C D D x + (, E x Where coeiciens are given by x' x' (,, ', ( ( x x' ( (5 C G x x d e d 4 π (5 ( xx' (,, ', ( ' (5 x' π( 4( D x x d xxe d x x ( x x' (, (, ',, (5 π 4( E x G x x d e d x x I he boundary inegral equaion (4b has hese approximaions applied a every node on he boundarys hen he ollowing se o linear algebraic equaions is obained. x' ' x' ' x' x' C i + Ci Di ( x' Ci + Ei ( x' (54, &&&&&,,% ' (,* Where he inluence marices + $,,+,,- $, and -, are deined by +./ $, + /,,., +./, + /,,. (55. ' ',. (56 Also, on applying he iniial condiion (48 a he cell nodes (, or &&&&&& $,% $, hen he values! are deermined, namely,!!4,, &&&&&&,% $ (57 Recasing ogeher equaions resuls in a sysem o equaions which in a generic orm can be wrien as (58 Where X is a nown 4 4 square marix which includes he inluence marices+ $,+,- $, - and E given by expression (4c-(44, 7 is a vecor o 4 unnowns ' ' which conains he unnowns! $.,!.,! $. 9#!. recased as 7.! $. or &&&&&,%.. Mehod o Soluion o he Problem This secion presens mehod o soluion o hea ranser problem, in paricular, hea low hrough a meallic bar (along is lengh which is analyzed by modeling ranser hrough a geomery (,x (,.Rearranging equaion (54 according o he speciied boundary condiions, BEM FORTRA 95 code or he compuaion o unnown values is developed. To validae he numerical implemenaion, soluions o he es problems are presened. The graphs are drawn wih he help o G-sharp soware.... Problem : Approximaed Values on he Boundaries Consider hea equaion!! which is modeled by hea ranser hrough a meallic bar wih analyical soluion given by Iniial condiions (x,, Boundary condiions are!, + (59 (, (6 (,+ (6!,!,!! (6 Assuming hea los on boh ends and insulaion on he sides, BEM code is implemened considering boundary discreisaion o odes. The resuls obained are illusraed in he graphs below..4. Exisence and Regulariy o Hea Equaion Soluion sing Iniial Daa, (x,(x The equaion or conducion o hea in one dimension or a homogeneous body has he orm u u xx The Cauchy problem or his equaion consiss in speciying (, x (x, where (x is an arbirary uncion. Exisence and uniqueness heorem is he ool which maes i possible or us o conclude ha here exiss only one soluion o a irs order dierenial equaion which saisies a given iniial condiion. Theorem The hea equaion wih bounded coninuous iniial daa (x has a soluion or < x < and >, deined by equaion (b, and u(x, saisies u(x, (x in he sense ha lim $, Acually, you can pu in uncions which aren coninuous or bounded; all we really need is ha doesn x grow oo as (aser hane or be so disconinuous ha he inegral doesn mae sense..4.. Speed o Propagaion The hea equaion has ineresing qualiaive eaures such as speed o propagaion. Recall ha or he wave equaion inormaion propagaed a a inie speed. sing equaion (b, suppose ha he iniial condiion (x is zero everywhere excep on some

10 64 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod small inerval (a, b. Because he exponenial uncion is never zero, he inegral ypically won be zero, even or small and x arbirarily ar rom (a, b. More generally, he iniial condiion aec he soluion u(x, or all x, no maer how small is. The hea propagaes wih ininie speed..4.. Smoohing Properies sing Class K Funcions In a mahemaical analysis a diereniabiliy class is a classiicaion o uncions according o he properies o heir derivaives. Higher order diereniabiliy classes correspond o he exisence o more derivaives. Funcions ha have derivaives o all orders are called smooh. Consider an open se on he real line and a uncion deined on ha se wih real values. Le be a non-negaive ineger. The uncion is said o be o class C i he ( derivaives exis and are coninuous ', ' ',..., (he coninuiy is auomaic or all he derivaives excep ( or. The uncion is said o be o classc, or smooh, i i has derivaives o all orders. The uncion is said o be o classc, or analyic i is smooh and i i equals is Taylor series expansion around any poin in is domain. In general, he classes C can be deined by declaring C o be he se o all coninuous uncions and declaring C or any posiive ineger o be he se o all diereniable uncions whose derivaive is in C. In paricular, C is conained in C or every, and here are examples o show ha his conainmen is sric. C is he inersecion o he ses C as varies over he non-negaive inegers. Firs, recall ha a uncion is said o be C i i is ininiely diereniable i.e. we can diereniae as many imes as we lie and he derivaives never develop corners or disconinuiies. For hea equaion soluion, he uncion ( xx' 4 is such a uncion, wih respec o all e π variables. Also, or any > his uncion decays o zero very rapidly away rom x x'. As a resul, we can diereniae boh sides o equaion ( wih respec o x as many imes as we lie, by slipping he x derivaive inside he inegral o ind ( x x' n e ( x' dx' n x (6 π 4π The inegral always converges (i, e.g., is bounded. In shor, no maer wha he naure o (disconinuous, or example he soluion (x, will be C in x, or any ime > ; he iniial daa is insanly smoohed ou..4.. niqueness and Sabiliy o Hea Equaion Soluion sing Lyapunov Funcion In he heory o ODEs, Lyapunov uncions are scalar uncions ha may be used o prove he sabiliy o equilibrium o an ODE. For many classes o ODEs, he exisence o Lyapunov uncions is a necessary and suicien condiion or sabiliy. A Lyapunov uncion is a uncion ha aes posiive values everywhere excep a he equilibrium in quesion, and decreases along every raecory o he ODE. The main advanage o Lyapunov uncion is ha he acual soluion (wheher analyical or numerical o he ODE is no required. This can be exended o PDES hea equaion which is converible o ODE via separaion o variable. I (x, saisies he hea equaion, deine he quaniy where ( x x dx (64 (, (, E x E, is a Lyapunov uncion, (assuming he inegral is inie similar o he energy or he wave equaion. I s no clear i E( above is physically any ind o energy, bu i can be used o prove uniqueness and sabiliy or he hea equaion. I s easy o compue de dx d (65 provided and decay as enough a ininiy which we ll assume. Replace in he inegral wih xx (since o ind, xx de xx dx d (66 ow inegraing by pars (ae an x derivaive o o xx, pu i on u o ind lim lim x x x x de dx d (67 We will assume ha limis o zero as x, in ac, impose his as a requiremen. Assuming x says bounded, we can drop he endpoin limis in de above (alernaively, d we could assume x limis o zero and says bounded. We obain de dx d (68

11 American Journal o Applied Mahemaics ; (4: de I s obvious ha d, so ( E is a non-increasing (and probably decreasing uncion o. Thus, or example, E or > E is simply he L E ( o course ( norm o he uncion ( x,, as a uncion o x a some ixed ime, and we ve shown ha his quaniy canno increase in size over ime. Here s a bi o noaion: we use., o mean hel norm o as a uncion o x or a ( ixed value o. Wih his noaion we can ormally sae he ollowing Lemma: Lemma Suppose ha ( x, saisies he hea equaion or < x < and > wih iniial daa (x,., we can wipe ou uniqueness (x. Then ( and sabiliy a he same ime. Consider wo soluions and o he hea equaion, wih iniial daa and. Form he uncion, which has iniial. Apply Lemma o and you immediaely daa obain Theorem I and are soluions o he hea equaion or < x < and > wih iniial daa ( x and ( x ( x and,, and boh decay o zero in x or all >,.,.,. hen ( ( To prove uniqueness ae all imes > we have Or ( ( ( o conclude ha a x, x, dx This or x and., so ( x, or all.4.4. niqueness and Sabiliy o Hea Equaion Soluion sing L orm For he wave equaion we measure how close wo uncions and g are on some inerval a [ < x < b] (eiher a or b can be ± using he supremum norm g SP a< x< b ( x g( x (we can also al abou he size or norm o a single uncion, as SP ( x a < x< b. There s anoher common way o measure he disance beween wo uncions on an inerval (a, b, namely b ( ( x a dx (69 The laer quaniy is called hel norm, so he disance beween wo uncions can be measured as he L norm o hen heir dierence. I s easy o see ha i mus be he zero uncion, and g implies g. The norm L is very much lie he usual Pyhagorean norm or vecors rom linear algebra (or us Calc I, which or a vecor X < x, x,..., xn > is deined as 4. Resuls and Discussion Problem X n ( x (7 When emperaures are speciied on boundary x and x respecively he resuls a various levels o ime are compued and illusraed in he graphs below or odes. Figure a: Approximaed and analyical lux on boundary x b a ( ( ( g x g x dx (69 We can also measure he norm o a uncion wih Figure b: Approximaed and analyical lux on boundary x

12 66 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod Problem When luxes are speciied on boh boundaries he resuls a various levels o ime ae compued and illusraed below. Figure b: Approximaed and analyical emperaures on he boundary x Figure a: Approximaed and analyical emperaures on he boundary x Figure b: Approximaed and analyical emperaures on he boundary x Figure 4a: Approximaed and analyical emperaures on he boundary x Problem When lux and emperaures are speciied on eiher boundary he resuls a various levels o ime ae compued and illusraed below. Figure 4b: Approximaed and analyical lux on he boundary x Figure a: Approximaed and analyical lux on he boundary x

13 American Journal o Applied Mahemaics ; (4: Table : Approximaed and analyical resuls or Problem Flux on boundary x Flux on BC x Table : Approximaed and analyical resuls or Problem Temperaure on boundary x Temperaure on boundary x I II III I II III I II III I II III I: Speciied values, II: Approximaed values, III: Analyical values. I: Speciied values, II: Approximaed values, III: Analyical values.

14 68 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod Table : Approximaed and analyical resuls or Problem Hea Flux on boundary x Temperaure on boundary x I II III I II III I: Speciied values, II: Approximaed values, III: Analyical values. Table 4: Approximaed and analyical resuls or Problem 4 Temperaure on boundary x Hea Flux on boundary x I II III I II III I: Speciied values, II: Approximaed values, III: Analyical values. We have analyzed Dirichle, eumann and Robin condiions in hea ranser problems (, ( and (, respecively and whose resuls are illusraed in he igures namely Figure (a and (b, Figure (a and (b, Figure (a and (b,a Figure 4(a and (b and respecively. These are he emperaure inside he domain a he nodes x,.5,.5 or ( (

15 American Journal o Applied Mahemaics ; (4: Problems is u (.5,.5.9, Problems is u (.5,.5.57, Problems is u (.5,.5.4, and Problems 4 u (.5,.5.489, respecively u.5,.5.5. From igure a lux is zero and he approximaed lux increases hen lucuaes as ime goes by. Aer some ime he approximaed lux increases and sar o converge owards analyical lux. From igure b lux on boundary x is wo and he approximaed values decrease a he beginning hen increases as ime increases o converge a he analyical soluion which is he equilibrium poin. Considering emperaure inside he domain a he node (5,.5 he approximaed emperaure, u.9 is lower compared o he analyical one u.5.comparing he approximaed and analyical values he dierence is small, hence BEM is accurae mehod hrough he use o Dirichle boundary condiions. In igure (a and igure (b, emperaure varies direcly proporional o ime indicaing energy ranser rom boundary x o x.sing he domain a he node (5, 5 he approximae emperaure, u.57 and analyical emperaure, u.5 is he same. Since he analyical and approximaed values agree hen BEM is eecive mehod using eumann boundary condiions. From igure a Flux is zero and he approximaed values increases and lucuaes a he beginning hen as imes moves he approximaed values increases and ends o diverge away rom he analyical lux. From igure b emperaure increases as ime increases on boundary x. Considering he emperaure inside he domain a he node (5,.5 he approximaed emperaure, u.4 is exremely lower compared o he analyical emperaure, u.5. The dierence beween he analyical and approximaed values is small showing he success o BEM Mehod hrough he use o Robin boundary condiions. In igure 4a on boundary x emperaure is direcly proporional o ime. From igure 4b on boundary x analyical hea lux is wo and he approximaed values decreases a he beginning he increases a ime increases and converges o he analyical lux. Considering he emperaure inside he domain a he node (.5,.5 he approximaed emperaure, u.489 is higher compared o he analyical emperaure, u.5. Comparing he analyical and he approximaed values, he dierence is very small and BEM is accurae mehod hough he use o Robin boundary condiions when he boundaries are inerchanged. Whereas he analyical soluion is ( 4.. Conclusion and Recommendaions 4... Conclusion This sudy analyzed one dimensional emperaure disribuion in a meallic bar o consan hermal conduciviy. Wih help o governing equaions in chaper wo undamenal soluions or hea equaion and BIE were successully derived. The approximaed emperaures and luxes on he boundaries converged wih exac showing he accuracy o BEM mehod Recommendaions Fuure wor can be direced o problems wih variable hermal conduciviy. Acnowledgemens The compleion o his research could no have been possible wihou he guidance, assisance, encouragemen and cooperaion o many people, whom I owe graiude and recogniion. Firs and oremos, I han he Almighy God or graning me good healh, srengh and guidance hroughou he sudy. I sincerely express my proound respec and graiude o my supervisors, Pro. Jacson Kwanza and Dr. Thomas Onyango or heir wisdom, proessional compeence, inellecual generosiy and commimen owards he compleion o his proec. I sincerely han he head o he deparmen, Dr. Karoi and all lecurers in he deparmen o Mahemaics and compuer science or heir organized eaching and or sharing heir ime and alens wih me. In special way, my respec goes o Dr. D.M Theuri, icholas Muhama and Dr. J.A. Oello all o JKAT or heir service o humaniy hrough eaching. I wish o exend my hans o he Dean o he aculy o Science and all he enire CEA communiy members or heir availabiliy, developmen and disseminaion o nowledge. Special hans go o my amily or heir paience, moral suppor and encouragemen hey provided during my course. omenclaure A q h T x q s e S L Area, > Hea ranser rae, W Convecional hea ranser coeicien, W/> K Thermal conduciviy, W/MK Temperaure, K Ininiy symbol Del operaor Dimensional ime, s Lengh variable, M Hea lux, K/s Surace Emissiviy Seen Bolzmann s consan Lengh o meallic bar, M, V Arbirary scalar ields a, b Disance, M H Heaviside uncion

16 7 Virginia Mwelu Kieu e al.: Deerminaion o One Dimensional Temperaure Disribuion in Meallic Bar sing Green s Funcion Mehod µ Closed Boundary ρ δ π div F Densiy, Kg/ Dela Pi Funcion Divergence Vecor ield Abbreviaions BEM BIE FEM SI FDM ODE PDE Reerences Enclosed region Ouward uni vecor normal o µ Boundary Elemen Mehod Boundary Inegral Equaion Finie Elemen Mehod Inernaional Sysem o ni Finie Dierence Mehod Ordinary Dierenial Equaions Parial Dierenial Equaions [] Barels, R. C., and Churchil, R. V. (94. Resoluion o Boundary Problems by se o a Generalized Convoluion. London: Bull Amer. Vol 48, pg [] Bec, J. V. (984. Green s Funcion Soluion or Transien Hea Conducion Problems. Inernaional Journal o Hea and Mass Transer, Vol 7, pg [] Brebia C. A. (984. The BEM or Engineers. London: Penech Press. [4] Chang, Y. P. and Tsou, R. C. (977. Hea Equaion in an anisoropic medium Homogenous in Cylindrical Regions-nseady Sae. Journal o Hea Transer, Vol 99, pg [5] Cannon K. and John R. (984. The One Dimensional Hea Equaion, s ed. London: Menlo Par. [6] Cooper M. and Jeery. (998. Inroducion o Parial Dierenial Equaions wih MATLAB. London: Academic press [7] Eduardo, D. G. (. Green s Funcions and umbering Sysem or Transien Hea Conducion. Journal on applied Mahemaics, pg [8] Greenberg, M.D. (986. Applicaion o Green s Funcions in Science and Engineering: ew Jersey, Prenice-Hall, Inc. Ediors. [9] James, M. H. and Jerey,. D. (987. Hea Conducion. Mine Ola: Blacwell Scieniic Publicaions, pg [] Misawo F. (.A soluion o One Dimensional Transien Hea Transer Problem by Boundary Elemen Mehod. Cuea. [] Onyango, T.M., Ingham, D.B. and Lensic, D.M. (8. Resoring Boundary Condiions in Hea Transer. Journal o engineering mahemaics, Vol 6,pg 85-. [] Ozisi,. M. (968. Boundary Value Problems o Hea Conducion. London: Consable and Company Ld, pg 55. [] Piis D. and Sissom L. (4. Hea Transer, nd ed. ewyor: Taa McGraw-Hill, pg -. [4] Praproni M., Ser R. and Trobe K. (. A new explici numerical scheme or nonlinear diusion problems, Parallel numerics: heory and applicaions, Joze Sean Insiue and niversiy o Salzburg. [5] Sephenson G. and Ardmore, M. P. (99. Advanced Mahemaical Mehods or Engineering and Science sudens.london: Cambridge niversiy Press, pg 9 [6] Venaaraman., Peres E. and Delgado I. (.Temperaure Disribuion in Space Mouning Plaes wih Discree Hea Generaion Source Due o Conducive Hea Transer. SA: Aca Asronauic, Vol, pg 9. [7] Viun L. (4.An inroducion o he BEM and Applicaions in modeling composie maerials. Research paper. [8] Weissein M. and Eric W. (999. Hea Conducion Equaion. Journal o Applied Mahemaics, Vol, pg -.