Entropy ISSN

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1 Enropy 003, 5, Enropy ISSN Full paper CONVECIVE HEAING OF SOLID SURFACE: ENROPY GENERAION DUE O EMPERAURE FIELD AND HERMAL DISPLACEMEN Yagoub Nassar Al-Nassar Mehanial Engineering Deparmen, KFUPM, P.O. Bo 876, Dhahran 36, KSA. ynassar@kfupm.edu.sa Reeived: Oober 00 / Aeped: 6 July 003 / Published: 3 Deember 003 Absra Conveive heaing and ooling of he surfaes find appliaion in proess indusry. During he heaing or ooling yle of he proess, hermodynami irreversibiliy whih an be assoiaed wih he proess parameers ours. Moreover, hermodynami irreversibiliy assoiaed wih he heaing yle an be quanified hrough enropy analysis. In he presen sudy, onveive heaing of he solid surfae is onsidered. A mahemaial formulaion of he emperaure rise and hermal sress developmen during he ransien heaing proess is presened. Enropy generaion due o emperaure field and hermal displaemen is also formulaed. he simulaion for emperaure rise, hermal displaemen, and enropy generaion are arried ou for seel subsrae. I is found ha hermal displaemen does no ealy follow he emperaure disribuion inside he subsrae maerial. he magniude of enropy generaion due o emperaure field is onsiderably higher han ha orresponding o he hermal displaemen. Keywords: onveive heaing, hermal sress, hermal displaemen, enropy generaion

2 Enropy 003, Lis of Symbols A dimensionless parameer, ( / γ ) A ross seional area, m hermal wave speed, m s - C speifi hea, kj kg - K - E Young's modulus, Pa h hea ransfer oeffiien, W m - K - k hermal onduiviy, W m - K - P pressure, Pa s enropy, kj kg - K - s* dimensionless enropy, (s* s / C ) 0 ime, s emperaure, K iniial emperaure, K * dimensionless emperaure, (- 0 ) / ( θ- 0 ) spaial oordinae along ais, m u inernal energy, kj kg - hermal diffusiviy, m s - hermal epansion oeffiien, K - β ompressibiliy oeffiien, Pa - γ h / k, m - ε displaemen, m ε* dimensionless displaemen ε θ srain hea soure emperaure, K ν Poisson's raio speifi volume, m 3 kg - ρ densiy, kg m -3 σ sress, Pa Inroduion Conveive ooling of surfae finds wide appliaion in indusry. In his ase, ooling of he surfae (i) improves mehanial and meallurgial properies of he subsrae maerial, suh as quenhing, and (ii) enhanes he life yle of he equipmen or pars suh as ooling of eleroni omponens. Moreover, during onveive heaing or ooling proess, hermal sress is generaed in he surfae region of he

3 Enropy 003, subsrae maerial. Depending on he emperaure gradien in he region heaed onveively, sress levels above or around yielding limi of he subsrae maerial an our. Consequenly, invesigaion ino hermal sress levels during he onveive heaing/ooling beomes neessary. Moreover, he emperaure and sress fields are involved wih hermodynami irreversibiliy in he sysem. herefore, sudy ino hermodynami irreversibiliy assoiaed wih he emperaure and sress fields gives insigh ino he hermodynami heaing of he subsrae maerial. Considerable researh sudies were arried ou o invesigae he onveive ooling/heaing of he surfaes. Lumped apaiane analysis for he quenhing of he mealli subsrae was arried ou by Campo and Hernandez-Morales []. hey showed ha for onveive ooling, wo primary dimensionless groups eised, whih inluded solid fluid hermal onduiviy raio and modified Rayleigh number. he ona problems wih friional hea generaion were sudied by Kul'hys'kyi Zhyhailo and Evushenko [] for semi-bonded bodies of revoluion by aking ino aoun he effe of onveive ooling of he free surfae. hey deermined he ondiions under whih he influene of onveive hea ehange on he level and disribuion of ona sresses ould be negleed. Numerial analysis of parially heaed verial plaes in naural onveive ooling was arried ou by Campo e al [3]. hey obained he orrelaions beween he indued mass flow rae as well as he maimum wall emperaures, he Rayleigh number, and he eension raio. ransien wo-dimensional hermal analysis of eleroni pakages by he boundary elemen mehod was eamined by Guven e al [4 ]. hey indiaed ha he resuls apured he singular flu field arising form he mismahed in he hermal onduion oeffiiens and geomeri disoninuiy. hree-dimensional onveive ooling in a verial hannel wih flush-mouned hea soures was arried ou by Gunes and Laikopoulos [5]. hey presened emperaure and veloiy field disribuions as well as maimum emperaure, maimum veloiy, and loal Nussel number disribuions in erms of Grashof number. hermal haraerisis of bak ooled impingemen ooler for pakaging of high power semionduor laser diode array were sudied by Mirji and Ahmad [6]. hey showed ha eperimenal hermal resisane values losely mah wih heoreial alulaions. Considerable researh sudies were arried ou o eplore hermal sresses developed in solid bodies. he numerial simulaion of hermal sresses due o grinding was arried ou by Moulik e al [7]. hey indiaed ha he predied hermal sress disribuions were onsisen wih eperimenally measured residual sresses on ground surfae. Green's funion for hermal sress mied boundary value problem of an infinie plae was inrodued by Hann and Hasebe [8]. hey obained an eplii soluion inluding a hybergeomerial funion by eplaining he mapping ehnique and omple variable mehod. he hermal sress around a penny-shaped rak in a hermopiezoeleiri solid was sudied by Shang and Kuna [9]. hey presened ea epression for he sress-inensiy faor. hermal sress analysis and residual sresses in a hermoplasi omposie beam were sudied by Sayman e al [0]. hey showed ha for he same plasi disribuion, he inensiy of he ransverse displaemen was he

4 Enropy 003, larges for ha orienaion angle. Finie elemen modeling of sress developmen during deposiion of ion assised oaing was arried ou by Ward and Arnell []. hey indiaed ha he predied rends of mean sress and disribuion were similar o hose observed in he lieraure. hermal sresses in an aively ooled wo-piee rail sruure were invesigaed by Saapahy and Persad []. hey showed ha use of hin resisive overlays minimized he hermal sresses on he sruure. Irreversibiliy of he hermal sysem an be quanified by he enropy analysis. A seond law analysis of he opimum design and operaion of hermal energy sorage sysems were onsidered by Rober e al [3]. He showed ha an enire yle of sorage and removal should be onsidered when opimizing he design and performane of a hermal sysem. hermodynamis analysis of hermo-mehanial oupling in Couee flow was sudied by Demirel [4]. He indiaed ha sine he irreversibiliy was relaed o loss of energy, he hermodynami analysis ould failiae he opimal ondiions whih produed less enropy for an eising design. he seond law analysis in fundamenal onveive hea ransfer problems was arried ou by Mahmud and Fraser [5]. hey derived analyial epression for enropy generaion and Bejan numbers from veloiy and emperaure profiles. Minimizaion of enropy generaion by asymmeri onveive ooling was invesigaed by Ibanez e al [6]. hey showed ha when he Bio numbers were differen, he enropy funion displayed a minimum for speifi ondiions. Yilbas [7] formulaed enropy generaion in nonequilibrium energy ranspor sysems. He indiaed ha enropy generaion enhaned as nonequilibrium energy ranspor inreased aross he wo-nonequilibrium sub-sysems. Conveive ooling of he surfae and sress field due o emperaure field were formulaed previously [8]. However, he irreversibiliy assoiaed wih he sysem was lef obsured. Consequenly, he presen sudy is ondued o presen losed form soluion for emperaure and sress fields due o onveive heaing of he surfae and previously obained soluion for he emperaure and sress disribuions are enhaned. Enropy generaion due o emperaure and sress fields is formulaed and he magniude of enropy generaion is predied for various hea ransfer oeffiiens. Mahemaial Model Consider a semi-infinie solid body heaed by onveion hea ransfer a he surfae as shown in Figure. A seep emperaure gradien is generaed in he surfae region of subsrae maerial during his proess. he problem of hermal shok on solid surfae an be eamined in half spae. he losed form soluion of he unseady emperaure field (,) in he half spae akes he form, as given by Shuja and Yilbas [8 ],

5 Enropy 003, 5 47 Fluid Solid θ 0 a 0 Figure A shemai view of heaing siuaion. 0 + erf ep( γ γ ) erf + γ () θ 0 where o,θ,, and γ are he iniial surfae emperaure ( 0, 0 ), hea soure emperaure, hermal diffusiviy, and he raio of he hea ransfer oeffiien (h) o he hermal onduiviy (k), respeively. erf sands for he omplemenary error funion, whih is - erf. From he sress analysis of one dimensional hermoelasiiy problem, he hermal srain, ε, in a solid subsrae along he ais as a funion of he sress, σ, and emperaure, (- o ), an be wrien as ( + ν )( ν ) ( + ν ) ε σ + () ( ν ) E ( ν ) where ν, E, and are he Poisson's raio, elasi modulus, and hermal epansion of a solid maerial, respeively. Upon solving he equaion of moion referened o [8] he losed form soluion of he sress field for whih he surfae ( 0 ) is sress free as well as when is very large ( ) is given as

6 Enropy 003, 5 47 > < < 0 σ σ σ (3) where E o γ γ γ γ γ γ ν θ σ erf ) ep( / erf ep / erf ep ) ( (4) erf ep erf ep ) ( 0 E γ γ γ γ ν θ σ (5) where E ( - ν ) / [ ( + ν) ( - ν ) ρ ] is he veloiy of propagaion of elasi longiudinal wave and ρ is he densiy of he solid maerial. In he ase of enropy alulaions, enropy generaion in he solid is a funion of emperaure and volume hange due o hermal epansion, in his ase s s (, ), herefore one an wrie ds M d + N d (6) where d S N S M and By definiion: P P β and

7 Enropy 003, where, and β are hermal epansion oeffiien and ompressibiliy oeffiien, respeively. Considering he Helmholz funion, F u s and df du ds s d, and knowing ha du ds Pd, herefore df -s d Pds. Using he following Mawell relaion P s (7) one an reah o he relaion β s (8) Also, he relaion: du ds Pd redues o s u and knowing ha C u herefore C s (9) Now equaion (6) beomes β d d C ds (0) whih is a unique soluion for enropy generaion in solid obained from Mawell relaions. Looking a he seond erm in equaion (0) ε ε β ρ ε ε β β d A d A d

8 Enropy 003, where ( ds C, speifi volume ρ d dε ρ β ε and ε displaemen ). herefore () Equaion () an be wrien in non-dimensional form by inroduing he following non-dimensional parameers s * s / C A ρ C β d * dε * ds * A () * ε * Inegraing equaion () yields: * ε * s * ln( ) Aln( ) (3) * ε * o o he displaemen, ε, an be obained afer inegraion of equaion (), i. e.: ε ε d 0 A mahemaia sofware is used o ompue emperaure, displaemen, and enropy generaion during onveive heaing of he surfae. Resuls and Disussions Heaing of solid subsrae due o onveive boundary a he surfae is formulaed. he losed form soluion for he hermal sress field is presened. he enropy analysis due o emperaure and sress fields are arried ou o demonsrae he emperaure, displaemen, and enropy fields, seel is employed in he simulaion. Figure shows emperaure profiles inside he subsrae maerial for hree differen hea ransfer oeffiiens. he sharp inrease of he emperaure ours in he surfae region of he subsrae maerial, whih is pariularly rue for high hea ransfer oeffiiens ( h > 0 5 W/m K). emperaure deay along he deph appears o be almos linear despie he fa ha he heaing siuaion is ransien. his is beause of: (i) long period of heaing duraion, seady heaing siuaion is resuled, and (ii) low

9 Enropy 003, hea ransfer oeffiien ( h 0 7 W/m K); i. e., impulsive like heaing (shok) a he surfae is omied. he influene of hea ransfer oeffiien on he emperaure rise is onsiderable, provided ha high hea ransfers oeffiien (0 7 W/m K) resuls in rapid hange of emperaure gradien in he subsrae maerial, i.e. emperaure gradien redues wih progressing heaing duraion. 500 EMPERAURE ( o C) s 0 s 5 s 0 s h 0 7 W/m K E+00.0E E E E-03.0E-0 DISANCE (m) (a) EMPERAURE ( o C) s 0 s 5 s 0 s h 0 5 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (b) EMPERAURE ( o C) s 0 s 5 s 0 s h 0 3 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) () Figure emperaure inside he subsrae maerial for differen heaing periods for hree differen

10 Enropy 003, values of onveive hea ransfer oeffiiens, namely 03,05, and 07 W/m K. Figure 3 shows he hermal displaemen inside he subsrae maerial for differen heaing duraions. he displaemen is maimum a he surfae and almos zero a some deph below he surfae. he hermal displaemen does no follow ealy he emperaure profiles despie he fa ha he subsrae maerial is onsidered o be elasi. his is beause of he srain developed in he surfae region whih does no vary linearly along he deph like emperaure profiles, i.e. epansion of he subsrae maerial below he surfae is suppressed by he hermal srain developed in his region. As similar o emperaure behavior, inreasing hea ransfer oeffiien resuls in large magniude of displaemen pariularly a he surfae of subsrae maerial. he displaemen gradien aains slighly higher values a low hea ransfer oeffiiens as ompared o ha orresponding o high hea ransfer oeffiiens. his is due o he emperaure profiles, provided ha he variaion in emperaure gradien is larger han is ounerpar orresponding o he displaemen. DISPLACEMEN (m) 6.0E E E-05.5E E+00 h 0 7 W/m K 5 s 0 s 5 s 0 s 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (a) 5.0E-05 DISPLACEMEN (m) 4.0E E-05.0E-05.0E E+00 5 s 0 s 5 s 0 s h 0 5 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (b)

11 Enropy 003, E-05 DISPLACEMEN (m).e E E E E+00 h 0 3 W/m K 5 s 0 s 5 s 0 s 0.0E+00.0E E E E-03.0E-0 DISANCE (m) () Figure 3: Displaemen inside he subsrae maerial for differen heaing periods for hree differen values of onveive hea ransfer oeffiiens, namely 03,05, and 07 W/m K. Enropy analysis overs he enropy generaion due o emperaure field and hermal srain. However, hermal srain an be formulaed hrough hermal displaemen. Consequenly, equaion (3) is wrien in erms of emperaure field and hermal displaemen. herefore, enropy generaion due o emperaure field and hermal displaemen are segregaed. Figure 4 shows enropy generaion due o emperaure field inside he subsrae maerial for differen heaing periods. Enropy aains maimum value in he surfae viiniy and as he disane inreases away from he surfae i redues. he enropy profiles follow almos he emperaure profiles provided ha he slopes of boh urves differ. In he early heaing periods enropy produion is low and as he heaing duraion progresses, i inreases. Moreover, for high hea ransfer oeffiiens (h 0 5 W/m K), enropy produion inreases. his is beause of he hea ransfer rae o he solid maerial, whih inreases wih improving hea ransfer oeffiiens..88e+03 ENROPY (J/kgK).85E+03.83E+03.80E+03.78E+03.75E+03 5 s 0 s 5 s 0 s h 0 7 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (a)

12 Enropy 003, E+03 ENROPY (J/kgK).85E+03.83E+03.80E+03.78E+03.75E+03 5 s 0 s 5 s 0 s h 0 5 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (b).70e+03 ENROPY (J/kgK).68E+03.65E+03.63E+03.60E+03 5 s 0 s 5 s 0 s h 0 3 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) () Figure 4:Enropy generaion due o emperaure field inside he subsrae for hree differen values of onveive hea ransfer oeffiiens, namely 03,05, and 07 W/m K. Figure 5 shows enropy generaion due o he sress field inside he solid subsrae for differen heaing periods. Enropy generaion aains high values in he surfae region and i redues wih inreasing deph below he surfae. Enropy profiles follow almos he displaemen profiles, provided ha heir magniudes are differen. Moreover, enropy generaion inreases wih progressive heaing periods. his ours beause of he hermal srain, whih enhanes wih progressive heaing period. Enropy generaion also inreases wih inreasing hea ransfer oeffiien. In his ase inreasing hea ransfer raes a he surfae enhanes he displaemen in he surfae region; onsequenly, enropy generaion inreases in his region. When omparing figures 4 and 5, i an be observed ha enropy generaion due o emperaure field aains onsiderably higher values in he surfae region as ompared o is ounerpar orresponding o sress field. his indiaes ha enropy generaed due o inernal energy gain of he subsrae maerial dominaes over he mehanial work done during he displaemen of he surfae.

13 Enropy 003, E-03 ENROPY (J/kgK).70E-03.80E E E+00 5 s 0 s 5 s 0 s h 0 7 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (a) 3.60E-03 ENROPY (J/kgK).70E-03.80E E E+00 5 s 0 s 5 s 0 s h 0 5 W/m K 0.0E+00.0E E E E-03.0E-0 DISANCE (m) (b) ENROPY (J/kgK) 8.00E E E-04.00E-04 h 0 3 W/m K 5 s 0 s 5 s 0 s 0.00E E+00.0E E E E-03.0E-0 DISANCE (m) () Figure 5:Enropy generaion due o sress field inside he subsrae for hree differen values of onveive hea ransfer oeffiiens, namely 03,05, and 07 W/m K.

14 Enropy 003, Conlusion emperaure and sress fields due o onveive heaing of he solid subsrae a surfae are formulaed analyially. he irreversibiliy assoiaed wih he hermal sysem is quanified hrough he enropy generaion. Seel is employed o simulae he emperaure field and hermal displaemen during he onveive heaing proess. I is found ha emperaure gradien is high in he surfae region of he subsrae maerial in he early heaing period, whih is more pronouned a high ransfer oeffiiens. hermal displaemen does no follow he emperaure profiles, whih is due o hermal srain developed in he region lose o he surfae. In his ase, hermal srain varies wih emperaure gradien. hermal displaemen inreases when boh hea ransfer oeffiien and heaing period inrease. Enropy generaed due o emperaure field follows almos emperaure profiles, provided ha he magniudes of he slopes of boh urves are differen. Enropy generaion in surfae viiniy is higher han ha orresponding o some deph below he surfae. Enropy generaion due o sress field follows hermal displaemen profiles, provided ha he magniudes of boh urves differ. When omparing enropy generaion due o emperaure and sress fields, i an be observed ha enropy generaion due o emperaure field is onsiderably higher han ha orresponding o sress field. his indiaes ha inernal energy gain of he subsrae maerial dominaes over he mehanial work done due o hermal displaemen of he surfae during he heaing proess. Aknowledgmens he auhors aknowledge he suppor of he King Fahd Universiy of Peroleum and Minerals, Dhahran, Saudi Arabia for his work. Referenes. Campo A. and Hernandez-Morales B., Lumped-apaiane analysis for he quenhing of workpiees: Limiaions in erms of Rayleigh numbers and solid/fluid hermal onduiviy raios, Journal of Maerials Proessing & Manufauring Siene, 997, 5(4): Kul'hys'kyi-Zhyhailo R. D. and Evushenko O. O., Conveive ooling in ona problems of hermoelasiiy wih friional hea generaion, Maerials Siene, 997, 33 (6): Campo A., Mana O., and Morrone B., Numerial analysis of parially heaed verial parallel plaes in naural onveive ooling, Numerial Hea ransfer Par A-Appliaions, 999, 36(): Guven I., Chan C.L., and Madeni E., ransien wo-dimensional analysis of eleroni pakages by he boundary elemen mehod, IEEE ransaions on Advaned Pakaging, 999 (3):

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