Study in Cylindrical Coordinates of the Heat Transfer Through a Tow Material-Thermal Impedance

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1 Research ural f Applied Scieces, Egieerig ad echlgy (): 9-63, 3 ISSN: 4-749; e-issn: Maxwell Scietific Orgaiati, 3 Submitted: uly 4, Accepted: September 8, Published: May, 3 Study i Cylidrical Crdiates f the Heat rasfer hrugh a w Material-hermal Impedace A. Diuf, I. Diage, M.S. Oul rahim, M.L. Sw, F. Niag ad G. Sissk Faculty f Sciece ad echlgy, Uiversity Cheikh Ata Dip i Dakar, P, Dakar-Fa, Seegal Uiversity Istitute f echlgy IU/U, Uiversity f hies, Seegal Abstract: We prpse i this study a characteriati f the temperature ad f the heat flux desity withi a material tw dispsed i cylidrical. he study is carried uder dyamic frequecy regime. y thermal-electrical aalgy, the thermal impedace f the material is determied. he prfiles f temperature curves, f heat flw desity ad thermal impedace allw us t appreciate the tw material behavir i isulati. he tw is made f plat fibers (sisal, ute, hemp). Keywrds: w - dyamic frequecy regime - hermal Impedace INRODUCION Feret mdels fr the study f heat trasfer (fiite medium r semi ifiite) are prpsed fr the characteriati f materials (Vumb et al., a). Measuremet techiques allw us t determie the therm physical parameters (thermal cductivity, thermal effusively, thermal diffusivity cefficiet...) (Vumb et al., b; Meukam et al., 4). We prpse a study i cylidrical crdiates f heat trasfer i a material f cylidrical shape, f fiite dimesis with axial symmetry i dyamic frequecy regime. he cylidrical system islated such as water pipes simple r caxial ustifies the purpse f this study. he prfiles f the curves f temperature ad heat flux desity allw t highlight the behavir f heat thrugh the material tw used as thermal isulati. Frm the electrical aalgy - hermal (at et al., 9) the thermal impedace f the material is determied i cylidrical crdiates. MAHEMAICAL MODELING Diagram f the study: Figure shws the tw material i cylidrical frm. It is subected at its base surfaces t stresses climate where the temperature varies i dyamic frequecy regime. he cefficiets f heat exchage at its faces ad the side face are attached. a ad a are the temperatures evirmetal, impsed i the directi () respectively at the frt ad rear, uder dyamic frequecy regime. r R O ; H ; R,m ; H,3m Fig. : Schematic f study desig C: maximum temperature f the evirmet he material is assumed t have a iitial temperature i C ω Excitati frequecy f temperature evirmetal Expressi f the temperature: Withut surce ad heat sik, the heat equati is expressed: Avec: α t λ α () ρ.c I cylidrical crdiates with axial symmetry, the temperature is t depedet the crdiateθ, the differetial equati is: Crrespdig Authr: G. Sissk, Faculty f Sciece ad echlgy, Uiversity Cheikh Ata Dip i Dakar, P, Dakar-Fa, Seegal 9

2 r + r r Z + α t able : Eige values 7, 8,, 4, 8 μμ 66, 7 8, 3 38, 3, 6 6 he sluti f Eq. uder dyamic frequecy regime, prvided the cditis impsed i Fig. ad applyig the methd f separati f variables leads t the sluti give by Eq.(4): ( r, C. r) A + ) ) e (4) 4 () () f( ) g( ) ( ) ( ) λ hr With -. r) k ( r. ) ( ).( r. ) k m m 4. m! m! m (.r) is a essel fucti with K y applyig the fllwig budary cditis: λ h Z λ h H λ hr r R [ ( r,, ] [ ( r, H, ] ( R, a a () (6) (7) (8) We btai expressis f the cstats A ad frm Eq. (6) ad (7): h + λh A λβ h h λ + λ + h h [ λβ. cch. + h. ] h λβ h h λ + λ. r).. r).. r).. r) h. λβ h... r).. r).. r). ( ) ( ). r h β. H Equati (8) pssible t btai the cstat C (9) () -4 Fig. : Graphical determiati f the Eige values ( ) λ ( ) hrr () R. (3) Equati () gives the Eige values (r ) we determied graphically frm Fig. : We get a few Eige values shw i able. At, we btai: ( r, C ( r), (4) Of the rmaliati cditi we btai C: C C R r r dr R r r dr R R R) [ R) + R) ] () (6) R Expressi f the heat flux desity: he vectr flux desity f heat cducti is give by the fllwig expressi: λ grad (7) r + (8) λ R ) hr. R ) (). 6 With

3 r Vectr desity f heat flw i the directi (Or) Vectr desity f heat flw i the directi (O) he desity f heat flw i bth directis (Gld) ad (O) are give respectively by: r r r φ. u (9) φ. u () he heat flux desity is btaied by calculatig glbal mdule: [ ] ( r, φ ( r, φ ( r, φ + () emperature mdule ( C) () (m) Fig. 3: Evluti f temperature i the tw material i fucti f depth Ifluece f excitati r 3/3m, h 3 w/ (m/ C), h w/ (m/ C), hr -3 w/ (m/ C); theure. ω 3 rad /s; () ω - rad/s; ω rad/s φ( r, + λ λ t[. r ). ). ) iω. C. e A + C. e β. A β.. r). ). ) } ] () emperature mdule ( C) Expressi f the thermal impedace: he thermalelectrical aalgy (Ould rahim et al., ) is used t defie the thermal impedace frm the relatiship: Z eq.φ Z eq hermal impedace f the material φ Heat flux desity thrugh the material emperature variati betwee the frt ad rear faces We btai: C. r). ). ) C e A e Z (4) φ RESULS AND DISCUSSION Evluti f temperature-fllwig the directi (OZ): Figure 3 shws the curves f temperature chages i the directi () i the tw material. We High light the ifluece f excitati fréquece. he curves i Fig. 3 shw the same prfile. At the frt, m, the material absrbs heat ad warms up (temperature rise). At iside the material, temperature decreases. whe depth icreases ad teds t disappear.. () r (m) Fig. 4: Evluti f temperature i the material as a fucti f r. Ifluece f excitati frequecy m; h 3 w/ (m/ C); h w/ (m/ C); hr 3 w/ (m/ C); t heure; ω 3 rad/s ω rad/s; () ω rad/s whe the thickess f the material becmes sigificat. he decrease i temperature is due t heat reteti by the material. Mre tha the strage capacity f eergy by the material is imprtat lw layer, the material is gd thermal isulatr. he cmparis f curves i Fig. 3 shws that the heat trasmitted t tw material is eve mre imprtat as the excitati frequecy is lw. Fllwig the radial directi (Or): he curves i Fig. 4 shw the evluti f temperature i a radial directi (Or) ctaied i the plae f the circular base. he curves highlight the ifluece f excitati frequecy. he curves i Fig. 4 shw the same prfile. he maximum temperature mdule at r, has a lw variati fr r <. m ad the decreases sigificatly i apprachig the lateral side. he heat trasfer cefficiet is assumed t virtually er the lateral side, the heat flw is impsed alg 6

4 Heat flux desity (W/m ) 4 3 () (m) Fig. : Evluti f the heat flux desity i the material as a fucti f depth. Ifluece f excitati frequecy, r m, h 3 W/(m/ C), h W/ (m/ C), hr 3 W/ (m/ C); t heure/ ω 3 rad/s, () ω rad/s, ω rad/s the directi (O) ad we bserve lateral heat lsses ad heat strage. We bserve a drp i temperature f the rder f C relative t the temperature f the evirmet, fr a thickess f abut.3 m. he cmparis f the curves i Fig. 4 cfirms the bservatis made i Fig. 3. Evluti f the heat flux desity: Fllwig the directi (O): he Fig. shws the evluti f the desity f heat flw thrugh the tw material. he ifluece f the excitati pulse f the exteral evirmet is emphasied. he curves i Fig. shw a verall reducti f the heat flux desity whe passig frm the frt,, withi the material. Fr a thickess f abut.3 m ad i the study cditis, the tw material retais a sigificat amut f heat reflectig a gd perfrmace i thermal isulati. he cmparis f the curves i Fig. shws that withi the material, the heat reteti is mre imprtat tha the excitati frequecy is imprtat. Fllwig the radial directi: I Fig. 6, we preset the evluti f the heat flux desity thrugh the material alg the radius f the base f the cylider. he ifluece f the excitati fréquece is Highlighted. he heat flux desity, maximum at r, has a lw variati fr r <. m ad the decreases sigificatly whe apprachig the side. Fr r>. m, the behavir f thermal isulati tw material is csiderable. hese curves als shw the imprtace f lateral heat lsses relative t the directi f heat flw impsed. he cmparis f curves i Fig. 6 shws that the heat reteti i tw material is eve mre imprtat that the frequecy excitati is high, curve ; te that the lw frequecy excitati allws ly lw heat exchages at the iterface f the material, curve. Heat flux desity (W/m ) 3. ()... r (m) Fig. 6: Evluti f the heat flux desity i the material i terms f r. Ifluece f excitati frequecy, r m, h 3 W/(m / C); h W/(m / C); hr - 3 W/ (m / C); t heure. ω 3 rad/s, () ω rad/s, ω rad s he themal impedace mdule ( C/m /W ) () Fig. 7: Evluti f thermal impedace mdule. Ifluece f psiti alg the radius r frequecy,3m; h 3 W/m / C - ; h W/ (m. C), hr 3 w/ (m. C); t heure. r m ; () r, m ; r, m the behavir f a thermal isulatig material. he curves f Fig. 7 shw the evluti f the thermal impedace f the tw material accrdig t the excitati frequecy f the temperature f the evirmet. Fr a frequecy bad 4 rad/s <ω<,. rad/s, the prfile f the curves d t deped the psiti r. he mdulus f the impedace is cstat fr lw frequecies, icreases fr itermediate frequecy ad reaches a plateau fr high frequecy. Fr a variati f temperature give betwee the tw faces f the material, the icrease i the mdulus f the impedace crrespds t a decrease i the flw f heat thrugh the material. his reducti i heat flw is partly due t the heat absrpti by the material ad the ther had t a lss f heat by the lateral face.. CONCLUSION Lg ( ω) hermal impedace f the tw material: he thermal impedace (Dieg et al., 7) is used t characterie 6 he study f heat trasfer i cylidrical crdiates has bee prpsed fr tw material characteriati.

5 Frm the mathematical mdel, the curves f chage f temperature, heat flux ad the dyamic impedace f the material have shw the ptimum rages f frequecies resultig i a high heat trasfer. he ifluece f heat trasfer cefficiet shws the eed t ctrl the exteral evirmet f material. ACKNOWLEDGMEN We thak all the authrs cited i refereces; their wrk we are a very big ctributi. We als thak the members f the iteratial grup f research reewable eergy f labratry f semicductrs ad slar eergy at the Uiversity Cheikh Ata Dip i Dakar. REFERENCES Dieg, A., L. Ould Habibulahy, A.S. Maiga, A. Dia ad G. Sissk, 7. Impedace spectrscpy methd applied t electrical parameters determiati bifacial silic slar cell uder magetic field.. Sci., 7: 48-. at, Y., A. Degivai ad G. Payet, 9. hermal cductivity measuremet f isulatig materials with a three layers device. It.. Heat Mass ras., : -. Meukam, P., Y. at, A. Numw ad.c. Kfa, 4. herm physical characteristics f ecmical buildig materials. Cstr. uild. Mater., 8: Ould rahim, M.S., I. Diage, S. amba, F. Niag ad G. Sissk,. Characteriati f the miimum effective layer f thermal isulati material twplaster frm the methd f thermal impedace. Res.. Appl. Sci. Eg. echl., 3(4): Vumb, M.L., A. Wareme, S. Gaye, M. Adi ad G. Sissk, a. Characteriati f the thermphysical prperties f kapk. Res.. Appl. Sci. Eg. echl., (): Vumb, M.L., A. Wareme ad G. Sissk, b. Characteriati f lcal isulatrs: Sawdust ad wl f kapk. Res.. Appl. Sci. Eg. echl., ():