Analytical solution of bioheat transfer equation with variable thermal conductivity in skin

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1 Irol Jourl Of Egrg Rrch Ad Dvlopm -ISSN: 78-67X, p-issn: 78-8X, Volum 4, Iu (Dcmr 7, PP.-7 Alycl oluo of oh rfr uo wh vrl hrml coducvy Rjh Kumr, Al K. Vhh d Su Ghg,, Dprm of Mhmc, Kuruhr Uvry, Kuruhr - 69 Hry, Id Corrpodg Auhor: Rjh Kumr, ABSTRACT: Tmprur dpd hrml propr of u mpor fcor o chv rlc modl hrml rm. Tu mprur dpd hrml coducvy ud o modl h P oh rfr uo wh o rlxo m. Thr dffr m-dpd urfc h flux, mly, couou, prodc d rmp yp r ppld o urfc whou h lo h oom. Lplc rform mhod ppld o olv h prolm d h gfc ffc of vrl coducvy prmr orvd o h u mprur druo. Kyword: Boh rfr; S u; Vrl hrml Coducvy; Lplc rform D of Sumo: 8--7 D of ccpc: I. INTRODUCTION Th rfr of h mly h coduco proc coupld o complcd phyologcl proc. Th hrml propr of vry cro dffr lyr; v wh h m lyr. Th propr of r flucd y mprur, dmg, prur, d g, c. A vry of hrml mhod hv dvlopd d ppld o h rm of d or jury volv hr rg or lowrg of mprur rgd r o ll or hrmlly durz croc cll. P modl [] wdly ud o modl uch prolm du o mplcy. Thrml wv modl w propod y Co [] d Vro [] o cpur h ffc of mcrorucurl rco h flux. Vrou ud hv do o cur d volvg u whou ffcg h urroudg hlhy u. Xu l. [4] d Rom l. [5] rvwd h ohrmomchcl hvor d mprur dpdc of lcrcl d hrml propr of u rpcvly. Poor l. [6] udd mprur rpo of u du o m-dpd urfc h flux. Lr, Tzou [7-8] roducd dul-ph-lg (DPL modl of h coduco. Arzdh l. [9] ulzd DPL modl rg h r h rfr prolm u. Lu l. [-] mployd h DPL modl o lyz h h rfr prolm hyprhrm rm, d lmd h coc hory dcovrd h lrur. D l. [] md h r umor chrcrc wh ow urfc mprur y ug f volum mhod. Agrwl l. [4] propod f lm modl o udy mprur druo d dp u of llpcl prd hp hum lm. I hrml lo rm. Kumr l. [5] vgd h hrml hvor lvg u ug m frcol dul-ph-lg oh rfr (DPLBH modl ujcd o drchl oudry codo durg hrml hrpy. Sho l. [6] dvlopd Rdofrucy lo (RFA mhmcl modl o udy h fluc of jcd oprcl rrgulrly hpd lvr umor. I wll lhd h phycl propry of grg d o-phycl mrl vry gfcly wh mprur. Th mprur dpdc of mrl propr l hrml coducvy d pcfc h c ffc ohrmomchcl hvor of vrou mrl. Thrfor, o chv ffc oluo of mprur chg prolm, h mprur dpdc of mrl propr hould o ccou, whch rur ddol modlg, xprmo d compuol ffor. A comprhv wor h do y vrou rrchr for vrl hrml coducvy couum mchc, ol mog hm r [7-8]. L l. [9-] udd h r rpo for hlf-pc wh vrl hrml coducvy d dffuvy h cox of h grlzd hrmolc dffuo hory. I p of ll h ud o wor h do o vg h ffc of vrl coducvy o P oh rfr uo. I h wor, h lycl oluo of h hrml wv modl of oh rfr wh mprur dpd vrl hrml coducvy od. Th rul how h h u hrml coducvy prmr ffc h u mprur druo. T u mprur, mprur, C h flux, C T l mprur of ody, Wm c pcfc h of u, C T rrl lood Jg K c pcfc h of lood,

2 Jg K vrl hrml coducvy of u, vrl coducvy prmr cd h flux mplud ( mplud u dy, gm lood m dy, ourc du o molc h gro h u, u hrml dffuvy, Wm K co hrml coducvy of u Wm dmol cd h flux gm lood prfuo r, m h Wm m, h flux rlxo m, m Lplc dom prmr l romwch coour gro l L u l lgh, m x coord vrl, m x dmol coord dmol u mprur dmol h flux rlxo m H Hvd fuco Drc dl fuco dmol h flux frucy dmol duro m of pul r h flux II. FORMULATION OF THE PROBLEM A o dmol f lgh u codrd who oom oudry uld. Thr dffr yp h flux, mly, ou, prodc d pul r r ppld o urfc. O dmol P oh uo c( = c ( T. ( = (,( ( h hrml coducvy wh vrl mprur uch h ( c ( =, ( Euo ( wh h u of uo ( d uo ( yld, h followg form of P uo ( ( ( = [ c ( T m]. (4 I c of hrml wv modl ( = (. (5 Ug uo (5, uo (4 gv ( ( [ c ( T Il codo T( x, =, T ( x, =. (7 m m. (6 ] = [ ( ] Boudry codo T T(, = f (, (, =. (8 For h hory of h d m rfr lvg u o of h crl u how o cr good modl h dcr h rpor phom. A mpor fcor o chv rlc modl h u of mhmcl fuco o dcr h mprur dpdc of hrml propr of u. A commoly ud pproch for modlg h mprur dpdc of hrml d lcrcl propr for mprur low C d o lr uo d mploy co mprur coffc uch : = (. (9 ( T Codrg h mppg (Krchhoff rformo [] T = ( d, ( wh = T T. ( Afr og, h u mprur T c od y olvg uo ( whch gv T =. (

3 E. (6 wh h u of E. (9 d E. (, yld h followg uo lr form ( ( [ c ( m] =. x III. SOLUTION OF THE PROBLEM Iroducg h dmol vrl d mlry crr x T T x =, = T, =, =, =, L T T T Wc m =, =, =, W =, =, (4 cl = Ug h o-dmol vrl d rmovg h dh, uo ( rduc o h followg form ( [ ] =. (5 wh l codo ( x, =, ( x, =, (6 d oudry codo whr C C C (, = f ( f (, (, =, (7 f ( ow fuco of dmol follow, f ( = (, (8 f ( = f, (9 ( = H ( H (. ( Applyg Lplc rform chu, uo (5 com, d =, ( dx whr = (, ( I Lplc dom, oudry codo r (, = VR, (, =, ( whr F ( Lplc rform of f ( d V R = F ( F(. (4 Soluo of uo (, y ug h oudry codo ( coh( ( x ( x, = ( VR. (5 coh( Ivr Lplc rform of ( x, c od from h followg Bromwch coour gro [] l ( x, = lm ( x, d. l l Ug h vro horm, h followg vr Lplc rform of uo (5 od for C l co( ( x = I m, (7 ( C m= = (6

4 coh( ( x co( ( x m ( x, = I coh( m= = ( m ( coh( ( x h( ( x ( x h( coh( ( x ( coh( coh( coh ( (8 C For <, for whr ( x, ( x, = ( = I d m h( coh( ( x coh( m= = coh( coh( ( x ( ( x m 4 h( m= = co( ( x ( m m coh( [ coh( coh( ( x 4 ], coh ( m ( x co( ( x = m m I ( m= = m ( ( co( ( x m m m (, ( m= = = r roo of uo ( m v, =,... m ( m v =. ( Ohr co r gv uo ( I l 4 coh( ( x = ( coh( =, = ( (, =, = =. m, m= = co( ( x ( m m, ( (9 IV. NUMERICAL RESULTS AND DISCUSSION Th followg pcfd vlu of rlv prmr r codrd mlr o h prmr gv lrur [-4]. Th hrml phycl propr of u r = gm, = gm, =.68 Wm K, c = 487 Jg K, c = 487 Jg K, =.87, m =.9 Wm, T = 7 C, T = 7 C, L =.5m, = 5 Wm, = 6. Th udy do for lmg c. Fgur dpc h u mprur rpo wh dffr vlu of vrl hrml coducvy prmr for h c. I clrly how h h vlu of gfcly ffcg h u mprur rpo. Tmprur cr for < <. d dcr for. < <.4 fr h com co. Fgur dpc h u mprur rpo wh dffr vlu of vrl hrml coducvy prmr for h c.. prodc h flux wh =. I ocd h h vlu of cr mplud of 4

5 mprur profl cr. Fgur dpc h u mprur rpo wh dffr vlu of vrl hrml coducvy prmr for h c.. pul r h flux. Fgur 4 dpc h u mprur rpo wh dmol dc for c d how h mprur profl ffcd y h prmr. Fgur 5 dpc h u mprur rpo ujcd o prodc h flux hr dffr dph udr h urfc wh = d =.5. I ocd h ocllory mplud com low for lrg dph from h hg. Fgur : Tmprur rpo wh m for h c wh dffr vlu of vrl coducvy Fgur : Tmprur rpo wh m for h c wh dffr vlu of vrl coducvy Fgur : Tmprur rpo wh m for h c wh dffr vlu of vrl coducvy 5

6 Fgur 4: Tmprur rpo wh dmol dc for h c wh dffr vlu of vrl coducvy Fgur 5: Tmprur rpo wh m dffr u dph for =.5 V. CONCLUSIONS Th lc of formo o mprur-dpd hrml propr prv u from ccurly prdcg h mprur druo of h rg u udrgog hrmohrpy. Thrfor, cry o codr mprur-dpd hrml propr o chv rlc modl of hrml hrp ordr o gr vld rul. I h udy, hrml wv modl of oh rfr uo modld wh vrl coducvy of u o chv rlc modl. Th prolm olvd y ug h Lplc rform mhod d ujcd o hr dffr oudry codo, ou, prodc d pul r. Tu mprur od for ll h hr c d orvd h u hrml hvor gfcly ffcd y h vrl hrml coducvy prmr. ACKNOWLEDGMENTS Th hrd uhor hful o h Coucl of Scfc d Idurl Rrch (CSIR, Id, for provdg fllowhp udr h JRF(/6/5/(EU-V chm. REFERENCES []. H.H. P, "Aly of u d rrl lood mprur h rg forrm," J Appl Phyol, vol.,pp. 9-, 948. []. C. Co, "A form of h coduco uo whch lm h prdox of ou propgo," Comp Rd, vol. 47, pp. 4-4, 959. []. P. Vro, "L prdox d l hor cou d l uo d l chlur," Comp. Rd., vol. 46, pp , 958. [4]. F. Xu, T.J. d K.A. Sff, "Bohrmomchcl hvor of u," Ac Mch S, vol. 4, pp. -, 8. [5]. C. Rom d D. Hmmrch, "Rvw of mprur dpdc of hrml propr, dlcrc propr, d prfuo of ologcl u hyprhrmc d lo mprur," Cr Rv Bomd Eg, vol. 4, pp , 4. [6]. H.Z. Poor, H. Moov d A. Mord, "Aly of h DPL o-h rfr uo wh co d m-dpd h flux codo o urfc," J Thrm Sc, dx.do.org/.98/tsci4857z, ISSN: [7]. D.Y. Tzou, "A ufd fld pproch for h coduco from mcro-o mcrocl", ASME J H Trf, vol. 7, pp. 8 6, 995. [8]. D.Y. Tzou, Mcro-o-Mcrocl H Trfr: h Lggg Bhvor. Tylor d Frc, Whgo, DC, 996. [9]. H. Arzdh d H. Ahmd, "Alycl ly of h dul-ph-lg modl of oh rfr uo durg r hg of u," H M Trf, vol. 5, pp , 4. []. K.C. Lu, Y. Wg d Y. Ch, "Ivgo of h oh rfr wh dul ph lg ffc," I J Thrm Sc, vol. 58, pp. 9-5,. []. K. Lu, d H. Ch, "Aly for h dul-ph-lg o-h rfr durg mgc hyprhrm rm," I J H M Trf, vol. 5, pp. 85-9, 9. []. K. Lu, H. Ch d P. Chg, "Ivr vgo of o-fourr h coduco u," J Thrm Bol, vol. 6, pp. -8, 6. []. K. D d Suh C Mhr, "Emo of umor chrcrc r u wh ow urfc mprur," J Thrm Bol, vol. 8, pp. -8,. 6

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