ANALYSIS OF THE DUAL PHASE LAG BIO-HEAT TRANSFER EQUATION WITH CONSTANT AND TIME-DEPENDENT HEAT FLUX CONDITIONS ON SKIN SURFACE

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1 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp ANALYSIS OF THE DUAL PHASE LAG BIO-HEAT TRANSFER EQUATION WITH CONSTANT AND TIME-DEPENDENT HEAT FLUX CONDITIONS ON SKIN SURFACE by Hamed ZIAEI POOR a, Hassan MOOSAVI a, and Amr MORADI b*, a Deparmen of Mechancal Engneerng, Isfahan Unversy of Technology, Isfahan, Iran b Deparmen of Mechancal Engneerng, Bu-Al Sna Unversy, Hamadan, Iran Orgnal scenfc paper DOI: 1.98/TSCI141857Z Inroducon Ths arcle focuses on emperaure response of skn ssue due o me-dependen surface hea fluxes. Analycal soluon s consruced for dual phase lag bo-hea ransfer euaon wh consan, perodc, and pulse ran hea flux condons on skn surface. Separaon of varables and Duhamel s heorem for a skn ssue as a fne doman are employed. The ransen emperaure responses for consan and me-dependen boundary condons are obaned and dscussed. The resuls show ha here s major dscrepancy beween he predced emperaure of parabolc (Pennes bo-hea ransfer), hyperbolc (hermal wave), and dual phase lag bo-hea ransfer models when hgh hea flux accdens on he skn surface wh a shor duraon or propagaon speed of hermal wave s fne. The resuls llusrae ha he dual phase lag model reduces o he hyperbolc model when τ T approaches zero and he classc Fourer model when boh hermal relaxaons approach zero. However for τ = τ T he dual phase lag model ancpaes dfferen emperaure dsrbuon wh ha predced by he Pennes model. Such dscrepancy s due o he blood perfuson erm n energy euaon. I s n conras o resuls from he leraure for pure conducon maeral, where he dual phase lag model approaches he Fourer hea conducon model when τ = τ T. The burn njury s also nvesgaed. Key words: dual phase lag model, Laplace ransform, skn ssue, hermal wave model, Fourer model Analyss of hea ransfer hrough bologcal maerals lke he skn ssue s very mporan no only for undersandng of bologcal processes bu also for many clncal applcaons such as cancer herapy, hyperherma and cryopreservaon [1, ]. The skn s an exensve organ of lvng ssues whch s made of hree man composons: epderms, derms, and subcuaneous ssue. The research of bohea ransfer was exended by advances n mcrowave, laser, and smlar echnologes. The accurae predcon of emperaure response hrough bologcal ssues s very dffcul due o he complex hermal neracon beween vascular and exra-vascular sysems. The accurae nerpreaon of bohea ranspor hrough bologcal maerals s very sgnfcan n many clncal procedures such as laser rradaon [3], hyperherma [4], Correspondng auhor; e-mal: amr.morad_hs@yahoo.com

2 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 1458 THERMAL SCIENCE, Year 16, Vol., No. 5, pp and emperaure-based dseases dagnoscs [5]. Durng he pas few decades, several researchers have presened analyss of bohea ransfer n he bologcal maerals. Pennes [6] developed frs bohea ransfer model of parabolc ype n bologcal ssues. Ths model assumes ha any hermal dsurbance on a body s nsananeously fel hroughou he body or he propagaon speed of hermal waves s nfne. Ths model ndcaed some unrealsc resuls where he hea conducon behavor showed a non-fourer or hyperbolc feaures lke hermal wave phenomenon. Afer some expermenal observaons [7], neres on he non- Fourer or wave lke feaure of hea conducon was exremely ncreased. Caaneo [8] and Verno [9] presened a modfcaon of Fourer s law as a lnear exenson of Fourer euaon o descrbe he hyperbolc euaon mahemacally. Lu e al. [1] frsly nroduced a general model of he hermal wave model of bohea ransfer n lvng organs. The parabolc (Pennes bohea ransfer) model was used by varous researchers o sudy bohea ransfer n bologcal sof ssues [11-13]. The hermal wave (hyperbolc) model of bohea ransfer n sof ssues was appled by some researchers o descrbe he hermal behavor of such organs. Dfferen expermens on processed mea wh dfferen boundary condons were carred ou by Mra e al. [14]. They observed wave-lke feaures of hea conducon. Lu e al. [15] developed a newly hermal wave model o nvesgae he hermal damage of skn ssue. They analycally and numercally solved he parabolc and hyperbolc bohea ransfer euaon for consan surface emperaure and consan surface hea flux boundary condons respecvely. Xu e al. [16] frs analycally solved he Pennes bohea ransfer euaon (PBTE), hermal sress and hermal damage for a snglelayer skn for dfferen boundary condons. They numercally solved he euaons for a mul-layer skn. Lu e al. [17] exended a hybrd numercal scheme o solve he non-fourer bohea ransfer euaon for a mul-layer skn ssue. Accordng o many expermenal observaons, hermal wave model produces more accurae ancpaon han ha of he Pennes bohea ransfer model. However some of s predcons do no agree wh he expermenal resuls [18, 19]. A sudy ndcaes ha hermal wave model only consdered he fas ransen process of hea ransfer bu no he mcrosrucural neracons. These wo effecs can be reasonably expressed by a dual phase lag (DPL) model and hus a phase lag for emperaure graden s nroduced []. The DPL, hyperbolc and parabolc bohea ransfer models were used by Xu e al. [1] o model bohea ransfer across he ssue. They appled he fne dfference scheme o solve he bohea conducon euaons numercally and fnally found large dscrepances amongs ancpaons of he Pennes, hermal wave and DPL models. Lu and Chen [] employed he DPL model o nerpre he non-fourer hermal behavor of ssue durng he hyperherma reamen. Analyss of magnec hyperherma reamen usng he DPL bohea ransfer model was carred ou by Lu and Chen [3]. They evenually concluded ha conrol of blood perfuson rae can help o have an deal hyperherma reamen. Zhang [4] performed an analyss of generalzed DPL bohea ransfer euaons based on non-eulbrum hea ransfer n lvng bologcal ssues. He found ha he lag mes for lvng ssues are very close o each oher. The DPL model was appled by Lu e al. [5] o analyze he bohea ransfer problem across skn ssue. They employed he DPL bohea ransfer model o sudy hea conducon across lvng ssues. As prevously dscussed, he emperaure dsrbuon and blood perfuson durng surface heang have suded by many researchers. In he clncal applcaons, hea ransfer ofen reures o smulaneously depend on ransen and spaal heang boh on he surface and nsde bologcal bodes. Varous researchers have carred ou some effors o analyze hermal behavor of lvng bologcal ssues for me-dependen surface hea flux condons. Lu and Xu [6] represened a closed

3 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp analycal soluon of Pennes bohea ransfer model for snusodal hea flux on he skn surface o esmae blood perfuson rae usng phase shf beween surface hea flux and surface emperaure. Shh e al. [7] analycally solved he PBTE o analyze hea conducon across skn ssue wh snusodal hea flux condon on skn surface. Ahmadka e al. [8] developed an analycal soluon va he Laplace ransform mehod for he hermal wave model o nvesgae bohea ransfer problem across skn ssue durng consan, pulse ran and cosne hea flux condons on skn surface. Horng e al. [9] nvesgaed he effec of pulsale of blood flow on hermal dsrbuon durng hermal herapy. They found ha pulsale velocy profle, wh varous combnaons of pulsale amplude and freuency has lle dfference n effec on he hermal performance of ssue compared wh unform or parabolc velocy profle. Shh e al. [3] numercally suded he coupled effec of pulsale blood flow and hermal relaxaon me durng hermal herapy. They found ha he hermal behavor s ue nsensve o pulsaon freuency n her sudy. In hs paper, we explore an analycal soluon for he DPL bohea ransfer euaon n skn ssue as a fne doman wh he consan, cosne, and pulse ran hea flux condons on he skn surface. We consruced analycal soluon by separaon of varables mehod and Dumamel s heorem. The burn njury s also nvesgaed here. Ths developmen n bohea ransfer problems of lvng bologcal ssues has no ye been pursued n he relaed leraure. Mos of repored analycal sudes appled he Pennes bohea ransfer euaon o sudy hea conducon problem across he skn wh me-dependen heang condons on skn surface. Bohea ransfer models The Pennes (Fourer) bohea ransfer model The PBTE for bologcal ssues are well-known and s expressed [6]: T T c k c ( T T) Q Q x b b b a me ex Here b and c b are he densy and specfc hea of he blood, respecvely,, c, and k are he densy, specfc hea, and hermal conducvy of skn ssue, respecvely, ϖ b s he blood perfuson rae, T and T a are skn ssue and blood emperaures, respecvely, Q me and Q ex and he meabolc hea generaon n skn ssue and he hea generaed by exernal heang sources, respecvely. In hs sudy, Q ex s zero. Thermal wave model of bohea ransfer The modfcaon of classcal Fourer s law by consderng he concep of fne propagaon speed of hermal waves was proposed by Verno [8] and Caaneo [9]: ( x, ) ( x, ) kt( x, ) Ths consuve relaon s wdely accepable. Here > s a maeral propery and s called he relaxaon me. Consderng e. () for hea flux ncludng relaxaon me,, as well as he Pennes euaon, a general form of hermal wave model of bohea ransfer n lvng bologcal ssues can be wren [1]: T T T Qme c ( c b bcb ) b bcb ( T Ta ) k Q me Qex Qex x (1) () (3)

4 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 146 THERMAL SCIENCE, Year 16, Vol., No. 5, pp where = /C, s he hermal dffusvy, and C he hermal wave speed n he medum [14, 18]. Dual phase lag model of bohea ransfer As dscussed n nroducon, he consuve relaon or hermal wave model has only consdered fas ransen process of hea ransfer and has gnored he mcrosrucural neracons. These wo affecs can be reasonably proposed by he DPL beween and T, a furher modfcaon of he classcal Fourer s model gves: ( x, ) T( x, ) ( x, ) k T ( x, ) T (4) where τ T s he relaxaon me whch s he phase-lag n esablshng he emperaure graden across he medum durng whch conducon occurs hrough s small-scale srucures. Based on e. (4) for emperaure graden ncludng he characersc me, τ T, as well as he hermal wave model of bohea ransfer euaon, a general form of he DPL model of bohea ransfer n lvng ssues s expressed by [16]: T T c ( c b bcb ) b bcb ( T Ta ) 3 T T Qme Qex (5) k T Q me Qex x x Analycal soluon of DPL model of bohea ransfer euaon Here a closed form analycal soluon of DPL bohea ransfer model s derved for he skn as a fne doman. Three ypes of boundary condons are used on skn surface. The soluon of DPL model of bohea ransfer euaon for consan surface hea flux condon on skn surface The nal and boundary condons for consan surface heang are expressed by: T T( x,) Ta, (6) T T k, x x (7) x The dmensonless varables are defned: W c T T W c W c,,,, xl b b a b b b b x kwbcb k c c W c W c T W L l b b me b b T,, b b b, c Wc k b b k Euaon (5) (wh respec o consan Q me and Q ex = ) afer arrangemen can be re-wren n erms of dmensonless varables: 3 (1 ) T (8) (9)

5 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp The nal and boundary condons n erms of dmensonless varables are: (,), 1, Ths problem consss of non-homogeneous dfferenal euaon and non-homogeneous boundary condon on surface. Subseuenly he problem s formulaed n erms of a seady par and a ransen par s: L,, Subsung e. (1) no e. (9) yelds: (1) (11) 1 (1) (1 ) 1 T The boundary condons for seady-sae par, e. (14) are: 1, L The soluon for e. (14) regardng he boundary condons (15) s obaned by: (13) (14) (15) cosh L ( ) (16) snh L Now he nal and boundary condons for he ransen par, e. (13), are gven by: 1 1(,), 1 1, L By consderng boundary condons (18), we expand he funcon followng Fourer seres: 1, T cos L Subsuon of e. (19) no e. (13) gves: T ( ) (1 T ) T( ) (1 ) T ( ), L The general soluon of dfferenal e. () s gven by: By seng: 1, (17) (18) no he (19) () T ( ) a f1, b f, (1) T (1 ) 4 (1 ) ()

6 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 146 THERMAL SCIENCE, Year 16, Vol., No. 5, pp he funcons f 1 and f are expressed n he followng fashon: f T T (1 ) (1 ) 1, e, f, e f f T T (1 ) (1 ) 1 f, e cos, f, e sn Employng he nal condons (17) one has: 1 1 a f, b f, cos 1 f1 f a, b, cos Solvng he algebrac euaon sysem (5) and (6) one arrves a: cosh L, a f1, b f, snh L, b f, cos a f 1 1 (7) The prevous closed form analycal soluon reduces o he hermal wave soluon when Λ T =. The soluon of DPL model of bohea ransfer for ransen hea flux condon on skn surface In he presen sudy, he cosne hea flux s consdered for ransen heang condon on he skn surface. Here, he boundary condon on he skn surface should be only adaped accordng o cosne hea flux, hus s expressed: T k cos( ) x x where s he hea flux on he skn surface, and ω s he heang freuency. Oher boundary condon and nal condons are smlar o ha of he prevous secon In hs ype of heang, he dmensonless varables are defned: k( T T ) W c x,, c, a b b 1 1 c,,, T T me, 1, L c c 1 k Euaon (5) wh respec o he consan Q me and Q ex = and can be rewren n erms of dmensonless varables: k l (3) (4) (5) (6) (8) (9)

7 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp (1 c 1 ) c 1 T The dmensonless form of surface boundary condon s: (3) cos( ) (31) Transen hea conducon problems wh me-dependen boundary condons or hea source canno be solved by employng he separaon of varables. Here he Durhamel s superposon negral wh separaon of varables s used. Accordng o hs mehod, connuous me-dependen dsurbance s reduced o he sum of he me sepwse dsurbances. So ha, he general soluon of, wh reference o he me-dependen boundary condon F cos( ) s obaned wh sum of sepwse soluon of, n each sep. Wh decreasng he me sep dτ, he oal effec a me s acheved by negrang he effec of F n he me nervals of d and summng wh he begnnng effec of F [31]: where, The funcon, df, F,, d d (3) s he soluon of e. (3) wh nal and boundary condons (1) and (11). s expressed by: cosh c1 ( L), a f1, b f, c snh c L c a f1, b f, cos (33) 1 whle he parameer and funcons f 1 and f are defned by: (1 c1 T ) 4 ( c1 ), L f f f f f f, e, e, e 1 T (1 c ) 1 T (1 c ) 1 T (1 c ) 1 T (1 c ) cos, e sn (34)

8 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 1464 THERMAL SCIENCE, Year 16, Vol., No. 5, pp The coeffcens aand bare calculaed smlar o ha of he prevous secon. Now he soluon of e. (3) wh me-dependen boundary condon (31) afer ulzng e. (3) s obaned by:,,, sn( )d The soluon of DPL model of bohea ransfer euaon for he pulse ran hea flux condon on skn surface Common burns usually occur when skn s exposed o he hgh nensy hea flux n a shor duraon. Therefore o sudy he hermal behavor of skn, s supposed he hgh nensy hea flux n a pulse me accdens o skn surface. The relevan boundary condon on skn surface s defned by: T k U U h( ) F( ) x x (35) (36) where U() s he un sep funcon and h() s an arbrary funcon whch s se eual o one n hs sudy. Here, agan, we apply Duhamel s heorem o solve e. (5) wh me-dependen boundary condon (36) whle he oher boundary and nal condons are smlar o ha of he prevous secon. For he pecewse connuous funcon F(), we have: T x, T x, h( ) T x, h d,, T x, T x, h( ) T x, h d, In he prevous euaon he funcon, T x s he soluon of e. (5) whch has been expressed by e. (7) f he dmensonless varables are replaced by he nal varables. The fnal soluon of T( x, ) wh respec o he boundary condon, e. (36), s: T x, T x, Ta d,, T x, T x, Ta d, (37) Wc b b cosh x L k Wbcb Wbcb T x, a f1, b f, kwbcb snh( L) c c Wbcb Wbcb Wc a f1, b f, b b x c c 1 k cos T a (38)

9 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp Analycal soluon of he PBTE The soluon of PBTE for consan heang on skn surface The soluon of he Pennes bohea ransfer model n erm of dmensonless varables (8) s obaned: cosh L, C f, C f, cos, snh( L) L 1 cosh( L) sn( L) 1 L 1 C, C 3, f, e L (39) The soluon of PBTE for ransen heang on skn surface The soluon of he Pennes bohea ransfer model n erm of dmensonless varables (9) s obaned:,,, sn( )d, 1 1 cosh c1 ( L), D f,, cos, c snh c L c D f (4) f, e, L and he coeffcens D and D are calculaed n he paern of he prevous secon. The soluon of PBTE for pulse ran heang on skn surface The soluon of he Pennes bohea ransfer model n erm of dmensonless varables (8) s obaned: T x, T x, Ta d,, T x, b b T x, T x, Ta d, kw c Wc b b cosh x L Wc b b k c C e snh L T Wc b b 1 c Wc b b C e cos x k 1 where he coeffcensc and C are represened n e. (39). Thermal damage In he burn evaluaon, s proven ha hermal damage begns when he emperaure a he basal layer, he nerface beween he epderms and derms ncreases above 44 ºC [3]. The evaluaon of hermal damage s very mporan n boengneerng scence of skn ssue a (41)

10 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 1466 THERMAL SCIENCE, Year 16, Vol., No. 5, pp and clncal applcaons. The uanave of hermal damage evaluaon was nally suggesed by Morz and Henrues [33]. Such calculaon was based on he fac ha he ssue damage could be represened as an negral of a chemcal process rae,. e.: AexpEa RT d (4) where A s a maeral parameer euvalen o a freuency facor, E a he acvaon energy, and R = J/mol. K he unversal gas consan. The consans A and E a are expermenally obaned. For he frs and second-degree of burns, T n e. (4) s he basal layer emperaure, whle for he hrd-degree of burn, s he emperaure a he nerface beween derms and subcuaneous layers (fa). The frs and second-degree burns occur when boh he condons T > 44 C and >.53 are sasfed n he basal layer. When = 1, he second-degree burn occurs [15]. Resuls and dscusson In hs sudy, he emperaure dsrbuon hrough skn ssue s nvesgaed for hree models (Pennes, hermal wave and DPL) of bohea ransfer for hree ypes of hea flux condons on skn surface. The properes of skn ssue and blood are smlar o ha of repored by Xu e al. [1]. The relaxaon me τ = 16 s [5] s consdered for all analycal resuls here. The areral emperaure and blood perfuson rae are consdered as T a = 37 C [1] and W b = ρ b ϖ b =.5 kg/m 3 /s [6], respecvely. A schemac doman of skn subjeced o he boundary condons s shown n fg. 1. Effec of he lag me, τ T, on emperaure profle when consan hea flux, = 5 W/m accdens o skn surface a he daa frame DF nerface s shown n fg.. I s noed ha he dscrepancy amongs bohea ransfer models ncreases, specally, a he nal mes of heang when he hea flux nensy ncreases. As shown n fg., for smaller relaxaon mes, τ T, he DPL resuls approach he hermal wave oucomes. Furher he Pennes bohea ransfer model gves hgher predcon han ha of hermal wave and DPL model due o nfne speed of hea propagaon. Moreover, unlke he hermal wave model, no wave-lke behavor s observed n he DPL model as expeced, bu a non-fourer dffuson-lke behavor s consdered due o he second hermal relaxaon, τ T, whose nfluence weaken he hermal wave behavor. Furher, unlke he pure conducon meda, he DPL model predcs dfferen emperaure wh ha of Pennes model when τ = τ T. Ths dfference resuls from he blood perfuson rae n he energy euaon of bologcal maerals. or [U() U( )] cos(ω) Fgure 1. A schemac doman of skn ssue subjeced o he boundary condons

11 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp Fgure. The emperaure response for consan surface heang a DF nerface when = 5 W/m,τ = 16, τ = 1 s There are many clncal applcaons where lvng bologcal ssues are exposed o he hgh nensy hea flux for he shor duraon of me. The emperaure dsrbuon for hree bohea ransfer model when skn subjecs o he hea flux = 83 W/m for me duraon = 1s a he basal layer s shown n fg. 3. I s evden ha here are major dscrepances among emperaure predcons of hree models n early mes of heang when skn s exposed o he hgh nensy hea flux n shor duraon. I s also observed ha when surface hea flux becomes zero afer a shor heang perod, he elevaed skn emperaure from he Pennes model sharply decreases whle mes lag exs for he dsappearance of hea flow from he hermal wave and DPL models. Furher when τ = τ T, he DPL resuls are subsanally dfferen from ha of calculaed by he Pennes model. Fgure 3. Temperaure response for hgh power hea flux and shor fre me duraon a he basal layer when = 83 W/m τ = 16 s, τ = 1 s Fgure 4. Temperaure response for low power hea flux and longme duraon a he basal layer when = 5 W/m, τ = 16 s, τ = s Fgure 4 provdes he emperaure dsrbuon for hree bohea ransfer model when skn accdens o he surface hea flux = 5 W/m for me duraon τ = s a he basal layer. I s found ha when hea flux s no nensve and duraon of me ncreases, he lower dscrepances amongs he bohea ransfer models are observed. I should be noced ha he Pennes and hermal wave models of bohea ransfer ancpae he hgher emperaure han ha of he DPL model durng duraon me. Bu afer ha he elevaed skn emperaure calculaed by he Pennes model sharply decreases whle he elevaed skn emperaures calculaed by he hermal wave and DPL models slowly decrease. Xu e al. [1] numercally solved he DPL bohea ransfer and found ha smlar resuls wh hose calculaed n hs sudy.

12 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 1468 THERMAL SCIENCE, Year 16, Vol., No. 5, pp Fgure 5. The varaon of emperaure a boh ED and DF nerfaces when he skn exposes o he propane gas flame for 3 s for = 83 W/m τ = 16 s, τ T = 1 s Bu hey numercally solved he energy euaon for a consan emperaure a he skn surface. Qualave comparson beween he presen DPL resuls (fgs. 3 and 4) and hose repored by Xu e al. [1] ndcaes ha accuracy of he presened analycal soluon n soluon of DPL bohea ransfer euaon. Comparson beween bohea ransfer models when skn s exposed o he hgh nensy hea flux = 83 W/m for he me duraon τ = 3 s a boh basal layer and DF nerface s depced n fg. 5. Ths suaon s smlar o one where skn s exposed o he propane gas flame for 3 seconds [15]. Influence of hermal relaxaon me, τ, s clearly observed because akes 8 seconds for he hermal wave model o rse n order o be fel a he DF nerface. Whle only akes abou seconds and 8 seconds for he Pennes and DPL models, respecvely. Here, he derved analycal resuls are also compared wh hose presened by Ahmadka e al. [8] for boh Pennes and hermal wave models and an excellen agreemen s observed beween he resuls (fg. 5). In hs sudy, o nvesgae bohea ransfer wh ransen boundary condon on skn surface, he cosne hea flux wh amplude = 5 W/m s consdered. Temperaure response for hree bohea ransfer models a he skn surface (x = ) for ω =.1 s llusraed n fg. 6. Fgure 6. Temperaure oscllaon of hree bohea ransfer model for boh basal layer and DF nerface when = 5 W/m, ω =.1, τ = 16 s = cos(ω) Fgure 7. Temperaure oscllaon of he skn surface and dfferen hea flux freuences when = 5 W/m, τ = 16 s, τ T = 1 s, = cos(ω) I s noed ha he emperaure amplude for he hermal wave bohea ransfer model s larger and smaller han ha of he DPL and Pennes models, respecvely. I s also

13 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp seen ha as he lag me, τ T, ncreases he emperaure amplude becomes smaller. Even for he ransen hea flux smlar o he consan one he DPL model predcs compleely dfferen emperaure wh ha of calculaed by he Pennes euaon when τ = τ T. Moreover, he resuls of fg. 6 exhb ha here s a good agreemen beween he presen resuls and hose repored by Ahmadka e al. [8] for he Pennes and hermal wave models. Generally, he hermal lag me, τ T, ges smooh he emperaure profle and decreases he effec of hermal wave (wave-lke behavor) n emperaure dsrbuon. Fg. 7 shows he DPL emperaure responses for dfferen freuences a he skn surface. I ndcaes ha by ncreasng he freuency of ransen hea flux, he emperaure amplude becomes smaller as well as he response cyclc me also becomes shorer. Fgure 8 demonsraes he DPL dmensonless emperaure oscllaon a he skn surface for dfferen values of he blood perfuson rae when ω =.1 and = 5 W/m from η = o η = 18. I s evden ha for he lower blood perfuson raes (c 1 =. and c 1 =.) here are no consderable changes beween he emperaure responses. Obvously, as he blood perfuson rae ncreases, he dscrepancy beween he emperaure oscllaons and hea flux ges more evden. I means ha hs mehod can be more suable for perfuson measuremen n hghly perfused ssues. The hgher freuency hea fluxes provdes smaller emperaure oscllaons, hus he lower freuency heang would be more preferable due o s hgh sensvy o blood perfuson rae. Effec of blood perfuson rae on he DPL emperaure Fgure 8. The varaon of DPL dmensonless emperaure for dfferen values of dmensonless blood perfuson when = 5 W/m, ω =.1, = cos(ω) τ = 16 s, τ T = 1 s Fgure 9. Temperaure response of DPL model a for dfferen values of blood perfuson rae when = W/m, τ = 16 s, τ T = 1 s dsrbuon of skn ssue a boh ED (basal layer) and DF nerfaces for consan surface hea flux = W/m s llusraed n fg. 9. Clearly, he larger perfuson raes ancpae emperaure response lower han ha of he smaller one because furher rae of blood perfuson can carry away he large amouns of hea. The hsory of emperaure whn skn deph when he skn s subjeced o he propane gas flame s depced n fg. 1. I s beleved ha he dscrepances amongs bohea ransfer models are more evden and large for he pons closer o he skn surface, whle hese devaons decreases severely when hea propagaes no he skn deph.

14 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo/Hea Transfer 147 THERMAL SCIENCE, Year 16, Vol., No. 5, pp Fgure 1. The varaon of emperaure along skn deph when he skn subjecs o propane gas flame a dfferen mes when = 93 W/m, τ = 3, τ = 16 s Fgure 11. The varaon of hermal damage a he basal layer when he skn subjecs o propane gas flame ( = 83 W/m τ = 3 s) for 3 seconds Furhermore, for early mes of heang, larger dfferences amongs hea ransfer models are observed, whle, as me goes on, hese dscrepances drop consderably. The burn evaluaon has always been and wll connually be a sgnfcan problem n boengneerng scence of skn ssue. The burn njury happens uckly and especally a he early sage of heang. Here, he hermal damage of skn ssue when skn s exposed o he propane gas flame = 83 W/m for 3 seconds for hree bohea ransfer models s demonsraed n fg. 11. I s ndcaed ha he burn me for DPL model s longer han ha of oher models. Furher he burn me becomes longer as he value of hermal relaxaon me, τ T, ges larger. Fgure 1 demonsraes he emperaure response of ssue along he skn deph and me when skn s exposed o he propane gas for 3 seconds. As dscussed n fgs. -4 and 9, for he neares pons o he skn surface and early me of heang, dscrepances beween he DPL and parabolc bohea ransfer models are grea and hese devaons decay as mes goes on and hea propagaes no he beneah layers of ssue. Hsory of skn emperaure along skn deph for hree bohea ransfer models a dfferen mes when skn s subjeced o consan hea flux = 5 W/m s llusraed n fg. 13. I can be clearly seen ha he parabolc model predc hgher emperaure ha ha of DPL and hermal wave model for pons closer o he surface. Whle hea propagaes no skn deph, emperaure obaned by he hermal wave model decreases abruply o he core emperaure, bu emperaure predced by he DPL model decays slowly. Fgure 1. Temperaure feld vs. deph and me when = 83 W/m, τ = 3, τ = 16 s, τ T = 1 s

15 Zae Poor, H., e al.: Analyss of he Dual Phase Lag Bo-Hea Transfer THERMAL SCIENCE, Year 16, Vol., No. 5, pp The ualave comparson beween fg. 13 wh hose calculaed by Xu e al. [1] (fg. 6 n her paper) shows a good agreemen and valdaes he presen DPL resuls. Conclusons In hs arcle, one dmensonal DPL model of bohea ransfer euaon for he sngle-layer skn ssue as a fne doman was analycally solved by employng he separaon of varables and Duhamel s superposon negral. The analycal resuls for he connuous consan hea flux ndcaed ha he larger devaon amongs predced emperaures of he DPL, hermal wave and Pennes euaons s found for nensve hea fluxes. The resuls demonsraed for fas heang wh hgh nensy hea flux, large dscrepances are observed among he predced emperaures by hree models of bohea ransfer a early mes of heang and pons closer o he skn surface. Conversely for lower hea powers wh long duraons, predced emperaure by hree bohea ransfer models are close o each oher. The resuls exhbed boh hermal relaxaons τ and τ T have sgnfcan effecs on skn emperaure. The hermal relaxaon, τ, enforces he wave fron phenomenon whle s smoohed by τ T. Unlke he pure conducon maerals, he DPL bohea ransfer model calculaes dfferen emperaure wh ha of he Pennes model when τ = τ T. The derved analycal resuls for he ransen hea flux condon showed ha for he lower freuences, he hgher emperaure ampludes are noced and such freuences are more preferable because of s sensvy o blood perfuson rae. I s found ha he surface emperaure of DPL model has larger phase shf wh he surface hea flux and ncreases as he value of τ T becomes longer. I s also found ha hree bohea ransfer models predc dfferen burn me when he skn ssue s exposed o he propane gas flame for shor duraon. References = 45 s = 15 s Fgure 13. Temperaure dsrbuon along skn deph for consan hea flux = 5 W/m a dfferen me when τ = 16 s [1] Cho, Y. I., Boengneerng Hea Transfer, Advances n Hea Transfer, Academc Press, London, Vol., 199 [] Xu, F., e al., Mahemacal Modelng of Skn Bohea Transfer, Appl. Mech. Rev., 6 (9), 5, pp [3] Jha, K. K., Narasmhan, A., Three-Dmensonal Bo-Hea Transfer Smulaon of Seuenal and Smulaneous Renal Laser Irradaon, In. J. Thermal Scences, 5 (11), 7, pp [4] Zhu, L., e al., Quanfcaon of he 3-D Elecromagnec Power Absorpon Rae n Tssue durng Transurehral Prosac Mcrowave Thermoherapy Usng Hea Transfer Model, IEEE Tran. Bomed. Eng., 45 (1998), 9, pp [5] Deng, Z. S., Lu, J., Mahemacal Modelng of Temperaure Mappng over Skn Surface and Is Implemenaon n Dsease Dagnosc, Compuers n Bology and Medcne, 34 (4), 6, pp [6] Pennes, H. H., Analyss of Tssue and Areral Blood Temperaure n he Resng Forearm, Journal of Appled Physology, 1 (1948),, pp [7] Peshkov, V., Second Sound n Helum II, Journal of Physcs, 11 (1994), 3, pp [8] Caaneo, C., A Form of Hea Conducon Euaon whch Elmnaes he Paradox of Insananeous Propagaon, Compe Rendus, 47 (1958), pp

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