Supplemeay Ifomaio No-ivasive, asie deemiaio of he coe empeaue of a hea-geeaig solid body Dea Ahoy, Daipaya Saka, Aku Jai * Mechaical ad Aeospace Egieeig Depame Uivesiy of Texas a Aligo, Aligo, TX, USA. * Coespodig Auho: email: jaia@ua.edu; 5 W Fis S, Rm, Aligo, TX, USA 769 Ph: + (87) 7-9338; Fax: + (87) 7 95 Theoeical Deivaio of Equaio () This secio deives a elaioship bewee he asie coe empeaue, T coe () a = ad he asie suface empeaue disibuio, T (θ,) of a hea-geeaig ifiie cylide. While simila soluios have bee deived i he pas usig he mehod of Gee s fucios (fo example, Özişik, M.N., Hea coducio, d Ed., Joh Wiley & Sos, 993), his deivaio akes io accou cicumfeeial vaiaio i T, as well as aisoopic hemal coducio wihi he cylide. This deivaio leads o a expessio fo T coe () ha foms he basis fo he echique o o-iusively deemie he coe empeaue. Cosa hea geeaio ae This sub-secio cosides a ifiie cylide of adius R geeaig hea a a uifom ad cosa ae Q, show schemaically i Supplemeay Figue S. The cylide is assumed o
have ohoopic hemal coduciviies, wih values of k ad k θ i he adial ad cicumfeeial diecios especively. The cicumfeeially-vayig, asie empeaue disibuio alog he oue suface a =R, give by T (θ,) is assumed o be kow, fo example, hough a ifaed measueme. The cylide is assumed o be a uifom empeaue iiially. The asie empeaue disibuio i he cylide ca be deemied by solvig he goveig eegy cosevaio equaio subjec o appopiae bouday codiios. Usig he sepaaio of vaiables appoach (Özişik, M.N., Hea Coducio, d Ed., Joh Wiley & Sos, 98), he empeaue field may be spli io wo pas ad assumed o be of he followig fom: Q R T(,, ) T (, ) T (,, ) A J e Bm ( ) J m k cos( m (S.) ) 4k m m m k whee T, compisig he fis wo ems, accous fo ieal hea geeaio, ad T, compisig he hid em accous fo ime- ad θ-depede empeaue a =R. Hee, J efes o he Bessel fucio of he fis kid, ad he adial om N, is give by N, R J( R) (S.) ad he eigevalues λ m ae obaied fom he oos of J m k k. Specifically, λ ae obaied fom he oos of J. Usig he ohogoaliy piciple (Özişik, M.N., Hea Coducio, d Ed., Joh Wiley & Sos, 98), A ae give by A R R J Q 4k N, d (S.3)
The coefficie fucios B m () mus be deemied such ha he soluio saisfies he goveig equaio ad bouday codiios. Noe ha he coe empeaue T coe () may be expessed i ems of he ukow coefficie fucios B m () by puig = i equaio (S.) T coe ( ) T QR, ( ), ( ) coe T coe Ae B ( ) 4k (S.4) Equaio (S.4) shows ha eve hough all coefficie fucios B m () ae eeded o deemie he geeal empeaue disibuio T(,θ,), oly he coefficie fucios B () ae eeded fo measuig he coe empeaue. To do so, he expessio fo T (,θ,) fom equaio (S.) is iegaed wih espec o θ o elimiae all B m () fo m>, followed by use of he mehod of udeemied paamees (Myes, G.E., Aalyical Mehods i Coducio Hea Tasfe, d Ed., AMCHT Publicaios, 998) o deive he followig odiay diffeeial equaio fo B () db d B RJ N R, T, d (S.5) Fuhe, based o he iiial codiio fo he empeaue field, he followig iiial codiio applies fo B () B () (S.6) A soluio fo equaio (S.5) subjec o (S.6) is give by RJ R B ( ) w I ( ) exp d N, (S.7) 3
whee w I (τ) is he cicumfeeially aveaged value of he measued empeaue a he ouside suface, give by w I ( ) T, d (S.8) This complees he deemiaio of he coe empeaue T coe (), which is give by equaio (S.4), whee he coefficie fucios ae give by equaios (S.) ad (S.7). Effec of ime-vayig hea geeaio ae The pevious sub-secio assumed a ime-ivaia ieal hea geeaio ae i he cylide. I his sub-secio, a moe geeal case is cosideed whee he hea geeaio vaies wih ime, give by Q(). Time-vayig hea geeaio is ecoueed i seveal egieeig applicaios, whee he measueme of coe empeaue i espose o Q() is of iees. To develop a echique fo ieal empeaue measueme fo his case, he goveig eegy equaio eeds o be evisied. As show i equaio (S.), he compoes T ad T of he oveall empeaue disibuio accou fo empeaue ise due o ieal hea geeaio ad due o he ime-depede suface empeaue especively. As a esul, vaiaio i Q wih ime affecs oly he T compoe of he empeaue field. A Laplace asfom appoach is used o deive a soluio fo T (,) fo whe Q vaies wih ime. Takig Laplace asfom of he goveig equaio ad bouday codiio fo T esuls i he followig: d T dt s T d d Q( s) k (S.9) 4
5 Whee ) (s Q is he Laplace asfom of he hea geeaio ae Q(). T also saisfies d dt a = (S.) ad T a =R (S.) The soluio o his odiay diffeeial equaio ca be show o be give by R s I s I sk Q T (S.) whee I is he modified Bessel fucio of he fis kid. As a esul, he coe empeaue T coe () fo a ifiie cylide ca be wie as,, ) ( ) ( ) ( ) ( coe coe coe B R s I sk Q L T T T (S.3) whee L - efes o he ivese Laplace asfom. Noe ha B () is give by equaio (S.7). Fo a give Q(), he ivese Laplace asfom may be deemied aalyically whe possible, o umeically usig ivese Laplace asfom umeical algoihms. This povides a echique o pedic he coe empeaue of he cylide i pesece of a ime-vayig hea geeaio wihi he body.
Supplemeay Figues 6
Supplemeay Figue S: Schemaic of he geomey of a hea-geeaig ifiie cylide. The goal of he measueme is o deemie he coe empeaue T(=,) as a fucio of ime fom measued suface empeaue T (θ,) as a fucio of ime. 7
Supplemeay Figue S: Compaiso of asie coe empeaue pediced by he heoeical model wih fiie-eleme simulaio esuls fo wo suface empeaue codiios, (a) cosa T = C, (b) liealy iceasig T ()= C. Boh model ad ( s) fiie-eleme simulaios assume a 3 mm adius cylide wih 57,954 W/m 3 heaig, ad he same hemal popeies as he expeimeal hemal es cell. 8
Supplemeay Figue S3: Measued suface empeaue disibuio aoud he cylide T (θ,τ) a a umbe of imes followig he sa of heaig fo.86 W heaig powe i he hemal es cell i fee covecio coolig codiios. 9
Supplemeay Figue S4: Plo of he maximum esidual ove he eie expeime duaio as a fucio of heaig powe fo Figue 3.
Supplemeay Figue S5: Pediced vaiaio of coe empeaue as a fucio of ime fo diffee umbe of eigevalues cosideed fo he B () em i equaio (). Fo compaiso, he embedded hemocouple measueme is also show as a boke lie.
Supplemeay Figue S6: Covegece of he ifiie seies i equaio (7) wih espec o he umbe of eigevalues cosideed, showig slow seies covegece.
Supplemeay Figue S7: Compaiso of IR camea measued empeaue of a suface wih a kow suface empeaue as a fucio of ime, duig a empeaue amp fom oom empeaue o 6 C a a ae of 5 C/mi. IR daa ae show boh wih ad wihou calibaio i ed lies. The kow suface empeaue is also ploed i black fo compaiso. 3