A MATHEMATICAL MODEL OF THE HUMAN THERMAL SYSTEM

Transcription

1 A MAHEMAICAL MODEL OF HE HUMA HERMAL SYSEM A hess Subed o he Graduae School of Egeerg ad Sceces of zr Isue of echology aral Fullfle of he Requrees for he Degree of MASER OF SCIECE Mechacal Egeerg by Eda Dde YILDIRIM Jauary 5 ZMR

2 We approve he hess of Eda Dde YILDIRIM Dae of Sgaure... 7 Jauary 4 Assoc. rof. Dr. Barı ÖZERDEM Supervsor Depare of Mechacal Egeerg zr Isue of echology... 7 Jauary 4 rof. Dr. Zafer LKE Depare of Mechacal Egeerg zr Isue of echology... 7 Jauary 4 Asss. rof. Dr. Ayuç EREK Depare of Mechacal Egeerg Douz Eylül Uversy... 7 Jauary 4 Assoc. rof. Dr. Barı ÖZERDEM Head of Depare zr Isue of echology... Assoc. rof. Dr. Seaha OZDEMR Head of he Graduae School

3 ACKOWLEDGEMES he auhor would le o ha her supervsor, Assoc.rof. Dr. Barı Özerde, for hs for hs help ad gudace durg he projec. he auhor also wshes o express her specal has o Hal Aya for he ay hours he spe helpg her every seps of he sudy. I addo, he auhor would le o ha her roo aes Selda Alpay, Leve Blr ad uç Er for her coes ad paece. Lasly, he auhor wshes o express her graefuless o her faly for her help ad suppor durg her sudes.

4 ABSRAC Maheacal odel of he hua heral syse, whch has bee grealy developed rece years, has applcaos ay areas. I s used o evaluae he evroeal codos buldgs, car dusry, exle dusres, he aerospace dusry, eeorology, edce, ad lary applcaos. I hese dscples, he odel ca serve for research o hua perforace, heral accepay ad eperaure sesao, safey ls. rese sudy vesgaes he aheacal odelg of he passve par of he hua heral syse. he Bo-Hea Equao s derved order o solve he hea rasfer pheoea he ssue ad wh evroe. I s assued ha he body s exposed o cobao of he coveco, evaporao ad radao whch are ae o accou as boudary codos whe solvg he Bo-Hea Equao. Fe dfferece echque s used order o fd ou he eperaure dsrbuo of hua body. he derved equao by uercal ehod s solved by wre sofware called Bo-heral. Bo-heral, s used o deere eperaure dsrbuo a succeedg e sep of he vscera, lug, bra all ssue ype of he orso, ec, head, leg, foo, ar, had, ad ea eperaure of orso, ec, head, leg, foo, ar, had. Addoally, for overall body, ea eperaure of he boe ssue, uscle ssue, fa ssue, ad s ssue ad ea eperaure of he oal body ca be obaed by Bo- heral Sofware. Also, he sofware s o be capae of deosrag he secoal vew of he varous body lbs ad full hua body. I order o verfy he prese sudy, predcos of he prese syse odel are copared wh he avalae expereal daa ad aalycal soluo ad show good agreee s acheved. v

5 ÖZ Güüüzde, sa eral sse aea odellees brço alada ullaılaa ve bu alalar ggde araadır. Balıca alalar se; esl seörü, baları ısıa ve havaladıra sseler, uzay çalıaları, araba saay, laç saay ve ordu uygulaalarıdır. Bu çalıada sa eral sse aeasel odellees celeeedr. Hücre çde ve çevresde ola ısı rasfer baıılarıı çözee ç Bo-Isı Dele çıarılıır. Çalıada sa vücuduu ovesyo, ııı ve buharlaa eler oplaıa aruz aldıı abul edlr. Sıcalı daılııı bula ç solu farlar e ullaılıır. Elde edle deleler Bo-heral adıda br yazılı ullaara çözülüür. Yazılı, daha öce hazırlaı dele aılarıı çözere ora oulları belrl br saı bey, as, ya, der, acer, ç orga ve e dousuu sıcalı daılııı vereeedr. Ayı zaada afa, boyu, gövde, baca, aya, el, oluda sıcalı daılııı hesaplayaee ve büü bu uzuvları ç aalarda ya, as, e gb doularıda sıcalı daılıııda bulaeedr. Ayı zaada Bo-heral yazılııı ullaara celee sedgz uzvu es olara sıcalı daılıııda görersz. Çalıaı doruluguu spalaa ç, çalıaı souda elde edle deerler, uygu ola deeysel daalar ve aal çözülerle arılaırılıır. Bu çalıada odel souçlarıı gere deeysel souçlar le gerese aal çözü le uyulu olduu görülüür. v

6 ABLE OF COES ABSRAC... LIS OF FIGURES. v LIS OF ABLES... v OMECLAURE. x CHAER. IRODUCIO CHAER. LIERAURE SURVEY... 3 CHAER 3. HUMA HERMAL SYSEM Maheacal Modelg Quaave Modelg... a heorecal Modelg... b Expereal Modelg Qualave Modelg Hua heral Syse Hea Loss Mechass Hea roduco Mechass Maheacal Modelg of Hua heral Syse Fudaeal Equaos Hea rasfer wh he ssue Coduco Meabolc Hea Geerao Coveco by he Crculaory Syse Sorage of heral Eergy Hea Exchage wh Evroe. 3 v

7 3.4.. Radao Coveco 5 a Free Coveco... 6 b Forced Coveco Evaporao... 7 a Hea Loss by Dffuso of Waer hrough he S.. 8 b Hea Loss by Swea Secreo. 9 CHAER 4. MODEL DEVELOME Geoery of Orgas herophyscal ad hysologcal roperes of Varous Orgas ad ssues Meabolc Hea roduco Role of Blood Hea rasfer he Ieraco wh he Evroe Radao Evaporao Hea Loss by Dffuso of Waer hrough he S Hea Loss by Swea Secreo Coveco Free Coveco Forced Coveco Hea Coduco heroregulaory Mechass Hua Body assve Syse Equao... 4 CHAER 5. SOLUIO ECHIQUE Fe Dfferece For of Bo-Hea Equao he Fe-Dfferece For of Bo-Hea Equao for ozero Values of r he Fe-Dfferece Fors Of Bo-Hea Equao For r Boudary Codos... 5 v

8 CHAER 6. BIO-HERMAL SOFWARE.. 54 CHAER 7. RESULS AD DISCUSSIO Verfcao of he Bo-heral rogra 6 CHAER 8. COCLUSIOS... 7 REFERECES AEDICES Appedx A- Equaos of he Model. Appedx B- Code of he Bo-heral Sofware v

9 LIS OF FIGURES Fgure age Fgure.. he scheac vew of Wydha ad As Model 4 Fgure.. he scheac vew of Gagge Modelg 5 Fgure 3.. he scheac vew of Seps of Maheacal Modelg Fgure 3.. he scheac vew of Maala eperaure Corol Ceer ad Is Fucos... 4 Fgure 3.3. Corol Volue of ssue Elee 7 Fgure 3.4. Relave Corbuos of Orgas o Hea roduco. Fgure 3.5. Couer Curre Hea Exchage Exrees.. Fgure 3.6. erceage of Hea Loss a Dffere Evroeal eperaures... 4 Fgure 3.7. Scheac Vew of Hea Loss by Coveco. 6 Fgure 3.8. Schee of Hea Loss by Evaporao.. 7 Fgure 4.. A Scheac Vew of he Modelg... 3 Fgure 4.. Spaal Dvso of Leg 3 Fgure 5.. Dscrzao of he Doa for Vercal ad Horzoal Lbs Fgure 5.. A r, φ,z ewor he Cyldrcal Coordae Syse 45 Fgure 5.3. Ierseco Face Bewee wo Layers Fgure 5.4. eperaure of he Four odes of a Crcle Wh Ceer r.. 5 Fgure 5.5. Dvso of z Axs of a Lb... 5 Fgure 6.. Scheac Vew of he Ipu Daa For. 54 Fgure 6.. Scheac Vew of he Secoal Vew For. 55 Fgure 6.3. Scheac Vew of he uercal Value For Fgure 7.. Coparso of he Hua Forear eperaure wh ees Daa.. 63 Fgure 7.. Coparso of he Uclohed S eperaure of Ar wh aaa e al Daa.. 64 Fgure 7.3. Coparso of he Muscle eperaure of Leg wh Eberhar Aalycal Soluo.. 65 Fgure 7.4. Sulao of eperaure Chages Leg wh e Fgure 7.5. Coparso of he Mea S eperaure of Hua Body wh Ar ad Shzer s Daa.. 67 x

10 Fgure 7.6. Sulao of eperaure Chages Hua Body wh e. 68 Fgure 7.7. Sulao of eperaure Chages Hua Body wh e. 7 x

11 LIS OF ABLES ae age ae 4.. heral roperes of Body ssues 33 ae 4. Evaporao Hea rasfer Coeffce fro ude erso Ar 37 ae 5. he Boudary Codos of Bo-Hea Equao for Each Lb. 53 ae 5. he Boudary Codos of Bo-Hea Equao for Soe Lb 53 ae 7.. Codos of Expereal Resuls ad Aalycal Soluo.. 6 ae 7. Codos of Roo he Scearo.. 69 x

12 OMECLAURE c p : Specfc hea J/gK D: Exeral daeer of he lb g: Accelerao of gravy 9.8 /s Gr: Grashof uber h: Covecve hea rasfer coeffce W/ K : heral coducvy W/K K: Coeffce for Forced Coveco Evaporao Hea Loss u: ussel uber r: radl uber v : Vapour ressure of Waer a Ar eperaure q: Hea flux W/ Re: Reyolds uber RH: Relave Hudy of ar : e s : eperaure C v: Velocy of ar /s V: Corol volue : heral dffusvy α : heral expaso coeffces ε : Essvy of he Hua Body σ : Sefa-Bolza Cosa W. -.K W. -.K -4 ν : Keac vscosy /s : Desy g/ 3 φ : Relave Hudy Subscrps a: abe ar ar: arery : ood c: coveco x

13 d: dffusoal e: evaporao e: Corol volue face bewee ad E f: fl free: free coveco forced: forced coveco : Desgao of he r locao of dscree odal pos. j: Desgao of he φ locao of dscree odal pos : Desgao of he z locao of dscree odal pos. : eabolc r: radao s: surface s: swea v: vapour Superscrp : e level x

14 CHAER IRODUCIO I s very pora o ow how he hua body wll behave uder dffere evroeal codos of ar eperaure, hudy ad wd velocy. Wh he rapd develope of copuer echology, he sudy of sulag hua heral behavor s ovg owards ore ad ore precse ehods. I he areas of aerospace, auoove dusry ad heag, velag ad ar-codog egeerg, predcos of hua heral respose are grealy dead. Sulag he hua heral behavor s doe by wo ajor ways: sulag he passve syse of hua heral syse ad sulag he acve syse of hua heral syse. he acve syse s sulaed by acve odelg, whch predcs regulaory resposes such as shverg, vasooo, ad sweag. he a purpose of he acve odel s ha regulaes he passve hea rasfer odel ad s respose for he aeace of he hua body s eperaure. he passve syse s odeled by passve odelg, whch sulaes he physcal hua body ad he hea rasfer pheoea occurrg hua body ad a s surface. Descrpo of he passve odel s doe by equaos resula fro he applcao of he hea ad ass balaces o a ssue corol volue.by applyg he heores of hea rasfer ad herodyacs processes, we ca predc he heral behavour of he ere hua body or a par of. he purpose of prese sudy s o develop a uclohed hua passve syse odel whch ca uercally calculae he eperaure dsrbuo of hua body a dffere evroeal codos. Sofware called Bo-heral s developed o solve he aheacal equaos ad calculae he eperaure dsrbuo of vscera, lug, bra, ad boe, uscle, fa, s eperaure of orso, ec, head, leg, foo, ar, had. I addo, ea eperaure of orso, ec, head, leg, foo, ar ad had ca be calculaed. For overall body, ea eperaure of he boe ssue, he uscle ssue, fa ssue, s ssue ad oal body are calculaed. Also, he sofware s o be capae of deosrag he secoal vew of he varous body lbs ad full hua body.

15 rese sudy gves he aheacal odelg of hua heral syse fro geeral forao o specfc forao. I Chaper, he prevous pora aheacal odels abou he hua heral syse doe by scess are gve. I addo o hs, he expereal ad heorecal sudes used verfcao of our odel are roduced hs chaper. I Chaper 3, frsly, he geeral forao abou he aheacal odelg ad s ypes are gve brefly. he, hua heral syse s explaed. he echass called hea-loss echass ad hea producg echass whch are acvaed whe hua body s exposed o ho ad cold evroeal codos, are explaed. Lasly, aheacal odelg of hua heral syse s roduced by gvg he hea rasfers sde he ssue ad hea exchage wh evroe. I Chaper 4, prese sudy s roduced. he assupos, correlaos ad herophyscal properes, whch are used our odel, are gve. Apar fro ha, hs chaper, Bo-Hea Equao s dervaed dffereal for I Chaper 5, soluo echque whch s used for solvg he Bo-Hea Equao s explaed. Also, he fe dfferece fors of Bo-Hea Equao for every odes of he hua body are gve hs chaper. I Chaper 6, sofware whch s developed o solve he equao derved Chaper 5 s roduced. he process of he progra s explaed sep by sep. I addo, flow char of he progra s gve order o follow he seps easly. I Chaper 7, he verfcao of he progra s roduced by usg he expereal ad heorecal sudes doe by earler sces. he graphcs of he eperaure of varous lbs versus e are obaed. I addo, by usg he vsual par of our sofware soe secoal vews of he hua body are gve hs chaper. I Chaper 8, he cocluso of he sudy ad he fuure wor are gve.

16 CHAER LIERAURE SURVEY Sce he begg of he edcal research, he scess have bee eresg he physologcal syse ad s applcaos. hey have red o fd he os accurae odel of he hua heral syse for sulag he reacos o he dffere evroeal codos. Le he evaluao of he every odelg process, he aheacal odelg of he hua heral syse has bee developg fro easy o ore coplex oes. Frsly, ees odeled he sgle elee of he hua body. ees, who was oe of he earles worers hs area, developed he so-called BIO-HEA EQUAIO order o calculae he seady sae eperaure dsrbuo hua ar 948. he hua forear was reseed as a cylder. he odel cludes coduco of hea he radal dreco of he cylder, eabolc hea geerao he ssue, coveco of hea by he crculag ood, hea loss fro he surface of he s by coveco, radao ad evaporao. he porace of he odelg s ha s esseally applcae o ay cylder elee of he body Wssler, 998. Fro 96 s, he researchers have bee sarg o odel he ere hua body. Due o he develope copuer echologes, ore coplex odelg of he hua body could be doe. Mul layered odelg of he hua body s oe of he. he bes ow are Wydha ad A s Model, Crosbe Model, ad Gagge Model. Wydha ad A s Model was probay he frs odel, whch roduced a rase odel for a hua body 96. hey approxaed he hua body by a sgle cylder ad dvded he cylder o a uber of h cocerc layers. he fe dfferece echque was used o approxae he equao. A se of resulg frs-order dffereal equaos was solved by usg a aalog copuer. A he surface of he cylder he rae of hea loss o he surroudgs s he su of he hea loss due o he coveco, radao ad evaporao Fa e.al.999. he weaess of hs odel was ha Wydha ad As copared her odels predcaos wh a wde varey of pushed daa whch were led accuracy. I addo, hese wors dd o clude he effec of ood flow he rasfer of hea sde he ssue. 3

17 Fgure.. Scheac vew of Wydha ad As Model. Cea ad Clar, 98 Aoher ul layered odel s Crosbe Model, whch s also rase. However, dffere fro oher odels, Crosbe s he frs sces o sulae he physologcal eperaure regulao o a aalog copuer. he physologcal heral regulao echas had o bee ae o accou odelg ul Crosbe. He approxaed he values of he heral coducvy of ssue, eabolc rae, ad he rae of hea loss by evaporao as a fuco of he ea body eperaure Fa e.al.999. I addo, Crosbe assued he hea flow fro he core o he s o be udrecoal. hs oe-desoal odel s dvded o hree layers: he core layer, he uscle layer ad he s layer. he core layer s he source of basal eabols ad he uscle layer s he source of creased eabols caused by exercsg or shverg. A s layer, he hea s los by radao, coveco ad evaporao. I he er layer, oly he basal eabolc rae ad hea coduco are cosdered. A sulaor was desged o predc seady sae values of s ad deep body eperaure, eabolc rae, ad evaporave hea loss. If he e cosas for he varous heral chages are roduced, he sulaor ca also predc he dyac resposes o sudde shfs evroeal eperaure Cea ad Clar, 98 Gagge Model s aoher faous rase odel depcg he hua body as ul layer. he body s cosdered o be a sgle cylder whch s dvded o wo cocerc shells, a ouer s layer ad a ceral core represeg eral orgas, boe, uscle, ad subcuaeous ssue. herefore, s called CORE ad SHELL odel. Gagge assued ha he eperaure of each layer s ufor. Hea s geeraed sde he core ad rasferred o he s boh by he ood ad ssue coduco. 4

18 Eergy balaces equaos wre for he core ad s clude hea sorage effecs, coducve hea rasfer bewee adjace ssue layers, ad covecve hea rasfer due o he ood flow. I he core eergy balace sese ad lae hea loss caused by resprao ad he effecs of eabolc hea geeraed durg exercse ad shverg are cosdered. I s layers, hea exchage occurs bewee he s surface ad surroudgs due o he coveco, radao ad he evaporao of he osure. All of hese are cluded he s eergy balace. Fgure.. Scheac vew of Gagge Model.Sh, 99 Gagge developed a sofware progra o solve he eergy balace equaos for core ad s eperaure as a fuco of e usg he accopayg corol equaos, whch are derved fro acve odel. Evroeal codos ad acvy levels are assged by user. hs progra ca predc a s heral respose for sulaos bewee 5-6 ues log. I s geerally applcae durg oderae levels of acvy ad ufor evroeal codos descrbed by dry bulb eperaures he rage 5 C-45 C ad hudy dow o %. he sregh of he Gagge odel s ha, s less coplex ad easer o use ha Solwj odel ad or Wssler odel. However, hs splcy ls he aou of forao ca provde wh ay gve sulao ad resrcs he odel applcaos. Aoher weaess of he Gagge odel s foud he corol equaos. here s cosderae varao of he cosas hese equaos fro oe pucao o he ex. hs rases quesos as o 5

19 why such varaos appear, how hese cosas are calculaed, ad oe value should be used as opposed o aoher Sh, 99. Durg he develope of hese odels gve above, soe scess eresed ul elee odel raher ha ul layer odel. I ul elee odel, he hua body s depced as several pars, such as ar, leg; sead of depced as sgle ul layered elee. Frs ul elee seady sae odel called Frs Verso of Wssler Model was developed by Wssler 96. Wssler exeded he ees odel of he forear o oba he eperaure dsrbuo of he ere body. I addo o he ees odel, Wssler oo o accou he hea loss hrough he resprao ad he couercurre hea exchage bewee he large areres ad large ves. Model was coposed of ercoeced cylders. he hua body s subdvded o sx cylders: he head, orso, wo ars ad wo legs. Each elee s assued o have a uforly dsrbued eabolc hea geerao, o have a uforly ood supply, o be hoogeous ad a soropc cylders. he elees coss of four cocerc layers: boe, uscle, fa ad s. he seady sae bo-hea equao was solved aalycally for each cylder. A s layers, hea loss occurs by coveco, radao ad evaporao Fala, 999 I 963, Wssler developed he Secod Verso of he Wssler Model, whch s rase verso. hs verso s ore coplex ha he prevous odel. Secod verso of he Wssler s odel cosdered ay pora facors such as he varay of he local ood flow rae, he heral coducvy, he rae of local hea geerao by eabolc reacos, he geoery of he hua body, hea loss hrough he respraory syse, sweag. hese facors had o bee cosdered by prevous researchers. hs useady sae odel s a exeso of Wydha ad As odel gve by sgle elee oly. I was developed 963 Shzer ad Chao, 99. Dffere fro frs verso, he secod verso cluded addoal ers descrbg hea exchage bewee ood ad ssue, boh capllary bed ad he large ood vessels. Fe dfferece soluo schee used secod verso eaed Wssler o expad hs odel cosderay. he body was dvded o5 elees o represe he head, horax, abdoe ad proxal, edal ad dsal seges of he ars ad legs. Each elee s dvded o four layers of core, uscle, fa ad s. Also, each elee has hree par vascular syse, whch are areres, ves, ad capllares. he large areres ad ves are odeled usg a areral ood pool ad a veous ood 6

20 pool, respecvely. Meabolc reacos were cosdered he a source of he hea.. Sh,99. I Secod Verso of Wssler Model, eergy balace equaos are wre for each ssue each elee ad clude hea sorage, eabolc hea geerao, covecve hea rasfer due o he ood flow he capllares, areral ad veous pools, coducve hea rasfer he radal dreco, or he dreco oral o he body ceerle. Coduco hrough logudal ad ageal axs s egleced. erfec hea rasfer s assued bewee he capllares ad surroudg ssue. Sese ad lae hea losses due o resprao are dvded equally bewee he head ad horax. Hea exchage processes wh he evroe hrough coveco, radao ad he evaporao of he osure are cluded as boudary codos a he s surface Sh, 99. Wssler solved hs odel by usg Cra-cholso s plc fe dfferece ehod. he radal dsace of each elee was dvded o 5 pos ad sulaeous equaos of each elee were solved by repeaed use of he Gaussa elao ehod. Copued resuls fro a Forra progra were copared o expereal daa for varyg evroeal codos ad acvy levels. Recal ad ea s eperaures, voluerc oxyge cosupo raes, ad wegh losses were recorded usg subjecs exercsg ar a eperaures C o 3 C. Resuls fro Wssler odel closely approxaed he expereal daa. he secod verso of Wssler Model has ay advaages over prevous odels. Frsly, I proves he Gagge ad Solwj odels a uber of ways. he passve syse used s he dealed represeao of he eperaure profles he hua body. he crculaory syse, whch cosss of dvso of ood vessels o areres, capllares ad ves, s ore realsc ha ay oher odels. Addoally, each elee coucaes wh adjace elee hrough ood flow ule he Solwj odel where each ssue layer coucaes solely wh he ceral ood pool. Furher, he Wssler odel s desged for a wde rage of applcaos, for exaple deep sea dvg. Fally, he dervao of he corol syse equaos for ood flow raes ad eabolc raes s based o he body s oxyge requrees. hs approach s a ore accurae ha ay prevous ehod. However, s also led o applcaos volvg ufor evroeal codos surroudg each dvdual body par. Also he odel does o allow for alerave ood flow pahways ha occur durg vasodlao ad vasocosrco. Aoher weaess s ha s uclear wha 7

21 ehods were used o deere values for varaes ad cosas used Wssler odel Sh, 99. I 97, Solwj developed aoher odel. he Solwj odel used 5 cyldrcal elees o represe he body ru, ars, legs, hads, ad he fee ad a sphercal elee for he head. Each elee s dvded o four cocerc shells represeg core, uscle, fa ad s ssue layers. Meabolc hea geeraed durg exercse ad shverg s dvded aog he uscle ssue layers. A ewor ood vessel s used o raspor ood fro a ceral ood pool, or he hear, o each ssue layer. Eergy balace was gve a se of 4 equaos whch are wre for each ssue layer ad ood pool. I all ssue eergy balace hea sorage effecs, coducve hea rasfer wh adjace ssue layers, covecve hea rasfer due o he ood flow ad basal eabolc hea geerao are cluded. Equao couplg he varous body regos s he oe of he ceral ood pool whch assues ha veous ood reurg o hear s xed perfecly. Addoally, lae hea loss due o resprao s dvded equally bewee he head core ad ru core, whch sees approprae because hese losses occur predoaly he head. All s eergy balace equaos clude hea losses due o coveco, radao ad he evaporao of he osure a he body surface. he resuls of hs dervao were preseed he for of a Forra progra. Expereal daa, specfcally recal ad s eperaures, wegh losses, ad voluerc oxyge cosupo raes, were used o valdae he accuracy of he odel for sedeary subjecs hea ad cold sress codos ad for subjecs exercsg evroes descrbed by ar eperaures he rage C-3 C ad a relave hudy of 3%. he corol syse of he Solwj s odel s dvded o hree pars: he herorecepors, he egrave syses, ad he effecors. he effecors echass are sweag, vasodlaao, vasocosrco ad shverg. Each of hese echass ca be represeed by a equao Cea ad Clar, 98. he Solwj odel s sregh ay aspecs. Frsly, s capae of calculag he spaal eperaure dsrbuo he dvdual body elees. Secodly, Solwj odel coecs hese dvdual elees hrough ood flow he areres, ves, ad hus, offers a proved represeao of he crculaory syse ad s effec o he dsrbuo of hea wh he body. Wh hese provees, he Solwj odel ca provde forao descrbg he overall heral respose of he body as well as local resposes o varyg abe codos ad physcal acvy. 8

22 [R8] Lasly, he corol syse of Solwj s odel s que dealed. I s dvded o hree subsyses: he sesg-coparg echas, he error sgal suao echas, ad he regulaory echas Ar ad Shzer, 984. However, besdes hese advaages here are uber of drawbacs he descrpo of he Solwj odel. hs odel dsregards he effecs of he rae of chage of s eperaure o he regulaory syses. Also, he local flueces are descrbed raher geerally ad whou ay heorecal ad expereal verfcao. Fally, coducve hea rasfer s cosdered oly he radal dreco or he dreco oral o he body ceerle. By eglecg he spaal ssue eperaure grades he agular ad axal drecos, he resrced he use of hs odel o sulaos volvg ufor evroeal codos Sh, 99. Gordo Fa e al. 97 has developed a odel especally o sulae ad predc he physologcal resposes of he hua o cold exposure 977. I uses uch ew physologcal daa. he body s dealzed as 4 cyldrcal ad sphercal agular seges. Each sege cosss of several cocerc ssue layers, for sace fve for he abdoe core, boe, uscle, fa, s. Each ssue layer s subdvded o agular shells ad a fe-dfferece forulao s wre for he ceral odal po of each such shell. he corol syse of hs odel s characerzed by he pu sgals. hese sgals are based o he varao he head core eperaure, he s eperaure ad also of he hea flux hrough he s over he whole body. Ar ad Shzer,984, has developed odel of heroregulao he hua body. hey odeled boh he passve ad acve syse of hua heral syse. I hs odel, he body s dvded o 4 cylders whch are furher subdvded o four radal layers. hs odel s wo-desoal odel radal ad ageal drecos. he odel aes o cosderao ar veloces, radao hea loss, hea rasfer by coveco ad evaporao of swea. Fela e al, 999, has developed passve odel for he hua heral syse. he body s dvded o 5 cyldrcal ad sphercal body elees. he odel aes o accou he hea rasfer wh ood crculao, coduco, clohes, coveco, radao, resprao, evaporao. he equaos are solved by usg he fe dfferece echque. 9

23 CHAER 3 HUMA HERMAL SYSEM 3. Maheacal Modelg Maheacal odelg eas o fd ou he aheacal relaos ha characerse he eral srucure of he deled syse ad o descrbe he erdepedeces bewee he pu ad oupu varaes. I cosrucg a aheacal odel, frs job s o decde whch characerscs of he objec or syse of erese are gog o be represeed he odel. I order o ae such defos, s ecessary ha he purpose of ag he odel be defed as clearly as posse. Afer he recogo of he proe ad s al sudy, he ex sep s ag cera dealzaos ad approxaos o elae uecessary forao ad o splfy as uch as posse. he hrd sep s he expresso of he ere suao sybolc ers order o chage he real odel o a aheacal odel whch he real quaes ad processes are replaced by aheacal sybols ad relaos. Afer he proe has bee rasfored o sybolc ers, he resulg aheacal syse s suded usg approprae aheacal echques, whch are used o ae heores ad predcos fro he eprcal po of vew. he fal sep he odel-buldg process s he coparso of he resuls predced o he bass of he aheacal wor wh he real world. he os desrae suao s ha he pheoea acually observed are accoued for he coclusos of he aheacal sudy ad ha oher predcos are subsequely verfed by expere. Fgure 3. represes he seps of aheacal odelg. he sold les he fgure dcae he process of buldg, developg, ad esg a aheacal odel as we have ouled above. he dashed le s used o dcae a abbrevaed verso of hs process whch s ofe used pracce WEB_ 995.

24 Real World Model Splfy Real Model Ierpre Absrac Coclusos Maheacal Model Calculae Sulae rogra Copuer Model Fgure 3.. Schee of he seps of aheacal odelg WEB_ 995. Maheecal odelg ca be classfed o wo group as quaave odelg ad qualave odellg. 3.. Quaave Modelg Quaave odels of syses ca be obaed wo basc ways: by heorecal odelg ad by expereal odelg of ha syse. May es he wo ehods are cobed egeerg pracce.

25 aheorecal Modelg heorecal odelg presupposes he owledge of esseal physcal laws of ha process. hs has o be que profoud order o oba a sasfyg process odel. he odel s foud ou by deerg he depede ad depede varaes ad by sag bewee he soe balace equaos, pheoeologcal laws, sae equaos ha, geeral, lead o a syse of ordary ad/or paral dffereal equaos or syse of dfferece equaos. Is advaage les avalay of odelg durg he plag perod of he syse sce he ehod does o eed he exsece of a real process. I hs phase, s also pora o fd ou he fluece of dffere desg paraeers o he process perforace. I order o oba a heorecal odel ha ca be hadled easly ad copued explcly, soe proper assupos ad approxaos, wheever ecessary, are always ade. bexpereal odelg Expereal odelg s ow he leraure as process defcao. For defcao he process has o be ready for operao ad suae sgal geerag ad recordg devces us be avalae. he aheacal odel s deered fro he easured sgals by usg adequae defcao ehods. he defcao ehods wor wh dscree-e sgals, herefore hey always lead o dscree-e odels, ha ae furher copuer sulaos easy. Applyg process defcao, ofe hgher odel perforace ca be reached, especally for learzae processes wh less or poor owledge abou her eral behavour. 3.. Qualave Modelg Qualave odelg represes he os recely developed ad sll uder furher research doa fdg ou aheacal odels for syses. he qualave odelg approach s based o hua experece, o owledge, o paer recogo. Exper syses, fuzzy ses ad eural ewors are soe eywords ha requre a leas powerful copuer ools for successful odellg.

26 3. Hua heral Syse Lvg creaures ay be classfed o wo caegores based o her ood eperaure: cold ooded ad war ooded aals. Cold ooded aals, are hose whose body eperaure flucuae wh he eperaure of he evroe. War ooded aals, cosue a group whch exhb a edecy o say he oral body saes. he survval of aals depeds o her ay o corol ad aa body eperaures wh he rage esseal o lfe. hs regulao has o be acheved uder wdely evroe, exercse, dsease or codos. I s well ow ha he eperaure of each of her orgas ad ere orgas should rea wh he val rage of -4 C, wh os of he eral eperaures corolled wh he rage of C.Varay he edurace e o he exree of hs wder rage aog he dffere orgas s also grea. Soe orgas such as bra, ay suffer rreverse daage should her eperaure be allowed o be exceed a uch closer l. he fucog of he oher orgas such a hear ay be slowed dow or eve pared should her eperaure be ep oo low wh ha rage Gufger,975. he body eperaure of hua begs reas relavely cosa, despe cosderae chage he exeral codos. I order o aa a cosa core eperaure, he body us balace he aou of hea produces ad absorbs wh he aou loses; hs s heroregulao. heroregulao aas he core eperaure a a cosa se po, averages 36. C, despe flucuaos hea absorpo, produco ad loss. he core eperaure flucuaes by abou.6 C ad s lowes aroud 3 a., ad hghes aroud 6 p.. Despopoulos ad Slberagl, 3. he body s heroregulaory respose s deered by feedbac fro he herorecepors. he corol ceer for body eperaure ad ceral herosesors are locaed he hypohalaus. he hypohalaus receves affere pu fro perpheral herorecepors, whch are locaed he s, ad ceral herorecepors, whch s sesve o he eperaure of ood, locaed he body core. he hypohalaus aas he core eperaure val rage by ag approprae hea-producg or hea-loss reflex echass by coparg he acual core eperaure wh he se- 3

27 po value. here are wo ceers he hypohalaus whch are cocered wh eperaure corol; he poseror oe cocered wh proeco agas cold ad he aeror oe cocered wh proeco agas hea. hese wo ceers are depced scheacally Fgure 3.. Fgure 3.. Scheac Vew of Maala eperaure Corol Ceer ad s Fucos Gufger, Hea-Loss Mechass Hea-loss echass proec he body fro excessvely hgh eperaures, whch ca be daagg o he body. Mos hea loss occurs hrough he s va he physcal echass of he hea exchage, such as radao, coduco, coveco, ad evaporao. Whe he core eperaure arses above oral, he hypohalac hea-producg ceer s hbed. A he sae e, he hea-loss ceer s acvaed ad so rggers oe or boh of he followg echass. 4

28 a Vasodlao of cuaeous ood vessels: Ihbg he vasooor fbers servg ood vessels of he s allows he vessels o dlae. As he s vasculaure swells wh war ood, hea s los fro he shell by radao, coduco ad coveco. b Sweag: If he body s overheaed or f he evroe s so ho ha hea codo o be los by oher, hea loss by evaporao becoes ecessary. he swea glads are srogly acvaed by sypahec fbers ad spew ou large aous of persprao. Whe he relave hudy s hgh, evaporao occurs uch ore slowly. I such cases, he hea-lberag echass ca o wor well, ad we feel serae ad rrae. 3.. Hea roducg Mechass Whe he exeral eperaure s cold, he hypohalac hea-producg echass ceer s acvaed. I rggers oe or ore of he followg echass o aa or crease core body eperaure. a Vasocosrco of cuaeous ood vessels: Acvao of he sypahec vasocosrcor fbers servg he ood vessels of he s causes he o be srogly cosrced. As a resul, ood s resrced o deep body areas ad largely bypasses he s. Because he s s separaed for deeper orgas by a layer of sulag fay ssue, hea loss fro he shell s draacally reduced ad shell eperaure drops oward ha of he exeral evroe. b Icrease Meabolc Rae c Shverg: If he echass descrbed above are o eough o hadle he suao, shverg s rggered. Shverg s very effecve creasg body eperaure, because uscle acvy produces large aou of hea. [3] 5

29 3.3 Maheacal Modelg of Hua heral Syse A ajor proe heral physology odelg s he aheacal descrpo of he heral sae of he orgas. Fro he aheacal po of vew, he hua orgas ca be separaed o wo eracg syses of heroregulao: he corollg acve syse ad he corolled passve syse. Maheacal odelgs of hese syses are called acve odel ad passve odel, respecvely. he acve syse s sulaed by acve odelg, whch predcs regulaory resposes such as shverg, vasooo, ad sweag. he a purpose of he acve odel s ha regulaes he passve hea rasfer odel ad s respose for he aeace of he hua body s eperaure. he passve syse s odeled by passve odelg, whch sulaes he physcal hua body ad he hea rasfer pheoea occurrg hua body ad a s surface. For exaple, he eabolc hea produced wh he body, s dsrbued over body regos by ood crculao ad s carred by coduco o he body surface, where, sulaed by clohg. I order o loose he hea o he surroudgs, he body uses four echass of hea rasfer; radao, coduco, coveco, evaporao ad resprao. All of hese hea rasfer pheoea gve above are odeled by he passve odelg. Descrpo of he passve odel s doe by equaos resula fro he applcao of he hea ad ass balaces o a ssue corol volue. By applyg he heores of hea rasfer ad herodyacs processes, aely by usg he passve odelg of he hua heral syse, we ca predc he heral behavour of he ere hua body or a par of. Iforao he chage of body s esseal for he sudy of a s olerace ad reaco upo exposure o herally hosle evroe. 3.4 Fudaeal Equaos All heroregulao odels use he sae geeral equaos o descrbe he hea rasfer. Hea s rasferred fro core o shell ad hece o s surface by coduco ad ood coveco. 6

30 3.4. Hea rasfer Wh he ıssue Cosder a represeave corol volue of ssue. hs basc elee s couous heral coucao wh surroudg ssues. o aa equlbru, heral eergy whch s geeraed sde he elee q s coduced o adjace elees, raspored coveced by he ood srea or sored sde he elee. ˆ q Coduco hea flux q s q V f Blood Srea Fgure 3.3 Corol Volue of ssue Elee As llusraed Fgure. 3.3, eergy-balace equao for hs sall corol volue of ssue s suarzed as: Rae of hea Rae of eergy Rae of eergy Rae of sorage eerg raspored by geerao of eergy hrough he boudg ood srea V V surface of V V Coduco he fudaeals law of hea coduco was developed by Fourer ad he coducve hea flux vecor s assued o obey Fourer s law of coduco. Accordgly, qr, φ,z, r r, φ,z, W/ 3. 7

31 where he eperaure grade s a vecor oral o he soheral surface, he hea flux vecor q r, φ,z, represes hea flow per u e, per u area of he soheral surface he dreco of he decreasg eperaure ad r s referred as he heral coducvy of aala ssue whch s depede upo ssue eperaure ad locao. Sce he hea flux vecor q r, φ,z, pos he dreco of decreasg eperaure, he us sg s cluded Equao 3. o ae hea flow a posve quay. he dvergece of he eperaure, he cyldrcal coordae syse r, φ, z s gve by û r û φ r û φ z z where û r,û φ, û z are he u dreco vecors alog he r, φ, z drecos respecvely. he oal coducve hea hrough he corol surface of ssue s gve by q ˆ da 3.3 A where A s he surface area of he volue elee V, ˆ s he ouward-draw oral u vecor o he surface elee da, q s he hea flux vecor a da. he Dvergece heore s used o cover he surface egral o volue egral. A q ˆ da qr, φ,z, dv 3.4 V Afer egrag he equao 3.4 o over corol volue of he ssue elee, he rae of hea, q r, φ, z,, eerg hrough he boudg surface of V s obaed.. he dervao of q r, φ, z, s gve below. he dvergece of he hea flux vecor q, he cyldrcal coordae syse r, φ, z s gve by q r, φ,z r q r q q z r r r φ φ 3.5 z 8

32 Applyg he equao 3. o he hree copoes of he hea flux vecor he r, φ, z drecos, he below equaos ca be ae; q r, r qφ, r φ ad q z z o oba he dvergece of he hea flux vecor for, he hree copoes of he hea flux vecor gve above are subsued o equao 3.5 qr, φ, z r he 3.6a r r r r φ r φ z z qr, φ,z r 3.6b r r r r φ z q r, φ,z 3.7 r r r r φ z Whe we subsue qr, φ,z, o he equao 3.4, we ca ge he rae of hea q eerg hrough he boudg surfaces of V Meabolc Hea Geeraıo he rae of eergy geerao wh a orgas s defed as he rae of rasforao of checal eergy o hea ad echacal wor by aerobc ad aaerobc eabolc acves. hese acves are he su of he bochecal processes by whch food s broe dow o spler copouds wh he exchage of eergy. he facors whch fluece he eabolc hea geerao clude surface area, age, geder, sress, ad horoes. Alhough eabolc hea geerao s relaed o overall body wegh ad sze, he crcal facor s surface area raher ha wegh self. hs reflecs he fac ha as he rao of body surface area o body volue creases, hea loss o he evroe creases ad he eabolc hea geerao us be hgher 9

33 o replace he los hea. Hece, f wo people wegh he sae, he aller or her perso wll have a hgher eabolc hea geerao ha he shorer or faer perso. Accordg o above forao, he oal aou of hea geeraed he corol volue s gve by: q q dv 3.8 V where q s he specfc rae of hea produco whch ay geerally be a fuco of ssue eperaure ad locao. [6] he hea geerao rae he edu, geerally specfed as hea geerao per u e, per u volue, s deoed by he sybol q r, ad gve he us W/ 3. Fgure 3.4 Relave Corbuos of Orgas o Hea roduco Despopoulos ad Slberagl,3. he aou of hea produced s deered by eergy eabols. A res, approxaely 56% of oal hea produco occurs he eral orgas ad abou 8% he uscle ad s. Durg physcal exercse, hea produco creases several-fold ad he perceage of hea produced by uscular wor ca rase o as uch as 9%.

34 Coveco by he Crculaory Syse Hea produced he body should be absorbed by he oodsrea ad coveyed o he body surface. Because all body ssues are poor coducors of hea. If he hea rasfer he body depeded o coduco, very large eperaure grades would be eeded, ad he ay o adap o varyg evroeal codos would be poor. herefore, he covecve flow of ood hroughou he body s very pora eral hea rasfer Cooey, 976. Whe here s a sgfca dfferece bewee he eperaure of he ood ad he ssue hrough whch flows, covecve hea rasfer wll occur, alerg he eperaure of boh he ood ad he ssue Valvoo,. he effecs of he ood crculao o he eral hea dsrbuo wh he body ca be suarzed hree ajor ways Cooey, I zes eperaure dffereces wh he body. ssues havg hgh eabolc raes are ore hghly perfused, ad hus are ep a early sae eperaure as less acve ssues. Cooler ssues are wared by ood cog fro acve orgas.. I corols effecve body sulao he body s rego. Whe he body wshes o rejec hea, how war ood flow o he eghborhood of he s s creased by vasodlao, ad how he ood s bypassed fro areres o ves va deeper chaels hrough vasocosrco, whe coservao of body hea s val. hese auoac echass eher rase or lower he eperaure grade for hea rasfer by coduco he subs layers. Fgure 3.5 Couer Curre Hea Exchage Exrees Cooey, 976.

35 Fgure 3.5 dcaes ha whe valve s ope, ood flow s roued hrough superfcal capllary bed, allowg effce rasfer of hea o body surface. Blood reurg hrough superfcal ves does o exchage sgfca aous of hea wh deep areral ood. Whe valve s closed superfcal ood flow s reduced, ad os ood reurs va deep ves. 3. Couercurre hea exchage bewee ajor areres ad ves ofe occurs o a sgfca exe. If hea coservao s ecessary, areral ood flowg alog he body s exrees s precooled by loss of hea o adjace veous sreas. hs reduces he eperaure of he lbs ad lowers hea losses. Sce os areres le deep, whle ves occur boh superfcal ad deep regos, he exe of he areroveous hea exchage depeds o he roue ae bac o he body ru by he veous ood. hs s auoacally regulaed by he vasodlao- vasocosrco echass. he capllary bed fors he ajor se for he lve exchage of ass ad eergy bewee he ood srea ad surroudg ssue. he areral ood, ow eperaure ar, flows hrough he capllary bed where coplee heral equlbru bewee he ood ad ssue s aaed. hs exchage s a fuco of several paraeers cludg he rae of perfuso ad he vascular aaoy, whch vary wdely aog he dffere ssues, orgas of he body. he er whch represes he rae of eergy raspored by ood srea s based o Fc s prcple. Assug ha, hea s exchaged wh he ssue he capllary bed ad ha o hea sorage occurs he oodsrea. Hece, oal eergy exchage bewee ood ad ssue s gve q w 3.9 V ar where ad are he desy ad specfc hea of he ood respecvely. Boh properes ca be ae as cosa. W s he ood perfuso rae whch ay geerally be a fuco of locao.

36 Sorage of heral Eergy Whe he body eperaure s cosa he rae of hea sorage he ssue s zero, pracce s eglge over log e perods. However, over shor perods ad severe evroes, hea sorage he ssue, whch ca be a pora copoe of he hea balace, deeres he olerace e for wor. herefore, hea sorage s lely o be pora a for perods of up o few hours. I sall aals s pora for perods of jus a few ues Cea ad Clar, 98. I su, uder rase codos par of he heral eergy geeraed or rasferred o he corol volue ay go o aler he aou sored sde. hus, he rae of chage sorage of heral eergy s gve by: q s dv 3. V where C p s he specfc hea of he varous ssues coprsg he orgas ad s also geerally a fuco of ssue eperaure ad locao Hea Exchage Wh Evroe Hea flow sde he hua body occurs whe he eperaure of he body surface s lower ha ha of he body eror. he body supply o he s s he chef deera of hea raspor o he s. Hea loss occurs by he physcal processes of radao, coduco, coveco, ad evaporao Despopoulos ad Slberagl, 3. 3

37 Fgure 3.6 erceage of Hea Loss a Dffere Evroeal eperaures Despopoulos ad Slberagl, Radao Radao s he loss of hea he for of frared waves. All objecs coually radae eergy accordace wh he Sefa-Bolza law,.e., proporoaely wh he surface area, essvy, ad he fourh power of he absolue eperaure. Whe he surroudg s cooler ha he body, e radave hea loss occurs. Uder oral codos, close o half of body hea loss occurs by radao. I coras, whe oe s surroudg s hoer ha he oe s body, a e hea ga va radao occurs Cooey, 976. he aou of cde radao ha s capured by a body depeds o s area, he cde flux, ad he body s absorpvy. I s coo o esae he absorpvy of a body as equal o s essvy a he eperaure of he surroudgs, alhough hs s srcly rue oly whe he body s radave equlbru wh he surroudgs. he e rae of hea exchage by radao bewee a orgas ad s evroe, usually expressed ers of u area of he oal body surface. q r 4 s 4 a ε * σ * W/ 3.4 where s he essvy whch s approxaely equal o he absorpvy. For cde frared radao, he absorpvy of hua s s very hgh, abou.97, ad s depede of color. For vse lgh, he s has a absorpvy of abou.65-.8, 4

38 depedg o wheher s whe or dar, respecvely Cooey, 976. Sefa- Bolza Cosa, σ, s W/ K 4. s s he surface eperaure of he body or clohes ad a s he eperaure of surroudgs Coveco Whe he body shell rasfers hea o he surroudg ar, coveco also coes o play. Because war ar eds o exed ad rse ad cool ar falls, he wared ar evelopg he body s coually replaced by cooler ar olecules. hs process, called coveco, subsaally ehaces hea exchage fro he body surface o he ar, because he cooler ar absorbs hea by coduco ore rapdly ha he alreadywared ar. he ovees of flud whch carry hea away fro he body surface ay be drve by wo echass: free coveco due o desy dffereces he flud assocaed wh eperaure grades; or forced coveco, due o exeral forces such as wd Cea ad Clar, 98. ure free coveco occurs uder saga codos whe he velocy of he abe oward he perso s zero. Forced coveco hea rasfer occurs whe he abe s approachg he body wh a defe ad usually seady velocy Cooey, 976. Covecve hea losses fro he body are srogly depede o ar velocy. he sples equao for characerzg covecve losses s qc hc * s a W/ 3.5 s s he surface eperaure of he body or clohes ad a s he eperaure of surroudgs. Ad h c s he covecve hea rasfer coeffce. 5

39 Fgure 3.7 ad Slberagl, 3. Scheac Vew of Hea Loss by Coveco Despopoulos Accordg o he equao 3.5 he calculao of hea loss due o coveco requres he esao of coveco hea rasfer coeffce h c, whch s calculaed fro he equao gve below u h D c l b 3.6 a where u s he ussel uber, D lb s he exeral daeer of he lb, h c s he coveco hea rasfer coeffce ad a s he heral coducvy of he ar W/K. a Free Coveco I he suao of desy grades, he body force acs o a flud. he e effec s he boucy force, whch duces free coveco curres. I he os coo case, he desy grade s due o a eperaure grade, ad body force s due o he gravaoal feld Icropera ad Dew, 996. For a subjec wh a ea s eperaure lower ha ar eperaure, he ar adjace o he s surface wll becoe heaed by coduco ad wll rse due o buoyacy. hs s he echas by whch hea s los fro he body by free coveco. 6

40 b Forced Coveco Whe he body s exposed o a wd or s ovg hrough he ar, he aural covecve boudary-layer flow s dsplaced ad he body losses hea by forced coveco. he varaes ha flueced forced coveco are he ea ar velocy, he flow dreco ad he aure of he flow wheher s laar or urbule. he degree of urbulece ad s scale ca have a profoud effec upo he hea loss Evaporao Hea loss by radao ad hea loss by coveco aloe are uae o aa adequae eperaure hoeosass a hgh evroeal eperaures or durg sreuous physcal acvy. Because waer absorbs a grea deal of hea before vaporzg, s evaporao fro he body surfaces reoves large aou of body hea. he waer los by evaporao reaches he s surface by dffuso ad by euroacvaed swea glads. A eperaures above 36 C or so, hea loss occurs by evaporao oly. I addo o hs, he surroudg ar us be relavely dry order for hea loss by evaporao o occur. Hud ar resrcs evaporao. Whe he ar s exreely hud e.g. a ropcal ra fores, he average perso ca o olerae eperaures above 33 C, eve uder resg codos. Despopoulos ad Slberagl,3. Fgure 3.8 Schee Hea Loss by Evaporao Despopoulos ad Slberagl,3. 7

41 I hua body, evaporave hea losses occur by several echass: a Hea losses by dffuso of waer hrough he s b Hea losses by swea secreo c Hea losses by evaporao of waer o spred ar. a Hea Losses by Dffuso of Waer hrough he S Waer dffuso hrough he hua s s par of he sese persprao. hs dffuso oals abou 35 l/day a average perso ad s assued o be proporoal o he dfferece bewee he vapor pressure of waer a he s eperaure ad he paral pressure of waer vapor he abe ar. Iouye Cooey, 976 gave he correlao abou he dffusoal hea loss per u area. qd s a W/ where s he s s he vapor pressure of waer a s eperaure ad a s he paral pressure of waer vapor he abe ar. s ad a are lleers of ercury ad hey are calculaed fro equao 3.5 ad 3.6 respecvely he vapor pressure of waer a s eperaure, s, ad paral pressure of waer vapor he abe ar, a, ca be calculaed by gve forulas Cooey, 976. s.9s 5.3 Hg 3.5 a v %RH Hg 3.6 I equao 3.5 s s he surface eperaure of he lb. I s Celsus C. I equao 3.6 v s he vapor pressure of waer a ar eperaure he codo of a ad RH s he relave hudy of he ar. Dffusoal hea loss ca be calculaed afer subsug he s ad a value whch are calculaed fro equao 3.5 ad

42 b Hea Losses By Swea Secreo Whe he hea loss aou s o aple o aa he core eperaure a suae rage, a auoac echas of he body appears. hs echas for creasg he hea loss s he sweag respose, whch provdes secreo of a dlue elecrolye soluo fro uerous glads o he s surface. he, evaporao fro he weed surface he occurs Cooey, 976. Hea loss by swea secreo per u area s gve by: qs 484 Ke s a W/ where K e s he coeffce for evaporao hea loss fro ude perso ar. I s cal/ hr Hg. he correlaos for K e whch have bee expereally deered by several vesgaors. 9

43 CHAER 4 DEVELOME OF MODEL I hs chaper, passve odelg of he hua heral syse s vesgaed. Facors ha help o buld he odel are gve below. I order o avod he coplexy soe assupos are ade. Lasly, he dervao of Bo-Hea Equao for passve syse ad boudary codos ha ac o he hua body are gve. As eoed before, passve odelg of hua heral syse deals wh he hea rasfer pheoea occurrg hua body ad a s surface. Applcao of hea balace o a ssue corol volue resuls equaos whch sulae he passve syse he aheacal po of vew. Solvg hese equaos by chose ehod leads o predc he heral behavor of ere hua body or a par of for dffere evroeal codos. he objecve of hs sudy s o prese he aheacal odel of hua hea rasfer. he odel s a ul-segeal, ul-layered represeao of he hua body wh spaal subdvos whch sulaes he hea rasfer pheoea wh he body ad a s surface. o sulae adequaely all hese hea-raspor pheoea, he prese odel of he passve syse accous for he geoerc ad aaoc characerscs of he hua body ad cosders he herophyscal ad he basal physologcal properes of ssue aerals. I order o avod he coplexy of he passve syse hua heral syse he followg facors should be cosdered whe aepg such a odel. Geoery of orgas. herophyscal ad physologcal properes of varous orgas ad ssues. 3 Meabolc hea produco. 4 Role of ood hea rasfer. 5 Coduco of hea due o he heral grades. 6 heroregulaory echass he orgas ad her fucos. 7 he eraco wh he evroe ad ar codo. 3

44 4.. Geoery of Orgas he hua body pars resee cylders appearace; herefore s covee o use cylders o odel he hua body. he body s dvded o 6 cocerc cylders, whch depc he head, ec, abdoe lower orso, horax upper orso, upper ars, lower ars, hads, hghs, calves, ad fee. A scheac vew of he odel s gve Fgure 4.. Fgure 4.. Scheac Vew of he Model 4.. herophyscal ad physologcal properes of varous orgas ad ssues. I our odel, each of lbs s subdvded he radal dreco, descrbg he dsrbuos of he varous ssue ypes hroughou he body. Accordg o hs dvso, assug ha he ssue heral coducvy, specfc hea of he ssue, ssue desy ad he ood perfuso rae of he ssue are segeally ufor each layer. 3

45 Fgure 4.. Spaal Dvso of Leg Each ssue elee s assged specfc ypes, e.g. bra, boe, uscle, fa, s, lug or vscera. For exaple, head s dvded axally o four cocerc shells. I head, he er shell s he bra ssue whch s surrouded by a boe shell. hs boe shell s covered wh h layers of fa ad s respecvely. he lugs locag he core of he horax upper orso are surrouded by he rb cage. he oher eral orgas vscera, locag he abdoe lower orso, parally eclosed bac by pelvc boe. Layers of uscle, fa, ad s ecase boh horax ad abdoe regos respecvely. All reag body pars are odeled as a sgle boe core surrouded by uscle, fa ad s. I boe ssue, he vesgaors sugges ha he eergy corbuo of ood flow ad eabols are eglge. hus he ood perfuso rae ad eabolc hea geerao are ae zero Gufger, 975. he heral properes of he dffere body ssues are lsed ae Meabolc Hea roduco he odel accous eabolc hea geerao whch dffers fro he layer o layer. Meabolc hea s assued o be geeraed uforly by eabolc ad checal reacos each seco of he cylder, bu he raes are o ecessarly equal. Soe layers produce uch ore eabolc hea, such as bra, whch has he hghes 3

46 eabolc hea geerao aou 34 W/ 3 aog he oher ssue ypes. I coras, boe ssues have o ay o produce hea; herefore, boe s eabolc hea geerao value s zero. ssue ype of eral orgas vscera also has hgh eabolc hea produco ay. For exaple, lower orso eabolc hea producg vscera s 4 W/ 3. I each layer, eabolc hea geerao s segeally ufor ca be see ae 4.. Body Elees ae 4. heral roperes of Body ssues Fala e.al.999 ssues Head Bra Boe Fa S ec Boe Muscle Fa S hroax Lug Boe Muscle Fa S Abdoe Vscera Boe Muscle Fa Ars Hads Legs Fee S Boe Muscle Fa S Boe Muscle Fa S Boe Muscle Fa S Boe Muscle Fa S r W. -.K g/ 3,8,5 85,85,357,85 85,85 55,357,85 85,85,,357,85 85,85,357,85 85,85,357,85 85,85,357,85 85,85,357,85 85,85 C p J.g -.K - 3,85,59,3 3,68,7 3,768,3 3,68 3,78,7 3,768,3 3,68 3,697,7 3,768,3 3,68,7 3,768,3 3,68,7 3,768,3 3,68,7 3,768,3 3,68,7 3,768,3 3,68 w, x -3 l.s q, W/ 3 3, ,

47 4.4 Role of Blood Hea rasfer he odel accous oly hea exchage bewee he ood capllares ad he ssues. Furher exchage of hea bewee he ajor veous vessels ad he ssue surroudg he s also posse. However, hs exchage s o sgfca copared o he exchage he capllary bed. herefore, hea exchages bewee large ood vessels heselves ad large ood vessels o ssue are o cluded he Equao of heral eergy balace for he ssue. A coo assupo s ade, based o Fc s prcple ha ood eers capllares a he eperaure of areral ood, ar, where hea exchages occurs o brg he eperaure o ha of he surroudg ssue,. here s assued o be o eergy rasfer eher before or afer he ood passes hrough he capllares, so ha he eperaure a whch eers he veous crculao s ha of he local ssue. Aoher assupo s ha here s o hea sorage occurs he oodsrea whe he hea exchage occurs bewee he capllary bed ad he ssue. Hea exchage wh capllary bed ad ssue s occurred accordg o he Equao 3.9. I Equao 3.9, he ood perfuso er, w, s ae hoogeous ad soropc ad ha heral equlbrao occurs he crocrculaory capllary bed hrough he ssue layer. For exaple, abdoe, he vscera layer s hghly perfused ha he oher layer abdoe. Blood perfuso raes of dffere ssues each cylder are gve ae 4.. Also he deses of ood,, ad specfc hea of he ood,, are ae cosa. her values are 6 g/ 3 ad 365 J/gK, respecvely Rdeou he Ieraco wh he Evroe he odel aes o accou ar velocy aroud he body, he abe eperaure ad he relave hudy of he ar order o calculae he hea losses by coveco, evaporao, ad radao. I calculaos, he ufor evroeal codo s assued. hs eas ha he abe ar eperaure, relave hudy of ar, ad wd velocy are he sae for each lb. 34

48 4.6 Radao I hs sudy, he rae of hea exchage by radao bewee a orgas ad s evroe s calculaed by usg he Sefa-Bolza law as gve Equao 4.. I s assued ha he hua body s radave equlbru wh he surroudg. herefore, he absorpvy of he body s ae he sae wh he body essvy a he eperaure of he surroudgs. Durg he calculao, he value of essvy of he hua body s ae.97 whch s he sadard essvy value gve leraure. q r 4 s 4 a ε * σ * W/ Evaporao he aou of hea reoved by evaporao s a fuco of evaporao poeal of he evroe. I s assued ha, odel, evaporave hea losses occur oly by dffuso of waer hrough he s ad by swea secreo. Hea loss by evaporao of waer o spred ar, aely resprao, s egleced our odel Hea Losses by Dffuso of Waer hrough he S I order o calculae he dffusoal hea loss, Iouye s correlao s used Cooey 976. hs correlao gves he dffusoal hea loss per u area. q d s a W/ where s s he vapor pressure of waer a s eperaure ad a s he paral pressure of waer vapor he abe ar. s ad a are lleers of ercury ad hey are calculaed fro Equao 4.3 ad 4.4, respecvely. 35

49 he vapor pressure of waer a s eperaure, s, ad paral pressure of waer vapor he abe ar, a, ca be calculaed by gve forulas Cooey 976. s Hg 4.3 s %RH Hg 4.4 a v I Equao 4.3 s s he surface eperaure of he lb. I s Celsus C. I Equao 4.4 v s he vapor pressure of waer a ar eperaure he codo of a ad RH s he relave hudy of he ar. Dffusoal hea loss ca be calculaed afer subsug he s ad a values whch are calculaed fro Equao 4.3 ad 4.4 o Equao Hea Losses by Swea Secreo I hs sudy, he forulao of hea loss by swea secreo per u area s gve by Cooey 976: q s 484 Ke s a W/ where K e s he coeffce for evaporao hea loss fro ude perso ar. I s cal/ hr Hg. Correlaos for K e whch have bee expereally deered by several vesgaors are show ae

50 ae 4. Evaporao Hea rasfer Coeffce fro ude erso Ar Cooey 976 K e Auhors Codos cal/ hr Hg.7 v.634 Clfford v >.58 /s, sadg, cross flow 9.66 v.5 Clfford v<.5 /s.7 v.37 elso.5 < v < 3.5 /s 8.4 v.37 Machle ad Hach -.6 v.4 Wydha ad As - 9. v.66 Four ad owell - 3. v.6 Four ad owell Coveco he hea rasfer by coveco depeds o he ar veloces, ar properes ad sze of he lbs. herefore, hs sudy, covecve hea rasfer coeffce, h c, s dffere for each lb ad calculaed he progra whch s dscussed succeedg chaper. he equao used our odel for characerzg covecve losses s gve below. q h * W/ 4.6 c c s a s s he surface eperaure of he body or clohes, a s he eperaure of surroudgs, h c s he covecve hea rasfer coeffce. Hea rasfer coeffce s calculaed fro he forula gve below. As see, order o fd ou he h c, he ussel uber, whch vares accordg o he coveco ype, should be ow. hc Dl b u 4.7 a where u s he ussel uber, D lb s he exeral daeer of he lb, h c s he 37

51 coveco hea rasfer coeffce ad s he heral coducvy of he ar W/K. Accordg o he oule eperaure a, radl uber ad heral coducvy W/K ad eac vscosy of he ar /s are foud fro WEB_,3. ussel uber correlaos, whch are used our odel, ad calculao of coveco hea rasfer coeffce boh for free coveco ad forced coveco are gve below Free Coveco I odel, Moeh 973 ussel correlao s used o calculae he ussel uber. Accordg o Moeh, he ussel uber for free coveco fro a hua body s gve by Cea ad Clar 98. u.63 Gr.5 r where r s he radl uber of he ar a gve eperaure ad Gr s he Grashof uber whch s expressed Equao 4.9 s 3 ν a 3 g β - a L Gr 4.9 β 4. f f s a K 4. I Equao 4.9, g s he accelerao of gravy 9.8 /s ad s he heral expaso coeffces of he ar /K whch s gve Equao 4., ν s he eac vscosy of he ar /s, f d he fl eperaure ad s s he surface 38

52 eperaure of he body or clohes ad a s he eperaure of surroudgs ad lasly L s he legh of he lb. For free coveco, he calculao sep of free coveco coeffce hc free s gve below;. Fro Equaos 4., 4., 4.9, fl eperaure K, heral expaso coeffces of he ar /K, ad Grashof uber are calculaed, respecvely.. Calculaed Grashof uber ad he radl uber of he ar a a are subsued o Equao 4.8 order o fd ou ussel uber for free coveco. 3. Fro Equao 4.7, ussel uber, whch s he fuco of h c, s obaed. 4. he calculaed ussel uber fro he correlaos for free coveco labeled wh Equao 4.8 s equaled o he ussel uber, whch s he fuco of of h c, obaed fro Equao 4.7 order o derve h c. 5. Afer applyg hese seps we ca fd he covecve hea rasfer coeffce for free coveco as he fuco of wd velocy Forced Coveco he calculao of a rae of hea loss by forced coveco requres he esao of ussel uber, whch depeds o he sze ad shape of he body, he aure of s surface ad flud properes. Soe relaos are avalae leraure for coo geoerc shapes such as cylder or sphere. Hua body s assued as a sooh cylder wh accepae accuracy order o calculae ussel uber. I forced coveco he ussel uber for a sooh cylder s gve by Cea ad Clar 98 u.6 Re.6 r.33 for 3 Re u.6 Re.8 r.33 for 4 4 Re where r s he radl uber of he ar a gve eperaure ad Re s he Reyolds uber whch s expressed Equao

53 Re D l v b 4.4 νa I Equao 4.4, D lb s he exeral daeer of he lb, v s he velocy of he wd /s ad ν s he eac vscosy of he ar /s. For forced coveco, he calculao sep of forced coveco coeffce hc forced s gve below;. Fro Equao 4.4 Reyolds uber s calculaed.. Accordg o he value of Reyolds uber, he ussel correlao for forced coveco s chose fro Equao 4. or Equao Fro Equao 4.7, ussel uber, whch s he fuco of h c, s obaed. 4. he calculaed ussel uber fro he correlaos for forced coveco s equaled o he ussel uber, whch s he fuco of h c, obaed fro Equao 4.7 order o derve h c. 5. Afer applyg hese seps we ca fd he covecve hea rasfer coeffce for forced coveco as he fuco of wd velocy. ussel uber ca be calculaed accordg o he coveco ype. ussel correlaos for boh free ad forced coveco ca be foud he leraure Cea ad Clar 98 Cooey 976 Icropera ad Dew Hea Coduco Coduco of hea occurs due o he heral grades bewee he ssue ad surroudg ssues. I our odel, coducve hea rasfer bewee he ssue ad surroudg ssue obeys he Fourer s Law of Coduco whch s descrbed Chaper 3. Also, s assued ha, he ssue heral coducvy, whch s gve ae 4., s segeally ufor each layer. 4

54 4. heroregulaory Mechass he Orgas ad her Fucos I hs odel, heroregulaory echas he orgas ad her fucos are o cosdered. herefore; vasooor acvy, sweag, shverg, creased eabols due o gladular acvy, ad pag are egleced. hese paraeers are ae o accou acve odelg of he hua heral syse, whch deals wh aag he hua body s eperaure a a cosa level. 4. assve Syse Equao assve syse Equao s he Equao ha descrbes he passve syse of hua body wh he aheacal po of vew. Forg he passve syse Equao s he fal ad os pora sep o fsh he aheacal odelg of he hua heral syse. I order o udersad he hea rasfer pheoea he ssues he Bo-Hea Equao should be derved for he u voluerc ssue. Accordg he assupos gve pervous secos, he Bo-Hea Equao s wre. A applcao of he Fs Law of herodyacs, he prcple of coservao of eergy, requres ha he rae of hea ga us rae of hea loss s equal o he rae of sorage he ssue. he subsuo of Equaos 3.4, 3.8, 3.9, ad 3. o Equao 3. yelds V qr, φ, z, dv V rc p rdv q r dv V V w r ar dv 4.5 V qr, φ, z, q r r, φ, z, rc p r w r ar dv

55 Equao 4.6 s derved for a arbrary sall volue elee V wh he vvo ssue, hece he volue V ay be chose so sall as o reove he egral; we oba. rc p r r, φ, z, qr, φ, z, q r w r ar 4.7 Subsue he Equao 3.7 o above equao we ca ge he fudaeal equao, whch s called Bo-Hea Equao. hs equao expresses he fac ha, a ay e, he su of he hea rasfer hrough he hree drecos of he cylder, he hea produced by ad he hea raspored by ood, s equal o he rae of eperaure varao, a ay po. r, φ, z, r r r r r r q r r φ z z [ w r ] φ ar 4.8 Equao 4.8 s referred o as he Bo-Hea Equao whch was frs suggesed hs for by ees. hs dffereal equao dsspao a hoogeous, fe ssue volue. descrbes he hea Accordg o Equao 4.8, fro lef o rgh, he frs er s he sorage of hea wh he ssue. [ g/ 3 s he desy of he ssue, J/g K s he specfc hea of he ssue ad lasly, s he e.]. Secod er s he hea coduco er, whch shows he hea flow fro warer o colder ssue rego radal, ageal, ad axal drecos. [ s he ssue coducvy W. -.K -, s he ssue eperaure K]. hrd er, q W/ 3, s hea produced by eabols. I addo, las er s he rae of eergy raspored by ood srea [ ar s he areral eperaure, subscrp refers o ood ad, w, descrbe he desy of ood g/ 3, ood perfuso rae /s ad specfc hea of he ood J/g K respecvely.] hs cobe effec s balaced by he fourh er whch s he Bo-Hea Equao ca be appled o 4

56 all ssue odes by usg he approprae aeral cosa, ad, he basal hea geerao er, q, ad basal ood perfuso rae, w, for each ssue layer. I order o fd ou he eperaure dsrbuo of he overall body, he Bo-Hea Equao gve Equao 4., solved he radal, ageal ad axal drecos by usg he fe dfferece echque. he soluo ehod wll be dscussed succeedg chaper. A he ouer surface of he hua body, hea s reoved by a ler cobao of coveco, radao, ad swea evaporao. hus, for each lb he oal hea loss due o exposg o he evroe s calculaed by su of he radao hea loss, coveco hea loss, ad evaporao hea loss. OAL HEA h LOSS c s a.35 ε σ s a 4 s a K e s a 4.9 he boudary codos of each lb are gve Chaper 5 where he soluo ehod s descrbed. 43

57 CHAER 5 SOLUIO ECHIQUE he aheacal odelg of a physologcal syse resuls a descrpo he ers of equaos such as dffereal equaos whch ca be solved by copuers ad uercal aalyss sofware. he a objecve of hs sudy s o fd ou he eperaure dsrbuo of he uclohed hua body. o acheve hs objecve, paral dervave for of he Bo- Hea Equao should be solved. he fe-dfferece echque s used o solve ou he Bo-Hea Equao by usg explc ehod. I prese odel, each secor of each ssue shell was dvded o odes. User defes he esh sze of he each lbs ad he space of he wo odes r, ad z drecos by choosg he sesvy of he calculaos. Each cyldrcal body par was dvded o sall hree desoal ssue elees as show he fgure below. Fgure 5. Dscrzao of he doa for vercal ad horzoal lbs. 44

58 5. Fe Dfferece For of Bo-Hea Equao he Bo-Hea Equao derved Chaper 4 should be chaged o fe dfferece for order o calculae he eperaure of he odes. rc p r, φ, z, r r r r r r q r φ z z [ w r ] φ ar Le he coordae r, φ,z of a po B a a ode by a cyldrcal coordae syse. Fgure 5. A r, φ,z ewor he cyldrcal coordae syse I Fgure 5., he, j ad subscrps are used o desgae he r, φ ad z locaos of dscree odal pos. o oba he fe-dfferece for of Bo-Hea Equao, we ay use he ceral dfferece approxaos o he spaal dervaves. he ceral dfferece value of he dervaves a he, odal po ca be wre as follows. r,,, r 5. 45

59 ,,, r r, 5.,,, j, φ φ, 5.3,,, z z, 5.4 Ad coordaes of po B s represeed a a ode by r r 5.5 φ j φ 5.6 z z 5.7 A po B, eperaure r, φ, z,, geerao rae q r, desy of he ssue r, specfc hea of he ssue r, coduco hea rasfer coeffce of he ssue r, are ood perfuso rae of he ssue w r, are deoed by r, φ, z, r, jφ, z B, 5.8 r q B r q 5.9 q r r B 5. 46

60 r B r 5. r B r 5. w r B w r w 5.3 I addo o beg dscrezed space, he proe us be dscrezed e. he eger s used for hs purpose, where 5.4 Ad he fe dfferece approxao o he e dervaves Bo-Hea Equao s expressed as r, φ,z,,,, 5.5 he equao 5.5 s cosdered o be a forward-dfferece approxao o he e dervave. he subscrp s used o deoe he e depedece of, ad he e dervaves s expressed ers of he dfferece eperaures assocaed wh he ew ad prevous es. herefore, calculaos us be perfored a successve es separaed by he erval. 5.. he Fe-Dfferece Fors of Bo-Hea Equao for ozero Values of r Whe he equaos fro 5. o 5.5 are subsued o Bo-Hea Equao ad by usg he explc ehod of soluo, he fe dfferece for of he Bo-Hea Equao for o zero values of r ca be obaed. 47

61 ,, r, r, r, r,, φ,j,,, z, q w ar, C p,, 5.6 Whe he ode s a he erface of he wo dffere hoogeeous ssue ypes, he heral properes such as heral coducvy, desy, specfc hea, of hs erface ode ca be obaed by cosderg he eghbor ssue layers. Obag he heral coducvy of erface ode ca be gve as a exaple. If s assued ha he corol volue surroudg he grd po s flled wh a aeral of ufor coducvy ufor coducvy, ad he oe aroud E wh a aeral of E, hus he heral coducvy of erface e he cyldrcal coordae syse ca be wre by usg he haroc ea of ad E. Fgure 5.3 s used o deosrae he erseco face bewee he wo hoogeeous layers. Fgure 5.3 Ierseco face bewee wo layers he os sragh forward procedure for obag he erface heral properes s o use he haroc ea of po ad E. 48

62 e r l r l re re l r e r E E E r E l re r r l e l re re 5.6a Oher heral properes ad he physologcal properes, such as ood perfuso rae, ca be also obaed by usg he haroc ea of he wo eghbor ssue layers properes. 5.. he Fe-Dfferece Fors of Bo-Hea Equao For r rc p r, φ, z, r r r r r q r r φ z z [ w r ] φ ar As see above, he Bo-Hea Equao appears have a sgulares a r. o deal wh hs suao, he Laplaca operaor he cyldrcal coordae syse s replaced by he Caresa equvale so ha Bo-Hea Equao aes for Özıı 994b as r r cpr r q r w r x y z ar 5.7 We cosruc a crcle of radus r, ceer a r. Le be he eperaure a r ad,, 3, 4 be he eperaures a he four odes hs crcle ersecs he x ad y axes as see he Fgure

63 y, 4 r x 3 Fgure 5.4. eperaure of he four odes of a crcle wh ceer r he he fe dfferece for of hs equao abou r becoes 4 c 3 4 p q w ar 5.8 r z _ wh a error of he order of r. If we deoe as he arhec ea of he _ eperaures aroud he crcle of radus r. aely, as he arhec ea of, aroud he crcle of radus r wh ceer a r. _ he a r he equao 5.8 becoes _ c p 4 q w c p ar 5. r z Whe equao 5. s wre fe dfferece fro, he Bo-Hea Equao ca be wre as see equao 5.. 5

64 C p,,, 4 r,, z, q w ar, Boudary Codos I our proe, soe odes locae o a boudary surface. o fd ou he eperaure of hese odes, boudary codos of he should be gve. Because of he srucure of he fe dfferece for of he Bo-Hea Equao, o boudary surface here wll be a fcous ode whch should be gve he er of real ode. Each lb s dvded o several plaes z drecos. pos ou he plae uber z drecos ad for each lb he las value s. Fgure 5.5. Dvso of z axs of a lb. Due o he goeercal ad evroeal codoal syery of each elee s assued ha he hua bosy s syercal accordg o he j ad j M. herefore, he calculao of he eeraure dsrbuo of quarer hua body s eough o ow he whole body ep dsrbuo. 5

65 All boudary codos ha we eed o solve he proe ad he defo of he fcous ode he er of real ode are gve ae 5. ad ae 5.. he boudary codos gve ae 5. are used soluo of he fe dfferece equao of he each lb. Besdes o he boudary codos gve ae 5., he soluo of Bo-Hea Equao for soe lb eeds exra boudary codos. hese exra boudary codos ad he ae of he lb are gve below he ae 5. he followg oao s used boh ae 5. ad ae 5. EE h c a 4 ε σ 4 a s a 484 K 36 e s a By usg ae 5. ad ae 5., he equaos represeg he eperaure of uclohed hua body a e sep are obaed. hese equaos are gve AppedxA. 5

66 53 ae 5. he Boudary Codos of Bo-Hea Equao for each lb. o Boudary Codos Fe Dfferece For of BC Fcous eperaure, j,, j, BC a j φ φ, j,, j, BC φ BC3 a j M, j,, j, φ,, r EE a EE r ae 5. he Boudary Codos of Bo-Hea Equao for soe lb. o BC4 BC5 BC6 ae of lb orso Head Had orso Head Foo z, j,, r EE,, Boudary Codos Fe Dfferece For of BC Fcous eperaure,, EE a EE z z,, EE a EE z z a,, z z EE,, z EE,,,, 53

67 CHAER 6 BIO-HERMAL SOFWARE Sofware called Bo-heral s wre Vsual Basc laguage o solve he syse of equaos of eperaure of he hua body obaed fro Chaper 5. he progra cosss of wo pars called Vsual ad Module. he Vsual par of he progra s gve hs chaper; however he Module par s gve Appedx A because s very log o gve hs chaper. I Vsual par of he progra, frs page s IU page whch user assgs he radus ad legh of each lb, evroeal codos, al eperaure of he hua body ad sesvy of he calculao. he vew of he IU page s gve below. Fgure 6.. Scheac Vew of he Ipu Daa For 54

68 he defaul values see body deso par are ae fro he leraure. he gve szes are he sze of he average hua body. Accordg o he user prefereces, each elee s radus ad legh value ca be chaged. hs leads o our odel flexe body desos. Below body deso par, here s a par called evroeal codos. I hs par, he eperaure ad relave hudy of he ar ad also wd velocy ca be assged. O he rgh sde of he for here s a cursor whch leads o assg he sesvy of he calculaos. User defes he esh sze of he elee by selecg he hgh, edu or low sesvy buo. Afer assgg he body desos ad evroeal codos ad defg he calculao sesvy, IU DAA OK buo s pressed. User should wa few secods because he assged values are rasfored o he Module par of he progra. I a few secods, SECIOAL VIEW buo s acvaed. he user presses hs buo o go o he Secoal Vew age. Fgure 6.. Scheac Vew of he Secoal Vew For 55

69 SECIOAL VIEW page s used o see secoal vew of he eperaure dsrbuo of he full body or he chose lb. I hs page, u eperaure ad axu eperaure ex boxes are used o assg he eperaure rage whch you would le o use by drawg he secoal vew. Wh he help of hese ex boxes, user ca selec he color rage of he secoal vew. Below he u ad axu eperaure ex boxes, here s a eperaure scale whch defe he eperaure value correspodg o he color used secoal vew. As see Fgure 6., buos o he rgh had sde of he eperaure scale lead o us o selec he whch par of he hua body s secoal vew s waed o see he eperaure dsrbuo. I Secoal Vew page, here s a buo labeled GO O UMERICAL RESULS. ressg hs buo leads o show he for called uercal Calculaos whch user assgs he oal e for sulao. I Fgure 6.3 he scheac vew of he uercal Values For ca be see. Fgure 6.3. Scheac Vew of he uercal Values For User frsly, assg he e durao waed o see he eperaure dsrbuo o he ADDIIOAL IME DURAIO ex box, he press he CALCULAE buo order o sar he calculao for fdg he eperaure dsrbuo of he 56

70 hua body. I addo, he durao declarg he calculaed e ca be see by OAL DURAIO ex box. Durg he calculao, user ca see he axu ad u eperaure value he whole hua body ad ae of he lb ha has u or axu eperaure. I UMERICAL VALUES For, SHOW buo s used o see he eperaure value of gve plae of lb ha assged by user. User assgs he ae of he lb ad whch plae he would le o see he eperaure. Afer copleg he calculao durao, by pressg he save daa buo all hese eperaure value gve below are wre a folder called hess daa. I hs folder, he eperaure values of he vscera, lug, orso boe, orso uscle, orso fa, orso s, ec boe, ec uscle, ec fa, ec s, bra, head boe, head fa, head s, leg boe, leg uscle, leg fa, leg s, foo boe, foo uscle, foo fa, ar boe, ar uscle, ar fa, ar s, had boe, had uscle, had fa, had s, ea eperaure of orso, ea eperaure of ec, ea eperaure of head, ea eperaure of leg, ea eperaure of foo, ea eperaure of ar, ea eperaure of had, ea eperaure of he boe ssue for overall body, ea eperaure of he uscle ssue for overall body, ea eperaure of he fa ssue for overall body, ea eperaure of he s ssue for overall body, ea eperaure of he oal body are sored. he oule of he progra s gve below. Sep : Assg he body desos, evroeal codos, al eperaure of he body ad sesvy of he calculao Ipu age of he rogra. Sep : ress Ipu Daa O Buo Sep 3: Geerae elee esh ad calculae esh sze of each lb. Sep 4: Module par of he progra assgs he herophyscal properes of he ssues. Sep 5: Module par of he progra calculaes he properes of ar. Sep 6: ress Go o uercal Resuls buo. Sep 7: Assg oal e for sulao. Sep 8: ress Calculae buo. Sep 9: Chec he ype of coveco wheher s forced or free. Sep : Calculae he hea rasfer coeffce accordg o he coveco ype. Sep : Solve eperaure dsrbuo of he body Sep : Icrease he e sep. 57

71 Sep 3: Solve ew eperaure dsrbuo of he body. Sep 4: Chec he sulao perod f s expred or o Sep 5: If he sulao e s o expred go o Sep. Sep 6: If he sulao e s expred, save he eperaure value of he hua body ad s lbs a folder by usg he Save Daa buo uercal Value page. Sep 7: ress Go o Secoal Vew buo o go o he Secoal Vew page order o see he eperaure dsrbuo seco of he chose lb or full hua body. Sep 8: Evaluae he saved daa ad draw cocluso. Sep 9: Ed rogra. I order o follow he seps easly, he flow char of he progra s gve below. 58

72 - Ial eperaure Value of Hua Body. - Daeer of he Each Lb - Hegh or Legh of he Each Lb - Abe Ar eperaure - Relave Hudy of he ar - Wd Velocy - Sesvy of Calculao ress Ipu Daa O Buo e Geerae Elee Mesh Calculae Mesh Sze For Each Lb Assg herophyscal roperes of ssues Calculae he roperes of Ar - Keac Vscosy ν - radl uber r - heral Coducvy Ka - Vapor ressure of Waer a Ar eperaure v - aral ressure of Waer Vapor he Ar a Equao Coeffce of Evaporao Hea Loss K e ae 4. Go o uercal Calcualo age 59

73 Assg oal e for Sulao ress Calculae Buo Calculae Is forced coveco? Y - heral Expaso Coeffce Equao 4. Calculae for Each Lb - Grashof uber Gr Equao ussel uber u Equao 4.8 Calculae for Each Lb - Reyolds uber Re Equao ussel uber u Equao 4. or Equao 4.3 Calculae for Each Lb Coveco Hea rasfer Coeffce h c Equao 4.4 6

74 Calculae he eperaure Dsrbuo of he Body Fro Equao 5. o Equao 5.7 Icrease e Sep Solve ew eperaure Dsrbuo Chec Sulao erod Is expred? Y Y ress Save Daa Buo e Go o Secola Vew age Choose he Elee o See he Secoal Vew Evaluaed he Sored Daa Ed 6

75 CHAER 7 RESULS ad DISCUSSIO A odel of uclohed hua passve heral odel has bee developed. As every copuer odel, prese odel should be verfed order o esash he accuarcy of he odel. he verfcao process s doe by coparso of odel predcos wh he expereal daa ad he aalycal soluo. o oly does hs process esash he accuracy of he odel, bu s also helps o defe he rage of codos for whch he odel s applcae. 7. Verfcao of he Bo-heral rogra he eperaure dsrbuo of he hua whole body or par of are obaed a gve Cases ae 7. fro he Bo-heral rogra ad he resuls are copared wh he expereal daa or aalycal soluo or predced odel doe by earler sces. ae 7.. Codos of Expereal Resuls ad Aalycal Soluo. Case Ar Relave Wd Ial eperaure C Hudy Velocy /s eperaure C 6 % % % %. 34 he copuer progra s used o fd ou he eperaure dsrbuo of hua body o each cases gve ae 7.. For each copuer ru, he seps gve Chaper 6 are followed. 6

76 CASE Case gve ae 7. s he expere codo of he ees Wssler, 998. He red he foud ou he seady sae eperaure dsrbuo hua ar. As see ae 7., CASE correspods o 6 C ad 5% relave hudy aural coveco codos. he al eperaure of he ar s 36 C. he sae codos are used order o copare calculaed daa. he sudy doe by ees red o solve seady sae eperaure dsrbuo hua ar; herefore, prese odel, he daa s ae whe our syse reached he seady sae codo. Uclohed Hua Forear eperaure vs. e 36,5 36, eperaure C 35,5 35, 34,5 34, 33,5 33, Radus c ee's Expereal Daa rese Sudy Fgure 7.. Coparso of he Hua Forear eperaure wh ees Daa As see Fgure 7. prese sudy resuls follow he sae red ad show a good agreee wh ee s Expereal Daa. 63

77 CASE he expere codo of Case s doe by aaa e al,. Durg he expere, uclohed s eperaures a he ar are easured a roo a 35 C ad relave hudy 9% for half a hour. Ially, he subjec eperaure s easured 3 C. I addo o hs, he expereal resuls are aalyzed by usg he Gagge Model, whch s wo-ode odel of hua herophyscal resposes. he codos gve Case s used prese sudy ad as see Fgure 7. our odel shows beer agreee wh he expereal daa ha calculaed daa by aada e al. he reaso for hs ha, o sulae he s eperaure of ar aada e al used Gagge Model whch depcs he hua body as a cylder whch cosss of oly core ad s. However, prese odel s approxao has ore accuracy ha Gagge Model because prese odel he hua body s dvded o ul-layered 6 cylders whch lead o show good agreee wh expereal daa Uclohed S epeaure of Ar vs. e Uclohed S epeaure of Ar C Ref [] Expereal Daa rese Sudy Ref [] Calculaed Daa 3 3 e Fgure 7.. Coparso of he Uclohed S eperaure of Ar wh aaa e al Daa. 64

78 CASE 3 Eberhar Fala e al., 999 solves he eperaure wh a hoogeous cylder of uscle ssue wh a aalycal ehod. he codo as descrbed Case 3 s used. he cylder s ally heral equlbru wh he evroe a C. A he cylder suddely suppled wh a cosa rae of ood flow herefore he eperaure of cylder rses qucly before covergg asypocally o he fal seady sae. he calculaed eperaure of he uscle ssue leg s copared wh a aalycal soluo of Eberhar. As see he Fgure 7.3, calculaed uscle ssue eperaure leg by prese odel shows good agreee wh he aalycal soluo. Leg Muscle ssue eperaure vs. e eperaure C e Eberhar Aalycal Soluo rese Sudy Fgure 7.3. Coparso of he Muscle ssue eperaure of Leg wh Eberhar Aalycal Soluo 65

79 Soe secoal vew of he leg s ae a ues, ues, order o sulae he chages eperaure wh e. a ues b ues c 4 ues d 6 ues Fgure 7.4. Sulao of eperaure Chages Leg wh e 66

80 CASE 4 he uercal values Case 4 are he codo of a odel doe by Ar ad Shzer. hs odel s cobed odel of he hua heral syse, aely cludes boh acve ad passve syse sulaos. Alhough we oly odeled he passve syse of he hua heral syse ca be see ha s good agreee wh he calculaed daa by Ar ad Shzer. he devao bewee wo graphs s caused by he hua heral resposes such as vasocosrco, vasodlao, sweag or shverg. Mea S eperaur vs. e eperaure C 34, 34 33,8 33,6 33,4 33, 33 3,8 3, e rese Sudy Ref [7] Calculaed Daa Fgure 7.5. Coparso of he Mea S eperaure of Hua Body wh Ar ad Shzer s Daa. Accordg o prese odel s calculaed daa, he eperaure chages ea s eperaure of he hua body wh e are gve below. 67

81 a Afer 5 ues b Afer 3 ues c Afer 45 ues d Afer 6 ues Fgure 7.6. Sulao of eperaure Chages Hua Body wh e 68

82 SCEARIO Apar fro coparso wh oher sudes, a scearo s wre order o show he eperaure chages ea body ad s wh respec o e. I hs scearo, a a a eers he Roo wh codo of 8. C ad 43% relave hudy. hs roo codo s he rage of heral cofor zoe defed by ASHRAE. Afer ues, he eers a roo whch s hoer ha prevous roo wh 47.8 C roo eperaure ad 7% relave hudy. hs roo s codo s slar o he saua codo descred WEB_4, 3. Afer ues he eers a dffere roo wh sae eperaure value wh Roo bu hgher relave hudy whch s 99%. Afer ues, he eers he fourh roo whch s a 47.8 C ad % relave hudy. Afer ues, he eers aga Roo. ae 7. Codos of Roo he Scearo Roo Ar eperaure C Relave Hudy e Durao 8. 5% % % % 69

83 Fgure 7.7. Sulao of eperaure Chages Hua Body wh e As see he Fgure above, he eperaure of s s affeced draacally whe he evroeal codo chages. S eperaure reached s hghes value whe he a s Roo 3. he reaso for hs s ha, whe he eperaure ad he relave hudy of he evroe rses hea loss by dffuso of waer hrough s ad hea loss by evaporao by swea decreases. hs causes hgh eperaure value he body whch s very harful eve faal for hua. 7

84 CHAER 8 COCLUSIOS I prese odel, a rase hree desoal aheacal odelg of he hua passve syse has bee developed usg fe dfferece echque. I order o crease he accuracy of he odel, he hua body s dvded o 6 cylders ad each cylder s dvded o four radal layers whch represe he ssue ype such as boe, fa, uscle, s, vscera, lug, ad bra. Accordg o assupos, he Bo- Hea Equao s derved ad wre for each ssue elee. As a resul of hs, fro Equao 5. o 5.34, whch are gve Appedx A, are obaed. I prese sudy, s s assued ha he hua body dsspaes hea by he cobao of coveco, radao ad evaporao. A progra, whch s wre Vsual Basc called Bo-heral, s used o deere eperaure dsrbuo a succeedg e sep of he vscera, lug, bra all ssue ype of he orso, ec, head, leg, foo, ar, had, ad ea eperaure of orso, ec, head, leg, foo, ar, had. Addoally, for overall body, ea eperaure of he boe ssue, uscle ssue, fa ssue, ad s ssue ad ea eperaure of he oal body ca be obaed by Bo-heral Sofware. Also, he sofware s o be capae of deosrag he secoal vew of he varous body lbs ad full hua body. he sofware s gve Appedx B. Ule prevous odels, hs odel, sofware user defes he desos of he body lbs ad evroeal codos. Addoally, user ay choose he sesvy of he calculao accordg o desred sesvy. All hese properes of he sofware ae our odel flexe. Also, he odel s capae of showg he secoal vew of he chose par of he hua body or self. he vsual for of he calculao, secoal vew of he hua body, gves dea o he user abou whch par of he body s geg cold or ho faser. o verfy he accuracy of he odel, copuer sulao resuls are copared wh avalae expereal daa, aalycal soluo ad calculaed daa of prevous sudes for dffere evroeal codos. Based o hese coparsos, soe coclusos are draw regardg odel accuracy ad applcay. 7

85 I addo a scearo s wre for sulag he eperaure a aboral codos. I hs scearo, could be see ha he ea s eperaure of he hua body reaed alos cosa alhough, he s eperaure of he body flucuaed wh varyg evroeal codos. he prese odel ca be elarged by ag o accou he hea loss by resprao, clohg sulag facor ad eabolc hea geerao durg physcal acves. hese could be he fuure wor of oher sudes. 7

86 AEDIX A he equaos gve below represeg he eperaure of hua body a e sep s gve hs Appedx A. Secoal vew of he each lb s gve before explag he equaos. Frs secoal vew s a. he secoal vew of oher plaes are gve. he vew s uberzed rego. Each rego has s ow eperaure equao. he boudary codos whch help us o solve he uow eperaure are gve ae 5. ad ae 5.. he descrpos of he regos ad her odes are explaed below. I order o splfy he equaos he followg abbrevaos are used. I s pora o say ha for each lb he value of,, ad M are chaged. orsoervalr orsoervalz orsoervalφ M 4 ecervalr ecervalz - ecervalφ M 4 Headervalr Headervalz Headervalφ M 4 Arervalr Arervalz - Arervalφ M 4 Legervalr Legervalz - Legervalφ M 4 Hadervalr Hadervalz Hadervalφ M 4 Fooervalr Fooervalz Fooervalφ M 4 73

87 ORSO A A o - A 74

88 75 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.. ar W q z,,,,,, r,, 4,,,,,, 5. I Equao 5.,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. he eperaure of he ode,, a ex e sep s gve below.,,,, ar W q,, EE z,,,, z r,, 4,, α 5.3 Rego ad 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

89 I Equao 5.4,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. For Rego, Equao 5.4 s he fcous eperaure. I order o,, elae hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego, he eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,, α,,,,,, z EE,,,, r φ z q W,, ar For Rego 3, Equao 5.4,,,,,, 5.5 s he eperaure value of leg. herefore, sead of ode orso,,, Leg should be,,legervaz - subsued. A Rego 3, he eperaure of he ode,, a ex e sep s,,,,,,,,,, r r,, α Leg,,,,,,,, leg ervalz r φ z q W ar,,,,,,,, gve below

90 Rego 4, 5 ad 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r φ z 5.7 q W, ar For Rego 4 ad Rego 6, Equao 5.7 s he fcous, eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego 4, he eperaure of he ode, a ex e sep s gve below.,,,,, r r, α, j,,, j, z EE,, r φ z, 5.8 q W, ar, For Rego 5, Equao 5.7, s he eperaure value of leg. herefore, sead of ode orso,, Leg should be, legervaz - subsued. A Rego 3, he eperaure of he ode, a ex e sep s gve below. 77

91 ,,,,, r r, α, j,,, j, Leg,,leg ervalz,, r φ z 5.9 q W, ar, Rego 7 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6., M,, M,, M,, M,, M,, M,, M, r r, M,, M,, M,, M,, M,, M, r φ z 5.3 q W, M, ar I Equao 5.3, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. I addo, s also he fcous eperaure. Boudary Codo 4 gve ae 5. s,m, used o elae hs fcous eperaure. A Rego 7, he eperaure of he ode,m, a ex e sep s gve below.,,,,,,,,,, r r, M, α, M,, M,, M, z EE, M,, M, r φ z, M, 5.3 q W, M, ar, M, 78

92 79 Rego 8 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.3,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of leg. herefore, sead of ode orso,,, Leg,, legervaz - should be subsued. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 8, he eperaure of he ode,, a ex e sep s gve below.,,,, ar W q z,,,, - Leg,,legervaz r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α 5.33,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

93 Rego 9 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r,,,,,, r φ z 5.34 I Equao 5.34, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 4 gve ae 5. s used. Also s, he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. below. q W ar, A Rego 9, he eperaure of he ode, a ex e sep s gve r EE,, r, r EE, α,, r, j,,, j, z EE,, r φ z, 5.35 q W, ar, Rego Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao

94 I Equao 5.36,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode,m, a ex e sep s gve below. 5.37,M, ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,m,,m, φ,m,,m, ar W q,m, EE z,m,,m, z r,,m,m,,,m,m,,m, EE r r,m, EE r,m,,m, r,m, φ α

95 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5..,,,,,,,,,,,, 4 q r z W ar,, 5.38 I Equao 5.38 here s o fcous eperaure. he eperaure of he ode,, a ex e sep s gve below.,,,, α 4 r z,, z EE,,,, 5.39 q W,, ar,, Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6. C p,,,,,,, r r r, j,,, j,,, r φ z, 5.4 q, w ar, I Equao 5.4 here s o fcous eperaure. he eperaure of he ode, a ex e sep s gve below. 8

96 Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.4,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. he eperaure of he ode,, a ex e sep s gve below. 5.43,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,, φ α,, ar W q z r r r r,,,,, j,,, j,,,,,, φ α

97 Rego 4 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,M,,M,,M, M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z q W,M, ar 5.44 I Equao 5.44, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. he eperaure of he ode,, a ex e sep s gve below., M,, M,, M,, M,, M, r r, M, α, M,, M,, M,, M,, M,, M, r φ z q W, M, ar, M, 5.45 Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z q W,M, ar

98 85 I Equao 5.46,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 5, he eperaure of he ode,m, a ex e sep s gve below Rego 6 ad Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ 5.48 M,, M,, ar W q z M,, M,, M,, r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α

99 86 I Equao 5.48,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. For Rego 6,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I cocluso, a Rego 6, he eperaure of he ode,, a ex e sep s gve below For Rego, Equao 5.49,, s he eperaure value of ar. herefore, sead of ode orso,,, Ar,, arervaz - should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below. 5.5,,,, ar W q z,,,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α,, ar W q z,,,,,, r,,,,,,,, arervaz -, Ar r arerva z -, Ar EE r,,,, r,, φ α

100 Rego 7, 8, 9 ad Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,, r, j,,, j, r φ,, r,, z, 5.5 For Rego 7, 8 ad 9, Equao 5.5, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. For hese rego, he eperaure of he ode, a ex e sep s gve below. q W ar,, r, r EE r EE, α, r, j,,, j,, r φ,,,, z 5.5 q W ar,, For Rego, Equao 5.5, s he eperaure value of ar. herefore, sead of ode orso, subsued., Ar should be, legervaz - 87

101 below. A Rego, he eperaure of he ode, a ex e sep s gve, r, r EE, α Ar, arerva z -, r, j,,, j,, r φ Ar, arerva z -,, z 5.53 q W ar,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5..,,,,,,,,,,,, 4 q r z 5.54 W,, ar For Rego 3, Equao 5.54,, s he eperaure value of ec. herefore, sead of ode orso,,, ec should be subsued.,, A Rego 3, he eperaure of he ode,, a ex e sep s gve below. 88

102 Rego 4 ad 5 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.56,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. For Rego 4, Equao 5.56,, s he eperaure value of ec. herefore, sead of ode orso,,, ec,, should be subsued. A Rego 4, he eperaure of he ode,, a ex e sep s gve below. 6.57,,,, ar W q,,,,,, ECK z r,, 4,, α,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,, ECK,,,, r,,,,,, r,,,, r,,,,,,,, φ α

103 For Rego 5, Equao 5.56 s he fcous eperaure. I,, order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. gve below. A Rego 5, he eperaure of he ode,, a ex e sep s,,,,,,,,,, r r,, α,,,,,, z EE,,,,,, r z φ 6.58 q W ar,,,, Rego 6 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r φ z 5.59 q W ar, I Equao 5.59,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. 9

104 9 I addo,,, s he eperaure value of leg. herefore, sead of ode orso,,, Ar,, arervaz - should be subsued. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. A Rego 6, he eperaure of he ode,, a ex e sep s gve below. 5.6 Rego 7 ad 8 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.6,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used.,,,, ar W q,, EE z,,,, r,,,,,,,, arervaz -, Ar r arervaz -, Ar EE r,,,, r,, φ α,,m ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,,,m, φ

105 For Rego 7, Equao 5.6, M, s he eperaure value of ec. herefore, sead of ode orso, M,, ec should be subsued.,m, A Rego 7, he eperaure of he ode,m, a ex e sep s gve below., M,, M,, M,, M,, M, r r, M, α, M,, M,, M,, M,, M, ECK, M, r φ z q W, M, ar, M, 5.6 For Rego 8, Equao 5.6 s he fcous eperaure. I,M, order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. gve below. A Rego 8, he eperaure of he ode,m, a ex e sep s,m,,m,,m,,m,,m, r r,m, α,m,,m,,m, z EE,M,,M,,M, r φ z q W,M, ar,m, 5.63 Rego 9 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao

106 I Equao 5.64,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 9, he eperaure of he ode,m, a ex e sep s gve below. 5.65,M, ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,m,,m, φ M,, M,, ar W q M,, EE z M,, M,, z r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α

107 Rego 3 ad 3 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,, r, j,,, j, r φ,, r,, z, 5.66 q W ar, For Rego 3, Equao 5.66, s he eperaure value of ec. herefore, sead of ode orso,, ec should be subsued., A Rego 3, he eperaure of he ode, a ex e sep s gve below., α,, r,,, j, r φ,,,, r ec,, z 5.67 q W ar,, For Rego 3, Equao 5.66, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. below. A Rego 5, he eperaure of he ode, a ex e sep s gve 94

108 ,,, j, r φ,, α, z, r,,, z EE, r,, 5.68 q W ar,, Rego 3 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r φ z 5.69 q W ar, I Equao 5.69, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 5 gve ae 5. s used. I addo,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. gve below. A Rego 3, he eperaure of he ode, a ex e sep s 95

109 96 5.7,, ar W q, EE z,, z r, j,,, j,,, EE r r, EE r,, r, φ α

110 ECK A A o - A 97

111 98 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5..,, ar W q z,,,,,, r,, 4,,,, 5.7 I Equao 5.7,, s he eperaure value of orso. herefore, sead of ode ec,,, orso orsoervalz,, should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below. 5.7 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q,,,, orso z r,, 4,, valz orsoer,, α

112 I Equao 5.73, s he fcous eperaure. I order o elae hs,, fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of orso. herefore, sead of ode ec,,, orso should be subsued. A Rego, he eperaure of he ode,, orsoervalz,, a ex e sep s gve below.,,,, r,,,,,,,,,, α r φ orso, orso ervalz z,,,, r,,,, q 5.74 W ar,,,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r q,,,,,,,,,,,, r φ z 5.75 W ar,, 99

113 I Equao 5.75, s he fcous eperaure. I order o elae hs,, fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of orso. herefore, sead of ode ec,,, orso should be subsued. Also,, orsoervaz,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below.,, r r EE,, r EE,, α r,,,,,,,,,,,, orso,,orsoervaz- r φ z,,,, 5.76 q W ar,,,, Rego 4 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6., M,, M,, M,, M,, M,, M,, M, r r q, M,, M,, M,, M,, M,, M, r φ z W, M, ar 5.77

114 I Equao 5.77, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of orso. herefore, sead of ode ec,,m, orso should be subsued. A Rego 4, he eperaure of he, M, orsoervaz ode,m, a ex e sep s gve below.,,,,,,,,,, r r, M,, M,, M,, M, α r φ orso, M,, orsoer valz, M, z 5.78 q W ar, M,, M, Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.79 q W,M, ar

115 I Equao 5.79, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of orso. herefore, sead of ode ec,,m, orso should be subsued. Also, M, orsoervaz,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. gve below. A Rego 5, he eperaure of he ode,m, a ex e sep s r EE, M,, M,, M, r r EE, M, α, M,, M, r orso, M,, M,, M,,M,orsoervalz, M,, M, r φ z 5.8 q Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6. W ar, M,, M,,,,,,,, r r,,, j, r φ,, z, q 5.8 W ar, I

116 I Equao 5.8, s he eperaure value of orso. herefore, sead of ode ec, orso, should be subsued. A Rego 6, he, orsoervaz eperaure of he ode, a ex e sep s gve below., α,,,,, r r, j,,, j, r φ orso, orsoer valz,, z 5.8 q W ar,, Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r q,,,,,, r φ z 5.83 W, ar I Equao 5.83, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo,, 3

117 4 s he eperaure value of orso. herefore, sead of ode ec,, orso orsoervaz, should be subsued. A Rego 7, he eperaure of he ode, a ex e sep s gve below Rego 8 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.85 here s o fcous eperaure. he eperaure of he ode,, a ex e sep s gve below.,, ar W q z,, orsoervalz, orso r, j,,, j,,, EE r r, EE r,, r, φ α,, ar W q z,,,,,, r,, 4,,,,

118 Rego 9 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.87,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. A Rego 9, he eperaure of he ode,, a ex e sep s gve below. 5.88,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,, φ α,,,, ar W q z,,,,,, r,, 4,, α

119 6 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.89,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I cocluso, a Rego 6, he eperaure of he ode,, a ex e sep s gve below. 5.9,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,,,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α

120 Rego Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,M,,M,,M, M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z q W ar,m, Equao 5.9,,M, 5.9 s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. he eperaure of he ode,m, a ex e sep s gve below. I, M,, M,, M,, M,, r α, M,, M,, M,, r φ M,, M, r, M,, M, z M, 5.9 q W ar, M,, M, Rego Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.93 q W ar,m, 7

121 8 I Equao 5.93,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode,m, a ex e sep s gve below Rego 3 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao a r gve Equao M,, M,, ar W q z M,, M,, M,, r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α, ar W q z,,, r, j,,, j, r,, r,,,,, φ

122 I Equao 5.95 here s o fcous eperaure. he eperaure of he ode, a ex e sep s gve below.,,,,, r r r, α, j,,, j,,,, r z φ 5.96 q W, ar, Rego 4 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r φ z q W, ar 5.97 I Equao 5.97 s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 3 gve ae 5. s used. For hese rego, he eperaure of he ode, a ex e sep s gve below. 9

123 5.98 Rego 5 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.99,, s he eperaure value of head. herefore, sead of ode ec,,, Head,, should be subsued. A Rego 5, he eperaure of he ode,, a ex e sep s gve below. 5.,, ar W q z,,, r, j,,, j,,, EE r r, EE r,, r, φ α,, ar W q z,,,,,, r,, 4,,,,,,,, ar W q,, Head,,,, z r,, 4,, α

124 Rego 6 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of head. herefore, sead of ode ec,,, Head,, should be subsued. A Rego 6, he eperaure of he ode,, a ex e sep s gve below. 5. Rego 7 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,, Head,,,, r,,,,,, r,,,, r,,,,,,,, φ α

125 5.3 I Equao 5.3,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of head. herefore, sead of ode ec,,, Head,, should be subsued. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 7, he eperaure of he ode,, a ex e sep s gve below. 5.4, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,, Head,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α

126 Rego 8 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,, r r,m,,m,,m,,m,,m,,m, r φ z q W,,M ar 5.5 I Equao 5.5,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,, M, s he eperaure value of head. herefore, sead of ode ec, M,, Head should be subsued.,m, gve below. A Rego 8, he eperaure of he ode,m, a ex e sep s,m,,m,,m,,m,,m, r r,m, α Head,M,,M,,M,,M,,M,,M, r φ z q W ar,m,,m, 5.6 Rego 9 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao

127 4 5.7 I Equao 5.7,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of head. herefore, sead of ode ec,m,,,m, Head should be subsued. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 9, he eperaure of he ode,m, a ex e sep s gve below. 5.8,M, ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,m,,m, φ,m,,m, ar W q z,m, Head,M,,M, r,,m,m,,,m,m,,m, EE r r,m, EE r,m,,m, r,m, φ α

128 Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,, r, j,,, j, r φ,, r,, z, 5.9 q W ar, I Equao 5.9, s he eperaure value of head. herefore, sead of ode ec,, Head, should be subsued. A Rego, he eperaure of he ode, a ex e sep s gve below., α,, r,,, j, r φ,,,, r Head,, z 5. q W ar,, Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao

129 ,,,,,,, r r, j,,, j,,,, r φ z 5. q W ar, I Equao 5.,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo, s he eperaure value of head. herefore, sead of ode, ec,, Head, should be subsued. below. A Rego, he eperaure of he ode, a ex e sep s gve, r, r EE r EE,, r, α, j,,, j, r φ Head,,, z, 5. q W ar,, 6

130 HEAD A A o - A 7

131 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5..,,,,,,,,,,,, 4 q r z 5.3 W ar,, I Equao 5.3,, s he eperaure value of ec. herefore, sead of ode Head,,, ec should be subsued.,,ecervalz - below. A Rego, he eperaure of he ode,, a ex e sep s gve,,,, α 4 ec,,,,ecervalz -,, r z q W,, ar,, 5.4 Rego -3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ 5.5 q W ar,, 8

132 I Equao 5.5,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. For Rego, Equao 5.5 s he eperaure value of ec.,, herefore, sead of ode Head,,, ec should be,, ecervalz - subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,, α ec,,,,,,,,ecervalz -,,,, r z φ q W ar,,,, 5.6 For Rego 3, Equao 5.5 s he fcous eperaure. I order,, o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,, α,,,,,, z EE,,,, r φ z,, 5.7 q W ar,,,, Rego 4 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao

133 ,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ 5.8 q W ar,, I Equao 5.8,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. below. A Rego 4, he eperaure of he ode,, a ex e sep s gve,, r r EE,, r EE,, α r,,,,,,,,,, r φ z,,,, z EE,,,, 5.9 q W ar,,,,

134 Rego 5-6 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m,,m,,m, r z φ 5. q W ar,m, I Equao 5., s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. For Rego 5, Equao 5.,M, s he eperaure value of ec. herefore, sead of ode Head, ec, M,- should be, M, ecervalz - subsued. below. A Rego 5, he eperaure of he ode, M, a ex e sep s gve,,,,,,,,,, r r,m, α,m,,m,,m, ec,m,,m,ecervalz-,m, r φ z 5. q W,M, ar,m, For Rego 6, Equao 5.,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego 6, he eperaure of he ode,m, a ex e sep s gve below.

135 ,,,,,,,,,, r r,m, α,m,,m,,m, r φ z z EE,M,,M,,M, 5. q W ar,m,,m, Rego 7 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.3 q I Equao 5.3,,M, W,M, ar s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. below. A Rego 7, he eperaure of he ode,m, a ex e sep s gve

136 , M, α r EE, M,, M, r, M, r EE, M,, M, r, M,, M,, M, r φ z EE, M,, M,, M, z 5.4 q W ar, M,, M, Rego 8-9 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,, r z φ q W ar,, 5.5 For Rego 8, Equao 5.5, s he eperaure value of ec. herefore, sead of Head,, ec should be subsued., ecervalz - A Rego 8, he eperaure of he ode, a ex e sep s gve below.,,,,, r r, α ec, j,,, j,, ecervalz -,, r z φ q W ar,, 5.6 3

137 For Rego 9, s he fcous eperaure. I order o elae hs, fcous ode Boudary Codo 4 gve ae 5. s used. A Rego 9, he eperaure of he ode, a ex e sep s gve below.,,,,, r r, α, j,,, j, z EE,, r φ z, 5.7 q W, ar, Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r,,,,,, r φ z 5.8 q W ar, I Equao 5.8, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 4 gve ae 5. s used. Also, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode, a ex e sep s gve below. 4

138 5 5.9 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.3 here s o fcous eperaure. he eperaure of he ode,, a ex e sep s gve below. 5.3 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,, ar W q, EE z,, z r, j,,, j,,, EE r r, EE r,, r, φ α,, ar W q z,,,,,, r,, 4,,,,,,,, ar W q z,,,,,, r,, 4,, α

139 6 5.3 I Equao 5.3,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. A Rego, he eperaure of he ode,, a ex e sep s gve below Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao ,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,,,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,, φ α,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

140 I Equao 5.34,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I cocluso, a Rego 3, he eperaure of he ode,, a ex e sep s gve below. r EE,,,,,, r r EE,, α,,,, r,,,,,,,,,,,, r φ z 5.35 q W ar,,,, Rego 4 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao 5.6.,M,,M,,M, M,,M, r r,m,,m,,m,,m,,m,,m,,m,,m, r z φ 5.36 q W ar,m, I Equao 5.36,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. he eperaure of he ode,m, a ex e sep s gve below. 7

141 Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.38,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 5, he eperaure of he ode,m, a ex e sep s gve below M,, M,, ar W q z M,, M,, M,, r, M, M,,, M, r M,, M,, r M,, M,, M,, M,, φ α M,, ar W q z M,, M,, M,, r, M, M,,, M, r M,, M,, r M,, M,, M,, M,, M,, φ M,, M,, ar W q z M,, M,, M,, r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α

142 9 Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.4 here s o fcous eperaure. he eperaure of he ode, a ex e sep s gve below. 5.4 Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao , ar W q z,,, r, j,,, j, r,, r,,,,, φ,, ar W q z r r r r,,,,, j,,,,j,,,,, φ α, ar W q z,,, r, j,,, j, r,, r,,,,, φ

143 3 I Equao 5.4, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. For hese rego, he eperaure of he ode, a ex e sep s gve below Rego 8 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.44,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. A Rego 8, he eperaure of he ode,, a ex e sep s gve below. 5.45,, ar W q z,,, r, j,,, j,,, EE r r, EE r,, r, φ α,, ar W q z,,,,,, r,, 4,,,,,,,, ar W q,,,, EE z,, z r,, 4,, α

144 3 Rego 9 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.46,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,,,, ar W q,, EE z,,,, z r,,,,,, r,,,, r,,,,,,,, φ α,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

145 3 I Equao 5.48,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. I addo,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode,, a ex e sep s gve below Rego Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao ,,,, ar W q,, EE z,,,, r,,,,,,,, EE r,, r,, EE r,,,, r,, φ α,,m ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,,,m, φ

146 I Equao 5.5,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used.,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. gve below. A Rego, he eperaure of he ode,m, a ex e sep s,m,,m,,m,,m,,m, r r,m, α,m,,m,,m, z EE,M,,M,,M, r φ z q W,M, ar,m, 5.5 Rego Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.5 q W,M, ar I Equao 5.5,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. 33

147 34 I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode,m, a ex e sep s gve below Rego 3 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.54, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. M,, M,, ar W q M,, EE z M,, M,, z r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α, ar W q z,,, r, j,,, j, r,, r,,,,, φ

148 below. A Rego 3, he eperaure of he ode, a ex e sep s gve,,, j, r φ,, α, z, r,,, z EE, r,, 5.55 Rego 4 q Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6. W ar,,,,,,,,, r r, j,,, j,,,, r φ z 5.56 q W ar, I Equao 5.56,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 5 gve ae 5. s used. I addo,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. gve below. A Rego 4, he eperaure of he ode, a ex e sep s 35

149 ,, ar W q, EE z,, z r, j,,, j,,, EE r r, EE r,, r, φ α

150 ARM A A o - A 37

151 38 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.58,, s he eperaure value of had. herefore, sead of ode Ar,,, Had,, hadervalz should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below.,,,, ar W q,,,, Had z r,, 4,, alz haderv,, α 5.59 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,, ar W q z,,,,,, r,, 4,,,,

152 I Equao 5.6, s he fcous eperaure. I order o elae hs,, fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of had. herefore, sead of ode Ar,,, Had should be subsued. A Rego, he eperaure of he ode,, hadervalz,, a ex e sep s gve below.,,,,,,,,,, r r,,,,,,,, α r φ Had, had ervalz z,,,, q 5.6 W ar,,,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r q,,,,,,,,,,,, r φ z 5.6 W ar,, 39

153 I Equao 5.6,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of had. herefore, sead of ode Ar,,, Had should be subsued. Also,, hadervaz,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below.,, r r EE,, r EE,, α r,,,,,, Had,,,,,,,, hadervaz r φ z,,,, 5.63 q W ar,,,, Rego 4 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6., M,, M,, M,, M,, M,, M,, M, r r q, M,, M,, M,, M,, M,, M, r φ z W, M, ar

154 I Equao 5.64, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of had. herefore, sead of ode Ar,,M, Had should be subsued. A Rego 4, he eperaure of he, M,hadervaz ode,m, a ex e sep s gve below.,,,,,,,,,, r r, M,, M,, M,, M, α r φ Had, M, hadervaz, M,, M, z 5.65 q W ar, M,, M, Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.66 q W,M, ar I Equao 5.66 s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s 4

155 he eperaure value of had. herefore, sead of ode Ar, Had,M, should be subsued. Also, M, hadervaz,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 5, he eperaure of he ode,m, a ex e sep s gve below. r EE, M,,M,,M, r r EE, M, α,m,,m, r Had,M,, M,,M,, M, hadervaz, M, r φ z,m, 5.67 q W ar, M,, M, Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r,,, j, r φ,, z, q 5.68 W ar, 4

156 I Equao 5.68, s he eperaure value of had. herefore, sead of ode Ar, Had, should be subsued. A Rego 6, he, hadervaz eperaure of he ode, a ex e sep s gve below.,,,,, r r, j,,, j,, α r φ Had, hadervaz,, z q W ar,, 5.69 Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r q,,,,,, r φ z 5.7 W, ar I Equao 5.7, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo,, 43

157 s he eperaure value of had. herefore, sead of ode Ar,, Had should be subsued., hadervaz below. A Rego 7, he eperaure of he ode, a ex e sep s gve r, r EE, r EE, α r,,, Had,,, j,, hadervaz r φ z,, 5.7 q W ar,, Rego hese rego s equaos ca be obaed by usg he sae boudary codos ec Rego herefore, he eperaure equaos a are he sae wh ec Rego Rego 5 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5..,,,,,,,,,,,, 4 q r z 5.7 W,, ar 44

158 I Equao 5.7,, s he eperaure value of orso. herefore, sead of ode Ar,,, orso should orso ervalr,arrasph,arrasz be subsued. A Rego 5, he eperaure of he ode,, a ex e sep s gve below.,, 4 r,, α,,,, orso orsoervalr,arrasph,arrasz z 5.73 q W ar,,,, Rego 6 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r φ z 5.74 q W ar,, I Equao 5.74,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of orso. herefore, sead of ode Ar,,, 45

159 orso should be subsued. A Rego 6, he orso ervalr, arrasph, arrasz eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,,,,,,,, α r φ orso,,,, orso ervalr, arrasph, arrasz z 5.75 q W ar,,,, Rego 7 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r φ z 5.76 q W ar, I Equao 5.76,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he eperaure value of orso. herefore, sead of ode,, Ar,,, orso should be subsued. orso ervalr,arrasph,arrasz 46

160 47 Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 7, he eperaure of he ode,, a ex e sep s gve below Rego 8 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao ,,M ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,,,m, φ,,,, ar W q z orso ervalr, arrasph, arrasz orso,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α

161 I Equao 5.78,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,, M, s he eperaure value of orso. herefore, sead of ode Ar, M,, orso should be subsued. orso ervalr,arrasph,arrasz gve below. A Rego 8, he eperaure of he ode,m, a ex e sep s, M, α,, M,, M, r φ, M, M,, M, r, M,, M,, M, r, M,, M, Head orso ervalr, arrasph, arrasz z 5.79 q W ar, M,, M, Rego 9 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.8 q W,M, ar 48

162 49 I Equao 5.8,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of orso. herefore, sead of ode Ar,M,, ervalr,arrasph,arrasz orso orso should be subsued. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 9, he eperaure of he ode,m, a ex e sep s gve below. 5.8 Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M, ar W q z, arrasz, arrasph orsoervalr orso,m,,m, r,,m,m,,,m,m,,m, EE r r,m, EE r,m,,m, r,m, φ α

163 ,,,,, r, j,,, j, r φ,, r,, z, 5.8 q W ar, I Equao 5.8, s he eperaure value of orso. herefore, sead of ode Ar,, orso orso ervalr,arrasph,arrasz should be subsued. A Rego, he eperaure of he ode, a ex e sep s gve below., α,, j,,, j, r φ,, r,, r,, orso orso ervalr, arrasph, arrasz z 5.83 q W ar,, Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao

164 ,,,,,,, r r, j,,, j,,,, r φ z 5.84 q W ar, I Equao 5.84,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo,, s he eperaure value of orso. herefore, sead of ode Ar,, orso orso ervalr,arrasph,arrasz should be subsued. below. A Rego, he eperaure of he ode, a ex e sep s gve, r, r EE, r EE,, r, α, j,,, j, r φ orso orso ervalr, arrasph, z arrasz,, 5.85 q W, ar, 5

165 LEG A A o - A - 5

166 53 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.86,, s he eperaure value of foo. herefore, sead of ode Leg,,, Foo,,fooervalz should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below.,,,, ar W q,,,, ervalz,,foo Foo z r,, 4,, α 5.87 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ,, ar W q z,,,,,, r,, 4,,,,

167 I Equao 5.88, s he fcous eperaure. I order o elae hs,, fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of foo. herefore, sead of ode Leg,,, Foo should be subsued. A Rego, he eperaure of he ode,, fooervalz,, a ex e sep s gve below.,,,,,,,,,, r r,,,,,,,, α r φ Foo,, fooervalz z,,,, q 5.89 W ar,,,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r q,,,,,,,,,,,, r φ z 5.9 W ar,, 54

168 I Equao 5.9,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of foo. herefore, sead of ode Leg,,, Foo should be subsued. Also,,fooervaz,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below.,, r r EE,, r EE,, α r,,,,,, Foo,,,,,,,, fooervaz r φ z,,,, 5.9 q W ar,,,, Rego 4 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6., M,, M,, M,, M,, M,, M,, M, r r q, M,, M,, M,, M,, M,, M, r φ z W, M, ar

169 I Equao 5.9, s he fcous eperaure. I order o elae,m, hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of foo. herefore, sead of ode Leg,,M, Foo should be subsued. A Rego 4, he eperaure of he, M,fooervaz ode,m, a ex e sep s gve below.,,,,,,,,,, r r, M,, M,, M,, M, α r φ Foo, M, fooervaz, M,, M, z 5.93 q W ar, M,, M, Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.94 q W,M, ar I Equao 5.94,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I 56

170 addo,,m, s he eperaure value of foo. herefore, sead of ode Leg, Foo,M, should be subsued. Also, M,fooervaz,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. gve below. A Rego 5, he eperaure of he ode,m, a ex e sep s r EE, M,, M,, M, r r EE, M, α,m,, M, r, M,, M,, M, Foo, M, fooervaz, M,, M, r φ z 5.95 q W, M, ar, M, Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r,,, j, r φ,, z, q 5.96 W ar, 57

171 I Equao 5.96, s he eperaure value of foo. herefore, sead of ode Leg, Foo, should be subsued. A Rego 6, he, fooervalz eperaure of he ode, a ex e sep s gve below.,,,,, r r, j,,, j,, α r φ Foo, foo ervalz,, z 5.97 q Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6. W ar,,,,,,,,, r r q,,,,,, r φ z 5.98 W, ar I Equao 5.98, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo,, 58

172 s he eperaure value of foo. herefore, sead of ode Leg,, Foo should be subsued., fooervalz A Rego 7, he eperaure of he ode, a ex e sep s gve below. r EE,, r r EE, α r,,, Foo, j,,, j,, fooervalz r φ z,, 5.99 q W ar,, Rego hese rego s equaos ca be obaed by usg he sae boudary codos ec Rego herefore, he eperaure equaos a are he sae wh ec Rego Rego hese rego s equaos ca be obaed by usg he sae boudary codos Ar Rego herefore, he eperaure equaos a are he sae wh Ar Rego

173 HAD A A o - A 6

174 6 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.,, s he eperaure value of ar. herefore, sead of ode Had,,, Ar,, should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below. 5. Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,, 4,,,,,,,, ar W q,,,,,, Ar z r,, 4,, α,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

175 I Equao 5.,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he eperaure value of ar. herefore, sead of ode Had,,,,, Ar should be subsued. A Rego, he eperaure of he ode,, a,, ex e sep s gve below.,,,,,,,,,, r r,, α Ar,,,,,,,,,,,, r φ z 5.3 q W ar,,,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r φ z 5.4 q W ar, I Equao 5.4,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he eperaure value of ar. herefore, sead of ode Had,, 6

176 63,,, Ar,, should be subsued. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below. 5.5 Rego 4 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.6,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I,,M ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,,,m, φ,,,, ar W q z,, Ar,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α

177 addo,, M, s he eperaure value of ar. herefore, sead of ode Had, M,, Ar should be subsued.,m, A Rego 4, he eperaure of he ode,m, a ex e sep s gve below., M,, M,, M,, M,, M, r r, M, α, M,, M,, M,, M,, M, Ar, M, r φ z 5.7 q W, M, ar, M, Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.8 q W,M, ar I Equao 5.8,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he eperaure value of ar. herefore, sead of ode Had,M,, A r,m, should be subsued. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. 64

178 gve below. A Rego 5, he eperaure of he ode,m, a ex e sep s, r M,, M, r EE, r EE, M,, M, r, M, α, M,, M,, M, r φ, M, Ar, M,, M, z M, 5.9 q W, M, ar, M, Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,, r, j,,, j, r φ,, r,, z, 5. q W ar, I Equao 5. s he eperaure value of ar. herefore, sead, of ode Had,, Ar should be subsued., 65

179 below. A Rego 6, he eperaure of he ode, a ex e sep s gve, α,,, j, r φ,, r,,,, r Ar,, z 5. q W ar,, Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r φ z q W ar, 5. I Equao 5.,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo,, s he eperaure value of ar. herefore, sead of ode Had,, A r, should be subsued. below. A Rego 7, he eperaure of he ode, a ex e sep s gve 66

180 Rego hese rego s equaos ca be obaed by usg he sae boudary codos ec Rego herefore, he eperaure equaos a are he sae wh ec Rego Rego 5 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao I Equao 5.4,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego 5, he eperaure of he ode,, a ex e sep s gve below.,, ar W q,,, Ar z r, j,,, j,,, EE r r, EE r,, r, φ α ar W q z,,,,,, r,, 4,,,,,,

181 ,, z EE 4,, α,,,,,, r z q W ar,,,, 5.5 Rego 6 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ gve Equao 5.6. q W ar,, 5.6 I Equao 5.6,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he fcous eperaure. I order o elae hs fcous ode Boudary,, Codo 4 gve ae 5. s used. A Rego 6, he eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,, α,,,,,, z EE,,,, r φ z q W ar,,,,,,

182 Rego 7 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ 5.8 q W ar,, I Equao 5.8,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s also he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 7, he eperaure of he ode,, a ex e sep s gve below.,, r r EE,, r EE,, α r,,,,,,,,,, r φ z,,,, z EE,,,, 5.9 q W,, ar,, 69

183 Rego 8 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m,,m,,m, r z φ 5. q W ar,m, I Equao 5.,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. gve below. A Rego 8, he eperaure of he ode,m, a ex e sep s,,,,,,,,,, r r,m, α,m,,m,,m, r φ z z EE,M,,M,,M, 5. q W ar,m,,m, Rego 9 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao

184 7 5. I Equao 5.,,,M s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 9, he eperaure of he ode,m, a ex e sep s gve below. 5.3,M, ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,m,,m, φ M,, M,, ar W q M,, EE z M,, M,, z r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α

185 Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r z φ 5.4 q W ar, I Equao 5.4, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 4 gve ae 5. s used. A Rego, he eperaure of he ode, a ex e sep s gve below.,,,,, r r, α, j,,, j, z EE,, r φ z q W, ar,, 5.5 Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao

186 ,,,,,,, r r,,,,,, r φ z 5.6 q W ar, I Equao 5.6,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 4 gve ae 5. s used. Also, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode, a ex e sep s gve below. r EE,, r, r EE, α,, r, j,,, j, z EE,, r φ z, q W ar,,

187 FOO A A o - A 74

188 75 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao For Rego, Equao 5.8,, s he eperaure value of leg. herefore, sead of ode Foo,,, Leg,, should be subsued. A Rego, he eperaure of he ode,, a ex e sep s gve below. 5.9 Rego Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao ,, ar W q z,,,,,, r,, 4,,,,,,,, ar W q,,,,,, Leg z r,, 4,, α,, ar W q z,,,,,, r,,,,,, r,,,, r,,,,,,,,,, φ

189 I Equao 5.3,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he eperaure value of leg. herefore, sead of ode Foo,,,,, Leg should be subsued. A Rego, he eperaure of he ode,, a,, ex e sep s gve below.,,,,,,,,,, r r,, α Leg,,,,,,,,,,,, r φ z 5.3 q W ar,,,, Rego 3 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r φ z 5.3 q W ar, I Equao 5.3,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s he eperaure value of leg. herefore, sead of ode Foo,,, 76

190 77 Leg,, should be subsued. Also,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 3, he eperaure of he ode,, a ex e sep s gve below Rego 4 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao I Equao 5.34,, M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I,,M ar W q z,m,,m,,m, r,,m,m,,,m r,m,,m, r,m,,m,,m,,,,m, φ,,,, ar W q z,, Leg,,,, r,,,,,,,,,, EE r r,, EE r,,,, r,, φ α

191 addo,, M, s he eperaure value of leg. herefore, sead of ode Foo, M,, Leg should be subsued.,m, A Rego 4, he eperaure of he ode,m, a ex e sep s gve below., M,, M,, M,, M,, M, r r, M, α, M,, M,, M,, M,, M, Leg, M, r φ z 5.35 q W, M, ar, M, Rego 5 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.36 q W,M, ar I Equao 5.36, s he fcous eperaure. I order o,m, elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he eperaure value of leg. herefore, sead of ode,m, Foo,M,, L eg, M, should be subsued. Also s he fcous,m, 78

192 79 eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 5, he eperaure of he ode,m, a ex e sep s gve below Rego 6 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao , ar W q z,,, r, j,,, j, r,, r,,,,, φ M,, M,, ar W q z M,, Leg M,, M,, r, M, M,,, M, M,, M,, EE r r M,, EE r M,, M,, r M,, φ α

193 I Equao 5.38 s he eperaure value of ar. herefore, sead, of ode Had,, Ar should be subsued., A Rego 6, he eperaure of he ode, a ex e sep s gve below., α,,, j, r φ,, r,,,, r Ar,, z 5.39 q W ar,, Rego 7 Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,,, r φ z 5.4 q W ar, I Equao 5.4,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. I addo, s he eperaure value of leg. herefore, sead of ode, Foo,, L eg, should be subsued. 8

194 8 A Rego 7, he eperaure of he ode, a ex e sep s gve below. 5.4 Rego hese rego s equaos ca be obaed by usg he sae boudary codos ec Rego herefore, he eperaure equaos a are he sae wh ec Rego Rego 5 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao a r gve Equao ,, ar W q,,, Ar z r, j,,, j,,, EE r r, EE r,, r, φ α ar W q z,,,,,, r,, 4,,,,,,

195 I Equao 5.4,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 6 gve ae 5. s used. gve below. A Rego 5, he eperaure of he ode,, a ex e sep s,, α,, 4 r z,,,,,, Rego 6 q 5.43 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6. W ar,,,,,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ 5.44 q W ar,, I Equao 5.44,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo, s he fcous eperaure. I order o elae hs fcous ode Boudary,, Codo 6 gve ae 5. s used. A Rego 6, he eperaure of he ode,, a ex e sep s gve below.,,,,,,,,,, r r,, α,,,,,,,,,,,, r φ z q W ar,,,,

196 Rego 7 Subsug he ode,, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,,,,,,,,, r r,,,,,,,,,,,, r z φ 5.46 q W ar,, I Equao 5.46,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,, s also he fcous eperaure. I order o elae hs fcous ode Boudary Codo 6 gve ae 5. s used. Also,,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego 7, he eperaure of he ode,, a ex e sep s gve below. r EE,,,,,, r r EE,, α,,,, r,,,,,,,,,,,, r φ z 5.47 q W ar,,,, 83

197 Rego 8 Subsug he ode, M, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m,,m,,m, r z φ 5.48 q W ar,m, I Equao 5.48,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. Also,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 6 gve ae 5. s used. gve below. A Rego 8, he eperaure of he ode,m, a ex e sep s,,,,,,,,,, r r,m, α,m,,m,,m,,m,,m, r φ z,m, 5.49 q W,M, ar,m, Rego 9 Subsug he ode,m, o he Fe dfferece for of Bo-Hea Equao gve Equao

198 ,M,,M,,M,,M,,M,,M,,M, r r,m,,m,,m,,m,,m,,m, r φ z 5.5 q W,M, ar I Equao 5.5,,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo gve ae 5. s used. I addo,,m, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 6 gve ae 5. s used. Also,M, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. gve below. A Rego 9, he eperaure of he ode,m, a ex e sep s r EE, M,, M,, M, r r EE, M,, M, r, M, α, M,, M,, M, r φ, M,, M,, M, z 5.5 q W ar, M,, M, 85

199 Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6.,,,,,,, r r, j,,, j,,, r z φ, 5.5 I Equao 5.5, s he fcous eperaure. I order o elae, hs fcous ode Boudary Codo 6 gve ae 5. s used. below. q W ar, A Rego, he eperaure of he ode, a ex e sep s gve,,,,, r r, α, j,,, j,,,, r φ z 5.53 q Rego Subsug he ode, o he Fe dfferece for of Bo-Hea Equao gve Equao 5.6. W ar,,,,,,,,, r r,,,,,, r φ z 5.54 q W, ar 86

200 I Equao 5.54,, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 6 gve ae 5. s used. Also, s he fcous eperaure. I order o elae hs fcous ode Boudary Codo 3 gve ae 5. s used. A Rego, he eperaure of he ode, a ex e sep s gve below. r EE,,, r r EE, α,, r, j,,, j,,,, r z φ 5.55 q W ar,, 87