Study of the Transfer Function of a Thermocouple

Transcription

1 nernaiona Journa of Maerias Sciences. SSN Voume 3, Number 3 8, pp Research nia Pubicaions hp:// Suy of he ransfer Funcion of a hermocoupe A. Hasseine a, A. Azzouz, H. Binous b an M.S. Remana c a Deparmen of inusria chemisry, Universiy of Bisra B.P.5 Bisra, Agérie b Deparmen of Chemica Engineering, NSA, BP 676, 8 unis, unisia ca Deparmen of Civi Engineering, Universiy of Bisra B.P.5 Bisra, Agérie Corresponing auhor: hasseine@necourrier.fr Absrac n he presen paper, we suie he saionary an ynamic characerisics of a hermocoupe Cu-Consanan [,, 3] in he case of naura convecion of air or iqui nirogen. he oucome of his wor was o highigh wo hermoeecric effecs: he Seebec an Peier effecs. We fin ha he majoriy of he heoreica characerisics of he hermocoupe isribuion of he emperaure, response ime an ransfer funcion aow a juicious choice of he hermocoupe an mae possibe he correcion of he measure emperaure signas. Keywors: hermocoupe, Seebec effec, Peier effec, ransfer funcion nroucion hermocoupes are wiey use sensors, in inusry or aboraory, o eermine an conro emperaure changes in a gas or a iqui. Ahough oher echniques, such as he ones base on opics, were receny eveope, hese sensors remain frequeny use because of heir ease of empoymen an heir ow cos of impemenaion. n 8, German physicis homas Seebec iscovere ha a curren wi fow in he presence of a emperaure graien, as ong as he circui were mae of issimiar meas. his effec is nown as he Seebec Effec []. n 83, French wachmaer urne physicis, Jean Chares Peier, noe ha if an eecric curren passes from one subsance o anoher, hen hea wou eiher be reease or absorbe in he juncion beween he meas, epening on he irecion of curren fow []. Hea is effecivey pumpe hrough he circui by charge carriers, an his is nown as he Peier effec [5]. Wha Peier ha acuay iscovere, was ha he Seebec effec was compeey reversibe [5]. A Peier evice is a hermoeecric evice which convers boh eecrica curren ino a emperaure graien, an a emperaure graien ino eecric curren.

2 A. Hasseine e a Governing equaion of he hermocoupe Figure shows he geomerica moe use for he hermocoupe mae of copper an consanan. Figure : hermocoupe configuraion he curren suy assumes ha he juncion has he same iameer as wire; i.e. he soer of he wo wires is carrie ou wih a grea care an wihou fier. Energy baance: n he presen suy, we negec boh he hea ransfer ue o conversion of ineic energy of he fui ino caorific energy as we as he caayic effecs. n aiion, we assume ha he sensiive eemen of he hermoeecric probe is compare o a cyiner an he isribuion of emperaure is uniform aong a cross-secion of he hermocoupe. he governing equaions are hen given by: For he copper secion of he hermocoupe [ ] pw c ρ ρ f w w w h For he consanan secion of he hermocoupe [ ] pw c ρ ρ f w w w h For he concerne maerias, however, heir Seebec coefficiens are very sma, an wi be viewe as zero beow [6]. heir herma conuciviy are assume o be consan in he concerne emperaure range, whie he eecric resisiviy is escribe by a inear funcion wih respec o he emperaure. We inrouce he foowing non-imensiona efiniions: a ; ; Δ ; / ρ Ι Ι wih

3 Suy of he ransfer Funcion of a hermocoupe Δ c he emperaure,, is aen equa o ambien emperaure an C is he emperaure in he juncion i.e. a aen equa o he fui s emperaure. Ne, we repace w an wih heir epressions in Eqs. an an we aso efine he foowing consans: h Δ Κ f h h Δ Κ h f he ime-epenen governing equaions in non-imensiona form become: For he mea m Κ 3 For he mea m Κ shou be noe ha,, K an K, which appear, in he equaions 3 an, represen he inverse of he ime-consan an he sensiiviy of he hermocoupe, respecivey. he hea ransfer coefficien, h, is given by Kramer s correaion [] where ony naura convecion is consiere. f we ae w w, he bounary an iniia coniions, associae o Eqs. 3 an, are he foowing: a ;, a ;,

4 A. Hasseine e a a emperaure coninuiy wries :,, 3 a boh fues are se o be equa: S, S, a iniiay he wires are a a specifie emperaure Seay-sae souion of he equaion of he hermocoupe he souion of baance equaions a seay-sae, equaions 3 an, give he emperaure profie in he wires. a Seay-sae emperaure in he copper secion, sinh / cosh sinh sinh / cosh / cosh / sinh / cosh / / / 5 / / / / cosh sinh cosh sinh cosh sinh b Seay-sae emperaure in he consanan secion / cosh, / cosh sinh sinh / / / sinh / cosh cosh sinh / / / 6 / / / / cosh sinh cosh cosh sinh Wih / /

5 Suy of he ransfer Funcion of a hermocoupe 3 Suy of he ynamic response of he hermocoupe he response ime of he hermocoupe is a very imporan characerisic. he hermocoupe is generay regare as a sensor conroe by a firs orer ifferenia equaion. he raiiona process use for he eerminaion of he ime-consan of he hermocoupe is a uni sep; for his ype of sensor he seay-sae is ofen reache afer approimaey five ime-consans. Le us wrie he equaions 3 an when conucion aong wires is negece: For he mea m 7 For he mea m 8 We procee by carrying ou he foowing change of funcions: θ w θ an w 9 Eqs. 7 an 8 become: θ θ θ θ Where an are respecivey he ime-consans of he copper an consanan wires. Afer aing he Lapace ransform of each erm of equaions an, one obains he souions in he fie ime; as foows: e e, are he ime isribuions of emperaures in he wo wires. 3

6 A. Hasseine e a Cacuaion of he ime-consans he ime-consan of he hermocoupe is, by convenion, he ime necessary for he emperaure o reach 63% of he seay-sae vaue. he resus obaine by eiher heoreica or graphica mehos are as foows: n he case of he air: Copper wire: Consanan wire: 89,,8s 588,,s n he case of nirogen: Copper wire: 75 8s Consanan wire:, 3s where a, a Deerminaion of he ransfer funcion of wire by he meho using Lapace ransforms he souion areay obaine is given here: θ a e Le us epress in he reuce form, i comes hen: θ e 5 he Lapace ransform of θ is given by: θ p p Le us se: p j an, i foows hen ha: θ j j 6 f we ae he moue of he reaion 6, i comes: θ j Which becomes afer simpificaion : θ jw w w Le us se: /

7 Suy of he ransfer Funcion of a hermocoupe 5 F w 7 Resus an iscussion he seay-sae emperaure of he wire is given by he Fig. an highighs he Seebec effec, which can be epaine by he asymmery of his curve ue o he arge herma resisance of he consanan wire. Whie in Fig. 3, a symmeric curve is observe. We concue from hese resus ha he Seebec effec canno be observe for fuis a ow emperaures an meas wih herma resisance. Figure : Seebec effec- case: air. Figure 3: Seebec effec case: nirogen iqui.

8 6 A. Hasseine e a he seay-sae emperaures obaine for he Peier effec are represene in he Figs. an 5. he emperaures are poe versus he raio /. One noes an asymmery in he behaviour of wire an wha can be epaine by he absorpion of hea when he curren crosses he juncion. Fig. 5 enabes us o ceary highigh he Peier effec. Figure : Peier effec- case: air. Figure 5: Peier effec- case: nirogen iqui.

9 Suy of he ransfer Funcion of a hermocoupe 7 Various vaues of he raio / where consiere: 3, 5, an. is observe ha he seay-sae emperaure profies of boh wires, Figs. 6 an 7, are perfecy symmerica. Neverheess, we observe ha he emperaure in he cenre of wire epens on he vaue on he raio / an i is higher for arger / vaues. n concusion, for his in of sensor, one shou choose vaues of / higher han. Figure 6: Mea resisance in he air meium. Figure 7: Mea resisance in he nirogen meium. he Fig. 8 an 9 represen he ynamic response of he same hermocoupe copperconsanan wih he same iameer 5. in he case of air fow Fig. 8 an of a 6 iqui nirogen fow Fig. 9. in he case of naura convecion. We observe ha he

10 8 A. Hasseine e a vaues of, in he case of he air are respecivey higher han he vaues, in he case of iqui nirogen; his resu can be epaine by he fac ha he vaue of he hea ransfer coefficien of iqui nirogen is higher han he air s counerpar /h. We aso fin ha in boh cases i.e. air an iqui nirogen he vaue of is higher han which is epaine by he goo herma conuciviy of copper. Figure 8: Disribuion of he emperaure- case: air. Figure 9: Disribuion of he emperaure case: nirogen iqui. he ransfer funcion is represene by poing he ampiue of his funcion, Eq.

11 Suy of he ransfer Funcion of a hermocoupe 9 7. However, his represenaion aows ony quaiaive inerpreaions. is noice ha he curve ecreases sowy as he frequency increases; bu afer cerain vaue of frequency i rops quicy an ens owars zero. n concusion, we can sae ha he ransfer funcion of a inear sysem efines i compeey; i.e. aows envisaging is behaviour. Concusion n his wor, we suie he saionary an ynamics characerisics of a hermocoupe copper-consanan in he case of naura convecion of air or iqui nirogen. he seaysae emperaure isribuions in he wires of he hermocoupe aow us o highigh wo hermoeecric effecs: he Seebec an Peier effec. When a coninuous curren of fui is appie, he non-homogeneous heaing of he sensiive eemen of he hermocoupe wire an weing is moifie because of he Peier effec appearing wih he weing. his heerogeneiy of heaing generae, a he momen of he remova of he curren, a reorganizaion of he fie of emperaure by conucion aong he coupe, which noaby moifies he evouion of he ecrease of he emperaure. he heoreica suy on he hea ransfer beween he hermocoupe an he surrouning meium epens on he raio /. nee, various vaues of his raio where seece. We observe; in spie of is ow voume he wire has a herma ineria, ahough his ineria is very wea. his resus in a eforme image of he suie phenomenon. he souion is o use a compensae ampifier, i.e. whose profi increases wih he frequency o cach up wih he effecs of herma ineria. is cear ha for a sysemaic use of he hermocoupe, i wi be necessary o equip i wih a compensaing newor. Finay, one can concue ha his suy showe ha he majoriy of he heoreica characerisics of he hermocoupe isribuion of he emperaure, response ime an ransfer funcion aow a choice of he hermocoupe more suiabe o he probem consiere. Finay, i is possibe o correc he measure emperaure by use of a compensaing newor. Nomencaure a : wire herma iffusiviy a ; ρc p c p : wire hea capaciy; : wire iameer; h : hea ransfer coefficien; : wire sensiiviy parameer; : wire engh; p : Lapace omain variabe; a,: ime an reuce ime ;

12 5 A. Hasseine e a, : emperaure an reuce emperaure : ambien emperaure; Δ, Δ ;, C C : emperaure an reuce emperaure of he cener of he wire;, : posiion aong he sensor ; : emperaure coefficien, : inverse of a ime consan ; θ : reuce emperaure θ ; : wire herma conuciviy; ρ : wire voumeric mass; ρ : wire eecrica resisance a ;, : ime consans of he infiniey ong wire ; f : frequency. a c References [] Georges Asch, es capeurs en insrumenaion inusriee Paris 98. [] P.L. Wasrom, Spaia epenence of hermoeecric voages an reversibe heas, Am. J. Phys. Vo. 56, Oc. 988, pp [3] A. Azzouz, nernaiona Journa of Hea an Mass ransfer, Voume 3, ssue 9, Sepember 99, Pages 5-63 [] D.K.C. MacDona hermoeecriciy: an inroucion o he principes. John Wiey an Sons, 96. [5] rving B. Caoff an Ewar Mier hermoeecric Maerias an Devices. Reinho Pubishing, 96. [6] Xuan XC, Ng KC, Yap C, Chua H. Opimizaion an hermoynamic unersaning of conucion-cooe Peier curren eas. Cryogenics ; : 5.