HEAT FLUX ESTIMATION IN THIN-LAYER DRYING. Olivier Fudym Christine Carrère-Gée Didier Lecomte Bruno Ladevie

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1 Invese Poblems n Engneeng : Theoy and Pacce d In Coneence on Invese Poblems n Engneeng June -8, 999, Po Ludlow, WA, UA HT5 HEAT FLUX ETIMATION IN THIN-LAYER DRYING Olve Fudym Chsne Caèe-Gée Dde Lecome Buno Ladeve Ecole des Mnes d Alb, Cene Enegéue Envonnemen Roue de Telle, 8 Alb, Fance, ( udym@ensmac ABTRACT A mehod s developed n ode o deemne he dyng cuves o sludge hn-laye dyng nce mass balance s vey dcul o measue o hn-laye dyng, hese cuves ae obaned hough enegy balance An expemenal devce s bul up, n ode boh o povde and o esmae he hea lux densy a he neace beween a ho meallc plae and he dyng sample An analycal dec model s made usng he uadpole omalsm, and he sysem anse uncon s calculaed The nvese poblem s solved usng he beck's seuenal uncon speccaon mehod, and he coespondng dyng cuve s deduced by a smple enegy balance Real expemen esuls ae pesened INTRODUCTION Among conac dyng echnologes, dum dyng s wdely used n he ood ndusy, o ea hea-sensve poducs I also pesens a gea nees o sludge dyng The poduc s spnkled o coaed on a ho oang cylnde The wall empeaue s above bolng empeaue Absence o mxue and agaon consues an advanage o vscous poducs lke sludge, enablng a good conol o esdence me, aveaged wae conen and hemal ecency Fo conac dye s desgn, s mpoan o pedc dyng aes, bu hs ae s lmed by nenal conducve anses I s hen necessay o nd a maxmum deal eeence, no dven by nenal anses, bu by he nnsc poduc s hn lm bolng ae : hs s he man scope o hs pape The pesen sudy s applcaon s o sludge dyng In a pevous wok, was shown by Vasseu (98 ha vey hgh hea lux ae exchanged dung he s nsans o vscous oodsu hn lm dyng, due o he low poduc s hckness NOMENCLATURE A, B, C, D Quadpole elemens F(s Tanse Funcon X Mosue conen Y Measued empeaue M Dy mae load ( Invese anse uncon d Tme sep Ineace hea lux densy T Compued Tempeaue a Themal dusvy l v wae laen hea Fuue me seps numbe s Laplace vaable Tme b Themal eusvy e enso locaon e Plae hckness ε e e λ Themal conducvy ϕ Hea lux densy φ Laplace T hea lux θ Laplace T empeaue ρc p Volumec hemal capacy Mass ae dyng, when nvolved n a vapozaon hn lm pocess, s vey dcul o measue : Bolng anse s a ahe sochasc and volen phenomena, and hghly as o hn lm bolng Mass losses ae dcul o measue, because vey lle maeal s coaed on he wall Copygh 999 by AME

2 Poduc s samplng s made mpossble, because dyng ae s oo hgh nce mass balance can be decly known, a way o esmae he dyng ae s dong an enegy balance, assumng ha he amoun o enegy nvolved n vapozaon dyng s known I s hus necessay o oban elable values o he neace hea lux beween he dyng poduc and he heaed suace The expemenal devce, developed o hs sudy, s pesened n nex secon Expemenal devce The s obecve o hs expemenal devce s o poduce hemal condons o hn lm vapozaon conac dyng The poduc's hn-laye (abou 7 mm hckness s coaed on a ho meallc plae The plae mus be hck enough n ode o soe he enegy amoun necessay o complee dyng (58 mm hckness I mus be hghly dusve n ode o ensue hgh hea lux denses a he neace The hea s soed a a empeaue above bolng empeaue and suddenly dschaged when he poduc s coaed Inal empeaue s povded by placng he plae n a dyng loop The second man goal s o esmae he neace hea lux beween he dyng poduc and he heaed suace I mus be emphaszed ha no dec suace empeaue can be made : a senso locaed on he on sde would peub he maeal coang, as well as hea anse Thus wall empeaue and lux mus be esmaed usng neo locaon empeaue and an nvese hea conducon mehod A hn hemocouple wh sepaaed conacs s dulled nsde he plae, nea he suace, lad n a decon paallel o sohemal cuves Wh hs pacula mehod, he senso locaon s pecsely known (g Anohe senso s lad ou on he plae's boom sde (T on g mm,7 mm ludge laye Themocouple lead plae s suace The sysem s shown on g An analycal one dmensonal conducve model s bul, assumng ha he plae s boom s nsulaed The uadpole omalsm (Degovann, 988 s used, and he anse uncon F (s beween he neace hea lux φ and empeaue T s calculaed e e x Themocouple lead Plae Insulang meda ludge laye Back sde senso Fgue - chemac Expemen T, T, ϕ T, ϕ Hea anse s hee assumed o be puely conducve and he sample homogeneous : T x * * T a wh a known hea lux densy a he plae suace x and nsulaed boom a xe Inally, he whole sysem s assumed o be a unom empeaue T A new vaable s hen consdeed such as TT * - T To we he pevous sysem n a uadpole om, Laplace ansom s appled Euaon ( becomes an odnay deenal euaon : d θ dx s a θ wh hea lux denon such as dθ(x,s (x,s λ dx φ ( Expessons ( and ( ae hen euvalen o a uadpole pesenaon (Basale, 994 such as ( ( Coppe plae Fgue - enso dulled nsde he plae θ A φ C B θ D φ (4 Dec model A dec model s bul n ode o calculae he senso empeaue evoluon, when a gven hea lux s appled o he θ φ C A B θ D (5 Copygh 999 by AME

3 A B C whee (, : D ch(ke (6 sh(ke kλ kλsh(ke (8 (7 and s k (9 a I s hen easy o deemne he anse uncons F (s beween he Laplace ansomed supecal hea lux φ, and he empeaues θ o be calculaed by θ (s φ F ( Invese mehod allows o calculae he supecal hea lux densy ( om he measued empeaues Y ( a senso locaon, and coespondng empeaue T ( om he dec model Invese anse uncon ( ae obaned om F (s hough a numecal Laplace nveson (ehes, 97 The empeaue T ( s hen calculaed by a convoluon poduc : T ( ( τ ( τ dτ ( O pacula nees s he anse uncon beween ( and he compued empeaue T ( a senso locaon : F (s A ( CA DC The nsulaed bounday condon wll be valdaed om back sde measuemens Y (T locaon Neveheless, would be possble o negae hese values n he bounday condon s knowledge Invese mehod In he case o one dmensonal poblem wh as and hgh hea lux changes, a seuenal mehod s adaped The uncon speccaon mehod, wh a seuenal consan hea lux unconal om s used (Beck, 97 Fo ha mehod (Beck, 985, ( s appoxmaed n a dscee om : [ ] n ( whee ( (d Euaon ( can be expessed n a dscee om : T T( ( d d [ ] (4 wh ( Assumng ha and T T ae known, s seached by an odnay leas suae pocedue usng he nex Y Y measuemens, assumng a consan hea lux dung he uue me seps (Fg : (5 - Fgue - Tme dscezaon a me sep The leas suaes pocedue mnmzes he unconal om J wh espec o : ( Y T ( J ( (6 Euaon (6 s deenaed wh espec o and se eual o zeo : J( T ( Y T ( (7 The hea lux ncemen dung he me sep s seached : δ (8 Thanks o E (5, he uue compued empeaues T ( can be wen as : T ( T T ( δ (9 and o he same eason (, he sensvy coecens n E (7 and E (9 ae eual : T T ( Euaons (7 yeld o he hea lux ncemen δ o empoaly assumed consan (E Ths euaon s used n a seuenal manne by nceasng by one each me sep Copygh 999 by AME

4 ( Y T ( δ ( sensvy o e sensvy o e sensvy o lambda sensvy o a sensvy o The sensvy coecens dened n E ( ae known om E (4 wh eplaced by o Fo example, he empeaue T a me, wh assumpon o consan hea lux ae me, can be expessed as : [ ] T d ( (K Assumng E (5 yelds o he sensvy coecens, wh o : [ ] d ( I s shown he gea nees ha have a dec model wh a convoluon such as E ( coupled wh he speccaon uncon mehod : he sensvy coecens ae consan wh me sep, and only he s values o he nvese anse uncon ae o be calculaed The sum o nvese anse uncon values n E ( also mply a gowng behavo o hese sensvy coecens Ths ac mpoves he uue me seps sablzaon eec n E (, because each uue me sep numbe ncemen wll conbue wh a sensvy coecen gowh ensvy analyss s o be dealed n nex secon 4 ensvy analyss The dec model, descbed n secon, s used wh a hea lux unconal om smla o expemenal esuls Reduced sensvy coecens o he man paamees ae ploed on g (4 I s shown ha sensvy o T o e s vey low Ths s due o he senso locaon, nea o he suace ensvy coecens o hemal conducvy and hemal dusvy ae he same ode han sensvy o mean hea lux gan φ I s hen mpoan o ge good knowledge o he plae s hemal popees Euaon (4 can be wen n a max om, n ode o expess he sensvy max X o he unknown hea lux componens : emps (s Fgue 4 Reduced sensvy coecens A whole me doman esmaon pocedue would be dcul o egulaze : The componen a me can be esmaed only vey close o me d n X (5 The dac om o sensvy coecens o can be undesood wh E (5, and s due o he deep exponenal decease o he anse uncon ( ensvy coecens, obaned om E (4-5 ae pesened on a map by g (5 I s shown ha he sensvy coecen s non zeo only o a vey ew pons vey close o each hea lux componen a me T X (4 4 Copygh 999 by AME

5 Reduced sensvy coecens 6 4 paamees numbe 5 Tme (s mehod, and he esuls ae exhbed only o one case, wh nal empeaue T eual o 8 C Tme sep s d s, he esmaed paamees numbe s 58, and he uue me seps numbe s 5 Measued and compued empeaues evoluon ae ploed on g (7 The measued senso empeaue Y s peecly ed wh T, bu hs s decly due o he nvese mehod, vey close o an exac machng algohm Moe mpoan s he peec beween back sde measuemen Y and compued empeaue T, because hs means a good concodance beween he expemen and he model, and poves he valdy o back sde nsulaed bounday condon assumpon Wall empeaue T and T ae almos dencal, due o he low dsance e and he coppe s hgh hemal conducvy Fgue 5 ensvy coecens map The compason beween g (5 and g (6 makes clea he pesen algohm nees, and he uue me seps sablzaon eec, snce he sensvy coecens cuve shown by g (6 s a gowng uncon wh he uue me seps numbe T and Y T Y and T x uue me sep numbe Fgue 6 Fuue me sep sensvy coecens o he seuenal mehod :, o (E 5 Hea lux esmaon esuls ome mehod s valdaon ess have been mplemened wh numecal expemens, and have poved s ecency, bu he coespondng esuls ae no pesened heen In hs secon, only eal expemen esuls ae shown The plae s placed nsde a dyng loop, and a sable nal empeaue above he poduc s bolng empeaue s eached beoe he expemen s saed A me, he poduc s uckly coaed on he plae, whle Y and Y acuson goes on Alhough a lo o expemens have been made, he goal o he pesen pape s only o pesen he geneal wok and Fgue 7 Compued and measued empeaues Y s submed o a sagh dmnuon due o sudden bolng phenomena, and he coespondng supecal hgh hea lux densy The decease s sopped when he hea lux nensy alls down whle hemal duson nsde he coppe plae ends o s homogenzaon When ( s nea o zeo, on sde and back sde empeaues end o he same value, coespondng o a uas-pemanen sae The wall empeaue T s always above poduc's bolng empeaue Bolng anse sops pobably due o he poduc sucue evoluon : a hgh exenal hemal essance s ound beween he ded poduc and he plae neace Fgue (8 ndcaes supecal hea lux densy ( esmaed om Y measuemens and nvese pocedue A hgh value, abou 6 Wm - s ased ae a vey sho me smalle han one second, due o bolng phenomena The lux decease occus when bolng dyng sops, and s ollowed by an evapoave dyng peod, nvolvng smalles hea anse (abou Wm - The man obecve o hn-laye dyng s o dy mos o poduc wh bolng dyng The ansen peod beween bolng and evapoaon dyng s moe o less dened When 5 Copygh 999 by AME

6 nal empeaue T s nea o he poduc's bolng empeaue, hs ansen peod s lage upecal hea lux knowledge can be used o calculae he enegy densy E los hough he suace : E ( ( τdτ (6 The global enegy densy los by he coppe plae ae a new uas-pemanen sae s eached (when on sde and back sde empeaues ae eual o T s : E p ' ρc e(t T (7 Fgue 8 Esmaed supecal hea lux densy ( Assumng he classcal sascal descpon o empeaue measuemen eos (Beck, 977, s possble o esmae he coespondng sandad devaons Fo he pesen expemens, he sandad devaon σ T o empeaue measuemens s ound o be abou 7 C Esmaed hea lux sandad devaon σ s ploed on g (9 as a uncon o he uue me seps numbe The decay s due o he sablzaon eec o gowng (see on g 6 σ Wm x E s ound o be ue close o E(, whee s he nal me when T T T Ths means ha ( s he unue lux cusng plae walls Ths ac conm he nsulaed back sde assumpon, bu also seems o pove hee ae no laeal losses, hence he one dmensonal conducve anse assumpon s vald I s now possble o calculae he dy mae load : E' M (X (8 X l v wh X and X nal and nal mosue conen I s vey mpoan o know pecsely he dy mae load M, snce dyng ae s hghly dependan on hs uany, and s no possble o deduce s value wh dec expemenal measuemens The mosue conen X(, he dyng cuves, and nally he dyng ae can be deduced om E (6-8 : X( dx d E( X Ml (9 v ( ( Ml v Dyng cuves ae shown by g ( I s noceable ha bolng hn-laye dyng, n ha pacula case, s vey as, hanks o he non nenal anse lmaon The global dyng me s nceasng wh nal empeaue Fgue 9 Esmaed hea lux sandad devaon σ as a uncon o he uue me seps numbe 6 Enegy balance and dyng cuves 6 Copygh 999 by AME

7 Beck JF, Nonlnea esmaon appled o he nonlnea nvese hea conducon poblem, Jounal o Hea Mass Tanse, Vol, pp 7-76, 97 Beck JF, Blackwell B, Cla C, Invese hea condon, llposed poblems, Wley-Inescence, New Yok, 985 Blanc G, Raynaud M, oluon o nvese hea conducon poblem om hemal san measuemens, Jounal o Hea Tanse, Vol 8, pp , 996 Fgue - Dyng cuves - Inal empeaue T eec CONCLUION The pesen expemen has been developed n ode o deemne he dyng cuves o sludge hn-laye dyng nce mass balance s vey dcul o measue o hn-laye dyng, hese cuves ae obaned hough an nvese mehod used o esmae supecal hea lux densy hsoy om nenal empeaue measuemens and enegy balance These cuves ae used as an deal eeence o dum dye's desgn, snce ndusal dyes always have an nenal conducve anse lmaon REFERENCE Vasseu J, Eude du séchage d un podu vseux en couche mnce su une pao chaude, pemean de dén un modèle de sécho su cylnde, hèse, 98 Degovann A, Conducon dans un "mu" mulcouche avec souces : exenson de la noon de uadpôle, In J Hea Mass Tanse, vol, pp , 988 Basale JC, DMalle, ADegovann, Exenson de la méhode des uadpôles hemues à l ade de ansomaons négales: calcul de anse anse hemue au aves d un déau plan bdmensonnel, In J Hea Mass Tanse Vol 7, pp -7, 994 ehes H, Remaks on algohm 68 Numecal nveson o Laplace ansom, ACM, Vol 5, No, pp 64, 97 Beck JF, KJ Anold, Paamees esmaon n engneeng scence, John Wley & ons, New Yok, Copygh 999 by AME