TRANSIENT HEAT TRANSFER FOR CIRCULAR JET IMPACTING A HORIZONTAL SURFACE: APPLICATION OF THE ITERATIVE REGULARIZATION METHOD

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1 B Proceedgs of the 5 th Iteratoal Coferece o Iverse Problems Egeerg: Theory ad Practce, Cambrdge, UK, -5 th July 5 TRANSIENT HEAT TRANSFER FOR CIRCULAR JET IMPACTING A HORIZONTAL SURFACE: APPLICATION OF THE ITERATIVE REGULARIZATION METHOD J. B. BAONGA, H. LOUAHLIA-GUALOUS, E. ARTIOUKHINE ad P. K. PANDAY FEMTO ST, CREST departmet, CNRS-UMR 674, aveue Jea Moul, 9 Belfort, Frace e-mal: hasa.gualous@utbm.fr Abstract - Ths paper presets the expermetal results of the local thermal boudary codos obtaed by solvg the traset verse heat coducto problem. The expermetal study cocers the heat trasfer of a lqud crcular jet mpgg a horzotal surface. The ater s used as the test flud h the uform temperature. A electrc resstace heater s placed sde the cylder ad 5 thermocouples are stalled order to measure the temperature the sold. The eratve regularzato method s used to solve the traset verse problem uder study. The uko fuctos are approxmated by cubc B-sples. The gradet s computed by usg the adjot equatos. NOMENCLATURE f temperature fucto, K h heat trasfer coeffcet, W/m K J resdual fuctoal, K m N agular odes umber Q heat flux, W/m r radus, m R radus of the heated surface, m r co-ordate of measuremet pot T calculated temperature, K T meas measured temperature, K z axal co-ordate, m co-ordate of measured pot, m z Greek symbols λ thermal coductvy, W/mK θ temperature varato Ψ Lagrage multpler γ descet parameter δ Drac fucto Subscrpts measured pots all erato umber r local value. INTRODUCTION Jet mpgemet heat trasfer has bee employed may practcal applcatos for coolg ad dryg because provded hgh heat trasfer coeffcet. Jet mpgemet coolg has bee used a may applcatos from the metal sheet dustry to coolg laser ad electroc equpmets. Extesve umercal ad expermetal studes o heat trasfer ad hydrodyamcs of lqud jet mpgemet have bee reported the lerature [-4]. The applcatos of lqud jet cluded coolg teral combusto eges, quechg of metals ad other materals maufacturg process, ad thermal cotrol of hgh performace computer compoets. Whe a crcular lqud jet strkes a flat plate, spreads radally very th flm alog the heated surface. Follog may authors [5,6], the jet mpgemet flo s dvded to four regos defed as: () the stagato zoe here the heat trasfer s maxmal, () the vscous boudary layer zoe here the dyamc boudary thckess s loer tha the flm thckess, () the thermal boudary layer here the flm thckess s hgher tha the thermal boudary layer, (v) the fully thermal ad vscous boudary layer rego ad (v) the hydraulc jump here the lqud sheet thckess s creased ad the jump s assocated h a Raylegh-Taylor stably. Numerous studes are coducted order to evaluate the average heat trasfer coeffcets, but the estmato of the traset-state local heat trasfer hout eglectg the heat coducto the sold has ot retaed much

2 B atteto. The aalyss of thermal performace the lqud jet mpgemet requres the koledge of the thermal boudary codos for the coolg surface. I ths case, the drect measuremets of the velocy ad temperature dstrbutos the flog flm are very dffcult to carry out because the flm thckess s geerally less tha 5 µm. Oly the let bulk temperature of the flm ad the all temperature sde the expermetal cylder are easly to measure hout perturbg the lqud jet flo. I ths ork, the procedure for solvg the to-dmesoal usteady lear verse heat coducto problem (IHCP) s preseted. It s appled to the problem of the lqud jet mpgg the horzotal surface order to estmate the thermal boudary codo of a hollo half-cylder. The eratve regularzato method s used to solve the verse problem uder aalyss. The method s based o the cojugate gradet method that uses to mmze the resdual fuctoal ad the resdual dscrepacy prcpal as the regularzg stoppg crero. Ths paper presets the results of the expermetal study of heat trasfer coeffcet for a free ater jet mpactg a horzotal plate. The measured average heat trasfer coeffcets are compared to the values estmated by solvg IHCP.. EXPERIMENTAL SET UP The test facly sho Fgure s composed of to crcus. The frst oe s a closed crcu here the test lqud s forced crculato usg the pump (3). The flud the reservor () s cooled trough the heat exchager (4). The secod crcu cossts of the feed to the test secto. The flud s crculated usg the pump () from the reservor () to the adjustable costat head tak (5) equpped h a dra ad a overflo. A flter (3) s used to elmate ay dust or partcles the orkg flud. A electrcal heater of 5W (7) assocated h a temperature system cotroller (8) are stalled the head tak (5) to cotrol the temperature of the test lqud. A Chromel-Alumel thermocouple (6) s placed the lqud bath to measure the temperature of the flud order to estmate the lqud propertes. The flud from the overhead tak (5) flos uder gravy through the regulatg valves to the heated surface. The temperature of the orkg flud the reservor () s cotrolled by usg the heater cartrdge of W () ad the regulatg system (9). - reservor - pump 3- flter 4- test secto 5- head tak 6- thermocouple 7- heater 8, 9- regulatg systems - heater - thermocouple 3- pump 4- heat exchager Fgure. Expermetal apparatus Fgure a shos the heater assembly used ths study. The expermetal cylder as made of a brass block ad as heated by usg oe cartrdge heater of W. The electrc heater has a mm of dameter ad 4 mm of legth. It s placed sde the expermetal cylder as sho Fgure. The electrc poer put s measured usg a ammeter ad a voltmeter. The test sample s oreted vertcally ad s of 5 mm of dameter ad 6 mm of legth. The coolg crcular plae surface s of 5 mm of dameter. The heatg cylder s thermally sulated h Teflo (thermal coducto of.3 W/mK) o all faces except the coolg face order to prevet the heat loss. The test sample ca be moved alog the vertcal axs order to study the fluece of the space betee the ozzle ad the heated surface. The temperatures sde the expermetal cylder are measured h 5 Chromel-Alumel thermocouples of dameter. mm (ucertaty of ±. C), placed to secto defed at.6 mm ad 8 mm from the coolg surface (Fgure b). For the secto S, 5 thermocouples are placed at.6 mm from the etted surface at tervals of 3.5 mm. For the secto S, 5 thermocouples are used at 8 mm from the outer surface at tervals of 3.5 mm. The thermocouples are located at 4 cocetrc crcles of rad 8mm,.5mm, 5 mm ad mm as sho Fgure b.

3 B 3 Thermocouples (a) (b) Fgure. Istrumetato of the test secto. Because of the ucertaty the heat flux dstrbuto alog the legth of the cylder, the all temperature dstrbuto as measured at to dfferet sectos as sho Fgure b. A data acquso system s used to record all temperature measuremets for each mposed electrc poer sde the expermetal cylder. The IHCP s solved order to determe the local heat flux ad the surface temperature of the heated dsk. The local heat trasfer coeffcet s determed at the exchage crcular surface for dfferet flo rates. The expermets are coducted for let lqud jet temperature of 37 C, ad the total electrc poer mposed sde the cylder of 4W ad 7W. For each flo rate, the measured temperature sde the all of the expermetal dsk s used for solvg the IHCP. 3. INVERSE HEAT CONDUCTION PROBLEM The physcal model cosders a vertcal cylder of heght H ad exteral radus R (Fgure 3). The heatg cylder s thermally sulated h Teflo o all faces except the horzotal cooled face order to prevet the heat loss. The soluto of the verse problem s carred out for half of the cylder. 5 thermocouples are stalled sde the expermetal cylder order to measure the expermetal all temperature for each test. Uko heat flux Wetted surface Isulated H meas z r H Expermetal cylder R Fgure 3. Physcal model. The geometrcal ad physcal propertes of the cylder are preseted Table. The mathematcal model of a heat coducto process a hollo cylder treats the traset-state heat trasfer problem. It s gve by the follog system of equatos: ρc p T( r, z, t ) T( r, z, t ) T( r, z, t ) T( r, z, t ) = + +, here < r < R, < z < H () λ t r z T (, z, t) T (R, z, t) =, here < t tf, z H () =, here < t tf, z H (3)

4 T (r, z,) = T, here : r R, z H (4) T λ (r, H, t) = Q(r, H, t), here : < t tf, r R z (5) T (r,, t) = f(r, t), here : < t tf, r R (6) The mathematcal model s supposed symmetrcal about the vertcal axs of symmetry. B 4 Table : Physcal propertes ad geometrcal characterstcs of the tested cylder. Physcal propertes of brass cylder: λ =6W/ mk, C = 385 J.kg. K, p ρ= 85kg.m 3 Physcal propertes of copper cylder: λ = 389 W / mk, C = 4 J.kg. K, p ρ= 896kg.m 3 Geometrcal dmesos of the brass cylder: Geometrcal dmesos of the copper cylder: R = 5 mm, H = 8 mm, H meas = 7.4 mm R = 5 mm, H = mm, H meas = 8mm I the verse problem, the heat flux Q (r, H, t) at the exteral surface of the cylder s uko. The temperatures measured at odes (r, z ) sde the sold are used as a addoal formato o the temperature dstrbuto the cylder to estmate the fucto Q (r, H, t) : T (r, z, t) = f, =,,..., (7) meas The IHCP cossts estmatg the fucto Q (r, H, t) usg the codos ()-(7). The soluto of the verse problem s based o the mmzato of the resdual fuctoal defed as: N meas N meas f J(Q) = = t [ T(r, z, t; Q ) f ] dt (8) here T(r, z; Q) are the temperatures at the sesor locatos computed from the drect problem ()-(6). Formally, the verse problem cossts mmzg the resdual fuctoal uder costras ()-(6). The uko heat flux s approxmated the form of a cubc B-sple as follos: Q mt mr = ( r, t ) p φ ( r ) φ (t ) (9) j= here m t x m r s the umber of approxmato parameters, p are the uko approxmato parameters ad φ are the gve bass cubc B-sple. Therefore, the IHCP s reduced to the estmato of a vector of parameters p = [ p, p,..., p m ]. The cojugate gradet method s used order to mmze the resdual fuctoal. The cojugate gradet algorhm s eratve. At each erato, the successve mprovemets of desred parameters are bult as follos: p + = p + γ = d,j j, =,,, mt x m r () here s a erato umber. follog expresso : γ s the descet parameter. d s the descet drecto determed by the d = g + β d, =,,, mt x m r () here the parameter β s computed as follos: g g β = () g, g

5 here, s the scalar product, s the orm ad β =. To mplemet the eratve mmzato procedure, the vector g s determed by the follog equato: G g = J' ( p) (3) here G s the Gram s matrx for bass fuctos. { G = φ, φ,, j,,..., m} G = (4), j j = φ, φ j s the scalar product the L space. The Gram s matrx s symmetrc ad posve. Let J (p) be the vector gradet of the fuctoal J(p) defed as : here ψ s the adjot problem soluto. tf R J' = ψ(r, H, t) φ dr dt, =,,, m (5) 3. Descet parameter To compute the descet parameter, the lear approxmato of the descet parameter s used [7]: here θ ( r, z, t; δq) γ = t f N meas [ T (r,z, t;q) Tmeas(r,z, t) ] = t N f meas = θ (r,z, t; δq ) θ (r,z, t; δq dt s computed from the problem varatos at the erato by usg the varato of the uko heat flux ( δ Q ). ( r,z,t; δq s defed at the sesor locatos from a cubc B-sple as follos: ) dt θ ) (, z ) δq mt mr (6) r. δq s approxmated ( r, t ) = δp φ ( r ) φ ( t ) (7) j= = The problem varatos s defed by the follog equatos: ρc p θ( r, z, t ) θ( r, z, t ) θ( r, z, t ) θ( r, z, t ) = + +, here r R, z H, < t t f (8) λ t r z θ (, z, t) θ (R, z, t) =,j j =, here < t tf, z H (9), here < t tf, z H () θ ( r, z,) =, here : r R, z H () θ λ (r,h,t) =δq(r,h,t), here : < t tf, r R z () θ ( r,, t) =, here : < t tf, r R (3) 3. Adjot problem The ecessary codos of optmzato [7] are defed the form of a adjot problem hch s represeted by the follog boudary-value problem: here: S(r, z, t) = { δ(r, r; z, z) = ad r, z H, < t R N meas ( r, z, t ) = ρ ψ C p ψ ψ ψ + ψ + + S(r, z, t λ t r r z ) ψ tf [ T(r, z, t; Q ) T (r, z, t) ]} (, z, t) ψ = r (, z, t) B 5 meas (4), here < t tf, z H (5)

6 ψ (R, z, t) ψ = (R, z, t), here < t tf, z H (6) r ψ ( r, z, tf ) =, here : r R, z H (7) ψ λ (r, H, t) =, here : < t tf, r R (8) z ψ ( r,, t) =, here : < t tf, r R (9) The gradet of the resdual fuctoal s determed by solvg the adjot problem. The symbol δ( r, r, z, z) represets the Drac fucto. The drect problem, the adjot problem ad the problem varatos are solved usg the cotrol volume method [8] ad the mplc fractoal-step tme scheme proposed by Bra [9]. 3.3 Regularzato The verse problem s ll-posed ad the umercal results depeds o the fluctuato occurrg at the measuremets. To obta a stable soluto of the IHCP the eratve regularzato method s used by stoppg eratos at the optmal value of the resdual fuctoal hch satsfes the follog codo: J(Q ) δ here δ s the tegrated error of the measured data as follos : t f N meas = (3) δ = σ (r,z,t)dt (3) σ ( r,z,t) s the stadard devato of measuremet errors for the temperatures measured at the poso ( r, z). I the absece of the measuremet errors, the geeral stoppg crero s defed as: here the small umber ε s used equal to 4. Q (r, t) Q (r, t) ε (3) Q (r, t) 3.4 Algorhm The verse heat coducto problem s solved usg the follog eratve procedure: The calculatos are started by usg the parameter β ad the uko heat flux Q (r, H, t) equal to zero. () soluto of the drect problem, () calculato of the resdual fuctoal, () soluto of the adjot problem, (v) calculato of the compoets of the fuctoal gradet, (v) calculato of the parameter β, (v) calculato of the compoet of descet drecto, (v) soluto of the problem varato to determe the parameter γ, (v) the e value of the heat flux desy s corrected. If the covergece crero s ot satsfed the eratve procedure s repeated utl the fuctoal s mmzed. 4. EXPERIMENTAL DATA PROCESSING The umercal procedure as valdated by usg the temperature data obtaed from the drect problem to smulate the measured temperatures sde the cylder. The local heat flux requred to solve the drect problem s ko ad s take varable spatally ad temporally. For each umercal applcato, the tme step sze s choose h respect the delta Fourer umber codo based o the sesor depth that s defed by the follog equato: λ t Fo =. (33) ρc H H p B 6 ( ) 4. Numercal verfcato of the soluto procedure We cosder a copper vertcal cylder havg 5 mm of dameter ad mm of legth. We have choose to verfy the umercal procedure by usg a ko heat flux varyg h the tme ad the radus of the cylder. The heat flux s mposed at the etted surface of the cylder (z=h) as sho Fgure 4 by the cotuous curve. The doard surface (z=) s assumed to be sotherm (T(r,,t)= 4 C). The uko traset heat flux s assumed to vary as follos: meas

7 B (.4r r 6463 r r r )(.t Q (r,t) =. + ) (34) Q [kw/m ] 6 t = 5s 4 t = s t = 5s t = 35s t = 4s r [m] Fgure 4. Comparso of the calculated ad exact heat fluxes. T meas - T [ C] t = 5s t = s t = 5s t = 35s -.3 t = 4s r [m] Fgure 5. Radal dstrbutos of the temperature resduals J(Q ) Fgure 6. Evoluto of resdual fuctoal as a fucto of the umber of eratos. I order to valdate the verse estmato procedure, e have assumed that the temperatures calculated from the drect problem soluto at the measuremet pots are the measured temperatures ( Tmeas (r, z, t) = f(r, z, t) ) for solvg ICHP. Fgure 4 shos that the estmated heat flux practcally cocdes h the exact heat flux for dfferet tmes. Ths valdato s carred out for the umber of approxmato parameters equal to 9x9. Fgure 5 shos the dfferece betee the computed temperatures ad the smulated measured temperatures. The maxmum devato s of ±.3 C. Fgure 6 shos the evoluto of the resdual fuctoal J(Q ) as a fucto of the umber of eratos. The eratos are cotued tll the covergece - crera J< s satsfed. 4. Expermetal determato of the local thermal boudary codos A vertcal brass cylder havg 5 mm of dameter s cosdered. The sesors locatos are shoed Fgures a ad b. A grd of odes alog the heght of the cylder ad odes the radal drecto have bee used. The local thermal characterstcs are computed by solvg IHCP here the temperatures measured at doard surface are used as the boudary codo to solve the drect problem. The temperatures measured at

8 heght z = 7.4 mm are used to solve the verse problem. The computato s coducted by usg the umber of measuremet pots equal to the odes umber. Durg expermets, the fxed flo rate for the study s frst adjusted uder gravy usg the regulatg valves. Ially the cylder dsk s covered order to prevet the lqud to et the exchage surface. For a fxed total heat flux, the expermetal dsk s heated. Whe the measured all temperature reached a steadystate, the exchage surface s rapdly ucovered. The lqud jet the hs the heat trasfer surface perpedcularly. The tme-depedet local all temperatures s recorded, utl the expermetal dsk reached a e steady-state. T [ C] Q [kw/m ] T s [ C] Heated Heatg the the surface Coolg the surface 3 4 t [s] Fgure 7. Measured temperature sde the sold. B Surface temperature Surface heat flux t [s] Fgure 8. Estmated heat flux ad surface temperature. Fgure 7 shos the evoluto of the measured temperature sde the sold by the sesor placed the vertcal axs at.6mm far from the exchage surface. It shos to dfferet zoes, the frst oe here the surface s ot etted by the ater jet ad the all temperature creased cotually alog the tme. The steady state s attaed for t approxmately equal to 8s (see Fgure 7). After ths tme, the temperature sde the sold becomes uform alog the tme. Fgure 7 shos a secod zoe here the all temperature decreased alog the tme because of the jet mpactg the heated surface s at the temperature (= 37 C) loer tha the all temperature. Whe the ater jet mpacted the heated surface, the all temperature decreased ad attaed the stable value durg a quck perod (s approxmately) from 73.6 C to 39.4 C. Fgure 8 shos the estmato of the surface temperature ad the surface heat flux the stagato pot (r=) for the phase of the coolg surface. The heat trasfer s hghest the zoe of the stagato ad the mpgemet lqud flo suated at the ceter of the heated surface here the flm thckess does approach fy. After ths zoe, the heat trasfer decreases because the th flm becomes relatvely uform o the etted surface. Fgures 9 ad are obtaed for the electrc poer put of 4 W ad 7W; ad, the flo rate of ater jet of 9 g.s -. The estmated stadard devato of temperature resduals of ths computato s of.4 C. It s shoed that he the jet of the ater mpacted the heated surface, the dsspated heat flux creased durg s, decreased durg 8s, ad fally reached a asymptotc value at the steady state. I ths zoe, the surface temperature reaches the mmum ad becomes uform alog the tme T [ C] Q [kw/m ] W W W W t [s] t [s] Fgure 9. Measured temperature sde the sold. Fgure. Estmated heat flux. 7

9 B 9 Fgures 9 ad sho the fluece of the electrc poer mposed sde the sold. For these fgures, expermets are coducted for the ater let temperature of 37 C. For the total electrc poer of 4W ad 7W, Fgure 9 shos the measured all temperature obtaed from the thermocouples located the axal drecto at.6mm far from the surface. It ca be see that the all temperature decreased for each electrc poer ad reached the same temperature value at the steady state. Fgure shos the dstrbuto of the heat flux after the ater jet mpactg the heated surface. For 4 W, the heat flux dsspated at the surface reached the e steady state durg a tme loer tha for 7W. Fgure shos the radal dstrbuto of the local Nusselt umber for to fxed flo rate of the mpgg jet (5 g/s ad 4.4 g/s). I ths expermet, the poer put s of 5W ad the let temperature of the ater (T ) s of 4 C Nu r 5 g/s 4.4 g/s r [mm] Fgure. Radal dstrbuto of th e local Nusselt umber. The local thermal characterstcs are computed by solvg the IHCP here the temperatures measured at z= are used as the boudary codo to solve the drect problem. The temperatures measured at z=7.4mm are used to solve the verse problem. The local heat trasfer coeffcet s calculated from the surface heat flux desy (Q,r ), the local surface temperature (T s,r ) as follos: Q,r h r = (35) T T s,r The local Nusselt umber s deduced from hr as: = h r Nu D r (36) λ here D s the dameter of the heated surface, λ s the thermal coductvy. Fgure shos that the radal Nusselt umber creases h the mpgg ater flo rate because at hgher flo rate the lqud flm carres more eergy from the heated surface. It s oted that the heat trasfer s hghest the stagato zoe here the flm thckess does approach fy. After ths zoe, the heat trasfer decreases the th lqud flm thckess. 5. CONCLUSIONS A traset measuremet procedure as developed to combe measuremets of temperature, IHCP method h jet characterstcs ad electrc poer put. Ths artcle presets the results obtaed by solvg the IHCP appled to the cyldrcal geometry. The local heat flux ad the surface temperature are determed by usg the measured temperatures sde the heated sold. Ths artcle shos the advatage of usg IHCP to determe the thermal boudary characterstcs. I ths ork, the verse aalyss s appled to a partcular case of jet mpactg the heated surface. I ths case, the measuremet of the local heat flux or surface temperature s very dffcult hout perturbg the flo. I more, the comparso of the calculated ad exact heat fl ux ad the evoluto of calculated ad measured temperature sho the precse character of the method. REFERENCES. B. Elso ad B.W. Webb, Local heat trasfer to mpgg lqud jets the ally lamar, trasoal, ad turbulet regmes. It. J. Heat Mass Trasfer (994) 37, 7-6.

10 . Y. Pa ad B.W. Webb, Heat trasfer characterstcs of array of free-surface lqud jets. Tras. ASME, J. Heat Trasfer (995) 7, L.M. Jj ad Z.Daga, Expermetal vestgato of sgle-phase multjet mpgemet coolg of a array of mcroelectroc heat sources, Coolg Techology for Electroc Equpmet. Iteratoal Symposum Hoolulu, HA, 988, pp Y. Pa ad B.W. Webb, Vsualzato of local heat trasfer uder arrays of free surface lqud jets. Istuto of Chemcal Egeers Symposum Seres (994) 35, E.J. Watso, The radal spread of a lqud over horzotal plae. J. Flud Mech. (964), X. Lu ad J.H. Lehard V, Lqud mpgemet heat trasfer o a uform flux surface, Natoal Heat Trasfer Coferece, 989, pp O.M. Alfaov, E.A. Artyukh ad S.V. Rumyatsev, Extreme Methods for Solvg Ill-Posed Problems h Applcatos to Iverse Heat Trasfer Problems, Begell House, Ne York, S.V. Patakar, Numercal Heat Trasfer ad Flud Flo, Mc-Gra Hll, Ne York, 98. B 9. P.L.T. Bra, A fe-dfferece method of hgher order accuracy for soluto of three dmesoal traset heat coducto. A.I.Ch.E. J. (96) 7,