H05 A 3-D INVERSE PROBLEM IN ESTIMATING THE TIME-DEPENDENT HEAT TRANSFER COEFFICIENTS FOR PLATE FINS

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1 Proceedigs of te 5 t Iteratioal Coferece o Iverse Problems i Egieerig: Teory ad Practice, Cambridge, UK, t July 005 A 3-D INVERSE PROBLEM IN ESTIMATING THE TIME-DEPENDENT HEAT TRANSFER COEFFICIENTS FOR PLATE FINS C.-H. Huag 1, Y.-L. Tsai 1 ad H.-M. Ce 1 Departmet of Systems ad Naval Mecatroic Egieerig, Natioal Ceg Kug Uiversity, Taia, Taiwa 701, R.O.C. cuag@mail.cku.edu.tw Departmet of Iformatio Maagemet, Kug Sa Uiversity of Tecology, Taia Hsie, Taiwa 710, R. O. C. mce@mail.ksut.edu.tw Abstract - Te local time-depedet surface eat trasfer coefficiets for plate fied-tube eat excagers are estimated i a tree-dimesioal iverse eat coductio problem. Te iverse algoritm utilizig te Steepest Descet Metod (SDM) ad a geeral purpose commercial code CFX4.4 is applied successfully i tis study i accordace wit te simulated measured temperature distributios o te fi surface by ifrared termograpy. Two differet eat trasfer coefficiets for staggered as well as i-lie tube arragemets wit differet measuremet errors are determied. Results of te umerical simulatio sow tat te reliable estimated eat trasfer coefficiets ca be obtaied by usig te preset iverse algoritm. 1. INTRODUCTION Heat excagers are te workorse of idustry. Variously kow as codesers, coolers, evaporators, eaters, vaporizers, ad so fort. Fied surfaces of te plate fied-tube eat excagers ave bee i use over a log period of time for dissipatio of eat by covectio. Applicatios for fied surfaces are widely see i aircoditioig, electrical, cemical, refrigeratio, cryogeics ad may coolig systems i idustry. Kays ad Lodo [9] itroduced various types of eat trasfer surfaces. Te estimatio of te covective eat trasfer coefficiet is more difficult to perform ta oter commo termo-fluid-dyamic quatities, especially i te case of o-uiform distributios ad/or of coductiocovectio problems. Ay et al. [] applied a cotrol volume based fiite differece formulatio ad a ifrared termograpy based temperature measuremets to estimate te local eat trasfer coefficiets of a plate fi i a -D iverse eat coductio problem. Recetly, Huag et al. [7] used te tecique of Steepest Descet Metod (SDM) ad commercial code CFX4.4 [3] to estimate te local covective eat trasfer coefficiets over fied surfaces i a steady-state 3-D iverse eat coductio problem based o te simulated temperature measuremets by ifrared termograpy. However te 3-D iverse eat coductio problem i estimatig te time-depedet local covective eat trasfer coefficiets o fied surface as ever bee examied. Te tecique of utilizig te iverse algoritms togeter wit te commercial code CFX4. as bee developed successfully by Huag ad Wag [6], tey applied te algoritm to estimate te ukow surface eat fluxes i a 3-D solid. By followig a similar tecique, Huag ad Ce [4] estimated successfully te ukow boudary eat flux i a 3-D iverse eat covectio problem. More recetly, Huag ad Li [5] applied te algoritm to a optimal eatig problem i determiig te optimal surface eat fluxes for a 3-D forced covectio problem. It sould be oted tat all of te above applicatios are 3-D iverse problems, tis implies tat te algoritm is powerful sice te 3-D iverse problems are still very limited i te ope literature. Te objective of tis study is to exted a 3-D steady-state iverse problem [7] to a trasiet 3-D iverse problem i estimatig te time-depedet local covective eat trasfer coefficiets of fied surfaces for te plate fied-tube eat excagers. Te umber of ukow eat trasfer coefficiets will icrease tremedously uder te preset cosideratio ad tis will also icrease te difficulty i solvig te preset iverse problem.. DIRECT PROBLEM A typical plate fied-tube eat excager is sow i Figure 1(a). Te plate fis of staggered arragemet wit domai Ω(x,y,z) is illustrated i Figure 1(b). Te surfaces S i, i = 1 to 6, are subjected to a covective boudary coditio wit prescribed eat trasfer coefficiet (S i,t), i = 1 to 6, were i = 1 to 4 represet te edge boudaries; wile i = 5 ad 6 idicate te top ad bottom surfaces, respectively. Te ukow eat trasfer coefficiet (S i,t) could be a fuctio of te temperature i te preset study. Te tube boudary surfaces S i, i = 7 to (I+6), are subjected to a prescribed temperature coditio T = T(S i,t), were I represets te umber of tubes.

2 Te edge surface area S i, i = 1 to 4 is small eoug we comparig wit top ad bottom surfaces S i, i = 5 to 6. Tis implies tat te eat trasfer rate troug S i, i = 1 to 4 ca be eglected. For tis reaso we assume tat te boudary coditios o surface S i, i = 1 to 4 are adiabatic coditios. Meawile, sice te fi tickess is ti, te temperature distributio o S 5 sould be very close to S 6 for ay time t, terefore it is also reasoable to assume tat te eat trasfer coefficiets o S 5 ad S 6 are equal to eac oter, i.e. (S 5,t) = (S 6,t). Te direct problem becomes T( T( T( T( k[ + + ] = ρcp ; i Ω(x,y,z,t), t > 0 (1a) x y t T(S i,t) = 0 ; o fi surface S i i=1 to 4, t > 0 (1b) T(S,t) k 5 = (S5,t)(T T) ; o fi surface S 5, t > 0 (1c) T(S,t) k 6 = (S6,t)(T T ) ; o fi surface S 6, t > 0 (1d) T(Si, t) = To ; o tube surfaces, i = 7 to I+6, t > 0 (1e) T( Ω, t) = T ; for t = 0 (1f) Here k is te termal coductivity of te fi, ρ ad Cp are te desity ad eat capacity of te material, respectively. Te direct problem cosidered ere is cocered wit calculatig te plate fi temperatures we te eat trasfer coefficiet (S i,t), i = 5 ad 6, termal properties as well as te iitial ad boudary coditios o tube surfaces are kow. Te solutio for te above 3-D eat coductio problem i domai Ω is solved usig CFX4.4 ad its Fortra subroutie USRBCS. 3. THE INVERSE PROBLEM For te iverse problem cosidered ere, te local time-depedet eat trasfer coefficiets (S i,t), i = 5 ad 6, are regarded as beig ukow, but everytig else i eq (1) is kow. I additio, te simulated temperature readigs usig ifrared termograpy o te fi surfaces S 5 ad S 6 are assumed to be available. Let te temperature readig take by ifrared scaers o fi surfaces S 5 ad S 6 be deoted by Y(S i,t) Y(x m,y m,t) Y m (S i,t), m = 1 to M ad i = 5 ad 6, were M represets te umber of measured temperatures at te extractig poits. Tis iverse problem ca be stated as follows: by utilizig te above metioed measured temperature data Y m (S i,t), estimate te ukow local time-depedet eat trasfer coefficiets (S i,t). Te solutio of tis iverse problem is to be obtaied i suc a way tat te followig fuctioal is miimized: tf M J [(S = i, t)] [Tm t) Ym t)] dt ; i = 5 ad 6 () t= 0 m= 1 were T m (S i,t) are te estimated or computed temperatures at te measured temperature extractig locatios (x m,y m ) ad at time t. Tese quatities are determied from te solutio of te direct problem give previously by usig te estimated local eat trasfer coefficiets (S i,t). 4. STEEPEST DESCENT METHOD FOR MINIMIZATION A iterative process based o te steepest descet metod [1] is ow applied for te estimatio of ukow eat trasfer coefficiets (S i,t) by miimizig te fuctioal J[(S i,t)], amely + 1 (S, t) (S, t) P i = i β t) ; for = 0, 1, ad i = 5 ad 6 (3) β is te searc step size i goig from iteratio to iteratio +1, ad P (S i,t) is te directio of descet (i.e. searc directio) give by P (S,t) J' i = t) ; i = 5 ad 6 (4)

3 3 wic is te gradiet directio J' t) at iteratio. To complete te iteratios i accordace wit eq (3), te step size ad te gradiet of te fuctioal J' t) eed be computed. I order to develop expressios for determiig tese two quatities, a "sesitivity problem" ad a "adjoit problem" eed be costructed as described below. 4.1 Sesitivity problem ad searc step size It is assumed tat we (S i,t) udergoes a variatio, T is perturbed to T+ T. Te replacig i te direct problem by + ad T by T+ T, subtractig from te resultig expressios te direct problem ad eglectig te secod-order terms, te followig sesitivity problem for te sesitivity fuctio T is obtaied: β T( Ω, t) T( Ω, t) T( T( k[ + + ] = ρcp x y t ; i Ω(x,y,z,t), t > 0 (5a) T(S i,t) = 0 ; o fi surfaces S i, i =1 to 4, t > 0 (5b) T (S5,t) T + k = (S5, t)(t T ) ; o fi surface S 5, t > 0 (5c) T (S6, t) T + k = (S6, t)(t T) ; o fi surface S 6, t > 0 (5d) T(S i, t) = 0 ; o tube surfaces, i = 7 to I+6, t > 0 (5e) T( = 0 ; for t = 0 (5f) By followig te stadard process as described i [8], te searc step size β ca be determied as: t f M [Tm t) Ym t)] Tm t)dt β = t= 0 m= 1 ; i = 5 ad 6 (6) t f M [ Tm t)] dt t= 0 m= 1 4. Adjoit problem ad gradiet equatio To obtai te adjoit problem, eq (1a) is multiplied by te Lagrage multiplier (or adjoit fuctio) λ( ad te resultig expressio is itegrated over te correspodet space domai. Te te result is added to te rigt ad side of eq (). By followig te stadard process as described i [8], te followig adjoit problem for te determiatio of λ( ca be obtaied: λ( λ( λ( Ω, t) λ( Ω, t) k[ + + ] + ρcp = 0 ; i ( Ω, t), t > 0 (7a) x y t λ( = 0 ; o fi surfaces S i, i =1 to 4, t > 0 (7b) λ λ + k = k[t(s5,t) Y(S5,t)] δ(x xm) δ(y ym) ; o fi surface S 5, t > 0 (7c) λ λ + k = k[t(s6,t) Y(S6,t)] δ(x xm) δ(y ym) ; o fi surface S 6, t > 0 (7d)

4 4 λ ( S i, t) = 0 ; o tube surfaces, i = 7 to I+6, t > 0 (7e) λ( = 0 ; for t = t f (7f) Fially, te gradiet of te fuctioal J[(S i,t] ca be obtaied as: λ(s,t) J'[(S,t)] i i = [Tt) T ] ; o surfaces S i, i = 5 ad 6 (8) k 5. RESULTS AND DISCUSSIONS Te objective of tis study is to sow te validity of te SDM i estimatig te time-depedet local surface eat trasfer coefficiets for a 3-D plate fied-tube eat excagers wit o prior iformatio o te fuctioal form of te ukow fuctio. Te pysical model for tis problem is described as follows: Te termal coductivity for te plate fi is take as k = 0W/(m-K), ρ = 7850 kg/m 3, Cp = 440 J/kg-K; ambiet temperature is cose as T = 96 K ad te temperatures o all tube surfaces are assumed as T(S i,t) = 353 K, i = 7 to (I+6). Oe of te advatages of usig te SDM is tat te iitial guesses of te ukow eat trasfer coefficiets (S i,t) ca be cose arbitrarily. I all te test cases cosidered ere, te iitial guesses for eat trasfer coefficiets used to begi te iteratio are take as (S i,t) = 0.0. I order to compare te results for situatios ivolvig radom measuremet errors, we assume ormally distributed ucorrelated errors wit zero mea ad costat stadard deviatio. Te simulated iexact measuremet data Y ca be expressed as Y m = Y m,exact + ω σ (9) were Y m,exact is te solutio of te direct problem wit exact eat trasfer coefficiets; σ is te stadard deviatio of te measuremets; ad ω is a radom variable, tat geerated by subroutie DRNNOR of te IMSL [8], is witi to.576 for a 99% cofidece boud. I order to simplify te problem, te measuremet errors o te surfaces S 5 ad S 6 are assumed te same. We ow preset below te umerical experimets i determiig (S i,t) usig te iverse aalysis. Te geometry ad grid system for te first test case, i.e. staggered tube arragemet for a fi plate, are sow i Figures (a) ad 3(a), respectively. Te dimesios for te fi i te x, y ad z directios are 0 mm, 170 mm ad 1mm, respectively. Te radius of te tube is take as 1.7 mm ad te logitudial pitc of te tube, i.e te distace betwee ceter of te two tubes, is 60.7 mm. Te umber of grids i te z-directio is take as 5 ad te total grid umber o te x-y plae is Te measured temperature extractig locatios are at te grid poits. Te measuremet time period t is 150 secods ad te total measuremet time t f is 3750 secods, i.e. tere are 5 time steps. Terefore tere exist a total of ukow discrete eat trasfer coefficiets i tis study. Te simulated exact fuctio of te surface eat trasfer coefficiets o surfaces S 5 ad S 6 i tis umerical experimet is assiged i te followig maer: (a) Firstly, solve eq (1a) by assumig te followig boudary ad iitial coditios: (y y) T(S,t) 0 max 3 = + 69 ; o S 3, were y max = 0, t > 0 (10a) ymax T(S 4,t) = 0 ; o S 4, t > 0 (10b) T(S i,t) = 0 ; o te rest of surfaces, t > 0 (10c) T(Ω, t) = 0 ; at t = 0 (10d) (b) Secodly, te values of te calculated temperature distributios o S 5 ad S 6 are te take as te simulated exact eat trasfer coefficiets. Te tree-dimesioal iverse problem is first examied by usig exact measuremets, i.e. σ = 0.0. After 30 iteratios te iverse solutios coverged. Te exact ad estimated (or calculated) eat trasfer coefficiets (S 5,t) at time t = 3600 s are reported i Figure. Te estimated eat trasfer coefficiets are also close to te exact values. Te relative error betwee exact ad estimated eat trasfer coefficiets is calculated as ERR1 =.9 %, were ERR1 is defied as J M (S, j) ĥ (S, j) ERR1 % = m 5 m 5 (M J) 100 % j 1 m 1 m (S5, j) (11) = = were J represets te umber of te discreted times, M te umber of grids ad ĥm, j(s5) te estimated values. Te correspodig measured ad estimated temperature distributios at time t = 3600 s are sow i Figure 3. By comparig Figures 3(a) ad 3(b) we fid tat te estimated temperatures are almost idetical to te

5 measured temperatures sice te relative error betwee te measured ad calculated temperatures is calculated as ERR = 0.05%, were ERR is defied as J M T (S, j) Y (S, j) ERR % = m 5 m 5 (M J) 100 % j 1 m 1 Ym (S5, j) (1) = = Te iverse calculatio te proceeds to cosider te iexact temperature measuremets. Te stadard deviatio of te measuremets is first take as σ = 0.1, te it was icreased to σ = 0.3. For σ = 0.1, 10 iteratios are eeded to satisfy te stoppig criteria based o te discrepacy priciple, te estimated eat trasfer coefficiets at times t = 50 s ad 3600 s are sow i Figure 4. Te relative errors for te eat trasfer coefficiets ad te temperatures are calculated as ERR1 = 7.80 % ad ERR = %. For σ = 0.3, te umber of iteratios to satisfy te stoppig criterio is oly 8, te estimated eat trasfer coefficiets at times t = 50 s ad 3600 s are sow i Figure 5, ad te relative errors for te eat trasfer coefficiets ad te temperatures are calculated as ERR1 = 11.6 % ad ERR = %. Based o te above umerical results, we cocluded tat te estimated eat trasfer coefficiets are sesitive to te measuremet errors, ad terefore for tis reaso a accurate measuremet tecique is required for suc kid of problem. 6. CONCLUSIONS Te SDM wit a adjoit equatio was successfully applied i determiig te time-depedet local eat trasfer coefficiets for plate fied-tube eat excagers for a 3-D iverse eat coductio problem. Two test cases ivolvig differet arragemet of fis, differet type of eat trasfer coefficiets ad differet measuremet errors were cosidered. Te results sow tat te SDM does ot require a priori iformatio for te fuctioal form of te ukow fuctios ad reliable estimated values ca always be obtaied. Ackowledgmet Tis work was supported i part troug te Natioal Sciece Coucil, R. O. C., Grat umber, NSC H REFERENCES 1. O. M. Alifaov, Iverse Heat Trasfer Problem, Spriger-Verlag, Berli, H. Ay, J. Y. Jag ad J. N. Ye, Local eat trasfer measuremets of plate fied-tube eat excagers by ifrared termograpy, It. J. Heat Mass Trasfer (00) 45, CFX-4.4 User's Maual, AEA Tecology Plc, Oxfordsire, U.K., C. H. Huag ad W. C. Ce, A tree-dimesioal iverse forced covectio problem i estimatig surface eat flux by cojugate gradiet metod, It. J. Heat Mass Trasfer (000) 43, C. H. Huag ad C. Y. Li, A tree-dimesioal optimal cotrol problem i determiig te boudary cotrol eat Fluxes, Heat Mass Trasfer (003) 49, C. H. Huag ad S. P. Wag, A tree-dimesioal iverse eat coductio problem i estimatig surface eat flux by cojugate gradiet metod, It. J. Heat Mass Trasfer (1999) 4, C. H. Huag, I. C. Yua ad H. Ay, A tree-dimesioal iverse problem i imagig te local eat trasfer coefficiets for plate fied-tube eat excagers, It. J. Heat Mass Trasfer (003) 46, IMSL Library Editio User's Maual: Mat Library Versio 1.0, IMSL, Housto, TX, W. M. Kays ad A. L. Lodo, Compact Heat Excagers, 3rd ed., Mcgraw-Hill, New York, H05 5

6 6 Figure 1(a). A typical plate fied-tube eat excager. Figure 1(b). Te geometry of plate fi i staggered arragemet.

7 7 (a) (b) Figure. (a) Te exact, ad (b) estimated, eat trasfer coefficiets at t = 3600 s wit σ = 0.0.

8 8 (a) (b) Figure 3. (a) Te measured, ad (b) estimated, temperatures at t = 3600 s wit σ = 0.0.

9 9 (a) (b) Figure 4. Te estimated eat trasfer coefficiets at (a) t = 50 s, ad (b) t = 3600 s, wit σ = 0.1.

10 10 (a) (b) Figure 5. Te estimated eat trasfer coefficiets at (a) t = 50 s, ad (b) t = 3600 s, wit σ = 0.3.