Temperature and Heat Flux Estimation from Sampled Transient Sensor Measurements. Department of Mechanical and Aerospace Engineering

Transcription

1 Temperature and Heat Flux Etimation rom Sampled Tranient Senor Meaurement Z. C. Feng, J. K. Chen, Yuwen Zhang 3 Department o Mechanical and Aeropace Engineering and Stephen Montgomery-Smith 4 Department o Mathematic Univerity o Miouri Columbia, MO 65, USA Abtract Laplace tranorm i ued to olve the problem o heat conduction over a inite lab. The temperature and heat lux on the two urace o a lab are related by the traner unction. Thee relationhip can be ued to calculate the ront urace heat input (temperature and heat lux) rom the back urace meaurement (temperature and/or heat lux) when the ront urace meaurement are not eaible to obtain. Thi paper demontrate that the ront urace input can be obtained rom the enor data without reorting to invere Laplace tranorm. Through Hadamard Factorization Theorem, the traner unction are repreented a ininite product o imple polynomial. Conequently, the relationhip between the ront and back urace are tranlated to the time-domain without invere Laplace tranorm. Thee timedomain relationhip are ued to obtain approximate olution through iterative procedure. We elect a numerical method that can mooth the data to ilter out noie and at the ame time obtain the time derivative o the data. The moothed data and time derivative are then ued to calculate the ront urace input. Correponding author. eng@miouri.edu, Fax: , Phone: chenjnk@miouri.edu 3 zhangyu@miouri.edu 4 montgomerymith@miouri.edu Page

2 NOMENCLATURE c p ma peciic heat o the lab, J/(kg K) G Traner unction i k thermal conductivity, W/(m K) L thickne o -D lab, m p pole o a traner unction q heat lux, W/m Q t t c T Laplace tranorm o the heat lux time, characteritic time, temperature o the lab above the ambient temperature, K U x z Greek ymbol Laplace tranorm o temperature patial coordinate variable, m zero o a traner unction ρ denity o the lab, kg/m 3 τ ω ξ dimenionle time dimenionle requency dimenionle length variable Subcript b back urace quantity ront urace quantity Keyword: Invere problem, traner unction, Savitzky-Golay, enor compenation, temperature meaurement Page

3 . Introduction To conduct thermal meaurement under harh environment, it ha been propoed that enor be located away rom direct contact with the environment. The deired meaurement are obtained by conidering the heat traner path between the point o interet and the enor. Imagine a thin plate eparating the environment and the enor that are mounted on the back urace. The ront urace temperature can be determined indirectly by olving an invere heat conduction problem [-3] baed on the tranient temperature and/or heat lux meaured at the back urace. Among the many method propoed to olve the invere heat conduction problem, the Laplace tranorm method (i applicable) mot conciely capture the mathematical relationhip in term o traner unction [4-8]. To obtain the deired quantitie rom the enor meaurement, the invere Laplace tranorm mut be ound. Since cloed orm invere Laplace tranorm rarely exit or the heat equation, many algorithm or the approximate invere Laplace tranorm have been propoed or thi purpoe [8-4]. Mot o the approximate invere Laplace tranorm algorithm tart rom known unction in the Laplace domain. Thi preent a evere retriction ince the time-domain enor data mut be tranormed into the Laplace domain. Obtaining meaningul Laplace tranorm rom the enor data i no trivial matter conidering the unavoidable noie contamination. Moreover, mot enor data are dicretized data ampled at equal time interval. The nature o the ampled data add to the complexity o recontructing the invere Laplace tranorm. In our previou work, we have propoed polynomial approximation to the traner unction [8]. In thi paper, we ue the Hadamard Factorization Theorem to expre traner unction a ininite product. Through the ininite product expanion o the traner unction, time domain relationhip between the ront and back urace are obtained. Thee relationhip are implemented in iterative procedure. In the ollowing ection, we preent the impliied model or the heat equation and the traner unction relating the thermal quantitie between the ront and back urace. In ection 3, iterative procedure are developed. In ection 4, we preent the reult o recontructed ront urace heat lux and temperature by uing the back urace temperature rom the analytical olution a the meaurement data. Noie i added to the analytical data. A concluion i given in ection 5.. Mathematical model and the Laplace tranorm olution Conider one-dimenional heat conduction over a inite lab. The governing equation i given by the ollowing: Page 3

4 where ρ, k and T T ρ c p = k () t x c p are the denity, thermal conductivity and peciic heat o the olid; they are all aumed to be contant. Since the one dimenional model here repreent an approximation o a heet-like threedimenional body when the temperature gradient in the in-plane direction are ignored, it i thu more intuitive to regard x = 0 and x = L a the ront and the back urace o the heet-like olid. To impliy the problem urther, we chooe the heet thickne L a the characteritic length and the contant t c ρ c p L = () k a the characteritic time. Thu we impliy the governing equation to where τ = t / t and ξ = x / L c T T = τ ξ or 0 < ξ < (3) now repreent the dimenionle time and dimenionle poition acro the thickne. The boundary condition can be precribed temperature on the ront and back urace T (t) and T b (t). Or the heat luxe on the ront urace and back urace can be precribed, i.e. T ( ξ = 0) = q ( t) ξ T ( ξ = ) = qb( t) ξ (4a) (4b) where q (t) and q b (t) are normalized heat lux (K) at the ront and back urace. The actual ront urace heat lux i q ( t) k L (W/m ) or intance. / Aume zero initial condition. Applying the Laplace tranorm [4, 5] to equation (3), we obtain the ollowing equation d U ( ξ ) U ( ξ ) = dξ where U (ξ) i the Laplace tranorm o T ( ξ, τ). The olution o the reulting equation i written a (5) U ξ ξ = c e + ce (6) Page 4

5 I we aume the back urace meaurement are known, we have the ollowing relationhip [6, 8, 5] U Q ( ) = ( ) coh inh inh U coh Q where the two vector repreent the Laplace tranorm o the temperature and heat lux o the ront urace and back urace repectively. The above matrix equation etablihe algebraic relationhip in the Laplace domain between the temperature and heat lux o the two urace. For intance, i the back urace i adiabatic, i.e. Q b b b ( ) ( ) ( ) = 0, we have the ollowing relationhip between the temperature on two urace: U ( ) = coh U b ( ), (8) and the ollowing relationhip between the ront urace heat lux and the back urace temperature: Q ( ) = inh U b ( ) () (7) To obtain the temperature and heat lux in the time domain rom the known olution in the Laplace domain, invere Laplace tranorm mut be ound. The analytical orm o the invere Laplace tranorm i poible only or a ew very pecial cae [4, 5]. In the literature, olution in time domain are obtained through numerical invere o the traner unction. Becaue o the convenience o Laplace tranorm method or variou phyical problem, there are well over 00 dierent algorithm available or calculating the invere Laplace tranorm [9-4]. In our previou work [8], we have propoed the polynomial approximation to the traner unction by matching the zero and pole o the traner unction. We preent a dierent approach baed on the Hadamard Factorization Theorem decribed in Chapter XI o [4a] in the ollowing. 3. Ininite product repreentation o the traner unction and the iterative olution in time domain Hadamard Factorization Theorem tate that i G() i a unction analytic on the whole complex a plane, atiying a growth condition G( ) C e or a poitive contant C and 0 < a < (or retated in the language o [4a], G() i an entire unction o order le than one), then G() ha the repreentation Page 5

6 G ( ) = G(0) ( ) (4) p k= k where p are the root o G (), repeated i neceary in the cae o multiple root. Since coh and k inh / are entire unction o order le than one (peciically we can take a = / ), it ollow that coh = (k ) π + k= (5) and inh = [ + ( kπ ) ]. (6) k= I inite number o term are kept in (5) and (6), the reulting polynomial are thoe obtained in [8]. We have examined the magnitude and phae or = iω o thee unction againt their polynomial approximation in [8] graphically. In the ollowing, we develop an iterative procedure with which it i eay to implement polynomial approximation to any order. Figure how the comparion when the polynomial are truncated at 8 th order. The exact and approximate curve or magnitude exactly overlap. The dierence in phae angle i mall but increae at requency increae. A more term are included, the phae angle dierence i reduced. With the ininite product repreentation o the traner unction, equation (8) can be expreed in the time domain a ollow: (k ) π d T ( τ ) = + Tb ( τ ) k. (7) = dτ Equation (7) incorporate no approximation. Thi compact relationhip ha not been reported in the literature. Aide rom providing a unique perpective on the olution o the heat equation, it can be ued to obtain approximation to the exact olution. The above equation can be written in the iterative orm: with T ( k+ ) (k + ) π ( τ ) = + d T dτ ( k ) ( τ ), or k =,, L, (8) Page 6

7 () π d T ( τ ) = [ + ( ) ] Tb ( τ ). (9) dτ The exact olution i obtained in the limit: ( k ) T ( τ ) = limt ( τ ). (0) k Similarly, equation () can be written a inh Q ( ) = Ub ( ), () which i expreed in time domain a q d d ( τ ) = [ + ( kπ ) ] Tb ( τ ). () dτ dτ k= The exact olution or the heat lux can be obtained in the limit: ( k ) q ( τ ) = lim q ( τ ). (3) k In the above, the equence are given by the ollowing iterative proce: where q q ( k+ ) () d ( k ) ( τ ) = [ + ( kπ ) ] q ( τ ), or k =,, L, (4) dτ dtb ( τ ) ( τ ) =. (5) dτ In ummary, by expreing the traner unction a ininite product, we derived the relationhip in the time domain. Thee relationhip are urther expreed in iterative orm. The iterative orm in (8) and (4) are cloe to identity when k i large. Thereore, approximate olution can be achieved when iteration converge. In practice, the meaurement rom the back urace are dicrete data ampled at a contant rate. In order to apply the iterative proce to ind the olution on the ront urace, we adopt a method that can be ued when only the dicrete data are available. 4. Calculation o the ront urace temperature and heat lux rom back urace meaurement Page 7

8 Although there i a vat literature in digital ignal proceing in dealing with dicrete data, our olution alo require time derivative o the ampled data. We have elected the Savitzky-Golay method [6-9] or it proven atiactory perormance in handling imilar problem. Thi method i widely ued acro dierent dicipline. In act, the original paper ha received well over two thouand citation. The Savitzky-Golay method perorm leat-quare it o a point and m point on either ide o it with a choen order o polynomial. For point on either edge o the data, aymmetric m+ data point are choen or the it. Thi correpond to perorming a moving m+ leat-quare it acro the data. The weight can be eaily calculated uing Gram polynomial. The weight or a 7 point, quartic polynomial it are given in the Appendix. A illutration on uing the weight, or given data y, y, K y, the moothed value correponding to y i given by n Y = (456y + 5y 35y3 + 0y4 + 0y5 9y6 + 5y7 ), (6) 46 where the weight are given in the econd column o the table in the Appendix. The weight in the next two column are ued to calculate Y and Y3 repectively. From Y 4 to Y 3, i.e. the point away rom the edge, the ollowing ormula i ued: n Y i = (5yi 30yi + 75yi + 3yi + 75yi+ 30yi+ + 5yi ). (7) The derivative calculation are conducted imilarly uing the weight in the table given in the Appendix. We apply the method developed above to calculate the ront urace temperature and heat lux while the back urace i aumed to be adiabatic; ee Figure. Aume that the ront urace heat lux i a unit tep. The analytical olution i available in the literature [30]: where T ( τ ) = T ( τ,0) (8) T b ( τ ) = T ( τ,) (9) ξ co( nπξ) -( nπ ) τ T ( τ, ξ ) τ ξ e = + + (30) 3 π = n n Page 8

9 τ = i Our computation tart rom the ampled back urace data T b ( τ i ) given by (9), where 0 τ, δ τ,δ,... and δτ i the ampling interval. The ampling interval i et to 0. in the ollowing. To the ampled exact data, we have added noie a ollow: where α and α are two caling actor and yi ( + α n i ) Tb ( τ i ) + α ni = (9) n i and n i are peudorandom number ollowing tandard normal ditribution. Together they imulate enor noie. The noiy data are ed to Savitzky-Golay algorithm () () to obtain the moothing and dierentiation o the data and T ( τ ) and q ( τ ) are calculated uing (9) and () () (5). Thee reult are then ubtituted into (8) and (4) to obtain T ( τ ) and q ( τ ). The iteration i continued until equence converge. The two panel in Figure how the recontructed ront urace and temperature rom the exact back urace temperature with 8 and 6 iteration repectively. We note that the eect o doubling the number o iteration ha a mall though noticeable eect. Further doubling o the iteration number generate reult which are inditinguihable rom thoe with 6 iteration. Even with only 8 iteration, the ront urace temperature agree with the analytical olution very well. For the ront urace heat lux, the agreement i very good except the initial two point. Since the ront urace heat lux i dicontinuou at τ =0, larger error occur nearby. The error at the irt two point do not decreae much a the number o iteration i increaed urther. Other than thee two point, the heat lux i very cloe to the contant input aumed in the analytical olution. The ampled back urace temperature data are the only data ued in the calculation. Senor noie will unavoidably aect the accuracy o the recontructed ront urace temperature and heat lux. Since the iterative cheme in our approach involve numerical derivative, noie could potentially get ampliied in the proce cauing the iteration cheme to diverge. In [8], we have ound that to recover the ront urace heat lux, the noie in the back urace meaurement mut be o mall that it i not eaible uing today enor. The above iterative procedure ue Savitzky-Golay algorithm or data moothing and dierentiation. The Savitzky-Golay algorithm employ leat quare it to the data which ha iltering eect. A a reult, we ound that our iterative procedure can tolerate much larger noie than in [8]. Figure 3 how the α = reult when noie with =0.0 i included. Again the data ued are ampled temperature marked with ymbol +. Deviation o the noiy back urace temperature denoted by + rom the exact reult i noticeable. The top and bottom panel correpond to 8 iteration and 6 iteration repectively. The mall dierence between the two panel are hardly noticeable. α The two panel in Figure 3 demontrate that the iterative procedure can tolerate enor noie. With the enor noie, the reult noticeably deviate rom the analytical olution. The dierence in ront urace Page 9

10 heat lux are igniicant. However, it i not reaonable to regard the dierence a error ince the input data (with the enor noie) no longer correpond to thoe or the analytical olution. To urther illutrate thi point, we how the reult in Figure 4 when even large enor noie, =0.03, i included. We ound that the reult change little with the number o iteration. However, they deviate rom the analytical olution even more igniicantly. The noiy input data are 5 temperature value denoted by + ymbol in Figure 4. While α = α α = α =0.03 may appear to be very mall number, the correponding noie i o igniicant that it ha changed the back urace temperature qualitatively. In particular, the temperature at τ = 0. 7 i lower than at τ = 0.6. Since the back urace temperature data are the only input to our iterative procedure, reult are obtained accordingly. In other word, deviation rom the analytical reult correponding to the contant heat lux are not the correct meaure or the accuracy o the procedure. I act, the reult would be upiciou i they how no deviation. Note alo that our procedure make no aumption about the contant heat lux on the ront urace. It i uniquely uited or the tranient olution. Finally, we conider the implementation o our algorithm when the actual data are the back urace temperature y = y, y, L, y ] (3) i [ n with ampling time intervalδ t. We regard the ampled temperature a correponding to the recaled time τ i = [ 0, δτ, L, ( n ) δτ ], (33) where δt δτ = t c and t c i given in (). The recovered ront urace temperature retain the unit o the back urace meaurement. The ront urace i given by q (τ ) k L. i / 5. Concluion Baed on the Hadamard Factorization Theorem, we developed an iterative procedure to calculate the ront urace heat lux and temperature uing meaurement on the back urace. Combining our procedure with the well-known Savitzky-Golay method or moothing and dierentiation o the ampled data, our procedure i able to tolerate meaurement noie. The noie tolerance i a igniicantly improvement to the work in [8]. Our procedure i not limited to contant heat lux input on the ront urace. Page 0

11 Reerence [] E.M. Sparrow, A. Haji-Sheikh, T.S. Lundgren, The invere problem in tranient heat conduction, ASME J. Appl. Mech. 86 (964) [] J.V. Beck, B. Blackwell, and C.R. St-Clair, Invere Heat Conduction: Ill Poed Problem, Wiley, New York, 985. [3] R. Paquetti, C.L. Niliot, Boundary element approach or invere heat conduction problem: application to a bidimenional tranient numerical experiment, Numer. Heat Traner, Part B0 (99) [4] O.M. Alianov, Invere Heat Traner Problem, Springer-Verlag, Berlin/Heidelberg, 994. [5] C.-Y. Yang, C.-K. Chen, The boundary etimation in two-dimenional invere heat conduction problem, J. Phy. D: Appl. Phy. 9 (996) [6] C.-H. Huang, S.-P. Wang, A three-dimenional invere heat conduction problem in etimating urace heat lux by conjugate gradient method, Int. J. Heat Ma Traner 4 (999) [7] M.N. Öziik, and H.R.B. Orlande, Invere Heat Traner: Fundamental and Application, Taylor & Franci, New York, 000. [8] A.F. Emery, A.V. Nenarokomov, T.D. Fadale, Uncertaintie in parameter etimation: the optimal experimental deign, Int. J. Heat Ma Traner 43 (000) [9] M. Monde, H. Arima, Y. Mitutake, Etimation o urace temperature and heat lux uing invere olution or one-dimenional heat conduction, ASME J. Heat Traner 5 (003) 3-3. [0] X. Xue, R. Luck, J.T. Berry, Comparion and improvement concerning the accuracy and robutne o invere heat conduction algorithm, Invere Prob. Sci. Eng. 3 () (005) [] J. Zhou, Y. Zhang, J.K. Chen, Z.C. Feng, Invere heat conduction in a inite lab with meaured back heat lux, in: Proceeding o the 47 th AIAA Aeropace Science Meeting, Orlando, FL, January 5-9, 009. [] J. Zhou, Y. Zhang, J.K. Chen, Z.C. Feng, Invere heat conduction uing meaured back urace temperature and heat lux, J. Thermophyic and Heat Traner 4 (00) [3] J. Zhou, Y. Zhang, J.K. Chen, Z.C. Feng, Invere heat conduction in a compoite lab with pyrolyi eect and temperature-dependent thermophyical propertie, ASME J. Heat Traner 3 (00) [4] M.N. Öziik, 993, Heat Conduction, nd ed., Wiley-Intercience, New York. [5] G.E. Myer, Analytical Method in Conduction Heat Traner, nd Ed., AMCHT Publication, Madion, WI, 998. [6] D. Maillet, S. Andre, J.C. Batale, A. Degiovanni, C. Moyne, Thermal Quadrupole: Solving the Page

12 Heat Equation through Integral Tranorm, John Wiley & Son, New York, 000. [7] A. Sutradhar, G.H. Paulino, L.J. Gray, Tranient heat conduction in homogeneou and nonhomogeneou material by the Laplace tranorm Galerkin boundary element method, Engineering Analyi with Boundary Element, Vol. 6, 00, pp [8] Z.C. Feng, J.K. Chen, and Y. Zhang, Real-time olution o heat conduction in a inite lab or invere analyi, International Journal o Thermal Science, 49 (00), [9] J. Abate, P.P. Valko, Multi-preciion Laplace tranorm inverion, International Journal or Numerical Method in Engineering, Vol. 60, 004, pp [0] B. Davie and B. Martin, Numerical inverion o the Laplace tranorm: a urvey and comparion o method. Journal o Computational Phyic, Vol. 33, 979, pp. -3. [] G.V. Narayanan and D.E. Beko, Numerical operational method or time-dependent linear problem. International Journal or Numerical Method in Engineering, Vol. 8, 98, pp [] D.G. Duy, On the numerical inverion o Laplace Tranorm: comparion o three new method on characteritic problem rom application. ACM Tranaction on Mathematical Sotware, Vol. 9, 993, pp [3] D.Y. Tzou, A Uniied Field Theory or Heat Conduction rom Macro- to Micro-Scale, ASME Journal o Heat Traner, Vol. 7 (995) 8-6. [4] D.Y. Tzou, Macro- to Microcale Heat Traner: The Lagging Behavior. Taylor & Franci, Wahington DC, 997 [4a] J.B. Conway, Function o One Complex Variable, Second edition. Graduate Text in Mathematic, Vol., Springer-Verlag, New York Berlin, 978. [5] G.E. Coali, Dynamic repone o a non-homogeneou D lab under periodic thermal excitation, International Journal o Heat and Ma Traner, Vol. 50 (007), [6] A. Savitzky and M.J.E. Golay, Smoothing and dierentiation o data by impliied leat quare procedure, Anal. Chem. 36 (964), [7] P.A. Goray, General leat-quare moothing and dierentiation by the convolution (Savitzky-Golay) method, Anal. Chem. 6 (990), [8] M. Jakubowka and W.W. Kubiak, Adaptive-degree polynomial ilter or voltammetric ignal, Analytica Chimica Acta 5 (004), [9] J. Luo, K. Ying, P. He, and J. Bai, Propertie o Savitzky-Golay digital dierentiator. Digital Signal Proceing, 5 (005), -6. [30] A. Faghri, Y. Zhang and J.R. Howell, 00, Advanced Heat and Ma Traner, Global Digital Pre, Columbia, MO. Page

13 Appendix. Table o weight or even point, quartic it Convolution weight or quartic moothing Firt order derivative: i norm i norm Page 3

14 Figure Caption Figure Comparion o the traner unction in (6) with it polynomial approximation. The three magnitude curve exactly overlap. The dierence in phae angle decreae a more term a included in the product. Figure One dimenional heat conduction problem with an adiabatic back urace Figure 3 Comparion o the recontructed olution (point) ater 6 iteration with the analytical olution (line). No noie i included in the back urace temperature, i.e. α = α =0. Top: 8 iteration; bottom: 6 iteration. Figure 4 Comparion o the recovered data (point) and the analytical reult (line) with 8 iteration. The ymbol + repreent back urace temperature with noie added. The ymbol o repreent the calculated ront urace temperature and * repreent the calculated ront urace heat lux. Calculation i baed on data or τ = [0,.5]. Noie cale actor are α = α =0.0. Top panel: 8 iteration; bottom panel: 6 iteration. Figure 5 Comparion o the recovered data (point) and the analytical reult (line) with 6 iteration. The ymbol + repreent back urace temperature with noie added. The ymbol o repreent the calculated ront urace temperature and * repreent the calculated ront urace heat lux. Calculation i baed on data or τ = [0,.5]. Noie cale actor are α = α =0.03;. Top panel: 8 iteration; bottom panel: 6 iteration. Page 4

15 Magnitude ω 3 Phae A ngl e H r a d L exact 6 term approximation 8 term approximation ω Figure Page 5

16 q ( τ ) q ( τ ) 0 = b = ξ T ( τ ) T b ( τ ) Figure Page 6

17 .8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ.8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ Figure 3 Page 7

18 .8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ.8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ Figure 4 Page 8

19 .8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ.8 Front Surace Temperature.6.4. Front Surace Heat Flux Back Surace Temperature τ Figure 5 Page 9