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Optimal sensor strategy for parametric identification of a thermal system

Optimal sensor strategy for parametric identification of a thermal system
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  See discussions, stats, and author profiles for this publication at: Optimal sensor strategy for parametricidentification of a thermal system, CONFERENCE PAPER  · JUNE 2000 CITATIONS 3 READS 16 4 AUTHORS , INCLUDING:Laurent AutriqueUniversity of Angers 126   PUBLICATIONS   199   CITATIONS   SEE PROFILE Alain FerriereFrench National Centre for Scientific … 43   PUBLICATIONS   387   CITATIONS   SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate,letting you access and read them immediately.Available from: Laurent AutriqueRetrieved on: 04 February 2016  Copyright © IFAC System Identification Santa Barbara, California, USA, 2000 OPTIMAL SENSOR STRATEGY FOR PARAMETRIC IDENTIFICATION OF A THERMAL SYSTEM Laurent Autrique* Charles Chaussavoine* Jean Pierre Leyris* Alain Ferriere** * CNRS-IMP, Universitd de Perpignan, 52 avenue de Villeneuve, 66860 Perpignan, FRANCE, phone : 33 468 662 239, FAX: +33 466 672 166, e-maih * CNRS-IMP, BP 5, 66125 Odeillo cedex, FRANCE Abstract: Recent investigations on the solar hardening and on the induced stresses in solar-irradiated metallic materials have shown encouraging results and provide an alternative to more conventional processes (laser or plasma sources). In this paper, heat transfers in a specimen submitted to short flashes of concentrated solar energy are investigated. Estimnation of model parameters is a crucial step for the control of solar systems regarding some industrial requirements such as the control of the products and the reproducibility of the results. Thus the unknown influent parameters have to be carefully estimated. An optimal sensor strategy is also proposed based on the sensitivity analysis in order to minimize the measurement noise. Copyright 2000 IFAC Keywords: Partial differential equations, Sensitivity analysis, Optimal estimation, Parameter identification 1. INTRODUCTION Surface thermal treatments of materials offer an extended field of industrial applications, includ- ing the aeronautic, power generation, and engine manufacturing industries. Concentrated solar en- ergy has already demonstrated promising capabil- ities for some processes, like the hardening of steel and cast iron, the alloying of metallic materials, or the glazing of plasma-sprayed ceramic coasting. For further applications, an efficient predictive model is essential to achieve an optimal sensors lo- cation and to ensure the control of the process. For the hardening of steel process investigated in this paper, it was established that the thickness of the modified layer is mainly dependent on the tem- perature distributions (gradients and levels). It is important to notice that temperature can exceed 3000 K after a few seconds (see (Ferriere, 1999)).  The evolution of the temperature inside the sam- ple is described by a classical partial differential equation. Physical parameters are temperature dependent and the strong dynamic of the transfor- mation prevent linearization of the system. Con- vective and radiative conditions are considered on the boundaries of the material involving the solar absorption factor. This optic property is strongly dependent on the temperature, since it is modified by the physico-chemical evolution of the surface material. Identification of this influent parame- ter requires temperature measurements. However near the treated surface, high temperature spa- tial gradients lead to inadmissible measurement noises mainly due to imprecision in the sensors locations. Thus the dilemma can be stated as follows : locate the sensors in weak temperature spatial gradients region with low sensitivity or locate the sensors in good sensitivity region but with measurement noises due to strong temper- ature spatial gradients. In this paper a contribu- tion to optimal sensor location for this complex  situation is presented. First the experimental set- up is briefly exposed and a thermal modelling of the physical system is proposed. Then, it is shown that the solar absorption factor (also called absorptivity) is an influent parameter of the model and has to be carefully determined. A sensitivity analysis for the identification problem is proposed in order to identify a boundary parameter from state measurements. An optimal sensor location problem is stated and results are discussed in the last paragraph. 2. A THERMAL MODEL FOR TttE HARDENING PROCESS 2.1 Description of the process The processing of solid material is a field of appli- cation where concentrated solar energy presents some major advantages. The easy access to el- evated flux densities enables investigations on transformations resulting from rapid heating and cooling of the surface of the specimens. The ex- periments, presented in (Ferriere et al., 1994), are completed in a 2kW solar furnace (see fig. 1), which is composed of a flat heliostat reflecting the sunlight on a parabolic mirror 2m in diameter whose axis is vertical. Fig. 1.2kW solar furnace The concentration factor is 16000, and the focal spot is about 16ram in diameter. The single-spot process is controlled by a fast mechanical shutter composed of two flat screens sliding horizontally very close to the focal plane in order to rapidly deliver or cut-off the concentrated solar beam. On fig. 2, continuous scanning process experimental set up is illustrated. mox mg m parabolic rnirmr oncc~tra~cd ~elal r;~li0aion mot mg ~lder door r evice for sun tracking Fig. 2. Schema : experimental setup  X : (Xl,•2,X3) E ~ C ff~3 , the space variable [m],  t E T , the time variable is],  O(x,t), the temperature [K],  p(O), the volumic mass [kg.m-a],  cp(O) , the specific heat [,  A(O), thermal conductivity [W.m-t.K-q,  h , the convective exchange coefficient [W.m-~.h'-t],  e , the emissivity,  o = 5.67 10 -s , the stefan constant,  a , the absorptivity,  ~(x), the concentrated solar flux, [W.rn-2],  ((t) ,the function corresponding to the spot duration (~ = 1 during the spot, else ( = 0). A classical thermal equation is considered V(x, t) E ~×T - ) e x,t) di~ a(e)gr~de(~,O =o(1) p(o)~p(e/ ot Initial temperature in the sample is Vx E ~: o(x,0) = Oo (2) On the irradiated face, the following boundary condition is considered V(x, t) 6 0g2i,.,. × T : -A(O) af.~) = hCe(.) - Oo) + ca(Or(.) - ®~) (3) 2.2 State equations Let us consider the following notations :  £t C/R 3 , the space domain, • O~ = O~irr U 0~ex~ , the surface of ft where 0Air,- is the irradiated face, and on the other boundaries V(z, t) E 0~e~t x T : -:ge)oe () = h(e(.) - Co) + ~(e~(.) - e~) (4) On The thermal evolution of the material during the process is described by the partial differential equations system (,S):  I System (S) I state equation (1) initial state (2) boundaries conditions (3) (4) X irradiated ace -'7 .... ~x2 back ace Fig. 3. domaine ft Cylindrical specimen is 0.02m in diameter and 0.0123m height. Thermal treatment of NS30 steel is investigated and the following data are consid- ered :  p(O) = -0.4440 + 8121.33 [kg/m 3] f 0.220 + 432.7 si 273 _< ® _< 888 Cp(O) 0.46® + 219.6 si 888 < O [Z/Kg.~']  )t(O) = 0.01290 + 10.033 [W/m.K]  h = 15 W.m-2.K -1 • c=0,7  o = 5.67 10 -8 W.m-2.K -4  O0 = 291K  ~(t)=lif0<t < 1.75 and0ift>_ 1.75. The solar absorption factor is strongly dependent on both temperature and physico-chemical reac- tions at the surface of the material. From exper- iment observations, this parameter is assumed to verify: 0.2 < o(O,t,~) < 0.8. In (Ferriere et al., 1999), the solar flux distribution on the focal plane (x = (xl, x2, 0.0123) E 0fti~r) is correctly represented by the gaussian approxi- mation : ¢(x)=¢¢ exp( x~+x22 r2 where the peak flux ~ depends on the solar irradiation (under a standard direct irradiation of 103W.m -2 , ~ = 16 106W.rn -2) and where r = 5.1 10-3m. For specific treatments where absorptivity factor o~ is constant (measured before and after the treatment), a direct problem can be written as follows: direct problem Pdi~ : For an estimated o, find O solution of ($) A finite element method is implemented to solve Pdir. Complex non linearities due to large temper- ature gradients and levels involved in the process are carefully taken into account by iterative nu- merical methods. 3. SELECTION OF THE INFLUENT PARAMETERS Several physical coefficients are considered in (S):  thermophysical parameters which are known with a given indetermination : pCp, X,  thermophysical and experimental parameters which are not well-known : convective ex- change coefficient h, emissivity e, absorptiv- ity a,  experimental conditions such as the peak flux ~, or the spot duration which are measured at each treatment. However, it is obvious that these factors have to be taken into account in different ways. For example, based on the experimentation and the process knowledge, it seems that a, ~ and the spot duration are crucial. In the same way, even if the convective exchange coefficient h and the emissivity e are quite difficult to estimate, their influences are expected to be weak. Moreover, thermal characteristics (pcv, A) are generally well- known. Effect of the uncertainty of parameters is investigated in (Autrique et al., 1999a) using a design of experiment numerical procedure. It is shown that the indeterminancy of the absorptivity sharply reduced the simulation accuracy and the model adequacy. Moreover, the thermo-physical parameters of the materials are well determined and do not pertub the simulation results. Thus an inverse heat transfer problem can be stated in order to identify the absorptivity. It is important to notice that for such a problem, this parameter is not assumed to be constant and is in most cases temperature and surface dependent. 4. SENSITIVITY ANALYSIS From temperature measurements, an identifica- tion procedure can be implemented. Whatever the numerical approach is, identification can lead to erroneous estimates due to sensor errors (see (Emery and Fadale, 1997)). Methods for optimum sensor locations are generally based upon sen- sitivity analysis (see (Fadale el al., 1995)). The problem under interest is to locate one or several sensors such that the estimation of the absorp- tivity (c~) is not adversely affected by errors in the measured state (O), that is Oct/O0 is small. Thus the sensitivity function 0@/0a has to be as large as possible. In order to determine spatial location where sensitivity function are maximum, sensitivity equations can be considered (see (Beck and Arnold, 1977)). Let us denote by (5®)g6a the temperature varia- tion resulting of an absorptivity variation (#50~) : (~o)~,~(~, t) = e(x, t; ~ + ~6~) - o(x, t; ~)  The sensitivity function is defined as : 50(x,t;cO = lim ~ e(x, t; c~ + Ma) - o(x, t; a) Let us denote by : 0 + = O(x,t;a+Ma) O = O(x,t;a) c,+ = a(o +) ~(.) = p(.)~p(.) (5) Then, equation 1 is modified V(x, t) • ~ x T : ° °+) do+- °+)) :° , 6) Initial state is Vx  ~2: o* (~,o) = 00 (r) On the irradiated face, V(x, t) • 0fti~ x T : _~(0+)---~- = h(e+ - e0) + ¢~(0+4 _ 04o) (s) an - (~+ + .6~) ¢(~)((t) and on the other boundaries, V(x,t) • 0£Uxt x T : -,X(O+)at + = h(O + - 00) + ¢a(O +4 - O~) (9) 0n (10) Evolution of O + is described by system (8+) System (8+) 1 state equation (6) initial state (7) boundaries conditions (8) (9) Equations 1 and 6 lead V(x,t)  ~ x T to : 30 + _ a(O+ ) ---~ a(O)~--O div (A(O+)g-r-'ad ® +) - £(O)g'7-*ad(O)) =0 11) Equations 2 and 7 lead Vx  ~ to : o+(z,0) - e(x,0) = o Equations 3 and 8 lead V(x, t) • Oflir~ x T to : (12) ~(e+)L~- + :,(o)~ h(O+ - o) an an +~o(0+~ _ o ~) _ (~+ + .~ - ~) ¢(x)<(t) and on the other boundaries, V(x, t)  Ofl~t x T, equations 4 and 9 lead to : (13) ~,(o+)a~ + ~(e).-°-.-~ = 8n h(o+ - e) + ~,,(0 +4 0_ ,) For p converging to zero, let us consider : da = 0+) = ~ 0) + ,~e-- ~(e+) = ~(0) + .,~02_ 6. d~ ~(0+) = ~(0) + ~O-- dO 0 +4 04 = 403#50 ao + oo 060 - +it-- at at at ~d(+) -- -- = grad(O) +/ugrad(60) aO* 00 a5o a~ a~ a. Then equation 10 becomes V(x, t)  ~ x T : a5o 60 da 00 a e)-.~?- + de t (14) Equation 11 becomes Vx  ~ : 60=0 (is) Equation 12 becomes V(x, t) • Of~irr x T : _A(o)°~ _ 6o d~ao = a n dO a: h50 + ea4035e - (60 d~ + 6a~ ¢(x)((t) \ dO ] (16) and 13 becomes : -A(O) 0~_.~,5(9 _ 660 dA 00 _ a n de a~ h60 + ¢~40350 (17) Equations 14, 16 and 17 can be easily written as : • V(z,t)•~xT: ~(a(O)~O)- (~(0)6o) = (18) o  V(x,t)  O~tirr X T : 0A(O)50 _ h60 + ¢a40360 - + 6~/¢(x)dt) • V x,t) ES~xT: a,\(0)~0 = h60 + ~o-403~O (20) a: In order to determine spatial subdomains where sensitivity is good enough, an arbitrary absorptiv- ity is given according to experimental knowledges and corresponding temperatures are calculated. Then, for a given uncertainty, the following system (Sse) leads to the determination of the sensitivity function 60 : System (Sse) state equation (18) initial state (15) boundaries conditions (19) (20) On the following figures, sensitivity functions are determined at the end of a spot of 1.75s and after
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