Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method

Iden tification of Con v ection Heat T ransfer Co efficien t of Secondary Co oling Zone of CCM based on Least Squares Metho d and Sto c hastic Appro ximation Metho d G. O. Iv ano v a IAMM NAN of Ukraine, Donetsk iv ano v a@iamm.ac.donetsk.ua No vem b er 1, 2018 Abstract The detailed mathematical mo del of heat and mass transfer of steel ingot of curvi- linear con tinuous casting mac hine is prop osed. The pro cess of heat and mass transfer is describ ed by nonlinear partial differen tial equations of parab olic type. P osition of phase boundary is determined by Stefan conditions. The temp erature of co oling w ater in mould channel is described b y a special balance equation. Boundary con- ditions of secondary co oling zone include radiant and con vectiv e comp onen ts of heat exc hange and accoun t for the complex mechanism of heat-conducting due to airmist co oling using compressed air and water. Con vectiv e heat-transfer coefficient of sec- ondary co oling zone is unkno wn and considered as distributed parameter. T o solve this problem the algorithm of initial adjustment of parameter and the algorithm of op erativ e adjustment are dev elop ed. 1 In tro duction Impro ved computing significan tly increased role of mathematical mo deling in research of thermo-ph ysical processes. This, in turn, imp oses stricter requirements to wards accuracy and efficiency of mathematical mo dels. It is w ell known that successful mo deling mostly dep ends on the righ t c hoice of a mo del, which is directly affected by reliability of thermo-ph ysical parameters used. F re- quen tly , empirical data alone can not pro vide sufficien t information ab out one-v aluedness conditions. Therefore recently the big atten tion is given to the solution of inv erse problems of heat conduction, in which it is necessary to define thermoph ysical prop erties of an ob ject on a v ailable (frequently rather limited) information ab out temperature field. In particular th us it is p ossible to iden tify b oundary conditions. There are difficulties in c hoice of some parameters of process for developmen t of mathematical mo dels of tec hnological processes. 1 While mo deling pro cess for sp ecific industrial conditions it is necessary to determine some thermal or physical parameters each time, in particular conv ective heat-transfer co efficien t (CHTC) on a surface of an ingot in the secondary co oling zone which dep ends on many factors. It is connected by that the conv ective heat transfer co efficient v alue is influenced with set of v arious factors. Besides, CHTC v alue can v ary strongly enough in a time and on space co ordinates. Thus, there is a problem of identification of the CHTC as distributed parameter. In the given work algorithms of initial adjustmen t of parameter when at the disp osal of there is enough plen ty of p oin ts in which the temp erature on a surface of an ingot is measured, and op erativ e adjustment when the temperature is measured only in one p oin t on a surface are considered. 2 Statemen t of problem The thermal field of the mo ving steel ingot and mold w all in the system of co ordinates attac hed to motionless construction of CCM is considered [1]. In fig. 1 the diagram of CCM is introduced. Figure 1: The heat conduction in the steel ingot in the mold area is describ ed b y nonstationary , nonlinear heat and mass transfer equation: 2 ∂ T ( τ , x, z ) ∂ τ + v ( τ ) ∂ T ( τ , x, z ) ∂ z = = 1 c ( T , x, z ) ρ ( T , x, z )  ∂ ∂ x  λ ( T , x, z ) ∂ T ∂ x  + ∂ ∂ z  λ ( T , x, z ) ∂ T ∂ z  , 0 < x < l, 0 < z < Z (1) and the b oundary conditions: − λ ( T , x ) ∂ T ∂ z = 0 , 0 ≤ x ≤ l , ∂ T ∂ x     x =0 = 0 , 0 ≤ z ≤ Z , λ ( T , z ) ∂ T ∂ x     x = l = λ g z δ  T | x = l + δ − T | x = l  + σ n   T | x = l + δ 100 ! 4 −  T | x = l 100  4   , 0 ≤ z ≤ Z, (2) where v ( τ ) – withdraw al rate, 2 l – ingot thic kness, Z – height of ingot in the mould, T ( τ , x, z ) – metal temp erature, c ( T , x, z ) – metal sp ecific heat, ρ ( T , x, z ) – densit y , λ ( T , x, z ) – thermal conduction, δ – effective thic kness of air gap b etw een ingot and the mould wall, λ g z – thermal conduction co efficien t of gap gas mixture, T | x = l – surface temperature of the ingot, T | x = l + δ – surface temperature of mold wall, σ n – the resulted radiation coefficient. Conditions of equalit y of tempe ratures and Stefan conditions, and also b oundary and initial conditions for the phase b oundary are set: T ( τ , x, z ) | x = ξ − ( τ ,z ) = T ( τ , x, z ) | x = ξ + ( τ ,z ) = T kr , λ ( T , x, z ) ∂ T ∂ ¯ n     x = ξ − ( τ ,z ) − λ ( T , x, z ) ∂ T ∂ ¯ n     x = ξ + ( τ ,z ) = µρ ( T kr )  ∂ ξ ∂ τ + v · ∂ ξ ∂ z  , 0 ≤ z ≤ Z, ξ ( τ , 0) = l , ξ (0 , z ) = ξ 0 ( z ) , (3) where ξ – the phase b oundary function of tw o v ariables x = ξ ( τ , z ), µ – crystallization laten t heat, T kr – crystallization temperature (a verage of the interv al “liquidus – solidus”), ¯ n – normal to the b oundary of phases. Heat equation for mould w alls: 3 ∂ T ( τ , x, z ) ∂ τ = 1 c ( T , x, z ) ρ ( T , x, z )  ∂ ∂ x  λ ( T , x, z ) ∂ T ∂ x  + ∂ ∂ z  λ ( T , x, z ) ∂ T ∂ z  , z 0 < z < Z, l < x < d (4) Boundary conditions for mould w alls represent the c haracter of heat exchange on eac h sigh t of w all: λ ( T , z ) ∂ T ∂ x     x = d = α 1 ( T water ( τ , z ) − T | x = d ) , z 0 ≤ z ≤ Z, λ ( T , x ) ∂ T ∂ z     z = Z = α 2 ( T os. 2 − T | z = Z ) , l ≤ x ≤ d, z = Z , − λ ( T , x ) ∂ T ∂ z     z = z 0 = α 3  T os. 3 − T | z = z 0  , l ≤ x ≤ d, z = z 0 , λ ( T , z ) ∂ T ∂ x     x = l + δ = = λ g z δ  T | x = l + δ − T | x = l  + σ n   T | x = l + δ 100 ! 4 −  T | x = l 100  4   , 0 ≤ z ≤ Z, x = l + δ, − λ ( T , z ) ∂ T ∂ x     x = l + δ = α 4 ( T os. 1 − T | x = d ) + C n "  T os. 1 100  4 −  T | x = d 100  4 # , z 0 ≤ z ≤ 0 , x = l + δ, (5) where d – mold wall thickness, z 0 – mold wall altitude ov er meniscus level, α 1 – heat transfer co efficien t from the mould w all to co oling water, T water ( τ , z ) – co oling water temp erature in the mold channel, α 2 , 3 , 4 – heat transfer co efficien ts from other mould wall to en vironment, T os. 2 , 3 , 4 – en vironment temperature, C n – the resulted radiation co efficien t. The follo wing balance equation describ es distribution of cooling water temperature in the mold channel: c · S · v water ∂ T water ( τ , z ) ∂ z = P I α 1 ( T water ( τ , z ) − T | x = d ) − P E α E ( T water ( τ , z ) − T E ) , (6) where c – v olume heat capacit y of w ater, S – the cross-section area of the mold channel, v water – water velocity , P I – p erimeter of the interior mold w all, P E – p erimeter of the 4 external mold w all, α E – heat transfer coefficient from cooling w ater to the external mould w all, T E – external mould w all temp erature. The co oling water temperature on the entry in the mould channel is known: T water (0 , Z ) = T water 1 ( τ ) (7) and it’s initial distribution in the mold c hannel: T water (0 , z ) = T water 0 ( z ) (8) The follo wing equation describ es heat and mass transfer on the curvilinear sections of CCM: ∂ T ∂ τ + θ m ( τ ) ∂ T ( τ , r , ϕ ) ∂ ϕ = 1 c ( T , r , ϕ ) ρ ( T , r, ϕ ) × ×  ∂ ∂ r  λ ( T , r , ϕ ) ∂ T ∂ r  + 1 r 2 · ∂ ∂ ϕ  λ ( T , r , ϕ ) ∂ T ∂ ϕ  + λ ( T , r , ϕ ) r · ∂ T ∂ r  (9) where θ m – angular velocity of ingot driving on the m -th curvilinear section. The conditions for unkno wn b oundary on the curvilinear sections are T ( τ , r, ϕ ) | r = ξ 1 , 2 − ( τ ,ϕ ) = T ( τ , r, ϕ ) | r = ξ 1 , 2+ ( τ ,ϕ ) = T kr , λ ( T , r , ϕ ) ∂ T ∂ ¯ n     ξ 1 − − λ ( T , r , ϕ ) ∂ T ∂ ¯ n     ξ 1+ = µρ kr  θ m ( τ ) · ∂ ξ 1 ∂ ϕ + ∂ ξ 1 ∂ τ  , ξ 1 (0 , ϕ ) = ξ 1 0 ( ϕ ) , λ ( T , r , ϕ ) ∂ T ∂ ¯ n     ξ 2+ − λ ( T , r , ϕ ) ∂ T ∂ ¯ n     ξ 2 − = − µρ kr  θ m ( τ ) · ∂ ξ 2 ∂ ϕ + ∂ ξ 2 ∂ τ  , ξ 2 (0 , ϕ ) = ξ 2 0 ( ϕ ) , (10) where ξ 1 ( ϕ ) and ξ 2 ( ϕ ) – phase boundaries (in terfaces). The b oundary conditions of the secondary co oling zone include radian t and conv ective comp onen ts of heat exc hange and accoun t for the complex mec hanism of heat-conducting due to air-mist co oling using compressed air and water. The b oundary conditions on the curvilinear sections are − λ ( T , ϕ ) ∂ T ∂ r     r = r m = α I ( G m ( τ ) , ϕ ) ·  T I m − T | r = r m  + C I m  T 4 I m − ( T | r = r m ) 4  (11) λ ( T , ϕ ) ∂ T 2 ∂ r     r = r m +2 l = = α E ( G m ( τ ) , ϕ ) ·  T E m − T | r = r m +2 l  + C E m  T 4 E m − ( T | r = r m +2 l ) 4  , (12) 5 where α I ( G m ( τ ) , ϕ ), α E ( G m ( τ ) , ϕ ) – conv ectiv e heat transfer co efficients, C I m , C E m – the resulted radiation coefficients, T I m , T E m – en vironment temp eratures, G m ( τ ) – w ater disc harge on the m -th section. The following equation describ es the heat and mass transfer on rectilinear sections of CCM (analogously (1)): ∂ T ∂ τ + v ( τ ) ∂ T ( τ , x, z ) ∂ x = = 1 c ( T , x, z ) ρ ( T , x, z )  ∂ ∂ x  λ ( T , x, z ) ∂ T ∂ x  + ∂ ∂ z  λ ( T , x, z ) ∂ T ∂ z  (13) When the liquid phase passes the straigh tening p oin t on the rectilinear section of the secondary co oling zone, the conditions for the unknown phase b oundary are set: T ( τ , x, z ) | x = ξ 1 , 2 − ( x,z ) = T ( τ , x, z ) | x = ξ 1 , 2+ ( x,z ) = T kr , λ ( T , x, z ) ∂ T ∂ ¯ n     ξ 1 − − λ ( T , x, z ) ∂ T ∂ ¯ n     ξ 1+ = µρ kr  v ( τ ) · ∂ ξ 1 ∂ x + ∂ ξ 1 ∂ τ  , λ ( T , x, z ) ∂ T ∂ ¯ n     ξ 2+ − λ ( T , x, z ) ∂ T ∂ ¯ n     ξ 2 − = − µρ kr  v ( τ ) · ∂ ξ 2 ∂ x + ∂ ξ 2 ∂ τ  . (14) The b oundary conditions for the rectilinear section: − λ ( T , x ) ∂ T ∂ z     z = z p = α I ( G m ( τ ) , x ) ·  T I − T | z = z p  + C I 4  T 4 I − ( T | z = z p ) 4  λ ( T , x ) ∂ T ∂ z     z = z p +2 l = = α E ( G m ( τ ) , x ) · ( T E − T | z = z p +2 l ) + C E 4  T 4 E − ( T | z = z p +2 l ) 4  . (15) W e assume, that the thermal stream of the end of the rectilinear site is equal to zero: λ ( T , z ) ∂ T ∂ x     x = x f = 0 . (16) The initial conditions for all temp erature field (on the rectilinear and curvilinear sec- tions): T (0 , x, z ) = T 0 ( x, z ) T (0 , r , ϕ ) = T 0 ( r , ϕ ) . (17) It is required to define the con vectiv e heat transfer co efficien ts α I ( G m ( τ ) , ϕ ), and α E ( G m ( τ ) , ϕ ) using the av ailable information ab out ingot temp erature. 6 This is a boundary in verse problem and it is ill-posed in classical sense. W ell-p osedness in classical sense (or Hadamard w ell-p osedness) means p erformance of three conditions: an existence of a solution, its uniqueness and stability (input data contin uous dependence). In our case the third condition is not satisfied. This is easily to v erify using for the solution this problem the metho d of direct rev ersion [2]. Therefore other approaches are necessary to solve this problem. 3 CHTC iden tification b y least squares metho d Consider an ingot in first co oling section of secondary co oling zone. W e hav e ingot surface temp erature measuremen ts in some p oints. So w e hav e to solve the Diric hlet problem for in terior heat exchange. The finite-difference method was used to appro ximate the solution of this problem. The conv ective heat-transfer co efficien t (CHTC) has sp ecial distribution along the surface of the ingot. P arab olic function with a sufficient degree of accuracy appro ximates distribution of CHTC on the part of surface that is exp osed to w ater-air spra ying from one nozzle. This parab ola has maximal v alue in the p oin t that corresp onds to nozzle co ordinate. CHTC is considered as constan t on the parts of the surface not sub jected to the forced co oling (fig. 2). Figure 2: In one co oling section the same t yp e spray nozzles are installed. They give an iden tical w ater-air spray . Hence the CHTC is the same parabola shifted along the abscissa axis (fig. 2). All sites under spra y nozzles can b e reduced to the coordinate origin so that the p eak of eac h parab ola should b e o ver the coordinate origin. Hence, it is necessary to define only t wo parameters - α p and α c . So, α ( ϕ ) is given b y α ( ϕ ) = α c − α p w 2 ϕ 2 + α p . (18) Consider the parts of the section, on which α ( ϕ ) = α c = const . Let K b e the ensem ble of points ϕ i , in whic h CHTC is equal to constan t. Let B b e the ensem ble of other p oin ts. The finite-difference approximation of b oundary condition (11) is 7 λ i, 0 T i, 2 − 4 T i, 1 + 3 T i, 0 2 q = α c ( T I 1 − T i, 0 ) + C I 1 ( T 4 I 1 − T 4 i, 0 ) , (19) where q – step of finite-difference grid b y radius r 1 [3]. It follows that the discrepancy of heat flo ws on the b oundary is: ∆ = λ i, 0 T i, 2 − 4 T i, 1 + 3 T i, 0 2 q − C I 1  T 4 I 1 − T 4 i, 0  − α c ( T I 1 − T i, 0 ) . Let us denote P i = λ i, 0 T i, 2 − 4 T i, 1 + 3 T i, 0 2 q − C I 1  T 4 I 1 − T 4 i, 0  , Q i = T I 1 − T i, 0 . Then w e find a v alue α c , suc h that the sum of squares of discrepancies is minimum, i.e. the follo w condition is satisfied S = X i ( P i − α c Q i ) 2 → min , ∀ i : ϕ i ∈ K . A necessary condition of the extremum existence of the function S ( α c ) is: ∂ S ∂ α c = − 2 X i Q i ( P i − α c Q i ) = 0 . It follows that α c = P i Q i P i P i Q 2 i . T o the eac h point ϕ i from we will put in conformit y a point y i on the segmen t [ − w , w ] suc h that | y i | is equal to the distance from the corresp onding ϕ i to the co ordinate of the nearest spray nozzle. F rom (18) and (19) we gain a discrepancy ∆ = λ i, 0 T i, 2 − 4 T i, 1 + 3 T i, 0 2 q − C I 1  T 4 I 1 − T 4 i, 0  −  α c − α p w 2 y 2 i + α p  ( T I 1 − T i, 0 ) . Then we can find a v alue α p , such that the sum S = X i ( P i − ( α c − α p w 2 y 2 i + α p ) · Q i ) 2 → min . F rom the follo wing necessary condition of extremum existence ∂ S ∂ α p = 2 X i P i − α c − α p y 2 i w 2 − 1 !! P i ! Q i y 2 i w 2 − 1 !! = 0 w e obtain α p 8 α p = α c X i Q 2 i y 2 i w 2 − 1 ! − X i P i Q i y 2 i w 2 − 1 ! X i Q 2 i y 2 i w 2 − 1 ! 2 . On fig. 3 comparative results of calculations (1 – by the method of direct reversion, 2 – by the least squares metho d) are presen ted. F or steel grade st40, width of a slab is 1m, l = 0,1m and v = 1(m/minute). The decision obtained by the metho d of direct rev ersion is unstable and unsuitable for practical use. The second curv e represen ts a spline appro ximation, whic h is gained as a result of the decision of a problem of identification by the least squares method. Figure 3: Th us, we fined the spline approximation of the CHTC, which is distributed on the surface of the moving ingot. This approximation gives the minim um of mean-square deviation b etw een measured surface temperature and calculated one according to the mo del as the result of solving of the direct problem. The CHTC for other sections of the secondary co oling zone is analogously defined. It should b e noted that an adv antage of the offered method is that the estimation error of the least squares method is negligibly small b y relatively small n umber of abnormal measurements. It is very important in case of temp erature measurement of a partially oxide scaled ingot surface. 9 4 Op erativ e adjustment of con v ectiv e heat transfer co effi- cien t (CHTC) CHTC obtained by initial adjustmen t v aries under c hanges of v arious parameters of pro cess (for example, am bient temp eratures). Therefore, it is necessary to pro vide its operative adaptation during work CCM. The fine-tuning of parameters should be carried out in real time. But during usual w ork of CCM the information on a thermal condition of an ingot is limited to temp erature indications in small num b er of p oin ts of the surface of an ingot. Suc h algorithms can b e based on the sto c hastic appro ximation metho d [4]. The temp erature on the ingot surface is measured in ev ery equal small time in terv als. Let us denote the measuring temp erature data T ∗ j . The computer mo dels the casting pro cess using the presen ted mathematical model. The under mo del calculated temperature in the corresponding point w e denote b y T j . It is necessary to correct the mo del parameters using information ab out deviations b et w een measured and calculated temp erature data to reduce these deviations to minim um. The difficult y of the decision of the giv en problem is that temp erature measurements are deformed by a random telemetry error. Op erativ e fine-tuning consists in refinemen t of the constant v alue α c , which defines the distribution of the con vectiv e heat transfer co efficien t obtained b y the solving of the problem of the initial adjustmen t of parameters. F or using the algorithm of sto c hastic approximation it is necessary , that the random error of temp erature indications w ould hav e the zero av erage and the finite v ariance. The algorithm of parameter adjustmen t is α j +1 = α j − k j ( T ∗ j − T j ) , (20) where α j – j -th approximate v alue of α c , k j – sp ecial sequence of num b ers, which satisfies to the following conditions: lim j →∞ k j = 0 , ∞ X j =1 k j = ∞ , ∞ X j =1 ( k j ) 2 < ∞ . (21) F or example the follo wing elemen tary sequence satisfies to such conditions k j = a b + j , where a, b ∈ R , a > 0. Selecting num b ers a and b , and also other sequences satisfying to the conditions (21), it is possible to c hange speed of con v ergence of algorithm. In [3], for example, it is recommended to keep k j as constant while the sign of discrepancy T ∗ j − T j not v ary , and change then k j so that to satisfy to ab o ve men tioned restrictions. T runcation condition of the parameter fine tuning algorithm w ork is o ccurrence of m last received appro ximations α n +1 , α n +2 , , α n + m in a vicinity of α n serv es: | α n − α n + i | < ε, ∀ i = 1 , , m. 10 If the condition is executed, assume α c is equal α n . F or c heck we use v alues CHTC whic h ha ve b een pick ed up exp erimen tally at the decision of a direct problem of mo deling of thermal field CCM [1]. 5 Examples of realization of the sto c hastic appro ximation metho d Numerical modeling allo ws establishing the basic features of tra jectories of parameter fine- tuning process. On fig. 4 tra jectories of parameter fine-tuning, characterizing a deviation of the distributed parameter from true v alue, for the algorithm using sequence k j = a j , j = 1 , 2 , 3 , ... are presented at v arious v alues of factor a . When a < 1 very slo w conv ergence is observed. In this case the time of parameter tuning is inadmissible big. Figure 4: If to c ho ose a = 1 the v alue of the parameter is in enough small vicinity of true v alue approximately after 200th iteration. At a = 2 the tra jectory of parameter fine- tuning reflects oscillations with damp ed amplitude and frequency and not later than for 200 iterations the parameter is adjusted. At increase a > 2 the amplitude of oscillations 11 gro ws. In this case also oscillations with damp ed amplitude and frequency are observed, but for fine-tuning it is required considerably more iterations. F rom here we conclude, that for the c hosen sequence the b est v alues of the factor a is a num b er from in terv al 1 ≤ a ≤ 2. W e inv estigate now influence of v alue b on speed of the algorithm’s conv ergence. On fig. 5 tra jectories of parameter fine-tuning are sho wn for v arious v alues b . V alues b less than zero lead to that fine-tuning go in a ”wrong” direction while the denominator is negativ e and at i = − b the denominator is equal to zero. Increase b leads to decrease of a v elo cit y of con vergence of algorithm. The same results hav e b een obtained for sequences, whic h will b e describ ed further. Therefore further parameter b everywhere will b e c hosen equal to zero. Figure 5: The following sequence also satisfies to conditions (21) k j = a n j , n j +1 = ( n j , ( T ∗ j − T j )( T ∗ j +1 − T j +1 ) > 0 j + 1 , ( T ∗ j − T j )( T ∗ j +1 − T j +1 ) ≤ 0 . (22) Results of this algorithm w ork are presented on fig. 6. In this case factor a needs to b e chosen within 1 ≤ a ≤ 3. V alues out of this range give smaller sp eed of algorithm con vergence. Consider another sequence, whic h also satisfies to conditions (21) 12 k j = a n j , n j +1 = ( n j , ( T ∗ j − T j )( T ∗ j +1 − T j +1 ) > 0 n j + 1 , ( T ∗ j − T j )( T ∗ j +1 − T j +1 ) ≤ 0 . (23) It h as slo wer conv ergence than the previous t wo sequences. Results of calculations with use of this sequence are presented on fig. 7. F actor a can b e chosen within 0 . 5 ≤ a ≤ 2. And, if 1 . 2 ≤ a ≤ 1 . 5, than obtained appro ximations differ from the true v alue no more than on 6 % after 20 iterations already . Figure 6: In the conclusion also it is necessary to notice, that the adv antage of sto c hastic ap- pro ximation algorithm is its successful work for enough wide in terv al of initial v alues of the distributed parameter. References [1] Tkachenko V.N., Ivanova A.A. Mo deling and Analysis of T emp erature Field of Ingot of Curvilinear Contin uous Casting Mac hine. – Electronic Mo deling - 2008. V ol.30, 3. p.87-103. (in russian) 13 Figure 7: [2] Tkachenko V.N. Heat processes modeling in automatic system of information handling. //Visn yk Donetsk ogo Natsionalnogo Universytetu, Ser.A, ’Pryro dn yc hi nauky’, 2002, No 2, p.379 - 383. (in russian) [3] Mar chuk G.I. Metho ds of computational mathematics. – Mosk ow: Nauk a, 1980, 535 c. (in russian) [4] A ndr ew P. Sage, James L. Melsa System Iden tification. – System Iden tification. Aca- demic Press, 1971, New Y ork and London. 14