Identification of macro and micro parameters in solidification model

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vo. 55, No. 1, 27 Identification of macro and micro parameters in soidification mode B. MOCHNACKI 1 and E. MAJCHRZAK 2,1 1 Czestochowa University of Technoogy, 68 Dabrowskiego St., 42-2 Częstochowa, Poand 2 Siesian University of Technoogy, 18a Konarskiego St., 44-1 Giwice, Poand Abstract. In the paper the therma processes proceeding in the soidifying meta are anayzed. The basic energy equation determining the course of soidification contains the component source function) controing the phase change. This component is proportiona to the soidification rate f S /f S [, 1], is a temporary and oca voumetric fraction of soid state). The vaue of f S can be found, among others, on the basic of aws determining the nuceation and nucei growth. This approach eads to the so caed micro/macro modes the second generation modes). The capacity of interna heat source appearing in the equation concerning the macro scae soidification and cooing of domain considered) resuts from the phenomena proceeding in the micro scae nucei growth). The function f S can be defined as a product of nucei density N and singe grain voume V a inear mode of crystaization) and this approach is appied in the paper presented. The probem discussed consists in the simutaneous identification of two parameters determining a course of soidification. In particuar it is assumed that nucei density N micro scae) and voumetric specific heat of meta macro scae) are unknown. Formuated in this way inverse probem is soved using the east squares criterion and gradient methods. The additiona information which aows to identify the unknown parameters resuts from knowedge of cooing curves at the seected set of points from soidifying meta domain. On the stage of numerica reaization the boundary eement method is used. In the fina part of the paper the exampes of computations are presented. Key words: micro/macro mode of soidification, inverse probems, boundary eement method, numerica modeing. 1. Introduction The therma processes proceeding in domain of soidifying meta or more generay) in the system castingmoud are described by the Fourier-Kirchhoff equation, this means a noninear paraboic partia differentia equation or the system of such equations) suppemented by the adequate physica, geometrica, boundary and initia conditions. The energy equation for casting domain contains the term describing the capacity of interna heat source source function) and this term contros the evoution of atent heat connected with the phase change. The soidification modes basing on the Fourier-Kirchhoff equation can be divided according to the cassification proposed by Stefanescu [1]) into two groups, namey the macro modes and the micro/macro ones. The difference between macro and micro/macro modes consists in the way of source function modeing. The typica procedure in the case of macro approach reduces to the assumption that the dependence between a oca voumetric fraction of soid state f S and temperature T is known and then after the mathematica manipuations one obtains the energy equation referring to the whoe, conventionay homogeneous, meta domain in which the parameter caed a substitute therma capacity appears e.g. [2,]). In the case of micro/macro modes used in this paper) the function f S is determined on the basis of aws concerning the nuceation and nucei growth, this means the phenomena proceeding on the micro eve e.g. [4 8]). The direct probems concerning the soidification and cooing processes in the casting-moud system can be soved using the numerica methods and in iterature among others the books and papers quoted previousy) one can find the precise information in this scope both for the case of macro and micro/macro modeing. The inverse probems appearing in the therma theory of foundry processes consist in the identification of casting-moud interna parameters e.g. specific heats or therma conductivities [9,1]), boundary conditions or initia temperatures. In the case of micro/macro modes the nucei density [11], growth coefficient or grains shape coefficient can be aso identified. In this paper a mixed task is considered. On the basis of cooing curves at seected set of points from meta domain simutaneousy the macro and micro/macro interna parameters are determined. As an exampe the voumetric specific heat of meta a macro parameter) and the nucei density a micro one) are taken into account. The detais concerning the mathematica description of the probem, the method of inverse probem soution and the resuts of computation wi be presented in the next chapters. 2. Governing equations The equation describing the soidification process ony heat conduction is taken into account) can be written in e-mai: moch@imi.pcz.p 17

2 B. Mochnacki and E. Majchrzak the foowing form c T ) T x, t) = [λ T ) T x, t)] + Qx, t) 1) where c is a voumetric specific heat, λ is a therma conductivity, Q is a source term, T, x, t denote the temperature, geometrica co-ordinates and time. Function Q is proportiona to a soidification rate, this means Qx, t) = L f Sx, t) 2) where L is a voumetric atent heat, f S is a voumetric soid state fraction at the considered point from meta domain. Finay c T ) T x, t) = [λ T ) T x, t)] + L f Sx, t) On the outer surface of the system the condition in genera form [ ] T x, t) Φ T x, t), = 4) ) is given, where / denotes a norma derivative. The initia condition t = : T x, ) = T 5) is aso known. The equations above presented constitute as was mentioned previousy) a base of numerica simuation both in the case of macro modes of soidification and in the case of micro/macro ones. The micro/macro mode of soidification the second generation one [1]) basing on the assumption that the kinetics of nuceation and nucei growth is proportiona to the undercooing beow the soidification point is considered. So, the driving force of the process is a difference between soidification point T cr and temporary oca temperature Fig. 1. T x, t) = T cr T x, t) 6) At first, the foowing function is introduced ωx, t) = Nx, t)v x, t) 7) where N is a grains density [grains/m ], V is a singe grain voume. If one considers the spherica grains and u = R/ is a crystaization rate R is a grain radius) then V x, t) = 4 π u x, τ) dτ 8) In the case of the others types of crystaization e.g. dendritic growth) the shape coefficient v < 1 can be introduced [6] and then ω x, t) = 4 πνn x, t) u x, τ) dτ 9) In the case of so-caed inear [5] mode the function f S is assumed to be equa directy to ωx, t) f S x, t) = Nx, t)v x, t) 1) and if f S = 1 then the crystaization process stops. The derivative of f S with respect to time equas f S x, t) = 4πv [ R x, t) x, t) ] + R x, t) 2 R x, t) N x, t) 11) Assuming the constant number of nucei e.g. [8]) we obtain f S x, t) = 4πνNR x, t) 2 R x, t) 12) and this case wi be beow discussed. In iterature the exponentia mode resuting from the theory proposed by Meh, Johnson, Avrami and Komogoroff e.g. [4,7]) can be aso found. Then f S x, t) = 1 exp [ ω x, t)] 1) For the sma geometrica voumes exp ω) = 1 ω and then the formuas 1), 1) ead to the same resuts. In this paper the inear mode is considered because the numerica approximation of source function in the energy equation and equations resuting from the sensitivity anaysis see: next chapter) is essentiay simper, whie the resuts are very cose. Additionay it is assumed that the nucei growth is determined by the formua [7,8] Fig. 1. Undercooing T R x, t) = µ [T cr T x, t)] 2 = µ T x, t) 2 14) where µ is the growth coefficient. 18 Bu. Po. Ac.: Tech. 551) 27

3 Identification of macro and micro parameters in soidification mode Finay, the oca vaue of source term for N = const resuts from the foowing equation L f S x, t) t 2 = 4πνLNµ T x, t) 2 µ T x, τ) 2 dτ.. Sensitivity with respect to nucei density and specific heat 15) The presented soution of inverse probem bases on the sensitivity coefficients [12]. Because the nucei density N and voumetric specific heat ct ) = c = const are identified therefore the sensitivity modes concerning these parameters must be constructed the direct approach is used [12 14]). Differentiating the energy equation with respect to N one has c [ ] T x, t) = { [λ T ) T x, t)]} [ ] fs x, t). + L 16) Using the Schwarz theorem and denoting T/ = U 1 one obtains c U 1x, t) = [λ T ) U 1 x, t)] + Q U. 17) The source function Q U in equation 17) equas Q U x, t) = 4πνLµ T x, t) 2 µ T x, τ) 2 dτ [ T x, t) 2NU 1 x, t)] 16πνLNµ T x, t) 2 µ T x, τ) U 1 x, τ) dτ. Denoting one has R x, t) = R U x, t) = µ T x, τ) 2 dτ µ T x, τ) 2 dτ, µ T x, τ) U 1 x, τ) dτ Q U x, t) = 4πνLµ T x, t) R x, t) 2 [ T x, t) 2NU 1 x, t)] 18) 19) 16πνLNµ T x, t) 2 R x, t) R U x, t). 2) The expression determining source term Q U is rather compex, but in numerica reaization it does not cause the essentia difficuties. Sensitivity equation is suppemented by the initia condition U 1 x, ) =, and the boundary one in genera form [ Φ U 1 x, t), U ] 1 x, t) =. 21) For exampe, in the case of Robin boundary condition λ T/ = αt T a ) α is a heat transfer coefficient, T a is an ambient temperature) we have λ U 1 / = αu 1. Sensitivity mode concerning the voumetric specific heat requires the differentiation of energy equation and boundary-initia conditions with respect to c. So T x, t) + c U 2x, t) where U 2 = T/ c and = [λ T ) U 2 x, t)] + Q U 22) Q U x, t) = 8πνLNµ T x, t) R x, t) [U 2 x, t) R x, t) + 2 T x, t) R U x, t)] 2) at the same time R x, t) = R U x, t) = µ T x, τ) 2 dτ, µ T x, τ) U 2 x, τ) dτ. 24) The boundary and initia conditions are the same as previousy. Both the sensitivity mode with respect to N and c are strongy couped with the basic one and the sensitivity probems can be soved under the condition that the basic soution is known. 4. Identification of unknown parameters If T d x, t) is the measured postuated) temperature fied in the domain, whie T x, t) is a temperature fied found for the assumed vaues of unknown parameters then the best soution corresponds to the minimum of functiona [15,16] S = F [T x, t) T d x, t)] 2 ddt MIN 25) where [, t F ] is a time interva considered. Because, as a rue, the information concerning T d is given in discrete form the vaues of temperature at the set of contro points sensors x i, i = 1, 2,..., M for times t, t 1,..., t F ) therefore the criterion 25) is formuated in the form M [ S = T xi, t f ) T d xi, t f )] 2 MIN 26) f= i=1 Bu. Po. Ac.: Tech. 551) 27 19

4 B. Mochnacki and E. Majchrzak or more generay S = f= i=1 M [ γ i T xi, t f ) T d xi, t f )] 2 MIN 27) where γ i > are the tapering functions. The necessary condition of functiona 26) minimum eads to the equations S = 2 M S c = 2 M ) T f i ) T f i c N=N k c=c k = = 28) where T f di = T dx i, t f ), = T x i, t f ), N k, c k for k = are the initia vaues start point), whie for k > 1 resut from the previous computations. Introducing the sensitivity functions we have M M ) U f 1i) k = ) ) k 29) U f 2i = Now the function is expanded into Tayor series, namey ) k+ k N = U1i) f k+1 N k) k c + U2i) f k+1 c k). ) Introducing ) into 29) one obtains [ M ) ] k 2 M U f 1i U f 1i M ) k ) k U f 2i U f M 1i M [ ] N k+1 N k i=1 c k+1 c k = M f=1 U f 1i ) k U f 2i [ ) ] k 2 U f 2i ) k [T fdi ) k U f 2i [T fdi ) k ) ] k ) ] k. 1) This system of equations aows to determine N k+1 and c k+1. If the iteration process is convergent then the sequences { N k} and { c k} tend towards the rea vaues of N and c. 5. Boundary eement method The primary and aso the additiona probems resuting from the sensitivity anaysis have been soved using the 1st scheme of the BEM for transient heat diffusion [17 19]. So, the foowing Fourier equation wi be considered F x, t) c = λ 2 F x, t) + Zx, t) 2) where F x, t) denotes the temperature or functions resuting from the sensitivity anaysis, whie Zx, t) is the source function for primary probem: Zx, t) = Qx, t), for additiona probems: Zx, t) = Q U x, t)). One can see that both c and λ are assumed to be the constant vaues. Taking into account the rather sma temperature interva in which the process discussed proceeds, such assumption is entirey acceptabe. The detais concerning the BEM appication in the case c = ct ) and λ = λt ) can be found in [17]. So, at first, the time grid is introduced = t < t 1 <... < < t f <... < t F <, t = t f. ) If the 1st scheme of the BEM is taken into account then the boundary integra equation corresponding to transition t f is of the form B ξ) F ξ, t f ) + 1 c f Γ F ξ, x, t f, t ) J x, t)dγdt = 1 f J ξ, x, t f, t ) F x, t)dγdt c Γ + F ξ, x, t f, ) F x, ) d + 1 c f Z x, t)f ξ, x, t f, t ) ddt. 4) In Eq. 4) F is the fundamenta soution [17 19] and F ξ, x, t f, t ) [ 1 r 2 ] = exp [4πa t f d/2 t)] 4a t f t) 5) where d is the dimension of the probem, r is the distance from the point under consideration x to the observation point ξ, a = λ/c, whie J ξ, x, t f, t ) = λ F ξ, x, t f, t ), 6) F x, t) J x, t) = λ and Bξ) is the coefficient from the interva, 1). We use the constant eements with respect to time [17,18] and then the boundary integra Eq. 4) takes a form B ξ) F ξ, t f ) + J x, t f ) g ξ, x) dγ Γ Γ = F x, t f ) h ξ, x) dγ + J ξ, x, t f, ) F x, ) d+ Z x, ) g ξ, x) d where h ξ, x) = 1 c f 7) J ξ, x, t f ), t dt 8) 11 Bu. Po. Ac.: Tech. 551) 27

5 Identification of macro and micro parameters in soidification mode and g ξ, x) = 1 c f F ξ, x, t f ), t dt. 9) In numerica reaization the foowing discrete form of the boundary integra Eq. 7) is considered N G ij J f j N = L H ij F f j + =1 P i F f 1 + L =1 W i Z f 1 4) where and G ij = Γ j g ξ i, x ) dγ j, H ij = { h ξ i, x ) dγ j, i j Γ j.5, i = j 41) P i = F ξ i, x, t f, ) d, W i = g ξ i, x ) d. 42) Fig. 2. Cooing curves The system of Eq. 4) can be written in the matrix form, namey G J f = H F f + P F f 1 + W Z f 1. 4) After determining the missing boundary vaues of F and J, the vaues of function F at the interna points ξ i for time t f are cacuated using the formua i = N + 1,..., N + L): F f i = N N H ij F f j L G ij J f j + P i F f 1 + =1 L =1 W i Z f 1. 44) 6. Exampe of computations The agorithm above presented can be used both in the case od 1D probem and 2D or D ones. It is ony the probem of adequate computer program construction. Beow the soution concerning the auminium pate G = cm 1D task) wi be shown. The infuence of moud is taken into account by the Robin condition for x = 1.5 and x = 1.5 heat transfer coefficient α = 25 W/m 2 K)). Nucei density N = 1 1, voumetric specific heat c = MJ/m K), the others parameters of materia [2]: λ = 15 W/mK), L = 975 MJ/m, µ = 1 6 m/sk 2 ), soidification point T cr = 66 C, pouring temperature T = 67 C. The cooing curves corresponding to the basic soution are shown in Fig. 2. The inverse probem has been soved under the assumption that the cooing curves are known, at the same time the different initia vaues of estimated parameters have been taken into account, for exampe N = 1 8, c = 2 variant 1) and N = 1 8, Fig.. Simutaneous identification of N and c variant 1 1 N k /N d, 2 c k /c d ) =.5 variant 2). In Figs.,4 the resuts of identification this means the vaues of N k /N d and c k /c d N d = 1 1, c d = denote the rea vaues of parameters) for successive iterations are shown. It is visibe that the iteration process is convergent and rea vaues of identified parameters are obtained after a few iterations. The testing computations show that iteration process is convergent even for initia vaue of nucei N = 1 and from numerica point of view it is very important information. It appears that the vaues N = 1 and c [1, 6] assure the convergence of simutaneous identification of the parameters N and c. Figs. 5 and 6 iustrate the soutions c Bu. Po. Ac.: Tech. 551)

6 B. Mochnacki and E. Majchrzak of inverse probem for N = 1, c = 6 variant ) and N = 1, c = 1 variant 4). N = 1, c = 1). One can see that the iterative process is convergent and the fina vaues of N and c are sufficienty exact. Fig. 4. Simutaneous identification of N and c variant 2 1 N k /N d, 2 c k /c d ) Fig. 6. Simutaneous identification of N and c variant 4 1 N k /N d, 2 c k /c d ) Fig. 5. Simutaneous identification of N and c variant 1 N k /N d, 2 c k /c d ) In the next version of computations in the pace of T d x i, t f ) the disturbed data in reation to the basic soution) have been introduced. The exact soution has been transformed randomy using the procedure described in [2]. In Fig. 7 the exampe of cooing curves obtained in this way is presented. The resuts of identification corresponding to successive iterations are shown in Fig. 8 Fig. 7. Disturbed cooing curves σ = 1) 7. Concusions Summing up, the information concerning the cooing curves at the seected set of points from casting domain aows to reconstruct parameters determining the soidification process even in the case when they beong to different macro and micro) eves. The east squares criterion 112 Bu. Po. Ac.: Tech. 551) 27

7 Identification of macro and micro parameters in soidification mode in which the sensitivity coefficients are introduced constitutes a very effective too for numerica soution of inverse probems from the scope of therma theory of foundry processes the same approach has been used by the authors of this paper aso in the case of others probems). It shoud be pointed out that the mode of soidification presented here concerns a sma superheating of meta because ony heat conduction is considered. The simpification consisting in rejection of moud subdomain and approximation of the moud infuence by the Robin condition on the externa surface of casting is not necessary e.g. [8]). It was introduced here in order to simpify the theoretica considerations connected with the main subject of this paper. Fig. 8. Simutaneous identification of N and c disturbed data 1 N k /N d, 2 c k /c d ) Acknowedgements. The research has been done as a part of Project No T8B References [1] D.M. Stefanescu, Critica reviews of the second generation of soidification modes for casting, Modeing of Casting, Weding and Advanced Soidification Process VI, 2 199). [2] J. Crank, Free and Moving Boundary Probems, Caredon Press, Oxford, [] B. Mochnacki and J.S. Suchy, Numerica Methods in Computations of Foundry Processes, PFTA, Cracow, [4] N.A. Awdonin, Mathematica Description of Crystaization, Zinatne, Riga, 198, in Russian). [5] C. Chang, D.M. Stefanescu and D. Shangguan, Modeing of the iquid/soid and the eutectoid phase transformation, Meta. Transactions A 2 A4), ). [6] E. Fras, W. Kapturkiewicz and H.F. Lopez, Macro and micro modeing of the soidification kinetics of casting, AFS Transactions 9248), ). [7] W. Kapturkiewicz, Modeing of Cast Iron Soidification, Akapit, Cracow, 2. [8] R. Szopa, Appication of the boundary eement method in numerica modeing of soidification - Part II. The micro/macro approach, J. Theoretica and App. Mechanics 26), ). [9] E. Majchrzak and J. Mendakiewicz, Identification of cast iron substitute therma capacity, Archives of Foundry 622), ). [1] E. Majchrzak and J. Mendakiewicz, Estimation of cast stee therma conductivity on the basis of cooing curves from the casting domain, Archives of Foundry 515), ). [11] E. Majchrzak, J.S. Suchy and R. Szopa, Linear mode of crystaization identification of nucei density, Giessereiforschung, Internationa Foundry Research 2, ). [12] K. Kurpisz and A. Nowak, Inverse Therma Probems, Computationa Mechanics Pub., Southampton, Boston, [1] K. Dems and B. Rousseet, Sensitivity anaysis for transient heat conduction in a soid body. Part 1, Structura Optimization 17, ). [14] K. Dems and B. Rousseet, Sensitivity anaysis for transient heat conduction in a soid body. Part 2, Structura Optimization 17, ). [15] O.M. Aifanov, Inverse Heat Transfer Probems, Springer- Verag, London, [16] M.N. Ozisik and H.R.B. Orande, Inverse heat transfer: fundamentas and appications, Tayor and Francis, Pennsyvania, [17] E. Majchrzak, Boundary Eement Method In Heat Transfer, Pub. of the Czest. Univ of Technoogy, Częstochowa, 21. [18] C.A. Brebbia and J. Dominguez, Boundary Eements, An Introductory Course, McGraw-Hi, London, [19] P.K. Banerjee, Boundary Eement Methods In Engineering, McGraw-Hi, London, [2] D. Janisz, Inverse Probems in Transient Heat Conduction, Doctora Thesis, Giwice, 24. Bu. Po. Ac.: Tech. 551) 27 11