Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 2, pp. 257-268, Warsaw 2008 IDENTIFICATION OF SUBSTITUTE THERMAL CAPACITY OF SOLIDIFYING ALLOY Ewa Majchrzak Silesian University of Technology, Gliwice, Poland e-mail: ewa.majchrzak@polsl.pl Bohdan Mochnacki Czestochowa University of Technology, Czestochowa, Poland e-mail: moch@imi.pcz.pl Józef S. Suchy AGH University of Science and Technology, Cracow, Poland e-mail: jsuchy@agh.edu.pl In the paper, the problemof identification of substitute thermal capacity C(T) is discussed. This parameter appears in the case of modelling of the solidification process on the basis of one domain approach (fixed do- mainmethod). Substitute thermal capacity (STC) canbe approximated, among others, by a staircase function and this case is considered. So, it is assumed that in themathematicalmodel describing thermal processes in the system considered the parameters of STC are unknown. On the basis of additional information concerning the cooling (heating) curves at a selected set of points, the unknown parameters can be found. The inverse problem is solved by using the least squares criterion, in which the sensitivity coefficients are applied. On the stage of numerical simu- lation, the boundary element method is used. In the final part of the paper, examples of computations are shown. Key words: casting solidification, inverse problem, numerical methods 1. Introduction An inverse problem from the scope of thermal theory of foundry processes is discussed. The problem belongs to the group of parametric ones. From the mathematical point of view, a transient non-linear boundary-initial inverse 258 E. Majchrzak et al. task concerning non-homogeneous domains (casting andmould) is considered. Thermal parameters appearing in the Fourier-Kirchhoff equation correspon- ding to the casting area (substitute thermal capacity and thermal conductivi- ty) are temperature-dependent, and they are assumed in the formof piece-vice constant functions. Successive ’stairs’ of functions determine thermophysical parameters of the molten metal, the mushy zone and the solid body. The introduction of substitute thermal capacity (STC) to the mathema- tical model of solidification and cooling processes proceeding in the casting domain leads to the model called ”a one domain method” (Mochnacki and Suchy, 1995; Majchrzak and Mochnacki, 1995; Majchrzak and Szopa, 2001), because the energy equation concerns an artificially homogeneous object, whi- le in reality, it is a composition of three time-dependent sub-domains. The substitute thermal capacity of the molten metal (T > TL) and the solid body (T < TS), where TS and TL are the border temperatures resulting from the equilibrium diagram, corresponds to volumetric specific heat, while for [TS,TL], it is a sum of the mushy zone volumetric specific heat and the spectral latent heat controlling the solidification process (Mochnacki and Su- chy, 1995; Kapturkiewicz, 2003). The aim of investigations presented in this paper is simultaneous identification of all ’stairs’ determining the course of STC. Many theoretical and experimental methods for measuring thermophysi- cal properties have been developed in literature, they include, among others, the steady statemethod, the probemethod, the periodic heatingmethod, the pulse heating method, etc. However, all the above methods belong to either steady-state or constant parameters estimation. Typical simultaneous solu- tions concern, as a rule, the identification of several thermophysical or boun- daryparameters treated as constant values (e.g.Huang andWu, 1995;Kurpisz and Nowak, 1995; Yang, 1999; Ozisik and Orlande, 1999; Abou Khachfe and Jarny, 2001). The transient function estimation is an inverse heat conduction problem,which has never been examined in the open literature from the scope of thermal theory of foundry processes. 2. Governing equations Theenergyequationdescribing the casting solidificationhas the following form (Mochnacki and Suchy, 1995; Majchrzak andMochnacki, 1995) c(T) ∂T(x,t) ∂t =∇[λ(T)∇T(x,t)]+L ∂fS(x,t) ∂t (2.1) Identification of substitute thermal capacity... 259 where c(T) is a volumetric specific heat, λ(T) is a thermal conductivity, L is a volumetric latent heat, fS is a volumetric solid state fraction at a considered point from the casting domain, T , x, t denote temperature, geometrical co- ordinates and time, respectively. The form of equation (2.1) shows that only conductional heat transfer is considered and the convection in the molten metal subdomain is neglected. The considered equation is supplemented by the equation (or equations) concerning themould subdomain cm(T) ∂Tm(x,t) ∂t =∇[λm(T)∇Tm(x,t)] (2.2) where cm is themould volumetric specific heat and λm is the mould thermal conductivity. In thecaseof typical sandmoulds, on thecontact surfacebetween the casting andmould the continuity condition in the form −λ ∂T(x,t) ∂n =−λm ∂Tm(x,t) ∂n (2.3) T(x,t)=Tm(x,t) can be accepted (∂/∂n denotes the normal derivative). On the external surface of the system, the condition in the general form Φ [ T(x,t), ∂T(x,t) ∂n ] =0 (2.4) is given. For instance, on the outer surface of the mould the Robin condition −λm ∂Tm(x,t) ∂n =α[Tm(x,t)−Ta] (2.5) determines the heat exchange between the mould and environment. In equ- ation (2.5), α is the heat transfer coefficient, and Ta is the ambient tempera- ture. For t=0, the initial condition t=0 : T(x,0)=T0(x) Tm(x,0)=Tm0(x) (2.6) is also known. It should be pointed out that equation (2.1) constitutes a base for the nu- merical modelling of solidification both in the macro (Mochnacki and Suchy, 1995; Majchrzak and Mochnacki, 1995) and the micro/macro scale (Kaptur- kiewicz, 2003; Majchrzak et al., 2006). 260 E. Majchrzak et al. In the case of a typical macro model of alloy solidification, the knowledge of temperature-dependent function fS in the mushy zone T ∈ [TS,TL] sub- domain is assumed, and then ∂fs(x,t) ∂t = dfs dT ∂T(x,t) ∂t (2.7) Finally, energy equation (2.1) takes the form [ c(T)−L dfS dT ]∂T(x,t) ∂t =∇[λ(T)∇T(x,t)] (2.8) where C(T)= c(T)−L dfS dT is the substitute thermal capacity. One can see that for T >TL : fS =0,while for T TL cP + L TL−TS for TS ¬T ¬TL cS for T TL C2 for TS ¬T ¬TL C3 for T 0 result from the previous iteration. The function Tfi is expanded in the Taylor series about known values of Ckl T f i =(T f i ) k+ 3 ∑ l=1 (Uf li )k(Ck+1 l −Ckl ) (3.5) Putting (3.5) into (3.4), one obtains (e=1,2,3) M ∑ i=1 F ∑ f=1 [(Tfi ) k+ 3 ∑ l=1 (Uf li )k(Ck+1 l −Ckl )−T f gi](U f ei) k =0 (3.6) or M ∑ i=1 F ∑ f=1 3 ∑ l=1 (Uf li )k(Ufei) k(Ck+1 l −Ckl )= M ∑ i=1 F ∑ f=1 [Tfgi− (T f i ) k](Ufei) k (3.7) The system of equations (3.7) can be written in a matrix form (Uk)⊤UkCk+1 =(Uk)⊤UkCk+(Uk)⊤(Tg−T k) (3.8) where U k =                   (U111) k (U112) k (U113) k · · · · · · · · · (UF11) k (UF12) k (UF13) k (U121) k (U122) k (U123) k · · · · · · · · · (UF21) k (UF22) k (UF23) k · · · · · · · · · (U1M1) k (U1M2) k (U1M3) k · · · · · · · · · (UFM1) k (UFM2) k (UFM3) k                   Tg =                    T1g1 · · · TFg1 T1g2 · · · TFg2 · · · T1gM · · · TFgM                    T k =                   (T11) k · · · (TF1 ) k (T12) k · · · (TF2 ) k · · · (T1M) k · · · (TFM) k                   Identification of substitute thermal capacity... 263 while C k =    Ck1 Ck2 Ck3    C k+1 =    Ck+11 Ck+12 Ck+13    (3.9) This system of equations enables finding the values of Ck+1e . The iteration process is stopped when the assumed number of iterations K is achieved. It should be pointed out that in order to obtain the sensitivity coefficients, the governing equations must be differentiated with respect to Ce (direct approach – see Kleiber, 1997; Majchrzak et al., 2005). So, differentiation of equation (2.8) (on the assumption that λ= const) leads to the formula C(T) ∂Ue(x,t) ∂t =λ∇2Ue(x,t)− ∂C(T) ∂Ce ∂T(x,t) ∂t (3.10) where ∂C ∂C1 =      1 0 0 ∂C ∂C2 =      0 1 0 ∂C ∂C3 =      0 for T >TL 0 for TS ¬T ¬TL 1 for T