Journal of Applied Mathematis and Computational Mehanis 2014, 13(2), 141-147 MATHEMATICA AND NUMERICA BAI OF BINARY AOY OIDIFICATION MODE WITH UBTITUTE THERMA CAPACITY. PART II Ewa Węgrzyn-krzypzak 1, Tomasz krzypzak 2 1 Institute of Mathematis, Czestohowa University of Tehnology Czestohowa, Poland 2 Institute of Mehanis and Mahine Design, Czestohowa University of Tehnology Czestohowa, Poland 1 ewa.skrzypzak@im.pz.pl, 2 skrzyp@imipkm.pz.pl Abstrat. In this paper, the results obtained from five models of the solidifiation with substitute thermal apaity were ompared. The alulations were arried out for steel ontaining 0.35% arbon with using an in-home solver based on the finite element method (FEM). A omparison was made on the base of analysis of the ooling urves at seleted nodes. Keywords: solidifiation, binary alloy, substitute thermal apaity, numerial alulations Introdution The main assumptions of the mathematial and numerial models used in the presented paper were disussed in detail in [1]. Numerial modeling of solidifiation an be divided into solidifiation at a onstant temperature, for example in ase of pure metals [2, 3] or solidifiation in the range of temperatures. olidifiation of the binary alloy ours in a range of [T, T ] temperatures speifying the beginning and the end of the proess. Inside the area limited by liquidus (T ) and solidus (T ) the emission of heat is observed. In the ase of steel, for the simpliity of numerial implementation, models with so-alled substitute thermal apaity are widely used [4-7]. An introdution to the model with substitute thermal apaity omes down to a suitable modifiation of the funtion desribing inrease in thermal apaity in the mushy zone [1]. In this ase, regardless of the adopted funtion, the integration proedure within the limits of [T, T ] should lead to the same value of total thermal apaity. However, the results of numerial models an be different due to the adopted quality of the spatial disretization, time step, et. Using an in-home solver based on the finite element method the results obtained from implemented numerial models of the solidifiation with substitute thermal apaity were ompared.
142 E. Węgrzyn-krzypzak, T. krzypzak 1. Mathematial and numerial desriptions The basis of the mathematial model is an equation of energy [1]: T T T λ m + m = meff x x y λ y ρ (1) t where T [K] denotes temperature, eff [J/(kgK)] is substitute heat apaity, ρ m [kg/m 3 ] - averaged density, λ m [W/(mK)] - averaged oeffiient of thermal ondutivity, t [s] - time. Equation (1) is supplemented by the boundary onditions of the first and seond kind and appropriate initial ondition: Γ Γ 1 :T =T b (2) T n 2 : λ = λ T n = q b (3) ( t 0) T0 T = = (4) where T b [K] is temperature on the boundary, q b [W/m 2 ] - known heat flux, T n - diretional derivative of temperature, n - vetor normal to the boundary Γ 2, T 0 [K] - initial temperature. ubstitute thermal apaity is determined using the following hypotheses: Hypothesis 1 [4]: Hypothesis 2 [4]: eff ( T) = m+ (5) T T ( ) T T eff( T) = s+ max s (6) T T where max is determined from the following equation [4]: Hypothesis 3 [4]: 1 2 ( T T )( + ) = ( T T ) max s m + (7) T ( p + ) T T eff ( ) = s + 1 m + s T T T T (8) p
Mathematial and numerial basis of binary alloy solidifiation models. Part II 143 Hypothesis 4 - Borisow model [4, 7]: eff 2 k 1 k Tp T ( T ) = m + (9) ( 1 k)( Tp T) Tp T Hypothesis 5 - amojłowiz model [3, 6]: eff ( T T) [ m ( T T ) m ( T T )] 2 ( T) = m m (10) m where m, s [J/(kgK)] are average speifi heat and speifi heat in solid phase respetively, [J/kg] - latent heat of solidifiation, k - phase separation oeffiient, T p [K] - temperature of melting of pure iron, m - tangent of the slope of the liquidus line, m - tangent of the slope of the solidus line. Using the proedures of spatial and impliit time disretization [8] with respet to equation (1) leads to the global FEM equation [1]: 1 + t 1 t f+ 1 f f+ 1 f K M T = B + MT, t = t t (11) where K denotes the heat ondutivity matrix, M - heat apaity matrix, B - righthand side vetor, t [s] - time step, f - time level. 2. Examples of alulation Computer simulations of the solidifiation were made assuming the material properties of steel ontaining 0.35% arbon (Tab. 1). The alulations were arried out in the retangular area of dimensions 200 x 50 mm (Fig. 1). Fig. 1. Boundary and initial onditions
144 E. Węgrzyn-krzypzak, T. krzypzak They were made in five series eah time hanging the way of mathematial desription of substitute thermal apaity. Boundary and initial onditions are presented in Figure 1. uh a hoie of boundary onditions has made failitate omparison of of the solidus and liquidus isotherms at seleted moments of time for subsequent distribution hypotheses of eff. The time step and the material properties were onstant. Alloy material properties [4] and parameters used in the alulations Parameter olid phase iquid phase [J/(kgK)] 690 820 ρ [kg/m 3 ] 7500 7200 λ [W/(mK)] 35 20 T, T [K] 1743, 1778 [J/kg] 2.7 10 5 p 6 T p [K] 1811 k 0.49 m, m 94.826, 194.826 Table 1 In Table 2 the of the solidus-liquidus isotherms in seleted moments for five solidifiation hypothesis are olleted. As a result of the introdution of appropriate boundary ondition on the left wall of the tested area and the thermal insulation on the other walls, solid phase grows in a horizontal diretion from left to right side, with the result that T and T isotherms are arranged vertially, parallel to eah other. This allows easy and preise determination of their temporary s on the horizontal axis. Hypothesis Comparison of the temporary s of the isotherms T and T (T ) Time 250 s 500 s 750 s (T ) (T ) (T ) (T ) Table 2 (T ) 1 82 89 116 124 142 152 2 83 88 117 123 143 151 3 83 87 117 123 142 149 4 85 90 120 126 146 154 5 83 90 116 126 142 154
Mathematial and numerial basis of binary alloy solidifiation models. Part II 145 Temporary s of the solidus and liquidus isotherms are similar. This is due to the fat that the average speifi heat in the solid-liquid area in eah ase has a similar value, whih implies a omparable rate of solidifiation. For a more omprehensive analysis of the results a omparison of ooling urves at seleted nodes in the test area was made. In Table 3 oordinates of the seleted nodes are olleted, while in Figures 2-4 a omparison of the ooling urves is presented. Index of node Coordinates of seleted nodes x Coordinates y 297 50 50 5078 100 50 5573 150 50 Table 3 The graphs of ooling urves in seleted nodes (Figs. 2-4) learly show good agreement between ompared hypotheses of distribution eff. Fig. 2. Cooling urves for node 297 (x = 50 mm)
146 E. Węgrzyn-krzypzak, T. krzypzak Fig. 3. Cooling urves for node 5078 (x = 100 mm) Fig. 4. Cooling urves for node 5573 (x = 150 mm)
Conlusions Mathematial and numerial basis of binary alloy solidifiation models. Part II 147 Tested models of solidifiation with substitute thermal apaity give omparable results. Eah of the ompared hypotheses an be used for modeling solidifiation of steel. In the ase of other alloys suh as bronze, brass or aluminum alloys the agreement between the disussed models may be less [4]. In-home omputer program, thanks to its flexibility, allows easy modifiation of solvers and relatively omfortable adaptation to solve three-dimensional problems. Referenes [1] Węgrzyn-krzypzak E., krzypzak T., Mathematial and numerial basis of binary alloy solidifiation models with substitute thermal apaity. Part I, Journal of Applied Mathematis and Computational Mehanis 2014, 13(2), 127-132. [2] krzypzak T., Węgrzyn-krzypzak E., Mathematial and numerial model of solidifiation proess of pure metals, International Journal of Heat and Mass Transfer 2012, 55(15-16), 4276- -4284. [3] krzypzak T., harp interfae numerial modeling of solidifiation proess of pure metal, Arhives of Metallurgy and Materials 2012, 57(4), 1189-1199. [4] Mohnaki B., uhy J.., Modelowanie i symulaja krzepnięia odlewów, Wydawnitwo Naukowe PWN, Warszawa 1993. [5] iedleki J., Tuzikiewiz W., ubstitute thermal apaity of binary alloys. Review of hypotheses, ientifi Researh of the Institute of Mathematis and Computer iene 2012, 4, 11, 121-129. [6] Majhrzak E., Mohnaki B., uhy J.., Identifiation of substitute thermal apaity of solidifying alloy, Journal of Theoretial and Applied Mehanis 2008, 46, 2, 257-268. [7] Mohnaki B., Modele matematyzne kierunkowej krystalizaji stopów, Krzepnięie Metali i topów 1984, 7, 125-145. [8] Grandin H., Fundamentals of the Finite Element Method, Waveland Press, Paris 1991.