A Latent Heat Method for Macro-Micro Modeling of

Transcription

1 A Latent Heat Method for Macro-Micro Modeling of Eutectic Solidification* By C. S. KANETKAR,** In-Gann CHEN,** D. M. S T EFANESC U* * and N. EL-KADDAH* * Synopsis Two different numerical methods were used to simulate the solidification of eutectic alloys. The first one was the classical approach based on an implicit, one dimensional, axisymmetric finite difference method (EUCAST program), while the second one involved a two dimensional control volume implicit scheme with a boundary condition describing the heat flux at the casting-mold interface (BAMACAST program). To account for the heat evolution term in the conductivity equation, an original latent heat approach was introduced, consisting of the calculation of the latent heat evolved by the fraction of solid formed as a function of time. In turn, the fraction of solid has been calculated based on nucleation and growth kinetics. The heat transfer coefficient used for simulation was determined based on an analytical approach for hollow cylinders. Gray cast iron and aluminum-silicon alloys of eutectic composition were used for the validation experiments. Cylindrical bars of different diameters were poured and the cooling curves were recorded at various locations in the metal and the mold. Experimental and computed data were compared, with excellent results for solidification time, undercooling, width of the mushy zone and heat transfer coefficient. Key words; macro-micro modeling; solidification; eutectic; latent heat method; simulation; cast iron; aluminum-silicon; EUCAST; BAMA- CAST. micro modeling, was also extended to other eutectic systems, such as Al-Si,66 as well as to the eutectoid transformation.) In general, the problem of heat flow in solidification is non-linear and has to be solved numerically. Although the techniques used for solving the differential equation of energy balance in conventional heat transfer models8-10' and in macro-micro models are similar, there are some differences in the treatment of latent heat in solidification. The so called enthalpy or specific heat methods are used for the conventional models, while a latent heat method is used for the macro-micro model. Detailed discussion of this point can be found in the recent review paper by Rappaz and Stefanescu.'l~ The work described in this paper is part of an effort whose purpose is to develop a systematic approach to the computation of nonequilibrium solidification of eutectic alloys, ultimately leading to a strong basis for process control and forecasting of mechanical properties. I. Introduction The idea of simulation of microstructure evolution during solidification of eutectic alloys through the application of nucleation and growth concepts was first conceived by Oldfield1~ in A model for solidification of eutectic gray cast iron was developed, in which the growth kinetics was integrated with heat flow analysis to predict the cooling curves which, in turn, could provide information about the microstructure. Stefanescu and Trufinescu2~ and Yanagisawa and Maruyama3~ applied this technique to study the effect of silicon inoculation on the microstructure of cast iron. Fredriksson and Svensson4~ proposed an analytical approach which they applied to white and spheroidal graphite cast iron. A finite difference method (FDM) program including a growth model based on diffusion of carbon through the austenite shell was proposed by Su et a1.5~ Although the model fell short in predicting the experimentally measured eutectic undercooling, it was successful in predicting the general trend of cooling curves for spheroidal graphite (SG) cast iron. It is believed that a more complex physical model should be used for the solidification of SG iron. The above mentioned technique, called macro- II. Experimental Method Figure 1 shows a sketch of the mold assembly used in this investigation. A 160 mm long air set (pep set type) sand mold with an outer diameter of 110 mm was used for casting 30 and 50 mm diameter cylindrical bars. Temperatures in the melt and the mold during solidification were measured using a battery of 8 type K thermocouples. Two sets of 3 thermocouples were placed in the mold at a 90 angle to measure the heat transfer coefficient at the metal mold interface while the remaining two were used to record the cooling curves during solidification at 2 different locations. Data acquisition was carried out using an ISAAC 200 interfaced with an IBM/XT computer. Two eutectic alloys were used in this investigation, a low and high temperature system. They are Al- 12%Si and a 3.7%C and 2%Si cast iron. A 2 kg charge of Al-Si of eutectic composition was melted in a resistance heating furnace. The melt was poured into the mold at 700 C. Cast iron was melted in a high frequency induction furnace and poured at C. The temperature data was recorded after every 2 s for a period of about 25 min to allow for complete solidification. * * Manuscript received on December 14, 1987; accepted in the final form on May 13, 1988, 1988 ISIJ Department of Metallurgical Engineering, The University of Alabama, P,O. Box G, Tuscaloosa, AL 35487, U.S.A. ( 860 ) Research Article

2 4 Transactions Is", Vol. 28, 1988 (861) Fig. 2. Schematic sketch of a solidification front. same temperature. b) Equiaxed growth : the nuclei grow as spherical equiaxed eutectic grains. c) Growth rate is controlled by bulk undercooling, i.e., by the difference between the equilibrium eutectic temperature and the temperature in the casting. Within the framework of these assumptions, the fraction of solid at any location within the melt during solidification may be represented by an Avarami type equation: f S =1- exp [- ~NR3(t)... (1) Fig. 1. Mold assembly and boundary conditions. III. Theoretical Model Figure 2 shows a sketch of the solidification front in the eutectic model. It is seen that nucleation and growth occur near the solid-liquid interface in the region where the melt undercools to a temperature below the eutectic temperature. Based on nucleation and growth kinetics, the overall rate at which the melt is being solidified will depend on the degree of undercooling at the liquid-solid interface, which in turn is affected by the rate at which heat is removed from the system. Accordingly, in the development of a suitable model for the system, the following must be represented: 1) The nucleation and growth kinetics 2) The temperature field during solidification. 1. Nucleation and Growth Kinetics The following assumptions have been made in the development of the formulation of the eutectic kinetic model : a) Instantaneous nucleation and unique nucleation temperature : as the analysis of nucleation rate15 shows nucleation takes place in a very short range of temperature, and hence, it can be assumed that all the nuclei are formed instantaneously at the 3 where, N: the number of nuclei per unit volume R : the radius of the growing nuclei at time t. The radius of the eutectic grains may in turn be obtained from the parabolic growth law by the following equation : where, r=r t=t J dr = p[te- T (t)]2dt...(2) r=ro t=0 i : the growth rate constant To : the radius of the critical nucleus Te : the eutectic temperature T: the bulk temperature in the melt.?. Heat Transfer and Solid cation Model The overall solidification rate of castings is generally controlled by the rate of heat extraction from the system. The general statement of the energy conservation equation in the system including the solid and liquid phases, is given by the following equation : a pcp at =V.V(K.T)+Q...(3) where, p : the density Gp : the specific heat K: the thermal conductivity. The source term Q in the heat conduction equation (Eq. (3)) describes the latent heat evolution per unit volume resulting from phase transformation (liquid to solid), and may be written as Q = L a fs at...(4) where, L : the volumetric latent heat of fusion. The coupling between the macroscopic heat flow

3 ( 862 ) Transactions ISIJ, Vol. 28, 1988 and the microscopic growth kinetic is quite apparent through Eq. (4). It should be noted that the latent heat of fusion is calculated such as to remain finite in the solidification region. Therefore, no special treatment is required for the latent heat term in solving the heat conduction equation, Eq. (3), as those employed in specific heat and enthalphy formulations of heat conduction equations (8) to (10). This point will be discussed later in detail. 3. Boundary Conditions The initial condition for Eq. (3) is that the pouring temperature, To, of the melt has to be specified. T = To at 0<r<a at t = 0...(5) where, a : the radius of the cylindrical bar. The boundary conditions have also to represent the heat flux from the casting to the surrounding. Inspection of Fig. 1 shows 2 types of boundary conditions as follows. The first one describes the convective and radiative heat flux at the metal-air interface which may be given by: Q = (hconv+hrad)(tb- Tamb)...(6) where, h~onv, hrad : the convective and radiative heat transfer coefficients, respectively Tb: the metal temperature at the free surface. The boundary conditions at the mold-metal interface have to represent the heat flow resistance in the refractory walls and the convective boundary layer at the mold-air interface. Two different boundary conditions have been used in this investigation to specify the heat flux from the casting to the sand mold. The first one, assumes the continuity of the heat flux at the metal-mold interface : -Km at = K at a...(7) r r=a ar r=a where, the subscript m : the sand mold. This boundary condition is used in the EUCAST program which is a one dimensional finite difference program. In this program, the thermal energy equation for the casting, Eq. (3), and the transient heat conduction equation in the sand mold are solved simultaneously together with the boundary condition describing convective heat flux at the mold-air interface; -Km at at = hconv(tamb- Tb)...(8) where, subscript b : air-mold interface h~onv : the convective heat transfer at the air-mold interface Tamb: the ambient temperature. In order to avoid the solution of the heat conduction equation in the sand mold, particularly with the uncertainty of the values of the physical properties of the sand, the experimentally measured heat flux at the metal-mold interface was used as a boundary condition for Eq. (3). The measured heat flux at the surface of the casting was expressed by the following equation: q = heff(ta- Tamb)...(9) where, heff = A+B exp (-Ct)...(10) is an effective heat transfer coefficient and A, B and C are constants. It should be remarked that this functional form of the heat transfer coefficient was chosen so that the heat flux will be similar to that derived from the analytical solution of the heat conduction equation for a hollow cylindrical mold assuming constant wall temperatures12~ (also Appendix I) : q = K(Ta _ Tb) a In 1 (b/a) -~ 00 Jp(aan) Jo(ban)an ) [0(ba n=1 JOlaan)-JO(ban) n)yi(aan) - J1(aa K n)yo(ban)l exp -Cant Pp...(11) where, Ta, Tb : interface temperatures of metalmold and mold-air, respectively J, Y: Bessel functions of the first and second kind. an is the n-th root of the following equation: Jo(aa)Yo(ab)- Jo(ab)Yo(aa) = 0...(12) Figure 3 shows a typical plot of the experimentally measured sand mold temperature during solidification for an aluminum casting at different locations. The corresponding curves obtained from the analytical solution, using the melting point (600 C), and the room temperature (20 C) as boundary temperatures, are also given in the same plot. It is seen that the sand temperature near the metal mold interface reaches the steady state temperature in relatively short time, which suggest that the heat flux at the metal-mold interface is almost constant during solidification. The measured effective heat transfer coefficient, heff, is shown in Fig. 4. The theoretically predicted heat transfer coefficient is also given in Fig. 3. Temperature vs. time curves for thermocouples in the metal and mold and calculated curves (analytical) for thermocouple locations inside the mold.

4 T Transactions ISIJ, Vol. 28, 1988 (863) the same figure. It is interesting to see that the two curves are functionally similar and the analytical solution gives a reasonably good estimate of the heat flux at the mold-metal interface. The usefulness of this approach is to provide the heat transfer coefficient where experimental measurements are not available, notwithstanding the significant saving of computer time. This approach was used in the BAMACAST program which is a two dimensional computer program based on the volume integration method. Iv. Computational Technique Equations (1) through (4) constitute a general mathematical description of the macro-micro eutectic solidification model. It should be noted that these equations are non-linear and can only be solved numerically. Two different computational schemes were developed during the course of this investigation. Initially, the problem was solved for a one dimensional system. Here, the mold was assumed to be infinitely long and the radial differential heat conduction equation, Eq. (3), was solved using Crank-Nicolson finite difference method.13~ The solution thus corresponds to the region in the middle of the mold where thermocouples were located. To represent the real system, a two dimensional model was later developed. The computational scheme chosen for solving the two dimensional formulation of the problem uses a control volume approach to derive the discretization equations.14~ To avoid numerical instability and to ensure a converged solution, the latent heat term in the differential energy balance equation was Q- a ) where, ( a r ) exp linearized a a Q = -8p2rR*W.L as (T_T*) - 3 ~rr*3n (Te - T*) follows : T* : the previous iteration value of T. Detailed discussion ssi n of f this normal n izing techniq given by Patankar.14~ The property values and the growth kinetics used in the computation are given in Table 1. V. Results and Discussion ue (13) (14) data Figures 5 and 6 show plots of the cooling curves for cast iron and Al-Si eutectic alloys at the center of experimental castings. The cooling curves exhibit undercooling of the melt at the beginning of solidification and subsequent recalescence. In addition, the degree of undercooling and the arrest temperature depend on the solidification time, as evident from Fig. 6. It is seen that the macro-micro eutectic model not only accurately predicts the degree of undercooling and the arrest temperature, but also the solidification time. The very good agreement between the theory and experiment demonstrates the appropriateness of the model employed. Although the agreement between the theoretically predicted and the experimentally measured values at the center of the casting for the one dimensional model is not as good as the one obtained from the two dimensional model, the approximate one dimensional model is

5 (864) Transactions ISIJ, Vol. 28, 1988 Table 1. Values and boundary conditions used in simulations. Fig. 5. Comparison between experimental and simulated cooling curves for gray cast iron of eutectic composition. Fig. 6. Experimental and calculated (using two dimensional programs) cooling curves for eutectic Al-Si alloy poured in sand molds of different size. seems to predict the cooling curve reasonably well. Thus, the approximate model can be very helpful in providing a reasonable picture of microstructure evolution at minimal computational cost. However, when accurate information is needed on cooling curves and on the temperature and solid fraction fields during solidification, the two dimensional model has to be used. The departure from equilibrium solidification during solidification of cast iron and Al-Si eutectic alloys can be readily seen from the enthalpy-temperature diagram, Figs. 7 and 8. The equilibrium curves are also given in these plots. The deviation from the equilibrium enthalpy is quite evident. Furthermore, inspection of Figs. 6 and 8 show that the solidification time, and hence the cooling rate, has a significant effect on the enthalpy history. Higher the cooling rate, larger is the deviation from equilibrium solidification. To examine the effect of nonequilibrium solidification on the solidification sequence in cast iron, Figs. 9

6 Transactions ISII, Vol. 28, 1988 (865) Fig. 7. Calculated and theoretical enthalpy res. temperature curves for cast iron at eutectic composition. Fig. 8. Calculated an d theoretical enthalpy res. temperature curves for Al- Si eutectic alloy. and 10 show the corresponding maps of temperature and solid fraction contours after 200 and 250 s. Inspection of these figures shows that the melt solidifies with a solid-liquid mushy zone (distance between solid isotherm, fs=1, and the liquid isotherm, f =O). The width of the mushy region may be considered as a measure of the deviation from equilibrium solidification. It increases with increasing solidification time. Theoretical predictions of the width of the mushy zone for a cast iron sample of 5 cm diameter and 20 cm length, is shown in Fig. 11. Figure 12 shows the computed radial solid fraction profiles during solidification and the solidification time for cast iron at a vertical location, corresponding to the tip of the thermocouple. Finally, it is worthwhile to outline the novel contributions of this work. Unlike heat transfer models based on single solidification temperature where an arbitrary freezing range is assumed, the macro-micro model is a fundamental approach to calculate the la-

7 (866) Transactions ISIJ, Vol. 28, 1988 Fig. 11. Calculated start and end of solidification wave fronts for 5 cm diameter bar and experimental points for thermocouple placed at the center of the bar. Fig. 9. Calculated temperature contours for a 5 cm diameter cast iron bar poured in pepset sand mold. Fig. 12. Distance from the metal-mold interface res. fraction of solid curves for eutectic gray cast iron sample. castings. VI. Conclusions Fig. 10. Calculated progression of solidification front from a 5 cm diameter cast iron bar poured in pepset sand mold. tent heat. It should be stressed that the linearization of the latent heat mentioned in Chap. IV reduced significantly the computational cost for the coupled heat flow and the growth kinetic model, as compared with that for conventional heat transfer models. Furthermore, the model presented here provides, in addition to thermal history of the casting, a detailed information regarding undercooling, solidification temperature and solidification time, which is precisely the information needed to characterize the structure of A macro-micro solidification model has been developed for solidification of eutectic alloys. In essence, the model couples the macroscopic heat flow with the nucleation and growth kinetics through the disposition of latent heat of freezing. Efficient one and two dimensional algorithms have been developed for simulation of temperature history and cooling curves in castings. The theoretical prediction was found to be in good agreement with experimental measurements. Nomenclature a, b : Radius of metal-mold and mold-air interface, respectively Cp, Cpm : Specific heat of metal and mold, respectively

8 n Transactions ISIJ, Vol. 28, 1988 (867) fs : Fraction of solid hconv~ brad: Convective and radiative heat transfer coefficient, respectively heff : Effective heat transfer coefficient at the metal-mold interface J, Y: Bessel function of the first and second kind, respectively K, Km : Thermal conductivity of metal and mold, respectively L : Volumetric latent heat of fusion N: Number of nuclei per unit volume Q. Latent heat evolution rate per unit volume q : Heat flux R : Radius of the growing nuclei at time t ro : Critical radius of the nucleus T: Bulk temperature TTY : Previous iteration value of T Ta : Metal-mold interface temperature Tb : Mold-air interface temperature Te : Eutectic temperature To : Pouring temperature Tamb : Ambient temperature v, A : Volume and surface area of the casting, respectively p, pm : Density of metal and mold, respectively p : Growth rate constant Acknowledgement Support for this research was provided by NSF- Alabama EPSCoR Science and Engineering. REFERENCES 1) W. Oldfield : Trans. Am. Soc. Met., 59 (1966), ) D. M. Stefanescu and S. Trufinescu: Z. Metallkd., 65 (1974), ) 0. Yanagisawa and M. Maruyama: The 46th Int. Foundry Congress, (1979), Paper No ) H. Fredriksson and I. L. Svensson : The Physical Metallurgy of Cast Iron, ed. by H. Fredriksson and M. Hillert, North-Holland, New York, (1984), ) K. C. Su, I. Ohnaka, I. Yamauchi and T. Fukusako: The Physical Metallurgy of Cast Iron, ed. by H. Fredriksson and M. Hillert, North-Holland, New York, (1984), ) D. M. Stefanescu, C. Kanetkar: The 54th Int. Foundry Congress, Nov. 1987, New Delhi, Paper No ) D. M. Stefanescu, C. S. Kanetkar: Computer Simulation of Microstructural Evolution, D. J. Sroloivtz, ed., Metall. Soc., Warrendale, (1985), ) V. R. Vollor : IMA J. Numerical Anal., 5 (1985), ) G. B. Thomas, I. V. Samarasekera and J. K. Brimacombe: Metall. Trans. B., 15B (1984), ) G. Conini, Del Guidice, R. W. Lewis and C. Zeinkiewicz : Int. J. Numerical Methods in Eng., 8 (1974), ) M. Rappaz and D. M. Stefanescu: Solidification Processing of Eutectics, ed, by D. M. Stefanescu et al., Metall. Soc., Warrendale, (1988), ) H. S. Carslaw and J. C. Jaeger: Conduction of Heat in Solids, Oxford Univ. Press., Oxford, (1959), ) E. F. Nogotov: Applications of Numerical Heat Transfer, Mcgraw-Hill, New York, (1978), ) S. V. Patankar. Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp., New York, (1980), ) D. M. Stefanescu and C. Kanetkar: Computer Simulation of Casting and Solidification Processes, H. Fredriksson, where, ed., European Mater. Res. Soc., les editions de physique, Strasbourg, (1986), 255 and 266. Appendix I For a hollow cylinder initially at To, and if the inner and outer surfaces are maintained at constant temperatures, TTa and T5 respectively, the solution of heat conduction equation may be written as Ref. 12) : T (r, t) _ T'a in (b/r)+ in(b/a) Tb In (r/a) Jo(aa n) Uo(rIXn) ~"~0 ~ 1 J0(aIXn)-~--Jp(ban) exp (-Dat) o {rbjo(acr )-raj0(ban)}jo(aan)uo(nan) n =1 JO( 2 as n - JO( 2 ba n ) ) x exp (-Dant)...(A-1) a, b : the inner and outer radii of the cylinder, respectively D= K : the thermal diffusivit Y pp J, Y: Bessel functions of the first and second kind. an is the n-th positive root the Uo(ran)=0, and Uo(rcrn) = Jo(ran)Yo(bci )- Jo(ban)Yo(ran)...(A-2) The heat flux at the inner wall can be calculated analytically using the following equation : rom Eq. (A-1), q(t)r=a = -K( a T(r, t)...(a-3) at(r t) F ar r=a a T'(r, ar t) - 1 ar' T. can r ab be r+ written T. a ar a In (b/a) a b ar b r ar + ~.o 0 Jo(aan) exp (-Dant) n=1 Jo(acrn)+ Jo(ban) a[ Jo(ran)Yo(ban)- Jo(ban)Yo(ran)J ar - ~ {T5J0(an)- TaJo(ban)} Jo(aan) exp (-Dant) n=1 JO(aan)-JO(ban) a Uo(ran) ar (A-4) Since Jo(z)=-Ji(z) and Yo(z)=-Yi(z), the derivat ive of the Basset functions in the summation terms in Eq. (A-4) are given by a[jo(ran)yo(ban)- Jo(ban)Yo(ran)] ar = IXn[- Ji(ran)Yp(ban)-f- Jo(ban)Yi(ran)]...(A-5) and a Uo(YIXn) - ar = an[- Ji(ran)Yo(ban)+ Jo(ban)Yi(ran)]...(A-6) From Eqs. (A-4) to (A-6), and To= Ta the temperature Research Article

9 (868) Transactions ISIT, Vol. 28, 1988 gradient at the inner surface of the hollow cylinder, r = a, reads a T(r, t) (Tb- Ta) ar r_a b a In - a +(T a-?-b)7 J0(aan)J0(ban)an n=1 JO(aan)-JO(ban) [Jo(ban)Y1(aan)- J1(aan)Yo(ban)] exp (-Dant)...(A-7) The heat flux at the inner wall, r = a, may be written from Eq. (A-7) and (A-3) as: q=k(ta_t) a In 1 (b/a) -ir JJo(aan)Jo(ban)an o ba n=1 Jo(aa,c)-Jo(ban). J ( n ) Y1 ( aan ) - Ji(aa n)yo(ban)] ex p - P cp at n ) (A-8) Research Article