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Heat transfer-solidification kinetics modeling of solidification of castings

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Abstract

A close examination of the recent developments in the field of computer simulation of solidification process reveals that a combination of both macroscopic and microscopic models is necessary in order to accurately describe the solidification of castings. Currently available macroscopic models include models that describe heat transfer from metal to mold, fluid flow of liquid metal during mold filling, and stress field in the casting. At the microscopic level, the models should include more intricate issues such as solidification kinetics and fluid flow in the mushy zone. Although significant progress has been accomplished over the years in each field, the task of including all of these models into a comprehensive package is far from being complete. This paper describes the state of the art on coupling the macroscopic heat transfer (HT) and microscopic solidification kinetics (SK) models and introduces thelatent heat method as a more accurate method for solving the heat source term in the heat conduction equation. A new method for calculation of fraction of solid evolved during solidification based on computer-aided cooling curve analysis (CA-CCA), as well as a method based on nucleation and growth kinetics laws, is discussed. A new nucleation model based on the concept of instantaneous nucleation, which is used to describe equiaxed eutectic solidification of commercial alloys, has been introduced. It is demonstrated that the instantaneous nucleation model agrees well with the experimental results in terms of cooling curves and of evolution of the fraction of solid during solidification. Validation results are also shown for SK models that are based on CA-CCA coupled with HT models for eutectic Al-Si and gray cast iron alloys.

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Abbreviations

A :

surface area [m2]

C p :

specific heat [J kg-1 K-1]

D :

liquid diffusivity [m2 s-1]

K :

thermal conductivity [W m-1 K-1]

K 1 K2,K 3,K 4 :

nucleation constants

I het :

heterogeneous nucleation rate [m-3 s-1]

L :

volumetric latent heat of fusion [J m-3]

L′ total heat evolved at the end of transformation [J kg-1]:

Ll′

heat evolved due to transformation until timet [J kg-1]:

N

number of particles that have already nucleated per unit volume [m-3]:

N s

number of heterogeneous substrates originally available per unit volume [m-3]:

N(x, t)

number of grains per unit volume (grain density) [m−3]:

Q L

latent heat evolved during solidification [J]:

Q

source term or rate of evolution of latent heat [W m−3]:

R(x, t)

average equiaxed grain radius [m]:

T

temperature at a given location in casting [K]:

T cc

temperature at the center of the casting obtained from cooling curve [K]:

T zc

temperature at the center of the casting as calculated from the zero curve [K]:

T E

equilibrium temperature [K]:

T o

ambient temperature [K]:

T M

melting point [K]:

T n

nucleation temperature [K]:

V

volume of sample [m3]:

a, b a 1 b 1

constants:

a o , b o , c o

constants:

do, eo

constants:

d

atomic diameter [m]:

fs

volume fraction of solid:

f(δ)

function of wetting angle between liquid and substrate:

h

heat transfer coefficient [W m-2 K-1]:

k

Boltzman's constant [J K-1]:

n′

number of surface atoms of the substrate per unit volume of liquid [m-3]:

ns

number of atoms in the heterogeneous substrate per unit volume for SiO2 [m-3]:

qcc, qzc

rate of convective heat loss from the casting [W]:

rs

radius of substrate [m]:

p

density [kg m-3]:

t

time [s]:

to

time at the beginning of solidification [s]:

ΔGc het

energy of formation of the critical nucleus [J]:

ΔGA

diffusional activation energy for growth [J]:

ΔT

undercooling [K]:

δRN/δt

nucleation rate [m-3 s-1]:

δR/δt

growth rate [m s-1]:

γ

vibrational frequency [s-1]:

μ1

growth constant [m s"-1 K-2]:

μ2

nucleation constant [m-3 K-2]:

σ surface tension [N m-1]

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Authors and Affiliations

  1. Department of Metallurgical and Materials Engineering, The University of Alabama, 35487, Tuscaloosa, AL

    Doru M. Stefanescu (University Research Professor and Director of the Solidification Laboratory), G. Upadhya (Graduate Research Assistant) & D. Bandyopadhyay (Graduate Council Research Fellow)

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  1. Doru M. Stefanescu
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  2. G. Upadhya
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  3. D. Bandyopadhyay
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Stefanescu, D.M., Upadhya, G. & Bandyopadhyay, D. Heat transfer-solidification kinetics modeling of solidification of castings. Metall Trans A 21, 997–1005 (1990). https://doi.org/10.1007/BF02656584

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