Investigation of the Solidification Behavior of NH 4 Cl Aqueous Solution Based on a Volume-Averaged Method
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Macrosegregation during alloy solidification is closely related to the interaction of the temperature, flow, and solute concentration fields. Bennon and Incropera[1] established a continuum model for momentum, energy, and solute transport. Beckermann et al.[2–4] proposed a multiscale/multiphase model based on the volume-averaged method. Wu et al.[5–10] carried out further in-depth investigation of volume-averaged method and proposed a multiphase model to predict the macrosegregation that takes into account the effects of melt convection, growth of columnar grains, and sedimentation of equiaxed grains on macrosegregation. Baoguang et al. and Dongrong et al.[11,12] also investigated macrosegregation using the volume-averaged method. Based on the volume-averaged method, a mathematical model that couples the conservation equations for mass, momentum, energy, and solute was developed to calculate the solidification process of an NH4Cl-70 pct H2O ingot and comprehensively elucidate the various phenomena occurring in the solidification process, e.g.,
the columnar-to-equiaxed transition (CET), the sedimentation or drift of equiaxed grains, and convection types. In addition, the formation mechanism of segregation was further studied and was experimentally verified.
II.
MATHEMATICAL MODEL
A. Macrotransport Equations The macrotransport equations include the conservation equations for mass, momentum, energy, and solute; the microscopic model includes a continuous nucleation model and the growth of columnar and equiaxed grains. The source term in the macroscopic conservation equations couples the macrotransport equation and the microscopic model. The macrotransport equations are as follows: Mass conservation equation: @ fq qq þ r fq qq~ uq ¼ Mpq : @t
½1
Momentum conservation equation: RI LI, LIMING ZHOU, JIAN WANG, and YAN LI are with the School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China. Contact e-mail: [email protected] Manuscript submitted August 24, 2016. METALLURGICAL AND MATERIALS TRANSACTIONS B
@ ~ pq : fq qq~ uq þ r fq qq~ uq ~ uq ¼ r fq qq kq r ~ uq þ U @t
½2
Energy conservation equation: @ fq qq~ uq ¼ r fq qq kq r ~ uq þ Qpq : uq þ r fq qq~ uq ~ @t
½3 Solute conservation equation: @ fq qq Cq þ r fq qq~ uq Cq ¼ r fq qq Dq r Cq þ Cpq : @t ½4 Grain transport equation: @ n þ r ð~ ue nÞ ¼ Ne : @t
½5
In these equations, fq is the volume fraction of each phase (p and q, respectively, represent the liquid phase l and the equiaxed grain e or columnar grain c, and p „ q); the volume fraction of each phase in every control unit must satisfy this relation fl + fe + fc = 1. is the Laplace operator, t is the time, qq is the density of each phase, and ~ uq is the mean volume velocity of each phase in the control unit. kq is the thermal conductivity of equiaxed and columnar grains, hq is the latent heat released from each phase, Hq is the thermal exchange coefficient between each phase, Tq is the temperature of
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