Numerical Analysis of the NiTi Solidification Process Influence of Thermal Conductivity

The present study deals with the numerical analysis of the solidification process of a NiTi binary alloy. The physical medium is taken as an incompressible fluid where the heat is transferred by conduction and convection, including the thermal phase change phenomenon. The energy equation, which includes both convection-diffusion heat transfer and a mushy region for the phase-change (solidification), is modelled by using an enthalpy-based formulation. The numerical approach is based on the finite volume method in body fitted coordinates with a PISO scheme to couple the pressure and velocity fields. The results are presented for the temperature field, as well as for the NiTi mass fraction during the solidification process.


Introduction
The Shape Memory Alloy NiTi (NiTi SMA) belongs to a unique class of materials which are known as functional materials [1].This is because they have the ability to recover their shape when the temperature is increased (the so-called Shape Memory Effect (SME), which can be one-or two-way).Additionally, under specific conditions NiTi SMAs can absorb and dissipate mechanical energy by undergoing a reversible hysteretic shape change when subjected to an applied mechanical loading (super-elastic effect).Both phenomena are connected with the reversible phase diffusionless austenitic/martensitic transformation, where the parent phase is austenite (generally cubic) -stable at high temperatures and small stresses, and the product phase is martensite (tetragonal, orthorhombic or monoclinic) -stable at low temperatures and high stresses.Furthermore, NiTi SMAs possess excellent mechanical properties, good corrosion resistance and excellent biocompatibility.Therefore, their application is in a wide variety of industrial sectors such as aerospace, automotive, biomedical and oil exploration [2].For a NiTi SMA consisting of at least two elements, chemical composition becomes an important variable; the other variable commonly used is the temperature [3].For an alloy at fixed composition, the formation and disassociation of phases with the temperature could be predicted according to the Ni-Ti phase diagram (see Fig. 1) [4].
The understanding of the phenomenon of martensitic transformation did not develop rapidly.This is because the Ni-Ti alloy system [4] is quite a complicated system.Various precipitates, which formatted during casting or under certain heat-treatments, had not been understood well from the proposed phase diagram.NiTi alloys could exhibit a SME, super-elastic effect and even two-way SME under the right conditions, which makes this material ideal for a variety of applications.In the early 1960s, Buhler and his co-workers discovered the SME in an equiatomic NiTi alloy [5,6].Both special behaviours, SME and Super elasticity, are shown in Fig. 2 [7].Fig. 2. Stress-strain-temperature diagram exhibiting the Shape Memory and Super elastic effect of NiTi alloy.Crystal transformation is caused due to changes in temperature and stress [7].
The aim of this study was to analyse the solidification and castings process of a Ni-Ti equiatomic alloy [8] with a new approach where the time-dependent phase-change of NiTi alloy is included.With respect to this the solidification process was modelled by the enthalpy-porosity method accounting for the variable (Case I) and constant thermal conductivity (Case II).

Numerical methods. 2.1. Governing equations
NiTi alloy solidification using ANSYS Fluent was modelled with unsteady simulation.Governing equations consisted of mass (Eq.1), momentum (Eq.2) and energy (Eq.3) conservation equations.The conservation equation of mass and momentum are decoupled from the one of thermal energy.These equations are solved using a segregated solver with the second order accurate upwind scheme.The conservation equations in basic form are as follow: where t is time, ρ is density and is velocity.
Where p is the static pressure, τ is the stress tensor and g r ρ are the gravitational body forces.
• Finally, for the solidification problem, the energy equation reads as: where H is enthalpy (see Eq. 1), ρ is density and v r is velocity.
For solidification the enthalpy-porosity technique [9] was used, where the melt interface technique is not tracked explicitly.Instead, a liquid fraction, which indicates the fraction of the cell volume that is in liquid form, is associated with each cell in the computational domain.
The liquid-solid mushy zone is a region in which the liquid fraction values lie between 0 and 1 and its temperature ranges between the liquidus ( ) l T and solidus temperatures.The mushy zone is modelled as a "pseudo" porous medium in which the porosity decreases from 1 to 0 as the material solidifies.When the material has solidified fully the porosity becomes 0 and hence the velocities also drop to 0.
( ) In equation 3 the enthalpy of the material is computed as the sum of the sensible enthalpy and the latent heat ( : where is reference enthalpy, reference temperature, specific heat at constant pressure, liquid mass fraction and latent heat of fusion.The liquid fraction, , is defined as:

Geometry and boundary conditions
Geometry of the NiTi alloy solidification in the experimental tube of length L and diameter D is depicted schematically in Fig. 4. Due to the symmetrical geometry, the NiTi alloy solidification was modelled as an axisymmetrical problem.As boundary conditions on the upper horizontal and lower curved walls the constant temperatures (T H >T C ) were maintained, whereas the other boundaries were exposed (cooled) to the air with a temperature of 25 °C.On the bottom part of the geometry the temperature of the curved wall was lower than the solidus temperature in order to allow the solidification and provide the phase change.The physical properties of the melt are given in Tab.I.
Such a geometrical set-up will allow us to perform the directional solidification experiments further, in which the solidification processing parameters (e.g.temperature gradient and growth rate) can be controlled independently [10], so that one may study the dependence of the microstructural parameter on either temperature gradient at constant growth rate or growth rate at constant temperature gradient for the constant initial NiTi solute composition.To perform a numerical analysis with convective boundary conditions on horizontal walls it was necessary to define the heat transfer coefficient from the interior part of the geometry (solid or liquid alloy) to the outer air (Fig. 6).First we have to define the outer properties of the media which were air with dimensionless Prandtl (Eq. 6) and Grashof (Eq.7) numbers to obtain the Rayleigh number (Eq.8) and, finally, the Nusselt number (Eq. 9).  ) In general, we can use the Nusselt number (Eq.10) to define the convective heat transfer coefficient z α on the outer side of the tube and, accounting for the heat conduction in the tube wall, to determine finally the overall heat transfer coefficient U (Eq. 11).Because the heat transfer coefficient is dependent on the inner temperature of the solid or liquid alloy there exists a temperature dependency in the heat transfer coefficient as presented in Fig. 7.

Grid refinement and numerical accuracy assessment
The grid independence of the present results has been established on the basis of a detailed analysis using four different meshes (the elements were concentrated towards each solid wall).With each grid refinement the number of elements in a particular direction was doubled and the minimum element size was halved.Such a procedure is useful (and encountered in many numerical studies [11][12][13]) for applying the Richardson's extrapolation technique, which is a method for obtaining a higher-order estimate of the flow value (value at infinite grid) from a series of lower-order discrete values.
For a general variable φ the grid-converged value according to the Richardson extrapolation is given as: where MIII φ is obtained on the finest grid and MII φ is the solution based on the next level coarse grid, r is the ratio between coarse to fine grid spacing and p=2 is the order of accuracy.
The variation of liquid fraction and temperature with grid refinement is given in tabulated form in Table II.The "percent" numerical error as given in Table II  The results of numerical accuracy analysis indicate that, as the mesh is refined, there is a consistent improvement in the accuracy of the predicted values, and the agreement between predictions obtained with mesh MIII and the extrapolated value is extremely good.Based on this, the simulations in the remainder of the paper were conducted on mesh MIII, which provided a reasonable compromise between high accuracy and computational effort.

Flow field
The temperature gradients and resulting density variation induce the natural convection during the solidification.As shown in Fig. 8, the movement due to the natural convection is rather slow and reduces as the time progresses.Due to that, one can conclude that, in the case of NiTi solidification in an experimental tube, the heat conduction is the sole transport mechanism.Comparing the distributions of the vertical velocity for Case I and Case II reveal that the increased (and constant) thermal conductivity (Case II) increases the heat conduction mechanism and, therefore, reduces the velocity magnitude (i.e.convection) significantly.

Temperature field
Fig. 9 presents the temperature field evolution with the time.The temperature contours are mainly lines, which signify that the heat transfer takes place by the conduction mechanism.The exception is the region in the vicinity of the lower curved wall, where the contours of the temperature are slightly curved.However, even in this region, the heat conduction is the prevailing mechanism and the curved contours result from the curved geometry of the lower wall.Comparing the distributions of the temperature for Case I and Case II (Fig. 9 and Fig. 10) reveals that the increased (and constant) thermal conductivity (Case II) increases the heat transfer mechanism and reduces the temperature value at any given point (0 < z/h < 1).Furthermore, one can conclude that the increased thermal conductivity reduces the overall solidification time.Last but not least, the increased heat transfer mechanism due to the greater value of thermal conductivity (Case II) results in a wider region of the solid phase and greater solid mass fraction (Fig. 12), as well as in a faster solidification process (Fig. 13).

Conclusions
In the present study the time-dependent phase-change of NiTi alloy in an experimental tube was studied by numerical means.The solidification process was modelled by the enthalpy-porosity method accounting for the variable (Case I) and constant thermal conductivity (Case II).
The influence of computational grid refinement on the numerical predictions was studied throughout the examination of grid convergence.By utilising extremely fine meshes the resulting discretisation error levels are below 0.50 %.
The highly accurate numerical results pointed out some important points, such as: • The calculated flow field shows slight movement (convection) in the vertical direction, and reduces as the solidification time progresses.• The temperature is transported mainly by the conductive heat transfer mechanism.
• The solidification process in the experimental tube is unidirectional.
• The solidification front moves upward.
• Increased thermal conductivity increases the heat conduction mechanism, widens the solid phase region, increases the solid mass fraction and fastens the solidification process (reduces the solidification time).
Last but not least, the present numerical approach and results will be compared with the experimental one, which will give us enough confidence to continue with the numerical analysis of the NiTi alloy continuous casting process on the lab-scale.

Fig. 7 .
Fig. 7. Temperature dependent heat transfer coefficient on the cooling wall.

Fig. 8 .
Fig. 8. Variation of the vertical velocity component along the horizontal mid-plane (i.e.z/h=0.05)for Case I (left) and Case II (right).

Fig. 9 .
Fig. 9. Variation of the temperature along the vertical mid-plane (i.e.x/R=0.0)for Case I (left) and Case II (right).

Fig. 11
Fig. 11 presents the variations of NiTi solid mass fraction during the solidification.As the time progresses, the solid region extends upward from the lower curved wall, which is entirely consistent with the time evolution of the temperature field.

Fig. 11 .
Fig. 11.Variation of the solid mass fraction along the vertical mid-plane (i.e.x/R=0.0)for Case I (left) and Case II (right).

Fig. 13 .
Fig. 13.Time variation of the solid mass fraction.
is a quantification of the relative difference between the numerical predictions and the extrapolated value obtained with Richardson's extrapolation technique.