Regular Article

Simulation for the Effect of Wetting Conditions of Melt Puddle on the Fe–Si–B Ribbon Alloy in the Planar-Flow Melt-Spinning Process

2017 Volume 57 Issue 1 Pages 100-106

Details

Abstract

The wetting effect of melt puddle between nozzle and chilling wheel in the planar-flow melt-spinning process was simulated numerically. A two-dimensional model was developed for the puddle in which the inertial force, viscosity, surface tension, wettability, and heat transfer with phase transformation were incorporated. The wetting conditions include the static contact angle between the puddle and nozzle and the dynamic contact angle between the puddle and chill wheel, and their influence on the puddle shape, ribbon thickness and air-pocket frequency were evaluated. Results show that the puddle shape was affected significantly by the wetting condition on the nozzle surface rather than that on the wheel surface. The contact condition between the puddle and nozzle must be non-wetting in order to reach a steady puddle shape rapidly. On the other hand, a wetting contact condition is preferable between the puddle and wheel surface to reduce the amount of air entrainment and lower the air-pocket frequency on the ribbon surface.

1. Introduction

Amorphous ribbons can be obtained by a rapid solidification process (RSP) in which the melt is solidified quickly to a solid state at a very high cooling rate (usually 10^{6} K/s).^{1)} Ribbons produced by this method possess excellent mechanical strength^{2)} and favorable electromagnetic properties,^{3)} which have been widely used in transformers as iron cores to promote their conversion performance.^{4)} Among the various RSP methods, the planar-flow melt-spinning (PFMS) method^{5)} is the most popular technique both in the academic sector and the industrial community due to its mass production capability to produce wide, thin, and uniform amorphous ribbons continuously. **Figure 1**(a) illustrates the puddle region in the typical PFMS process. The melt in the crucible is exerted by an applied gas pressure to flow through a rectangular nozzle onto the rotating chill wheel, forming a puddle between the nozzle and chill wheel which is constrained by the surface tension of the melt. The rotating chill wheel cools the melt and pulls out the solidified melt continuously from the beneath portion of the puddle to form the metal ribbon at the same flow rate as that of the melt ejecting onto the wheel. Previous studies found that PFMS could manufacture amorphous ribbons successfully, but patterns usually appeared on the ribbon surfaces. These patterns depend heavily on the flow stability of the puddle between the nozzle and wheel.^{6)} An unsteady casting process was found to reduce the heat transfer rate and result in the occurrence of crystalline phase.^{7)} It is quite difficult to control the casting process and make it steady because the puddle stability was influenced by the complicated process parameters and the material properties. Experimental studies usually spend much time on the determination of successful operating parameters for the ribbon production. The exploring procedures are quite uneconomical in the savings of time and cost. Therefore, numerical simulation based on computational fluid dynamics (CFD) has been widely used to explore the puddle formation, melt flow, and heat transfer behaviors.^{8,9,10)}

Fig. 1.

(a) Schematic of the PFMS apparatus and the CFD computational domain (not to scale). (b) Geometry and dimensions of the simulation domain. (Online version in color.)

The study of Wu *et al.*^{8)} used the SOLA-VOF scheme^{11)} and CFD technique to develop a 2-D model, in which the transient behaviors during the formation of puddle were simulated by combining the energy equation and the temperature-dependent viscosity. Their model successfully simulated the evolution of puddle formation and the effects of casting conditions on the characteristics of fluid flow and the heat transfer in the puddle region were explored. Their results found that a faster melt injection velocity resulted in a smaller negative pressure at the upstream meniscus of the puddle, which may reduce the occurrence of air pockets on ribbon surface. Based on their model, Chen and Hwang^{12)} modified the treatments of the free surface boundary conditions to study this flow. Their modified model took more time for the puddle to reach steady state but the results were more consistent with experimental observations. Bussmann *et al.*^{9)} investigated the velocity and temperature fields within a steady melt puddle by the finite volume method (FVM). Their model also employed a correlation between temperature and viscosity and assumed the flow velocity of melt was equivalent to the wheel speed when the melt was cooled below the solidifying temperature to characterize the ribbon production process. Their results showed that the puddle length was influenced significantly by the wettability on the nozzle surface. Besides, a higher wheel speed or a lower melt flow rate through the nozzle may reduce the puddle length, ribbon thickness, and strength of vortices at both upstream and downstream of the puddle. Liu *et al.*^{10,13)} studied the initial development of melt flow and heat transfer of Fe–Si–B alloy numerically during the PFMS process. The heat transfer in the rotating chill wheel was incorporated into the model and the solidified behaviors were characterized by the variation of melt viscosity. They assumed the melt viscosity was a temperature-dependent exponential function when the melt temperature was above the glass temperature of the alloy and a constant of 1 Pa-s below the glass temperature. The purpose of this assumption was to force the solidified ribbon moving with the same velocity as the tangential velocity of the chill wheel. Their model may simulate the initial evolution of the puddle shape and the corresponding velocity and temperature fields within the puddle. The interfacial heat transfer coefficient between the puddle and rotating wheel was analyzed and the transient behaviors of the puddle were examined with various parameters in the casting process. Sowjanya and Reddy^{14)} proposed a numerical approach to simulate the heat transfer phenomenon during the ribbon formation with more realistic casting conditions. Their model may predict the produced ribbon was either amorphous or crystalline structure. They found that the effect of jetting temperature on the ribbon thickness was negligible at high wheel speeds. Nusselt number was also used to introduce a convective heat transfer coefficient between the wheel surface and surrounding air. The numerical model assisted the prediction for the occurrence of a continuous, broken, or no ribbon formation during the melt spinning process.

In general, the boundary condition at the inlet of puddle was set as a uniform velocity in the simulations, which was quite different from the practical condition that the melt flowrate was controlled by the applied gas pressure in the crucible. Only the work of Sowjanya and Reddy^{14)} employed the ejection pressure on the boundary to replace the assumption of uniform velocity. Besides, Huang and Fiedler^{15)} found that the wetting effects of the melt puddle play an important role in the factors affecting the alloy ribbon quality in their experimental observations. However, it was difficult to observe the wettability of melt puddle on the nozzle and wheel surfaces in the experiments and the relative numerical simulation is still absent now, which motivates the present investigation. In this work, we investigate the importance of wetting conditions of melt puddle in the PFMS process and construct a numerical simulation model in which a more practical boundary condition with applied gas pressure at the inlet of puddle is employed. The effects of wetting conditions of puddle on the nozzle and chill wheel surfaces are evaluated with temperature-dependent thermo-physical properties of the melt. The developed model takes the inertia force, surface tension, wettability, and temperature-dependent viscosity into consideration to simulate the phase transformation and forming of amorphous ribbons. The momentum and energy equations are solved from the upstream meniscus zone to the downstream region where the emergence of ribbon takes place. The effects of wetting and non-wetting contacts on the interfaces between the puddle/nozzle and puddle/wheel on the puddle shape, ribbon thickness and air-pocket frequency on the ribbon surface are analyzed in detail.

2. Numerical Model

2.1. Description of Physical System
The developed physical system is based on the laboratory-scale melt spinning apparatus,^{16)} in which the simulation domain includes the nozzle ejection zone, the gap between the nozzle and wheel, and the downstream region as shown in Fig. 1(a). Because the melt flows through the channel in the nozzle to the nozzle exit, the nozzle ejection zone is considered in the simulation domain. The geometrical dimensions of the domain are illustrated in Fig. 1(b), in which the nozzle-wheel gap *G* is 0.1 mm and the width of nozzle slot *B* is 0.4 mm. The following assumptions and simplifications are made to establish the simulation model:

(a) The ribbon width is much larger than the ribbon thickness. Hence, a 2D model is reasonable and appropriate to simulate the melt puddle.

(b) Because the radius of chill wheel is much larger than the puddle length, the curvature of chill wheel is ignored and the bottom of the simulation domain is assumed to be a flat surface.

(c) A two-phase flow model is employed and the puddle is considered to be a laminar incompressible flow.

(d) The chemical reactions and mass transfer at the interface of the two-phase flow are excluded.

(e) A large temperature gradient exists in the bottom of puddle due to the high cooling rate of chill wheel. Accordingly, the thermophysical properties of melt are assumed to be dependent of temperature. The properties of ambient air are all constants.

(f) No-slip boundary conditions are imposed on the puddle/nozzle and puddle/wheel interfaces.

(g) Due to the high cooling rate (usually 10^{6} K/s) in a very short solidification time for the PFMS process, the release of latent heat during the solidification process is ignored and all the produced ribbons are with amorphous structure.

(h) The boundary between the puddle and nozzle is adiabatic. Radiation heat transfer from the puddle is neglected.

(i) A convection boundary condition is imposed on the wheel surface. The heat transfer between the puddle and wheel is characterized by a constant heat transfer coefficient.

2.2. Governing EquationsThe governing equations for this problem involve the continuity equation, the momentum equation, and the energy equation. Because the two-phase flow includes the melt and air and both are immiscible fluids, the volume of fluid (VOF) method^{11)} is employed to characterize the behaviors of the multiphase flow. Equations used in the model are written as follows:

(1) Continuity equation:

(1) |

(2) Momentum equation:

(2) |

(3) |

(4) |

(3) Energy equations:

(5) |

(4) Models of material parameters:

The thermophysical properties of the melt and air are listed in **Table 1**. For the melt of Fe-based alloy, the density, viscosity, surface tension, and thermal conductivity are all functions of temperature. All the properties of air are constants and obtained from the material database of ANSYS FLUENT. Because the proposed model is a two-phase flow, the thermophysical properties are defined using the volume fraction in the mixture zone for the Eqs. (1), (2), (3), (4), (5) as follows:

(6) |

(7) |

(8) |

(9) |

Table 1. The thermophysical properties of Fe-based alloy and air.

Property | Fe_{78}Si_{9}B_{13} | Air |
---|---|---|

Density, ρ (kg/m^{3}) | T≥T, _{g}ρ = 8761−1.525(_{m}T−273) | ρ = 1.225_{a} |

T<T , _{g}ρ = 7922_{m} | ||

Viscosity, μ (Pa-s)^{18)} | T≥T, _{g}μ = 4.635×10_{m}^{−4}×e^{2607.1/(}^{T}^{−749.2)} | μ = 1.789×10_{a}^{−5} |

T<T or _{g}μ≥1, _{m}μ = 1_{m} | ||

Surface tension, σ (N/m) | T≥T, _{g}σ = 2.33−4.59×10_{m}^{−4}×(T−273) | – |

T<T, _{g}σ_{m} = 2.08 | ||

Specific heat, C (J/kg-K)_{p}^{12)} | C = 544_{pm} | C = 1006.43_{pa} |

Conductivity, k (W/m-K)^{12)} | T≥T , _{g}k = 29.8_{m} | k = 0.0242_{a} |

T<T , _{g}k = 8.99_{m} | ||

Glass temperature, T (K)_{g}^{18)} | T = 823_{g} | – |

The Eqs. (1), (2), (3), (4), (5) are solved together with the boundary conditions as listed in **Table 2**. The assigned operating parameters in Table 2 have been found to be able to fabricate the Fe_{78}Si_{9}B_{13} ribbon successfully.^{16)} Instead of the impractical assumption of uniform velocity at the inlet of the melt puddle, a pressure difference Δ*P* is used at the inlet boundary which makes it reasonable to compare the results with the experiments directly. Besides, the jetting temperature *T _{j}* of the melt is assumed to be greater than the melting point by 100°C. In general, the range of heat transfer coefficient at the wheel surface is about the order of 10

Table 2. The boundary conditions in the model.

Inlet | p=ΔP=20 kPa |

u·t = 0 | |

T = 1553 K | |

α = 1_{m} | |

Outlet | p = p_{atm} |

n·∇u = 0 | |

n·∇T = 0 | |

Nozzle inner wall | u = 0 |

n·∇T = 0 | |

Nozzle outer wall | u = 0 |

n·∇T = 0 | |

Chill wheel | u·n = 0 |

u·t = U = 25 m/s | |

−kn·∇T = h(T−T_{∞}) | |

T_{∞} = 300 K |

The simulation domain is divided into two zones (zone 1 and 2) as shown in Fig. 1(b). The initial conditions are respectively given as listed in **Table 3**. At the beginning, the nozzle ejection zone (zone 1) is filled with the melt and air occupies the gap between the nozzle and wheel and the downstream region (zone 2). After starting the casting process, the development and evolution of melt puddle, air flow, and the ribbon formation are simulated.

Table 3. The initial conditions in the model.

2.3. Numerical Method
zone 1 | α = 1_{m} |

u = 0 | |

T = 1553 K | |

zone 2 | α = 1_{a} |

u = 0 | |

T = 300 K |

The FVM method was used to solve the developed model. To obtain the convergent solution efficiently, the non-uniform quadrilateral cell system was used and the grid independence tests were performed to decide the proper mesh size. **Figure 2**(a) illustrates the variation of ejection velocity of melt at the nozzle exit with time at different ranges of mesh size. All the grid tests demonstrate excellent convergence. The simulations were run till 4 ms before the steady state was reached. Accordingly, the appropriate mesh size is chosen as the case of 0.005–0.010 mm and the total number of nodes is about fifty thousand. The mesh has an equal space in the x-direction, while the smallest grids in the y-direction are located at the bottom of the simulation domain corresponding to the ribbon formation region. The velocity and pressure fields are coupled accurately in solving the continuity and momentum equations by using the pressure-implicit with splitting of operators (PISO) algorithm^{19)} provided by the implicit unsteady solver in the FLUENT package. The choice of time step is quite important for solving transient problems, particularly when an implicit solver is used. According to the flow characteristics and the total mesh number, the time step of 10^{−7} s was employed in the calculations. The QUICK scheme^{20)} was used to calculate the convection term of the governing equations. The mixture phase is formed at the grids near the melt/air interface. A geometric reconstruction scheme was used to determine the phase locations in the VOF model. The solid ribbon is defined as the region where the melt velocity is equal to the wheel speed *U*. Accordingly, the ribbon thickness could be determined and measured at the location of melt outlet on the right boundary of the puddle. In order to verify the accuracy of the simulation results, a comparison was made for the ribbon thicknesses obtained by the present model and the experiments^{16)} as shown in Fig. 2(b) at different wheel speeds. The deviation is about 4 *μ*m in each case primarily due to the uncertain data in the experiments, for example, the heat transfer coefficient *h*. While the simulation results have exactly the same tendency as the experimental work and it is believed that the established numerical model could predict the characteristics of this melt flow successfully.

Fig. 2.

(a) The effect of non-uniform grid size on the ejection velocity of melt. (b) Comparison of ribbon thicknesses obtained by present simulation and experiments^{16)} at different wheel speeds.

3. Results and Discussion

We first investigate the wetting effect on the puddle shape, and then discuss the variation of ribbon thickness and its influence on the formation of air-pockets on the ribbon surface. It is noted that the viscosity of the melt plays an important role to characterize the ribbon formation. The viscosity increases with decreasing the melt temperature and is assumed to be 1 Pa-s below the glass temperature *T _{g}*, which is large enough to make the solidified alloy have the same velocity with the tangential velocity of rotating wheel and imitate the solid ribbon to be pulled out by the chill wheel.

The wetting conditions on the puddle/nozzle and puddle/wheel interfaces can be denoted respectively by the contact angles *θ _{n}* and

Fig. 3.

The evolution of puddle shape with time for the typical case with *θ _{n}* =

The puddle shapes at different wetting conditions of *θ _{n}* are shown in

Fig. 4.

The puddle shapes at different wetting conditions on the nozzle surface.

The uniformity of thickness is an important quality for the ribbon alloy produced by the PFMS process. It is generally controlled by the melt flow rate through the nozzle. Here we found that the wetting condition on the puddle/nozzle interface may affect the melt flow rate and thus influence the produced ribbon thickness. The variation of melt flow rate with time at several assigned values of *θ _{n}* is illustrated in

Fig. 5.

The variations of mass flow rate with time at the nozzle inlet for several assigned wetting conditions on the nozzle surface. (Online version in color.)

Fig. 6.

The variations of ribbon thickness with the wetting conditions of *θ _{n}* and

Although the contact angle *θ _{w}* is a minor role affecting the evolution of puddle shape and ribbon thickness, it is found that it is an important factor for the formation of air pockets on the air-side ribbon surface. The volume fraction of the upstream puddle underneath the nozzle is shown in

Fig. 7.

The volume fraction of upstream puddle underneath the nozzle at two typical wetting conditions on the wheel surface: (a) *θ _{w}* = 70° and (b)

The appearance of air pockets should be avoided to the utmost during the casting process because the air pockets increase the thermal resistance and lower the heat transfer rate between the ribbon and chill wheel, which may induce the occurrence of crystalline structure within the solidified ribbon. Hence, the frequency of air pockets should be as low as possible in order to produce a high quality alloy ribbon. The frequency of air pockets could be obtained from the wheel speed and the wavelength between two air pockets and the results are shown in **Fig. 8** for several different contact angles of *θ _{w}* and

Fig. 8.

The variations of the air-pocket frequency with the wetting conditions of *θ _{n}* and

4. Conclusions

The present time-dependent numerical model provides a simple and efficient technique to simulate the characteristics of melt puddle in the PFMS process. The effects of wetting conditions of melt puddle on the nozzle and chill wheel surfaces are explored in detail. Particularly, the melt with temperature-dependent thermophysical properties is assumed to be driven by applying a pressure difference across the crucible and nozzle which is more realistic to the practical condition in the casting process. The contact angle *θ _{n}* on the nozzle surface strongly affects the puddle shape and the time required for the puddle to reach a steady state. It is also a dominant factor that may affect the ribbon thickness significantly. The influence of contact angle

Acknowledgements

The support for this work from China Steel Corporation through the grant number 03T1D-RE009 is gratefully acknowledged.

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