ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
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
Yu-Guang SuFalin ChenChung-Yung WuMin-Hsing Chang
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2017 Volume 57 Issue 1 Pages 100-106

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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 106 K/s).1) Ribbons produced by this method possess excellent mechanical strength2) 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) method5) 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 scheme11) 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 Hwang12) 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 Reddy14) 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 Reddy14) employed the ejection pressure on the boundary to replace the assumption of uniform velocity. Besides, Huang and Fiedler15) 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 106 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 Equations

The 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) method11) is employed to characterize the behaviors of the multiphase flow. Equations used in the model are written as follows:

(1) Continuity equation:   

ρ t +( ρu ) =0, (1)
where t is the time, ρ the density, and u the flow velocity.

(2) Momentum equation:   

ρ Du Dt =-p+μ 2 u+ρg+F, (2)
where D Dt t +u , p is the pressure, μ the viscosity, g the acceleration of gravity, and F the source term of surface tension. The effect of surface tension at the interface of the multiphase flow uses the continuum surface force (CSF) model.17) The source term F is written as   
F= σρκ α a ( ρ a + ρ m ) /2 , (3)
where σ is the surface tension, κ=n represents the curvature at the interface, n= α a | α a | is the normal vector at the interface, and α the volume fraction (0<α<1). The suffixes a and m denote the air and melt, respectively. The unit normal vector at the three-phase point of air, melt, and solid is defined as:   
n= n c cos θ c + t c sin θ c , (4)
where nc and tc are the unit normal and tangential vectors at the solid boundary, and θc is the contact angle.

(3) Energy equations:   

ρ C p DT Dt =k 2 T, (5)
where Cp is the specific heat, k the thermal conductivity, and T the temperature.

(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:   

ρ= ρ a α a + ρ m α m , (6)
  
μ= μ a α a + μ m α m , (7)
  
C p = C pa α a + C pm α m , (8)
  
k= k a α a + k m α m , (9)
in which αa+αm = 1. If αa = 1 and αm = 0, the fluid in this zone is full of air. Conversely, if αa = 0 and αm = 1, the zone is full of melt or ribbon. The case that αa and αm are both between 0 and 1 indicates the region at the interface of the two-phase flow.
Table 1. The thermophysical properties of Fe-based alloy and air.
PropertyFe78Si9B13Air
Density, ρ (kg/m3)TTg, ρm = 8761−1.525(T−273)ρa = 1.225
T<Tg , ρm = 7922
Viscosity, μ (Pa-s)18)TTg, μm = 4.635×10−4×e2607.1/(T−749.2)μa = 1.789×10−5
T<Tg or μm≥1, μm = 1
Surface tension, σ (N/m)TTg, σm = 2.33−4.59×10−4×(T−273)
T<Tg, σm = 2.08
Specific heat, Cp (J/kg-K)12)Cpm = 544Cpa = 1006.43
Conductivity, k (W/m-K)12)TTg , km = 29.8ka = 0.0242
T<Tg , km = 8.99
Glass temperature, Tg (K)18)Tg = 823

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 Fe78Si9B13 ribbon successfully.16) Instead of the impractical assumption of uniform velocity at the inlet of the melt puddle, a pressure difference ΔP is used at the inlet boundary which makes it reasonable to compare the results with the experiments directly. Besides, the jetting temperature Tj 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 105 to 106 W/m2 K.10,14,15) Accordingly, here we assume that the heat transfer coefficient h is 106 W/m2 K which is the maximum value reported in the literature, and could be thought appropriate because the melt is cooled to an amorphous state. The ambient temperature T is 300 K.

Table 2. The boundary conditions in the model.
Inlet 1 2 ρ m | un | +pP=20 kPa
u·t = 0
T = 1553 K
αm = 1
Outletp = patm
n·∇u = 0
n·∇T = 0
Nozzle inner wallu = 0
n·∇T = 0
Nozzle outer wallu = 0
n·∇T = 0
Chill wheelu·n = 0
u·t = U = 25 m/s
kn·∇T = h(TT)
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.
zone 1αm = 1
u = 0
T = 1553 K
zone 2αa = 1
u = 0
T = 300 K

2.3. Numerical Method

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) algorithm19) 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 scheme20) 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 experiments16) as shown in Fig. 2(b) at different wheel speeds. The deviation is about 4 μm 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 experiments16) 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 Tg, 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.

3.1. Puddle Shape

The wetting conditions on the puddle/nozzle and puddle/wheel interfaces can be denoted respectively by the contact angles θn and θw, where the subscripts n and w indicate the nozzle and wheel, respectively. The contact angle in the range of 0°–90° indicates the wetting contact condition, and in the range of 90°–180° implies the non-wetting contact condition. The static contact angle of 110° is assigned as the reference angle of θn on the stationary nozzle surface.10) The contact line between the puddle and wheel is variational and the contact angle is usually a function of the speed of contact line. Here we first choose a typical contact angle θw of 110° to demonstrate the evolution of melt puddle. Figures 3(a) to 3(f) shows the successive development of the puddle shape for the typical case with the reference contact angles of θn and θw. The melt is indicated by the grey area and the black area is the region filled with ambient air. The evolution of puddle shapes are qualitatively similar to the high-speed images in the experiment.16) First, the melt ejects straightly from the nozzle slit and deforms in the middle part due to the interaction of surface tension and driving force as shown in Fig. 3(a). At the moment about 0.5 ms in Fig. 3(b), the melt touches the chill-wheel surface and is dragged toward the right downstream region by the rotating chill wheel. The melt puddle gradually forms at the time of 1.0 ms as displayed in Fig. 3(c). As the casting process proceeds, a crescent shape appears on the upstream boundary of the puddle and a meniscus also forms in the downstream region as illustrated in Figs. 3(d) to 3(f). The separation point of the puddle from the nozzle surface initially moves away from the nozzle slit with time. After 2.0 ms, the puddle shape gradually approaches a steady state and remains stable as shown in Figs. 3(e) to 3(f). The puddle shapes are almost the same for the cases of t = 4.0 ms and 5.0 ms. The evolution process of puddle shape was also examined for the other cases with different wetting conditions. Results show that the time of 4.0 ms is sufficient for the puddle to reach a steady state and exhibit a stationary shape. Thus, the steady puddle shapes and conditions in the following discussion are all determined at the time of 4.0 ms.

Fig. 3.

The evolution of puddle shape with time for the typical case with θn = θw =110°.

The puddle shapes at different wetting conditions of θn are shown in Figs. 4(a) to 4(e), in which the contact angle of θw was fixed at the reference condition of 110°. The results reveal that when the puddle and nozzle are in the wetting contact condition with θn<90°, the melt moistens the nozzle surface easily and there is no meniscus shape appearing at the upstream region. The distance between the separation point of puddle from nozzle surface in the downstream region and the nozzle slot decreases gradually with increasing θn. That is, a decrease in the wettability on the nozzle surface tends to shorten the puddle length. When the puddle/nozzle interface is in the non-wetting contact condition with θn>90°, the puddle shape resembles the experimental observations, in which the meniscus appears on the upstream boundary with short distance between the separation point and nozzle slot. In addition, the effect of wheel surface wettability on the puddle shape also has been explored at different contact angles of θw, including θw = 30°, 70°, 90°, 110°, and 150° at the assigned reference condition of θn = 110°. Unlike the effect of θn, however, the resultant puddle shapes are almost totally the same as the shape shown in Fig. 4(d). This result indicates the wettability on the wheel surface is a minor role in the factors affecting the puddle shape. The reason is that the capillary force on the wheel surface is relatively smaller than the inertial and viscous forces, which had been discussed in the previous work by Bussmann et al.9)

Fig. 4.

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

3.2. Ribbon Thickness

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 θn is illustrated in Fig. 5. The results reveal that a lower contact angle θn tends to increase the melt flow rate eventually, while it takes more time for the puddle to reach a steady state. For example, as shown in the cases of θn=30°, it requires approximately 4 ms for the melt flow to be steady, which also indicates the uniformity of ribbon thickness may be reduced during the casting process. On the contrary, for the typical non-wetting case of θn=150°, it takes 2 ms only for the flow rate reaching steady, implying that one can obtain a more uniform ribbon thickness. The effect of wetting condition on the resultant ribbon thickness is shown in Fig. 6 for several assigned values of θn and θw. The wetting contact condition on the puddle/nozzle interface (i.e. θn < 90°) tends to increase the puddle length and produce a thicker alloy ribbon. However, the non-wetting contact condition (i.e. θn > 90°) exhibits shorter puddle length and result in a thinner ribbon. That is, the ribbon thickness decreases gradually with increasing θn. It is also found that the wetting condition on the puddle/wheel interface indicated by θw has no apparent effect on the ribbon thickness. Accordingly, we may conclude that the state of the puddle and ribbon thickness are primarily dominated by the contact angle θn rather than θw. The contact angle θn must be non-wetting to ensure that the puddle can reach steady state rapidly.

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 θn and θw.

3.3. Formation of Air Pockets

Although the contact angle θw 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 Figs. 7(a) and 7(b) for two typical wetting conditions of θw. As shown in Fig. 7(a), when the puddle and chill wheel are in the wetting contact condition with θw<90°, the air is relatively difficult to be entrained into the puddle/wheel interface. In opposite, when the puddle and wheel are in the non-wetting contact condition with θw>90°, the air is easier to be captured at the puddle/wheel interface and form the air pockets on the ribbon surface as shown in Fig. 7(b).

Fig. 7.

The volume fraction of upstream puddle underneath the nozzle at two typical wetting conditions on the wheel surface: (a) θw = 70° and (b) θw = 110°. (Online version in color.)

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 θw and θn. It is reasonable that the wettability of puddle on the nozzle surface has no significant influence on the air-pocket frequency, since the entrainment of air occurs on the puddle/wheel interface. The effect of θw on the air-pocket frequency is quite obvious as displayed that the air-pocket frequency increases significantly with increasing θw. That is, a lower contact angle of θw benefits the reduction of air-pocket frequency and the enhancement of heat transfer rate by reducing the thermal contact resistance at the puddle/wheel interface. Accordingly, it is potentially an efficient way to reduce the formation of air-pockets on the ribbon surface by enhancing the wettability of melt puddle on the chill wheel surface.

Fig. 8.

The variations of the air-pocket frequency with the wetting conditions of θn and θw.

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 θn 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 θw on the puddle shape and ribbon thickness is insignificant; however, the air-pocket frequency on the wheel-side ribbon surface depends heavily on θw. In summary, a non-wetting condition on the nozzle surface and a wetting condition on the chill wheel surface are preferable in order to obtain a high quality alloy ribbon with amorphous structure, less air pockets, and uniform thickness. The present model also could be extended further in future to investigate the effects of other critical parameters such as the gap height between nozzle and wheel, width of nozzle slot, and injection temperature of the melt.

Acknowledgements

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

References
 
© 2017 by The Iron and Steel Institute of Japan
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