Studies on Transport Phenomena during Solidification of a Binary

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Trans Indian Inst Met (December 2012) 65(6):801–807 DOI 10.1007/s12666-012-0186-x

TECHNICAL PAPER

TP 2631

Studies on Transport Phenomena during Solidification of a Binary Solution (NH4Cl + H2O) on an Inclined Cooling Plate D. Mohanty • A. K. Nayak • N. Barman

Received: 30 June 2012 / Accepted: 11 September 2012 / Published online: 9 October 2012 Ó Indian Institute of Metals 2012

Abstract In the present work, a numerical study is considered to predict the transport phenomena during solidification of a transparent binary solution (NH4Cl ? H2O) on an inclined cooling plate. During solidification, a shear layer is developed near the mush. The present work involves the prediction of the transport phenomena during solidification of the binary solution under this shear flow. In the mathematical model of this solidification process, the free surface of the solution is traced by the volume-of-fluid method and the solidification phenomenon is represented by a set of volume averaged single phase mass, momentum, energy and species conservation equations. The governing equations are solved based on the pressure-based finite volume method according to the SIMPLER algorithm using TDMA solver along with the enthalpy update scheme. Finally, the simulation predicts the temperature, velocity, solid fraction and the species distributions in the computational domain. Keywords Solidification  Inclined cooling plate  Shear flow  Macrosegregation List of symbols C Concentration of solute (wt% of Si) f Mass fraction p Pressure (Pa) T Tempe´rature (°C) t Time (s) D. Mohanty  N. Barman (&) Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India e-mail: [email protected] A. K. Nayak Department of Mechanical Engineering, Trident Academy of Technology, Bhubaneswar 751024, India

 U(u,v)

Velocity vector (m/s)

Greek Symbols l Viscosity (Pa-s) Subscripts l Liquid n Iteration number p pth cell s Solid

1 Introduction Nowadays, semisolid metal forming (SSM) is a beneficial manufacturing technique in which the metallic alloys are cast in presence of induced fluid flow/stirring [1–3]. During casting, the strong induced fluid flow arrests the growth of dendrites by detaching them from the solid–liquid interface and carries them into the mould to form a semisolid slurry. The suspended dendrites in the slurry become rosette or globular particles after coarsening [3] in the slurry. This slurry offers less resistance to flow even at a high solid fraction. Based on the behavior, many casting methods are developed such as rheocasting and thixocasting [3]. These methods have several advantages over the other conventional casting techniques, such as redistribution of the macrosegregation, reduction of porosity, low forming efforts, and increased strength and/or elongation. Because of the above advantages, the semisolid metal forming is ideally suited for producing a wide range of components for automotive, aerospace, defence and structural applications. However, the essence of the semisolid metal forming lays on the formation mechanism of the slurry during solidification under shear flow, which indeed demands a detail study. Hence, in the present work, a study is considered on the

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transport phenomena during solidification under shear flow. Here, the solidification under shear flow is achieved by considering the melt flow on an inclined cooling plate. As the liquid metals are opaque in nature, the direct observation of the transport phenomena during solidification is not feasible. In the present work, hence, a metal analogous transparent binary solution (NH4Cl ? H2O) is considered [4]. For fundamental understanding of the solidification process in presence of shear flow, many research works have been carried out. Spencer et al. [1], Joly and Mehrabian [2] have studied the solidification process under shear flow using mechanical stirrer. The stirred slurry is either used for preparing feedstock in thixocasting or directly formed into parts in rheocasting. They found that the stircast structures are typically duplex in nature, consisting of globular solid particles those are formed under stirring and embedded in a finer dendritic matrix formed without shearing action. Pasternak et al. [5] and Ji et al. [6, 7] also studied the solidification process under shear flow using either single or twin helicoidal screws. Gabathuler et al. [8] and Blais et al. [9] studied the solidification process in presence of electromagnetic stirring. Kiuchi and Sugiyama [10] and Wang et al. [11] developed the shear cooling roll (SCR) type mechanical stirring process. In this method, slurry forms in the gap between a rotating roll and a stationary cooling ‘shoe’. As the metal solidifies in the gap, a continuous slurry stream is produced by the shearing and cooling action of the rotating roll. Buncka et al. [12] produced the aluminium slurry on an inclined cooling channel. It is a simple semisolid slurry forming technique where the liquid metal with a slight superheating is poured over the inclined cooling channel. During processing, the liquid melt falls quickly into the semisolid range by removing heat and consequently nucleates. The newly developed grains get sheared under the shear flow and carried out with the flow. Indeed, the transport phenomena during solidification under shear flow are complex in nature, and demand a detail study for systematic and effective design of the semisolid facilities. Since the experimentation is very expensive, the present work considers a numerical study to investigate the transport phenomena of a binary solution (NH4Cl ? H2O) during solidification on an inclined cooling plate. In context to the modeling of solidification process, Chakraborty and Dutta [13] formulated an enthalpy based model for solidification of binary alloys. Kumar and Dutta [14] and Barman et al. [15] proposed models for the solidification of binary alloys under shear flow.

Trans Indian Inst Met (December 2012) 65(6):801–807

binary solution (NH4Cl ? H2O) under shear flow. The flow is considered on an inclined cooling plate. The twodimensional computational domain of the system is shown in Fig. 1. During processing, the aqueous ammonium chloride solution (8 wt% of NH4Cl) is poured on the inclined cooling plate. It is mentioned here that an adiabatic length is considered for the flow stabilization and then, the solution is cooled at a constant temperature of -30 °C. As the liquid moves down over the cooling plate due to the gravitational force, a shear layer is developed in the flow domain. Concurrently, as heat is removed from the cooling plate, the solidification starts at the surface of the cooling plate. Since the solidification of a binary solution occurs over a range of temperatures; hence, a mushy zone containing dendrites appears during the solidification process. It also involves the rejection of solute. In the present work, therefore, the distributions of velocity, temperature and solute (NH4Cl) during processing are studied under shear flow. The fragmentation of the dendrites during shearing is neglected in this work. The thermo-physical properties of the NH4Cl ? H2O solution and the system data used in simulation are given in Table 1.

3 Mathematical Modelling The flow of the liquid solution over an inclined cooling plate requires the tracking of free surface between the liquid solution and the surrounding air. The present work adopts the commonly used volume-of-fluid (VOF) method [17] to describe the free surface of the solution. The fluid flow is represented by the mass and momentum equations where the single-domain-volume-average properties are considered in the entire domain [13]. The solidification process is accounted by coupling the energy and species equations [13] with the flow equations. For simplicity, the solidification shrinkage is neglected in the present work and also the flow is considered incompressible. The corresponding governing equations are given as. Conservation of fluid fraction function: oa  Þ¼0 þ r:ðUa ot

where a is a fluid fraction function. The a is equal to 1 in fluid region and equal to 0 in air.

2 Description of the Physical Problem The present work considers a numerical study on the solidification behavior of a metal analogous transparent

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ð1Þ

Fig. 1 A schematic of the physical system

Trans Indian Inst Met (December 2012) 65(6):801–807

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Table 1 The thermo-physical properties of NH4Cl ? H2O and system data [16] Value

Specific heat (Cp)

3,000 J/kg K

Thermal conductivity (k)

0.4 W/m K

Density (q)

1,000 kg/m3

Liquid viscosity (l)

1.0 9 10-3 kg/m s

Species diffusion coefficient (Dl)

4.8 9 10-9 m2/s

Latent heat of fusion (La)

3.0 9 105 J/kg

Thermal expansion coefficient (bT)

4.0 9 10-5/K

Solutal expansion coefficient (bC)

0.025 -15.4 °C

Eutectic concentration (Ce) (wt% NH4Cl)

19.7

Melting point of pure H2O (Tm) Equilibrium partition coefficient (kp)

0 °C 0.3

ð7Þ

The source terms are given as 1. Sb represents the thermal and solutal buoyancy sources and given as

Thermo-physical properties

Eutectic temperature (Te)

 u ¼ a fl uliquid þ fs usolid þ ð1  aÞuair

Sb ¼ qref g ½bT ðT  Tref Þ þ bs ðC  Cref Þ

System data Solution composition at inlet (Cin) (wt% NH4Cl)

8

Solution temperature at inlet (Tin)

25 °C

ð8Þ

where bT and bs are thermal and solutal expansion coefficients, respectively. 2. The source terms Su and Sv in the momentum equations accounts for mushy-region morphology and given ð1fl Þ2 [19] as Su ¼ A ðf 3 þbÞ u and l ð1  fl Þ2 Sv ¼ A 3 v ð9Þ ðfl þ bÞ where A is a constant (*1.6 9 106) and b is a computational constant introduced to avoid division by zero (*1 9 10-3). 3. Sh is the source term in the energy equation and given as   o ðq fl DHÞ þ r:ðq uDHÞ Sh ¼  ð10Þ ot

Top surface temperature(Tin)

25 °C

Bottom surface temperature (Tcold)

-30 °C

Cavity height (H)

40 mm

Cavity length (L) Length of adiabatic section (Lad)

300 mm 50 mm

Sc ¼ 

Height of the solution at inlet (Hsol)

16 mm

where non-equilibrium solidification is accounted using the Scheil equation for solute redistribution.

Conservation of mass: o  ¼0 ðqÞ þ r:ðq UÞ ot

ð2Þ

o op ðq uÞ þ r:ðq U uÞ ¼ r:ðl ruÞ  þ Su þ Sb sin h ot ox ð3Þ ð4Þ

Energy conservation:   o k  ðq TÞ þ r:ðq UTÞ ¼ r: rT þ Sh ot cp

ð5Þ

ð11Þ

Initially, the domain contains air at Tin temperature. Then, the aqueous solution with 8 wt% of NH4Cl composition and Tin temperature is poured on the inclined surface. The boundary conditions are as follows: Left surface:For y  Hsol , u ¼ Up , v ¼ 0,a ¼ 1, T ¼ Tin and Cl ¼ Cin For y [ Hsol , ou ox ¼ 0, v ¼ 0, a ¼ 0, T ¼ Tin and Cl ¼ 0 oCl ov oa oT Right surface:ou ox ¼ 0, ox ¼ 0, ox ¼ 0, ox ¼ 0 and ox ¼ 0 oCl Bottom surface:u ¼ 0, v ¼ 0, oa oy ¼ 0 and oy ¼ 0 For x  Lad , oT ¼0 oy For x [ Lad ,

Species conservation: o  l Þ ¼ r  ðDþ rCl Þ þ Sc ðqCl Þ þ r  ðqUC ot

o o ðq fs Cl Þ  kp Cl ðq fs Þ ot ot

3.1 Initial and Boundary Conditions

Conservation of momentum:

o  ¼ r:ðl rvÞ  op þ Sv þ Sb cos h ðq vÞ þ r:ðq UvÞ ot oy

4. Sc is the source term in the species equation and given as

ð6Þ

T ¼ Tcold Top surface:u ¼ v ¼ 0, a ¼ 0, T ¼ Tin and

þ

where D ¼ q fl Dl . The above continuum equations are valid over the entire domain, the specific nature of the four regions (air, liquid, solid and mush) are accounted through the volume average properties and the source terms. Each property (u) in the entire computational domain is calculated as

oCl oy

¼0

4 Numerical Modelling The governing equations (1–11) are coupled with the boundary conditions and solved with a pressure-based finite

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volume method according to the SIMPLER algorithm using TDMA solver Patankar [18]. To calculate the latent heat content (DH) in the energy equation, the well known enthalpy update scheme, proposed by Brent et al. [19], is adopted. Equation (12) shows the equation for enthalpy update scheme. In the method, the latent heat content of each control volume is updated according to the temperature values predicted from the energy equation in previous iteration.  ap ½DHp nþ1 ¼ ½DHp n þ 0  k  ½hp n  cp  f 1 ½DHp n ap ð12Þ where ap, a0p are the coefficients of the discretized energy equation, k is a relaxation factor and f 1 is the inverse of the latent heat function depending on systems (pure or binary system). For binary system, f 1 is given (Chakraborty and Dutta [13]) as  ðkp 1Þ DH 1 f ðDH Þ ¼ Tm  ðTm  TL Þ ð13Þ La The liquidus temperature (TL ) is updated based on the linearised phase diagram of the solution. Fig. 2 shows a typical binary phase diagram of an alloy system. Subsequently, the liquid fraction and solid fraction in the pth cell is calculated as fl ¼ DH La where 0\DH\La

ð14aÞ

ð14bÞ fs ¼ 1  fl During simulation, the convergence is declared when j(/-/old)//maxj \ 10-5 where / stands for the solved variables at a grid point at the current iteration level, /old represents the corresponding value at the previous iteration level and /max is the maximum value of the variable at the current

iteration level in the entire domain. Also, the work conducts a comprehensive grid-independence study. It is found that a 122 9 82 non-uniform grid is suitable for the present simulation and a time step of 0.1 s offers a better convergence.

5 Results and Discussion In the present work, the transport phenomena during solidification of an aqueous ammonium chloride (with 8 wt%) solution over an inclined cooling plate are studied numerically. At first, the numerical code is validated against the standard Blasius solution. In Fig. 3, the velocity profiles of the present prediction and the standard Blasius solution are compared. It is found that the present prediction agrees well with the Blasius solution. Hence, this code is used for prediction of the flow field, temperature and species distributions during solidification of the aqueous solution. The cooling plate is inclined at an angle of 30° with the horizontal plane and the liquid solution of 16 mm thickness is poured on the plate. Figure 4a shows the air/ liquid interface and regions of liquid solution, mush and the solidified solution at steady state condition. As the aqueous solution flows over the cooling plate, the temperature of the solution decreases. The corresponding temperature distribution near to the cooling plate is shown in Fig. 4b. When temperature reaches to the liquidus temperature, the solidification of the solution starts. However, there are three phases of the aqueous solution at steady state condition: solid, mush and liquid as seen in Fig. 4a. The solid phase formed near to the cooling plate and the mush occurs over the solid phase based on the temperature distribution in the domain (see Fig. 4b). It is also found that the width of the mush region increases along the plate length. Hence, the liquid column is displaced upward in the Y-direction. Accordingly the air/ 1.4 1.2 1

u/U

0.8

Present Work Blasius Solution

0.6 0.4 0.2 0

Fig. 2 A typical phase diagram of a binary system

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0

1

2

3

4

5 eta

Fig. 3 Comparison of velocity profiles

6

7

8

9

10

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liquid interface is displaced up along the length of the plate as seen in Fig. 4a. The fluid is flowing through the mush and over it. A zoomed view of the velocity distribution near to the cooling plate is shown in Fig. 5a. It is found that the velocity in the solid region is vanished. The velocity has a small value in the mush and reaches a uniform flow in the bulk aqueous solution. As width of the mush increases along the length of the cooling plate, the fluid shifted up and the magnitude of the uniform flow, over the mush, also increases along the length of the plate. Figure 5b shows the corresponding velocity distribution at different plate lengths. It is seen that the uniform flow, over the mush, has higher magnitude at the trailing edge of the plate. The present work also predicts the solute (NH4Cl) distribution at different plate lengths of the cooling plate as shown in the Fig. 6. As solidification of the binary solution involves the rejection of solute (NH4Cl) because of its solubility difference

between the solid and liquid phases, the solute concentration increases in the mush as found in the Fig. 6. In the vicinity of the solid interface (fs = 0.99), the solute is less and it increases along Y-direction in the mush width. It is found that a small amount of solute is diffused in the bulk liquid close to the mush. During solidification under shear flow, some of the solute from upstream of the mush is transported along the plate length with the flow. Thus, the solute in the mush increases along the plate length. In addition, the total rejection of the solute increases along the plate length due to the increase of the mush width. In the Fig. 6, a nose like distribution is found in the vicinity of the solid interface near to the trailing edge of the mush. Near to the end of the plate, the mush width is more. Thus, the rejected solute gets enrich near to the solid interface where convective effect is nil. In the present work, the effect of plate inclination angle (h) on the transport phenomena is also studied where the

Fig. 4 a Various region in the computational domain, b temperature distribution near to the cooling plate

Fig. 5 a A zoomed veiw of the velocity distribution near the cooling plate b Velocity distribution at different lengths (x = 0.025, 0.1, 0.2 and 0.3 m) of the cooling plate

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Fig. 6 Solute (NH4Cl) distribution at different lengths (x = 0.025, 0.1, 0.2 and 0.3 m) of the cooling plate

aqueous solution with inlet velocity of 0.1 m/s is poured on the cooling plate at different inclination angles (h = 10°, 30° and 50°). Figure 7a shows the velocity distribution at x = 0.2 m for different inclination angles. It is found that velocity of the fluid over the mush increases when the inclination angle increases. At higher velocity, the heat transfer from the bulk liquid increases. Hence, the width of the mush increases at high value of the inclination angle. The fact is observed in the Fig. 7a also. With increasing velocity, the solute redistribution in the mush occurs well. This solute redistribution reduces the solute rejection in the mush during solidification. According a low solute distribution is observed in the mush at higher inclination angel of the plate as shown in the Fig. 7b.

6 Conclusions The present work predicts the transport phenomena during solidification of a binary solution (NH4Cl ? H2O) on an inclined cooling plate. The work considers the VOF method for tracing the free surface of the solution and the volume averaged single phase mass, momentum, energy and species conservation equations represent the solidification process along with the enthalpy update scheme. Finally, the distributions of the temperature, velocity and the species are predicted in the computational domain. It is found that the mush width increases along the plate length and also increases with the increasing angle of inclination of the plate. The rejected solute in the mush increases along the Y-direction and also along the plate length. It also found that the solute rejection reduces with increasing plate inclination angle.

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Fig. 7 a Velocity and b solute (NH4Cl) distributions for different plate inclination angles (h = 10°, 30° and 50°) at plate length of x = 0.2 m

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