Solute convection effects on a bubble entrapped as a pore during unidirectional upward solidification

https://doi.org/10.1016/j.ijheatmasstransfer.2019.01.060Get rights and content

Highlights

  • Strong horizontal flow deflects the pore shape toward the downstream direction.

  • Upward flow near the bubble cap is dominated by pressure difference across the cap.

  • Vertical velocity component at the bubble cap is responsible for time for bubble entrapment.

  • Time for bubble entrapment is scaled to reflect dominant effects of solute transport.

  • There exist local maximum concentration regions in triangle shapes near the triple- phase line.

Abstract

A transport model is proposed to predict entrapment of a bubble during unidirectional upward solidification. Pore formation and its shape in solid influence not only microstructure of materials, but also contemporary issues of various sciences of biology, engineering, foods, geophysics and climate change, etc. In this study, COMSOL computer code is used to solve conservation equations of mass, momentum, energy and solute concentration in liquid, gas and solid phases satisfied by their interfacial conditions. It shows that strong convection deflects the pore growth toward the downstream direction. Time for bubble entrapment, however, depends on the ratio between square of concentration boundary layer thickness and solute diffusivity. Concentration boundary layer thickness is proportional to solute diffusivity divided by difference in bubble growth rate and vertical component of fluid velocity at the bubble cap. The latter is almost independent of incoming horizontal flow. Solute diffusion is therefore the mechanism responsible for bubble entrapment, where the bubble growth rate and vertical velocity component of fluid at the bubble cap are of the same order of magnitude. A triangle region with local high concentration is also found to occur in solid region near the triple-phase line. Predicted contact angle agrees with that obtained from the Abel’s equation of the first kind during solidification. This work provides a general model for a fundamental and systematical understanding of mechanisms of a bubble entrapped as a pore in solid.

Introduction

Pore formation and its shape in solid influence not only microstructure of materials, but also contemporary issues of sciences of biology, engineering, foods, geophysics and climate change, etc. Pore formation has been studied in different areas including welding [1], [2], [3], [4], [5], casting [6], [7], [8], phase change materials [9], [10], lotus-type porous materials [11], [12], [13], [14], selective laser melting [15], and so on. Actual pore morphology strongly controls properties such as fatigue strength [16], Young’s moduli, tensile strength, compressive yield strength, energy absorption, thermal and electrical conductivities of the alloy castings, etc. [13]. Influence of morphology of the pore enhances as product decreases its size.

The predictions of the pore shape should account for different length scales between the pore size and wavelength of deformation of the solidification front, resulting from different transport processes. In the cases of constrained growth of the pore in the mushy zone [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], or pore radius around the same magnitude as wavelength of deformation of the solidification front, the pore shape and development during casting of aluminum alloys from a number of models, ranging from analytic solutions to highly complex simulations of evolving porosity and microstructure with stochastic nucleation and growth were reviewed by Lee et al. [22]. Each of models has limitations such as: (1) analytic solutions are applicable only to directional solidification [23]; (2) criteria functions cannot be extrapolated to new alloys or processes [24]; (3) computational models using Darcy's law only show a good correlation to experiment for percentage porosity, not pore morphology [25]; (4) hydrogen diffusion models only predict average values [26]; and (5) continuum-stochastic models [27], which predicted the distribution of porosity and maximum pore size are computationally very expensive.

As wavelength increases or deformation of the solidification front reduces, pore formation is resulting from engulfment of a bubble by a solidification front rather than constrained growth of a bubble in the mushy zone [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. Entrapment of a bubble during unidirectional solidification in the absence of the mushy zone is a process often encountered in manufacturing lotus-type porous materials [38], [11], [39], crystal growth [40], isolated pores in the welding or joining of pure metals or alloys containing minor impurities [1], [4], freezing of sea ice [41], etc. Hadji [30] thus proposed a steady-state model including the lubrication equation of a thin liquid layer and heat conduction equations in the liquid and particle to examine bubble engulfment or relative magnitude between bubble size and wavelength of the deformed solidification front. By accounting for a spherical bubble in a fixed shape (and a SiC solid particle), interface morphology depended on solidification rate, thermal gradient, gap thickness, and thermal conductivities of the bubble and liquid. In the case of a large gap thickness of 10−8 m, for example, the solidification front was convex due to vanished thermal conductivity of the bubble. An increase in solidification rate greater than 40 μm/s resulted in depression on the elevated part of the solidification front. The trough widened as solidification rate increased, indicating that an increase in solidification rate strongly deformed the solidification front and decreased bubble radius. The analysis also showed that bubble engulfment or rejection depends primarily on competition between hydrodynamic and thermal forces. Park et al. [32] also analytically studied capture of a spherical bubble (or an insoluble particle, a solid particle) submerged in a liquid and approached by an advancing solidification front. Using asymptotic analysis and lubrication approximation, the behavior of the bubble in a fixed spherical shape and deformation of the solidification front were determined by an interplay among gap thickness, thermal conductivity differences between the bubble and liquid, solid–liquid interfacial energy, density change due to liquid–solid phase transition, and Marangoni effect at the liquid–gas interface. A small gas-to-liquid thermal conductivity ratio resulted in repulsion of the bubble away from the solidification front. Deformation of the solidification front depends on the capillary length-to-bubble radius ratio or relative magnitudes between bubble size and wavelength of deformation of the solidification front, and gas-to-liquid thermal conductivity ratio. Capillary length represents the ratio between the variation in temperature due to surface tension and imposed temperature across the bubble radius. There exist convex and concave-convex solidification fronts for gas-to-liquid thermal conductivity ratio less than unity. A small capillary length-to-bubble radius ratio gives rise to a convex solidification front. The concave-convex solidification front is confined to a narrow range close to unity value of gas-to-liquid thermal conductivity ratio. As the capillary length-to-bubble radius ratio increases, this region becomes wider. The effects of density change, convection due to density change, Gibbs-Thomson effect, thickness of the liquid layer between the bubble and solid, etc. on surface deformation and critical speed responsible for bubble engulfment were also presented. Kao et al. [35] examined the interaction of a spherical bubble in a fixed shape (and a solid or liquid particle) with a propagating solidification front in a binary alloy. Numerical boundary integral and continuation methods were used to determine the critical speed for particle capture, as a function of the system parameters. In view of constitutional undercooling, the presence of solute decreases particle speeds by an order of magnitude below those for a single-component system. Thermal and solutal Marangoni effects played a large role.

A further increase in wavelength of the solidification front, fluid flow plays a significant important role in porosity [1]. Kou [1] and Kou and Wang [42] proposed that bubbles nucleated on the solidification front can be entrapped or escaped, depending on upward and downward directions of fluid flow near the solidification front. Wei and Hsiao [43] then proposed a general equation to predict the shape of an entrapped bubble as a pore during solidification. The governing equations were derived by taking time derivative of the equation of state and substituting solute transport rate across the cap. Instead of solving solute concentration equation, an empirical mass transfer coefficient was introduced to evaluate solute transport across the bubble cap during solidification. The single model can be generated to two- and three-dimensional development of the pore shape in the presence or absence of the mushy zone, provided that significant effects of solidification rate and mass transfer coefficient are relevantly specified. The empirical mass transfer coefficient, however, depends on fluid flow and solidification rate [44].

In the case of small deformation of the solidification front, Karagadde et al. [45] is the first to propose a rather detailed model to numerically trace the growth and movement of a hydrogen bubble in micro-scale under isolated and free growth conditions in a melt having a hydrogen input around the physical domain at a constant temperature during solidification in aluminum castings. Velocity, temperature and concentration fields were predicted from conservation equations of mass, momentum and concentration of liquid phase, whereas the variation of multiphase domains with time was predicted by the level-set equation. The growth rate of bubble surface in the level-set equation was determined by flow velocity and hydrogen diffusion from liquid into bubble. Extending previous work [45], Karagadde et al. [8] further predicted the growth and engulfment of gas microporosity in the mushy zone during aluminum alloy solidification in downward and horizontal directions. Energy equation were introduced. An explicit enthalpy scheme together with Gibbs-Thomson equation were used to predict the shape of mushy zone, coupled with a level-set method for tracking the hydrogen bubble evolution. The predicted average bubble radius and cooling rate in qualitative were in agreement with that provided by Fang and Granger [46], whereas the predicted solidification front speed versus initial bubble radius were in accordance with scale analysis proposed. Panwisawas et al. [15] also proposed a thermal fluid dynamics model and conducted experiments to study the evolution of pores during selective laser melting of Ti-6Al-4V. The complete fluid flow and energy equations were solved, using the volume-of-fluid (VOF) equation to trace pore shape. The predicted pore shape qualitatively agreed with observed morphologies of the pores affected by fluid flow.

This study is to propose a systematical model to predict solute convection resulting in deformation of a bubble cap entrapped as a pore during unidirectional upward solidification. The model is applicable to the bubble size much smaller than radius of curvature of the solidification front encountered in the absence of constitutional supercooling [47], [48] or morphological instability [48], [49]. Even though the Reynold number (ucR/ν) is smaller than unity [50], high Schmidt number (ν/D) [51] indicates that concentration field can be affected by fluid flow around the bubble cap above the solidification front. A more general and relevant model to investigate development of the pore shape affected by fluid flow, heat transfer and concentration coupling with a self-consistent deformation of solidification front is still limited [8], [45]. In this work, the commercial software packages COMSOL is used to solve transport equations of mass, momentum, energy, species and phase field equations in liquid, solid and gas phases. This work provides a fundamental and critical step to examine the effects of different working parameters such as solidification rate, directions of solidification, liquid concentration and bubble size at the initial state, etc on controlling of pore shapes and their formation in solid encountered in different manufacturing, materials, medical technologies, etc. in the near future.

Section snippets

System model and analysis

The present model as illustrated in Fig. 1 predicts transport processes during upward unidirectional solidification of water containing carbon dioxide. Since solubility of solid is much less than that of liquid, carbon dioxide is accumulated ahead of the solidification front. A bubble due to supersaturation readily initiates heterogeneously at a concave region of the solidification front denoted by line BE [52], [38], [53], [54], because solute concentration in the concave area of the

Results and discussion

In this study, the effects of convection on the shape of a pore resulting from an entrapped bubble during upward unidirectional solidification of water containing carbon dioxide are numerically and theoretically investigated. Two velocities profiles are considered at entrance location denoted by line AB (see Fig. 1 and Table 1). They are, respectively, 10-7 velocity and a parabolic velocity profile with the maximum velocity of 3×10-2 m/s at the top. Initial temperature profiles in liquid, gas

Conclusions

Conclusions drawn are the followings:

  • 1.

    This study provides a rigorous model accounting for mass, momentum, energy and concentration transport in distinct phases to predict and reveal mechanisms of development of the pore shape in solid during solidification. Scale analysis is also presented to interpret computed and experimental results.

  • 2.

    The flow becomes upwards as the bubble is approached. In the case of stronger flow, the bubble is deviated in the downstream direction.

  • 3.

    Time scale for bubble

Conflict of interest

The authors declared that there is no conflict of interest.

Acknowledgement

The work is financially supported by NSC 102-2221-E-110-038.

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