2.2. Experimental set-up, instrumentation and measurements

The bottom-cooled solidification cell was made of 8 mm thick perspex glass, as shown in figure 2. The inner volumetric dimension of the cell was 110 × 70 × 50 mm³. To prevent the condensation problem during in situ experiments, a cover cell was placed on the solidification cell. The bottom copper plate of the cell was maintained at −22 °C using an assembly of peltier, thermoelectric cooler (TEC) controller, DC power supply, heat exchanger and a julabo circulating bath (Kumar et al. Reference Kumar, Srivastava and Karagadde2018a; Kumar, Srivastava & Karagadde Reference Kumar, Srivastava and Karagadde2018c). Nine K-type thermocouples (accuracy of ±0.15 °C) were attached to each of the vertical sidewalls to measure the temperature. The temperature data was acquired through data loggers (PICO ® TC-08).

Figure 2.

Schematic of the solidification cell.

The density is known to be a function of temperature and composition. For observing the buoyant convection patterns during the experiments, the density-gradient-based shadowgraph technique was employed. In the shadowgraph, a white light source (250 W) was collimated using a plano-convex lens. The collimated beam was passed through the solidification cell, and density changes in the cell were captured by a CCD camera (Imperx CLB-B2520M). For the velocity measurements of the liquid, particle image velocimetry (MicroVec PIV ®) was employed. The PIV system consists of a DPSS (diode pumped solid state) laser (532 nm), synchronizer and a CCD camera (Imperx CLB-B2520M, resolution 2456 × 2048). Neutrally buoyant hollow glass particles (average diameter 10 μm) were used as seeding particles. The details of the employed PIV set-up can be found in the references (Kumar et al. Reference Kumar, Kumawat, Srivastava and Karagadde2017; Kumar, Srivastava & Karagadde Reference Kumar, Srivastava and Karagadde2018b).

The calibration curve of the refractive index with known composition was established using a refractometer (Anton Paar ®). Samples (approximately 0.5 ml) were extracted using a thin needle-syringe from the cell during the experiment, and the refractive index was measured. Using the calibration curve and measured refractive index of the samples, the composition of KNO₃ and NH₄Cl was evaluated. Since the extraction of samples disturbs the flow slightly, several repititions (~10) were performed to obtain the compositional difference needed for evaluating the Rayleigh number. The error bars of the Rayleigh numbers were based on these repititions.

3.1. Convective flows in faceted and dendritic growth cases

Figure 3 shows the spatio-temporal evolution of the flow field obtained from the shadowgraphs during bottom-cooled solidification of water-23 wt% KNO₃ (a–c) and water-24 wt% NH₄Cl (d–f). In water-23 wt% KNO₃, the flow of rejected (low density) mixture was random (figure 3a), not in specific patterns, such as a plume, whereas in water-24 wt% NH₄Cl, the plume formation and localized solutal convection (near the bottom section) were observed (figure 3d). Evolving microstructures were captured using a portable microscope (Dinolite ®, Model AM7515MZT). The faceted and dendritic solids were observed in water-23 wt% KNO₃ (inset of figure 3b) and water-24 wt% NH₄Cl (inset of figure 3e), respectively.

Figure 3.

Observation of solutal buoyancy-induced convective flow using shadowgraph for (a–c) water-23 wt% KNO₃ and for (d–f) water-24 wt% NH₄Cl: (a) random solutal convection at 60 min and (b) at 260 min (faceted microstructure is shown in the inset of figure), (c) absence of convection at 355 min after liquid becomes nearly homogenous (at eutectic composition), (d) plume formation at 60 min and (e) at 600 min (dendritic morphology captured at solid–liquid interface presented in the inset of figure), (f) localized solutal convection at 970 min. (Note: The region marked by a rectangular box (yellow) on the interface in (c), is used for intensity measurements to be reported later).

The random convective flow essentially homogenized the composition and temperature of the liquid in the cell (particularly ahead of the mushy zone) and reduced the bulk composition to near-eutectic composition within 300 min. Note that in figure 3(c), the macroscopic freezing interface is slightly inclined, which could be attributed to slight variations in the compositional distribution induced by the random convective flow patterns. The composition of the bulk fluid and the strength of convection were found to further reduce with time, and was measured to be water-11.8 wt % KNO₃ at 350 min. At 600 min, the composition reached the uniform eutectic composition water-9.9 ± .3 wt% KNO₃, and the interface became flat (figure 4a–c) compared to previous time instants (figure 3c).

Figure 4.

Observations of the reducing convective flow behaviour are shown here using shadowgraphs. Panels (a–c) show absence of convection in the liquid after the liquid reaches eutectic composition in water-23wt% KNO₃, at (a) 740 min (b) 900 min and (c) 1280 min. However, in the water-24wt% NH₄Cl case (d–f), presence of weak, localized solutal convection with composition reaching the eutectic was observed at (d) 1410 min, further experiencing eutectic growth at (e) 1515 min and at (f) 1760 min. (Note: The region marked by a rectangular (yellow) box, on the interface in (a and e), is used for intensity measurements).

In the case of dendritic growth, the solutal convection dominated in the form of plumes. Plumes transport the composition to the top of the cell, while simultaneously exchanging heat with the warmer bulk liquid. At the top, plumes mix with the bulk, which then gradually moves downward (bulk velocity scale is 10⁻⁵–10⁻⁶ m s⁻¹). As a result, a solutal gradient is set up in the direction of gravity, which delays the creation of the eutectic composition ahead of the mushy region.

At 600 min, the compositions at the top of the cell and near the solidifying interface were water-21.3 wt% NH₄Cl and water-22.47 wt% NH₄Cl, respectively. The growth was comparatively slower, between 970 min (figure 3f) and 1410 min (figure 4d), due to continuous change in composition and the liquidus temperature. Composition reached closer to the eutectic composition (water-20.05 wt% NH₄Cl) at 1410 min. Furthermore, eutectic solid was observed much later, at 1410 min (figure 4e,f), compared to 600 min in the faceted case.

3.2. Evolution of the solid

In the two hyper-eutectic binary mixtures (water-23 wt% KNO₃ and water- 24 wt% NH₄Cl), the primary solid (α) phase is primarily the salt system. At temperatures below the eutectic line, the solid exists as a mixture of α-solid and the eutectic (existing in the inter-granular regions). In the present experiments, such a solid phase was observed at the bottom of the cell and is termed as α+eutectic solid (figure 5a,b). On comparing the solidifying structures of the two binary mixtures, the formation of a larger amount of eutectic solid in the faceted case was observed, as the liquid ahead of the freezing mixture was able to reach near-eutectic composition uniformly due to stronger convective mixing (figure 5c). Similar solid evolution was also observed for a lower bottom-cooled condition (−11 °C) of water-23 wt% KNO₃ (figure 19a). In water-24 wt% NH₄Cl, the liquid composition was much slower in reaching near-eutectic composition, and, therefore, the amount of pure eutectic solid was considerably less. Moreover, the rate of growth of the uppermost location of the freezing interface was found to be considerably larger in the eutectic growth for the faceted case.

Figure 5.

Evolution of the solid during solidification of (a) water-23 wt% KNO₃ at 660 min (b) water-24 wt% NH₄Cl at 1200 min. (c) Comparison of the progress of the height of the interface between water-23 wt% KNO₃ and water-24 wt% NH₄Cl.

The existence of a fully eutectic phase and the mixture of primary and eutectic phases were ascertained by measuring the composition of the solid at the end of the experiments. The change in the mass of the solid sample was evaluated by evaporating the water from the solid mixture. In the faceted case, the average composition of α+eutectic and the eutectic was 55 ± 3 wt% and 14 ± 1 wt% KNO₃, respectively, whereas in the dendritic case, the average composition of the α+eutectic solid was 32 ± 2 wt% NH₄Cl.

3.3. Velocity field in bulk fluid

In the faceted growth, the maximum velocity magnitude was 4.1 mm s⁻¹ at 100 min (figure 6a), and it reduced with time to 3.3 mm s⁻¹ (at 120 min) (figure 6b). As the solidification progressed, the convection strength decreased due to the shifting of composition towards the eutectic composition, which led to a decrease in velocity magnitude with time (figure 6a,b).

Figure 6.

Velocity field shown as vectors superimposed on the contours representing the magnitude of the velocity. Panels (a,b) show those for faceted growth (water-23wt% KNO₃) at (a) 100 min and (b) 120 min, where random but higher magnitude flow field was observed, compared to those of dendritic growth (water-23wt% NH₄Cl) showing stabilized plumes at 100 min (c).

After the initial solutal convection during dendritic growth, stabilized plumes were observed in the cell (figure 6c). The approximate average velocity of the plumes was 0.6 mm s⁻¹ and that of the bulk fluid was much lower (~10⁻² mm s⁻¹). However, in the faceted case, the velocity field in the bulk fluid was uniform but larger in magnitude (2–4 mm s⁻¹).

3.4. Uncharacteristic temperature rise in the faceted case

Figure 7 shows the temperature measurements at different locations along with the height of the cell for the faceted (water-23 wt% KNO₃, figure 7a) and dendritic (water-24 wt% NH₄Cl, figure 7b) growth cases. Significantly dissimilar temperature profiles were observed for the two cases, with the faceted solidification showing drastic initial cooling, followed by an unusual temperature rise. However, the temperature profiles of the dendritic case showed slow but gradual temperature decrease, on par with the nominal cooling rate. Moreover, both cases demonstrated a slight temperature rise towards the end of the experiment (marked by the grey box in figure 7a,b), which is very likely to have been associated with the heat gained from the surroundings. Based on the changes in the slope of temperature versus time measurements of the faceted case, the following three modes of heat transfer in the liquid have been identified: (a) convection-dominated regime, (b) transition regime and (c) conduction-dominated regime. Note that similar regimes were identified in the dendritic case as well, but the changes in the slopes were relatively less severe.

Figure 7.

Temperature measurement at the different heights within the cell showing (a) higher cooling rate, unconventional gain in temperature and conduction-dominated cooling in the liquid during solidification of water-23 wt% KNO₃ and (b) slow cooling in liquid for water-24 wt% NH₄Cl.

3.4.2. Transition regime

At the end of the convection-dominated regime (the magnitude of convective velocity can be seen from figure 8(a), where the velocity is very low compared to figure 6a,b), low thermal gradients were observed in the bulk fluid. The bulk-fluid composition approached the eutectic composition, where localized solutal convection (the local convective flow was in the order of 0.01 mm s⁻¹, as shown in figure 8b), eutectic growth and a gain in the temperature were observed. Weaker and localized compositional convection maintains the low thermal gradient near the interface. This region was termed the transition regime (300–500 min) (figure 7a). In this regime, a sudden increase in growth rate was observed at the eutectic interface (due to conduction-dominated solid growth) (as shown in figure 5c). Similar gain in the bulk-liquid temperature was observed by Thompson et al. (Reference Thompson, Huppert, Worster and Aitta2003) during bottom-cooled solidification of water-KNO₃-NaNO₃ where KNO₃ was the primary solidifying component. However, the increase was attributed to the heat gain from the surrounding environment.

Figure 8.

Velocity vectors and contours for the faceted case, (a) at the end of the convection-dominated regime, (b) at the start of the transition regime, (c) at the end of the transition regime.

In the present work, we further observed gain in the temperature with different initial compositions of water-KNO₃ systems (experimental details of water-(15 and 20) wt% KNO₃ is presented in appendix A.1 (figures 16 and 17)). Similar gain was also observed with a higher bottom-cooled condition (bottom was maintained at −11 °C) solidification of water-23 wt% KNO₃ (appendix A.2, figures 18–20). In order to verify the rise in temperature in different faceted-forming mixtures, experiments with the hyper-eutectic mixtures of water-Na₂SO₄ were performed, where similar random convection patterns and gain in temperatures were observed during bottom-cooled solidification (data in appendix A.2, figures 21 and 22).

In the water-NH₄Cl (dendritic growth) case, a positive thermal gradient in the liquid existed throughout the experiment (figure 7b). The gain in temperature, and hence the transition regime, did not exist, unlike the cases of water-KNO₃ and water-Na₂SO₄ (faceted-growth) systems.

3.4.3. Conduction-dominated regime

In the conduction-dominated regime, the temperature decreased gradually, with continuous growth of the eutectic solid (no convective flow was observed, as shown in figure 8c). As a result, a larger thermal gradient in the liquid was set up at the interface, which appeared to deflect the collimated light in the shadowgraphs by showing a high greyscale intensity. The intensity measurements at different locations near the solid–liquid interface (marked as rectangular boxes in figures 3 and 4) are shown in figure 9. For the faceted case, a high-intensity (greyscale intensity magnitude ~255) uniform band was observed in figure 9(b) at the interface (740 min), unlike figure 9(a), when the localized convection was present at 355 min. Similarly, in the water-NH₄Cl (figure 4d–f) case, the eutectic growth led to a high-gradient region at the interface after 1410 min. From the temperature and the intensity measurements, the gradient near the interface was observed to be higher in water-NH₄Cl (figures 7b and 9c) than the water-KNO₃ (figures 7a and 9b) system.

Figure 9.

Figure shows the shadograph images and the corresponding grey-scale intensity variation in the liquid ahead of the mushy zone. The whitish grey region (high intensity) corresponds to the water-rich liquid. These variations demonstrate the trends of compositional distribution during random convection and conduction-dominated regimes. (a) convection-dominated regime in water-23 wt% KNO₃, where random mixing homogenizes the concentration field, and (b) conduction-dominated regime in water-23 wt% KNO₃, where a peak concentration ahead of the interface is seen. (c) conduction-dominated regime in water-24 wt% NH₄Cl, which shows a typical diffusion profile.

5.1. Role of permeability on the convective scales

A scale analysis is performed to obtain the velocity, temperature/composition scales in the mushy/porous, and the bulk zones during convection. For the analysis, we consider a three-layer (solid, porous mushy zone and liquid) system in a bottom-cooled configuration, as shown in figure 12.

Figure 12.

Schematic representation of problem description for the scale analysis.

The solid is of height H_S with a mushy zone of height l above the solid. The mushy zone is characterized by a porous zone with an overall permeability K, which is assumed to be constant in the present analysis. A liquid layer of height h is available on top of this porous zone. The system can be envisaged to be a unidirectional solidification case, where the height of the mushy zone is constant during the convection process. The left and right walls are considered adiabatic. Note that a thermally stable condition is assumed, and compositional gradient is considered as the driving factor for the Rayleigh–Bénard convection. The scale analysis using the Rayleigh number in the (a) porous and (b) bulk-fluid zones are discussed separately below.

5.1.1. Porous zone: H_S ≤ y ≤ H_P

The governing equations for the mass, Darcy flow, energy and species conservation, and respective boundary conditions in the porous media are written as

conservation of mass:

(5.1a)\begin{equation}\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0,\end{equation}

Darcy flow in x direction:

(5.1b)\begin{equation}u ={-} \frac{K}{\mu }\frac{{\partial P}}{{\partial x}},\end{equation}

Darcy flow in y direction:

(5.1c)\begin{equation}v ={-} \frac{K}{\mu }\left( {\frac{{\partial P}}{{\partial y}} + \rho g} \right),\end{equation}

conservation of energy:

(5.1d)\begin{equation}\frac{{\partial T}}{{\partial t}} + u\frac{{\partial T}}{{\partial x}} + v\frac{{\partial T}}{{\partial y}} = {\alpha _{th}}\left( {\frac{{{\partial^2}T}}{{\partial {x^2}}} + \frac{{{\partial^2}T}}{{\partial {y^2}}}} \right),\end{equation}

conservation of species:

(5.1e)\begin{equation}\frac{{\partial C}}{{\partial t}} + u\frac{{\partial C}}{{\partial x}} + v\frac{{\partial C}}{{\partial y}} = D\left( {\frac{{{\partial^2}C}}{{\partial {x^2}}} + \frac{{{\partial^2}C}}{{\partial {y^2}}}} \right),\end{equation}

boundary conditions:

(5.1f)\begin{equation}\left. {\begin{array}{c@{}} {u,v = 0,T = Th,C = {C_S}\quad \textrm{at}\,y = {H_S}},\\ {u = {u_l}, v = {v_l},T = {T_{interface}},C = {C_{interface}}\quad \textrm{at}\,y = {H_P}}, \end{array}} \right\}\end{equation}

where u and v are velocity in x and y directions, respectively, $\mu$ is the dynamic viscosity of liquid and P is pressure. Density $(\rho )$ is established with Boussinesq approximation $\rho = {\rho _0}[1 - {\beta _T}(T - {T_0}) - {\beta _C}(C - {C_0})]$, where $\beta_T$ is thermal expansion coefficient in $\text{K}^{-1}$. For the scale analysis in the porous zone, flow is assumed to be predominant along the y-axis $(v \gg u)$. From that, the velocity scales can be written as (5.2a,b). Note that the ratio of the two length scales in the system, namely the primary arm spacing of the microstructural features and the length of the mushy region, given by ${\lambda _1}/l$, is negligible. This reaffirms the assumption of $v \gg u$.

(5.2a,b)\begin{equation}v \approx \frac{{gK\Delta \rho }}{\mu }\sim \frac{{gK{\beta _T}\Delta T}}{\upsilon },\quad u \approx \frac{{gK\Delta \rho }}{\mu }\frac{{{\lambda _1}}}{l}\sim \frac{{gK{\beta _T}\Delta T}}{\upsilon }\frac{{{\lambda _1}}}{l}.\end{equation}

The convective velocity scale can be written in terms of the composition

(5.2c,d)\begin{equation}{v_P}\sim \frac{{gK{\beta _C}\Delta C}}{\upsilon },\quad {u_P}\sim \frac{{gK{\beta _C}\Delta C}}{\upsilon }\frac{{{\lambda _1}}}{l}.\end{equation}

Subscript P stands for the porous (mushy) zone. From the scale relations, it can be clearly observed that, as the permeability decreases, v and u show lower velocities for a given buoyancy-driven system. This implies that the faceted cases have lower vertical velocities than those of the dendritic cases in the mushy zone. At the base of the plumes (in dendritic growth), low solid fraction (and higher primary arm spacing) leads to a high value of the permeability (K), which indicates a higher velocity in porous zone.

5.1.2. Bulk-fluid zone: H_P < y ≤ H

For the scale analysis in the bulk fluid, the conservation of mass, momentum and energy in the bulk fluid is given in (5.3). The conservation of momentum in the bulk is solved through the Navier–Stokes equation.

Mass conservation:

(5.3a)\begin{equation}\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0,\end{equation}

momentum balance:

(5.3b)\begin{equation}\frac{{\partial \boldsymbol{v}}}{{\partial t}} + \boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla v} ={-} \frac{1}{{{\rho _0}}}{\boldsymbol{\nabla}}\boldsymbol{P} + \upsilon {\nabla ^2}\boldsymbol{v} - \frac{{\rho g}}{{{\rho _0}}},\end{equation}

energy conservation:

(5.3c)\begin{equation}\frac{{\partial T}}{{\partial t}} + u\frac{{\partial T}}{{\partial x}} + v\frac{{\partial T}}{{\partial y}} = {\alpha _{th}}\left( {\frac{{{\partial^2}T}}{{\partial {x^2}}} + \frac{{{\partial^2}T}}{{\partial {y^2}}}} \right),\end{equation}

species conservation:

(5.3d)\begin{equation}\frac{{\partial C}}{{\partial t}} + u\frac{{\partial C}}{{\partial x}} + v\frac{{\partial C}}{{\partial y}} = D\left( {\frac{{{\partial^2}C}}{{\partial {x^2}}} + \frac{{{\partial^2}C}}{{\partial {y^2}}}} \right).\end{equation}

The vector v represents u and v velocities in x and y directions, respectively. The vertical component of velocity at the interface of porous media and bulk liquid will be equal (${v_{P - }} = {v_{P + }}$, where ${v_{P - }}$ is the vertical velocity magnitude in the porous zone and v _P+ is the vertical velocity magnitude in the bulk liquid; point P is the interface of porous media and bulk liquid). The horizontal component of velocity at this interface was proposed by Beavers & Joseph (Reference Beavers and Joseph1967), and is given by

(5.4)\begin{equation}\mathop {\frac{{\partial u}}{{\partial y}}}\nolimits_{y = {H_P}} = \frac{\chi }{{\sqrt K }}({u_{P + }} - {u_P}),\end{equation}

where ${u_{P + }}$ indicates horizontal velocity at the interface of porous and bulk fluid, $\chi$ is a unit order dimensional quantity, which characterizes the structure of permeable material within the porous zone (Beavers & Joseph Reference Beavers and Joseph1967). From the relation, it can be concluded that the horizontal velocity gradient is inversely proportional to permeability. The high velocity gradient in the low permeable case implies more vorticity and circulation which can lead to low gradient of temperature and species (vorticity/circulation is inversely proportional to thermal or compositional gradient (Meunier & Villermaux Reference Meunier and Villermaux2003)). The lower permeability in faceted growth leads to a higher horizontal velocity gradient in the vertical direction when compared to the dendritic case.

As a secondary confirmation, we compare the scaling relations with the measured velocity gradient and vorticity near the interface experimental PIV data, as shown in figures 6 and 13. Figure 13(b) shows that the vorticity in the x–y plane near the interface is mainly dominated by the horizontal gradient of vertical velocity, compared to the vertical gradient of horizontal velocity. At the plume location (high permeable dendritic case), the vertical gradient of horizontal velocity is negligible, which indicates that, in the bulk, there is low circulation, and the plume is the only dominant convective path. Whereas, in the faceted case having low permeability (figure 13a), the vorticity is controlled by the vertical gradient of horizontal velocity and horizontal gradient of vertical velocity, which is known to create considerable mixing in the bulk. The mixing in the faceted case leads to the homogenization of thermal and composition field, as observed in figure 7(a), whereas the plume creates a thermal and composition gradient in bulk with a lower magnitude of vorticity. In the following section, this behaviour is represented through a numerical simulation in order to capture the average temperature rise in the bulk.

Figure 13.

Experimentally captured data of the vertical gradient of horizontal velocity, the horizontal gradient of vertical velocity and vorticity near the solid–liquid interface at 100 min during solidification of (a) water-23 wt% KNO₃ (low permeable faceted growth) and (b) water-24 wt% NH₄Cl (high permeable dendritic growth). Vorticity fields for the faceted case (c), and its corresponding velocity field from figure 6(a), and for the dendritic case (d), and its corresponding velocity field from figure 6(c). These data are provided as supporting information to theoretical scaling estimate.

5.2. Gain in temperature during the transition regime

A numerical model to simulate solidification was developed with an aim to generalize the plausible mechanism behind the gain in the liquid temperature. The aim of the simulation is to demonstrate the evolution of the thermal field during the onset of eutectic phase growth after the reduction of convection. A one-dimensional simulation of solidification was performed with the following assumptions: (a) the solid–liquid interface is flat during eutectic growth (this assumption is well justified by the experimental data presented earlier) and the interface is maintained at the eutectic temperature, and (b) the system is in a bottom-cooled configuration with a fixed temperature at the bottom and an adiabatic top surface. The conditions of the simulation are based on the actual experimental measurements to enable the generalization in both high and low Prandtl number (Pr) fluids. The simulation methodology involves explicitly solving the heat equation with a Dirichlet-type temperature boundary at the bottom, an adiabatic boundary at the top and the Stefan condition at the solid–liquid interface. The heat equation for solid and liquid and the boundary conditions are given by

(5.5a)\begin{gather}\rho {C_p}\frac{{\partial T}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {k\frac{{\partial T}}{{\partial x}}} \right),\end{gather}

(5.5b)\begin{gather}T{|_{x = 0}} = {T_{Bottom}},\end{gather}

(5.5c)\begin{gather}{\left. {\frac{{\partial T}}{{\partial x}}} \right|_{x = L}} = 0,\end{gather}

(5.5d)\begin{gather}{\left. {{k_s}\frac{{\textrm{d}{T_s}}}{{\textrm{d}x}}} \right|_{x = {x^ \ast }}} - \rho HV = {\left. {{k_l}\frac{{\textrm{d}{T_l}}}{{\textrm{d}x}}} \right|_{x = {x^ \ast }}},\end{gather}

where $\rho$ is the density, t is the time, and ${x^ \ast }$ is the position of the solid–liquid interface.

In systems that experience dendrite formation, convection in the liquid is non-existent at the onset of the formation of the eutectic phase. Thus, the system can be assumed to solidify under purely conductive heat transfer conditions, which is well described by the above model. However, systems that exhibit faceted growth experience almost negligible thermal gradients in the liquid phase due to vigorous random convection in the bulk liquid prior to the onset of eutectic growth, which gradually reduces with time through the transition regime. The heat transfer in this regime is through conduction and localized random convection. The convective effects enhance the net heat flow into the liquid phase. Therefore, the net heat flow into the liquid is considered to be higher than that of pure conduction alone, by approximately 2–4 times. This is justified by considering the orders of magnitudes of conductive (~20 W m⁻²) and convective (~80 W m⁻²) heat flux on the liquid side of the interface. Note that this exercise was performed with a primary objective of capturing an order of magnitude of the temperature change, with the help of a simplified one-dimensional model. For accurate predictions of temperature change, more exact, three-dimensional numerical simulations would be performed; these have not been included here to avoid restricting the focus of the present investigation. The initial temperature gradient condition in the cell has been used from the experimental data, as shown in figure 7.

At the end of the transition regime, the system experiences a purely conductive heat transfer. Simulations were performed on NH₄Cl and KNO₃ materials (figure 14). Further, simulations were also performed on low-Prandtl-number fluids, such as alloys of aluminium (dendritic) and silicon (faceted), in order to observe the corresponding thermal fields (figure 15).

Figure 14.

Temperature plot at different positions in the cell after the end of convection during (a) faceted growth of the water-KNO₃ system and (b) dendritic growth in the water-NH₄Cl system.

Figure 15.

Temperature plot at different positions in the cell in the metallic system during solidification of (a) faceted growth in Si-alloy and (b) dendritic growth in the Al-alloy system.

The temperature history at various locations along the height is evaluated, mimicking the experimental measurement locations. The comparison shows a clear agreement between the two sets of results (figures 7 and 14). The NH₄Cl (dendritic) case predicts monotonically decreasing temperature with time, while the KNO₃ (faceted) case predicts a gain in the temperature profile in the transition regime. The extent of temperature rise is of similar magnitude as observed in the experimental results. Therefore, the mechanism of random convective flow causing an earlier eutectic transformation resulting in an increased liquid temperature is also ascertained with this numerical analysis. Similarly, numerical simulations on low-Prandtl-number fluids, such as alloys of aluminium and silicon also predicted a similar temperature profile (figure 15), thus indicating that the hypothesis could be extended to the realm of low-Pr fluids.