Solidification of binary aqueous solutions under periodic cooling. Part 2. Distribution of solid fraction

Abstract

We report an experimental study of the distributions of temperature and solid fraction of growing $\text{NH}_{4}\text{Cl}$–$\text{H}_{2}\text{O}$ mushy layers that are subjected to periodical cooling from below, focusing on late-time dynamics where the mushy layer oscillates about an approximate steady state. Temporal evolution of the local temperature $T(z,t)$ at various heights in the mush demonstrates that the temperature oscillations of the bottom cooling boundary propagate through the mushy layer with phase delays and substantial decay in the amplitude. As the initial concentration $C_{0}$ increases, we show that the decay rate of the thermal oscillation with height also decreases, and the propagation speed of the oscillation phase increases. We interpret this as a result of the solid fraction increasing with $C_{0}$, which enhances the thermal conductivity but reduces the specific heat of the mushy layer. We present a new methodology to determine the distribution of solid fraction $\unicode[STIX]{x1D719}(z)$ in mushy layers for various $C_{0}$, using only measurements of the temperature $T(z,t)$. The method is based on the phase behaviour during thermal modulation, and opens up a new approach for inferring mushy-layer properties in geophysical and engineering settings, where direct measurements are challenging. In our experiments, profiles of the solid fraction $\unicode[STIX]{x1D719}(z)$ exhibit a cliff–ramp–cliff structure with large vertical gradients of $\unicode[STIX]{x1D719}$ near the mush–liquid interface and also near the bottom boundary, but much more gradual variation in the interior of the mushy layer. Such a profile structure is more pronounced for higher initial concentration $C_{0}$. For very low concentration, the solid fraction appears to be linearly dependent on the height within the mush. The volume-average of the solid fraction, and the local fluctuations in $\unicode[STIX]{x1D719}(z)$ both increase as $C_{0}$ increases. We suggest that the fast increase of $\unicode[STIX]{x1D719}(z)$ near the bottom boundary is possibly due to diffusive transport of solute away from the bottom boundary and the depletion of solute content near the basal region.

Solidification of binary aqueous solutions under periodic cooling. Part 1. Dynamics of mushy-layer growth

“…The phase lag increases with height due to the diffusive time required for the propagation of the changing boundary temperature through the medium. In Part II (Ding et al 2018) we show that perturbations of this form can be justified for small amplitude and high-frequency modulation with ωh 2 /α ≫ 1 about the late-time steady state. However, in the present early-time limit the phase lag δφ shows some nonlinear variation with height and the amplitude δT shows faster than exponential decay near the mush-liquid interface.…”

“…During the initial modulation periods, the mushy-layer undergoes rapid growing and its porosity structure and solid-fraction distribution evolve significantly in time. In Part 2 of this series of papers (Ding et al 2018), we will extend our experimental studies and analysis to investigate the distribution of the mush temperature during late modulation periods when the fine mush structure becomes approximately time-independent. From this we determine the distribution of the solid fraction in mushy layers.…”

“…We show that the theoretical model provides reasonably accurate predictions for the growth and temperature distributions of mushy layers under various conditions of thermal modulation. A companion paper (Part II, Ding et al 2018) considers modulation of late time growth, where the mushy layer oscillates about a steady state thickness, and presents a method to measure the solid fraction using the measured thermal response to fluctuating boundary temperature.…”

Stability of Vertical Double-Diffusive Interfaces In The Presence Of Material Diffusion

“…The melt started to solidify in the vertical direction (directional solidification), and this leads to a region of mixed phase (solid and liquid) which is the mushy layer and it is unstable. Then this process leads to appearance of thin plumes from the mushy layer and rose to the liquid layer and they concluded that these plumes are related to freckles [27,[29][30].…”