Perturbation solutions are obtained for one-dimensional phase change problem in a finite region subject to convective and radiative boundary condition at the fixed boundary. The radiative term is first approximated by Taylor’s series expansion, then the perturbation technique is used. Analytical expressions for total solidification time and the rate of solidification as well as the temperature distributions are obtained. Close agreement is observed between the present analysis and that of an early work.
We model hydrothermal convection using a partial differential equation formed by Darcy velocity and temperature—the velocity formulation. Using the Elder problem as a benchmark, we found that the velocity formulation is a valid model of hydrothermal convection. By performing simulations with Rayleigh numbers in the non-oscillatory regime, we show that multiple quasi-steady-state solutions can be one of the reasons that caused the Nusselt–Rayleigh discrepancy found in previous experiments. The results reveal more understandings about the nature of uncertainty of convection modes in porous media.
Abstract. Reactive transport processes in natural environments often involve many ionic species. The diffusivities of ionic species vary. Since assigning different diffusivities in the advection-diffusion equation leads to charge imbalance, a single diffusivity is usually used for all species. In this work, we apply the Nernst–Planck equation, which resolves unequal diffusivities of the species in an electroneutral manner, to model reactive transport. To demonstrate the advantages of the Nernst–Planck model, we compare the simulation results of transport under reaction-driven flow conditions using the Nernst–Planck model with those of the commonly used single-diffusivity model. All simulations are also compared to well-defined experiments. Our results show that the Nernst–Planck model is valid and particularly relevant for modeling reactive transport processes with an intricate interplay among diffusion, reaction, electromigration, and density-driven convection.
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