Frozen barrier evolution in saturated porous media

A numerical model capable of simulating the freezing of aqueous solution flow in saturated porous media is presented. This model is based on a finite-difference approximation of the coupled equations for liquid water flow, heat and solute transport and phase change. The phase change equation facilitates the condition for the special case when liquid water and ice can reside in the pore space simultaneously, leading to a 'mushy' zone. Results are presented to show the evolution of multiple frozen regions growing by a chain of freezing pipes. Two different regimes for the evolution of frozen bodies are distinguished based on system parameters. For the regime with lower freezing rate separate frozen bodies exist at steady-state, while for higher freezing rate the regime is characterized by linked frozen bodies. The numerical solution for the first regime is tested by a semi-analytical solution for the case of fresh water. For the second regime the model is able to simulate the process up to the point when linking of the separate frozen bodies occurs. For both regimes freezing is hindered downgradient of the freeze pipe where solute becomes highly concentrated, and a wedge of unfrozen media forms. For the first regime the wedge eventually forms into a liquid 'island' surrounded by ice-bearing porous media. © 2002 Elsevier Science Ltd. All rights reserved

Extending the lining life in circulatory vacuum units at OAO EVRAZ NTMK

Practical methods for extending the life of submersible tubes in vacuum chambers are considered. The structure of periclase-chromite components is studied. Refractories corresponding to optimal vacuum-chamber operation in the converter shop at OAO EVRAZ NTMK are selected. © 2013 Allerton Press, Inc

Stability analysis of traveling wave solution for gravity-driven flow

A linear stability analysis was performed for three models of flow in unsaturated porous media to determine the conditions for growth of small perturbations. The models considered include the conventional Richards equation (RE), a sharp front Richards equation (SFRE) and an extended Richards equation (RRE). The first two models are based on the use of an equilibrium capillary pressure-saturation function, while the third model is derived using a dynamic capillary pressure-saturation function represented by a relaxation coefficient. A traveling wave solution was formulated for each of the governing equations and used as the basic solution of each model. The stability analysis was based on imposing a small perturbation to the basic solution. The RE model yields only the well-known monotonically decreasing saturation profile toward the wetting front, and the wetting front is unconditionally stable. The SFRE model by its nature has a monotonically increasing saturation profile toward the front and an abrupt drop to the initial saturation. This flow is unconditionally unstable. The RRE model is distinct from the others in that it is the only model that is able to produce truly non-monotonic saturation profiles. The wetting front for the RRE model is conditionally stable, i.e. stable for high frequency perturbations, and unstable otherwise. This leads to the existence of a wave-number for maximum amplification, which should relate to the dimensions of fingers in unstable flow. © 2002 Elsevier B.V. All rights reserved

Dynamic capillary pressure mechanism for instability in gravity-driven flows; Review and extension to very dry conditions

Several alternative mathematical models for describing water flow in unsaturated porous media are presented. These models are based on an equation for conservation of mass of water, and a generalized linear law for water flux (Darcy's law) containing a term called the dynamic capillary pressure. The distinct form of each alternative model is based on the specific form of expression used to describe the dynamic capillary pressure. The conventional representation arises when this pressure is set equal to the equilibrium pressure given by the capillary pressure-saturation function for unsaturated porous media, and this conventional approach leads to the Richards equation. Other models are derived by representing the dynamic capillary pressure by a rheological relationship stating that the pressure is not given directly by the capillary pressure-saturation function. Two forms of rheological relationship are considered in this manuscript, a very general non-equilibrium relation, and a more specific relation expressed by a first-order kinetic equation referred to as a relaxation relation. For the general non-equilibrium relation the system of governing equations is called the general Non-Equilibrium Richards Equation (NERE), and for the case of the relaxation relation the system is called the Relaxation Non-Equilibrium Richards Equation (RNERE). Each of the alternative models was analyzed for flow characteristics under gravity-dominant conditions by using a traveling wave transformation for the model equations, and more importantly the flow described by each model was analyzed for linear stability. It is shown that when a flow field is perturbed by infinitesimal disturbances, the RE is unconditionally stable, while both the NERE and the RNERE are conditionally stable. The stability analysis for the NERE was limited to disturbances in the very low frequency range because of the general form of the NERE model. This analysis resulted in what we call a low-frequency criterion (LFC) for stability. This LFC is also shown to apply to the stability of the RE and the RNERE. The LFC is applied to stability analysis of the RNERE model for conditions of initial saturation less than residual. © 2005 Springer

Simulation of two-dimensional gravity-driven unstable flow

The coupled equations for flow in unsaturated soil as proposed by Beliaev and Hassanizadeh [6] are described. These equations account for mechanism of dynamic capillary pressure via a first order relaxation function. A form of the relaxation function for the dynamic capillary pressure-saturation relation is proposed based on physical reasoning and a semi-analytical solution to the flow equations. A mass conservative and computational efficient numerical solution to the coupled equations in two space dimensions is derived and applied to the simulation of gravity-driven unstable flow. Simulated fingers have all the morphological features of fingers observed in laboratory experiments. The results demonstrate that the dynamic capillary pressure mechanism causes initial destabilization of the flow, while the mechanism of capillary hysteresis leads to finger persistence. © 2002 Elsevier B.V. All rights reserved

Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension to very dry conditions

Several alternative mathematical models for describing water flow in unsaturated porous media are presented. These models are based on an equation for conservation of mass of water, and a generalized linear law for water flux (Darcy's law) containing a term called the dynamic capillary pressure. The distinct form of each alternative model is based on the specific form of expression used to describe the dynamic capillary pressure. The conventional representation arises when this pressure is set equal to the equilibrium pressure given by the capillary pressure - saturation function for unsaturated porous media, and this conventional approach leads to the Richards equation. Other models are derived by representing the dynamic capillary pressure by a rheological relationship stating that the pressure is not given directly by the capillary pressure - saturation function. Two forms of rheological relationship are considered in this manuscript, a very general non-equilibrium relation, and a more specific relation expressed by a first-order kinetic equation referred to as a relaxation relation. For the general non-equilibrium relation the system of governing equations is called the general Non-Equilibrium Richards Equation (NERE), and for the case of the relaxation relation the system is called the Relaxation Non-Equilibrium Richards Equation (RNERE). Each of the alternative models was analyzed for flow characteristics under gravity-dominant conditions by using a traveling wave transformation for the model equations, and more importantly the flow described by each model was analyzed for linear stability. It is shown that when a flow field is perturbed by infinitesimal disturbances, the RE is unconditionally stable, while both the NERE and the RNERE are conditionally stable. The stability analysis for the NERE was limited to disturbances in the very low frequency range because of the general form of the NERE model. This analysis resulted in what we call a low-frequency criterion (LFC) for stability. This LFC is also shown to apply to the stability of the RE and the RNERE. The LFC is applied to stability analysis of the RNERE model for conditions of initial saturation less than residual. © Springer 2005

Non-equilibrium model for gravity-driven fingering in water repellent soils: Formulation and 2D simulations

The instability of infiltrating flow is studied using the mass balance equation coupled with a first-order relaxation equation relating the rate of change of saturation to the difference between the dynamic water pressure and the saturation-dependent equilibrium water pressure. A numerical solution of the mass balance equation, based on a mass conservative scheme, is applied to the simulation of infiltrating flows in a vertical, two-dimensional plane region. Both water wettable and water repellent soils are considered in the analysis. The effect of water repellency is introduced by modification of the equilibrium saturationpressure relationship, in which water repellency causes the relation to become flatter. Conditions of even slight water repellency are found to be sufficient to cause infiltrating flows to become unstable. A sensitivity analysis related to the width of the surface source shows that the number of fingers generated increases with increasing source width. The sensitivity analysis also indicates that the non-equilibrium model approach can provide a physically plausible reason for flows becoming stable when the surface flux becomes vanishingly small. © 2003 Elsevier Ltd. All rights reserved

Stability analysis of gravity-driven infiltrating flow

Stability analysis of gravity-driven unsaturated flow is examined for the general case of Darcian flow with a generalized nonequilibrium capillary pressure-saturation relation. With this nonequilibrium relation the governing equation is referred to as the nonequilibrium Richards equation (NERE). For the special case where the nonequilibrium vanishes, the NERE reduces to the Richards equation (RE), the conventional governing equation for describing unsaturated flow. A generalized linear stability analysis of the RE shows that this equation is unconditionally stable and therefore not able to produce gravity-driven unstable flows for infinitesimal perturbations to the flow field. A much stronger result of unconditional stability for the RE is derived using a nonlinear stability analysis applicable to the general case of heterogeneous porous media. For the general case of the NERE model, results of a linear stability analysis show that the NERE model is conditionally stable, with lower-frequency perturbations being unstable. A result of this analysis is that the nonmonotonicity of the pressure and saturation profile is a requisite condition for flow instability

Microstructure of Complex Silicon-containing Modifier

Various research methods show that the microstructure of the complex siliconcontaining modifier ”Insteel 7” consists of six phases: TiFeSi2, Ca1