First, different porous media theories are presented. Some approaches are based on the classical mixture theory for fluids introduced in the 1960s by Truesdell and Coworkers. One of the first ...researchers who extended the theory to porous media (thus mixtures containing at least one solid constituent) and also accounting for chemical reactions was Bowen. Another important branch of porous media theory goes back to Biot. In the beginning, he dealt with classical geotechnical problems and set up his model empirically. Mathematicians often use reaction–diffusion equations which are limited in comparison with continuum models by several restrictive assumptions and very often only applicable to special problems. In this paper, the focus lies on approaches based on the mixture theory which incorporate chemical reactions. Different strategies to describe the chemical potential for mixtures are presented, and different opinions about the exploitation of the second law of thermodynamics for mixtures are put forward. Finally, several works of different types including chemical reactions in porous media are summarized.
•In this article models for porous media containing one, two or three pore fluids, e.g. water, oil and air, are shown. The inherent macroscopic material parameters follow from their measurable ...microscopic counterparts by micro-macro considerations which, in turn, follow from gedanken experiments. All models are intended to describe the propagation of sound waves. Capillary pressures in mixtures of immiscible fluids in the pores are taken into account and relative permeabilities are considered.
In this article models for porous media containing one, two or three pore fluids, e.g. water, oil and air, are shown. The newly presented model for three pore fluids is an extension of the established ones for one and two pore fluids. The inherent macroscopic material parameters of all models follow from their measurable microscopic counterparts by micro-macro considerations which, in turn, follow from gedanken experiments. The models are intended to describe the propagation of sound waves. Capillary pressures in mixtures of immiscible fluids in the pores are taken into account and relative permeabilities are considered.
We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a ...real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.
A model for lime consolidation of porous solids Detmann, Bettina; Gavioli, Chiara; Krejčí, Pavel ...
Nonlinear analysis: real world applications,
June 2022, 2022-06-00, Letnik:
65
Journal Article
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We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical ...reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection–diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations.
In this study, a 1000 L pilot scale internal loop airlift bioreactor was operated and compared to a mathematical model to determine the best design for optimal supply of oxygen for nitrification and ...sufficient air for biomass fluidization. The design model is based on parameters such as geometry, carrier density, and airflow of the 1000 L pilot scale bioreactor. The model predicts a range of superficial air velocities (0.009–0.013 m/s) under which the airlift bioreactor was fluidized. Three superficial air velocities (0.009 m/s, 0.011 m/s and 0.013 m/s) were experimentally tested in the pilot plant and the obtained circulation velocities were compared with the predicted design scenarios. The predicted velocity was in agreement with the measured velocity. The aim of the mathematical model and the calculations of different geometry scenarios was to define the optimal geometry design for the physical model. The results show that the ratio of the cross-sectional area between the riser and the downcomer of 1.33 resulted in the lowest superficial liquid velocity of 0.076 m/s in the riser at a relative low superficial air velocity of 0.011 m/s and a carrier density of 1030 kg/m3. This bioreactor design enabled longest retention time of particles in the oxygenated riser.
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•Low growth rate and competing heterotrophs continue to challenge nitrification.•Low liquid velocity and higher carrier fraction in riser enhance nitrification.•A three phase ...mathematical model was applied for predicting low liquid velocity.•Geometry, aeration, and carrier characteristics were major design criteria.•Construction of a 1000L pilot plant verified the design and liquid velocity.
In this study, a 1000L pilot scale internal loop airlift bioreactor was operated and compared to a mathematical model to determine the best design for optimal supply of oxygen for nitrification and sufficient air for biomass fluidization. The design model is based on parameters such as geometry, carrier density, and airflow of the 1000L pilot scale bioreactor. The model predicts a range of superficial air velocities (0.009–0.013m/s) under which the airlift bioreactor was fluidized. Three superficial air velocities (0.009m/s, 0.011m/s and 0.013m/s) were experimentally tested in the pilot plant and the obtained circulation velocities were compared with the predicted design scenarios. The predicted velocity was in agreement with the measured velocity. The aim of the mathematical model and the calculations of different geometry scenarios was to define the optimal geometry design for the physical model. The results show that the ratio of the cross-sectional area between the riser and the downcomer of 1.33 resulted in the lowest superficial liquid velocity of 0.076m/s in the riser at a relative low superficial air velocity of 0.011m/s and a carrier density of 1030kg/m3. This bioreactor design enabled longest retention time of particles in the oxygenated riser.
The identification of earth pressures acting on in-service transportation tunnel linings is essential for their health monitoring and performance prediction, especially for those exhibiting poor ...structural performance. Since pressure gauges incur substantial costs, the inversion of pressures based on easily observed structural responses, such as deformations, is desirable. The inherent challenge in this inverse problem lies in the non-uniqueness of solutions, which arises from the fact that various pressures can yield structural responses fitting equally well with the observed data. However, existing approaches for pressure inversion predominantly rely on a deterministic framework, often neglecting a detailed discussion on this non-uniqueness. In addressing this gap, this study introduces a Bayesian approach. The proposed statistical framework enables the quantification of uncertainty induced by non-uniqueness in inversion results. The analysis identifies the uniform component in distributed pressures as the primary source of non-uniqueness. The mitigation of solution non-uniqueness can be achieved by increasing the quantity of deformation data or incorporating an observation of internal normal force in a tunnel lining -- the latter proving to be notably more effective. The practical application in a numerical case demonstrates the effectiveness of this approach and the associated findings. In addition, our investigation recommends maintaining deformation measurement accuracy within the range of -1, 1 mm to ensure satisfactory outcomes. Finally, deficiencies and potential future extensions of this approach are discussed.
We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical ...reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection-diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations.
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