A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6

Cell-element simulations to optimize the performance of osmotic processes in porous membranes

We present a new module of the software tool PoreChem for 3D simulations of osmotic processes at the cell-element scale. We consider the most general fully coupled model (see e.g., Sagiv and Semiat (2011)) in 3D to evaluate the impact on the membrane performance of both internal and external concentration polarization, which occurs in a cell-element for different operational conditions. The model consists of the Navier–Stokes–Brinkman system to describe the free fluid flow and the flow within the membrane with selective and support layers, a convection–diffusion equation to describe the solute transport, and nonlinear interface conditions to fully couple these equations. First, we briefly describe the mathematical model and discuss the discretization of the continuous model, the iterative solution, and the software implementation. Then, we present the analytical and numerical validation of the simulation tool. Next, we perform and discuss numerical simulations for a case study. The case study concerns the design of a cell element for the forward osmosis experiments. Using the developed software tool we qualitatively and quantitatively investigate the performance of a cell element that we designed for laboratory experiments of forward osmosis, and discuss the differences between the numerical solutions obtained with the full 3D and reduced 2D models. Finally, we demonstrate how the software enables investigating membrane heterogeneities

3D morphology design for forward osmosis

We propose a multi-scale simulation approach to model forward osmosis (FO) processes using substrates with layered homogeneous morphology. This approach accounts not only for FO setup but also for detailed microstructure of the substrate using the digitally reconstructed morphology. We fabricate a highly porous block copolymer membrane, which has not been explored for FO heretofore, and use it as the substrate for interfacial polymerization. The substrate has three sub-layers, namely a top layer, a sponge-like middle layer, and a nonwoven fabric layer. We generate a digital microstructure for each layer, and verify them with experimental measurements. The permeability and effective diffusivity of each layer are computed based on their virtual microstructures and used for FO operation in cross-flow setups at the macro-scale. The proposed simulation approach predicts accurately the FO experimental data

On convergence of a discrete problem describing transport processes in the pressing section of a paper machine including dynamic capillary effects: one-dimensional case

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system of equations which describe filtration process in the pressing section of a paper machine. Two flow regimes appear in the modeling of this problem. The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards approach together with a dynamic capillary pressure model. The finite volume method is used to approximate the system of PDEs. Then the existence of a discrete solution to proposed finite difference scheme is proven. Compactness of the set of all discrete solutions for different mesh sizes is proven. The main Theorem shows that the discrete solution converges to the solution of continuous problem. At the end we present numerical studies for the rate of convergence

A one-dimensional model of the pressing section of a paper machine including dynamic capillary effects

This work presents the dynamic capillary pressure model (Hassanizadeh, Gray, 1990, 1993a) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure-saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure-saturation relation shows significant influence on the distribution of water pressure. The behaviour of the solution agrees with laboratory experiments (Beck, 1983)

A Two-Dimensional Model of the Pressing Section of a Paper Machine Including Dynamic Capillary Effects

The paper production is a problem with significant importance for the society and it is a challenging topic for scientific investigations. This study is concerned with the simulations of the pressing section of a paper machine. A two-dimensional model is developed to account for the water flow within the pressing zone. Richards’ type equation is used to describe the flow in the unsaturated zone. The dynamic capillary pressure–saturation relation proposed by Hassanizadeh and co-workers (Hassanizadeh et al., 2002; Hassanizadeh, Gray, 1990, 1993a) is adopted for the paper production process. The mathematical model accounts for the co-existence of saturated and unsaturated zones in a multilayer computational domain. The discretization is performed by the MPFA-O method. The numerical experiments are carried out for parameters which are typical for the production process. The static and dynamic capillary pressure–saturation relations are tested to evaluate the influence of the dynamic capillary effect

A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6

On convergence of a discrete problem describing transport processes in the pressing section of a paper machine including dynamic capillary effects: one-dimensional case

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system of equations which describe filtration process in the pressing section of a paper machine. Two flow regimes appear in the modeling of this problem. The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards approach together with a dynamic capillary pressure model. The finite volume method is used to approximate the system of PDEs. Then the existence of a discrete solution to proposed finite difference scheme is proven. Compactness of the set of all discrete solutions for different mesh sizes is proven. The main Theorem shows that the discrete solution converges to the solution of continuous problem. At the end we present numerical studies for the rate of convergence

A one-dimensional model of the pressing section of a paper machine including dynamic capillary effects

This work presents the dynamic capillary pressure model (Hassanizadeh, Gray, 1990, 1993a) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure-saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure-saturation relation shows significant influence on the distribution of water pressure. The behaviour of the solution agrees with laboratory experiments (Beck, 1983)

The influence of porous media microstructure on filtration

We investigate how a filter media microstructure influences filtration performance. We derive a theory that generalizes classical multiscale models for regular structures to account for filter media with more realistic microstructures, comprising random microstructures with polydisperse unidirectional fibres. Our multiscale model accounts for the fluid flow and contaminant transport at the microscale (over which the media structure is fully resolved) and allows us to obtain macroscopic properties such as the effective permeability, diffusivity, and fibre surface area. As the fibres grow due to contaminant adsorption this leads to contact of neighbouring fibres. We propose an agglomeration algorithm that describes the resulting behaviour of the fibres upon contact, allowing us to explore the subsequent time evolution of the filter media in a simple and robust way. We perform a comprehensive investigation of the influence of the filter-media microstructure on filter performance in a spectrum of possible filtration scenarios