Hollow fibre membrane bioreactors provide a fast and efficient method for engineering functional tissue for use in medical treatments. Flow is utilised to overcome mass transport limitations by perfusing a nutrient-rich culture medium through the fibre lumen, which can then transport along the fibre lumen or across the porous membrane wall. Cells seeded at the outer membrane wall consume the nutrient and subsequently produce waste metabolites, which are transported away through an external extra-capillary space (ECS) along with excess nutrient. We present and investigate a two-dimensional axisymmetric model for fluid flow and solute transport through a single-fibre bioreactor configuration, with cells seeded to the external fibre wall. Fluid flow is modelled by steady lubrication and Darcy equations, which are coupled to the solute transport problem modelled by a system of advection–diffusion equations, supplemented with a reaction term to model the cell layer. Our model analysis reveals how spatially varying wall permeability distributions can be utilised to provide uniform nutrient delivery to a spatially uniform, homogeneous cell population. We also reveal how maximising the transmural pressure drop across the membrane wall is the dominant mechanism for waste removal rather than traditional experimental methods of flushing the ECS.

Membranes define the outer boundary of the living protoplast and the internal compartmentation of plant cells. From a structural point of view, membranes consist of a lipid bilayer and proteins essential for functions such as solute transport, signal transduction, and numerous metabolic reactions. While membranes can represent a significant barrier to the free movement of many solutes, those with sufficient lipid solubility may move across membranes by dissolving into the lipid bilayer. However, selective membrane transport is generally observed for hydrophilic solutes such as mineral nutrients and cell metabolites. Such selective transport requires an input of metabolic energy, and in plants this occurs via the production of proton electrochemical gradients across the membrane by substrate- (ex. ATP, PPi) driven proton pumps. Selective solute transport is then mediated by membrane-associated secondary transport systems which utilize the proton electrochemical gradient to drive the transport process. This review of membrane structure and transport system function provides a background for a further examination of herbicide interactions with plant membranes.

A groundwater model was set up to study far-field transport for the potential of a radioactive waste repository the Boom Clay in the Netherlands. The existing national groundwater model, the Netherlands Hydrological Instrument, was extended in the vertical direction to include geological formation up to and beyond the Boom Clay. As the amount of hydrogeological data in the deeper subsurface is limited, simplifications in the model schematisation were necessary. Moreover, nationwide data about the tops and bottoms of many of the deeper geological formations and their members are lacking and required interpolation. Finally, values for hydrogeological parameters, such as porosity and hydraulic conductivity, are also lacking for the deeper formations. These values were estimated using relationships with depth and lithology. Moreover, no quantitative data about heterogeneity within the deeper geological formations or its members were available.

In the Dutch research programme on the geological disposal of radioactive waste (OPERA), the post-closure safety of a generic repository is assessed in either Boom Clay or rock salt. Disposal of Dutch radioactive waste is not foreseen in the next decades and a choice of host rock has not been made. In the early, conceptual phase of the radioactive waste disposal process in the Netherlands no potential repository locations were selected and a groundwater flow model for the entire Netherlands was build. As a starting point a geological disposal facility is assumed to be present at a depth of at least 500 m within a Boom Clay formation of 100 m in order to be able to make an assessment of post-closure safety with this geological formation in a disposal concept. With these assumptions, a general idea of potential flow patterns has been obtained and broken down into pathline trajectories. These trajectories were calculated to achieve input for the potential transport of radioactive isotopes (radionuclides) from this waste in the Netherlands after the closure of a disposal facility in Boom Clay.

The groundwater flow patterns in the deeper subsurface strongly resemble the larger scale flow patterns in the shallow subsurface, with flow from infiltration areas in the east and the south of the Netherlands towards to seepage areas of the polders in the west and the northern part of the country or towards the river valleys of the Rhine and IJssel. Groundwater flow velocities, however, are much lower in the deeper part of the model and consequently travel times are much larger. The conservative travel times from the pathlines range from a few 1000 years to more than 10,000,000 years depending on the location for the repository. Longer travel times are obtained for locations with a downward groundwater flow in the Boom Clay.

Because of the simplifications in the model schematisation and the uncertainty in the model parameters, the present results should only be considered as a first indication. Moreover, the model could not be validated due to a lack of validation data. However, the insight gained with the model may help to design a data collection strategy for dedicated model validation, such as measuring the hydraulic gradient over the Boom Clay to validate downward flow in the Boom Clay to obtain the necessary data for a post-closure safety assessment.

This study presents a numerical solution to the three-dimensional solute transport in heterogeneous media by using a layer-integrated approach. Omitting vertical spatial variation of soil and hydraulic properties within each layer, a three-dimensional solute transport can be simplified as a quasi-three-dimensional solute transport which couples a horizontal two-dimensional simulation and a vertical one-dimensional computation. The finite analytic numerical method was used to discretize the derived two-dimensional governing equation. A quadratic function was used to approximate the vertical one-dimensional concentration distribution in the layer to ensure the continuity of concentration and flux at the interface between the adjacent layers. By integration over each layer, a set of system of equations can be generated for a single column of vertical cells and solved numerically to give the vertical solute concentration profile. The solute concentration field was then obtained by solving all columns of vertical cells to achieve convergence with the iterative solution procedure. The proposed model was verified through examples from the published literatures including four verifications in terms of analytical and experimental cases. Comparison of simulation results indicates that the proposed model satisfies the solute concentration profiles obtained from experiments in time and space.

We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.