A recent experiment showed that, contrary to theoretical predictions, beyond a cutoff point grinding coffee more finely results in lower extraction. One potential explanation for this is that fine grinding promotes non-uniform extraction in the coffee bed. We investigate the possibility that this could occur due the interaction between dissolution and flow promoting uneven extraction. A low dimensional model in which there are two possible pathways for flow is derived and analysed. This model shows that, below a critical grind size, there is a decreasing extraction with decreasing grind size as is seen experimentally. In the model this is due to a complicated interplay between an initial imbalance in the porosities and permeabilities of the two pathways which is increased by flow and extraction, leading to the complete extraction of all soluble coffee from one pathway.

Moisture measurement in bauxite ore as it is being offloaded from a ship is a challenge that was brought to a European Study Group with Industry at the University of Limerick in 2017. A refinery asked the Study Group to confirm if it is possible to calculate the moisture content once every second, by passing microwaves through the ore as it is being offloaded at about 2 m/s. The microwaves experience a phase shift and signal attenuation, depending on the amount of moisture present. We review the theory of microwave transmission through a polarising medium, and we present and study the data produced by the microwave analyser. We explore the consequences of the method for measuring phase change, and the effects of noise on the phase shifts observed. We provide an algorithm for lifting the measured phase shifts from their restricted range to an unlimited range for the true phase shift. Concurrent measurements of ore height are central to the lifting algorithm. We note the effect of interference on the attenuation seen in data.

A viable data driven approach for determining dynamical systems describing engineering processes would be a valuable tool in condition monitoring. The application of the SINDy algorithm for dynamical system discovery is investigated in the context of a reciprocating compressor. A feasibility study was carried out in which an attempt was made to recover a model of the compressor from synthetic data obtained from that model. A simplified model of the compressor with two degrees of freedom was developed from an existing model. Following the SINDy approach a parsimonious model was constructed from a large library of functions using sparse regression. This model has the same structure as and similar coefficients to the original model thus demonstrating the potential of this approach.

When frying potato snacks, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the snacks are properly cooked. Here we propose a multiphase mathematical model for the frying of potato snacks, where water inside the dough is evaporated from both the top and bottom surfaces of the snack at two receding evaporation fronts. The vapor created at the top of the snack bubbles away to the surface, whereas the vapor released from the bottom surface forms a buoyant blanket layer. By asymptotic analysis, we show that the model simplifies to solving a one-dimensional Stefan problem in the snack coupled to a thin-film equation in the vapor blanket through a nonlinear boundary condition. Using our mathematical model, we predict the change in the snack density as a function of time and investigate how lift-off time depends on the different parameters of the problem.

Low Earth Orbit is becoming crowded with satellites. Updating estimates of collision probabilities is important as new deployments are authorised but is difficult because only limited information is given. This report investigates developing analytic estimates of collision probabilities. A survey of approaches reported in the literature is carried out. A collision involving a satellite from the Iridium cluster is reviewed. A simple analytic expression for the collision probability between two satellites is derived using the smallness of several dimensionless ratios appearing in the problem. Single collision probabilities are then extended to orbital planes populated by n satellites with the aim of finding the optimal point at which to traverse such an orbit. This report demonstrates that analytic estimates relevant to the problem can be made. Further work should focus on: making these estimates rigorous by using a formal asymptotic approach, considering multiple orbital planes and introducing time dependence

Coffee is a beverage enjoyed worldwide and an active area of current research. The brewing of coffee also presents an opportunity to demonstrate the model, simulate and optimise paradigm in action. This is done in the context of brewing cafetiere coffee. Using a published model and experimental data we show that an existing model can be parameterised to simulate the results of that experiment. From here an optimisation step can be applied in which the experimental recipe can be changed to improve the quality of the coffee produced in the experiment. This activity is implemented in the NUMBAS platform.

(a) Espresso coffee is made by forcing hot water under high pressure througha compacted bed of finely ground coffee. (b) Drip filter brewing involves pouring hotwater over a loose bed of coarser coffee in a filter. In either method water flows through the bed, leaching soluble coffee components from the grains. Any undissolved solids in the fluid are filtered from the extract as the liquid leaves the filter. Credit: Kevin M. Moroney

Espresso is a beverage brewed using hot, high-pressure water forced through a bed of roasted coffee. Despite being one of the most widely consumed coffee formats, it is also the most susceptible to variation. We report a novel model, complimented by experiment, that is able to isolate the contributions of several brewing variables, thereby disentangling some of the sources of variation in espresso extraction. Under the key assumption of homogeneous flow through the coffee bed, a monotonic decrease in extraction yield with increasingly coarse grind settings is predicted. However, experimental measurements show a peak in the extraction yield versus grind setting relationship, with lower extraction yields at both very coarse and fine settings. This result strongly suggests that inhomogeneous flow is operative at fine grind settings, resulting in poor reproducibility and wasted raw material. With instruction from our model, we outline a procedure to eliminate these shortcomings.

When frying potato snacks, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the snacks are properly cooked. Here we propose a multiphase mathematical model for the frying of potato snacks, where water inside the dough is evaporated from both the top and bottom surfaces of the snack at two receding evaporation fronts. The vapour created at the top of the snack bubbles away to the surface, whereas the vapour released from the bottom surface forms a buoyant blanket layer. By asymptotic analysis, we show that the model simplifies to solving a one-dimensional Stefan problem in the snack coupled to a thin-film equation in the vapour blanket through a non-linear boundary condition. Using our mathematical model, we predict the change in the snack density as a function of time, and investigate how lift-off time depends on the different parameters of the problem.

Achieving a uniform extraction of soluble material from a porous matrix is a generic problem in various separation and filtration operations, with applications in the food processing, chemical and pharmaceutical industries. This paper describes models of fluid flow and transport of soluble material within a packed granular bed in the context of coffee extraction. Coffee extraction is described by diffusion of soluble material from particles of one or more representative sizes into fluid flowing through the packed bed. One-dimensional flow models are compared to computational fluid dynamics (CFD) models. A fine and a coarse coffee grind are considered. Model results are compared to experimental data for a packed cylindrical coffee bed and the influence of a change in geometry to a truncated cone is considered. Non-uniform flow in the truncated cone causes significant variation in the local extraction level. Coffee extraction levels during brewing are analysed using extraction maps and the degree of variation is represented on the industry standard coffee brewing control chart. A high variation in extraction yield can be expected to impart bitter flavours into the brew and thus is an important variable to quantify.

The aim of a drug eluting stent is to prevent restenosis of arteries following percutaneous balloon angioplasty. A long term goal of research in this area is to use modelling to optimise the design of these stents to maximise their efficiency. A key obstacle to implementing this is the lack of a mathematical model of the biology of restenosis. Here we investigate whether mathematical models of cancer biology can be adapted to model the biology of restenosis and the effect of drug elution. We show that relatively simple, rate kinetic models give a good description of available data of restenosis in animal experiments, and its modification by drug elution.

We provide a simple two-dimensional model of bubbly two-phase flow which can be used to investigate why waves form and propagate downward while a pint of Guinness is settling. We start out with the basic equations of the two-phase flow and use the large timescale difference of beer convection and rising bubbles in order to treat the convection flow as quasistatic. Using this argument we further simplify the two-phase mixture equations to that of a single liquid whose density varies with bubble concentration. A stability analysis shows that waves can occur through an instability analogous to the Kelvin-Helmholtz instability which forms in parallel shear flow. We provide a description of the form of these waves, and compare them to observations. Our theory provides a platform for the description of waves in more general bubbly two-phase shear flows.

A surprising phenomenon witnessed by many is the sinking bubbles seen in a settling pint of stout beer. Bubbles are less dense than the surrounding fluid so how does this happen? Previous work has shown that the explanation lies in a circulation of fluid promoted by the tilted sides of the glass. However, this work has relied heavily on computational fluid dynamics (CFD) simulations. Here, we show that the phenomenon of sinking bubbles can be predicted using a simple analytic model. To make the model analytically tractable, we work in the limit of small bubbles and consider a simplified geometry. The model confirms both the existence of sinking bubbles and the previously proposed mechanism.

We report progress towards developing a mathematical model that can be used to optimise the design of a novel power take off unit for a wave energy generator. We show that the power take off unit can be considered as a non-smooth, dissipative dynamical system. We derive equations of motion using the Lagrangian framework, incorporating a Rayleigh dissipation function and discuss a procedure for generating approximate analytical solutions.

A model for the growth of lead sulphate particles in a gravity separation system from the crystal glassware industry is presented. The lead sulphate particles are an undesirable byproduct, and thus the model is used to ascertain the optimal system temperature configuration such that particle extraction is maximised. The model describes the evolution of a single, spherical particle due to the mass flux of lead particles from a surrounding acid solution. We divide the concentration field into two separate regions. Specifically, a relatively small boundary layer region around the particle is characterised by fast diffusion, and is thus considered quasi-static. In contrast, diffusion in the far-field is slower, and hence assumed to be time-dependent. The final system consisting of two nonlinear, coupled ordinary differential equations for the particle radius and lead concentration, is integrated numerically.

In this paper, we outline the first model to describe drug elution from a drug-filled stent. This novel polymer-free drug-eluting stent stores the therapeutic drug in the inner layer of a tri-layer wire. This inner layer acts a reservoir releasing the drug through laser-drilled holes on the outer surface of the stent struts. We simplify the general model using the assumption of low drug solubility and consider a special case where the dissolution occurs in a uniform downward direction. The main advantage of our simplified model is the ability to achieve analytical solutions. These solutions allow for calculating the drug release profile rapidly and for identifying the dependence of the various parameters of the system. We find that the duration of drug elution is prolonged when the solubility or diffusivity of the drug decreases.

Technology for promoting nucleation is important in a number of contexts, for instance degassing carbon dioxide lakes, designing champagne glasses and stout beer widgets. A new design of stout beer widget has recently been proposed which makes use cellulose fibres to initiate foaming in canned stout beers. However, our current scientific understanding of the nucleation of bubbles by cellulose fibres is incomplete, making it impossible to optimise this technology. One particularly poorly understood aspect is the detachment of bubbles from a gas pocket in the fibre. We report experimental and theoretical results towards a model of the detachment based on a model of Rayleigh-Plateau instability including a disjoining pressure.

Drug-eluting stents have been used widely to prevent restenosis of arteries following percutaneous balloon angioplasty. Mathematical modelling plays an important role in optimising the design of these stents to maximise their efficiency. When designing a drug-eluting stent system, we expect to have a sufficient amount of drug being released into the artery wall for a sufficient period to prevent restenosis. In this paper, a simple model is considered to provide an elementary description of drug release into artery tissue from an implanted stent. From the model, we identified a parameter regime to optimise the system when preparing the polymer coating. The model provides some useful order of magnitude estimates for the key quantities of interest. From the model, we can identify the time scales over which the drug traverses the artery wall and empties from the polymer coating, as well as obtain approximate formulae for the total amount of drug in the artery tissue and the fraction of drug that has released from the polymer. The model was evaluated by comparing to in-vivo experimental data and good agreement was found.

Many Feed-in Tariff designs exist. This paper provides a framework to determine the optimal design choice through an efficient allocation of market price risk. Feed-in Tariffs (FiTs) incentivise the deployment of renewable energy technologies by subsidising remuneration and transferring market price risk from investors, through policymakers, to a counterparty. This counterparty is often the electricity consumer. Using Stackelberg game theory, we contextualise the application of different FiT policy designs that efficiently divide market price risk between investors and consumers, conditional on risk preferences and market conditions. Explicit consideration of policymaker/consumer risk burden has not been incorporated in FiT analyses to date. We present a simulation-based modelling framework to carry this out. Through an Irish case study, we find that commonly employed flat-rate FiTs are only optimal when policymaker risk aversion is extremely low whilst constant premium policies are only optimal when investor risk aversion is extremely low. When both policymakers and investors are risk averse, an intermediate division of risk is optimal. We provide evidence to suggest that the contextual application of many FiT structures is suboptimal, assuming both investors and policymakers are at least moderately risk averse. Efficient risk allocation in FiT design choice will be of increasing policy importance as renewables deployment grows.