On the efficient computation of high-order derivatives for implicitly defined functions

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order derivatives. A technique based on algorithmic differentiation is presented which allows for a precise calculation of higher-order derivatives. The method can be widely applied even for the case of only numerically solvable, implicit dependencies which totally hamper a semi-analytical calculation of the derivatives. As a demonstration the method is applied to a quantum field theoretical physical model. The results are compared with standard numerical derivative methods.Comment: 11 pages, 4 figures, to appear in Comput. Phys. Commu

Trace Contaminant Removal using Hybrid Membrane Processes in Water Recycling

Water recycling plays an essential role in integrated water management, especially in an arid country like Australia but also worldwide [1]. Water recycling, however, has suffered extensive constraints due to "toilet to tap" media campaigns and "yuck factor" attitudes in the community. The support of the community for water recycling projects generally decreases as the personal contact with the recycled water increases [2]. Some of the very valid concerns of the community stem from uncertainties involved in water recycling, such as the issue of persistent organic pollutants (POPs) potentially present in recycled waters or the ever growing group of endocrine disrupting chemicals have been of particular concern to sections of the community. Endocrine disrupters have the potential to interfere with our normal growth, development and reproduction. Modulation of that system could cause severe adverse health effects. Industrial chemicals, consumer chemicals and chemicals in the environment can be endocrine disrupters that mimic, enhance or inhibit the action of hormones [3, 4]. Sewage disposal to water sources may be a major exposure pathway for pharmaceuticals, synthetic and natural hormones, industrial chemicals to humans and wildlife, directly and via the food chain. This concerns disposal of treated effluents and applications of recycled water. This paper aims to address some of the uncertainties and risks involved in recycling technology and aims to stress caution and the need for well designed recycling projects. This risk expands to water treatment in situations where contaminated waters are treated

Diagnosing fouling of hollow fibre MF membranes in wastewater reclamation

Fouling in membrane filtration processes is problematic but inevitable as it occurs with the retention of contaminants that accumulate on the membrane surface. The causes of fouling are often specific, depending upon feed water constituents, the membrane, and the operation regime. Therefore, it is desirable that a thorough investigation is performed on fouled membrane elements of the affected plant. This technique is known as membrane autopsy , which identities the cause of poor membrane performance, and hence, gives the opportunity to rectify or mitigate the problem and improve future plant design. In this study, the cause of membrane fouling at a small water recycling plant using a hollow fibre micro filtration (MF) system is investigated. A membrane autopsy protocol has been developed for water recycling applications that consists of four major steps: I) tensile testing to investigate the membrane mechanical integrity, (II) direct visual inspection, III) membrane surface analysis using field-emission environmental scanning electron microscopy (FESEM) (as well as atomic force microscopy (AFM) although it is not used in this case) techiques, and IV) foulant constituent analysis. Results obtained from this study indicate that the membrane has been fouled by a mixture of colloids and organic matters, enhanced by the presence of multivalent cations. Possible measures to mitigate fouling in this particular case have also been suggested

Removal of adsorbing estrogenic micropollutants by nanofiltration membranes in cross-flow – experiments and model development

Nanofiltration (NF) can be used in water and wastewater treatment as well as water recycling applications, removing micropollutants such as hormones. Due to their potential health risk it is vital to understand their removal mechanisms by NF membranes aiming at improving and developing more effective and efficient treatment processes. Although NF should be effective and efficient in removing small molecular sized compounds such as hormones, the occurrence of adsorption onto polymeric membranes results in performances difficult to predict and with reduced effectiveness and efficiency. This study aims firstly at defining, understanding and quantifying the relevant filtration operation parameters and, secondly, in identifying the physical mechanisms of momentum and mass transfer controlling the adsorption and transport of hormones onto polymeric NF membranes in cross-flow mode. The hormones estrone (E1) and 17-b-estradiol (E2) were chosen as they have very high endocrine disrupting potency. The NF membranes used and tested were the NF 270, NF 90, BW30, TFC-SR2 and TFC-SR3 since they have a wide span of pore sizes. The first step is to experimentally acquire the knowledge of how fluid flow hydrodynamics and mass transfer close to the membrane affect hormone adsorption. The focus will be particularly on the effect of operating pressure, circulating Reynolds numbers (based on channel height, Reh) and hormone feed concentration. These hydrodynamic parameters play an important role in concentration polarisation development at the membrane surface. A Reh increase from 400 to 1400 for the NF 270 membrane caused the total mass adsorbed of E1 and E2 to decrease from 1.5 to 1.3 ng.cm-2 and 0.7 to 0.5 ng.cm-2, respectively. In contrast, a pressure increase from 5 to 15 bar yielded an increase in the adsorbed mass of E1 and E2 from 1.0 to 1.8 ng.cm-2 and 0.5 to 0.7 ng.cm-2, respectively. Moreover, increasing hormone feed concentration caused an increase in the mass adsorbed for both hormones. These observations led to the conclusion that adsorption is governed by the initial concentration at the membrane surface which, in turn, depends on the hormone feed concentration, operating Reh and pressure. Membrane retention, however, depends on the initial polarisation modulus, defined as the ratio between the initial concentration at the membrane surface and the initial feed concentration. The same trends were obtained for the TFC-SR2 membrane. However, this membrane has a much lower permeability compared to the NF 270 one (7.2 vs 17 L.h-1.m-2.bar-1, respectively) and concentration polarisation is less severe. The experimental variations in mass adsorbed and retention as a function of the operating filtration parameters (Reh and pressure) were therefore lower. Based on these experimental results, a sorption model was developed. This model predicts well both feed and permeate transient concentrations for both hormones and membranes (NF 270 and TFC-SR2) in the common range of operating pressures and Reh of spiral-wound membrane modules. The model was further applied for E2 in the presence of background electrolyte, yielding good predictions. These findings are an important advancement in determining which membrane would be more suitable to effectively remove hormones with a substantial reduction of experimental work. The above-mentioned developed model does not give insight into the phenomena occurring inside the membrane since it focuses on the feed conditions. However, membrane characteristics, such as material and pore radius were found to have an impact in adsorption and retention of hormones. It was found experimentally that polyamide, from which the active layer of the NF membranes is made, adsorbs three times more mass of hormone than any other polymers constituting the membranes. Since this active layer is the membrane selective barrier of the membrane that is in contact with the largest hormone concentration (due to concentration polarization in the feed solution) it is concluded that the active layer adsorbs most of the hormones. Further experimental work carried out in this thesis showed that increasing the pore radius from 0.32 nm to 0.52 nm increased the E2 mass adsorbed from 0.17 ng.cm-2 to 1.1 ng.cm-2 and decreased the retention from 88% to 34%. These results show that the wider the pore, the larger the quantity of hormone that penetrates (i.e. partitions) inside the membrane and, therefore, the more the membrane adsorbs the hormone. For membranes of similar pore radius, the membrane with larger internal surface area was found to adsorb more. All the previous results led to the establishment of a new model for the hormone transport inside the membrane pore taking convection, diffusion and adsorption into account. Since the differential equation describing the transport with adsorption inside the pore has no analytical solution, a numerical model based on the finite-difference approach was applied. With such a model, its validation against experiments and parametric studies it was possible to understand the transport mechanisms of adsorbing hormones through NF membranes. The results show that for low pressures the hormone transport is diffusion dominated. In contrast, for higher pressures (above 11 bar) the transport is convection dominated, showing that a purely diffusion transport model does not describe well the actual transport phenomena of hormones in NF membranes. Furthermore, it was found that two similar molecules can behave very differently in terms of adsorption on the membrane. E1, which adsorbs 20% more than E2 in static mode, being slightly smaller than E2, partitions more inside the membrane pore and adsorbs double under filtration conditions. This study contributes to illuminating the adsorption mechanisms of hormones onto NF membranes by understanding what parameters control adsorption such as hydrodynamics, materials, structure, etc. This not only identifies a potential problem in large scale applications, but it also provides an understanding of the mechanisms involved in the removal of these hormones and a tool that can be used to design future membranes for the improvement of micropollutant removal

Distributed task rescheduling with time constraints for the optimization of total task allocations in a multirobot system

This paper considers the problem of maximizing the number of task allocations in a distributed multirobot system under strict time constraints, where other optimization objectives need also be considered. It builds upon existing distributed task allocation algorithms, extending them with a novel method for maximizing the number of task assignments. The fundamental idea is that a task assignment to a robot has a high cost if its reassignment to another robot creates a feasible time slot for unallocated tasks. Multiple reassignments among networked robots may be required to create a feasible time slot and an upper limit to this number of reassignments can be adjusted according to performance requirements. A simulated rescue scenario with task deadlines and fuel limits is used to demonstrate the performance of the proposed method compared with existing methods, the consensus-based bundle algorithm and the performance impact (PI) algorithm. Starting from existing (PI-generated) solutions, results show up to a 20% increase in task allocations using the proposed method.EPSRC Grant EP/J011525/