Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

Dynamic simulation of red mud washers used in aluminum industries

Clarifier-Thickener equipment is used in a wide range of continuous sedimentation and sludge thickening processes where solid particles from continuous inflow mixtures are separated from the liquid. In this operation, the concentration of solids increases due to settling, so that the formation of a thicker bed is inevitable with time. Under optimal operating conditions, it is always possible to obtain two discharges from these vessels: a highly concentrated suspension at the bottom (underflow), and a clarified liquid stream at the top of the equipment (overflow or effluent). In the Bayer Process an insoluble sub-product is formed as a result of the digestion of the bauxite ore with caustic soda. This product is called “red mud” and it has to be continuously removed by settlers or thickeners/clarifiers. This project proposes the simulation of a continuous thickener/clarifier in order to predict the concentration profile and the height of the mud level (process controlled variable) as alternative to current measurement system, that contains long delay discrete sampling time(15 minutes each measurement). The simulation also, can provide an option of creating a knowledge base for off-line control. The project essentially involves two methods of simulation, namely mathematical modelling and neural networks. The model based applies a kinematic model to approximate the process behaviour using both, equipment and suspension characteristics. On the other hand, due to the large amount of historical data, neural network is proposed for system identification. The first method is based on the solution of a highly nonlinear model, based on kinematic modelling of sedimentation extended to flocculated suspensions. This approach uses a conservative finite difference scheme of the upwind type for solving an initial boundary value problem (IBVP). The successful simulation of the equipment and further validation of the mathematical model are then achieved once the properties of the suspensions have been determined. These properties are related to flux batch settling and solid stress functions, whose parameters can be obtained experimentally.On-site testing of the characteristics of red mud was conducted at the Rio Tinto Yarwum Alumina Refinery in Queensland. The settling properties of the suspension were determined via batch settling. For measurement of the rheology properties, the vane technique was used employing a Haake VT 550 rheometer. The results of the simulation showed that the concentration profile and height of the heavy mud level can be determined via a steady state model for a given underflow concentration. These results, however, were not in good agreement with measured data. The second method of simulation involved the use of Rio Tinto Yarwun historical data to develop a neural model in order to obtain a relationship between process variables. This approach has the advantage that no mathematical model is needed. With this method, historical data (continuous data) are obtained and analysed, and daily averages of the variables involved in the process are calculated. Different network architectures are tested according to the washer process. Ultimately, two networks were developed to describe washer dynamics.

Convection-diffusion-reaction models of sedimentation : Numerical approximation, analysis of solutions and inverse problems

The core of this Doctoral thesis is mainly based in the studies of one-dimensional initial-boundary value problems, which are given by a single non-linear hyperbolic partial differential equation (PDE) with non-convex flux function, or by a system of strongly degenerate parabolic PDEs, for the simulation of sedimentation processes of solid particles immersed in a fluid. Particular attention is paid to the case of settling in vessels with varying cross-sectional area. Sedimentation processes are widely used in wastewater treatment (WWT) and mineral processing, where accurate model calibration and reliable simulators are needed. Among the topics covered in the research presented in this thesis are the construction of entropy solutions, the development and implementation of reliable numerical schemes for hyperbolic PDEs (and systems of PDEs), the solution of inverse problems of flux identification, and the disseminationof results to the applied sciences.The outputs of this thesis can be divided into three parts. The first part (Papers I to III) contains the construction of the entropy solutions for the PDE modeling the batch sedimentation in vessels with non-constant cross-sectional area(Paper I and II) and for the PDE modeling centrifugal sedimentation (Paper III). The problem is in both cases solved by the method of characteristics and the types of solutions are distinguished mainly depending on the initial value.Paper II contains the description and solution of the inverse problem of flux identification for the model of sedimentation in conical vessels due to gravity, and Paper III the inverse problem for the model of centrifugal settling. In bothcases, the solution of the inverse problem has the advantage that almost the entire flux function can be identified from only one experiment. These identification methods mean a significant advantage in comparison with the classic one, made by standard tests in cylindrical vessels, in terms of the portion of flux identified. An algorithm necessary for the identification from discrete data is also presented in each problem (Papers II and III).The second part (Papers IV to VI) includes the development of numerical methods for the simulation of sedimentation in WWT. In Paper IV, a numerical scheme for the case of continuous and batch sedimentation in vessels withvarying cross-sectional area is studied. An advantageous CFL condition is derived as an improvement over other numerical methods for the same kind of application. Simulations of continuous and batch settling are also included.Papers V and VI consider reactive settling, where the unknown is a vector of solid and liquid components, and each model is described by a coupled system of convection-diffusion-reaction PDEs. In Paper V, a method-of-lines formulation for the approximation of the model equations is introduced. This formulation has the advantage that it can be solved by any time stepping solver, such as those commonly used in the WWT community where ordinary differentialequations (ODEs) should be solved simultaneously with the PDE system. Additionally, an invariant-region property is proved for the scheme and simulations of interesting scenarios are presented. In Paper VI, sequencing batch reactors (SBRs) are studied. The model equations for the SBRs are derived following Paper V, but with the addition that in this case, the extraction and filling of mixture lead to a moving-boundary problem. The movement of the boundary is described by an ODE which can be precomputed. A reliable numerical scheme that preserves the mass is proposed and numerical simulations for the case of denitrification are shown.The third part (Papers VII and VIII) is related to applications and dissemination of the flux identification methods to the applied sciences. The validation of the inverse problem for batch settling in conical vessels is presented in Pa-per VII. The validation was carried out with data taken from activated sludge collected from the WWT plant in Västerås, Sweden. Paper VIII contains a review of flux identification methods related to PDE models for sedimentation processes. Advantages and disadvantages are discussed, and simulations of identified fluxes with the methods under study are presented.In Chapter 4 the numerical simulation of multidimensional batch sedimentation is discussed and two-dimensional simulations are presented

Development of Advanced Control for Paste Thickeners

A paste thickener produces underflow with a high solids concentration by removing water from a dilute suspension. Paste thickening technology has been gradually gaining acceptance in the mining industry. Some of the main challenges in thickening are the complex dynamics due to severe interactions between operating parameters and the existence of a large range of process time constants. Additionally, being a downstream process, paste thickeners suffer from fluctuating feed conditions from upstream processes. Furthermore, the operation of paste thickeners involves a tight operating window as the underflow properties are sensitive to small changes in concentration. The above challenges motivate this work to develop a systematic advanced control strategy, in particular, model predictive control (MPC) to effectively and efficiently regulate this complex, multivariable process. In this thesis, the sedimentation-consolidation model is adopted and analysed to identify important dynamical features and the key process parameter which will be considered in control system design. The model is then validated using industrial plant data. An extended Kalman filter is developed to estimate in real-time the key process parameter which affects thickening dynamics significantly. Through extensive simulation studies, it has been demonstrated that the proposed MPC approach can deliver a higher underflow solids concentration and a better regulated underflow removal rate. In addition to the basic MPC, several important extensions are developed based on three practical considerations. Firstly, it is illustrated that incorporating ``future'' time-varying constraints in the MPC can help deal with the control difficulty arising from the co-disposal of underflow and coarse reject, leading to improved control performance. Secondly, an MPC with a non-uniformly spaced optimisation horizon is proposed to deal with the timescale multiplicity of thickening. The proposed approach provides similar levels of performance as compared to a conventional MPC with the advantage of reduced computational complexity. A stability ensuring condition for the proposed MPC is also developed. Thirdly, an MPC is developed to control both underflow solids concentration and rake torque. The nonlinear rake torque control problem is converted into a linear MPC problem, reducing the computational complexity of the optimisation problem. Simulation studies have shown that the proposed approach can effectively control the underflow solids concentration while minimising the rake torque, potentially preventing serious operational problems such as underflow blockage

Optimization of treatment options to enable the use of Abandoned Mine Drainage (AMD) for hydraulic fracturing in Marcellus Shale

Abandoned Mine Drainage (AMD) is a promising make-up water source for hydraulic fracturing in Marcellus Shale as it is often available in the vicinity of planned natural gas wells and can be used to reduce fresh water utilization. When mixing AMD with Marcellus Shale produced water, the high sulfate concentration in AMD could precipitate a significant amount of barium and strontium in produced water as a pre-treatment for produced water prior to reuse for hydraulic fracturing. In order to promote the recycling of AMD and produced water, it is necessary to optimize treatment processes to remove solids created by mixing these waters. Conventional coagulation/flocculation process was optimized with respect to mixing/settling time, coagulant dosage, pH, sludge handling and clarifier design using two different Marcellus Shale flowback waters and actual AMD that is available in their vicinity. This conventional process is compared against ballasted flocculation that has smaller footprint and may be more suitable as a mobile treatment system. The mixing ratio of flowback water and AMD was studied in order to meet potential effluent quality requirement for sulfate of 100 mg/L that may be imposed by gas companies and was compared to chemical equilibrium calculations by Phreeqc and MINEQL models. The effluent quality from the conventional and ballasted flocculation processes are comparable with turbidity below 5 NTU despite the fact that the contact time required for the ballasted flocculation is just 10 min compared to 1 hour required for conventional treatment process. Careful optimization of the AMD : Marcellus Shale Produced water mixing ratio is needed to ensure effluent quality with respect to sulfate concentration. The solids produced by ballasted flocculation had better settling characteristic and higher sludge density than the solids produced by conventional process

Effect of Sludge Age and Concentration on the Settling Velocity of Sludge

Bioenvironmental Engineerin

Hyperbolic Conservation Laws

[no abstract available

Design and development of an on-line sedimentation analyser

This thesis describes the design and development of a multiple vibrating reed analyser for the routine on-line procurement of sedimentation kinetic data in two phase media. Briefly, each reed system comprises a stainless steel rod pinned at an intermediate point along its length. One end is exposed to the settling suspension whilst the other is driven into transverse vibration at resonance using an alternating current electromagnet. The principle of operation of the device relies on the fact that the resonance frequency of a stiff reed performing simple harmonic motion in a fluid medium is directly related to the fluid bulk density. In the case of a settling solid/liquid suspension, or a two-phase liquid dispersion containing a heavier phase, the fluid bulk density and hence the hydrodynamic head decreases with time as the heavier phase settles. Sedimentation kinetic data including settling velocities and flux profiles are in turn obtained by continuous monitoring of the resonance frequencies of a number of reeds positioned at set levels along a settling tank. The feasibility of operation of the analyser has been successfully verified in conjunction with a variety of model and industrially relevant systems. The former include mono and polydisperse glass ballotini / water mixtures with solids particle size and concentration ranges 55 - 200 μ and 1.75 - 2.81 % v/v respectively. Measured settling velocities are in excellent accord (ca. ± 0.1 %) with those obtained from direct visual observation of suspension-clear liquid interfaces. The industrially relevant systems include kaolin/water suspensions with solids concentrations as high as 20 % v/v and oil / water emulsions with light phase concentrations in the range 30 - 50 % v/v. The performance of a number of empirical models including those proposed by Richardson and Zaki (1954), Garside and Al-Dibouni (1977), Bamea and Mizrahi (1973), Reed and Anderson (1980), Batchelor (1972) and Happel (1958) has been evaluated by comparing predicted settling velocities with those obtained using the analyser. We find that the model proposed by Richardson and Zaki produces the best agreement (ca ± 1.3 %) whilst Happel's model (1958) produces the worst results (ca ± 24 %). For polydisperse systems we observe that measured interface velocities correspond to Stokes (1851) particle sizes which are in reasonable agreement (ca. ± 6 %) with the experimentally determined smallest significant particle size in the sample distribution. Also, as the initial solids concentration increases, the Stokes diameter decreases indicating a greater tendency for particles to segregate resulting in more diffuse interfaces. Therefore, the behaviour of such systems is characterised by differential rather than hindered settling. By measuring the time delay between the onset of sedimentation and that resulting in a change in reed resonance frequencies at various locations along the settling tank we have been able to obtain estimates of propagation wave velocities marking the transition between steady and unsteady state behaviour. For the systems tested we find that the wave propagation velocity is, in general, independent of solids concentration but increases with the mean particle size