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Uncertainty is inherent in experimentation, modeling, and analysis. Variations and errors in parameter estimates or physical processes are unavoidable and can affect the reliability of model predictions. Therefore understanding the role of uncertainty is embedded in the process of modeling and approximating the real world. In this manuscript we consider uncertainty propagation in a theoretical model of water/wastewater treatment. In dead-end microfiltration contaminated water is fed through a membrane that filters out colloids, bacteria, and protozoa. However, these particles foul the membrane reducing the filter productivity, which is alleviated by periodically reversing the flow, i.e. backwashing. We investigate how uncertainty in sensitive parameter estimates propagates to the estimates of the optimal amount of volume of water that is filtered in a fixed time period and the associated backwashing timing and duration. We find that the model provides conservative estimates for the total volume since the uncertainty is not propagated symmetrically with respect to over and underestimating specific measurable quantities. The uncertainty in the timing is more symmetric implying that there is essentially an equal amount of uncertainty for increasing or decreasing the frequency and duration of backwashing. We identified biofilm production as propagating the most uncertainty in the volume estimate. The fouling rate has the most effect on the timing estimates. Additionally we explored the affect of asymmetric parameter distributions and find that, for most parameters, asymmetry does not lead to increased asymmetry in predicted optimal regimes, implying that uncertainty in the skewness is likely not an issue. [ABSTRACT FROM AUTHOR]

One of the central problems in membrane implementation is that foulants deposit on surfaces of microfilters employed for water and wastewater treatment and reuse, reducing their productivity. In an effort to remove as much of the deposited materials as possible and maintain the flux, the direction of flow across these membranes is regularly and frequently reversed. However, backwashing only removes the loosely or reversibly bound materials necessitating the entire system to be taken off-line for chemical cleaning when fouling increases beyond a threshold value. We recently presented a mathematically rigorous methodology to determine timing and duration of backwashes so as to delay membrane chemical cleaning and reduce associated operational complexity and costs. By considering an optimal control formulation, using the flow direction as the control variable, the optimal timing and maximum volume of water that can be microfiltered in a given time period were predicted during dead-end filtration of bacterial suspensions. The value of these predictions depend on our ability to accurately and precisely estimate input parameters such as the clean membrane resistance, influent water quality estimates, and biofilm and exo-polymeric substance formation rates and effects. These estimates are inherently uncertain leading to uncertainty in the predictions, which can be rigorously quantified using Sobol indices. These provide a global method for assessing the sensitivity of model predictions (e.g. optimal timing and volume filtered) with respect to variations in the inputs. [ABSTRACT FROM AUTHOR]

Microfiltration and ultrafiltration are methods of removing colloidal impurities from water and wastewater. One of the major issues when dealing with the practical implementation of membranes is the reduction of water productivity as the foulants accumulate on the membrane surface or within the pores. Membrane regeneration by periodic backwashing is an effective method of reducing fouling; however, to date the timing and duration of the backwashing for effective fouling control is largely only empirically determined. In this manuscript, we present an optimal control formulation to determine the timing and duration for membrane regeneration by backwashing. In this formulation, we use the direction of the flow as the control variable and make predictions regarding the optimal protocol. We explicitly include irreversible attachment due to bacterial deposition and biofilm formation on the membrane and demonstrate that irreversible attachment of bacteria has important ramifications for the effective timing of hydraulic backwashes as well as the efficiency in producing clean water. In particular, we find that irreversible attachment and additional fouling due to exo-polymeric substance (EPS) production and biofilm formation decreases the maximum filtration volume. Additionally, as the effectiveness of membrane regeneration declines, the timing of the cycling is also altered. In general, including the role of EPS in biofouling substantially changes predictions of backwashing timing and implies important considerations for practical predictions. [ABSTRACT FROM AUTHOR]

Microfiltration is used in a variety of industrial and municipal water purification settings where one of the main concerns is fouling from the particulate matter that is removed from the water. Our focus has been on developing a unified model that captures fouling behavior in a consolidated manner rather than describing individual blocking regimes using power law models. The unified model provides greater insights into fouling mechanisms so that a deeper understanding of flux decline can be obtained. Moreover, by characterizing both forwards and backwashing behavior together, mathematical theory is available to develop strategies that increase the effectiveness of microfiltration in conjunction with backwashing used to regenerate the filter. We present a very simplified model that was developed to provide details regarding the mathematical analysis and how optimal control theory can be used to predict the timing and duration of backwashing that will optimize the overall water flow through the membrane. We use optimal control theory to derive an analytic solution to the optimal problem and develop a strategy to implement the solution. The model estimates of forward operation are compared with experimental data for constant pressure filtration and indicate that the model is able to capture the basic processes. More interestingly, the optimal control solution and proposed implementation strategy are consistent with empirical demonstrations but provide mathematical evidence that the flux may be increased dramatically by precise timing of the forward and backwashing cycles. Model predictions can be evaluated during pilot-testing that often precedes microfilter regulatory approval and plant design. [ABSTRACT FROM AUTHOR]

Abstract: The instantaneous transmembrane pressure needs to be continuously increased to compensate for foulant accumulation during constant flux microfiltration. Herein, we compare predictions of a unified mathematical model and conventional blocking laws with laboratory data obtained during constant flux operation. Foulants employed included single species cultures of bacteria and coagulated natural colloids, both of which are known to form compressible cakes. The first principles model unifies fouling arising from pore blocking by individual cells, cake formation, as well as bacterial secretion of exopolymers. It also incorporates non-uniform spatial deposition patterns that have been observed during unstirred dead-end filtration. Mechanistically, these heterogeneities arise either from non-uniform membrane surface porosity or stochastic initial binding of foulants. Previous studies have shown that the initial patchy or uneven deposit morphology is magnified over longer time-scales during bacterial filtration by differential extracellular polymeric substances (EPS) secretion through quorum sensing. However, in this study, we are primarily interested in the averaged behavior (mainly flux and pressure). By spatially averaging the microscale variables we are able to compare with classical blocking law models. We show that blocking laws and the unified model both accurately model fouling under our experimental conditions. The unified model provides mechanistic insights into (bio)colloid deposition and associated fouling during constant flux microfiltration particularly since it is obtained excellent predictive agreement with experimental data using parameters taken exclusively from our recent study of constant pressure filtration. [Copyright &y& Elsevier]

Abstract: Microfiltration is used in a wide range of municipal and industrial settings to remove particulate matter including pathogenic microorganisms such as bacteria and protozoa. As more water is filtered at constant pressure, the accumulation of retained particles on the membrane decreases the filtration rate; a process commonly referred to as fouling. Mathematical treatment of flux decline has proved to be a useful tool in diagnosing filtration data even though the mathematical underpinnings are not completely understood. In particular, little is known about the transition between fouling phenomena (e.g. pore blocking to cake filtration). Moreover, less is known about the effect of extracellular polymeric substances (EPS) production by bacteria when they accumulate on a membrane over extended durations. In this manuscript, we develop a novel approach to model bacterial microfiltration by considering the effects of both differential binding and exopolymer production. Spatial gradients in bacteria concentrations initially occur due to the non-uniform membrane surface porosity and differential deposition caused by the stochastic nature of microorganism adhesion. These heterogeneities in bacterial deposition and associated pore blocking result in variable secretion of extracellular polymers. Long-term fouling is quantified as the cumulative resistance posed by both bound bacteria and EPS. We compare numerical simulations quantitatively and qualitatively to previously published experimental data and investigate variations of microbial deposition patterns across the membrane. We find substantial agreement between the model and experimental observations. We are also able to conclude that fluid dynamics must be important if the dominant variability is in the membrane structure, rather than in bacterial adhesion. However, variation in bacterial adhesion alone can also induce substantial spatial heterogeneity. [Copyright &y& Elsevier]

One tool in efforts to tackle the ever growing problem of water scarcity is municipal wastewater reclamation to produce drinking water. Microfiltration (MF) is a central technology for potable reuse because it is highly effective in removing pathogenic protozoa, bacteria, and other colloids and for reverse osmosis pretreatment. However, as microfiltered materials accumulate at the membrane surface, its productivity is reduced requiring periodic removal of foulants. A mathematical model of MF is described in the context of hollow fiber filtration that focused on optimizing constant flux operation with backwashing. Design curves were also proposed for determining backwash timing. The model analysis is evaluated against real-world MF fouling for membranes that range in age from a few weeks to three years, observed at the world's largest water reuse facility operated by the Orange County Water District. The presented model compares well with the full-scale operational data, and model parameters accurately capture variations in fouling kinetics with membrane age, providing clues to changes in optimal regeneration timing and frequency as membrane performance declines over long time scales. • Comparison against three years of data provided by a full-scale water reuse facility. • Quantification of effects of aging on fouling and efficiency. • Parameter estimation for full-scale operation based on aging of microfilters. [ABSTRACT FROM AUTHOR]

Abstract: Dead-end microfiltration is an effective method for removing particulate matter including protozoa and bacteria in a variety of settings. As the filtered components accumulate on the filter, the flux declines during constant pressure operation necessitating periodic back-washing. It has been recently reported empirically that the filtered components are not homogeneously distributed on the filter. This must be accounted for in any mathematical theory describing the filtering process. We present a method for determining the coupled fluid dynamics and bacterial transport for dead-end filtration. Results of numerical simulation indicate that the model is able to closely capture experimental data; moreover, the model predicts that the coupled system induces the experimentally observed patchy deposition patterns through spatial variability in the pore structure and porosity of the membrane. [Copyright &y& Elsevier]