Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

Meadows, Tyler; Wolkowicz, Gail S. K.

doi:10.1016/j.nonrwa.2020.103157

Cited by 3 publications

(3 citation statements)

References 12 publications

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Section: Resultsmentioning

confidence: 98%

“…Note that it was recently demonstrated analytically that the reactor will fail (AOB and NOB washout) if the initial biomass concentrations X j (t 0 ) are less than a minimum value that is dependent on the initial nutrient concentrations (X j, min = f(r, S 1 (t 0 ), S i + 1 (t 0 ))), where r is the refill fraction (see Eq. 26 in ref 30). From this fixed point, we then performed a one-parameter continuation of the curve of fixed points of the form P ̂using dissolved oxygen or the AMX activity constant.…”

Section: ■ Results and Discussionmentioning

confidence: 99%

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Bifurcation Analysis of an Impulsive System Describing Partial Nitritation and Anammox in a Hybrid Reactor

Wade

Wolkowicz

2021

Environ. Sci. Technol.

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Low-energy nitrogen removal under mainstream conditions is a technology that has received significant attention in recent years as the water industry drives toward long-term sustainability goals. Simultaneous partial nitritation-Anammox (PN/A) is one process that can provide substantial energy reduction and lower sludge yields. Mathematical modeling of the PN/A process offers engineers insights into the operating conditions necessary to maximize its potential. Laureni et al. ( Laureni Laureni Water Res.201914) have recently published a simplified mechanistic model of the process operated as a sequencing batch reactor that investigated the effect of three key operating parameters on performance (Anammox biofilm activity, dissolved oxygen concentration and fraction of solids wasted). The analysis of the model was limited, however, to simulation with relatively few discrete parameter sets. Here, we demonstrate through the use of bifurcation theory applied to an impulsive dynamical system that the parameter space can be partitioned into regions in which the system converges to different fixed points that represent different outcomes: either the washout of nitrite-oxidizing bacteria or their survival. Mapping process performance data onto these spaces allows engineers to target suitable operating regimes for specific objectives. Here, for example, we note that the nitrogen removal efficiency is maximized close to the curve that separates the regions in parameter space where nitrite-oxidizing bacteria washout from the region in which they survive. Further, control of solids washout and Anammox biofilm activity can also reduce oxygen requirements while maintaining an appropriate hydraulic retention time. The approach taken is significant given the possibility for using such a methodology for models of increasing complexity. This will enable engineers to probe the entire parameter space of systems of higher dimension and realism in a consistent manner.

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Section: Resultsmentioning

confidence: 98%

Section: ■ Results and Discussionmentioning

confidence: 99%

Bifurcation Analysis of an Impulsive System Describing Partial Nitritation and Anammox in a Hybrid Reactor

Wade

Wolkowicz

2021

Environ. Sci. Technol.

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“…Motivated by these results, we will address the problem of exact tracking of second-order impulsive systems in infinite-dimensional spaces. At present, the theory of impulsive partial differential systems has been applied in biological sciences [17,18], methanol poisoning, flexible mechan-ical structure [19], detoxification experiments [20], and other fields.…”

Section: Introductionmentioning

confidence: 99%

Iterative learning tracking control for second‐order nonlinear hyperbolic impulsive partial differential systems

Dai

Zhang

et al. 2023

IET Control Theory & Appl

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This paper studies the iterative learning control (ILC) for a class of second‐order nonlinear hyperbolic impulsive partial differential systems. Firstly, to follow the discontinuous desired output, a P‐type learning law is adopted, and sufficient conditions for the convergence of the tracking error is established under identified initial state value. The rigorous analysis is also given using the impulsive Gronwall inequality. Secondly, the tracking error of output trajectory is considered in systems with state initial values shifting based on an initial learning algorithm. These results of this paper show that the tracking error on the finite time interval can uniform converge to 0 as the iteration index goes to infinity if impulse number of the systems is only a finite numbers. Finally, two numerical simulation examples are given to verify the effectiveness of the theoretical results.

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Competition in the nutrient-driven self-cycling fermentation process

Smith,

Meadows,

Wolkowicz

2024

Nonlinear Analysis: Hybrid Systems

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Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

Cited by 3 publications

References 12 publications

Bifurcation Analysis of an Impulsive System Describing Partial Nitritation and Anammox in a Hybrid Reactor

Bifurcation Analysis of an Impulsive System Describing Partial Nitritation and Anammox in a Hybrid Reactor

Iterative learning tracking control for second‐order nonlinear hyperbolic impulsive partial differential systems

Competition in the nutrient-driven self-cycling fermentation process

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