Discontinuous phenomena in bioreactor system

Hany A. Hosham; Eman D Abou Elela

doi:10.3934/dcdsb.2018294

Article Contents

2019, Volume 24, Issue 6: 2955-2969. Doi: 10.3934/dcdsb.2018294

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Discontinuous phenomena in bioreactor system

Hany A. Hosham^1,2, , and
Eman D Abou Elela^1,

1.
Department of Mathematics, Faculty of Science, Taibah University, Yanbu 41911, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

* Corresponding author: Hany A. hosham
* Corresponding author: Hany A. hosham

Received: November 2017

Revised: April 2018

Early access: October 2018

Published: June 2019

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Abstract

This paper critically examines discontinuous bifurcation and stability issues in model of methane gas production from organic waste via decaying process in two cases, namely sliding and non-sliding flow. The presence of certain types of discontinuities in Monod curve lead to discontinuous system and therefore the criteria for the existence and stability of equilibrium points are established. The analysis highlights the presence of several types of border collision bifurcations depending upon the effect of the dilution factor, biomass concentration and solid-liquid-gas separator efficiency, like nonsmooth fold, persistence and grazing-sliding scenarios. In addition, numerical simulations are carried out to illustrate and validate the results.

Keywords:

Mathematics Subject Classification: Primary: 34A36, 34C23, 34K18, 34K20, 92B05; Secondary: 34K21, 74H60.

Citation:

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Figure 1. Structural frame of Upflow anaerobic sludge blanket(UASB)

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Figure 2. Equilibrium transition due to the effect of SLG separator deficiency $\alpha_6$, admissible (solid line) and virtual (dashed line): (a) Two equilibrium points of $\ominus$-system (b)Two equilibrium points of $\oplus$-system.

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Figure 3. Persistence bifurcation of CPWS (8) when $\alpha_6 = 0.1213$

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Figure 4. Persistence bifurcation of CPWS (8) when: (a) $m = 0$ where $\alpha_1^{max} = 2.639$ and (b) $m = 200$ where $\alpha_1 = 0.7572$

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Figure 5. Existence of nonsmooth bifurcation of sliding flow (5) at $\lambda = 0, \alpha_6 = 0.04802$

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Figure 6. Existence of nonsmooth bifurcation of sliding flow (5) at $\lambda = 1, \alpha_6 = 0.04905$

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Figure 7. Existence of persistence bifurcation of sliding flow (5) at $\lambda = 0, \alpha_6 = 0.0255$

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Figure 8. Existence of persistence bifurcation of sliding flow (5) at $\lambda = 1, \alpha_6 = 0.02496$

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Figure 9. Numerical simulation illustrating a grazing-sliding bifurcation occurring at $\alpha_6 = \alpha_6^{graz}$ in DS (5).

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