Modeling and Mathematical Analysis of the Clogging Phenomenon in Filtration Filters Installed in Aquaria

Abstract

This paper proposes a mathematical model for replicating a simple dynamics in an aquarium with two components; bacteria and organic matter. The model is based on a system of partial differential equations (PDEs) with four components: the drift-diffusion equation, the dynamic boundary condition, the fourth boundary condition, and the prey-predator model. The system of PDEs is structured to represent typical dynamics, including the increase of organic matter in the aquarium due to the excretion of organisms ($e.g$. fish), its adsorption into the filtration filter, and the decomposition action of the organic matter both on the filtration filter and within the aquarium. In this paper, we prove the well-posedness of the system and show some results of numerical experiments. The numerical experiments provide a validity of the modeling and demonstrate filter clogging phenomena. We compare the feeding rate with the filtration performance of the filter. The model exhibits convergence to a bounded steady state when the feed rate is reasonable, and grow up to an unbounded solution when the feeding is excessively high. The latter corresponds to the clogging phenomenon of the filter.

Figures (4)

• Figure 1: Outline of filtering system of an aquarium. There are dust and bacteria in the aquarium. Both dust and bacteria are taken in from the right boundary by the action of pumps within the filtration filters installed on both sides. A proportion in the filtration filter is absorbed, and the portion not absorbed is expelled from the left side.
• Figure 2: Time series of two cases. We compared with two cases: $f=0.5, 2.0$. The left-hand and right-hand sides four figures indicate the result when $f=0.5$ and $f=2.0$, respectively. The top graphs are time series of the spatial average to $v_1$ and $v_2$. The second graphs from the top are time series of $\sigma_1$ and $\sigma_2$. The second heatmap from the bottom describes the time series of $v_1$. The horizontal axis describes domain $I$. The vertical axis describes time interval $(0,T)$ for $T=500$. The second heatmap from the bottom describes the time series of $v_2$.
• Figure 3: Comparison of parameters for the transition. The horizontal axis represents the capacity parameter $C_\rho$. The vertical axis represents the feeding rate $f$. The height corresponds to the value of the time derivative of the time-average of $v_1$ at time $5000$. The zero-height area indicates a non-clogging case. In cases where the time derivatives are non-zero, continuous growth of $v_1$ (organic matter) is observed. For the green points, we illustrate the time series in Fig. \ref{['fig_time_series_for_transition']}.
• Figure 4: Time series graphs for points crossing the transition boundary. The horizontal axis represents $log_{10} t$ for time $t \in [0, 5000]$, and the vertical axis represents $log_{10} v_1$. We set $C_{\rho}=0.50$ and conducted numerical experiments for five cases of $f$ with increments of $0.25$, such that $f = 0.25, 0.50, \ldots, 1.25$. The time derivative of the spatial average at $t=5000$ is calculated for each case.

Theorems & Definitions (70)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1: Theorem 4.1-4.2 in KatoTanabe1962
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof