Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved undera Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.

翻译：序批式反应器（SBRs）是广泛应用于废水处理、化学工程等领域的设备。该类反应器可在固体生物质颗粒沉降与压缩的同时，与溶解于液相中的营养物质发生生化反应。这些反应的动力学可由既定的活性污泥模型（ASMx）之一描述。SBR分阶段运行，配备可移动的提取与填料装置及排出口。该单元的一维模型可表述为退化对流-扩散-反应方程组构成的移动边界问题，其未知量分别对应固相与液相各组分的浓度。通过变换至固定计算域，采用显式单调格式与替代的半隐式变体进行离散化。半隐式变体基于每个时间步内求解总固体浓度的非线性方程组，随后求解固体组分百分比与液相组分浓度的线性方程组。研究表明，半隐式格式是适定的，且两种变体产生的近似解均满足不变区域原理：固体浓度非负且不超过设定最大值，百分比非负且总和为1，底物浓度非负。这些性质在库朗-弗里德里希斯-列维（CFL）条件下得以实现，其中半隐式变体的约束条件较显式变体更宽松。基于实际参数的数值算例表明，半隐式变体因而具有更高的计算效率。