Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially 1D model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is developed by first deriving a method-of-lines formulation (discrete in space, continuous in time) and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in WRRFs illustrate the model and its discretization.

中文翻译：

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具有不同横截面积的罐中反应沉降模型的线法公式

反应沉降是指分散在粘性流体中的小固体颗粒沉降的过程，同时构成固相和液相的组分之间发生反应。该过程对于水资源回收设施 (WRRF)（以前称为废水处理厂）中二沉池 (SST) 的模拟和控制特别重要。通过将沉降与压缩的机械模型与生物动力学反应模型相结合，制定了 SST 中反应沉降的空间一维模型。此外，允许罐的横截面积作为高度的函数而变化。最终模型是一个强退化抛物线非线性偏微分方程系统，其中包括用于描述进料的不连续系数，下溢和上溢机制，以及对馈送机制建模的奇异源项。最终模型的有限差分方案是通过首先导出线法公式（空间离散，时间连续）然后通过时间离散化传递到完全离散方案来开发的。此公式的优点是它与开发 WRRF 软件的常见做法兼容。主要的数学结果是不变区域属性，这意味着产生了物理相关的数值解。WRRF 中 SST 中的反硝化模拟说明了该模型及其离散化。最终模型的有限差分方案是通过首先导出线法公式（空间离散，时间连续）然后通过时间离散化传递到完全离散方案来开发的。此公式的优点是它与开发 WRRF 软件的常见做法兼容。主要的数学结果是不变区域属性，这意味着产生了物理相关的数值解。WRRF 中 SST 中的反硝化模拟说明了该模型及其离散化。最终模型的有限差分方案是通过首先导出线法公式（空间离散，时间连续）然后通过时间离散化传递到完全离散方案来开发的。此公式的优点是它与开发 WRRF 软件的常见做法兼容。主要的数学结果是不变区域属性，这意味着产生了物理相关的数值解。WRRF 中 SST 中的反硝化模拟说明了该模型及其离散化。这意味着产生了物理相关的数值解。WRRF 中 SST 中的反硝化模拟说明了该模型及其离散化。这意味着产生了物理相关的数值解。WRRF 中 SST 中的反硝化模拟说明了该模型及其离散化。