Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization

Abstract

a b s t r a c tThe bio-kinetic and sedimentation processes of wastewater treatment plants can be modelled by a large system of coupled nonlinear ordinary and partial differential equations (ODEs and PDEs). The subprocess of continuous sedimentation, which contains concentration discontinuities, is modelled by a degenerate parabolic conservation PDE with spatially discontinuous coefficients. A spatial discretization of this PDE described in Bürger et al. (2013) results in a large system of method-of-lines ODEs for the entire plant and simulation can be performed by integration in time. In practice, standard time integration methods available in commercial simulators are often used. Shortages of such methods are here shown, such as the smearing of shock waves by Runge-Kutta methods and long execution times. A semi-implicit time discretization, which is described in detail, provides substantially shorter computational times and is more efficient than standard methods.

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

“…Furthermore, since b is continuous and the solution on both sides of the discontinuity r = b(t) is continuous in a neighbourhood of (r c , t c ), Equation (23) implies that b ′ is continuous. (This can also be established from the formulas (24) and (53). )…”

“…The solution for t ≥ t c is qualitatively the same as in cases L and Mb. The discontinuity r = b(t) may be nonmonotone before t c , as is shown in Figure 7; however, it is decreasing (directed upwards) in t c < t < t s where it is given by (53) with the time points t = t c and t = t s satisfying (51) and (52).…”

“…For gravity sedimentation, the similar constitutive relation (4) with 2 r replaced by the acceleration of gravity g was derived by Bürger et al 49 The resulting degenerate parabolic equation has been used extensively for the modelling and simulation of batch and continuous sedimentation with compression in clarifier-thickeners with applications to both mineral processing and wastewater treatment; see, eg, previous studies. 8,[50][51][52][53] To be able to construct solutions in detail, we here neglect the extra solids stress e and substitute (4) into the conservation law (2a), which becomes (1a). A constitutive assumption is needed for the hindered-settling function.…”

“…We refer to previous studies for analysis, numerical methods, and the related inverse problem of identifying the functions and . For gravity sedimentation, the similar constitutive relation with replaced by the acceleration of gravity was derived by Bürger et al The resulting degenerate parabolic equation has been used extensively for the modelling and simulation of batch and continuous sedimentation with compression in clarifier‐thickeners with applications to both mineral processing and wastewater treatment; see, eg, previous studies …”

A conservation law with multiply discontinuous flux modelling a flotation column

“…Well-established spatially one-dimensional models of clarifier-thickeners can be formulated as a scalar conservation law for the local solids concentration as a function of depth and time, where the flux is discontinuous as a function of spatial position due to upward-and downward-directed bulk flows, transitions to overflow and underflow transport, and a singular source term marking the feed [8,10,15]. Clarifier-thickener models have motivated in part the mathematical research on conservation laws with discontinuous flux [3,4,6,10,[14][15][16][17][18][19][20].…”