Modeling floc size distribution of suspended cohesive sediments using quadrature method of moments

Shen, Xiaoteng; Maa, Jerome P.‐Y.

doi:10.1016/j.margeo.2014.11.014

Cited by 35 publications

(51 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This part could be improved by selecting a better fragmentation distribution function in the future. Note that, usually, the same α should be used for all shear rates in similar environments if the same

{\overset{true¯}{b_{i}}}^{(k)}

is selected (e.g., Maggi et al, ; Shen & Maa, ). Therefore, a value of α = 0.92 with binary breakup are applied for all shear rates under NFBA conditions to represent narrow spread FSDs throughout (Figure b).…”

Section: Resultsmentioning

confidence: 99%

“…The PBE is essentially a conservation equation on the number density n ( ξ , t ) of aggregates by a single or a combination of properties ξ at time t , in which ξ can be various internal features of the particles such as size, density, mass, volume, shape, and composition. In this study, the number density function defined based on particle size L (i.e., ξ = L ) is focused on (e.g., Maggi et al, ; Shen & Maa, ; Winterwerp, ), since it is the fundamental property to provide critical clues of the floc settling and organisms attaching. After neglecting advection, diffusion, and settling, the governing equation of PBE in terms of the number density n ( L , t ) of flocs with size L and at time t in a homogenous system can be expressed as (Marchisio, Pikturna, et al, ):

\frac{\partial n ((), L, t)}{\partial t} = - \frac{\partial}{\partial L} ([], G_{L} ((), L, t) \cdot n ((), L, t)) + A ((), L, t) + B ((), L, t),

in which G L ( L , t ) is the growth rate (in units of meter per second) and A ( L , t ) and B ( L , t ) are the aggregation and breakage source and sink terms, respectively.…”

Section: Flocculation Modelmentioning

confidence: 99%

“…The shear rate determines also the carrying capacity K for biomass growth through equation . The aggregation correction factor α is a model fitting parameter accounting for particle geometry corrections β 0 (

β_{0} = β_{n_{f}} / β

in which the collision frequency β for spheres is expressed by equation and β nf is the collision frequency for fractal particles), contact efficiency E 0 ( E 0 ≤ 1), and attachment probability α 0 ( α 0 ≤ 1) and can be written as α = β 0 ⋅ E 0 ⋅ α 0 (Shen & Maa, ). Note that for two particles with volume 1,000 times different (i.e., v j / v i = 1000), β 0 may be in order of 10 for n f ≈ 2.0 and in order of 10 2 for n f ≈ 1.5 (Jiang & Logan, ; Lee et al, ).…”

Section: Flocculation Modelmentioning

confidence: 99%

“…By applying the moment transformation (equation ) for the governing equation (equation ), the moment equation can be simplified to (Shen & Maa, ; Su et al, ):

\begin{matrix} lefttrue \\ \frac{\partial m_{k / p}}{\partial_{t}} = \frac{1}{2} {false\sum}_{i = 1}^{N_{d}} ω_{i} {false\sum}_{j = 1}^{N_{d}} α (L_{i}, L_{j}) \cdot β (L_{i}, L_{j}) \cdot ω_{j} \cdot {((), {L_{i}}^{3} + {L_{j}}^{3})}^{\frac{k}{3 p}} \\ - {false\sum}_{i = l}^{N_{d}} {L_{i}}^{k / p} ω_{i} {false\sum}_{j = 1}^{N_{d}} α (L_{i}, L_{j}) \cdot β (L_{i}, L_{j}) \cdot ω_{j} + {false\sum}_{i = 1}^{N_{d}} a_{i} {\overset{true¯}{b}}_{i}^{((), k / p)} ω_{i} - {false\sum}_{i = 1}^{N_{d}} {L_{i}}^{k / p} a_{i} ω_{i} \\ + \frac{k}{p} {false\sum}_{i = 1}^{N_{d}} w_{i} \cdot G_{L} (L_{i}) \cdot {L_{i}}^{(k / p) - 1}, \end{matrix}

in which

{\overset{true¯}{b_{i}}}^{(k / p)} = \int_{0}^{\infty} L^{k / p} b (()|, L, λ) italicdL .

…”

Section: Flocculation Modelmentioning

confidence: 99%

“…Due to the complexity of biological effects, most numerical models only account for the mineral fraction. Examples of this approach include the one‐class population balance equation (PBE) by Winterwerp (), the two‐class model by Lee et al (), the three‐class model by Shen, Lee, et al (), and multiclass models such as by Maggi et al () and Shen and Maa (, , ). One exception is Maggi (), where the organic and inorganic fractions of aggregates are represented using two single‐class equations with an additional growth term to identify biomass coating, which is among the early contributions to illustrate the effects of microorganism on particle aggregation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Approach to Modeling Biofilm Growth During the Flocculation of Suspended Cohesive Sediments

et al. 2019

Self Cite

View full text Add to dashboard Cite

The floc size distribution (FSD) is crucial to predict cohesive sediment dynamics in aquatic environments. Recently, increasing attention has been given to biofilm effects on the FSDs of suspended particles since the presence of biofilms on particle surfaces may lead to larger flocs and thus higher settling velocities. In this study, results from a settling column experiment conducted by Tang and Maggi (2018; https://doi.org/10.1002/2017JG004165) under nutrient‐free and biomass‐free, nutrient‐affected and biomass‐free, and nutrient‐affected and biomass‐affected conditions, with different suspended sediment concentrations, shear rates, and nutrient concentrations, have been used to validate modeled FSDs that is based on the population balance equation solved by the quadrature method of moments. In addition to the processes of aggregation and breakage, the effects of biofilm are expressed in the growth term of the population balance equation. The logistic growth pattern is used to account for an increase in biomass, which is primarily controlled by the specific growth rate and the carrying capacity. In this study, the biofilm growth rate is assumed nutrient dependent, and the carrying capacity of floc size is hypothesized to be proportional to the Kolmogorov microscale. With eight size classes to interpret a simulated FSD, the predicted and observed FSDs exhibit a reasonable match for all nutrient‐free and biomass‐free, nutrient‐affected and biomass‐free, and nutrient‐affected and biomass‐affected conditions. This simplified bioflocculation model fills the gap between the simulations of the FSDs of cohesive sediments without and with biofilms and has the potential to be included in large‐scale models in the future.

show abstract

{\overset{true¯}{b_{i}}}^{(k)}

Section: Resultsmentioning

confidence: 99%

\frac{\partial n ((), L, t)}{\partial t} = - \frac{\partial}{\partial L} ([], G_{L} ((), L, t) \cdot n ((), L, t)) + A ((), L, t) + B ((), L, t),

in which G L ( L , t ) is the growth rate (in units of meter per second) and A ( L , t ) and B ( L , t ) are the aggregation and breakage source and sink terms, respectively.…”

Section: Flocculation Modelmentioning

confidence: 99%

β_{0} = β_{n_{f}} / β

Section: Flocculation Modelmentioning

confidence: 99%

“…By applying the moment transformation (equation ) for the governing equation (equation ), the moment equation can be simplified to (Shen & Maa, ; Su et al, ):