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Equations and their physical interpretation in numerical modeling of heavy metals in fluvial rivers

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Abstract

Based on the previous work on the transport-transformation of heavy metal pollutants in fluvial rivers, this paper presented the formulation of a two-dimensional model to describe heavy metal transport-transformation in fluvial rivers by considering basic principles of environmental chemistry, hydraulics, mechanics of sediment transport and recent developments along with three very simplified test cases. The model consists of water flow governing equations, sediment transport governing equations, transport-transformation equation of heavy metal pollutants, and convection-diffusion equations of adsorption-desorption kinetics of particulate heavy metal concentrations on suspended load, bed load and bed sediment. The heavy metal transport-transformation equation is basically a mass balance equation, which demonstrates how sediment transport affects transport-transformation of heavy metals in fluvial rivers. The convection-diffusion equations of adsorption-desorption kinetics of heavy metals, being an extension of batch reactor experimental results and a major advancement of the previous work, take both physical transport, i.e. convection and diffusion and chemical reactions, i.e. adsorption-desorption into account. Effects of sediment transport on heavy metal transport-transformation were clarified through three examples. Specifically, the transport-transformation of heavy metals in a steady, uniform and equilibrium sediment-laden flow was calculated by applying this model, and results were shown to be rational. Both theoretical analysis and numerical simulation indicated that the transport-transformation of heavy metals in sediment-laden flows with clay-enriched riverbed possesses not only the generality of common tracer pollutants, but also characteristics of transport-transformation induced by sediment motion. Future work will be conducted to present validation/application of the model with available data.

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Abbreviations

x and y:

coordinates of two dimensions in plane

dx and dy :

differential lengths in x and y directions

h1, h2 and h3:

vertical lengths occupied by suspended load (flowing water), bed load (flowing water) and bed deformation

h :

water depth (h 1 + h 2)

c :

vertically averaged dissolved heavy metal concentration

u and v:

vertically averaged flow velocities in x and y directions (flowing water and suspended load

w x and w y :

vertically averaged velocities of bed load in x and y directions (flowing water and bed load)

ss and sb:

suspended load concentration and bed load concentration

Ns, Nb and Nm:

vertically averaged particulate concentrations of heavy metal on suspended load, bed load and sediment of the riverbed perimeter (or heavy metal contents sorbed by unit weights of suspended load, bed load and unit area of the riverbed sediment)

Δh :

active thickness of bed sediment for adsorption-desorption

ρ s :

specific weight of sediment particles

p′:

the porosity of bed

E x and E y :

turbulent diffusion coefficient components of flow

E s x and E s y :

turbulent diffusion coefficient components of suspended load

N :

particulate heavy metal concentration in batch reactor experiments at any time

C :

dissolved heavy metal concentration in batch reactor experiments at any time

k1, k2 and b:

coefficient of adsorption rate, coefficient of desorption rate and maximum adsorption capacity

E Ns x and E Ns y :

turbulent diffusion coefficients of particulate heavy metal concentrations on suspended load in x and y directions

k1b, k2b and bb:

coefficient of adsorption rate, coefficient of desorption rate, and maximum adsorption capacity for suspended load

k1m, k2m and bm:

coefficient of adsorption rate, coefficient of desorption rate and maximum adsorption capacity for bed load

k1m, k2m and bm:

coefficient of adsorption rate, coefficient of desorption rate and maximum sorption capacity for bed sediment

z 0 :

bed elevation

c 0 :

designated dissolved pollutant concentration

δ(t):

δ function with time

L :

length of the numerical flume

s*:

suspended sediment-carrying capacity

ks and ms:

coefficients in suspended sediment-carrying capacity formula

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Authors and Affiliations

  1. Numerical Simulation Group for Water Environment, Tianjin Key Laboratory of Environmental Remediation and Pollution Control, Key Laboratory of Pollution Processes and Environmental Criteria of the Ministry of Education; College of Environmental Science and Engineering, Nankai University, Tianjin, 300071, China

    SuiLiang Huang

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Huang, S. Equations and their physical interpretation in numerical modeling of heavy metals in fluvial rivers. Sci. China Technol. Sci. 53, 548–557 (2010). https://doi.org/10.1007/s11431-009-0389-5

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