3. Theoretical transport model of a single suspension
For a one-dimensional case, the transport equation of a single suspension (e.g., only heavy metal ions or SPs) can be written as (Ahfir et al., 2017; Bedrikovetsky et al., 2017):
(1)
where C is the suspension concentration [ML−3], t is the time [T], x is the transport distance [L], D is the hydrodynamic dispersion coefficient [L2T−1], which is defined as D =α d·u , whereα d is the hydrodynamic dispersivity [L], andu is the average internal velocity [LT−1],n is the porosity [-], σ is the attachment concentration [MM−1], which refers to the mass of the suspension adsorbed per unit mass of the solid matrix of the porous medium, and ρ s is the solid matrix density [ML−3].
The deposition rate (or the release rate) in Eq. (1) has the following form (Bai et al., 2019):
(2)
where λ is the reaction rate constant [T−1] and S is the equilibrium adsorption concentration, which can be expressed as (Bai et al., 2019):
(3)
where κ d is the transient equilibrium constant [M−1L3] andκ d=β 1·S max/C L.β 1 is the dimensionless attenuation factor of the initial deposition process (i.e., σ =0 when t =0),S max is the maximum attachment concentration, andC L is the characteristic concentration, which is assumed to be C L=1 (whose dimension is the same as that of C ).
From Eq. (2), the attachment concentration can be deduced as for the initial deposition process. Actually, the reaction rate constantλ essentially reflects the mass conversion between the flowing water and deposited suspension on the solid matrix under hydrodynamic forces, which is closely related to the seepage velocity. For a column test, the ratio of the mass of migrating matter flowing through cross-section x at time t to the total mass injected into the porous medium is defined as:
(4)
where M b (%) is the penetration rate, Ais the cross-sectional area of the sand column [L2], and m is the total mass of the suspended matter injected into the porous medium [M].
The deposition rate is defined asM a=1−M b (%), which denotes the amount of suspended matter deposited in the range of (0,x ) in the column. The dimensionless timeV P is defined as the ratio of the total volume of water flowing through the column to the void volume of the column (i.e., the pore volume; V P=(unAt )/(nAL )=ut /L ), where L is the column length [L]).