Sedimentation

Theoretically, small spheres in a dilute suspension will settle with a velocity given by Stokes' law:

V = g{r? - f d2/18v where Vis the settling velocity [LT-1], g is the acceleration of gravity [LT- ], rp is the density of particles [ML-3], rf is the density of the fluid [ML-3], d is the diameter of the particles [L], and v is the kinematic viscosity [L2T-1]. The law is valid for particles with Reynolds numbers = drfV/v <1.

Since absolute spheres rarely exist, a form resistance factor f can be applied:

V = g (rp - rf )d2/l8vf where f can have a value up to 5 for planktonic algae.

The factors rf, d, v, and f of the formulas can be regulated by the phytoplankton themselves. d varies from 2 to 500 mm and due to the second power dependence it is by far the strongest regulator of the sinking rate when various phytoplankton species are compared. rp varies from <1 for cyanobacteria with air vacuoles to >1.2 for diatoms with silica frustules. Often the individual species are able to regulate their densities by accumulation of lipids with low densities, ionic regulation, excretion of mucilaginous matter, forming of gas vacuoles, etc. The individual species can regulate their sinking rate with a factor of up to 5 apparently related to their nutritional status. By nutrient limitation, a faster sinking rate will provide a faster transportation of the phytoplankton to deeper water layers with higher nutrient concentrations and an increase in buoyancy can transport the organisms back to the photic layers. Diurnal migrations of phytoplankton are for some species assisted by swimming by flagellas. Combined with the buoyancy regulation, both grazing and nutrient depletion can be reduced.

However, the fastest sinking rates which can be obtained for planktonic algae are <1 mm s-1 and even light winds of 3-4 ms- are able to create turbulence exceeding the intrinsic movements ofthe algae by a factor of 10. But at the lower boundary of a mixed layer the turbulent movements are decreased and the actual loss of particles from a mixed layer is more a dis-entrainment from the mixed layer. Thus the critical factor is not the intensity of the turbulence itself, but rather the absolute depth of the mixed layer. The rate of sinking loss can be expressed as

Steady state

DPs DT

Time dependent, one sediment pool

K,Ps

DPs D7

K,Ps

K,PSeT

DPs D7

DPs D7

K,Ps

K>PeVs

DPs D7

K,PS

K,PEvSe7

DPs D7

Time dependent, several sediment pools

K3Pe0t

D7 DP

DPs DT DP,

Time dependent, multilayer model

Equations for the first layer: DP 1

DP,1

— = K31 Pe1 VS19T-20-K2(PS - P I1) + K21(P I1 - P ,2)

Equations for the Nth layer:

=K3NPENVSN9T-20- K2n\P ,n-1 - P N) + K2N(P,N- P ,n+1)

Figure 1 Sediment phosphorus models with increasing complexity. Ps = dissolved phosphate in water column; PE = concentration of exchangeable, sediment phosphorus; PI = concentration of phosphate in pore water; PB = concentration of sorbed phosphate in sediment; CS = concentration of exchangeable phosphorus in active layer; VS = volume of active layer; K1 = sedimentation rate constant; K2 = diffusion rate constant; K3 = mineralization rate constant; © = temperature coefficient; T=temperature (°C); S = sedimentation rate; R = release rate. Superscripts indicate the number of the layer in the sediment.

where Ks is the loss rate [T x], V is the sinking velocity [LT_1], and h is the thickness of the mixed layer [L]. The use of sediment traps have demonstrated that the sedimentation process is rather a trapping of horizontally transported particles being caught in the diffusive boundary layer above the sediment or above the mouth of a sediment trap. In models of aquatic ecosystems, Stokes' law is seldom used, since the behaviors of single particles are aggregated over size classes and species. In models, phytoplankton is treated as groups of phytoplankton which settle, dependent on averages of shapes, densities, nutritional status, and temperatures. One-dimensional aquatic models typically use settling velocities an order of magnitude lower than the measured velocities and the values used in two- or three-dimensional models.

Sediment Transport

Particles settled on the sediment surface do not necessarily stay where they are deposited. Waves and currents create shear stresses acting on the sediment and cause resuspension and transport of particles which again can resettle at places with less stress. An aggregated approach can be a separation of the sediment area in subareas with dominance of erosion, transport, or accumulation.

The distribution of subareas is affected by the potential, effective fetch, and water depth, and a classification can be applied using the relation between area type and the water content in the upper centimeter of the sediment:

• transportation areas: 50-75% water

• accumulation areas: >75% water

According to Figure 2, the critical water depth for separation of accumulation area and transport + erosion area for a lake is given by

where D is the critical depth in meters and f is the potential effective fetch in kilometers.

A mapping of the sediments according to water content can be used to identify the various areas. The distribution of sediment types is only valid on a long-term scale (years) and the empirical approaches should be carefully used, since the diagram can be modified by local effects for example, tidal currents, river inlets, vegetation cover, and steep bottom slopes. If a daily or seasonal resolution is necessary, the actual wind directions and speeds must be used as forcing functions for more refined three-dimensional hydrodynamic models. Instant resuspension and transport of sediment are driven by unidirectional currents supplemented with the orbital velocity generated by waves. The following three steps

Effective fetch, PFeff 0 10 20 30 40 50 60 km

Effective fetch, PFeff 0 10 20 30 40 50 60 km

Figure 2 The distribution of sediment types in relation to water depth and effective fetch.

are carried out: (1) wave height and wave periods are measured or calculated from empirical or theoretical models; (2) the maximum orbital velocity is calculated from wave height and wave period; and (3) the threshold grain size for transportation is calculated for the estimated or measured sum of velocities.

Sorption

Sorption of dissolved elements to sediment particles includes both physical adsorption to particle surfaces due to electrostatic forces, absorption of elements diffusing into particles, and chemosorption, where dissolved elements after adsorption or absorption react with the particles by chemical reactions like ion exchange and precipitation. At the sediment surface, the sinking particles and dissolved elements meet another environment with abrupt changes in concentration levels, redox potential, and pH. Sorption and desorption of elements may play an important role in the total flux across the sediment-water interface.

Sorption can be described as an instant equilibrium or as a rate dependent of the temporal resolution wanted. The most simple equilibrium expression is a linear adsorption isotherm:

where A is the sorbed element concentration, C is the element concentration in the liquid phase, and k1 and k2 are constants. If k2 is insignificant, k1 expresses the number of times the concentration of the sorbed element is greater than the concentration in the liquid and is then called the partition coefficient. The Langmuir sorption isotherm is given by

where Amax is the maximum sorbed concentration and k is a constant relating to bonding energy. The equation assumes that the sorption energy does not vary with surface saturation which is the case in the Freundlich isotherm:

A = kCn where k and n are constants.

The Freundlich isotherms can be modified by the Temkin equation,

where R is the gas constant, T is the absolute temperature, and a and b are constants. Another alternative is the Redlich-Petersen equation:

where b and a are constants.

The selection of sorption model can be done by goodness of fit to experimental data, but in some cases hysteresis phenomena occur. The models can be even more complicated if the hysteresis is time dependent. This irreversibility is due to chemical reactions occurring simultaneously with the sorption reaction. For some elements, the best models may vary when the redox potential or concentration levels change.

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