Deposition-Erosion Criterion for Sediment Transport in Sewers based on Shear Stress Calculations
Authors: Ir. Raf Bouteligier, Dr. ir. Guido Vaes, Prof. dr. ir. Jean Berlamont FACULTEIT TOEGEPASTE WETENSCHAPPEN DEPARTEMENT BURGERLIJKE BOUWKUNDE LABORATORIUM VOOR HYDRAULICA KASTEELPARK ARENBERG 40 B-3001 HEVERLEE |
Katholieke Universiteit Leuven |
Technical Paper January 2002 |
The idea of the deposition – erosion criterion based on shear stress is that if the actual shear stress θ is below the critical shear stress value for deposition θcr, deposition, i.e. if θ < θcr, deposition, then deposition will occur. If the actual shear stress value θ is in-between the critical shear stress value for deposition and the critical shear stress value for erosion, i.e. if θcr, deposition < θ < θcr, erosion, then no erosion or deposition will occur and all suspended sediment is transported along the conduit. If the actual shear stress θ exceeds the critical maximal shear stress value for erosion θcr, erosion, i.e. if θ > θcr, erosion, then erosion will occur.
Figure 1: Illustration of the deposition – erosion criterion based on shear stress
Deposition - Erosion calculations can be made according to the following algorithm:
Calculation of the actual shear stress θ
The actual shear stress θ [N/m2] can be calculated according to eq. (1):
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where: λc composite friction factor [-] ρ density of water [kg/m3] V flow velocity [m/s] |
Calculation of the critical shear stress values for deposition and erosion
Calculate the critical shear stress values for deposition according to eq. (2)
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where: γdeposition deposition parameter [-] g gravitational acceleration [m/s2] s specific sediment density [-] ρ density of water [kg/m3] d50 sediment particle size [mm] |
Calculate the critical shear stress values for erosion according to eq. (3)
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where: γerosion erosion parameter [-] g gravitational acceleration [m/s2] s specific sediment density [-] ρ density of water [kg/m3] d50 sediment particle size [mm] |
Note: |
γdeposition can not exceed γerosion, i.e. γdeposition ≤ γerosion |
Calculation of the deposition / erosion rate qs
If θ < θcr, deposition then deposition occurs. The deposition rate qs [kg/s] can be calculated according to eq. (4)
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where: θ actual shear stress [N/m2] θ < θcr, deposition critical deposition shear stress [N/m2] αdeposition deposition parameter [kg/s] (αdeposition ≥ 0) βdeposition deposition parameter [-] (βdeposition ≥ 1) |
Note: |
qs ≤ 0 (deposition) |
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αdeposition ≥ 0, maximal deposition rate (i.e. for θ equal to zero) |
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βdeposition ≥ 1 |
Care must be taken that no more sediment will be deposited than the amount of sediment that is carried by the flow. Therefore the deposition rate qs is limited according to eq. (5).
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where: Cold concentration prior to deposition – erosion calculations [kg/m3] Q flow [m3/s] |
If θcr, deposition < θ < θcr, erosion then there will be no erosion and no deposition. All suspended sediment is transported and therefore qs [kg/s] will be zero and Cnew will be equal to Cold (see eq. 6).
If θ > θcr, erosion then erosion occurs. The erosion rate qs [kg/s] can be calculated according to eq. (6)
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where: θ actual shear stress [N/m2] θcr, erosion critical erosion shear stress [N/m2] αerosion erosion parameter [kg/s] βerosion erosion parameter [-] |
Note: |
qs ≥ 0 (erosion) |
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αerosion ≥ 0 |
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βerosion ≥ 1 |
Care must be taken that no more sediment will be eroded than the amount of sediment that is stored in the bed. This is expressed by eq. (7).
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where: Δt timestep [s] Mbed sediment mass stored in the bed [kg] |
The sediment mass that is stored in the bed Mbed can be calculated according to eq. (8).
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where: Abed cross-sectional area of the sediment bed [m2] Δx mesh size [m] ρ density of water [kg/m3] sb bulk specific gravity [-] |
The bulk specific gravity sb can be calculated assuming a 60% void ratio (see eq. (9)).
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Combining eq. (7) and eq. (8) results in an upper erosion limit expressed by eq. (10).
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Calculation of the resulting sediment concentration in the flow
The new sediment concentration in the flow Cnew (i.e. after deposition – erosion calculations) can be calculated according to eq. (11)
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where: Cold original sediment concentration [kg/m3] qs deposition - erosion rate [kg/s] Q flow [m3/s] |
Six user editable parameters are introduced, three of which (αdeposition [kg/s] ; βdeposition [-] ; γdeposition [-]) apply to deposition processes and three of which (αerosion [kg/s] ; βerosion [-] ; γerosion [-]) apply to erosion processes. An example of the application of these six parameters is shown in figure 2. Figure 3 shows an example of the deposition – erosion criterion in its most simple form, i.e. when αdeposition = αerosion, βdeposition = βerosion = 1 and γdeposition = γerosion.
Figure 2: Example of the deposition – erosion rate criterion representing the deposition – erosion rate qs as a function of the shear stress θ
Figure 3: Example of the deposition – erosion criterion in its most simple form representing the deposition – erosion rate qs as a function of the shear stress θ
Article © Raf Bouteligier et al 2002