# Design of Desanding Facilities for Hydropower Schemes Based on Trapping Efficiency

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{cr}depending on mean longitudinal basin flow velocity and still-water settling velocity. Garde et al. [4] used their own and third party experimental data to propose a relationship between trapping efficiency and the ratio of basin length to basin depth. The accuracy of the proposed approach is stated to be within ±50%. Ortmanns and Minor [5] measured vertical flow velocity fluctuations in the basins of three facilities and proposed a settling velocity retardation term that includes geometric and hydraulic parameters. However, application of this approach is limited to a small range of input parameters underlying the investigated prototype configurations. In contrary to the aforementioned approaches, Velikanov [6] suggests calculating the required basin length based on settling probabilities, resulting in slightly smaller basin lengths than with the classical approach. In principle, these approaches are reasonable, but solely (i) the settling basin and (ii) ideal boundary conditions (i.e., uniform, homogeneous approach flow and outlet flow) are considered, often diverting from prototype conditions. Hence, actual basin hydraulics may entail facility performance that is below design expectations, as is often encountered in prototype structures.

_{in}(g/L) and outlet C

_{out}(g/L), the concentration-related trapping efficiency becomes

## 2. Materials and Methods

#### 2.1. Numerical Model

^{2}s

^{−1}), $d$ = sediment particle diameter (m), ${d}_{*}$ = dimensionless particle diameter (–), $g$ = gravitational acceleration (m s

^{−2}), ρ

_{s}= sediment density (kg m

^{−3}) and ρ

_{f}= fluid density (kg m

^{−3}). To account for the effects of particle–particle interaction on the settling velocity, the approach of Richardson and Zaki [18] is used in the model. Sediment entrainment (resuspension) is modeled as a function of excessive dimensionless shear stress $\theta -{\theta}_{cr}$ and is calculated using the pick-up function of [19]. Here, the dimensionless critical Shields parameter ${\theta}_{cr}$ can be set as constant or computed using the Soulsby–Whitehouse equation [17], and the dimensionless local bed shear stress is $\theta ={u}_{*}^{2}/\left(gdG\right)$. The bed shear velocity ${u}_{*}$ is calculated from the law-of-the-wall velocity profile for turbulent flows [20],

_{w}= flow velocity parallel to wall (m/s), κ = von Kármán constant (–), υ = kinematic viscosity of water (m

^{2}s

^{−1}), and the bed surface roughness k

_{s}is assumed to be proportional to the local median grain diameter of packed bed sediment. The eddy viscosity ${\nu}_{t}$ is defined as

^{2}s

^{−2}) and ε = turbulent kinetic energy dissipation rate (m

^{2}s

^{−3}); both quantities are obtained by the turbulence model.

#### 2.2. Modeling of Tranquilizing Racks

#### 2.3. Parameter Study

#### 2.3.1. Reference Configuration

_{tz}=10 m (≥2W

_{bs}according to [25]) and horizontal and vertical expansions ending at the same streamwise position. The transition zone horizontal expansion angle is α = 5.7, following the recommendation in [9]. At the basin end, a semi-circular weir with vertical front wall impounds the flow. For the reference configuration, discharge Q

_{d}= 4 m

^{3}/s, width W

_{bs}= 4 m and flow depth h

_{bs}= 5 m are assumed, meeting the recommendation W

_{bs}/h

_{bs}≈ 0.8 given in [26]. This agrees well with existing desanding facilities in Switzerland. The resulting longitudinal mean basin flow velocity is v

_{x}= Q

_{d}/(W

_{bs}h

_{bs}) = 0.2 m/s.

_{cr}= 330 µm was chosen. The PSD is discretized by four grain classes (Figure 4b).

_{cr}= 330 µm can be computed according to Soulsby [17] from Equation (2) and results in w

_{s,0}= 0.043 m/s for ρ

_{s}= 2650 kg m

^{−3}and water at 8°C with υ = 1.39∙10

^{−6}m

^{2}s

^{−1}. The basic basin length can be calculated for configuration V0 according to the classical design approach [3],

_{bs}/W

_{bs}≥ 8 given in [26]. It must be pointed out that this approach only considers a linear trajectory of settling particles in a straight channel with flat bed and does not account for any geometrical variation or facility components, such as transition zone or outflow weir, respectively. Furthermore, the still-water settling velocity w

_{s,0}is a crucial desanding facility design parameter, and its related uncertainty directly affects the calculation of the basin length. For example, for a water temperature of T = 10 °C the calculated still-water settling velocity for particles with d = 300 µm may vary by about 50% if calculated with the approach of Wu and Wang [27] (w

_{s,0}= 0.028 m/s) or Zanke [28] (w

_{s,0}= 0.042 m/s).

#### 2.3.2. Parameter Variation

_{bs}, basin flow depth h

_{bs}and weir characteristics are constant for all simulations. Furthermore, sediment input including PSD and the critical diameter d

_{cr}= 330 µm are identical for all simulations. For the comparative parameter study, the following parameters were varied: basin length L

_{bs}, inlet channel Froude number F

_{o}= v

_{o}(g h

_{o})

^{−1/2}, with v

_{o}= inlet channel velocity and h

_{o}= inlet channel flow depth, inlet channel radius r and conjunction angle γ, respectively, transition zone horizontal expansion angle α and vertical expansion angle β and transition zone length L

_{tz}. To achieve constant basin flow depth h

_{bs}for varying values of F

_{o}, the inlet channel flow depth h

_{o}was accordingly adapted by shifting the inlet channel bed level. These parameters were chosen because they are (i) considered to essentially control basin approach flow conditions and consequently basin hydraulics and (ii) not considered in the classical, commonly used design approach. Table 1 gives an overview of the range of investigated parameters. All in all, the parameter study comprised some 70 simulation runs.

#### 2.3.3. Evaluation

- Trapping efficiency η
_{c}as ratio of trapped sediment to sediment input, see Equation (1). - Relative trapping efficiency$${\eta}_{c,rel}=\frac{{\eta}_{c,sim}}{{\eta}_{c,V0}},$$
_{c,sim}resulting from corresponding parameter study simulations and the trapping efficiency η_{c,V0}of the reference configuration V0 for the critical particle diameter d_{cr}= 330 µm. Note that values of η_{c,rel}above unity indicate that performance of the configuration with given parameter set is better than the setting of the reference configuration, and vice versa. - Mean basin turbulent kinetic energy (TKE) k
_{m}is used as an indicator for flow turbulence and presence of eddy structures and hence as a simplified measure for flow inhomogeneity. - Mean basin bed shear velocity u
_{*m}is an estimate for potential resuspension of particles after having settled to the basin bed. - The size of the recirculating flow zone in the basin is quantified by the length x
_{r}representing the extent (in x–z plane) of the transition zone recirculating flow into the basin (Figure 6). The extent of recirculating flow is identified by the area permanently characterized by basin longitudinal flow velocities v_{x}≤ 0 m/s. - Extent of weir approach flow, i.e., the effect of the weir on basin flow, is estimated by the distance x
_{up}for which incipient streamline upwards deflection occurs over most of the flow depth and the whole basin width upstream of the weir (Figure 7).

## 3. Results and Discussion

#### 3.1. Validation of Numerical Model

_{1}(4 pm), t

_{2}(6:15 pm) and t

_{3}(8:15 pm), while the mean discharge was nearly constant at 2.2 m

^{3}/s. Four acoustic Doppler velocimetry probes (ADVs) with side-looking heads were used for the 3-D flow velocity measurements in the basin. Measurements were carried out at 9 to 12 cross sections corresponding to 3 to 4 m intervals along the basin. Each cross section contained about 50 to 70 single flow velocity measuring points [10]. The measurement time was 90 s for each measuring point with a sampling frequency of 100 Hz.

#### 3.1.1. Flow Field

_{x}, which is the driving component in the desander basin. Figure 8 shows contour plots for selected sections (cross sectional, longitudinal, horizontal) of measured and simulated v

_{x}. In view of the involved uncertainties in the experimental measurements in the order of 10 to 15% (see [10]), the agreement was judged satisfactory. The model reproduced the homogenization of the flow in the transverse direction with a more even distribution of streamwise flow velocities with increasing distance from the inlet (see cross sections in Figure 8). Additionally, in the longitudinal plots the general axial velocity decay in the streamwise direction was visible both in the experiments and the simulations. The flow acceleration in the gap between the tranquilizing racks and the side walls as well as the deceleration in the central flow column just after the rack were well reproduced by the model. While the elevated flow at the surface at the basin end was also given by both approaches, the bottom jet passing below the tranquilizing racks was more pronounced and extended longer in the experimental data. This also holds for the streamwise velocity components close to the basin sidewalls, which were similar in flow pattern but more pronounced in the experiments. Furthermore, the evaluation of the mean cross sectional longitudinal flow velocity [23] shows good agreement with the simulation results.

#### 3.1.2. Sediment Transport

_{1}, t

_{2}and t

_{3}(Table 2). The simulation results show that model performance with respect to the concentration-related trapping efficiency η

_{c}(Equation (1)) increases with increasing median diameter of the suspended sediment load. For low SSC input at t

_{1}, the simulated trapping efficiency was more than 50% below the measurement data, while for t

_{3}with a pronouncedly coarser PSD the deviation was only 10%. It was noted that the model underestimated the trapping efficiencies for all three times of measurements, which is conservative in terms of required basin length. Evaluating the mean outlet PSD of the three recordings, the simulation results are in good agreement with the measurements, indicating that the particle size refinement in the basin is well reproduced by the model (Figure 9).

_{1}in particular shows a high share of cohesive particles. For the subsequent parameter study, only non-cohesive particles with diameter larger than 76 µm were considered (see Figure 4). (ii) In situ, the tranquilizing racks locally hinder the sediment-laden flow. Hereby, particles can collide with rack bars, depending on particle inertia. This results in deceleration of particle speed or deflection of the particle trajectory. As a consequence, particles may deposit earlier, thus trapping efficiency increases. However, the interaction of particles and rack bars was not considered in the model, which may lead to a general difference in trapping efficiency. Moreover, PSD and SSC measurement uncertainties must also be considered, which are not further discussed in the present paper but are discussed by the authors in [10,11].

#### 3.2. Correlation of Desanding Facility Configuration and Trapping Efficiency

#### 3.2.1. Basin Length

_{bs}= 32 m resulted in a trapping efficiency for the critical particle diameter of η

_{c,sim}= 0.87 (Figure 10a). For similar flow conditions in the main part of the basin but in absence of a transition zone and end weir, a basin length of L

_{bs}= 32 m was sufficient to achieve complete settling of particles with diameter d

_{cr}= 330 µm according to the classical design approach (Figure 10b). This has been reproduced by numerical simulations of a configuration with a geometry according to the classical design approach (η

_{c,sim}= 1 for L

_{bs}= 32 m) to verify the numerical model robustness.

_{bs}markedly affects the trapping efficiency, i.e., trapping efficiency increases with increasing basin length. For basins shorter than the reference configuration (V7.1 to V7.5, Table A1, with L

_{bs}= 8 m and 16 m, respectively), i.e., for relative basic basin length χ = L

_{bs}/L

_{bs}

_{,V0}< 1, no clear delineation of the recirculation zone is possible, because the recirculating flow extends along the entire basin to the weir, which leads to lower trapping efficiencies than for V0, resulting in relative efficiencies (Equation (6)) below unity, i.e., η

_{c,rel}< 1.

_{bs}≥ 32 m, the mean basin bed shear velocity levels were at about 1.0 cm/s. This is because the bed shear velocity in the upstream part of the basin was strongly affected by the flow conditions in the transition zone, whose effect diminishes for longer basins. This agrees with the fact that x

_{r}and x

_{up}became independent of L

_{bs}and remained constant at 13.1 m and 6 m, respectively. For L

_{bs}= 40 m (25% longer than V0, i.e., χ = 1.25), almost all particles equal to or larger than the critical diameter were trapped, i.e., η

_{c,sim}= 0.95. Complete trapping of the critical diameter d

_{cr}= 330 µm was achieved for L

_{bs}= 56 m (75% longer than V0, i.e., χ = 1.75). Particles remained in suspension for such a long distance because the recirculation triggered by the transition zone led to a compression of streamlines in the upper part of the water column, and thus uniform flow can hardly be established in the basin (Figure 7). The weir at the end of the basin also affected basin flow but with minor effect. Thus, with an increase of 25% in basin length, the trapping efficiency increased from 0.87 to 0.95, but to achieve 1.0 an additional 75% of basin length relative to the basic length was necessary (Figure 11).

_{c,sim}for configurations of varying basin length (V0 and V7.1 to V7.5, Table A2), the relative basic basin length χ = L

_{bs}/L

_{bs}

_{,V0}for expected prototype trapping efficiency η

_{c,d}can be expressed by the basin length factor (R

^{2}= 0.99)

#### 3.2.2. Inlet Channel and Approach Flow Conditions

**Froude Number:**In case of supercritical approach flow as observed, e.g., below bottom intakes in steeply sloped rivers (parameter study configurations V5.3 and V5.4, with F

_{o}= 2.0 and 3.5, respectively), a hydraulic jump formed in the transition zone region that led to a recirculation zone at the water surface and a near-bed jet, both extending nearly to the middle of the basin. The mean basin turbulent kinetic energy k

_{m}was about doubled, and the mean basin bed shear velocity u

_{*m}about quadrupled compared to V0 (F

_{o}= 0.35), resulting in a reduction in trapping efficiency η

_{c,sim}of about 12% for the investigated configurations, and thus

**Channel course:**At excessive approach flow asymmetry in terms of small inlet channel radii r and inlet channel conjunction angles γ, the resulting trapping efficiency η

_{c,sim}was 2 to 6% lower compared to V0 (Figure 12). In contrast, a slightly curved (r > 10 m) or slightly angled (130° < γ < 180°) inlet channel favored the trapping efficiency according to the simulation results. This could be due to the weakened flow momentum in basin direction, which is accompanied by decreased flow recirculation at the vertical expansion. With increasing r or γ, the volumetric fraction of recirculating flow Λ also increased linearly. The combination of both reveals that the more asymmetric the approach flow, the shorter the overall basin flow recirculation zone. Moreover, the mean basin bed shear velocity u

_{∗m}increased about linearly at increasing approach flow asymmetry. Furthermore, for particle sizes smaller than the critical diameter, the trapping efficiencies were markedly more reduced, indicating a particle-size-specific response to the flow field. For curved (R

^{2}= 0.86) or angled (R

^{2}= 0.98) inlet channel (units are omitted in all correlations hereafter), the parameter study yielded:

#### 3.2.3. Transition Zone Geometry and Recirculating Flow

**Horizontal expansion angle:**The overall effect of the transition zone horizontal expansion angle α on the basin flow field and trapping efficiency η

_{c,sim}was minor. Values of turbulent kinetic energy k

_{m}and recirculation zone length x

_{r}were virtually constant for α = 90, 60, 45 and 30°, while they increased for α = 22.5, 15, 10 and 5.7°. Values of u

_{∗m}were almost constant across the whole range of investigated angles (configurations V2.1 to V2.7, Table A2). Figure 13a shows the correlation between α and x

_{r}. The increasing values can be explained by the location of recirculating flow, which was shifted towards the basin with decreasing α. In contrast, for large α values lateral flow recirculation mainly occurred within the transition zone. For the recirculation zone length x

_{r}the parameter study yielded (R

^{2}= 0.96)

**V**

**ertical expansion angle:**An increase in the vertical expansion angle β from 19.3° (configuration V0) to 90° (configuration V3.6) led to a consistent increase in k

_{m}, u

_{∗m}and x

_{r}of up to 50%. As shown in Figure 13a, the correlation between β and x

_{r}can be approximated by the hyperbolic tangent regression function (R

^{2}= 0.95)

**Transition zone length:**The systematic variation of the transition zone length L

_{tz}(configurations V4.1 to V4.4) defines values of the horizontal and vertical expansion angles (cf. Table A1). Thus, the effect of L

_{tz}on the recirculation zone length x

_{r}is in line with the above findings, but the resulting range of relative trapping efficiencies η

_{c,rel}was slightly extended (Figure 13b). The relative trapping efficiency η

_{c,rel}increased with decreasing x

_{r}values.

_{r}, which in turn results in different trapping efficiency η

_{c,sim}. As shown in Figure 13b the correlation between x

_{r}and η

_{c,rel}can be approximated by a polynomial function. Based on the parameter study (R

^{2}= 0.95) it follows

**Offset in bottom level of inlet channel and basin:**Complementary, to account for the offset in bottom level of inlet channel and basin, the approach of Durst and Tropea [29] for recirculating flow over a backwards-facing step, i.e., for β = 90°, may be adapted. Their findings allow for the estimation of the recirculation zone length x

_{r}based on a vertical expansion ratio ${h}_{bs}/{h}_{o}$ and step height ${h}_{bs}-{h}_{o}$ (Figure 1). For a fully turbulent approach flow, i.e., an approach flow with a Reynolds number R

_{o}> 2 × 10

^{4}, they found that the expansion ratio is independent of R

_{o}. At prototype desanding facilities, R

_{o}is usually in the order of 10

^{5}to 10

^{6}.

^{2}= 1) can be defined (Figure 14) to calculate the recirculation zone length x

_{r,β = 90°}due to flow over a backwards-facing step:

#### 3.2.4. Tranquilizing Racks

_{m}decreased by 53 to 83%, and u

_{∗m}decreased by 5 to 60%. Moreover, under asymmetric approach flow conditions a slight reduction in the recirculating flow zone could be identified (configurations V1.1_r and V8.3_r). The effect on basin hydraulics was also expressed by an increase in trapping efficiency η

_{c,sim}by up to 8% relative to the basic configuration. The increase in trapping efficiency was more pronounced at configurations with asymmetric approach flow conditions, where the flow became aligned and more uniform due to the tranquilizing racks.

#### 3.2.5. Weir

_{up}can hardly be estimated for configurations, that lead to partially swirled and helicoidal basin flow. In particular, this holds for configurations with asymmetric approach flow conditions. Thus, to evaluate the effect of the end weir on the trapping efficiency, the following configurations allowing for a distinct identification of x

_{up}were selected: V0, V2, V3, V4 and V7.3 to V7.5.

_{up}≈ 3 to 6 m (Table A2), desanding facility geometries with higher x

_{r}values show lower x

_{up}values and vice versa. In this state, it was found by supplementary simulations that x

_{up}(i) showed almost no variation with basin flow velocity v

_{x}and (ii) correlated with the ratio of weir overflow depth h

_{wo}to weir height h

_{w}. The corresponding correlation reads

#### 3.2.6. Special Configurations

_{c,sim}≈ 1.0) for particles with critical diameter can be achieved, and recirculating flow is reduced or vanishes, i.e., x

_{r}becomes zero (Table A2).

_{m}were achieved among all investigated geometries.

## 4. Design Guideline

#### 4.1. General Recommendations

#### 4.2. Enhanced Design Concept

- Definition of critical diameter d
_{cr}of the sediment load: The definition of the critical diameter may be based either on concentration-related or mass-related approaches. The smaller the critical grain size, the lower the sediment input into the power waterway and the lower hydro-abrasion at turbine components. However, the lower the critical grain size, the more sediment settles in the basin, potentially increasing flushing water volumes lost for power production. The latter largely depends on the flushing system (e.g., continuous vs. discontinuous). The application of the mass-related approach is only possible when a particle size distribution curve of suspended sediment is available. Typical design values of the critical particle diameter are 0.25 to 0.30 mm for water intakes, from which water is conveyed via a desanding facility directly to turbines, i.e., without passing reservoirs or headwater storage basins that would act as settling ponds. - Calculation of still-water settling velocity ${w}_{s,0}$ for critical diameter d
_{cr}using Equation (2). (Note: The present study and concept are based on the approach of Soulsby [17]; using other approaches is inconsistent and may lead to incorrect results). - Definition of basin cross section and calculation of longitudinal basin flow velocity: The definition of the basin cross section follows the common recommendation W
_{bs}≈ 0.8 h_{bs}. Here, h_{bs}is the overall basin mean flow depth, which is typically taken in the middle of the basin in streamwise direction for a sloping basin invert. Often, the cross section is rectangular in the upper part, followed by a constriction in the lower part with slopes of 4:5 (V:H) to facilitate sediment flushing. For the rectangular cross section, the longitudinal mean basin flow velocity v_{x}= Q_{d}/(W_{bs}h_{bs}) can be calculated, where Q_{d}is the basin design discharge. For non-rectangular cross section, the mean wetted basin flow area A_{mean,bs}is considered to result in v_{x}= Q_{d}/A_{mean,bs}. - Calculation of adjusted basin length ${L}_{bs}$: First, the basic basin length ${\widehat{L}}_{bs}$ has to be computed based on the classical design approach using Equation (5). Second, select a target trapping efficiency η
_{c,d}, which may be estimated considering economic aspects and site-specific construction constraints. The larger the target trapping efficiency, the longer the basin, but the lower the sediment input into the power waterway and hydro-abrasive wear at hydraulic machinery. To quantify the latter, turbine erosion models should be applied [30,31,32], allowing for a quantitative cost–benefit assessment of the whole system from desanding facility to hydraulic machines. A design trapping efficiency of η_{c,d}= 95% is considered meaningful in most cases, resulting in 35% to 40% longer basins compared to the classical approach. Consequently, the scaling factor χ = L_{bs}/L_{bs,V0}(Figure 11) for the target trapping efficiency η_{c,d}can be determined using Equation (7), and the adjusted basin length results in ${L}_{bs}=\chi \xb7{\widehat{L}}_{bs}$. - Calculation of adjustment terms $\Delta {L}_{i}$ according to Section 3.2:
- Inlet channel geometry and approach flow conditions: $\Delta {L}_{a}=\left(1-{\eta}_{c,rel}\right){\widehat{L}}_{bs}$ with relative trapping efficiency ${\eta}_{c,rel}$ for supercritical approach flow according to Equation (8) and for subcritical approach according to Equation (9);
- Transition zone geometry: $\Delta {L}_{r}=\left(1-{\eta}_{c,rel}\right){\widehat{L}}_{bs}$ with relative trapping efficiency ${\eta}_{c,rel}$ according to Equation (12), where the recirculation zone length x
_{r}is the maximum value out of Equation (10) (horizontal expansion α), Equation (11) (vertical expansion β) and the complementary Equation (13) (offset in bottom level); - Tranquilizing racks: $\Delta {L}_{tr}=\left(1-{\eta}_{c,rel,mean}\right){\widehat{L}}_{bs}$ with mean increase in relative trapping efficiency ${\eta}_{c,rel,mean}$ according to Equation (14) for subcritical frontal, subcritical curved or angled, or supercritical approach flow;
- Weir: $\Delta {L}_{w}={x}_{up}$ with extent of weir approach flow in terms of distance x
_{up}according to Equation (15).

- Calculation of total basin length ${L}_{tbs}={L}_{bs}+\sum \Delta {L}_{i}$, i.e., the sum of adjusted basic basin length ${L}_{tbs}$ plus all adjustment terms $\Delta {L}_{i}$.

#### 4.3. Application Example

_{bs}= 35 m, W

_{bs}= 5.8 m, α = 11.3°, β = 18.7° and γ = 158°. The field measurements at the Saas Balen facility yielded h

_{bs}= 3.28 m, h

_{o}= 1.54 m, h

_{w}= 3.92 m, h

_{wo}= 0.40 m, F

_{o}= 0.15 and basin design discharge Q

_{d}= 2.24 m

^{3}/s. The mean basin flow velocity is v

_{x}= Q

_{d}/(W

_{bs}h

_{bs}) ≈ 0.12 m/s. Further, the trapping efficiency was found to be η

_{c}= 0.74 for particles of d

_{cr}≈ 200 μm. Now follow the six main steps of the enhanced design concept:

- The critical diameter is an unknown a priori and has to be reckoned back based on the classical design approach. Solving Equation (5) resulted in the still-water vertical settling velocity w
_{s}_{,0}for the given basin length. Consequently, Equation (2) could be solved which resulted in the design particle diameter, representing the critical diameter of the target particle size yielded d_{cr}= 203 μm ≈ 200 μm for water at 5 °C. - The settling velocity for the critical diameter was obtained in step 1.
- The basin cross section and longitudinal basin flow velocity are given and were determined by field measurements.
- In this example, the basic basin length ${\widehat{L}}_{bs}$ was equal to the current prototype basin length, corresponding to an arithmetical trapping efficiency of the critical particle size of 87% (see Section 3.2.1, Figure 11). Assuming that maintenance costs due to turbine abrasion are too high for the current facility layout and should be reduced by extending the desander basin in the streamwise direction, to achieve a target trapping efficiency of 95%, the target basic basin length became ${L}_{bs}=\chi \xb7{\widehat{L}}_{bs}$ = 1.39 ∙ 35 m = 48.65 m.
- Adjustment terms:
- Inlet channel geometry and approach flow conditions: for subcritical flow and γ = 158°, ΔL
_{a}= (1 − 1.031) ∙ 35 m = −1.09 m; - Transition zone geometry: with ${x}_{\mathrm{r},\alpha}$ = 10.7 m from Equation (10),${x}_{\mathrm{r},\beta}$ = 12.7 m from Equation (11) and ${x}_{\mathrm{r},\mathrm{h}}$ = 15.0 m from Equation (13) for an expansion ratio ${h}_{bs}/{h}_{o}=$ 2.13 and a step height ${h}_{bs}-{h}_{o}=$ 1.74 m, the critical relative trapping efficiency due to recirculating flow became ${\eta}_{c,rel}\left({x}_{r,max}=15.0\mathrm{m}\right)=0.992$ from Equation (12), so that ΔL
_{r}= (1 − 0.992) ∙ 35 m = 0.27 m; - Tranquilizing racks: for a subcritical, asymmetric (angled) approach flow from Equation (14) ΔL
_{tr}= (1 − 1.09) ∙ 35 m = –3.15 m; - Weir: for ${h}_{wo}/{h}_{w}=$ 0.10 from Equation (15) ΔL
_{w}= ${x}_{up}$ = 6.30 m.

- The total basin length required to achieve 95% trapping of particles d
_{cr}= 200 μm resulted from L_{tbs}= ${L}_{bs}+\sum \Delta {L}_{i}=$48.65 m − 1.09 m + 0.27 m − 3.15 m + 6.30 m ≈ 51 m.

_{cr}= 200 μm.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variation | r | γ | L_{tz} | α | β | L_{bs} | F_{o} |
---|---|---|---|---|---|---|---|

(m) | (°) | (m) | (°) | (°) | (m) | (–) | |

V0 | ∞ | 180 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V1.1 | 2 | – | 10 | 5.7 | 19.3 | 32 | 0.35 |

V1.2 | 10 | – | 10 | 5.7 | 19.3 | 32 | 0.35 |

V1.3 | 20 | – | 10 | 5.7 | 19.3 | 32 | 0.35 |

V1.4 | 5 | – | 10 | 5.7 | 19.3 | 32 | 0.35 |

V2.1 | ∞ | 180 | 10 | 90 | 19.3 | 32 | 0.35 |

V2.2 | ∞ | 180 | 10 | 60 | 19.3 | 32 | 0.35 |

V2.3 | ∞ | 180 | 10 | 45 | 19.3 | 32 | 0.35 |

V2.4 | ∞ | 180 | 10 | 30 | 19.3 | 32 | 0.35 |

V2.5 | ∞ | 180 | 10 | 15 | 19.3 | 32 | 0.35 |

V2.6 | ∞ | 180 | 10 | 10 | 19.3 | 32 | 0.35 |

V2.7 | ∞ | 180 | 10 | 22.5 | 19.3 | 32 | 0.35 |

V3.1 | ∞ | 180 | 10 | 5.7 | 90 | 32 | 0.35 |

V3.2 | ∞ | 180 | 10 | 5.7 | 60 | 32 | 0.35 |

V3.3 | ∞ | 180 | 10 | 5.7 | 45 | 32 | 0.35 |

V3.4 | ∞ | 180 | 10 | 5.7 | 30 | 32 | 0.35 |

V3.5 | ∞ | 180 | 10 | 5.7 | 25 | 32 | 0.35 |

V3.6 | ∞ | 180 | 10 | 5.7 | 37.5 | 32 | 0.35 |

V4.1 | ∞ | 180 | 0 | 90 | 90 | 32 | 0.35 |

V4.2 | ∞ | 180 | 5 | 11.3 | 35 | 32 | 0.35 |

V4.3 | ∞ | 180 | 15 | 3.8 | 13.1 | 32 | 0.35 |

V4.4 | ∞ | 180 | 20 | 2.9 | 9.9 | 32 | 0.35 |

V5.3 | ∞ | 180 | 10 | 5.7 | 19.3 | 32 | 2.0 |

V5.4 | ∞ | 180 | 10 | 5.7 | 19.3 | 32 | 3.5 |

V6.3 | based on V0, but with continuously expanding basin area | ||||||

V6.5 | based on V0, but with impact wall at end of transition zone (TZ) | ||||||

V6.9 | based on V0, but with longitudinal vertical walls in TZ | ||||||

V6.10 | based on V0, but with both-sided vertical inclined plates in basin | ||||||

V6.13 | based on V0, but with horizontal flow deflection plates at beginning of TZ | ||||||

V6.14 | based on V0, but with horizontal flow deflection plates and tranquillizing racks in TZ | ||||||

V7.1 | ∞ | 180 | 10 | 5.7 | 19.3 | 8 | 0.35 |

V7.2 | ∞ | 180 | 10 | 5.7 | 19.3 | 16 | 0.35 |

V7.3 | ∞ | 180 | 10 | 5.7 | 19.3 | 40 | 0.35 |

V7.4 | ∞ | 180 | 10 | 5.7 | 19.3 | 48 | 0.35 |

V7.5 | ∞ | 180 | 10 | 5.7 | 19.3 | 56 | 0.35 |

V8.1 | ∞ | 160 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V8.2 | ∞ | 140 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V8.3 | ∞ | 120 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V8.4 | ∞ | 170 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V8.5 | ∞ | 150 | 10 | 5.7 | 19.3 | 32 | 0.35 |

V8.6 | ∞ | 130 | 10 | 5.7 | 19.3 | 32 | 0.35 |

## Appendix B

Variation | k_{t,m} | u_{*m} | x_{r} | x_{up} | η_{c,sim} | η_{c,sim/}η_{c,V0} |
---|---|---|---|---|---|---|

(cm^{2}/s^{2}) | (cm/s) | (m) | (m) | (–) | (–) | |

V0 | 249 | 1.0 | 13.1 | 6 | 0.87 | 1.00 |

V0_r | 62 | 0.5 | 25.2 | n/a | 0.92 | 1.06 |

V1.1 | 198 | 3.1 | 10.9 | n/a | 0.85 | 0.98 |

V1.1_r | 66 | 2.6 | 9.8 | n/a | 0.94 | 1.08 |

V1.2 | 185 | 1.8 | 14.4 | n/a | 0.91 | 1.05 |

V1.3 | 211 | 1.2 | 15.3 | n/a | 0.90 | 1.03 |

V1.4 | 203 | 3.0 | 11.4 | n/a | 0.85 | 0.98 |

V2.1 | 232 | 0.9 | 9.0 | 6 | 0.89 | 1.02 |

V2.1_r | 83 | 0.5 | 21.3 | n/a | 0.94 | 1.08 |

V2.2 | 233 | 0.9 | 9.0 | 6 | 0.89 | 1.02 |

V2.3 | 233 | 0.9 | 9.0 | 6 | 0.90 | 1.03 |

V2.4 | 234 | 0.9 | 9.0 | 6 | 0.89 | 1.02 |

V2.5 | 247 | 1.0 | 10.0 | 6 | 0.88 | 1.01 |

V2.5_r | 63 | 0.4 | 23.2 | n/a | 0.93 | 1.07 |

V2.6 | 248 | 1.0 | 11.7 | 6 | 0.86 | 0.99 |

V2.7 | 238 | 1.0 | 9.4 | 6 | 0.88 | 1.01 |

V3.1 | 365 | 1.5 | 19.6 | 5 | 0.85 | 0.98 |

V3.1_r | 83 | 0.7 | 32.0 | n/a | 0.91 | 1.05 |

V3.2 | 354 | 1.3 | 19.2 | 5 | 0.86 | 0.99 |

V3.3 | 345 | 1.3 | 18.8 | 5 | 0.86 | 0.99 |

V3.4 | 302 | 1.1 | 14.6 | 6 | 0.86 | 0.99 |

V3.5 | 277 | 1.1 | 14.3 | 6 | 0.87 | 1.00 |

V3.6 | 340 | 1.2 | 16.4 | 6 | 0.86 | 0.99 |

V4.1 | 400 | 1.6 | 27.8 | 3 | 0.80 | 0.92 |

V4.2 | 336 | 1.2 | 19.9 | 5 | 0.85 | 0.98 |

V4.3 | 198 | 0.9 | 10.2 | 6 | 0.9 | 1.03 |

V4.4 | 149 | 0.8 | 6.7 | 6 | 0.94 | 1.08 |

V5.3 | 437 | 3.9 | 15.0* | n/a | 0.77 | 0.89 |

V5.4 | 528 | 4.0 | 15.1* | n/a | 0.76 | 0.87 |

V5.4_r | 246 | 3.8 | 21.3* | n/a | 0.91 | 1.05 |

V6.3 | 175 | 0.4 | 0.0 | n/a | 0.89 | 1.02 |

V6.5 | 90 | 1.9 | 7.4 | n/a | 0.91 | 1.05 |

V6.9 | 276 | 1.4 | 26.7 | n/a | 0.93 | 1.07 |

V6.10 | 246 | n/a | 6.5 | n/a | 0.90 | 1.03 |

V6.13 | 64 | 1.9 | 0.0 | n/a | 0.98 | 1.13 |

V6.14 | 14 | 0.8 | 0.0 | n/a | 0.98 | 1.13 |

V7.1 | 284 | 1.7 | 8.0 | n/a | 0.54 | 0.62 |

V7.2 | 356 | 1.2 | 16.0 | n/a | 0.71 | 0.82 |

V7.3 | 181 | 1.0 | 13.1 | 6 | 0.95 | 1.09 |

V7.4 | 174 | 1.0 | 13.1 | 6 | 0.98 | 1.13 |

V7.5 | 165 | 1.0 | 13.1 | 6 | ≈1.0 | 1.15 |

V8.1 | 222 | 1.2 | 14.3 | n/a | 0.90 | 1.03 |

V8.2 | 226 | 1.7 | 12.3 | n/a | 0.89 | 1.02 |

V8.3 | 277 | 3.2 | 11.6 | n/a | 0.82 | 0.94 |

V8.3_r | 100 | 2.9 | 10.6 | n/a | 0.89 | 1.02 |

V8.4 | 269 | 0.9 | 10.6 | n/a | 0.88 | 1.01 |

V8.5 | 208 | 1.8 | 12.9 | n/a | 0.89 | 1.02 |

V8.6 | 253 | 2.9 | 11.9 | n/a | 0.86 | 0.99 |

## Appendix C

Parameter | Description | Investigated Values | Units | |
---|---|---|---|---|

Lower | Upper | |||

L_{bs} | Basin length | 8 | 56 | m |

F_{o} | Inlet channel Froude number | 0.35 | 3.5 | – |

r | Inlet channel radius | 2 | 20 | m |

γ | Inlet channel conjunction angle | 120 | 180 | ° (angle) |

α | Transition zone horizontal expansion angle | 5.7 | 90 | ° (angle) |

β | Transition zone vertical expansion angle | 19.3 | 90 | ° (angle) |

d_{cr} | Critical particle diameter | 0.06 | 2 | mm |

h_{wo}/h_{w} | Relative weir overflow depth | 0.14 | 0.67 | – |

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**Figure 1.**Schematic of a typical desanding facility with inlet channel after the intake, transition zone, basin and outlet with end weir: plan view of facility with (

**a**) curved and (

**b**) angled inlet channel, (

**c**) longitudinal section; main flow direction from left to right.

**Figure 2.**Details of tranquilizing racks at prototype facilities with view in flow direction, i.e., the open part of the rack profiles are opposed to the flow.

**Figure 3.**Modeling of tranquilizing racks with plan view (upper) and frontal view (lower part): fully blocked baffles (P = 0) in combination with permeable parts incorporating head loss (P = 1, ζ > 0) are used for efficient simulation.

**Figure 4.**(

**a**) PSD with particles in the non-cohesive fine and medium sand size range used for the numerical simulations and (

**b**) discretization of the PSD by four grain size classes and corresponding still-water settling velocities w

_{s,}

_{0}for ρ

_{s}= 2650 kg m

^{−3}and water at 8 °C.

**Figure 5.**Special configurations with arrows indicating applied modification: (

**a**) continuously expanding basin area (configuration V6.3), (

**b**) impact wall (configuration V6.5), (

**c**) longitudinal vertical flow guiding walls (configuration V6.9), (

**d**) both-sided vertical inclined plates (configuration V6.10), (

**e**) horizontal flow deflection plates (configuration V6.13) and (

**f**) horizontal flow deflection plates and racks (configuration V6.14); main flow direction is from left to right (in x-direction).

**Figure 6.**Longitudinal profile of recirculation zone length x

_{r}; the line representing v

_{x}= 0 m/s is the dividing streamline that intersects with the bed at the reattachment point; main flow direction is from left to right.

**Figure 7.**Longitudinal profile of simulated flow in basin for reference configuration V0 illustrated by streamlines along basin middle axis (x-z plane); main flow direction from left to right (in x-direction).

**Figure 8.**Contour plots of (

**a**) measured and (

**b**) simulated streamwise flow velocity component v

_{x}for the Saas Balen facility. The indicated non-dimensional locations X, Y and Z are normalized by basin length, width and depth, respectively; main flow is in x-direction.

**Figure 9.**Mean PSD of suspended sediment in inlet channel measured at prototype and mean outlet PSD curves from measurements and numerical simulations.

**Figure 10.**Longitudinal trajectory of surface particle with diameter d

_{cr}= 330 µm under similar flow conditions in main part of basin: (

**a**) desanding facility according to reference configuration V0 and basin length L

_{bs}= 32 m resulted in trapping efficiency η

_{c,sim}= 0.87, while (

**b**) complete settling of particles was achieved for configuration according to classical design approach.

**Figure 11.**Relation (Equation (7)) between trapping efficiency η

_{c,sim}, and relative basin length χ = L

_{bs}/L

_{bs,V}

_{0}resulting from the parameter study.

**Figure 12.**Correlations (Equation (9)) between relative trapping efficiency η

_{c,rel}and (

**a**) inlet channel radius r and (

**b**) inlet channel conjunction angle γ resulting from parameter study.

**Figure 13.**Correlations between recirculation zone length x

_{r}and (

**a**) horizontal expansion angle α (Equation (10)) and transition zone vertical expansion angle β (Equation (11)), respectively and (

**b**) relative trapping efficiency η

_{c,rel}(Equation (12)), red markers indicate results from configurations V4.1 to V4.4.

**Figure 14.**Correlation (Equation (13)) between vertical expansion ratio ${h}_{bs}/{h}_{o}$ and recirculation zone length x

_{r}to step height ${h}_{bs}-{h}_{o}$ for an approach flow with a Reynolds number R

_{o}> 2 × 10

^{4}, adapted according to [29]. Experimental results indicate that ${x}_{r}/\left({h}_{bs}-{h}_{o}\right)$ = 8.6 remained constant for ${h}_{bs}/{h}_{o}$ > 2.

Parameter | Description | Investigated Values |
---|---|---|

L_{bs} (m) | Basin length | 8, 16, 32, 40, 48, 56 |

F_{o} (–) | Inlet channel Froude number | 0.35, 2.0, 3.5 |

r (m) | Inlet channel radius | 2, 5, 10, 20 |

γ (°) | Inlet channel conjunction angle | 180, 170, 160, 150, 140, 130, 120 |

α (°) | Transition zone horizontal expansion angle | 5.7, 10, 15, 22.5, 30, 45, 60, 90 |

β (°) | Transition zone vertical expansion angle | 19.3, 25, 30, 37.5, 45, 60, 90 |

L_{tz} (m) | Transition zone length | 0, 5, 10, 15, 20 |

**Table 2.**Comparison of prototype measurements (p) and simulation results (s) at three different times t

_{1}, t

_{2}and t

_{3}: suspended sediment concentration at inlet channel (C

_{in}) and at the outlet weir (C

_{out}), diameter d

_{50}at inlet and resulting trapping efficiencies (η

_{c}).

Data Set | C_{in}(g/L) | d_{50,in}(µm) | C_{out,p}v(g/L) | η_{c,p}(–) | C_{out,s}(g/L) | η_{c,s}(–) |
---|---|---|---|---|---|---|

t_{1} | 0.34 | 22 | 0.08 | 0.76 | 0.27 | 0.32 |

t_{2} | 2.26 | 67 | 0.67 | 0.70 | 1.17 | 0.59 |

t_{3} | 1.10 | 121 | 0.25 | 0.77 | 0.45 | 0.70 |

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**MDPI and ACS Style**

Paschmann, C.; Vetsch, D.F.; Boes, R.M.
Design of Desanding Facilities for Hydropower Schemes Based on Trapping Efficiency. *Water* **2022**, *14*, 520.
https://doi.org/10.3390/w14040520

**AMA Style**

Paschmann C, Vetsch DF, Boes RM.
Design of Desanding Facilities for Hydropower Schemes Based on Trapping Efficiency. *Water*. 2022; 14(4):520.
https://doi.org/10.3390/w14040520

**Chicago/Turabian Style**

Paschmann, Christopher, David F. Vetsch, and Robert M. Boes.
2022. "Design of Desanding Facilities for Hydropower Schemes Based on Trapping Efficiency" *Water* 14, no. 4: 520.
https://doi.org/10.3390/w14040520