Experiment and fitting calculation of migration critical velocity of small-sized sediment particles erosion in rainwater pipeline

Sediments in rainwater pipelines can be easily washed up and carried away during rainy days. Among the various substances of urban stormwater pollutants, the suspended solids, heavy metals, and polycyclic aromatic hydrocarbons (PAHs) are considered the major causes of contamination in receiving environments (Zoppou 2001; Fletcher et al. 2013).

Previous research on the erosion process of sediments in drainage pipes focused on the exploration of the dynamic migration of sediments in pipes under different conditions. Sediment transport regularity in pipelines was simulated over time with an artificial neural network (Ebtehaj & Bonakdari 2013). Nonlinear regression and MINITAB software were used to predict sediment migration under different conditions, involving few parameters and rapid calculation (Ebtehaj et al. 2014). Gene expression programming was used to simulate the movement of suspended particles or sediments in pipelines (Ab. Ghani & Md. Azamathulla 2011). Sediment particle and sphere velocity measurements were carried out in two pipe channels, and a semi-theoretical equation was established for sediment transport at the limit of deposition in sewers (Ota & Perrusquia 2013). The raw acoustical turbidity was used as an online monitoring tool, and the accurate suspended solids concentration and water height could be obtained through this approach (Pallarès et al. 2016). The erosion or deposition process in a sewer channel in Paris were monitored for a long period of time during rainfall. Results showed that sediments with small particle size (<4.15 mm) were likely to be washed out (Shahsavari et al. 2017).

The sediment in pipelines was continuously washed, and the change in the height of sediment was measured to determine the conditions of sediment erosion and transport (Campisano et al. 2004). The critical shear stresses of sediment at different erosion velocities in a sewer were determined (Hrissanthou & Hartmann 1998). The transport capacity of the sediment in the pressure drainage pipe was poor. The sediment showed signs of starting to migrate at a velocity of 0.35 m/s, and significant migration occurred at 0.50 m/s (Gunkel & Pawlowsky-Reusing 2017). In studying the ‘starting’ process of sediment particles in drainage pipelines, the movement of riverbed sediments and the erosion of ground particles by rain can be identified. A small amount of sediment movement was observed on the bed during the evaluation of the criteria for the riverbed sediment to start (He et al. 2002).

When the flow velocity exceeds a certain limit, the sediment particles will migrate. This limit is the ‘starting velocity’, which is called the ‘critical velocity’ of sediment particle migration. In theory, the particles cannot be washed out when the flow velocity is lower than the ‘critical velocity’.

In this study, the erosion process of sediment particles in a rainwater pipeline was simulated in the laboratory during rainfall. The migration critical velocity of small-sized sediment particles was investigated under different particle sizes, pipe wall roughness, and sediment thickness. On this basis, mathematical models were established to calculate the critical velocity of sediment particles when being eroded in sewers under certain conditions. This approach helps control sediment pollution in urban drainage pipes.

Figure 1 presents the device used for simulating the erosion migration process of sediment particles in a rainwater pipeline. The plexiglass pipe has a length of 2 m and a diameter of 150 mm. The test pipe was connected to the bottom of a water tank, and the pipe flow was controlled through the valve and flow meter at the beginning of the pipe. The water flow velocity was measured by a velocity instrument. To facilitate the placement of sediment particles in the pipe, grooves were made at the top of the pipe. The sand paper, with a width of 12 cm, was laid on the inner wall of the pipe to simulate the commonly used drainage pipe materials. The grit size of sand paper was determined according to the roughness of different pipes. Table 1 illustrates the pipe, wall roughness, and the corresponding grit size of sand paper. Roughness is measured with a roughness tester.

The sediments used in the experiment were taken from outdoor rainwater drains. Such sediments were air-dried to constant weight, and the sediment particles with different sizes were sieved with a mesh sieve. At the beginning of the experiment, sediment of a certain mass and particle size was weighed and evenly mixed with a certain volume of water. The wet sediments were evenly smeared on the inner wall within the range of 1.00–1.25 m at the beginning of the pipeline, with a thickness between 0.20 cm and 0.35 cm. The valve was opened to a certain flow level, and sampling was conducted at 0, 15, 30, 45, and 60s at the outlet of the pipe. Then, the suspended solids (SS) concentration of the samples was measured. The experiments were completed considering changes in particle sizes, pipe wall roughnesses, and sediment thicknesses.

The ‘critical velocity’ is defined in this study as the water flow velocity corresponding to an erosion rate of approximately 10% (specifically in the range of 9.50%–10.50%) when the erosion time is 1 min.

The critical velocity at which the sediment particles are going to start eroding in the rainwater pipeline is affected by many factors: the three factors, namely, particle sizes (d₅₀), pipe wall roughness (μ), and sediment thickness (h), have a great impact (Hong et al. 2016; Aksoy et al. 2017; Liu et al. 2018). The parameter selection of this study is based on the data from the literature (Table 2).

The effect of sediment particle size on the migration critical velocity is shown in Figure 2(a). Under the same conditions, the critical velocity is small when the particle size is large; that is, within the smaller range of 0.33 mm to 0.83 mm. When the particle size decreases from 0.83 mm to 0.33 mm, the critical velocity increases from 0.441 m/s to approximately 0.530 m/s.

At the same particle size, the large flow velocity has large flow energy. Therefore, the large flow velocity has a high erosion rate. For example, under the particle size of 0.83 mm, the flow velocity increases from 0.387 m/s to 0.481 m/s, and the erosion rate increases from 6.16% to 13.22%. When the particle size is 0.33 mm, the flow velocity increases from 0.497 m/s to 0.553 m/s, and the erosion rate increases from 6.75% to 12.88%.

Rainwater pipes have different pipe wall roughnesses due to diverse pipe materials, rainwater quality, or service life. The four roughnesses in Figure 2(b) correspond to the four commonly used pipes (Table 1). The pipe wall roughness has a great impact on the migration critical velocity. The critical velocity significantly decreases with the decrease of roughness. The pipe wall roughnesses of 250, 150, 38, and 0.041 μm correspond to the critical velocities of 0.864, 0.781, 0.553, and 0.441 m/s, respectively.

The different sediment thickness in the pipeline will also bring about the changes in the migration critical velocity. Figure 2(c) studies the relationship between the two. Evidently, the critical velocity changes in the same direction as the sediment thickness, that is, the critical velocity is small when the sediment thickness is also small. In this study, the change of sediment thickness is small. When the thickness reduces from 0.35 cm to 0.20 cm, the critical velocity decreases from 0.497 m/s to 0.418 m/s.

When the particle size, pipe wall roughness, and sediment thickness are within a certain range, the range of the critical velocity in pipelines is obtained through experiments. Figure 3 lists the critical velocity ranges of the four types of pipes when the particle size is between 0.33 mm and 0.83 mm, and the sediment thickness is between 0.20 cm and 0.35 cm. In summary, the minimum and maximum values of the critical velocity gradually increase with the roughness increase. When the roughness increases from 0.041 μm (plexiglass pipe) to 250 μm (cast iron pipe), the minimum value increases from 0.418 m/s to 0.820 m/s, and the maximum value increases from 0.561 m/s to 1.195 m/s, both of which almost have been doubled.

Figure 3 shows that the range of the critical velocity also increases with the roughness increase. The variation rate between the maximum and the minimum values in the critical velocity range of each pipe has been calculated. The variation rates of the four pipes, namely, plexiglass pipe, steel/PVC pipe, galvanized/clay pipe, and cast iron pipe, are 34.21%, 34.70%, 39.62%, and 45.73%, respectively. Under large roughness, when the particle size or sediment thickness changes, more complex erosion characteristics are shown, thereby making the critical velocity span large.

The calculation models of the migration critical velocity for each pipe under four particle sizes were established by statistical analysis and data fitting (Table 3).

In the same way, the mathematical models of the other three pipes were established as Equations (9)–(11).

The migration critical velocity under a certain sediment thickness and particle size could be calculated according to Equations (8)–(11).

Equations (8)–(11) were used to calculate the critical velocity of the erosion under different conditions and to compare with the experimental measured values under the same conditions. The results shown in Figures 4–7 correspond to the above-mentioned four types of pipes.

Figures 4–7 display the calculated and measured values under different conditions, and the difference rates between the two are also shown. Under different particle sizes, roughnesses, and sediment thicknesses, the difference rates between the calculated and the measured values are in the range of −3.60% to 5.33%. Globally, the simulation results show a good agreement with the experimental data. The established mathematical models can efficiently simulate the critical state of erosion in the rainwater pipeline.

Highly aggregated soils have been proven to be resistant to water erosion (Shi et al. 2010; Mahmoodabadi & Ahmadbeigi 2013). When the sediment particles are fine, the adhesive force is strong, and the agglomerates that they form can greatly resist erosion (Eshel et al. 2004). The results showed that the high water flow dynamic and rainfall intensity often lead to high water erosion (Defersha & Melesse 2012; Shi et al. 2012; Guo et al. 2018). It was also proposed that rainfall intensity had a greater impact on the distribution of sediment particles than slope (Berger et al. 2010). When other conditions remain unchanged, the high water flow or rainfall intensity could bring great water flow velocity, which can efficiently erode the sediment on the slope or in the pipeline.

When the water flow dynamic is sufficiently large, the aggregates formed by the sedimentary particles can be disintegrated and transported. A threshold for rainfall intensity was found, beyond which the disintegration of aggregates (particle size > 0.25 mm) would be serious (Wang et al. 2016). The high rainfall intensity could release fine particles by breaking up large aggregates (Hao et al. 2019). In this study, the sediment aggregates are presumed to have been disintegrated or partially disintegrated when the erosion rate has reached 10%. Therefore, the erosion rate of 10% was used as an evaluation index.

The sediment particles in this study belong to the category of small particle size (d₅₀ = 0.33–0.83 mm), and all have strong viscosity. When the median particle size is small, the content of fine particles is high, the cohesive force is strong, and aggregates are difficult to be eroded and transported. Under the same conditions, small-sized particles need high water flow dynamic to be washed away and floated; thus, the critical velocity is large (Figure 2(a)). The effects of the hydraulic conditions, river bed roughness, and particle composition were considered during the study of the starting conditions of non-uniform sediment particles on the river bed. The coarse particles were easy to start, and the fine particles were difficult to start (He et al. 2002).

Once disintegration occurred, the transport mechanism between the fine and the coarse particles differed. During the erosion of red soil by rainfall runoff, sediments with a particle size of <0.1 mm were transferred in the form of suspension/jump, whereas large sediments with a particle size of 0.5–2 mm were transferred in the rolling form (Wu et al. 2017). It was pointed out that the particle size of sediment transported in the form of suspension/jump was approximately <0.05 mm under rainfall conditions (Hao et al. 2019). In this study, the median particle size was in the range of 0.33 mm to 0.83 mm, and the transport of both forms should exist after the aggregate collapse.

When the bed roughness increased, the energy loss increased due to the exchange of the fluid and the boundary (Cheng et al. 2016). The frictional resistance of the sedimentary particles on the pipe wall was great with the increase of the friction coefficient (Cheng et al. 2016). The soil surface roughness had a significant delayed effect on runoff generation during the study of the effect of soil surface roughness on runoff; the surface runoff was difficult to form when roughness increased (Ding & Huang 2017). The soil surface roughness also affected the threshold flow of wind and water erosion, which increased with the increase of roughness (Bullard et al. 2018).

When the flow or velocity increases to a certain extent, the particles begin to ‘start’. Therefore, a larger wall roughness has a great critical velocity, which will also consume a substantial amount of energy (Figure 2(b)). It was found that the particle's movement on the pipe wall was blocked due to the pipe roughness, and a substantial amount of energy was consumed to maintain the state of continued movement (Li et al. 2019). The motion of large particles in the drainage pipe was studied, and it was concluded that increasing the flow rate would easily cause the particles to move (Walski et al. 2009).

When the erosion rate of a large deposition thickness reaches the same 10%, the absolute amount of particles floating by eroding is great, and the water flow dynamic is needed (Caviedes-Voullieme et al. 2017).

In general, the minimum and maximum values of the critical velocity gradually increase with the increase of pipe roughness (Figure 3). The above-mentioned analysis indicated that when the roughness of the pipe is large, it will cause a large resistance to the water flow, thereby resulting in the decline of the water flow dynamic (Celmer et al. 2008; Cheng et al. 2016; Bullard et al. 2018). A high flow velocity is required to make the particles ‘start’. The erosion characteristics become complicated with the roughness increase, thereby making the range span of the critical velocity large.

The critical velocity ranges for the four pipes are calculated on the basis of the studied particle size and sediment thickness by using the mathematical models (Equations (8)–(11)): 0.413–0.573 m/s for the plexiglass pipe, 0.525–0.697 m/s for the steel/PVC pipe, 0.719–1.021 m/s for the galvanized/clay pipe, and 0.790–1.179 m/s for the cast iron pipe (Figures 4–7).

The starting conditions of sediment in the bottom bed of natural rivers were simulated and studied; the results indicated that the starting velocity of sediment with a particle size between 0.75 mm and 3.5 mm was within the range of 0.1–0.5 m/s under different slopes (He et al. 2003). When the particle size is in a large range (i.e. 0.001–10 mm), the starting velocity is in the range of 0.02–1.0 m/s (He et al. 2002).

The flow velocity in the drainage pipes that can continuously move large particles (referring to the vertical height of the particles being greater than the water depth in the pipe) is in the range of 0.6–1.0 m/s (Walski et al. 2009). A sediment erosion test was carried out in a channel with a length of 1.24 km (sediment particle size distribution: d₃₀ = 0.3 mm, d₅₀ = 1 mm, and d₉₀ = 10 mm; and sediment thickness of 1–10 cm). When the flow velocity was approximately 1 m/s, 50–75% of the inner surface area of the middle and lower reaches of the channel would be covered with sedimentary particles from upstream erosion (Caviedes-Voullieme et al. 2017).

In this study, the sediment particles in the rainwater pipeline may migrate due to water erosion. The flow velocity corresponding to the erosion rate of 10% is taken as the migration critical velocity. The critical velocity of sediment particles with small size presents regularity under different particle sizes, pipe wall roughnesses, and sediment thicknesses.

The sediment particles in the rainwater pipeline are viscous. The cohesive force of particles with a small size is relatively strong. The formed aggregates also have strong erosion resistance. The critical velocity is great when the particle size is small (d₅₀ = 0.33–0.83 mm). The critical velocity significantly decreases with the decrease in the pipe wall roughness. When the deposition thickness is large, the absolute number of particles floating and transporting corresponding to the erosion rate of 10% is great; thus, the migration critical velocity is also large. Under the research conditions, the migration critical velocities of small-sized sediment particles in pipes with four types of wall roughness have been determined: 0.413–0.573 m/s for the plexiglass pipe, 0.525–0.697 m/s for the steel/PVC pipe, 0.719–1.021 m/s for the galvanized/clay pipe, and 0.790–1.179 m/s for the cast iron pipe. The mathematical models for calculating the migration critical velocity of small-sized sediment particles in rainwater pipelines were established through quadratic fitting. Under the conditions of this study, the difference rate between the calculated and the measured values is in the range of −3.60% to 5.33%, thereby showing a good consistency between the two.

This work was supported by the National Natural Science Foundation of China (51808285).

All relevant data are included in the paper or its Supplementary Information.