Abstract

Multiple models from the literature and experimental datasets have been developed and collected to predict sediment transport in sewers. However, all these models were developed for smaller sewer pipes, i.e. using experimental data collected on pipes with diameters smaller than 500 mm. To address this issue, new experimental data were collected on a larger, 595 mm pipe located in a laboratory at the University of los Andes. Two new self-cleansing models were developed using this dataset. Both models predict the sewer self-cleansing velocity for the cases of non-deposition with and without deposited bed. The newly developed and existing models were then evaluated and compared on the basis of the most recently collected and previously published datasets. Models were compared in terms of prediction accuracy measured by the root mean squared error and mean absolute percentage error. The results obtained show that in the existing literature, self-cleansing models tend to be overfitted, i.e. have a rather high prediction accuracy when applied to the data collected by the authors, but this accuracy deteriorates quickly when applied to the datasets collected by other authors. The newly developed models can be used for designing both small and large sewer pipes with and without deposited bed condition.

INTRODUCTION

Understanding sediment transport is important for designing self-cleansing sewer systems. Sewer deposits are the source of several problems, such as the reduction of hydraulic capacity, blockage and premature overflows (Shirazi et al. 2014; Ebtehaj et al. 2016; Torres et al. 2017; Kargar et al. 2019; Montes et al. 2019; Safari 2019). Traditionally, conventional minimum velocities and shear stress values have been suggested to define self-cleansing conditions, both in academic literature (Yao 1974; Ackers et al. 1996) and industry design manuals (British Standard Institution 1987; Great Lakes 2004). Several authors (Yao 1974; Nalluri & Ab Ghani 1996) have shown that the use of these traditional criteria and conventional values is likely to lead to overdesigning the slope for small diameter pipes (i.e. pipes with diameter D smaller than 500 mm). To address this issue, laboratory investigations have been carried out (e.g. May et al. 1989; Ab Ghani 1993; Vongvisessomjai et al. 2010; Safari et al. 2017a; Alihosseini & Thamsen 2019). These studies focused on estimating the self-cleansing conditions and developing corresponding predictive models in which the minimum self-cleansing velocity () is a function of several input variables, such as the mean particle diameter (), the hydraulic radius (), the specific gravity of sediments (), the dimensionless grain size () or the volumetric sediment concentration ().

According to Safari et al. (2018), the above and similar experimental works have studied two self-cleansing design criteria: (i) criteria for bed sediment motion and (ii) criteria for sediment non-deposition in sewer pipes. Both criteria are useful for predicting the self-cleansing conditions. In this paper, the non-deposition design criterion is studied using an experimental approach.

Traditionally, non-deposition self-cleansing design criteria have been classified in two general groups (Vongvisessomjai et al. 2010; Safari et al. 2018): (i) non-deposition without deposited bed and (ii) non-deposition with deposited bed of sediments.

The first group, non-deposition without deposited bed, is a conservative and frequently used criterion for designing self-cleansing sewer systems. In this context, Robinson & Graf (1972) defined critical mean velocity (or minimum self-cleansing velocity, as presented in this study) as the condition in which particles begin deposition and form a stationary deposit at the bottom of the sewer pipe, i.e. the particles do not form a permanent deposit.

Several studies have been carried out in this field, in which models are proposed for the prediction of a minimum self-cleansing velocity that guarantees the non-deposition of particles in sewer pipes. In this context, Mayerle (1988) analysed the sediment transport in a 152 mm diameter pipe using uniform sand ranging from 0.50 mm to 8.74 mm, and sediment concentration between 20 and 1,275 ppm. May et al. (1989) analysed sediment transport in a 300 mm diameter concrete pipe using non-cohesive material with a mean particle diameter of 0.72 mm. May (1993) used a 450 mm diameter concrete pipe to study the transport of sands with a mean particle diameter of 0.73 mm. Ab Ghani (1993) studied the non-deposition sediment transport without deposited bed in three sewer pipes of 154 mm, 305 mm and 450 mm, varying the particle diameter from 0.46 mm to 8.3 mm. Ota (1999) carried out experiments in a 305 mm sewer pipe varying the particle diameter from 0.714 mm to 5.612 mm. Vongvisessomjai et al. (2010) developed two models for bedload transport and two models for suspended load transport using data collected in two pipes of 100 mm and 150 mm diameter. Safari et al. (2017a) conducted experiments in a trapezoidal channel and proposed an equation which includes the cross-section shape factor (). Recently, Montes et al. (2018) collected experimental data from Ab Ghani (1993) and using an evolutionary polynomial regression multi-objective genetic algorithm (EPR-MOGA) developed new self-cleansing models.

The above studies resulted in a series of predictive models for the estimation of self-cleansing velocity but none of them analysed self-cleansing velocity in the context of larger sewer pipes. As a result, all non-deposition self-cleansing models are only useful for designing small sewer pipes ( < 500 mm).

Usually, the equations reported in the literature for non-deposition without deposited bed criterion are in the form of: 
formula
(1)
where g the gravitational acceleration; the Darcy's friction factor; the dimensionless grain size ; the specific gravity of sediments; the kinematic viscosity of water; D the pipe diameter; and a, b, , , are coefficients, which vary with each study. For example, in the Ab Ghani (1993) model, a = 3.08, b = 0.21, c1 = −0.53, c2 = −0.09 and c3 = −0.21: 
formula
(2)

The second group, non-deposition with deposited bed, is a less conservative criterion used for the design of large self-cleansing sewer systems (D > 500 mm) (Safari et al. 2018). In this criterion, a small permanent sediment bed is allowed at the bottom of the pipe. Several investigations (May et al. 1989; El-Zaemey 1991; Ab Ghani 1993; Butler et al. 1996) have found that a permanent sediment bed, with mean proportional sediment depth () close to 1.0%, increases the sediment transport capacity. However, strong supervision of the systems is required because it is close to critical condition (Vongvisessomjai et al. 2010).

Based on the aforementioned, several studies have been carried out for describing this phenomenon using predictive numerical models based on experimental data. Experiments by El-Zaemey (1991) were carried out in a 305 mm diameter pipe using bed sediment thicknesses of 47 mm, 77 mm and 120 mm, and granular sediments ranging from 0.53 mm to 8.4 mm in size. Perrusquía (1992) studied the sediment transport in a 225 mm diameter concrete pipe using uniform-sized sands of 0.9 mm and 2.5 mm. May (1993) conducted experiments in a 450 mm diameter pipe using two uniform sands with a mean particle diameter of 0.73 mm and 0.47 mm. Ab Ghani (1993) used a 450 mm diameter pipe varying the deposited bed width () from 47 mm to 384 mm. Nalluri et al. (1997) used the data collected from El-Zaemey (1991) and modified the May et al. (1989) model to predict self-cleansing conditions in deposited bed sewers. Safari et al. (2017b) used the particle swarm optimization (PSO) algorithm to improve the May (1993) model; good results were obtained with this new model. Recently, Safari & Shirzad (2019) defined an optimum deposited bed thickness, and proposed a new self-cleansing model for sewers with deposited bed.

Models found in the literature to predict the non-deposition bedload transport with deposited bed are in terms of the deposited bed width or the mean proportional sediment bed. As an example, a model was outlined by El-Zaemey (1991) in the following form, where Y is the water level and the deposited bed width: 
formula
(3)

As can be seen from the aforementioned, several authors have studied the sediment transport modes to develop new self-cleansing criteria. Each author has developed predictive models which are useful for designing new sewer infrastructure. However, various limitations have been identified in the use of self-cleansing models. For example, Safari et al. (2018) pointed out that non-deposition without deposited bed is useful only in small sewers; for large pipe diameters, the non-deposition with deposited bed criterion must be applied. However, models developed for deposited bed conditions present poor accuracy when different datasets are used (Nalluri et al. 1997). Recently, Safari et al. (2018) highlighted the poor performance of the equations found in this criterion and recommend further experimental research in this field. In addition, Perrusquía (1992) suggested further experimental work, especially in large sewer pipe diameters (i.e. pipe diameter large than 500 mm).

In this study, new self-cleansing models for non-deposition without deposited bed and deposited bed were developed. A 595 mm diameter PVC was used to collect experimental data. The aim was to improve sediment transport prediction in large sewer pipes, based on a new experimental dataset.

EXPERIMENTAL METHODS

Experimental data were collected on a 595 mm diameter and 10.5 m long PVC pipe, located in the Hydraulics Laboratory of the University of los Andes, Colombia. This pipe was supported on a variable steel truss, allowing pipe slopes between 0.042% and 3.44%. The pipe was directly connected to a 30 m3 upstream tank which was supplied through a 40 HP pump. The flow rate was controlled using a manually operated valve, allowing it to vary from 0.6 L s−1 to 67.3 L s−1. The pipe had four-point gauges to measure the water depth along the entire length of the flume. A sediment feeder was used to supply granular material with a mean particle diameter ranging from 0.35 mm to 2.60 mm to the PVC pipe. The specific gravity of sediments varied from 2.64 to 2.67, which was calculated using the pycnometer method, according to ASTM D854-10 (ASTM D854-14 2014). Figure 1 shows the general scheme of the experimental setup.

Figure 1

Schematic diagram of the experimental setup.

Figure 1

Schematic diagram of the experimental setup.

The experiments were carried out under uniform flow conditions, i.e. no variations in flowrate and water depth, for both non-deposition criteria. The data collection strategies were similar for both cases; however, the main difference related to the sediment supply to the PVC pipe, which depended on the criterion to be studied. In this context, for the non-deposition without deposited bed criterion, the sediment feeder supplied the material until the particles barely moved with the water and did not form a permanent deposit at the bottom of the pipe. In contrast, for non-deposition with deposited bed, sediment was supplied to form a deposited loose bed along the entire length of the flume. This methodology followed the guidelines of several previous experimental works carried out by different authors (e.g. Novak & Nalluri 1975; Ota 1999; Perrusquía 1991; Ab Ghani 1993; Vongvisessomjai et al. 2010; Safari et al. 2017a; Alihosseini & Thamsen 2019). The methodology used to collect the data in both cases is described below.

Non-deposition without deposited bed

The first case considered in this paper is the non-deposition without deposited bed condition. The collection of experimental data was as follows. Firstly, the pipe slope was mechanically adjusted and the value was measured using a dumpy level. Secondly, the flow control valve was opened and a constant flow of water was supplied to the pipe. The flowrate was measured with a real-time electromagnetic flowmeter which was connected directly to the pipe feeding the upstream tank. Thirdly, the water levels were measured using the four-point gauges. The downstream tailgate was adjusted until the water depth varied less than ±2 mm between the four-point gauges, which is the condition in which uniform flow conditions could be assumed (Ab Ghani 1993). Using the values recorded of flowrate and water level, the mean velocity was computed. Fourthly, when uniform flow conditions were achieved, the sediment was supplied to the pipe. The sediment feeder was slowly opened until the non-deposition condition was obtained. This condition, also known as ‘flume traction’, (i.e. no separated dunes present and no deposition of stationary material at the bottom of the pipe) was checked by visual inspection. Finally, the sediment supply rate () was estimated by weighing the amount of material that passed in a given time at the outlet of the sediment feeder. The sediment discharge was estimated as , where is the particle density. The calculated sediment discharge was used to compute the volumetric sediment concentration ().

The experimental procedure above was repeated for several flowrates, pipe slopes and sediment sizes. A total of 107 data for the non-deposition without deposited bed condition were collected using above experimental approach, as shown in Table 1.

Table 1

Non-deposition without deposited bed experimental data collected in the 595 mm PVC pipe

Run no.
(mm)(−)(ppm)(mm)(%)(m/s)
1.51 2.66 10,119 9.88 1.78 0.61 
1.51 2.66 11,609 7.27 1.78 0.51 
1.51 2.66 3,940 11.83 1.57 0.67 
1.51 2.66 3,803 14.41 1.57 0.84 
1.51 2.66 3,892 18.89 1.22 1.02 
1.51 2.66 3,681 14.41 0.96 0.77 
1.51 2.66 19,957 7.92 3.43 0.63 
1.51 2.66 14,854 9.23 3.43 0.77 
1.51 2.66 16,731 10.53 3.43 0.97 
10 1.51 2.66 13,608 12.48 2.74 0.75 
11 1.51 2.66 13,841 10.53 2.74 0.75 
12 0.35 2.65 8,720 9.88 2.70 0.80 
13 0.35 2.65 6,431 10.53 1.43 0.73 
14 0.35 2.65 588 14.41 0.25 0.45 
15 0.35 2.65 736 16.98 0.25 0.56 
16 0.35 2.65 700 20.16 0.25 0.62 
17 0.35 2.65 726 23.32 0.68 0.71 
18 0.35 2.65 1,227 25.82 0.68 0.77 
19 0.35 2.65 2,499 19.53 1.23 0.85 
20 0.35 2.65 2,280 20.79 0.89 0.93 
21 0.35 2.65 1,909 27.38 0.89 0.93 
22 0.35 2.65 4,155 14.41 1.36 0.71 
23 0.35 2.65 3,279 18.89 1.36 0.84 
24 0.35 2.65 2,498 22.06 1.36 0.97 
25 0.35 2.65 2,051 25.51 1.36 1.02 
26 0.47 2.66 4,012 13.77 1.36 0.74 
27 0.47 2.66 2,804 18.89 1.36 0.88 
28 0.47 2.66 3,153 22.06 1.36 0.98 
29 0.47 2.66 3,410 25.20 1.36 1.02 
30 0.47 2.66 1,837 27.07 0.89 0.91 
31 0.47 2.66 1,658 24.26 0.89 0.84 
32 0.47 2.66 1,668 20.16 0.89 0.80 
33 0.47 2.66 3,276 14.41 0.89 0.66 
34 0.47 2.66 796 28.93 0.42 0.82 
35 0.47 2.66 667 33.85 0.42 0.87 
36 0.47 2.66 913 40.80 0.42 0.98 
37 0.47 2.66 79.69 0.04 0.45 
38 0.47 2.66 17 95.27 0.04 0.56 
39 0.47 2.66 20 107.70 0.04 0.65 
40 0.47 2.66 47 119.29 0.08 0.73 
41 0.47 2.66 43 100.77 0.17 0.79 
42 0.47 2.66 88.37 0.17 0.60 
43 1.22 2.67 955 22.37 0.68 0.77 
44 1.22 2.67 1,043 25.20 0.68 0.81 
45 1.22 2.67 1,150 28.00 0.68 0.85 
46 1.22 2.67 1,341 30.78 0.68 0.91 
47 1.22 2.67 1,130 33.24 0.68 0.90 
48 1.22 2.67 1,421 38.40 0.68 1.02 
49 1.22 2.67 943 39.90 0.42 0.96 
50 1.22 2.67 826 33.85 0.42 0.86 
51 1.22 2.67 745 24.89 0.42 0.71 
52 1.22 2.67 13 72.82 0.17 0.50 
53 1.22 2.67 14 88.12 0.17 0.62 
54 1.22 2.67 20 93.57 0.08 0.60 
55 1.22 2.67 44 106.11 0.08 0.67 
56 1.22 2.67 30 103.58 0.08 0.58 
57 1.22 2.67 1,748 28.93 0.89 1.01 
58 1.22 2.67 1,639 25.82 0.89 0.94 
59 1.22 2.67 1,099 19.84 0.89 0.83 
60 1.22 2.67 3,322 18.89 1.10 0.90 
61 1.22 2.67 2,123 14.41 1.10 0.71 
62 1.22 2.67 2,185 23.00 1.10 1.02 
63 1.22 2.67 2,645 22.69 1.40 1.04 
64 1.22 2.67 2,791 18.25 1.40 0.95 
65 1.22 2.67 3,692 14.41 1.40 0.71 
66 2.60 2.64 83 80.73 0.21 0.75 
67 2.60 2.64 129 90.37 0.21 0.87 
68 1.51 2.66 21 90.86 0.04 0.60 
69 1.51 2.66 62 89.12 0.04 0.79 
70 1.51 2.66 44 87.37 0.04 0.74 
71 1.51 2.66 68 86.36 0.13 0.75 
72 1.51 2.66 54 74.69 0.13 0.66 
73 1.51 2.66 70 72.02 0.21 0.70 
74 1.51 2.66 96 78.91 0.21 0.76 
75 1.51 2.66 66 84.84 0.21 0.78 
76 1.51 2.66 76 86.61 0.04 0.76 
77 1.51 2.66 80 88.37 0.04 0.78 
78 1.51 2.66 2,729 17.62 1.19 1.10 
79 1.51 2.66 1,701 20.48 0.72 0.87 
80 1.51 2.66 2,086 18.89 0.93 0.99 
81 1.51 2.66 4,066 9.23 1.19 0.62 
82 1.51 2.66 6,869 7.92 1.91 0.78 
83 1.51 2.66 6,253 7.92 1.78 0.78 
84 2.60 2.64 18 92.83 0.04 0.59 
85 2.60 2.64 23 101.71 0.04 0.64 
86 2.60 2.64 527 48.77 0.47 1.14 
87 2.60 2.64 903 38.10 0.47 1.00 
88 2.60 2.64 1,068 29.55 0.47 0.88 
89 2.60 2.64 541 57.39 0.47 1.24 
90 2.60 2.64 1,373 41.69 1.23 1.41 
91 2.60 2.64 2,800 33.24 1.23 1.22 
92 0.35 2.65 83 42.88 0.04 0.41 
93 0.35 2.65 86 50.52 0.04 0.57 
94 0.35 2.65 176 55.97 0.04 0.64 
95 0.35 2.65 188 63.01 0.04 0.74 
96 0.35 2.65 32 82.28 0.04 0.61 
97 0.35 2.65 85 103.34 0.04 0.80 
98 0.35 2.65 500 54.55 2.54 1.21 
99 0.35 2.65 843 42.88 2.54 1.09 
100 0.35 2.65 963 33.85 2.54 1.00 
101 2.60 2.64 3,025 11.51 0.89 0.61 
102 2.60 2.64 1,945 19.53 0.89 0.88 
103 2.60 2.64 1,869 26.14 0.89 1.06 
104 2.60 2.64 1,726 31.71 0.89 1.11 
105 2.60 2.64 999 32.93 0.59 1.05 
106 2.60 2.64 994 40.20 0.59 1.13 
107 2.60 2.64 824 48.77 0.59 1.19 
Run no.
(mm)(−)(ppm)(mm)(%)(m/s)
1.51 2.66 10,119 9.88 1.78 0.61 
1.51 2.66 11,609 7.27 1.78 0.51 
1.51 2.66 3,940 11.83 1.57 0.67 
1.51 2.66 3,803 14.41 1.57 0.84 
1.51 2.66 3,892 18.89 1.22 1.02 
1.51 2.66 3,681 14.41 0.96 0.77 
1.51 2.66 19,957 7.92 3.43 0.63 
1.51 2.66 14,854 9.23 3.43 0.77 
1.51 2.66 16,731 10.53 3.43 0.97 
10 1.51 2.66 13,608 12.48 2.74 0.75 
11 1.51 2.66 13,841 10.53 2.74 0.75 
12 0.35 2.65 8,720 9.88 2.70 0.80 
13 0.35 2.65 6,431 10.53 1.43 0.73 
14 0.35 2.65 588 14.41 0.25 0.45 
15 0.35 2.65 736 16.98 0.25 0.56 
16 0.35 2.65 700 20.16 0.25 0.62 
17 0.35 2.65 726 23.32 0.68 0.71 
18 0.35 2.65 1,227 25.82 0.68 0.77 
19 0.35 2.65 2,499 19.53 1.23 0.85 
20 0.35 2.65 2,280 20.79 0.89 0.93 
21 0.35 2.65 1,909 27.38 0.89 0.93 
22 0.35 2.65 4,155 14.41 1.36 0.71 
23 0.35 2.65 3,279 18.89 1.36 0.84 
24 0.35 2.65 2,498 22.06 1.36 0.97 
25 0.35 2.65 2,051 25.51 1.36 1.02 
26 0.47 2.66 4,012 13.77 1.36 0.74 
27 0.47 2.66 2,804 18.89 1.36 0.88 
28 0.47 2.66 3,153 22.06 1.36 0.98 
29 0.47 2.66 3,410 25.20 1.36 1.02 
30 0.47 2.66 1,837 27.07 0.89 0.91 
31 0.47 2.66 1,658 24.26 0.89 0.84 
32 0.47 2.66 1,668 20.16 0.89 0.80 
33 0.47 2.66 3,276 14.41 0.89 0.66 
34 0.47 2.66 796 28.93 0.42 0.82 
35 0.47 2.66 667 33.85 0.42 0.87 
36 0.47 2.66 913 40.80 0.42 0.98 
37 0.47 2.66 79.69 0.04 0.45 
38 0.47 2.66 17 95.27 0.04 0.56 
39 0.47 2.66 20 107.70 0.04 0.65 
40 0.47 2.66 47 119.29 0.08 0.73 
41 0.47 2.66 43 100.77 0.17 0.79 
42 0.47 2.66 88.37 0.17 0.60 
43 1.22 2.67 955 22.37 0.68 0.77 
44 1.22 2.67 1,043 25.20 0.68 0.81 
45 1.22 2.67 1,150 28.00 0.68 0.85 
46 1.22 2.67 1,341 30.78 0.68 0.91 
47 1.22 2.67 1,130 33.24 0.68 0.90 
48 1.22 2.67 1,421 38.40 0.68 1.02 
49 1.22 2.67 943 39.90 0.42 0.96 
50 1.22 2.67 826 33.85 0.42 0.86 
51 1.22 2.67 745 24.89 0.42 0.71 
52 1.22 2.67 13 72.82 0.17 0.50 
53 1.22 2.67 14 88.12 0.17 0.62 
54 1.22 2.67 20 93.57 0.08 0.60 
55 1.22 2.67 44 106.11 0.08 0.67 
56 1.22 2.67 30 103.58 0.08 0.58 
57 1.22 2.67 1,748 28.93 0.89 1.01 
58 1.22 2.67 1,639 25.82 0.89 0.94 
59 1.22 2.67 1,099 19.84 0.89 0.83 
60 1.22 2.67 3,322 18.89 1.10 0.90 
61 1.22 2.67 2,123 14.41 1.10 0.71 
62 1.22 2.67 2,185 23.00 1.10 1.02 
63 1.22 2.67 2,645 22.69 1.40 1.04 
64 1.22 2.67 2,791 18.25 1.40 0.95 
65 1.22 2.67 3,692 14.41 1.40 0.71 
66 2.60 2.64 83 80.73 0.21 0.75 
67 2.60 2.64 129 90.37 0.21 0.87 
68 1.51 2.66 21 90.86 0.04 0.60 
69 1.51 2.66 62 89.12 0.04 0.79 
70 1.51 2.66 44 87.37 0.04 0.74 
71 1.51 2.66 68 86.36 0.13 0.75 
72 1.51 2.66 54 74.69 0.13 0.66 
73 1.51 2.66 70 72.02 0.21 0.70 
74 1.51 2.66 96 78.91 0.21 0.76 
75 1.51 2.66 66 84.84 0.21 0.78 
76 1.51 2.66 76 86.61 0.04 0.76 
77 1.51 2.66 80 88.37 0.04 0.78 
78 1.51 2.66 2,729 17.62 1.19 1.10 
79 1.51 2.66 1,701 20.48 0.72 0.87 
80 1.51 2.66 2,086 18.89 0.93 0.99 
81 1.51 2.66 4,066 9.23 1.19 0.62 
82 1.51 2.66 6,869 7.92 1.91 0.78 
83 1.51 2.66 6,253 7.92 1.78 0.78 
84 2.60 2.64 18 92.83 0.04 0.59 
85 2.60 2.64 23 101.71 0.04 0.64 
86 2.60 2.64 527 48.77 0.47 1.14 
87 2.60 2.64 903 38.10 0.47 1.00 
88 2.60 2.64 1,068 29.55 0.47 0.88 
89 2.60 2.64 541 57.39 0.47 1.24 
90 2.60 2.64 1,373 41.69 1.23 1.41 
91 2.60 2.64 2,800 33.24 1.23 1.22 
92 0.35 2.65 83 42.88 0.04 0.41 
93 0.35 2.65 86 50.52 0.04 0.57 
94 0.35 2.65 176 55.97 0.04 0.64 
95 0.35 2.65 188 63.01 0.04 0.74 
96 0.35 2.65 32 82.28 0.04 0.61 
97 0.35 2.65 85 103.34 0.04 0.80 
98 0.35 2.65 500 54.55 2.54 1.21 
99 0.35 2.65 843 42.88 2.54 1.09 
100 0.35 2.65 963 33.85 2.54 1.00 
101 2.60 2.64 3,025 11.51 0.89 0.61 
102 2.60 2.64 1,945 19.53 0.89 0.88 
103 2.60 2.64 1,869 26.14 0.89 1.06 
104 2.60 2.64 1,726 31.71 0.89 1.11 
105 2.60 2.64 999 32.93 0.59 1.05 
106 2.60 2.64 994 40.20 0.59 1.13 
107 2.60 2.64 824 48.77 0.59 1.19 

Non-deposition with deposited bed

The methodology used to collect the experimental data for the non-deposition with deposited bed case was similar to the one used for the non-deposition without deposited bed case. The main difference related to the supply of sediment into the pipe, as the non-deposition with deposited bed case required constant sediment thickness throughout the entire length of the test. The whole data collection strategy was as follows. Firstly, an initial pipe slope was mechanically adjusted, and the flow control valve was opened. As a result, a constant water flow was supplied to the pipe, and its value was recorded with the real-time electromagnetic flowmeter. Secondly, the sediment feeder was slowly opened until the material formed a permanent deposited loose bed, which was continuously monitored by visual inspection. Thirdly, the water levels were recorded using the four-point gauges, and uniform conditions were checked for. If non-uniform conditions were observed, the downstream tailgate was varied until water level differences were smaller than ±2 mm between the four-point gauges. In this step, if the non-deposition with deposited bed condition changed (because a permanent deposit or dunes formed by the change in water level), the pipe slope and the tailgate were iteratively adjusted until uniform flow conditions and a constant sediment width had been observed for at least 15 min. Fourthly, the water level, the pipe slope and the sediment width values were recorded, and the sediment thickness (using the sediment width value) and flow velocity (using flowrate and water level) were calculated. Finally, the sediment supply rate was measured at the outlet of the pipe. The sediment that passed in a given time was collected, dried and weighed, and the sediment discharge was calculated, as described in the ‘Non-deposition without deposited bed’ section. Five samples of sediments were collected to validate that the sediment supply rate was constant during the entire test. The volumetric sediment concentration was computed using the sediment discharge and the flowrate.

The experimental procedure described was repeated for several flowrates, pipe slopes and sediment sizes. A total of 54 experiments were carried out to collect data for the non-deposition with deposited bed case. The experimental data collected this way is presented in Table 2.

Table 2

Non-deposition with deposited bed data experimentally collected in the 595 mm PVC pipe

Run no.
(mm)(-)(ppm)(mm)(%)(m/s)(%)(mm)
1.51 2.66 786 23.46 0.975 0.73 0.94 115 
1.51 2.66 763 22.76 0.720 0.80 0.13 43 
1.51 2.66 744 26.57 0.763 0.83 0.25 60 
1.51 2.66 982 28.63 0.763 0.96 0.21 55 
1.51 2.66 389 35.25 0.508 0.86 0.38 73 
1.51 2.66 702 32.62 0.763 0.93 1.12 125 
1.51 2.66 939 39.54 0.805 1.05 0.86 110 
1.51 2.66 632 51.01 0.720 0.90 0.58 90 
1.51 2.66 1,214 20.87 0.975 0.87 0.61 93 
10 1.51 2.66 3,283 14.96 1.822 0.82 0.51 85 
11 1.51 2.66 9,596 20.34 2.076 1.12 1.03 120 
12 1.51 2.66 4,419 22.08 1.992 1.15 0.51 85 
13 1.51 2.66 10,275 9.63 5.424 0.87 0.30 65 
14 1.51 2.66 2,980 29.03 1.525 1.16 0.86 110 
15 1.51 2.66 2,249 23.84 1.525 1.00 0.30 65 
16 1.51 2.66 6,227 15.90 2.500 1.06 0.58 90 
17 1.51 2.66 2,128 35.73 0.847 1.06 1.12 125 
18 1.51 2.66 7,400 22.25 2.034 1.21 0.71 100 
19 1.51 2.66 3,702 23.67 2.034 1.11 0.45 80 
20 1.51 2.66 4,172 25.03 2.034 1.21 0.78 105 
21 2.6 2.64 2,951 28.40 1.525 1.16 0.86 110 
22 2.6 2.64 4,435 23.02 1.992 1.23 0.58 90 
23 2.6 2.64 4,962 20.49 2.119 1.04 0.45 80 
24 2.6 2.64 9,101 14.96 2.585 1.07 0.51 85 
25 2.6 2.64 2,213 40.97 1.314 1.18 0.58 90 
26 2.6 2.64 4,995 33.33 1.568 1.21 0.64 95 
27 2.6 2.64 3,432 36.12 1.398 1.24 0.58 90 
28 2.6 2.64 2,408 44.25 1.271 1.39 1.12 125 
29 2.6 2.64 1,968 52.01 1.059 1.26 0.86 110 
30 2.6 2.64 1,615 55.59 1.017 1.29 0.71 100 
31 1.22 2.67 2,327 15.26 1.653 0.90 0.35 70 
32 1.22 2.67 4,759 17.26 1.653 1.11 0.45 80 
33 1.22 2.67 3,162 22.01 1.653 1.17 0.64 95 
34 1.22 2.67 1,710 30.22 1.229 0.97 0.40 75 
35 1.22 2.67 987 31.51 1.229 1.17 0.51 85 
36 1.22 2.67 1,052 20.90 0.890 0.81 0.38 73 
37 1.22 2.67 1,660 31.19 0.466 0.80 0.45 80 
38 1.22 2.67 488 27.58 0.636 0.89 0.55 88 
39 1.22 2.67 3,365 9.01 1.525 0.88 0.18 50 
40 1.22 2.67 2,527 29.46 1.144 1.28 0.67 97 
41 1.22 2.67 652 34.59 0.720 1.01 0.51 85 
42 1.22 2.67 460 37.32 0.678 0.90 0.45 80 
43 1.22 2.67 1,504 17.05 1.059 0.75 0.25 60 
44 1.22 2.67 5,697 12.11 2.203 1.20 0.33 68 
45 0.47 2.66 2,516 8.43 1.398 1.39 0.49 83 
46 0.47 2.66 2,594 9.46 1.610 1.20 0.33 68 
47 0.47 2.66 8,522 10.34 2.373 1.05 0.29 64 
48 0.47 2.66 6,424 14.12 2.373 1.53 0.32 67 
49 0.47 2.66 5,317 15.06 1.822 1.36 0.71 100 
50 0.47 2.66 2,572 17.63 1.314 1.10 0.39 74 
51 0.47 2.66 547 19.78 0.847 0.92 0.35 70 
52 0.47 2.66 764 27.60 0.890 0.89 0.30 65 
53 0.47 2.66 1,918 24.86 1.229 1.05 0.35 70 
54 0.47 2.66 5,131 21.53 1.780 1.30 0.38 73 
Run no.
(mm)(-)(ppm)(mm)(%)(m/s)(%)(mm)
1.51 2.66 786 23.46 0.975 0.73 0.94 115 
1.51 2.66 763 22.76 0.720 0.80 0.13 43 
1.51 2.66 744 26.57 0.763 0.83 0.25 60 
1.51 2.66 982 28.63 0.763 0.96 0.21 55 
1.51 2.66 389 35.25 0.508 0.86 0.38 73 
1.51 2.66 702 32.62 0.763 0.93 1.12 125 
1.51 2.66 939 39.54 0.805 1.05 0.86 110 
1.51 2.66 632 51.01 0.720 0.90 0.58 90 
1.51 2.66 1,214 20.87 0.975 0.87 0.61 93 
10 1.51 2.66 3,283 14.96 1.822 0.82 0.51 85 
11 1.51 2.66 9,596 20.34 2.076 1.12 1.03 120 
12 1.51 2.66 4,419 22.08 1.992 1.15 0.51 85 
13 1.51 2.66 10,275 9.63 5.424 0.87 0.30 65 
14 1.51 2.66 2,980 29.03 1.525 1.16 0.86 110 
15 1.51 2.66 2,249 23.84 1.525 1.00 0.30 65 
16 1.51 2.66 6,227 15.90 2.500 1.06 0.58 90 
17 1.51 2.66 2,128 35.73 0.847 1.06 1.12 125 
18 1.51 2.66 7,400 22.25 2.034 1.21 0.71 100 
19 1.51 2.66 3,702 23.67 2.034 1.11 0.45 80 
20 1.51 2.66 4,172 25.03 2.034 1.21 0.78 105 
21 2.6 2.64 2,951 28.40 1.525 1.16 0.86 110 
22 2.6 2.64 4,435 23.02 1.992 1.23 0.58 90 
23 2.6 2.64 4,962 20.49 2.119 1.04 0.45 80 
24 2.6 2.64 9,101 14.96 2.585 1.07 0.51 85 
25 2.6 2.64 2,213 40.97 1.314 1.18 0.58 90 
26 2.6 2.64 4,995 33.33 1.568 1.21 0.64 95 
27 2.6 2.64 3,432 36.12 1.398 1.24 0.58 90 
28 2.6 2.64 2,408 44.25 1.271 1.39 1.12 125 
29 2.6 2.64 1,968 52.01 1.059 1.26 0.86 110 
30 2.6 2.64 1,615 55.59 1.017 1.29 0.71 100 
31 1.22 2.67 2,327 15.26 1.653 0.90 0.35 70 
32 1.22 2.67 4,759 17.26 1.653 1.11 0.45 80 
33 1.22 2.67 3,162 22.01 1.653 1.17 0.64 95 
34 1.22 2.67 1,710 30.22 1.229 0.97 0.40 75 
35 1.22 2.67 987 31.51 1.229 1.17 0.51 85 
36 1.22 2.67 1,052 20.90 0.890 0.81 0.38 73 
37 1.22 2.67 1,660 31.19 0.466 0.80 0.45 80 
38 1.22 2.67 488 27.58 0.636 0.89 0.55 88 
39 1.22 2.67 3,365 9.01 1.525 0.88 0.18 50 
40 1.22 2.67 2,527 29.46 1.144 1.28 0.67 97 
41 1.22 2.67 652 34.59 0.720 1.01 0.51 85 
42 1.22 2.67 460 37.32 0.678 0.90 0.45 80 
43 1.22 2.67 1,504 17.05 1.059 0.75 0.25 60 
44 1.22 2.67 5,697 12.11 2.203 1.20 0.33 68 
45 0.47 2.66 2,516 8.43 1.398 1.39 0.49 83 
46 0.47 2.66 2,594 9.46 1.610 1.20 0.33 68 
47 0.47 2.66 8,522 10.34 2.373 1.05 0.29 64 
48 0.47 2.66 6,424 14.12 2.373 1.53 0.32 67 
49 0.47 2.66 5,317 15.06 1.822 1.36 0.71 100 
50 0.47 2.66 2,572 17.63 1.314 1.10 0.39 74 
51 0.47 2.66 547 19.78 0.847 0.92 0.35 70 
52 0.47 2.66 764 27.60 0.890 0.89 0.30 65 
53 0.47 2.66 1,918 24.86 1.229 1.05 0.35 70 
54 0.47 2.66 5,131 21.53 1.780 1.30 0.38 73 

Literature data

Other datasets were collected from the literature for the self-cleansing models shown in Table 3. A total of 483 and 400 data for non-deposition without deposited bed and with deposited bed, respectively, were collected. These data were used to evaluate the performance of the self-cleansing models proposed in this study.

Table 3

Literature self-cleansing models for predicting the non-deposition sediment conditions in sewer pipes

ReferenceModelNon-deposition criterionNo. dataPipe diameter (mm)Particle diameter (mm)Sediment concentration (ppm)
Mayerle (1988). Data collected from Safari et al. (2018)   Without deposited bed 106 152 0.50–8.74 20–1,275 
May et al. (1989)   Without deposited bed 48 298.8 0.72 0.31–443 
Perrusquía (1991)  Only experimental data With deposited bed 38 225 0.9 18.7–408 
El-Zaemey (1991)   With deposited bed 290 305 0.53–8.4 7.0–917 
Ab Ghani (1993)   Without deposited bed 221 154, 305 and 450 0.46–8.30 0.76–1,450 
Ab Ghani (1993)   With deposited bed 26 450 0.72 21–1,269 
May (1993)  Only experimental data Without deposited bed 27 450 0.73 2–38 
May (1993)   With deposited bed 46 450 0.47–0.73 3.5–8.23 
Ota (1999)   Without deposited bed 36 305 0.71–5.6 4.2 –59.4 
Vongvisessomjai et al. (2010)   Without deposited bed 45 100 and 150 0.20–0.43 4–90 
Safari et al. (2017b With deposited bed Data from May (1993)  
Safari & Shirzad (2019)   With deposited bed Data from El-Zaemey (1991), Perrusquía (1991), May (1993) and Ab Ghani (1993)  
Montes et al. (2018)   Without deposited bed Data from Ab Ghani (1993)  
ReferenceModelNon-deposition criterionNo. dataPipe diameter (mm)Particle diameter (mm)Sediment concentration (ppm)
Mayerle (1988). Data collected from Safari et al. (2018)   Without deposited bed 106 152 0.50–8.74 20–1,275 
May et al. (1989)   Without deposited bed 48 298.8 0.72 0.31–443 
Perrusquía (1991)  Only experimental data With deposited bed 38 225 0.9 18.7–408 
El-Zaemey (1991)   With deposited bed 290 305 0.53–8.4 7.0–917 
Ab Ghani (1993)   Without deposited bed 221 154, 305 and 450 0.46–8.30 0.76–1,450 
Ab Ghani (1993)   With deposited bed 26 450 0.72 21–1,269 
May (1993)  Only experimental data Without deposited bed 27 450 0.73 2–38 
May (1993)   With deposited bed 46 450 0.47–0.73 3.5–8.23 
Ota (1999)   Without deposited bed 36 305 0.71–5.6 4.2 –59.4 
Vongvisessomjai et al. (2010)   Without deposited bed 45 100 and 150 0.20–0.43 4–90 
Safari et al. (2017b With deposited bed Data from May (1993)  
Safari & Shirzad (2019)   With deposited bed Data from El-Zaemey (1991), Perrusquía (1991), May (1993) and Ab Ghani (1993)  
Montes et al. (2018)   Without deposited bed Data from Ab Ghani (1993)  

: Darcy's friction factor with sediment, .

: Dimensionless grain size, .

: Grain friction factor, , where is the kinematic viscosity of fluid.

: Transition factor, , where is the particle Reynolds number, .

: Incipient motion threshold velocity, .

: Dimensionless parameter of transport.

NEW SELF-CLEANSING MODELS

The least absolute shrinkage and selection operator (LASSO) (Tibshirani 1996) regression method was used in this study to develop new self-cleansing models. The LASSO method can be seen as an extension of ordinary least squares (OLS), because it minimizes the value of the residual sum of squares (RSS). However, this is a shrinkage method for feature selection which itself solves the problem of multicollinearity by increasing the bias of the regression in search of decrease in the variance. Additionally, it uses the absolute value of the coefficients in the shrinkage penalty, which allows this method to reduce some of the regression coefficients to an exact value of zero. This helps to avoid problems related to model interpretation and overfitting (James et al. 2013). The LASSO method coefficients minimize the following expression: 
formula
(4)
where are the observed values; n the number of data; the intercept value; the model parameter j; the input variable set and the shrinkage penalty (James et al. 2013).

Selection of model input variables to represent the particle Froude number are made based on the variables that have the greatest impact on sediment transport. Several authors (May et al. 1996; Ebtehaj & Bonakdari 2016a, 2016b) found that the size and roughness of the pipe (represented by the Darcy friction factor and the pipe diameter), the relative flow depth, the diameter of particle size, the specific gravity of sediments and the volumetric sediment concentration are the input variables that best predict sediment transport. These input variables can be divided into four dimensionless groups called: (i) transport: defined by the volumetric sediment concentration; (ii) sediment: defined by the dimensionless grain size, the specific gravity of sediments and the variable; (iii) transport mode: defined by , , , and ; and (iv) flow resistance: defined by the Darcy friction factor. Based on these, the input variables vector should include the previous variables to predict the particle Froude number.

Two new self-cleansing models were developed for the two sediment non-deposition conditions already mentioned. The R package ‘glmnet’ (Friedman et al. 2010) was used to apply the LASSO method. In both cases the model output variable was the threshold particle Froude number and the model input variables were selected automatically from the set by solving the following regression problem: 
formula
(5)
 
formula
(6)
where and are the observed and estimated particle Froude number, defined as: 
formula
(7)
 
formula
(8)
where VL is the self-cleansing velocity, g is gravitational constant, is the specific gravity of the sediment, the pipe slope, D the pipe diameter, A the wetted area, R the hydraulic radius, the dimensionless grain size, the Darcy friction factor, d is mean particle diameter, Y the water level, the volumetric sediment concentration and the bed sediment width. By applying the LASSO method to 107 experimental data collected, the following model was obtained for the non-deposited conditions (linearized version shown in Equation (9) and non-linear in Equation (10)): 
formula
(9)
 
formula
(10)
The same analysis was carried out for the non-deposition with deposited bed condition. The 54 data collected in the laboratory were used as observed information. The model obtained was similar to the one for non-deposition without deposited bed condition (see Equations (9) and (10)) with the difference being that the input variables and appear in the final expression: 
formula
(11)
 
formula
(12)

VALIDATION OF SELF-CLEANSING MODELS

The self-cleansing models shown in Equations (10) and (12) were tested with the datasets obtained from the literature (as shown in Table 3) with the aim of (a) further evaluating the accuracy of the self-cleansing models shown here and (b) comparing these to literature models, all under the different hydraulic conditions and sediment characteristics, used in the literature. In addition, the literature self-cleansing models shown in Table 3, all of which were developed with the data collected on smaller pipes (i.e. less than 500 mm), were tested with the data collected on the 595 mm PVC pipe to further assess their prediction accuracy under these conditions.

Model prediction accuracy is estimated using two performance indicators, root mean squared error (RMSE) and mean absolute percentage error (MAPE): 
formula
(13)
 
formula
(14)

Note that a value of RMSE and MAPE close to 0 indicates high model prediction accuracy, i.e. good fit between the observed and predicted data. The RMSE and MAPE values obtained for the case of non-deposition without deposited bed are presented in Table 4.

Table 4

Performance of models found in the literature and the new self-cleansing model (Equation (10)) obtained for non-deposition without deposited bed criterion

DatasetPerformance indexSelf-cleansing model
Mayerle (1988) May et al. (1989) Ab Ghani (1993) Ota (1999) Vongvisessomjai et al. (2010) Montes et al. (2018) New model, Equation (10)
Mayerle (1988)  RMSE 4.119 3.273 3.376 3.502 3.310 3.170 3.147 
MAPE 10.079 15.194 9.636 10.439 10.762 14.500 12.504 
May et al. (1989)  RMSE 4.321 3.433 3.545 3.652 3.472 3.330 3.302 
MAPE 12.400 17.822 16.637 16.593 17.657 21.657 21.810 
May (1993)  RMSE 4.151 3.291 3.392 3.511 3.328 3.189 3.167 
MAPE 37.349 9.706 10.738 8.110 9.536 9.226 8.331 
Ab Ghani (1993)  RMSE 1.598 0.567 0.603 0.762 0.569 0.500 0.510 
MAPE 26.965 9.338 10.350 11.930 10.278 8.730 9.435 
Ota (1999)  RMSE 4.068 3.210 3.306 3.424 3.234 3.093 3.066 
MAPE 19.632 12.396 9.644 10.313 7.461 7.174 6.807 
Vongvisessomjai et al. (2010)  RMSE 3.956 3.132 3.222 3.332 3.159 3.031 3.007 
MAPE 24.764 8.274 6.748 4.626 2.036 5.337 2.012 
Current study RMSE 4.041 3.177 3.276 3.387 3.208 3.072 3.047 
MAPE 40.327 29.304 23.307 28.990 19.203 15.639 14.471 
DatasetPerformance indexSelf-cleansing model
Mayerle (1988) May et al. (1989) Ab Ghani (1993) Ota (1999) Vongvisessomjai et al. (2010) Montes et al. (2018) New model, Equation (10)
Mayerle (1988)  RMSE 4.119 3.273 3.376 3.502 3.310 3.170 3.147 
MAPE 10.079 15.194 9.636 10.439 10.762 14.500 12.504 
May et al. (1989)  RMSE 4.321 3.433 3.545 3.652 3.472 3.330 3.302 
MAPE 12.400 17.822 16.637 16.593 17.657 21.657 21.810 
May (1993)  RMSE 4.151 3.291 3.392 3.511 3.328 3.189 3.167 
MAPE 37.349 9.706 10.738 8.110 9.536 9.226 8.331 
Ab Ghani (1993)  RMSE 1.598 0.567 0.603 0.762 0.569 0.500 0.510 
MAPE 26.965 9.338 10.350 11.930 10.278 8.730 9.435 
Ota (1999)  RMSE 4.068 3.210 3.306 3.424 3.234 3.093 3.066 
MAPE 19.632 12.396 9.644 10.313 7.461 7.174 6.807 
Vongvisessomjai et al. (2010)  RMSE 3.956 3.132 3.222 3.332 3.159 3.031 3.007 
MAPE 24.764 8.274 6.748 4.626 2.036 5.337 2.012 
Current study RMSE 4.041 3.177 3.276 3.387 3.208 3.072 3.047 
MAPE 40.327 29.304 23.307 28.990 19.203 15.639 14.471 

Values in bold type show the best performing model in each dataset analysed.

The following observations can be made from Table 4:

  • The Mayerle (1988) model seems to be overfitted as it has high prediction accuracy (RMSE = 4.119; MAPE = 10.079) only for the data collected in their own experiments. When this model is applied to other datasets, the results are not satisfactory. For example, when the Mayerle (1988) model is applied to the data collected in our experiments, poor performance is obtained (as shown in Figure 2). This is due to the inability of this model to extrapolate predictions beyond the range of data that was used for its development.

  • Results obtained by using the May et al. (1989) model were similar to the Mayerle (1988) model results. If the May et al. (1989) model is used for designing large self-cleansing sewer pipes, the model tends to overestimate the minimum velocity required to avoid particle deposition. Additionally, an incipient motion threshold velocity is required to use this model. This value needs to be estimated on the basis of experimental data and regression equations obtained for certain sediment characteristics which is not pragmatic. In this context, Safari et al. (2018) outlined several studies that attempt to predict incipient motion threshold velocity using equations based on experimental data.

  • The Ab Ghani (1993) model presents better results in comparison with Mayerle (1988) and May et al. (1989) models. The model includes two additional input variables (the dimensionless grain size and the Darcy friction factor) to predict the particle Froude number. However, the value of the exponent related to the dimensionless grain size is low (−0.09), which shows that this variable is not a significant input for this model. In addition, this model has good prediction performance when the 595 mm pipe diameter data (for < 8.0) is used (as shown in Figure 2), for the same abovementioned previously.

  • The Ota (1999) model uses a similar group of input variables to estimate the self-cleansing velocity. This model has similar prediction results to the Mayerle (1988) and May et al. (1989) models, with acceptable accuracy for small particle Froude numbers and poor prediction accuracy for larger particle Froude number values ( > 7.0), as shown in Figure 2.

  • The Vongvisessomjai et al. (2010) model shows good performance in general for all datasets. However, when this equation is applied to the 595 mm PVC pipe diameter data, the model tends to overestimate the particle Froude number (as shown in Figure 2). In comparison with the Ab Ghani (1993) model, this model is simpler and does not consider the dimensionless grain size and the Darcy friction factor in the estimation of the modified Froude number (structure is similar to Ota (1999) equation) which is an advantage. This model seems to be more general and good in the prediction on self-cleansing conditions for pipe diameters of less than 500 mm.

  • The Montes et al. (2018) model tends to represent the observed data for all the datasets evaluated better than previous self-cleansing models. This model has the same structure as the Vongvisessomjai et al. (2010) and Ota (1999) models, with values of exponents of different input variables being slightly different. The model shows high accuracy for all datasets but is still inferior to the new model shown in Equation (10) (see below).

  • The new model shown in Equation (10) has high prediction accuracy for all datasets, especially for the data collected using larger sewer pipes. Even when this model is applied to existing data in the literature, better results are obtained than those obtained using literature self-cleaning models (as shown in Figure 3 and Table 4). This model has a similar structure to the Vongvisessomjai et al. (2010) and Montes et al. (2018) equations.

Figure 2

Comparison of performance of non-deposition without deposited bed models using the experimental data collected for the 595 mm PVC pipe. (a) Mayerle (1988); (b) May et al. (1989); (c) Ab Ghani (1993); (d) Ota (1999); (e) Vongvisessomjai et al. (2010); (f) Montes et al. (2018); and (g) Equation (10).

Figure 2

Comparison of performance of non-deposition without deposited bed models using the experimental data collected for the 595 mm PVC pipe. (a) Mayerle (1988); (b) May et al. (1989); (c) Ab Ghani (1993); (d) Ota (1999); (e) Vongvisessomjai et al. (2010); (f) Montes et al. (2018); and (g) Equation (10).

Figure 3

Comparison of performance of Equation (10) using the experimental data collected in the literature. Data from: (a) Mayerle (1988); (b) May et al. (1989); (c) Ab Ghani (1993); (d) May (1993); (e) Ota (1999); and (f) Vongvisessomjai et al. (2010).

Figure 3

Comparison of performance of Equation (10) using the experimental data collected in the literature. Data from: (a) Mayerle (1988); (b) May et al. (1989); (c) Ab Ghani (1993); (d) May (1993); (e) Ota (1999); and (f) Vongvisessomjai et al. (2010).

As the previous results show, all the traditional self-cleansing models found in the literature presents poor performance/accuracy when tested with the new experimental dataset. As Figure 2 shows, all the models tend to overestimate the threshold velocity. This confirms the assumption that traditional self-cleansing models can make accurate predictions only for small sewer pipes, i.e. pipes with diameter <500 mm.

The results obtained for the case of non-deposition with deposited bed data are shown in Table 5.

Table 5

Performance of models found in the literature and the new self-cleansing model (Equation (12)) obtained for non-deposition with deposited bed criterion

DatasetPerformance indexSelf-cleansing model
El-Zaemey (1991) Ab Ghani (1993) May (1993) Safari et al. (2017b) Safari & Shirzad (2019) New model, Equation (12)
Perrusquía (1991)  RMSE 0.786 0.576 2.669 2.883 0.521 0.464 
MAPE 17.411 10.833 63.261 71.279 10.550 10.348 
El-Zaemey (1991)  RMSE 0.494 0.814 2.580 2.749 0.757 0.659 
MAPE 10.436 13.408 60.744 71.963 14.251 11.922 
May (1993)  RMSE 3.409 1.153 3.561 3.562 1.409 1.014 
MAPE 49.757 11.702 45.381 47.177 18.734 11.154 
Ab Ghani (1993)  RMSE 5.105 2.407 3.724 3.722 1.316 1.161 
MAPE 72.772 33.614 47.580 48.831 16.544 14.178 
Current study RMSE 4.217 2.117 2.753 2.696 3.059 1.565 
MAPE 54.510 27.483 27.487 26.186 21.047 10.355 
DatasetPerformance indexSelf-cleansing model
El-Zaemey (1991) Ab Ghani (1993) May (1993) Safari et al. (2017b) Safari & Shirzad (2019) New model, Equation (12)
Perrusquía (1991)  RMSE 0.786 0.576 2.669 2.883 0.521 0.464 
MAPE 17.411 10.833 63.261 71.279 10.550 10.348 
El-Zaemey (1991)  RMSE 0.494 0.814 2.580 2.749 0.757 0.659 
MAPE 10.436 13.408 60.744 71.963 14.251 11.922 
May (1993)  RMSE 3.409 1.153 3.561 3.562 1.409 1.014 
MAPE 49.757 11.702 45.381 47.177 18.734 11.154 
Ab Ghani (1993)  RMSE 5.105 2.407 3.724 3.722 1.316 1.161 
MAPE 72.772 33.614 47.580 48.831 16.544 14.178 
Current study RMSE 4.217 2.117 2.753 2.696 3.059 1.565 
MAPE 54.510 27.483 27.487 26.186 21.047 10.355 

Values in bold type show the best performing model in each dataset analysed.

The following can be observed from Table 5:

  • The El-Zaemey (1991) model tends to correctly represent the self-cleansing conditions for Perrusquía (1991) data and their own data. However, for Ab Ghani (1993) and our data collected on the 595 mm PVC pipe, this model's performance is poor, with low fitting levels obtained (as shown in Figure 4). This model tends to overestimate the minimum self-cleansing velocity, which leads to installing steeper and hence more costly pipes.

  • The Ab Ghani (1993) model has the same structure as the El-Zaemey (1991) model, as both models consider the same group of input variables to calculate the threshold self-cleansing velocity. The results obtained tend to present good accuracy for all datasets. The Ab Ghani (1993) model has acceptable accuracy even on our data collected on the 595 mm PVC pipe (as shown in Figure 4), with RMSE and MAPE values of 2.117 and 27.483, respectively. Having said that, this model is still inferior to the new model shown in Equation (12) for the data collected on a large diameter pipe.

  • The May (1993) model tends to underestimate the minimum self-cleansing values on large sewer pipes, as shown in Figure 4(c). As a result, particle deposition problems could arise in real sewer systems. Additionally, this model has as an input the dimensionless transport parameter (), which was calculated for limited sediment and hydraulic conditions. Based on the above, this transport parameter is difficult to estimate, and its prediction does not present good accuracy with experimental data. Full details can be found in May (1993).

  • The Safari et al. (2017b) model results are similar to the May (1993) and Ab Ghani (1993) models when compared for large sewer pipes, i.e. our data. These models tend to underestimate the minimum self-cleansing velocity in large sewer pipes. However, the results are better than for El-Zaemey (1991), as shown in Table 5.

  • The Safari & Shirzad (2019) model results are similar to May (1993) and Safari et al. (2017b), i.e. the self-cleansing calculation tends to be underestimated in large sewer pipes. In contrast, this model presents a simpler structure because it does not consider the dimensionless parameter of transport () and the calculation of velocity is explicit. Results tend not to be satisfactory for large sewer pipes (as shown in Figure 4).

  • The new model shown in Equation (12) estimates the self-cleansing conditions across all experimental datasets with acceptable accuracy, as shown in Figure 5. This model is explicit for calculating self-cleansing velocity and considers similar group of parameters than the models in the literature. Based on the results obtained, this model can be used to design new self-cleansing sewer pipes considering the non-deposition with deposited bed criterion.

Figure 4

Comparison of performance of non-deposition with deposited bed models using the experimental data collected for the 595 mm PVC pipe. Models from: (a) El-Zaemey (1991); (b) Ab Ghani (1993); (c) May (1993); (d) Nalluri et al. (1997); (e) Safari et al. (2017b); and (f) Equation (12).

Figure 4

Comparison of performance of non-deposition with deposited bed models using the experimental data collected for the 595 mm PVC pipe. Models from: (a) El-Zaemey (1991); (b) Ab Ghani (1993); (c) May (1993); (d) Nalluri et al. (1997); (e) Safari et al. (2017b); and (f) Equation (12).

Figure 5

Comparison of performance of Equation (12) using the experimental data collected from the literature. Data from: (a) Perrusquía (1991); (b) El-Zaemey (1991); (c) May (1993); and (d) Ab Ghani (1993).

Figure 5

Comparison of performance of Equation (12) using the experimental data collected from the literature. Data from: (a) Perrusquía (1991); (b) El-Zaemey (1991); (c) May (1993); and (d) Ab Ghani (1993).

CONCLUSIONS

In this study the non-deposition criteria was applied to large sewer pipes. A set of 107 data and 54 data, for non-deposition without deposited bed and deposited bed, respectively, was collected at laboratory scale. These experiments were carried out varying steady flow conditions and sediment characteristics. The data collected were used to test the performance of typical self-cleansing equations found in the literature. In addition, based on the LASSO technique, two new self-cleansing models were obtained for each non-deposition criterion. These new models were tested with data collected from the literature and their performance was measured by using RMSE and MAPE.

The following conclusions are based on the results obtained:

  • (1)

    The two new self-cleansing models developed and presented here have overall best predictive performance for two different sediment non-deposition criteria when compared to a selection of well-known models from the literature. This is especially true for predictions made on larger diameter pipes (500 mm and above).

  • (2)

    The existing self-cleansing models from the literature tend to be overfitted, i.e. demonstrate a rather high prediction accuracy when applied to the data collected by the authors, but this accuracy deteriorates quickly when applied to the datasets collected by other authors. For large sewer pipes, these models, being developed for datasets collected on smaller diameter pipes, tend to overestimate the threshold self-cleansing velocities, especially in the case of non-deposition without deposited bed.

Further research is recommended to test the performance of new models in larger sewer pipes and with different pipe materials, sediment characteristics and hydraulic conditions. In addition, experiments under non-steady conditions are essential to test the sediment dynamics in real sewer systems.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/wst.2020.154. Supplementary material 1: https://youtu.be/YC_AEBMqYC0. Supplementary material 2: https://youtu.be/ivyoBba8V-c.

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