Predicting bedload sediment transport of non-cohesive material in sewer pipes using evolutionary polynomial regression – multi-objective genetic algorithm strategy

Figures & data

Table 1. Traditional self-cleansing models used to evaluate the bedload sediment transport in sewer pipes.

Self-cleansing models	Reference	d (mm)	D (mm)	Equation
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=6.37{C_v}^{1/3}{\left(\frac{d}{D}\right)}^{-0.5}$	Craven (1953)	0.25–1.65	152	(4)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=4.32{C_v}^{0.23}{\left(\frac{d}{R}\right)}^{-0.68}$	Mayerle (1988)	0.50–5.22	152	(5)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=3.08{D_{gr}}^{-0.09}{C_v}^{0.21}{\left(\frac{d}{R}\right)}^{-0.53}{\lambda }^{-0.21}$	Ab Ghani (1993)	0.46–8.30	154, 305 and 450	(6)
$C_v=0.0303\left(\frac{D^2}{A}\right){\left(\frac{d}{D}\right)}^{0.6}{\left[1-\frac{v_t}{v_l}\right]}^4{\left[\frac{{v_l}^2}{gD\left(SG-1\right)}\right]}^{1.5}$	May et al. (1996)	0.16–8.30	77–450	(7)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=4.31{C_v}^{0.226}{\left(\frac{d}{R}\right)}^{-0.616}$	Vongvisessomjai, Tingsanchali, and Babel (2010)	0.20–0.40	100–150	(8)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=4.49{C_v}^{0.21}{\left(\frac{d}{R}\right)}^{-0.54}$	Ebtehaj, Bonakdari, and Sharifi (2014)	Ab Ghani (1993) and Vongvisessomjai, Tingsanchali, and Babel (2010) data		(9)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=0.404{\left(\frac{R}{d}\right)}^{0.5}+23.25{\left(\frac{R}{d}\right)}^{0.5}{C_v}^{0.5}$	Najafzadeh, Laucelli, and Zahiri (2017)	Ab Ghani (1993) data		(10)
$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=7.34{C_v}^{0.13}{D_{gr}}^{-0.12}{\left(\frac{d}{R}\right)}^{-0.44}{\beta }^{-0.91}$	Safari et al. (2017)	0.15–0.83	Trapezoidal channel	(11)

Figure 1. Experimental apparatus used to collect bedload sediment transport data.

Figure 2. Grading curve of material used on experimental setup.

Table 2. Bedload experiments in the 242 mm acrylic pipe.

Experiment	Model development stage	d (mm)	So (m m^−1)	Cv (ppm)	R (mm)	FR*	vl (m s^−1)
1	Training	1.510	0.0080	13.0	15.31	1.54	0.24
2	Training	1.510	0.0020	13.2	63.20	3.61	0.56
3	Training	0.351	0.0080	0.5	64.89	5.86	0.46
4	Training	1.510	0.0060	176.8	55.17	5.48	0.85
5	Training	1.510	0.0060	109.7	51.12	5.21	0.81
6	Training	1.510	0.0070	160.2	51.33	5.79	0.90
7	Training	1.510	0.0060	139.0	46.08	4.96	0.77
8	Training	1.510	0.0020	3.2	65.50	3.31	0.52
9	Training	1.510	0.0050	4.0	60.91	2.89	0.45
10	Training	1.510	0.0050	101.2	52.38	5.25	0.82
11	Training	0.351	0.0025	2.2	70.27	6.75	0.53
12	Training	1.510	0.0050	70.1	52.65	5.21	0.81
13	Training	1.510	0.0050	81.1	42.78	4.45	0.69
14	Training	1.510	0.0050	103.1	42.69	4.37	0.68
15	Training	1.510	0.0070	67.3	25.69	3.41	0.53
16	Training	1.510	0.0050	87.0	55.51	5.27	0.82
17	Training	1.510	0.0050	94.0	47.36	4.80	0.75
18	Training	1.510	0.0050	33.4	19.97	2.77	0.43
19	Training	1.510	0.0050	82.8	52.10	5.08	0.79
20	Training	0.351	0.0025	41.4	59.95	9.81	0.77
21	Training	1.510	0.0020	9.8	70.47	3.48	0.54
22	Training	1.510	0.0080	632.3	34.72	5.79	0.90
23	Training	1.510	0.0020	6.4	58.90	2.95	0.46
24	Training	1.510	0.0050	128.9	39.40	4.18	0.65
25	Training	0.351	0.0050	10.1	47.60	7.90	0.62
26	Training	1.510	0.0030	44.7	66.13	4.48	0.70
27	Training	1.510	0.0050	51.3	30.03	3.76	0.59
28	Training	1.510	0.0060	80.6	54.24	5.34	0.83
29	Training	1.510	0.0070	226.4	55.57	6.11	0.95
30	Training	0.351	0.0050	12.1	49.08	8.28	0.65
31	Training	1.510	0.0050	104.8	46.28	4.63	0.72
32	Training	1.510	0.0060	94.2	27.61	3.73	0.58
33	Training	1.510	0.0050	62.5	26.76	3.60	0.56
34	Testing	1.510	0.0070	424.4	43.86	5.60	0.87
35	Testing	1.510	0.0030	24.4	59.83	4.28	0.67
36	Testing	1.510	0.0060	87.6	50.15	5.08	0.79
37	Testing	1.510	0.0030	21.1	57.74	3.99	0.62
38	Testing	0.351	0.0050	20.5	53.97	9.04	0.71
39	Testing	0.351	0.0050	1.8	23.89	3.82	0.30
40	Testing	1.510	0.0050	70.9	38.69	3.99	0.62
41	Testing	1.510	0.0060	100.1	40.23	4.18	0.65
42	Testing	1.510	0.0080	715.0	41.41	6.76	1.05
43	Testing	1.510	0.0050	103.5	44.75	4.67	0.73
44	Testing	1.510	0.0030	32.0	60.51	4.80	0.75

Table 3. Dataset used to evaluate the performance of self-cleansing models.

Experimental data	Model development stage	No. of runs	D (mm)	d (mm)	So [%]	Cv (ppm)	vl (m s^−1)
Present study	Training	33	242	0.35–1.51	0.20–0.80	0.26–875.62	0.24–0.95
Present study	Testing	11	242	0.35–1.51	0.30–0.80	1.77–715.01	0.30–1.05
Mayerle (1988)	Testing	106	152	0.50–8.74	0.14–0.56	20.00–1275.00	0.37–1.10
Ab Ghani (1993)	Testing	221	154, 305, 450	0.46–8.30	0.04–2.56	0.76–1450.00	0.24–1.22
Ota (1999)	Testing	36	305	0.71–5.61	0.20	4.20–59.40	0.39–0.74
Vongvisessomjai, Tingsanchali, and Babel (2010)	Testing	36	100, 150	0.20–0.43	0.20–0.60	4.00–90.00	0.24–0.63

Table 4. Optimization strategies adopted to derive new self-cleansing models.

Optimization strategy	Group of parameters	Parameters	Functional relationship
1	Hydraulic characteristics	R, A	FR* = f(R, A, λ, Cv, d)
	Pipe material	λ
	Sediment characteristics	Cv, d
2	Hydraulic characteristics	R, A	FR* = f(R, A, λ, D, So, Cv, d)
	Pipe material	λ
	Pipe characteristics	D, So
	Sediment characteristics	Cv, d
3	Hydraulic characteristics	R, A	FR* = f(R, A, λ, D, So, Cv, d, Dgr)
	Pipe material	λ
	Pipe characteristics	D, So
	Sediment characteristics	Cv, d, Dgr

Table 5. Models obtained using EPR for different optimization strategies.

OS	Expression	Equation
1	$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=3.35{C_v}^{0.20}{\left(\frac{d}{R}\right)}^{-0.60}$	(13)
2	$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=6.20{S_o}^{0.15}{C}_v^{0.13}{\left(\frac{d}{R}\right)}^{-0.50}$	(14)
3	$\frac{v_l}{\sqrt{gd\left(SG-1\right)}}=5.60{S_o}^{0.14}{C}_v^{0.16}{D_{gR}}^{0.02}{\left(\frac{d}{R}\right)}^{-0.58}$	(15)

Table 6. Performance of models returned by EPR-MOGA-XL and literature self-cleansing models/equations. Bolded values show best performing models.

Data source	Stage	Performance measure	Traditional Models								EPR-MOGA models
			Eq. (4)	Eq. (5)	Eq. (6)	Eq. (7)	Eq. (8)	Eq. (9)	Eq. (10)	Eq. (11)	Eq. (13)	Eq. (14)	Eq. (15)
Present Study	Training	SSE	4.25	0.96	1.21	0.46	0.41	0.98	1.65	0.63	0.61	0.12	0.06
		CoD	0.00	0.65	0.56	0.83	0.85	0.64	0.40	0.77	0.78	0.96	0.98
		AIC	53.72	4.53	16.41	−15.87	−23.24	5.26	22.62	−5.15	−10.32	−62.57	−82.17
Present Study	Testing	SSE	3.56	0.80	0.91	0.40	0.28	0.88	1.75	0.66	0.55	0.11	0.06
		CoD	0.00	0.64	0.59	0.82	0.87	0.60	0.21	0.70	0.75	0.95	0.97
		AIC	19.98	3.56	8.98	−0.03	−7.99	4.58	12.15	5.48	−0.50	−16.23	−20.15
Mayerle (1988)	Testing	SSE	2.86	0.22	0.46	0.67	0.67	1.23	2.28	0.61	1.09	0.79	0.56
		CoD	0.43	0.96	0.91	0.87	0.87	0.76	0.55	0.88	0.78	0.84	0.89
		AIC	117.57	−152.47	−72.71	−32.08	−35.74	28.09	93.31	−43.11	15.20	−17.43	−52.24
Ab Ghani (1993)	Testing	SSE	2.88	2.55	0.38	0.44	0.32	0.36	0.75	0.84	0.25	0.33	0.18
		CoD	0.38	0.45	0.92	0.90	0.93	0.92	0.84	0.82	0.95	0.93	0.96
		AIC	239.81	213.22	−202.55	−168.96	−243.61	−221.39	−58.67	−27.49	−300.53	−237.82	−363.39
Ota (1999)	Testing	SSE	1.63	0.78	0.09	0.07	0.07	0.13	0.26	0.09	0.07	0.04	0.04
		CoD	0.09	0.57	0.95	0.96	0.96	0.93	0.86	0.95	0.96	0.98	0.98
		AIC	23.69	−2.97	−76.10	−85.07	−89.09	−66.19	−42.59	−78.11	−90.27	−107.40	−105.57
Vongvisessomjai, Tingsanchali, and Babel (2010)	Testing	SSE	4.18	2.42	0.16	0.02	0.02	0.59	1.80	0.76	0.15	0.27	0.10
		CoD	0.00	0.00	0.93	0.99	0.99	0.75	0.23	0.68	0.93	0.89	0.96
		AIC	57.48	37.75	−56.32	−135.57	−134.89	−13.10	27.14	0.02	−61.54	−39.24	−71.42

Figure 3. Fitting of traditional equations and EPR-MOGA models, using (a) Present study training data; (b) Present study testing data; (c) Mayerle (1988) data; (d) Ab Ghani (1993) data; (e) Ota (1999) data and (f) Vongvisessomjai, Tingsanchali, and Babel (2010) data.

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