Ebtehaj, I., Bonakdari, H. (2017). No-deposition Sediment Transport in Sewers Using of Gene Expression Programming. Soft Computing in Civil Engineering, 1(1), 29-53. doi: 10.22115/scce.2017.46845

Isa Ebtehaj; Hossein Bonakdari. "No-deposition Sediment Transport in Sewers Using of Gene Expression Programming". Soft Computing in Civil Engineering, 1, 1, 2017, 29-53. doi: 10.22115/scce.2017.46845

Ebtehaj, I., Bonakdari, H. (2017). 'No-deposition Sediment Transport in Sewers Using of Gene Expression Programming', Soft Computing in Civil Engineering, 1(1), pp. 29-53. doi: 10.22115/scce.2017.46845

Ebtehaj, I., Bonakdari, H. No-deposition Sediment Transport in Sewers Using of Gene Expression Programming. Soft Computing in Civil Engineering, 2017; 1(1): 29-53. doi: 10.22115/scce.2017.46845

No-deposition Sediment Transport in Sewers Using of Gene Expression Programming

^{1}Ph.D. Candidate, Department of Civil Engineering, Razi University, Kermanshah, Iran

^{2}Professor, Department of Civil Engineering, Razi University, Kermanshah, Iran

Receive Date: 26 May 2017,
Revise Date: 13 June 2017,
Accept Date: 14 June 2017

Abstract

The deposition of flow suspended particles has always been a problematic case in the process of flow transmission through sewers. Deposition of suspended materials decreases transmitting capacity. Therefore, it is necessary to have a method capable of precisely evaluating the flow velocity in order to prevent deposition. In this paper, using Gene-Expression Programming, a model is presented which properly predicts sediment transport in sewer. In order to present Gene-Expression Programming model, firstly parameters which are effective on velocity are surveyed and considering each and every of them, six different models are presented. Among the presented models the best is being selected. The results show that using verification criteria, the presented model presents the results as Root Mean Squared Error, RMSE=0.12 and Mean Average Percentage Error, MAPE=2.56 for train and RMSE=0.14 and MAPE=2.82 for verification. Also, the model presented in this study was compared with the other existing sediment transport equations which were obtained using nonlinear regression analysis.

Transmitting flow through sewage channel is often accompanied by solid materials. Sediment deposition takes place because of solid materials wide range entry into the sewer as well as the intermittent and variable nature of flow regimes within the sewer. Therefore, the management of sediment transport in sewers is considered as one of the most important items in sewer designing and operation. During wet weather flow, the flow rate is enough to suspend the solid sediments. Solid materials deposition in sewers takes place especially in low flow rate cases such as the beginning of the layout period, low consumption hours or warm seasons of the year. Permanent deposition on the pipe bed causes cross sectional variation and bed roughness and therefore velocity and shear stress distribution change and sewer hydraulic resistance consequently influences sediment transport capacity and finally causes operation maintenance cost increments. In order to convey minimum entry flow into the sewer, the slope ought to be as much to be able to prevent sediment deposition or for a fixed channel slope, the minimum transmitting flow rate shall be as much to be able to transport solid materials. In addition to design the velocity that is somehow capable of transmitting no-deposition solid materials, pipe diameter shall be selected in a way that transmitting maximum flow rate becomes possible.

Therefore, methods are needed to manage deposit transmission in a way that the transmitting flow would be capable of cleansing deposited sediments. Also, hand design process needs to be economical and optimized (Butler and Clark, 1995). The traditional method of designing sewage channels to prevent sediment deposition in the flow uses minimum velocity or minimum shear stress. In this method, sewer designing was done by presenting a fixed velocity or minimum shear stress at a determined flow depth or specified period of time. For example, ASCE (1970) proposes the constant velocity for full and semi-full flow equal to 0.6 m/s for sanitary sewer and 0.9 m/s for storm sewer. British Standard (1987) proposes 0.75 m/s for full flow and storm sewer and 1 m/s for combined sewer. European Standard (1997) considers the constant velocity for pipes with diameters less than 300 mm equal to 0.7 m/s. While this criterion has not presented any suggestion for larger diameter pipes, flow conditions are not denoted in this standard. Also, for constant shear stress criterion, ASCE (1970) has proposed shear stress within the range of 1.3 to 12.6 N/m^{2} and Lysne (1969) has proposed shear stress between 2 to 4 N/m^{2}. Therefore, we can conclude that velocity or minimum shear stress values are not equal in different conditions and countries. This is related to implemented experiments, size of sediments in different region and other parameters. So, in order to determine self-cleansing velocity, one has to achieve factors effective on sediment transport such as sediment concentration and size, flow hydraulic depth or radius, pipe roughness and diameter, so that the designer can achieve minimum required velocity according to regional conditions.

To survey sediment transmission in no-deposition case within sewers, sediment transport has been presented in two general ways: using dimensional analysis and semi-experimental relations. In order to present sediment transport relations with the use of dimensional analysis, dimensionless parameters are determined after implementing various experiments and studying the effect of effective parameters on sediment transport, and finally sediment transport relations are presented. In order to present semi-experimental relations with the use of effective forces exerted on a particle in equilibrium state, relations are presented. Using dimensional analysis, presented relations are given in three different states. The first approach, evaluates densimetric Froude Number (Fr) with the use of volumetric sediment concentration (C_{V}), relative flow depth (d/R) and overall sediment friction factor (λ_{s}) (Pedroli 1963; Graf and Acaroglu 1968; Novak and Nalluri 1975; Nalluri 1985). The second approach calculates Fr similar to the first case, but the difference is that in this approach in addition to the presented parameter in the first case, dimensionless particle number (D_{gr}) is also used, (Mayerle 1988; Mayerle et al. 1991; Ab Ghani 1993; Azamathulla et al. 2012). The third approach of the presented relations which uses dimensional analysis evaluates Fr by using volumetric sediment concentration (C_{V}) and flow proportional depth (d/R or d/y) (Mayerle et al. 1991; Ota and Nalluri 1999; Vongvisessomjai et al. 2010). Semi-experimental relations are also presented in different ways and will be briefly presented. May (1982) obtained his model of bed load transport based on effective loads which are exerted on particles transmitted at limit of deposition. Using dimensional analysis the author simplified the theoretical model in order to present his model and fit it with experimental data. May et al. (1996) modified the relation of May (1982) by using seven different sets of data. This relation is considered as the best sediment transport relation at limit of deposition, which is achieved semi-experimentally (Ackers et al. 1996). Correcting the relation by Ackers and White (1973) and in order to consider flow cross section form in pipes, Ackers (1991) presented his relation. May (1993) presented his relation in a semi-experimental way to transport at the limit of deposition based on effective shear stress on the sediments surface. To develop a new practical methodology for sewer, a comprehensive research project conducted in the UK based on available experimental knowledge. The results of this project were offered by Butler et al. (2003). The harvest of this study is presented as a self-cleansing sewer design methodology based on a new definition of self-cleansing. The authors considered an efficient self-cleansing sewer which has sediment transport capacity by considering a minimum amount of deposited bed to balance between consolidated expenses of construction, operation, and maintenance. Banasiak (2008) investigated the behavior of non-cohesive and partly cohesive deposited sediment in a partially full sewer pipes and its effect on the hydraulic performance of sewer. They found the presence of cohesive- like beds is more desirable than granular ones in terms of the bed roughness. Because the attendance of fine sediments as deposited sediment results in partly cohesive deposited solids so that decrease or in such cases prevent bed forms development. Ota and Perrusquia (2013) conducted several experimental tests in two sewer pipes at the limit of deposition condition to the measurement of sediment particle and sphere velocity. As regards, sediment transport depends on sediment repose angle, the authors developed a new semi-theoretical equation based on a reclaimed non-dimensional bed shear stress. Safari et al. (2017) carried out a series of experiment tests on trapezoidal channel cross-section. Using these samples and collected a wide range of experimental data of U-shape, rectangular and circular channel cross sections from the literature, the authors developed a self-cleansing midels based on definition of a shape factor to consider the effect of channel cross section.

In recent years, using soft computing (SC) in different sciences has led to desirable results (Mondal et al. 2012; Gad and Khalaf 2013; Gorani et al. 2014; Al-Abadi 2014; Khoshbi et al. 2016; Ahmadianfar et al. 2016; Azimi et al. 2017). To overcome the uncertainty and complexity accompained with bed load sediment transport estimation in sewers, Azamathulla et al. (2012) presented multi-ninlinear regression-based model and adaptive neuro-fuzzy inference systems (ANFIS). They found that the offered ANFIS model could employed as a strongth alternative tool in sedimen transport prediction at clean pipe. Ebtehaj and Bonakdari (2013) evaluated the performance of artificial neural network (ANN) in estimation of sediment transport using self-cleansing concept. They found the superior results of ANN in compared with existing regression-based methods. Ebtehaj and Bonakdari (2014) employed two different algorithms; back-propagation (BP) and hybrid of back-propagation and least-square (BP-LS); to train ANFIS in predicting of sediment transport in sewers. Moreover, to the generation of fuzzy inference systesm (FIS), sub-clustering (SC) and grid partitioning (GP) were utilized. Based on these methods, they introduced four different methods in ANFIS training. The results illustrated that a combination of GP and Hybrid results in the most precise sediment transport prediction.

All computational methods have different advantages and disadvantages depending on the type of problems, the decision on whether or not to use it done. In ANN, the learning and computations are easy but the major drawbacks of this approach are as arriving at the local minimum, less generalizing performance, over-fitting problem and slow convergence speed. Moreover, attaining the optimal structure of a constructed ANN is not simple (Rezaei et al. 2017). The main shortcoming of fuzzy logic (FL) is in finding the shape of each variable and suitable membership functions are untangled by trial and error (Singh et al. 2012, 2013). To overcome the disadvantage of ANN and fuzzy logic, ANFIS has been introduced which are knowning as a most popular strong SC tool. ANFIS is an adaptive fuzzy system which allows to utilization of ANN topology with FL simultaneously. It not only contains the features of both approaches, but also removes some shortcomings of their lonely-utilized case. Indeed, ANFIS consists of ANN advantages such as understanding mathematical details is not obligate and acquaintance with the job data is enough, employed different algorithm within learning course and solving nonlinear complex problems with strong capacity (Isanta Navarro 2013). Moreover, the advantages of ANFIS in comparison with ANN are attained highly nonlinear mapping, better learning capacity, and involves fewer tuneable parameters. However, the most constraints in ANFIS are more complex than FIS, not existe for all types of FIS (Rezaei et al. 2017) and there is no law for tunning the membership functions.

In addition to these drawbacks, the main problem in both of ANN and ANFIS is the existence of a black-box and don’t provide a certain equation to apply in practical applications. Therefore, it needs to a technique to overcome to this shortcoming. One of the newest presented models in soft computing topic is Gene Expression Programming (GEP). The main shortcomings of this method are premature convergence due to derivation of this method from genetic programming and genetic algorithm, preservation of best individual based on roulette-wheel selection method with elitism so that results in losing other better individuals (Gan et al. 2007) and CPU time consuming. Azamathulla and Ab. Ghani (2010) predicted pipeline scour depth with the use of GEP and concluded that in comparison with existing models, the presented model provides better results. Khan et al. (2012) used GEP in order to predict bridge pier scour. The authors compared their presented model with artificial neural network and regression relations and concluded that the presented model leads to more satisfactory results when compared to existing models. Chang et al. (2012) compared three different methods available in soft computing, adaptive neuro-fuzzy inference system, feed forward neural network and GEP, to survey bed load in the rivers. Azamathulla and Ahmad (2012) used GEP model to predict transverse mixing coefficient in open channels flow. Using laboratorial results mostly, the authors presented a relation in order to estimate transverse mixing coefficient which presented the results with more precession compared with the existing relations. With the use of Gene-Expression Programming (GEP) in this study, sediment transport in sewerage channels has been studied. The presented model is applicable for no-deposition case.

To increase the accuracy of the presented model in this study – in comparison with the existing models (Ab Ghani and Azamathulla, 2011) which only used the four basic mathematical operations multiplication, subtraction, division and addition – various functions which can be seen in Table 1 were used. Firstly, considering the effective parameter on sediment transport, six different models have been presented. Comparing the presented models with data sets which were not used in presenting models, the best model has been selected. To assess the accuracy of the models presented through GEP algorithm versus the existing equations, the experimental results of Ota and Nalluri (1999) which had no role in the training of the GEP were used.

2. Non-deposition sediment transport equations

May et al. (1996), with the use of seven different data sets (Mack 1982; May et al. 1989; Mayerle et al. 1991; May 1993; Nalluri and Ab Ghani 1993; Ab Ghani 1993; Nalluri et al. 1994) studied the existing sediment transport relations. Laboratorial data was used to evaluate these relations. Results of studying the relations showed that each relation presents good results only for data sets which have been used for relation presenting, thus in order to present a relation for sediment transport studying at the limit of deposition, they presented following relation :

(1)

(2)

where C_{V} is volumetric sediment concentration, D pipe diameter, A Cross-sectional area of the flow, d median diameter of particle size, g gravitational acceleration, s specific gravity of sediment (=ρ_{s}/ρ), V flow velocity, V_{t}the required velocity for incipient motion of sediment (Eq. 2) and y flow depth.

In order to sediment transport at limit of deposition Ackers et al. (1996) considered the above relation as the best existing relation for designing usage and Vongvisessomjai et al. (2010) too used Eq. 1 for verification of his relation. Considering volumetric sediment consideration (C_{V}) and relative flow depth (d/R), Ebtehaj et al. (2014) presented the Fr in the form of following relations:

(3)

Ab. Ghani and Azamathulla (2011) used GEP to predict bed load transport in sewers. The authors presented their equation by considering the parameters of volumetric sediment concentration (C_{V}), relative depth of flow (d/R), dimensionless particle number (D_{gr}) and Overall sediment friction factor (λ_{s}=1.13D_{gr}^{0.01}C_{V}^{0.02}λ_{C}^{0.98}, λ_{C} clear water friction factor) as follows:

(4)

3. Data Collection

In this research a combination of thelab test results by Vongvisessomjai et al. (2010) and Ota and Nalluri (1999) was used. The model is proposed using experimental results presented by Vongvisessomjai et al. (2010) and the results of lab experiments are used to verify the feasibility of the model proposed by Ota and Nalluri (1999). Vongvisessomjai et al. (2010) conducted their tests on pipes in two sizes of 100 and 150 mm in diameter and 16 m in length. They employed two sections to measure the flow: one at a distance of 4. 5 m upstream, and the other at the distance of 5.5 m downstream. These two points were 6 m apart. In each section the velocities were measured at flow surface, middle depth and near bottom and their mean average was taken as the average velocity. For the air/water phase of the flow, the Manning coefficient of roughness (n) was equal to 0.0125. Vongvisessomjai et al. (2010) tests were conducted in a non-deposited bedstate. More details are given in Vongvisessomjai et al. (2010). To validate the accuracy of results presented in this article, Ota and Nalluri (1999) data were used for limit of deposition. For the purpose of their tests at limit of deposition, Ota and Nalluri (1999) used six different dimensions of d (ranging from 0.71 mm to 5.61 mm). They conducted 24 tests in total. Moreover, to test the impact of granulation on sediment transport, they conducted 20 further experiments using five different ranges of sediments with an average diameter of d = 2 mm. More details are given in Ota and Nalluri (1999). Table 1 shows the range of the data used in their tests.

Table 1. Range of data in Ota and Nalluri (1999) and Vongvisessomjai et al. (2010) studies

y/D

V (m/s)

R (m)

C_{V}(ppm)

d (mm)

Ota and Nalluri (1999)

0.39-0.84

0.515-0736

0.005-0.076

16-59

0.6-6.3

Vongvisessomjai et al. (2010)

0.2-0.4

0.24-0.63

0.012-0.032

4 to 90

0.2-0.43

4. Overview of Gene Expression Programming

Gene expression programming (GEP) is an expansion of genetic programming (GP) (Koza 1992). GEP belongs to the family of evolutionary algorithms and is closely related to genetic algorithms and genetic programming. From genetic algorithms it inherited the linear chromosomes of fixed length; and from genetic programming it inherited the expressive parse trees of varied sizes and shapes (Ferreira 2001). The GEP procedure is such that initially required functions for model creation and terminal set are being selected. In the next step, in order to evaluate the aimed parameter (in this study Fr) and comparing it with the real value, existing data sets are being recalled. Afterwards, in order to randomly present the initial population, chromosomes are being produced. In the next step, for population mass production with the use of existing chromosomes, the program is run and the fitness of target function is surveyed. If we arrive at pause conditions, program is stopped, otherwise with the use of new chromosomes - which have been corrected via genetic operators - as well as new population; again target function is being evaluated. This action continues until program pause conditions are present.

Fitness of an individual program (i) for fitness model (j) has been presented by Ferreira (2006) in the following form:

(5)

where p precision and E(ij) the error of program i for fitness case (j). For absolute error, it is being stated as in following form:

(6)

Also the fitness value (f_{i}) for an individual program is stated in the following form:

(7)

where R is selection range, p_{(ij)}the predicted value by individual program (i) for fitness case (j) and T_{j}the target value for fitness case (j). After fitness function determination, the terminal set (T) and function set (F) have to be determined in order to selecting chromosomes.

5. Methodology

In order to survey sediment transport in pipes, effective parameters on flow and sediment particles movement have to be recognized. According to laboratorial studies by researchers (Ab Ghani 1993; May et al. 1996; Vongvisessomjai et al. 2010), the most important surveyed and utilized parameters to present their relations, include parameters like flow velocity (V), dimensionless particles number (D_{gr}), volumetric sediment concentration (C_{V}), median diameter of particles size (d), pipe diameter (D), flow depth (y), hydraulic radius (R), cross sectional area of the flow (A), overall sediment friction factor (λ_{s}) and special gravity of sediment (s). Thus dimensionless parameters could be considered in the form of movement, transport, sediment, transport mode, and flow resistance. Movement parameters are respectively stated as densimetric Froude number (Fr) or (ψ) which uses shear stress instead of velocity. Transport parameter contains volumetric sediment concentration (C_{V}) or the presented transport parameter (φ), dimensionless particle number (D_{gr}), proportional average size of particles (d/D) and specific gravity of sediment (s). Transport form parameter includes the ratio of hydraulic radius to the median diameter of particles size (R/d), the ratio of squared pipe diameter to the flow cross sectional area (D^{2}/A), relative flow depth (y/d) - instead of which usually R/d is being used - and the flow resistance parameter that considers flow overall frictional coefficient (λ_{s}). Based on these explanations, in order to study the effect of each and every parameter in different dimensionless groups, dimensionless parameters can be presented in order to predict Fr in the form of Table 2.

Table 2. Dimensionless sediment transport parameters in clean pipes

Dimensionless groups

Parameter type

Movement

Transport

Sediment

Transport mode

Flow resistance

It is necessary to use different statistical indexes to verify the feasibility of the proposed model. The statistical indexes used in this study include dimensionless coefficient criteria called R-Squared (R^{2}), the relative criteria of Mean Average Percentage Error (MAPE) and absolute criteria of Root Mean Squared Error (RMSE). The R-Squared (R^{2}) index is the ratio of the combined dispersion of the estimated model and the observed value to the dispersion of the estimated and observed models. The MAPE expresses the estimated value in relation to the observed value. MAPE is a non-negative index which has no higher limit. The RMSE is a criterion of mean error that has no upper limit and has the lowest possible value of zero, representing the best estimation by the model.

(8)

(9)

(10)

The above-mentioned indexes present the estimated amounts as the average of the forecasted error and do not present any sort of information on the forecasted error distribution of the suggested models. It is obvious that a high correlation coefficient (80- 90%) is not always considered as an indication of the high accuracy of a model; on the contrary, this index may lead to showing high accuracy for mediocre models (Legates and McCabe 1999). In addition RMSE index indicates the model’s ability to predict a value away from the mean (Hsu et al. 1995). Therefore, the presented model must be evaluated using other indexes such as average absolute relative error (AARE) and threshold statistics (Jain et al. 2001; Jain and Ormsbee 2002; Rajurkar et al. 2004; Maghrebi and Givehchi 2007). TS_{x}index indicates forecasted error distribution by each model for x% of the anticipations. This parameter is determined for various amounts of average absolute relative error. The amount of the TS index for x% of the predictions is determined as explained below:

(10)

(11)

where Y_{x} is the number of the forecasted amounts of all the data for each amount of AARE is less than x%.

6. Derivation densimetric Froude number based on GEP

This section concentrates on GEP method to calculate densimetric Froude number (Fr). The training set must be selected from amongst all the existing data for the purposes of presenting a model. To that end, the data presented by Vongvisessomjai et al. (2010) was selected as training set and the data presented by Ota and Nalluri (1999) was selected as testing set. The training environment of the system has been defined after selecting the training set. After classifying the data, various parameters must be defined to make a model. To create the generation the initial population of the individuals, multi-genic chromosomes, are used which include four genes. We must now determine the number of the initial population. Considering Ferreira’s (2001) suggestion stating that using the size population within the range of 30- 100 can lead to good results, the size of the used population include 50 chromosomes in this study which was selected through trial and error. After selecting the population size, the individuals are evaluated and their fitness function is calculated using MSE as follows:

(12)

where Q_{ij} is the amount observed for fitness case, and P_{ij} is the amount predicted through using i individual chromosome for fitness case j. The best state is when the equation E_{ij}= 0 is obtained. This means that the amounts predicted using i individual chromosome for fitness case j is equal to the amount observed for fitness case j (P_{ij}= E_{ij}). The set of terminals and the set of function must be determined for each gene in the chromosome after selecting fitness function. The function sets used in this study include {×, -, ÷, ×, Gau2} while the set of terminals are as follows:

(13)

Afterwards the number of genes and their head and tail length must be determined for each gene in the chromosome. By using trial and error and the succeeding rate, four genes were selected in the present study in each chromosome. The head length was selected to be 5 (h=5) and while the maximum number of arguments per function is equal to 2 (n_{max}= 2) the length of the tail turns out to be equal to 6 (t= 5 × (2-1) +1). The genetic operator rate must now be determined. Genetic operators such as mutation, inversion, transportation (IS, RIS, gene transportation), recombination and crossover (one point, two points and gene recombination) were used. The rates of the mentioned parameters are presented in Table 3. We must finally determine the linking function. Considering the fact that using four different sub-expressions in this study has led to having 4 genes, the genes must be bound in order for us to reach the final result. Therefore, {+} operator has been used as the linking function among the genes in this study. Simulating the model begins after determining the essential parameters. Gau2{x, y} function presented in Table 2 returns exp (-(x+ y)^{2}) amount.

Table 3. Parameters of GEP model

Parameter

Setting

Population size

50

Number of generations

250000

Number of chromosomes

50

Number of genes

4

function set

×, -, ÷, ×, Gau2

Linking function

Addition

Mutation rate

0.03

Inversion rate

0.15

IS transposition rate

0.1

RIS transposition rate

0.1

Gene transposition rate

0.15

One-point recombination rate

0.3

Two-point recombination rate

0.3

Gene recombination rate

0.15

7. Result and Discussion

To study sediment transport, different parameters in no-deposition stage and to present a model which could estimate the best results in comparison with actual values, dimensionless parameters in Table 2 have been used. As considered in this table, dimensionless parameters effective on sediment transport in no-deposition mode are categorized in five groups. In order to present a model, the effect of four groups of transport, deposition, and transport form and flow resistance on movement group was surveyed. Thus, six different models are listed in Table 4. In the presented models, volumetric sediment concentration (C_{V}) which is related to transport dimensionless group and overall sediment frictional coefficient (λ_{s}) which is related to flow resistance dimensionless group have been considered constant. For sediment group D_{gr} and d/D parameters and for transport form group d/R، D^{2}/A and y/D have been considered.

Table 4. Dependent parameters in predicting Fr considering the effect of dimensionless group parameters

Model

Dependent

Independent

Train

Test

parameter

parameters

R^{2}

MAPE

RMSE

R^{2}

MAPE

RMSE

1

Fr

C_{V}, D_{gr}, d/R, λ_{s}

0.98

2.66

0.16

0.96

2.94

0.12

2

Fr

C_{V}, D_{gr}, D^{2}/A, λ_{s}

0.89

6.58

0.73

0.81

11.09

0.30

3

Fr

C_{V}, D_{gr}, y/D, λ_{s}

0.89

5.33

0.53

0.90

7.84

0.32

4

Fr

C_{V}, d/D, d/R, λ_{s}

0.99

2.56

0.12

0.99

2.82

0.14

5

Fr

C_{V}, d/D, D^{2}/A, λ_{s}

0.92

5.70

0.65

0.85

9.39

0.27

6

Fr

C_{V}, d/D, y/D, λ_{s}

0.97

2.94

0.20

0.96

3.05

0.19

Table 3 shows sextet presented models with the use of Table 1. In order to present models, laboratorial results by Vongvisessomjai et al. (2010) have been utilized. After presenting different models to evaluate the estimated results, via each model, the Fr has been surveyed with the use of Ota and Nalluri (1999) laboratorial results. According to verification criteria presented in Table 3, model 4 which uses volumetric sediment concentration (C_{V}), relative flow depth (d/R), proportional average size of particles (d/D) and overall frictional factor (λ_{s}) to estimate Frdelivers the best result. The MAPE index in Fr evaluation with the use of this model is about 2.56% for test and 2.82% for train and RMSE is 0.12 for train and 0.14 for test. It is considered that the effect of data sets alternations on the precision of this model is about less than 1%. Other presented models in this table, compared to the mode with train data, show better results than test data, and this is an indication that using these models (models 1, 2, 3, 5 and 6) would not be trustworthy. Therefore, it could be said that, to present a model which could well estimate Fr in sewer at limit of deposition state, effective parameters are able to be considered like model 4 in Table 3. This means that using C_{V} as transport parameter, d/R as transport form parameter, d/Das sediment parameter and λ_{s} as flow resistance parameter in Fr evaluation, leads to good results. The presented equation through using the parameters of model 4 and expression tree presented in Figure 1 can be presented as follows. The amounts of the parameters presented in this figure have been shown in Table 5.

(14)

We can rewrite the above-mentioned formula as follows:

(15)

Figure 1. Expression Tree (ET) for GEP formulation

Table 5. Values of the parameters used in ET (Figure 1)

d0

d1

d2

d3

G1C5

G2C2

G2C0

G2C5

G3C8

G3C6

G4C7

G4C0

G4C6

C_{v}

d/R

d/D

λ_{s}

89.66

33.10

4.23

15.45

132.42

67.08

92.40

-5.90

9.54

Figure 2 shows the Fr results predicted by model 4 (Eq. 15) in both training and testing stage. Due to the fact that the accuracy of GEP model presented in Table 4 in this research (Eq. 15) has been studied quantitatively for both test (MAPE= 2.82 & RMSE= 0.14) and train (MAPE= 2.56 & RMSE= 0.12) states, in this figure, we will attend to studying the prediction results of the GEP model. The figure indicates that the forecasted Fr which were obtained through using GEP presented fairly good results in both train and test states while almost all estimated amounts have a relative error of less than 10%. The data used for the purpose of test and train of equation 15 have different ranges of Fr in such manner that the Fr used in training the model was within the range of 4 to 9 while the Fr range used in testing the model is 3 to 6. Therefore, it could be stated that while studying the model accuracy in test state all the Fr are not within the range which was used in training the model, thus, considering the qualitative results (Table 4) and quantitative results (Figure 2), it proves the accuracy of the presented results obtained by this equation.

Figure 2. Comparison of GEP result for both of train and test stages with actual values

Figure 3 compares the prediction results of Fr through using the GEP model presented in this study (Eq. 15) and the existing regression equation with the actual values. The figure shows that the results of the presented predictions using GEP are almost accurate in a way that all forecasted points have an error less than ten percent and taking into consideration Figure 4 which shows the accumulative distribution of the error, we can see that the maximum relative error in estimating Fr through using GEP is almost equal to 6 %. Also, this figure indicates that approximately 90% of the anticipated amounts have a relative error less than 5%. Now in case we intend to study the results of the presented model through statistical indexes, referring to Table 6 shows us that the amounts of the presented statistical indexes for this model with an R^{2}= 0.99, MAPE= 2.82 and RMSE= 0.14 is minimum in amount in comparison to other equations presented in this table. The equation presented by Ab. Ghani and Azamathulla (2011) is less accurate (R^{2}= 0.74, MAPE= 13.18 and RMSE= 0.49) considering Table 6 and Figure 4. The figure indicates that in the majority of the points the results are presented with an error more than 10 percent. Figure 4 shows that only 25% of the amounts estimated by this model have a relative error less than 10%. Also, it indicates that some of the Fr forecasted by this equation have a relative error more than 30 percent which indicates the uncertainty of the equation presented by Ab Ghani and Azamathulla (2011). Therefore, using this equation for the purpose of estimating Fr cannot be that much confident. Ebtehaj et al. (2014) equation is fairly accurate because it estimates the majority of Fr with a less-than-10-percent relative error, but it is less accurate in comparison with the equation presented in this study. This is in a way that considering Figure 4, which indicates the distribution of the estimation error by different models, we can see that approximately 70 percent of the estimation results of this model have an error less than 5% while for the model presented in this study the predicted amounts have an error of less than 5% for almost 90% of the Fr. At times, May et al. (1996) equation which has been obtained through semi-experimental method and has been known as one of the best sediment transport equations in limit of deposition (Vongvisessomjai et al. 2010) presents the estimated amounts with a more- than-15% relative error according to Figure 5 while the equation presented in this study has a maximum relative error of 6%. Also, considering Figure 6, the amounts of statistical indexes presented by this equation (R^{2}= 0.93, MAPE= 5.74 and RMSE= 0.24) indicates lesser estimation accuracy of this equation in comparison with that of the presented equation.

Figure 3. Comparison of proposed equation and existing equations

Figure 4. Error distribution of for GEP and existing equations

Table 6. Validation of proposed equation and existing equations with statistical indexes

Equation

R^{2}

MAPE

RMSE

Proposed Equation (Eq. 15)

0.99

2.82

0.14

Ab. Ghani and Azamathulla (Eq. 4)

0.74

13.18

0.49

Ebtehaj et al. (Eq. 3)

0.97

3.70

0.18

May et al. (Eq. 1)

0.93

5.74

0.24

Accordingly, in this study, the effects of GEP model output on the variations of dimensionless particle number (D_{gr} in this study) were investigated. The discrepancy ratio (DR) (ratio of predicted to actual values) was employed to measure the sensitivity of the GEP model to D_{gr}parameter. A DR value of 1 shows a perfect agreement, while values smaller (or greater) than 1 indicate under (or over) prediction of discharge coefficient in side weir. The result of the GEP model for variations of the discrepancy ratio (DR) values is plotted versus the dimensionless particle number (D_{gr}) in Figure 5. The maximum, mean and minimum DR values for GEP model were obtained 1.06, 1.005 and 0.93, respectively. As Figure 5 shows it, for almost all the Fr estimated, the DR is close to 1. When GEP predicts the model using the over design way, the dimensionless particle number will be equal to 1.7 (DR= 1.06) and when it uses the under-design way to predict Fr, dimensionless particle number is equal to 2.15 (DR= 0.93).

Figure 5. DR values versus D_{gr} for GEP model

Based on explanations given on Figures 3 and 5, and Table 6, the equation presented in this study is more accurate than the existing regression equations. While it is more accurate in studying the estimation accuracy through using statistical indexes, it is also more accurate in studying the estimation error distribution in Figure 4 and can be utilized as a substituting method in Fr estimation for sediment transport in no-deposition mode.

8. Conclusion

Transmitting flow from sewerage systems often contains suspended materials. Therefore, transporting suspended materials and preventing their sedimentation are important matters in flow transport through sewerage networks. Different methods have been presented for sediment transport in sewages, but due to the lack of recognition of effective factors on sediment transport these methods show different results in different conditions. Hence, in recent years, soft computations have been utilized in order to estimate densimetric Froude number (Fr) in these systems. In this paper, with the use of presented model by Gene-expression programming (GEP), Fr has been estimated. In order to present the effective factor on Fr estimation, six different models were presented. In these models, the effect of movement, transport, sediment, transport mode and flow resistance parameters have been considered. After Fr estimation, the precision of all sextet models has been studied. The results indicated that among the three parameters provided by “Transport mode” group, the best and the worst accuracy were achieved by using d/R and D^{2}/A (respectively) as improper use of the parameters of this group, up to two-fold increase relative error. In addition to, Also, with the constant parameters in the groups “transport”, “flow resistance” and “transport mode”, the parameter d/D in all input combinations, leading to better results than when used D_{gr} as “sediment” parameter. Therefore, it was revealed that the model which considers volumetric sediment concentration (C_{V}), relative flow depth (d/R), proportional average size of particles (d/D), overall friction factor (λ_{s}) for Fr estimation, shows the best results. The presented model estimates Fr with an average error value about 2.82%. The comparison of existing methods illustrated the high level of accuracy of Ebtehaj et al. (Eq. 3) method in comparison with others. It should not be inappropriate use of GEP functions such as Eq. (4) results in weak performance of model. The presented model with existing values was also studied and the results showed that in proportion with existing relations the model well estimates the Fr. Incidentally making use of the proposed GEP-based technique in form of the most superlative formulations has a dominant role to experience in the attaining astonishing and remarkable successes for real-world application. Another plus aspect of this study is the use of extracted mathematical expressions as realistically valuable technique for practical engineering as an alternative for existing methods.

Notation:

A Cross-sectional area of the flow

C_{V} Volumetric sediment concentration

D Pipe diameter

d Median diameter of particle size

D_{gr} Dimensionless particle number

E_{(ij)} Error of program i for fitness case j (Eq. 5)

Fr Densimetric Froude number

P Precision (Eq. 5)

P_{(ij)} Value predicted by individual program i for fitness case j (Eq. 6)

R Hydraulic radius, Selection Range (Eq. 6)

s Specific gravity of sediment (=ρ_{s}/ρ)

V Velocity of flow

V_{t} Incipient flow velocity which follows from equation (2)

y Flow depth

λ_{c} Clear Water friction factor

λ_{s} Overall sediment friction factor

ψ Flow parameter

φ Transport parameter

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