No-Deposition Sediment Transport in Sewers Using Gene Expression Programming

Journal of Soft Computing in Civil Engineering 1-1 (2017) 29-53
How to cite this article: Ebtehaj I, Bonakdari H. No-deposition sediment transport in sewers using gene expression programming.
J Soft Comput Civ Eng 2017;1(1):29–53. https://doi.org/10.22115/scce.2017.46845.
2588-2872/ © 2017 The Authors. Published by Pouyan Press.
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
No-Deposition Sediment Transport in Sewers Using Gene
Expression Programming
I. Ebtehaj1
, H. Bonakdari2*
1. Ph.D. Candidate, Department of Civil Engineering, Razi University, Kermanshah, Iran
2. Professor, Department of Civil Engineering, Razi University, Kermanshah, Iran
Corresponding author: bonakdari@yahoo.com
https://doi.org/10.22115/SCCE.2017.46845
ARTICLE INFO ABSTRACT
Article history:
Received: 26 May 2017
Revised: 13 June 2017
Accepted: 14 June 2017
The deposition of the flow of 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 the sewer. In order to
present Gene-Expression Programming model, firstly
parameters which are effective on velocity are surveyed and
considering 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.
Keywords:
Bed load;
Sediment transport;
Sewer;
No-deposition;
Gene-expression programming.

30 I. Ebtehaj, H. Bonakdari/ Journal of Soft Computing in Civil Engineering 1-1 (2017) 29-53
1. Introduction
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 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 [1]. 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. For example, ASCE [2] 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 [3] proposes 0.75 m/s for full flow and storm sewer and 1 m/s for
combined sewer. European Standard [4] 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 [2] has proposed shear stress within the range of 1.3 to 12.6 N/m2,
and
Lysne [5] has proposed shear stress between 2 to 4 N/m2
. 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 a 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,

I. Ebtehaj, H. Bonakdari/ Journal of Soft Computing in Civil Engineering 1-1 (2017) 29-53 31
pipe roughness, and diameter, so that the designer can achieve minimum required velocity
according to regional conditions.
To survey sediment transmission in the 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 an 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 (CV), relative flow depth
(d/R) and overall sediment friction factor (λs) [6–9]. 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 (Dgr) is also used [10–13]. The third approach of the
presented relations which uses dimensional analysis evaluates Fr by using volumetric sediment
concentration (CV) and flow proportional depth (d/R or d/y) [11,14,15]. Semi-experimental
relations are also presented in different ways and will be briefly presented. May [16] obtained his
model of bed load transport based on effective loads which are exerted on particles transmitted at
the 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. [17] modified the relation
of May [16] by using seven different sets of data. This relation is considered as the best sediment
transport relation at the limit of deposition, which is achieved semi-experimentally [18].
Correcting the relation by Ackers and White [19] and in order to consider flow cross section
form in pipes, Ackers [20] presented his relation. May [17] 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. [21]. 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 [22] 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 [23] 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. [24] 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 model based on the 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 [25–
29]. To overcome the uncertainty and complexity accompanied with bed load sediment transport
estimation in sewers, Azamathulla et al. [13] presented multi-nonlinear regression-based model
and adaptive neuro-fuzzy inference systems (ANFIS). They found that the offered ANFIS model
could employ as a strength alternative tool in sediment transport prediction at the clean pipe.
Ebtehaj and Bonakdari [30] evaluated the performance of artificial neural network (ANN) in the
estimation of sediment transport using self-cleaning concept. They found the superior results of
ANN in compared with existing regression-based methods. Ebtehaj and Bonakdari [31]
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 systems (FIS), sub-clustering (SC) and grid partitioning (GP)
were utilized. Based on these methods, they introduced four different methods of 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. 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 [32]. 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 [33,34]. To overcome the disadvantage of ANN and fuzzy logic,
ANFIS has been introduced which are known as a most popular strong SC tool. ANFIS is an
adaptive fuzzy system which allows to the 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 [35]. Moreover, the advantages of ANFIS in comparison with ANN have attained highly
nonlinear mapping, better learning capacity, and involves fewer tunable parameters. However,
the most constraints in ANFIS are more complex than FIS, not exist for all types of FIS [32] and
there is no law for tuning 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 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 the 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 [36] and CPU time-
consuming. Azamathulla and Ab. Ghani [37] predicted pipeline scoured depth with the use of
GEP and concluded that in comparison with existing models, the presented model provides better
results. Khan et al. [38] used GEP 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. [39] 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 [40] used GEP model to predict transverse mixing coefficient in open
channels flow. Using laboratory results mostly, the authors presented a relation 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 to
the no-deposition case.
To increase the accuracy of the presented model in this study – in comparison with the existing
models [41] 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
[14] which had no role in the training of the GEP were used.
2. Non-deposition sediment transport equations
May et al. [17], with the use of seven different data sets [11,12,42–46] studied the existing
sediment transport relations. Laboratory 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:
4
t
1.5
2
0.6
2
2
V
)
V
V
(1
)
1)D
g(s
V
(
)
D
d
)(
A
D
(
10
3.03
C 


 
(1)
0.47
0.5
t
]
d
y
[
1)d]
0.125[g(s
V 
 (2)
where CV 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 (=ρs/ρ), V flow velocity, Vt the required velocity for incipient motion of sediment (Eq.
2) and y flow depth.
In order to sediment transport at the limit of deposition Ackers et al. [18] considered the above
relation as the best existing relation for designing usage and Vongvisessomjai et al. [15] too used
Eq. 1 for verification of his relation. Considering volumetric sediment consideration (CV) and
relative flow depth (d/R), Ebtehaj et al. [47] presented the Fr in the form of following relations:
0.54
0.21
V
R
d
4.49C
1)d
g(s
V
Fr









 (3)
Ab. Ghani and Azamathulla [41] used GEP to predict the bed load transport in sewers. The
authors presented their equation by considering the parameters of volumetric sediment
concentration (CV), the relative depth of flow (d/R), dimensionless particle number (Dgr) and
Overall sediment friction factor (λs=1.13Dgr
0.01
CV
0.02
λC
0.98
, λC clear water friction factor) as
follows:









































 d
R
D
8.43λ
λ
λ
0.014
D
1
5.91
C
)
d
R
(
0.41
1.425
1)
gd(s
V
gr
1.5
s
s
s
gr
V
(4)
3. Data collection
In this research, a combination of the lab test results by Vongvisessomjai et al. [15] and Ota and
Nalluri [14] was used. The model is proposed using experimental results presented by
Vongvisessomjai et al. [15], and the results of lab experiments are used to verify the feasibility of
the model proposed by Ota and Nalluri [14]. Vongvisessomjai et al. [15] 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 were 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. [15] tests were conducted in a non-deposited bed
state. More details are given in Vongvisessomjai et al. [15]. To validate the accuracy of results
presented in this article, Ota and Nalluri [14] data were used for a limit of deposition. For the
purpose of their tests at the limit of deposition, Ota and Nalluri [14] 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 [14]. Table 1 shows the range of the data used in their
tests.
Table 1
Range of data in Ota and Nalluri [14] and Vongvisessomjai et al. [15] studies.
y/D V (m/s) R (m) CV (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) [48]. 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 [49].
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. Afterward, 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, the 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.
The fitness of an individual program (i) for fitness model (j) has been presented by Ferreira [50]
in the following form:
0
,
1
,
)
( )
(
)
( 

 ij
ij f
else
f
then
p
ij
E
If (5)
Where p precision and E(ij) the error of program i for fitness case (j). For the absolute error, it is
being stated as in the following the form:
j
(ij)
T
p
E(ij) 
 (6)
Also, the fitness value (fi) for an individual program is stated in the following form:
 

 )
T
p
|
(R
f j
(ij)
i
(7)
where R is selection range, p(ij) the predicted value by the individual program (i) for fitness case
(j) and Tj 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 select 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 laboratory studies by researchers
[12,15,17], the most important surveyed and utilized parameters to present their relations,

include parameters like flow velocity (V), dimensionless particles number (Dgr), volumetric
sediment concentration (CV), 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 (λs)
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 (CV)
or the presented transport parameter (φ), dimensionless particle number (Dgr), 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 (D2
/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 (λs). 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
1)d
ρg(s
τ
ψ
1
,
1)
gd(s
V
Fr 0




Movement
3
V
V
1)d
g(s
VR
C
φ
,
C


Transport
s
d/D,
,
Dgr
Sediment
y/D
d/y,
/A,
D
d/R, 2
Transport mode
)/D
k
(k
,
λ s
0
s 
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 (R2
), the relative criteria of Mean Average Percentage Error (MAPE) and absolute
criteria of Root Mean Squared Error (RMSE). The R-Squared (R2
) 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.
  
   
2
n
1
i
2
i
GEP
i
EXP
n
1
i
2
i
EXP
i
EXP
n
1
i
i
GEP
i
GEP
i
EXP
i
EXP
2
Fr
Fr
Fr
Fr
Fr
Fr
Fr
Fr
R























(8)




n
1
i
2
i
GEP
i
EXP
)
Fr
(Fr
n
1
RMSE
(9)
)
Fr
|
Fr
Fr
|
(
)
n
100
(
MAPE
n
1
i
i
EXP
i
GEP
i
EXP




(10)
The indexes mentioned above present the estimated amounts as the average of the forecasted
error and do not present any 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 [51]. In addition, RMSE index indicates the model’s ability to
predict a value away from the mean [52]. Therefore, the presented model must be evaluated
using other indexes such as average absolute relative error (AARE) and threshold statistics [53–
56]. TSx 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:
100
n
Y
TS x
x

 (11)

 






 







n
1
i
i
EXP
i
GEP
i
EXP
Fr
Fr
Fr
n
1
AARE
(12)
Where Yx is the number of the forecasted amounts of all the data for each amount of AARE is less
than x%.

6. Derivation of 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 to present a model. To that end,
the data presented by Vongvisessomjai et al. [15] was selected as training set, and the data
presented by Ota and Nalluri [14] was selected as a 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 [49] suggestion stating
that using the size population within the range of 30- 100 can lead to good results, the size of the
used population includes 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:
j
ij
i
i
i O
P
E
for
E
f 



1
1000
(13)
Where Qij is the amount observed for fitness case, and Pij is the amount predicted by using i
individual chromosome for fitness case j. The best state is when the equation Eij= 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 (Pij= Eij). The set of terminals and the set of the
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:






 s
2
gr
V
r λ
,
D
R
,
A
D
,
R
d
,
D
d
,
D
,
C
,
F
T (14)
Afterward, 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 (nmax= 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 four genes, the genes must be bound 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 that effect on
the sediment transport in no-deposition mode are categorized into 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 (CV) which is related to transport
dimensionless group and overall sediment frictional coefficient (λs) which is related to flow
resistance dimensionless group have been considered constant. For sediment group Dgr and d/D
parameters and transport form group, d/R ،D2
/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 R2
MAPE RMSE R2
MAPE RMSE
1 Fr CV, Dgr, d/R, λs 0.98 2.66 0.16 0.96 2.94 0.12
2 Fr CV, Dgr, D2
/A, λs 0.89 6.58 0.73 0.81 11.09 0.30
3 Fr CV, Dgr, y/D, λs 0.89 5.33 0.53 0.90 7.84 0.32
4 Fr CV, d/D, d/R, λs 0.99 2.56 0.12 0.99 2.82 0.14
5 Fr CV, d/D, D2
/A, λs 0.92 5.70 0.65 0.85 9.39 0.27
6 Fr CV, d/D, y/D, λs 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,
laboratory results by Vongvisessomjai et al. [15] 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 [14] laboratory results. According to verification criteria presented in Table 3,
model 4 which uses volumetric sediment concentration (CV), relative flow depth (d/R),
proportional average size of particles (d/D) and overall frictional factor (λs) to estimate Fr
delivers the best result. The MAPE index in Fr evaluation with the use of this model is about
2.56% for the test, and 2.82% for train and RMSE is 0.12 for train and 0.14 for the 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 a sewer at the limit of deposition state, effective parameters can be considered like
model 4 in Table 3. This means that using CV as transport parameter, d/R as transport form
parameter, d/Das sediment parameter, and λs 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.
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9.54
5.9
D
d
92.4
D
d
D
d
λ
C
exp
132.42
D
d
λ
C
67
D
d
C
D
d
R
d
4.23
λ
15.45
D
d
C
33.1
R
d
C
R
d
λ
D
d
R
d
89.66
D
d
r
F
2
s
V
s
V
V
s
V
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s
(15)
We can rewrite the above-mentioned formula as follows:
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0.62
D
d
92.4
D
d
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d
λ
C
exp
C
67
D
d
132.42
D
d
λ
C
D
d
R
d
4.23
λ
15.45
D
d
C
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d
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89.66
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d
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2
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2
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(16)

Fig. 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
Cv d/R d/D λs 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.
Fig. 2. Comparison of GEP result for both of train and test stages with actual values.

Figure 3 compares the prediction results of Fr by 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 of 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 R2
= 0.99, MAPE= 2.82
and RMSE= 0.14 is minimum in an amount in comparison to other equations presented in this
table. The equation presented by Ab. Ghani and Azamathulla [41] is less accurate (R2
= 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 of
less than 10%. Also, it indicates that some of the Fr forecasted by this equation have a relative
error of more than 30 percent which indicates the uncertainty of the equation presented by Ab
Ghani and Azamathulla [41]. Therefore, using this equation for the purpose of estimating Fr
cannot be that much confidence. Ebtehaj et al. [47] equation are 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. [17] 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 [15] 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
(R2
= 0.93, MAPE= 5.74 and RMSE= 0.24) indicates lesser estimation accuracy of this equation
in comparison with that of the presented equation.

Fig. 3. Comparison of proposed equation and existing equations.
Fig. 4. Error distribution of for GEP and existing equations.

Table 6
Validation of proposed equation and existing equations with statistical indexes.
Equation R2
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 (Dgr 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 Dgr
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 inside weir. The result of the GEP
model for variations of the discrepancy ratio (DR) values is plotted versus the dimensionless
particle number (Dgr) 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).
Fig. 5. DR values versus Dgr for GEP model.

Based on explanations given in 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 inflow
transport through sewerage networks. Different methods have been presented for sediment
transport in sewage, 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 the 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 D2
/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 Dgr as “sediment” parameter. Therefore, it was revealed
that the model which considers volumetric sediment concentration (CV), relative flow depth
(d/R), proportional average size of particles (d/D), overall friction factor (λs) for Fr estimation,
shows the best results. The presented model estimates Fr with an average error value of 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 an inappropriate use of GEP
functions such as Eq. (4) results in weak performance of the 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 the 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 a realistically valuable technique
for practical engineering as an alternative for existing methods.
Notation
A Cross-sectional area of the flow
CV Volumetric sediment concentration
D Pipe diameter
d Median diameter of particle size
Dgr 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 (=ρs/ρ)
V Velocity of flow
Vt Incipient flow velocity which follows from equation (2)
y Flow depth
λc Clear Water friction factor
λs Overall sediment friction factor
ψ Flow parameter
φ Transport parameter
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