a predictive model on air entrainment by plunging water jets using gep and ann

KSCE Journal of Civil Engineering (2014) 18(1):304-314

Copyright ⓒ2014 Korean Society of Civil Engineers

DOI 10.1007/s12205-013-0210-7

− 304 −

Water Engineering

pISSN 1226-7988, eISSN 1976-3808

www.springer.com/12205

A Predictive Model on Air Entrainment by Plunging Water Jets

using GEP and ANN

Tamer Bagatur* and Fevzi Onen**

Received April 26, 2012/Revised December 10, 2012/Accepted February 20, 2013/Published Online November 1, 2013

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Abstract

Plunging water jet flow situations are frequently encountered in nature and environmental engineering. A plunging liquid jet hasthe ability to provide vigorous gas-liquid mixing and dispersion of small bubbles in the liquid, and enhances mass transfer rate byproducing larger gas-liquid interfacial area. This process is called air-entrainment or aeration by a plunging water jet. Advances infield of Artificial Intelligence (AI) offer opportunities of utilizing new algorithms and models. This study presents Artificial NeuralNetwork (ANN) and Gene-Expression Programming (GEP) model, which is an extension to genetic programming, as an alternativeapproach to modeling of volumetric air entrainment rate by plunging water jets. A new formulation for prediction of volumetric airentrainment rate by plunging water jets using GEP is developed. The GEP-based formulation and ANN approach are compared withexperimental results, Multiple Linear/Nonlinear Regressions (MLR/NMLR) and other equations. The results have shown that theboth ANN and GEP are found to be able to learn the relation between volumetric air entrainment rate and basic water jet properties.Additionally, sensitivity analysis is performed and it is found that nozzle diameter is the most effective parameter on the volumetricair entrainment rate among water jet velocity, jet length and jet impact angle.

Keywords: water jet, air entrainment, modeling, gene-expression programming, artificial neural network

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1. Introduction

A plunging water jet passing through the surrounding air

entrains a large amount of air bubbles into a pool and forms a

large submerged two-phase (gas-liquid) contacting area (Fig. 1).

A plunging liquid jet has the ability to provide vigorous gas-

liquid mixing and dispersion of small bubbles in the liquid, and

enhances mass transfer rate by producing larger gas-liquid

interfacial area. Thus, this principle can be applied to introduce

air into a body of water regulate the oxygen content and

effectuate mixing. Air entrainment process by plunging water jet

is frequently encountered in nature and engineering.

In hydraulic structures, it is often a primary cause of air

entrainment (e.g., venturi, weirs, siphons and spillways). Also,

plunging water jets are encountered in a wide variety of

industrial and environmental situations. In many industrial

processes, plunging water jets are used such as those involving

air bubble flotation (e.g., mineral, oil and grease). In sewage and

water treatment plants, water jet aeration system is applied to

activated sludge treatment of livestock wastewater such as

domestic wastewater.

Various researchers have been reported aeration process and

oxygen transfer studies by conventional plunging water jets. Van

de Sande and Smith (1973) studied surface entrainment of air

*Associate Professor, Civil Engineering Dept., Dicle University, Diyarbakir 21280, Turkey (E-mail: [email protected])

**Assistant Professor, Civil Engineering Dept., Dicle University, Diyarbakir 21280, Turkey (Corresponding Author, E-mail: [email protected])

Fig. 1. Photograph of Air Bubble Entrainment by Plunging Water

Jet (VN = 4.0 m/s, DN = 0.0125 m, Lj = 0.05 m)

A Predictive Model on Air Entrainment by Plunging Water Jets using GEP and ANN

Vol. 18, No. 1 / January 2014 − 305 −

by high velocity water jets. McCarty and Molloy (1974)

reviewed of stability of liquid jets and the influence of nozzle

design. Van de Sande and Smith (1975) investigated mass transfer

from plunging water jets. Then, Van de Sande and Smith

(1976) discussed jet break up and air entrainment by low

velocity turbulent water jets. Avery and Novak (1978) studied

oxygen transfer at hydraulic structures. Jennekens (1979) discussed

water jet technique according to aeration and mixing. Djkstra

et al. (1979) concentrated on development and application of

water jet aeration for wastewater treatment. McKeogh and

Elsawy (1980) studied air entrainment in pool by plunging

water jet. McKeogh and Ervine (1981) investigated air entrainment

rate and diffusion pattern of plunging liquid jets. Tojo and

Miyanami (1982) studied oxygen transfer in jet mixers. Tojo et

al. (1982) investigated oxygen transfer and liquid mixing

characterizes of plunging jet reactors. Ohkawa et al. (1986a)

studied some flow characteristics of a vertical liquid jet system

having downcomers. Also, Ohkawa et al. (1986b) studied

flow and oxygen transfer in a plunging water system using

inclined short nozzles and performance characteristics of its

system in aerobic treatment of wastewater. Nakasone (1987)

studied aeration at weirs and cascades. Sene (1988) investigated air

entrainment by plunging jets. Detsch and Sharma (1990)

studied the critical angle for gas bubble entrainment by

plunging liquid jets. A critical review of the various aspects of

the gas entrainment by plunging jets is given by Bin (1993).

Evans et al. (1996) studied free jet expansion and gas entrainment

characteristics of a plunging liquid jet. Cummings and Chanson

(1997) investigated air entrainment in the developing flow

region of plunging jets. Bagatur et al. (2002) studied the effect

of nozzle type on air-entrainment by plunging water jets.

Bagatur and Sekerdag (2003) studied air entrainment characteristics

in a plunging water jet system using rectangular nozzles with

rounded ends. Baylar and Emiroglu (2003) investigated air

entrainment and oxygen transfer in a venturi. Emiroglu and

Baylar (2003) studied role of nozzles with air holes in air

entrainment by a water jet. Chanson et al. (2004) investigated

physical modelling and similitude of air bubble entrainment at

vertical circular plunging jets. Deswal and Verma (2007)

studied air-water oxygen transfer with multiple plunging jets.

Also, Deswal and Verma (2008) concentrated on performance

evaluation and modeling of a conical plunging jets aerator.

Kiger and Duncan (2012) reviewed air entrainment mechanisms in

plunging jets and breaking waves.

Advances in field of Artificial Intelligence (AI) offer opportunities

of utilizing new algorithms and models (Taskiran, 2010). Artificial

intelligence is the area of computer science focusing on creating

machines that can engage on behaviors that humans consider

intelligent. In the past few years, the applications of artificial

intelligence methods have attracted the attention of many

investigators. Many artificial intelligence methods have been

applied in various areas of civil, geotechnical and environmental

engineering. Hsu et al. (1995) used artificial neural network

modeling in the rainfall-runoff process. Dorado et al. (2003)

studied on prediction and modeling of the rainfall-runoff

transformation of a typical urban basin using Artificial Neural

Network (ANN) and Genetic Programming (GP). Ferreira (2001)

suggested gene-expression programming as a new adaptive

algorithm for solving problems. Habib and Meselhe (2006) used

Artificial Neural Networks (ANNs) and nonparametric regression

to develop stagedischarge relationships for coastal low-gradient

streams. Guven and Gunal (2008) used genetic and gene-

expression programming approach for estimating of local scour

in downstream of hydraulic structures. Guven and Aytek (2009)

presented new approach for stage-discharge relationship with

gene-expression programming. Eldrandaly (2009) studied integrating

gene expression progra-mming and geographic information

systems for solving a multi site land use allocation problem.

Azamathulla et al. (2011) used gene-expression programming

for the development of a stage-discharge curve of the Pahang

River. Baylar et al. (2011) predicted oxygen transfer efficiency

of cascades using GEP modeling. Kisi et al. (2012) used soft

computing approaches (such as ANN and GEP) for prediction of

lateral outflow over triangular labyrinth side weirs under

subcritical conditions.

The objective of this current study is to develop a new model

for prediction of volumetric air entrainment rate (QA) by plunging

water jets using ANN and GEP models. The performance of the

proposed GEP and ANN models is compared with experimental

results, Multiple Linear/Nonlinear Regressions (MLR/NMLR)

and other equations.

2. Genetic Expression Programming (GEP)

Gene Expression Programming (GEP) is an algorithm based

on Genetic Algorithms (GA) and Genetic Programming (GP).

This algorithm develops a computer program encoded in linear

chromosomes of fixed-length. The main aim of GEP is to

develop a mathematical function using a set of data presented to

GEP model. For the mathematical equation the GEP process

performs the symbolic regression by means of the most of the

genetic operators of GA. The flowchart of GEP is given in Fig. 2.

The process starts with the generation of the chromosomes of a

certain number of individuals (initial population). Then these

chromosomes are expressed and the fitness of each individual is

evaluated against a set of fitness cases. Then, the individuals are

selected according to their fitness to reproduce with modification.

These new individuals are subjected to the same developmental

processes such as expression of the genomes, confrontation of

the selection environment, selection, and reproduction with

modification. The process is repeated for a certain number of

generations or until a good solution is found (Ferreira, 2001;

2004; 2006).

3. GEP Model Components

GEP is a new evolutionary AI technique developed by Ferreira

(2001). This technique is an extension of GP, developed by Koza

Tamer Bagatur and Fevzi Onen

− 306 − KSCE Journal of Civil Engineering

(1992). The two main elements of GEP are the chromosomes

and Expression Trees (ETs). The chromosomes may be consisted

of one or more genes which represents a mathematical

expression. The mathematical code of a gene is expresses in two

different languages called Karva Language such as the language

of the genes and the language of the Expression Trees (ET). The

GEP genes composed of two parts called the head and tail. The

head includes some mathematical operators, variables and

constants (+, −, *, /, sin, cos, 1, a, b, c) and they are used to

encode a mathematical expression. Terminal symbols which

are variables and constants (1, a, b, c) are included in the tail.

If the terminal symbols in the head are inadequate to explain a

mathematical expression, additional symbols are used. A

simple chromosome as liner string with one gene is encoded

in Fig. 3. Its ET and the corresponding mathematical equation

are also shown in same figure. The translation of expression

tree to Kara Language is performed by starting to read from

left to right in the top line of the tree and from top to bottom.

The sequences of genes used in this method are similar to

sequences of biological genes and have coding and non-

coding parts.

In GEP, the main operators are the selection, transposition, and

cross-over (recombination). The chromosomes are modified to

get better fitness score for the next generation by means of these

operators (Ferreira, 2006). At the beginning of the model

constructions, the operator rates which are specified show a

certain probability of a chromosome. In common, recommended

mutation rate is ranging from 0.001 to 0.1. Furthermore,

recommended transposition operator and cross-over operator are

to be 0.1, 0.4, respectively.

4. Artificial Neural Networks (ANN)

ANN is a flexible mathematical structure which is capable of

identifying complex nonlinear relationships between input and

output data sets. ANN models have been found useful and

efficient, particularly in problems for which the characteristics of

the processes are difficult to describe using physical equations

(Hsu et al., 1995). Due to difficulties in solutions of the complex

engineering systems, researchers have started to study on ANN

inspired by the behavior of human brain and nervous system.

Each ANN model can be differently organized according to the

same basic structure. There are three main layers in ANN

structure; a set of input nodes, one or more layers of hidden

nodes, and a set of output nodes (Freeman and Skupura, 1991).

Each layer basically contains a number of neurons working as an

independent processing element and densely interconnected with

each other. The neurons using the parallel computation algorithms

are simply compiled with an adjustable connection weights,

summation function and transfer function. The methodology of

ANNs is based on the learning procedure from the data set

presented it from the input layer and testing with other data set

for the validation. A network is trained by using a special

learning function and learning rule. In ANNs analyses, some

function called learning functions is used for initialization;

training, adaptation and performance function. During the

training process, a network is continuously updated by a training

function which repeatedly applies the input variables to a

network till a desired error criterion is obtained. Adapt functions

is employed for the simulation of a network, while the network is

updated for each time step of the input vector before continuing

the simulation to the next input. Performance functions are used

to grade the network results. In this study, Gradient descent with

momentum and adaptive learning rate (traingdx), gradient

descent with momentum weight and bias learning function

(learngdm) and Mean Square Error (MSE) were used for training

function; adapt function and performance function, respectively. In

the learning stage, network initially starts by randomly assigning

Fig. 2. Algorithm of GEP Technique (Teodorescu and Sherwood,

2008)

Fig. 3. Chromosome with One Gene and Its Expression Tree and

Corresponding Mathematical Equation (Kayadelen et al.,

2009)


Vol. 18, No. 1 / January 2014 − 307 −

the adjustable weights and threshold values for each connection

between the neurons in accordance with selected ANNs

model. After the weighted inputs are summed and added the

threshold values, they are passed through a differentiable

non-linear function defined as a transfer function. This

process is continued, until a particular input captures to their

output (i.e., target) or as far as the lowest possible error can be

obtained by using an error criterion.

An ANN model can be differently composed in terms of

architecture, learning rule and self organization (Freeman and

Skupura, 1991). The most widely used ANNs are the feed-

forward, multilayer perceptions trained by back-propagation

algorithms based on gradient descent method (FFBP). This

algorithm can provide approximating to any continuous function

from one finite-dimensional space to another for any desired

degree of accuracy. The superiority of FFBP is that it sensitively

assigns the initial weights values and therefore it may yield

closer results than the each other. Also this algorithm has easier

application and shorter training duration.

5. Data Collection

In this study, some typical experimental data were preferred

due to different features of studies. Thus, experimental measurements

of Vande Sande (1975, 1976) (14 data sets), Ohkawa et al.

(1986a) (127 data sets), Bagatur et al. (2002) (14 data sets),

Bagatur and Sekerdag (2003) (4 data sets), Baylar and Emiroglu

(2003) (6 data sets), Emiroglu and Baylar (2003) (18 data sets)

are used as training and testing sets of the proposed GEP. The

80% of the data (144 data) were used to training. The remaining

20% of the data (35 data) were extracted at random and used for

the test stage. The proposed GEP formulation is trained with

these experimental data taken from six different experimental

studies (Table 1).

6. The Volumetric Air Entrainment Rate (Qa)

The great variety of physical mechanisms involved in the air-

entrainment process by a plunging liquid jet makes any

quantitative prediction in terms of basic control variables: the

nozzle diameter, the liquid jet flow rate, the falling height of the

plunging jet, as well as the physical properties of the fluids used.

In general, the volumetric air entrainment rate (QA) is influenced

by the variables characterizing water jet velocity (VN), nozzle

diameter (DN), jet length (Lj) and jet impact angle (Van de Sande

and Smith, 1976; Bin, 1993). Fig. 4 illustrates air entrainment

depend on parameters of water jet system.

The following relationship describes the volumetric air entrainment

rate as a function of its independent parameters:

(1)

Figure 5 shows the typical effects of jet velocity on the

volumetric air entrainment rate (QA) in which QA is plotted versus

VN. The increase in QA with increases in VN may be described to

the increased momentum of the jet flow. In addition, Fig. 5

emphasizes the existence of the various air entrainment regions.

Air entrainment curve is divided approximately into three

regions. These regions are defined as: i) The low velocity region

(region I) ii) The normal velocity region (region II) iii) The high

velocity region (region III).

Van de Sande and Smith (1974, 1975, and 1976) used four

main parameters: water jet velocity, jet length, nozzle

diameter and jet impact angle as (Sinθ). Thus, the general

form of the relationship between air entrainment and the input

parameters stated in the above mentioned reviews can be

written as:

(2)

Where,

DN = Nozzle diameter, m

Lj = Jet length, m

QA = Volumetric air entrainment rate, m3/s

VN = Flow velocity, m/s

QA F VN DN Lj θ, , ,( )=

QA aVN

bDN

cLj

dsinθ

e=

Table 1. Experimental Studies for Water Jet System

AuthorNozzle velocity (VN),

m/sNozzle diameter (DN),

mJet length

(Lj), mJet impact angle (θ),

deg.

1 Vande Sande (1975, 1976) 3.15-13.76 0.0039-0.012 0.100-0.200 300

2 Ohkawa (1986a) 2.0-14.3 0.007-0.013 0.025-0.750 900

3 Bagatur et al. (2002) 2.0-13.0 0.008 0.150 450

4 Bagatur and Sekerdag (2003) 2.0-13.0 0.0047-0.0075 0.150 450

5 Baylar and Emiroglu (2003) 2.5-15.0 0.020 0.300 600

6 Emiroglu and Baylar (2003) 2.5-15.0 0.010-0.0195 0.300 450

Fig. 4. Air Entrainment by a Plunging Jet and Typical System

Parameters: (a) Inclined Jet, (b) Vertical Jet



The volumetric air entrainment rate (QA) is estimated by the

following equations for air entrainment regions (Ohkawa et al.,

1986a):

In region I (DN = 0.007 − 0.011 m, VN = 2.0 − 5.0 m/s

and Lj = 0.025 − 0.750 m),

(3)

In region II (DN = 0.007 − 0.011 m, VN = 5.0 − 10.0 m/s),

(4)

In region III (DN = 0.007 − 0.011 m, VN = 10.0 − 14.3 m/s),

(5)

In region I (DN = 0.011 − 0.013 m, VN = 2.0 − 5.0 m/s),

(6)

In region II (DN = 0.011 − 0.013 m, VN = 5.0 − 10.0 m/s),

(7)

In region III (DN = 0.011 − 0.013 m, VN = 10.0 − 13.2 m/s),

(8)

These equations agree with the experimental data with an

accuracy of ±30% for QA. Van de Sande and Smith (1976)

derived from experimental studies following equation with

an accuracy of ±10−20% for region I of air entrainment (DN

= 0.0039 − 0.012 m, VN = 3.125 − 5.0 m/s and Lj = 0.10−

0.20 m):

(9)

7. GEP model development

To generate the mathematical function for the prediction of air

entrainment by water jet was the main aim of development of GEP

models. For that reason, a development of GEP model was

realized. The GEP model has three input parameters (flow velocity,

nozzle diameter and jet length). The parameters of GEP models are

presented in Table 2. There are five major steps in preparing to use

gene expression programming, and the selection of the fitness

function is the first step. For this problem, we measured the fitness fiof an individual program i by the following expression:

(10)

Where,

C (i, j) = Value returned by the individual chromosome i for

fitness case j (out of Ct fitness cases)

M = Range of selection

Tj = Target value for fitness case j

If (the precision) is less than or equal to 0.01, then

the precision is equal to zero, and fi = fmax = CiM. For our case, we

used an M = 100 and, therefore, fmax = 1,000. The advantage of

this kind of fitness function is that the system can find the

optimal solution for itself (Ferreira, 2001).

The second major step consists in choosing the set of terminals

T and the set of function F to create the chromosomes. In this

problem, the terminal set consists obviously of the independent

variable, i.e. QA = f{VN, DN, Lj, θ}. The choice of the appropriate

function set is not so obvious, but a good guess can always be done

to include all the necessary functions. In this case, we used the four

basic arithmetic operators (+, −, *, /), and some basic mathematical

functions (√, 1/x, x2, x3, x1/3, x1/4, x1/5). The third major step is to

choose the chromosomal architecture, i.e. the length of the head and

the number of genes. In this study, it is firstly used a length of the

head, h = 8 and a gene per chromosome. Secondly, it is used a

length of the head, h = 8 and two genes per chromosome. The

fourth major step is to choose the linking function. In this case, we

QA 6.96 102–VN

2.3DN

1.9Lj

0.5=

QA 3.03 101–VN

1.4DN

1.9Lj

0.5=

QA 6.70 102–VN

2DN

1.9Lj

0.5=

QA 6.18 105–VN

2.3DN

1 3⁄Lj

0.5=

QA 2.48 104–VN

1.4DN

1 3⁄Lj

0.5=

QA 5.83 105–VN

2DN

1 3⁄Lj

0.5=

QA 2.10 102–VN

2DN

1.5Lj

0.33=

fi M C i j,( ) Tj––( )j 1=

cj

∑=

C i j,( ) Tj–

Fig. 5. Variation between the Volumetric Air Entrainment Rate and

Water Jet Velocity (Bagatur et al., 2002)

Table 2. Parameters of the Optimized GEP Model

Parameter Description of parameterSetting of parameter

P1 Chromosomes 30

P2 Fitness function error type R2

P3 Number of the genes 1-3

P4 Head size 8

P5 Linking function *

P6 Function set +, −, *, /, √, x2

P7 Mutation rate 0.044

P8 One-point recombination rate 0.3

P9 Two-point recombination rate 0.3

P10 Inversion rate 0.1

P11 Transposition rate 0.1


Vol. 18, No. 1 / January 2014 − 309 −

linked the sub-ETs by multiplication. And finally, the fifth major

step is to choose the set of genetic operators that cause variation and

their rates. It is used a combination of all genetic operators

(mutation, transposition, and recombination) with parameters of the

optimized GEP model (Guven and Gunal 2008). Fig. 6 shows the

ET of the formulation which actually is:

dblTemp = ((((d[1] × d[1]) × (d[0]−d[2])) × pow(d[2],

(1.0/5.0))) × pow((d[0] × d[2]), (1.0/4.0))) (11)

In the GEP formulation (Model 1), the actual parameters are

d[0] = VN, d[1] = DN and d[2] = Lj. After putting the corresponding

values, the final equation becomes (R2 = 0.94):

QA=VN0.25 DN

2 Lj0.45 (VN−Lj) (12)

Where,


Lj= Jet length, m

QA = volumetric air entrainment rate, m3/s


For two genes per chromosome, Fig. 7 shows the ET of the

formulation which actually is:

dblTemp=(d[1] × (d[0]−(d[0] × ((G1C1 − d[2]) − d[1]))))

× (((sqrt(d[2]) × d[2])-d[2]) × (pow(d[0],

(1.0/3.0)) × d[1])) (13)

The constant in the GEP formulation (Model 2) is G1C1 =

4.75 and the actual parameters are d[0] = VN, d[1] = DN and d[2]

= Lj. After putting the corresponding values, the final equation

becomes (R2 = 0.95):

QA=VN1.33 DN

2 (Lj1.5−Lj)−(Lj+DN−3.75) (14)

Where,


Lj = Jet length, m



For two genes per chromosome, Fig. 8 shows the ET of the

formulation which actually is:

dblTemp=((pow((d[0]+G1C0),2)/(sqrt(d[0])+d[2]))+sqrt((d[0]/d[2])))

× (pow(d[1], 2)/((pow(d[3], 3)+(d[2]+d[2]))+(d[3]/d[2])))

(15)

The constant in the GEP formulation (Model 3) is G1CO =

0.516, G1C1 = 4.49, G2C0 = -8.47, G2C1 = -8.46, G3CO = 3.25

and the actual parameters are d[0] = VN, d[1] = DN, d[2] = Lj and

d[3] = Sinθ. After putting the corresponding values, the final

equation becomes (R2 = 0.97):

(16)QA

0.516 VN+( )2

VN

0.15Lj+( )

-------------------------------VN

Lj

------⎝ ⎠⎛ ⎞

0.5

+DN

sinθ32Lj sinθ Lj⁄+ +

-----------------------------------------------=

Fig. 6. Expression Tree (ET) for the Proposed GEP Formulation

(Eq. 11)


(Eq. 13)


(Eq. 15)



Where,


Lj = Jet length, m



It should be noted that the proposed GEP formulations (Eqs.

12 and 16) are valid for the ranges of DN = 0.0039 − 0.020 m, VN =

2.0 − 15.0 m/s, Lj = 0.025 − 0.750 m and θ = 30 − 90o in estimating

the volumetric air entrainment rate (QA).

8. Training and Testing Results of GEP Modeling

The training and testing patterns of the proposed GEP

formulation are based on well established and widely dispersed

experimental results from the literature. The prediction of the

proposed GEP formulation versus experimental values for training

sets is given in Figs. 9 and 10. As seen from figures, GEP model

prediction is well-agreed with the experimental data.

Also, the prediction of the proposed GEP formulation

versus experimental values for testing sets is given in Figs.

11 and 12.

The performance of ANN and GEP in training and testing sets

is validated in terms of the common statistical measures R2

(coefficient of determination), Mean-Square Error (MSE) and

Root-Mean-Square Error (RMSE).

(17)

(18)

(19)

Where,

n = Number of samples

Qo = Observed values

Qom= Mean of Qo

Qp = Predicted value

Qpm= Mean of Qp

Qx = (Qo−Qom)

Qy = (Qp − Qpm)

R2 Qx∑ Qy

Qx

2Qy

2

∑∑

-------------------------------⎝ ⎠⎜ ⎟⎛ ⎞ 2

=

MSEQo Qp–( )2

∑n

-----------------------------=

RMSEQo Qp–( )2

∑n

-----------------------------1 2⁄

=

Fig. 9. The Measured and Estimated Volumetric Air Entrainment

Rate in the Train Period

Fig. 10. Comparison between Estimated (by GEP) and Measured

Values of QA for Train Set


Rate in the Test Period

Fig. 12. Comparison between Estimated (by GEP) and Measured

Values of QA for Test Set


Vol. 18, No. 1 / January 2014 − 311 −

Table 3 compares the GEP models, with one of the independent

parameters removed in each case, and deleting any independent

parameter from the input set yielded larger RMSE and lower R2

values. These three independent parameters have non-negligible

influence on QA and so the functional relationship given in eqs. (1)

and (2) is used for GEP modeling in this study. The performance of

proposed GEP model (Model 3) showed a high generalization

capacity with MSE = 0.0000000410, RMSE = 0.000202 and

R2 = 0.97.

9. Results of ANN Modeling

For this study 20% of the data (35 data) were extracted at

random and used for the test stage. The remaining 80% of the

data (144 data) were used to train the neural network. Testing

data set were not used during development of the network, so

they could form a good indicator to test the accuracy of the

developed network. The feed-forward neural networks which

consist of multilayer perceptions trained back-propagation

algorithms were employed for this study. In Fig. 13 and Fig. 14,

the architecture of the ANN model formed by using feed-

forward artificial neural networks with three-four inputs is

shown. To evaluate the results of the developed ANN model, the

coefficient of determination (R2), the Mean-Square Error (MSE)

and Root-Mean-Square Error (RMSE) were used as statistical

verification tool.

Estimated volumetric air entrainment rate values obtained with

ANN model are graphically compared with the measured

Table 3. Sensitivity Analysis for Independent Parameters for GEP

Model number Model type MSE RMSE R2

GEP Model 1(1 gene)

QA = f (VN, DN, Lj) 0.0000000863 0.000293 0.94

GEP Model 2(2 genes)

QA = f (VN, DN, Lj) 0.0000000810 0.000284 0.95

GEP Model 3 QA = f (VN, DN, Lj, sin) 0.0000000410 0.000202 0.97

GEP Model 4 QA = f (DN, Lj) 0.000000772 0.000838 0.53

GEP Model 5 QA = f (VN, Lj) 0.000000531 0.000729 0.64

GEP Model 6 QA = f (VN, DN) 0.000000110 0.000332 0.93

Fig. 13. The Architecture of Feed-Forward Artificial Neural Net-

works with Three Inputs

Fig. 14. The Architecture of Feed-Forward Artificial Neural Net-

works with Four Inputs


Rate in the Test Period (by ANN)

Fig. 16. Comparison between Estimated (by ANN) and Measured

Values of QA for Test Set



volumetric air entrainment rate values in Fig. 15 and Fig. 16. As

seen from figures, ANN model prediction is well-agreed with the

experimental data.

The statistical performances of both ANN and GEP models are

summarized in Table 4. As far as Table 4 is concerned, satisfactory

agreement between the model predictions and experimental data

is observed for of models.

The ANN simulations were conducted using a program code

written in MATLAB language. The appropriate model structure

was determined after trying different ANN architectures. The

ANN models with four and five hidden layers are used. The

sigmoid and linear activation functions are used for the hidden

and output nodes. Sensitivity analysis is also performed to see

how much the input parameters are effective on output the

volumetric air entrainment rate parameter (Fig. 17). As can be

seen from Fig. 17, nozzle diameter is found to be the most

effective parameter on the volumetric air entrainment rate and

then water jet velocity and jet length, respectively.

10. Comparison of ANN and GEP with Regres-sion Model and other Equations

Multiple Linear Regressions (MLR) generalize the simple linear

regression model by allowing for many terms in a mean function

rather than just one intercept and one slope. The general Multiple

linear regression (MLR) models with response y and terms x1,…,

xi have the form:

(20)

Where,

the bi = Model coefficients

xi = Independent parameters

y = Predicted variable

The model coefficients (bi) were calculated by least square

regression. Multiple Linear Regression (MLR) analysis yielded

the following equations:

(21a)

(21b)

Multiple Non-Linear (NMLR) functions of interest are as

follows:

Y = aX1b1X2

b2… Xk bk (22)

Nonlinear relationships between the dependent variables and

the independent variables based on multivariate power function

were considered. The following equations were derived:

(23a)

(23b)

Table 5 shows comparison of ANN and GEP with regression

model and other equations for predicting of volumetric air

entrainment rate in air entrainment regions. The superior

performance of GEP, compared to other methods, is attributed to

the powerful Artificial Intelligence (AI) techniques for computer

learning inspired by natural evolution to find the appropriate

mathematical model (expression) to fit a set of fits.

11. Conclusions

A new model for prediction of volumetric air entrainment rate

by plunging water jets using Gene Expression Programming

(GEP) and ANN has been developed for air entrainment regions

in the present study. The GEP-based formulation and ANN

approach are compared with experimental results, Multiple

Linear/Nonlinear Regressions (MLR/NMLR) and other equations.

y b0 b1x1 … bixi+ + +=

QA 0.000695 0.000131VN 0.00449DN 0.00101Lj+ + +–=

QA 0.00120 0.000127VN 0.205DN 0.000808Lj+ + +–=

0.00143sinθ–

QA 0.63VN

1.38DN

2Lj

0.35=

QA 0.47VN

1.25DN

1.89Lj

0.43sinθ 1.09–

=

Fig. 17. Sensitivity Analysis of ANN for Volumetric Air Entrainment

Rate

Table 4. Statistical Performance of ANN/GEP Models

Model MSE RMSE R2

GEP model 3 0.0000000410 0.000202 0.97

ANN model 1 0.000000104 0.000324 0.93

ANN model 2 0.000000070 0.000259 0.96

Table 5. Comparison of GEP/ANN with Regression Model and

Other Equations

Method R2 Entrainment region

Eq. (3) 0.57 Region I

Eq. (4) 0.73 Region II

Eq. (5) 0.93 Region III


Eq. (7) 0.94 Region II

Eq. (8) 0.94 Region III


GEP model 3 (eq. 16) 0.97 Region I, II, III

ANN model 2 0.96 Region I, II, III

MLR (eq. 21a) 0.63 Region I, II, III (3 parameters)

MLR (eq. 21b) 0.75 Region I, II, III (4 parameters)

NMLR (eq. 23a) 0.93 Region I, II, III (3 parameters)

NMLR (eq. 23b) 0.95 Region I, II, III (4 parameters)


Vol. 18, No. 1 / January 2014 − 313 −

The results have shown that the both ANN and GEP are found to

be able to learn the relation between volumetric air entrainment

rate and basic water jet properties. Additionally, sensitivity

analysis is performed and it is found that nozzle diameter is the

most effective parameter on the volumetric air entrainment rate

among water jet velocity, jet length and jet angle. The proposed

GEP formulation (Eq. 16) is valid for the ranges of DN = 0.0039 −

0.020 m, VN=2.0 − 15.0 m/s, Lj = 0.025 − 0.750 m and θ= 30 − 90o

for estimating the volumetric air entrainment rate (QA). The overall

results support the use of GEP and ANN as an effective tool for

forecasting and estimating of the volumetric air entrainment rate.

References

Avery, S. T. and Novak, P. (1978). “Oxygen transfer at hydraulic

structures.” J. Hydr. Div. Proc., ASCE, Vol. 104, No. HY11, pp.

1521- 1540.

Azamathulla, H. M., Ghani, A. A., Leow, C. S., and Chang, C. K.

(2011). “Gene-Expression programming for the development of a

stage-discharge curve of the Pahang river.” Water Resour. Manage.,

Vol. 25, No. 11, pp. 2901-2916.

Bagatur, T., Baylar, A., and Sekerdag, N. (2002). “The effect of nozzle

type on air-entrainment by plunging water jets.” Water Qual. Res. J.

Canada, Vol. 37, No. 3, pp. 599-612.

Bagatur, T. and Sekerdag, N. (2003). “Air entrainment characteristics in

a plunging water jet system using rectangular nozzles with rounded

ends.” Water SA, Vol. 29, No. 1, pp. 35-38.

Baylar, A. and Emiroglu, M. E. (2003). “Air entrainment and oxygen

transfer in a venturi.” Proc. of the Inst. of Civil Eng-Water & Maritime

Eng., Vol. 156 (WM3), pp. 249-255.

Baylar, A., Unsal, M., and Ozkan, F. (2011). “GEP Modeling of oxygen

transfer efficiency prediction in aeration cascades.” KSCE J. of Civil

Eng., KSCE, Vol. 15, No. 5, pp. 799-804.

Bin, A. K. (1993). “Gas entrainment by plunging liquid jets.” Chem.

Eng. Sci., Vol. 48, No. 21, pp. 3585-3630.

Chanson, H., Aoki, S., and Hoque, A. (2004). “Physical modelling and

similitude of air bubble entrainment at vertical circular plunging

jets.” Chem. Eng. Sci., Vol. 59, No. 4, pp. 747-754.

Cummings, P. D. and Chanson, H. (1997). “Air entrainment in the

developing flow region of plunging jets-Part 1: Experimental.” J.

Fluids Eng., Vol. 119, No. 3, pp. 603-608.

Deswal, S. and Verma, D. V. S. (2007). “Air-water oxygen transfer with

multiple plunging jets.” Water Qual. Res. J. Canada, Vol. 42, No. 4,

pp. 295-302.

Deswal, S. and Verma, D. V. S. (2008). “Performance evaluation and

modeling of a conical plunging jets aerator.” Intn. J. Math. Physl.

Eng. Sci., Vol. 2, No. 1, pp. 33-37.

Detsch, R. M. and Sharma, R. N. (1990). “The critical angle for gas

bubble entrainment by plunging liquid jets.” Chem. Eng. J., Vol. 44,

No. 3, pp. 157-166.

Djkstra, F., Jennekens, H., and Nooren, P. (1979). “The development

and application of water jet aeration for wastewater treatment.”

Prog. Water Technol., Vol. 11, No. 2, p. 181.

Dorado, J., Rabunal, J. R., Pazos, A., Rivero, D., Santos, A., and

Puertas, J. (2003). “Prediction and modeling of the rainfall-runoff

transformationof a typical urban basin using ANN and GP.” Appl.

Artif. Intell., Vol. 17, No. 4, pp. 329-343.

Eldrandaly, K. (2009). “Integrating gene expression programming and

geographic information systems for solving a multi site land use

allocation problem.” American Journal of Applied Sciences, Vol. 6,

No. 5, pp. 1021-1027.

Emiroglu, M. E. and Baylar, A. (2003). “Role of nozzles with air holes

in air entrainment by a water jet.” Water Qual. Res. J. Canada, Vol.

38, No. 4, pp. 785-795.

Evans, G. M., Jameson, G. J., and Rielly, C. D. (1996). “Free jet expansion

and gas entrainment characteristics of a plunging liquid jet.” Exper.

Thermal and Fluid Science, Vol. 12, No. 2, pp. 142-149.

Ferreira. C. (2001). “Gene expression programming: A new adaptive

algorithm for solving problems.” Complex Systems, Vol. 13, No. 2,

pp. 87-129.

Ferreira, C. (2004). “Gene expression programming and the evolution

of computer programs.” Recent Developments in Biologically

Inspired Computing, L.N. de Castro, F.J. Zuben (Eds.), Idea Group

Publishing, pp. 82-103.

Ferreira, C. (2006). Gene-expression programming: Mathematical

modeling by an artificial intelligence, Springer, Berlin.

Freeman, J. A. and Skupura, D. M. (1991). Neural network algorithms,

applications and programming techniques, Addison-Wesley, New

York.

Guven, A. and Aytek, A. (2009). “A new approach for stage-discharge

relationship: Gene-expression programming.” J. Hydrologic Eng.,

Vol. 14, No. 8, pp. 812-820.

Guven, A. and Gunal, M. (2008). “Genetic programming approach for

prediction of local scour downstream hyraulic structures.” J. Irrig.

and Drain Eng., Vol. 134, No. 2, pp. 241-249.

Habib, E. H. and Meselhe, E. A. (2006). “Stage-discharge relations for

low-gradient tidal streams using data driven models.” J. Hydraul.

Eng., Vol. 132, No. 5, pp. 482-492.

Hsu, K., Gupta, H. V., and Sorooshian S. (1995). “Artificial neural

network modeling of the rainfall-runoff process.” Water Resour.

Res., Vol. 31, No. 10, pp. 2517-2530.

Jennekens, H. (1979). Water jet technique combines aeration and

mixing, Water and Sewage Works, pp. 71-73.

Kayadelen, C., Gunaydin, O., Fener, M., Demir, A., and Ozvan, A.

(2009). “Modeling of the angle of shearing resistance of soils using

soft computing systems.” Expert Systems with Applications, Vol. 36,

No. 9, pp. 11814-11826.

Kiger, K. T. and Duncan, J. H. (2012). “Air-Entrainment mechanisms in

plunging jets and breaking waves.” Ann Review of Fluid Mech, Vol.

44, No. 1, pp. 563-596.

Kisi, O., Emiroglu, M. E., Bilhan, O., and Guven, A. (2012). “Prediction

of lateral outflow over triangular labyrinth side weirs under subcritical

conditions using soft computing approaches.” Expert. Syst. Appl.,

Vol. 39, No. 3, pp. 3454-3460.

Koza, J. R. (1992). Genetic programming: On the programming of

computers by means of natural selection, MIT, Cambridge.

McCarty, M. J. and Molloy, N. A. (1974). “Review of stability of liquid

jets and the influence of nozzle design.” Chem. Eng. Sci., Vol. 7, No.

1, pp. 1-20.

McKeogh, E. J. and Elsawy, E. M. (1980). “Air retained in pool by

plunging water jet.” J. Hydr. Div., ASCE, Vol. 106, No. 10, pp.

1577- 1593.

McKeogh, E. J. and Ervine, D. A. (1981). “Air entrainment rate and

diffusion pattern of plunging liquid jets.” Chem. Eng. Sci., Vol. 36,

No. 7, pp. 1161-1172.

Nakasone, H. (1987). “Study of aeration at weirs and cascades.” J.

Envir. Eng., Vol. 113, No. 1, pp. 64-81.

Ohkawa, A., Kusabiraki, D., Kawai, Y., and Sakai, N. (1986a). “Some



flow characteristics of a vertical liquid jet system having downcomers.”

Chem. Eng. Sci., Vol. 41, No. 9, pp. 2347-2361.

Ohkawa, A., Kusabiraki, D., Shiokawa, Y., Sakai, N., and Fujii, M.

(1986b). “Flow and oxygen transfer in a plunging water system

using inclined short nozzles and performance characteristics of its

system in aerobic treatment of wastewater.” Biotechnol Bioeng., Vol.

28, No. 12, pp. 1845-1856.

Sene, K. J. (1988). “Air entrainment by plunging jets.” Chem. Eng. Sci.,

Vol. 43, No. 10, pp. 2615-2623.

Taskiran, T. (2010). “Prediction of California Bearing Ratio (CBR) of

fine grained soils by AI methods.” Advances in Eng. Software, Vol.

41, No. 6, pp. 886-892.

Teodorescu, L. and Sherwood, D. (2008). “High energy physics event

selection with gene expression programming.” Comp. Physics

Commun., Vol. 178, No. 6, pp. 409-419.

Tojo, K., Naruko, M., and Miyanami, K. (1982). “Oxygen transfer and

liquid mixing characterizes of plunging jet reactors.” Chem. Eng. J.,

Vol. 25, No. 2, pp. 107-109.

Tojo, K. and Miyanami, K. (1982). “Oxygen transfer in jet mixers.”

Chem. Eng. J., Vol. 24, Issue 1, pp. 89-97.

Van de Sande, E. and Smith, J. M. (1973). “Surface entrainment of air

by high velocity water jets.” Chem. Eng. Sci., Vol. 28, No. 5, pp.

1161-1168.

Van de Sande, E. and Smith, J. M. (1975). “Mass transfer from plunging

water jets.” Chem. Eng. J., Vol. 10, No. 3, pp. 225-233.

Van de Sande, E. and Smith, J. M. (1976). “Jet break up and air

entrainment by low velocity turbulent water jets.” Chem. Eng. Science,

Vol. 31, No. 3, pp. 219-224.

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