prediction of penetration depth in a plunging water jet using soft computing approaches

ORIGINAL ARTICLE

Prediction of penetration depth in a plunging water jet using softcomputing approaches

Fevzi Onen

Received: 19 September 2012 / Accepted: 22 August 2013

� Springer-Verlag London 2013

Abstract The flow characteristics of the plunging water

jets can be defined as volumetric air entrainment rate,

bubble penetration depth, and oxygen transfer efficiency.

In this study, the bubble penetration depth is evaluated

based on four major parameters that describe air entrain-

ment at the plunge point: the nozzle diameter (DN), jet

length (Lj), jet velocity (VN), and jet impact angle (h). This

study presents artificial neural network (ANN) and genetic

expression programming (GEP) model, which is an

extension to genetic programming, as an alternative

approach to modeling of the bubble penetration depth by

plunging water jets. A new formulation for prediction of

penetration depth in a plunging water jets is developed

using GEP. The GEP-based formulation and ANN

approach are compared with experimental results, multiple

linear/nonlinear regressions, and other equations. The

results have shown that the both ANN and GEP are found

to be able to learn the relation between the bubble pene-

tration depth and basic water jet properties. Additionally,

sensitivity analysis is performed for ANN, and it is found

that DN is the most effective parameter on the bubble

penetration depth.

Keywords Penetration depth � Genetic expression

programming (GEP) � Artificial neural network

(ANN) � Regression analysis

1 Introduction

It is well known that a water jet that after passing through a

gas layer (e.g., atmosphere) plunges into a pool of water at

rest, entrains into an important amount of air, carries a way

below the pool free surface, and forms a submerged two-

phase region. The plunging jet flow situations are

encountered in nature (e.g., at impact of waterfalls). In

hydraulic structures, it is often a primary cause of air

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

Plunging water jets are used in a wide variety of industrial

and environmental situations. In many industrial processes,

plunging water jets, such as those involving air bubble

flotation, are used (e.g., mineral, oil, and grease). In sewage

and water treatment plants, water jet aeration system is

applied to activated sludge treatment of livestock waste-

water such as domestic wastewater.

Various researchers have reported aeration process and

oxygen transfer studies by conventional plunging water

jets. Van de Sande and Smith [1] studied surface

entrainment of air by high-velocity water jets. McCarty

and Molloy [2] reviewed the stability of liquid jets and

the influence of nozzle design. Van de Sande and Smith

[3] investigated mass transfer from plunging water jets.

Then, Van de Sande and Smith [4] discussed jet breakup

and air entrainment by low-velocity turbulent water jets.

Avery and Novak [5] studied oxygen transfer at hydraulic

structures. Jennekens [6] discussed water jet technique

according to aeration and mixing. Djkstra et.al [7] con-

centrated on development and application of water jet

aeration for wastewater treatment. McKeogh and Elsawy

[8] studied air entrainment in pool by plunging water jet.

McKeogh and Ervine [9] investigated air entrainment rate

and diffusion pattern of plunging liquid jets. Tojo and

Miyanami [10] studied oxygen transfer in jet mixers.

F. Onen (&)

Civil Engineering Department, Engineering Faculty,

Dicle University, 21280 Diyarbakir, Turkey

e-mail: [email protected]

123

Neural Comput & Applic

DOI 10.1007/s00521-013-1475-y

Tojo et al. [11] investigated oxygen transfer and liquid

mixing characteristics of plunging jet reactors. Bin [12]

studied air entrainment by plunging liquid jets. Ohkawa

et al. [13] studied some flow characteristics of a vertical

liquid jet system having downcomers. Also, Ohkawa et al.

[14] 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. Sene [15] investigated air entrainment by

plunging jets. Detsch and Sharma [16] 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 [17].

Cummings and Chanson [18] investigated air entrainment

in the developing flow region of plunging jets. Bagatur

et al. [19] studied the effect of nozzle type on air

entrainment by plunging water jets. Bagatur and Sekerdag

[20] studied air entrainment characteristics in a plunging

water jet system using rectangular nozzles with rounded

ends. Emiroglu and Baylar [21] investigated air entrain-

ment and oxygen transfer in a venturi. Chanson et al. [22]

investigated physical modeling and similitude of air

bubble entrainment at vertical circular plunging jets.

Deswal and Verma [23] studied air–water oxygen transfer

with multiple plunging jets. Kiger and Duncan [24]

reviewed air entrainment mechanisms in plunging jets and

breaking waves.

Artificial neural network (ANN) is a mathematical

system having an interconnected assembly of simple

elements, which emulates the ability of biological neural

network. ANN technique can represent a complex non-

linear relationship between the input and the output of

any system. Many AI methods have been applied in

various areas of civil and environmental engineering. The

ANN method has been successfully used in modeling of

water resource systems in areas such as stream-flow

forecasting rainfall–runoff modeling, suspended sediment

modeling, and modeling combined open-channel flow.

ANNs have also been applied extensively in the field of

hydrology for estimation and forecasting of hydrologic

variables [25], [26]. The neural network approach has

been applied to many branches of science, including

aspects of hydraulic and environmental engineering. More

recent applications of ANNs in the field of hydraulic

engineering have been studied by Azamathulla et al. [27].

Genetic programming (GP) is a new technique used in the

field of water resource engineering. Ferreira [28] sug-

gested gene-expression programming as a new adaptive

algorithm for solving problems. Guven and Gunal [29]

used genetic and gene-expression programming approach

for estimating of local scour in downstream of hydraulic

structures. Guven and Aytek [30] presented a new

approach for stage–discharge relationship with gene-

expression programming. Azamathulla et al. [31] used

gene-expression programming for the development of a

stage–discharge curve of the Pahang River. Baylar et al.

[32] predicted oxygen transfer efficiency of cascades

using genetic expression programming (GEP) modeling.

Unsal [33] predicted penetration depth in sharp-crested

weirs using GEP modeling. Azamathulla [34] studied

gene-expression programming to predict friction factor for

southern Italian rivers. Kisi et al. [35] used soft com-

puting approaches (such as ANN and GEP) for prediction

of lateral outflow over triangular labyrinth side weirs

under subcritical conditions.

Advances in the field of artificial intelligence influence

many science topics as well as civil engineering applica-

tions. New algorithms and models, especially those based

on soft computing, enable researchers to solve the most

complex systems in different ways. The use of forecast

methods not based on physics equation, such as ANN and

GEP methods, are becoming widespread in various engi-

neering fields.

The objective of this study is to develop a new formu-

lation technique for the prediction of bubble penetration

depth (Hp) by plunging water jets using ANN and GEP

models. The performance of the proposed GEP and artifi-

cial neural network (ANN) models is compared with

experimental results, multiple linear/nonlinear regressions

(MLR/MNLR), and other equations. This paper is the first

study that investigates the accuracy of GEP and ANN in

the bubble penetration depth modeling based on system

parameters.

2 Bubble penetration depth

Penetration depth (HP) of the bubbles produced by the jet,

which was defined as the vertical distance from the water

surface to the lower end of the submerged biphasic region

in the water, was measured by a scale fitted to the tank wall

(Fig. 1). Bubbles entrained by a vertical plunging jet pen-

etrate the pool liquid to a maximum depth. This point is not

strictly defined since the lower limit of the bubble swarm

fluctuates continuously, but a time average can be esti-

mated. Several authors measured the maximum depth of

bubble penetration in the vertical or inclined plunging jet

systems. At the maximum depth of bubble penetration, the

local liquid velocity in the submerged jet at that point is

assumed to be equal to the bubble-free rise velocity. This

led to a direct linear relationship between the maximum

penetration depth and the product of the jet diameter and

flow velocity. Figure 2 shows effect of flow velocity on

bubble penetration depth.


123

Bin [12] mentions a simple pure empirical relationship

for the maximum penetration depth:

Hp ¼ 2:1V0:775N D0:67

N ð1Þ

Late, Bin [17] modified the earlier formula for the

maximum penetration depth:

Hp ¼ 2:4V0:66N D0:66

N ð2Þ

where Hp = bubble penetration depth, m; VN = flow

velocity, m/s; DN = nozzle diameter, m.

In this study, the following relationship describes the

bubble penetration depth as a function of its independent

variables:

Hp ¼ f VN;DN; Lj; h� �

ð3Þ


velocity, m/s; DN = nozzle diameter, m; Lj = jet length,

m; h = jet impact angle (degree).

3 Data collection

In the bubble penetration depth (Hp), experimental mea-

surements of Ohkawa et al. [13] (16 data sets), Bagatur

et al. [19] (20 data sets), Bagatur and Sekerdag [20] (12

data sets), Baylar and Emiroglu [21] (15 data sets) are used

as training (50 data sets) and testing sets (13 data sets) of

the proposed GEP, ANN, MLR, and MNLR methods.

The proposed GEP formulations are trained with these

experimental data taken from four different experimental

studies (Table 1).

4 Multiple linear regression

Multiple linear regression (MLR) is a method used to

model the linear relationship between a dependent variable

and one or more independent variables. The dependent

variable is sometimes also called the predictand, and the

independent variables the predictors. MLR is based on least

squares: The model is fit such that the sum of squares of

differences in observed and predicted values is minimized.

Multiple linear functions of interest are as follows:

Y ¼ aþ b1x1 þ b2x2 þ b3x3 ð4Þ

where Y is the value of the dependent variable; a is the

constant; b1 is the slope for X1; X1 is the first independent

variable that is explaining the variance in Y; b2 is the slope

for X2; X2 is the second independent variable that is

explaining the variance in Y; b3 is the slope for X3; X3 is the

third independent variable that is explaining the variance in

Y.

Values of root-mean-square error (RMSE) were used to

find the fitting degrees of linear models with experimental

Fig. 1 Air entrainment parameters of a plunging water jet system

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

Hp

(m)

VN (m/s)

Bagatur et al. (2002)

Baylar and Emiroglu (2003)

Ohkawa et al.(1986)

Bagatur and Sekerdag (2003)

Fig. 2 Effect of flow velocity on bubble penetration depth

Table 1 Experimental studies for plunging water jet system in present study

Authors Nozzle velocity, VN, m/s Nozzle diameter, DN, m Jet length, Lj, m Jet impact angle (h), degree

Ohkawa et al. [13] 2.0–14.3 0.007–0.013 0.025–0.750 90�Bagatur et al. [19] 2.0–13.0 0.008 0.150 45�Bagatur and Sekerdag [20] 2.0–13.0 0.0047–0.0075 0.150 45�Baylar and Emiroglu [21] 2.5–15.0 0.020 0.300 60�


123

data. The best model selection process stopped when the

lowest RMSE value was reached. Multiple linear regres-

sion analysis yielded the following equation:

Hp ¼ �0:5þ 0:032VN þ 40:3DN � 0:83Lj þ 0:44SinhR2 ¼ 0:829

ð5Þ



m; and h = jet impact angle (degree).

5 Multiple nonlinear regression

Multiple nonlinear functions of interest are as follows:

Y ¼ aXb11 Xb2

2 . . .Xbkk ð6Þ

Nonlinear relationships between the dependent variables

and the independent variables based on multivariate power

function were considered. The following equations were

derived:

Hp ¼ 4:84V0:73N D0:93

N L�0:21j ðSinhÞ0:73

R2 ¼ 0:960 ð7Þ

6 Artificial neural networks (ANN)

Artificial neural network is a flexible mathematical struc-

ture that is capable of identifying complex nonlinear rela-

tionships between input and output data sets. ANN models

have been found useful and efficient, particularly in prob-

lems for which the characteristics of the processes are

difficult to describe using physical equations [36]. Due to

difficulties in solutions of the complex engineering sys-

tems, 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. 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 algo-

rithms 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 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

ANN analyses, some functions called learning functions

are used for initialization, training, adaptation, and per-

formance 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. Adaptive functions are

employed for the simulation of a network, while the net-

work is updated for each time step of the input vector

before continuing the simulation to the next input. Per-

formance functions are used to grade the network results.

In this study, gradient descent with momentum and adap-

tive 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 the adjustable weights and the

threshold values for each connection between the neurons

in accordance with selected ANN model. After the

weighted inputs are summed and added to the threshold

values, they are passed through a differentiable nonlinear

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. 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

weight values, and therefore, it may yield closer results

than the other. Also, this algorithm has easier application

and shorter training duration [37].

7 Development of ANN model

For this study, 20 % of the data (13 data) were extracted at

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

the data (50 data) were used to train the ANN. Testing data

set was 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 that

consist of multilayer perception-trained back-propagation

algorithms were employed in this study. In Fig. 3, the

architecture of the ANN model formed by using feed-for-

ward ANN with four inputs is shown. To evaluate the

results of the developed ANN model, R2 and RMSE were

used as statistical verification tools.

The ANN simulations were conducted using a program

code written in MATLAB language. The appropriate

model structure was determined after trying different ANN

architectures. An ANN normally consists of three layers:

an input layer, a hidden layer, and an input layer. Three


123

layers with four input neurons, five hidden neurons, and

one output neurons were used. The ANN with five hidden

layers is used. The sigmoid and linear activation functions

are used for the hidden and output nodes, respectively. The

initial weights used in the ANN model were generated

randomly to values close to zero. The training may require

many epochs, being carried out until the training sum of

square error reaches a specified error goal. The ANN networks

training were stopped after 23 epochs since the variation in

error was too small after this epoch. The feed-forward neural

networks that consist of multilayer perception-trained

back-propagation algorithms were employed in this study.

The error graph for the optimum ANN model during

training is shown in Fig. 4. To evaluate the results of the

developed ANN model, the coefficient of determination

(R2) and root-mean-square error (RMSE) were used as

statistical verification tools.

Estimated bubble penetration depth values obtained with

ANN model are graphically compared with the measured

prediction of bubble penetration depth in Fig. 5. As seen in

figures, ANN model prediction fits the experimental data.

8 Genetic expression programming (GEP)

Genetic expression programming is an algorithm based on

genetic algorithms (GA) and GP; it was invented by

Ferreira [34] and incorporates both the simple, linear

chromosomes of fixed length similar to the ones used in

genetic algorithms and the ramified structures of different

sizes and shapes similar to the parse trees of genetic pro-

gramming. So, the phenotype of GEP is made of the same

kind of ramified structures used in genetic programming,

but the ramified structures created by GEP are the

expression of a totally autonomous genome. The main aim

of GEP is to form a mathematical function, and it can adapt

a set of data presented to GEP model. For the mathematical

equation, the GEP process performed the symbolic

regression by means of most of the genetic operators of

GA. The main difference between GA, GP, and GEP is

belonging to nature of the individuals. GA individuals are

symbolic strings of fixed length (chromosomes); on the

other hand, GP individuals are made up of different sizes

and shapes. GEP individuals are also (expression) trees of

different sizes and shapes, encoded as strings of fixed

length using Karva notation. Therefore, GEP maintains the

benefits of GAs and GP, while it overcomes some of their

limitations. GAs chromosomes are easy to manipulate

genetically, but they lose in functional complexity, whereas

GP trees exhibit functional complexity, but are computa-

tionally expensive. GEP genetic operators always made

valid expression. Therefore, the basis for the novelty of

GEP resides on the revolutionary structure of GEP genes

[38].

The process starts with the generation of the chromo-

somes of a certain number of individuals (initial popula-

tion). Then, these chromosomes are expressed, and the

fitness of each individual is evaluated against a set of fit-

ness cases. Then, the individuals are selected according to

their fitness to reproduce with modification. These new

individuals are subjected to the same developmental pro-

cesses such as expression of the genomes, confrontation of

the selection environment, selection, and reproduction with

Hp

Lj

VN

DN

Input Layer

Hidden Layer

Output Layer

Fig. 3 Architecture of feed-forward ANN with four inputs

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Tra

inin

g E

rro

r (M

SE

)

Iteration

Fig. 4 Training error graph for the ANN models

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13

Hp

, m

Sample

ANN

measured

Fig. 5 Measured and estimated bubble penetration depths in the test

period (by ANN)


123

modification. The process is repeated for a certain number

of generations or until a good solution is found [24]. The

flowchart of GEP is given in Fig. 6.

The two main players of GEP are the chromosomes and

expression trees (ETs). The expression of the genetic

information is encoded in the chromosome. As in nature,

the process of information decoding is named translation,

and this translation implies a code and a set of rules. The

genetic code of GEP is very simple: a one-to-one relation

between the symbols of the chromosome and the nodes

they represent in the trees. The rules determine the spatial

organization of nodes in the expression trees and the type

of interaction between sub-ETs. Thus, there are two lan-

guages in GEP: the language of the genes and the language

of expression trees and, thanks to ETs structures which are

determined from simple rules and their interactions. It is

possible to at once infer the expression tree given the

sequence of a gene and vice versa. This is termed Karva

language [39].

The GEP genes are 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 expres-

sion. Terminal symbols that 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. 7. Its ET and the corresponding mathematical equation

are also shown in same figure. The translation of expres-

sion tree to Kara language is performed by reading 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 noncoding parts.

In GEP, the main operators are the selection, transpo-

sition, and crossover (recombination). The chromosomes

are modified to get better fitness score for the next gener-

ation by means of these operators. At the beginning of the

model constructions, the operator rates that are specified

show a certain probability of a chromosome. In common,

recommended mutation rate ranges from 0.001 to 0.1.

Furthermore, recommended transposition operator and

crossover operator are 0.1 and 0.4, respectively.

9 Development of GEP model

To generate the mathematical function for the prediction of

bubble penetration depth was the main aim of development

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

developed. The GEP model has four input variables (the

nozzle diameter, jet length, jet velocity, and jet impact

angle). 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 fi of an individual program i by the following

expression:

fi ¼Xci

j¼1

M � Cði;jÞ � Tj

�� ð8Þ

where M = range of selection; C (i, j) = value returned by

the individual chromosome i for fitness case j (out of Ct

fitness cases); and Tj = target value for fitness case j.Fig. 6 Gene-expression programming (GEP) algorithm [40]

Fig. 7 Chromosome with one gene and its expression tree and

corresponding mathematical equation


123

If C i;jð Þ � Tj

�� (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 [28].

The second major step consists in choosing the set of

terminals T and the set of function F to create the chro-

mosomes. In this problem, the terminal set consists obvi-

ously of the independent variables, i.e., VN;DN; Lj; sin h� �

.

The choice of the appropriate function set is not so obvi-

ous, 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 mathe-

matical functions (1/x, Hx, x1/3, x1/4, x2, x3).

The third major step is to choose the chromosomal

architecture, i.e., the length of the head and the number of

genes. We used a length of the head, h = 8 and three genes

per chromosome. The fourth major step is to choose the

linking function. In this case, we linked the sub-ETs

(expression trees) by multiplication. And finally, the fifth

major step is to choose the set of genetic operators that

cause variations and their rates. The combination of all

genetic operators were used (mutation, transposition, and

recombination) with parameters of the optimized GEP

model [25]. Figure 8 shows the ET of the formulation

which actually is:

Hp¼��

1=��

d�0��d�1��þpow

�d�1�;2��

þ�pow

�G1C0;2

��d�3��d�2��

��

1=�pow

��pow

�d�3�;2�þd�2��;3��

þ�d�3��d�0��pow

��pow

�pow

�d�3�;�1:0=3:0

��;2�

��pow

�d�1�;�1:0=3:0

��G2C0

��;2�=d�0��

ð9Þ

The constants in the formulation are G1C0 = 4.19 and

G2C0 = 1.66, and the actual variables are d[0] = VN,

d[1] = DN, d[2] = Lj, and d[3] = Sinh. After substituting

the corresponding values in GEP formulation for bubble

penetration depth, the final equation becomes

Hp ¼1

VNDN þ D2Nð Þ

þ 17:64LjSinh� ��

� 1

Lj þ Sinh2� �3

!

þ VNSinhð Þ" #

�2:75D0:66

N Sinh1:34� �

VN

ð10Þ



m; sinh = jet impact angle (degree).

It should be noted that the proposed GEP formulation

(Eq. 9) 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 jet impact

angle h = 30–90� in estimating bubble penetration depth

(Hp).

10 Training and testing results of GEP modeling

for bubble penetration depth

The training and testing patterns of the proposed GEP

formulation are based on well-established and widely dis-

persed 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.

Also, the prediction of the proposed GEP formulation

versus experimental values for testing sets is given in

Fig. 11.

The performance of GEP in training and testing sets is

validated in terms of the common statistical measure

coefficient of determination (R2) and root-mean-square

error (RMSE). The RMSE and R2 statistics are used for

evaluating the accuracy of the models (Eqs.11, 12). The

RMSE describes the average difference between model

values and observations in units of the bubble penetration

depth (Hp).

R2 ¼P

QxQyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPQ2

x

PQ2

y

q

0

B@

1

CA

2

ð11Þ

RMSE ¼P

Qo � Qp

� �2

n

" #1=2

ð12Þ

where Qx = (Qo-Qom); Qy = (Qp-Qpm); Qo = observed

values; Qom = mean of Qo; Qp = predicted value;

Qpm = mean of Qp; and n = number of samples.

Table 3 compares the GEP models, with one of the

independent variables removed in each case, and deleting

Table 2 Parameters of the optimized GEP model

Parameter Description of parameter Setting of parameter

P1 Chromosomes 30

P2 Fitness function error type R2

P3 Number of the genes 3

P4 Head size 8

P5 Linking function *

P6 Function set ?, -, *, /, H, x1/3, x1/4, x2,

x3

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


123

Fig. 8 Expression tree (ET) for

the proposed GEP formulation

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1 5 9 13 17 21 25 29 33 37 41 45 49

Hp

, m

Sample

Gep

Measured

Fig. 9 Measured and estimated

bubble penetration depths in the

train period


123

any independent variable from the input set yielded larger

RMSE and lower R2 values. These four independent vari-

ables have non-negligible influence on Hp, and so the

functional relationship given in Eq. 9 is used for GEP

modeling in this study. The testing performance of pro-

posed GEP model (Model 1) showed a high generalization

capacity with R2 = 0.962 and RMSE = 0.0425

11 Results and discussion

The objective of this section of the paper is to examine,

discuss, and compare the results obtained from GEP, ANN,

and regression models for predicting of bubble penetration

depth in air entrainment regions. To evaluate how accurate

the results of the developed models are, the coefficient of

determination (R2) was used as a statistical verification tool.

In statistics, the overall error performances of the rela-

tionship between two groups can be interpreted from

coefficient of correlation (R) values. If a proposed model

gives R [ 0.8, there is a strong correlation between mea-

sured and predicted values for the overall data available in

the database. In addition to this, the statistical performance

of any model is evaluated in terms of some error criteria

such as root-mean square error (RMSE), which is a

significant criterion as well as R value, since sometimes a

model with high R2 value may exhibit high RMSE [37].

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 models.

Table 5 shows comparison of GEP/ANN with regression

model and other equations for predicting of bubble pene-

tration depth in air-entrainment regions. The performance

of the tested methods was analyzed by computing the R2

and RMSE values for bubble penetration depth using GEP,

ANN, and regression methods, as summarized in Table 5.

Here, a low RMSE value implies a good performance of the

applied method. Referring to Table 5, GEP model outper-

forms in high-value predictions (RMSE = 0.0425 and

R2 = 0.963) as reflected in lower RMSE and higher R2. The

superior performance of GEP, compared with other meth-

ods, is attributed to the powerful AI techniques for com-

puter learning inspired by natural evolution to find the

appropriate mathematical model (expression) to fit a set of

fits. Multiple nonlinear regressions (MNLR) are quite close

to GEP for predicting the bubble penetration depth in air

entrainment regions in Table 5.

Sensitivity analysis is also performed to see how much

the input variables are effective on output bubble pene-

tration depth variable (Fig. 12). When the parameters are

relative to one another, it would be difficult to get definite

results using sensitivity analysis. Therefore, it is significant

to ensure that each parameter is independent in the process

of sensitivity analysis. As can be seen from Fig. 12, nozzle

diameter is found to be the most effective variable on the

bubble penetration depth and then water jet velocity, jet

impact angle, and jet length, respectively.

12 Conclusion

In this study, the use of ANN and GEP model for prediction

of bubble penetration depth was investigated. The bubble

penetration depth (Hp) was first evaluated based on four

y = 0.9804x + 0.006R² = 0.9631

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Hp

(G

EP

)

Hp (Measured)

GEP

Best fit

Fig. 10 Comparison between estimated (by GEP) and measured

values of bubble penetration depth (Hp) for train set

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8 9 10 11 12 13

Hp

, m

Sample

Gep

Measured

Fig. 11 Measured and

estimated bubble penetration

depths in the test period


123

major variables that describe air entrainment at the plunge

point: the nozzle diameter (DN), jet length (Lj), jet velocity

(VN), and jet impact angle (h). A new formulation for bubble

penetration depth has been developed using GEP. The GEP-

based formulation and ANN approach are compared with

experimental results, multiple linear/nonlinear regression

equation (MLR/MNLR), and other equations. Comparison

results indicated that GEP performs better than the ANN

models, MLR, and equations. MNLR were quite close to

GEP, which serves much simpler model with explicit for-

mulation (Eq. 9). The proposed GEP formulation (Eq. 9)

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 jet impact

angle h = 30–90� in estimating bubble penetration depth

(Hp). Additionally, sensitivity analysis is performed for

ANN, and it is found that nozzle diameter is the most

effective parameter on bubble penetration depth rate among

water jet velocity, jet impact angle, and jet length. The study

suggests that GEP techniques can be successfully used in

modeling bubble penetration depth from the available

experimental data.

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Model 6 Hp = f(VN, DN) 0.0782 0.897

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ANN model 0.0890 0.901

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Method RMSE R2

Eq. (1) Bin [12] 0.121 0.799

Eq. (2) Bin [17] 0.105 0.823

GEP (Eq. 10) 0.0425 0.963 (train)

0.0485 0.962 (test)

ANN 0.0890 0.901 (test)

MLR (Eq. 5) 0.101 0.829

MNLR (Eq. 7) 0.0493 0.961

0

0.02

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0.08

0.1

0.12

VN DN Lj Sin

Sen

siti

vity

parameters

Fig. 12 Sensitivity analysis of ANN for bubble penetration depth


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prediction of penetration depth in a plunging water jet using soft computing approaches

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