Estimation of Friction Factor in Pipe Flow Using Artificial Neural Networks
Abstract
Estimation of the friction factors is very important for analysis of fluid mechanic behavior especially in the pipes and open channels. The friction factor in turbulent flow of pipe depends on Reynolds number and relative roughness. There is an implicit relationship between Reynolds number, relative roughness and Darcy friction factor that called Colebrook-White equation. Because of time consuming and iterative nature of previous methods for the solution of this equation, the attention has been oriented widely toward using artificial neural networks (ANN) in recent years. For getting the best network structure that minimizes the errors and having a good speed, we propose a multilayer perceptron network with feed forward algorithm that trained with Levenberg-Marquardt method. The dataset used for training the network consists of 2000 input/output values. Each prototype serves the Reynolds number and relative friction factor as the input and Darcy friction factor as output. In this paper, a combinational normalization strategy was also used to improve the performance of the network. The acceptance indexes for designing the network structure are Maximum percentage error, mean squared error and Regression value. If the acceptance criterion for training neural networks is 'Maximum percentage error' and it is less than 1% then the best choice is '2-10-20-1' structure. However when the training time is not important, '2-20-25-1' structure gives the best performance indexes. Neural network toolbox of MATLAB R2009a was used for all simulations.
Canadian Journal on Automation, Control & Intelligent Systems Vol. 2 No. 4, June 2011
52
Estimation of Friction Factor in Pipe Flow Using
Artificial Neural Networks
Mehran Yazdi
1
, Ayub Bardi
2
1. Department of Communications and Electroniques, Faculty of Electrical and computer Engineering,
Shiraz University, Shiraz, Iran. Email: yazdi@shirazu.ac.ir
2. Faculty of E-Learning, Shiraz University, Shiraz, Iran. Email: Ayub.bardi@yahoo.com
Abstract
—
Estimation of the friction factors is very
important for analysis of fluid mechanic behavior especially in
the pipes and open channels. The friction factor in turbulent
flow of pipe depends on Reynolds number and relative
roughness. There is an implicit relationship between Reynolds
number, relative roughness and Darcy friction factor that
called Colebrook-White equation. Because of time consuming
and iterative nature of previous methods for the solution of
this equation, the attention has been oriented widely toward
using artificial neural networks (ANN) in recent years. For
getting the best network structure that minimizes the errors
and having a good speed, we propose a multilayer perceptron
network with feed forward algorithm that trained with
Levenberg-Marquardt method. The dataset used for training
the network consists of 2000 input/output values. Each
prototype serves the Reynolds number and relative friction
factor as the input and Darcy friction factor as output. In this
paper, a combinational normalization strategy was also used
to improve the performance of the network. The acceptance
indexes for designing the network structure are Maximum
percentage error, mean squared error and Regression value.
If the acceptance criterion for training neural networks is
‘Maximum percentage error’ and it is less than 1% then the
best choice is ‘2-10-20-1’ structure. However when the
training time is not important, ‘2-20-25-1’ structure gives the
best performance
indexes. Neural network toolbox of
MATLAB R2009a was used for all simulations.
KeyWords
—
Pipe, Friction factor, Artificial Neural network,
Colebrook-White equation, Levenberg-Marquardt algorithm
I. INTRODUCTION
The energy of fluid flow continuously dissipates because of
flow resistance. In flow through pipes, the main reasons of
flow resistance and head loss are shear stress and roughness of
inner pipe wall. The Darcy-Weisbach equation is the accepted
formula for prediction of head loss in pipes as follows:
(1)
Where
l
h
is the head loss, f is the Darcy friction factor,
L is the length of pipe, V is the flow velocity, D is the inner
pipe diameter and g is the gravity constant. Although there are
differences between open channels and closed pipes such as
calculation of boundary shear stresses and different shape of
cross sections, but the principles of head loss in pipes are the
same as those in open channels [1].
Since 18
th
century, many experiments have done by
researchers to estimate Head loss during fluid flow in pipes
and channels. About 1770, Chezy [2] suggested an empirical
equation:
(2)
Where C
chezy
is the Chezy coefficient, R is the hydraulic
radius and
S
is the channel slope. In equation (2), C
chezy
is
an empirical coefficient that varies among streams. Several
empirical formulas provide values for different condition of
streams like Bazin formula and Keulegan formula [3]. Writing
the momentum equation in steady uniform equilibrium in open
channel flow indicates the relation between C
chezy
and Darcy
friction factor:
(3)
This equation expresses that if we have a good estimation
of Darcy friction factor then calculation of Chezy coefficient
is possible. This dependence is very useful in the estimation of
head loss in open channels in a series of iterative calculations
[3].
Moreover, there are many experiments corresponding to
computing Darcy friction factor in pipes and existence of
several relations depending on the region of flow (i.e. laminar
flow or turbulent flow). In laminar flow (Re<2000) there is a
simple equation that describes the relation between darcy
friction factor and Reynolds number. Nikuradse experimental
data used for derivation the implicit Colebrook-white formula
is valid in all types of turbulent flow (smooth, transition and
fully rough) [3 , 4].
g
V
D
fL
h
l
2
.
2
=
SRCV
chezy
.=
f
g
C
chezy
8
=
Canadian Journal on Automation, Control & Intelligent Systems Vol. 2 No. 4, June 2011
53
Re<2000 (e.g.Streeter and Wylie, 1981) (4)
Re>2000~3000 (5)
(Colebrook- White’s formula) [5]
Where Re is the Reynolds number and ε is the roughness
height. Colebrook-White equation indicates a causal relation
among Darcy friction factor, Reynolds number, pipe wall
roughness and the diameter of pipe. A method to solve this
equation is numerical Newton’s iteration method preferably
with a quadratic rate of convergence [4]. The error in this
method is 0.01% after 4-6 iterations [4]. Since the iterative
solution for this implicit equation is so time consuming,
Moody [6] ploted a diagram known as Moody chart (Fig. 1).
However, for more accuracy there are other solution methods
in Darcy friction factor in turbulent region like Genetic
algorithm or Artificial Neural Network (ANN). ANN is
widely used for open channels and pipes. [7,8,9,10] were used
this method for prediction of friction factor in open channels
or pipes. The most reason for tendency of researchers to
calculate neural network in hydraulic engineering problems is
self- organized, self-adopted and self-trainable algorithms that
is suitable for non-linear and complex problems.
In this paper, a multi-layer perceptron is used as an
artificial neural network with computing performance
determination based on correlation coefficient (R-value),
maximum absolute error, mean square error (MSE), minimum
time of training and minimum neurons of network trained for
the definition of Darcy friction factor in turbulent region.
Fig. 1. Moody chart
II. PROPOSED MULTILAYER PERCEPTRON
NEURAL NETWORK
With training an artificial neural network using
experimental conditions as prototypes and the results as
targets, one can predict the result of unexamined conditions in
a fast and reliable way. The advantage of ANN is its high
capability in simulation of complicated functions and non-
linear relationships by use of neurons. The ANN’s ability in
the self-adaption and self-organization has caused broad use of
this method in simulation of hydraulic problems.
Since there is non-linearity in Colebrook-White equation, the
multilayer perceptron (MLP) network with nonlinear
transform functions could minimize the mean square error as
performance determination [11].
In this paper, we develop an MLP network with ‘tansig’
function as the non-linear activation function in its hidden
layers. Each input data for this network has two values, one
for Reynolds number and one for relative roughness and target
of this input data is its corresponding Darcy friction factor.
The feed forward Back propagation (BP) algorithm is used as
the appropriate method for training this MLP network. The BP
algorithm is only suitable for those problems which require
superior stability and accuracy [9]. The Neural network
toolbox of MATLAB R2009a was used for all simulations.
A. Data Acquisition
Darcy friction factor in turbulent pipe flow depends on
Reynolds number and relative roughness, which are related as
follows:
(6)
Where ρ is the fluid density and µ is the dynamic
viscosity. In an open channel, D as the pipe diameter is
replaced by D
H
as the hydraulic diameter of the channel and ε
is replaces by k
s
as the equivalent sand roughness height.
Many researchers have reported that the friction factor in open
channels is nearly 10% higher than that in pipes [12].
However, when there is the lack of suitable experimental data
in special open channels, the assumption that
f
is merely
varied by Reynolds number and relative roughness makes
reliable answers that can be corrected in iterative mode when
comparing with Chezy formula. Generally, in an open channel
flow we can write below equation for Chezy coefficient:
(7)
Where A
c
is the open channel cross sectional geometric
shape, N
c
is the Non-uniformity of the open channel for both
profile and plan, Fr is the Froude number and U
c
is the degree
of flow unsteadiness. In equations (6) and (7)
f
and
chezy
C
Re
64
=f
+−=
f
D
fRe
51.2
7.3
log0.2
1
10
ε
=
D
VD
ff
ε
µ
ρ
,
),,,,(Re,
cccchezy
UFrNA
D
fC
ε
=
Canadian Journal on Automation, Control & Intelligent Systems Vol. 2 No. 4, June 2011
54
values are respectively related to two and six parameters.
Researchers determine these parameters empirically and then
relate each friction factor to corresponding conditions.
Since data acquisition from Moody chart is not accurate and
takes a long time, here we use the Haaland formula [13] to
create input and output matrices. The Haaland formula is an
approximation relationship for Colebrook-White formula and
suggested by S.E. Haaland in 1983 as follows:
(8)
Reynlods number domain is selected in the interval of 3000
to 1×10
8
. Twenty types of relative roughness from Fig. 1 are
picked out as the second arrays of each input matrix .The
number of data for training the network in Zeng Yuhong,
Huai Wenxin [9] is 250, in D.A. Fadare and U.I. Ofidhe [7] is
2560 and in Azimian A.R [10] is 572. In our work, 2000
input-output data were used such that 1600 prototypes were
applied for training the MLP, 200 for validation and 200 for
testing the trained network.
B. Dataset Normalization
Since the Reynolds number is in an extensive domain of
numbers, it can cause a large amount of Maximum percentage
error after training the network. Thus Azimian [10] suggested
a logarithmic normalization method before applying a training
procedure in order to decrease the error. To do that, we have:
Maximum percentage error= (9)
Normalized (Re) = (10)
In equation (9) is the Darcy friction factor which can
be calculated using Haaland equation and
ANN
f
is the Darcy
friction factor that is calculated using the neural network. In
equation (10) ‘Ln’ indicates natural logarithm, Re
min
is the
minimum number of Reynolds number domain and Re
max
is
the maximum number of Reynolds number domain. There is
another kind of normalization that can be used by ‘premnmx’
MATLAB
®
command. D.A. Fadare and U.I. Ofidhe [7] used
this function to normalize input and output dataset.
We used both of these normalizations and train an ‘2-10-20-
1’ structure as a typical network with levenberge-Marquardt
algorithm to compare the effect of these pre-processing
normalizations with structures without normalization. As
shown in Table I, all of theperformance indexes have been
significantly improved after pre-processing of dataset.
In Table I, for instance , ‘2-10-20-1’ structure indicates that
Network has 2 input values, 10 neurons in the first hidden
layer, 20 neurons in the second hidden layer and 1 output
value as
ANN
f
.
TABLE
I
E
FFECT OF
N
ORMALIZATION
O
N
P
ERFORMANCE
I
NDEXES
Network
structure
Regression
value
Max.
error
MSE* Number of
iterations
Without
normalization
2-10-20-1 0.75 574% 4.38×10
-5
1000
Normalization 2-10-20-1 1 0.98% 3.14×10
-8
1000
*MSE=mean square error
C. Design of MLP Feed forward network
Designation of network is done by the Neural network
Toolbox of MATLAB
®
. In this network an input vector with
2×1 element was used for inserting the Reynolds number and
relative roughness. Hyperbolic Tangent sigmoid (‘tansig’)
transfer function was used as non-linear function for hidden
layers and purelin transfer function was used as a linear
function for output layer (see Fig. 2).
Fig. 2. the schematic sketch of a feed forward MLP with two hidden
layers to estimate friction factor.
D. Experimental results
There are various methods for training the ANN using BP
algorithm such as Levenberg-Marquardt (LM), Bayesian
Regulation and Quasi Newton [14]. In [7,9,10], Levenberg-
Marquardt method prefers for fulfilling the simulation because
of the fast training and less memory occupying. Here these
three mentioned methods are investigated.
The performance indexes for some previous works have
listed in Table II. The major causes of wide distinction
between these works are differences in the amount of
input/output dataset, the dissimilarity in the pre-processing
normalization, number of iterations and eventually unlikeness
in performance indexes.
+
−= Re
9.6
7.3
/
log8.1
1
11.1
10
D
f
ε
%100)( ×
−
haaland
ANNhaaland
f
ff
(
)
( )
)(ReRe
)(ReRe
minmax
min
LnLn
LnLn
−
−
haaland
f
Canadian Journal on Automation, Control & Intelligent Systems Vol. 2 No. 4, June 2011
55
TABLE
II
P
ERFORMANCE
I
NDEXES FOR SOME OF
P
REVIOUS
S
IMULATIONS
Network
structure
Acceptance
criteria
Azimian A.R. [10] 2-7-1 Minimum
training time;
Maximum
percentage
error smaller
than 1%
D.A. Fadare and U.I.Ofidhe [7]
2-20-31-1 MAPE*,MSE
,SSE**,R-
value
Zeng Yuhong , HuaiWenxin [9]
2-20-1 Unknown
* MAPE: mean absolute percentage error
** SSE: Sum of squared error
In this paper, the evaluation criteria for the network
structures are the least possible value for maximum percentage
error, minimum of MSE, the nearest R-value to 1 and the least
cost time for the training process after 1000 iterations.
Consequences for some typical network structures trained by
LM, Bayesian and Quasi Newton algorithms are presented in
Tables III through V.
TABLE
III
L
EVENBEG
-M
ARQUARDT
A
LGORITHM
Iteration *Time Max.
percentage
error (%)
MSE R-
value
Network
structure
220 0:00:07 12.03 7.39e-5 0.99985
2-7-1
1000 0:01:08 11.30 1.26e-5 0.99997
2-20-1
1000 0:01:07 1.17 1.72e-7 1
2-7-7-1
1000 0:03:04 1.05 5.80e-8 1
2-10-15-1
1000 0:04:25 0.98 3.14e-8 1
2-10-20-1
1000 0:16:14 0.35 3.39e-9 1
2-20-25-1
TABLE
IV
B
AYESIAN REGULARIZATION
Iteration *Time Max.
percentage
error (%)
SSE R-
value
Network
structure
1000 0:00:30 8.63 0.0518 0.99993
2-7-1
1000 0:01:11 8.40 0.00868 0.99999
2-20-1
1000 0:01:13 1.20 0.000131 1
2-7-7-1
1000 0:03:20 1.16 6.92e-5 1
2-10-15-1
1000 0:04:56 1.15 6.56e-5 1
2-10-20-1
1000 0:20:53 1.25 7.71e-5 1
2-20-25-1
TABLE
V
BFGS
Q
UASI
-N
EWTON
Iteration
*
Time
Max.
percentage
error (%)
MSE
R
-
value
Network
structure
294 0:00:09 29.3 3.89e-4 0.99921
2-7-1
132 0:00:07 61.9 1.42e-3 0.99708
2-20-1
184 0:00:09 24.8 1.98e-4 0.99959
2-7-7-1
665 0:01:19 17.5 3.00e-5 0.99994
2-10-15-1
331 0:00:56 27.1 1.76e-4 0.99965
2-10-20-1
245 0:04:38 29.7 2.14e-4 0.99956
2-20-25-1
*Time in above tables means the total cost time for training, validation and
test of the network. The computer used for this operation has 1.86 Giga Hz
frequency for CPU.
III.
D
ISCUSSION
The obtained results in Tables III through V demonstrate
that LM algorithm gives the optimum solution, i.e. the shortest
time with the least error and suitable R-values. The
comparison between ‘2-10-20-1’ structure results in Tables III
and IV shows that the maximum percentage error and training
cost time of structures trained using LM algorithm are much
better than those of the same structures when they are trained
by Bayesian regularization algorithm.
As shown in Table III through IV the most accurate
network structure is 2-20-25-1 in Table III. This network was
trained using LM algorithm and has the smallest error among
other network structures at same number of iterations. In
addition, its Regression value is 1. But training of such heavy
structure is too time consuming (see Table III).
By considering ‘2-20-25-1’ structure, the modification in
normalization of dataset used in this paper causes that the
number of hidden layers becomes less than that used by D.A.
Fadare and U.I. Ofidhe [7] and consequently the poroposed
algorithm is much faster. Moreover, the combinational
normalization used here causes smaller errors as well as
greater R-value than ‘2-20-31-1’ structure suggested in [7].
Increasing the neurons of hidden layers is not necessarily
efficient, because the resulted network takes more time for
being trained.
If the only acceptance criterion for neural structures is
‘Maximum percentage error’ with the constraint of being
smaller than 1%, then the best choice is ‘2-10-20-1’ structure
trained by LM algorithm. The training time for this structure is
also reasonable.
The obtained results in Table V also confirm that BFGS
Quasi-Newton is not appropriate for our network at all. The
errors of trained structure in Table V are greater than those
using LM and Bayesian regularization algorithms as the
techniques of training.
IV.
C
ONCL USION
In this paper, a method for estimating the Darcy friction
factor in turbulent pipe flow was suggested. The efficient
calculation with less time consuming for training was
achieved by the new method which suggestes applying a
normalization process for datasets. This normalization is
Canadian Journal on Automation, Control & Intelligent Systems Vol. 2 No. 4, June 2011
56
composed of non-linear logarithmic normalization associated
by premnmx function.
We also used ‘Maximum percentage error’ with the
constraint of being less than 1% as an acceptance criterion for
neural structures. Since ‘Maximum percentage error’ of ‘2-10-
20-1’ structure trained by applying LM algorithm is 0.98%, so
it can be preferably selected as the desirable neural structure.
However, if having the shortest training time is not very
important, we can suggest ‘2-20-25-1’ structure with using
LM algorithm as the technique of training.
R
EFERENCES
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(Washington D.C. 2002)
[3] H. Chanson, The Hydraulics of Open Channel Flow:An
Introduction.Second Edition. (Department of Civil Engineering,
The University of Queensland, Australia. 2004).
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(2003) 75-98
[5] C.F. Colebrook, Turbulent flow in pipes with particular reference
to the transition region between the smooth and rough pipe laws,
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- CitationsCitations7
- ReferencesReferences11
- However, the developed methodology for training can be used with appropriate dataset or appropriate equations to produce relevant solution in such cases where the aforementioned ANN cannot be used [14][15][16]. Application of ANN for simulation of other types of friction factor rather than Colebrook, namely, Hazen– Williams friction coefficient for small-diameter polyethylene pipes, can also be found in the literature [17], while more recently other attempts of ANN usage for modeling friction factors in pipes have been reported [18, 19]. Nowadays, not only can the ANN approach be used in hydraulics and for simulation of fluid flow, but also it can be widely applied in the various branches of engineering, such as for the control systems [19, 20], as an auxiliary tool in medicine [21][22][23][24][25], a flow pattern indicator for gas-liquid flow in a microchannel [26], and an extension of structural mechanics tools for fast determination of structural response [27].
[Show abstract] [Hide abstract] ABSTRACT: Nowadays, the Colebrook equation is used as a mostly accepted relation for the calculation of fluid flow friction factor. However, the Colebrook equation is implicit with respect to the friction factor. In the present study, a non iterative approach using Artificial Neural Network (ANN) was developed to calculate the friction factor. To configure the ANN model, the input parameters of the Reynolds Number (Re) and the relative roughness of pipe (휀/퐷) were transformed to logarithmic scales. The 90,000 sets of datawere fed to the ANN model involving three layers: input, hidden, and output layers with, 2, 50, and 1 neurons, respectively. This configuration was capable of predicting the values of friction factor in the Colebrook equation for any given values of the Reynolds number (Re) and the relative roughness ranging between 5000 and 108 and between 10−7 and 0.1, respectively. The proposed ANN demonstrates the relative error up to 0.07% which had the high accuracy compared with the vast majority of the preciseexplicit approximations of the Colebrook equation. Intelligent Flow Friction Estimation. Available from: https://www.researchgate.net/publication/301228160_Intelligent_Flow_Friction_Estimation [accessed Apr 13, 2016].- This work used the feed-forward (FF), multi layer perceptron network (MLP). Given the non-linearity in the Colebrook equation, the MLP network with non-linear transform functions could minimize the mean square error as performance determination [13]. The hidden layer consists of 2 layers with a predefined amount of neu-rons.
- Levenberg–Marquardt LM method was used for training the FFBP artificial neural network. LM method is preferred for fulfilling the simulation because of the fast training and occupying less memory [17].
[Show abstract] [Hide abstract] ABSTRACT: Weirs are small overflow dams used to alter and raise water flow upstream and regulate or spill water downstream watercourses and rivers. This paper presents the application of artificial neural network (ANN) to determine the discharge coefficient (Cd) for a hollow semi-circular crested weirs. Eighty five experiments were performed in a horizontal rectangular channel of 10 m length, 0.3 m width and 0.45 m depth for a wide range of discharge. The results of examination for discharge coefficient were yielded by using multiple regression equation based on dimensional analysis. Then, the results obtained were also compared using ANN techniques. A multilayer perceptron MLP algorithm FFBP network was developed. The optimal configuration of ANN was [2,10,1] which gave mean square error (MSE) and correlation coefficient (R) of 0.0011 and 0.91 respectively. Performances of ANN model reveal that the Cd could be better estimated by ANN technique in comparison with Cd obtained using statistical approach.- In the study of laminar flow in pipes several analytical tools have been used to model and analyze the behavior of laminar flow in pipes Some of the tools are very good in the analyses of some specific parameters in the flow fluid, while some other tools are very effective in the analyses of some other flow parameters. Mehran et al. [16] used Artificial Neural Network (ANN) to estimate the friction factor in pipe. Artificial Neural Network (ANN) is used to solve the Colebrook- White equation.
[Show abstract] [Hide abstract] ABSTRACT: This paper investigate some important works done on numerical analysis and modeling of laminar flow in pipes. This review is focused on some methods of approach and the analytical tools used in analyzing of the important parameters to be considered in laminar flow; such as frictional losses, heat transfer etc. in laminar flow in pipes of different shapes, and the importance of laminar flow in its areas of applications. Prominent researchers have approached this from dif-ferent perspectives. Some carried out analysis on the pressure drop as a function of permeability, some worked on fric-tion factor analysis, some discussed heat transfer effects of laminar flow in the entrance region, while some discussed its applications in various industries. Some of these works were done considering a given form of pipe configuration or shape which is circular pipes. Only a few, of the literature reviewed have related their considerations to different forms of pipes. Most consider pipes to be majorly circular in shape, but in industries today some circular pipes have become elliptical in shape due to long time usage of the pipes, which would have contributed to increase in some different forms of losses in the industries. In engineering, efficiency and effectiveness improvement is the major goal, if a research work has been done, considering the important parameters in laminar flow showing their effects on different forms of pipe configuration as a result of pipe deformation due to usage, huge amount of money will be saved. This will show clearly how the efficiency of a given circular pipe has seriously been affected due to deformation, and the level of loss this has resulted to.- [Show abstract] [Hide abstract] ABSTRACT: Estimation of Darcy friction factor and pipe network analysis are essential ingredients in the design and distribution of potable water. Common formulae for friction factors estimate include Colebrook-White, Moody, Swamee and Jain, Barr, Haaland, Tsal and Wood formulae. Accuracy of pipe network analysis depends on Darcy friction factor, but little is known on update of these formulae and their performance in developing countries. In this paper, as a follow up on our previous studies, Oke (2007); Babatola et al. (2008) an overview and performance evaluation of these formulae is presented using statistical methods (model of selection criterion and statistical errors). Darcy Friction factor formulae were obtained from archive. These formulae were used to estimate friction factors in pipes at various Reynolds number and relative roughness. Estimated friction factors were evaluated statistically using absolute error, total error, mean error and model of selection criterion using Colebrook-White friction factor as the reference friction factor. Colebrook-White was used as reference because it is widely recommended formulae and has a wide range of Reynolds number. The study revealed that friction factor in pipes varies with the formulae and varies from 0.0157 to 0.0727. In all cases Tsal formula has the smallest friction factors. Based on the mean error, accuracy were in order of Newton Raphson > Prandtl and Nikurdse > Zingrang and Sylvester > Serghide > Barr > Swamee and Jain > Eck > Haaland > Brkic > Wood > Moody > Chen (1979) > Buzzelli > Sonnad and Goudar > Vatankhah and Kouchakzadeh > Monzon et al > Churchill (1973)> Jain > Round > Manadilli > Evangleids et al > Avci Kargoz > Tsal > Churchill (1977) > Chen (1985). It is concluded that Newton Raphson ; Prandtl and Nikurdse; Zingrang and Sylvester ; Serghide ; Barr; Swamee and Jain; Eck ; Haaland ; Brkic ; Wood and Moody are first choice friction formulae based on the values of model of selection criterion. ABSTRACT 075
- [Show description] [Hide description] DESCRIPTION: O presente artigo se propõe a modelar matematicamente o processo e aparato de elevação artificial através do bombeamento mecânico linear de duplo efeito, utilizando sistema hidráulico projetado para operar parcialmente ou inteiramente submerso na zona de produção, em poços com diferentes configurações (lineares ou com desvios), fluídos com densidades similares à do petróleo, além de ser uma alternativa viável para bombeamento em aquíferos com grandes profundidades.
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The "fluid friction factor" (f) should be abandoned because it is a mathematically undesirable parameter that complicates the solution of fluid flow problems. f is the dimensionless group 2g PD5/8LW2. This group is mathematically undesirable because it includes P, W, and D. Therefore if f is used in the solution of a problem, the problem must be solved with P, W, and D in the same... [Show full abstract]
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The main parameters in the analysis of water distribution networks are the friction coefficients, lengths and diameters of pipes. Estimation of the friction coefficient of pipes is very important in the field of water and wastewater engineering, such as distribution of velocity and shear stress, erosion, sediment transport and head loss. Several relations have been proposed to estimate the... [Show full abstract]
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The first published work on fluid flow patterns in pipes and tubes was done by Reynolds. He observed the flow patterns of fluids in cylindrical tubes by injecting dye into the moving stream. Reynolds found that as he increased the fluid velocity in the tube, the flow pattern changed from laminar to turbulent at a Reynolds number value of about 2100. Later investigators have shown that under... [Show full abstract]
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Determination of friction factor is an essential prerequisite in pipe flow calculations. The Darcy-Weisbach equation and other analytical models have been developed for the estimation of friction factor. But these developed models are complex and involve iterative schemes which are time consuming. In this study, a suitable model based on artificial neural network (ANN) technique was proposed... [Show full abstract]
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