Žarko Ćojbašić

Dejan Brkić
email: dejanbrkic0611@gmail.com

Very accurate explicit approximations for calculation of the Colebrook friction factor

Keywords: Colebrook equation, Colebrook-White equation, Flow friction, Hydraulic resistance, Genetic algorithms, Optimization technique

In section 2, page 3, line 56: why lambda and lambda_0 are evaluated in MS Excel? The part in

Introduction

The Colebrook equation is nowadays mostly used for computing of the turbulent flow friction in pipes [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF]. It was developed solely by Colebrook in 1939 [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF], but he also used the data from his joint experiment with White from 1937 [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]. Therefore, his equation is sometimes referred as the Colebrook-White equation. Unfortunately, in this equation unknown friction factor λ is given implicitly (1):

D 71 . 3 Re 51 . 2 log 2 1 10 (1)
where λ is the Darcy friction factor (known also as the Darcy-Weisbach or Moody friction factor), Re is the Reynolds number, and ε/D is relative roughness of inner pipe surface. All mentioned values are dimensionless. The practical range for the Reynolds number is between 2320 and 10 8 while range for the relative roughness is between 10 -6 and 0.05.

The unknown friction factor cannot be extracted from the Colebrook equation in explicit way with no approximation involved (exception is approach with the Lambert W-function, but even then further evaluation of this function is approximate [START_REF] Brkić | W solutions of the CW equation for flow friction[END_REF][START_REF] Keady | Colebrook-White formula for pipe flow[END_REF][START_REF] More | Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes[END_REF][START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF][START_REF] Brkić | Comparison of the Lambert W-function based solutions to the Colebrook equation[END_REF]). Therefore, the Colebrook equation has to be solved in an iterative procedure, which is the main reason that numerous explicit approximations have been developed by many authors [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF][START_REF] Brkić | Determining friction factors in turbulent pipe flow[END_REF][START_REF] Gregory | Alternate to standard friction factor equation[END_REF]. A number of alternatives to the Colebrook equation along with a discussion about their use are provided in Giustolisi et al. [START_REF] Giustolisi | Some explicit formulations of Colebrook-White friction factor considering accuracy vs. computational speed[END_REF]. The Colebrook equation is also used as the base for the turbulent part of the widely known Moody plot [START_REF] Moody | Friction factors for pipe flow[END_REF].

In this preliminary paper, two new very accurate approximations are presented (in addition to this preliminary paper improvement of coefficients for all other available approximations of Colebrook's equation is planned to be shown in the future). This two new improved approximations shown here, were developed using the genetic algorithm technique for optimization of parameters in the already existing models proposed by Serghides in 1984 [START_REF] Serghides | Estimate friction factor accurately[END_REF] and Romeo et al. in 2002 [14]. Genetic algorithms are one of the evolutionary computational intelligence techniques inspired by Darwin's theory of biological evolution [START_REF] Goldberg | Genetic algorithms in search, optimization and machine learning[END_REF]. They provide solutions for different types of problems, searching the most suitable among chromosomes that make the population in the potential solutions space, which makes them suitable even for difficult problems of finding the global optimum for nonlinear and multimodal optimization problems (e.g. gene expression programming analysis of implicit Colebrook's equation in turbulent flow friction factor calculation can be found in Samadianfard [START_REF] Samadianfard | Gene expression programming analysis of implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF]).

Elaboration of error

All errors shown in this paper are calculated using iterative solution where the coefficients in the Colebrook equation are 2.51 and 3.71. Used relative error is defined as [(λ-λ 0 )/λ 0 ]•100%, where λ 0 is assumed as the accurate friction factor that is obtained from iterative solution of the explicit Colebrook equation while λ is friction factor calculated using the selected explicit approximation of the Colebrook equation. The iterative solution of the implicit Colebrook equation after sufficient number of iterations can be used for further comparisons as the most accurate. This approach is used in this paper for the evaluation of the accuracy of the explicit approximations of the Colebrook equation, i.e. for the calculation of λ 0 . The iterative solution is obtained after 32767 iterative cycles which is more than enough (required number of iteration is usually no more than ten) [START_REF] Brkić | Determining friction factors in turbulent pipe flow[END_REF]. Also, Matlab was used for the optimization of the parameters of the models A and B shown in the next section of this paper (e.g. a Matlab code for the calculation of Colebrook's flow friction can be found in Clamond [START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF]). The Matlab codes provided in next section were used to evaluate λ using approximations with improved parameters presented here.

Of course, the Colebrook equation is empirical, where the numerical constants 2.51 and 3.71 were obtained by numerical fitting from the experimental data. Sometimes constant 3.71 can be changed with 3.7 (as is in original paper by Colebrook [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF]). This change produces deviation of 0.138%. To date, the most accurate approximations to the explicit Colebrook equation has the relative error in range between 0.13% and 0.15% [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF]. This error is very small and it is in the margin of the error caused by changed coefficient 3.7 to 3.71. On the other hand, American Gas Association (AGA) recommended use of coefficient 2.825 instead of 2.51 for calculation of flow of natural gas through pipes [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF]. This causes deviation of even 3.2%.

As already noted, the Moody chart [START_REF] Moody | Friction factors for pipe flow[END_REF] was developed using Colebrook's equation [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF] and not opposite. Using Moody's chart for the estimation of error instead of the iterative solution of the implicit Colebrook equation, i.e. using different digitalization techniques for determination of λ 0 introduces additional errors in calculation [START_REF] Yıldırım | Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF][START_REF] Brkić | New explicit correlations for turbulent flow friction factor[END_REF][START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF]. The methodology for estimation of error is the same as from the study of Brkić [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF]. Other papers with the related estimation of error are also available [START_REF] Gregory | Alternate to standard friction factor equation[END_REF][START_REF] Samadianfard | Gene expression programming analysis of implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF][START_REF] Yıldırım | Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF][START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF][START_REF] Zigrang | A review of explicit friction factor equations[END_REF][START_REF] Abdolahi | The effect of major parameters on simulation results of gas pipelines[END_REF].

It should be noted that the relative error changes dramatically over the domain of relative roughness and the Reynolds number [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF]. The error shown here is the maximal relative error over the entire practical range for the Reynolds number from 2320 to 10 8 and for the relative roughness ranging from 10 -6 to 0.05. This area is covered with 740 points uniformly sampled over the ranges of input variables, which means that estimated error can be slightly above (somewhere among four adjacent tested points of the matrix which consists of 740 points).

Also it has to be noted that uniform sampling might sometimes result into misleading conclusions [START_REF] Giustolisi | Some explicit formulations of Colebrook-White friction factor considering accuracy vs. computational speed[END_REF].

Proposed explicit approximations

As noted in the study by Brkić [8], the five most accurate approximations to the Colebrook equation are by Vatankhah and Kouchakzadeh from 2008 [START_REF] Vatankhah | Discussion of "Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF][START_REF] Vatankhah | Discussion: Exact equations for pipe-flow problems[END_REF], Buzzelli from 2008 [START_REF] Buzzelli | Calculating friction in one step[END_REF],

Romeo et al. from 2002 [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF], Serghides from 1984 [START_REF] Serghides | Estimate friction factor accurately[END_REF], and by Zigrang and Sylvester from 1982 [START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF]. They are with the maximal relative error of no more than 0.15% [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF].

Using the existing model for approximations proposed by Serghides in 1984 [START_REF] Serghides | Estimate friction factor accurately[END_REF] and Romeo et al. in 2002 [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF], two new approximations with optimized parameters will be introduced.

Respectively, the first one, here labeled as the model A, is now with the relative error of no more than 0.0026% while the second one, labeled as model B, is with the relative error of no more than 0.0083%. This means that the relative error is decreased from 0.1385% to only 0.0026% for the model A, and from 0.1345% to only 0.0083% for the model B. 2) is proposed by Serghides [START_REF] Serghides | Estimate friction factor accurately[END_REF] and his coefficients are changed here using genetic algorithms techniques. His model is with three internal cycles additionally accelerated by Steffensen's method [START_REF] Johnson | On Steffensen's method[END_REF]. Coefficient 12 is changed to 12.585 while 3.7 is changed to 3.71, compared to original approximation [START_REF] Serghides | Estimate friction factor accurately[END_REF].

Model A. Model A (

S D S ( 5
)
Distribution of error for the model A approximation is shown in figure 1.

Figure 1. Distribution of error for the model A approximation

Matlab code for the model A with the improved parameters is given between the next two dash lines:

- ------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------3.2. Model B. Model B ( 6) is proposed by Romeo et al. [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF] and their coefficients are also changed here using genetic algorithms techniques. Approximation by Romeo et al. [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF] is similar with the model proposed by Zigrang and Sylvester [START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF] with three internal cycles and with the model proposed by Chen [START_REF] Chen | An explicit equation for friction factor in pipes[END_REF] with two internal cycles. In model B, coefficient ---------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------As already noted, and as also can be seen from figures 1 and 2, relative error is not uniformly distributed. This fact is characteristic of all available explicit approximations of the Colebrook equation [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF].

Comparative analysis

Using model A and parameters improved by genetic algorithm optimization technique, maximal relative error decreased 53.26 times compared to coefficients proposed by Serghides [START_REF] Serghides | Estimate friction factor accurately[END_REF]. Using model B and the same optimization technique, the relative error decreased 16.2 times compared to the parameters proposed by Romeo et al. [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF]. Improvement best can be seen from figure 3.

Conclusion

Existing approximations of the Colebrook equation are complex by the rule. Some of them even look simple, but essentially they are not since they contain non-integer exponent or many logarithmic expressions [START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF]. The computation of logarithm in many computer languages is based on series expansions that require several powers of the argument to be computed and added to each other. In particular, all of them require the computation of some logarithms to achieve a sufficiently accurate solution, without caring how such an operation may affect real computer applications [START_REF] Abdolahi | The effect of major parameters on simulation results of gas pipelines[END_REF]. In this paper, observed models were not simplified, but the accuracy of approximations has been increased by improving parameters using genetic optimization technique, namely 53.26 times when the model A is used and 16.2 times when the model B is used (genetic algorithms can be also successfully used in other branches of engineering [START_REF] Ristanović | Intelligent control of DC motor driven electromechanical fin actuator[END_REF][START_REF] Ćojbašić | Computationally intelligent modeling and control of fluidized bed combustion process[END_REF]). In other words, computational burden stays unchanged compared with the approximations by Serghides [START_REF] Serghides | Estimate friction factor accurately[END_REF] and Romeo et al. [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF], but accuracy of these model increases with the introduction of here presented new parameters [START_REF] Danish | Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in smooth pipes using Adomian decomposition method[END_REF].

Additionally, curve fitting or surface fitting tool from Matlab can be used for fine tuning of coefficients after improvements made using genetics algorithm optimization techniques [23,

( 6 )

 6 Distribution of error for the model B approximation is shown in figure2.

Figure 2 .

 2 Figure 2. Distribution of error for the model A approximation

  function Lambda = modelB(Re,RelEpsilon) % Re is Reynolds number and RelEpsilon is relative roughness A = log10((RelEpsilon./7.646).^0.9685+(4.9755./(206.2795+Re)).^0.8759); B = log10((RelEpsilon./3.8597)-(4.795./Re).*A); Lambda = (-2.*log10((RelEpsilon./3.7106)-(5./Re).*B)).^-2;

Figure 3 .

 3 Figure 3. Improvement of coefficients in the existing models A and B using genetic algorithm

Figure 1

 1 Figure 1 Click here to download high resolution image

Figure 2

 2 Figure 2 Click here to download high resolution image

Figure 3

 3 Figure 3 Click here to download high resolution image

  -

	function Lambda = modelA(Re,RelEpsilon)
	% Re is Reynolds number and RelEpsilon is relative roughness
	S1 = -2*log10(RelEpsilon./3.71+12.585/Re);
	S2 = -2*log10(RelEpsilon./3.71+2.51.*S1./Re);
	S3 = -2*log10(RelEpsilon./3.71+2.51.*S2./Re);
	Lambda = (S1-((S2-S1).^2)/(S3-2.*S2+S1)).^-2;

  3.7065 is changed to 3.7106, coefficient 5.0272 to 5, coefficient 3.827 to 3.8597, coefficient 4.567 to 4.795, coefficient 7.7918 to 7.646, exponent 0.9924 to 0.9685, coefficient 5.3326 to 4.9755, addend 208.815 to 206.2795 and finally exponent 0.9345 to 0.8759, compared to original approximation[START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF].

	1	2	log	10	7106 . 3	D	Re 5	log	10	8597 . 3	D	Re 795 . 4	log	10	646 . 7	D	9685 . 0	2795 . 206 9755 . 4	Re	8759 . 0

Acknowledgement

The work of Žarko Ćojbašić was partially supported by the research grant No. TR35016 from the Ministry of Education and Science of Republic of Serbia. The work of Dejan Brkić was supported by the research grant No. 11-51-2322/2012-03 from the Provincial Secretariat for Science and Technological Development of Autonomous Province of Vojvodina (Republic of Serbia).

 [START_REF] Vatankhah | Discussion: Exact equations for pipe-flow problems[END_REF]. Using only fitting tool of Matlab (with no use of genetic algorithms), Vatankhah and Kouchakzadeh [START_REF] Vatankhah | Discussion of "Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF][START_REF] Vatankhah | Discussion: Exact equations for pipe-flow problems[END_REF] improved coefficients from the approximation of Sonnad and Goudar [START_REF] Sonnad | Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF][START_REF] Sonnad | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF] (in that way maximal relative error decrease from 0.8003% to 0.1472%).

Beyond the most accurate optimized approximations that we have presented here, genetic optimization methodology can be applied to the wider set of the approximations for explicit pipe flow friction calculation published to date, in order to offer improved accuracy with possibly other computational benefits.


PhD Dejan Brkic
email: dejanrgf@tesla.rcub.bg.ac.rs

Dejan Brkić

W solutions of CW equation for flow friction

Keywords: Colebrook-White equation, Lambert W-function, Turbulent flow friction, Hydraulic resistance, Analytical approximations 00A06, 33B99, 33F05, 65D20, 65H05, 76F06 Colebrook-White equation, Lambert W-function, Turbulent flow friction, Hydraulic resistance, Analytical approximations

Empirical Colebrook-White (CW) equation belongs to the group of transcendental functions. CW function is used for determination of hydraulic resistances associated with fluid flow through pipes, flow of rivers, etc. Since CW equation is implicit in fluid flow friction factor, it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. Alternate mathematically equivalents to the original expression of the CW equation, but now in explicit form developed using Lambert W-function, are shown (with related solutions). W function is also transcendental, but it is used more general compared with CW function. Hence, solution to W function developed by mathematicians can be used effectively for CW function which is of interest only for hydraulics.

Thank you for your comments. I appreciate them. They were useful for me and I have improved my text according to your suggestions.

1. I have noted in text that CW equation is empirical (first word in abstract also). I have noted this fact several times through text. 2. I have noted that coefficient 2.51 is replaced with 2.825 for gas flow as recommended by American Gas Agency AGA and I have associated the paper of Haaland from 1983 with this fact 3. I have corrected my error regarding electrical resistance (word current is replaced with resistance). Word resistance is more suitable, but with constant voltage, change of resistance implies also change of current. But you are note is absolutely good, changes of resistance is causing changes in current (or flow in case of fluid flow). It is not acceptable to replace cause and its consequence. 4. Paper of D.J. Zigrang, N.D. Sylvester, (1985) is also cited. 5. Definition of Lambert W-function is now added in text (second row in section 2). 6. According to solution for observed branch of W function by Barry et al I have added improved solution of CW function. These are equations ( 16) and [START_REF] Sonnad | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF]. Also I have added solution for CW function based on solution of W function proposed by Winitzki (section 3.5). I have also added graph of error distribution of one solution of CW based on solution of W by Barry et al. 7. Name of author is now added after title. 8. I have rearranged term 'in our paper'to 'here'or similar…. 9. Now, I use only numbers for equations, i.e. [START_REF] Hayes | Why W?[END_REF] and not (10a) 10. I have not found graphical abstract in any paper in Applied Mathematics Letters. But I have added graph of W and CW function in text (figure 1). This figure can be also used as graphical abstract if suitable.

                 D 71 . 3 Re 51 . 2 log 2 1 10 (1)
Empirical implicit CW equation itself can produce an error of more than 5% but today even apropos this fact it is accepted standard for calculation of flow friction factor in hydraulically smooth and rough pipes [START_REF] Zigrang | A review of explicit friction factor equations[END_REF]. Many researchers adopt a modification of the CW equation, using the 2.825 constant instead of 2.51 especially for gas flow calculation [START_REF] Haaland | Simple and explicit formulas for friction factor in turbulent pipe flow[END_REF][START_REF] Coelho | Considerations about equations for steady state flow in natural gas pipelines[END_REF]. This adoption for gas flow produced deviation of maximal 3.2% compared with classical CW equation.

As an alternative to the implicit CW equation, many approximate explicit formulas were given.

Gregory and Fogarasi [START_REF] Gregory | Alternate to standard friction factor equation[END_REF], Yıldırım [START_REF] Yıldırım | Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF] and Brkić [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF] made comparisons of in that time available approximations of CW equation. "W" for Lambert function is not as old as the related function [START_REF] Hayes | Why W?[END_REF]. The modern history of Lambert W began in the 1980s, when a version of the function was built into the Maple computer-algebra system and given the name W. Corless et al [START_REF] Corless | On the Lambert W function[END_REF] proposed name Lambert W for this function and this name is also used here. But in formulas is used only letter W for related function because this notation is shorter. Lambert W-function is somewhere known as Omega [START_REF] Corless | On the Lambert W function[END_REF]. For CW equation, only the positive part of the principal branch of the Lambert W-function is considered (Figure 1) because the other branches correspond to nonphysical solutions of CW equations, so the simplified notation W is not ambiguous. Fortuitously, the letter W has additional significance because pioneering work on many aspects of W by Wright [START_REF] Wright | Solution of the equation ze z =a[END_REF]. Although White was not actually a co-author of the paper in which CW equation was presented [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF],

Explicit reformulation of the CW equation based on Lambert W-function

Colebrook made a special point of acknowledging important contribution of White to the development of the equation [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]. So letter W has additional symbolic value in here reformulated CW equation (2):

                                      D 71 .
In the papers of Moore [START_REF] More | Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes[END_REF], Nandakumar [START_REF] Nandakumar | Analytical calculation of Fanning friction factor[END_REF] and Goudar and Sonnad [START_REF] Goudar | Explicit friction factor correlation for turbulent flow in smooth pipes[END_REF][START_REF] Sonnad | Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF][START_REF] Sonnad | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF], other possible transformations of CW equation using Lambert-W function are shown. But relations shown in these papers have limitation in applicability for high values of Reynolds number and relative roughness because today available computers cannot operate with extremely large numbers [START_REF] Sonnad | Constraints for using Lambert W function-based explicit Colebrook-White equation[END_REF].

In the paper of Keady [START_REF] Keady | Colebrook-White formulas for pipe flow[END_REF], CW equation is expressed in Maple notation using Lambert W-function. Clamond [START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF] also provides Matlab and FORTRAN codes for CW relation expressed in term of the Lambert W-function.

Possible solutions of CW equation based on Lambert W-function

Besides the relative simplicity of an explicit from of CW equation transformed using Lambert W-function, it allows highly accurate estimation of friction factor as the Lambert W-function can be evaluated accurately [START_REF] Gautschi | The Lambert W-functions and some of their integrals: a case study of highprecision computation[END_REF].

Formal solution

Since the Lambert W is transcendental function, formal solution of the Lambert W-function can be expressed only in endless form (4):

                         ) ln( x ln x ln x ln x W  (4)

Solution using series expansion

Functions such as exponential, logarithmic or square root are useful tools in solving broad classes of mathematical problems. With just the four basic operations of arithmetic any linear equation can be solved. Adding square roots quadratic equations can be solved as well. For some class of problems are useful trigonometric functions. These functions can be classified as elementary. But for solution of implicit equations such as CW, best function is Lambert W.

Logical question which can arise is why Lambert W-function is not elementary function, while trigonometric, logarithmic, exponential, etc are. Note that Taylor series is used in principle in pocket calculators. This means that W button can be added very easy. Using series expansion, principal branch of the Lambert W-function can be noted as [START_REF] Haaland | Simple and explicit formulas for friction factor in turbulent pipe flow[END_REF]:

      n 1 1 n 7 6 5 4 3 2 0 x ! n n x O x 5 54 x 24 125 x 3 8 x 2 3 x x x W                  (5)
Whether W ultimately attains such canonical status will depend on whether the mathematical community at large finds it sufficiently useful [START_REF] Hayes | Why W?[END_REF].

Solution using Boyd shifted function

It is convenient to define a new function and new parameter [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] such that both the domain and range are the nonnegative real axis as ( 6):

        1 y x W 1 x W y        (6)
With also "shifted" argument of the function [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] (7):

              (7)
Then, the Lambert W function can be transformed in ω function [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] (8):

1 y e ) 1 (       (8)
Approximate solution 1 of ω function can be expressed [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] (9):

               
))

In previous equation ( 9) hyperbolic tangent function can be defined tanh(ξ)=(e ξ -e -ξ )/(e ξ +e -ξ ).

Improved solution after Boyd [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] is also available (10):

) 1 ( 0 0       (10)
Where Ω is [START_REF] Corless | On the Lambert W function[END_REF]:

3                    (11)
When the improved approximation is used as the first guess for Newton"s iteration scheme [START_REF] Wright | Solution of the equation ze z =a[END_REF] only four iterations reduce the relative error significantly over the entire domain of "shifted" function [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF]

: i ) 1 y ( ) ( i i 1 i e ) 1 (              (12)
This error does not have uniform distribution for entire domain of "shifted" function [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF]. Values of errors presented for "shifted" function i.e. ω(y) have limited meaning, because our interest is in evaluation of error for W(x).

Solution proposed by Barry et al.

Barry et al. [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF] gives their approximation for the upper branch of the Lambert W-function valid for CW [START_REF] More | Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes[END_REF]: For CW equation, previous equation can be arranged [START_REF] Nandakumar | Analytical calculation of Fanning friction factor[END_REF]:

                                                         
                         D S
   ( 14
)
Where S is [START_REF] Goudar | Explicit friction factor correlation for turbulent flow in smooth pipes[END_REF]:

               Re 1 . 1 1 ln Re 1 . 1 ln 816 . 1
Re ln S [START_REF] Goudar | Explicit friction factor correlation for turbulent flow in smooth pipes[END_REF] Presented relation produce maximal relative error no more than 3% compared to original implicit CW equation. More accurate procedure, but also more complex is available in the paper of Barry et al. [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF]. To reduce error below 2%, S should be replaced with S * [START_REF] Sonnad | Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF] or S ** [START_REF] Sonnad | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF]: Since the error derived using presented procedures and real errors produced by using implicit CW equation are of both signs the errors will add in some cases and cancel in other cases.

                 917365

Solution proposed by Winitzki

According to Winitzki [START_REF] Winitzki | Uniform approximations for transcendental functions[END_REF] approximation for Lambert W-function can be obtained by using [START_REF] Sonnad | Constraints for using Lambert W function-based explicit Colebrook-White equation[END_REF]:

                          x 1 ln 2 x 1 ln 1 ln 1 x 1 ln x W (18)
Using ( 14) and previous equation one new approximate formula for CW equation can be obtained [START_REF] Keady | Colebrook-White formulas for pipe flow[END_REF]: [START_REF] Keady | Colebrook-White formulas for pipe flow[END_REF] This approximation is accurate as those developed by using procedure proposed by Barry et al. [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF].

                          

Conclusion

  Dear editor/reviewers, This is revised version of my manuscript: W solutions of CW equation for flow friction Dejan Brkić, Ph.D., Petroleum and Natural Gas Engineer, Strumička 88, 11050 Beograd, Serbia Tel. +381642543668, e-mail: dejanrgf@tesla.rcub.bg.ac.rs Sincerely, Dejan Brkić, PhD Cover Letter DB

  Lambert W and CW are transcendental functions. The (real-valued) Lambert W-function is solution of the nonlinear equation W·e W =x. The range of the lower branch of inverse Lambert function is -1≤W -1 , while the upper branch W 0 is divided into -1≤  0 W ≤0 and 0≤  0 W . W 0 is referred to as principal branch of the Lambert W-function. Only  0 W part of the principal branch of the Lambert W function will be used for solution of here presented problem. The Lambert function is implemented in many mathematical systems like Mathematica by Wolfram Research under the name ProductLog or Matlab by MathWorks under the name Lambert. Note that name

  of Lambert W-function can be noted as[START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]

  error for approximate equation (17) over the entire practical domain of relative roughness (ε/D) and Reynolds number (Re) is shown in figure 2.

  CW equation is widely used in the petroleum industry for calculations of oil and gas pipelines, in civil engineering for calculation of water distribution systems, in chemical engineering, and in all fields of engineering where fluid flow can be occurred. The problem is that, since CW equation is implicit, containing the friction factor, the Reynolds number and the pipe roughness, it has to be solved iteratively. Even today in the era of advance computer technology, explicit approximations of the implicit CW relation are very often used for calculation of friction factor in pipes. Presented methods for solution of CW equation based on Lambert W-function provide equally satisfied accuracy of solution for observed problem. Details about work and life of Johann Heinrich Lambert can be read in the paper of Gray and Tilling [25]. λ-Darcy friction factor (dimensionless) Re-Reynolds number (dimensionless) D-inner pipe diameter (m) ε-roughness of the inner surface of pipe (m) ε/D-relative roughness of the inner surface of pipe (dimensionless) W-Lambert function ω-shifted, auxiliary function proposed by Boyd x-argument of the Lambert W-function y-argument of the shifted, auxiliary function proposed by Boyd n-positive integer number  , Ω, ζ, S-auxiliary terms defined in text

Response to Reviewers DB Figure 1 DB Click here to download high resolution image Figure 2 DB Click here to download high resolution image 1. Introduction

	Problem of flow in pipes and open conduits was one which had been of considerable interest to
	engineers for nearly 250 years. Even today, this problem is not solved definitively [1]. The
	difficulty to solve the turbulent flow problems lies in the fact that the friction factor is a complex
	function of relative surface roughness (ε/D) and the Reynolds number (Re). Precisely, hydraulic
	resistance depends on flow rate. Similar situation is with electrical resistance when diode is in

*Detailed *

Manuscript DB Click here to view linked References

	circuit. To be more complex, widely used, empirical Colebrook-White (CW) equation valuable
	for determination of hydraulic resistances for turbulent regime in smooth and rough pipes is
	iterative (implicit in fluid flow friction factor). The unknown friction factor appears on both sides
	of the equation i.e., both the right and left-hand terms contain friction factor [2,3]. Colebrook-
	White equation is also known as Colebrook equation or simple CW equation (1):

  D

	1	2	log	10	10	4343 . 0	S	71 . 3	2	log	10	Re 18 . 2	71 . 3

Note that log is actually ln, i.e. log e in the paper of Boyd[START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF]

List of Figures


Mourad Abbas
email: m_abbas04@yahoo.fr

Kamel Smaïli
email: kamel.smaili@loria.fr

D Berkani
email: dberkani@hotmail.com

K Smaili

Comparing TR-Classifier and KNN by using Reduced Sizes of Vocabularies

Keywords: TR-classifier, k Nearest Neighbors, Arabic corpus, topic vocabulary. I

INTRODUCTION

Topic identification has been sufficiently studied for Indo-European languages. Generally, the methods used are those of text categorization: Bayesian classifiers [START_REF] Tzeras | Automatic Indexing Based on Bayesian Inference Networks[END_REF][START_REF] Lewis | Comparison of two Learning Algorithms for Text Categorization[END_REF][START_REF] Moulinier | Is Learning Bias an issue on Text Categorization Problem?[END_REF], decision tree [START_REF] Lewis | Comparison of two Learning Algorithms for Text Categorization[END_REF][START_REF] Moulinier | Is Learning Bias an issue on Text Categorization Problem?[END_REF][START_REF] Fuhr | A rule-based Multistage Indexing Systems for Large Subject fields[END_REF], neural networks [START_REF] Wiener | A neural network approach to topic spotting[END_REF][START_REF] Ng | Feature selection perceptron learning, and a usability case study for text categorization[END_REF], kNN "k Nearest Neighbors" [START_REF] Creecy | Trading Mips and Memory for Knowledge Engineering: Calssifying Census Returns on the Connection Machine[END_REF][START_REF] Yang | Expert Network: Effective and Efficient Learning from Human Decisions in Text Categorization and Retrieval[END_REF], etc. Nevertheless, for Modern Standard Arabic, few works have been carried out [START_REF] El-Kourdi | Automatic Arabic Document Categorization Based on the Naïve Bayes Algorithm[END_REF][START_REF] Sawaf | Statistical Classification Methods for Arabic News Articles. Arabic Natural Language Processing[END_REF][START_REF] Abbas | Comparison of Topic Identification Methods for Arabic language[END_REF][START_REF] Abbas | Topic Identification by Statistical Methods for Arabic language[END_REF].

The aim of this study is to evaluate two text categorization methods on Arabic corpora. The first one is TR-classifer [START_REF] Abbas | Topic Identification for Automatic Speech Recognition[END_REF], a new method based on computing triggers, and the second one is the k Nearest Neighbors.

The motivation behind the conception of the TRclassifier is that the information present in the longerdistance history is significant [START_REF] Rosenfeld | Adaptive Statistical Language Modeling: A Maximum Entropy Approach[END_REF]. Indeed, for a topic identification task, the presence of the term "guitar" could trigger another list of terms: "music", "dance", etc. So, the main idea is to represent each topic by a number of triggers which allow characterizing each topic, and then facilitating the identification.

The second method that we have used, is the wellknown kNN. As this method is considered in [START_REF] Yang | A re-examination of text categorization methods[END_REF] as one of the top-performing classifiers, we selected it to evaluate TR-classifier by comparing performances of the two methods.

We should note that small sizes of topic vocabularies are chosen for TR-classifier. Whereas, a general vocabulary is needed for kNN, consequently it has been constructed by the concatenation of the topic ones.

The Arabic corpus used in our experiments is downloaded from the website of the Omani newspaper Alwatan1 ; it is composed of more than 9000 articles.

In section II, we give some details both about TRclassifier and kNN method. We talk, in Section III, about the corpus, the representation of documents and the vocabulary construction. Experiments and results are exposed in section IV.

II. METHODS DESCRIPTION

A. TR-Classifier

We start by giving a definition of Triggers as TRclassifier is based on computing them. So, triggers of a word w k are the ensemble of words that have a high degree of correlation with it [START_REF] Abbas | Topic Identification for Automatic Speech Recognition[END_REF][START_REF] Rosenfeld | Adaptive Statistical Language Modeling: A Maximum Entropy Approach[END_REF][START_REF] Haton | Speech Recognition from signal to its interpretation[END_REF]. The main idea of the TR-classifier is based on computing the average mutual information of each couple of words which belong to the vocabulary V i . Couples of words or "triggers" that are considered important for a topic identification task, are those which have the highest average mutual information (AMI) values [START_REF] Guodong | Interpolation of n-gram and mutual information based trigger pair language models for Mandarin speech recognition[END_REF][START_REF] Tillman | Selection criteria for word trigger pairs in language modeling[END_REF]. Each topic is then endowed with a number of selected triggers M, calculated using the training corpus of the topic T i . Identifying topics by using TR-method consists in: • Giving corresponding triggers for each word w k Є V i , where V i is the vocabulary of the topic T i .

• Selecting the best M triggers which characterize the topic Ti.

• In the test step, we extract for each word w k of the test document, its corresponding triggers.

• Computing Q i values by using the TR-distance given by ( 1):

(1) Where i stands for the i th topic. The denominator presents a normalization of AMI computation. B. k Nearest Neighbors kNN has been applied to text categorization before two decades [START_REF] Masand | Classifying news stories using memory based reasoning[END_REF][START_REF] Yang | Expert Network: Effective and Efficient Learning from Human Decisions in Text Categorization and Retrieval[END_REF][START_REF] Iwayama | Cluster-based text categorization: a comparison of category search strategies[END_REF][START_REF] Yang | A re-examination of text categorization methods[END_REF]. Indeed, Yang compared it to a set of text categorization methods using the benchmark Reuters corpus (the 21450 version, Apte set) [START_REF] Yang | A re-examination of text categorization methods[END_REF]. It has been found that KNN is one of the top-performing methods after SVM [START_REF] Yang | A re-examination of text categorization methods[END_REF]. Many other researches have found that the kNN method achieves very good performances by using different data sets [START_REF] Yang | A re-examination of text categorization methods[END_REF][START_REF] Joachims | Text Categorization with Support Vector Machines: Learning with Many Relevant Features[END_REF][START_REF] Baoli | A Comparative Study on Automatic Categorization Methods for Chinese Search Engine[END_REF].

The strategy of the kNN algorithm is quite simple, so that, to identify a topic-unknown document d, kNN ranks the neighbors of d among the training document vectors, and uses the topics of the k Nearest Neighbors to predict the topic of the test document d. The topics of neighbors are weighted using the similarity of each neighbor to d. In order to measure this similarity, the cosine distance is used, although other measures are possible, as the Euclidean distance. The cosine similarity is defined by [START_REF] Lewis | Comparison of two Learning Algorithms for Text Categorization[END_REF].

( ) ( ) ( ) ∑ ∑ ∑ = = = = V k V k ik jk V k ik jk i j d d d d D D sim 1 1 2 2 1 , (2)
where D j is the j th training document and D i is the i th test document. | V | is the size of the general vocabulary. d j and d i represent the weights of the words belonging respectively to D j and D i . To assign the test document d to the correct topic, a cutoff threshold is needed [START_REF] Yang | A re-examination of text categorization methods[END_REF].

III. CORPUS REPRESENTATION

We started by downloading Arabic texts from the archives of the Omani newspaper Alwatan of the year 2004. The size of the extracted corpus is about 10 millions terms which correspond to 9000 articles, distributed over six topics, in this case: Culture, religion, economy, local news, international news and sports. 90 % of these articles are reserved to training and the rest to the evaluation. We should note that we have realized some elementary operations for topic identification, as eliminating insignificant words that do not bring any information, as function words, and also words whose frequencies are less than a definite threshold. We dress in table 1 the size of the entire corpus, before and after removing insignificant words.

The construction of the vocabulary has been made by using the term frequency method which gives good results though its simplicity [START_REF] Yang | A comparative study on feature selection in text categorization[END_REF]. Other terms selection methods as Mutual Information [START_REF] Seymore | Nonlinear interpolation of topic models for language model adaptation[END_REF] and Document Frequency lead also to satisfactory performances. The kNN method uses a general vocabulary, whereas the TR-classifier, uses a vocabulary per topic, i.e., six topic vocabularies are built, in our case. We should note that these vocabularies are very small; indeed the size of each topic vocabulary is 300 terms. Nevertheless, they are composed of terms ranked from the maximum, to the minimum according to their frequencies. The reason behind the vocabularies size reduction is to make the topic identification process faster.

Documents need to be transformed to a compact vector form, and the dimension of the vector corresponds to the size of the vocabulary. Each word of the document is weighted by a definite value. The weights or vector components are those commonly used in text categorization, particularly for the TFIDF classifier [START_REF] Joachims | A probabilistic analysis of the rocchio algorithm with tfidf for text categorization[END_REF]. Hence, after removing insignificant words, we calculated both the frequency of each word, which is called Term Frequency, and the Document Frequency of a word w, that means the number of documents in which the word w occurs at least once. The weight of each term results then from the product of Term Frequency and Inverse Document Frequency [START_REF] Joachims | A probabilistic analysis of the rocchio algorithm with tfidf for text categorization[END_REF][START_REF] Salton | Developments in Automatic Text Retrieval[END_REF][START_REF] Seymore | Using Story Topics for Language Model Adaptation[END_REF].

IV. EXPERIMENTS AND RESULTS

A. TR-Classifier Performances

As we mentioned in section III, TR-classifier uses a vocabulary per topic, and the words of each vocabulary are ranked according to their frequencies. In these experiments we used much reduced sizes of the six topic vocabularies, in this case 300 terms. The evaluation of the TR-classifier has been made by varying both topic vocabularies sizes and triggers number N. So, we present performances on average in figure 1. The number of triggers is important to achieve good performances. According to figure 1, we notice that average Recall (R) and Precision (P) rates improve when increasing the triggers number. Indeed, for N=20, R and P are respectively equal to 54.88 % and 57.30 %. These values continue to be enhanced when N is augmented to 250, in this case R=89.67 % and P=86.44 %.

B. kNN Evaluation

The main computation made by kNN is the sorting of training documents in order to find the k Nearest Neighbors for the test document. The value of k is usually optimized by several trials on the training and validation sets.

A general vocabulary is used by the kNN method. Thus we constructed a vocabulary by concatenating the six topic vocabularies used in the previous experiment. The resulted size is 800 words.

Performances are lower than those of TR-Classifier by 14 % which is considered as an important difference between the two methods. We present in figure 2, performances in terms of Recall of the two classifiers, for the six studied topics.

CONCLUSION

In this paper, two methods of topic identification have been presented. Their performances have been tested on an Arabic corpus that we have constructed using many thousands of texts, downloaded from an online newspaper. One of these methods is the TR-Classifier: a new technique that we exposed in this paper, and the second one is the well-known kNN. The strong point of the TR-Classifier is its ability to realize better performances by using reduced sizes of topic vocabularies, compared to kNN. The reason behind that, is the significance of the information present in the longer-distance history that TR-Classifier uses. Undoubtedly, kNN is one of the best methods which give best performances; nevertheless in the case of small vocabularies, as shown in the aforementioned experiments, its performances didn't exceed 76 % in terms or Recall. In perspectives, we aim to enhance TR-Classifier performances by using superior sizes of vocabularies, though it outperforms kNN by 14 %, which is considered as a satisfactory result.

wFigure 1 .

 1 Figure 1. TR-Classifier performances using a vocabulary size 300 in terms of Recall and Precision

Figure 2 .

 2 Figure 2. TR-Classifier performances compared to kNN ones

TABLE I .

 I NUMBER OF TERMS BEFORE AND AFTER ELIMINATING INSIGNIFICANT WORDS

	Topics	N. words before	N. words after
	Culture	1.359.210	1.013.703
	Religion	3.122.565	2.133.577
	Int. news	855.945	630.700
	Economy	1.460.462	1.111.246
	Loc. news	1.555.635	1.182.299
	Sports	1.423.549	1.067.281
	Total	9.813.366	7.139.486

http://www.alwatan.com


PhD Dejan Brkic

    Comments on "Experimental study of the pressure drop in the cathode side of air-forced open-cathode proton exchange membrane fuel cells" by Barreras et al., Int.

Commentary

Comments on "Experimental study of the pressure drop in the cathode side of air-forced Open-cathode proton exchange membrane fuel cells" by Barreras (5) of the commented paper]. The friction factor obtained by authors' equation differs from Colebrook's results up to 60%. Both equations are empirical and developed for the different conditions of flow, for different fluids, different geometry and materials of conveying channels and therefore, the different results are acceptable and expected. On the other hand, the Colebrook equation is well established and therefore it is already implemented in the numerous computing codes [START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF]. This is the main reason to express this new equation from the commented paper using Colebrook's expression [START_REF] Barreras | Experimental study of the pressure drop in the cathode side of air-forced Open-cathode proton exchange membrane fuel cells Hydrogen Energy[END_REF]. In the next equation, Colebrook's expression is actually noted as 'x'. (dimensionless)

x D                            5
Eq. ( 18) of the commented paper is developed for the specific value of relative roughness of inner surface of conveying channels. Therefore, in Eq. ( 1), the relative roughness should be set as ε/D=0.03086 to preserve complete equality between here presented Eq. ( 1) and Eq. ( 18)

of the commented paper. This value of relative roughness can be changed in a presence of certain liquid fluids [START_REF] Lu | Water management studies in PEM fuel cells, part IV: Effects of channel surface wettability, geometry and orientation on the two-phase flow in parallel gas channels[END_REF][START_REF] Brkić | Can pipes be actually really that smooth?[END_REF].

Of course, the Colebrook and the equation from the commented paper are implicitly given for the unknown friction factor. Therefore they have to be solved using iterative procedure [START_REF] Brkić | Determining friction factors in turbulent pipe flow[END_REF].

The Colebrook equation, 'x', is not supposed to be solved separately in iterative procedure if the goal is in solving of Eq. ( 1). In that way the convergence properties of Eq. ( 1) will be disturbed. So it is not allowed to calculate

 / 1
in iterative procedure using only the Colebrook expression 'x' and then to calculate new value of

 / 1 using x     5 17 . 14 / 1 
. To conclude, Eq. ( 1) has to be solved as the single equation in an iterative procedure. There are numerous explicit approximations of the Colebrook equation [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF], but equally, they cannot be used for the calculation of the friction factor of Eq. ( 1). New explicit approximations should be developed based on Eq. ( 1).

Using Eq. ( 16) and Eq. (18) [or Eq. (1) of this commentary], the diagram similar to the Moody's can be plotted [START_REF] Moody | Friction factors for pipe flow[END_REF]. Of course, the boundary, between the laminar regime [presented by Eq. ( 16)] and the turbulent regime [presented by Eq. (18) of the commented paper or Eq.

(1) of this commentary] has to be corrected compared with the Moody diagram as already described by in the commented paper.

Finally, it has to be noted that the authors of the commented paper misquote the source of the Colebrook equation. The Colebrook equation was developed by the Colebrook solely [START_REF] Colebrook | Turbulent flow in with particular reference to the transition region between the smooth and rough pipe laws[END_REF], but also, he used data from his joint experiment with White [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF].

In this commentary, the Darcy friction factor is used (noted as λ). The authors of commented paper use f or f D for the Darcy friction.

  PhD, Strumička 88, 11050 Belgrade, Serbia, cell: +381642543668, e-mail: dejanrgf@tesla.rcub.bg.ac.rs, dejanbrkic0611@gmail.com In their recent paper [1], Barreras et al. developed the equation for the friction factor in the transitional turbulent regime for the air flow in the cathode side of air-forced open-cathode proton exchange membrane fuel cells [Eq. (18) of the commented paper]. As the authors clearly noted, their equation can be compared with the widely known Colebrook equation [Eq.


Dr Dejan Brkic

New explicit correlations for turbulent flow friction factor

Two new correlations of single-phase friction factor for turbulent pipe flow are shown in this paper. These two formulas are actually explicit approximations of iterative Colebrook's relation for calculation of flow friction factor. Calculated friction factor are valid for whole turbulent flow including hydraulically smooth and rough pipes with special attention on transient zone of turbulence between them. Hydraulically smooth regime of turbulence does not occur only in total absence of roughness of inner pipe surface, but also, four new relations for this theoretical regime are presented. Some recent formulas for turbulent flow friction calculation are also commented.

I wish to thank reviewer to his or her valuable comments. 1. Abstract is now changed. Two new explicit approximations to the Colebrook approximation are shown. I did not see that Fang et al. favor their approximation for smooth pipes. In their paper is equally important approximation to the Colebrook equation. Also as I explained, the Colebrook equation is not only for "rough" pipes. It is for whole regime of turbulence which includes smooth and rough law and transient zone between them. Smooth law is not only valuable in absence of roughness and therefore equation of Fang et al. for smooth pipes has limited application as I explain in my text. 2. I do not use anymore Zigrang and Sylvester method from 1985 to estimate the complexity of approximations. But to be clear, Fang et al. approximation has few not simple power terms (1.1007, 1.1105 and 1.0712). These are not simple input parameters for hand calculators. 3. I examined approximation in 740 points in my paper "Review of explicit approximations to the Colebrook relation for flow friction". My results can be compared with results by [START_REF] Gregory | Alternate to standard friction factor equation[END_REF] and [START_REF] Zigrang | A review of explicit friction factor equations[END_REF]. Both papers are in my reference list. Results by [START_REF] Yıldırım | Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF] are different and I have comments in my paper on this. [START_REF] Gregory | Alternate to standard friction factor equation[END_REF] and [START_REF] Zigrang | A review of explicit friction factor equations[END_REF] are 25 years old research. My results with newly developed approximations can be compared with results from S. Genić, I. Arandjelović, P. Kolendić, M. Jarić, N. Budimir, V. Genić, A review of explicit approximations of Colebrook's equation, FME Transactions 39 (2011), 67-71. My analysis was done using MS Excel and I am preparing new research in MATLAB with much more points than 740. 4. Moody diagram starts from 0 with laminar zone. Of course Colebrook's equation is only for turbulent zone. Turbulent zone is for Re>2300 or in the Moody chart for Re>4000. But laminar zone can exist up to Re=10 4 . To avoid any misunderstandings, my charts are for Re>10 4 . For example, from figure 1, for [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] equation, it can be seen that relative error is up to 0.62% for Re>10 4 (or about 0.73% for Re=3000). I do not claim anymore that my analysis covers the entire Re range of the

New explicit correlations for turbulent flow friction factor

Dejan Brkić, Ph.D., Petroleum and Natural Gas Engineer, Strumička 88, 11050 Beograd, Serbia Tel. +381642543668, e-mail: dejanrgf@tesla.rcub.bg.ac

            D 71 . 3 Re 51 . 2 log 2 1 10    (1)
Turbulent part of the famous Moody chart was drawn using Colebrook's results [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF][START_REF] Moody | Friction factors for pipe flow[END_REF]). Colebrook's formula can be used for whole turbulent regime including the smooth and the rough zone of turbulence and also for the transient zone between them.

Colebrook's formula has two parts which presents "smooth" and "rough" portion of turbulence.

Only

                 (2)
"Rough" part in Colebrook's formula is presented by von Karman's formula (3) which is not implicitly given.

                    D D    10 10 log 2 14 . 1 71 . 3 log 2 1
(3) Prandtl's and von Karman's formula are also known as NPK (Nikuradse-Pandtl-von Karman) formulas.

Using logarithmic rules, one can found that Colebrook's formula is incorrect because log(A)+log(B)≠log(A+B). But Colebrook's equation solves not problem from mathematics, but from hydraulics. Colebrook's formula is empirical and make fine transitional curve between smooth and rough turbulent law. NPK equations used separately for smooth law (Prandtl's Colebrook's equation includes not only these two extremes presented by NPK equations but also smooth law where relative roughness is very small (ε/D→0) with significant laminar boundary layer and also rest of turbulent regime where relative roughness smoothly increases to its physical maximum while thickness of laminar boundary layer decreases.

The Colebrook equation is also known under the Colebrook-White name, but this equation should be quoted in a correct way from source [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF]. Although, White was not actually a co-author of the paper in which considered equation was presented, Colebrook made a special point of acknowledging important contribution of White to the development of the equation based on his previous joint work [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]. Paper of Colebrook is from 1939 (as can be seen from doi: 10.1680/ijoti.1939.13150 of this paper) and not from 1938-39 as often erroneously cited (volume 11 has only issues 4, 5 and 6 all from 1939).

Colebrook's formula can be solved in iterative procedure since it contains implicitly given flow friction factor. Also, numerous explicit approximations can be used to substitute implicit Colebrook's equation which cannot be solved for unknown flow friction factor with no approximate calculus [START_REF] Moody | An approximate formula for pipe friction factors[END_REF][START_REF] Wood | An explicit friction factor relationship[END_REF][START_REF] Eck | Technische Stromungslehre[END_REF][START_REF] Churchill | Empirical expressions for the shear stress in turbulent flow in commercial pipe[END_REF][START_REF] Swamee | Explicit equations for pipe flow problems[END_REF][START_REF] Jain | Accurate explicit equation for friction factor[END_REF][START_REF] Churchill | Friction-factor equation spans all fluid flow regimes[END_REF][START_REF] Chen | An explicit equation for friction factor in pipes[END_REF][START_REF] Round | An explicit approximation for the friction factor-Reynolds number relation for rough and smooth pipes[END_REF][START_REF] Barr | Solutions of the Colebrook-White function for resistance to uniform turbulent flow[END_REF][START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF][START_REF] Haaland | Simple and explicit formulas for friction factor in turbulent pipe flow[END_REF][START_REF] Serghides | Estimate friction factor accurately[END_REF][START_REF] Manadilli | Replace implicit equations with signomial functions[END_REF][START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF][START_REF] Sonnad | Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF][START_REF] Buzzelli | Calculating friction in one step[END_REF][START_REF] Avci | A novel explicit equation for friction factor in smooth and rough pipes[END_REF][START_REF] Vatankhah | Discussion: Exact equations for pipe-flow problems[END_REF][START_REF] Papaevangelou | A new explicit equation for the friction coefficient in the Darcy-Weisbach equation[END_REF]Brkić, 2011a,b;[START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF][START_REF] Ghanbari | Newly developed friction factor correlation for pipe flow and flow assurance[END_REF]. Two novel explicit approximations to the Colebrook's formula will be shown in further text.

Some recent explicit formulas for turbulent flow friction

There are many different explicit formulas developed to substitute implicit Colebrook's relation.

Most recent formula is by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF]. [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] estimated very accurately maximal relative error up to 0.5% for their approximation of Colebrook's formula. From figure 1, it can be seen that this error is slightly bigger, up to 0.62% for Re>10 4 (or about 0.73% for Re=3000). This is very accurate which means that their approximation is in first ten sorted by accuracy (Brkić, 2011c). Also, approximations proposed by [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF], [START_REF] Buzzelli | Calculating friction in one step[END_REF], more complex form of approximation by [START_REF] Serghides | Estimate friction factor accurately[END_REF], more complex form of approximation by [START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF] and approximation by [START_REF] Vatankhah | Discussion: Exact equations for pipe-flow problems[END_REF] will always give more accurate results than those calculated by approximations proposed by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] as shown in figure 2. On the other hand as shown in figure 3, inconclusive conclusion of accuracy in general can be found by using of approximation by [START_REF] Barr | Solutions of the Colebrook-White function for resistance to uniform turbulent flow[END_REF], simpler form of approximation by [START_REF] Serghides | Estimate friction factor accurately[END_REF], approximation by [START_REF] Chen | An explicit equation for friction factor in pipes[END_REF], [START_REF] Sonnad | Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF], [START_REF] Papaevangelou | A new explicit equation for the friction coefficient in the Darcy-Weisbach equation[END_REF] and by using simpler form of approximations by [START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF], all versus approximation by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF]. Approximation proposed by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] will give better prediction of implicit Colebrook equation compared with other approximations from Brkić (2011c). Relative error of certain approximations to the Colebrook equation was also presented in [START_REF] Gregory | Alternate to standard friction factor equation[END_REF] and [START_REF] Zigrang | A review of explicit friction factor equations[END_REF]. As noted by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF], [START_REF] Yıldırım | Computer-based analysis of explicit approximations to the implicit Colebrook-White equation in turbulent flow friction factor calculation[END_REF] conducted comprehensive analysis of existing correlations for single-phase friction but he used Techdig 2.0 software which caused remarkable reading errors factor.

Findings by Brkić (2011c) as well as findings by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] do not support Yıldırım's error estimations because he greatly overestimated errors and offered different accuracy-based rank order for the correlations evaluated than in [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] or in Brkić (2011c). One can be always aware that Moody diagram was constructed using Colebrook's equation and not opposite [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF][START_REF] Moody | Friction factors for pipe flow[END_REF]. It is not acceptable to replace cause and its consequence such was done by reading of points from Moody diagram using Techdig 2.0 software. [START_REF] Ghanbari | Newly developed friction factor correlation for pipe flow and flow assurance[END_REF] also made their approximation of Colebrook's equation based on data which was obtained by digitizing the Moody's diagram. This approach produces approximation with error up to 3% as can be seen from figure 4.

New explicit approximations to the Colebrook's formula

In some recent papers, few new explicit approximations of Prantdl's relation were shown [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF][START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF][START_REF] Danish | Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in smooth pipes using Adomian decomposition method[END_REF]. As noted before, Prantdl's relation is actually Colebrook's equation in the case of total absence of relative roughness (ε/D). Approximations to the Prandtl equation by [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF] and [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF] are suitable for transformation in approximations to the Colebrook equation. Approximations to the Prandtl equation given in suitable form (with its smooth part "x" transformed in explicitly given form) can be easily transformed in approximation to the Colebrook equation ( 4).

                 D x x 71 . 3 log 2 1 log 2 1 10 10    (4)
Using approximations to the Prandtl equation proposed in [START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF], new approximation to the Colebrook equation can be obtained (5).

            D

 

(5) After few simple transformations new approximation to the Colebrook equation ( 6) can be obtained from approximations to the Prandtl equation proposed in [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF].

                             D 71 . 3
  (6)
Distribution of relative error for both presented approximations (marked as new in figure 6) to the Colebrook equation can be seen in figure 5. Presented equations with relative error up to 2% can be used to substitute implicit Colebrook's equation. This error is comparable with other available approximations as can be seen from figure 6.

Figure 6. Two new approximations compared with some other available approximations to the Colebrook equation

Both equations are presented in figure 6 are presented with one line "new" since they produce identical error. The Colebrook equation is empirical and therefore error of 2% can be neglected.

New explicit approximations to the Prandtl's formula

Hydraulically smooth regime of turbulence does not occur only in total absence of roughness of inner pipe surface (Brkić, 2011d), but also, four new relations for this theoretical regime are presented. As noted before, Colebrook's equation in total absence of relative roughness is actually Prandtl's equation, and vice versa. These four approximations to the Prandtl equation was developed using inverse procedure from previous section, i.e. from appropriate approximations to the Colebrook formula, "rough" part was simply removed (Table 1).

Table 1: Equations for hydraulically smooth regime developed for total absence of roughness (when ε/D=0)

Approximations to the Prandtl equation (new 1-4) were developed using approximations to the Colebrook formula shown in [START_REF] Romeo | Improved explicit equation for estimation of the friction factor in rough and smooth pipes[END_REF], [START_REF] Zigrang | Explicit approximations to the solution of Colebrook friction factor equation[END_REF], [START_REF] Manadilli | Replace implicit equations with signomial functions[END_REF][START_REF] Serghides | Estimate friction factor accurately[END_REF], respectively (new 1-4).

Comparison of presented approximations of Prandtl's relation is given in figure 7.

Figure 7. Accuracy of some approximations to the Prandtl equation

With relative error of up to 0.15% to the Prandl equation, all approximations from table 1 can be used instead of implicit form of Prandtl's equation (or Colebrook's when ε/D=0).

Conclusion

Two new explicit approximations to the standard implicit

                                   

  .rs Abstract: Two new correlations of single-phase friction factor for turbulent pipe flow are shown in this paper. These two formulas are actually explicit approximations of iterative Colebrook's relation for calculation of flow friction factor. Calculated friction factor are valid for whole turbulent flow including hydraulically smooth and rough pipes with special attention on transient zone of turbulence between them. Hydraulically smooth regime of turbulence does not occur only in total absence of roughness of inner pipe surface, but also, four new relations for this theoretical regime are presented. Some recent formulas for turbulent flow friction calculation are also commented. Nomenclature: λ-Darcy, Darcy-Weisbach or Moody friction factor (dimensionless), Re-Reynolds number (dimensionless) ε/D-Relative roughness of inner pipe surface (dimensionless) s equation (1) is implicit in flow friction factor.

  for rough law (von Karman's equation) do not make such transitional curve. NPK equations represent only extremes of Colebrook formula. This is because the relative roughness of inner pipe surface (ε/D) is not only physical characteristic of pipe surface. It also depends on thickness of laminar boundary layer which occurs near pipe wall during the turbulent flow of fluid. So one extreme of turbulence is smooth flow in total absence of roughness (ε/D=0) covered by Prandtl's equation while other extreme is rough flow where boundary layer does not exist (ε/D→max for observed inner pipe surface) which is covered by von Karman's equation.

Figure 1 .

 1 Figure 1. Distribution of estimated relative error of approximation by Fang et al. (2011) compared with iterative solution of implicit Colebrook equation

Figure 2 .

 2 Figure 2. Approximation of Fang et al. (2011) compared with the most accurate approximations

Figure 4 .

 4 Figure 4. Distribution of estimated relative error of approximation by Ghanbari et al. (2011) compared with iterative solution of implicit Colebrook equation

Figure 5 .

 5 Figure 5. Distribution of estimated relative error of two new approximations compared with iterative solution of implicit Colebrook equation

Figure 1 .Figure 7 .

 17 Figure1. Distribution of estimated relative error of approximation by[START_REF] Fang | New correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing single-phase friction factor correlations[END_REF]

  Figure 1 DB Click here to download high resolution image

Detailed Response to Reviewers DB

  Moody diagram. I do not have deeper comments on Haaland or Barr approximation. 5. Colebrook paper is from Vol. 11, Issue 4 which was published in February 1939. Note that previous Issue 3 is in Vol. 10 which was published in January 1939. Issue 2 in Vol. 10 is the last from 1938. It is not usual practice, but this was in Journal of the ICE (Vol. 10 with issues 1 and 2 in 1938, Vol. 10 with issue 3 in 1939, Vol. 11 with issues 4, 5 and 6 in 1939, Vol. 12 with issue 7 and 8 in 1939) 6. My text is now with no informal expression. I hope that my English expression is now improved.

*

  Colebrook equation are shown. Their relative error over the entire practical domain of Reynolds number and relative roughness are up to 2%. Since the Colebrook equation is empirical this is good results. In that way, main goal which is avoiding of iterative procedure is fulfilled. Also, there are many explicit approximations to the implicit Prandtl equation in literature. The Prandtl equation is actually the Colebrook equation in total absence of inner pipe roughness. In some cases, an explicit approximation to the Colebrook equation can be easily used to develop an explicit approximation to the Prandtl equation. Procedure for this is explained in text, and vice versa, using some of the explicit approximations to the Colebrook equation known from literature, for new explicit approximations to the Prandtl equation were developed. With relative error of no more than 0.15%, these for equations are very accurate.

	List of figures:

Table 1 :

 1 Equations for hydraulically smooth regime developed for total absence of roughness (when ε/D=0) Equation Name

List of tables:

Table 1: Equations for hydraulically smooth regime developed for total absence of roughness (when ε/D=0)