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

Ph.D Dejan Brkić

Karman Pnk Prandtl-Nikuradse-Karman

Keywords: Accuracy, Comparison, Flow rate, Friction, Hydraulics, Piping Accuracy, Comparison, Flow rate, Friction, Hydraulics, Piping Mots clés: Precision, Comparaison, Debit, Frottement, Hydraulique, Tuyauterie NPK Nikuradse-Prandtl-

In some recent papers a few approximations to the implicit Nikuradse-Prandtl-Karman equation were shown. The Nikuradse-Prandtl-Karman equation for calculation of the hydraulic friction factor is valid for the hydraulically smooth regime of turbulence. Accuracy of these approximations for the friction factor in so called smooth pipes is checked and related problems from the hydraulics are analyzed in the spotlight of the recently developed equations. It can be concluded that pipes can be treated as smooth below certain value of the Reynolds number but after that even new polished pipes with a minor roughness follow the transitional and subsequently the rough law of flow at a higher values of the Reynolds number.

Introduction

Perfectly smooth surfaces do not exist [START_REF] Taylor | Characterization of the effect of surface roughness and texture on fluid flow-past, present, and future[END_REF]. Hydraulically smooth regime does not occur only in absence of the roughness (i.e. only when ε/D=0). This means that smooth regime can occur even if the relative roughness exists (if it is minor, i.e. if ε/D→0). This problem is shown in the spotlight of some recent new formulas.

Different hydraulic regimes

In their recent paper [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF] analyze the flow friction factor with the special attention to so called "smooth" pipes. They note that the implicit equation developed by [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF] is valid for rough pipes which should imply that its accuracy for "smooth" pipes can be disputed. The Colebrook equation is valid for the entire turbulent regime which includes the turbulent regime in the hydraulically smooth pipes, the transient (partially) turbulent regime and the fully turbulent regime in the hydraulically rough pipes. This is obvious from the title of the paper of Colebrook "Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws". The Colebrook equation is not valid for the laminar regime which occurs for approximately Re<2320. It is valid for 2320<Re<10 8 (the turbulent regime). It has to be noted that for the laminar regime, there are no smooth and rough pipes (Figure 1). Furthermore, in the laminar regime, all pipes are hydraulically smooth. If the pipe roughness (protrusions of inner pipe surface) is completely covered by the laminar sub-layer, the surface is smooth from the hydraulic point of view. In the laminar flow there is no laminar sub-layer, or better to say the main and only layer of flow is laminar, hence, the prefix "sub" is sufficient (there is no turbulent layer). In other words, in the laminar regime, all pipes are "smooth" as mentioned before. With further increasing of the Reynolds number, thickness of the laminar sub-layer decreases baring the protrusions and fluid flow through a pipe becomes consequently hydraulically smooth, and then gradually roughs, both from the hydraulic point of view (Figure 1). Hence the introductory turbulent flow through the rough pipes (because the perfectly smooth pipes do not exist) can be noted as the hydraulically smooth. In the turbulent regime a rough pipe can be treated as smooth or rough which depends on the circumstances (Figure 1).

Figure 1. Different hydraulic regimes

Accuracy of the Colebrook equation can perhaps be disputed, but up to date it has been an accepted standard for the calculation of the friction factor in the turbulent flow both in, "smooth" and rough pipes. The well known Rouse and Moody diagrams (or better to say, their turbulent part) had been constructed using Colebrook"s formula (1):

                 D 71 . 3 Re 51 . 2 log 2 1 10 (1)
For flow of natural gas or other gaseous fluids the coefficient 2.51 should be replaced with 2.825 according to the recommendation by AGA (American Gas Association) and American Bureau of Mines.

All equations in this short report are presented using the Darcy friction factor where the Darcy-Weisbach or the Moody friction factors are synonyms (here noted as λ). In the other hand, some researchers use the Fanning friction factor (f). This is also correct where the connection between these two factors is (λ=4•f). The Fanning friction factor is C f in [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF].

Determination of hydraulic regime

As noted before, the turbulent regime can be divided into the three sub-regimes; i.e. in the "smooth" (introductory) turbulent regime, the partially (transient) turbulent regime and the rough (fully) turbulent regime.

The turbulent regime usually occurs when Re>2300 or slightly above. The Reynolds number was introduced by Reynolds (1883a,b) first in the "Proceedings of the Royal Society" followed by a longer paper in the "Philosophical Transactions of the Royal Society". Special issue of the "Philosophical Transactions of the Royal Society" dedicated to these papers was published in 2008 with title "Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds" paper". Further about the history of the Reynolds number can be seen in [START_REF] Rott | A note on the history of the Reynolds number[END_REF].

As shown in Brkić (2011a), the smooth regime of turbulence occurs only if ξ<16 and if Re>2320 while the rough turbulent regime occurs if ξ>200. Between is the transient (partial) turbulent regime (16<ξ<200). The Reynolds number (Re) is a well known parameter while a parameter ξ is defined by (3):

      Re D (3)
The value of parameter ξ as defined is valid for the Darcy friction factor (regarding the value of ξ for the Fanning friction factor readers can consult Abodolahi et al. (2007) where the hydraulically smooth regime occurs if ξ f <8 and if Re>2320).

The NPK, i.e. the Nikuradse-Prandtl-Karman equation is Colebrook"s equation in the total absence of roughness (when ε/D=0). The implicit NPK equation ( 2) cannot be derived from the Colebrook equation if the relative roughness is very small (ε/D→0). But hydraulically smooth regime also occurs in the technical systems when the relative roughness is significantly low (ε/D→0) and not only in the absence of roughness as can be seen from the figure 2 (the hydraulically smooth regime exist not only when ε/D=0 and Re>2300, but also when ξ<16 and Re>2300).

  8 . 0 Re log 2 51 . 2 Re log 2 1 10 10                   (2)
The NPK equation is actually the PNK (Prandtl-Nikuradse-Karman) in [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF].

The Colebrook, as well as the NPK equation can be calculated only by using an iterative calculus or using approximate formulas. So, the main problem is not to find an approximate formula for the NPK equation which is valid only for the smooth part of turbulent regime, or better to say, for the turbulent regime in the absence of roughness (when ε/D=0). Problem is to find an approximate formula for the implicit Colebrook equation which is valid for the whole turbulent regime, including the smooth, transient and the rough portion of the turbulent regime.

Today, the NPK equation can be used only as an approximation for the Colebrook equation valid only for the "smooth" portion of the turbulent regime when the roughness can be neglected entirely (ε/D=0). This is a standard since 1939 when the paper of Colebrook was published and especially since 1944 when the paper of Moody was published [START_REF] Moody | Friction factors for pipe flow[END_REF].

Turbulent smooth regime

The NPK equation can be used only if ξ<16, calculated using the Darcy friction factor (and even then with disputed accuracy) and if Re>2300. As can be seen from figure 2, for ε/D=0.01, the upper limit for "smooth" regime is for Re≈6500 (the lower is 2300). This means that the NPK relation produces a relative error for this value of the relative roughness compared to the standard Colebrook equation of up to 24% (δ 1 from figure 2 with additionally Δδ 1 compared with the equation for the hydraulically smooth regime by Buzzelli). Similar, for e.g. ε/D=0.0005, "smooth" regime is up to Re≈2•10 5 (and not below 2300) where the relative error compared to Colebrook is up to 17% (δ 2 from the figure 2 with additionally Δδ 2 compared to the equation for the hydraulically smooth regime by Buzzelli). In theory, as relative roughness decreases (ε/D→0), the relative error also decreases rapidly (δ→0).

Figure 2. Examined hydraulic problem

When the turbulent smooth regime is indicated, it is better to use an equation with the relative roughness included, such as those by [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF] and [START_REF] Buzzelli | Calculating friction in one step[END_REF], and not those developed for use in the total absence of roughness. The bigger problem is how to measure or estimate the value of roughness accurately [START_REF] Sletfjerding | Friction factor directly from roughness measurements[END_REF][START_REF] Farshad | New developments in surface roughness measurements, characterization, and modeling fluid flow in pipe[END_REF]. [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF] have examined few equations valid for the "smooth" turbulent regime such as Blasius, Filonenko, etc (Table 1). Also, in their recent paper [START_REF] Danish | Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in smooth pipes using Adomian decomposition method[END_REF] used the Adomian decomposition method (ADM) and the Restarted Adomian Decomposition Method (RADM) to develop their explicit approximation of the NPK equation (Table 1). Also [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] have analyzed correlations of a single-phase friction factor for the turbulent pipe flow, also with the special attention to so called "smooth" pipes (Table 1). 2011) can be used to substitute the implicit NPK equation very accurately (these are actually non-iterative explicit approximations to the implicit NPK equation). But also, they are very complex, hence, the equation by Filonenko or even simple power-law equations such Blasius can be used as an adequate and not that complex substitution to the implicit NPK equation.

Turbulent smooth regime in the absence of roughness

Also as shown in Brkić (2011b), equations for the smooth regime in total absence of the roughness can be used as the base for the new approximations to the Colebrook equation if they are in suitable form for such transformation. Equations 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 in such suitable form.

Turbulent smooth regime with presence of roughness

The equations by [START_REF] Buzzelli | Calculating friction in one step[END_REF] are very accurate explicit approximation of the Colebrook equation. Also, in addition, [START_REF] Buzzelli | Calculating friction in one step[END_REF] presented his equation specially developed for the "smooth" conditions (3):

                       B B A A 18 . 2 1 Re log 2 1 10  (3)
Where parameters A and B are defined as (3a,b):

    41 . 1 Re ln 774 . 0    A (3a) A D B           51 . 2 7 . 3 Re  (3b)
Approach by [START_REF] Buzzelli | Calculating friction in one step[END_REF] is good one because he uses the relative roughness even for the "smooth" regime of turbulence (Figure 2).

New equations developed at the Princeton and the Oregon University

According to [START_REF] Barenblatt | Scaling laws for fully developed turbulent flow in pipes: discussion of experimental data[END_REF] and [START_REF] Cipra | A new theory of turbulence causes a stir among experts[END_REF], the new reexaminations of classical and historically adopted relations for determination of hydraulic friction factor show that some of them are off by as much 65%. As noted in [START_REF] Cipra | A new theory of turbulence causes a stir among experts[END_REF], it seems that the many classical textbooks for hydraulics will have to be revised. For example, recent experiments at the Princeton University have revealed aspects of the smooth pipe flow behavior that suggest a more complex scaling than previously noted. The Princeton research shows that in the partially turbulent regime friction factor relationship follows an inflectional rather than the monotonic relationship given in the Moody diagram. Researchers from the Princeton concluded that friction factor behavior of a honed surface in the transitional regime does not follow Colebrook relationship and that for all conditions of roughness, logarithmic scaling was apparent at the higher Reynolds numbers with the same constants determined for smooth pipes. Another team at the Oregon University, working with a completely different type of facility have come to a similar conclusion. Note that the difference in scale of the Oregon and the Princeton devices is dramatic: for example, Princeton"s Superpipe weighs about 25 tons, whereas the Oregon tube weighs about 30 grams [START_REF] Mckeon | Friction factors for smooth pipe flow[END_REF]. Another interesting conclusion from [START_REF] Cordero | An improved experimental correlation for Darcy friction factor[END_REF] is that the power-law represents the velocity profile better than the logarithmic law for the Reynolds numbers below approximately 98 thousands.

Blasius's law (related to the power-law velocity profile) is considered more accurate than the NPK log-law in that region. From a practical point of view, it is best to apply Blasius's law up to Re=66,964 where it coincides with results by [START_REF] Mckeon | Friction factors for smooth pipe flow[END_REF][START_REF] Mckeon | A new friction factor relationship for fully developed pipe flow[END_REF]

Comparison of different formulas

From figure 3 and4, it can be seen that different equations produce the different results. But from figure 2, it also can be seen that these differences have a minor or none influence on the calculation. From figure 2, it can be clearly seen that only the effect of roughness can make an influence on the final results (to increase the accuracy of the final results). [START_REF] Churchill | Friction factor equation spans all fluid-flow regimes[END_REF] is valid for the turbulent and even the laminar flow of the Newtonian fluid (including the zone between them). According to the recent paper by Brkić (2011c), the approximations of the Colebrook equation 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], [START_REF] Serghides | Estimate friction factor accurately[END_REF], [START_REF] Zigrang | Explicit approximations to the solution of the Colebrook"s friction factor equation[END_REF] and [START_REF] Vatankhah | Discussion of "Turbulent flow friction factor calculation using a mathematically exact alternative to the Colebrook-White equation[END_REF] are among the five most accurate up to date.

Their relative error is no more than 0.15% compared to the iterative solution of the implicit Colebrook equation (for the whole turbulent regime). The other three approximations mentioned in the paper of [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF] are not among the most accurate. These mentioned approximations are by [START_REF] Haaland | Simple and explicit formulas for friction factor in turbulent pipe flow[END_REF] with the relative error of no more than 1.5%, [START_REF] Swamee | Explicit equations for pipe flow problems[END_REF] with the relative error of no more than 2.5% and [START_REF] Avci | A novel explicit equation for friction factor in smooth and rough pipes[END_REF] with the relative error up to 5%.

Measuring the CPU time is a good approach for a comparison of the formulas in hydraulics [START_REF] Giustolisi | Some explicit formulations of Colebrook-White friction factor considering accuracy vs. computational speed[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][START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF]. Also, one has to be aware that computational speed does not depend only on the problem size but also on the computing environment (the type of CPU or other hardware components). [START_REF] Giustolisi | Some explicit formulations of Colebrook-White friction factor considering accuracy vs. computational speed[END_REF] observe that the computation of logarithm in the computer languages is based on the series of expansions that require several powers of the argument to be computed and added to each other. Note that the explicit equations proposed by [START_REF] Danish | Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in smooth pipes using Adomian decomposition method[END_REF] and [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF] contain many logarithmic expressions. Approximations to the Colebrook equation examined by Brkić (2011c) are sorted based on their accuracy and the complexity as the criterions. The measurement of the CPU time can be a further step forward.

The Colebrook equation can also be transformed and approximately solved by using the Lambert W-function as shown in Brkić (2011d,e). The Lambert W function is also mentioned in [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF].

Conclusion

It can be concluded that pipes can be treated as the smooth below certain value of the Reynolds number but after that even the new polished pipes with a minor roughness follow the transitional and subsequently the rough law of flow at the higher values of the Reynolds number. Today, the Colebrook equation is a standard for the calculation of flow friction factor with the particular reference to the transition region between the smooth and the rough pipe laws. It is implicit in the flow friction factor, but nowadays it can be solved easily using an iterative procedure or some of the very accurate approximations.

The Colebrook equation can be replaced with some of the new formulas like those developed from the data of the Princeton or Oregon pipe facility [START_REF] Cordero | An improved experimental correlation for Darcy friction factor[END_REF], but the effect of the roughness cannot be neglected. Unavoidable the effect of roughness is the main reason why the Colebrook equation is a standard even for the smooth portion of turbulence. The roughness effect can be minor, but with slightly increased value of the Reynolds number, it will inevitable appear. a Also note that ln(Re) 2 and ln(Re) 3 are actually ln 2 (Re) and ln 3 (Re), respectively, i.e. (ln(Re)) 2 and (ln(Re)) 3 . According to equation 7 in [START_REF] Li | A new explicit equation for accurate friction factor calculation of smooth pipes[END_REF], one can assume that using logarithmic rule, ln(Re) 2 and ln(Re) 3 can be rearranged to 2•ln(Re) and 3•ln(Re), respectively which produce error (because 2 and 3 are power of whole logarithm and not only of Re)

References:

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

  thus providing a continuous friction factor. McKeon et al. (2004) adjusted parameter in the NPK equation to fit the results obtained from the Princeton and the Oregon pipe facility (4): with the coefficients adjusted by McKeon et al. (2004) is also implicit in flow friction factor, hence the methodology proposed by Li et al. (2011) and Danish et al. (2011) can be applied to it. The Princeton and the Oregon equations for the fully developed pipe flow can be seen in McKeon et al. (2005). These equations for the fully developed pipe flow can be used as an improved substitution for the Colebrook equation valid for the whole regime of turbulence.

Figure 3 .

 3 Figure 3. Maximal relative error of presented equations for hydraulically smooth regime

Figure 4 .

 4 Figure 4. Comparison of the most accurate explicit approximations of the NPK equation

  Figure 1 DB Click here to download high resolution image

Table 1 :

 1 Equations for hydraulically smooth regime developed for the total absence of roughness According to figure 2, equations proposed by Li et al. (2011), Danish et al. (2011), and Fang et al. (

Table 1 :

 1 Equations for hydraulically smooth regime developed for the total absence of roughness Equation Name

			25 . 0 316 Re . 0

Table 1 DB Graphical abstract DB Click here to download Electronic Annexes: Graphical abstract DB.tif

 1


Comparison of the Lambert W-Function Based Solutions to the Colebrook Equation

Keywords: Fluid flow, Friction factor, Turbulent flow, Hydraulic resistances, Colebrook equation, Lambert W-function Fluid flow, Friction factor, Turbulent flow, Hydraulic resistances, Colebrook equation, Lambert W-function

Comparison of the Lambert W-function based solutions to the Colebrook equation

Introduction

The empirical Colebrook equation is valid for turbulent regime in smooth and rough pipes with special attention on transitional regime between them (1).

      ⋅ + ⋅ ⋅ - = D 71 . 3 Re 51 . 2 log 2 1 ε λ λ (1)
The Colebrook equation has two parts, i.e. turbulent smooth and turbulent rough part which can be noted separately. The Colebrook equation unites these two parts in one coherent equation but also Colebrook was not attempting to find some sort of approximation to a sum of the smooth and rough expressions. Actually, Colebrook was seeking a combined equation that is asymptotic to the smooth and rough expressions. This new coherent equation covers transitional zone between smooth and rough regime. First addend in bracket of previous equation is implicitly

given and it represents smooth regime (2).

( ) 8 . 0 Re log 2 Re 51 . 2 log 2 1 - ⋅ ⋅ =       ⋅ ⋅ - = λ λ λ (2)
Similarly the rough portion of turbulence is represented by second addend of the Colebrook equation (3).

      ⋅ - =       ⋅ ⋅ - =       ⋅ ⋅ - = D D D ε ε ε λ log 2 14 . 1 2 log 2 74 . 1 71 . 3 log 2 1 (3)
Mathematically, shown unification is incorrect i.e. log(A+B)≠log(A)+log(B), but physically this approach gives good results. Problem can be treated as inverse; according to the logarithmic [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF] who used data from the experiment conducted by his colleague, Professor White and himself [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]. Later in 1940's, Rouse and Moody used Colebrook's equation for their famous charts [START_REF] Moody | Friction factors for pipe flow[END_REF]. Since then, the Colebrook equation has been historically adopted as sufficiently accurate for engineering applications although it is often incorrect. Even [START_REF] Moody | Friction factors for pipe flow[END_REF] stated the accuracy of his diagram is about ±5% for smooth pipes and ±10% for rough pipes. In spite of that, the Colebrook equation is replaced very slowly by new equations such as those developed using the data from the Princeton super-pipe or the Oregon facility [START_REF] Cordero | An improved experimental correlation for Darcy friction factor[END_REF]. Actually, many possible equations can be found that are asymptotic to the smooth and rough expressions. Colebrook chose one and compared it with limited experimental data available to him. Since different types of roughness cause different forms of transition from rough to smooth, it is clear that the Colebrook equation should not be regarded as special. It was the first and that it is now widely used even though hardly anybody seems to acknowledge its considerable limitations. Alternate mathematically equivalents to the original expression of the Colebrook equation in explicit form with no approximation involved actually exist [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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF][START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF], Brkić 2011a). These alternate equations were developed using the Lambert W-function. Some of the existed transformations cannot be applied for high values of relative roughness of inner pipe surface and the Reynolds number [START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF]). This limitation is applied only for computer computations. Other presented transformations do not sufferer of such limitation as will be shown in this paper.

Derivate of explicit Lambert W-function based Colebrook equation

The Lambert W-function is define by relation W(x)•e W(x) =x, where W is the Lambert function and x is its argument [START_REF] Corless | On the Lambert W function[END_REF][START_REF] Hayes | Why W?[END_REF]. For the real values of the argument x, the W-function has two branches, W 0 (the principal branch) and W -1 (the negative branch). Only upper branch W 0 is used for the transformations of Colebrook's equation in explicit form (Figure 2).

             = ⋅ ⋅ ⇒         ⋅ - =         ⋅ ⋅ =       ⋅ ⋅ = ⇒       ⋅ ⋅ = ⇒       ⋅ = Φ ⇒ = +       ⋅ = Φ + Φ ⇒       ⋅ = Φ ⋅ ⋅ + Φ ⋅ ⇒ ⋅ = Φ ⇒ =         ⋅ -       ⋅ - =         ⋅ ⋅ - =         ⋅ ⋅ - =         ⋅ ⋅ -       ⋅ - λ λ λ λ λ λ λ λ
W a W a a W x x W x W a a a a a a a a a a (4)
After that implicit expression in the Colebrook equation can be replaced with the Lambert Wfunction while its second so called "rough" addend remains untouched (5).

         ⋅ =             ⋅ + ⋅       ⋅ ⋅ ⋅ - =
         ⋅ = ⋅ = ⋅ ⋅ -       ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ D x D x e x x x D e x W 71 . 3 1 71 . 3 02 . 5 ln(10) Re 51 . 2 Re 71 . 3 ) 10 ln( 2 1 ε ε ε λ (6)
In previous equation ( 6) argument of the Lambert W-function has changed. It is now noted as x 1 where this new argument contains the old argument x from the set of equations ( 5). However, the numerical solution of eq. ( 6) using computers is not possible for all values of relative roughness and the Reynolds number and their combinations which can be occurred in engineering practice.

Problem is in the exponential term in the argument x 1 of the W-function which increases rapidly.

In fact, infinite combinations of relative roughness and the Reynolds number values exist for which the argument x 1 of the W-function becomes too large even for today very powerful computers. Argument of the Lambert W-function has to be smaller than 1.79•10 308 because registers in memory of computers have limited capacity [START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF].

Solutions of alternate equations, accuracy requirements and error distribution

The main aim of this paper is not to investigate accuracy or computational burden (i. Since the Lambert W is transcendental function, its formal solution can be expressed only in endless form (7). Similar as using iterative solution, accuracy of formal solution depends on number of terms used to form "tower" in eq. ( 7).

( )

                      = ) ln( x ln x ln x ln x W O (7)
After [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF], instead W=W(x) it is convenient to define a new function ω=ω(y) as auxiliary tool (8).

( ) (

)( ) [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF] proposed approximate solution for the upper branch of the Lambert Wfunction which is valid for solution of here presented problem (9). Also, improved version eq. ( 9) can be seen in [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF].

( )                    - - - = Ω + ⋅ = ⋅       - = Ω ⋅ + - + = + - = - ⋅ = ⋅ + ≈ ⋅ + = + = - ⋅ - +       - ⋅ - - - i y i i i
( )                                                         ⋅ +       ⋅ ≈ + 5 12 1 ln 5 12 ln 5 6 ln 0 x x x x W (9)
Winitzki ( 2003) also proposed global approximation to the Lambert W-function ( 10).

( ) ( ) ( ) ( ) ( )         + + + + - ⋅ + ≈ x 1 ln 2 x 1 ln 1 ln 1 x 1 ln x W (10)
Two numerical examples are solved in tables 1 and 2 using formulas shown in this section of the text. These two examples are marked also in figure 1 and 3 to 9. The implicit Colebrook equation today can be easily solved using some kind of spreadsheet solvers (iterative solution). Here will be used as standard, solution calculated using MS Excel ver. 2007. Since iterative solution after enough number of iterations can be accepted as accurate, it will be used for comparisons in examples 1 and 2. Approach with set of equations (5) will be used in table 1 while approach with set of equations ( 6) will be used in table 2. Main difference between the calculation in table 1 and table 2 is in the argument of the Lambert W-function x and x 1 , respectively. Argument x is never too large for computer register while in other approach, argument x 1 is. Of course, argument x is not too large only if the Reynolds number and relative roughness are in the range which occurs in engineering practice.

Error distribution for solution of equations ( 5) and ( 6) using procedure by [START_REF] Boyd | Global approximations to the principal real-valued branch of the Lambert Wfunction[END_REF] is shown in figures 3 and 4, respectively. Similar for the procedure proposed by [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF] and [START_REF] Winitzki | Uniform approximations for transcendental functions[END_REF] error distribution can be seen in figures 5 and 6, and 7 and 8. 6) by [START_REF] Winitzki | Uniform approximations for transcendental functions[END_REF] The distribution of error shown in figures 3, 5 and 7 are for the set of equations ( 5). Distribution of error is sensitive and therefore similar set of equations ( 11) produce different distribution of error (Figure 9) but calculation is still possible for all practical range of the Reynolds number and relative roughness. 11) by [START_REF] Barry | Analytical approximations for real values of the Lambert W-function[END_REF]

       ⋅ =         ⋅ + ⋅ - =       ⋅ -

Conclusion

The Colebrook equation can be expressed in explicit form using the Lambert W-function.

Related approximate solutions of the Lambert W-function are shown in this paper. Accuracy of some available Lambert W based forms of the Colebrook equation from literature is investigated [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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF][START_REF] Nandakumar | Analytical calculation of fanning friction factor[END_REF], Brkić 2011a). Also, related approximate solutions of the Lambert W-function are shown. Some already known limitation in use of some of available formulas is confirmed [START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF]. Other available formulas do not suffer of these limitations for practical use (Brkić 2011a). The maximal error of these new formulas compared with iterative solution of Colebrook equation is up to 2% or [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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF], [START_REF] Nandakumar | Analytical calculation of fanning friction factor[END_REF] cannot be used for all practical combination of the Reynolds number and relative roughness (Figures 4,6,8). Reason is that typical IEEE-compliant machines cannot operate with numbers larger than about 10 308 (Figure 10). (2011b). It has to be said that use of an explicit approximation do not necessary mean that the computational burden is reduced in comparison to iterative approach [START_REF] Giustolisi | Some explicit formulations of Colebrook-White friction factor considering accuracy vs. computational speed[END_REF].

Actually, main purpose of this paper is to compare the different formulations of Lambert W based Colebrook equation available in literature [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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF][START_REF] Nandakumar | Analytical calculation of fanning friction factor[END_REF], Brkić 2011a). The paper compares various implementations of the Lambert W methodology and shows that some of these are less able than others to yield solutions using modern computer hardware. This is because the functions require the evaluation of terms with numerical values outside the ranges that can be expressed on most computers. Approach by Brkić (2011a) does not suffer of that constraints and hence it can be recommended since it can be use for the whole range of the Reynolds number and relative roughness that can be occured in engineering practice. Approach by [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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF] (2007) suffers of certain limitations as it is explained in this paper as well as in [START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF]. Decision whether someone will use the Lambert W based solution proposed by Brkić (2011a), iterative solution or some of explicit approximate formulas presented in [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF], requires deeper analysis.

Nomenclature

  incorrect to split the Colebrook relation into two pieces. How well Colebrook's equation smoothing contact between the turbulent smooth and the turbulent rough regime can be best seen in the graphical interpretation (Figure1).

Figure 1 .

 1 Figure 1. Colebrook relation make transitional curve between hydraulically smooth regime and

Figure 2 .

 2 Figure 2. Detailed view of the Lambert W-function

  (5) was developed using the original idea of Colebrook to bridge the gap smoothing the line between hydraulically "smooth" turbulent flow and turbulent flow in rough pipes as shown before in figure1. No constraints in ranges for values of the Reynolds number and relative roughness in comparisons to the original Colebrook relation do not exist (as will be shown in this paper). But, beside presented reformulations of the Colebrook equation, in the papers of[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 | Explicit reformulation of the Colebrook-White equation for turbulent flow friction factor calculation[END_REF] and[START_REF] Nandakumar | Analytical calculation of fanning friction factor[END_REF], another possible transformation of the Colebrook equation using the Lambert-W function is shown. But relation (6) shown in these papers has limitation in applicability for high values of the Reynolds number and relative roughness as shown in[START_REF] Sonnad | Constraints for using Lambert W Function-based explicit Colebrook-White equation[END_REF] because today available computers cannot operate with the extremely large numbers.

  e. speed) of presented procedures for solution of the Lambert W based Colebrook equation. Accuracy of alternate representations of the Colebrook equation depends only on accuracy of used approximate solution of the Lambert W-function. Besides the relative simplicity of an explicit form, presented equations allow highly accurate estimation of friction factor as the Lambert Wfunction can be evaluated accurately. But, the main purpose of this paper is to compare different Lambert W based solutions of the Colebrook equation and to make comparisons among them. Or

  say, main purpose is to identify some constraints in applicability of certain solutions whereas other solutions do not suffer of such limitations.[START_REF] Clamond | Efficient resolution of the Colebrook equation[END_REF] provides MATLAB and FORTRAN codes for the Colebrook relation expressed in term of the Lambert W-function.[START_REF] Keady | Colebrook-White formulas for pipe flow[END_REF] also provides Maple interpretation of here presented problem.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 345678 Figure 3. Distribution of error of solution of explicit Lambert W-function based Colebrook

Figure 9 .

 9 Figure 9. Distribution of error of solution of explicit Lambert W-function based Colebrook

  Figures 3, 5 and 7). Error distribution of new formulas (set of equations 5 and 11) is stabile or better to say, distribution of error does not change with change of procedure for solution of the Lambert W-function. Transformation of the Colebrook equation based on the Lambert W-function by

Figure 10 .

 10 Figure 10. Constraints for using of a possible Lambert W function-based explicit Colebrook

Figure 1 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 1345678910 Figure 1. Colebrook relation make transitional curve between hydraulically smooth regime and turbulent rough regime 65x82mm (300 x 300 DPI)

Table 1 .

 1 Solutions of the explicit, Lambert W-function based Colebrook equation (5)

	Limitation of formulas presented by set of equations (6) for high values of the Reynolds number
	and relative roughness is shown in table 2.

Table 2 .

 2 Solutions of the explicit, Lambert W-function basedColebrook equation (6)

Table 1 .

 1 Solutions of the explicit, Lambert W-function basedColebrook equation (5)

				example 1 Re=316227, ε/D=5·10 -4	example 2 Re=10 6 , ε/D=10 -2
	Eq. (7)
	x			145047.72	458682.29
	W λ -0.5 after eq. (5)	9.620883753 7.393176451	10.66879129 5.13128464
	λ			0.018295229	0.037979373
	Relative error δ %	0.8815%	0.1294%
	Set of eqs. (8)
	x			145047.72	458682.29
	y			394281.59	1246828.73
	ω 0		10.32879411	11.39448775
	Ω			0.000065130	0.0000088393
	ω	0		10.32946683	11.39458847
	ω 1		10.67279093	11.7140380025
	ω 2		10.62232833	11.6698835813
	ω 3		10.62088328	11.6687910258
	ω 4		10.62088214	11.6687903773
	ω 5		10.62088214	11.6687903773
	W λ -0.5 after eq. (5)	9.62088214 7.39317599	10.6687903773 5.131284633
	λ			0.018295232	0.037979373
	Relative error δ %	0.8815%	0.1294%
	Eq. (9)
	x W + 0	( ) x	145047.72 9.743385337	458682.29 10.79550445
	λ -0.5 after eq. (5)	7.389556742	5.131196922
	λ			0.018313157	0.037980671
	Relative error δ %	0.9803%	0.1328%
	Eq. (10)
	x			145047.72	458682.29
	W(x) λ -0.5 after eq. (5)	9.696953700 7.390934613	10.74585341 5.131231501
	λ			0.018306330	0.037980159
	Relative error δ %	0.9427%	0.1315%
	Iterative solution of eq. (1) λ -0.5	7.425691007	5.134604649
	λ o		0.018135362	0.037930274
	Relative error δ %	-	-
	δ % =100•|λ o -λ|/λ o

Table 2 .

 2 Solutions of the explicit, Lambert W-function based Colebrook equation (6) % =100•|λ o -λ|/λ o where λ o is iterative solution of eq. (1) from Table1

			example 1 Re=316227, ε/D=5·10 -4	example 2 Re=10 6 , ε/D=10 -2
	Eq. (7)
	x 1	44791026258619.00
	W λ -0.5 after eq. (6) λ Relative error δ %	28.09735354 7.425691011 6.39•10 -8 % 0.018135363	Computer calculation not possible
	Set of eqs. (8)
	x 1	44791026258619.00
	y		121754632756837.00
	ω 0 Ω	28.95385161 1.3882775•10 -31
	ω	0	28.95385161
	ω 1 ω 2 ω 3	29.1085521825 29.0974181464 29.0973535390	Computer calculation not possible
	ω 4	29.0973535368
	ω 5	29.0973535368
	W λ -0.5 after eq. (6)	28.0973535368 7.42569100698
	λ Relative error δ %	0.01813536298 4.63•10 -8 %
	Eq. (9)
	x 1 W + 0 λ -0.5 after eq. (6) ( ) x λ	44791026258619.00 28.253824116 7.5615996248 0.0174893086	Computer calculation not possible
	Relative error	3.5624%
	Eq. (10)
	x 1	44791026258619.00
	W(x) λ -0.5 after eq. (6) λ	28.161979769 7.4818246387 0.0178642566	Computer calculation not possible
	Relative error δ %	0.4949%
	δ

http://mc.manuscriptcentral.com/engcom Engineering Computations


Dr Dejan Brkić
email: dejanrgf@tesla.rcub.bg.ac.rs

Review of explicit approximations to the Colebrook relation for flow friction 1

Keywords: Colebrook equation, Approximations, Fluid mechanics, Turbulent flow, Friction coefficient, Darcy friction factor, Hydraulic resistances, Excel spreadsheet Colebrook equation, Approximations, Fluid mechanics, Turbulent flow, Friction 21 coefficient, Darcy friction factor, Hydraulic resistances, Excel spreadsheet

Because of Moody's chart as demonstrated applicability of the Colebrook equation over a very wide range of Reynolds number and relative roughness values, this equation become the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can causes problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is main reason for attempting to develop a relationship that is a reasonable approximation for the Colebrook equation but which is explicit in friction factor. Review of existing explicit approximation of the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative solution of implicit Colebrook equation is shown for entire range of turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that most of available approximations of the Colebrook equation, with few exceptions, are very accurate with deviation of no more than few percentages.

Because of Moody's chart as demonstrated applicability of the Colebrook equation over a very wide range of Reynolds number and relative roughness values, this equation become the accepted standard of accuracy for calculated hydraulic friction factor. Colebrook equation suffers from being implicit in unknown friction factor and thus requires an iterative solution where convergence to 0.01% typically requires less than 7 iterations. Implicit Colebrook equation cannot be rearranged to derive friction factor directly in one step. Iterative calculus can causes problem in simulation of flow in a pipe system in which it may be necessary to evaluate friction factor hundreds or thousands of times. This is main reason for attempting to develop a relationship that is a reasonable approximation for the Colebrook equation but which is explicit in friction factor. Review of existing explicit approximation of the implicit Colebrook equation with estimated accuracy is shown in this paper. Estimated accuracy compared with iterative solution of implicit Colebrook equation is shown for entire range of turbulence where Moody diagram should be used as the reference. Finally, it can be concluded that most of available approximations of the Colebrook equation, with few exceptions, are very accurate with deviation of no more than few percentages.

I wish to thank reviewers and editors for valuable comments.

Reviewer 1:

1. I hope that my expression of English is now better. 1. Abstract was rearranged now as you suggested (last two sentences). Please, see also Research highlights. My text has now graphical abstract also….

2. I disagree with your suggestion to reduce number of approximation shown in my review. Honestly, I have considered your suggestion seriously, but in the meantime I have found new approximations such as Papaevangelou et al (2010) and Vatankhah andKouchakzadeh (2008, 2009). They are very accurate. Therefore, I have added these two approximations... 3. I agree with you that problem of range of applicability of certain approximation is very important. This problem is very complex. So to avoid any further speculations I have added diagram of accuracy of each approximation over the entire practical range of Reynolds number and relative roughness. According to these diagrams, one can choose will he/she use in certain case this particular approximation or not.

4. Detailed discussion with results is now presented in more appropriate way. 5. Conclusion is now rearranged after your suggestions.

Introduction 1

Difficulty of solving turbulent flow problems in pipes lies in the fact that hydraulic friction factor 2 is a complex function of relative surface roughness and Reynolds number. Precisely, hydraulic 3 resistance depends on flow rate. Similar situation is with electrical resistance when diode is in a 4 circuit. Furthermore, being more complex, widely used empirical Colebrook equation is 5 transcendental which means that it cannot be solved by using only elementary functions and 6 basic arithmetic operations in definitive form. Problem is that, since the Colebrook equation is 7 implicit (i.e. both the right and left-hand terms contain friction factor), containing the unknown 8 friction factor in implicit form, the Reynolds number and the pipe roughness, it has to be solved In laminar flow, all pipes behave as smooth but Colebrook equation is not valid for this regime.

9

One of the presented approximations in this paper includes laminar regime (Churchill 1977).

                 
(1)

3

The development of approximate equations for the calculation of friction factor in rough pipes

6                                 D log 2 14 . 1 D 2 log 2 74 . 1 D 71 . 3 log 2 1 10 10 10
(2)

7
Greek ε is the equivalent Nikuradse"s sand-grain roughness value for the inner surface of pipe (or 8 so called uniform roughness). Prandtl"s and von Karman"s relations are also known as NPK

17                  D 71 . 3 Re 51 . 2 log 2 1 10 (3) 18
Colebrook equation also can be noted as (3a): 4) recommended by the 18 American Gas Association (AGA) in case of natural gas pipelines calculations, using 2.825 19 constant instead of 2.51 (Haaland 1983, Coelho andPinho 2007). This procedure produces 20 maximal deviation up to 3.2% (Figure 3). Keady 1998, More 2006, Nandakumar 2007, Sonnad and Goudar 2004, 2005, 2006, 2007).

19                Re

21

                 D 71 . 3 Re

16

Approximations proposed by Brkić (2011a) were also developed using Lambert W-function and 17 its solution proposed by Barry et al (2000). Further about Lambert-W function, readers can see in How well Colebrook equation fits the experimental data is beyond the scope of presented 1 approximations. Perhaps one of these equations even fits the available data better than the 2 Colebrook equation. Until the comparison is made with real, measured values, however, this will 3 not be known. According to Cipra (1996), some of the key formulas of turbulence are off by as 4 much as 65%. Yoo andSingh (2004, 2010) found that the Colebrook equation produced an 5 average error of more than 11% while the roughness height of commercial pipes varied quite 6 significantly, depending on the pipe size and type. 6). Its accuracy is not improved compared with Moody approximation.

                           
9 V 44 . 0 225 . 0 Re D 88 D 53 . 0 D 094 . 0                               (6) 10
Where V is (7):

11 134 . 0 D 62 . 1 V          (7)
12 Estimated error of Wood approximation is up to 23.79% compared with implicit Colebrook 13 equation (Figure 5).

Eck approximation 19

Approximation proposed by Eck (1973) is most simple but not very accurate (8), but better than 20 those by Moody (1947) andWood (1966). 6). (1976) andJain (1976). It is first approximation with improved accuracy (9): proposed by Churchill (1973) and Swamee and Jain (1976): (including unstable zone between them) with error up to 2.19% (12):

21             
9                     
6                     
13   12 1 5 . 1 2 1 12 C C 1 Re 8 8                    (12) 14
Where C 1 is (13):

15 16 9 . 0 1 D 27 . 0 Re 7 1 ln 457 . 2 C                          (13) 16
And C 2 is (14): First, really accurate approximation (15) was developed by Chen (1979). 1980) is relative simple but not very accurate ( 16):

6                                 
18                            5 . 6 D Re 135 . 0 Re log 8 . 1 1 10 (16)
12 Estimated error of Round approximation is up to 10.92% compared with implicit Colebrook 1 equation (Figure 9). 18) is less accurate than ( 19), since the 2 first one is based on two internal iterations while the second one uses three internal iterations.

                                               
19                          Re 13 D 7 . 3 log Re 02 . 5 D 7 . 3 log 2 1 10 10 (18) 20 13                                     
3 Estimated error of more complex but also more accurate approximation by Zigrang and Sylvester 4 (1982) is up to 0.13% compared with implicit Colebrook equation (Figure 11). For simpler form 5 of Zigrang and Sylvester approximation error is up to 1% (Figure 11).

13                             n n
12   2 1 2 3 2 1 2 1 S S 2 S S S S                 (21) 13   2 1 2 2 1 781 . 4 S 2 S 781 . 4 S 781 . 4                 (22) 14
Where S 1 is (23):

15             Re 12 D 7 . 3 log 2 S 10 1 (23) 16
Where S 2 is (24): 17 1 Cronologically, after approximation by Serghides (1984) are approximations proposed by Chen (1984,1985). But these approximations are similar with Altshul, Russian power-law equation from Soviet practice and therefore they will be presented with this equation later in the text. Note also that author of Chen approximation from 1979 is Chen N.H. (Chen 1979), while the author of Chen approximations from 1984 is Chen J.J.J. (Chen 1984).

             Re S 51 . 2 D 7 . 3 log 2 S 1 10 2 (24) 1 Where S 3 is (25): 2              Re S 51 . 2 D 7 . 3 log 2 S 2 10 3 (25) 3
Form of approximation by Serghides (1984) ( 21) is more accurate than ( 22), since the first one is 4 based on three internal steps while the second one use two internal steps. More complex version 5 is with accuracy up to 0.13% while less complex one is up to 0.35% (Figure 14). is up to 0.13% compared with implicit Colebrook equation (Figure 16). 2006) as (29a, 29b), to increase its accuracy with error up to 0.15% (Figure 18). These 5 two improved equations here will be noted as approximations by Vatankhah and Kouchakzadeh 6 (2008,2009).

2                                                     
7                   9633 . 0 G / G 31 . 0 G Re 4587 . 0 ln 8686 . 0 1 (29a) 8     2 98 . 0 G / G 28 . 0 G Re 4587 . 0 ln 8686 . 0                        (29b)
                                      Re Re
                   (31)
1 Φ(Re) can be neglected in most cases. Some additional details can be seen in paper of Rao and 2 Kumar (2009). Estimated error of Rao and Kumar approximation is up to 82% compared with 3 implicit Colebrook equation (Figure 19).

11                           2 2 10 1 1 B 18 . 2 1 Re B log 2 B B 1 (32) 12
Where B is (33): Colebrook equation (Figure 20).

13                 D 32
                                   (35) 7
Estimated error of approximation by Avci and Karagoz ( 2009) is up to 4.7% compared with 8 implicit Colebrook equation (Figure 21).

                      (
5                 D 71 . 3 10 log 2 1 4343 . 0 10 (37) 6                D 71 . 3 Re 18 . 2 log 2 1 10 (38) 7
Where β is (39):

8                 Re 1 . 1 1 ln Re 1 . 1 ln 816 . 1
Re ln (39) 9 Estimated error of Brkić approximation is up to 2.3% compared with implicit Colebrook 10 equation (Figure 23). Additionally (Barry et al. 2000), error can be reduced using (39a) or (39b): There is no special explanation for different coefficients in some equations (e.g. 3.7065 or 3.707 11 instead of 3.71 etc). Only reasonable explanation can be that this changed coefficients maybe [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. This comparative study is based on some particular conditions of test grid 17 points which means that presented relative error could be different using different check points.

16                    917365 . 0 Re 1 ln 917365 . 0 Re ln 4586887 . 0 4586887 . 1 1 (39a) 17                             
                            Re 458

18

But here presented analysis with 592 and 740 check points give good general picture of accuracy 19 for the presented approximations. Relative error is not distributed systematically over the entire 20 range of Reynolds number and relative roughness which means that real maximal relative error 21 can be slightly above here reported. MS Excel file is available as electronic appendix2 to this 1 paper and in this file any other values for Reynolds number and for relative roughness can be 2 used as input parameters. Modern software, especially in the case of spreadsheets, can be 3 conveniently programmed to solve any implicit equation, with a minimum of programming and 4 details. Some of MS Excel effectiveness for numerical computations comes from a module 5 "Solver". It was originally designed for optimization problems, where one has to find values of a 6 number of different parameters such that some quantity is minimized, usually the sum of errors 7 of a number of equations. With this tool one can find such optimal solutions, or solutions of one 8 or many equations, even if they are nonlinear. In more details, to allow iterative computations in 9 MS Excel 2007, one has to choose "Excel options", and then in "Formulas" to tick box "Enable 10 iterative calculation". In this case, Excel is set to terminate the calculation after maximum 3·10 4 11 iterations or 1·10 -7 difference or less between the values of two successive iterations.

12 Consequently, deviations involving explicit equations have been reported to 1·10 -5 %. This means 13 that real relative error is presented by sum of calculated relative error and deviation. So deviation 14 in "Excel options" has to be set to be significantly smaller compared to estimated error of 15 observed approximations ("Maximum Change" in "Formulas"). 25). Approximations proposed by Moody (1947), Wood (1966), Eck 3 (1973) and Round (1980) should not be used because they produce significant relative error 4 (Figure 26). Moderate accurate approximations such as proposed by Jain (1976), Swamee and 5 Jain (1976( ), Churchill (1973( , 1977)), Manadilli (1997), Brkić (2011a) and Avzi and Karagoz 6 (2009), can be used since they made maximal relative error up to 5% (Figure 27). any desired degree of precision. For many applications, the simpler but less accurate explicit 8 equation will be sufficed. Sometime, simplicity is sacrificed for excessive accuracy. To find 9 balance between these two extremes it is appropriate to introduce concept of complexity or 10 complexity index of explicit approximations. Zigrang and Sylvester (1985) defined complexity 11 as the number of algebraic notation calculator key strokes required to solve the equation for 12Re=10 5 and ε/D=0.001 (Table 2). Complexity index is defined as quotient of key strokes 13 required for an observed approximation and the least complex one (Table 2).

5 4 1 Re 68 D 11 . 0            (41) 6 4 1 Re 100 D 46 . 1 1 . 0             (42) 7 8
Altshul formula was eliminated from the last wording of the Russian norms. However, it is used 9 as before since other recommendations were not proposed (Sukharev et al 2005). Readers also can see paper by Chen (1985). In general, approximations by Moody (1947) and by 3 Wood (1966) also belong to power-law formulas.

1 3164 . 0                     (43) 16 3 . 0 67 . 0 D 7 . 0 Re 1 184 . 0                     (
2                     

Conclusion 6

Maybe, it is difficult for many to recall for the time as recently as the 1970"s where there were no 7 personal computers or even calculators that could do much more than add or subtract. In that 8 environment an implicit relationship such as Colebrook (1939), which was well-known then, was 9 impractical and some simplification was essential. Today, it is not difficult to solve single 10 Colebrook equation by iteration. But solution of complex looped pipeline problem in such case 11 requires double iterative procedure where first is for the standard implicit Colebrook equation 12 while second one is for Hardy Cross method or similar iterative method used to solve simulation 13 problem in a looped pipe network (Brkić 2009b(Brkić , 2011b)). This double procedure can be serious 14 burden even for today very powerful computers. has to be chosen and finally box "Enable iterative calculation" have to be ticked. This allows 1 implementation of so called "Circular references" into a calculation. -Implicit Colebrook equation cannot be rearranged to derive friction factor directly.

-Colebrook equation has to be solved iteratively or using approximations.

-Iterative calculus can cause problem in simulation of flow in a pipe system.

-Error of almost all explicit approximations of the Colebrook relation is up to 3%.

-An explicit approximation of the Colebrook relation can be very complex but also it can be easily implemented in a computer code. .33 a number of estimated algebraic notation calculator key strokes required to solve the approximation, i.e. to find value of friction factor λ (estimated, i.e. average number of strokes) b Complexity index is defined as quotient of key strokes required for an observed approximation and the least complex one; here Eck (8) c the least complex approximation presented here

9

  iteratively. Even today in the era of advance computer technology, explicit approximations of the 10 implicit Colebrook relation is very often used for calculation of friction factor in pipes. The 11 reason that so many researchers propose approximate solutions to Colebrook equation is that 12 these correlations are necessary to calculate the pressure drop and average velocity in conduits in 13 one step. Friction factor can be derived using logarithmic or power law formulation (Zagarola et 14 al 1997). Colebrook equation belongs to the logarithmic law. The laws of resistances to fluid 15 flow through rough pipe are of great significance. Colebrook equation is valid for turbulent 16 regime in rough pipes including so called rough and smooth turbulent regime with special accent 17 on transient regime between them. There is no perfectly smooth inner surface of pipe. All pipe 18 walls have physically rough surfaces (Figure 1). Degree of roughness varies depending on the 19 manufacturing process, surface finish, type of pipe material (Hammad 1999), age, conditions of 20 exploitation, etc. In turbulent flow, thin layer of fluid very close to inner pipe surface in which 21 flow is laminar is called "laminar sub-layer". If the pipe roughness (protrusions of inner pipe 22 surface) is completely covered by the sub-layer, the surface is smooth from the hydraulic point of 3 view. With increasing of Reynolds number, thickness of laminar sub-layer decreases baring 1 protrusions and fluid flow through pipe become consequently rough from the hydraulic point of 2 view. At very low Reynolds number, relative roughness does not have influence on friction 3 factor and it depends only on value of Reynolds number. But, on the contrary, at very high 4 Reynolds number, Reynolds number does not have influence on friction factor and it depends 5 only on relative roughness. Between these two opposite regime friction factor depends on both, 6 Reynolds number and relative roughness. Colebrook equation was developed to cover this 7 transient zone of turbulence, but it also covers completely smooth and completely rough regime.

  8

Figure 1 .

 1 Figure 1. Hydraulic regimes; A) Hydraulically "smooth", B) Partially turbulent, and C)

(

  when Colebrook equation was published, for turbulent regime in smooth pipes 21 widely was used Prandtl equation also implicit in friction factor (Colebrook 1939). Prandtl 4 derived a formula from the logarithmic velocity profile and experimental data on smooth pipes 1

4

  began with Nikuradse"s turbulent pipe flow investigations in 1932 and 1933 (Hager and Liiv 5 2008). For turbulent regime in rough pipes widely was used von Karman"s relation (2):

9(

  Nikuradse-Prandtl-Karman) equations. Colebrook later performed experiments on sixteen spun 10 concrete-lined pipes and six spun bitumastic-lined pipes ranging in diameter from 101.6 mm to 11 1524 mm with average surface roughness values between 0.04318 mm and 0.254 mm (Taylor et 12 al 2006).

  to classify the data available at the time and those from experiment conducted by 15 himself and his colleague White (Colebrook and White 1937), Colebrook (1939) developed a 16 curve fit to describe transitional roughness (3):

  a monotonic change in the friction factor from smooth to fully 1 rough (Figure 2). It is valid especially for commercial steel pipes. Strictly mathematically is 2 incorrect what Colebrook had done, i.e. log(A+B)≠log(A)+log(B), but physically this relation 3 gives good results. Problem can be treated as inverse; according to logarithmic rules equally is 4 incorrect to split the Colebrook relation into two pieces.

Figure 2 .9

 2 Figure 2. Colebrook relation make transitional curve between hydraulically "smooth" regime

Figure 3 . 3 Figure 3

 333 Figure 3. Distribution of estimated deviation of implicit Colebrook equation modified by AGA

9

  researchers use Fanning factor which is not the same as the Darcy friction factor 6 (here noted as λ). Darcy friction factor is 4 times greater than the Fanning friction factor, but 7 physical meaning is equal. Darcy, Darcy-Weisbach and Moody friction factors are synonyms. 8 Colebrook equation is somewhere known as Colebrook-White equation (CW equation). White 10 was not actually a co-author of the paper in which this equation was presented (Colebrook 1939). 11 But, Colebrook made a special point of acknowledging important contribution of White to the 12 development of the equation (Colebrook and White 1937, Colebrook 1939). Letter W has 13 additional symbolic value because alternate explicit reformulations of Colebrook equation with 14 Lambert W-function involved exist (Brkić 2011a, Clamond 2009, Goudar and Sonnad 2003,15

7 8

 7 As it will be shown, there were some early expressions of Colebrook equation in explicit form 9 which were not particularly accurate, but in the years 1973-1984 there was a flurry of activity 10 obtaining more accurate approximations that appeared mainly in the chemical engineering 11 literature. Note that some of presented approximations exist in several versions. Here has to be 12 very careful because typographical errors are always possible(Concha 2008, Brkić 2009a).13Approximations will be presented starting from the oldest. Estimated accuracy compared with 14 iterative solution of implicit Colebrook equation will be shown in figures 4-23 for entire range of 15 turbulence where Moody diagram should be used as the reference.

  20

  to 21.49% compared with implicit Colebrook equation (Figure4), it has today only 1 historical value.

Figure 4 .7

 4 Figure 4. Distribution of estimated error of Moody approximation compared with implicit

Figure 5 .

 5 Figure 5. Distribution of estimated error of Wood approximation compared with implicit

Figure 6 .7

 6 Figure 6. Distribution of estimated error of Eck approximation compared with implicit

  Churchill approximation (valid only for turbulent regime) is up to 2.18% 11 compared with implicit Colebrook equation (Figure7).

Figure 7 .

 7 Figure 7. Distribution of estimated error of approximation by Churchill (valid only for turbulent

  on this approximation readers can see in paper of Swamee and Rathie 1 (2007).

4

  Approximation proposed by Jain (1976) (11) with error up to 2.05% is comparable with those

  5

7

  Distribution of estimated error of Jain approximation over turbulent part of Moody"s chart is 8 shown in figure7.

  approximation (full range of turbulence including laminar regime) 11 Approximation proposed by Churchill (1977) covers entire laminar and turbulent regime 12

  error of Churchill approximation (full range of turbulence including 1 laminar regime and unstable zone between them) over turbulent part of Moody"s chart is shown 2 in figure 7.

7

  Estimated error of Chen approximation is up to 0.35% compared with implicit Colebrook 8 equation (Figure8).

Figure 8 .

 8 Figure 8. Distribution of estimated error of Chen approximation compared with implicit

Figure 9 .7

 9 Figure 9. Distribution of estimated error of Round approximation compared with implicit

  9

  Barr approximation is up to 0.27% compared with implicit Colebrook 11 equation (Figure10).

Figure 10 .

 10 Figure 10. Distribution of estimated error of Barr approximation compared with implicit

Figure 11 .

 11 Figure 11. Distribution of estimated error of Zigrang and Sylvester approximations compared

  Haaland equation (20) is valid for flow of liquid. Haaland (1983) suggested that n=3 15 yields friction factors in consonance with those recommended for use in gas transmission lines. 16 17 Estimated error of approximation by Haaland (1983) valid for liquid flow is up to 1.4% 18 compared with implicit Colebrook equation (Figure 12).

Figure 12 .1

 12 Figure 12. Distribution of estimated error of Haaland approximation for liquid flow compared

Figure 13 .

 13 Figure 13. Distribution of estimated error of Haaland approximation for gaseous flow compared

Figure 14 .

 14 Figure 14. Distribution of estimated error of Serghides approximations compared with implicit

  of functions appearing in mathematical models of many processes is the 14 signomial functions. A signomial function is defined as the sum of signomial terms, which in 15 turn are products of power functions multiplied with a real constant. Estimated error of 16 approximation byManadilli (1997) is up to 2.06% compared with implicit Colebrook equation 17 (Figure15).

Figure 15 .1

 15 Figure 15. Distribution of estimated error of Manadilli approximation compared with implicit

Figure 16 .

 16 Figure 16. Distribution of estimated error of approximation by Romeo, Royo and Monzón

Figure 17 .3

 17 Figure 17. Distribution of estimated error of Sonnad and Goudar approximation compared with

  Figure 18. Distribution of estimated error of Vatankhah and Kouchakzadeh approximation

  19

Figure 19 .

 19 Figure 19. Distribution of estimated error of Rao and Kumar approximation compared with

  Figure 20. Distribution of estimated error of Buzzelli approximation compared with implicit

Figure 21 .(

 21 Figure 21. Distribution of estimated error of Avci and Karagoz approximation compared with

  36)18 Estimated error of approximation by Papaevangelou, Evangelides and Tzimopoulos is up to 19 0.85% compared with implicit Colebrook equation (Figure22).

  Figure 22. Distribution of estimated error of approximation by Papaevangelou, Evangelides and

Figure 23 .

 23 Figure 23. Distribution of estimated error of Brkić approximation compared with implicit

18

  For parameter β (39), solution for Lambert W-function byBarry et al. (2000) is used. Procedure 19 byWinitzki (2003) gives parameter β with similar accuracy (40):

1

  Parameter β calculated by using procedures byBarry et al. (2000) and byWinitzki (2003) gives 2 similar results in accuracy.

  formula itself is not very accurate (Cipra 1996), its accurate resolution 7 is nonetheless an issue for numerical simulations because a too crude resolution may affect the 8 repeatability and comparisons of calculation (Clamond 2009).

12

  better fit experimental data. Examples for this are e.g. Churchill (1973) approximation (9) and 13 Jain (1976) approximation (11) with slightly different coefficients. 14 15 Churchill (1977) relation (12) holds for all values of Reynolds and relative roughness, including 16 laminar regime (Figure24).

  Figure24. Churchill (1977) approximation includes laminar and highly unstable transient zone

3 4(

 3 figures 25-30) are used for relative roughness (ε/D) and 37 points for Reynolds number (Re).

7

  Approximations byZigrang and Sylvester (1982) (18), Haaland (1983), Sonnad and Goudar 8 (2006) and Papaevangelou et al (2010) produce maximal error up to 1.5% (Figure 28). Very 9 accurate (Figure 29), with estimated error up to 0.5%, are approximations by Chen (1979), Barr 10 (1981), Zigrang and Sylvester (1982) (19), Serghides (1984), Romeo et al (2002), Buzzelli 11 (2008), and approximations proposed by Vatankhah and Kouchakzadeh approximations (2008, 12 2009). Note that approximations proposed byVatankhah and Kouchakzadeh (2008, 2009) are 13 actually very successfully improved approximation bySonnad and Goudar (2006).

Figure 25 .Figure 27 . 21 Figure 28 . 25 Figure 29 .2

 252721282529 Figure 25. Inaccuracy of approximation byRao and Kumar (2007)

Figure 30 .6

 30 Figure 30. Extremely accurate approximations presented in higher resolution

  profiles in boundary layers (Zagarola et al 1997). As mentioned in 2 introduction, Colebrook equation is based on the logarithmic formulation. Example of the power-3 law formula, valuable for the same flow regimes as Colebrook"s is old Altshul equation from the 4 Soviet era (Figure 31) (41, 42):

Figure 31 .

 31 Figure 31. Distribution of deviation of Altshul formula (41) compared with implicit Colebrook

  44) 17 Deviation of presented power-law formulas from Russian practice and by Chen (1984) is shown 18 in figure 32.

Figure 32 .

 32 Figure 32. Power law formulas as substitution for implicit Colebrook equation

  approach can be used for determination of friction factor. Good example for the 7 era of computerization is approach ofÖzger and Yildlrim (2009). They use adaptive neuro-fuzzy 8 computing technique for determination of turbulent flow friction coefficient. In the paper of Yoo 9 and Singh (2005) are shown two new methods for the computation of commercial pipe friction 10 factor. Today, main problem is not how to calculate friction factor. Problem is how to measure or 11 estimate roughness of pipe (Farshad et al 2001). Most pipes usually have rough inner pipe 12 surface. Resistance to fluid flow offered by rough boundaries is larger than that for smooth one 13 due to the formation of eddies behind protrusions. Colebrook equation is valid for both, smooth 14 and rough turbulent regime including transient zone between them. In principle, a system of 15 partial differential equations known as Navier-Stokes equations describes the exact behavior of 16 the fluid flow in so-called boundary layer, but solving these equations remains beyond current 17 theory and computations. Sletfjerding and Gudmundsson (2003) proposed also methodology for 18 determination of friction factor directly from roughness measurements. In that way they 19 eliminated roughness as a parameter in Colebrook equation (only Reynolds number and pipe 20 diameter are necessary as input parameters). Using a similar approach to that of in Nikuradse"s 21 experiment, Sletfjerding and Gudmundsson (2003) related measured roughness values with 22 friction factor, but their equation is implicit in friction factor. In formulation given by 1 Sletfjerding and Gudmundsson (2003) equation is implicit and valid for average steel pipe (45):

3

  For other materials of pipes readers can consult paper ofSletfjerding and Gudmundsson (2003).

  equations give the friction factor explicitly as a function of Reynolds 17 number and relative roughness. Comparative analysis indicates that almost all approximate 18 equations give a very good prediction of the friction factor and can reproduce the Colebrook 19 equation and its Rouse and Moody plot. Therefore, these approximations for the friction factor 20 provide a rational, accurate, and practically useful method over the entire range of the Moody 21 chart in terms of Reynolds number and relative roughness. Apropos relative complexity at first 22 sight, these approximations can be very easily implemented in a computer code. 30

  ., Mesbah, A., Boozarjomehry, R.B., Svrcek, W.Y. 2007. The effect of major 5 parameters on simulation results of gas pipelines. Int. J. Mech. Sci. 49(8), 989-1000. 6 Avci, A., Karagoz, I. 2009. A novel explicit equation for friction factor in smooth and rough 7 pipes. J. Fluid. Eng. ASME 131(6), 061203 (1-4). 8 Barr, D.I.H. 1981. Solutions of the Colebrook-White function for resistance to uniform turbulent 9 flow. Proc. Inst. Civil. Eng. 71(2), 529-536. 10 Barry D.A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C.J., Stagnitti F. 2000. Analytical 11 approximations for real values of the Lambert W-function. Math. Comput. Simulat. 53(1-2), 95-. 2009a. Comments on "Settling velocities of particulate systems 15: Velocities in 14 turbulent Newtonian flows". Int. J. Miner. Process. 92(3-4), 201-202. 15 Brkić, D. 2009b. An improvement of Hardy Cross method applied on looped spatial natural gas 16 distribution networks. Appl. Energ. 86(7-8), 1290-1300.

  17

  Brkić, D. 2011a. An explicit approximation of the Colebrook equation for fluid flow friction 18 factor. Petrol. Sci. Tech. 29(xx), xxx-xxx. accepted, in press 19 Brkić, D. 2011b. Gas distribution network hydraulic problem from practice, Petrol. Sci. Tech. ., Tanasković, T.I. 2008. Systematic approach to natural gas usage for domestic heating 22 in urban areas. Energy 33(12), 1738-1753. 23 1 Chen, J.J.J. 1984. A simple explicit formula for the estimation of pipe friction factor. Proc. Inst. 2 Civil. Eng. 77, 49-55. 3 Chen, J.J.J. 1985. Systematic explicit solutions of the Prandtl and Colebrook-White equations for 4 pipe flow. Proc. Inst. Civil. Eng. 79, 383-389. 5 Chen, N.H. 1979. An explicit equation for friction factor in pipes. Ind. Eng. Chem. Fundam. .H. 1980. Comments on: "An explicit equation for friction factor in pipe". Ind. Eng. 8 Chem. Fundam. 19(2), 229-230. 9 Churchill, S.W. 1973. Empirical expressions for the shear stress in turbulent flow in commercial 10 pipe. AIChE J. 19(2), 375-376. 11 Churchill, S.W. 1977. Friction-factor equation spans all fluid flow regimes. Chem. Eng. 84(24), . 1996. A new theory of turbulence causes a stir among experts. Science 272(5264), 951. 14 Clamond, D. 2009. Efficient resolution of the Colebrook equation. Ind. Eng. Chem. Res. 48(7), 15 3665-3671. 16 Coelho, P.M., Pinho, C. 2007. Considerations about equations for steady state flow in natural gas 17 pipelines. J. Braz. Soc. Mech. Sci. Eng. 29(3), 262-273. 18 Colebrook, C.F. 1939. Turbulent flow in pipes with particular reference to the transition region 19 between the smooth and rough pipe laws. J. Inst. Civil. Eng. (London) 11(4), 133-156. 20 Colebrook, C.F., White C.M. 1937. Experiments with fluid friction in roughened pipes. Proc. 21 Roy. Soc. Ser. A Math. Phys. Sci. 161(906), 367-381. 22 33 Manadilli, G. 1997. Replace implicit equations with signomial functions. Chem. Eng. 104(8.F. 1944. Friction factors for pipe flow. Trans. ASME 66(8), 671-684. 3 Moody, L.F. 1947. An approximate formula for pipe friction factors. Trans. ASME 69(12), A. 2006. Analytical solutions for the Colebrook and White equation and for pressure 6 drop in ideal gas flow in pipes. Chem. Eng. Sci. 61(16), 5515-5519. 7 Nandakumar, V. 2007. Analytical calculation of Fanning friction factor. Hydrocarb. Process. 8 86(1), 97-100. 9 Ouyang, L.B., Aziz, K. 1996. Steady-state gas flow in pipes. J. Petrol. Sci. Eng. 14(3-4), 137-, Yıldırım G. 2009. Determining turbulent flow friction coefficient using adaptive 12 neuro-fuzzy computing technique. Adv. Eng. Softw. 40(4), 281-287. 13 Papaevangelou G, Evangelides C, Tzimopoulos C. 2010. A new explicit equation for the friction 14 coefficient in the Darcy-Weisbach equation, Proceedings of the Tenth Conference on Protection 15 and Restoration of the Environment: PRE10, July 6-9, 2010, Greece, Corfu, 166, pp. 1-7. 16 Rao, A.R., Kumar B. 2007. Friction factor for turbulent pipe flow. Division of Mechanical 17 Science, Civil Engineering, Indian Institute of Science, Bangalore, India, ID Code 9587, 18 http://eprints.iisc.ernet.in/9587/ 19 Rao, A.R., Kumar B. 2009. Transition of turbulent pipe flow. J. Hydraul. Res. IAHR 47(4), 529-., Royo, C., Monzon, A. 2002. Improved explicit equation for estimation of the friction 22 factor in rough and smooth pipes. Chem. Eng. J. 86(3), 369-374.

  smooth pipes. Can. J. Chem. Eng. 58(1), 122-123. 2 Rouse, H. 1943. Evaluation of boundary roughness. Proc. 2nd Hydraulics Conf. University of 3 Iowa Studies in Engineering, Bulletin 27, Wiley: New York, 105-116. 4 Serghides, T.K. 1984. Estimate friction factor accurately. Chem. Eng. 91(5), 63-64. 5 Schorle, B.J., Churchill, S.W., Shacham, M. 1980. Comments on: "An explicit equation for 6 friction factor in pipe". Ind. Eng. Chem. Fundam. 19(2), 228-229. 7 Sletfjerding, E., Gudmundsson, J.S. 2003. Friction factor directly from roughness measurements. 8 J. Energ. Resour. ASME 125(2), 126-130.9 Sonnad, J.R., Goudar, C.T. 2004. Constraints for using Lambert W function-based explicit 10 Colebrook-White equation. J. Hydraul. Eng. ASCE 130(9), 929-931. 11 Sonnad, J.R., Goudar, C.T. 2005. Explicit friction factor correlation for pipe flow analysis. 12 Hydrocarb. Process. 84(6), 103-105. 13 Sonnad, J.R., Goudar, C.T. 2006. Turbulent flow friction factor calculation using a 14 mathematically exact alternative to the Colebrook-White equation. J. Hydraul. Eng. ASCE 15 132(8), 863-867. 16 Sonnad, J.R., Goudar, C.T. 2007. Explicit reformulation of the Colebrook-White equation for 17 turbulent flow friction factor calculation. Ind. Eng. Chem. Res. 46(8), 2593-2600. 18 Sukharev, M.G., Karasevich, A.M., Samoilov, R.V., Tverskoi, I.V. 2005. Investigation of the 19 hydraulic resistance in polyethylene pipelines. J. Eng. Phys. Thermophys. 78(2), 350-359. 20 Swamee, P.K., Jain A.K. 1976. Explicit equations for pipe flow problems. J. Hydraul. Div. 21 ASCE 102(HY5), 657-664.

1 Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 . 11 Figure 6 . 13 Figure 7 .Figure 8 . 18 Figure 9 . 20 Figure 10 . 22 Figure 11 .Figure 12 .Figure 13 .Figure 14 .Figure 15 . 11 Figure 16 .Figure 17 .Figure 19 .Figure 20 . 21 Figure 21 .Figure 22 .Figure 23 . 4 Figure 24 .Figure 25 .Figure 27 .Figure 28 .Figure 29 .Figure 30 .Figure 31 . 14 Figure 32 .

 112345116137818920102211121314151116171920212122234242527282930311432 Figure 1. Hydraulic regimes; A) Hydraulically "smooth", B) Partially turbulent, and C)

  Figure 1 DB Click here to download high resolution image

  The calculation of the parameters ofRomeo et al (2002) approximation was done through non-4 linear multivariable regression. Estimated error of approximation by Romeo, Royo and Monzón

	1	2	10 log	7065 . 3	D	Re 0272 . 5	10 log	827 . 3	D	Re 567 . 4	10 log	7918 . 7	D	. 0	9924	815 . 208 3326 . 5	Re	9345 . 0	(27)
	5

3

Table 1 .

 1 Maximal relative error for available approximations for test check points Approximation by Rao and Kumar (2007) is extremely inaccurate compared with standard

	16
	17
	18	Maximal percentage (relative) error of presented approximations over the entire range of
	19	applicability of Colebrook equation is shown in table 1.
	20

2

Colebrook equation (Figure

Table 2 .

 2 Complexity and complexity index of available explicit approximations

	16
	17
	18	Eck (1973) approximation is the least complex with only 27 key strokes required for solution and
		hence this equation has complexity index 1 (Table 2).

Table 1 .

 1 Maximal relative error for available approximations for test check points

	δ max (%)	Relative roughness (ε/D)	Reynolds number (Re)

Table 1 DB Click here to download Table: Table 1 DB.docTable 2 .

 12 Complexity and complexity index of available explicit approximations a Complexity b Complexity index

	c Eck (8)	27	1
	Moody (5)	29	1.07
	Churchil (9)	31	1.14
	Haaland (20)	35	1.29
	Jain (11)	35	1.29
	Swame and Jain (10)	36	1.33
	Round (16)	36	1.33
	Manadilli (26)	44	1.62
	Zigrang and Sylvester (18)	47	1.74
	Avzi and Karagoz (35)	47	1.74
	Rao and Kumar (30)	61	2.25
	Brkić (38)	67	2.48
	Sonnad and Goudar (28)	67	2.48
	Papaevangelou, Evangelides and Tzimopoulos (36)	67	2.48
	Brkić (37)	69	2.55
	Zigrang and Sylvester (19)	69	2.55
	Vatankhah and Kouchakzadeh (28a)	77	2.85
	Barr (17)	80	2.96
	Chen (15)	91	3.37
	Wood (6)	98	3.62
	Buzzelli (32)	104	3.85
	Churchil (12)	106	3.92
	Serghides (22)	107	3.96
	Romeo, Royo and Monzon (27)	125	4.62
	Serghides (21)	144	5

Table 2 DB Click here to download Table: Table 2 DB.doc

 2

MS Excel can be also successfully used in other engineering fields; see electronic appendix inBrkić and Tanasković (2008)

Acknowledgment

7

1

Most available approximations of the Colebrook equation are very accurate. Exceptions are 2 Round (1980), Eck (1973), Moody (1947), Wood (1966), andRao andKumar (2007) 3 approximations. The average error of almost all explicit approximations of the Colebrook (1939) 4 relation is up to 3%.