Reconsideration of the friction factor data and equations for smooth, rough and transition pipe flow

The key element in design of pipelines is the friction factor estimation. After the brief review of the experimental data and friction factor correlations for isothermal single phase flow, we have checked the validity of well-known correlations through statistical criteria. During this process it was statistically proved that some of the well-known and permanently cited friction factor equations can be improved. Moreover we have prepared, for practical engineering purposes, equations that cover the entire range of laminar, critical and turbulent pipe flow.


Introduction
Transportation of various fluids absorbs a huge amount of energy in any project concerning chemical engineering and energy production and distribution plants. The key part in design of pipelines is the friction factor estimation. Although the best known handbooks in chemical, thermal and hydraulic engineering, like [1][2][3][4][5][6], cite "classical work" in the field of fluid mechanics, numerous continuously published papers about the fluid flow through the circular pipes indicate that this is still a very interesting topic. Generally speaking there are two phases in the development of the equations for the friction factor. Up to the middle of 20 th century Prandtl, von Karman, Nikuradse, Colebrook and others set the equations that are still in use by virtue of their comprehensiveness and compliance with numerous experimental data. Since that time experimental and theoretical studies on liquid and gas flow in pipeline was carried out in order to: improve the results and equations obtained by Prandtl, von Karman, Nikuradse, Colebrook and others achieved during the first half of the 20 th century; extend the friction factor equation in order to cover high Reynolds number flow (of relevance especially for natural gas pipelines); explore the friction factor for a new class of tubes with very small diameter (micro tubes).
In this paper we have tried to gather all of the available experimental data and correlations for friction factor and through the statistical analysis to clear all of their advantages and shortages. At the very end we made original equations that cover complete range of laminar, critical and turbulent flow in pipes in a form that is very practical for design purposes. 1

Brief review of the data and calculations of pressure drop in the pipes
Hereby we will restrict the review to isothermal flow of Newtonian fluids. For calculations of pressure drop for incompressible flow Weisbach [7] proposed in 1845 the equation where: p (Pa) is total friction loss; is a friction factor coefficient; L (m) is the length of the pipe; D (m) is the pipe internal diameter; u (m/s) is the average fluid velocity; (kg/m 3 ) is the fluid density. Pipe friction factor, in general, depends on Reynolds number (Re) that correlates the inertia and internal friction forces and relative pipe roughness (Rr).
Osborne Reynolds [8] was the first to describe the transition from laminar to turbulent flow in smooth pipes and found that non-dimensionless group D u Re (2) that we nowadays call Reynolds number plays critical role in this transition ( , Pa·s, is the fluid viscosity). Citation from [4]: "Reynolds numbers below 2000 to 2100 correspond to laminar or viscous flow; numbers from 2000 to 3000 ÷ 4000 correspond to a transition region of peculiar flow, and numbers above 4000 correspond to turbulent flow." Darcy [9], according to a large number of experiments, noticed that pressure drop differs for "new" and "old" pipes, so he provided separate equations for each case. Later on, a concept of relative pipe roughness was introduced by von Mises [10], in the following form D Rr (3) where (m) is the absolute roughness of pipe internal surface.

Period of classical research
This era ends with the Colebrook's friction factor equation for turbulent flow and Moody's diagram.

Laminar flow in pipes
In laminar flow viscous forces prevail the inertia forces and fluid flows in layers. In this case pipe friction factor is calculated from Hagen-Poiseuille equation and depends only on the Reynolds number [11][12] Re 64 (4)

Turbulent flow in pipes
As the Reynolds number increases, the inertia forces (proportional to the velocity squared) prevail the influence of fluid viscosity. Turbulent flow is characterized by the development of transverse velocity component that causes the mixing of fluid perpendicular to the flow direction. Prandtl and von Karman developed a theoretical analysis for pipe flow with suitable formulas for smooth and rough pipe. Smooth pipes are defined as those having small surface irregularities when compared with the thickness of the boundary layer. Rough pipes are characterized with irregularities of the walls sufficient to break up the laminar boundary layer. So, for turbulent flow the influence of the pipe wall roughness plays an important role and has been introduced in the friction factor computations. Three cases of turbulent flow are divided as follows [13][14]:  [15]; in [16] it is said that "the flow showed the first symptoms of roughness when according to [17]

Turbulent flow in hydraulically smooth pipes
For hydraulically smooth pipes Blasius [19] found the relation 4 / 1 Re 3164 . 0 (5) and this was the first correlation ever made for pipe friction versus Reynolds number.

Turbulent flow in transitional range between hydraulically smooth and rough pipes
Generally speaking, the friction factor transition between turbulent flow in smooth and rough pipes is not well understood. Von Mises [10] published the first equations using experimental data known at his time Colebrook and White [15], on the basis of measurements on commercial pipes, concluded that non-uninform roughness of commercial pipes produces somewhat different behavior than inflectional friction factor behavior found by Nikuradse [24]. They pointed out that there is a smooth (aka gentle, monotonic) transition between PKN equations (8) and (9). Using and PKN equations, as well as their own data and data from other researchers gathered on commercial pipes, Colebrook [26]  Rr Rr (12) that became widely accepted design formula in for turbulent friction in range Re = 4000 ÷ 10 8 and Rr = 0 ÷ 0.05. Equation (12) consists of two parts: equation (8) for smooth pipes when Rr 0 and equation (9)  Rr Rr (13) but it fills the gap between smooth and rough pipe flow, and up to a certain extent it provides satisfactory results. The transition between smooth and rough pipe flow according to (12) is named "smooth transition". Colebrook's equation (12) published almost 80 years ago has been recognized by the engineers as the very useful, despite its implicit form. Among the first, Rouse [27] confirmed Colebrook's equation (12) by his own measurements and provided a friction factor diagram using (12) with abscissa Re and ordinate 1 (he placed and Re on auxiliary axes of diagram). Equation (12) was also plotted in 1944 by Moody [28] into what is now called the "Moody chart" for pipe friction (in this diagram Re and are placed on main axes). Since the publication of mentioned diagrams Colebrook's equation became the basis for friction factor estimations in chemical engineering, HVAC, energetics as well as in many other fields of engineering. Moody chart is probably the most famous and useful one in engineering fluid mechanics.  (8) and (9) and Colebrook's equation (12) were published without any quantitative comparison with experimental data, so the reader cannot have any idea what the authors themselves thought or knew about their accuracy. Since the end of World War II equations (8), (9) and (12) were reviewed and only some of the important results of this process will be presented hereby.

The nature of pipe wall roughness
In 7 decades since Colebrook published his work [26] the important fact emerges. Nikuradse used meshed sand grains of 0.8 ± 0.02 mm to produce a relatively uniform roughness on his pipes (sand grain was glued to the inner pipe surface). This kind of roughness, named sand-grain roughness (k s ), cannot be found in commercial pipes which are characterized with a "natural roughness". The Colebrook's correlation and Moody's chart were also developed using the k s . In engineering books and handbooks like [1-6] Colebrook's correlation and/or Moody's chart are accompanied with values of pipe wall roughness (most commonly tabulated) for copper, steel, iron and other pipes. These k s values were determined by fitting a roughness height to match a measured pressure drop, so mentioned tables of pipe wall roughness contain values of "equivalent sand-grain roughness".
The important question arises: how is k s related to a physical roughness measured from surface topography. This is especially important since many of the commercial pipe surfaces listed along with Colebrook's correlation or Moody's diagram don't have a consistent values of roughness. Some surfaces are listed with a roughness height that spans over an order of magnitude -for example concrete pipes have roughness in range 0.3 to 3 mm. Additionally, the equivalent sand-grain roughness height is unknown for numerous other surfaces of engineering interest. As a result, one can only expect to obtain accurate results using the Moody diagram if k s is known for the surface of interest and the flow is fully rough. Commercial pipes are characterized with random natural roughness, and in recent years two parameters were found to be of the special interest [29]: root-mean-square roughness height (k rms ) and skewness of the roughness elevation distribution (s k ). Recently published research such as [16,[29][30][31] show that k s is 1.5 to 6 times greater than k rms and that pretty simple correlations in form of can be obtained [32]. (4), (8), (9) and (12) The U. S. Bureau of Reclamation [33] reported large amounts of field data on commercial pipes: concrete, continuous-interior, girth riveted and full-riveted steel pipes. The conclusion in [33] is that some of the collected field data could not be explained by the Colebrook's equation (12).

Comments on the accuracy of equations
In [34] estimation is given that "the probable variation of for commercial pipe is about ±10%, but this variation is masked by the uncertainty of quantifying the surface roughness." One of the (rare explicit) comments on Colebrook's equation (12) is given in [13] in which White comments that "the equation (12) is accurate to ±15% percent for design calculations over the full range of Re and Rr. It can be used for circular and noncircular Zagarola and Smits [30] stated that, based on experimental data form [23], the standard deviation of equation (8) is approximately ±4% and the largest discrepancy exceeds ±6%. This means that the same error is to be expected for Colebrook's correlation (12).
Recent studies on micro tubes have shown that there are some (almost neglectable) deviations from laminar flow equation (4) and authors presume that these deviations are present due to the certain role of pipe roughness [35] or compressibility of the fluid [36]. It must be noted that almost any set of experimental data gathered on non-micro pipes showed similar deviations. In turbulent region for Re < 100·10 3 experiments also agree with calculations done with (12), but in [37] it is stated that surface roughness should be more carefully studied.
Schroeder [38] states that the Colebrook equation (12) is nothing else but simple combination of (8) and (9). He notices that (12) always predicts a higher friction factor than when (8) and (9) are used separately and that "the maximum difference in friction factor is about 17% which translates to an 8.5% difference in flow". Schroeder claims that, from engineering point of view, this difference "is not large".

Extending of the range of Reynolds numbers in recent experimental research
Nikuradse [24] conducted his experiments in the range Re = 3100 -10 6 . Without any explanation Moody drew his chart [28] with the upper limit Re = 10 8 , assuming that the Colebrook's equation (12) holds true for extended range. This assumption was seriously tested in past few decades because in magisterial natural gas pipelines Reynolds numbers is often greater than 10 7 due to gas pressure in range 80 -180 bar. Also, natural gas pipelines are generally coated so, having in mind that their diameter is usually large, their relative roughness is very small. This is the reason that more than a few research teams tried to provide experimental data for Re > 10 6 . For example, in [39] it was stated that the PKN correlation for smooth pipes (8) predicts too low values of the friction factor in case of high pressure natural gas flow, especially at high Reynolds numbers. Flack and Schultz [32] commented that all recent experimental results gathered in range Re > 10 6 show that the Moody diagram overestimates the pressure drop in the transitional flow regime for honed and commercial steel pipes (from an engineering standpoint Moody diagram provides a conservative estimate of pressure drop for a given roughness). This clearly indicates that the Colebrook roughness function used in the formulation of the Moody diagram may not be applicable to a wide range of roughness of engineering interest.
Using the experimental data from [20], as well as from [30] (15) and states that for Reynolds numbers between 10 4 and 30·10 6 equation (14) predicts the friction factor to within 1.4%. Some researchers, as referred by Smith [42], made the following modification of (12) Rr (16) in order to obtain better agreement with experimental data at higher Reynolds numbers.

Inflectional behavior and abrupt transition
As said before, Nikuradse's experiments show that the inflection of friction factor curve exists in transitional turbulent region. Experiments on a rough honed pipe published in [43] confirmed inflectional friction factor behavior, similar to Nikuradse but not so pronounced.
Using the large set of experiment data (even recently gathered) Afzal [ Rr j Rr (17) For j = 11 equation (17) fits well with the inflectional roughness data, and for j = 0 equation (17) provides similar values to (12).
The American Gas Association (AGA) presented two comprehensive reports analyzing the flow of natural gas in real pipelines [44,45]. One of main conclusions was that friction factor shows a more abrupt transition from smooth to rough turbulent flow than the smooth and gentle transition predicted by Colebrook's equation (12).
Results from a joint research project involving four European natural gas transmission companies were presented in [46], and later also discussed in [47]. The new proposed friction factor formula is where n is used to control the shape of the transition region. For n = 1 a transition is smooth and results are close to equation (12), while for n = 10 there is a more abrupt transition (socalled point transition). The reader is not provided with any further advice about how the value of this parameter should be selected -in [47] it is said that any value in range n = 1 ÷ 10 is possible, and an average value n = 3 is pointed out in [48].

Critical regime -transition between laminar and turbulent flow
As the Reynolds number increases over 2000, the flow regime becomes unstable and inertia forces begin to dominate, and this causes a general increase of the friction factor. Experimental data in this range of Re show a large scatter and many of books show this range hatched. Concha [50] concludes that in the region Re = 2000 -4000 friction factor is independent of the wall roughness so, using a straight line in the log( ) versus log(Re) he obtained the relation

Explicit forms of turbulent flow friction factor equations
For smooth pipe flow Filonenko [51] (21) and Incropera and DeWitt [52] gave this equation applicable range of Re = 3·10 3 -5·10 6 . Explicit form of (16) is given in [31] Re 91 . Rr j Rr (22) and explicit form of (18)  Rr (24) It has be noted that Altshul' s equation is a kind of combination of (5) and (13), i.e. for Rr 24) becomes equal to (5 qual to (10). ASHRAE Fundamentals [6] suggests the use of the equation (24)  Rr (25) ASHRAE Fundamentals [6] states that "Friction factors obtained from the Altshul-Tsal equation are within 1.6% of those obtained by Colebrook's equation", but it was shown in [55] that (24) and (25) show significant deviation from (12), greater than 27%. As well as in few other papers, it was shown in [55] that according to various statistical criteria, one of the best explicit approximations of Colebrook's equation is the one published by Haaland [56] who proposed the following equation 2 (26) For n = 1 equation (26) provides good approximation of Colebrook's equation (25), and according to author for n = 3 or n = 4 equation (26) shows so-called abrupt transition between smooth and rough pipe flow.

Equations that cover the whole laminar-turbulent region
There were two kinds of attempt to cover the whole laminar-turbulent region.
In [57] authors gave the simple equation for smooth pipes b a Re (27) where a and b depend on Re as reproduced in Table 1.

Region of developing turbulent flow
Besides aforementioned problems there is one more. With the exception of the smooth pipe flow, other available experimental data does not support the friction factor values calculated by (Colebrook) in region Re = 4000 -20000. This is the region where the fluid flow develops from the unstable critical flow to stable turbulent flow. As Bratland [48] says "The traditional Moody diagram shows -in clear contrast both to Nikuradse's measurements and the theoretical diagrams in figures 2.8.8 and 2.8.9 -an abrupt jump in the friction factor close to Re = 3000. Although the literature contains numerous claims that such a jump is supposed to exist, it is difficult to find hard evidence in the form of measurements to support it." Using Kolmogorov's theory [59] Bratland developed a kind of "theoretically based" diagram in Moody's style [48] and showed that: "theoretical" curve shows the existence of inflection of friction factor curve exists in transitional turbulent region, although the numerical values are not the same as in Nikuradse's experiment; "theoretical" curve has no abrupt transition in region Re = 2000 ÷ 4000; "theoretical" values of friction factor in range Re = 4000 ÷ 20000 are lesser than values obtained by Colebrook's equation, especially for larger relative roughnesses.

Friction factor correlations vs experimental data
Having regard to the presented status of the problem, correlation and experiments, as well as the commitment of engineers to perform calculations as accurately as possible, we've collected numerical values of experimental research and created a reference database for: smooth pipes from [23,30,40,41,60] and previously unpublished data [61], that contains z = 231 points and sandgrain rough pipes from [24] that contains z = 181 points. Afterwards we reviewed some of the correlations listed above and the results are shown in Tables 2, 3 and 4. Statistical parameters are defined in Appendix.

Estimation of relative roughness according to experimental data
Let's use the data from [16] where Langelandsvik et al reported measurements in a commercial steel pipe covering the smooth, transitionally rough, and fully rough regimes. Their data are shown in Figure 2. Any of hereby presented equations can be applied to measured data in order to obtain equivalent sandgrain roughness, and results are shown in Table 6. It is obvious that (38) shows smooth transition and in order to cover the abrupt transition we have corrected the correlation (39) (40) According to statistics equation (40) can be used to predict abrupt transition of real pipelines (Table 7).

Conclusion
As stated earlier, for calculations of pressure drop in pipes the combination of Weisbach and Colebrook equations, (1) and (12), are cited and recommended in most of the chemical engineering and HVAC textbooks and handbooks. Despite its ambiguities there are numerous papers that treat Colebrook's equation as the best that can be recommended for smooth/transition/rough turbulent flow. In last two decades in a certain number of articles a great attention is devoted to the problem of turbulent flow and attempts have been made to explain the nature of turbulence in commercial pipes: the database for flow in smooth pipes is expanded, data from commercial pipeline with low relative roughness (characterized by the abrupt transition) were analyzed, and so on.
In this paper we have tried to mention the basic unsolved problems regarding to turbulent flow and to establish practical correlation (on the basis on available literature data) that covers turbulent flow in all its forms. Statistical parameters of (38) show that it correlates the experimental results very well and ensure that the calculations of friction factor can be done with verifiable accuracy. Furthermore, we have connected (38) with the equation for laminar flow (4) through a simple expression (39), which enables the computations in the critical zone of the flow for various relative roughness of the pipe. While (4) covers the range range Re = 2000 ÷ 35.5·10 6 and Rr = 0 ÷ 0.0333. In has to be noted that (38) provides smooth transition between smooth and rough pipe flow, and that (40) supports the inflectional behavior and abrupt transition.
We also conducted a statistical analysis of known and often cited equations for the friction factor in turbulent flow and found that some forms of modified equations have far better statistical indicators than the original equations. At the very end we will once again repeat hereby suggested equation for calculations of the friction factor coefficient in the pipes 6 (41) that cover the range of relative roughness between Rr = 0 and Rr = 0.0333.