## Abstract

The discharge in a full flow regime represents the discharge capacity of a vertical pipe, and the Darcy–Weisbach friction factor () is an important variable to calculate discharge. Since all existing equations for contain the Reynolds number (Re), it is problematic if the velocity is unknown. In this study, the performance of existing equations collected from studies on vertical pipes is assessed, and an approximation for the of vertical pipes in the full flow regime, without Re, is proposed. The performance of the Brkić and Praks equation is the best, with a maximum relative error (MRE) of 0.003% (extremely accurate). The MRE of the new approximation is 0.43%, and its assessment level is very accurate. This work is expected to provide a reference for the design and investigation of the drainage of vertical pipes.

## NOTATION

The following symbols are used in this paper:

• discharge coefficient (–);

•
• diameter of the vertical pipe (m);

•
• diameter of the barrel (m);

•
• gravitational acceleration (m s−2);

•
• head above the crest of the vertical pipe (m);

•
• length of the vertical pipe (m);

•
• length of the barrel (m);

•
• projection length of the vertical pipe over the tank floor (m);

•
• discharge (m3 s−1);

•
• Re

Reynolds number (–);

•
• angle between the horizontal plane and centerline of the barrel ();

•
• viscosity (m2 s−1);

•
• relative roughness of pipe (–);

•
• entrance loss coefficient (–);

•
• Darcy–Weisbach friction factor (–);

•
• mean velocity (m s−1).

## INTRODUCTION

A vertical pipe, which can be distinguished as a vertical drain or an overflow pipe depending on whether the pipe extends above the tank floor (Kalinske 1940), is widely used in different types of drainage systems, such as roof rain leaders (Padulano & Del Giudice 2018), manholes (Banisoltan et al. 2015) and tank drains. In addition, such pipes are used in spillways of dams after performing certain geometric optimizations, such as in morning glory spillways (Leopardi 2014). In general, existing studies define three flow regimes of a vertical pipe, which can also be subdivided further (Banisoltan et al. 2017; Padulano & Del Giudice 2018). The three flow regimes, with the head varying from low to high, are weir-like, transition and full flow regimes. The discharge in a full flow regime corresponds to the conveying capacity of a vertical pipe, and it is an essential parameter for a drainage device. In addition, the full flow regime is the only regime in which the discharge coefficient can be derived using the energy conservation equation.

A vertical pipe is considered to be fully filled, and the swirl on the water surface is considered to be infinitesimal or even nonexistent, in the full flow regime, as shown in Figure 1. The discharge of full flow regimes can be defined as ; here, the discharge coefficient C can be calculated as , where is the Darcy–Weisbach friction factor and is the entrance loss coefficient.

Figure 1

Schematic of the full flow regime in (a) an overflow pipe, (b) a vertical drain.

Figure 1

Schematic of the full flow regime in (a) an overflow pipe, (b) a vertical drain.

The Darcy–Weisbach friction factor () has been investigated in several studies, and is believed to be affected by the material and size of pipe, and the velocity of the flow in the pipe as well (Fanning 1877). During the development of solving , the relative roughness () and the Reynolds number (Re) are introduced to describe the three characteristics; and two important implicit equations, the Colebrook equation (a function of and Re) and the Nikuradse–Prandtl–von Karman (NPK) equation (a function of Re), are proposed for solving of the rough pipes and smooth pipes, respectively. Furthermore, explicit equations abound that have been developed based on them. For example, Samadianfard (2012) proposed an explicit solution of based on the Colebrook equation with gene expression programming analysis; Brkić & Praks (2018) derived an accurate explicit approximation of the Colebrook equation with the Wright ω-Function (a shifted Lambert W-function); Brkić (2011b) and Heydari et al. (2015) took the complexity of equations into consideration for the comparison of explicit equations; analogously, Li et al. (2011) compared the computation time of explicit solutions and proposed an explicit determination of based on the NPK equation with a ternary cubic polynomial of for the smooth pipe case. The details of the explicit equations are collected and discussed in the third section of this paper. However, since , it is problematic if the flow velocity in the vertical pipe () is unknown, such as in the design stage. Alazba et al. (2012), in their study on the friction head-loss of center-pivot irrigation machines, proposed a simple equation for with a constant velocity value based on the field data.

In this study, the accuracy of the existing equations for in the case of the full flow regime in a vertical pipe is assessed, and a new equation for that does not involve the variable v is proposed based on the data from Anwar (1965), Padulano et al. (2013), Padulano et al. (2015) and Banisoltan et al. (2017).

## STATE-OF-THE-ART REVIEW OF FULL FLOW IN A VERTICAL PIPE

In this study, the literature pertaining to the flow in a vertical pipe was investigated, and the details of the experiments performed in these studies are listed in Table 1; the sketch of the inflow condition is shown in Figure 2. Figure 3 shows the discharge coefficient C as a function of the non-dimensional water head ; the corresponding values of and Re are shown in Figure 4. The value of Re recorded in the literature is within the range of , and that of is within .

Table 1

Details of experiments reported in the literature

SourceInflow conditionVertical pipe
Barrel
Diameter d (m)Length l (m)Projected distance P (m)Angle ()Diameter (m)Length (m)
Anwar (1965)  radial 0.0663 0.609 0.15 –
0.1016
0.0384
Padulano et al. (2013); Padulano et al. (2015)  unilateral 0.07 1.5 –
0.1
Banisoltan et al. (2017)  radial 0.076 1.219 –
Humphreys et al. (1970)  unilateral 0.0758 1.78 0.105 0.053 2.79
0.1265 0.38 0.152 17.5 0.076 7.62
0.2015 0.61 0.379
Zhang (2017)  unilateral 0.1 0.96 0.19 0.573 0.080 7.40
SourceInflow conditionVertical pipe
Barrel
Diameter d (m)Length l (m)Projected distance P (m)Angle ()Diameter (m)Length (m)
Anwar (1965)  radial 0.0663 0.609 0.15 –
0.1016
0.0384
Padulano et al. (2013); Padulano et al. (2015)  unilateral 0.07 1.5 –
0.1
Banisoltan et al. (2017)  radial 0.076 1.219 –
Humphreys et al. (1970)  unilateral 0.0758 1.78 0.105 0.053 2.79
0.1265 0.38 0.152 17.5 0.076 7.62
0.2015 0.61 0.379
Zhang (2017)  unilateral 0.1 0.96 0.19 0.573 0.080 7.40
Figure 2

Sketch of inflow condition: (a) radial flow; (b) unilateral flow.

Figure 2

Sketch of inflow condition: (a) radial flow; (b) unilateral flow.

Figure 3

Discharge coefficient C as a function of the non-dimensional water head .

Figure 3

Discharge coefficient C as a function of the non-dimensional water head .

Figure 4

Ranges of the relative roughness and Reynolds number of the collected data.

Figure 4

Ranges of the relative roughness and Reynolds number of the collected data.

## SOLUTIONS FOR THE DARCY–WEISBACH FRICTION FACTOR

### Friction factor for turbulent flow in rough pipes

For turbulent flow in rough pipes, the Colebrook equation is regarded as a transcendental expression of (Brkić 2011b). Since the Colebrook equation is an implicit function, many researchers have proposed explicit approximations to avoid the iteration solution. Among these, the most popular approximations were collected, as presented in Table 2. The friction factor () of turbulent flow in rough pipes is believed to be affected by Re and . In general, the relationship among these parameters can be classified into the following three categories: logarithmic, power (No. 1, No. 2, No. 4, No. 19 and No. 32 in Table 2) and the combination of the fronts (No. 8 in Table 2).

Table 2

Summary of the approximation equations for the friction factor of rough pipes

No.Author and sourceYearEquation for rough pipe
Moody
(Brkić 2011b
1947
Altshul
(Olivares et al. 2019
1952
Altshul II
(Olivares et al. 2019
1952
Wood
(Brkić 2011b
1966

Eck
(Brkić 2011b
1973
Jain
(Brkić 2011b
1976
Swamee and Jain
(Olivares et al. 2019
1976
Churchill
(Brkić 2011b
1977

Chen
(Brkić 2011b
1979
10 Round
(Olivares et al. 2019
1980
11 Shacham
(Zigrang & Sylvester 1985
1980
12 Shacham II
(Olivares et al. 2019
1980

13 Barr
(Olivares et al. 2019
1981
14 Pavlov
(Olivares et al. 2019
1981
15 Zigrang–Sylvester
(Zigrang & Sylvester 1982
1982
16 S. E. Haaland
(Haaland 1983
1983
17 Serghides I
(Brkić 2011b
1984

18 Serghides II
(Brkić 2011b
1984

19 Tsal
1989

(Olivares et al. 2019
1997
21 Romeo
(Romeo et al. 2002
2002
(Olivares et al. 2019
2006

23 Rao and Kumar
(Brkić 2011b
2007

24 Buzzelli
(Olivares et al. 2019
2008

(Brkić 2011b
2008

26 Avci
(Brkić 2011b
2009
27 Papaevangelo
(Olivares et al. 2019
2010
28 Brkić
(Brkić 2011b
2011

29 Brkić II
(Brkić 2011b
2011

30 Fang et al.
(Fang et al. 2011
2011
31 Ghanbari
2011
2012
33 Winning and Coole
(Winning & Coole 2015
2014
34 Heydari et al.
(Heydari et al. 2015
2015
35 Mikata and Walczak
(Vatankhah 2018
2015

36 Shaikh
(Shaikh et al. 2015; Brkić 2016
2015
37 Biberg
(Biberg 2016
2016
38 Offor and Alabi
(Offor & Alabi 2016
2016
39 Brkić and Praks
(Brkić & Praks 2018
2018

40 Vatankhah
(Vatankhah 2018
2018
No.Author and sourceYearEquation for rough pipe
Moody
(Brkić 2011b
1947
Altshul
(Olivares et al. 2019
1952
Altshul II
(Olivares et al. 2019
1952
Wood
(Brkić 2011b
1966

Eck
(Brkić 2011b
1973
Jain
(Brkić 2011b
1976
Swamee and Jain
(Olivares et al. 2019
1976
Churchill
(Brkić 2011b
1977

Chen
(Brkić 2011b
1979
10 Round
(Olivares et al. 2019
1980
11 Shacham
(Zigrang & Sylvester 1985
1980
12 Shacham II
(Olivares et al. 2019
1980

13 Barr
(Olivares et al. 2019
1981
14 Pavlov
(Olivares et al. 2019
1981
15 Zigrang–Sylvester
(Zigrang & Sylvester 1982
1982
16 S. E. Haaland
(Haaland 1983
1983
17 Serghides I
(Brkić 2011b
1984

18 Serghides II
(Brkić 2011b
1984

19 Tsal
1989

(Olivares et al. 2019
1997
21 Romeo
(Romeo et al. 2002
2002
(Olivares et al. 2019
2006

23 Rao and Kumar
(Brkić 2011b
2007

24 Buzzelli
(Olivares et al. 2019
2008

(Brkić 2011b
2008

26 Avci
(Brkić 2011b
2009
27 Papaevangelo
(Olivares et al. 2019
2010
28 Brkić
(Brkić 2011b
2011

29 Brkić II
(Brkić 2011b
2011

30 Fang et al.
(Fang et al. 2011
2011
31 Ghanbari
2011
2012
33 Winning and Coole
(Winning & Coole 2015
2014
34 Heydari et al.
(Heydari et al. 2015
2015
35 Mikata and Walczak
(Vatankhah 2018
2015

36 Shaikh
(Shaikh et al. 2015; Brkić 2016
2015
37 Biberg
(Biberg 2016
2016
38 Offor and Alabi
(Offor & Alabi 2016
2016
39 Brkić and Praks
(Brkić & Praks 2018
2018

40 Vatankhah
(Vatankhah 2018
2018

*No. 33 and are constants, and their values are determined using Re; these values can be found in a determination table proposed by Winning & Coole (2015).

No. 36 The recommended value of is related to Re and , and the details can be found in Brkić (2016). In the calculation performed in the present study, is regarded as −0.75, for which the applicable range is and .

### Friction factor for turbulent flow in smooth pipes

For a smooth pipe, the Nikuradse–Prandtl–von Karman (NPK) equation is used to evaluate the accuracy of the explicit approximate equations. The most popular approximations were collected, as presented in Table 3. Since the NPK equation can be regarded as a particular case of the Colebrook equation in which the roughness is completely absent, Brkić (2011a) believed that an approximation of the NPK equation in a suitable form can be transformed into an approximation of the Colebrook equation, i.e. . The approximations with matching forms were thus transformed and are presented in Table 3.

Table 3

Summary of approximation equations for the friction factor for smooth pipes and transformed equations for rough pipes

No.Author and sourceYearEquation for smooth pipeEquation transformed for rough pipe
Blasius
(Brkić 2012
1913
Konakov
(Olivares et al. 2019
1950
Filonenko
(Olivares et al. 2019
1954
Techo et al.
(Techo et al. 1965
1965
Danish et al.
(Brkić 2012
2011
Fang et al.
(Fang et al. 2011
2011
Li et al.
(Li et al. 2011
2011
Taler
(Taler 2016
2016
No.Author and sourceYearEquation for smooth pipeEquation transformed for rough pipe
Blasius
(Brkić 2012
1913
Konakov
(Olivares et al. 2019
1950
Filonenko
(Olivares et al. 2019
1954
Techo et al.
(Techo et al. 1965
1965
Danish et al.
(Brkić 2012
2011
Fang et al.
(Fang et al. 2011
2011
Li et al.
(Li et al. 2011
2011
Taler
(Taler 2016
2016

## ANALYSIS AND RESULT

### Assessment of accuracy of the existing approximations

The Darcy–Weisbach friction factor () corresponding to the data from Anwar (1965), Padulano et al. (2013), Padulano et al. (2015) and Banisoltan et al. (2017) was calculated using the Colebrook equation and the approximations mentioned previously, and the maximum relative errors (MRE) of the approximations compared with the Colebrook equation were determined, as shown in Figure 5. Offor & Alabi (2016) classified the accuracy of approximations considering the MRE, and the threshold value was considered to be 5%. It is thus inadvisable to accept an approximation that has an MRE greater than 5%, and approximations with an MRE of up to 0.14% and 0.5% are assessed as extremely accurate and very accurate, respectively. Accordingly, 34 approximation equations exist for rough pipes, which are acceptable among the 40 approximations mentioned (the MRE of the equation of Rao and Kumar, listed as No. 23 in Table 2, is 26.88%, and it is not shown in Figure 5 considering the scale of the chart). Among these 34 equations, 24 approximations were assessed as very accurate and three were assessed as extremely accurate. The MREs of the extremely accurate approximations were 0.003%, 0.010%, and 0.014% for the Brkić and Praks (No. 39 in Table 2), Biberg (No. 37 in Table 2) and Serghides I (No. 17 in Table 2) approximations, respectively.

Figure 5

Maximum relative errors of the approximations. Note: The approximation of Rao and Kumar (No. 23 in Table 2) is not shown in Figure 5; and the approximation of Blasius does not have a corresponding transformed equation.

Figure 5

Maximum relative errors of the approximations. Note: The approximation of Rao and Kumar (No. 23 in Table 2) is not shown in Figure 5; and the approximation of Blasius does not have a corresponding transformed equation.

For the equations transformed for rough pipes, the MREs of all the approximations decrease after transformation, as observed from Figure 5. The values of all MREs are less than 5%, which indicates that these values are acceptable. Among the considered approximations, four can be judged as very accurate. Thus, the transformed equations from the approximations for smooth pipe can be suitably employed to calculate the value for a rough pipe.

### Proposed equation for the friction factor not including Re

To propose a new approximation for , the data from Anwar (1965), Padulano et al. (2013), Padulano et al. (2015) and Banisoltan et al. (2017) were employed. According to the brief review of existing approximations for the of a pipe, can be represented using logarithmic and power law equations. Since a logarithmic equation can be approximated with a power law form and considering the convenience of further analysis, the power law form was selected for deriving the new approximation, i.e., . The variables , Re, Re, Re and their logarithms were used in the approximations mentioned above. In the full flow regime of a vertical pipe, the flow velocity is determined by (for the overflow pipe, l should be replaced with ); accordingly, was used to replace Re in the new approximation equation of . In particular, the variables , ,, , , , and were used to establish the new equation. The resulting expression can be defined as Equation (1); this approximation had an MRE of 0.43%, which corresponds to an assessment level of very accurate. The comparison of the values calculated using the Colebrook equation and Equation (1) is shown in Figure 6. Although the accuracy of Equation (1) is lower than that of certain existing approximations, the error is acceptable.
(1)
Figure 6

Comparison between the value of the friction factor calculated using the Colebrook equation and that obtained using Equation (1).

Figure 6

Comparison between the value of the friction factor calculated using the Colebrook equation and that obtained using Equation (1).

The partial derivative sensitivity analysis (PDSA), in which a formula is differentiated by input variables, is believed to be capable of assessing the effect of input variables on an equation. According to the approximations for of smooth pipes (Table 3), it is clear that decreases with an increase in Re, and according to the transformations for of rough pipes (Table 3), increases with an increase in . The form of approximations for of rough pipes is more complicated, and thus, a partial derivative analysis must be performed to assess the sensitivity of variables. The partial derivatives of the Brkić and Praks equation (No. 39 in Table 2) were calculated as an example. It was noted that Re ranges from −1.76E−8 to −3.57E−9, and ranges from 6.27 to 1.31E + 1; i.e., increases with a decrease in Re or an increase in . For Equation (1), ranges from 6.79E−2 to 2.49E−1, and ranges from −2.16E + 4 to −9.42E + 3; i.e., increases with a decrease in or an increase in . This trend is similar to that of the existing approximations.

## DISCUSSION

For the data from Humphreys et al. (1970) and Zhang (2017), which corresponds to the full flow regime in a vertical pipe with a joint barrel, the feasibility of calculating of the vertical pipe with Equation (1) was verified, and the result is shown in Figure 7. Owing to the existence of the head loss in the joint section (transition loss coefficient) and friction loss in the barrel, the velocity in this case is less than that in a vertical pipe without a joint barrel; thus, the calculated using Equation (1) is less than that calculated using the Colebrook equation, as shown in Figure 7. The value calculated using Equation (1) is within the range of ±20% of the value calculated using the Colebrook equation; therefore, it is acceptable to approximate using Equation (1) in the full flow regime of the vertical pipe with a joint barrel, when the discharge is unknown and the length of the barrel is not excessively large.

Figure 7

Comparison between the value of the friction factor calculated using the Colebrook equation and that obtained using Equation (1).

Figure 7

Comparison between the value of the friction factor calculated using the Colebrook equation and that obtained using Equation (1).

## CONCLUSION

We investigated the Darcy–Weisbach friction factor of a full flow regime in vertical pipes based on data from the literature. The existing approximations of for a rough pipe were reviewed, and their performance considering the collected data was assessed and classified. The existing approximate equations of for a smooth pipe were reviewed and transformed to approximations for a rough pipe; the transformed approximations were assessed, and the results were found to be satisfactory. Furthermore, a new approximation not including the variable Re was proposed and shown to be very accurate.

## ACKNOWLEDGEMENTS

This work was supported by the Natural Science Foundation of Shaanxi Province (2017JZ013); the Leadership Talent Project of Shaanxi Province High-Level Talents Special Support Program in Science and Technology Innovation (2017); the National Key Research and Development Program of China (2016YFC0402404); the National Natural Science Foundation of China (51679197, 41330858 and 51679193).

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