Approximations of the Darcy–Weisbach friction factor in a vertical pipe with full flow regime

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);

R_e

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).

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.

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 (R_e) are introduced to describe the three characteristics; and two important implicit equations, the Colebrook equation (a function of and R_e) and the Nikuradse–Prandtl–von Karman (NPK) equation (a function of R_e), 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).

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 R_e are shown in Figure 4. The value of R_e recorded in the literature is within the range of ⁠, and that of is within ⁠.

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 R_e 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).

For a smooth pipe, the 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.

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.

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.

For the data from Humphreys et al. (1970) and Zhang (2017), which correspond 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.

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 R_e was proposed and shown to be very accurate.

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).