Accurate Explicit Equations for the Determination of Pipe Diameters

The determination of diameter in p ipe flow problems requires the use of diagrams or iterat ive solutions of the Colebrook – White equation. Diagrams are inaccurate because of reading errors and are impossible to use when the whole problem is going to be solved by a computer. Iteration type solutions can be very time consuming when large water distribution networks are involved. In this paper, accurate explicit equations for the determination of pipe diameter are developed. Two equations are presented, a simple and a more complex one. The maximum relative errors in the computation of the diameter, D, for 4x10≤ Re ≤10 and 10≤ ε/D ≤5x10 are 0.36% and 5.65x10 % respectively. It was found that the complex equation is far superior to other explicit equations available in the literature.


Introduction
Among other problems the following three problems arise in the design of pipe distribution systems: I) determination of discharge, Q; II) determination of the head loss, h f ; and III) computation of the pipe diameter, D.
To solve these problems, the following three equations are required: Continuity equation which states that Q is constant and given by: (1) Darcy-Weisbach equation (2) Colebrook -White equation (3) in wh ich D=internal p ipe diameter; V=average velocity; f=Darcy-Weisbach dimensionless friction factor; L=pipe length; g=acceleration of gravity; ε=equivalent sand grain roughness of the pipe wall; Re=Reynolds number (Re=VD/ν); ν=kinemat ic viscosity.
For s mooth pipes, the Co lebrook-White equation [6] yields: (4) while for large Reynolds numbers (Re→∞) it is asymptotic to the equation: (5) Equations (4) and (5) were derived by Karman [25] and Prandtl [18] respectively with a slight modificat ion of the numerical factors to agree with the experimental data of Nikuradse [17]. It should be noted that the roughness ε in Karman and Prandtl's equations is the diameter of the uniform sand used by Nikuradse to create the artificial roughness in his experimental p ipes. Since roughness of commercial pipes is not uniform in size, distribution and shape, Colebrook suggested the "equivalent sand roughness' which is the roughness ε that would satisfy Eq. (5) for large Reynolds numbers. Colebrook [6] The computation of h f and D however requires the estimation of f. Iterative type solutions of Eq. (3) are time consuming, especially when large water distribution networks are involved. The Moody diagram [16] has been used very extensively, especially in the past. The use of this diagram has two main disadvantages: a) accuracy is limited because of reading errors in logarithmic scale; and b) it is impossible to be used in computer co mputations. An additional way of solving pipe flow problems without the previous disadvantages is the derivation of equations which give f explicitly in terms of the known variab les.
A general review of explicit functions of f and a comparative study among them is given in Yıldırım [26] and Fang et al. [9]. Explicit equations of f, however, wh ich can be used for the solution of type III problems (co mputation of D) are really scarce.
It is the purpose of this paper to develop explicit and very accurate equations of f for the solution of pipe diameter problems.

Review of Available Solutions for the Estimation of the Pipe Diameter
Swa mee and Jain [21] presented an equation which in terms of the parameters T and R (Eqs. (8) and (10)), is written as: (11) According to the authors this equation covers smooth turbulent flow, rough turbulent flow and the transition in-between. It is valid for 10 -8 ≤ 1/R ≤10 -3 , wh ich corresponds to 3x10 8 ≥ Re ≥ 3x10 3 , and 10 -6 ≤T ≤10 -2 , which corresponds to 2x10 -6 ≤ε /D≤ 2x10 -2 . For these ranges, D is estimated with an error ±2%.
Gu lyani [11] presented a simple method that takes into account the effect of surface roughness on pipe diameter. For each case under study he proposes two bounding equations which apply for s mooth and extremely rough pipes. He estimates then the diameter for a s mooth pipe and a factor which depends on the real roughness of the pipe. By mu ltip lying this factor with the d iameter of the smooth pipe he receives an appro ximate value of the real diameter o f the pipe.
Swa mee and Rathie [22] using the Lambert Function W(x) and Langrange's theorem derived a rather cu mberso me exact equation for the computation of D consisting of a series with infinite terms. According to the authors the use of only three terms ensures a sufficient accuracy for all practical purposes. In terms of the parameters T and R this equation is the following: (12) where: (13)
Bo mbardelli and Garc ia [2] also investigated the hydraulic design of large d iameter p ipes studied by the Hazen-Williams fo rmula . This equation, in contrast to the Darcy -Weisbach equation, includes a conveyance coefficient, which is constant for the p ipe material and independent of the flow reg ime. They concluded that the use of the Hazen -Williams equation can lead to serious practical and conceptual imp lications in otherwise straightforward computations.

Proposed Explicit Equations
A data set of 2853 exact values of f was generated by solving numerically the Colebrook-White equation (9) for 4x10 3 ≤ R ≤ 4x10 7 . For each value of R, T was varied in the range 4x10 -6 ≤ T ≤ 3x10 -2 . The p revious ranges of R and T correspond to 10 4 ≤Re≤10 8 and 10 -5 ≤ε/D≤5x10 -2 . The coefficients for the two equations that are presented in this paper were estimated with the Least Squares Method in Matlab.
Absolute relative errors were estimated by the formu la: (15) where f cw is the value obtained by the Colebrook -White equation.
Depending on the desired accuracy, two explicit equations for f are proposed here. The first equation has the form: (16) This equation has a mean absolute error of 1.8%, and a coefficient of determination, r 2 , greater than 0.999. When f fro m Eq. (16) is used for the co mputation of the diameter D according to Eq. (7), the mean absolute error in D is 0.36%.
The second equation is more complex in form, but also even more accurate than Eq.     This equation has a mean absolute error of 0.028%, and a coefficient of determination, r 2 , also greater than 0.999. The mean absolute error in D as it is computed by (7) is 5.65x10 -3 . The accuracy of (17) is shown in Figures 1 to 3. Figure 1 shows exact values of f vs. predicted values of f and Figure 2 shows the distribution of the relat ive errors of f when it is computed by Eq. (17). The values of f are generated for 4x10 3 ≤R≤4x10 7 and 4x10 -6 ≤T≤0.03. Figure 3 shows exact values of D vs. predicted values of D computed by Eqs. (7) and (17).
A summary of the errors of Eqs. (16) and (17) is given in Table 1. Tab le 1 shows also the errors of the aforementioned equations available in the literature. It is shown that Eq. (11) of Swa mee and Jain [21] and Eq. (14) of Swa mee and Swa mee [23] g ive identical errors and they are the least accurate equations. Equation (12) of Swa mee and Rathie [22] is substantially better than the previous two equations. It is obvious that equation (17), proposed in this paper, is the most accurate equation of all the previous ones.
To further assess the accuracy of Eqs. (16) and (17), a larger data set of 7957 values was generated by numerically solving the Colebrook -White equation (9) in the range 4*10 3 ≤ Re ≤ 10 8 and 10 -5 ≤ε/D ≤ 5x10 -2 . It was determined that the errors never exceeded the corresponding values given in Table 1.

Conclusions
An exp licit method for the computation of the diameter of pipes in pipe flow problems and in designing water distribution networks is very important. The two equations presented here for the computation of the friction factor, f, and, consequently, for the co mputation of the diameter, D, are very precise and superior to the ones presented in the literature, and can be used very easily for all practical cases.