Discussion of “Gene expression programming analysis of implicit Colebrook–White equation in turbulent flow friction factor calculation” by Saeed Samadianfard 92-93 (2012), 48-55]

Abstract Maximal relative error of the explicit approximation to the Colebrook equation for flow friction presented in the discussed paper by Saeed Samadianfard [J. Pet. Sci. Eng. 92–93 (2012) 48–55; doi. 10.1016/j.petrol.2012.06.005] is investigated. Samadianfard claims that his approximation is very accurate with the maximal relative error of no more than 0.08152%. Here is shown that this error is about 7%. Related comments about the paper are also enclosed.


Introduction
presents an approximate explicit formula as a replacement for the implicitly given Colebrook equation for fluid flow pipe friction. The empirical Colebrook equation (1) [Eq. 1a of the discussed paper] relates the unknown flow friction factor (ʄ 0 ) with the known Reynolds number (Re) and the known relative roughness of inner pipe surface (ɸ/D). Widely used empirical and nonlinear Colebrook's equation for calculation of Darcy's friction factor is iterative i.e. implicit in fluid flow friction factor since the unknown friction factor appears on the both sides of the equation [ʄ 0 =f(ʄ 0 , Re, ɸ/D)] (Colebrook 1939 Equation (1) [Eq. 1a of the discussed paper] is from Colebrook (1939)  (2) Samadianfard (2012) used genetic programming to develop his approximation. ojbašiđ and Brkiđ (2013) also used genetic algorithms to improved accuracy of some of the available approximations (Romeo et al. 2002;Serghides 1984). Samadianfard (2012) claims that his approximation (2) [Eq. (29) of the discussed paper] produces maximal relative error, ɷ max of no more than 0.08152% compared with the very accurate iterative solution of the implicit Colebrook equation, ʄ 0 . Using the procedure from Brkiđ (2011a), it can be shown that this error, ɷ max in the practical range of applicability of the Reynolds number (Re) and the relative roughness of inner pipe surface (ɸ/D) is about 7.4289% 1 [mean (average) relative error is about 1.6198%]. Distribution of the relative error of approximation by Samadianfard (2012) [(2); Eq.

Error analysis
(29) of the discussed paper] can be seen in Figure 1. Relative error is defined as ɷ max = (ʄ 0 Ͳ ʄ)/ʄ 0 ͼ100% as shown by Eq. (26) Table 1 [Table 2 of discussed paper] and using 43 values of the Reynolds number (Re); from 4ͼ10 3 to 10 4 with pace 10 3 , from 10 4 to 10 5 with pace 10 4 , from 10 5 to 10 6 with pace 10 5 , from 10 6 to 10 7 with pace 10 6 , and from 10 7 to 10 8 with pace 10 7 . Similarly as in Samadianfard (2012), results are shown in Table 1 and compared with the results from Table 2 of the original paper by Samadianfard (2012). According to this second check, the maximal relative error ɷ max is about 6.9107% and mean (average) relative error in the range of applicability of equation is about 1.2595%. Table 1. Distribution of maximal relative error of the approximation by Samadianfard (2012) [(2); Eq.
(29) of the discussed paper] Mean square error as defined by Eq. 27 of the discussed paper, for the mesh of 740 points is about 1.1460ͼ10 Ͳ7 and for the mesh of 860 points is about 2.2711ͼ10 Ͳ7 . According to Winning and Coole (2013), the approximation by Samadianfard (2012) is in the group with medium value of mean square error (very small is lower than 10 Ͳ11 , small is between 10 Ͳ11 and 10 Ͳ8 , medium is between 10 Ͳ8 and 5ͼ10 Ͳ6 , and large is above 5ͼ10 Ͳ6 ). Samadianfard (2012) reported value of mean square error of 1.95ͼ10 Ͳ10 .

Source of the error
Regarding the error from Table 1 and from Figure 1 of this discussion, it is useful to mention that actually three level of the accuracy can be introduced; 1. Colebrook's equation is empirical (other maybe more accurate equations can be used to describe related physical processes), 2. Colebrook's equation can be solved very accurately using iterative procedure (term "accurate by default" can be used or better to say, this error can be neglected in many cases; this is ʄ 0 in this discussion), 3.
Relevant explicit approximations of the Colebrook equation (this is ʄ in this discussion) can be used to avoid iterative procedure (their error can be estimated very accurately compared with the error of iterative solution). The estimated error from Table 1 and from Figure 1 of this discussion belongs to the category explained in point 3 (where friction factor from point 2 assumed as accurate). Using "philosophy" from point 2, value ʄ 0 of friction factor is calculated, while friction factor ʄ is calculated using point 3. Finally, the maximal relative error was calculate as ɷ max = (ʄ 0 Ͳʄ)/ʄ 0 ͼ100%. It is also useful to note that the Moody diagram (Moody 1944) is only graphical interpretation of the Colebrook equation (Colebrook 1939). Moody's diagram is not less or more accurate compared with Colebrook's equation. So, inaccurate reading from Moody's diagram can introduce one more source of error (YŦldŦrŦm 2009, Brkiđ 2011b, Fang et al 2011, which is maybe possible cause of the value of the error estimated in Samadianfard (2012). In this case, source of the error is not then calculative, but rather systematic, where the main source of the error is in inaccuracy of ʄ 0 (which should be accurate by default).

Analysis of the equation
It is worth to mention that the approximation by Samadianfard (2012) does not contained a single logarithmic expression. This structure of the equation is valuable contribution since the logarithmic expression uses a lot of computational resources in a computer environment (Giustolisi et al. 2011;Clamond 2009). Only, three other available approximations do not use logarithms; Moody (1947), Wood (1966) and Chen (1984), and all three are less accurate than the approximation by Samadianfard (2012). But also, Samadianfard (2012) uses nonͲinteger power such as Re ɸ/D which in computer environment usually means exp[(ɸ/D)ͼln(Re)] where 'ln' is natural (Napierian) logarithm (Clamond 2009).

Reference notes
Finally, it has to be mentioned that source of the review article of Brkiđ (2011a)  Also, it can be mentioned that Colebrook equation is published solely by Colebrook (1939) and not in collaboration with White as often cited. Colebrook (1939) acknowledged contribution of White in a footnote of his paper with the reference of their previous joint work (Colebrook and White 1937).

Notation remark
In this discussion, the Darcy friction factor is labeled as ʄ while in the discussed paper as f. This is because f should be rather used for Fanning friction factor. This can make confusion since the physical meaning is equal but where the Darcy friction factor is 4 times greater than Fanning's friction factor.

Conclusion
According to the classification proposed by Samadianfard (2012), his approximation with the relative error of about 7% (7.43% after first evaluation and 6.91% after second evaluation; both in MS Excel), have to be characterized as less accurate method (group of methods with relative error of more than 3%) and should be placed in Figure 3 of the discussed paper (or even in Figure 2 of the discussed paper with the nonͲadvisable approximations) and not in Figure 6 of the discussed paper with the extremely accurate approximations.