### Colebrook’s equation and Moody’s graph

Colebrook’s equation is as follows^{1,3,4,6,7},

$$frac{1}{sqrt f } = – 2.0log left( {frac{2.51}{{Resqrt f }} + frac{varepsilon /d}{{3.7}}} right).$$

(1)

Based on the Re numbers and ε/d values from the Moody plot, the *F* values were calculated using Eq. (1) and shown in Figs. 1a and 2a. the *F* data at ε/d = 0.004 read from Refs.^{1,3,4,5,6} are also given in Fig. 2a, and the discrepancies between the data in the references and the actual roughness calculation are shown in Fig. 2b.

The definition of gap is, ({text{relative}} ;{text{error}} = left| {f – f_{{_{{{text{Colebrook}}}} }} } right|/f_{{ _{{{text{Colebrook}}}} }} { times }100,) or *F* is the reference datum, and *F*_{Colebrook} is the calculation with Eq. (1). The obvious error is in the transition roughness region of Figure 2b, and the maximum error is less than 3%, which is similar to the reported maximum error of reading the Moody chart data in refs.^{11.15}. The results of FIG. 2 indicate that the current calculation of *F* is reasonable.

### Pseudo-interpolation error

The Moody diagram contains twenty-one ε/ds, and the smooth pipe at ε/d = 0 has not been calculated and shown in Fig. 1a or 2a. Thus, the twenty ε/d have nineteen midpoints, which are different from the pseudo midpoints, like the ε/d of the dotted line in Fig. 1. The wrong reading *F* using the pseudo-middle of both *F* curves at any adjacent ε/ds was evaluated by comparing *F*s at pseudo-middle nineteen ε/ds at true *F*s at the nineteen midpoint ε/ds at nine numbers Re, i.e. 10^{4}3×10^{4}ten^{5}3×10^{5}ten^{6} 3×10^{6}ten^{7}3×10^{7}and 10^{8}. Many relative errors between the two types of *F*s are less than 1%. The locations at which the relative errors are greater than 1% are shown in Figure 3a, and the corresponding relative errors are given in Figure 3b.

Figure 3b indicates that the maximum error of the misread *F* at the pseudo-middle ε/d = 2.24 × 10^{−5} and D = 10^{8} is about 4% and most errors are less than 1.5%. These errors are apparently less than the 15% inaccuracy of using the Moody chart introduced by White^{3}. The influence of the pseudo interpolation is not significant. There are two regions in which the pseudo-interpolation could have a large relative error. One is near the *F* curve at ε/d = 2.24 × 10^{−5} with a large Re, and another is at Re = 10^{8}. We can see in figure 3b that the error increases with Re, which is also valid for all the other curves of error as a function of Re not shown here. Therefore, the location at Re = 10^{8} must contain the maximum error for the pseudo interpolation phenomenon. However, the *F* at Re = 10^{8} and ε/d = 2.24 × 10^{−5} is rarely used in practice, for example in white textbooks^{3} and Ding^{6}.

Figure 4 compares the relative errors of the pseudo interpolation with those of the manual method, in which the error of the manual method is noted “by hand”. The manual method means that the *F*s at the two pseudo-middles ε/ds, i.e. 0.00283 and 2.24 × 10^{−5}were read directly from the Moody table in Ding’s manual^{6} using a pencil and ruler. At a high Re number, the *F* of the manual method approximates that of the pseudo interpolation in Figure 4, which indicates that the pseudo interpolation used here is similar to the manual method usually adopted. At the relatively low Re number in the transition roughness region, the error of the manual method is large and approaches 1-3%, which is similar to the error in Fig. 2b. This indicates that the manual method may cause a relative error of 1-3% in the transition roughness area.