Model/Reference	Model before optimization	Model after optimization with GA	Model after optimization with SA
Chen [14]	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2 \log [\frac{E}{3.7065 D} - \frac{5.0452}{R e} \log_{10} [\frac{1}{2.8257} \\ \cdot {(\frac{E}{D})}^{1.1098} + \frac{5.8506}{R e^{0.8981}}]] \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2.002 \log [\frac{E}{3.7 D} - \frac{4.975}{R e} \log_{10} [\frac{1}{3.185} \\ \cdot {(\frac{E}{D})}^{1.078} + \frac{5.495}{R e^{0.908}}]] \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2.002 \log [\frac{E}{3.707 D} - \frac{5.262}{R e} \log_{10} [\frac{1}{3.111} \\ \cdot {(\frac{E}{D})}^{1.02} + \frac{5.504}{R e^{0.869}}]] \end{array}$
Barr [15]	$\frac{1}{\sqrt{f}} = - 2 \log_{10} [\frac{E}{3.7 D} + \frac{4.518 \log (\frac{1}{7} R e)}{R e (1 + \frac{1}{29} R e^{0.52} {(\frac{E}{D})}^{0.7})}]$	$\frac{1}{\sqrt{f}} = - 2.001 \log_{10} [\frac{E}{3.69 D} + \frac{4.571 \log (\frac{1}{7.114} R e)}{R e (1 + \frac{1}{30.407} R e^{0.605} {(\frac{E}{D})}^{0.842})}]$	$\frac{1}{\sqrt{f}} = - 2.001 \log_{10} [\frac{E}{3.697 D} + \frac{4.56 \log (\frac{1}{7.013} R e)}{R e (1 + \frac{1}{29.224} R e^{0.586} {(\frac{E}{D})}^{0.821})}]$
Papaevangelou et al. [16]	$f = \frac{0.2479 - 0.0000947 {(7 - \log R e)}^{4}}{{[\log [\frac{E}{3.615 D} + \frac{7.366}{R e^{0.9142}}]]}^{2}}$	$f = \frac{0.246 - 0.0000982 {(7.192 - \log R e)}^{3.677}}{{[\log [\frac{E}{3.556 D} + \frac{6.756}{R e^{0.905}}]]}^{2}}$	$f = \frac{0.247 - 0.0000982 {(7.008 - \log R e)}^{3.889}}{{[\log [\frac{E}{3.593 D} + \frac{7.174}{R e^{0.912}}]]}^{2}}$
Avci and Karagoz [17]	$f = \frac{6.4}{{[I n (R e) - I n (1 + 0.01 R e \frac{E}{D} (1 + 10 \sqrt{\frac{E}{D}}))]}^{2.4}}$	$f = \frac{6.609}{{[I n (R e) - I n (1.082 + 0.009 R e \frac{E}{D} (1.286 + 9.896 \sqrt{\frac{E}{D}}))]}^{2.418}}$	$f = \frac{6.831}{{[I n (R e) - I n (1.076 + 0.01 R e \frac{E}{D} (1.289 + 8.098 \sqrt{\frac{E}{D}}))]}^{2.434}}$
Offor and Alabi [4]	$\begin{array}{l} f = [- 2 \log_{10} [\frac{E / D}{3.71} + \frac{- 1.975}{R e} [I n {[(\frac{E / D}{3.93})}^{1.092} \\ + {\frac{7.627}{R e + 395.9}]]]]}^{- 2} \end{array}$	$\begin{array}{l} f = [- 2 \log_{10} [\frac{E / D}{3.712} + \frac{- 1.972}{R e} [I n {[(\frac{E / D}{3.992})}^{1.086} \\ + {\frac{7.541}{R e + 402}]]]]}^{- 2} \end{array}$	$\begin{array}{l} f = [- 2 \log_{10} [\frac{E / D}{3.707} + \frac{- 1.972}{R e} [I n {[(\frac{E / D}{3.861})}^{1.088} \\ + {\frac{7.533}{R e + 397.41}]]]]}^{- 2} \end{array}$
Shacham [18]	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{E}{3.7 D} - \frac{5.02}{R e} \log (\frac{E}{3.7 D} + \frac{14.5}{R e})]$	$\frac{1}{\sqrt{f}} = - 2.007 \log [\frac{E}{3.68 D} - \frac{5.133}{R e} \log (\frac{E}{3.71 D} + \frac{14.028}{R e})]$	$\frac{1}{\sqrt{f}} = - 2.008 \log [\frac{E}{3.65 D} - \frac{5.125}{R e} \log (\frac{E}{3.72 D} + \frac{13.5}{R e})]$
Sousa et al. [12]	$\frac{1}{\sqrt{f}} = - 2 \log [\frac{E}{3.7 D} - \frac{5.16}{R e} \log (\frac{E}{3.7 D} + \frac{5.09}{R e^{0.87}})]$	$\frac{1}{\sqrt{f}} = - 2.001 \log [\frac{E}{3.697 D} - \frac{5.15}{R e} \log (\frac{E}{3.71 D} + \frac{5.06}{R e^{0.873}})]$	$\frac{1}{\sqrt{f}} = - 2.001 \log [\frac{E}{3.7 D} - \frac{5.203}{R e} \log (\frac{E}{3.7 D} + \frac{5.078}{R e^{0.867}})]$
Manadilli [19]	$\frac{1}{\sqrt{f}} = - 2 \log_{10} [\frac{95}{R^{0.983}} - \frac{96.82}{R} + \frac{ε}{3.7 D}]$	$\frac{1}{\sqrt{f}} = - 2.015 \log_{10} [\frac{95.01}{R^{0.981}} - \frac{98.623}{R} + \frac{ε}{3.693 D}]$	$\frac{1}{\sqrt{f}} = - 2.012 \log_{10} [\frac{94.913}{R^{0.983}} - \frac{96.742}{R} + \frac{ε}{3.72 D}]$
Zigrang and Sylvester [20]	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2 \log_{10} (\frac{ε}{3.7 D} - \frac{5.02}{R e} \cdot \log_{10} (\frac{ε}{3.7 D} \\ - \frac{5.02}{R e} \cdot \log_{10} (\frac{ε}{3.7 D} - \frac{13}{R e}))) \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 1.999 \log_{10} (\frac{ε}{3.72 D} - \frac{5.001}{R e} \cdot \log_{10} (\frac{ε}{3.63 D} \\ - \frac{4.91}{R e} \cdot \log_{10} (\frac{ε}{3.9 D} - \frac{11.5}{R e}))) \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 1.999 \log_{10} (\frac{ε}{3.719 D} - \frac{5.034}{R e} \cdot \log_{10} (\frac{ε}{3.65 D} \\ - \frac{5.11}{R e} \cdot \log_{10} (\frac{ε}{3.715 D} - \frac{12.5}{R e}))) \end{array}$
Ghanbari et al. [21]	$f = {- 1.52 \log [{(\frac{E}{7.21 D})}^{1.042} + {(\frac{2.731}{R e})}^{0.9152}]}^{- 2.169}$	$f = {- 1.463 \log [{(\frac{E}{7.568 D})}^{1.047} + {(\frac{2.639}{R e})}^{0.93}]}^{- 2.2}$	$f = {- 1.482 \log [{(\frac{E}{6.335 D})}^{1.098} + {(\frac{2.876}{R e})}^{0.96}]}^{- 2.153}$
Swamme and Jain [22]	$\frac{1}{\sqrt{f}} = - 2 \log_{10} [\frac{ε}{3.7 D} + \frac{5.74}{R^{0.9}}]$	$\frac{1}{\sqrt{f}} = - 2.009 \log_{10} [\frac{ε}{3.78 D} + \frac{5.398}{R^{0.895}}]$	$\frac{1}{\sqrt{f}} = - 2.017 \log_{10} [\frac{ε}{3.708 D} + \frac{5.219}{R^{0.889}}]$
Fang et al. [23]	$f = 1.613 {[\ln (0.237 {(\frac{ε}{D})}^{1.1007} - \frac{60.525}{R e^{1.1105}} + \frac{56.291}{R e^{1.0712}})]}^{- 2}$	$f = 1.617 {[\ln (0.231 {(\frac{ε}{D})}^{1.1004} - \frac{60.5}{R e^{1.1103}} + \frac{56.69}{R e^{1.072}})]}^{- 2}$	$f = 1.603 {[\ln (0.231 {(\frac{ε}{D})}^{1.095} - \frac{60.355}{R e^{1.108}} + \frac{56.14}{R e^{1.069}})]}^{- 2}$
Romeo et al. [9]	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.7065} - \frac{5.0272}{R e} \log (\frac{ε / D}{3.827} - \frac{4.567}{R e} \\ \cdot \log ({(\frac{ε / D}{7.7918})}^{0.9924} + {(\frac{5.3326}{208.815 + R e})}^{0.9345}))) \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.7105} - \frac{5}{R e} \log (\frac{ε / D}{3.886} - \frac{4.792}{R e} \\ \cdot \log ({(\frac{ε / D}{7.74})}^{0.969} + {(\frac{5.01}{212.2 + R e})}^{0.876}))) \end{array}$	$\begin{array}{l} \frac{1}{\sqrt{f}} = - 2 \log (\frac{ε / D}{3.71} - \frac{5.002}{R e} \log (\frac{ε / D}{3.807} - \frac{4.376}{R e} \\ \cdot \log ({(\frac{ε / D}{7.858})}^{1.008} + {(\frac{5.422}{206.658 + R e})}^{0.973}))) \end{array}$
Sonnad and Goudar [24]	$\frac{1}{\sqrt{f}} = 0.8686 \ln (\frac{0.4587 \cdot R e}{s^{\frac{s}{s + 1}}})$ where $s = 0.124 \cdot R e \cdot \frac{ε}{D} + \ln (0.4587 \cdot R e)$	$\frac{1}{\sqrt{f}} = 0.868 \ln (\frac{0.468 \cdot R e}{s^{\frac{s}{s + 1}}})$ where $s = 0.126 \cdot R e \cdot \frac{ε}{D} + \ln (0.422 \cdot R e)$	$\frac{1}{\sqrt{f}} = 0.868 \ln (\frac{0.466 \cdot R e}{s^{\frac{s}{s + 1}}})$ where $s = 0.125 \cdot R e \cdot \frac{ε}{D} + \ln (0.41 \cdot R e)$
Serghides [8]	$f = s_{1} - \frac{{(s_{2} - s_{1})}^{2}}{s_{3} - 2 s_{2} + s_{1}}$ where $s_{1} = - 2 \log_{10} (\frac{ε}{3.7 D} + \frac{12}{R e})$ $s_{2} = - 2 \log_{10} (\frac{ε}{3.7 D} + \frac{2.51 s_{1}}{R e})$ $s_{3} = - 2 \log_{10} (\frac{ε}{3.7 D} + \frac{2.51 s_{2}}{R e})$	$f = s_{1} - \frac{{(s_{2} - s_{1})}^{2}}{s_{3} - 2 s_{2} + s_{1}}$ where $s_{1} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{12.97}{R e})$ $s_{2} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{2.51 s_{1}}{R e})$ $s_{3} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{2.51 s_{2}}{R e})$	$f = s_{1} - \frac{{(s_{2} - s_{1})}^{2}}{s_{3} - 2 s_{2} + s_{1}}$ where $s_{1} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{11.919}{R e})$ $s_{2} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{2.51 s_{1}}{R e})$ $s_{3} = - 2 \log_{10} (\frac{ε}{3.71 D} + \frac{2.51 s_{2}}{R e})$
Buzzelli [25]	$\frac{1}{\sqrt{f}} = A - \frac{A + 2 \log_{10} (\frac{B}{R e})}{1 + \frac{2.18}{B}}$ $A = \frac{0.774 I n (R e) - 1.41}{1 + 1.32 \sqrt{\frac{E}{D}}}$ $B = \frac{E R e}{3.7 D} + 2.51 A$	$\frac{1}{\sqrt{f}} = A - \frac{A + 2.002 \log_{10} (\frac{B}{R e})}{1.001 + \frac{2.033}{B}}$ $A = \frac{0.765 I n (R e) - 1.486}{1.139 + 1.491 \sqrt{\frac{E}{D}}}$ $B = \frac{E R e}{3.69 D} + 2.522 A$	$\frac{1}{\sqrt{f}} = A - \frac{A + 2 \log_{10} (\frac{B}{R e})}{1 + \frac{2.26}{B}}$ $A = \frac{0.783 I n (R e) - 1.383}{0.952 + 1.298 \sqrt{\frac{E}{D}}}$ $B = \frac{E R e}{3.708 D} + 2.511 A$