Table 2 The presented correlations in the research to approximate biodiesel surface tension using GEP, and GMDH networks.

From: On the evaluation of surface tension of biodiesel

Models

Equation

GEP

\(\begin{aligned} {\text{ST}} = & SQRT\left( {\left( {\left( {3.77590719321268} \right.} \right.} \right. \\ & \times \,SQRT\left( {SQRT\left( {\left( {\left( {\left( {SQRT\left( {9.46592608417005} \right) + \left( {T - Mw_{3} } \right)} \right)^{2} } \right)} \right.} \right.} \right. \\ & \left. {\left. {\left. {\left. {\left. {\left. { + \,9.46592608417005} \right)} \right)} \right)} \right)^{2} } \right)} \right) + \,\left( {\left( {\left( {SQRT\left( {SQRT\left( {Mw_{3} } \right)} \right)} \right.} \right.} \right. \times \,\left( {\left( {\left( {2.50160222174749} \right.} \right.} \right. \\ & \left. {\left. { - \left( {\left( { - 0.679647206030457 - \left( { - 0.679647206030457} \right)^{2} } \right) - 5.46555986205634} \right)} \right)^{2} } \right) \\ & \left. {\left. {\left. {\left. { \times \,\left( {\left( {\left( {Mw_{2}^{2} } \right) + 0.679647206030457} \right) \times \left( {Mw_{3}^{2} } \right)} \right)} \right)} \right) - 5.46555986205634} \right)^{2} } \right) \\ & + \,\left( {\left( {SQRT\left( {Mw_{2} } \right) + 7.72576067384869} \right)^{2} } \right) + \left( {\left( {\left( {SQRT\left( {Mw_{2} } \right)^{2} } \right) - \left( {SQRT\left( {T) + Mw_{3} } \right)} \right)} \right.} \right. \\ & \left. { \times \,\left( {\left( { - 2.76589251380963} \right)^{2} + Mw_{1} } \right)} \right) \\ \end{aligned}\)

GMDH

\(\begin{gathered} N2 = 38.5165 + T \times Mw_{1} \times 27.7516 - T \times Mw_{3} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, \times \,2.85178 - (T)^{2} \times 9.99647 \times 10^{ - 5} - Mw_{1} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, \times \,8704.75 + \,Mw_{3} \times 1066.18 - (Mw_{3} )^{2} \times 2869.59 \hfill \\ N1 = 2.08003 + T \times Mw_{1} \times 43.664 - Mw_{1} \times 45111.7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \,Mw_{1} \times N2 \times 1094.33 + (Mw_{1} )^{2} \times 15771.5 + N2 \times 0.924756 \hfill \\ ST = - \,18.1792 + Mw_{3} \times N2*93.2258 - Mw_{3} \times N1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, \times \,93.3023 - N2 \times N1 \times 10.0947 + N2^{2} \times 5.03386 \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \,N1 \times 2.32169 + N1^{2} \times 5.03685 \hfill \\ \end{gathered}\)

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